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Observation of the dipole-dipole interaction between cold Rydberg atoms by microwave spectroscopy

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Observation of the dipole-dipole interaction between cold
Rydberg atoms by microwave spectroscopy
Hyunwook Park
Sacheon, South Korea
B.S.. Pusan National University, 2003
M.S., Pusan National University, 2005
A Dissertation presented to the Graduate Faculty of the
University of Virginia in Candidacy for the Degree of
Doctor of Philosophy
Department of Physics
University of Virginia
May. 201 2
UMI Number: 3515532
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Hyunwook Park
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May 2012
Abstract
We measured the dipole-dipole interaction between nsnp pairs of atoms by the line
broadening technique. The broadening rate relies on the atomic density, equivalently the
average internuclear spacing R av , and principal quantum number n. This measurement of
the dipole-dipole broadening can be expressed in terms of n and provides a simple measure
of line broadening due to increased atomic density in laboratory units.
Calculation of
the dipole-dipole interaction was compared to the observations. It was realized that the
observations, which have smaller broadening rates than the calculation, stem from the spinorbit coupling, which results in the shift-free and small-shift dipole-dipole energy levels as
well as normal shift levels.
As a result of the dipole-dipole interaction, the nsnp molecules form attractive and re­
pulsive dipole-dipole potentials in which atoms are forced to move toward each other and
farther apart, respectively. These motions of the atoms in the dipole potentials induce
collisional ionization and trigger plasma formation from Rydberg atoms. The collisional
ionization was systematically investigated by comparing the effects of the attractive, re­
pulsive, and almost flat potentials. It turned out that atoms transferred to the attractive
potential are ionized in a few microseconds, while those on the repulsive potential are not
significantly ionized, similar to the flat, potential case. Essentially the same result was ob­
served again with an enhanced ion signal by extending the sampling to a broader range of
internuclear separation via high microwave power.
We also detected plasma fields by using the exaggerated property of Rydberg atoms
responding to external electric fields. Rydberg atoms were injected into a plasma cloud,
and the ns — np microwave transition was driven to detect the plasma fields by measuring
Stark shifts. We were able to measure a microscopic field as small as 0.1 V/cm. In the
presence of a strong macroscopic field, the resonances are not. only shifted but also doublepeaked. The time evolution of the plasma fields was also studied.
]
Acknowledgments
I would like to thank all my family in Korea, who have shown their utmost and un­
changing support since I came into this doctoral program. I am deeply indebted to my
parents for their lifelong dedication and care toward their youngest child.
I would like to thank not only my colleagues who have directly contributed to this
dissertation, but all my friends who have indirectly contributed by helping me lift up my
spirit in tough times. My special thanks to Stella, Pablo, and Haeri, from whom all of my
emotional strength and comfort have sprung during my six years in the States. All the
laughs and cries we have shared like a family were all poured into this paper.
I would like to acknowledge my staffs, Tammie, Dawn, Beth, Chris, Larry, and Rick for
rendering their assistance in no time whenever I needed them.
I am very grateful for Professor Bob Jones and Professor Jongsoo Yoon in the annual
research review for their efforts to keep me on track in my study.
I would also like to thank Professor Bob Jones, Professor Louis Bloonifield. and Professor
Ian Harrison for sharing their invaluable insights and time serving on my Ph.D dissertation
committee.
Last but not least, I am so grateful to have Professor Tom F. Gallagher as my advisor
for 5 years. The insurmountable knowledge and enthusiasm he possesses have strengthened
my devotion toward the field of physics and inspired me to become a better researcher every
day. His dedication to his students encouraged them to always knock on his door with new
ideas and suggestions, with anticipation that they would be given positive feedbacks coupled
with constructive criticism. He has been my role model and a mentor in life: he will remain
ii
as is as long as I continue my journey in academia.
Contents
1 Introduction
1.1 The dipole-dipole interaction between cold Rydberg atoms
1.2 Thesis outline
1
1
4
2
8
9
9
11
16
23
26
30
30
34
37
38
39
41
Experimental Approach
2.1 Magneto-Optical Trap
2.1.1 How the Rb-MOT works
2.1.2 780nm-lasers
2.1.3 Magnetic field
2.1.4 Ultra High Vacuum(UHV) chamber
2.1.5 Properties of the MOT
2.2 Excitation laser for Rydberg atoms
2.2.1 960nm beam and amplification
2.2.2 480nm laser pulse
2.2.3 Measurement of the number of Rydberg atom
2.3 Microwave field pulse
2.4 Detection and data collection
2.5 Summary of daily operation
3 The dipole-dipole interaction of Rb nsnp atoms
3.1 Introduction
3.2 Theory
3.2.1 The dipole-dipole interaction of spinless atoms
3.2.2 The dipole-dipole interaction including spin-orbit, coupling
3.3 Experimental Approach
3.4 Experimental results
3.5 Lineshape model of ns - npj(j=l/2 and 3/2) transitions
3.5.1 Transition strength
3.5.2 Finding the inter-atomic distance R
3.5.3 Lineshape fitting to the observations
3.6 Concluding remarks
4
44
44
46
46
54
58
62
68
68
73
76
82
The ionization of Rb Rydberg atoms in the attractive n s n p dipole-dipole
potential
87
4.1 Introduction
87
4.2 Experimental Approach
90
CONTENTS
4.3
4.4
4.5
4.6
4.7
Observations
Modeling
High Microwave Power Observations
Conclusions
NOTE
iv
92
97
105
107
110
5 Probing the fields in an ultracold plasma by microwave spectroscopy
5.1 Introduction
5.2 Experimental Approach
5.3 Observations and analysis
5.4 Conclusions
118
118
120
123
137
6 Future Directions
140
7 Conclusions
155
V
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
A schematic diagram of the Zeeman energy levels of J=0 and 1 in the pres­
ence of inhomogeneous magnetic field. Due to the Doppler shift, counterpropagating photons transfer momentum to atoms via scattering, resulting
in a damping force to escaping atoms(Doppler cooling). Linearly varying
magnetic field provides restoring force towards the intersection of lasers. The
combination of the Doppler cooling and gradient-field confinement holds cold
atoms
The hyperfine structure of 85 i?6 5Sj/ 2 and 5p 3 / 2 . A small fraction of excitation
to 5p3/2 F=3 off-resonantly by the Trap laser and its spontaneous decay to 5s
F=2 cause rapid loss of trapped atoms. The Repump laser drives the fallen
atoms back to 5/ j 3/ / 2 F=3 level from which atoms can fall again back to 5 si / 2
F=3 level
The saturated absorption spectra for the Rb 5sj/ 2 — 5j>3/2 hyperfine transi­
tions and their crossover peaks taken by Anderson [4]. The broken arrows
indicate the side-lock points for the trap and repump lasers, respectively. . .
The resonances of MW transitions of the 38s-38pi/ 2 transition with different
switch-off delay times, (a) shows two peaks in the presence of the magnetic
field. From (b), the two peaks start merging and becoming narrower with
longer switching-off delay times in (c)-(h). A clear Rabi feature on the side
lobe is observable in delay times longer than 3 ms
The variation of the number of Rydberg atom as a function of the switchingoff delay time. Due to the increasing loss of trapped atoms, the production
of Rydberg atom decreases as the delay time is increased
The linewidth of 38s —38pi/ 2 resonances as a function of a nulling coil current.
The resonances reach the minimum linewidth at 0.14 Amp where the residual
field is minimized
The 38s - 38pi/ 2 resonances with different nulling coil currents. From (a) to
(d), the Rabi lineshape is enhanced by adjusting nulling coil currents with
10 mA step, while the currents of the other coils are fixed
The sorption pump diagram. P1-P3 and VI-V4 represent the batteries of the
sorption pump and the valves. A thermocouple vacuum gauage is attached
to measure the vacuum pressure
A typical measurement of the rising curve of MOT
The amplitude of the MOT signal from the CCD linear array, vacuum pres­
sure, and MOT rising time as a function of letter
A schematic diagram of the alignment of 960nm laser and TA
10
12
15
18
19
21
22
23
27
29
31
LIST OF FIGURES
2.12 The measurement of the bandwidth of the 480 nm laser. While scanning the
piezo voltage of the laser grating to drive 5p3/2 — 38sj/ 2 transition, (a) the
spectrum analyzer signal and (b) the integrated signal of Rydberg atom are
simultaneously recorded
2.13 The measurement of the waist size of the 480 nm laser by the knife-edge
method, (a) The transmission of the 480 nm laser beam is measured as a
function of position displacement of the knife, (b) Derivative of (a) and its
fitting to a Gaussian function yield the size of waist of 480 nm laser, which
is 162 pm in this case
2.14 Oscilloscope traces of field ionization pulse(solid line) and its trigger (dotted
line) pulses. 15 V of DC input generates a pulse with 2.2 (j.s rise time and
600 V amplitude
(a) Two atoms with dipoles pi and p 2 aligned in the z direction are separated
by R. R is at an angle 6 relative to the z axis, (b) Two quantum atoms,
1 and 2, have their internuclear axis along the z axis and are separated by
R. Their dipoles can be either parallel or perpendicular to the z axis. The
linearly polarized microwave field is at an angle 9 relative to the 2 axis. . .
3.2 Energy levels vs R for a pair of spinless atoms. There is one nsns state, which
is symmetric, and both symmetric (solid line) and antisymmetric (broken
line) nsnp states. Transitions from the symmetric nsns state are only allowed
to the symmetric nsnp states, as shown by the arrows
3.3 (a) Energy level diagram. The 780 nm excitation of the 5p 3 / 2 state is con­
tinuous, from the MOT beams. The 480 nm excitation of the ns\j2 state
is pulsed, and a mm wave pulse drives the ns to np transition, (b) Timing
diagram showing the timing sequence on each shot of the pulsed laser. The
laser pulse is 8 ns long, the mm wave pulse is 500 ns long, and the field
ionization pulse has a rise time of 2.16 ps. The time-resolved signals due to
field ionization of the ns and np states are detected
3.4 Density measrements of the trapped 5p S / 2 and ns Rydberg atoms. A power
meter is used to measure the fluorescence power of MOT. To measure the
radius of the MOT, the power meter is replaced by a linear CCD array. The
density of the trapped atoms, which is assumed to be a spherical Gaussian
cloud, can be determined from these two measurements. Once the number
of trapped atoms is known, the number of Rydberg atoms is obtained by
measuring the filling time of the MOT and the reduced fluorescence power
when the 20 Hz-480 nm excitation laser is sent through the MOT to produce
Rydberg atoms. The waist of 480 nm beam at the focus and the density
distribution of atoms in the MOT determines the density distribution of the
Rydberg atoms
3.5 Recordings of the 39s to 39p^ 2 transition at average densities p a v , of 0.04,
0.16, 0.42, and 0.59 xlO 9 cm" 3 , showing the asymmetric broadening of the
resonance. The broken line is the off resonance background 39pi/2 signal.
(left inset) Width vs. average density, (right inset) Observed resonance at
0.59 x 10 9 cin - ' 1 (solid line) and Lorentzian fit to the resonance (dotted line)
showing the cusped shape of the observed resonances
vi
35
36
40
3.1
47
53
59
61
03
LIST OF FIGURES
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
The widths (FWHM) of the resonances as a function of the average density,
p a v , showing the linear increase with density, (a) ns — np\/2 (b) ns — np 3 / 2 Observed broadening rates of the ns — npj/ 2 and ns — np3/<2 transitions vs.
n 4 /10 6 . They are obtained from the slopes of the width vs. density plots
shown in Fig.3.6. The broadening rates exhibit the expected n 4 scaling but
are lower than the initial estimate given by Eq.(3.23), shown by the solid line
Resonance widths (FWHM) of ns — np 3 / 2 transitions in a magnetic field of
~ 0.5 G vs. density
Stick spectra expected for nsns to nsnp transition of spinless atoms, the
nsns to nsnp 1/2 transition, and the nsns to nsnp 3 / 2 transition. The atoms
are assumed to be at fixed R such that the maximum shift of a spinless atom
state is ±10 MHz. The solid bars are transitions between symmetric states,
and the shaded bars are transitions between asymmetric states. There are
no asymmetric transitions for spinless atoms. The weighted average shift is
zero for both the symmetric and antisymmetric states in all cases
Geometry of the MOT and the 480 nm laser beam, which together define the
volume of Rydberg atoms
Relative number of atoms \)n{po/p)) 1 ^ 2, as a function of density relative to
the density at the center of the trap, p/po
The effect of averaging over density on the line of Fig.3.9 with a frequency
shift of -10 MHz for the internuclear spacing R = R av - (a) The line for
R — R av . (b) Averaging over the range of internuclear spacings at the density
p(r) = 3/4nR av , showing the broadening to larger shift, (c) Averaging over
the different densities p(r) at different positions r in the trap. The low density
regions of the trap substantially reduce the shift
Observed (solid lines) and calculated (broken lines) 44s — 44pi/2 resonance
lineshapes at the experimental average densities 0.04, 0.17, and 0.47 xlO 9
cm -3 . The atomic frequency vo =45114.68 MHz. In all cases the broken lines
are at the level of the background signal far off resonance. To match the ob­
served lineshapes requires densities a factor of two higher in the calculations.
Specifically, the densities of 0.07. 0.35, and 0.88 xlO 9 cm"" 3 are used in the
calculation
Observed (solid lines) and calculated (broken lines) 44s - 44p 3 / 2 resonance
lineshapes at the experimental average densities 0.04, 0.15, and 0.29 xlO 9
cm -3 . The atomic frequency vq =46345.42 MHz. In all cases the broken
lines are at the level of the background signal far off resonance. As in Fig.
12, to match the observed lineshapes requires densities a factor of two to four
higher in the calculations. Specifically, the densities of 0.05, 0 28, and 0.63
x 10 9 cuf 3 are used in the calculation
Observed (solid lines) and calculated (broken lines) lineshapes for ns — npi/2
transition of n= 28, 34, 39. and 44 at relatively high densities. In all cases
the broken lines are at the level of the background signal far off resonance.
For rt =28, 34, 39, and 44, the experimental average densities are 0.75, 0.79,
0.76. and 0.47 xlO 9 cm -3 , respectively, and densities used in the calculations
are 1.31, 1.55. 1.4. and 0.88 xlO 9 cm" 3 , respectively
vii
64
66
67
72
73
75
78
79
80
SI
LIST OF FIGURES
4.1
4.2
.
4.3
4.4
4.5
(a)Typical energy levels for the experiment. The 480 nm laser excites pairs
of ns atoms to the nsns potential over a range of internuclear spacings, as
shown by the slanted solid line arrows. A microwave pulse, shown by the
dotted line arrows, is used to drive the transition to either the nsnp state or
the ns{n—l)p state, which can either be attractive or repulsive, depending on
the microwave frequency. Pairs excited to a repulsive potential move apart,
and those excited to an attractive potential collide, resulting in an ion and a
more deeply bound atom, (b) Timing diagram for the nsns—nsnp transition.
The 8 ns long 480 nm laser pulse excites the atoms to the ns state, and is
immediately followed by a 500 ns long microwave pulse which drives nsns
pairs to the nsnp state. After a time delay r, a field ramp is applied to
observe the time-resolved ion, np, and ns signals
Field ionization signal with the excitation laser tuned to the 5/>3/2 — 41s
transition and no delay between the laser and PFI pulse. As the atomic
density is increased, ions and atoms in high-lying states as well as 41s atoms
are observed. The broken lines indicate the gates to measure the number of
ion and atoms
The number of ions and 41p atoms with no delay as a function of the number
of Rydberg atoms measured in Fig. 4.2. Both graphs of ions and 41p atoms
vs. total Rydberg atoms are not quite linear but slightly quadratic
Ion signals obtained with a delay of 5 ps and a microwave power producing
a 10 MHz linewidth after excitation to the 40s state, (a) Ion signal obtained
in the vicinity of the 40s - 40p transition at 61332 MHz. (b) Ion signal in
the vicinity of the 40s - 39p transition at 70262 MHz. Although the former
transition is to a state of higher energy and the latter to a state of lower
energy, the ions in both cases are observed only in the attractive potentials.
Ion signals vs microwave frequency observed subsequent to a delay r after
driving the 41s41s to 41s41p transition. The peak Rydberg atom density is
Po = 6 x 10 9 cm -3 . The delay time r is raised from 0 to 11 /us. Traces for
later times are offset vertically and are on coarser scales. The left hand side
of each trace shows the zero signal level. The off-resonant, or background,
ion signal rises monotonieally with delay time, and for r > 5 fJ.s there is a
visible increase in the number of ions at frequencies just below the atomic
41s - 41p transition at 56629 MHz. For delays of 9 and 11 ps the ion signals
are distinct. Also distinct in the 9 and 11 ps traces are the random spikes in
the ion signal. They originate from the spontaneous evolution to a plasma
on intense shots of the laser leading to an abnormally large number of 41s
atoms
viii
91
93
94
96
98
LIST OF FIGURES
41 p signals vs microwave frequency observed simultaneously with the ion
signals of Fig. 4.5. The time delay r increases from 0 to 11 ps. Traces
for later times are offset vertically. The left hand side of each trace shows
the zero signal level. The off resonant, or background, 41p signal increases
until the delay reaches 5 p,s, then it decreases. With short delay times, the
observed lineshape is nearly symmetric, but when the delay is longer than
5 ps the low frequency side of the lineshape begins to disappear since pairs
excited to the attractive potential are lost to ionization. For reference, we
include a thin trace of the rescaled low density signal obtained with the same
microwave power for r = 0 ps
4.7 Construction of the ion signal after a 9 ps delay for the 41s41p state with
a — —2/9. (a) Infinite resolution spectrum expected for the distribution of
atoms in our trap (thin line). The ionization signal comes only from atoms
which collide in 9 ps, i. e. only from frequencies u < v cut (bold line), (b) 4
MHz Lorentzian representing the experimental resolution, (c) Convolution
of (a) and (b), which is the contribution to the ion signal from the 41s41p
state with a — —2/9. There is an analogous contribution for the state with
a = -4/9
4.8 Observed and calculated signals obtained using the 41s state after a delay
of 9 ps before the field ionization pulse, (a) Observed (solid line) and calcu­
lated (dotted line) ion signals. The random spikes are due to a spontaneous
evolution to plasma on unusually intense shots of the laser, (b) Observed
differences between the scaled zero delay and 9 ps delay 41p signals (solid
line) and calculated ion signal(dotted line), (c) Observed (solid line) and
calculated (dotted line) 41p signals
4.9 Observed ion and 41p signals after a delay of 7 //s at three microwave powers
leading to linewidths of 4, 8, and 16 MHz with low density-atomic samples,
showing the evolution to plasma following the high power microwave pulse.
The peak Rydberg atom density is po = 1 x 10 10 cm" 3 . In all cases the atomic
41s — 41p transition frequency, 56629 MHz, is shown by the broken vertical
line. (a)Ion signal with a microwave power of 4 MHz-linewidth. There is
greater ionization at frequencies below the atomic transition frequency , but
only on intense shots of the laser, (b) 41p signals taken with a microwave
of 4 MHz-linewidth with 0 (dotted line) and 7 ps (solid line) delays. The
lineshape is symmetric with no delay, but there is a noticeable asymmetry
after 7 ps. (c) Ion signal with a microwave power of 8 MHz-linewidth. There
is a clearly visible ion signal over a frequency band 50 MHz wide below the
atomic frequency, (d) 41p signals taken with a microwave power of 8 MHzlinewidth with 0 (dotted line) and 7 ps (solid line) delays, (e) Ion signal with
a microwave power of 16 MHz-linewidth. The ion signal now extends over a
100 MHz wide frequency band, and it is too large to be explained by only
the atoms which have followed attractive potentials and undergone collision.
The collisional avalanche creating the plasina is underway, (g) 41p signals
taken with a microwave power of 16 MHz-linewidth with 0 (dotted line) and
7 //s (solid line) delays, the lineshape is symmetric with no delay, but has a
pronounced asymmetry after 7 /us
ix
4.6
99
100
104
106
LIST OF FIGURES
x
4.10 Ionization signals obtained after 7 fjs with no microwaves(O), +20 MHz,
repulsive, detuning (•), and -20 MHz, attractive detuning (•) and the three
microwave powers producing linewidths of 4, 8, and 16 MHz with low density
atomic samples vs the number of Rydberg atoms initially excited, (a) With
a microwave power of 4 MHz-linewidth, a barely discernible difference exists
between tuning to the attractive potential and the other two cases, (b) With
a microwave power of 8 MHz-linewidth, there is a clear difference between
excitation to the attractive potential and the the oterh two cases, which are
indistinguishable, (c) With a microwave power of 16 MHz-linewidth, there is
now a large difference between excitation to the attractive potential and the
the other two cases, which axe still indistinguishable
108
4.11 The ion signal as a function of the driving microwave field strength. The
broken line indicates the 41s - 41p transition frequency. Eq represents a rela­
tive microwave field strength. As the field increases, increased ion production
extends over the low frequency side of the atomic transition. When power is
extremely increased, two peaks of ion production are observed as indicated
by the arrows
Ill
4.12 The 41s — 41p resonances simultaneously measured with Fig. 4.11 as a func­
tion of the driving microwave field strength. The broken line indicates the
41s — 41p transition frequency. Two arrows indicate the frequencies where
the ion production reaches maximum
112
4.13 The sum of the ion and 41p signals as a function of the microwave field. . . 113
4.14 The power broadening of the total signal, which is the sum of the ion and
41p signals
114
4.15 The ionization signal as a function of the microwave frequency observed with
extremely high microwave power by Li [21]. Two peaks of ionization from
39s39p 3 /2 attractive potential are observed similar to Fig. 4.11
115
5.1
Timing and energy diagrams for the experiment. The first, plasma laser pulse
produces the ultracold plasma and the plasma temperature is determined
by the laser frequency. The second, 42s laser pulse, which is delayed by
time t, produces the probe 42s atoms. The microwave pulse immediately
after the second laser pulse is used to drive the 42s to 42p transition, which
is detected by selective field ionization of the 42p atoms during the field
ionization pulse. As shown., the 42p signal is earlier than the 42s signal. The
shift and broadening of the microwave transition are used to determine the
fields in the plasma
120
5.2 Schematic diagram of the apparatus. The six 780nm MOT beams and the
vacuum envelope are not shown, and the microwave horn is outside the vac­
uum envelope. The 480nm laser beams produce cylindrical volumes of plas­
mas and Rydberg atoms
122
•5.3 The resonance shifts of 42s - 42p transition in the presence of various static
fields. The peak shifts to lower frequency as the static field is raised. ... 124
5.4 The plot of the peak shift, vs. dc field observed in Fig.5.3. The peak shifts
quadratically and is fit to 6v — BE 2 yielding 6 = 1.18 MHz/(V/cm) 2 . . . . 125
LIST OF FIGURES
5.5
5.6
5.7
5.8
5.9
5.10
5.11
6.1
6.2
XI
42sj/2 to 42pi/2 resonances in a nearly neutral plasma. The dash-dot lines
represent Lorentzian fits of the experimental data. The laser producing the
plasma is tuned 1 cm -1 above the ionization limit to produce a nearly neu­
tral plasma and the number of ion in the plasma is varied by adjusting the
intensity of the laser. As more ions are produced, the resonant peak shifts
further to the low frequency side due to the increasing microscopic field. . . 126
42si/2 to 42pi/ 2 resonances in the presence of plasmas with different electron
temperatures. In all cases the number of ions is 4 x 10 4 . The dash-dot lines
represent the Iineshape model. In (a) T e = lK, there is only a microscopic
field. For all higher temperatures, the microscopic field exists as background
for the macroscopic field produced by the excess ions. As the laser frequency
is tuned further above the ionization limit, the resonant peak tends to shift
further to the low frequency side and shows a non-Lorentzian profile due to
the increasing charge imbalance between the electrons and ions in the plasma. 128
Radial ion and electron distributions. The width of the electron cloud is
determined by the best fit of the Iineshape model in Fig.5.6. As the laser is
tuned further above the ionization limit, there are fewer electrons left in the
plasma since more electrons escape from the ion cloud due to their higher
kinetic energy
131
The microscopic and macroscopic fields of plasmas with various temperatures.
The microscopic field is dominant in the T e =lK plasma since the plasma is
nearly neutral. In the other plasmas, the macroscopic field is stronger than
the microscopic field
133
42sj/2 to 42pi/2 resonances as a function of delay time between the plasma
laser and the probe laser. The dash-dot lines represent the best fits of the
Iineshape model to the observed resonances. At early times, until 2.5 /is, the
resonant peak shifts to lower frequency due to the creation of the excess ions.
For delays in excess of 2.5 fj.s the peak shifts back to the atomic frequency
due to the expansion of the ion cloud
134
The size of the ion cloud as a function of the delay time, r. The square dot
represents the observed size of the expanding ion cloud and the solid line is
the best fit to the observation
135
The sum of the calculated microscopic and macroscopic fields for the best
fits to the observed data in Fig.5.9. The total field reaches its maximum in
the central region where the probe atoms (42s atoms) are located at 4 ^zs.
At later times the field at the location of the atoms decreases although the
peak field, at larger radial distance, continues to increase
136
The energy and timing diagrams for Ramsey experiment. The experimen­
tal procedure is very similar to the linebroadening technique, but here two
identical separated pulses are used
The observed Ramsey interference fringes, (a) The Ramsey fringes from
the low atomic density sample shows 100 % contrast whereas the increased
density washes out the contrast of the fringes, (b) The variation of the
contrast as a function of the atomic density for n=33, 36. 39, and 41
141
143
LIST OF FIGURES
The energy and timing diagrams of the dressed-state experiment. The dress­
ing field is tuned at the resonance of the np — (n + l)s and its pulse length
should be slightly longer than the probe field
6.4 The resonances of the 34s — 34p!/ 2 transition with a low density sample as
a function of the field strength of the dressing field. The length of the probe
field is 2 fj,s and the dress field is 2.4 p,s
6.5 The linebroadening of the 34s — 34pj/ 2 transition with the low dressing field
strength
6.6 The expected line broadening (thick solid line) due to the high density atoms
and the observation (light solid line)
6.7 The resonances of the 34s - 34p 3 / 2 transition with a low density sample as a
function of the field strength of the dressing field
6.8 The energy levels of the 40si/240p 3 / 2 molecular state in the presence of elec­
tric field E. The internuclear axis and the polarization of the field are aligned
(0=0) and the field is set such that m,j — 1/2'is above mj=3/2 at R = oo.
M— 1 states form a well with equilibrium position Ro—5-5 /j.m. The bound
state is indicated by an arrow
6.9 The energy and timing diagrams of the filtering-atom experiment. The filter­
ing field is tuned at the ns-(n-l)p resonance and applied before the probe
field arrives so that only close atoms participate in the ns - np transition. .
6.10 The resonances of the 34s - 34p x / 2 transition, (a) In the presence of the
filtering field tuned at the 34s - 33p 3 / 2 transition, the cusp is substantially
suppressed compared to that of whole atomic sample, (b) The vertical scale
is adjusted to directly compare the linewidth between the original and filtered
atomic samples
6.11 The signal from the removed atoms by the first pulse (light line) and the
signal from the whole atoms (thick line)
xii
6.3
144
145
146
147
148
150
151
153
154
xiii
List of Tables
3.1
3.2
3.3
3.4
3.5
3.6
3.7
Basis states for the spinless atoms of Fig. 3.1(b)
Symmetries, eigenvectors, and eigenvalues of nsnp states for the spinless
case of Fig. 3.1(b) "s" represents a symmetric state, and "as" represents an
asymmetric state
Basis states for spin-orbit coupled nsi^npj states
Symmetries, eigenvectors, and eigenvalues of the nsnp states in terms of the
basis states of Table III. "s" represents a symmetric state, and "as" represents
an asymmetric state
Broadening rates of the ns - npj transition
Initial nsi/2^ s i/2 states, "s" represents a symmetric state, and "as" repre­
sents an asymmetric state
Possible transitions, level shifts, and squared transition matrix elements . .
49
52
55
58
65
69
71
1
Chapter 1
Introduction
1.1
The dipole-dipole interaction between cold Rydberg atoms
A gas of Rydberg atoms is a fascinating system for the study of a broad range of
quantum phenomena [1, 2]. Rydberg atoms are highly excited atoms which have one or
more electrons with a large principal quantum number n. Due to the large n, Rydberg
atoms exhibit remarkable characteristics. Since the outer electron is far away from the
nucleus, the size of the atoms is huge compared to the Bohr radius, increasing like ~ n 2 .
For an alkali atom, when excited to a Rydberg state, it is a valence electron and ionic core.
In this sense, alkali Rydberg atoms are similar to hydrogen atoms. In a Rydberg state, the
dipole moment increases proportional to n 2 , the geometrical cross-section rises proportional
to n 4 , and the binding energy falls as ~ n~ 2 . Also, the radiative lifetime scales as n 3 . These
properties result in exaggerated response of the loosely bound electron to perturbations such
as electromagnetic fields and blackbody radiation.
On the other hand, since the adjacent energy gap decreases as rt -3 , higher resolution
excitation lasers are required to explore the Rydberg states. With developing laser tech­
niques, this difficulty has been resolved using monochromatic dye lasers. Furthermore, the
high resolution microwave field makes it possible to detect transitions between Rydberg
states with extremely high accuracy. This precise control of Rydberg state atoms can be
used to observe coupling between nearby Rydberg atoms and their motion on a relatively
Chapter 1. Introduction
2
long time scale. If Rydberg atoms are excited from a magneto-optical trap (MOT), since
the MOT not only confines but also cools down atoms, it provides a desirable cold atomic
sample when studying atom-atom interactions. For example, Rb atoms are cooled down to
300 /iK in this work, so they only move 0.24 /jm in 1 fis. Since the time scale of interest here
is microseconds, such a small thermal energy does not affect the atom-atom interaction in
most cases. Moreover, if the trapping magnetic field is switched off, we can minimize any
dephasing effects from the inhomogeneous trapping field.
One of the most interesting aspects of cold Rydberg atoms is their potential applications
for quantum information physics. For the first step towards quantum logic operations in a
quantum computer, intensive efforts have gone into making a quantum phase gate. As a
potential candidate for the quantum gate, a notion of dipole blockade has been proposed
and actively studied from theoretical and experimental perspectives. The idea is that, in
short, due to the dipole-dipole interaction of a pair of Rydberg atoms, it is possible to
excite one atom to a Rydberg state but not both atoms of the pair [3, 4, 5]. Accordingly,
understanding the dipole-dipole interaction is crucial to applying this mechanism to the
quantum gate and overcoming any realistic problems. That is, prior to the application,
intensive fundamental studies on the dipole-dipole interaction between atoms should be
conducted. Also, above all, since the dipole-dipole interaction plays the most important
role in the atom-atom interaction, its study will provide useful knowledge for atom and
molecular systems in a broad range of science from physics and chemistry to biological
fields.
In general, the dipole-dipole interaction is an electrostatic force between permanent
dipoles such as polar molecules, given by
T/
V
dd =
A • m - 3(^1 • R)(fT 2 • R)
^
, . ,%
(1.1)
where jix is the i t i, dipole, and R is distance between two dipoles [6]. As two dipoles approach
closer to each other, the interaction becomes stronger and the two dipoles tend to rotate
to minimize energy, exerting a torque depending on the angle between the dipoles. From
3
Chapter 1. Introduction
the quantum mechanical point of view, the notion of the dipole-dipole interaction can be
extended to neutral atoms, which have no permanent dipole moment. Instead, the transition
dipole moment of an atom can be thought of as the permanent dipole. The transition
dipole moment is a vector quantity which depends on the phase factor between two states
and charge distribution, and determines the properties of dipole-coupled states. When
two atoms are closely spaced, the transition dipole moments behave like the permanent
dipoles, forming a molecular state, and their relative orientation gives the polarization of
the transition. This polarization of the transition dipole moment determines how the atoms
interact with radiative fields [7j.
For example, the dipole-dipole interaction can be observed via Forster resonant energy
transfer(FRET) [8]. Consider the 49s41d atom pair. 49s is coupled to 49p by the transition
dipole element fi^Qsidp and 4Id is coupled to 42p by M4id42p- However, the energy spacings
are not the same, that is, W4g s _49 p
^ / 4jd-42 P - Using the dc St.ark effect, however, these
spacings can be easily tuned to resonance. As a result, these two atoms exchange energy,
and they oscillate between 49s and 49p and 41d and 42p, respectively. By observing the
production of either 42p or 49p as a function of interatomic distance, Ditzhuijzen et al. ob­
served the atomic transition by FRET induced by the dipole-dipole interaction. Apart from
the FRET example, the effects of the dipole-dipole coupling have been observed in other
types of experiment, due to the strong coupling in Rydberg states by the large transition
dipole moments [9, 10, 11].
Due to the importance of the dipole-dipole interaction, particularly in the large untapped
territory of Rydberg atom physics, this thesis work is largely focused on discovering the
properties of the dipole-dipole interaction and its effects on the Rydberg atom or molecular
system. This basic study on the dipole-dipole interaction is anticipated to be used in future
studies such as connections between Rydberg states and plasmas, ionization mechanism
from Rydberg states, and long range Rydberg-Rydborg molecules, etc.
Chapter 1. Introduction
1.2
4
Thesis outline
Subsequent chapters describe the experimental approach and several independent projects
that were performed during this thesis work. Each chapter includes an experimental de­
scription, but more details and common parts for all experiments can be found in chapter
2. Throughout the thesis, the atomic units are used unless noted otherwise.
In Chapter 2, fundamental information about the apparatus and experimental approach
is provided. The first year of the thesis work is focused on characterizing and improving the
density and stability of the MOT. This chapter is focused on operating the MOT at high
density and building a high intensity 480 nm laser so that a high density Rydberg sample is
achieved. Also, minimizing magnetic field is a mandatory task to perform high resolution
microwave spectroscopy, which is employed as a probe for most experiments in this thesis.
Determining the field-off timing, achieving a high density MOT and a high power 480 nm
laser are the most important experimental steps, since high density and a field-free Rydberg
sample are crucial in this thesis.
In Chapter 3, the most basic and fundamental work on the dipole-dipole interaction
is conducted. Theoretical approach describes the dipole-dipole interaction between nsnp
molecules introducing spin to atoms, starting from a classical picture. In this theoretical
work, we found that nsnp dipole-coupled state splits into several levels, some of which
are not shifted and some of which shift depending on internuclear spacing. Based on the
calculation, line broadening as a function of atomic density is computed and furthermore, a
lineshape model of nsns — nsnp transition is developed. Observations by the line broadening
technique are compared to the calculation of the broadening rate and the lineshape model.
In Chapter 4, the collisional ionization mechanism of Rydberg atoms is described. In the
experiment, atoms are transferred to both attractive and repulsive dipole-dipole potentials
and time delays are allowed to see which potential induces collisions and subsequently
ionization. Applying high microwave power makes the difference between the two potentials
more distinct. Based on the calculations in Chapter 3. we present a lineshape model for the
increased ion signal to compare to the observations.
Chapter 1. Introduction
5
In Chapter 5, we use Rydberg atom spectroscopy as an application of Rydberg atoms
to probe the plasma field. Using the idea that the Rydberg state is very sensitive to the
electromagnetic field, we produce a few Rydberg atoms within the plasma cloud to detect
the electric field from plasma. Due to the high sensitivity to small fields, we attempt to
measure microscopic field as well as the macroscopic field. This high resolution probe is
used to explore time evolution of plasma field.
In Chapter 6, the thesis work is summarized and conclusions are briefly discussed.
Finally, in Chapter 7, some experiments, which were not finished but worth continuing
will be shortly introduced.
6
Bibliography
[1] T. F. Gallagher. Rydberg Atoms. Cambridge University Press, 1994.
[2] R. F. Stebbings and F. B. Dunning. Rydberg States of Atoms and Molecules. Cam­
bridge University Press, 1983.
[3] M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P.
Zoller, Phys. Rev. Lett. 87, 037901 (2001).
[4] E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker and M.
Saffman, Nature Phys. 5, 110(2009).
[5] A. Gatan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet,
A. Browaeys, and Ph. Grangier, Nature Phys. 5, 115(2009).
[6] John David Jackson. Classical Electrodynamics, Third Edition. John Wiley & Sons,
Inc. 1998.
[7] Bruce W. Shore. The Theory of coherent Atomic Excitation, John Wiley & Sons, Inc.
1990.
[8] C. S. E. van Ditzhuijzen, A. F. Koenderink, J. V. Hernandez, F. Robicheaux, L. D.
Noordam, and H. B. van Linde nvan den Heuvell, Phys. Rev. Lett. 100, 243201 (2008).
[9] A. Reinhard. K. C. Younge. T. C. Liebisch. B. Knuffman, P. R. Berman, and G.
Raithel, Phys. Rev. Lett. 100, 233201 (2008).
[10] R. Heidemann, U. Raitzsch, V. Bendkowsky, B. Butscher, R. Low, L. Santos, and T.
Pfau, Phys. Rev. Lett. 99, 163601 (2007).
BIBLIOGRAPHY
7
[11] T. J. Carroll, K. Claringbould, A. Goodsell, M. J. Lim, and M. W. Noel, Phys. Rev.
Lett. 93, 153001 (2004).
8
Chapter 2
Experimental Approach
In the early stages of conducting research, my greatest challenge was to achieve useful
experimental conditions and more important, to maintain them daily. Except for the optical
tables, all laboratory equipment degraded with time. For example, the dye solution had to
be replaced every 6 months (depending on the type of dye), distilled cooling water had to be
re-filled every 3 months, and the diodes were easily damaged; there was never a day to start
an experiment without prior prep work. The trickiest part in setting up the experiment
was opening the chamber to repair or replace parts in it. Despite such painful efforts, it
was worth opening the chamber, since I learned from it. In that sense, I was lucky to have
had the chance to handle the vacuum chamber myself to gain experience.
In this chapter, I provide a thorough analysis of the experimental set up as well as a few
recommendations to maintain the equipment in good shape. However, I have skipped some
details in the procedures to avoid unnecessary repetition of well known techniques written
in former dissertations of the lab and instead have provided the references that best describe
them. In managing the magneto-optical trap(MOT), I would especially like to recommend
Anderson's dissertation, which provides good theoretical explanation as well as details of
the equipment.
This chapter consists of several sections, which come in the order of the experimental
procedures- I. preparation of sample(MOT), 2. excitation of Rydberg atoms, 3. probing
the interaction between Rydberg atoms, and 4. collecting data. At the end of this chapter,
Chapter 2. Experimental Approach
9
a summary of daily operation and some suggestions will be given.
2.1
Magneto-Optical Trap
The magneto-optical trap(MOT) basically consists of two parts, an inhomogeneous mag­
netic field which has a zero field point and laser beams that cross at the zero field. In the
laboratory, the MOT requires many more subparts to produce a high performance stabilized
atomic trap. The optical part of the MOT requires a narrow band laser centered at 780 nm,
the laser frequency held stable to within 1 MHz long enough to perform an experiment, and
precise alignment of the lasers. The magnetic field of the MOT requires the anti-Helmholtz
coils to generate a gradient magnetic field, three pairs of nulling coils with proper currents
to cancel the earth's and stray magnetic fields, and a field-switching-off circuit for field-free
experiments. Apart from these requirements, other practical and small components are
required for better performance.
In the following subsections, I first describe how the Eb-MOT works and then provide
detailed information for the subparts.
2.1.1
How the Rb-MOT works
Since Anderson [1] discussed the full details of the MOT, a simplified description is given
here. Since the Rb-MOT works using the energy level shifts caused by a gradient magnetic
field, learning about the Zeeman energy level is critical. Fig. 2.1 shows a simple example
where only two participating J-levels are considered for simplicity. In the presence of the
magnetic field, the excited state J=1 level splits into three levels, the MJ= 0, ±1. The
± 1 levels, respectively, shift up or down linearly depending upon the spatially varying field
strength. We shall use these two linearly varying levels and one flat ground J=0 level.
Initially, Rb atoms are in the ground level moving in a random direction and with a
random velocity. Let us take as an example atoms, shown in Fig.
2.1, with position x
> 0 and traveling in the +x direction with velocity v + (escaping from the trapping region)
and the trapping la.ser is red-detuned, slightly below the m=0 level of the excited state.
Chapter 2. Experimental Approach
10
energy
0 J= 1
12 MHz
an
ja -mm
position X
^ field B
Figure 2.1: A schcmatic diagram of the Zeeman energy levels of J=0 and 1 in the presence of inhomogeneous
magnetic field. Due to the Doppler shift, counter-propagating photons transfer momentum to atoms via
scattering, resulting in a damping force to escaping atorns(Doppler coolmg). Linearly varying magnetic field
provides restoring force towards the intersection of lasers. The combination of the Doppler cooling and
gradient-field confinement holds cold atoms.
Chapter 2. Experimental Approach
11
When the atoms encounter photons, which are circularly a~ polarized and traveling in the
-x direction, they see a slightly higher frequency than that of the laser, due to the Doppler
shift. If the Doppler shift of the atoms compensates the lack of energy between the laser
and the m=-l level, atoms absorb the photons of the laser and emit spontaneously. At that
moment, the atoms feel a damping force as they gain photons in the -x direction and emit,
on average, isotropically, resulting in a momentum transfer from the photon to the atom in
the opposite direction of the atom's initial motion. Similarly, atoms escaping towards the
-x direction scatter photons traveling in the -fx direction with cr + polarization, and as a
result, the atoms are slowed by the damping force. This mechanism applies to the y and z
directions in the same way, and the result is a spherical atomic cloud.
In a realistic Rb-MOT, the energy levels axe not a simple 3-level system. The more
complicated hyperfine levels are involved as shown in Fig. 2.2. To apply the above mech­
anism, we choose the 5s!/ 2 F=3 level for the ground state and 5p 3 / 2 F=4 for the excited
state. Both split according to the linearly varying magnetic field, but the ideas of cooling
and confining are the same as the description in Fig. 2.1.
2.1.2
780nm-lasers
Trapping laser Due to the relatively small spacings between hyperfine levels of 5p 3 / 2 ,
the laser should be narrow enough in its frequency band to drive only the correct transition
as shown in Fig.
2.2. Beside the transitions from 5F=3 to 5p3/ 2 F=2, 3, and 4,
crossover peaks between them exist in the saturated absorption spectrum(SAS), which is
adopted for frequency stabilization, and thus the structure of the SAS is more complicated
than Fig. 2.2. To distinguish the closely spaced levels and precisely red-detune the F=3-4
transition, the laser bandwidth is required to be less than 5 MHz. The red-detuning of the
trap laser is about 12 MHz from the 5sj/ 2 F=3 to 5p 3 / 2 F=4 transition, and it varied daily
in a range of 11 ~ 13 MHz, depending upon which detuning maximized the MOT signal.
To obtain that narrow la.ser. an external cavity diode laser is employed. The diode
that we use is a Sanyo DL7140-2G1 diode centered af 785nm with power 70mW when
free running. Mounting a grating in front of the diode at a certain angle forms a cavity,
Chapter 2. Experimental Approach
12
120
MHi
Trap beam
780.030 nm
Repumpbeam
780.024 nm
F=2
Figure 2.2: The hyperfine structure of 85 Rb 5 si / 2 and 5p 3 / 2 . A small fraction of excitation to 5p 3 / 2 F=3
off-resonantly by the Trap laser and its spontaneous decay to 5s F=2 cause rapid loss of trapped atoms.
The Reputnp laser drives the fallen atoms back to 5p3/2 F=3 level from which atoms can fall again back to
OS1/2 F=3 level.
Chapter 2. Experimental Approach
in which the first order of the diffracted beam goes back into the diode.
13
With proper
adjustment of the grating angle(horizontal and vertical), a laser beam of 1 MHz-width with
28 mA threshold current can be obtained. Once satisfactory alignment is obtained, only the
horizontal knob is used to change the crude wavelength of the laser. For precise adjustment
of the wavelength, temperature and current controllers are used.
As shown by Fig. 2.2 the trap laser drives the F=3 to F=4 transition, which is a cycling
transition. The trap laser can not only drive the F=3 to F=4 transition but also drive F=3
to F=3 (a small fraction about 1 %). The problem is that excited atoms in the 5p 3 /2 F=3
level spontaneously decay to 5s F=3 and F=2. From the latter the fallen atoms cannot be
driven again to 5p3/2 by the trap laser, and they are lost. To fix this, we use the repump
laser, which drives 5s F=2 atoms back to 5p 3 / 2 F=3 through the transition 5s F=2 to 5p 3 /2
F=3 so that 5p 3 / 2 F—3 atoms can spontaneously decay back to bsi/ 2 F—3 to supply ground
state atoms. Thus the F=3-4 driving laser is called the trap laser and the F=2-3 one is
called the repump laser.
Saturated Absorption Spectroscopy (SAS) Although narrow band lasers are ob­
tained, well defined frequencies matching the atomic transitions are needed; 5s F=3 to 5p 3 / 2
F=4: 780.030 nm and 5s F=2 to 5p 3 /2 F=3: 780.024 nm. For starters, a wavelength meter
is used to reach 780.030 nm and 780.020 nm with 0.001 nm resolution, but a more sophis­
ticated method is required to determine the exact wavelength. Therefore, SAS is adopted
for two purposes. First, precise transition frequencies can be determined by removing the
1st order Doppler broadening. Second. SAS generates a reference signal to lock the laser
frequencies.
Let us review shortly how the saturated absorption spectrum can be produced.
A
laser beam tuned at about 780 nm is sent, through a Rb vapor cell and a mirror reflects
the out-coming beam back through the cell again along the same path. At the end, the
beam, which experienced the medium twice, is delivered to a photo detector to generate
the saturated absorption spectrum. The first pass of the light produces a normal Doppler
broadened spectrum about 1 GHz wide, which is an absorption curve from atoms with
Chapter 2. Experimental Approach
14
random velocities. But the second pass of light generates a completely different spectrum
due to the existence of zero-velocity atoms. When photons with the Doppler free frequency,
vo, travel through the medium, they are absorbed by the zero-velocity atoms, and in the
second pass, the photons see the same steady atoms again reducing the absorption. However,
photons with any other frequencies see different atoms on the two passes. As a result, the
light absorption efficiency by the zero-velocity atoms is less than by other atoms when
the absorption spectrum is observed on the photo detector. The smaller absorptions at
atomic transition frequencies appear as sharp spikes within a broad Doppler spectrum.
This technique not only separates the hyperfine transitions but also determines the absolute
transition frequencies.
Although the lasers are set to the exact frequencies, they still need further care to
maintain frequency. Frequencies easily drift away from predetermined values, since the
diode laser, which is the light source, is very sensitive to mechanical vibrations, temperature,
current fluctuations, and more. It is impossible to keep the lab silent, steady, and stable
enough to hold the diode frequency stable even for a second. To overcome this difficulty,
the laser frequencies were locked to one of six SAS peaks, three hyperfine transition peaks
and three crossover peaks as shown in Fig. 2.3 [1]. The crossover peak arises from non-zero
velocity atoms at mean frequency of each pair of the real hyperfine transition peaks. For
the trap laser, the laser frequency was side-locked to the high frequency side of the crossover
peak, which is located half way between the F=3-3 and F=3-4 resonances. The trap laser is
side-locked where a broken arrow is drawn in Fig. 2.3. Since we are using an acousto-optic
modulator(AOM), the laser frequency is shifted down by 42 MHz before passing through the
SAS cell, and the actual laser frequency is red-detuned by 12 MHz from F=3-4 transition.
The repuinp laser frequency is locked near to the F=2-3 resonance, also indicated by a
broken arrow. For each laser a servo-locking box, which is connected to a piezo behind the
grating in the laser, sends a voltage to tune the drifting wavelength back to the original value
by adjusting the grating angle. Once the laser is locked, acceptable stability is maintained
typically for more than an hour. However, the slightest clapping sound nearby can break
the lock, so the room is kept as silent as possible during experiments.
Chapter 2. Experimental Approach
15
Saturated Absorption Spectra
Ftt>
Rb
F=3— F'
F=»3— F'
Bb fls F=2— F'
Rb°
Fs
o
200
Relative Frequency (MHz)
Figure 2.3: The saturated absorption spectra for the Rb 5si/2~ 5p^/2 hyperfine transitions and their crossover
peaks taken by Anderson [4], The broken arrows indicate the side-lock points for the trap and repump lasers,
respectively.
Chapter 2. Experimental Approach
16
Master/Slave lasers We have always searched to achieve a higher density atomic
cloud, or more closely spaced atoms, since the strength of the interaction is proportional
to R~ 3 , assuming that we are interested in the dipole-dipole interaction. Here R is the
distance between atoms. In order for a high density experimental sample to be obtained,
making a high density MOT is the top priority because the trapped atoms are the source
of Rydberg atoms. Putting aside the magnetic field gradient and the magnitude of laser
detuning, the number of trapped atoms increases as the laser intensity increases, until it is
saturated.
The power loss of the MOT lasers comes first from the external cavity of the diode laser.
Out of 60 mW free running power, only about 10 mW of the narrow spectral line beam
comes out of the cavity and a fraction of the outcoming beam is sent to the Rb vapor cell
to produce a SAS signal. Afterwards, the multiple reflections from mirrors, transmission
through a beam-shaping prism, and several optical isolators, etc. consume another 50% of
the beam power. Finally, a beam power of 4 mW, at most, reaches the vacuum chamber,
where the beam is split into three ways for a three dimensional MOT, with 1 mW for each
split beam. To overcome the power loss, a slave-master laser system is employed for both
the trap and repump lasers. Most of the processes resulting in power loss are done using
the master beam, and the rest of the master beam is sent into the slave diode to produce
exactly equal narrow band light with about 50 mW of power. Before the 3-way splitting,
the beam power is measured to be about 30 mW, which is sufficient to saturate the number
of trapped atoms.
2.1.3
Magnetic field
anti-Helmholtz coils: The anti-Helmholtz coil is employed to generate a gradient of
magnetic field(theoretical description is provided in the earlier section). It consists of two
coils that carry the same amount of current, but in opposite directions, so that the fields
from each coil point in opposite directions and cancel to zero at the midpoint of the two
coil planes. Each coil frame, which is aluminum holding 95 turns of wire, contains a water-
Chapter 2. Experimented Approach
17
jacket and a plastic wedge. Water runs through the coil frame to cool the coils. Otherwise,
the heat destroys the insulation of the coils. The plastic wedge in the middle of the frame
works to suppress eddy currents when the current is in the switching mode, which will be
described in the following paragraphs.
The current is supplied in two different ways. One is simply a constant current so that
the field gradient is present at all times, and it is called "non-switching mode". The other
current mode is the "switching mode", which diverts the current from the coils at a 20
Hz repetition rate. The non-switching mode is used in experiments that demand a high
density MOT, for instance, in the ionization experiment due to an attractive dipole-dipole
potential(Chapter. 4). The non-switching mode can also be used in order to investigate the
magnetic, field effects.
In most experiments, however, the presence of the magnetic field is unwanted because
it, causes complications and the magnetic, field effects are not of interest in the present
research. Accordingly, a switching circuit, which was designed by a former researcher [2],
is used. Using a trigger pulse synchronized with the excitation laser, the switching circuit
diverts the current 4 ms before the laser shot arrives and turns the current back after 1 ms
to recover the MOT. So, the current is off for 5 ms and on for all other times. The delay
time of 4 ms was chosen in the following way. By driving 38s — 38pi/ 2 MW transitions
at various delay times with a low atomic density sample, the residual magnetic field as a
function of delay was determined. As shown in Fig. 2.4, due to the presence of a significant
amount of magnetic, field at shorter delay times(up to 2 ms), the resonances do not show
Rabi-lineshape features on the side lobes, whereas delays longer than 3ms produce the
narrow Rabi-lineshapes. Although the longer delay time can form a more field-free MOT,
the demand for a high density MOT limits the time delay. To determine the optimal time,
the number of Rydberg atom is measured as a function of the delay time. Since the intensity
of the excitation laser is fixed, the measurement in Fig.
2.5 indirectly shows the number
of trapped atoms as a function of delay time. Based both on Fig. 2.5 and Fig. 2.4, 4 ins
is the optimum delay time for field-free, high density MOT.
Chapter 2. Experimental Approach
18
(h) 10 ms
(g) 6 ms
w
'c
•Q
I—
03
(d)3
,
Q.
00
CO
ft
k
J I
•-]
'1
W2
A
:
...
MW frequency [X4 MHz]
Figure 2,4: The resonances of MW transitions of the 38s — 38pi/2 transition witli different switch-off delay
times, (a) shows two peaks in the presence of the magnetic field. From (b), t he two peaks start merging and
becoming narrower with longer switching-off delay times in (c)-(h). A clear Rabi feature on the side lobe is
observable in delay times longer than 3 ms.
Chapter 2. Experimental Approach
19
1.5x10 -
£
o
TO
B
Q>
-Q
"O
1.2x10 -
ii i
EC
.2
9.0x10 H
iS
3
Cl
O
CL
6.0x10 4
10
M-switching delay [ms]
Figure 2.5: The variation of the number of Rydberg atom as a function of the switching-off delay timeDue to the increasing loss of trapped atoms, the production of Rydberg atom decreases as the delay time is
increased.
Chapter 2. Experimental Approach
20
Nulling coils: Besides the trapping magnetic field, other magnetic fields exist mostly
due to the Earth's field. Although these are very small in magnitude compared to the
trapping magnetic field, they affect microwave transitions significantly, especially in the
Rydberg state of interest. To get rid of the unwanted fields, three sets of Helmholtz coils
are put in the x, y, and z directions, and their currents are supplied by three independent
power supplies, respectively. Essentially, these minor unwanted fields are probed by driving
one of ns — np 1 / 2 microwave transitions.
The process to determine each current setting is as follows.
1. Turn the main magnetic field to the switching mode.
2. Turn on one of the three currents and turn up by small step, typically 20 mA. If there
is no information about the current value, the step may be larger, for example, 50~100 mA.
3. For each current step, drive a microwave transition, for example, 40s to 40pi/ 2 • A
long microwave pulse is recommended, for example, 2 ps, so that details of spectral features
are resolvable. The laser intensity(480 nm) should be attenuated to populate approximately
isolated atoms.
4. Plot the width of resonance as a function of the current, as shown in Fig. 2.6.
5. Fix the current at the minimum width. Turn on another current and repeat the steps
of 2~4.
6. When proper currents are found to minimize the transition linewidth, the linewidth
should be 0.5 MHz, for instance, for 2 ps long pulses. The side lobes of the Rabi lineshape
should be found on either side of the main peak.
If the Rabi-line shape is not symmetric, the above procedures are recommended to be
repeated with smaller steps(~10 mA) of current. Finally, when all the unwanted fields are
removed, a nice 0.5 MHz Rabi-lineshape should be observed. Fig. 2.7 shows the different
Rabi-sido lobes with very small differences in fields, although the widths of the central peak
are more or less the same. For reference, the current values are 0.17. 0.90, and 1.18 A, for
the nulling coils marked 1, 2, and 3 on the coil frames, respectively, which are from the
latest calibration on 7-26-2011 -
Chapter 2. Experimental Approach
21
0 55 -
0.50-
0.45-
N
X
1 0.40"O
0.35-
0.30-
0 25 •
0.04
T"
~r
~T~
0.06
0.08
0.10
!
0.12
0.14
T"
~r
T
0.16
0.18
0 20
0.22
current [A]
Figure 2,6: The liriewidt.h of 38s - 38pi/ 2 resonances as a function of a nulling coil current. The resonances
reach the minimum linuwidth at 0.14 Amp where the residual field is minimized.
Chapter 2. Experimental Approach
22
(VJVWHVYA
(A
+••
C
3
.q
V.
.<0.
a
CO
CO
(d)
'
,
1
1
1
1
1 ^
1
'
1
>
1
18103
18104
18105
18106
18107
18108
MW frequency [X4 MHzJ
Figure 2.7: The 38s - 38pi/ 2 resonances with different nulling coil currents. From (a) to (d), the Rabi
lineshapc is enhanced by adjusting nulling coil currents with 10 niA step, while the currents of the other
coils are fixed.
Chapter 2. Experimental Approach
23
MOT
chamber
VI
V2
V3
i
i
r
r
Thermocouple
vacuum guage
Three pump
batteries
Figure 2.8: The sorption pump diagram. P1-P3 and V1-V4 represent the batteries of the sorption pump
and the valves. A thermocouple vacuum gauage is attached to measure the vacuum pressure.
2.1.4
Ultra High Vacuum(UHV) chamber
UHV is a requirement to form the MOT. The number of trapped atoms drops as the
vacuum pressure increases. Vacuum pressures higher than 10~ 8 Torr did not allow the MOT
to trap atoms in the present setup. In October 2010, the UHV chamber had to be opened to
replace the Rb getters. It is worth noting how to open, get all work inside done properly, and
close the UHV chamber with a vacuum pressure back down to 10~ 9 Torr successfully. The
entire process from opening to closing the UHV chamber with low pressure and calibrating
the detector took a month to finish. The following is a step-by-step procedure for future
opening of the chamber.
Sorption pump: First, let us learn how to operate the sorption pump and check if
Chapter 2. Experimental Approach
24
there is a leaking point between flanges of the sorption pump, so that when closing the
chamber, the required work can be done without any delay, because closing tasks should
be performed more carefully and rapidly. For a roughing pump to get the pressure ~ 10 -3
Torr, the sorption pump is used since it is oil and vibration free. The sorption pump system
is drawn in Fig. 2.8 and is operated with the following procedure.
1. Attach sorption pumps to the MOT chamber (but do not open yet the valve between
the sorption pump and the chamber).
2. Cool down the sorption pumps to absorb materials and trap them into sieve tube
containers(call it the pump battery), PI, P2, and P3. When the temperature of the batteries
reaches equilibrium, open the battery valve and then the pressure will go down. To cool
pumps, liquid nitrogen is poured into styrofoam buckets, in which the pump batteries are
placed.
3. Cool for about 20 minutes, and when the pressure stops dropping, close the valve
of sieves. At this point, check if there is a leak by monitoring the speed of rising pressure.
The pressure is measured by a thermocouple vacuum gauge which is attached between the
chamber and the pump manifold, and is accurate from 1 to 10 -3 Torr.
4. If the pressure does not go as low as 10~ 3 Torr, repeat steps 2 and 3.
Note that after repeating the procedures 2~3 three times, the pumps need to be refreshed
by baking out trapped materials.
5. To clean the pump manifold, wrap heating tapes around the pump battery and
release gas inside. Heat up the tape upto 250 °C for 2~3 hours.
6. When the pumps have cooled to room temperature, repeat procedures 2~3.
When the pressure reads ~10^ 3 Torr on the thermocouple vacuum gauge and the pres­
sure stays there, it is time to move on to the ion pump for lower vacuum pressures.
Opening: Now since any possible leak from the sorption pump and its manifold have
been checked, open the MOT chamber by opening the valve, V4. Be sure to turn off the ion
pump before opening any window or flange. When unscrewing bolts especially for the first
flange to be opened, release only a few threads first for all bolts to avoid any damage to
the threads. Then open the chamber flanges which hold the MCP detector and Rb getters.
Chapter 2. Experimental Approach
25
Put gloves on your hands when you handle the inner surface of the chamber components
Closing: To begin, all the windows and flanges that were opened are cleaned with
Methanol throughly, closed and the sorption pumps are cooled down. Operating the sorption
pump is summarized in the sorption pump section. When the pressure reaches ~10 -3 Torr
on the thermocouple vacuum gauge, the valve, V4, between the sorption pump and the
MOT chamber is closed. Note that the torque for all flange screws is 96 IN/lb, but 60
IN/lb for the ion pump valve.
1. Turn on the ion pump. The ion pump should be powered by a high power supply(ULTEK ion pump power unit, Model: 60-057). When turning on the pump, you might
be able to observe some arcs from the chamber. If the arc does not go away in a second,
turn off the pump. This happens when there is still too much material inside. In this case,
go back to the sorption pump again.
2. If the arcs disappear right away, leave the system working for hours. But the pressure
should be checked every 30 minutes.
3. When the ion pump power unit reads 20 pA, it is disconnected and the pump
connected to the "DIGITEL SPC" ion pump controller.
4. Since significant amount of gas adheres to the surface of the chamber and body parts,
bake the chamber to remove the gas molecules adhered to the inner surfaces of the chamber.
5. The Rb getter, MCP, and PFI introduce additional outgassing when heated by
operating current and voltages. To prevent the surge of outgassing, slowly increase the
operating currents and voltages. These have to be done one by one.
Replacements and repairs: Replacement and repair of parts should be done with
efficiency and accuracy. When one of the dual-microchannel-plate(MCP) or Rb getters
needs replacement, it is recommended to replace both at the same time even if the other
is relatively new. 10 Rb dispensers purchased from SAES Advanced Technologies are spotwelded on the insides of the flanges. Three flanges share 10 dispensers. In other words, two
flanges each contain 4 dispensers, and the third flange contains 2 dispensers.
Extra caution is needed when replacing the MCP, since the MCP is extremely fragile(and
expensive). Assembling MCP and its housing is the most difficult part of the setup. The
Chapter 2. Experimental Approach
26
wiring, sealing, and grounding have to be done properly, since each layer of the MCP
assembly goes to a different electrode or ground, and there is high chance of contact between
the layers. Possibilities of unwanted contacts are tested by measuring resistance between
different layers and between layers and ground. Verify that each BNC connector on the
output flange is connected to the proper layer. When putting the MCP assembly back in
place, no contact should be made between the chamber and MCP. Unwanted contact can
occur because the space for the MCP assembly is limited. The gain of the present pair of
MCP's is 1.7 x 10 7 at 2000 V with bias current 9.3 /uA, according to test of the manufacturer.
For all parts going into the chamber except for the Rb dispensers and MCP, thorough
cleaning is strongly recommended.
2.1.5
Properties of the MOT
First of all, it is recommended to record the vacuum pressure, getter current, and MOT
signal on the CCD linear array every day so that the standard operating parameters of the
MOT become familiar to the user. This is valuable, since it is not possible to immediately
find the cause of a malfunction of the MOT due to the fact that it consists of so many sub­
parts. Recording as much information as possible on a daily basis provides helpful clues
to find the cause of the trouble. For future reference, as of January 21 st 2012, the MOT
intensity is 1 V on the CCD linear array, vacuum pressure 1.5xlO~ 10 Torr, diode powers
45 and 42 mW for trap and repump lasers, respectively, at getter current 3.2 Amp.
Investigating the filling time, r, of the MOT, is necessary when calibrating the MCP
detector to determine the number of atoms. For more information about the detector
calibration and measuring the number of atoms, refer Jianing's dissertation [4]. r may vary
depending on the detuning and power of the trapping laser, gradient of magnetic field, and
speed of Rb supply. The detuning of the laser and the gradient of magnetic field arc const ant,
after being set at specific values, and although the power of the laser slightly changes with
time, the variation is negligible. However, the current of the Rb dispenser, I ge ,t ter , by which
the speed of Rb supply varies, can not be fixed, because the Rb getter requires stronger
heating daily to supply a constant amount of Rb. In this sense, measurements of r as a
Chapter 2. Experimental Approach
0.010-
27
seed beam injection timing
0.009
"in
0.008
3
J[
>>
0.007 -
c
0.006 -
a>
c
0.005
0.004 0.003
—i—
-10
—T~
-5
r~
10
time [sec]
Figure 2.9: A typical measurement of the rising curve of MOT.
function of I indicate when the Rb getter needs to be replaced.
To measure r, a photodiode is put at the focused fluorescence of the MOT, and the
amplitude of the photodiode signal is monitored over time with an oscilloscope. The seed
beam of the trapping laser is blocked to record the background signal of the MOT, and it
is unblocked to record a rising curve. The curve is fit to the exponential function of
I ( t ) = /o + C(l - e ~ t / T ) ,
(2.1)
where Jo is the background level and C the scaling constant, in order to extract r. Fig. 2.9
shows a typical measurement of the rising curve and its fitting. The risetime measurement
is repeated with different getter currents, letter and the corresponding vacuum chamber
pressures are also recorded, as shown in Fig.
2.10. As I getter is increased, r drops quite
linearly, while the vacuum pressure rises quadratically. The operating current is chosen
between 3.2 and 3.5 Arnps since shorter rising times, lower vacuum pressure, and higher
Chapter 2. Experimental Approach
28
MOT amplitude are preferred. These measurements of rising time, vacuum pressure, and
MOT amplitude as a function of I ge tter were taken on August 31 st 2011.
Chapter 2. Experimental Approach
<n
« 0.04 •
3
29
MOT amplitude
fa­
ro
0.02
T3
Q.
i
ro o.oo
A
J
>_
J
i
I
.—I
i
I—i
I
.—I—i—I—i—I
I
I
I
,
F 30vacuum pressure
15.
CD
V—
3
W
W
<D
I
I ^ i
J
L
I
I
O
Q)
(fl.
<D
E
•*—«
o>
c
I
I
,
I
I
I
MOT rising time
"«
0—!
2.4
,
,
2.6
,
,
,
2.8
j
3.0
,
,
3.2
,
,
3.4
,
[
3.6
1
1
3.8
,
1
40
,
1
4.2
dispenser current [ A
Figure 2.10: The amplitude of the MOT signal from the CCD linear array, vacuum pressure, and MOT
rising time as a function of I getter.
Chapter 2. Experimental Approach
2.2
2.2.1
30
Excitation laser for Rydberg atoms
960nm beam and amplification
The light source to excite Rydberg atoms from the 5p 3 / 2 state of Rb is a 960nm diode
mounted in a Toptica DL100 laser (960nm laser beam is frequency-doubled on a subsequent
step). The original diode was replaced by a QLD-960-50s(QPhotonics) diode for higher
power(QLD-960-100s is also available for higher power of 100 mW). The new diode produces
50 mW centered at 960 nm in free running mode. An external cavity is built in the laser
head, and its power output is about 20 mW with vertical polarization.
When changing the laser frequencies, a wavelength meter, Wavemaster(Coherent) refur­
bished in March 2011, is used while monitoring a Fabry-Perot Interferometer(FPI) signal.
For coarse tuning, the horizontal knob of the grating, G in Fig.
2.11, is adjusted. Once
the wavelength is set close to the resonance between 5p3/ 2 and a Rydberg state on the
wavelength meter, the bias voltage to the piezo of G is swept between -7 and +7 V to
check if the diode is in a single mode range or if it needs further fine tuning. Then, the
temperature and current are varied until the exact resonant wavelength is obtained (in a
single mode, increasing temperature and current increases the wavelength). For the final
step, the laser beam is directed to the MOT through amplification stages and a frequency
doubling crystal to excite the Rydberg atoms. On this step, a stable(less than 5% shot-toshot fluctuation) and large Rydberg signal should be observed by adjusting the bias voltage
of the servo-locking box.
If an unstable Rydberg signal is encountered, the cause probably lies in either the MOT
fluctuation or mode-hopping of the 960nm laser. If the MOT seems stable, the diode is
most likely in the mode-hopping range. First, if any sign of jittering FPI peaks is observed,
the diode is in the middle of mode hopping, and the injection current should be adjusted. If
the problem is not fixed, the coarse tuning knob on the grating should be checked. Second,
if the strong and stable Rydberg signal disappears all of a sudden, while the bias voltage of
the servo-locking box is being tuned, the diode is very close to the margin of stable mode,
and the temperature should be changed to shift the diode mode to the middle of the single
Chapter 2. Experimental Approach
L2
f=150 mm
Ml
31
H
ISL2
ISll
f=150 mm
25.7 cm
W — .Kl
>
12.3 cm
DC
20 mwy.
->
10.2 cm 10.2 cm
diodco.-
Ramp
Voltage
switch
FP cavity
39 cm
Servo Box
TAdiodf
M2
15 mW
-o
300 mW
29.5 cm
Figure 2.11: A schematic diagram of the alignment of 960nm laser and TA.
mode range.
If the problem of mode-hopping continues, the spatial matching between the 1st order
retro-reflecting and the Oth order out-coming beams in the external cavity should be exam­
ined. The detailed procedure for this is described in the manual of the Toptica laser. In
short, the current needs to be reduced close to the threshold, and by adjusting both hori­
zontal and vertical knobs a sudden flash should be seen at the proper alignment. Repeating
the previous steps leads to lower currents for flashing and better alignment of the feedback
beam to the diode.
In the case of diode replacement, the laser control unit should be shut down initially.
Then the old diode is removed from the circuit board and replaced with a new one after
correct pin connections are ensured(Takirtg multiple pictures of the laser head from different
angles before disassembly is highly recommended). Instruction is provided in the manu­
facturer's manual for the correct combination of diode pins and jumpers. After the new
diode is set correctly, the control unit is turned on, and the diode current is increased to
Chapter 2. Experimental Approach
32
the normal operating value (DL 100 Diode Laser System Manual, 5.4 Replacement of the
laser diole/change of wavelength, page 23).
To check if the diode is properly mounted and working, the grating is removed, and
measurements of wavelength and power axe compared to the specification from the manu­
facturer. Then, the beam is collimated by adjusting the focus lens in the diode housing.
Finally, the grating is put in place and the above steps of the alignment of the feedback
beam are followed.
During experiments, the 960 nm laser is side-locked to the Fabry-Perot (FP) cavity
resonance signal for long term stabilization of the laser frequency. A simplified loop of
this locking operation is drawn in Fig.
2.11. A fraction of the output beam from TA
amplifier is directed to FP cavity to generate the diode laser spectrum, while the FP cavity
is modulated by a voltage ramp of the FP driver. First, the frequency of the diode laser is
tuned close to the target frequency by adjusting the bias voltage of the servo-locking box,
which goes to the grating piezo of the 960 nm laser. Then the voltage ramp of the driver for
the FP cavity is switched by a DC voltage in order to set the tunings of the FP cavity to
the atomic frequency. That is, the FP cavity length is now adjusted manually by adjusting
the DC voltage. When the FP cavity length is tuned to the atomic frequency by the DC
voltage, a jittering error signal around the set level of the lock should be seen. At this point,
the lock switch of the servo-box should be switched on quickly. The grating, G, is adjusted
according to the error signal. The locking of the 960 nm laser lasted about 10~30 minutes
in the usual environment,, and when it goes off-resonant, due to the FP cavity fluctuation,
the voltage of the DC power supply is adjusted to push the FP cavity length back to the
original value.
tapered amplifier (TA): The 960 nm beam is amplified by two consecutive methods,
since its 20 mW power is not, strong enough to perform the experiments. The first method
is the TA. The TA consists of a diode, focal lenses, and temperature and current controllers.
The output power is extremely sensitive to its alignment to the injection beam. The injection
beam needs to be well aligned as well as tightly focused on the TA diode. Fig. 2.11 shows
the simplified view of the 960nm-TA alignment.
Chapter 2. Experimental Approach
33
A TA diode, LD1611 with output power 500 mW centered at 970 nm (Power Technolo­
gies), is put in a copper mount, which is connected to temperature and current controllers.
To extend the lifetime of the device, the temperature is kept to about 18 °C and the current
at 1.5 Amps. With good installation and alignment, 30 mW of seed laser power can pro­
duce 900 mW according to the manufacturer, although about 15 mW injection laser power
produced 300 mW in this lab. The optimal input coupling is achieved when ASE from the
back, or input, face of the TA and seed laser beam are mode matched.
Once the seed beam and TA beams are well matched, re-alignment is unnecessary when
changing the laser colors. However, in the case of a substantial change in the laser wave­
length, for example, from 961.576nm to 964.883nm corresponding to a change in the final
state from 40s to 30s, a minor adjustment of the mirrors of the seed laser beam is needed.
When re-aligning, Ml and M2 are adjusted until the power meter placed behind the TA
reads a maximum value. In the rare case of low output power, the focal length of L2 mounted
on a translation stage needs to be adjusted, to recover the mode matching condition.
Dye amplifier: The second amplification is carried out using a two-stage side pumped
dye amplifier pumped by Nd:YAG laser pulses. The CW 960 nm beam out of the TA is
focused again and sent through the dye cell, in which dye solution flows constantly to change
the dye molecules pumped by the 532 nm pump beam at 20 Hz. The seed beam stimulates
dye molecules excited by the pumping laser, producing amplified light of identical color. The
most sensitive skill required for this technique is putting the seed beam along the pumping
beam; both beams must be focused along the same spatial line. The dye solution is LDS
925 dye in a solvent of 85% ethylene glycol and 15% propylene carbonate at a concentration
of 0.125// [21].
Other possibilities for further amplification: 1. Third stage of dye amplifier. It
turns out that the pump light has more than enough power for two-stage amplification so
the third stage of dye amplification was attempted. The configuration is basically the same
as two-stage dye amplification but an additional dye cell is placed between the second dye
cell and the frequency-doubling crystal. The density of dye solutions is varied to produce
better amplification efficiency, but the third stage amplification did not work very efficiently.
Chapter 2. Experimental Approach
34
The brightness of the blue light out of the crystal is nearly equal to that obtained with twostage amplification. Nevertheless, better results should be obtained if the third stage was
handled with greater care.
2. Amplifying the pulse of 480 nm light using LD489 dye solution and UV pump light.
With the same notion of dye amplification of CW beam, the 480 nm pulse light is amplified
using a dye solution, which is a mixture of 0.2435g LD489 and 600ml Methanol, pumped by
the Nd:YAG third harmonic. The blue beam is focused and sent through the dye solution
cell as a thin and straight beam line, which is spatially overlapped with the focused UV pump
beam. The outcoming beam is found to be much brighter than it is without amplification.
In terms of the production of Rydberg atoms, the light amplified by this approach saturates
the excitation of the Rydberg state. In this case, if we make a denser MOT, this blue light
is able to excite more Rydberg atoms.
2.2.2
480nm laser pulse
The energy between 5p 3 / 2 and the Rydberg state of interest is about 480 nm in wave­
length. The laser color is obtained by frequency-doubling the 960 nm beam in a KNbO%
crystal. The CW 960 nm beam needs to be first amplified before this step, since this non­
linear process demands a high power of the injection beam. The amplification of the 960nm
laser pulse via the TA and the two-stage dye amplifier is discussed in the earlier sections.
The final 960 nm pulse is 10 ns long and sent through the crystal at the phase matching
angle, generating a 480 nm laser pulse with 100 fii of energy. Note that the incident 960
nm laser beam is focused at the crystal and the angle of the crystal must be adjustable.
When changing Rydberg state to a neighboring n state by changing the 960 nm laser fre­
quency, only minor adjustment of the crystal angle is needed, while substantial angle tuning
is expected when changing n by 10 or more.
The measurement of the frequency bandwidth of the laser is performed in the following
way. To scan the laser frequencies, a voltage ramp is applied to the piezo which tunes
the grating of the 960 nm laser. The voltage is provided by National Instruments BNC2110, which is controlled by a LabView computer program. As the piezo-voltage is scanned
Chapter 2. Experimental Approach
35
FSR=3 GHz= 4.8 V
(a) Etalon signal
to
xt
c
3
€
ra
o>
TJ
!3
o.
E
<
Gaussian fit
SV= 0.17 V
8v= 106.3 MHz
(b) Rydberg atom
j
•6
•3
-1.5
-1.0
0
| -Q.5
3
0.0
6
Piezo Voltage [ V ]
Figure 2.12: The measurement of the bandwidth of the 480 nm laser. While scanning the piezo voltage of
the laser grating to drive 5P3/2 — 38si/ 2 transition, (a) the spectrum analyzer signal and (b) the integrated
signal of Rydberg atom are simultaneously recorded.
from 0 to 10 V, the integrated Rydberg and etalon signals are recorded in the computer
simultaneously. Fig. 2.12 shows the etalon spectrum, which has free spectral range(FSR)
of 1.5 GHz (corresponding to 3 GHz at 480 nm laser scan) and 4.8 V of piezo voltage, and
the Rydberg production on the same scale. The voltage width of the Rydberg signal is
0.17 V, which corresponds to 106 MHz in frequency. This result of 106 MHz bandwidth is
consistent with the time duration, about 10 ns, of the 532 nm Nd:YAG pump beam.
The measurement of the 480 nm beam size was also carried out to find the the density
and geometry of the Rydberg atomic cloud. We use the knife-edge scanning method. In
this measurement the 480 nm beam is redirected outside the vacuum chamber, and the
knife edge, mounted on a 10 /xm-resolution translation stage, is scanned through the focus
of the beam while the transmission of laser power past the knife edge is measured as shown
in Fig. 2.13(a). The derivative of the laser transmission with respect to distance is fit to a
Gaussian function, as shown in Fig. 2.13(b), yielding the beam radius of the 480 nm beam.
Chapter 2. Experimental Approach
36
2.53
(0
F
(a)
2.01.5-
CQ
©
a
o
1.0-
&
0.5-
9
0.0-
(A
C
C
—,—.—,—.—|—-—i—•—i—>—i—«—i—
0.0
0.2
0.4
0.6
0.8
1.0
1.2
12-i
10
c
3
4
S*
(b)
8
6
4
2
0-2
-0.6
1
-0.4
'
1
j
1
-0.2
0.0
0.2
"
1
0.4
«
1—
0.6
distance[mm]
Figure 2.13: The measurement of the waist size of the 480 nm laser by the knife-edge method, (a) The
transmission of the 480 nm laser beam is measured as a function of position displacement of the knife, (b)
Derivative of (a) and its fitting to a Gaussian function yield the size of waist of 4S0 nm laser, which is )G2
/xm in this case.
Chapter 2. Experimental Approach
2.2.3
37
Measurement of the number of Rydberg atom
Throughout the entire dissertation, measuring the number of Rydberg atom is one of
the most important tasks to analyze data after the experiments, since it determines the
interatomic distance, R, and R determines the interaction strength between the atoms.
The method employed here is previously described in more detail by Singer et al. [3] and
Han [4], so the procedure will be discussed briefly.
The number of Rydberg atoms is
basically determined by measuring the reduction in the number of atoms in the MOT by
the excitation lasers. First, the maximum number, N ma x, of trapped atoms is measured
by measuring the fluorescence power from the MOT through a 95% transmission focusing
lens into a solid angle of 1.86 x 1(T
3
sr without the excitation laser. The total fluorescence
power is divided by the power, 9.3 x 10~ 12 W, radiated by a single Rb atom. Second, the
filling time of the MOT is measured by observing the time dependent fluorescence after
the trapping laser is turned on, which is described earlier in this chapter. Finally, the time
average number of atoms, N avg , in MOT is measured when the excitation pulses are present.
The number of Rydberg atoms produced by each laser shot is given by
NRydberg ~~ (^rnax
N'avg)R^"jilly
(^-2)
where R is the repetition rate of the pulsed laser. However, it is time consuming to repeat
this measurement daily, so we record the integrated area of the MCP Rydberg signal,
A Rydberg, on an oscilloscope when estimating NRydberg in the presence of the 480 nm laser.
The idea is that the Rydberg atoms excited by the 480 nm laser are all ionized by the
field ionization pulse resulting in the Rydberg signal. Once the conversion factor between
A Rydberg and NRydberg is achieved, the number of Rydberg atom can be easily computed by
measuring the waveform of the MCP signal.
Using the number of Rydberg atoms and the measurements of the sizes of the MOT and
the 480 nm laser waist(which are described in the earlier parts of this chapter), the densities
of the MOT and Rydberg atom cloud are determined. If the laser beam propagates in the z
direction, x and y are the perpendicular directions, and the origin of the coordinates system
Chapter 2. Experimental Approach
38
is the center of MOT. The density in the trap is given by
p(x,y, z) =
(2.3)
where the distance from the center of the trap r is defined by r 2 = x 2 + y 2 + z 2 , p 0 is
the density at the center of the trap, and r« and ri, are the radii of MOT and the laser
respectively.
2.3
Microwave field pulse
As a probe field or an intermediate excitation step, a microwave(MW) pulse(or two
pulses) is utilized. To generate a fundamental MW frequency, the Agilent 83622B Synthe­
sizer is used. It has a frequency range of 2 - 20 GHz with resolution 2 kHz and power up
to 20 dBm(100 mW). When using an internally generated pulse, the pulse rise/fall time is
25 ns with a delay of 15 ns from the trigger pulse(positive). Here the pulse rise/fall time
indicates the time for the pulse to rise(fall) from the zero(top) to the top(zero) level. In
the experiments, both internal and external switches are used depending on the operating
program, and both produce consistent data. Yet, it should be noted that when using a pulse
shorter than 500 ns, the internal switch needs a long power-build up time, which means
the short pulse takes a long time to reach its full amplitude, for instance, a 200 ns long
pulse takes longer than 1 minute to produce its full power. Therefore, the external switch
is recommended. The external switch, a General Microwave model F9120, triggered by a
negative logic pulse (+5 V to 0 V), demands +5 and -15 V supply voltages and precisely
forms the continuous wave MW power into a pulse with a rise/fall time less than 10ns.
The MW frequency must be multiplied, since the transition frequency between Rydberg
states of interest is 20-200 GHz. First, an active doubler(DBS 2640X220, input 13.25-20
GHz) or quadrupler(DBS 4000X410, input 10-15 GHz) doubles or quadruples the frequency,
respectively, and amplifies the output power. Both need an input power of 10-17 dBm
and a power supply of +12V, 500 mA. For further frequency multiplication, a passive
doubler(V2W r O, input frequency 25 - 37.5 GHz) or tripler(W3WO, input frequency 25 -
Chapter 2. Experimental Approach
39
36.67 GHz) are used.
Finally, the power of the frequency-multiplied MW pulse is precisely adjusted through
attenuators before it propagates from a horn. All MW components are placed outside the
vacuum chamber, and the final MW pulse propagates through one of the 4-inch diameter
windows of the chamber.
In the beginning of most experiments, we look for an optimum MW power so that
power broadening of the spectrum is ruled out and perturbation theory can be applied.
The MW power is reduced until the resonances of ns — np transition yield a transform
limited linewidth. The transform limited linewidth stems from the Fourier transformation
of the MW pulse, assuming that the MW pulse is a perfect square with A t in temporal
length. The observed linewidth is supposed to be Av = 1/ A t with the optimum MW power.
It should be, however, noted that the MW output pulse is not quite square shaped, since
the pulse inevitably has rising and falling time. Furthermore, the effect of the deviation
becomes more significant, as the shorter pulses are generated. For example, a 100 ns long
pulse looks like a trapezoid. In our experiments, pulses longer than 2 fj.s seem fine to use,
while particular attention should be given to short pulse experiments.
2.4
Detection and data collection
The detection of the final state of atoms begins with pulsed field ionization(PFI). The
ionization pulse rises almost linearly until it reaches its maximum in 2 /is and ionizes atoms
from high to low lying energy' states in order of binding energy, due to the fact that a higher
state demands a lower ionization field than a more tightly bound lower state. Oscilloscope
traces of the field pulse itself with 15 V DC input and its positive trigger pulse with 10
V amplitude are shown in Fig.
2.14. It is assumed that PFI simply projects atoms onto
atomic states(which is not always true). Depending on the polarity of the field pulse, either
an ion or an electron is kicked towards the MCP, which collects a time resolved ion or
electron signal. The data are stored in a computer for further analysis.
The typical MCP operating voltage is 1800 V, and the PFI input voltage depends on
Chapter 2, Experimental Approach
0.003
40
trigger pulse
Fl pulse
2.2 us
0.002
3
0.001 -
ro,
>.
c
<D
0.000 -
-0.001
-0.002
-0.000005
0.000000
-r
~r
0.000005
0.000010
0.000015
0.000020
time (second)
Figure 2.14: Oscilloscope traces of field ionization pulse(solid line) and its trigger(dotted line) pulses. 15 V
of DC input generates a pulse with 2.2 /^s rise time and 600 V amplitude.
Chapter 2. Experimental Approach
41
the ionization field of the excited Rydberg states under study. Note that when ionizing
the lower states, higher PFI input voltage is needed. In the present setup, the lowest state
which could be ionized is 22d with 140 V DC input. Increasing the DC input voltage
higher than 120 V introduced some arcs within the vacuum chamber, which is unacceptable
environment to generate Rydberg signals in experiments.
2.5
Summary of daily operation
MOT: 1. Turn on the AOM driver, the photo detector of SAS, and voltage ramp of
the diode grating. Try to find SAS by adjusting the piezo voltage of the diode. If the
wavelength has deviated too far from the desired value, adjust the horizontal knob of the
grating.
2. Slowly(0.5 A for one minute) increase the getter current up to a normal operating
value.
3. Turn on and increase the current of the anti-Helmholtz coils upto 13 Amps(8 Amps
for the switching mode), and turn on the nulling coils(these currents values were determined
earlier in the Nulling coils section). Before this, check if the cooling water is flowing by
looking at the flow-indicator.
4. Lock the laser frequencies.
At this point, the MOT signal should be seen from both the linear CCD detector and the
infra-red camera. In the switching mode, a blinking MOT should be observed on the linear
CCD detector and the infra-red camera monitor, otherwise, check the switching circuit and
the trigger pulse.
Rydberg atom:
1. Turn on and slowly increase the voltage of the micro channel plate(MCP) detector
up to 1.8 kV in 10 minutes to prevent any possible damage on the MCPs.
2. Turn on the voltage going to the PFI circuit.
3. Turn on the current of TA(1.5 Amps).
Chapter 2. Experimental Approach
42
4. Turn on the pump to circulate the dye solution.
5. Turn on the Nd:YAG laser.
6. Lock the 960nm-laser.
At this point, a stable and strong Rydberg signal should be observed, otherwise, check
the frequency of the 960nm-laser using the wavelength meter and the alignment. The angle
of the frequency-doubling crystal needs to be slightly adjusted daily.
Bibliography
[1] W. Anderson, Ph. D thesis, University of Virginia (1996).
[2] W. Li, Ph. D thesis, University of Virginia (2005).
[3] K. Singer, M. Reetz-Lamour, T. Amthor, L. G. Marcassa, and M. Weidemuller, 93,
163001 (2004).
[4] J. Han, Ph. D thesis, University of Virginia (2009).
44
Chapter 3
The dipole-dipole interaction of Rb
t
nsnp atoms
3.1
Introduction
The dipole-dipole interaction is the most primitive and fundamental force between neu­
tral atoms and molecules, given by Vm ~
where m is the dipole moment of ith
atom and R is the distance between atoms. Despite being weaker than other inter particle
forces, the dipole-dipole interaction plays significant role in the regime of closely spaced
atomic and molecular systems due to its R~ 3 dependence. In fact, the dipole-dipole inter­
action shows prominent effects amongst high n state atoms as well as closely spaced atoms,
since dipole moments rise in proportion to n 2 , where n is the principal quantum number.
Thus a dense Rydberg gas is proposed
ELS
an effective system to investigate the dipole-dipole
interaction.
As a result of the dipole-dipole interaction, resonant energy transfer is observed in diverse
disciplines [1, 2, 3], A concept widely known as Forster resonant energy transfer(FRET) is
when two neighboring atoms are dipole-dipole coupled to higher and lower states with equal
energy spacings, the atoms exchange energy as donor and acceptor in a noncollisional and
nonradiative way, resulting in one excited atom and the other de-excited. FRET is applied
to chemical and biological systems such as finding a biological structure and measurement
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
45
of distances between molecules. In a pair of Rydberg atoms, FRET is observed by tuning
the energy levels into the resonance with an electric field [4, 5, 6, 7]. For example, the
process [5]
Cs23p3/2
Cs23p3/2 —^ Cs23s 4- Cs24s
(3.1)
is resonant at a field of 80 V/cm. Cs 23p3/2 and 24s, and Cs 23p3/2 and 23s are dipole
coupled respectively, and when the levels are tuned to the resonance, a donor Cs 23p3/2
atom transfers energy to an acceptor Cs 23p3/2 producing Cs 23s24s. Besides this, there
have been other manifestations of the strong dipole-dipole coupling in the frozen Rydberg
gas [4, 5, 8, 9, 10, 18, 12, 6, 13, 14, 7],
Another effect of the dipole-dipole interaction is the formation of attractive and repulsive
potentials between atoms [7, 18, 16, 11]. Atoms in a magneto-optical trap(MOT), initially
almost frozen, attract and repel each other as a result of the interaction and are forced to
move. The motion of the atoms is partially responsible for the ionization and, further, the
evolution of Rydberg atoms to plasma. This topic will be discussed in a subsequent chapter.
An increasingly popular and attractive application of the dipole-dipole interaction is the
dipole blockade [18]. The dipole blockade is a concept in which one of two closely spaced
atoms can be excited but the other cannot be driven to the same state due to energy level
shifts by the dipole-dipole interaction. The dipole blockade is being broadly studied for its
possibile use as a quantum gate. To understand the dipole blockade and to proceed with
its applications, extensive study on the dipole-dipole interaction is crucial. For instance,
studies by us and Walker et al. consistently show that very weakly shifted energy levels
and even zero shifted levels by the dipole-dipole interaction are present, meaning that the
dipole blockade may not be as efficient as expected.
In spite of the importance of the dipole-dipole interaction, only a few measurements
have been reported. Afrousheh et al. reported the line-broadening of the 45d5/2 - 46d5/2
two photon transition by adding 45p3/2 and 46p:j/2 atoms, which are dipole coupled to
45rf5/2 [19], Since 45d5/2 and 45p3/2 are weakly coupled, a very small broadening, 10 kHz,
is observed, whereas the relatively stronger coupling of 45d5/2 and 46p3/2 leads to 40 kHz
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
46
broadening. They also observed that the broadening is suppressed by a magnetic field [20].
However, the density range(~ 107 cm -3 ) of those measurements was so limited that detailed
spectral features and strong broadening were not observed.
In this chapter, the line-broadening technique is employed to extend the observation
of the dipole-dipole interaction to higher densities, up to ~ 109 cm -3 . Unlike Afrousheh's
experimental apporach, the microwave ns — rip transitions are driven for n — 28 ~ 51,
since a dense ns atomic sample is readily produced without considerable sign of ionization.
By switching the trap magnetic field on and off, the inhomogeneous broadening due to the
magnetic field effect is explored. In the theoretical section, the calculation of the dipoledipole interaction is performed with both spinless and spin-orbit coupled atoms. Based
on the calculation including spin-orbit coupling, the ns - np resonances are reproduced to
compare to the observed non-Lorentzian lineshape. Lastly, the implications of the results
are discussed.
3.2
Theory
In this section, the dipole-dipole interaction of the nsnp atom pair is calculated. The
calculation begins with a classical approach and is then converted to a quantum mechanical
picture. The calculation is conducted for spinless atoms first and extended to the atoms
with the spin-orbit coupling.
3.2.1
The dipole-dipole interaction of spinless atoms
A classical picture of the dipole-dipole interaction:
Prior to the dipole-dipole
interaction between neutral atoms, let us discuss that of permanent dipoles. In general,
when refering to the dipole-dipole interaction, it implies attractive or repulsive forces be­
tween polar molecules or permanent dipoles. Although this type of interaction is not exactly
the same as what will be thoroughly discussed in this chapter, the classical dipole-dipole
picture has useful similarities to the interaction between quantum atoms. Since the clas­
sical picture is, furthermore, easy to imagine and can be extended to a quantum picture,
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
47
z
z
atom2 0
atom2
atom:*.
MI
atoml0
(a)
(b)
Figure 3.1: (a) Two atoms with dipoles m and y.2 aligned in the z direction are separated by R. R is at an
angle 8 relative to the z axis, (b) Two quantum atoms, 1 and 2, have their internuclear axis along the z axis
and are separated by R. Their dipoles can be either parallel or perpendicular to the z axis. The linearly
polarized microwave field is at an angle 6 relative to the z axis.
it is chosen as an analogy to address the complicated concept of the quantum mechanical
dipole-dipole interaction. As shown in Fig. 3.1(a), consider two classical dipoles with static
electric dipole moments p\ and p2, which are both polarized along the 2-axis and separated
by R. 6 indicates the angle between the polarization vector z and the displacement vector
R of the two dipoles. Then the classical dipole-dipole interaction is given by
V d d = fTi • (T 2 - 3(fTi • R)(fl2 • R)
R3
ViM
HH-2
„
£ 3 ~ ( 1 - 3 COS
2 n\
0).
(3.2)
Depending upon 6, the interaction can be either attractive or repulsive, and its strength
also varies. With a fixed R, the dipoles experience the strongest repulsive force p-ipv/R3
when the dipoles are aligned at 6 = 90°, while the dipoles experience the strongest attractive
force —2fj.itj.2lR3 when the dipoles are aligned on the z-axis at 8 — 0°. Explicitly, the
resulting force is attractive for 0 < 54° and 6 > 126°, and repulsive for 54° < 6 < 126°.
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
48
Returning to quantum atoms, we axe interested in the interaction between a neutral
atom pair. Specifically, consider a pair of atoms, one in the ns and the other in the np
state. These atoms in nsnp axe strongly dipole-coupled and can exchange energy as follows,
nsnp —> npns
(3.3)
Instead of the permanent dipole in this case, these quantum atoms have transition dipole
moments fj.sp and fxps- These atoms behave similarly to the atoms in the classical picture.
Therefore, the transition dipole moment simply substitutes for the permanent dipole mo­
ment of the classical picture in order for a quantum mechanical dipole-dipole interaction.
This type of dipole-dipole interaction is responsible for FRET of Eq. (3.1) and the off res­
onant van der Waals interaction. In these cases the dipole-dipole interaction derives from
the dipole transition matrix elements. In the Forster resonance case, one can think of the
interaction as being between a pair of dipoles synchronously oscillating at the transition
frequency.
The conventions of the term symbols:
Labeling conventions of a diatomic state
of interest for a spinless atom is given by nsnp. A diatomic state is a direct product of
two atomic states and the state of atom 1 is given first followed by the state atom 2, for
example, a pair of atom 1 in ns m and atom 2 in np m is written as ns m np m . The subscripts
represent the magnetic quantum number m for each atomic state. In the later section of the
dipole-dipole interaction with spin-orbit coupling, essentially the same conventions apply,
but for the expression of a single atom, we put two subscriptions j and rrij in order, where j
is total angular momentum and m,j is its projection on the quantization axis. For instance,
npi-i represents that the n p atom has j = i and m.j = —* .
2
2
A choice of the quantization axis: When computing the dipole-dipole interaction,
we need a well-defined quantization axis typically to be the z-axis. There are two possible
quantization axes as shown in Fig. 3.1(a) and (b), one is the polarization direction of
the MW field and the other is the intcrnuclear axis. Assuming spinless atoms, if we take
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
49
the polarization direction of the MW field as the quantization axis as in Fig. 3.1(a), the
Selection rules only allow sopo(or
pqsq )
transition dipole, since the initial nso atoms can
only go to npo state by 7r-transition(Am=0). In the regime in which the microwave Rabi
frequency is large compared to the dipole-dipole interaction, the microwave field direction
might be the logical choice.
However, in the line broadening experiment, since the Rabi frequency is set in the
opposite regime to apply the perturbation theory, we take the internucleax axis as in Fig.
3.1(b), an approach adopted by King and van Vleck some time ago[23]. Accordingly, since
the microwave field is randomly polarized relative to the atoms,
a-
as well as tt-transitions
are allowed and the p atoms can have m=±l as well as 0, which produces M—0 and ±1
states listed in Table. 3.1. Here M is the total azimuthal angular momentum of the atom
pair. This approach simplifies the calculation of the energy eigenvalues and eigenvectors
and is easily generalized to include spin.
Evaluation of matrix elements using Edmonds' C-tensor:
To compute the
matrix elements, the dipole-dipole interaction is expressed in terms of Edmonds' C-tensor.
Operations of the tensor operators in the matrix elements, however, demand complicated
and long calculations. Therefore, the matrix elements are broken into independent parts by
the Wigner-Eckart theorem and then evaluated as the product of the components. Specifi­
cally, the matrix element can be written as a product of 3-j symbol and a reduced matrix
element,
(s'l'j'rn'\C{kq)\sljm) =
(~1)J
k
q
j
\
mj
Tabic 3.1: Basis states for the spinlcss atoms of Fig. 3.1(b)
M
0
±1
basis states
riSQnpo, nponso
nsonp ± 1 ,np ± inso
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
50
The reduced matrix element can also be written as,
(s'l'j'\\C(k)\\slj) = ( - i r ^ k V ( 2 j + D(2j' + l ) -
where <
I'
f
s
j
I
k
I'
j'
j
I
s\ ,
\ (i ||C(fc)||£),
k
(3.5)
> is a 6- j symbol and
( l ' \ \ C ( k ) \ \ l ) = (-l)'V(2i' + l)(2i + l)
U'
k
l\
^0 0
0^
(3.6)
Since these will be used later in the spin included system, the formula is given in a general
form.
To write the dipole-dipole interaction between two atoms in terms of C-tensors, let us
begin with Eq. (3.2) in Cartesian coordinates as follows,
Vdd = xix2 + y m - 2zxz2
R3
(3.7)
where Xi, yt, and Zi specify the position of the Rydberg electron in the ith atom relative
to the center of the atom. Using the relations between the Cartesian coordinates and the
spherical harmonics(Y/m),
r(-y u + yi-i)
y
~ i y Tr^n
z =
and
+ Yl ~
^
(3.8)
VT
4tt
X1X2 +yiV2 = -~g-'"jr2( 1 yii 2^l-i + l Y\-\ 2 y\i)
(3.9)
2ziZ2
= ^nr 2 l Y X { s 2 Y 1 Q ,
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
51
the dipole-dipole interaction can be written in terms of the spherical harmonics as follows,
(
y
+' y j - i V n +21Y120y10)),
(3.10)
where 'Yim indicates the spherical harmonics acting on the Rydberg electron of the ith atom
and ri indicates the distance between the Rydberg electron of the ith atom and the center
of the ith atom. Finally, the relation between the spherical harmonic tensors and C-tensors
of
/
C•,*' = VsTT
n'
<3U)
47r
leads to the following form of the dipole-dipole interaction,
vdd =
+ Cl.Cl + 2 d e l ) .
(3.12)
For simplicity, the index of ith atom is omitted in the C-tensors. Instead, a pair of C's
implies that the first C acts on the first, atom and the second C does on the second atom.
For the spinless atoms, there are six nsnp or npns states listed in Table 3.1, and these
states form a 6 x 6 Hamiltonian matrix. This 6x6 matrix can be split into three 2x2 matrices
according to M—0 and ±1, and each M matrix can be handled independently to find the
eigenvalues and eigenvectors. To find the Hamiltonian matrix elements, each matrix element
coupling ns and np is computed. For example, for M=0, ns^npo and np 0 nsu are coupled,
and the matrix element is given by
(ns0npo|Vdd|nponso>
2
=
--^(ns 0 |2iN?o)(npo|*2|nso)
(3.13)
=
~ ^(solnQjlpoXpoNColso)
( 3 ' 14 )
2
"snp(so|C(5|po)(po|C,(5|so)
R3
(3.15)
2r;nsnp
(3.16)
3i?3 '
where r n s n p is the radial matrix element of ns - np, and (so|Co(pa) is
a
purely angular
matrix element of Edmonds' C tensor [24]. We follow the convention that when there is no
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
52
principal quantum number indicated for a matrix element, it is an angular matrix element.
With the internuclear axis as the quantization axis, only one of the pairs of C tensors enters
into any dipole-dipole matrix element. Then the Hamiltonian matrix for M=0 is
/
H M =o(nsonpo)
0
(3.17)
-2/3
Similarly, computing the matrix element for M=± 1,
(nsonp-tilVddlnp-tinso) =
' nsnp
(3.18)
3R 3 '
gives the Hamiltonian matrix as,
0
1/3
1/3
0
(3.19)
HM^±i(nsonp ± i) =
Here the common factor of r^ s n p /R 3 for the Hamiltonian matrices is omitted.
Eigenvalues and eigenvectors axe obtained by diagonalizing each of the 2x2 Hamiltonian
matrices, and the results are listed in Table 3.2. Each M has symmetric and antisymmetric
states in the interchange of the two atoms. Since the matrix elements and eigenvalues have
the common factor of r^ snp /R? it is convenient to write an eigenvalue W as
W
nsnp
(3.20)
x>
where x> s a numerical factor of order one.
Table 3.2: Symmetries, eigenvectors, and eigenvalues of nsnp states for the spinless case of Fig. 3.1(b) "s"
represents a symmetric state, and "as" represents an asymmetric state.
M
0
0
±1
±1
symmetry
s
as
s
as
eigenvector
(ns 0 npo + nponso) / y/2
(usqupq - np 0 ns 0 )/\/2
(ns 0 np± i + np±inso)/V2
(ns 0 np±i - np ± 1 nso)/y/2
eigenvalue(x)
-2/3
2/3
1/3
-1/3
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
M=o
<
53
symmetric states
\
antisymmetric states
nsns
Interatomic distance R [arb. units]
Figure 3.2: Energy levels vs R for a pair of spinless atoms. There is one nsns state, which is symmetric, and
both symmetric (solid line) and antisymmetric (broken line) nsnp states. Transitions from the symmetric
nsns state are only allowed to the symmetric nsnp states, as shown by the arrows.
In Table 3.2 and in all subsequent tables, energy eigenvalues are given in terms of x In Fig.3.2 we show the energies of the nsns and nsnp states vs. the internuclear spacing
R. Since the nsns state is symmetric, only the symmetric nsnp states are accessible by
a microwave transition. The allowed transitions are indicated in Fig.3.2. Both M—0 and
M=± 1 states are accessible since the microwave polarization is random in the molecular
frame of Fig.3.1(b). For a given internuclear spacing, the negative shift of the M — 0
symmetric state is twice as large as the positive shift of the M—±1 states, but there are
two M = ± 1 states, so the average shift is again zero.
Since the broadening of the resonances as a function of the atomic density is practically
useful information, the dipole-dipole broadening rate is estimated. The magnitude of the
typical energy shift by the interaction between two atoms is r^ snp /R 3 . If we assume that
all atoms are spaced by average distance R av in the atomic cloud, the energy shift can be
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
54
written as,
4?rrnsnpP".v
(3.21)
where pav is the average atomic density. In the line broadening experiment, what is actually
measured is double the energy shift W, and so the broadening rate T, the ratio of the
broadening to the average density, is given by
nsnp
(3.22)
Useful Cs and Rb radial matrix elements are given by Walker and Saffman, and for the
Rb ns — np transitions of n w 40, T"£snp=n2 [13]. Converting Eq. (3.22) to laboratory units
yields a broadening rate
r = 8.38 x 10~15n4(MHz • cm3).
For n = 4 0, a density of 109 cm
3.2.2
3
(3.23)
implies a broadening of 21.5 MHz.
The dipole-dipole interaction including spin-orbit coupling
Including the spin of the electron introduces an interaction between the spin and orbital
motion of the electron, so the spin-orbit coupling is included when finding eigen states
and energies. One of the results of the spin-orbit interaction is energy level splitting (fine
structure), otherwise the fine structure levels are all degenerate. For example, the Rb n=40p
level splits into 40pi/2 and 40j>3/2 with a spacing of 1.7 GHz. This shift is much larger than
the 21.5 MHz dipole-dipole interaction estimated in the previous section. Since the spinorbit splitting for the alkali atoms is proportional to n*~3 (n*:effective principal quantum
number) and the dipole-dipole interaction is proportional to n 4 , the spin-orbit interaction
should be taken into account in the n-range of interest. Contrary to this, hyperfine structure
arising from nuclear spin is negligible due to the fact that the hyperfine interval of 33s is
0.55 MHz and the interval decreases as n*~ 3 .
Except for the spin-orbit coupling, the general procedure of the calculation is the same as
for the spinless atom. The internuclear axis is chosen as the quantization axis of z described
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
55
in Fig.3.1(b). A molecular state is written as a direct product of two atoms, one of which
is given first, followed by the second atom. The subscript j indicates the total angular
momentum of each atom, and m indicates the projection of j on the quantization axis z.
Like the spinless atom, the the projection of the total angular momentum of a molecule
on the z-axis, M, is a good quantum number. When evaluating the matrix element, the
dipole-dipole interaction is given in the same form as Eq. (3.12) and the first C of the pair
acts on the first atom and the second C acts on the second atom.
Including spin of the electron creates more nsns initial states than are found in the spinless molecule. Two M=0 states and one each of M=± 1 states are effectively all degenerate
due to the fact that the hyperfine interval is small compared to the dipole-dipole interaction
and the magnetic field is minimized. Since the circularly polarized trapping lasers provide
random polarization, it is assumed that the initial state nsns is evenly populated in two
M=0 and one each of M=± 1 states. This assumption is applied when computing transition
probabilities and lineshape-modeling.
The spin-orbit interaction also generates a lot more nsnp states than are found in the
spinless molecule. The calculation of the dipole-dipole interaction is conducted in both
nsnpi/2 and nsnp :i / 2 molecular states. Possible nsnp or npns basis states are listed in
Table 3.3. The basis states for each \M\ would all be degenerate, but the dipole-dipole
interaction lifts the degeneracy. As an example, the dipole-dipole matrix element between
Tabic 3.3: Basis states for spin-orbit coupled ns\/2npi states
n s \/2 n Pi/2
M
0
basis
(•ns i i npi i, np i
±1
(ns i
2 2
i i
, 1 ,
2
np i
2
. i ,
2
i ns i i, ns i
i np i i, np 11 ns i i)
2
2
2 2
2
2 2
2 2
2
2
np i . i
?
x
n s i ^ i )
2
2 - ?
ns 1 / 2 np 3 / 2
M
0
basis
(nsii np3 i, np3 1 ns11, ns 1 1 nps1, np31 ns1 1)
±1
(ns 1 , i np3 , 1, np-i . 1 ns 1 , 1, ns 1._ 1 n»3 , 3 , rtp3 , 3 ns 1 _ 1)
22
22
22
22
2 + 2
22
22
2 +2
( n s ' 1 , 1 n p3 , 3 , n p 3 , 3 n s i . i )
9" 2
9.3-7
2
2
±2
2 2
2
2
r
2
2
2 2
2
2
2 2
2 2
2
2
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
56
ns i i np i _ i and np 11 ns i _ i is evaluated by
5
5
2
2
2
2
2
2
(3.24)
2%* x (1/3) x (-1/3)
=
(2/9)
The matrix elements (s 11\C}Api i) and (vi_i |Cr> Is i _ i) are evaluated using Eqs.(3.4)-(3.6).
2
2
5
5
2
2
2
5
Similar computations are performed for non-zero matrix elements coupling the nsnp basis
states. These matrix elements will be off-diagonal components in the Hamiltonian matrix.
The Hamiltonian matrix can be broken into blocks according to M and it is easier to handle
r2
them separately. The common factor of
will be omitted in the Hamiltonian matrix
for simplicity. Following the ordering of the basis states given in Table 3.3, we write the
Hamiltonian blocks for n-Sj/2nPi/2 pairs as
^
0
2/9
0
2/9
2/9
0
2/9
0
0
2/9
0
2/9
0
2/9
0
N
(3.25a)
Hm=0 ( n s ll2 n Pl/ 2 ) =
\ 2/9
)
and
(3.25b)
HM^±i(nsi/ 2 np l / 2 )
Similarly for nsi/ 2 np^/ 2 pairs, the Hamiltonian blocks are given by
/
HM=o(ns 1 / 2 np 3 i 2 )
0
1/9
0
-4/9 \
1/9
0
-4/9
0
0
-4/9
0
1/9
0
1/9
0
(3.26a)
=
^ -4/9
/
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
57
0
—4/9
0
1/3\/3
-4/9
0
l/3\/3
0
0
l/3\/3
0
0
0
0
0
HM^±i(ns 1 / 2 np 3 / 2 )
\ 1/3\/3
(3.26b)
and
/
#M=±2(ns 1/2 np 3/2 ) =
\
0
1/3
1/3
0
\
(3.26c)
/
The diagonalization of each Hamiltonian block yields eigenvectors and eigenvalues, which
are given in Table 3.4. The common factor of
r2
is omitted here again. The eigenvec­
tors are written based on the basis states in Table 3.3, and they are either symmetric or
antisymmetric with respect to the interchange of the two atoms. The symmetry s stands
for the symmetric states and as for the antisymmetric states.
For example, the eigen­
vector (1,1,1,1)/2 with the eigenvalue 4/9 for n.s 1 / 2 np 1 / 2 M—0 symmetric state represents
I (nsi inp3_ i + n » 3 _ i nsi i + nsi _ i np31 +np3insi_i ] • These degenerate states split
1
\
22
2
2
2
2
22
2
2
22
22
2
2/
into several energy levels with different shifting rates due to the dipole-dipole interaction,
similar to the spinless case. The most noticeable result from the spin-orbit coupled states is
that zero-shift levels also exist in the nsi^npi^ M=0 states contrary to the spinless case.
This is consistent with the report by Walker and Saffman [22], in which the off resonant
dipole-dipole interaction, or van der Waals interaction, is calculated in second order pertur­
bation theory. In the range where i?~6 dependent van der Waals interaction is dominant,
they found non-shifted energy levels, very small shift levels, and generally shifting levels.
In the dipole blockade applications, these weakly or non shifted levels may be a serious
problem, since blocking the excitation of the second atom is only possible if the excited
energy, level is shifted enough to be off-resonant for the excitation. In this particular line
broadening spectroscopy, small- and non-shifted levels appear as non-Lorentzian resonances
and exhibit no dipole-dipole interaction. The spectral line analysis based on these energy
levels will be discussed in a later section.
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
58
Table 3.4: Symmetries, eigenvectors, and eigenvalues of the nsnp states in terms of the basis states of Table
III. "s" represents a symmetric state, and "as" represents an asymmetric state.
nsl/2nPl/2
M
0
0
0
0
±1
±1
symmetry
s
s
as
as
s
as
eigenvector
(l,l,l,l )/2
(l,l,-l,-l)/2
(lrU,-l)/2
(-l,l,l,-l )/2
(1,1)/N/2
(h-l)/V2
M
0
0
0
0
±1
±1
±1
±1
±2
±2
symmetry
s
s
as
as
s
s
as
as
s
as
eigenvector
(-l,-l,l,l)/2
(1,1, l,l)/2
(l,-l,l,-l)/2
(-l,l,l,-l)/2
eigenvalue (x)
4/9
0
-4/9
0
-2/9
2/9
n s l/2 n P3/2
3.3
(1,1,2.68,2.68)/>/4.05
(-2.68,-2.68,1,1)/V4.05
(1,-1,2.68,-2.68)/V4.05
(-2.68,2.68, l,-l)/%/4.05
(1,1)/V2
(-1,1)/v / 2
eigenvalue(x)
5/9
-1/3
1/3
-5/9
0.0718
-0.5162
-0.0718
0.5162
1/3
-1/3
Experimental Approach
The fundamental idea of the experiment is shown in Fig.3.3. A magneto-optical trap(MOT)
provides cold Rb atoms in the &P3/2 state. Although it is not shown in the timing diagram,
for a field-free sample, the MOT field is switched off 6 tns before the 480 nm laser pulse
fires, and it turns back on 1 ms after the laser pulse. The steady state §P3/2 atoms are
excited to a ns Rydberg state by a 480 nm laser pulse at a 20 Hz repetition rate. The
480 nm light has 110 MHz bandwidth. It is generated by amplifying the 960 nm beam
with a tapered amplifier and a two-stage dye amplifier pumped by a 10 ns-long 532 nm
Nd:YAG laser pulse, and by frequency-doubling in the KNbO 3 crystal. The tightly focused
480 nm beam passes through the center of MOT, making a nsns Rydberg atomic cloud. A
microwave pulse is sent to the nsns atoms to drive the nsns — nsnp transitions, followed
by the PFI pulse. The PFI pulse selectively ionizes ns and np atoms, and the time gated
signal of np atoms is detected and stored in a computer for further analysis.
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
59
np
mm wave
ns
laser (ns)
480 nm\
N
laser
mm wave pulse
vA/WNAAAAAAAAAA
780 nm
laser
ns
np
time
(a) Energy level diagram
(b) Timing diagram
Figure 3.3: (a) Energy level diagram. The 780 nm excitation of the 5p 3 / 2 state is continuous, from the
MOT beams. The 480 nm excitation of the nsx/2 state is pulsed, and a mm wave pulse drives the ns to
np transition, (b) Timing diagram showing the timing sequence on cach shot of the pulsed laser. The laser
pulse is 8 ns long, the mm wave pulse is 500 ns long, and the field ionization pulse has a rise time of 2.16
/js. The time-resolved signals due to field ionization of the ns and np states are detected.
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
60
Since the number of Rydberg atoms is an important factor in this experiment and
subsequent chapters, we briefly discuss how to measure it here. We measure the fluorescence
radiated from the MOT with and without the 480 nm laser. The reduced fluorescence by
the laser is assumed to be due to transfer to a Rydberg state, and the measured photon
power is readily converted to the number of Rydberg atoms. As shown in Fig.3.4, a fraction
of the MOT fluorescence at 780 nm is collected in a power meter through 95% transmission
focusing lens. The measured power divided by 9.3xl0~ 12 W, the photon power radiated
by a single atom, yields the number of trapped atoms. The reduced amount of the trapped
atoms is the number of Rydberg atoms. Taking into account the solid angle subtended by
the lens and the trap refilling time gives the averaged number of the Rydberg atoms excited
by the 480 nm laser pulse.
The geometry of the Rydberg atom cloud is determined by the MOT and the 480 nm
laser. We record the fluorescence from the center of the trap on a CCD linear array, and its
image is fit to a Gaussian. Assuming the MOT to be spherically symmetric, we obtain the
MOT density distribution as p(r) = poe~r2/rM, where p(r) is the density at distance r from
the center, po is the peak density at the center, and rm is the radius of the MOT cloud.
tm
is measured to be tm=380
pm
from the image of the MOT. The focused 480 nm beam
is assumed to be cylindrical with the radius ri=70 ^m, since the beam divergence in 380
lim is approximately flat. The 70 //m is taken by measuring the waist of the focused 480
nm beam.
The MW pulses are generated from a HP83622B synthesizer using the internal switch.
The doubled MW frequencies from an active doubler are multiplied again by 2 or 3 using
a passive doubler or tripler. One difficulty is that in lower states than n=33, the driving
frequencies for the s — p transition are out of range. According to [21], the passive doubler
and tripler not only produce 2 nd and 3 rd harmonics but also 4 th and 6 th harmonics, respec­
tively, although the output power is not precisely adjustable. The power adjustment is not
problematic since the MW power does not need to change once the optimal power is found.
However, the 4 t h and 6 t h harmonics demand greater MW power input than the 2 n d and y d
do.
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
61
480 nm laser beam
Power meter,
Fast photodiode,
or Linear CCD array
Focusing lens
A
.MOT
V
Rydberg
atoms
Figure 3.4: Density measrements of the trapped 5j>3/2 and ns Rydberg atoms. A power meter is used to
measure the fluorescence power of MOT. To measure the radius of the MOT, the power meter is replaced by
a linear CCD array. The density of the trapped atoms, which is assumed to be a spherical Gaussian cloud,
can be determined from these two measurements. Once the number of trapped atoms is known, the number
of Rydberg atoms is obtained by measuring the filling time of the MOT and the reduced fluorescence power
when the 20 Hz-480 nm excitation laser is sent through the MOT to produce Rydberg atoms. The waist
of 480 nm beam at the focus and the density distribution of atoms in the MOT determines the density
distribution of the Rydberg atoms.
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
62
MW frequency sweeps through the ns — np resonances are carried out over many laser
shots with a very low density atomic sample to determine the optimum MW power. While
sweeping the MW frequencies, the gated np signal is saved in a computer for further analysis.
In most of the experiments, the MOT magnetic field is switched off to make a field-free cold
Rb sample, and the MW power is set such that the extracted FWHM from the resonances is
as narrow as the inverse of the pulse length, 2 MHz. For the experiment with the magnetic
field, the MOT field is kept on and the low density resonances have 5 MHz widths with the
same MW power as in the field-off experiment.
Once the MW power is set, the intensity of the 480 nm laser is adjusted in the neutral
density filters to provide atomic samples with various densities. In the routine density
dependent measurements, the atomic density is measured by integrating the total Rydberg
signal immediately after the MW sweep.
3.4
Experimental results
A typical density dependent measurement, for example, the 39s - 39p\ji resonance, is
shown in Fig. 3.5, in which the dotted line indicates the base line Of the off-resonant back­
ground and the vertical scales of the spectra are differently magnified for clear comparison
of the broadenings. In the low density limit (approximately isolated atoms), a transform
limited linewidth is observed. As the atomic densities are raised, the linewidth progres­
sively increases. The right inset of Fig. 3.5 illustrates how to extract the linewidth from
the resonances. The linewidth as a function of the average density, p av , is plotted in the
left inset. The exact definition of the average density will be given in a subsequent section.
Although the Lorentzian function does not fit well to a high density spectrum, as shown in
the right inset, it is still useful since it is easy to fit by a computer and it does not introduce
any bias due to the choice of an exotic line shape. The linewidth of the resonances grows
with increasing atomic density, and this trend fits quite well to the linear growth. The
linear increase of the dipole-dipole broadening is consistent with Eq. (3.21). The slope of
the linear function is the broadening rate T.
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
63
•p
c
3
•e
frequency [MHz}
average density [10s cm'1]
0.04X109cm3
0.16
-0.42
.0.59
66520
66540
66560
66580
66600
66620
frequency [MHz]
Figure 3.5: Recordings of the 39s to 39pj/2 transition at average densities p a v , of 0.04, 0.16, 0.42, and 0.59
xlO9 era "3, showing the asymmetric broadening of the resonance. The broken line is the off resonance
background 39signal, (left inset) Width vs. average density, (right inset) Observed resonance at
0.59 x 109 cm -3 (solid line) and Lorentzian fit to the resonance (dotted line) showing the cusped shape of
the observed resonances.
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
64
7-
5-
2 -
0.0
0.1
0.2
0.3
0.4
0.5
average density [109 cm'3]
Figure 3.6: The widths (FWHM) of the resonances as a function of the average density, p a v , showing the
linear increase with density, (a) ns — np l / 2 (b) ns - np 3 / 2 -
To generalize the dipole-dipole broadening, the experiments are performed for ns - np^/ 2
transitions as well as for other n states,
28, 29, 34, 39, 44, and 50. As shown in Fig. 3.6,
the interaction for all n grows linearly. This confirms the linear density dependence and
that a higher n has a larger broadening rate. From Eq. (3.23), it is expected that states
with higher radial matrix element have larger T, equivalently higher n has larger F. As an
example of T, the broadening of 39s — 39pjy2 transition is exptected to be 1.9 MHz with
pav = 108 cm 3 but the measurement is about half of the expectation.
The observed broadening rates are plotted as a function of n 4 /10 6 in Fig. 3.7 and also
shown in Table 3.5 to compare them with the calculation of T. First, the n 4 scaling of the
broadening rate is as expected. Second, the broadening for the ns - np3(/2 transitions is
slightly bigger than for the ns — np1(/2 transitions due to the slightly bigger radial matrix
elements of the ns — np 3 / 2 coupling. Lastly, the broadening is substantially smaller than the
calculation. The discrepancy may come from the density measurement of which uncertainty
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
65
Table 3.5: Broadening rates of the ns - np 3 transition
*O
t —r
S
OC
1
OC
transition
28s — 28pi /2
29s - 29pin
39s — 39pi/2
44s - 44pi/ 2
50s - 50pi /2
29s — 29p3/2
34s - 34p 3/2
39s - 39p 3 / 2
44s - 44p3/2
50s - 50j>3/2
atomic ns - npj frequency [MHz]
197992.56
176101.79
104107.41
66570.64
45114.68
29978.00
180849.97
106928.24
68381.08
46345.42
30799.14
broadening rate [10 9MHz-cma]
3.1(3)
3.4(9)
7.1(6)
10.9(11)
17.9(3)
27.2(25)
3.9(3)
7.6(10)
11.3(20)
18.5(47)
30.7(38)
is typically 100%. Alternative reasons for the discrepancy is the existence of the zero and
small shifted level as mentioned in the theory section. Since the spin-orbit coupling is not
taken into account for the T calculation, F appears to be larger than the observation, in
which the spin-orbit coupling suppresses the dipole-dipole interaction. It is evident that
simply estimating the linewidth cannot extract the full information about the dipole-dipole
interaction, as shown in the right inset of Fig. 3.5. Therefore, spectral lineshape analysis
has been done with the dipole-dipole interaction model including the spin-orbit coupling in
the subsequent section.
Even after the currents of the trapping coils are turned off, it takes a few milliseconds
for the magnetic field to disappear due to the eddy currents in the metal of the vacuum
chamber. In spite of the presence of optical molasses, the MOT loses a considerable number
of the trapped atoms during the field-off time in the field switching mode. Consequently,
reaching the minimized magnetic field, less than 50 mG, limits the atomic density. To
raise the densities above those in Fig.3.6 and to explore the effect of magnetic fields at
high densities, Paul Tanner et al (a former graduate student in the Gallagher's lab) made
measurements using the same line broadening technique but with a longer MW pulse and a
shorter waiting times after turning off the current in the trap coils. In these measurements
the minimum linewidths were ~2 MHz, suggesting the presence of fields of ~0.5 G. As
in the experiments with 50 mG fields, they observed broadening from the symmetric low
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
66
60-,
calculation
ns-np slope:8.38
50-
E
40observation
• ns-np1I2 slope: 4.69
o
r*
*
30-
ns-npM slope: 4.89
0)
Q.
O
V)
20-
10-
0
1
2
3
4
5
6
7
8
9
10
n4/106
Figure 3.7: Observed broadening rates of the ns — np j/2 a nd ns - np 3 / 2 transitions vs. n 4 /10 6 . They are
obtained from the slopes of the width vs. density plots shown in Fig.3.6. The broadening rates exhibit the
expected n 4 scaling but are lower than the initial estimate given by Bq.(3.23), shown by the solid line.
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
20-,
n=51: 29.22
n=41: 14.66
n=39: 6.00
n=34: 5.17
n=30: 1.45
15-
N
x
2
67
10-
•£
s
0.0
0.5
1.0
average density [
1.5
2.0
109 cm'3]
Figurg 3.8: Resonance widths (FWHM) of ns - npj/2 transitions in a magnetic field of ~ 0.5 G vs. density.
density resonance into an asymmetric cusp shaped resonance at high density. The observed
resonances are fit to Lorentzians to extract their widths, which are shown as a function
of density in Fig.3.8. At low densities the widths increase linearly with density, and at
essentially the same broadening rate, T = 4.25 x 10~ 15 n 4 (MHzcm 3 ), as the 50 mG data of
Fig.3.6.
As shown by Fig.3.8, for n — 51, the linewidths are not broadened as the density is raised
above 0.25 x 109 cm' 3 We attribute this phenomenon to the atoms which are closest to
each other ionizing, removing them from the observed 51p signal [7]. Ions are detected
when the 51s — 51p transitions at high densities are observed. The density scale of Fig.3.8
has been normalized to that of Fig.3.6 in the following way. We measured the broadening
of the 39s - 39pi/2 transition in the same MOT used to obtain the data of Fig.3.6 with
and without turning off the trapping magnetic fields. The broadening rates, which are the
slopes of the linewidth vs. density graphs, are the same to within 10%. Therefore we adjust
the density scale of Fig.3.8 to give the same broadening rate for the 39s — 39pj/2 transition
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
68
as shown in Fig.3.6.
3.5
Lineshape model of ns — n p j ( j = 1/2 and 3/2) transitions
It is established in the previous section that the dipole-dipole interaction of spinless
atoms does not adequately explain the observed ns — np transition broadenings. The ob­
served spectrum is narrower than the calculation of the spinless case by a factor of 2 and
deviates from a general Lorentzian resonance lineshape. It is presumed that the spin-orbit
coupling suppresses the dipole-dipole interaction, and various energy levels and their un­
even transition strengths cause the abnormal lineshape. To verify this, the nsns — nsnp
resonances are reproduced by applying the dipole-dipole interaction model including the
spin-orbit coupling. The transition strength is first computed and then the unique geom­
etry and /i-distribution of the Rydberg cloud axe taken into account in the calculation.
Finally a simple lineshape model is developed to compare with the observations.
3.5.1
Transition strength
In the experiments, what is actually detected is the np atoms as a result of the ns — np
transitions by the microwave pulse. The starting state is the nsns molecular state and
only one of the two atoms is assumed to participate in the transition. In other words, the
nsns —nsnp or nsns — npns transition has occured. Contrary to the spinless diatomic state,
the nsns state with spin has antisymmetric states as well as symmetric states with respect
to the interchange of the two atoms. There is one antisymmetric state for M=0 and one
symmetric state for each M=0 and ±1. These four possible states are equally excited by
the 480 nm laser and given in Table 3.6. The transition matrix elements between nsns and
nsnp are calculated based on the basis states given in Table 3.5 and 3.6.
To compute the transition strengths and the lineshape in a manageable format, two
approximations are made, both of which are reasonable if there is no power broadening.
First, it was assumed that the microwave field only drives the atoms from the initial nsns
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
69
Table 3.6: Initial nsi/ 2 nsi/2 states, "s" represents asymmetric state, and "as" represents an asymmetric
state.
symmetry
s
M
0 ,
s
±1
as
0
state vector
(nsii nsi i Hb ns i 1 nsi i )/y/2
22
2 2
2 2
22
nsi . i n s i i i
2 2
2 2
(nsii nsi i - ns i i nsi i )/y/2
9 9
77
22
2 2
state to the nsnp state. In fact, the microwave field can drive the atoms to the npnp state
as well by the process [27, 31]
nsns —» nsnp —» npnp.
(3.27)
Like the nsns state, the npnp state does not exhibit a dipole-dipole energy shift. Conse­
quently, it can only be reached from the nsns state when the microwave frequency is at
the atomic ns — np frequency. When the ns and np atoms are at any finite internuclear
separation, the intermediate nsnp state of the two-photon nsns — npnp transition is out
of resonance, and the two photon transition is suppressed. This point is verified by calcu­
lations based on a three-level model analogous to Eq. (3.27). Second, it is assumed that
the transitions are in the perturbation theory limit. In this approximation the transition
probabilities are proportional to the squared transition matrix elements.
The transition probability, P, is given by
P=\(ns\V c \np)\ 2 ,
(3.28)
where V c is the coupling operator and couples the initial and the final states, although the
exact form of P is
P = \{nsns\V c \nsnp)\ 2 ,
(3.29)
since one of two n s atoms is just a spectator and does not affect the transition probability,
and we can write the transition matrix element in the form of Eq. (3.28). The coupling
operator can be written in terms of the dipole moment and the microwave field as V c = p.-E.
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
70
The transition matrix element is given by
(alVclp) =
(nslfii
• E \np)
= (ns\xiE x + ViE y
+ ZiE z \np).
(3.30)
Since the molecule is symmetric about the internuclear axis 2, the microwave field can be
assumed to lie in the x — z plane. As shown by Fig.3.1(b), the microwave field is at an angle
of 9 with respect to the internuclear axis. Thus Eq.(3.30) can be reduced to
(3.31)
ns Pi • £?| np^ = Ecoscot {ns |XiSinO + Zicos0\ np}.
This matrix element can be rewritten using Edmonds' C-tensor as
ns |/Z • E np^ = Ecosutr.nsnp
4- C\ x )sin9 -I- C QCO sB np
ns
(3.32)
V2
For AM=0 and ±1 transitions only the Cq and C\_A matrix elements, respectively, are
nonzero. For example, the two M — 0 basis states nsi ins 1 1 and nsi 1 npi _ 1 are coupled
2 2 2 " 2
2 2 2 2
by the z-component of the microwave field by the matrix element
n s 1 i n s 1 _ 1 p • E n s i i n p i _ i ) = E c o s c o t 'r .nsnpCOs6 (s 1 _ 1 | C n | p i _ l
25
2
\
2
2
2 '
1
2
(3.33)
2
Since the factor E cosujtr n s T i p is common to all the transitions, it is omitted and our
attention is focused on the angular factor.
The square of the angular matrix element
is averaged over the orientation of the microwave field to obtain the relative transition
probabilities.
The result is that they are simply proportional to the squared C-tensor
matrix elements,
P oc |(ns|C4|np)| 2 .
Table 3.7 shows the energy shifts in terms of
/R 3 , or x, and the squared transition
matrix element (ns |C^| np) 2 . These values are provided for
nsnp
(3.34)
nsns
transitions to spinless
states, ns 1 /2 n Pi/2 states, and nsj/2iP3/2 states. In Fig.3.9 we present stick spectra for
these three cases. The intensities shown in Fig.3.9 are given by the products of the squared
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
71
Table 3.7: Possible transitions, level shifts, and squared transition matrix elements
from M
symmetric
antisymmetric
0
0
0
0
nsns —» nsnp (spinless)
to M
shift (x)
ns l/2 ns l/2
symmetric
antisymmetric
antisymmetric
-+ nsl/ 2 nPl/ 2
from M
to M
shift(x)
0
0
±1
±1
±1
0
0
0
±1
±1
0
0
0
±1
0
-2/9
-2/9
0
4/9
-4/9
2/9
nsl/2nsl/2
symmetric
-2/3
1/3
2/3
-1/3
0
±1
0
±1
squared
transition
matrix element
1/3
1/3
0
0
•
squared
transition
matrix element
2/9
2/9
2/9
2/9
2/9
2/9
2/9
n s l/2 n P3/2
from M
to M
shift (x)
0
0
0
±1
±1
±1
±1
±1
0
0
0
0
±1
±1
0
0
±1
±1
±2
0
±1
±1
-1/3
0.0718
-0.5162
5/9
-1/3
0.0718
-0.5162
1/3
-5/9
-0.0718
0.5162
squared
transition
matrix element
4/9
0.42
0.1
1/9
1/9
0.05
0.38
2/3
4/9
0.42
0.26
transition matrix elements and the multiplicities of the transitions. In all cases, the average
of the shifts weighed by the strengths is zero.
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
•symmetric states
Y///X asymmetric states
72
nsnp
3
si
nsnp,,
S
c
"T
•
r-
spinless
r~
—
10
-10
shift [MHz]
Figure 3.9: Stick spectra expected for nsns to nsnp transition of spiniess atoms, the nsns to nsnp\/ 2
transition, and the nsns to nsnp^/2 transition. The atoms are assumed to be at fixed R such that the
maximum shift of a spiniess atom state is ±10 MHz. The solid bars are transitions between symmetric
states, and the shaded bars are transitions between asymmetric states. There are no asymmetric transitions
for spiniess atoms. The weighted average shift is zero for both the symmetric and antisymmetric states in
all cases.
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
73
MOT
LASER ->
Figure 3.10: Geometry of the MOT and the 480 nrn laser beam, which together define the volume of Rydberg
atoms.
3.5.2
Finding the inter-atomic distance R
In the the spectral line model from the dipole-dipole interaction, finding the internuclear
distance, R, of the atomic pair is crucial since the strength of the dipole-dipole interaction
is very sensitive to R, which is randomly distributed over the Rydberg atom cloud even
with uniform density. For this reason, the atomic sample is sliced into infinitesimal volume
elements of constant density and the probable R distribution within the small volume is
calculated.
To begin, the geometry of the Rydberg atom cloud is found by assuming the MOT to
be spherical and the 480 nm laser to be a cylindrical beam. The laser propagates through
the center of the MOT along the x axis and the intersecting volume forms a cigar-shaped
cloud as shown in Fig.3.10. Assuming a spherical Gaussian MOT density of radius
a Gaussian light intensity cylinder of radius
and
aligned along the z-axis, the density of the
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
74
Rydberg atom cloud is given by
_w*
e "T
p(w,z) = poe
(3.35)
in the cylindrical coordinates w and z, and it can be written as
_»2
_ *2
p ( w , z ) - p 0 e "re
(3.36)
where po is the peak density at the center of MOT, and the transverse radius rx is defined
by
.1 = 1. + i_.
4
r
r
L
(3.37)
M
After taking the logarithm, Eq. (3.36) can be written as
Mpo/P) - (~2 + ~2~)
\rT
t
(3-38)
M)
and then
w2
z2
~2~j
i
/
\
+
2
i
( —T \
r x ln(po/p)
r^ln(p 0 /p)
=
(3.39)
making it apparent that surfaces of constant density are ellipsoids of revolution about the
2 axis.
The total number of Rydberg atoms within the ellipsoid is given by
_ 1
roo
N = on /
2irwe
tt
roo _ *1
e
dz
J-oo
dw /
Jo
(3.40)
= P ^ ,2T \ t m -
Since an average density is measured in the experiments, it is useful to define the relationship
between the average density and peak density as follows.
PO
2 ^ pQv
(3.41)
Chapter 3. The clipole-dipole interaction of Rb nsnp atoms
75
Figure 3.11: Relative number of atoms [ln(po/p)] 1/2 as a function of density relative to the density at the
c e n t e r o f t h e t r a p , p/po-
The volume enclosed by the ellipsoid of density p is
V ( p ) = y^T r iw( ln (Po//3)) 1/2 -
(3-42)
The change in volume with density is
dV _
dp
2-nr^r M
(In(p 0 /P) 1 / 2 p
(3.43)
The number of atoms contained in the ellipsoidal shell between volume V and V + dV is
dN = pdV = 2nr^r M (ln(p 0 /p)) l / 2 dp,
(3.44)
from which the variation in number with density is found. Explicitly,
dN
dp
which has the form shown in Fig.3.11.
2irr^r M (\n(p 0 /p)) 1 / 2 ,
(3.45)
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
3.5.3
76
Lineshape fitting to the observations
Using Eq.(3.45) the atomic cloud can be broken into -well defined density shells of dis­
crete, A(p/po). However, the internuclear spacing R is not definitive even in the miniscule
volume of the shell. Let us consider first the variation of R at a fixed density p. The average
spacing, R av (Wigner-Seitz radius), is defined by 47r/?|v/3 = IJ p. On average the volume
per atom is 47ri?^/3, so given one atom, there is on average, a second atom in a sphere of
radius R av around it. Since there are many atoms, it is reasonable to assume that the second
atom is always found in the sphere of radius R a v An alternative and more sophisticated
expression of the most probable R is, Rnn = (27rp)-1/3, based on the nearest-neighbor dis­
tribution function [35]. However, we use the Wigner-Seitz radius since it is simpler. Before
the atoms are excited to the Rydberg state, their interactions are negligible on the length
scale of interest, so it is assumed that the probability of finding the second atom is uniform
within the sphere of radius R av . Accordingly, the probability of having its nearest neighbor
between R and R -I- dR is
dP(R) =
tints
^av
,R< Rw
(3-46a)
and
= 0,R>Rav.
(3.46b)
The frequency shift, v { R ) , for a particular atom pair is given by
» { R ) = ^
<3-47)
and
dv{R)
dR
=
3n A \
W
(3.48)
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
77
where x is an eigenvalue obtained in the theory section. The probability of the frequency
being between v and v+dv is then given by
d P _ d P d R
di/
dR
du
- (^)
h
("9>
- fE.
i>2
In short, R a v is the upper limit to the spacing of the nearest neighbor atoms. If a single
line of a stick spectrum of Fig.3.9 is considered, for example, x—-4/9 at frequency shift
v c =-10 MHz, as shown in Fig.3.12(a), the distribution of spacings given by Eq.
(3.46a)
translates into a frequency band beginning at v c and falling off to larger frequency shifts
with a 1/v 2 dependence for frequency shift | v |>| v c |. All frequency components of Fig.3.9
behave in the same way; the only difference being that the breadth of the band shown in
Fig.3.12(b) scales with v c .
The spectrum in Fig.3.12(b) turns into one in Fig.3.12(c) after averaging over the density
distribution. Each of the elements in the stick spectra of Fig.3.9 can be treated in the same
manner. When the contribution for all the lines, Xi, of the stick spectra are added together,
the lineshape function is given by
\ ^ftp{r)r2dr
f°°nI
This final spectrum is obtained assuming infinite instrumental resolution. The instrumental
resolution is easily taken into account by convoluting the infinite resolution spectrum with
the optimum power Rabi lineshape for a 500 ns long microwave pulse. The Rabi lineshape
function is given by
Lflabi
n2
/
\2 ^
(i/0 - vy
.
2
- v)t
:
o
2
(3.51)
where iv0 is the atomic resonance frequency, il is the Rabi frequency, t is the microwave
pulse length. The model described here is restricted to the perturbation theory limit.
The nsns - nsnpj ( j = 1/2 and 3/2) transition spectrum is reproduced by the model
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
78
(a) *=-4/9
(b)X=-4/9
at a certain density
(c)x=-4/9
over the whole sample
r~
-60
•
r
-60
•
t,
j\
1
1
r
-40
-20
0
frequency [MHz]
Figure 3.12: The effect of averaging over density on the line of Fig.3.9 with a frequency shift of -10 MHz for
the internuclear spacing R = R a v (a) The line for R = R av . (b) Averaging over the range of internuclear
spacings at the density p(r) = 3/4-rr/?<,„, showing the broadening to larger shift, (c) Averaging over the
different densities p(r) at different positions r in the trap. The low density regions of the trap substantially
reduce the shift.
Chapter 3-
Tie dipole-dipole interaction of Rb nsnp atoms
79
003-
average density
0.02-
0.04X10 9 cm
0.01 -
0.07X10 9 cm"
0.00
c
0-10-
0.17X10 cm"
~ 0 05 -
0.35X10 cm"
0.150.10-
0.050.00-
-40
-20
0
20
40
v-y0 [MHz]
Figure 3.13: Observed (solid lines) and calculated (broken lines) 44s — 44pj/ 2 resonance lineshapes at the
experimental average densities 0.04, 0.17, and 0.47 xlO 9 cm -3 . The atomic frequency fo =45114.68 MHz.
In all cases the broken lines are at the level of the background signal far off resonance. To match the observed
lineshapes requires densities a factor of two higher in the calculations. Specifically, the densities of 0.07,
0.35, and 0.88 xlO 9 cm -3 are used in the calculation.
Chapter 3, The dipole-dipole interaction of Rh nsnp atoms
0.03-
80
average density
0.04X109 cm'3
0.02-
0.05X109 cm 3
0.01 0.00-
0.15-
i 0.10-
0.15X10 cm
S 0.05 -
0.28X10B cm'3
0.00-
0.150.10-
0.29X109 cm
0.05-
0.63X109 cm'
000
-40
0
-20
v-v 0
20
40
[MHz]
Figure 3.14: Observed (solid lines) and calculated (broken lines) 44s - 44p 3 / 2 resonance lineshapes at the
experimental average densities 0.04, 0.15, and 0.29 x 10 9 cm" 3 . The atomic frequency vo =46345-42 MHz. In
all cases the broken lines are at the level of the background signal far off resonance. As in Fig. 12, to match
the observed lineshapes requires densities a factor of two to four higher in the calculations. Specifically, the
densities of 0.05, 0.28, and 0.63 x!0 9 cm -3 are used in the calculation.
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
0.15
~n~ 44
t
average density
R
0.10
Ji
0.05
81
0.47X10* cm3
V
o.eaxio'cm3
0.00
0.21
• rj=39
0.14
n
0,07
/
0.00
0.21
0.76X10® cm
1.4X10* cm1
V
~i
.
r
.
»
.
•n= 34
0.14
jI
0.07
/
0.00
*
"
T
.
I
\
i
r
0.79X10* cm'1
1.58X10* cm *
r
.
i
0,28
0.21
fi
0.14
0.07
J l
J
\
0 75X10" cm
1.31X10* cm
0.00
v-v0 [MHz]
Figure 3.15: Observed (solid lines) and calculated (broken lines) lineshapes for ns — npi/2 transition of n=
28, 34, 39, and 44 at relatively high densities. In all cases the broken lines are at the level of the background
signal far off resonance. For n =28, 34, 39, and 44, the experimental average densities are 0.75, 0.79, 0.76,
and 0.47 xlO 9 cm" 3 , respectively, and densities used in the calculations are 1.31, 1.55, 1.4, and 0.88 xlO 9
cm -3 , respectively.
Chapter 3• The dipole-dipole interaction of Rb nsnp atoms
82
and n=44 spectra are presented in Figs.3.13 and 3.14 as examples. The model fits well to
the non-Lorentzian observed lines including the cusp and wings. The least square fitting
is employed for the best fitting curve.
The fit parameter is the density of the atomic
sample, and discrepancies of the density (about by a factor of two) between the model and
the observation have occurred mainly due to the uncertainty of the density measurement.
Similar results are shown in Fig.3.15 for various n states at relatively high densities. The
model works well over a range of the density and n.
3.6
Concluding remarks
The line broadening experiments are performed to measure the dipole-dipole interac­
tion. The density dependent measurement provides a quantitative measure of the dipoledipole interaction in laboratory units, although the cusp-shaped lines deviate from a general
Lorentzian transition profile. The cusp-shaped resonances are attributed to two factors: the
existence of the very weakly and non shifted nsnp levels and the distribution of the atomic
density in the Rydberg atomic cloud. According to the theoretical approach, the spin-orbit
interaction suppresses the dipole-dipole interaction; in fact, several nsnp levels have small,
even zero, dipole-dipole shifts. These weakly dipole-dipole coupled pairs of atoms are driven
at the atomic transition frequency, only contributing to the center of the spectrum. Fur­
thermore, the fraction of nearly isolated atoms also contributes only to the atomic frequency
in the signal.
In these measurements the line broadening is less than expected on the basis of a naive
classical model. These results appear to contradict measurements of dipole-dipole processes
through the Forster resonant energy transfer [4, 5], in which the breadth of the observed
resonances substantially exceeded the naive expectations. In those measurements, obser­
vation of the transition requires that atoms be closely spaced so that they are strongly
dipole-coupled to exchange transition energies. About 20% of the Rydberg atoms in the
MOT, which are the most closely spaced, contribute to the transition signal. Consequently,
a dipole-dipole shift is observed, resulting in wider resonances than expected on the basis
Chapter 3. The dipole-dipole interaction of Rb nsnp atoms
83
of the average density.
Contrary to the resonant energy transfer process, all atoms of the atomic sample equally
participate in the transition in the line broadening experiment. Since the other 80% of
atoms, which have relatively weak dipole-dipole interaction, contribute to the central fre­
quencies of the spectrum and thus the observed resonances are sharply magnified at the
center as a cusp. Moreover, a fraction (for example, 25% for nsns - nsnpi/ 2 transition) of
the atoms has zero dipole-dipole energy shift regardless of atoms' spacing and its transition
occurs only through the atomic frequencies. For these reasons the line broadening is less
than expected on the basis of the average density.
In the experiment with the presence of the magnetic field, essentially the same broaden­
ing rate is observed in comparison with the field-free measurements, although the magnetic
field seems to slightly suppress the broadening rate from T — 4.8 to 4.25 x 10~ 1 5 n' i (MHzcm 3 ).
These measurements appear to be inconsistent with Afrousheh's results [20] in which the
broadening is considerably suppressed by a magnetic field. In the measurements of Afrousheh
et al. the dipole-dipole broadening is less than the energy shift caused by the magnetic field.
As a result, the magnetic field lifted the degeneracy of the dipole-dipole coupled states, lead­
ing to less broadening than occurs in zero field. However, in our experiments the atomic
density is raised further to increase the dipole-dipole interaction and thus the broadening is
always larger than the breadth due to the magnetic field, so in the density range of interest
to us the broadening is not influenced by the magnetic field.
It would be useful to explore other probes of the dipole-dipole interaction such as the use
of the Ramsey method of separated oscillatory fields, echoes, or spectroscopy of states at
or near a Forster resonance. In the future direction, the Ramsey method will be introduced
as an alternative and more sensitive way to probe the dipole-dipole interaction.
84
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87
Chapter 4
The ionization of Rb Rydberg
atoms in the attractive nsnp
dipole-dipole potential
4.1
Introduction
Studies on cold plasmas have been actively conducted due to their easy-to-make and
easy-to-control properties compared to a typical plasma of 10 4 ~10 7 K temperature [2,
1,3]. In particular, a plasma produced from Rydberg atoms is a convenient channel to
access the plasma phase and characterize its properties. Since the valence electrons in
Rydberg atoms are loosely bound, Rydberg atoms spontaneously evolve into a plasma in
some circumstances. Although the formation of a plasma from Rydberg atoms has been
observed to depend on experimental conditions such as the allowed interaction time; n, the
principal quantum number, and density of atomic sample [4, 5, 6], the precise mechanism
has not been completely understood.
According to recent studies, Rydberg atoms evolve into a plasma in several steps [6]. In
the very beginning of the evolution, ionization occurs and electrons leave the atomic cloud.
The excess ions, forming a macroscopic Coulomb potential, attract subsequently produced
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
88
electrons, and the energetic electrons collide with the remaining Rydberg atoms to trigger
an avalanche of ionization. Depending on the density of the atomic sample and its quantum
state, plasma formation occurs in tens of nanoseconds to tens of microseconds [4, 7].
Among the processes of plasma formation, the early stage of ionization is the trickiest
part to understand and thus most actively studied, since the ionization is a consequence of
rapid collective effects. Several plausible notions have been proposed to explain the early
ionization. Direct photoionization, the simplest explanation, is possible by blackbody radi­
ation. Blackbody radiation is able to ionize some atoms and redistribute the population of
remaining atoms to nearby states. However, blackbody photoionization is not so persuasive
an explanation in the regime of high n states, since its ionization rate drops as n~ 2 [8].
A more likely form of ionization at high n is collisional ionization of atoms induced by
the attractive or repulsive force of neighboring atoms. The origin of the force that causes
the motion of atoms is the dipole-dipole interaction. Let us take examples directly relevant
to this experiment. First, consider two atoms in the dipole coupled nsnp state. The basic
and fundamental features of the nsnp interaction are discussed in the previous chapters.
As a result of the dipole-dipole interaction between two dipole coupled atoms, the pair of
atoms forms potentials in which atoms either attract or repel with a strength proportional
to 1/i? 3 , where R is the distance between the two atoms. Especially for Rydberg atoms,
since the dipole-dipole force soars in proportion to n 4 , the collisional process occurs easily
and quickly.
Second, let us take a slightly different, example. When two atoms form the nsns molec­
ular state, these two atoms are not dipole coupled, but the atom pair still creates repulsive
or attractive potentials depending on n, although the interaction is weaker than that of
nsnp pair. This type of interaction, known as the van der Waals interaction, is the off-
resonant dipole-dipole interaction, which has a 1 /R e dependence. However, the off-resonant
dipole-dipole interaction becomes the 1/i? 3 type interaction when two atoms get closer than
a critical distance, R c where the dipole-dipole coupling equals the detuning between the
molecular levels. Li et al. observed the effect of R,: by observing the ionization process in
46d46d and 48s48s molecular states [7]. They found that due to its strong dipole-eoupling
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
89
to nearby state, l/R 6 -type 46d46d interaction turns into to 1/R 3 type interaction with large
R c compared to the 48s48s interaction, for which R c is smaller.
These interaction-induced ionizations are observed in both the dipole-dipole potential
and the van der Waals potential, independently. The atoms located in the dipole-dipole
potential are forced to move and accelerate obtaining much higher speed than their initial
speed [9]. Although the motion caused by the dipole-dipole force can either be attractive
or repulsive, the ionization differs based on the type of potential [7, 10, 11]. In a few
microseconds, a substantially increased ion production is observed from the atoms placed
in the attractive dipole-dipole potential, while no sign of increased ionization is observed
by the repulsive potential. Similar results are observed from van der Waals potentials as
well. Like the dipole-dipole attractive potential, atoms excited to the attractive van der
Waals potential are clearly ionized in a few microseconds due to the subsequent collision.
However, unlike the dipole-dipole potential, ionization is observed in the van der Waals
repulsive potential as well [12]. It should be noted that the ionization from the van der
Waals repulsive potential was observable only in long delay times, ~ 20/is, in which atoms
were transferred to nearby attractive dipole-dipole potential.
In the range of small R, typically R < 1/im, the high order multipole interactions
become important [13]. In this regime, the potential curves are not simply proportional
to ~ 1/R?. Strongly multipole coupled levels at small R lead to many avoided crossings
and occasionally wells, resulting in a complicated structure. Nevertheless, these multipole
couplings have little effect on collisional ionization. When atoms reach the small R regime
from Ri of interest, the atoms are moving so fast that they only traverse the possible
crossings and wells diabatically and undergo ionizing collisions. Therefore, the assumption
of ~ 1/R 3 potential curve is not too oversimplified for estimating the collision time. What
actually affects the collision time is the initial thermal energy of the system. For example,
if the initial thermal energy of atoms is zero, the motion of the atoms only depends on the
interaction force, but if they are heated to 300 fiK, the collision time is reduced by a factor
of 2 with an atomic density of ~ 10 9 cm" 3 .
In this chapter, the early ionization mechanism is explored with attempt to separate the
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
90
effects of the nsnp attractive and repulsive dipole-dipole potential by comparing with that of
nearly flat nsns van der Waals potential for n=40 and 41. The experiment first shows that
the ionization occurs only in the attractive potential. The following experiment describes
the time evolution of atoms in dipole-dipole potentials, and its results are compared with the
ionization model. According to the collisional ionization model, the initial thermal energy
plays a significant role in reducing the collision time. A high microwave power experiment
and its results are presented to emphasize the difference of ionization from the attractive
and repulsive dipole potentials. At the end, a short discussion and conclusions are given.
4.2
Experimental Approach
The broad sketch of the experiment is shown in Fig.5.1. Similar to previous chapters,
a 480 nm laser pulse excites trapped 5p 3 / 2 atoms to the ns state, producing nsns atomic
pairs. The trapping magnetic field is continuously applied in order to keep the Rydberg
atom density as high as possible. The nsns energy level is nearly flat, although it forms a
slightly repulsive van der Waals potential. Due to the negligible nsns shift, the 110 MHz
bandwidth of the 480 nm laser is enough to excite atoms with random interatomic spacings
in the MOT. These nsns atoms are then driven to nsnp or ns(n — l)p state by 500 ns long
microwave pulses. Both nsnp and ns(n — l)p states are dipole-coupled to the nsns state
and form more or less the same depth of potentials. Atoms transferred to attractive and
repulsive potentials are allowed to move for a delay time, r, and then are ionized by the
PFI pulse. As the microwave frequency is swept, the time resolved ion and np signals are
recorded and stored in a computer for further analysis. In measuring the absolute number
of Rydberg atoms, the same procedure as in Chapter. 3 is conducted.
Note that the ionization from the dipole-dipole potential only occurs when particular
experimental conditions are fulfilled. First, the trapping magnetic field has to be constantly
applied so that the MOT provides enough atoms to be excited to a Rydberg state. With the
switching mode of the magnetic field, the average MOT density drops by a factor of three
and no sign of ionization is observed. Second, atomic population transfer from nsns to nsnp
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
91
microwave
480nm laser (ns)
fteM ionization ptiLs
480noi
laser
microwave pulu
delay r
microwave
500 iw
nsfrt'lip
np f \(n-l)p
interatomic distance R farb. units}
(a)
(b)
Figure 4.1: (a)Typical energy levels for the experiment. The 480 nm laser excites pairs of ns atoms to the
nsns potential over a range of internuclcar spacings, as shown by the slanted solid line arrows. A microwave
pulse, shown by the dotted line arrows, is used to drive the transition to either the nsnp state or the
ns(n — 1)p state, which can either be attractive or repulsive, depending on the microwave frequency. Pairs
excited to a repulsive potential move apart, and those excited to an attractive potential collide, resulting in
an ion and a more deeply bound atom, (b) Timing diagram for the nsns — nsnp transition. The 8 ns long
480 nm laser pulse excites the atoms to the ns state, and is immediately followed by a 500 ns long microwave
pulse which drives nsns pairs to the nsnp state. After a time delay r, a field ramp is applied to observe the
time-resolved ion, np, and ns signals.
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
92
should be efficient by the microwave pulse. There are two ways to transfer more atoms.
First, since the variable microwave pulse length gives a control of the sampling width of the
random interatomic spacings, a shorter pulse can be used to excite a broader range of atoms
and therefore, larger number of atoms can participate in the ionization process. Second,
increasing microwave power also leads to the broader sampling range. In the experiment,
the microwave pulse length is fixed at relative short 500 ns but the power is continuously
raised to increase the atom number in the nsnp state.
Time-resolved field ionized signals from different laser intensities are presented in Fig.
4.2. After the 41s41s excitation, an unexpected amount of ion and 41 p atoms are measured
at no time delay when the number of Rydberg atoms is increased. The initial production
of ion and 41p atoms raises the background signals above zero in the far-off resonant range
in the subsequent section, although the exact mechanism of this rapid redistribution is not
understood. However, the 5^3/2 — 38s transition is not power broadened and the linewidth
of the 480 nm laser bandwidth is 110 MHz, as presented in Chapter. 2. Thus, the effects
of ac Stark shift due to the laser can be precluded [14]. For reference, the change in the
number of ions and atoms produced as a function of total Rydberg atom number is shown
in Fig. 4.3.
In this experiment, different microwave powers are used. First, a substantially reduced
power is used to apply the perturbation theory in the ionization model. In this power
regime, the microwave pulse produces a transform limited linewidth with a low density
sample. A 2 MHz-width spectrum produced by 500 ns pulse is broadened to 4 MHz in
the presence of the trapping magnetic field. Second, the microwave power is increased to
improve the efficiency of the excitation rate to the dipole-dipole potentials so that ionization
from different potentials is clearly compared.
4.3
Observations
First, ionization from attractive and repulsive potentials is compared in Fig. 4.4. The
ion signal is recorded over wide range of the microwave frequency below and above the
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
0.0000
41p
ion
41s
-o.ooos - !V^v)ytMC^iNV»
93
40p
...
!
-0.0010 -
number of Rydberg atom
1.96X10
g -0.0015-
o>
'35
6.68X10
£ -0.0020 Q.
•27.88X10
-0.0025
51.44X10
-0.0030
-0.0035
-0.0040
0000000
1
i
1
1
0.000001
0.000002
0.000003
0.000004
time
Figure 4.2: Field ionization signal with the excitation laser tuned to the 5p 3 / 2 - 41s transition and no delay
between the laser and PFI pulse. As the atomic density is increased, ions and atoms in high-lying states
as well as 41s atoms are observed. The broken lines indicate the gates to measure the number of ion and
atoms.
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
94
5x10 3
w
4x10 3
•
#of41p
A
# of ions
E
o
,
o5 3x10
Q.
3 2x103
o
*
1x103
0
1.0x10 3
8.0x10 2
CO
c=6.0x10
o
2
-^4.0x10 2
2.0x10 2
A
0.0
A
•
1
0
,
1
1x10"
.
1
2x10 4
.
1
3x10 4
.
1
1
4x10 4
1
5x10 4
>
[
6x10 4
number of Rydberg atoms
Figure 4,3: The number of ions and 41 p atoms with no delay as a function of the number of Rydberg atoms
measured in Fig. 4.2. Both graphs of ions and 41p atoms vs. total Rydberg atoms are not quite linear but
slightly quadratic.
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
'
95
atomic resonance so that atoms are transferred to both attractive and repulsive potentials.
The resonance frequency, which is indicated by a broken line in the figure, is the boundary
between the two potentials. The microwave power is raised enough for the distinct ion
signal to appear in the vicinity of the resonance frequency. As shown in Fig. 4.2, ions are
already produced from the high density 40s atomic cloud before they are influenced by the
dipole-dipole potentials. The background offset from the initial ions is subtracted to zero in
the figure. Fig. 4.4(a) shows the ion signal when atoms in 40s40s are transferred to 40s40p
potentials. The number of ions is not influenced by far-off resonant frequencies and starts
rising at a frequency below the resonance. After reaching its maximum near the resonance
frequency, the ion production sharply decreases to the background level. The ionization
from the attractive potential is in stark contrast to the lack of ionization from the repulsive
potential; no sign of increased ionization is observed on the repulsive side of the potential.
Although the absorption of the microwave photon increases the energy of the atoms,
the increased energy has nothing to do with the increased ion production. To verify this
point, 40s atoms are driven down to 40s39p dipole-dipole potentials. In Fig. 4.4(b), the
reversed signal of Fig. 4.4(a) is shown due to the fact that the distance of the microwave
frequency from 40s40s to the 40s39p attractive potential is now farther than to the repulsive
potential as described in Fig. 5.1(a). The increased number of ions is again only caused by
the attractive potential, and the ionization is not due to increased energy by the microwave
photon.
The ionization process is also influenced by the allowed time r, since collision time differs
depending on the initial separation of atoms. For a systematic study of the time-dependent
ionization, r is varied from 0 to 15 fj.s after the microwave population transfer from 41s41s
to 41s41p, while recording both ion and 41p atom signals simultaneously. In this time
dependent experiment, the microwave power is reduced to compare the observations with
the ionization model, which is based on perturbation theory. In Fig. 4.5, observed ion
signals are presented at delay times, r=0 ~ 11 fis. The broken line indicates the atomic
resonance frequency of the 415 — 41p transition at 56629 MHz, and the zero background
level is shown on the left at the beginning of the microwave sweep by blocking the excitation
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
96
(a) ion from 40s40p
v>
"E
.ci
«B
TO
C
g>
c/>
60500
61000
62000
61500
(b) ion from 40s39p
c
o
69800
i
70000
>
r
70200
1
70400
70600
'
1
70800
MW frequency [MHz]
Figure 4.4: Ion signals obtained with a delay of 5 /is arid a microwave power producing a 10 MHz linewidth
after excitation to the 40s state, (a) Ion signal obtained in the vicinity of the 40s - 40p transition at 61332
MHz. (b) Ion signal in the vicinity of the 40s - 3 9p transition at 70262 MHz. Although the former transition
is to a state of higher energy and the latter to a state of lower energy, the ions in both cases are observed
only in the attractive potentials.
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
97
laser. For r=3 /xs, no significant increase in ion production is observed at any frequency.
At
T =5
fj,s, the increased ion signal emerges around the resonance frequency and grows
progressively afterwards. Again, the prominent ion signal is on the low frequency side
which is within the attractive potential regime. The increased ion signal is distinct at r=9
and 11 fj,s, but at longer delays random ionization spikes appear in all frequencies due to
the fact that a dense cluster of Rydberg atoms spontaneously evolves into plasma if enough
time is allowed. Therefore, our discussion is limited to about 11 /xs, since the spontaneous
evolution to a plasma for any delays beyond 11 /xs creates a fairly big background noise
which covers up all other processes of interest.
Since 41p atoms are the source for increased ions, the loss of 41jt> atoms becomes an
indirect measure of the ionization. Fig. 4.6 shows the 41s — 41p resonances as a function
of delay time simultaneously measured with the ion signal of Fig. 4.5. The broken line
indicates the atomic resonance frequency, and the steps in the beginning indicate the zero
line. At zero delay, the low density trace is presented to show the resonance frequency. Until
T= 3
/is, when no ionization occurs from 41s41p molecules, the resonances are symmetric.
After r=3 /xs, however, the low frequency side of the resonance is cut off as ionization
becomes increasingly active.
4.4
Modeling
The experimental data of Figs. 4.4 and 4.5 apparently show that when pairs of atoms
are placed on an attractive dipole-dipole potential and are allowed time to collide, ionization
occurs. The notion that cold atoms move due to attractive dipole-dipole potentials is not
new [15, 16], and other examples of motion on Rydberg-Rydberg potentials have been
observed [17, 18]. It is straightforward to compute the time required to collide and ionize,
assuming atoms at rest initially. According to the calculation, delay times greater than 10/xs
are required for atoms to collide [7, 19]. Most of the time elapses when atoms are far apart
and moving slowly, an observation which suggests that the initial thermal motion of atoms
is important. When we take it into account, our simple model, similar to Robicheaux's,
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
98
delay time
0.040.03
0.02
x=11uS
0.01 •
0.04 •
=9ns
0.03
0.02
0.01
0.02
Cfl
c
3 0.01 -
.a
ro
CO
g>
f
w
c
x=5|iS
0.02
c
8:8^
x=3^s
o
0.01
0.020
T=1fiS
0.015
x=0|aS
0.015
0.010
j
56400
,
(
56500
r-
1
1
56600
(
56700
.
|
56800
MW frequency [MHz]
Figure 4.5: Ion signals vs microwave frequency observed subsequent to a delay r after driving the 41s41s to
41s41p transition. The peak Rydberg atom density is p 0 = 6 x 10 9 cm" 3 . The delay time r is raised from
0 to 11 )xs. Traces for later times are offset vertically and are on coarser scales. The left hand side of each
trace shows the zero signal level. The off-resonant., or background, ion signal rises monot.onically with delay
time, and for r > 5 fis there is a visible increase in the number of ions at frequencies just below the atomic
41s - 41p transition at 56629 MHz. For delays of 9 and 11 fis the ion signals are distinct. Also distinct in the
9 and 11 /is traces are the random spikes in the ion signal. They originate from the spontaneous evolution
to a plasma on intense shots of the laser leading to an abnormally large number of 41s atoms.
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
99
delay time
0.04-1
T=11 us
0.02-
0.06-
0.03 0.06-|
0.03-
XI
0.10
c
oi
0.05 -
£ 0.00-
31
0.10-1
0.05-
0.00-1
0.10-1
0.05-
0.00-1
0.10-t
0.05 0.00
56400
56500
56600
56700
56800
MW frequency [MHz]
Figure 4.6: 41p signals vs microwave frequency observed simultaneously with the ion signals of Fig. 4.5.
The time delay r increases from 0 to 11 (is. Traces for later times are offset vertically. The left hand side
of each trace shows the zero signal level. The off resonant, or background, 41p signal increases until the
delay reaches 5 fxs, then it decreases. With short delay times, the observed lineshape is nearly symmetric,
but when the delay is longer than 5 /js the low frequency side of the lineshape begins to disappear since
pairs excited to the attractive potential are lost to ionization. For reference, we include a thin trace of the
rescaled low density signal obtained with the same microwave power for r = 0 //s.
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
100
cut J
41s-p)(J transition line for a=-2/9
•if: ionization
Lorentzian function
convolution of ip and <p
-150
-100
0
-50
50
100
150
vQ-v [MHz]
Figure 4.7: Construction of the ion signal after a 9 fis delay for the 41s41p state with a = -2/9. (a)
Infinite resolution spectrum expected for t.he distribution of atoms in our trap (thin line). The ionization
signal comes only from atoms which collide in 9 (is, i. e. only from frequencies v <
(bold line), (b)
4 MHz Lorentzian representing the experimental resolution, (c) Convolution of (a) and (b), which is the
contribution to the ion signal from the 41s41p state with a = —2/9. There is an analogous contribution for
the state with a = —4/9.
agrees well with experimental data [19].
The model is straightforward and its summarized procedure is shown in Fig. 4.7. The
collision time, r, is first calculated for two atoms separated by R based on the calculation
of the dipole-dipole potential discussed in Chapter. 3. Since R can also be translated into
frequency shift, u(R), a certain collision time defines a cutoff frequency, v c , which specifies
the maximum R, Rmax- The transition line from nsns to nsnp is cut off at vc and only the
low frequency side of uc leads to ionization. Finally, the convolution with the Lorentzian
function leads to the ionization lineshape model.
It is assumed that the attractive dipole-dipole interaction is the dominant mechanism
to induce collisions, and the initial thermal velocity from MOT temperature 300 fiK is
taken into account. While two atoms are in motion, they are assumed to remain on the
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
101
same potential. The hyperfine structure is ignored in the model since its 1 MHz broadening
is negligible compared to the 50 MHz dipole-dipole broadening. Taking into account the
spin-orbit interaction, three attractive nsi/ 2 np1/2 potentials axe present, given by
There axe two potentials with x = —2/9, and one with x — —4/9. In both cases |x| < 1
since the dipole-dipole potentials axe not as strong as those of spinless atoms. To compute
the motion on the attractive dipole-dipole potential we use the Lagrangian of the two atom
system, given by
L=T - V = l
M
R?+l-Mv
2
t
+^,
(4.2)
where T is the kinetic energy, V the potential energy, M the reduced mass of the two
atom pair, half the mass of a Rb atom, and v t the relative velocity perpendicular to the
internuclear axis. The second term on the right hand side can be written in terms of the
angular momentum £, which is given by
t = MRv t
(4.3)
and is conserved. Using £, the equation of motion can be written as
+
^=
(4 - 4 »
and the equivalent potential is
Using the fact that H is conserved, we write it in terms of the initial value of R, R^, and the
r kT
initital value of v t , vu- Taking the most probable vu as JS,
M • <p(R) can be written as
<«>
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
102
The time to collide for the pair of atoms in the potential 4>(R) is given by
/•* dt
.
/T ^
T= I
= yj7
,
J r =o
v M 7 h= o ^
dR
{Ri)
_ ^R)
(4-7)
+
i_ Mv 2 Ri
where VRi is the initial relative velocity along the internuclear axis, which we assume to be
in the —R direction. It is convenient to specify \Mv 2 Ri =fjkT, where /3 ranges from 0 to
oo. Finally, the average collision time r aw (R t ) is given by
fg roo
e-<V k T d0dR
rR,
Y mJ0JR=0
Tav(Rii
j™e-M k T di3
'
( 4
'
8 )
Eq. (4.8) yields the most probable collision time for a pair of atoms which are initially
separated by Ri in an attractive dipole-dipole potential. In terms of the experiment, the
allowed time for pairs of atoms to collide is fixed by a delay time r. In other words, only
atoms spaced close enough can experience a collision in time r, and the boundary distance
Rmax is given by Eq. (4.8). This distance in turn defines the minimum frequency shift, or
cutoff frequency, i>c(Rmax)> at which an ionizing collision will occur with the delay time r.
The frequency is given by
Vc(Rrnax)
=
•
(4-9)
•**'max
The full details of the procedure to generate the nsns - nsnp transition lineshape are
described in Chapter. 3 [20]. In short, the basis states of nsns and nsnp are first found
including spin. Using nsnp basis states, the Hamiltonian matrix for the dipole-dipole in­
teraction is constructed. Diagonalizing the Hamiltonian matrix yields the eigenvectors and
eigenvalues. The eigenvalues,
provide the information on how the nsnp levels shift.
Putting the nsns and nsnp eigenstates into the transition matrix element coupled by the
microwave field, the transition strength is calculated. Combining the energy level shifts,
transition strength, and density distribution yields the infinite resolution lineshape of the
nsns — nsnp transition.
In this model, only attractive potentials, x=-2/9 and -4/9, are considered. For example,
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
103
the transition line from nsns to x=-2/9 nsnp is drawn in Fig. 4.7(a) by the thin solid line.
The allowed delay time, 9 /us, cuts off the transition lineshape at -4.5 MHz, and only the
lower frequency part indicated by the bold line in Fig. 4.7(a) contributes to the ionization
signal. The truncated bold line is convoluted with a Lorentzian function of 4 MHz linewidth,
shown in Fig. 4.7(b), since a Lorentzian line with 4 MHz width is observed at r=0 and
low atomic density. Fig. 4.7(c) shows the resulting lineshape of the ionized signal from
the
—-2/9 nsnp potential. A similar procedure is conducted for x=~4/9. Since x=-2/9
is two fold degenerate, it is weighted by a factor of 2. Taking into account the transition
strength from nsns to nsnp for each x based on perturbation theory, the final lineshape of
the ionization signal is generated by summing the signals due to the attractive potentials
(negative x' s )For comparison, the lineshape model is plotted by a broken line against the observation
in Fig. 4.8(a). The observed line is from T—9 Ns and the peak density 6 x 10 9 cm -3 of the
atomic sample. As shown, the graph of the model and the observation are well aligned. Also,
we hypothesized that 41p atoms are consumed to produce the ion signal, so the decreased
41s — 41p transition signal on the low frequency side of the resonance after 9 fjs time delay
should match the increased ion signal at T—9 /xs. To verify this, we subtracted the 41p
signal at r=9 fj.s from the scaled 41p signal at r=0. As shown in Fig. 4.8(b), the agreement
between the model and the decreased 41p signal is desirable.
The 41p signal represents atoms leftover from the ionization at delayed times. Accord­
ingly, reproducing the 41p signal by the model is attempted. The procedure is similar to
the reproduction of the ion signal. First of all, all existing dipole-dipole levels are taken
into account including the repulsive potentials. Then from the 41s41s — 41s41p transition
line, the high frequency side of vc now represents for the truncated 41p signal. That is, the
signal shown by bold line in Fig. 4.7(a) is removed in this case. The subsequent procedures
are the same as those of the ion signal reproduction. Fig. 4.8(b) compares the final 41p
lineshape model at r=9 fis with the corresponding observation. Again, the agreement is
good.
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
104
0.02
observation
model
o>
c 0.00 -
0.02
(b) 41P(T=0) - 41p(x=9jis)
o>
Q.
5 0.00 -
T
0.04 -
? 0.02 Q.
0.00-
-r
T
-150
-100
T
-50
T
0
T
50
1 00
150
v0-v [MHz}
Figure 4.8: Observed and calculated signals obtained using the 41s state after a delay of 9 /is before the field
ionization pulse, (a) Observed (solid line) and calculated (dotted line) ion signals. The random spikes arc
due to a spontaneous evolution to plasma on unusually intense shots of the laser, (b) Observed differences
between the scaled zero delay and 9 (is delay 41p signals (solid line) and calculated ion signal(dotted line),
(c) Observed (solid line) and calculated (dotted line) 41p signals.
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
105
4.5
High Microwave Power Observations
In this section, the ionization process is observed again when the microwave power is
raised above that used for Figs. 4.5 and 4.6 and the modeling. While high microwave powers
are not useful for quantitative measurements of ionization on attractive potentials, they are
nonetheless interesting. Power broadening extends the sampled range of R so that more
atoms are transferred to the dipole-dipole potential. Since the increased number of atoms
on the attractive or repulsive potential have an increased probability to collide and ionize,
the resultant ionization rate is increased. The sampling frequency width expands linearly
with the microwave field. In this experiment where ionization from attractive and repulsive
potentials is compared, the enhanced ionization rate underscores the huge difference in
ionization rates.
The microwave power in the regime of perturbation theory is determined in the low
power experiment of the previous section, producing a 2 MHz linewidth with the field
switching mode and a 4 MHz linewidth with the field constantly present from a low density
atomic sample. Let us call the low microwave field strength
raised up to
2E
q and
AE$
E
q . The field strength is now
to obtain the results of Figs. 4.9 and 4.10 with r fixed at 7 /j.s.
The microwave pulse length is fixed at 500 ns. The atomic density is maintained at about
po — 1 x 1010cm~3.
Figs. 4.9(a), (c), and (e) show the ion signals with the field
E
q,
2E
q , and
4E
q , re­
spectively. The broken line indicates the 41s - Alp atomic transition frequency. With the
field
E
q , the amplitude of the increased ion signal is not adequate to overcome the noisy
background by the random high intensity laser shots, although the increased ion signal still
shows a peak at slightly below the resonance frequency. With the field
2£'o,
the increased
ion production on the low frequency side of the resonance is quite obvious and sharply cuts
off at the high frequency side. With the field
4Eo,
the ionization signal is too large to be
explained simply by the motion in the dipole-dipole potentials.
Simultaneously recorded 41 p signals as a function of the sweeping frequency are shown
in Figs. 4.9(b), (d), and (f). For comparison, zero-delay resonances are also presented. In
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
106
s
1.2
•O
h_
03
0.9
D)
tfi
0.6
MW field E0
(a)
MW field 2E0
(c)
MW field 4E
x=7[xs
C
o
0.3
1.2
c
13
(0)
0.9
re.
ro
c
g>
0.6
to
Q.
0.3
r
56560
56630
56700 56560
56630
56700 56560
56630
56700
MW frequency [MHz]
Figure 4.9: Observed ion and 41p signals after a delay of 7 /is at three microwave powers leading to linewidths
of 4, 8, and 16 MHz with low density-atomic samples, showing the evolution to plasma following the high
power microwave pulse. The peak Rydberg atom density is po = 1 x 10 10 cm 3 . In all cases the atomic
41s - 41p transition frequency, 56629 MHz, is shown by the broken vertical line. (a)Ion signal with a
microwave power of 4 MHz-linewidth. There is greater ionization at frequencies below the atomic transition
frequency , but only on intense shots of the laser, (b) 41p signals taken with a microwave of 4 MHz-linewidth
with 0 (dotted line) and 7 us (solid line) delays. The lineshape is symmetric with no delay, but there is a
noticeable asymmetry after 7 /us. (c) Ion signal with a microwave power of 8 MHz-linewidth. There is a
clearly visible ion signal over a frequency band 50 MHz wide below the atomic frequency, (d) 41p signals
taken with a microwave power of 8 MHz-linewidth with 0 (dotted line) and 7 /is (solid line) delays, (e)
Ion signal with a microwave power of 16 MHz-linewidth. The ion signal now extends over a 100 MHz
wide frequency band, and it is too large to be explained by only the atoms which have followed attractive
potentials and undergone collision. The collisional avalanche creating the plasma is underway, (g) 41p
signals taken with a microwave power of 16 MHz-linewidth with 0 (dotted line) and 7 fjs (solid line) delays,
the lineshape is symmetric with no delay, but has a pronounced asymmetry after 7 /is.
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
107
the low field measurement, only a few 41 p atoms on the low frequency side axe removed
compared to the no delay signal, resulting in a slightly asymmetric lineshape. Consistent
with the ion signals as a function of the field strength, the 41s - 41p resonance progressively
loses its signal only in the low frequency side. In particular, the high field AEq ionization
decreases the low frequency side of the 41p signal below the background level, which cannot
be explained only by the motion of atoms in the attractive potential.
As another way to compare the effects of ionization by the attractive, repulsive, and
nearly flat potentials, the ion production is directly measured with different microwave
fields, Eq, 2Eq, and 4Eq as shown in Fig. 4.10. The delay time is fixed at 7 /xs and the
microwave frequency is detuned by ±20 MHz to form the repulsive and attractive potentials
respectively. For the nearly flat potential, no microwave pulse is applied, since the 41s41s
state forms a slightly repulsive van der Waals potential. Fig. 4.10 shows the results. The
spread of the data points is due to laser shot-to-shot fluctuations.
As shown in Fig. 4.10(a),
the moderate field Eq produces a bit more ions from the attractive potential than from
the repulsive and nearly flat potentials. In Fig. 4.10(b) and (c), the differences between
increasing microwave fields become more distinct. While the number of increased ion from
the attractive potential is gradually increasing, that from the repulsive potential does not
increase considerably. The production of ion from the repulsive potential seems more or
less the same as the nearly flat potential.
With the quantitative observations and the model, these high power experiments show
that the attractive dipole-dipole potential contributes to the early stage (about up to 11
/is) of the plasma formation from Rydberg atoms, but the repulsive potential does not play
a significant role in it.
4.6
Conclusions
The interaction-induced ionization of Rydberg atoms is observed. The primary purpose
of the experiment is to separate the effects of the nsnp dipole-dipole potentials and nsns van
der Waals potentials. Atom pairs excited to slightly repulsive nsns and strongly repulsive
Chapter 4. The ionization ofRb Rydberg atoms in the attractive nsnp dipole-dipole
potential
108
MW detuning
• Sv=-20 MHz
MW field strength
5.0x10-
(cHE0
a
5v=+20 MHz
O
no MW
x=7 jiS
a
0.0
5.0x10 -
(b)2E0
w
c
o
-O
•
E
3
C
O DO
o
0.0
•
o
5.0x10 -
(a)E0
0.0
2x10
3x10
4x10
5x10
number of Rydberg atoms
Figure 4.10: Ionization signals obtained after 7 /is with no microwaves (O). +20 MHz, repulsive, detuning
(O), and -20 MHz, attractive detuning (•) and the three microwave powers producing linewidths of 4, 8,
and 16 MHz with low density atomic samples vs the number of Rydberg atoms initially excited, (a) With a
microwave power of 4 MHz-linewidth, a barely discernible difference exists between tuning to the attractive
potential and the other two cases, (b) With a microwave power of 8 MHz-linewidth, there is a clear difference
between excitation to the attractive potential and the the oterh two cases, which are indistinguishable, (c)
With a microwave power of 16 MHz-linewidth, there is now a large difference between excitation to the
attractive potential and the the other two cases, which arc still indistinguishable.
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
109
nsnp potentials are barely ionized in a time scale of —10 ps and a density range of ~
10 9 cm~ 3 . In contrast, the collisional ionization of atom pairs within the attractive nsnp
dipole-dipole potential is clearly observed. The distinct difference is reconfirmed by the
high power experiment in which the high driving microwave power substantially enhances
the ionization rate by extending the sampling range of R, leading to the conclusion that
the long range attractive dipole-dipole potential plays a dominant role in the early stage of
evolution of Rydberg atoms to plasma.
The increased ion signal by the attractive potential is reproduced in the simple model
based on the dipole-dipole interaction including the spin-orbit interaction. It is realized
that the initial thermal velocity plays a significant role in reducing collision time, and thus
lowering temperature can retard the ionization process. The ionization model agrees well
with the observations.
The unexpected observation of no additional ionization from the repulsive potential can
be addressed as follows. The energy gain from the motion of atoms in the repulsive potential
is very small, practically equal to the initial thermal energy. Furthermore, two neighboring
atoms in the repulsive potential repel each other and a relatively long travel time is required
to reach a third atom to collide. Specifically, the receding atom with thermal energy 300 pK
takes about 30 ps to encounter the third atom at a typical atomic density of ~ 109cm"~3.
However, longer than 15 ps is too long to observe a distinct effect of the repulsive potential
due to the avalanche of ionization in this experimental environment.
There is one essential difference between the dipole-dipole repulsive potential and the
van der Waals repulsive potential. Consider atom pairs of ns^ns# and nsAnPB- First, nsA
and USB repel each other and nsg approaches to a third atom nsc• Since the nsgnsc pair
is also in a repulsive potential, nsB and nsc can only experience an elastic collision. To have
an ionizing collision, one of the atoms must make a transition to np state. As pointed out by
Amthor et al. [12], such a transition can be driven by black body radiation [12], a process
which occurs at a rate of ~ 103 s _1 at n = 40 [8]. In contrast, TISA and npg similarly repel
each other but npg has an equal chance to encounter either nsc or npo- In the case of the
npgnsc pair, they again have equal opportunity to form either an attractive or repulsive
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
110
potential. Then a npsnsc attractive pair will ionize by the induced collision. Therefore, if
a long interaction time is given, different ionization rates from the two repulsive potentials
would be observable. For instance, increased ion production from the nsns potential was
observed with longer delay times, ~20 us, and higher n states, n—60 and 80 [12].
4.7
NOTE
Note that it is worth doing even higher power experiments. In fact, an extremely high
microwave power that drives more than enough number of atoms introduces other processes
such as a rapid avalanche of the ionization. The measurements with power greater than
4E(j suggest more study on the ionization mechanism and the nsnp dipole-dipole potentials.
Fig. 4.11 and 4.12 show the simultaneously measured ion and 41p signals with increasing
microwave power. The microwave pulse length is 500 ns and the delay time is 7 /is (the
experimental conditions are the same as those for Fig. 4.9, except for the microwave power).
With the field strength 64Uo, the ion signal and 41 p signal show two maxima indicated by
the arrows, unlike other observations, but the mechanism is not completely understood.
As discussed in an earlier section, we assume that the loss of the 41p signal is transferred
to the ion signal. Based on this idea, the 41p and ion signals are added to generate a total
41p signal, as shown in Fig. 4.13. The lineshapes seem to be symmetric and are powerbroadened as the microwave power is raised. The linewidth, extracted from Fig. 4.13, as
a function of the microwave field is plotted in Fig. 4.14. The dotted lines represent the
Lorentzian fit to the observations.
A similar ionization at very high power was observed by Li [21] as shown in Fig. 4.15.
These data were measured in the same experimental setup. The atoms were initially excited
to 39s39s and driven to 39s39p3/2 dipole-dipole potentials by 500 ns long microwave pulse.
The allowed time for the ionization is 3.5 fis. Like our observation, the ionization from the
attractive potential only shows equivalent features of the two peaks at a very high driving
power.
Fig. 4.2 raises another question about the initial productions of ion, 40p, 41p, and high
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
111
64 E,
32E
c
3
-16E,
CO,
CO
c
o>
4E,
'55
c
o
—
56300
2E o
56400
56500
56600
56700
56800
MW frequency [MHz]
Figure 4.11: The ion signal as a function of the driving microwave field strength. The broken line indicates
the 41s - 41p transition frequency. Eo represents a relative microwave field strength. As the field increases,
increased ion production extends over the low frequency side of the atomic transition. When power is
extremely increased, two peaks of ion production are observed as indicated by the arrows.
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
112
64E
32E
16E
4E;
qT
2E,
56300
56400
56500
56600
56700
56800
MW frequency [MHz]
Figure 4.12: The 41s - 41p resonances simultaneously measured with Fig. 4.11 as a function of the driving
microwave field strength. The broken line indicates the 41s - 41p transition frequency. Two arrows indicate
the frequencies where the ion production reaches maximum.
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
113
32E
16E
c
3
8E
Q.
4E,
+
c
o
2E,
56300
56400
56500
56600
56700
56800
MW frequency [MHz]
Figure 4.13: The sum of the ion and 41 p signals as a function of the microwave field.
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
114
160
120
N
z
•C
*D
s
80-
40 -
~1
0
'
1
10
'
1
20
'
1
30
1
1
1
40
1
SO
'
1
60
'
1
70
MW field strength [ E ]
Figure 4.14: The power broadening of the total signal, which is the sum of the ion and 41p signals.
lying states. Since the field ionization pulse arrives immediately after the 480 nm laser pulse
in this measurement, we do not expect to observe any other states except for 41s, especially
because the laser bandwidth was measured to be 100 MHz with the high power of the 480nm
laser. Nascimento et al. [13, 14] claim that the initial production of the np atom, when laser
exciting ns atoms, is due to a Stark effect of the excitation laser, but this assertion cannot
explain our observation. A possible explanation is that the field ionization pulse may drive
atoms to diverse states other than ns, although through experimental and theoretical works
are needed. Experimentally, various field ionization pulse length can be applied to examine
the effect of diabatic passage of energy levels. For this, the field ionization pulse circuit
needs to be modified to control the pulse length.
These two questions will be motivations of future work in the lab.
Chapter 4. The ionization of Rb Rydberg atoms in the attractive nsnp dipole-dipole
potential
115
1.6-
£
c
39s - 39p3
1.4-
z>
-8
to
1.2-
"5
c
O)
in
c
£>
1.0-
o
J)
UJ
0.8-
0.6-
,
670
6/5
68.0
68.6
69.0
r
69.5
Microwave frequency (GHz)
Figure 4.15: The ionization signal as a function of the microwave frequency observed with extremely high
microwave power by Li [21]. Two peaks of ionization from 39s39p3/2 attractive potential are observed similar
to Fig. 4.11.
116
Bibliography
[1] T.C. Killian, S. Kulin, S. D. Bergeson, L. A. Orozco, C. Orzel, and S. L. Rolston, 83,
4776 (1999).
[2] T. Killian, T. Pattard, T. Pohl, and J. Rost, Phys. Rept. 449, 77 ( 2007).
[3] S. L. Rolston, Physics 1 2 (2008).
[4] M. P. Robinson, B. LaburtheTolra, M. W. Noel, T. F. Gallagher, and P. Pillet, Phys.
Rev. Lett. 85, 4466 (2000).
[5] S. K. Dutta, D. Feldbaum, A. Walz-Flannigan, J. R. Guest, and G. Raithel, Phys. Rev.
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LaburtheTolra, N. Vanhaecke, T. Vogt, N. Zahzam, P. Pillet, and D. A. Tate, Phys.
Rev. A 70, 042713(2004).
[7] W. Li, P. J. Tanner, and T. F. Gallagher, 94, 173001 (2005).
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[10] T. Amthor, M. Reetz-Lamour, S. Westermann, J. Denskat, and M. Weidemuller, 98,
023004 (2007).
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Rev. A 78, 040704(R) (2008).
BIBLIOGRAPHY
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(2007).
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J. P. Shaffer, New J. Phys. 12, 093023 (2010).
[14] V. A. Nascimento, L. L. Caliri, A. Schwettmann, J. P. shaffer, and L. G. Marcassa,
Phys. Rev. Lett. 102, 213201 (2009).
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[16] S. D. Gensemer and P. L. Gould, Phys. Rev. Lett. 80, 936 (1998).
[17] A. Fioretti, D. Comparat, C. Drag, T. F. Gallagher, and P. Pillet, Phys. Rev. Lett.
82, 1839 (1999).
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limeter Waves, Ph. D. dissertation, University of Virginia, Charlottesville, Virginia
(2005).
118
Chapter 5
Probing the fields in an ultracold
plasma by microwave spectroscopy
5.1
Introduction
Plasmas, called the fourth state of matter, are omnipresent in the universe. They cannot
be defined by a single category since they are produced from diverse physical conditions [1].
Textbook, plasmas are ionized gases in which temperatures range from ~ 104 to ~ 1010 K
and the densities are up to ~ 1027cm~3. Many forms of plasmas are practically impossible
to generate in laboratories, and those which are generated are often difficult to characterize.
However, the introduction of ultracold plasmas, with T~1 K, has enabled us to generate
extremely well characterized plasmas. [1, 2, 3].
A cold plasma is typically produced from cold atoms held in magneto-optical trap(MOT)
by lasers. The cold atoms in the trap are photoionized, forming a plasma with well defined
initial parameters [1, 3]. During the laser excitation the first photo electrons leave, but the
majority are bound by the macroscopic slightly positive charge of the cloud. The result is
a nearly neutral ultracold plasma. This simple method provides us with a plasma which is
easily manipulated and has well defined characteristics. The variable laser frequency defines
the initial kinetic energy of the electrons, and adjusting the intensity of the laser enables
us to control the density of the plasma. Aside from the aspect of being easy to control, the
Chapter 5. Probing the fields in an ultracold plasma by microwave spectroscopy
119
cold plasma has attracted a great deal of attention due to its potential for strong coupling
[4, 5].
The coupling of plasma is defined by the ratio of potential energy to the kinetic energy
of the plasma components, which is generally written as Ti = E p /Ek =
^ ere
i represents the components of the plasma, ions or electrons and Ri = (3/47rTVa)1/3. In
general, the plasma is strongly coupled if T > 1, that is, the Coulomb force exceeds the
thermal energy of the electrons or ions so that plasma particles are tightly bound to each
other. Since the cold plasmas produced from a MOT in laboratory are easily controllable,
it might be possible to produce a strongly coupled plasma by reducing the temperature and
increasing the density.
Although a small amount of neutral atoms does not affect the physical state of a plasma,
a plasma can significantly influence the state of atoms. In fact, since the Rydberg atoms
are rather sensitive to electromagnetic fields, the physical state of the Rydberg atom is
readily perturbed by plasma fields [7, 8]. If a probe atom is placed among plasma particles,
the atom detects the microscopic field from the nearest neighboring ion [9, 10] and the
macroscopic field from the net charge of the plasma particles. As a result of the plasma
field, the energy state of the atom is perturbed and the energy level shifts (Stark shift),
depending on the field strength of the plasma. This approach is suggested as a sensitive
method of measuring fields in the plasma [11].
When measuring fields by a Stark shift, the limitation of its precision is the resolution of
the probe [7, 8]. Laser spectroscopy, a general method for the measurement, wholly relies on
the bandwidth of the probing laser field, yet the bandwidth of lasers which have been used
ranges from 100 MHz to 10 GHz [7, 8]. Lasers have been used to measure the macroscopic
field of plasmas, ~10 V/cm. However, the microscopic field, ~0.1 V/cm, is too small to be
observed using the lasers.
In our experiment, a cold plasma is generated by a 480 nm dye laser pulse and probe
atoms are injected into the plasma to measure both microscopic and macroscopic fields. This
technique is essentially equivalent to the laser spectroscopy measuring the Stark shifts, but
here we use a microwave field to obtain higher resolution. The time evolution of the plasma
Chapter 5. Probing the fields in an ultracold plasma by microwave spectroscopy
120
is observed by allowing various delay times between the plasma formation and the injection
of the probe atoms. The measurements will be compared to a simple lineshape model.
5.2
Experimental Approach
(a) The energy level diagram
3yE*/l<E
Ionization threshold
Rydbergstates
Laser,42s
' -x
Laser,,,
--plasma
5P,/2
MOT beam
780 nm
5s
(b) The timing diagram
Lascrr,|J5ril, Liso r.^
•Jis
t
mm wave
Field Ionization
/
, -42p 42s
hk
. time
Figure 5.1: Timing and energy diagrams for the experiment. The first, plasma laser pulse produces the
ultracold plasma and the plasma temperature is determined by the laser frequency. The second, 42s laser
pulse, which is delayed by time t, produces the probe 42s atoms. The microwave pulse immediately after the
second laser pulse is used to drive the 42s to 42p transition, which is detected by selective field ionization of
the 42p atoms during the field ionization pulse. As shown, the 42p signal is earlier than the 42s signal. The
shift and broadening of the microwave transition are used to determine the fields in the plasma.
The energy level and timing diagrams are shown in Figs. 5.1(a) and (b) respectively.
Cold Rb atoms are held in vapor-loaded MOT and excited to the 5p3/2 state by the trapping
lasers. The trapping coil currents are on at all times, forming inhomogeneous magnetic
fields. A pulse of 480 nm dye laser light ionizes the cold atoms, producing a cold plasma
at a 15 Hz repetition rate. The excess of laser energy above the ionization limit is varied
to change the electron temperature. At a delay time, t, after the plasma dye laser, a
second pulse of 480 nm laser generated by frequency-doubling a 960 nm diode laser excites
Chapter 5. Probing the fields in an ultracold plasma by microwave spectroscopy
121
a small fraction of the 5p3/2 atoms to the 42s state, and the a subsequent 200 ns long
microwave pulse drives the 42s — 42p transition to probe the Stark shifts of the 42p level.
To be precise, it is the 42pi/2 level, which does not have a tensor polarizability. In this
experiment, the relatively short microwave pulse is used to minimize the loss of electrons
during the microwave transitions. Strictly speaking, the difference of the Stark shifts of
the 42s and 42p states is measured but since the 42s level is practically flat, the shift of
42p level mostly contributes to the resonance shift. The number of the Rydberg atoms in
the 42s state is kept, less than 5% of the number of ions in the plasma by reducing the
laser intensity. Thus, they can be treated as isolated atoms, leaving the plasma properties
unaffected. Finally the field pulse selectively ionizes 42p and 42s atoms, and the gated 42p
signal is recorded and stored in computer as the microwave frequency is swept over many
shots of the lasers.
Fig. 5.2 shows a simplified picture of the vacuum chamber and the laser beams. The
two blue lasers are focused and spatially overlapped at the center of the MOT, producing
a cylindrical shaped plasma and Rydberg atom clouds with diameters of 160 pm. One
laser beam enters vertically from the top window of the chamber and the other from the
bottom window.
To align the two laser beams, several overlapping points outside the
chamber are checked, and the mirrors which direct the laser beams into the chamber are
precisely adjusted to maximize the ion and Rydberg signal while monitoring both signals
independently. The four rods allow application of a field pulse for pulsed field ionization
and a dc static, field for measuring the dc. Stark shifts.
Depending on whether ion or
electron detection is used, the polarity of the ionization field can be switched. During a
microwave frequency sweep, only ions are detected, but for measuring the number of plasma
particles, both are detected. It must be noted that when switching detection between ions
and electrons, the polarity of three components should be checked: MCP detector, signal
collecting circuit, and PFI pulse. The power supplies for the MCP detector and the PFI
circuit should be turned down to zero before the polarity is switched to avoid possible
damage to the apparatus. Throughout the experiment, the microwave power is kept low
enough to exclude the power broadening of the 42s - 42p spectra.
Assuming a 2-level
Chapter 5. Probing the fields in an ultracold plasma by microwave spectroscopy
122
system, a 5 MHz wide resonance is expected with optimum microwave power and a low
density atomic sample.
Microwave Horn
HV
MOT
AJ
480 nm laser beam
Figure 5.2: Schematic diagram of the apparatus. The six 780nm MOT beams and the vacuum envelope
are not shown, and the microwave horn is outside the vacuum envelope. The 480nm laser beams produce
cylindrical volumes of plasmas and Rydberg atoms.
The measurement of the number of Rydberg atoms is discussed in Chapter 1, and the
number of ions is measured in a similar way. However, the geometry of ion and atom clouds
are assumed to be a cylindrical shape for simplification. In cylindrical coordinates, the
density of ions is a function of radial position in the trap perpendicular to the direction of
480 nm beam propagation, r = ^ fx 2 + y2, and the position in the propagation direction^.
Explicitly, it is given by
W (r,z)=p /0 e-<
rJ/r » + ' 2/r ™>,
(5.1)
where p/o = is the ion density at the center of the MOT, and r w and r r n are the radii of
the lasers and MOT respectively. The relation between the total number of ions in the trap
and the density at the center of the trap is obtained by integrating the density over the
Chapter 5. Probing the fields in an ultracold plasma by microwave spectroscopy
123
trap volume;
Ni
= (^-)PlQ r h r m-
(5.2)
In the measurement of the microscopic field, the frequency of the plasma laser is tuned
slightly above the ionization limit about, 1 cm -1 , and the delay time is fixed at 10 ns to
produce a nearly neutral cold plasma. The number of plasma particles is varied by progres­
sively reducing the intensity of the plasma laser. The neutral density filters are placed in
the laser beam to adjust the laser intensity. In the measurement of the macroscopic field,
the frequency of the plasma laser is varied to produce plasmas with different temperatures
from 1 to 206 K and the laser intensity is kept at its maximum. In observations of the time
evolution of the plasma, the delay time is varied from 0 to 15 /xs with the full intensity of
the laser.
5.3
Observations and analysis
Prior to the measurement of the plasma field, the difference of the polarizabilities of the
42s and 42p states is measured. To measure the polarizabilities, the microwave frequency
is slowly swept through 42s — 42p resonance in static fields from 0 to 1.7 V/cm. Fig. 5.3
shows the 42s — 42p resonances with various static fields. The frequency shift versus the
static field is plotted in Fig. 5.4. The shifts arise from the difference of the polarizability
between the 42s and 42p states. The frequency shift, Au, is fit to the quadratic function,
Av = ~0E 2 , where E is the static field, yielding (3 = 1.18MHz/(V/cm) 2 .
When a plasma is produced from the MOT by photoionization just above the ionization
limit, minimal macroscopic field is present because the plasma is almost neutral and elec­
trically balanced. However, a microscopic field always exists due to local charge imbalance.
To measure the microscopic field, a nearly neutral plasma is produced by tuning the plasma
laser to 1 cm -1 above the ionization limit. To maintain the plasma charge balance until
seen by the probe atom, the delay time is fixed at minimum delay, T =10 ns. While driving
42s - 42pi/ 2 transition of the probe atom, the intensity of the plasma laser is reduced using
neutral density filters to decrease the number of plasma particles. As shown in Fig. 5.5,
Chapter 5. Probing the fields in an ultracold plasma by microwave spectroscopy
124
c
3
n
w
c*
o
o
T
q
CM
A
w^V>,>V*-vN" :
A
7f
1
52340
1
1
52360
•
1
no field resonance
52380
"T~
~r
52400
52420
52440
MW frequency [MHz]
Figure 5-3: The resonance shifts of 42s - 42p transition in the presence of various static fields. The peak
shifts to lower frequency as the static field is raised.
the resonance shifts progressively to the low frequency side as the ion number is gradually
increased. The resonances are fit to a Lorentzian function to extract the peak frequencies,
and 1 MHz shift is observed with 2.4 x 104 ions, which corresponds to a field of 0.3 V/cm.
We assume that the microscopic field only from the nearest neighboring ion, becausae
approximately, rapidly moving electrons do not effectively screen the ion field. Increasing
the density of the ion leads to a reduced nearest neighbor distance (R NN ), which is a distance
from the ion to the probe atom. The microscopic field is estimated to be proportional to
2/3
Pj
based on E ~
9
2/3
The expected p/
dependence agrees well with the observed
pf 64 from Fig. 5.5.
To observe macroscopic field, a charge imbalanced plasma is produced by tuning the
plasma laser above the limit and so that more of the electrons escape from the plasma
cloud. The laser intensity is kept at its maximum, producing 4 x 104 photoions, to observe
a noticeable field strength. Since the delay time is fixed at r=10 ns, only the microscopic
Chapter 5. Probing the fields in an ultracold plasma by microwave spectroscopy
125
Sf = pE
o-
P=1.18 MHz/(V/cm) 2
-5 -
N
-10-
X
Ss -15•C
in
>.
o
c
<u
3
O"
a>
-20-
-25-
-30-
-35—I—
0.0
—I— —I— —I—
0.4
02
0.6
—I— —I— ~l—
0.8
1.0
1.2
1.4
1.6
1.8
E [V/cm]
Figure 5.4: T h e plot of the peak shift vs. dc field observed in Fig.5.3. The peak shifts quadratically and is
fit to 5v = PE 2 yielding /3 = 1.18 MHz/(V/cm) 2 .
Chapter 5, Probing the fields in an ultracold plasma by microwave spectroscopy
126
experimental data
Lorentzian fit
number of ions
, 2.4X104
0.6X10'
0.1X10'
without plasma
1MHz
52380
52390
52400
52410
Frequency (MHz)
Figure 5.5: 42si/ 3 to 42pi/ 2 resonances in a nearly neutral plasma. The dash-dot lines represent Lorentzian
fits of I.he experimental data. The laser producing the plasma is tuned 1 cm -1 above the ionization limit to
produce a nearly neutral plasma and the number of ion in the plasma is varied by adjusting the intensity of
the laser. As more ions are produced, the resonant peak shifts further to the low frequency side due to the
increasing microscopic field-
Chapter 5. Probing the fields in an ultracoid plasma by microwave spectroscopy
127
field exists in Fig. 5.6(a) when the laser is tuned slightly above the limit by 1 cm -1 . If
laser frequency is increased further, it produces more energetic electrons, and more leave
the plasma cloud rapidly before the laser of probe atom arrives. Depending on the excess
photon energy of the 480 nm laser over the ionization limit, from 43 to 143 cm -1 , the number
of excess ions is increased, producing stronger macroscopic fields. Immediately after the
microwave sweep, the numbers of ions and electrons are measured to determine the number
of excess ions. The ratios of the number of electrons to that of ions are 99(5), 89(7), 81(8),
and 73(10) % at the electron temperatures 1, 62, 134, and 206 K, respectively. Fig. 5.6(b)(d) shows the gradual shifts of the 42s - 42 p l / 2 resonances to lower frequencies due to the
increasing field. Moreover, the resonance becomes broader and finally double-peaked with
the highest number of charge imbalanced ions. The broad double-peaked signal stems from
the difference in spatial distributions of the probe atoms, the ions, and the electrons.
Due to these inhomogeneous charge and atom distributions, the resulting macroscopic
fields not only shift the microwave resonances but also distort and broaden the lineshapes
of the 42s - 42pj/2 transition. Under this condition, it is not possible to simply measure
and define the macroscopic field in the plasma cloud. However, it is possible to compute
the macroscopic field as a function of the radial position, r, since the distributions of the
charges and the atoms can reasonably be assumed to be a Gaussian along the r-axis and the
number of the particles can be easily measured. To simplify the calculation, the clouds of
the plasma and probe atom are assumed to be an infinitely long cylinder to apply Gauss's
law, since the MOT diameter, r m , is much larger than the beam waist of the 480 nm laser,
r w . The plasma ions and atoms have radial Gaussian distribution given by
P i ( r ) = (wo/a)e _r2/r ™
(5.3a)
and
PA
{T ) = (P4o/a)e^r2/r™,
(5.3b)
where pio/a is the density of the ion and P A O /Q is that- of the atom at r = 0. Here a is a
geometrical factor, which is given by a = 1.42 in our case. If the electron temperature is
Chapter 5. Probing the fields in an ultracold plasma by microwave spectroscopy
128
microscopic field
(a) 1K
calculation
experimental data
(1cm1)
5!
CM
52360
52370
52380
52390
52400
52410
Frequency (MHz)
Figure 5.6: 42 si/2 to 42pj / 2 resonances in the presence of plasmas with different electron temperatures. In
all cases the number of ions is 4 x 10 4 . The dash-dot lines represent the lineshape model. In (a) T e = lK, there
is only a microscopic field. For all higher temperatures, the microscopic field exists as background for the
macroscopic field produced by the excess ions. As the laser frequency is tuned further above the ionization
limit, the resonant peak tends to shift further to the low frequency side and shows a lion-Lorentzian profile
due to the increasing charge imbalance between the electrons and ions in the plasma.
Chapter 5. Probing the fields in an ultracold plasma by microwave spectroscopy
129
zero, the electrons have exactly the same distribution as the ion's. However, the electrons
initially have a thermal energy in this experiment, so the electrons in the outer region can
escape the ion potential with relative ease whereas the ions are almost stationary on the
time scale of interest. Thus, the cluster of the electrons forms a smaller cylindrical volume
than that of the ions, and the macroscopic field is stronger in the outer region of the plasma
cloud. The radial distribution of the electrons is assumed to be PE( T ) — Peoe _r2 ^ r s where
r c is the adjustable width of the electron distribution determined by the measured number
of electrons at various electron temperatures, and PEQ is the electron density at r—0. In
a special case, when the temperature of the electrons is zero, the distributions of ions and
electrons are identical [13, 14] and the macroscopic field is zero at all locations, although the
microscopic field is always present due to the nearest neighboring ion to the probe atom.
The plasma fields are calculated based on the distribution of particles. First, the magnitude
of the microscopic field from the nearest-neighbor ion is explicitly given by
i Ejnicro I
~'
(^*^)
KNN
Since ^•R^ lN =pj(r), the microscopic field c.an be written as
'micro
(5.5)
Since the microscopic field only depends on the density of the ions, its strength is expected
to be highest at r=0 and drops with increasing r. Second, the macroscopic field arises from
the charge imbalance, so both the ion and electron distribution are required to compute
it. Applying Gauss' law, the charge distributions lead to the macroscopic field and its
magnitude is given by
'macro
(5.6)
which is in the radial direction, whereas the microscopic field points in a random direction
due to the random orientation of the ion relative to the probe atom.
Chapter 5. Probing the fields in an ultracold plasma by microwave spectroscopy
130
The measurements of the number of particles and the dimensions of the clouds are
crucial, since the only parameter to determine the fields is the density of the particles.
These measurements give a well defined ion distribution, plotted as a solid line in Fig.
5.7, and the probe atoms are distributed in the same way. However, the electrons are not
necessarily confined to the same density curve as the ions although they still form a radial
Gaussian distribution. In Fig. 5.7, the electron distributions are also plotted based on
the measurement of the electron number at various electron temperatures. As the electron
temperature is increased, a smaller number of electrons are held and their density curves are
shrinking within the ion density curve, since greater charge imbalance is required to retain
more energetic electrons. The peak density, pso, of the electrons at r—0, or equivalently r c ,
is the only variable that determines the density curve because the total electron number is
already measured and the curve is assumed to be a Gaussian.
The lineshape of the 42s — 42pi/2 transition is calculated by putting the shifts due to
the calculated fields into a Lorentzian function, which is a reasonable choice because the
lineshape of the plasma-free resonance is close to it, as shown on the bottom of Fig. 5.5.
Including the term for the dc Stark shift, the lineshape is given by
W2
L(U'r) =
4(1/ - v 0 + /3E 2 (r}) 2 + W 2 '
(5'?)
where W—5.2 MHz and v o=52396.5 MHz are the full width at half maximum and the center
frequency of the resonance observed in the absence of a plasma, and the squared total field
is given by
E 2 (r) = E l l c r o ( r ) + E 2 m a c r o (r).
(5.8)
Averaging L{v,r) over the volume occupied by the atoms gives the lineshape,
r, x
L
(") =
I L(",r)pA(r)dr
r
J
TT5
PA{r)dr
'
^5'9)
The results of the lineshape fitting are drawn on top of the observed spectra as broken
lines in Fig. 5.6. The fit, parameter is t,hc clcctron peak density and the best fits are obtained
Chapter 5. Probing the Gelds in an ultracold plasma by microwave spectroscopy
131
1.0x109
•v \
<s\
\v\
8.0x1 Oe
\\\
\\\
W'\
\v\
c?
I
\v\
\\\
6.0x10s
't/i
S
ion distribution
\v\
\\A
\v\
\y.\
\V.\
V\V
\ '\
4.0x108
electron distribution
— Te=1K
\ \ A.
\\^V
N
*N,
W\
2.0x106
Te-62K
- - - T e-134K
- T-206K
e
0.0
0
100
200
300
distance (um)
Figure 5.7: Radial ion and electron distributions. The width of the electron cloud is determined by the best
fit of the lineshape model in Fig.5.6. As the laser is tuned further above the ionization limit, there are fewer
electrons left in the plasma since more electrons escape from the ion cloud due to their higher kinetic energy.
Chapter 5. Probing the fields in an ultracold plasma by microwave spectroscopy
132
by minimizing the least-squaxes. The lineshape model agrees well with observations at
different electron temperatures. Accordingly, from the best spectral line fits, the electron
distributions are defined as shown in Fig. 5.7, and subsequently the plasma fields at the
various temperatures are extracted from them as shown in Fig. 5.8.
The microscopic
field is always present and its strength falls from its the maximum at r—0. At T e —l K,
the microscopic field is dominant but as T e is increased, the macroscopic field becomes
dominant. The macroscopic field rises from zero at r=0, reaches its maximum around the
outer boundary of the ion cloud, and diminishes for larger r. The density of the probe 42s
atoms weighted by r is also shown in Fig. 5.8. With a low T e plasma, most probe atoms
feel the same field over the volume. However, as the macroscopic field becomes stronger
and dominant, a divorce between the locations of the high population of the atom and the
strong field becomes apparent, which is the cause of the double peaked lineshape in Fig.
5.6(d).
The evolution of the plasma is explored by allowing various delay times between the
plasma laser and the probe atom laser. An initially near-neutral plasma, which is produced
by the laser tuned 1 cm' 1 above the limit with the full laser power for the maximum number
of photoions, 4 x 104, is allowed to freely expand for r=0~15 jis. Since the electrons escape
rapidly from the plasma cloud, the macroscopic field progressively builds up from the excess
charge of the ions, but again the microscopic field is present regardless of the electrons'
action. To probe the plasma fields as a function of allowed evolution time r, the probe
42s atoms are placed in the plasma cloud, subsequently driving the 42s — 42pi/ 2 transition.
The resulting resonances are shown in Fig. 5.9. At r=10 ns, the expected 1 MHz peak
shift to the low frequency of. the resonance is observed due to the microscopic field. As r
is increased, the resonance shifts farther by another 1.5 MHz until 4 fis. Between r= 2.5
and 4 jj.s, the lineshape becomes slightly broader and asymmetric due to the developing
macroscopic field. After 4/is, the resonant peak shifts back to the plasma-free resonance
due to the fact that the ion component in the plasma cloud slowly expands while the cold
probe atoms are still in the original place.
To compute the fields from the expanding plasma, the size of the ion cloud is measured
Chapter 5. Probing the fields in an ultracold plasma by microwave spectroscopy
ion/ 42S atom
133
Macroscopic field of plasma with T
«—206K
•—..W)
a— 134K
o—62K
• — 1K
Microscopic field
0.8 -
<13
0.6
300
distance (pm)
Figure 5.8: Tlie microscopic and macroscopic fields of plasmas with various temperatures. The microscopic
field is dominant, in the T' e =lK plasma since the plasma is nearly neutral. In the other plasmas, the
macroscopic field is stronger than the microscopic field
Chapter 5. Probing the fields in an ultracoJd plasma by microwave spectroscopy
experimental data
calculation
J
134
delay time
15)iS
7 ns
4(is
04
qT
CM
1|iS
10ns
no plasma
52380
52390
52400
52410
frequency (MHz)
Figure 5.9: 42. Si/2 to 42pi/2 resonances as a function of delay time between the plasma laser and the probe
laser. The dash-dot lines represent the best fits of the lineshape model to the observed resonances. At early
times, until 2.5 (is, the resonant peak shifts to lower frequency due to the creation of the excess ions. For
delays in cxcess of 2.5 lis the peak shifts back to the atomic frequency due to the expansion of the ion cloud.
Chapter 5. Probing the fields in an ultra.coM plasma by microwave spectroscopy
135
800
700-
•o
3
O
o
c
o
600-
o
400-
Q)
N
V)
<D
sz
b-
500-
300-
200-
100
0
2
4
6
8
10
12
14
16
Delay time r [^sj
Figure 5.10: The size of the ion cloud as a function of the delay time, r. The square dot represents the
observed size of the expanding ion cloud and the solid line is the best fit to the observation.
as a function of delay time. The temporal broadening of the ion signal is recorded with r to
determine the size of the expanding cloud, r exp . The correlation between the time when we
detect an ion and its position before the field pulse ionization is determined by translating
the plasma laser beam and recording the change in the time. Fig. 5.10 shows that the size
of the ion cloud increases slowly at very early times and then rises quadratically. To verify
the measurement, it is compared to the previous measurement by Kulin et at. [15], in which
the ion cloud expands in the following way.
r exp(<)
= r l + K*) 2 '
(5'10)
where r w is the cloud size at zero delay and V Q is the initial velocity of the plasma. Our
observation is best fit to Eq. (5.10) with
uq
=40 m/s. which is close to the Kulin's result,
wo=35~45 m/s.
Using the radius of the expanding plasma and putting the modified distributions of
Chapter 5. Probing the fields in an ultracold plasma by microwave spectroscopy
2.0-
136
rp(r) ion/ probe atom(42s)
1.6-
— °— 7ns
1.4 -
— 15ns
— 4|is
CD
0.8-
T)
0.6-
— 2.5ns
0.40.2
0.0
0
100
200
300
400
500
600
distance (Mm)
Figure 5.11: The sum of the calculated microscopic and macroscopic fields for the best fits to the observed
data in Pig.5.9. The total field reaches its maximum in the central region where the probe atoms (42s atoms)
are located at 4 [is. At later times the field at the location of the atoms decreases although the peak field,
at larger radial distance, continues to increase.
the plasma into Eq. (5.5) and (5.6), the plasma field at the probe atoms is calculated.
Fig. 5.11 shows that, as longer delay times are allowed, the plasma field is developed by
losing electrons from the plasma core, where the probe atoms are placed. These results are
consistent with the shifts of the microwave resonances in Fig. 5.9 in that the probe atoms
feel an increasingly strong field in the early stage of the developing macroscopic field and
then a fading field afterwards, since the probe atoms are confined in the initial location.
In fact, the probe atoms can move only about 3.5 /iin while the ions move 700 fxm in 15
/j.s. Based on the calculation of the expanding plasma field, the linshape of 42s — 42pi/2
transition is computed by following the same procedures as the line-fitting in Fig. 5.6. As
shown in Fig. 5.9, the reproduced lineshape agrees well with the observation at various
delay times.
Chapter 5. Probing the fields in an ultracold plasma by microwave spectroscopy
5.4
137
Conclusions
As an application of Rydberg atom physics, isolated atoms in a Rydberg state were
injected into plasma cloud to probe the plasma electric field. The electric fields generated
from the low density plasma were measured by high resolution microwave spectroscopy.
From the nearly neutral plasma, a microscopic field was measured as small as 0.1 V/cm
and up to 0.3 V/cm, which was made possible by combining the high polarizability of the
Rydberg state and the high resolution of microwave spectroscopy. The macroscopic field
arises from the charge imbalance caused by the loss of energetic electrons. The microscopic
and macroscopic fields are spatially resolved along the radial axis by finding the distributions
of ions and electrons. The microscopic field is dominant in the low T e plasma, but the
macroscopic field develops and becomes dominant as the T e is raised. In the temporal
analysis, it is found that the plasma field migrates from the origin of its formation outwards
along the ion expansion. In the early stage of the expansion, the rapid loss of electron and
relatively slow expansion of the ion cloud lead to the development of the macroscopic field,
which finally moves outside the atomic cloud. A simple lineshape model was constructed by
calculating the fields with the density distributions of the ions and electrons, and it agreed
well with the observations. Improvement of this technique is expected by improving the
sensitivity of the Rydberg state to the field. In fact, the nd — nf microwave transition may
be more efficient due to its larger Stark shift than the ns — np transition, although it has a
tensor polarizability.
138
Bibliography
[1] T. Killian, T. Pattard, T. Pohl, and J. Rost, Phys. Rept. 449, 77 (2007).
[2] T.C. Killian, S. Kulin, S. D. Bergeson, L. A. Orozco, C. Orzel, and S. L. Rolston, 83,
4776 (1999).
[3] S. L. Rolston, Physics 1 2 (2008).
[4] D. J. Wineland, J. C. Bergquist,W. M. Itano, J. J. Bollinger, and C. H. Manney, 59,
2935 (1987).
[5] F. Diedrich, E. Peik, J. M. Chen, W. Quint, and H.
[6] S. Iehimaru, Rev. Mod. Phys. Vol. 54 No. 4 (1982).
[7] B. N. Ganguly and A. Garscadden, Appl. Phys. Lett. 46, 540 (1985).
[8] D. K. Doughty and J. E. Lawler, Appl. Phys. Lett. 45, 611 (1984).
[9] V. L. Jacobs, J. Davis, and P. C. Kepple, 37, 1390 (1976).
[10] Henry Margenau and William W. Watson, Rev. Mod. Phys. 8, 22 (1936).
[11] H. R. Griem, Plasma Spectroscopy. (McGraw-Hill, New York, 1964) pp.72-78.
[12] D. Feldbaum, N. V. Morrow, S. K. Dutta, and G. Raithel, 89, 173004 (2002).
[13] D. J. Wineland, J. J. Bollinger, W. M. Itano, and J. D. Prestage, J. Opt. Soc. Am. B
2, 1721 (1985).
[14] R.D. Knight and M.H. Prior, J. Appl. Phys. 50, 3044 (1979).
BIBLIOGRAPHY
[15] S. Kulin, T.C. Killian, S. D. Bergeson, and S. L. Rolston, 85, 318 (2000).
139
140
Chapter 6
Future Directions
During this thesis work, we have tried many more projects. Some of them turned out to
be physically meaningless or hard to reach a conclusion. However, some were worth studying
further. Here I would like to briefly discuss several experiments which are in progress or
probably worth pursuing. These are all related to the dipole-dipole interaction, and the
experiments are being performed or can be performed using the MOT.
Ramsey method
We used the line broadening technique to measure the ns — np
dipole-dipole interaction in Chapter 3. There are alternative methods to observe the inter­
action. First, as a more sensitive way, the Ramsey method could be employed. In Ramsey
experiment, two identical pulses separated in time are used to generate Ramsey interference
fringes. The essence of the experiment is shown in Fig. 6.1. The idea is that the contrast
of these interference fringes is very sensitive to dephasing effect such as stray fields and
interactions between atoms. Accordingly, we first must get rid of stray fields; then only the
dipole-dipole interaction is left to cause loss of the contrast of the fringes. When a high
density atomic sample is used, the Ramsey patterns for pairs of atoms with different R are
centered at different frequencies, decreasing the contrast, since the atomic sample in the
MOT contains atom pairs with a broad range of internuclear spacings.
As a measure of the dipole-dipole interaction, the contrast, C, is defined as follows,
min
max
max
(6.1)
Chapter 6. Future Directions
np
141
480 nm laser pulse (ns)
MW pulse
field
ionization
480 ran laser
pulse
MW pulses
5P
780 ran CW
MOT laser
5s
time
(a) Energy diagram
(b) Timing diagram
Figure 6.1: The energy and timing diagrams for Ramsey experiment. The experimental procedure is very
similar to the linebroadening technique, but here two identical separated pulses are used.
Chapter 6. Future Directions
where
I max
142
is the intensity of the first maximum and
I m in
is the intensity of the first, mini­
mum indicated in Fig. 6.2(a) where Ramsey interference patterns with low and high density
atom samples are shown. Fig. 6.2 is obtained with the pulse length, r=200 ns and the sep­
aration between two pulses, T=300 ns. For example, in a very low density measurement,
the minima are zero (/ m in=0), equivalently C— 1, but in a high density measurement, C
is always less than 1, since 0 < 7 mm < Imax- The variation of C as a function of the
atomic density in states of different principal quantum numbers n was observed as shown
in Fig.6.2(b). For all n studied, C decays from 1 at low density to a minimum non zero
contrast value, at which it saturates. Due to the larger dipole moment, the energy shift in
a higher n is larger than that in a lower n, resulting in a faster loss of C. For instance, the
n=41 transition loses its contrast most rapidly compared to others.
The observations of Fig.6.2(b) raise a question, " Why does the contrast saturate at
a non zero value?". To explain this, we have to consider several factors simultaneously
involved. First, there are very small- or non-shifted energy levels, which generate high
contrast fringes even if the atoms are closely spaced. Second, the relatively isolated atoms
in low density region also produce the 100 % contrast fringes. Lastly, the transition process,
nsns —» nsrip —> npnp
(6.2)
is also possible. In Chapter 3, we considered only the nsns —> nsnp process because the
microwave power was reduced to the perturbation theory limit, so ignoring the npnp is a
reasonable approximation. However, in this Ramsey experiment, it is not certain that the
microwave power is low enough not to produce the npnp pairs. The problem is, if the npnp
pairs of atoms are produced, that the npnp atoms, the result is two np atoms, whereas the
nsnp levels only result in one np atom. As a result, the nsns - npnp transition adds 100 %
contrast fringes into the nsns — nsnp fringes.
It is experimentally difficult to separate these factors, but a preliminary analysis suggests
that the plateau in the contrast shown in Fig.6.2(b) and the cusp shaped lines of Chapter
3 arise largely from the zero or small shift levels.
143
Chapter 6. Future Directions
0.24 •
0.22-
w 0.20c 0.18 •
3
£ 0.16•2, 0.14
-2.05X10 cm
0.07
max
CM
c£ 0.12
CO 0.10
0.08-I
0.06-
0.041.31.2-
_L
_1_
114810
114820
114830
114840
MW frequency [MHz]
1.1 -
(b)
1.0-
0.9-
• flO
0.8 •
co 0.7
to
0.6
o 0.5
o
0.4
0.3
0.2
0.1
0.0
o
•
D
O
n=33
n=36
n=39
n=41
—i—,—|—,—[—.—|—i—|—i—|—i—|—>—|— —i
4.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
average density [10s cm"3]
Figure 6.2: The observed Ramsey interference fringes, (a) The Ramsey fringes from the low atomic density
sample shows 100 % contrast whereas the increased density washes out the contrast, of the fringes, (b) The
variation of the contrast as a function of the atomic density for rt=33, 36, 39, and 41.
Chapter 6. Future Directions
144
(n+l)s
480 nm laser pulse (ns)
field
ionization
pulse
dress
J dress
i/WWWWWWM
WWWWWWWVAi
np
ns
probe
np
probe
ns
time
(a) Energy diagram
(b) Timing diagram
Figure 6.3: The energy and timing diagrams of the dressed-state experiment. The dressing field is tuned at
the resonance of the np — (n + l)s and its pulse length should be slightly longer than the probe field.
The dipole-dipole broadening in a dressed state
We also tried to observe the dipole-dipole interaction using dressed states of Rydberg
atoms. In this approach, we assume that there are dipoles aligned along the dressing
microwave field, in contrast to the randomly oriented dipoles in Chapter 3. As a first step,
in the presence of the dressing microwave field we have observed the Autler-Townes splitting.
The experimental approach is shown in Fig. 6.3. Using a low density atom sample, narrowlinewidth peaks were observed for the 34s — 34p]/2 transition in the presence of the dressing
microwave field, as shown in Fig. 6.4(b) and (c), but as the power of the dressed field is
raised further, the two peaks split again into four and power-broaden with higher powers.
For t.he moment,, since we do not. understand the peak splitting into four, the dressing field
is reduced such that only two peaks are observed.
Although we do not entirely understand the splitting into more than two with high-power
Chapter 6. Future Directions
145
^ f'eW strength
(a) no dress field
(b) En
M
c
3
.Q
J V \
(0 2En
a
,/ • i
i - i
(e)BE0
yiOAL
16E0
VyA
\»AvVv
104050
104100
1041SO
Microwave frequency [MHz]
Figure 6.4: The resonances of the 34s - Zip\/2 transition with a low density sample as a function of the field
strength of the dressing field. The length of the probe field is 2 [is and the dress field is 2.4 /is.
dressing fields, we have observed the dipole-dipole interaction with a moderate power of the
dressing field such as Fig. 6.4(b) and (c). As shown in Fig. 6.5, at the high density, a
broadened spectrum was observed. However, the broadening of each of the two peaks does
not seem to be symmetric. The peak at 104103 MHz is broadened more on the low frequency
side, whereas the peak at 104107 MHz is broadened more on the high frequency side. If two
peaks were broadened in a symmetric way, the double peak would look like the thick line
in Fig. 6.6 where the valley between the peaks is filled much more than the observation.
It seems that there is a kind of interference effect in the frequency range between the two
peaks. One possible explanation is that two different dipoles may exist in this system, which
cause an interference due to two different oscillating frequencies. In fact, there are dipole
couplings
l'-np(n+i)s
between
np
and (n + l)s as well as / i n s n p between
ns
and
rip.
Chapter 6. Future Directions
146
0.14
atomic density
0.12
3X109 cm"3
5X107 cm"3
c
3
X2
0.10
Urn
$
0.08
<*>
0.06
0.04
!
104090
,
,
I
104095
104100
1
I
1
104105
104110
1
104115
104120
Microwave frequency [MHz]
Figure 6.5: The linebroadening of the 34s - 34pi/ 2 transition with the low dressing field strength.
It is also worth trying to find what causes the peak to split into four components in a
high dressing field as shown in Fig. 6.4. For comparison, the 34s - 34p 3 / 2 transitions are
presented in Fig. 6.7 in the presence of the dressing field. The pulses are 2 and 2.4 /is long
for the probe and dress fields, respectively. In this case, the peak splits into three ways first
and then becomes less obvious what happens.
Chapter 6. Future Directions
147
0.14
0.20
expected
0.12
observation
c
3
•Q
fa.
Q£
0.08
0.12
0.06
104095
104100
104105
104110
104115
Microwave frequency [MHz]
Figure G.6: The expected line broadening (thick solid line) due to the high density atoms and the observation
(light solid line).
Chapter 6. Future Directions
148
fdrssi fiefd strength
(a) no dress field
(b) E 0
(c)2E0
(d)4E0
«•>
-3
<») 8E0
h
j I ^^ g ) 3 2 E "
,
106700
«
,
106800
.
,
106900
»
,
107000
1
,
107100
f
,
107200
Microwave frequency [MHz]
Figure 6.7: The resonances of the 34s —34p3/a transition with a low density sample as a function of the field
strength of the dressing field.
Chapter 6. Future Directions
Long range Rydberg-Rydberg molecules
149
An interesting question in the dipole-
dipole interaction is whether it is possible to make a long range Rydberg-Rydberg molecule
bound by the dipole-dipole interaction. In calculating the ns\/ 2 n P3/2 dipole-dipole potential
in the presence of an electric field, we discovered that a long range potential well can be
formed when the dipole-dipole interaction is comparable to the energy separation between
the np3/2 rrij = 1/2 and m 3 =3/2 levels shifted by the electric field. This well is shown in
Fig. 6.8, in which the electric field is set such that rrij = 1/2 is above rrij = 3/2 at R = oo,
and the internuclear axis, R, is parallel to the electric field, E. If we define M as the total
angular momentum, M =0 and 1 levels converge to the upper R — oo asymptote and M=1
and 2 levels converge to the lower R— oo asymptote side. For R |! E, M is conserved, and
t.he energy curves of different M cross due to no coupling for AM ^0. In contrast, the energy
curves of M=1 repel each other due to the AM=0 dipole-dipole coupling. The potential
well for n=40 with the electric field detuning <5=10 MHz has an equilibrium at R=5.5 ^m
and its vibrational frequency is about 15 kHz, which is comparable to the radiative decay
rate, ~ 10 kHz, that we may not be able to observe its oscillation. An advantage of this
well is that the equilibrium distance is adjustable by the electric field.
A possible approach to observing homonuclear Rydberg molecules confined in the dipoledipole well is to detect the bound atoms in the well by the microwave spectroscopy. First,
atoms are excited to the well by a microwave pulse, and then the bound atoms are driven
to another detectable state by another pulse of the microwave. For this experiment, a short
pulse of the microwave for the first pulse may be more effective in order for a wide range of
sampling.
Filtering atoms In cold atom experiments, one constraint is always present, which
is the density distribution of the MOT. As discussed in Chapter 3, many atoms, which
are fax apart from each other, are located in the outer region of the atomic cloud and
some have little or no shift. As a result, in the transition lineshape, the amplitude near
the resonance frequency is very pronounced compared to the off-resonant frequency region
where information about the dipole-dipole interaction is found. If we were able to filter
Chapter 6. Future Directions
150
40 -,
n=40
6=10 MHz
30-
6=0
20-
N
X
2
10-
2>
6=10 MHz
bound statt
M=0
|M|=1
|M|=2
-10-
-20 -
-30
2
4
6
8
10
12
Figure 6.8: The energy levels of the 4 0 s i / 2 4 0 p 3 / 2 molecular state in the presence of electric field E . T h e
internuclear axis and the polarization of the field are aligned (9=0) and the field is set such that rrij=1/2 is
above mj=3/2 at R = oo. M=1 states form a well with equilibrium position i?o=5.5 fJ-m. The bound state
is indicated by an arrow.
Chapter 6. Future Directions
151
probe
fdepleting
(n-l)p
»*
Figure 6.9: The energy arid timing diagrams of the filtering-atom experiment. The filtering field is tuned at
the ns — (n — 1 )p resonance and applied before the probe field arrives so that only close atoms participate
in the ns - np transition.
Chapter 6. Future Directions
152
out the distant or non interacting atoms, it might be possible to observe a spectrum only
from strongly interacting atoms. For example, we could rule out the effect of the density
distribution from the cusp lineshape and contrast-saturation of Ramsey fringes, if we filtered
out the distant atoms.
To remove a certain range of atoms, we can add a microwave pulse before the probe
microwave pulse arrives. For instance, as shown in Fig. 6.9, the first microwave pulse
transfers widely spaced ns atoms to the lower (n — l)p state by tuning the frequency to the
ns — (n — l)p resonance. Immediately after the first pulse, the probe microwave pulse drives
the ns — np transition. Fig. 6.10 compares the spectrum of the filtered atomic sample
with that of the original atomic sample. The first pulse drives 34s atoms to the 33p3/2
and the second pulse drives the 34s — 34pj/2 transition. The pulse length is 1 /is for both
pulses, and the first pulse is tuned to 117115.98 MHz. As shown in Fig. 6.10(a), the cusp
is substantially removed. Fig. 6.10(b), which is rescaling of Fig. 6.10(a) for comparison,
emphasizes the difference in the linewidth of which the full width at the half maximum is
stretched.
It is also simple to see what frequency range of the lineshape is removed by subtracting
the signal of the filtered atoms from that of the whole atomic sample. In Fig. 6.11, the
light line represents the removed atoms by the first pulse, and the thick solid line obtained
with the whole atomic sample is also plotted for comparison. It seems that the first pulse
effectively removes the isolated atoms or those with frequency shift < 2 MHz. If we increase
the length of the first pulse, the filtering efficiency may be enhanced.
Chapter 6. Future Directions
153
0.20
(a)
whole atoms
filtered atoms
0.15-
3
n
V.
g
0.14
(b)
0.12
0.10
0.08
,w*.
j
,
,
,
1
,
1
,
1
.
1
.
f
i
1—
104040 104060 104080 104100 104120 104140 104160 104180
Microwave frequency [MHz]
Figure 6.10: The resonances of the 34s —34pi /2 transition, (a) In the presence of the filtering field tuned at
the 34s - 33p 3 / 2 transition, the cusp is substantially suppressed compared to that of whole atomic sample,
(b) The vertical scale is adjusted to directly compare the linewidth between the original and filtered atomic
samples.
Chapter 6. Future Directions
154
whole atoms
removed portion of atoms
0.06 4-
—i
104040
1
1
104060
1
1
104080
1
1
104100
1
[—-i
104120
[
104140
'—-i
104160
•
1
104180
Microwave frequency tMHz]
Figure 6.11: The signal from the removed atoms by the first pulse (light line) and the signal from the whole
atoms (thick line).
155
Chapter 7
Conclusions
The dipole-dipole interaction between Rb nsnp atom pair was discussed from both
theoretical and experimental perspectives in Chapter 3. First, the theoretical prediction
was presented by calculating the dipole-dipole interaction of spinless atoms and extending
it to the spin-orbit coupled states- The estimation of the line broadening rate provides a
crude measure of the strength of the interaction in laboratory units. In the experiment, the
dipole-dipole interaction was measured using the line broadening technique. The observed
broadening rate is smaller than the calculation by a factor of two. This is attributed to
the suppression of the dipole-dipole interaction by introducing the spin-orbit coupling. The
detailed effects of the spin-orbit coupling were studied by generating the lineshape model,
which reproduces a cusped asymmetric lineshape. It was found that the cusp lineshape
stems partially from the density distribution of atoms in the MOT and partially from the
existence of small- and zero-shifted dipole-dipole energy levels. The results of this chapter
will be a useful background to study many phenomena arising from dipole-coupled states.
As a result of the dipole-dipole interaction, two closely spaced atoms attract or repel
each other forming the dipole-dipole potentials. In Chapter 4, we explored the effects of
the interaction on the ionization process when atoms are transferred to either attractive or
repulsive potentials. The experimental results showed that atoms placed on the attractive
potential are ionized within 15 ^s, while no significant sign of ionization was observed from
the repulsive potential. To observe clearer ionization, we used the increased microwave
Chapter 7. Conclusions
156
power which is efficient in increasing the number of atoms driven to the dipole-dipole po­
tentials, producing an enhanced ionization signal. With the enhanced signal, the repulsive
potential still is not observed to contribute to the ionization, in contrast to the attractive
potential. Unless enough time is given, longer than 20 /xs, the collisional ionization induced
by the repulsive dipole-dipole potential is not very different from the ionization by the
almost fiat nsns van der Waals potential.
In Chapter 5, we measured plasma fields using the property of Rydberg atoms as a
sensitive probe to electromagnetic fields. Ultracold neutral plasmas were produced by photoionization, and their temperature and density were easily varied by adjusting the fre­
quency and intensity of the laser, respectively. With the help of high resolution microwave
spectroscopy, we were able to detect a microscopic field as small as 0.1 V/cm through the
injected probe Rydberg atoms. As the charge imbalance in the plasma is intensified by
losing more electrons, the development of a stronger macroscopic field was observed. The
macroscopic field was determined by the distribution of the remaining electrons within the
ion cloud, which is almost stationary in the first a few microseconds. By using various delay
times, the expansion of the plasma field was observed. A simple model of the plasma field
was made based on the distribution of the charge imbalance, and it agrees well with the
observed plasma fields at different temperatures and delay times.
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