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Short-pulse microwave excitation and ionization of Rydberg atoms

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Short Pulse Microwave Excitation and Ionization of Rydberg Atoms
Robert Bryans Watkins, Jr.
Marietta, Georgia
B.S, Massachusetts Institute of Technology, 1985
A Dissertation Presented to the Graduate
Faculty o f the University o f Virginia
in Candidacy for the Degree of
Doctor of Philosophy
Department of Physics
University of Virginia
May, 1996
( d lX :
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Abstract
This thesis describes two experiments with alkali atoms excited to
approximately n = 20, where n is the principal quantum number. Both involve very
short pulses of microwave radiation, on the order o f a few cycles.
In the first, microwave multiphoton transitions between two states in
potassium are driven. The amplitude, phase and duration of the radiation driving the
transitions is precisely controlled. The duration ranges from one half-cycle to about
50 cycles. If the transitions are not driven too strongly, it is found that the pattern
obtained is analogous to optical diffraction, with the number of microwave cycles
analogous to the number of slits used for diffraction.
In the second experiment, sodium atoms are ionized with microwave pulses
ranging in duration from about 5 cycles to 25,000 cycles. It is found that the
microwave electric field amplitude required for ionization depends very strongly on
the pulse duration. In the cases where ionization is incomplete, significant trapping
is observed in states higher in energy than the one originally excited.
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Acknowledgements
It goes without saying.
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That's a lie. It’s just one o f the many light moments my friends and I have
shared in graduate school. The rest, o f course, has been pure drudgery. That's a lie
too. Why am I lying so much? It’s because I'm giddy. I'm giddy because I’m tom.
And I'm tom because...Wow, this could go on forever. Which is sadly unlike the
time that I have to spend with my friends. That, in many ways, now comes to an
end. I could try to list all of you, but I'd certainly fail and be filled with regret. So
I’ll just say: so long for now.
I do want to thank my family for allowing me to be a part of you (I realize
you didn't have a big choice!): my parents Robert and Gertrude, my sisters Donna,
Barbara, Carolyn and Andrea, and my brother Hugh.
I also want to thank the Master o f Ceremonies, my thesis advisor Tom
Gallagher. He's the best in the business.
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iv
Table of Contents
Chapter 1: Introduction .....................................................................................................
1
Chapter 2: Potassium Excitation - In tro d u ctio n ............................................................ 3
Stark states o f potassium ..................................................................................... 3
Stark energy m a p ..................................................................................................... 4
Microwave multiphoton tran sitio n s...................................................................... 6
Landau-Zener theory ..................
6
Accumulated phase ................................................................................................ 9
Schrodinger’s equation ........................................................................................ 11
Chapter 3: Potassium Excitation - Experimental
.................................................................................................................................................
Parallel-plate transmission l i n e ...........................................................................
Impedance Z of two parallel s t r i p s ...................................................................
Phase-locking the m ic ro w a v es...........................................................................
General experimental s e t u p ................................................................................
14
14
17
18
22
Chapter 4: Potassium Excitation - Results and Discussion
...............................................................................................................................
1/2 cycle p u ls e s .....................................................................................................
3/2 cycle p u ls e s .....................................................................................................
More than 3/2 cycles
Optical diffraction ................................................................................................
Resonances inside the anticrossing
..............................................................
25
27
29
30
34
44
Chapter 5: Sodium Ionization - In tro d u ctio n ..............................................................
Combined Coulomb-Stark potential .................................................................
Classical ionization limit o f 1/ 16n4 .............
Number of Cycles Required to Ionize
.................................................................................................................................................
50
51
53
58
Chapter 6: Sodium Ionization - Experimental P rocedure..........................................
Arrangement o f microwave equipment.
..................................................
Amplification characteristics of the Litton TWTA .......................................
Approximating the 600 ps pulse a n a ly tic a lly .................................................
Calibrating the p o w e r ...........................................................................................
62
64
66
67
72
Chapter 7: Sodium Ionization - Results and D is c u s sio n .......................................... 78
Ionization T h re s h o ld s........................................................................................... 78
Pulse lengths used for the s-state d a ta .............................................................. 82
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V
Ionization thresholds forseveral d -s ta te s ........................................................ 84
Bound state red istrib u tio n ................................................................................... 91
Conclusion ............................................................................................................ 104
Appendix A
......................................................................................................................
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106
vi
List of Figures
Fig. 2-1.
Stark energy map for the m = 0 stateso f potassium near n = 19...............
4
Fig. 2-2.
Isolation of the 21s and 19,3 levels of p o ta ss iu m .........................................
5
Fig. 2-3.
Energy difference between the 19,3and 21s states with time.......................
8
Fig. 3-1.
Capacitor transmission line.............................................................................
15
Fig. 3-2.
Septum transmission line.................................................................................
16
Fig. 3-3.
Microwave phase-locking scheme.................................................................
19
Fig. 3-4.
Microwave apparatus........................................................................................
21
Fig. 3-5.
General experimental setup.............................................................................
22
Fig. 4-1. Evolution of resonances with RF power.......................................................
26
Fig. 4-2. Stuckelberg oscillations.....................................................................................
28
Fig. 4-3. Comparison of 1/2 and 3/2 cycle pulses ....................................................
31
Fig. 4-4. Evolution of resonances with number o f cycles..........................................
33
Fig. 4-5.
..........................................
34
Fig. 4-6a. Evolution of resonances, 2-8 cycles, data....................................................
36
Fig. 4-6b. Evolution of resonances, 2-8 cycles, sim ulation........................................
37
Fig. 4-6c. Evolution of resonances, 10-21 cycles, data.
..........................................
38
Fig. 4-6d. Evolution of resonances, 10-21 cycles, simulation....................................
39
Fig. 4-7.
Phase dependence of resonances....................................................................
43
Fig. 4-8.
Multiphoton resonances inside the 21s- 19,3 anticrossing........................
45
Fig. 4-9.
Broadening of 1-photon resonance inside c ro ssin g ...................................
47
Optical diffraction and interference pattern
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vii
Fig. 5-1.
Combined Coulomb-Stark p o ten tial.............................................................
51
Fig. 5-2a. Stark energy diagram for Na in the vicinity of n = 1 5 ..........................
54
Fig. 5-3.
Methods by which Na may ionize.................................................................
57
Fig. 6-1.
Arrangement o f microwave equipment.........................................................
64
Fig. 6-2.
Amplification characteristics of the Litton TW TA
....................
66
Fig. 6-3.
Shape o f the 600 ps pulse, from the sampling oscilloscope....................
68
Fig. 6-4.
Analytical model of the 600 ps pulse...........................................................
69
Fig. 6-5.
Fourier transform of the pulse shown in fig. 6-4a. .................................
70
Fig. 6-6.
Diagram of the WR137 waveguide with copper septum..........................
73
Fig. 7 -1(a). Number of full cycles needed for ionization at l/3n5 ..........................
79
Fig. 7-2. Microwave ionization thresholds, 24s and 26s............................................
80
Fig. 7-3. Microwave ionization thresholds, 29s and 3 4 s .......................................... 81
Fig. 7-4. Microwave ionization thresholds, 2 6 d .........................................................
85
Fig. 7-5. Microwave ionization thresholds, 3 2 d .........................................................
86
Fig. 7-6. Microwave ionization thresholds, 3 8 d .......................................................... 87
Fig. 7-7. Microwave ionization thresholds, 4 4 d .........................................................
88
Fig. 7-8. Microwave ionization thresholds, 5 0 d .........................................................
89
Fig. 7-9. Bound state redistribution o f 24s state........................................................... 92
Fig. 7-10. Close-up of bound state redistribution of 24s state
.....................
95
Fig. 7-11. Variation of bound state redistribution with RF pulse duration
98
Fig. 7-12. Bound state redistribution for the 39s state.....................................
99
Fig. 7-13. Variation of bound state redistribution with I m I, 38d state..............
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101
Fig. 7-14. Variation o f bound state redistribution with I m I 44d state.
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I
Chapter 1: Introduction
Given rapid advances in technology, it is now possible to make very short
pulses of optical radiation. It is thus important to study experimentally the effects of
very short pulses of radiation on atomic transitions. A practical approach to this
utilizes alkali atoms excited to a relatively high principal quantum number n. Atoms
prepared in this manner are commonly referred to as Rydberg atoms.
Properties of Rydberg atoms are generally exaggerated. For example, the size
o f the atom, as measured by the mean radius o f an electron's orbit, increases as n2.
An atom in the n = 20 state will therefore have linear dimensions 400 times as great
as the same atom in the ground state. It is easy to believe that the applied electric
field needed to ionize such an atom will be much smaller than that needed to ionize
the ground state, since the electron is already quite distant from the nucleus.
The electric dipole moment is similarly exaggerated. Thus a Rydberg atom
will exhibit a large Stark shift of its energy levels with only a modest applied electric
field. Rydberg atoms are therefore ideal for a study o f transitions between Starkshifted energy levels.
In this thesis we look at two experiments that use the above properties of
Rydberg atoms to advantage. First, in chapters 2-4, we look at multiphoton
transitions between Stark levels of potassium where the radiation required is in the
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microwave regime, near 500 MHz. The duration, phase and amplitude of the
microwave radiation is very precisely controlled. The duration ranges from one half­
cycle up to about fifty cycles.
Second, in chapters 5-7, we detail an experiment using microwave radiation at
8 GHz to ionize sodium. These pulses are as short as 5 cycles and as long as 24,000
cycles.
In both experiments it is found that the results depend very strongly on the
duration of the radiation.
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3
Chapter 2: Potassium Excitation - Introduction
Resonance phenomena, particularly the two-state problem with a sinusoidally
varying interaction, have great fundamental importance. Our goal in this experiment
is to study transitions in a two-state system as the duration, phase and amplitude of
the radiation driving the transitions are carefully controlled.
Since the spacings between energy states of an atom become smaller as the
principal quantum number n is increased, two states may be chosen with a
convenient separation. If too large a separation is chosen, the energy spacing, and
hence the frequency of the electromagnetic radiation needed to drive transitions, will
be large. Radiation with such a high frequency, especially in the visible portion of
the spectrum, is difficult to precisely control. On the other hand, if the energy
spacing is very small, the radiation may be easy to control, but it becomes more
difficult to differentiate between two states.
W e have chosen to work with the Stark states of potassium. Specifically, the
energy spacing between an (n+2)s state and the lowest-energy member of the n Stark
manifold can be chosen to be in the microwave regime. These states are shown in
fig. 2-1 for the case of n = 19, where we have chosen to operate. The two relevant
states are shown in isolation in fig. 2-2, taken from Appendix A. W e label the
lowest energy member of the n = 19 manifold as the 19, 3 state, since it is
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4
-290
21p
Energy (1 /c m )
-300 H
-310
2ls
19d
-320
-330
0.0
0.2
0.4
0.6
0.8
E lectric F ield (k V /cm )
Fig. 2-1. Stark energy map fo r the m —0 states o f potassium near n
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=
19.
5
-309.90
19.3
•309.95
-310.00
-305 X 19 3
-308 X ’3
-307
\
-308
IS
211
-310---------0 100 200 300 400
£ -310.05
-310.10
21s
•310.15
300
298
302
Electric Field (V/cm)
304
306
Fig. 2-2. Isolation o f the 21s and 19,3 levels o f potassium, with an applied electric
field o f about 300 V/cm.
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adiabatically connected to the 19f state (n = 19, / = 3) in zero field.
Two features are immediately apparent in fig. 2-2. First, the energy spacing
between the two states varies with applied electric field. Second, because of
interactions of the outer electron in K with the inner core of electrons, the two states
exhibit an anticrossing whose size is indicative o f the coupling between the states.
In this case the spacing is 339 MHz. In other words, one photon with this frequency
will just match the closest energy spacing between the two states.
Microwave multiphoton transitions1 may occur at certain static field values, as
shown in the inset of fig. 2-2. These multiphoton transitions can be thought of as
arising in two distinct ways. In the photon picture, a resonance transition will occur
if the combined energy of an integral number N o f photons exactly matches the
energy difference between the two states at a given static field. Since the probability
of higher-order transitions (those requiring a large number of photons) depends on a
high density of photons, the microwave power, and hence the amplitude of the
microwave electric field, must also be high.
To discuss the electric field picture, we m ust introduce the theory of
transitions at avoided level crossings, or anticrossings. This problem was treated by
Landau, Zener and Stiickelberg and the theory is known as Landau-Zener theory.2
Traversal of the anticrossing is assumed to begin at infinite detuning on one side and
end at infinite detuning on the other side of the anticrossing and is assumed to occur
at a constant rate. Referring to fig. 2-2, if we begin in the lower state and traverse
the anticrossing, an adiabatic transition would keep the population in the lower state
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and a diabatic transition would transfer all population to the upper state. The
probability of a diabatic transition is given by
e -2*r
(2 - 1)
where
In our case, oo^ = 339 MHz, k = (546 MHz/(V/cm)), and dE/dt is the slew rate of the
electric field in (V/cm)/sec. Note that the avoided crossing gap is 339 MHz and k is
the electric dipole moment of the 19,3 state.
There exists a very helpful intuitive model o f these transitions. It is easiest to
refer again to our specific case as detailed in fig. 2-2. In this diagram the
anticrossing is located at 304.2 V/cm. If we begin in the lower state on either side
of the anticrossing and the electric field slews rapidly through the anticrossing, a
transition will be made to the upper state. That is, a "jump" is made to the other
state. If, on the other hand, the electric field slews very slowly through the
anticrossing, no jump is made, and all population will remain in the lower state. In
fact, all slew rates are somewhere in between these two extremes, so that a
superposition o f the two states is created. Since these transitions depend on the
curvature o f the states, which is present primarily in the vicinity o f the anticrossing,
we can say approximately that a Landau-Zener transition3 can occur if the total field
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8
if-
a
h
m
Fig. 2-3. Energy difference between the 19,3 and 21s states as a function o f time
when microwave pulses are applied. The anticrossing is ignored here.
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(static plus microwave) reaches the avoided crossing.
Taking this viewpoint, higher-order transitions, those for which more photons
are absorbed, occur at static fields further away from the avoided crossing, so again
we arrive at the conclusion that the amplitude of the microwave electric field must
be higher so that the total electric field will reach the avoided crossing. Indeed, it
has been observed that the maximum number of photons absorbed is proportional to
the microwave field amplitude, with an offset.1
We can apply this electric field picture to a one-cycle pulse as shown in fig.
2-3a where we plot the energy difference between the 19,3 and 21s states, versus
time, for a static field somewhat less than the crossing field plus a microwave field
of amplitude E^inCmt). Note that one half-cycle adds to the static field. When a
Landau-Zener transition occurs at time tt shown in the figure, a superposition of the
two states is created. This superposition evolves according to the energy difference
between the states, until the states recombine at time tj. The accumulated phase is
given by the following4
(2-3)
where AW(t) is the energy difference between the states as a function o f time. If 0
= 2tcN, where N is an integer, there is constructive interference. If N is half-integer,
there is destructive interference. W e define the symbol E, to be the static electric
field relative to the crossing field Ec, and to be positive when the static electric field
is less than Ec. With these definitions, AW(t) = k(Es- ErfSinCcat)).
With the one-cycle pulse shown in fig. 2-3a, we find the phase accumulation
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10
is given by:
=
TiE.S
k
(0
2£
, ( E a)
IE.
- .—- s in '1 S _ __ 2
<0
\ if)
(
1
E,J ) 2
{ * * )
If we set eq. 2-4 equal to 2jcN and plot as a function o f the static field Es, we obtain
a pattern resembling Stiickelberg oscillations, which is compared to data in the next
chapter. We will see that the match with data is only approximately correct This is
because in deriving eq. 2-4, we have chosen the limits o f integration by assuming
that the two states split instantaneously at the anticrossing. This ignores significant
curvature of the states in the vicinity of the crossing. Note that no resonances appear
with one cycle. Resonance comes from inter-cycle interference, as we will see
shortly.
In fig. 2-3b there are several cycles o f the microwave field. If we integrate
over one period we find
e2
=
f AW(t) dL
(2-5)
o
Upon integration this gives
(2-6 )
independent of Erf. Note that ©2 equals d>+ - <&. in fig. 2-3b. We integrate over one
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period because the response o f the system to many cycles may be obtained by
multiplying the time evolution operator for one cycle, as discussed in Appendix A.
Now setting eq. 2-6 equal to 2jcN for constructive interference, we find that
N o = kE ,.
(2-7)
Since kEs is the separation between the energy levels at a static field value Es, and
N o is the energy of N photons of angular frequency to, eq. 2-7 is ju st the
multiphoton resonance condition. Thus we arrive at the resonance condition using
both the photon and electric field pictures.
Now take another look at fig. 2-3b. If at time t, we are completely in the 21s
state and the phase accumulation 4>. is equal to it, 3tc, 5tc, etc., then at time tj all
population will still be in the 21s state. Then the phase accumulation 0 + does not
matter and at time t t + 27C/to all population will, again, still be in the 21s state. This
condition on O. is precisely what gives us zeros in the Stiickelberg oscillations
derived from fig. 2-3a. Thus, if no population is transferred after one cycle, no
population will be transferred after many cycles, even if the resonance condition is
met. We shall see this in the data in chapter 4.
In order to properly treat the two state problem, we must use Schrodinger's
equation in the time-dependent form, H*F = i(h/2rc)(d'F/dt). We can write the
Hamiltonian as follows:
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12
'
H
,
-kE b
'
=
, 6
0
(2 - 8 )
where b = 2 (llvfe) = 2rt(339 MHz/2) and V represents the core coupling between
the states.3 The energy of the 21s state is taken as zero, and the energy of the 19,3
state is then -kE(t) where k = 2rc(546 MHz/(V/cm)) and E(t) =
+ Erf sin (tot).
If we write the total wavefunction vF(t) = T 1(t)'F1 + ^ ( t ) ^ where 4*, and
represent the static field energy eigenstates, then we can write Schrodinger’s equation
as
1 -kE b 1 '
,
b
0
,
\
t2i
= I•
2 tc
2/
(2-9)
A
\
T2
i
2/
which leads to two coupled equations:
bT, - kET. = i(— )f,
2
1
2 it
1
(2- 10)
bT l * i ( l i )t2
We then integrate these equations numerically using the Runge-Kutta 4th order
algorithm. This method is used for all simulations in chapter 4.
Our goal, then, is to connect the photon picture with the electric field picture
of resonant transitions. We do this by observing how resonances build up as
progressively more cycles of radiation are used.
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13
References
1. R.C. Stoneman, D.S. Thomson, and T.F. Gallagher, Phys. Rev. A 37, 1527
(1988).
2. L. Landau, Phys. Z. Sowjetunion 1, 46 (1932); C. Zener, Proc, Roy. Soc. London
A 137, 696 (1932); E.C.G. Stuckelberg, Hel. Phys. Acta 5, 369 (1932).
3. Jan R. Rubbmark, Michael M. Kash, M ichael G. Littman, and Daniel Kleppner,
Phys. Rev A 23, 3107 (1981).
4. M.C. Baruch and T.F. Gallagher, Phys. Rev. Lett. 68, 3515 (1992).
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14
Chapter 3: Potassium Excitation - Experimental
Since our goal in this experiment is to drive transitions within a two-state
system using very short radio frequency (RF) pulses of well-characterized shape and
duration, we need an apparatus to guide these pulses. W e shall discuss the
microwave equipment first, followed by details o f other components of the
experiment.
Coaxial cable is used for all the connections in the microwave setup,
including the phase-locking scheme discussed below. Coaxial cable such as RG-58
exhibits excellent bandwidth characteristics, having very low loss from DC to about
1.5 GHz. But it cannot be used to apply the RF to the atoms, since the amplitude of
the electric field in coaxial cable varies with the distance from the center conductor.
Waveguide also is not useful because the dimensions for the frequency range of
interest would be large, giving a low power density inside the waveguide. For
example, a waveguide for the 500 MHz radiation used in this experiment would have
transverse dimensions on the order o f 1/2 meter. With moderate microwave power,
the electric field amplitude inside would be small.
We have instead constructed a
parallel-plate transmission line. The advantage is that the plate spacing can be
chosen to give the best compromise between good field homogeneity and sufficient
electric field amplitude. The two versions used in these experiments are shown in
figs. 3-1 and 3-2. All parts were machined from brass, except for the septum in fig.
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Fig. 3-1. Capacitor transmission line.
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16
I*- 2 * s
"•*
i.r
Coo ti <I c i l f c
Fig. 3-2. Septum transm ission line.
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17
3-2, which is copper.
In order to match the standard 50Q. impedance o f the rest of the components
of the system, the transmission line should have a 50£2 impedance also. The
impedance Z o f two parallel strips o f metal o f width W and separation S is defined
as the ratio o f the voltage between the two strips divided by the current through one
of the strips. This is given by the expression1
3-1
where |i0 and Eg indicate, respectively, the permeability and permittivity of free
space. If Z is set to 50 ohms in eq. 3-1, we find that W/S = 377/50. The
transmission line must therefore have a (width)/(separation) ratio equal to (377)/(50)
throughout its length. In order to better impedance-match the BNC connectors, which
have dimensions on the order of millimeters, to the coaxial cable, the end portions
are tapered while maintaining the (377)/(50) scale. The width and separation are
simultaneously tapered, as can be seen in the top and side views o f figs. 3-1 and 3-2.
In the version of the transmission line shown in fig. 3-1, high-voltage
capacitors (ATC lOOpF, 3600 V) are placed at each end o f the ground plane. These
allow the microwaves to pass while isolating the static field and severely attenuating
the ionization pulse, which has a 1 ps risetime. Both o f these are applied to the
ground plane. Attenuation of the ionizing pulse is necessary to avoid damage to the
microwave amplifier. We find that about 1/1000 o f the ionizing voltage is present
outside of the capacitors. Typically 4000 volts is used to ionize the atoms, so that
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18
the microwave amplifier is exposed to less than 5 volts, which it regularly survives.
For the version shown in fig. 3-2, a thin metal plate or septum is inserted
into the line. The static field and the field ionization pulse are then applied to this
pi ate. The static field is then still isolated, and the ionizing pulse is reduced by
approximately the same factor of 1/1000.
The capacitor version of the transmission line has better field homogeneity
because of the greater separation of the plates. However, because o f the capacitors,
lower RF frequencies are not transmitted as well. Since these capacitors are absent
in the septum version, it has excellent transmission all the way down to DC, but does
not produce as homogeneous a field.
Imperfect impedance matching causes some reflection of the microwave
pulses. Using a pickup probe we observe distortions of the pulse between the plates
consistent with a voltage reflection coefficient of 0.19. The transmission line
attenuates the pulses by 2 dB, but does not change their shape. The reflection
coefficient and attenuation is nearly the same for each version. Specifically, the
capacitor version exhibits no more reflection than the septum version.
An important part of the experimental apparatus is the equipment needed to
produce short RF pulses in the 50-1000 M Hz range. In order that the exact shape of
the pulses may be chosen, and to ensure that the pulse is the same each time the
laser fires, we have phase-locked the microwaves using the scheme outlined in fig. 33. The Philips pulse generator (PM5785B) has an internal clock speed o f up to 125
M Hz and produces output pulses as short as 3.5ns with a risetime o f Ins. W e can
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
19
RF 0*C.
tjP ? 7 * o 8
T#
f
+raA**itsi t*\
/.7te
Ltwlfmr*
(rtA.
false See.
Re 71194
fl;l,-ft
rnrf-Tss
\
/
pr»*1
lifer
Fig. 3-3. M icrowave phase-locking schem e.
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phase-lock the (continuous) RF from the HP 8350B oscillator as follows. The clock
output from the Philips is passed through a step-recovery diode, which produces a
comb o f harmonics. This is mixed (mixer A in fig. 3-4) with 50% o f the oscillator
output, whereby the sum and difference frequencies are obtained. Only the low
frequencies are amplified, so that the difference frequency provides an error signal
for the FM input. At integral multiples of the clock frequency, which can be chosen,
the RF is phase-locked. We find that, at 500 MHz, the oscillator may be tuned by 23 MHz and the RF will remain locked. The remaining 50% of the power is double­
mixed (Watkins-Johnson M IA ), first with one o f the Philips outputs, and then with a
pulse from the HP 8112A, which is triggered at 20 Hz by the laser. The resulting
pulses are then amplified by a broadband Amplifier Research 1W1000 (100kHz-1000
MHz, 30 dB gain, 1 watt minimum with less than 1 dB compression into 50 ohms).
The characteristics of the amplifier are important, since a 3.5 ns, 500 MHz pulse has
significant Fourier components from almost DC to 1000 MHz. With the microwave
pulses phase-locked, we can directly observe them on a sampling oscilloscope.
The complete configuration of the microwave components is shown in fig. 34. Notice that the continuous RF is applied to the local oscillator port of mixer B.
The square pulse from the pulse generators is applied to the intermediate frequency
ports of mixers B and C, and a train of RF pulses is then present at the radio
frequency port of mixer B. A single pulse for each laser firing is present at the radio
frequency port of mixer C.
To summarize, in order to observe the evolution o f resonant transitions as a
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T ,* tf
Fig. 3-4. M icrowave apparatus.
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22
Fig. 3-5. General experim ental setup.
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23
function of the number o f cycles of microwave radiation, we need to be able to do
the following: apply a static electric field (remains throughout the experiment), excite
the K atoms to the lower eigenstate (which is the 21s state away from the crossing),
apply the RF pulse, and selectively ionize to detect only the upper state. Then one
of the parameters is changed (usually either the static field or the RF frequency), and
the experiment is repeated. The parameter is scanned many times throughout the
desired range in order to increase the signal-to-noise ratio.
The procedures used to supply and excite the atoms are similar to those
previously used.2 The potassium atoms are obtained from a resistively-heated oven,
and pass between the plates of the transmission line, as in fig. 3-5. Two Nd:YAGpumped tunable dye lasers, counterpropagating to the atoms, then excite the 4s-4p
and 4p-21s transitions. After the RF pulse is applied, the atoms are selectively fieldionized. It is worth repeating that all of this occurs in a static electric field. The ions
are then extracted through a 0.4 mm hole in the top plate of the transmission line,
and amplified by a microchannel plate detector. This signal is gated, integrated,
digitized, and sent to a computer.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
24
Esfgrgpg.es
1. J.D. Jackson, Classical Electrodynamics (John W iley & Sons, 1975).
2. M. Gatzke, M.C. Baruch, R.B. Watkins, and T.F. Gallagher, Phys. Rev. A 48,
4742 (1993).
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25
Chapter 4: Potassium Excitation - Results and Discussion
Because of the nearly linear dependence of the energy levels shown in fig. 22, multiphoton resonances are evenly spaced as a function of static field. As the RF
power is increased, higher-order resonances appear and those already present tend to
broaden. This can be seen in fig. 4-1 for microwave pulses of a relatively long
duration o f 100 ns. Recall that the 21s-19,3 anticrossing occurs at 304.2 V/cm. As
discussed before, the appearance of higher-order resonances with increasing RF
power can be thought of in two ways. In the photon picture, more power means a
higher density of photons. This is necessary if a greater number is to be absorbed
simultaneously. In the electric field picture, more power means a larger amplitude
field. We can then choose a static field further from the crossing and still meet the
condition that the total electric field, static plus microwave, extends close to the
crossing. Notice that, even though the 3-photon resonance is nearly 3 V/cm from the
crossing, an RF amplitude of 0.95 V/cm is enough to drive this transition easily.
This has also been seen by Bloomfield e t a l.1 With an anticrossing spacing of 900
Mhz and a microwave frequency o f 10 GHz, they found that a given resonance
could be driven if the microwave field amplitude reached approximately a third the
way to the crossing. We will see shortly that this result is strongly dependent on the
number of cycles of RF used to drive the transition. In fig. 4-1, the RF pulse
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RF freq. = 500 MHz
duration x 100 ns = 50 cycles
RF amplitude =
0.14 V/cm
2-photoos
C
o
_C0
3
Q.
O
Q.
0
C8
0.36 V/cm
CO
©
Q.
Q.
3
3-photoas
0.95 V/cm
aoi
308
n
Static Electric Field (V/cm)
Fig. 4-1. Evolution o f resonances w ith R F power.
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3M
duration is long enough that the resonances are fully formed, in contrast to data
presented later in this chapter in which the pulse duration varies.
Notice also in fig. 4-1 that the 1- and 2-photon resonances get very broad as
the microwave electric field amplitude is increased. This is an example of the
familiar effect of power broadening: the amplitude term in Rabi's formula approaches
unity even off-resonance as the Rabi frequency increases.2
These microwave multiphoton resonances have been studied before with
microwave pulses of long duration. However, since we are able to form RF pulses
of variable duration, we have studied the effects of driving these resonances with
progressively more cycles.
We begin with the shortest pulse that causes any population transfer, a 1/2
cycle pulse that adds to the static field. As the RF field traverses the crossing, the
atom, orginally in the lower state, is put into a coherent superposition of the lower
and upper states. Since these states have different energies, the coherent state evolves
in time. As the RF field traverses the crossing again, the coherent state recombines,
and we detect the population in the upper state. This result is given quantitatively by
eq. 2-4. Thus for a given RF amplitude, a pulse that is wider in time and therefore
allows more phase accumulation will cause Ram sey interference fringes, or
Stuckelberg oscillations, that are more closely spaced as a function of static field.
We show results for 1/2 cycle pulses o f three different frequencies in fig. 4-2. The
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28
Stuckelberg Oscillations from 1/2 Cycle
RF amplitude = 7.9 V/cm
146 MHz
~1
-a
!
-«
■ -I---
f
-4
i
-T- -2
' ■
Static Field relative to Crossing (V/cm)
Fig. 4-2. Stuckelberg oscillations.
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- I
0
29
93 MHz 1/2 cycle pulse is widest in time. The noisy lines are the data, and the
smooth curves are simulations calculated from the phase-accumulation result stated in
eq. 2-4. These simulations have been shifted by as much as 0.5 V/cm along the xaxis in order to better match the data. In any case, the dependence o f the fringe
spacing on RF frequency seems to be well reproduced. Notice that fringes tend to
wash out near the crossing. This is due to a combination o f field inhomogeneity in
the transmission line and the fact that the upper state is populated progressively more
by the lasers as the static field approaches the crossing. Notice also that the position
of the leading edge o f the Ramsey fringes does not depend on the frequency, so this
is not a resonance phenomenon. Rather, the location of the leading edge depends on
amplitude, as discussed below. Not shown is the effect of a half-cycle which
subtracts from, rather than adds to, the static field. This showed essentially no
population transfer, which is not surprising given the energy level configuration. The
curvature o f the states is exceedingly small away from the crossing, so the
probability o f a Landau-Zener transition is negligible. Also, a one-cycle pulse gives
results very similar to those of the half-cycle pulse adding to the field.
3Z2-sycte-Pulses
If we use a 3/2 cycle pulse, with two of the half-cycles adding to the static
field, we find a much different result. In short, we find resonances with the 3/2
cycle pulse. It is instructive to look explicitly at the difference between one half­
cycle and two half-cycles adding to the static field. These results are shown in fig.
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30
4-3. With the 1/2 cycle pulses, the leading edge of the Ramsey fringes moves as .ne
RF amplitude is increased. This is evidence o f a non-resonant phenomena. We
expect that, if resonances are formed, their location in static field would remain
constant as the RF amplitude is changed. This is because the static Held at which a
given integral number of microwave photons fits between the states is only
dependent on the energy of the photons, and therefore the frequency of the
microwave field, and not dependent on the microwave field amplitude. Note that, in
contrast to fig. 4-2, the microwave amplitude is quite small, so that only about two
fringes are seen before the crossing is reached. The fringes begin further from the
crossing as the pulse amplitude increases from 1.45 V/cm to 2.55 V/cm. The
location of the large peak also changes in the same manner.
In contrast, when a 3/2 cycle pulse is used, resonances appear. The spacing
between these resonances is given by (RF frequency)/(slope o f 19,3 state) = (250
MHz)/(546 MHz/(V/cm)) = .46 V/cm, which matches the experimentally observed
spacing in fig. 4-3. As expected, their locations do not change with a small increase
in RF power. However, the Rabi frequency is large enough that even the population
at the 4-photon resonance at about 302.2 V/cm undergoes oscillations. We should
note that, if the RF amplitude was large enough, the resonances could move due to a
significant AC Stark shift.3
More Hwn.3/2 .cycles
As progressively more cycles are added to the pulse, we see the
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Static Field (V/cm)
31
Fig. 4-3. Comparison o f 1/2 and 3/2 cycle pulses. 3/2 cycles is the m inim um
needed to produce resonances.
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32
resonances narrow up. A very helpful intuitive picture of this is to equate the
number of microwave cycles with the number of slits in an optical diffraction
pattern. In both cases, adding more paths to interference increases the sharpness of
the resonances obtained.
In fact, it has been shown4 that, in the perturbative limit, the pattern obtained
is exactly the same as optical diffraction. For the convenience of the reader, this
reference has been reproduced in Appendix A.
In fig. 4-4, which is taken from this reference, we see the pattern obtained
with I, 2, 3, and 5 half-cycles adding to the static field. The noisy curves are the
data, and the smooth curves are numerical simulations described in Appendix A. We
can compare four important features in this figure with the optical diffraction pattern
reproduced in fig. 4-5.5 These are detailed in the following four paragraphs.
The one-cycle pulse (referring to the number o f half-cycles that add to the
static field) causes Ramsey interference fringes as discussed above. Because the RF
frequency is relatively high, the fringes are spaced very far apart, so we see less than
two fringes. This single-cycle response corresponds to the diffraction pattern with
one slit.
As the pulse duration is increased so that 2 cycles add to the static field,
resonances are formed. The spacing of the resonances after 2 cycles exactly matches
the condition imposed by the energy of N photons, shown at the top of the graph,
equaling the energy spacing between the states. In this case it's given by 454 MHz/
(546 MHz/(V/cm)) = 0.83 V/cm. These are analogous to the interference peaks seen
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33
8.
297
298
299 300 301 302
Static Reid (V/cm)
303
304
Fig. 4-4. Evolution o f resonances w ith num ber o f cycles. From A ppendix A .
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Fig. 4-5. Optical diffraction and interference pattern. From reference 5.
S
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35
in the optical pattern with 2 slits. The heights of these peaks are modulated by the
single-cycle response, as with the optical pattern.
The resonances get narrower as the number o f microwave cycles is increased.
This corresponds to increasing the number of slits when producing the optical
diffraction and interference pattern. However, this only holds true when the
resonances are weakly driven. When the Rabi frequency becomes great enough, the
pattern becomes more complicated. This is easily seen in the lower-order
resonances.
Finally, the subsidiary peaks that appear in the optical pattern are also present
in the microwave resonance data. These occur between the resonance peaks and
number n - 2, where n is the number o f slits or cycles. Again, these are only clearly
visible in the microwave data when the system is weakly driven. If the Rabi
frequency is large, for example near the 5-photon resonance in fig. 4-4, the
population in the upper state can begin to decrease. In this case, the resonance peak
may actually be smaller than the subsidiary peaks. This does not occur in the optical
pattern, since there is no Rabi flopping, so the brightness of the one of the primary
maxima is unlimited.
In figs. 4-6a, b, c, d we see the results o f the same experiment at a somewhat
higher RF frequency, so that the spacing between resonances is greater, and at lower
power. The Rabi pattern in the vicinity o f the resonances is more apparent in this
data. It's easy to follow the development o f the 2- and 3-photon resonances located
at about -2 and -3 V/cm respectively, but the Rabi frequency for the 1-photon
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36
RF Amp. = 1.0 V/cm
freq = 570 MHz
8 cydes
5 cycles
e
o
■a
«a
-4
2
0
4 cycles
Vi
3 cycles
-2
2 cycles
-5
-4
3
-2
-1
Static Field (V/cm)
Fig. 4-6<z E volution o f resonances, 2-8 cycles, data
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0
37
8 cycles
RF amp = 1.0 V/cm
frcq = 570 MHz
5 cycles
4 cycles
3 cycles
2 cycles
-5
-4
3
2
Static Field (V/cm)
1
Fig. 4-6b. E volution o f resonances, 2-8 cycles, sim ulation.
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0
38
RF Amp = 1.0 V/cm
freq = 570 MHz
21 cycles
T
“I
-4
-2
0
19 cycles
4M^V
T
■
T
1
15 cycles
T
T
-2
12 cycles
T
T
T
l
-4
10 cycles'
T
T
-5
-4
T
-3
T
T
-2
-1
Static Field (V/cm)
Fig. 4-6c. Evolution o f resonances, 10-21 cycles, data
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T
39
21 cycles
RF amp = 1.0 V/cm
fineq = 570 MHz
*5
-4
-3
-2
-1
0
19 cycles
15 cycles
Q.
Q.
3
12 cycles
5
4
5
*4
*3
2
2
-3
Static Field (V/cm)
1
0
10 cycles
-1
Fig. 4-6d. Evolution o f resonances, 10-21 cycles, simulation.
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0
40
resonance is so large that the pattern in the vicinity of -1 V/cm quickly bears little
resemblance to a resonance.
We can use a known analytical method6 to find the Rabi frequencies for these
resonances at this microwave power. For an n*-order multiphoton resonance, using a
microwave electric field intensity Erf, the Rabi frequency is given by:
4-1
I/
where in our case Qg is the 2 Is -19,3 anticrossing spacing, 2tt(339 MH z ), k is the
slope of the 19,3 state, 546 MHz/(V/cm), f is the microwave frequency, and Ja is the
n'b-order Bessel function. For the data of fig. 4-6, Erf = 1.0 V/cm, and f = 570 MHz.
Using these numbers, we find that the 1-, 2-, and 3-photon Rabi frequencies,
are 145 MHz, 36 MHz, and 6 M Hz respectively. This information is
tabulated in table 4-1.
Resonance
Rabi frequency, Q ^ J 2 n
Rabi period
(Rabi
period)/2
in
microwave
cycles
1-photon
145 M Hz
6.9 ns
4 cycles
2-photon
36 M Hz
28 ns
8 cycles
3-photon
6 M Hz
167 ns
48 cycles
Table 4-1. On-resonance Rabi frequencies fo r the data in f i ^ 4-6. N ote that 1/2
Rabi period is the tim e for com plete population o f the upper state to occur.
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41
Note that eq. 4-1 is strictly only valid5 when f »
Q J ln , which in our case is not
well met since 570 MHz is not much greater than 339 MHz. However, numerical
integration o f the Schrodinger equation, as described at the end o f chapter 2, yields
Rabi frequencies that agree within 3% with the numbers in table 4-1.
The data shown in fig. 4-6 were taken in a slightly different manner than the
data in fig. 4-4. In order to avoid exciting the upper state during laser excitation of
the 21s state, the lasers were fired in a static field significantly less than that of the
crossing. Then a pulse was applied that adiabatically increased the static field close
to the crossing value. The microwave pulse was then applied, and the static field
adiabatically returned to its original value before Pulsed Field Ionization was used to
detect the upper state. This is why the data in fig. 4-6 do not show the overall
increase in upper state population shown by the dashed lines near the crossing at
304.2 V/cm in fig. 4-4.
Looking at fig. 4-6 we can see clearly these Rabi oscillations occurring as the
resonances build up. It's difficult to see a clear picture o f the buildup of the 1photon resonance in this data, since it's driven so strongly. Population on resonance,
at -0.84 V/cm, has been completely driven to the upper state after only 4 microwave
cycles. When more cycles are added, oscillations are visible even off resonance, so
that a nice peak is not seen.
The 2-photon resonance, at -2.0 V/cm, is not driven so strongly. From the
table above, 8 microwave cycles are needed to drive population on resonance
completely into the upper state. When the number of cycles is less than this, we can
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42
see a peak evolving in the data. W ith more than about 8 cycles, Rabi oscillations
begin to occur and the structure is complicated.
The 3-photon resonance, at -3.1 V/cm, is relatively weakly driven. Even after
21 cycles, the on-resonance population is only half in the upper state. We can see a
well-defined peak form as more cycles are added. A very broad peak can clearly be
discerned after 5 cycles. After 21 cycles, a narrow peak has been formed.
One consequence of determining the number o f cycles needed to build up
resonance peaks concerns Floquet theory.7 Floquet theory considers the analog of
modulating the frequency of a radio wave. Just as a radio wave is broken into a
carrier and sidebands, the 19,3 state in our two-state problem is broken up into a
carrier and sidebands. Each o f these sidebands is spaced by the energy o f one
microwave photon. Floquet theory contains no information about the length o f the
microwave pulse. In order to determine when this approximation breaks down, we
need only ask how many cycles are needed to complete the resonances. We find that
on the order o f 10 cycles are needed, which is consistent with the results o f Breuer.8
Since Floquet theory also contains no information about the phase of the RF
field, it's important to know to what extent this approximation is valid. Results are
shown in fig. 4-7. For a 3/2 cycle pulse, as expected, there is a large difference
depending on whether one half or two half cycles add to the static field. With a pulse
as short as 5/2 cycles, the phase makes no apparent difference. As a reference, the
long-pulse result is included at the bottom of Fig. 4-7. Apparently, the phase of the
microwave pulse has little effect on the results with a duration as short as 3 cycles.
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43
RF amplitude = 4.2 V/cm
freq. = 493 MHz
1 half-cycle adds,
2 half-cylces subtract
s
o
•a
s
Q.
£
a
3
VJ
u
|
3
50 cycles
296
298
300
302
304
Static Field (V/cm)
Fig. 4-7. Phase dependence o f resonances. Show n at the le ft are the num ber
o f half-cycles adding to or subtracting fro m the static field .
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44
Resonances inside the anticrossing
To further study the effects of short pulses, we have investigated the
resonances inside the 2 Is -19,3 anticrossing. The first 6 o f these multiphoton
resonances are shown in fig. 4-8. The static field for this data is constant at 304.2
V/cm, the location of the anticrossing. Because o f field inhomogeneities, we see all
of the resonances, not ju st the transitions from an odd number o f photons as
predicted for states o f good parity.9 We do however only see the odd ones in our
numerical integration o f the Schrodinger equation, if the static field is precisely the
same as the avoided crossing field. But any small deviation from this static field
value causes the even-numbered resonances to appear in these calculations.
If we set the RF frequency to the value for the 1-photon resonance, 339 MHz,
and decrease the pulse length while at the same time increasing the RF power
enough that the upper state population on resonance remains approximately the same,
we see the resonance broaden. A plot of this effect is shown in fig. 4-9. The noisy
curves are the data. The smooth curves are Fourier power spectrums of the
microwave pulses obtained as follows.
For a truncated sinusoidal oscillation f(t) = sinfo^t) o f duration T, the Fourier
transform is given by10
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45
'------1------»------1------
r
•
r
100
150
r
i
■
r ■■
1
i
aoijHindo^ z m s -raddfl
Fig. 4-8. M ultiphoton resonances inside the 21s- 19,3 anticrossing. The R F pow er
used fo r each segm ent o f data is noted betw een the bold markers.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
RF Frequency (M H z)
2(90
250
300
350
400
t
The power spectrum is then given by g(to)2. If we vary co^, while keeping to constant
at 2it(339 MHz), we are essentially asking how much power exists at 339 MHz in
each of these pulses. These are the smooth curves in fig. 4-9. They have been
shifted along the x-axis by as much as 10 MHz in order to align their peaks with
those of the data.
The broadening of the peaks is then essentially due to uncertainty in the
frequency of the microwave field driving the transitions. This uncertainty varies as
1/T where again T is the duration o f the pulse. So we expect the width o f the data
traces to vary in this manner, and this is what we observe.
Conclusion
We have seen that there is a smooth connection between the two apparently
different pictures of multiphoton resonance described by photons or an oscillating
electric field. Resonances appear with two half-cycles adding to the static field, so
that intercycle interference may occur. As progressively more cycles are added, the
resonances obtained narrow up. In the perturbative limit, the pattern obtained is
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47
RF pulse duration = 200 ns
RF amplitude = 5 mV /cm
100 ns
6 mV/cm
40 ns
18 mV/cm
20 ns
60 mV/cm
10 ns
105 mV/cm
'
290
i
,
"
,
300
,
-
, ■
3*0
1
40P
- ■
>" '
'
"
1|
4S0
RFfreq(MHz)
Fig. 4-9. Broadening o f 1-photon resonance inside crossing w ith pulse duration.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
identical to optical diffraction, as discussed in the paper reproduced in Appendix A.
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49
Rgfegpc.es
1. L.A. Bloomfield, R.C. Stoneman, and T.F. Gallagher, Phys. Rev. Lett. 57, 2512
(1986).
2. J.J. Sakurai, M odem Quantum M echanics (Addison-Wesley, 1985).
3. T.F. Gallagher, Rydberg A tom s (Cambridge University Press, 1994).
4. R.B. W atkins, W.M. Griffith, M.A. Gatzke, and T.F. Gallagher, submitted to
Phys. Rev. Lett.
5. A.A. M ichelson, Studies in Optics (Univ. of Chicago Press, 1927).
6. M. Gatzke, M.C. Baruch, R.B. Watkins, and T.F. Gallagher, Phys. Rev. A 48,
4742 (1993).
7. C.H. Townes and F.R. Merritt, Phys. Rev. 72, 1266 (1947).
8. H.P. Breuer, K. Dietz, and M. Holthaus, Z. Phys. D 8, 349 (1988); 10, 13
(1988).
9. J.H. Shirley, Phys. Rev. 138B, 979 (1965).
10. George Arfken, M athem atical M ethods For Physicists (Academic Press, 1985).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
Chapter 5: Sodium Ionization - Introduction
Despite the fact that electric field ionization has long been studied and is
generally well understood,1 recent technical advancements have shown the importance
of studying ionization with short pulses of oscillating electric fields. For example, it is
now possible to make optical radiation pulses containing only a few cycles.2
If we look at ionization classically, we begin with the combined Coulomb-Stark
potential which in atomic units is given by,3
V=-J_+Ez
(5-1)
1*1
and is approximated in 1 dimension in fig. 5-1. This is valid for an I m 1= 0 state
with an electric field applied along the z-axis. Here m is the azimuthal orbital angular
momentum quantum number. We ignore the electron's spin. In three dimensions the
local maximum of the potential is a saddle point which occurs at
z = ~
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(5-2)
51
40-
20-
-
20-
-4 0 -
-6 0 -
-8 0 -
-0.4
-
0.2
0.0
0.2
0.4
z (a.u.)
Fig. 5-1. Com bined Coulomb-Stark potential seen by an electron
w hen an electric fie ld is applied along the z-axis.
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52
where the potential has the value
V = - 2 y/E .
(5-3)
If the electron is bound by an energy W, then at an electric field given by E = W 2/4
the electron is no longer bound, and the atom ionizes. If we ignore the Stark shifts o f
the energy levels they are given by -l/2 n 2 in atomic units, and we can calculate this
classical field for ionization as
E=—
.
1 6n4
For
a typical Rydberg
(5-4)
state o f n* 30, this field turnsout to beabout 400 V/cm,
is fairly easy to attain in the laboratory. Note that this wasderived assuming m = 0,
but for I m I «
2n, the fields are only a few percent higher.3
If we take the Stark shifts into account, the energy of the extreme red state is
given to a good approximation by
(5-5)
W =-—
2n2
2
which leads to an ionization field of
E = -L .
9n4
(5-6)
For the red Stark states o f hydrogen, these classical calculations turn out to be very
close to experimental values for static and low frequency rf fields such that © «
1/n3. 4 However, the calculation above does not apply to the blue Stark states.
Because the wavefunctions for these states lie on the uphill side o f the atom, away
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which
53
from the saddle point, somewhat larger fields are required for ionization, on the order
of twice that needed for the red Stark states. There is a very clear picture of this
effect on page 234 of Bethe and Salpeter5, in which the Stark manifolds of hydrogen
are shown as a function of electric field. As the electric field increases, the red states
begin to disappear at lower fields than do the blue states.
For non-hydrogenic atoms, static and pulsed electric fields have been shown to
ionize with an amplitude close to the classical limit o f 1/ 16n4.. In contrast, a series of
experiments has shown that non-hydrogenic atoms, in particular the low I m I states
of sodium, have exhibited a smaller ionization threshold when oscillating fields are
applied.6 It is worth noting here that we may safely ignore the effects of the magnetic
fields to which the atom is exposed in many experiments involving time-varying
electric fields, since the atom responds primarily to the electric field component.7 This
can be understood most simply by recalling that the forces FB and FE on a charged
particle in combined magnetic and electric fields respectively, obey the following
relationship:
Fb
qE
c
where q and v are the particle's charge and velocity, respectively. Thus FB is less than
F e by the factor o f v/c, which equals 1/100 for the first Bohr orbit in hydrogen, and
decreases as 1/n for Rydberg atoms.
The differences in ionization thresholds may be understood by referring to a
map of the Stark states of sodium as shown in figure 5-2. Sodium has a total of
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-4 7 0 —
<-41
-911
t.a
1 .0
9 .0
■Jit
0 .0
1.0
9 .0
Fig. 5-2 cl Stark energy diagram fo r N a (m = 0) in the vicinity o f n - 15. N ote
the prom inent avoided level crossings.
b. Stark energy diagram fo r N a ( l m 1=2) in the vicinity o f n = 15. The levels
appear alm ost hydrogenic.
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55
eleven electrons, ten o f which are tightly bound in a core. Because of interactions of
the outer electron with this core, degeneracies in the energy level structure are
removed and the Stark states do not cross, but rather anticross. These anticrossings
are clearly visible in fig 5-2a, as opposed to fig 5-2b which is close to a hydrogenic
Stark map. If a sodium atom is produced with the outer electron in one o f the states
shown in fig 5-2a and the electric field slowly increased, all anticrossings will be
traversed adiabatically. Thus we can visually follow one o f the continuous lines to
determine the energy o f the state at any value of the field. When the field crosses the
classical ionization limit o f l /l 6n4, the atom will ionize. This explains the hydrogenic
l/9n4 threshold, as one can see by looking at figure 5-2b and imagining that all the
states cross. If the reddest Stark state is excited and the field increased, all
anticrossings will be traversed diabatically, and a larger field of l/9n4 is required to
encounter the classical ionization limit.
However, if an oscillating field is applied whose frequency is close to the
energy separation at the anticrossings encountered, Landau-Zener transitions can occur
which will be partially adiabatic and partially diabatic.8 Referring to fig. 5-2a,
imagine starting in the 16s state, and subsequently exposing the atom to an oscillating
electric field whose peak amplitude is about 2.3 kV/cm. If the frequency is in the
regime discussed above, then when the electric field increases beyond the amplitude at
which the 16s-15,2 anticrossing occurs, the population will split and the atom will now
be in a superposition of both states. The 15,2 state is adiabatically connected to the
zero-field. 15d state. When the field decreases again, the states will split again. The
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56
net effect is that some of the 16s population has now been transferred to the 15,2
state. As the electric field continues to oscillate, this process continues, and eventually
there is population in all the states of the manifold.9 The bluest state of the n-manifold
can then make a Landau-Zener transition to the reddest state of the n+1 -manifold.
Since the n++n+l manifold crossing occurs for lower field amplitudes as n increases,
population will diffuse upwards to higher n. W hen n is large enough that the
microwave field amplitude is sufficient for classical ionization, the atom will ionize.
That is, the microwave field equals l/16n4 for that high n-level. The apparent
requirements for all this to occur is that the microwave field be in the correct
frequency regime, and that its amplitude be equal or greater than the field at which the
n- and n+1- manifolds cross. This value is given by 1/3n5 in general. In our case,
n=l5 and
(5.142 109 V/cm) = 2260 V/cm
3n3
(5-8)
3(15)3
which can be seen in fig. 5-2 to be the lowest field at which the n=15 and n= 16
manifolds cross. Keep in mind, however, that we have implicitly assumed that the
number of microwave cycles is large so that, even if the probability for a given
transition is low the atom can still reach a high enough n for ionization to occur. As
discussed in Chapter 7, anticrossing spacings decrease with increasing n, so that we
can never choose the "perfect" microwave frequency. That is, the probability for
transition will always be low for some inter-manifold crossings in the ionization
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57
0.0
1.0
2-0
3.0
F (kV/cml
Fig. 5-3. M ethods by which N a may ionize. Adiabatic ionization fro m the 20d state
results in intersection w ith the classical ionization lim it at point d. Hydrogenic
ionization ends at point d2. Ionization by a m icrowave fie ld fo llo w s a typical path
ending at d0. The l/2 1 n 4 threshold is also indicated.
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58
process.
Note that this discussion has assumed a linearly polarized microwave field. If
instead circular polarization is used, a field amplitude close to the classical ionization
field is required.10
A schematic o f this process is shown in fig 5-3. The l/21n4 threshold has been
observed for the bluest I m I = 2 Stark states o f sodium.11
Number o f Cycles Required to Ionize
Consider now the minimum number o f cycles required to ionize the atom.
According to our model, if we start in a state n and apply a microwave field whose
amplitude equals l/3n5, the atom will ionize when a higher state n,. (for n ^ ^ ) is
reached such that the microwave field is sufficient to ionize this state classically.
Then we have the following condition:
1
1
3n 5
l& i^
(5-9)
which gives approximately
n =
c 3
(5-10)
If we call An the number of n levels we must pass through on our way to
ionization, this is equal to n,. - n which gives
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59
3
If we start out in the n = 30 state and apply eq. 5-11, we find that An = 17.
This
gives an indication o f the minimum number of half cycles required for ionization.
This is only approximate however, since, for example, as can be seen from a careful
study of fig. 5-3, at a high enough n the microwave field may be sufficient to reach a
junction o f the n and n+2 manifolds. In this case, a half cycle may cause an n—m+2
manifold transition, and the minimum number o f cycles needed for ionization may be
reduced.
Now consider what might happen if we make the microwave pulse only a few
cycles long. If the number of half cycles in the pulse is less than An, and the field
amplitude is close to l/3n5, then we would expect that transitions induced by the
microwave pulse would take the atom to an insufficiently high n to allow classical
ionization to occur. In this case, the field amplitude might need to be raised well
beyond l/3n5 to achieve significant ionization probability. Also, the sharp threshold
for ionization noted above may no longer exist, since the probability for ionization
may increase gradually as the amplitude of the microwave pulse increases.
In addition, since significant transitions to higher n may occur, even though
insufficient to induce ionization, we might expect to find population "trapped" in
higher n levels than the one initially excited.
The purpose o f this experiment, then, is to investigate the effects of microwave
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60
ionization with these very short pulses. In the next chapter we explain the apparatus
used.
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61
References
1. R.F. Stebbings, C.J. Latimer, W.P. West, F.B. Dunning, and T.B. Cook,
Phys. Rev. A 12, 1453 (1975).
2. J. Zhou, G. Taft, C.P. Huang, M.M. Mumane, H.C. Kapteyn, and I.P. Christov,
Opt. Lett. 19, 1149 (1994).
3. T.F. Gallagher, Rydberg A tom s (Cambridge University Press, 1994).
4. J.E. Bayfield and D.W. Sokol, Phys. Rev. Lett. 61, 2007 (1988).
5. Hans A. Bethe and Edwin E. Salpeter, Quantum M echanics o f One- and Two-Electron
A tom s (Plenum Publishing, 1977).
6. D.R. Mariani, W. van de Water, P.M. Koch, and T. Bergeman, Phys. Rev. Lett. 50, 1261
(1983).
7. David J. Griffiths, Introduction to Quantum M echanics (Prentice Hall, 1995).
8. Jan R. Rubbmark, Michael M. Kash, Michael G. Littman, and Daniel Kleppner, Phys.
Rev. A 23, 3107 (1981).
9. P. Pillet, H.B. van Linden van den Heuvell, W.W. Smith, R. Kachru, N.H. Tran, and T.F.
Gallagher, Phys. Rev. A 30, 280 (1984).
10. Chung Yi Lee, Circularly and Elliptically Polarized M icrowave Ionization o f Sodium
Rydberg A to m s (Dissertation, University o f Virginia, 1994).
11. H.B. van Linden van den Heuvell and T.F. Gallagher, Phys. Rev. A 32,
1495 (1985).
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62
Chapter 6: Sodium Ionization - Experimental Procedure
The general setup is similar to that described in chapter 3 for the potassium
excitation experiment, so details are given here only for the differences.
Sodium atoms from a resistively heated oven pass into a piece o f WR137
waveguide, shown in figure 6-6, and described below. Two Nd:YAG-pumped dye
lasers are used to excite the sodium from its ground, 3s state to an intermediate 3pw
level, and then from the 3pw to either an ns or nd Rydberg level, with n in the range
24 to 50. The microwave pulse is applied to the waveguide approximately 200 ns
after the lasers fire. About 100 ns after the end of the microwave pulse, a high
voltage pulse is applied to the septum in order to ionize the atoms not already ionized
by the microwaves. Na+ ions are collected by using a positive pulse o f ~ 5000 V/cm,
with a risetime o f 1 (is. As explained in Pillet et al,[ the field-ionization and
microwave-ionization signals arrive at the detector at different times, so the signals
from each are easily separated. Since the two signals are complements o f each other,
we only look at the field-ionization signal. The ions leave the waveguide through a
0.4 mm diameter hole in the top, and the signal is amplified by microchannel plates.
The ionization probability is measured as a function o f peak microwave field by
integrating the field ionization, or remaining atom, signal while scanning the
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63
attenuation of the microwave pulses.
The unique aspect of this experiment is the production of very short microwave
(RF) pulses, so this will be described in detail. The underlying idea for the production
of RF pulses is to use a mixer to multiply together two signals: (1)CW microwaves at
the desired frequency and (2) a square wave whose temporal width is equal to the
desired duration of the RF pulse.
We use a terminated section o f W R137 waveguide operated at 7.98 Ghz in the
T E 10 mode. The experimental arrangement is shown in figure 6-1. The following
pulse lengths are used for various parts o f the experiment, all given in nanoseconds:
0.6, 1.2, 1.8, 2.8, 4.0, 50, 100, 200, 500, 1000, 3000. To obtain the pulses in the
range o f 600ps to 4.0 ns, full width at half maximum, we use an Avtech AVM -l-PS
pulse generator to drive the intermediate frequency ports o f two Watkins-Johnson
M14A mixers. Using two mixers in sequence gives greater contrast between the on
and off states of the microwave pulse. This is followed by another M14A mixer
whose intermediate frequency port is modulated by a Philips PM 5785B pulse
generator, which serves to remove most o f a small tail from the Avtech-generated
pulses. This train of pulses, with a typical repetition rate o f 5 MHz, is the input to a
Hughes CW Travelling Wave Tube Amplifier (TWTA), the output of which is
monitored on a sampling scope in order to precisely set the pulse length. One of these
pulses is selected and amplified to approximately 500 watts by a Litton M624 pulsed
TW TA, modulated by a Precision Instruments Digital Delay Generator, before being
attenuated by a variable coaxial attenuator and transmitted to the waveguide section.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
f Ht*£~
> •« .'
" — ' I f ’ UC
/'- j~ * k .' w ^ « " r « « « •'» j
""rtTTzial
*.-* <X««-»
g « ^ TW«
1
tm f t m i
r»
/•v* Tfi3
.>
T.
’r. u t y **MH
A
m u < y ‘-r)
( I oc ?••** /
IHf*
fclifj r y n t
*/W t j.r»>
■ srl
4a ^
a*
[TVMVx
%
Mf
**lCdim
Lili**
TWTA
x-»
Fig. 6-1. Arrangem ent o f m icrowave equipment.
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owV
*
r
65
The 50 ns pulses are formed by modulating all mixers with the Philips pulse generator,
while pulses o f 100 ns and longer remain CW until reaching the Litton TWTA, which
is pulsed by the Precision Instruments delay generator.
When electrons are observed, a negative pulse of about 2800 V/cm is applied
to the septum, sufficient to ionize atoms in a state as low as 20s, but allowing a range
of arrival times at the microchannel plate detector for different n-levels. Thus we are
able to time-resolve the electron signal, allowing us to analyze the bound states as a
function of n-level. Since the interaction region is between the septum and the top o f
the waveguide, Na+ ions travel to the septum as the electrons move out through the
collection hole. The ions liberate electrons from the septum, which arrive at the
detector -100ns after the Na electrons. This later signal is large only when the Na ions
are bom in a large electric field, that is for low n. This can be seen in the data in
chapter 7. The front of the microchannel plate detector is held at +175 volts, so that
electrons hit the detector with at least 175 eV of energy, ensuring that the detector is
equally efficient for all electrons, independent of the field in which they are bom.
Let's look more carefully at the production o f the shortest, 600 ps, pulses. In
order to get full use of the Litton amplifier, we need to apply about 15 dBm (15 dB
relative to one milliwatt of power), as shown in figure 6-2 and explained below. But
the maximum output o f the HP oscillator is only about 18 dBm, and there are losses
through the mixers of approximately 6-8 dB each. Thus we must use the Hughes
TWTA as an intermediate amplifier. It's maximum output is 20 watts, so we easily
stayed within its linear regime by driving it with only -10 dBm, with a corresponding
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66
65
Litton Amplifier Linearity Check
60-
E
m
■o 5e c5 ®
5
o
Q.
3
a.
3
o
50-
45-
0
5
10
15
20
input pow er (dBm)
Fig. 6-2. A m plification characteristics o f the Litton TW T A .
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25
67
output of about 1.5 watts. In any case, we are able to observe the pulses directly after
the Hughes amplifier, on the sampling oscilloscope. This is possible because the
Hughes is a CW (Continuous Wave) amplifier. The output is then attenuated and
applied to the input o f the Litton amplifier.
Since the maximum duty cycle of the Litton TW TA is only 2%, it is
impossible to observe its output directly on the sampling oscilloscope, since the
repetition rate is not fast enough. To be certain that we are operating in its linear
regime, we have measured its gain. In fig. 6-2 we plot the amplification
characteristics o f the amplifier. The solid line shows a linear response, corresponding
to a gain o f 42 dB. The dotted lines show the point at which we operate the
amplifier. It is clearly well within the linear regime. Apparently, we could have
obtained more output power, about 3 dB, and still been below saturation, but because
of the extreme importance of linearity, it was decided to give a large safety margin.
Shown in figure 6-3 is the actual 600 ps pulse, taken from the "vertical signal
out" of the sampling oscilloscope. Because the pulses were not phase-locked, the
sampling scope displayed points of essentially random amplitude within the envelope
of the pulse. A small tail can be seen in the later portions of the pulse. However, this
was found to have no observable effect on ionization, even when the duration of the
tail was increased to 100 ns.
To approximate the 600 ps pulse analytically, we multiply 8 GHz RF with a
Gaussian envelope, as shown in fig. 6-4a. The Fourier transform o f this pulse is
shown in figure 6-5. Since the bandwidth o f the Litton amplifier is only 7-11 GHz,
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68
u
•o
3
V
>
2 —
13
tim e; sc a le sh o w n a b o v e
Fig. 6-3. Shape o f the 600 p s pulse, fro m the sam pling oscilloscope.
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69
1. 0
-
Approxunate shape of
0.6 ns pulse at 8.0 GHz
0 .5 -
0.0
-0 .5 -
- 1. 0 -
-1
0
1
2
3
1.0 after bandwidth is
limited to 7-11 GHz
0 .5 -
0.0
-0 .5 -
- 1. 0 -
-1
0
1
2
time in ns
Fig. 6-4. A nalytical m odel o f the 600 ps pulse.
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3
70
0 .8
-
0.6 o>
■o
3
a.
£
<
>
0.2 -
0.0
4
5
6
7
8
9
10
11
Frequency in GHz
Fig. 6-5. Fourier transform o f the pulse shown in fig . 6-4a The dotted lines
show the bandwidth o f the L itton TW TA.
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12
71
we removed all frequencies outside of this range, and inverse-transformed to obtain the
pulse in figure 6-4b. This should closely approximate the pulse obtained at the output
of the Litton amplifier. Apparently, the limited bandwidth of the Litton may create
some ringing in the pulse. However, the falloff in amplification of the Litton is not
actually as stringent as this. Also, the amplitude of the ringing is less than that of the
tail noted above. For these reasons, we believe the 600 ps pulse was of high enough
quality to give reliable data. O f course, the pulses of longer duration will have an
even smaller bandwidth, and should be even more faithfully amplified by the Litton
amplifier.
The waveguide itself also has a limited bandwidth of transmission, because
there is a cutoff frequency below which no propagation can occur. For WR137
waveguide this frequency is 4.3 GHz. As detailed below, all our power lies well
above this frequency.
Since we are using a waveguide and not a cavity, we may directly read the
average power deposited on the terminating end of the system using an HP 8473B
detector diode. We can then find the peak electric field
for the TE10 mode as
follows:
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72
<power>
(6- 1)
cross-sectional area
(6 - 2 )
Notice that
the effective longitudinal wavenumber for the waveguide, approaches
2k /X as the width of the waveguide a goes to infinity.
The measured power out o f the system at the exit from the vacuum chamber
was 55.6 dBm. However there was a measured loss o f 2.0 dB from the entrance to
the exit o f the vacuum chamber, so we will assume a 1 dB loss from the entrance to
the center of the waveguide. Thus the average power in the interaction region was
56.6 dBm, or 457 watts. Since no field can exist inside the copper septum, as shown
in figure 6-6, we subtract its cross-sectional area from that of the waveguide to find
the effective area.
Putting in the following numbers:
co =2n(1.9%GHz)
c =30cm GHz
a =3.48cm
e0=8.854xlO'12—
m
area =(1.58)(3.48) -(0.31)(2.83) cm 2
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(6-3)
73
0
o.Hma
0
l l . S ’ c /'l
Fig. 6-6. Diagram o f the W R137 waveguide w ith copper septum.
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74
gives kj = 1.41 cm '1 and therefore Eoy = 420 V/cm.
We don’t expect any error leading to the above microwave field calibration to
exceed 5%, which is the magnitude of the calibration corrections supplied with the
power meter. The other sizable error is likely to be the estimated loss between the
interaction region in the waveguide and the HP detector diode measuring the power.
Estimating this at ± 0.5 dB implies an error o f 6%, so we believe our calibration is
correct to within 10% or better, equal to ± 0.8 dB.
It is worth noting that a cylinder of sodium atoms is excited by the lasers.
Since the thermal beam moves at about 105 cm/sec, or lm m /psec, an atom that is
detected was excited a short distance from the center of the waveguide. For a lpsec
delay, used for all but the longest microwave pulses, the variation in microwave field
amplitude over a 1mm distance from the antinode amounts to only 0.5%. For the
longest delay used, 3psec, the variation is about 4%. This is less than 0.4 dB, and is
well within the error bars implied by the size of the symbols used to plot the data in
the next chapter.
It is important to look at the waveguide modes that might be excited, as among
the modes with low cutoff frequencies, only the TE10 mode has an antinode in the
center of the waveguide, which we are utilizing as the interaction region. The lowest
cutoff frequency in WR137 waveguide is 4.3 Ghz, and this is for the TE10 mode.2 The
next lowest cutoff frequency, 8.6 GHz, is for the T E ^ mode. As can be seen in
figure 6-5, some power in our shortest pulse does lie above 8.6 GHz, so it should be
possible for some power to reside in the TEjo mode and thus upset our power
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calibration. However, the T E jq mode has a node o f the electric field at the center of
the waveguide,3 and thus would be very inefficiently excited by the waveguide
couplers we employ, if at all. These couplers have an antenna positioned at the center
of the broad side o f the waveguide. The next cutoff frequency encountered is
approximately 10.4 GHz, for both the TM n and the TEU. None o f the power in our
pulse resides above this frequency, so these modes are not excited. It should be noted
that the TE q, mode, whose cutoff is 9.5 GHz, is barely in reach based purely on
frequency, but will not be excited because the electric field in this case is
perpendicular to the antenna in our waveguide coupler. Thus we can be confident that
most, if not all, o f the power in our pulse resides in the TEl0 mode.
This mode has the added benefit that the slots in the top and bottom o f the
waveguide do not affect the fields inside, since no current flows across the center of
the broad face.4 Also, the copper septum does not upset the field because the electric
field is everywhere perpendicular to it.
Dispersion is also a well-known phenomena in waveguides. The group
velocity of a particular frequency component is given by:
/
vg = c
\
c
1-
I
(6-4)
w )
where 0)c is the cutoff frequency for a given mode, and c is the speed of light in free
space. The cutoff frequency for the TE10 mode is 4.3 GHz, so we can calculate the
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76
group velocities of 7.0 GHz and 9.0 GHz waves as 0.79c and 0.88c respectively.
Almost all of our power lies within this frequency range. For our piece o f waveguide
20 cm long these components arrive 86 picoseconds apart, which is a small fraction
of our 600 ps pulse length. In any case, we have viewed directly on the sampling
scope the 600 ps pulse both before and after traversing the waveguide and have
observed no substantial difference in pulse shape.
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77
References
1. P. Pillet, H.B. van Linden van den Heuvell, W.W. Smith, R. Kachru, N.H. Tran,
and T.F. Gallagher, Phys. Rev. A 30, 280 (1984).
2. J.D. Jackson, Classical Electrodynam ics (John W iley & Sons, 1975).
3. Dwight E. Gray, editor, Am erican Institute o f Physics Handbook (McGraw-Hill,
1972).
4. Charles H. Townes and A.L. Schawlow, M icrowave Spectroscopy (McGraw-Hill,
1955).
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78
Chapter 7: Sodium Ionization - Results and Discussion
If each half cycle of a microwave pulse causes a transition to one n-level
higher, then as defined in eq. 5-11, An/2 gives approximately the number of full
cycles needed for ionization when the microwave amplitude equals l/3n5, where n
denotes the initial state excited by the lasers. In fig. 7 -la we have plotted An/2 for a
range of n values somewhat beyond that used in this experiment. The shortest pulse
we can make is 600 ps FWHM (full width at half maximum), which corresponds to
about 5 cycles at our frequency o f 8 GHz. However, since this is measured FWHM,
there are a few more cycles available, albeit with reduced field amplitude, as can be
seen in the sketch o f the pulse in fig 7 -lb. These cycles may cause intermanifold
transitions once a sufficiently high n has been reached partway through the ionization
process.
Ionization Thresholds
In figs. 7-2 and 7-3 are plotted the microwave ionization thresholds for several
s states, over a range of pulse lengths. These pulses are listed in table 7-1, with the
corresponding num ber of cycles. We observed the remaining atom population, which
is the complement of the ionization signal. We define zero signal as that obtained by
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
79
This is a plot o f An/2 versus n, where
An=(2/3)n5M - n
30
25
fall cycles needed for ii
20
15
10
5
0
10
20
30
40
50
SO
Initial n level
Fig. 7-1(a). N um ber o f fu ll cycles needed fo r ionization at l/3 r f. (b) Sketch o f 600ps
pulse.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
80
100
ns
— 4.0 ns
— 2.8 ns
0.6 ns
I /3 .7 n
0. 8 0.60 .4 -
0.20.0i
0
T "
20
RF attenuation (dB)
10 j
l/3 .7 n
Fig. 7-2. M icrowave ionization thresholds.
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81
0.8
-
13
c
op
S3
a
o
l/3.7n
<
oo
c
'2
95 V/cm
100 ns
2
<D
04
4.0 ns
2.8 ns
<— 0.6 ns
'Y T ^ v
s^
v S N ^
\Y
x
xv
♦ "A JT v
\
♦ ++
a H
1/3.7*
T "
“T "
20
10
RF attenuation (dB)
Fig. 7-3. M icrow ave ionization thresholds.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i
0
1/9n
82
blocking the excitation lasers, and a signal of one as the signal obtained when the
microwave power is off. Microwave power increases from left to right, and as the
signal decreases from one, more microwave ionization is occuring. The increased
signal at 7 dB and 10 dB on the 24s and 26s graphs, respectively, is caused by
increased integrator efficiency when the signal is broadened in time. Apparently, the
integrator cannot accurately integrate a signal with a very large amplitude and a very
short time duration, possibly because of brief saturation. Also, recall that completely
diabatic ionization o f the reddest member of the Stark manifold implies the hydrogenic
ionization threshold of l/9n4. The longest pulse used for the s-state thresholds was
either 100 or 200 ns, which we found to give essentially identical thresholds.
Pulse length
Number of full cycles
600 ps
5
1.2 ns
10
1.8 ns
14
2.8 ns
22
4.0
32
100 ns
800
Table 7-1. Pulse lengths used for the s-state data, with, the
corresponding number of full cycles, rounded to the nearest integer.
With these points in mind, we can see that the lowest field at which ionization
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
83
occurs is approximately the well-known threshold of l/3n5, but the probability for
ionization depends strongly on pulse length and initial n-level. When a long pulse is
used, either 100 or 200 ns, a relatively sharp ionization threshold is observed near
1/3.7n5, which agrees with a previous study.1 However, as previously observed2, this
threshold becomes broader as n is increased. For the shortest pulse length o f 600 ps,
there is significant ionization of the 24s state at l/3n5 , for which the number of full
cycles required = 6, whereas there is essentially no ionization of the 34s state, for
which 11 cycles are required. Since the pulse contains less than 11 cycles, no
ionization occurs until a higher microwave field amplitude is used. In this case,
ionization begins when the amplitude reaches about 95 V/cm, as shown in fig. 7-3. If
we set l/16n4 = 95 V/cm, we find that n = 43, which is the level to which the atom
must climb before classical ionization occurs. Since it takes a minimum o f 9 half­
cycles, or about 5 full cycles, to climb from n = 34 to n = 43, it is not surprising
that ionization o f the 34s state begins at 95 V/cm with the 600 ps pulse.
For all n-levels studied, the thresholds become broader with increasing n for a
given pulse length, a consequence o f the fact that An increases. For example, notice
the very sharp threshold that a 4 ns pulse gives for the 24s state, in contrast to the
broad threshold obtained for the 34s state. This occurs in spite o f the fact that the
minimum number o f cycles required for ionization increases only slightly, compared to
the length of the 4.0 ns pulse, for 34s as compared to 24s. This is probably due to the
decrease in the n-to-n+1 transition probability as n increases. This has been calculated
by Pillet et al.3 For approximately the energy-level configuration encountered in this
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84
experiment, and at an RF frequency o f 8 GHz, the maximum half-cycle transition
probability, 54%, was found to occur when the n-to-n+l anticrossing spacing was 1
cm '1. These spacings decrease as 1/n4. For the sodium I m 1= 0, 1 states of n = 20,
the n-to-n+l anticrossing is about 0.7 cm '1, implying that the corresponding couplings
for n = 24 and 34 are 0.34 cm '1 and 0.08 cm '1respectively. With these couplings, the
transition probabilities are only -2 0 % and -2% , respectively, accounting for the change
in threshold shape when n is changed from 24 to 34.
For a given n, the thresholds become sharper as the pulse length, and hence the
number of cycles, is increased. This is because of the increased number of chances for
a given Landau-Zener transition to occur. The coherence o f the pulse also plays a role
here. As discussed by Mahon et al* the coherent effect o f many cycles increases the
probability for a given inter-manifold transition above what would be expected in the
single cycle Landau-Zener model. Thus we expect that a relatively sharp threshold
can be achieved at l/3n5 if the microwave pulse duration is increased enough. This is
even more apparent for the d-state data, discussed below. For short pulses, however,
we find that not until the microwave field amplitude approaches l/9n4 is the ionization
always near completion.
Ionization thresholds for several d-states are shown in figures 7-4 through 7-8.
For a given d-state and pulse length, ionization thresholds were taken with the lasers
polarized parallel and perpendicular to the microwave field. Parallel polarization
suppresses excitation of the I m I =2 states, so that theoretically only I m 1= 0, I
states are excited. When perpendicular polarization is used, a significant number of
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
85
1.2-1
26d state
parallel polarization
\ +++
0.6-
13
e
W)
S3
+ ++
u.6ns
50 ns
200 os
500 ns
1000 ns
3000 ns
2
2
<
ojq
c
•2
•2
2
c§
0 .8 -
0.6 -
26d state
perpendicular polarization
0.4 -
0.2
l/3 .7 a 5
0.0
•
40
~T“
30
I
20
T "
10
RF attenuation (dB)
Fig. 7-4. M icrowave ionization thresholds.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
~\
0
86
1-2-1
32d state
parallel polarization
’W
-
v
. . .
\
++
—▼—200 ns
500 ns
1000 ns
3000 ns
10
°-8 1
*AY+++++t<
32d state
perpendicular polarization
++++++
0.6
\
0.4-
0 .2
V *
T
\
\
* '
-
_
♦
Y —Y \ %
TY
■■■■%
A
AA
-
w
l/3.7n-
0.0 -
I l/21n4
T"
40
* ‘ i»
30
~r20
"T"
10
RF attenuation (dB)
Fig. 7-5. M icrowave ionization thresholds.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I
0
87
1 .2 -i
n
c
+++
i— 0.6 ns
♦ — 50 ns
—▼— 200 ns
— ± — 500 ns
1000 os
3000 ns
> +
0 .2 -
■ ■ ■ !•£ * > ▼ -
38d state
parallel polarization
_
\ +v
T
10
n
o
T
10
0
+ + + + /-+ + +v
38d state
perpendicular polarization
l/3.7a
F"
20
RF attenuation (dB)
l/21n
Fig. 7-6. M icrowave ionization thresholds.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
l/9n
88
44d state
parallel polarization
0.6 ns
200 ns
500 ns
1000 ns
—■— 3000 ns
- V
O
0.0-
—
44d state
perpendicular polarization
+
....
V
f j «
1/3.7n
40
30
—r 20
T
10
RF attenuation (dB)
l/21n
l/9n
Fig. 7-7. M icrowave ionization thresholds.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
89
SOd state
parallel polarization.
\^ w
0.6 -
°'4
sr °-2
s
o
*a
£
u
04
A
++++++
+
■■
0.6 ns
200 ns
500 ns
1000 ns
3000 ns
&
o.o H
00
e
•a
1
V\
V \\\ \
0. 8 -
73
a
4 a
T“
30
40
1.2-1
T “
T “
20
10
“I
0
SOd state
perpendicular polarization
1.0
0 .8 0.6
-
0.4
■ .................
0 .2 0.0 -
>■»
1/3.7n
40
T"
30
i
10
120
RF attenuation (dB)
l/21n
l/9n
Fig. 7-8. M icrowave ionization thresholds.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i
0
90
the atoms are excited to I m 1=2. This is important because the Na I m 1 = 2 Stark
states are composed o f I m 1 > 2 zero-field eigenstates, which all have very small
quantum defects. The avoided crossings o f these states are smaller and thus more
hydrogenic.5 In summary, the inter-manifold anticrossing size decreases with
increasing n and increasing m. As previously reported, the ionization thresholds are
quite different for the two laser polarizations.6 Specifically, for the 32d state with
intermediate duration microwave pulses of 200ns and 500ns, multiple thresholds are
evident for the case o f perpendicular laser polarization, as shown in fig. 7-5. Since
I m I = 2 states have very small anticrossings, the probability of a Landau-Zener
transition occuring during a particular microwave cycle is low. Thus the I m 1 = 2
population generally ionizes at higher fields o f either l/21n4 or l/9n4. Apparently,
however, a pulse duration of 3000ns, corresponding to 24,000 cycles, is sufficient to
ionize most of the I m 1= 2 population before 1/2In4 is reached.
For all pulse lengths, the probability for ionization is greater for the case of
parallel polarization. In this case we have only I m 1= 0, 1 states, which have larger
anticrossings than the I m I =2 states. The I m I =2 states therefore have thresholds
that are closer to hydrogenic in nature. As n increases, all anticrossings become
smaller, and this distinction evidently becomes less important: the thresholds for the
44d and 50d states become nearly identical for the two different polarization schemes.
Most striking is the very clear l/9n4 threshold for the shortest pulse. This can
be seen in the data for the 38d, 44d and 50d states.
In the most hydrogenic case
shown, the SOd state with perpendicular polarization, essentially no ionization occurs
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91
until the microwave field reaches l/9n4. Notice, however, that with the shortest pulse,
ionization is not complete until the microwave field amplitude is significantly greater
than l/9n4. For example, notice in fig. 7-7 the 44d state for the case o f perpendicular
polarization. With the SO ns pulse, ionization is complete when a field amplitude of
l/9n4 is reached. In contrast, with the 600 ps pulse, ionization has just begun at this
point, or, equivalently, the probability for ionization is very low with a field amplitude
o f l/9n4. Only when the amplitude is significantly greater, by about 4 dB, does 100%
ionization occur. A likely explanation is that not all of the atoms are in the red Stark
states. The bluer Stark states require a microwave field amplitude greater than l/9n4
in order to ionize. A larger number of cycles allows more chances for transitions into
the reddest Stark states.
Bound state redistribution
As discussed in chapter 5, we expect that some atoms that are exposed to but '
not ionized by the microwaves would be left in n levels higher than the one initially
excited. We have found evidence o f this. By observing electrons instead o f ions, we
can achieve good enough time resolution in the field ionization signals to infer the
final state distribution.
As shown in figure 7-2, ionization o f the 24s state with a 600 ps pulse is
incomplete, even with microwave fields significantly above l/3n5. In figure 7-9 we
show a graph of signal due to field ionization of electrons, versus time, o f the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(c) 3 dB (> I/3n5)
co
s
00
cw
E
45s
20s
34s
(b) 7 dB (< l/3n5)
o
<
00
s
‘S
•a
s
u
0*
(a) noR F
24s state
0.6 ns RF pulse
0.0
0.2
0.4
0.6
0.8
time ( p.s)
24s
Fig. 7-9. B ound state redistribution o f 24s state.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
93
remaining population after the 24s state has been exposed to a 600 ps microwave
pulse o f different amplitudes. Note that a more negative signal indicates a larger
number of electrons arriving at the microchannel plates at a given time.
When the
microwave amplitude is zero, shown in figure 7-9a, the resulting signal is primarily
due to adiabatic ionization of the 24s state. The signal appearing at later times is due
to a combination of diabatic ionization7 and electrons liberated from the septum by the
Na+ ions, as discussed in chapter 6. The latter is the cause o f the small peak
approximately 100 ns after the large peak. Notice that an arrow is drawn below the
large peak, labelling the horizontal axis with the time that adiabatic ionization of the
24s state arrives. This is in fact our definition of all arrows labelled with a particular
state: that state is excited by the lasers, and then the atoms are ionized by the field
pulse. An identical field pulse is used each time. No microwaves are applied when
this calibration of the time axis is done. The arrival time o f the peak o f the resulting
signal is then labelled with that initial state. W hen the microwave amplitude is
increased so that the s state mixes with the nearby manifold, figure 7-9b results. The
microwave amplitude is still below the l/3ns threshold, but approximately equal to
.7/3n5, the field at which the s-state joins the nearby manifold.
The resulting trace is
similar to figure 7-9a, except that there is some earlier signal due to adiabatic
ionization o f states in the nearby manifold. These states have higher energy and are
ionized by a smaller electric field, and thus earlier in time as the field pulse ramps up
to its peak.
The integrated areas o f figures 7-9a and 7-9b are the same. This is
expected since no microwave ionization has occurred, but rather only redistribution to
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
94
the neighboring manifold.
In figure 7-9c, the microwave amplitude o f 297 V/cm is
significantly above the l/3n5 value o f 287 V/cm. There is an evident increase in the
signal at much earlier times, and hence at much smaller field ionization amplitudes.
For convenience, we have labelled the location of the adiabatic field-ionization peaks
for several n-levels, using the method discussed above. This is clear evidence that
some population is left in states higher than the one originally populated by the lasers.
The integrated area of fig 7-9c is about 75% o f the area o f fig 7-9a, which is
consistent with the ionization threshold data in fig 7-2.
In figure 7-10 we show a trace similar to figure 7-9c, but with a slightly higher
microwave amplitude and an expanded vertical scale. Again we have labelled the
location of several nearby n-levels. Somewhat surprising is the population to n-levels
far higher than needed for ionization at a field of l/1 6 n \ The microwave field
amplitude in figure 7-10 is 330 V/cm. Setting this equal to 1/16n4 ^ n = 31;
however, some population is left in states as high as n = 45. To summarize, we excite
sodium atoms to the 24s state with the lasers. Then a 600 ps microwave pulse is
applied whose amplitude is sufficient to classically ionize n = 31 states. The atoms
make transitions to higher states as the microwave field oscillates. We would expect
that any atom which climbs to the n = 31 state would be ionized by the microwave
field, and clim b no higher. However, because of the finite amount of time spent
above the classical ionization limit during each microwave cycle, the possibility of
climbing beyond it exists and evidently occurs. The structure in the trace of fig. 7-10
was reproduced many times and contains several peaks corresponding very nearly to
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
95
24s state
electron signal
2 dB (> l/3n5), 0.6 as RF pulse
■ca
45*
00
i/3
40S
S
o
<
00
s
s
‘<3
E
<o
30s
29S
as
27$
26s
-
0.2
0.0
0.2
0 .4
time ( *is)
Fig. 7-10. Close-up o f bound state redistribution o f 24s state.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
96
the location o f the s-states. Since an ns-state ionizes adiabatically at nearly the same
field as the reddest member of the (n-l)-m anifold and the bluest member of the (n-2)manifold, we believe we are observing population that is preferentially trapped in
extreme or near-extreme states o f the Stark manifolds. This is consistent with the
work o f Hettema et al, during which it was observed that transitions between the
extreme and middle members of a Stark manifold are relatively unlikely to occur.
As seen in figure 7-9, there is a signal from the septum electrons that arrives
somewhat after the main peak for a given n, and with reduced amplitude. These
signals somewhat cloud the data o f figure 7-10. However, the septum electron signal
is delayed more as n is increased, and its intensity compared to the main peak also
decreases. The latter effect is due to the fact that atoms in a higher n state give birth
to ions in a smaller electric field. These ions are thus accelerated more slowly
towards the septum and remove fewer electrons. In fig. 7-10, the first n-level whose
septum electron signal arrives earlier than the 24s main peak is the 30s, and this signal
is less than 15% o f the intensity o f the main 30s peak. Since these signals due to the
Na+ ions liberating electrons from the septum are generally quite small, we do not
believe the overall picture of figure 7-10 is significantly affected.
W e see similar structure in the bound state populations o f n=34 and 39 after
exposure to the 600 ps pulse, although the s-state peaks are not as pronounced as with
the 24s state. In both cases, we observe redistribution to states above the level at
which classical ionization could occur.
As the microwave pulse duration is increased beyond 600ps, while the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
97
microwave field strength remains somewhat above l/3n5, the population in excited
states decreases, as does the total bound-state signal. This can be seen in the
ionization threshold data earlier in this chapter. However, the fraction o f the boundstate signal itself that is in excited states, and the distribution among those higher
states, we find to be only weakly dependent on pulse length. For the 24s state, fig. 711 shows the bound state signal for three o f the five pulses used. Notice that as the
pulse duration is increased, the total bound state signal decreases, indicating o f course
that more atoms have been ionized. But the distribution to final states, and the highest
n-level reached, remains nearly the same.
For the 34s state, we find the excited portion o f the bound state is weighted
more towards lower-n for the 600ps pulse. This weighting towards lower-n is true for
the 600ps and 1.2 ns pulses in the case of the 39s state. For the 39s state, traces for
the 600 ps pulse, as well as the 1.8 ns and 4.0 ns pulses, are shown in fig. 7-12.
Notice with the 600 ps pulse, the highest n-level reached is about the same as with the
other pulses, but a greater proportion of population is left below n - 45.
In figure 7-13 we show electron field ionization traces for the 38d state, for
microwave amplitudes of zero and 15 dB using a 600ps pulse. Note from figure 7-6
that 15 dB is greater than 1/3n5 for this n-state. Not shown is a trace with the
microwave amplitude less than l/3n5. In this case, the result is similar to figure 7-9,
in that there is a small amount o f early signal due to population of the adjacent
manifold. Recall that polarizing the lasers parallel to the microwave field suppresses
excitation of the I m 1= 2 states, which have smaller anticrossings. As expected,
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98
24s state
RF> l/3n5
4.0,1.8,0.6 ns from top down
time (|xs)
Fig. 7-11. Variation o f bound state redistribution with R F pulse duration.
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99
39s state
RF > l/3n5
4.0, 1.8,0.6 os pulses
CQ
S
op
c/3
£
o
55s
<
00
s
'£
*E
50s
E
0)
o*
45s
600 ps
0.0
0.2
time (ps)
0 .4
Fig. 7-12. B ound state redistribution fo r the 39s state.
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100
there is more diabatic field ionization for the no-RF trace with perpendicular laser
polarization than with parallel polarization. Diabatic ionization is seen as a large peak
about 50 ns after the adiabatic ionization peak. The small peak about 250 ns after the
adiabatic ionization peak is due to electrons liberated from the septum.
The lighter traces are taken with the microwave amplitude above l/3n5.
We
define the amount of population trapped in excited levels as the percentage of the
signal that arrives before the beginning o f the large adiabatic ionization peak. This
cutoff is labelled on the graph. This population is greater for the parallel polarization
case. For perpendicular polarization, the ionization process is much more hydrogenic,
and we expect very little transition to higher states compared to vertical polarization,
and this is what we observe.
In figure 7-14 are plotted similar results for the 44d state, again using a 600 ps
pulse. Notice again that the amount o f population trapped in states higher than 44d is
greater for the case of parallel laser polarization, when primarily I m 1= 0, I states
are excited. The large peak in the no-rf traces that arrives about 100 ns after the
adiabatic peak is probably due to partially diabatic Held ionization.
Both the 38d and 44d traces were repeated with 50 ns and 200 ns pulses.
With the 50 ns pulse, the percentage o f the total bound state left trapped in excited
states was close to the 600 ps result. However, with the 200 ns pulse, results indicate
that the trapped population was much smaller than with the 600 ps and 50 ns pulses.
For most o f these traces with the 200 ns pulse the total bound state was small, and the
portion o f this left in excited states smaller still, which is difficult to discern above the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
101
-
10-
-
20-
NoRF
45d
38d state
parallel laser polarization (a)
■ca
No RF (darker trace) and 0.6 ns RF > l/3n:
op
£
£
o
-30
-
0.2
-
0.1
0.0
0.1
0.2
0.3
0.4
<
oo
c
'E
•a
£
as
-
10-
45d
38d state
p erpendicular laser polarization (b)
-20-
38d
-30
-
0.2
-
0.1
0.0
0.1
0.2
0.3
time (p.s)
Fig. 7-13. Variation o f bound state redistribution w ith Im I, fo r the 38d state.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.4
102
cutoff for 44d state
0.000
•0.005 -
SOd
-0.010-
-0 .0 1 5 -
44d stale
No RF (darter trace) andRF > l/3n:
parallel laser polarization
op
£
£
- 0.020
-0.4
-
0.2
0.0
0.000
-0.005 -
- 0 . 010 -
SOd
diabatic field ionization
-0 .0 1 5 44d
adiabatic
-
perpendicular laser polarization
0.020
-0.4
-
0.2
time (jlls)
Fig. 7-14. Venation o f bound state redistribution w ith Im I, fo r the 44d state.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.0
103
digitizing noise of the oscilloscope. For this reason, we do not include the 200 ns data
in the averages below.
Since this trapped population is only weakly dependent on pulse length for
almost all cases studied (24s, 34s, 39s with 600 ps, 1.2 ns, 1.8 ns, 2.8 ns, 4.0 ns, and
38d, 44d with 600 ps, 50 ns), we have placed the results in table 7-2. These are the
average values for the percentage o f total bound state that remains in a state higher
than the one originally excited by the lasers. It is worth reiterating that the total
bound state signal itself, of course, decreases as the microwave pulse is made longer,
since the probability for ionization increases. Table 7-2 lists the percentage of this
total signal that is left in excited states.
State initially excited
Percentage of total
bound state trapped is
excited states
24s
50 %
34s
40 %
39s
20 %
38d, parallel
10 %
38d, perpendicular
4 %
44d, parallel
4 %
44d, perpendicular
2 %
Table 7-2. Average percentage o f atoms not ionized by the
jnicipw^yes that, a re ie ftin excited states. As. explained in the
text, the 200ns pulse is not included in the averages for the 38d or 44d,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Recall that anticrossing spacings decrease with increasing n and increasing I m L
Thus in table 7-2, the 24s state is the least hydrogenic and the 44d, perpendicular state
is the most hydrogenic. Apparently, the states that are more hydrogenic undergo
fewer transitions to higher n during the ionization process.
Conclusion
We find that microwave ionization of sodium Rydberg atoms depends strongly
on the duration o f the microwave pulse applied. With a field amplitude equal to l/3n5,
the pulse will cause essentially no ionization unless a requirement for the minimum
number of cycles is met. For a given initial state and microwave field amplitude, the
probability for ionization increases with the pulse duration. We also find that a
significant percentage of atoms that are not ionized by the microwaves are left in
excited states. This percentage is strongly dependent on anticrossing spacings, but is
only weakly dependent on the pulse duration.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
105
References
1. H.B. van Linden van den Heuvell and T.F. Gallagher, Phys. Rev. A 32, 1495 (1985).
2. C.R. Mahon, J.L. Dexter, P. Pillet, and T.F. Gallagher, Phys. Rev. A 44, 1859 (1991).
3. P. Pillet, H.B. van Linden van den Heuvell, W.W. Smith, R. Kachru, N.H. Tran,
and T.F. Gallagher, Phys. Rev. A 30, 280 (1984).
4. C.R. Mahon, J.L. Dexter, P. Pillet, and T.F. Gallagher, Phys. Rev. A 44, 1859 (1991).
5. M.G. Littman, M.L. Zimmerman, T.W. Ducas, R.R. Freeman, and D. Kleppner, Phys.
Rev. Lett. 36, 788 (1976).
6. H.B. van Linden van den Heuvell and T.F. Gallagher, Phys. Rev. A 32, 1495 (1985).
7. T.H. Jeys, G.W. Foltz, K.A. Smith, E.J. Beiting, F.G. Kellert, F.B. Dunning, and R.F.
Stebbings, Phys. Rev. Lett. 44, 390 (1980).
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106
A ppendix A
Mnltiphotoo Resonance with One to Many Cycles
R J . Watkins, W .M . G riffith, M jL Getzke1, and T S . Gallagher
D ^ u tB M L d F ifd a
Unisanity
(M d w a ^ V A flM
Alntnet
Using radio frequency pubes we connect the relatively unstructured response of a two
level system to a half cycle pulse to the resonant response to a monochromatic wave. There is
an obvious change from the non resonant effect of a half cycle pulse to an interference pattern
a m in y » a Young's two slit interference psttera with two cycles. The analogue of
illuminating more slits, adding progresavety more cydea. hada to a gradsal evoidiae of the
interference
into sharp resonances. The soucinwd response to M cycles is implicit in
the response to one cycle.
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Recently 10* V/an half cycle poises with center frequencies of up to a THz, have been
used both to demonstrate a new form of field
of Rydberi atoms and to redistribute
the population from an initially excited Rydberg sate [1,2]. These processes have h—i
described classically by Mona Carlo Methods, P] semidassically, [4] and quantum
mecfcaeaily [5,6]. la the pvt ja r redutributioo of Rydberg sate population by a GHz half
cycle pulse baa beee shown a o t iit Shchdbeq orillarir— . as aught be sea ia a charged
particle collision with a well defined impact patameto [7J.
Numerical integration of the Schroedinger equation has ben used to describe effects of
half cycle pulses and short laser pulses [8]. Although this approach appears unsuited for
describing the intetactian of an atom with monochromatic radiation, [9] as pointed out by
Moloney and Meath, [10] the responae of an atomic system to an arbitrary number of cycles is
implicit in the response to a single field cycle: if the tune evolution operator to a single cycle
is represented by the unitary transformation U the time evolution operator to M cycles is UM.
Connecting descriptions of a single cycle to long poises is a longeracademic rince laser
pulses with only a few optical field cycles are now being generated [11,12]. Sere we report the
experimental observation of the responae of a two level system to pulses containing from one
to fifty radio frequency (rf) cycles. Specifically, we have driven transitions between the K 21s
and 19,3 states in a static electric field E,. The 19,3 state is adiabatically connected to the zero
field n ■ 19, f * 3 stale. As shown by Fig. 1 the 19,3 state has a large linear Stark shift and
an avoided crossing with the 21s state at the static field E,. ■ 304.2 V/cm, where the two states
are 339 MHz apart We drive traritinm b m w the two states by adding an rf field pulse,
parallel to the static field, with a chosen number of cycles. As shown ia Fig. 1, for E, < Eg
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the positive half cycle adds to the sacic field bringing the sates closer to the avoided cxosting
while the negative half cycle moves the sates away from the avoided crossing.
We fix the amplitude E,* and the number of cycles and observe the 2Is -» 19,3 iim b im i
probability as we slowly sweep the static field, which alters the energy —
between the
two states. With a single cycle the transition probability increases tnonotooically as E, +
the peak meal field, sppreacbes Eg. Thiadrpmrirawqnhcureteramdbycnetiaaag Fjg. 2a.
a plot of the energies of the 2b and 19.3 n a n vs. tian ins single cydrpdbe for static and
rf fields such that E, + Erf - Eg. The transition probability arises from the mixing of the two
stales at the peak of the field, reflected in the deviations of the 21s and 19,3 energies from a
constant and a sinusoidal oscillation. For a fixed amplitude E^ with smaller static fields the
distortion of the energy levels and the transition probability are lower. When E, + Erf exceeds
Ee the avoided craasinf is traversed twice, as shown in Fig. 2b, and there are two paths by
which atoms can go from the 21s to the 19,3 state. These two paths interfere constructively or
destructively depending on their phase difference I, leading to pronounced Stuckelburg
oscillations in the transition probability [7]. Our interest here is in the intercycle interference
which occurs with more than one if cycle. Sparifieafiy, we here ohaerred changes in the
transition probability as progressively mom cydea me added. A aaeond cycle reenduoes a
sinusoidal modulation onto the single cycle transition probability, roughly analogous to Young’s
two slit interference pattern. As more cycles are added die maxima develop into progressively
sharper resonancesjust as illuminating more slits produces a sharper optical diffraction pattern.
In the experiment, K atoms pass midway between the two plates of a parallel plate
transmission line where they are excited by two dye lasers from the ground 4s state to the 21s
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state ia a static dearie field. The atoms are subsequently
rf to the rf pni— to drive the
2Is -• 19,3 transition. Fussily, a field ionization pete is applied with ae
«ufKriw.t
to ionize the 19,3 state but not the 21s state. The ions pea through a 0.4 mm <tiam<w hole in
the upper plate of the transmission line and are detected.
To generate the short if pulses we use a Philips PM 5785B pulse generator with dock
and pulse outputs at the same frequency, typically 20 MHz. The clock output drives a step
recovery diode to generate harmonics, and a Hewlett Packard (HP) 5340B sweep ««natnr ia
phase locked to the desired harmonic. Putting the sweep wwti«w output into the local
oscillator port of a mixer and the pride gnaenmnr ootpui ieto the
frequency port
produces a train of phase-locked rfp u ta at the mfie frequency port. The rf prises tage from
3.5 ns to 100 ns long and contain integral numbers of half cycles. Using a second mixer, we
select the first rf pulse after the laser pulse and amplify it The amplified pulses are brought by
50Q coaxial cable to the parallel plate transmission line inside the vacuum system. To have a
50Q impedance in the line, the ratio of the plate width to spacing is maintained at 377/50 [13].
Where the transmission line joins the cable the plate spacing is I mm, and over a length of 10
cm it is increased to 8.5 mm. Far 22 cm this separation is maintained, and in the middle of this
region the atomic beam crosses the transmission line. At the end of the parallel plate line the
spacing is reduced to 1 mm over a 10 cm length and command to a SQQload.
The lower plate of the transmission line is isoiamd with |1|T ■■'■■■ '•T *"*"". allowing
the dc voltage and ionization pulse (1 ps rise time) to be applied to it Rf pulses are attenuated
by 2 dB on passing through the entire line but are not changed in shape. Using a pickup probe,
we have observed small distortions in the pulse in the parallel plate section which are consistent
with voltage reflection coefficients of 0.19 at the capacitors.
In Fig. 3 we show the observed and calculated 19,3 signals as functions of static field
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110
for a 454 MHz pul* of amplitude Erf - 4.4 V/cm and several pulse u»yttn Thee are two
contributions to the experimental signals, the 21s -• 19,3 « m Hem* driven hy the rf p"1—and
the direct laaer excitation of the upper eneriy eigenstate as E, approaches % [14], Due to
overiappinf field ionization signals we can not obtain absolute ex^rimgini
probabilities, and the experimental traces are normalized to the
19,3 signals, shown
by the smooch solid curves. The portion of a calculated signal due to laser
upper energy eigensmae is those bythe broken tines (14] and dm
of the
is dee to the rf
transitions. The transition probebifity is cafaahwdby mmpun^g die writery nine ewhaian
matrix U farasingle cyde andtising be sasrir U**to cakailatrthe tesponae to Mcycles, la
Fig. 3a we show the trace obtained with a 3/2 cycle pulse, specifically a negative half cycle
before and after the positive half cycle. Traces essentially identical to those shown in Fig. 3a
are obtained with pulses conaining one positive and one negative half cycle as well as only one
positive half cycle. Given the energy levels shown in Fig. 1 it is not surprising that it is only
the number of positive half cycles which matters. The signal monotonically increases until E,
* 301 V/cm, at which point the peak field E, + Erfslightly exceeds the avoided crossing field
Eg. For larger values of E, Stuckelburg oscillations, haring the origin depicted ia Fig. 2b, are
clearly visible in the calculated signal. We attribute the djscrepaary hfflunrn the awuurrd and
ralnilatnd curves of Fig. 3a for E, > 301 V/cm to the small reflections in the transaasBoa line.
The regime Erf + E, > Eg, the Stuckelburg oscillation regime, is particularly sensitive to the
precise shape of the individual field cycle. This issue is lesa important in the region Erf + E,
< Eg where the multiphoton resonances build up from many cycles.
A pulse with two positive half cycles (and three negative half cycles) produces the trace
of Fig. 3b. The most striking difference from Fig. 3a is that the structureless region between
297 and 301 V/cm in Fig. 3a now exhibits a pronounced sinusoidal modulation similar to
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Young's two slit interference pattern. The origin of the modulation is intercycle interference
which is easily imdenanud by retiming B Fig. 2c, which shows the time dependence of the two
energy levels in a two cycle pulse for which E, + E * « Eg. There is a transition
from the 21s to the 19,3 sate on each of the field cycles resulting in the two wMatina pathways
shown in Fig. 2c. The amplitudes add oonstnretively if their phase difference 4 * 2rN and
destructively if * * 2t (N+'A) where N is an integer. * is the energy difference between the
two dates integrated over one cycle, i.e. the area indicated in Fig. 2c. Since the W21l and
W jj j are mtwBst and li— urtrffy varying rapecttvdy, • ■
where
*are red ^ « s are the time average caevesnf tie ram rerere, Le. thmranregies in the static
field alone. Interference maxima, where ♦ «■ 2tN, occur for static fields re which the
separation of the two levels is an integer multipie N of the rf frequency. Since the maxima
occur at the fields of the N photon resonances it is convenient to a ll them the resonant maiima
Note that in Fig. 3b the transition probability vanishes at 302.4 V/cm, as in Fig. 3a. If the
transition probability is zero after one cycle, it remains so after any number of cycles.
When there are three positive (and (bur negative) half cycles we obtain the trace of Fig.
3c. Each of the resonant maxima of Fig. 3b becomes sharper, and an additional set of
appears between them. In a three slit optical experiment the new maxima would be subsidiary
maxima smaller than the resonant maxuna,(15] and such is tire case far E, < 300 V/cm, where
the single cyde responae of Fig. 3a is weak. When the transition probability approaches one
the optical analogy breaks down. For example, the N ■ 4 resonant maximum at 301 V/cm is
smaller than the new maxima on either side of ^ • 301 V/cm. At the resonant maximum the
single cyde response of Fig. 3a is strong enough that ia only three cycles the atoms can make
approximately the fall Rabi oscillation from 21s to 19,3 and back to 21s again. A pulse with
five positive (and six negative) half cycles leads to the trace of Fig. 3d in which the observed
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112
features are generally sharper than ia Fig. 3c, as might be
Only where the single
cycle transition probability of Fig. 3a is small do the resonant
stand out dearly.
The fact that the Rabi frequency exceeds the inverse of the pulse length for much 0f Fig.
3 obscures an interesting point:
the multipboaon transition probability in lowot onto
perturbation theory (LOPT) is identical to optical diffraction. If p is the 2Is -• 19,3 transition
amplitude for a single cycle the transition amplitude from the mth cyde of an M cycle pulse is
pe“ * where * is the phase shift between cycles defined previously aed stem ie Rg. 2c. In
LOPT tbe transition prabsbtty F far the M cyde pate is fiv « by
(1)
precisely the same expression used to describe an optical diffraction pattern from M slits. This
expression is only valid for P < < 1, a condition which is not well satisfied in much of Fig. 3.
The similarity to optical diffraction is more apparent in the traces shown in Fig. 4 afam
with slightly higher frequency, 370 MHz, and lower amplitude, 1 V/cm. With this low field
amplitude the single cyde response is weak enough that the features which develop too tbe two
and three photon resonances remain clearly scpanasd. To acquire these data we excited the
atoms in a static fidd well below the avoided crossing and brought dmatoms to tbe tobc fields
given in Fig. 4 with a 1 ps risetime 5 V/cm pulse, before applying the rf pulse. This procedure
ensured that the observed signal was due only to the rf transition, not to laser excitation of the
upper state in the anticrossing.
In Fig. 4 we show calculated curves for 1, 2, 5, 10, and IS cycles as well as
experimental curves for all but one cycle. We cannot make a single cyde pulse at 370 MHz,
but as shown by Fig. 4a, the calculated single cyde transition probability is almost structureless.
As shown in Fig. 4b, with two cycles a broad resonant maximum appears at the location of the
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113
two photon transition. With the five cycle pulse shown ia Fig. 3c the two photon resonance is
clearly visible above the adjacent subsidiary mavimum on the high smtic field side. In
this subsidiary maximum is reduced in intensity from the resonant
by the square of
the number of field cycles, as is true in an optical diffiaction pataern.[15] With the ten cycle
pulse of Fig. 3d the three photon resonance appears, and the subsidiary madmi near the two
photon resonanceare clearly visible but are only a factor of ten weaker than the resonance itself,
not a factor of 100 as would be the esse for diffraction. LOPT and therefore the optical
description, are breaking down for the aw photon renancr Witii IS ejekt, as shown in Fig.
3e, the three photon resonance is more viable, and (he two photon resonance is broken ep inns
a group of mamma of comparable size, indicating the complete breakdown of LOPT.
In conclusion, we have shown that as rf pulses are increased in length from one to many
cycles the resulting transition probability evolves from an almost structureless function of the
energy level sparing to oneexhibiting the sharp featurescharacteristic of resonance experiments.
This evolution is implicit in the responae to a single rf cyde and is nmii«r to dm evolution of
a diffraction pattern as more slits are illuminated. The dKftactian analogs dona our thinking
in useful ways. For example, as pointed out by Jones [16], the analog to blazing a grating,
choosing the shape of a field cyde i.e. its harmonic content, can enhance the transition
probability for specific high order transitions [17-19].
It is a pleasure to acknowledge «*imtii»n'«g conversation with R.JL Jones and D.W.
Schumacher. Hus work has been supported by the Air Force Office of Scientific Research.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
114
tPresent address; Physics Division. NIST, Gaithersburg, MD 20899
1. R.R. Jones, D. You, and P.H. Bucksfaaum, Phys. Rev. T*a. 70. 1236(1993).
2. N.E. TieUdng and R.R. Janes, Phys. Rev. A52. 1371 (1995).
3.
C.O. ReinhnM. 1C. tUBm, S. Shno, and J. Bwgjurfa , J. Phys. B2fi L6S9 (1993).
4.
N.F. Ramsey, MolCBllKJGUL (Ctdned Unhand? PWm, O iW . 1955).
5.
KJ. LaGanua and P. Lenier, Phys. Rev. A49. R1547 (1994).
6.
A. Bugacov, B. Pinuz, M. Pont, and R. Shakeshaft. Phys. Rev. A51. 1490 (1995).
7.
G.M. Lankfaujjzen and L.O. Nooidam, Phys. Rev. Lett. 24, 355(1995).
8.
K.C. Kulander, K. J. Scfaaftr, and J.L. Kiauae, in Atnmi in inteMe
edited by M. Gavrila (Academic, New York, 1992).
9.
S.-I. Chu, Adv. A t Mol. Phys. 2L 197 (1985).
Fi»M.
10. J.V. Moloney and WJ. Meath, Mol. Physics 1L 1337 (1976).
11. J. Zhou, G. Taft, C.P. Huang, M.M. Muraane, H.C. Kapteyn, and LP. Oixistov, Opt
Lett. 12, 1149 (1994).
12. A. Stingl, M. Lenzner, Ch. Spielmann, F. Krause, and R- Sapbca, Opt Lett. 20. 602
(1995).
13. S. Ramo, J.R. Whinnery and T. van Ouaer, Weidi « ih Waves in Communication
Electronics (Wiley, New York, 1984).
14. R.C. Stoneman, G.R. Janik, and T.F. Gallagher, Phys. Rev. A24, 2952 (1986).
15. M.V. Klein and T.E. Furtak, Qfltia (WOey, New. York, 1986).
16. R.R. Jones, private communication.
17. C. Chen and O.S. Elliot Phys. Rev. Lett, f ii 1737 (1990).
18. D.W. Schumacher, F. Weihe, H.G. Muller, and P.H. Budafanum, Phys. Rev. Lett. 21,
1344(1994).
19. V. Veniard, R. Taleb, and A. Maquet Phys. Rev. Lett 24. 4161 (1995).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
115
Figure Caption*
Fig. I. Energy level diagram showing the K 19,3 and 21s «mn vs.
field E, (shown in
the figure at 297 V/cm). The levels have aa avoided craning at E, ■
• 304.2 V/cm. For
E, < Ee the positive half cycle of the added if field bring the atoms to the avoided crossing,
as shown by the single cycle of the rf field of amplitude E^.
Fig. 2. Temporal variations of the energies of the 21x and 19,3 antes showing the evolution
starting from an initially populated 21s state in several rf pulses, (a) a single cycle pulse for
which E, + Erf - E^ The single possible 21s -• 19,3 path is shown by the broken line, (b)
a single cycle pulse for which E, +
> Eg. There are two 21s -» 19,3 paths, down by dm
broken lines, with a phase difference 9 between them leading to Stuckeiberg oscillations. 6 is
the indicated area between the two paths, (c) a two cycle pulse with E, +
» Eg. There are
two 21s -• 19,3 paths with, phase difference ♦ leading to intercycle interference, or resonance.
Fig. 3. Experimental and calculated signals for 454 MHz pulses of amphande 4.4 V/cm. (a)
3/2 cycles (1 positive half cycle) (b) 5/2 cycles (2 positive half cycles) fc) 7/2 cycles (3
positive half cycles) (d) 11/2 cycles (5 positive half cycles). The noisy curves are the
experimental data. The smooth solid curves are the total calculated signals and the broken lines
the calculated contribution due to excitation of the upper state at the avoided crossing. All
experimental traces have been normalized to the range [0,1], and the calculations have been
offset by -0.25, one scale division, for clarity.
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116
Fig. 4. Experimental and ralcularrd transition probabilities for 370 MHz pulses of
1 V/cm. (a) I cycle (b) 2 cycles (c) S cycles (d) 10 cycles (e) 15 cycles. As in Fig. 3 the
noisy traces are the experimental data and the smooth curves the
probabilities, and the y-axis is drawn similarly.
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transition
117
19.3
-309.95
- 3 1 0 .0 0
0
i
100 200 300 400
- 3 1 0 .0 5
- 3 1 0 .1 0
339 MHz
21s
- 3 1 0 .1 5
294
296
£
302
300
296
Electric Reid (V/cm)
304
306
Figure i. Watkins etal
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118
19.3
2 Is
19.3
21s
19.3
2P
B
u
v
21s
t
Figure 2. Watkins, et al
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296
297
298
299 300 301 302
Static Raid (V/cm)
303
304
Figure 3. Watkins etai
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
N
300
301
302
Static Raid (V/cm)
Figure 4. Watkins e ta l
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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