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A classical interpretation of observed switching statistics in microwave-driven Josephson junction systems

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A C la ssica l In te r p r e ta tio n o f O b served S w itc h in g S ta tistic s
in M icro w a v e-d riv en J o se p h so n J u n c tio n S y ste m s
By
JEFFREY EUGENE MARCHESE
B.S. (University of Colorado, Boulder) 1983
M.S. (University of California at Davis) 2002
DISSERTATION
Submitted in partial satisfaction of the requirements for the degree of
DOCTOR OF PHILOSOPHY
in
Engineering — Applied Science
in the
OFFICE OF GRADUATE STUDIES
of the
UNIVERSITY OF CALIFORNIA
day:
Committee in Charge
2007
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UMI Number: 3261177
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Jeffrey Eugene Marchese
December 2006
Engineering - Applied Science
A Classical Interpretation of Observed Switching Statistics in Microwave-driven
Josephson Junction Systems
Abstract
We present a classical interpretation of phenomena th at have been attributed to macroscopic
quantum tunneling. In this study, we apply classical analytical and numerical techniques to
microwave-driven Josephson junction systems. The phenomena considered are referred to
in published experiments as multi-peaked switching distributions, Rabi oscillations, Ramsey
fringes, and spin-echo oscillations. In short, we find that a transient, modulating response of
the system to microwave signals determines the oscillatory nature of the switching response.
ii
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C ontents
List o f Figures
v
1
Introduction
1
2
Background
2.1 Josephson junction b a s i c s ......................................................................................
2.1.1 The superconducting quantum s t a t e .....................................................
2.1.2 The Josephson r e la tio n s ...........................................................................
2.1.3 Equation of m o tio n.....................................................................................
2 .2
N orm alization............................................................................................................
2.3 Resonance a n a ly sis..................................................................................................
2.4 Phase lo c k in g ............................................................................................................
2.5 Switching distribution basics ...............................................................................
2 .6
Quantum transition te m p e ra tu re .........................................................................
6
7
7
13
15
20
22
23
3
M ulti-peak sw itching distributions
3.1 Overview ..................................................................................................................
3.2 Two-junction interferometer (S Q U ID )...............................................................
3.3 R esults........................................................................................................................
3.4 C onclusions...............................................................................................................
25
25
26
30
33
4
R abi-type oscillations
4.1 O v e rv ie w ..................................................................................................................
4.2 Classical m o d e l.........................................................................................................
4.3 Perturbation analysis...............................................................................................
4.4 Simulation of tr a n s ie n ts .........................................................................................
4.5 Rabi-type oscillations ............................................................................................
4.6 C onclusions...............................................................................................................
34
34
35
36
41
44
48
5
Tem perature dependence in R abi-type oscillations
5.1 O v e rv ie w ..................................................................................................................
5.2 Classical m o d e l.........................................................................................................
5.3 Simulation details ..................................................................................................
50
50
51
52
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10
5.4
5.5
5.6
Simulation results .................................................................................................
Experimental r e s u lt s ..............................................................................................
C onclusions.............................................................................................................
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53
54
6
R am sey-typ e fringes and spin-echo-type oscillations
6.1 Overview ................................................................................................................
6.2 Classical m o d e l.......................................................................................................
6.3 Simulation details.....................................................................................................
6.4 Simulation results.....................................................................................................
6.5 C onclusions.............................................................................................................
55
55
56
59
62
65
7
R esonances in Josephson ju n ction system s
7.1 Classical model for the three-junction lo o p ........................................................
7.2 Simulation results.....................................................................................................
7.2.1 Dispersion response for the three-junction l o o p .....................................
7.2.2 Oscillatoryphenomena for the three-junction l o o p ..............................
7.2.3 Off-resonance p h e n o m e n a ........................................................................
7.3 C onclusions.............................................................................................................
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67
71
71
73
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76
8
C onclusions and future work
8.1 Two v iew s................................................................................................................
8.2 Future w o rk .............................................................................................................
8.3 In closing....................................................................................................................
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78
79
80
A N um erical m ethods
82
B
84
R abi-m odulation frequency detailed derivation
Bibliography
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List o f Figures
1.1 Potential diagram.......................................................................................................
Josephson junction schematic..................................................................................
Contours of integration C l and C r ........................................................................
Simple Josephson junction circuit...........................................................................
Tilted-washboard potential......................................................................................
Ansatz validation. Plotted here vs. response amplitude are (a) resonance
frequency, (b) average phase, and (c) energy. [ 3 6 ] ............................................
2 .6
Ansatz validation. Plotted here vs. signal amplitude are [for up — up] (a)
Response amplitude and (b) energy; [for up < up] (c) response amplitude
and (d) energy. [35]
2.7 Experimental data for quantum transition temperature T* (from Ref. [1 ]).
Panel (a) shows the switching distribution. Panel (b) indicates T*. D ata for
two junctions are presented, with I c = 1.62 pA and I c = 162 nA, respectively.
2
2 .1
8
2.2
2.3
2.4
2.5
10
13
15
3.1 Interferometer schematic..........................................................................................
3.2 Multi-peak switching distribution as a function of driving frequency. Mark­
ers represent numerical integration results; dashed curve indicates Eq. (3.14);
solid line from Ref. [2], Parameters were: I c = 11.05 pA, up = 40 n s~ l , (3l =
0.07, a = 0.01, r\ ~ 10~8, and T = 370 m K , or © = 1.4 x 10~3. [34] . . .
3.3 Comparison of simulations to experiments. Open markers correspond to
simulations; filled markers represent experimental results; dashed curves rep­
resent Eq. (3.12); solid curves reflect Eq. 3.17. [ 3 4 ] .........................................
4.1 Signaling protocol schematic....................................................................................
4.2 Amplitude and energy response. Panels (a,b) show results of the perturba­
tion theory Eqs. (2.32), (2.35), (2.41), and (2.42). Panels (c,d) show results
of direct simulations of Eq. (4.1) for the same parameters as (a and b). Pa­
rameters were: a = 0.00151477, rj = ^ /l —cof = 0.904706, es = 0.00108, and
T = ep{t) = 0..............................................................................................................
4.3 Linear transient responses to microwave onset as calculated from Eqs. (4.21)(4.23) for the parameter values of Fig. 4.2...........................................................
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24
27
30
32
36
41
42
Normalized modulation frequency Q r (at T = 0) as a function of signal
amplitude gs. P anel (a) shows a range of damping a values, with rj = 0.94259
(us = y /l —rj2 = 0.577886). Gray line with “star” markers indicates the
asymptotic form given in Eq. (4.20). Arrows indicate the frequency for which
small oscillations would be overdamped for respective values of a. Panel (b)
shows a range of dc bias rj values, with a — 0.001. Curves represent solutions
to Eq. (4.9). Markers represent data from numerical simulations of Eq. (4.1)
with £ p { t ) = T — 0....................................................................................................
4.5 Characteristic attenuation (3 as a function of damping a and signal amplitude
es for T = 0. Lines represent the derived relationship 0 = a /2 and markers
represent data from the simulations of Fig. 4.4...................................................
4.6 Simulation study of Rabi-type oscillations. Panel (a) depicts escape proba­
bility P as a function of signal duration At s. Numerically simulated quanti­
ties are represented by dots (each of which represents 25,000 escape events.)
Panel (b) depicts ensemble average of normalized energy, simulated through
Eq. (4.1), as a function of signal duration (ensemble size N = 50,000, ran­
domly chosen 6 S.) Parameters were: T — 30 mK, a = 0.00151477, r] —
y / l - u j = 0.904706, u s = 2irv0 i/u o = 0.652714, es = 0.00217, ep = 0.08474.
These parameters were inspired by those reported in Ref. [3]. Panel (c) Sin­
gle trajectory of energy versus time for parameters listed in (b). Panel (d)
Single trajectory of energy versus time for T = 0 mK. Panels (e-g) are ex­
panded views of the time-averaged, zero-temperature, and 30 mK energy
curves, respectively...................................................................................................
4.7 Modulation frequency VIr as a function of signal amplitude ss for a =
0.00151477, 77 = 0.904706, T = 30 mK. Lines represent calculations using
Eq. (4.9), the open markers represent statistical data from simulations, and
the filled markers are the measurements copied from Ref. [3]..........................
4.4
5.1 Single-junction loop schematic................................................................................
5.2 Single-j unction loop potential energy for: (a) Mjc = 0.32; (b) M ([c — 0.50; (c)
Mdc = 0.68. Note th at here tp in Eq. (5.3) has been rescaled to p* = (p —7r)
5.3 Simulation results: Temperature effects on escape probability. Panel (a)
shows our Rabi-type oscillations for I c = 2.1 pA, C — 0.4 pF, L = 580 pH,
= 0.780,
= 0.00135, u s = 0.706409, and ep = 0.0675. Panel (b)
indicates the relationship between oscillation amplitude and temperature as
measured from the first complete period in panel (a). Line added as an aid
to the eye. Note log scale on left axis...................................................................
5.4 Experiment results: Temperature effects on escape probability. In this figure
the notation for the vertical axis, P (|l)), refers to the tunneling probability
from the |1) state. Reprinted with permission of Ref. [4]..................................
6 .1
Signalling diagrams for simulating (a) Ramsey-type fringes and (b) spin-echotype oscillations.........................................................................................................
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6.2
Direct simulation of the Ramsey-type fringe switching response. Josephson
junction response to the sequential application of two 7t/2 microwave pulses
followed by a probe field. Panels (a,b) show phase-difference and energy
for a non-switching sequence in the case of A = t± — t\ = 400. Panels
(c,d) indicate a switching event for A = t<± —t\ — 70. Parameters were
a = 10~4, rj = 0.904706, es = 2.17 x 10~3, ujs = uit = ^ 1 - r]2 = 0.652714,
ep = 8 . 2 x 1 0 —2, and 0 = 0 .....................................................................................
6.3 Direct simulation of the spin-echo-type oscillation switching response. Joseph­
son response to the application of two 7t/2 microwave pulses, with an inter­
vening 7r-pulse, followed by a probe field. The delay between the tt / 2 pulses
is Atd = ti — t\ — 2182. The 7r-pulse offset is for (a,b) (non-switching)
t 2 ~t] = 1 0 0 0 . For (c,d) (switching) t 2 —t\ — 900. Parameters were a = 1 0 “ 4,
t] = 0.904706, £s = 2.17 x 10~3, £p = 7.5 x 10~2, and 0 = 0............................
6.4 Switching probabilities for the Ramsey-type fringe (a-d) and the Spin-echotype oscillation (e,f) simulations. Each point represents statistics of ~ 2,500
events at © = 2.00 x 10~4. The horizontal axis in (a-d) represents the ir/2pulse separation (At j = t± — t\). Panels (a,b) have a = 10~ 3 and e = 0.085.
In (a) the driving frequency u>s — lji = \ / l —rf = 0.652714. Panel (b) shows
nearly vanishing Ramsey-type fringe frequency at ujs = 0.997w/. Panels (c,d)
have a = 1 0 “ 4 and e — 0.082. Panel (c) uses the same driving frequency as
(a), (d) shows vanishing Ramsey-type fringe frequency near ujs — 0.990a;;.
Other parameters not noted in (a-d) are the same as used in Fig. 6.2. Panels
(e,f) reflect data from spin-echo-type simulations for At,; = 2750 time units.
The 7r-pulse offset ( £ 2 —h ) is shown on the horizontal axis. Parameters used
in (e,f) are the same as in Fig. 6.3 with (e) having u>s = a>; and (f) having
u)s = 0.990a;;. The lines are an aid to the eye.....................................................
6.5 Frequency response as a function of applied microwave frequency for two
different dissipation parameters: (a) a = 10~4, and (b) a = 10-3 . Filled
diamonds depict Ramsey-type fringe frequencies, d p , with parameters cor­
responding to Fig. 6.2. The open circles in (a) represent spin-echo-type fre­
quencies with parameters as in Fig. 6.3. The gray lines represent integration
of Eq. (6.5) to ti — 1 ; dashed lines to 1 4 = 10,000; solid black to 14 = 100,000.
7.1 Three-junction loop schematic. Junction 3 is smaller than the other two and
is indicated by the factor of k applied to the capacitance, resistance, and
critical current...........................................................................................................
7.2 Three-junction loop potential energy for several values of M<;c: (a) M*. = 0.5,
(b) Mdc — 0.52, and (c) M c/C — 0.542. Other parameters were k = 0 . 6 8 and
/? = 0.09994................................................................................................................
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67
70
7.3
7.4
7.5
7.6
7.7
Resonant frequency response in the three-junction loop. Lines represent the
predicted frequency response from Eq. (7.15). The open circles indicate the
resonances from the results of our simulations. The inset is a switching
distribution indicating the resonance at the box marked ’A’. This provides
an example for determining the placement of open circles in the larger plot.
Parameters for the simulation were Pl = 0.09994, k = 0.68, a — 0.00004,
= 0.0004, and 0 (T ) = 0.00912...........................................................................
Direct simulation of Ramsey-type fringe switch events. Panels (a,c) represent
the reduced phase-difference for a trial event which does not switch and one
th at switches, respectively. Panels (b,d) display the resulting potential energy
for panels (a,c), respectively. Squares labelled “lo” and “hi” indicate the low
and high potential energy wells, respectively. Parameters were: ojs —0.42528,
ep = 0.0178 and 0 = 0.0..........................................................................................
Simulated switching phenomena for the three-junction loop: Rabi-type oscil­
lations and Ramsey-type fringes. Panel (a) shows Rabi-type oscillations for
M dc = 0.52, PL = 0.09559, k = 0.68, a = 0.00015, tos = 0.443, e, = 0.00182,
sp = 0.0149, and 0 (T ) = 0.003. Statistics were gathered for ~ 20,000 es­
cape events. Panel (b) provides the Rabi-type frequency as a function of
microwave amplitude, with £s for u>s = 0.45629 ~ uir. Panel (c) is the re­
sulting switching distribution for Ramsey-type fringes with u>s = 0.42528
and £p = 0.0195. The inset indicates the driving frequency which achieves a
fringe frequency of zero; here ojs = 0.45629 ~ tor . ep = 0.0148. Panel (d) pro­
vides the relationship between fringe frequency and driving frequency. The
arrow indicates measurement of u>i = 0.46557 by direct simulation. Unless
indicated, all other parameters were the same as those for panel (a).............
Oscillation phenomena for three-junction loop, spin-echo-type oscillations.
Parameters were: M dc = 0.52, Pl — 0.09559, k = 0.68, a = 0.00015, uis =
0.419787, £s = 0.00182, ep = 0.0195, and 0 (T ) = 0.003. Each dot represents
22,000 escape events.................................................................................................
Simulations of off-resonance Rabi-type oscillation frequencies. Panel (a) de­
picts the off-resonance modes for the three-junction loop. Parameters were:
M dc = 0.52, pL = 0.09559, k = 0.68, a;s = 0.443, £s = 0.00182, ep = 0.063,
and ©(T) = 0.003. Two values of characteristic damping are given. The
filled squares correspond to a = 0.00015, empty circles:® = 0.0. The reso­
nance frequency, cjr (as shown in Fig. 7.5(d)) is 0.45629. Panel (b) shows
the off-resonance modes for the single-j unction circuit. Parameters were
a = 0.00151477, rj =
1 - cof, ujs = 0.99coi = 0.646188, and © = 0.0002.
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Acknowledgem ents
Two persons stand out with regard to all the accomplishments documented herein. My wife
Hsiao-Fen Marchese has been steadfast in both her emotional and financial support of my
endeavors. My thesis advisor Professor Niels Grpnbech-Jensen has, from the very first day
of my graduate career, never wavered in his support.
W ith regard to technical assistance, Matteo Cirillo, Alexey Ustinov, John Clarke,
Juergen Lisenfeld, Paul Reichardt, and Travis Hime have contributed significantly to the
progress reported herein.
Also, my work was supported in part by the UC Davis Center for Digital Security
under the AFOSR grant FA9550-04-1-0171.
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C hapter 1
Introduction
This chapter provides an overview of the historical events leading to our efforts
presented here. Included is some discussion of the overall structure of the thesis. The
chapter following this introduction gives detailed background regarding the systems studied.
There we present fundamental definitions and derivations which are used here and in the
subsequent chapters.
This dissertation is concerned with the classical theory and numerical simulation of
experimentally-demonstrated phenomena associated with Josephson junction devices. The
semi-classical description of a simple Josephson junction circuit utilizes the equation of a
driven pendulum with damping. As such, these equations describe a potential energy curve
known as a tilted-washboard potential. Figure 1.1(a) illustrates a region surrounding a local
energy-minimum of this potential.
In the early 1980’s, a group of seminal papers by A. J. Leggett, et al., employed the
Josephson junction in a thought experiment to determine a means for observing macroscopic
quantum tunneling [5-8]. They proposed that since the Josephson junction maintains two
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Possible
escape event
Possible
escapeevent
k T
•S
K
AE
<P
9
Figure 1.1: Potential diagram.
coherent states (representing a macroscopic number of electrons), and since there is a finite
energy barrier, the system state should have some finite probability to tunnel through the
barrier (as expected by quantum mechanical theory.) Further, predictions were developed
of experimentally visible oscillations, as the system state evolved in the presence of an
oscillating driving field. In Fig. 1.1(b) it can be seen that the probability of tunneling is
greater at energy
E\
than for
E
q.
Thus, as the system evolves, the probability of escape
oscillates as the system is driven from one state to the other. These processes are defined
to occur well below the quantum transition temperature [1 ]
k s T * « tkvo/2-iT,
(1.1)
where k s is Boltzmann’s constant, h is Planck’s constant, and too is the frequency of natural
vibrations for the potential well with zero-bias current, known as the Josephson plasma
frequency.
Subsequently, several landmark studies were published, which reported fundamen­
tal precursors to the quantum phenomena. These early efforts concerned the occurrence of
switching distributions consisting of many sharp peaks - called multi-peak switching distri­
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butions [9,10].
Later, other oscillatory phenomena were reported, including those known as Rabi
oscillations, Ramsey fringes, and spin-echo oscillations [3,11-29]. These take their titles
from analogous quantum phenomena found in atoms and atomic particles.
Validating the quantum nature of these effects is important to establishing the
Josephson junction as a key element in devices for quantum computing. Switching distri­
butions with multiple peaks across a given amplitude and frequency of driving field would
seem to be evidence of distinct states. Oscillations in response to these driving fields also
show possible links to state-like behavior. Currently, various researchers and several em­
bryonic firms are striving to extend these effects into a working prototype for a quantum
computing device.
Previous research into classical Josephson junction models explored basic behavior
of the classically-described systems [30-32]. These developments applied the theory of
phase-locking to the sine-Gordon equation, which for a “small” Josephson junction reduces
to the pendulum equation. The theory of phase-locking describes the process by which a
perturbed system adjusts its own phase to match that of the driving signal.
Our research developed these ideas further as alternative explanations for the afore­
mentioned state-like behavior. Our first efforts addressed multi-peak switching distributions
from a classical viewpoint and are documented in Refs. [2,33,34], These papers recorded
both experimental collaborations and our numerical work. The work done in Ref. [34] is
discussed in Chapter 3.
Next, we examined the Rabi oscillation phenomena in Refs. [35,36], where we use
the term Rabi-type oscillation to describe the classically produced variety. The work per­
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formed in the latter reference is presented in Chapter 4. As a follow-up to this work, a
European experimental group performed temperature studies which established that these
oscillations can be detected for temperatures well above th at for which quantum phenom­
ena should be obscured by temperature effects (i.e., the quantum transition temperature).
Chapter 5 documents our contributions to th at work in the form of numerical data using
a classical thermal model. The main result of this collaboration was to underscore the
agreement between experimental evidence and our classical model.
Chapter
6
discusses Ramsey fringe and spin-echo oscillation phenomena, and re­
flects the work performed in Refs. [37,38]. Again, we changed the notation to differentiate
from the quantum phenomena, using the terms: Ramsey-type fringes and spin-echo-type
oscillations. We consider their nature by extending the classical theory for Rabi-type oscil­
lations: Various signals are presented to the system in the experimentally prescribed fashion
which “interrupt” the system and force a re-establishing period to regain phase-locking.
As our investigation progressed, we learned of specific resonance behaviors which
were studied in experiments such as those in Refs. [14,21,25,27,29,39]. The phenomena
examined were associated with loop circuits composed of one or more Josephson junctions.
These devices are more sophisticated than the simple one junction circuit in that the energy
profile consists of double potential wells which can be skewed or biased such that the minima
of the wells are at the same or differing energies. In addition to the above oscillatory
phenomena, we also examined the resonance characteristics of these wells. This material,
as documented in Ref. [40], is covered here in Chapter 7 along with a brief discussion of
off-resonance behavior in both the single-j unction circuit and the multi-junction loop.
The last Chapter 8 , summarizes our work and offers an outline of work for future
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consideration.
Finally, there are two appendixes. The first describes the primary method of nu­
merical integration for our simulations. The second contains a detailed, alternative deriva­
tion of the Rabi-type oscillation frequency.
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C hapter 2
Background
This chapter presents an introduction to the physics and applications for a simple
Josephson junction circuit. The first section covers the very basics of superconductors and
their configuration to create a Josephson junction. The Josephson relations are developed
followed by a derivation of the equation of motion.
The equation of motion can be formulated in a more manageable form through
normalization using a characteristic time and energy. This analysis is offered below in the
Normalization section.
The concepts of resonance and phase locking are key elements in understanding
the Josephson junction response to perturbing influences such as damping and driving.
The Resonance section in this chapter applies basic harmonic analysis techniques to derive
equations for the linear and anharmonic resonance in the simple Josephson junction circuit.
Next, phase locking is covered as an extension of the resonance analysis.
Following phase locking, the next section of this chapter details the procedures
used in capturing data to produce switching distributions. The final section of this chapter
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discusses the quantum transition temperature, a key reference point in identifying quantum
effects.
2 .1
2 .1 .1
J o s e p h s o n j u n c t io n b a s ic s
T h e su p e r c o n d u c tin g q u a n tu m s ta te
In a superconductor, a macroscopic number of electrons exist in a condensed form
such that they share the same quantum state [41,42]. This condensed form is characterized
by pairing of the electrons in momentum space such th at each electron in a given pair has
the opposite momentum vector from its mate. These pairs are known as Cooper pairs. Due
to the large number of electrons involved, the superconductive quantum state is considered
to be a macroscopic quantum state and is represented here by
ip- In equilibrium the electron
pairs are distributed with uniform density p and share a common alignment or phase (b such
that
(2 .1)
The electric current density (J = [current/normal surface area]) associated with
the electron pairs can be expressed
J - = -m I 2t
~
~
~c A M 2 1 >
(2 .2)
where A is the vector potential, m and e are the mass and charge of a single electron,
the “2 ” in the last term reflects the paired nature of the electrons (the 2 ’s in the prefactor
cancel), and c is the speed of light.
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Substituting Eq. (2.1) reduces to
J a = p— (HV(f> - — a ) .
m \
c )
(2.3)
These are the fundamental equations which are used to derive the Josephson relations and
the equation of motion.
It is important to note th at the above superconductive properties are exclusively
observed at temperatures below a characteristic temperature for specific materials. Thus,
there is a constitutive property known as the critical temperature Tc which signals the
production of large numbers of Cooper pairs relative to single electrons as temperature
is dropped below Tc. The unpaired, single electrons which exist in this state are called
quasiparticles', in keeping with standard solid-state and condensed m atter nomenclature.
2 .1 .2
T h e J o se p h so n rela tio n s
9
9
SL 1
11
SR
^
%
k
i.—
Z
Figure 2 .1 : Josephson junction schematic.
Following Refs. [41,42], a Josephson junction employs two superconductive ele­
ments as fashioned in Fig. 2.1.2.
Sr
and
Sr
are labels for the left and right superconductors
respectively. The \Ec are labels for the wavefunction states of each superconductor. If K is
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the coupling energy between the two states, then the Schrodinger’s equation ih ^ n - = H \ip)
can be written [41]
ih~ t t
=
eV ^L + KxpR
ih ^ W
=
- cV-iPr + K iPl
(2.4)
where the energy level has been chosen such that E l = —E r = eV. Substitution of Eq. (2.1)
reveals
=
^ - V P L p R s m ^ L - <P r )
=
-^Y^PLPRS\n{(pL - <pR)
9<Pl
ot
=
K [p Z
,,
, , eV
~E\
cosVPL - <Pr) + —
h y—
pR
n
d(pR
~ zr
at
=
K [ pl
t,
± \ eV
— cos(<Pl ~ <Pr ) - - r n y pr
h
(2.5)
/„ ^
(2 .6 )
The current density of electron pairs is given by
and it follows that
2K
J = -^-^PLPRsm((pL ~ (Pr )-
A reasonable assumption is that p l =
PR
=
P*
(2 .S)
= constant, so for J* = 2Kp* /h
J — J* sin(<£>),
(2.9)
where we have combined the two phases, <p — <pi - <Pr Note that p* is a constant. However, we denote the externally supplied
current
with p*, which isnot zero, due to the presence of a current source whichcontinuously
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replenishes pairs as they tunnel across the barrier. Also from the above it follows that
d(p
2eV
1 = — ■
<21°)
These last two equations are the constitutive relations of the Josephson effect and are called
the Josephson relations.
2 .1 .3
E q u a tio n o f m o tio n
X
Z
® H,
Figure 2.2: Contours of integration C l and C r .
This derivation involves first developing a differential relationship between <p and
an imposed magnetic field using a contour integration of the system and then further ex­
amining the electrodynamics (Maxwell’s equations) of the Josephson junction.
First consider the integration paths in Fig. 2.1.3 which are applied to Eq. (2.3) in
the presence of the total magnetic field H = Hyy
__
2
e ( me
v<w~R = R 2 i v
\
"+
)'
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where V x A = H. Integrating along the contours gives
VR.(X) -
+ dx)
<pLb(x + dx) - ipLa(x)
=
g
^
(
=
Y c jc
A
+
~
J , ) . d
l
( A + ^ ^ J s) ' d1'
^2'11')
In the figure, the light gray region in the center represents a dielectric barrier, the dark gray
indicates the penetration area of the magnetic field in the superconductors.
Next, Ref. [42] makes two assumptions to ensure the integrals involving J s vanish.
First, the thickness of each superconductor is much larger than the depth of penetration of
the magnetic field
(d
— t<*)/2. This allows the portion of the contours
Cl
and
C
r
parallel
to J s to be extended outside the magnetic field. Second, the integration paths inside the
penetration region are such that they can be made perpendicular to J s.
Then the integrals can be evaluated
<p(x + dx) - <p(x) =
=
[ipLb(x + dx) - (pRb(x + dx)} - [ipLa(x) - p Ra(x)}
^ \ [
hc UcL
f
A ■dl +
JcR
A di
( 2 . 12 )
If the barrier thickness isnegligible, then
if(x + dx) —<p(x) =
2e
2e
f
f
— ffl A ■dl = —/ (V x A) • da
fie J
nc Js
= d ^-H y dx,
(2-13)
(2-14)
where the theorem: j> A ■dl = f s (V x A) • da, is used.
A similar deduction can be made for a field in the x-direction. Together these
yield
%
%
f a
=
=
-
f
a
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<2
1
5
>
These relations together with Eq. (2.9) can be combined with the Maxwell equation
__ 47t
1 <9D
V x H = — J -\----- —
c
c dt
.
.
(2.16)
where D is the electric displacement. This results in
f)Hv dH x
Air
1 0D Z
~ox
jrL - - oy
ir L= —
c Jz +c— id
a tL
2 -1 7
giving
d 2y \
dy2) ~
He2 ( d 2<p
8t t e d \ d x 2
.
T*
s m < ^
+
,dV
^ ,
.
.
( 2 -1 8 )
where C' — er/4nrtd is the junction capacitance per unitarea; er is the relative dielectric
constant and t j is the dielectric barrier thickness.
Using Eq. (2.10) we can write
d2(p
dx2
d 2 y> 1 d2<p _
dy 2 c2 d t 2
1
Aj
sin ip,
(2-19)
where
-2 =
C2
A-kC d
=
C2 t d
erd
and
A /=
^
RiredJ*
Equation (2.19) is the sine-Gordon equation. In the case of a small Josephson
junction, </? varies little spatially (relative to the overall junction size) and this reduces to
d?w
9
-jjj+ u .)j sm<^ =
0
,
(2 .2 0 )
with loj2 = (c/A j ) 2 = (2e/hC) Ic, where C is the junction capacitance and I c is the critical
current; both of which are constitutive properties of the junction.
This is the equation for an undamped, undriven pendulum. The full equation
(plus our normalization) is described in the next section.
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13
Figure 2.3: Simple Josephson junction circuit.
2 .2
N o r m a liz a tio n
The full equation of motion for the simple Josephson junction circuit, shown in
Fig. 2.3, adds terms for damping, driving, and thermal noise to the simple description in
Eq. 2.20, and is written
hC d2ip
~2 e~dr2 +
2
h dip
_
,
e # d r + cSm<^ = dc + ^ smw“ T + N (T)
.
(2-21)
where r now represents time, C is the junction capacitance, R is the shunting resistance, I c
is the critical current, I&. is a dc bias current, I ac and toac are the alternating current and
driving frequency respectively. The
N (
t
)
term represents a thermal noise current given by
the thermodynamic dissipation-fluctuation relationship [43]
(
n
(N(r))
=
0
(t ) N ( t ' ) )
=
™jLLs(T-T),
(2.22)
with T being the temperature and 8 ( t — t ) is the Dirac delta function.
Normalization is performed with respect to the critical current (Ic) and time r —
(wo-1 ) t, where uiq — y/2eIc/h C is the zero-bias Josephson plasma frequency. W ith this
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normalization, the coefficient of the first-order derivative of <p becomes the normalized
dissipation a = hioo/2eRIc. It is also convenient to scale the energies to the Josephson
energy H j — I ch/2e — I c$ o/2 tt, where 4>o = h/2e = 2.07 x 10~15V ■s is the flux quantum.
The normalized equations can be expressed
ip + oap + sin (p
=
r) + es smu>st + n(t)
(2.23)
(n(f))
=
0
(2.24)
(n(t)n(t)^
where the characteristic temperature is
= 2aQ5(t
—t ) ,
(2.25)
= k B T / H j and the normalized bias current is
0
V ~ Idc/Ic and es = Iac/Ic■ This is known as the equation for the resistively and capacitively
shunted junction (RCSJ). Note th at for various types of signaling used in experiments and
simulations rj and es can be functions of time and can include discontinuous signals through
the use of Heaviside step functions.
From the above normalization, we define the undamped, zero-temperature system
energy
H = H k + Hp.
(2.26)
Here we have chosen H k and Hp to represent the kinetic and potential energies, respectively
Hk =
^<p2
Hp =
1
—cos 9? —
(2.27)
r/ip.
(2.28)
Hp describes a tilted-washboard potential, where the periodic function creates multiple,
regular wells in <p. Referring to Fig. 2.4, the bias current r] acts as a force which tilts the
whole curve such that rj =
0
gives maxima (or minima) of equal energies, and higher values
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Figure 2.4: Tilted-washboard potential.
create greater energy separation between the subsequent maxima. In the figure (adapted
from Ref. [44]), A and B indicate the minima and maxima respectively and E indicates the
local energy height of the barrier.
2 .3
R e s o n a n c e a n a ly s is
This analysis follows th a t of Refs. [33,35]. (Note that the simpler case of the
harmonic oscillator is presented in Ref. [45].) We use an ansatz to extract information from
the equation of motion regarding resonance and system parameters. Then we validate the
ansatz with simulations.
Start with the zero-temperature equation of motion
ip + aip + sin ip = r) + es sin ujst
(2.29)
with energy
(2.30)
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A simple approximate description of the solution to (2.29) can be found in the
lowest order Fourier component with frequency ta3. We consider the zero-voltage, steadystate monochromatic ansatz
ip = ipo + A sin(uit + 0) =
+ A ^ s,
(2-31)
where the zero-voltage state is defined by (ip) = cpo = 0, and A is the oscillation amplitude
which may explore the anharmonicity of the potential well. 'I',, is a shorthand for the
sinusoidal term.
For the time-independent case, the ansatz gives
=
J $ A )’
( 2 ' 3 2 )
where Jn is the n th order Bessel function of the first kind and the following identities are
used [46]:
OO
cos(asinx)
=
Jo(a) +
Jvnja) cos[2 rax]
2
71—1
sin(asin:c)
=
2
OO
^ J 2 n+i(a) sin[(2 n + l)x].
The dynamic components yield
A'Fs 4- a A \i/s + Jo(A)Asin<po + 2Ji(A )\ks cos <po = rj + es sinuqf,
where we wish only to consider solutions oscillating with frequency
los
(2.33)
.
Using Eqs. (2.32) and (2.33) reduces to
<2 -34>
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For small damping (a « 1), the resonance of this system is [30]
v
A
\T
(2.35)
\J o (A )j
This is referred to as the anharmonic resonance frequency.
For small oscillations (A
1), we get the linear resonance frequency [42]
ui‘ = V I
(2.36)
T,
which is consistent with the result for a similar perturbation expansion using the simple
ansatz ip = <po + Sip.
Equation (2.34) is then expressed
+ uir'&s =
'Fs +
(2.37)
sinuist.
Expanding the derivatives along with T s and collecting terms results in
[(wr 2 —lu2) cos# —aw sin#] sinwt + [(wr 2 — ^ 2) sin# + aw cos#] cos cot — ^ sin uist.
(2.38)
In the steady-state the response frequency matches the signal frequency, w = ws. We now
have two equations
(wr
u>r 2 —
- w.,
LUS2)) cos a — aws sm o
—
uX
[(wr 2 —ws2) sin # + aojs cos #] cosu>st
=
0
.
(2.39)
This leads to useful relations for e, and
(ws 2 - wr 2) + (aws)'
tan#
aw..
=
Wo
— Wr
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(2.40)
(a)l
0.6
0 .4
0.2
0 .0 0 6
fc;
0 .0 0 4
3
0 .002
I
0 .0 6
0 .0 4
<•—
gsS'-..->
'
0.0 2
0.0
0.0
0 .2
0 .4
0.6
A
Figure 2.5: Ansatz validation. Plotted here vs. response amplitude are (a) resonance fre­
quency, (b) average phase, and (c) energy. [36]
Notice that the ansatz dictates J q(A) > rj as a condition for existence of the zero-voltage
state. Also notice that uy < loq and that uir —>luq for A
0.
The average system energy of the monochromatic ansatz is found by inserting
Eqs. (2.31) and (2.32) into (2.30), then averaging over the period given by (2.35),
(H h * ,u =
\ a 2 ^ 2
The minimum energy for a given
H0
17
+
1
- ^y 2 -/o(/l) 2J^ A j - Wo-
(2-41)
is
— 1 — \ / 1 —?7 2 —rjsm^r].
(2.42)
The applicability of the simple monochromatic ansatz Eq. (2.31) is illustrated in
Figs. 2.5 and 2.6. In the first figure we examine the validity of the ansatz with regard to the
response amplitude A. Thus a comparison is made of the numerical results of Eq. (2.29)
(markers) with the monochromatic ansatz Eq. (2.31) (curves) for a = £s(t) — £p(t) = T — 0 .
Results for r] = 0.904607 are displayed with o and solid curves, while results for rj — 0.98 are
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19
0.50
0.40
a
0.30
0.20
0.10
0.00
o
0.02
H; 0.01
0.00
0
0.01 0.02
0 0.01 0.02 0.03
e s
6 S
Figure 2.6: Ansatz validation. Plotted here vs. signal amplitude are [for u>s — toi] (a)
Response amplitude and (b) energy; [for ujs < u>i\ (c) response amplitude and (d) energy. [35]
displayed with • and dashed curves. Panel (a) shows the anharmonic relationship between
amplitude A and frequency u r Eq. (2.35); panel (b) gives amplitude A and average phase
difference ipo Eq. (2.32); and panel (c) illustrates amplitude A and energy H —H q (2.41)
and (2.42).
In Fig. 2.6 we examine the validity of the ansatz with regard to the driving signal
amplitude es. As in the first figure, markers represent the numerical simulations and the
curves represent the approximations. The parameter values are a — 10~3,
77
= 0.94259.
In panels (a,b) the frequency of the driving signal is set to that of the linear resonance
ujs
— 0.577886(= uji). In panels (c,d) the frequency is offset from linear resonance uis =
0.5237246. The insets show detail of the minimum values of £s near the resonance. Note
the multi-valued response at low driving amplitudes.
The agreement between the results of the ansatz and the simulations is certainly
suitable [except at the highest amplitudes (when Jo (A) —> 77), where higher harmonics of the
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dynamics contribute significant components to the dynamics.] Thus we have demonstrated
the utility of this ansatz. In Chapter 4, we use a modified version of this basic ansatz to
develop a perturbation theory for the system response to the application of microwaves and
damping.
2 .4
P h a s e lo c k in g
As mentioned above, the response frequency (phase) of a steady-state system
matches the signal frequency (phase). The process of approaching this state is described as
phase locking. Once the steady-state is reached the system is considered to be phase-locked.
Stability of the phase-locked state can be evaluated by following the treatm ent outlined for
the breather solution of a driven sine-Gordon system [31].
The basis for the phase-locked state is understood through the following energy
analysis, found in Ref. [35], The driven and damped system in steady-state can be described
as a state where the driving power is completely dissipated by damping. Over the course
of a signal period the rate of energy going in must equal that exiting
J rZ7r/us
f 2 ir/bJs
AH —
1
Hdt = 0 .
(2.43)
o
For the Josephson junction system, combining this with equations (2.30) and (2.31) gives
8
rrZ
2T
iT
x/L
j uO
js
rZ
27
Tr/ujs
s
l*
T/LOs
/
(p sin u stdt — a
ip2dt = —esA7rs in 0 —airA 2 ujs = 0.
Jo
Jo
(2.44)
Using Eq. (2.40), the phase-locked state can then be characterized
aAu>,
sin 0 — -------- -.
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(2.45)
A perturbation to a phase-locked state may introduce slow modulation frequencies
to the dynamics. To illustrate we introduce a slow modulation frequency in both phase
and amplitude of the phase-locked state. The locked phase is given in terms of averaged
and dynamic parts 9 = 9q + 89 with |<50| <C
1
and 89 -C ws. Thus 89 carries the time-
dependent portion of the frequency response, adding an extra degree of freedom to represent
the predisposition for the system to match the driving frequency such that ujs + 89 =>■ cor .
In this approximation, we write the change in total energy over one driving period
as
. TT
=
d H dA .
2-7T -
2-7T
los
u)s a A ou>r
2tt d H dA ■■
A
~
— j j = — - w - r o — UJr = — - £ r r «— M = - e s A w s m (6>0 + 89) - oltxA^Ws
ujs
o A oujr
= —esAir (sin 9q + 89 cos 9q) — airA 2 u>s = —esAir89 cos 9q
2 dH dA ■
■
89
u)s dA dujT
- A \ j e s 2 — (aAu;s)289,
(2.46)
where we have made liberal use of Eq. (2.45). The derivatives can be expressed
dH
dA
~
A 2 1 + 1 A du)r
r
2 u)r dA
m
(247)
Equation (2.46) leads to the conclusion that a slow modulation frequency
r
can be ex­
pressed
Mr = ~ w aA y /e s 2 ~ (a A u js)2~
,
(2.48)
where A and es are related through Eq. (2.40) and A is related to cur through Eq. (2.35). It
should also be noted that dcor/d A < 0, so that Eq. (2.48) is a real and positive value. This
modulation frequency Q,r exists only for signal amplitude ss larger than a threshold value
aAojs. Thus es —* 0 leads to Qr —>0.
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2 .5
S w itc h in g d is tr ib u t io n b a s ic s
An important procedural element to both the numerical simulations and exper­
imental work presented here is the production of switching distributions. In general, a
switching distribution is developed by watching for events from a process which has some
random or stochastic stimulating element. After an event (or some defined limiting result e.g., exceeding a given time duration) the relevant initial and final conditions are recorded.
Each of these cycles is called a trial, and each trial is repeated to collect enough events to
provide suitable statistics. The results are then combined into a histogram. The shape of
the resulting image can often reveal the underlying physical basis which governs the process.
An example of one recipe we used in our study of Josephson systems involved
(for each trial) initializing the system in the zero-voltage, steady-state with an applied
microwave signal. Here thermal excitation fills the requisite stochastic role. The amplitude
of the bias current is then raised (or swept) very slowly (adiabatically) until the phasedifference escapes from the tilted-washboard potential and a non-zero voltage ((<p) /
0
) is
observed. The bias current at the moment of escape is recorded. The resulting histogram
of these events reveals a switching distribution with two or more peaks. One peak at the
bias current which distorts the potential beyond the minimum tenable well-depth. The
other peaks occur at the bias current which correspond to the potential well with the
system resonant frequency and related harmonics. This distribution is called a multi-peak
switching distribution and is detailed in the next chapter (Chapter 3).
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2 .6
Q u a n tu m tr a n s it io n t e m p e r a t u r e
References [1,47] demonstrate and define, respectively, the reference point known
as the quantum transition temperature (also called the quantum crossover temperature.
This mark is considered (from the quantum viewpoint) to be the point at which [for de­
creasing temperature] the probability of escape by means of thermal fluctuations becomes
significantly reduced, relative to th at for escape by means of quantum tunneling. Note
that there is some slight nuance to usage of the term “quantum crossover temperature” ,
as the word “crossover” denotes a sudden change in character of the system at a specific
coordinate.
A theoretical formulation of the quantum transition temperature in a dissipative
system is found in Ref. [48] and was there shown to be estimated by
h r
* ^lix
(1 + a 2 f , 2 - a
.
(2.49)
This expression can be further refined by applying the resonance frequency at the bias point,
uq —> 1 —r/21/4. For (a —>0), this expression approaches Eq. (1.1). Since ojq is the unbiased
linear resonance frequency, the accuracy of this expression can be improved by including
the effects of biasing on the potential through the substitution of cuq —» up2 = \ / l — rf2.
The quantum transition temperature can be found experimentally using a proce­
dure similar to that mentioned in the previous section. Reference [1] produced the distri­
butions illustrated in Fig. 2.7(a) (Note: The inset diagram in panel (a) was used by [1]
to demonstrate hysteresis in the junction system.) In this diagram, I is the applied bias
current which was swept slowly until an escape was recorded. From these distributions
the standard deviations of the distribution widths AI = (/ ( / —{/)) 2 j\ X/ 2 were calculated in
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24
35
30
25
I
1 mV
•
1.62/iA measured PO) width
*
162nA measured Pfl) width
—
MQT theory, no damping
MQT theory, with damping
20
15
10
5
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0
(b)
Figure 2.7: Experimental data for quantum transition temperature T* (from Ref. [1]). Panel
(a) shows the switching distribution. Panel (b) indicates T*. D ata for two junctions are
presented, with I c — 1.62 fiA and I c = 162 nA, respectively.
Fig. 2.7(b). From this latter figure, the quantum transition temperature can be identified
as the point on the curve for which the slope approaches zero.
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C hapter 3
M ulti-peak sw itching distributions
This chapter describes the work performed in Ref. [34] as it pertains to this dis­
sertation. Two separate efforts were coordinated and combined in th at work. One effort
was a series of numerical simulations, the other was experimental in nature. The emphasis
here is on the numerical results and related theory. References [2,33,49] contain earlier
developments in the group’s work and are also reflected here.
3 .1
O v e r v ie w
References [9,10] proposed and tested a theory regarding the driving of a Joseph-
son junction system with a constant source of microwave radiation which stimulates the
discrete energy levels in the system. These energy levels are functions of the shape of the
static potential well, which is formed by the phase difference <p between the two junction
superconductors separated by a thin dielectric barrier and biased with an constant force.
In this quantum mechanical view (Fig. 1.1(b)) the energy levels each have different
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tunneling probabilities and thus, on a statistical basis, the aggregated escape events from
the potential should reflect this. Also, the low temperature at which these experiments were
conducted ensures that k s T <C foujQ, so that classical modes of escape can be eliminated
from consideration.
Other experiments documenting various aspects of this effect include Refs. [50-57],
Of these, Ref. [54] is notable for their observation of the multi-peak switching distribution
at temperatures higher the quantum transition temperature T*.
The classical view for a Josephson junction circuit employs the Kramers one­
dimensional model for thermally-activated escape from a potential well [58]. Escape from
the well corresponds to (an abrupt) transition from a zero-voltage state to a non-zero voltage
state (Fig.
. (a)). Statistics of these switching events have been shown to be consistent
1 1
with the Kramers model. In addition, continuous application of a microwave signal triggers
a series of resonances as the bias current is slowly (adiabatically) swept through a range of
values. Thus one observes the appearance of sharp peaks in escape probability as a function
of dc (or constant) bias current.
3 .2
T w o -ju n c tio n in te r fe r o m e te r (S Q U I D )
We consider two identical Josephson junctions coupled by a superconducting loop
as sketched in Fig. 3.1. The equations describing this system are written
(pi + aapi + siny>i
=
rj\ + \/2ni(t)
(f>2 + atp2 + sin if 2
= r]2 T a/2n 2 (t).
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(3.1)
(3.2)
27
®(t)
Figure 3.1: Interferometer schematic.
Here an exception to our standard normalization has been used for the characteristic energy,
which is twice that described for the noise terms in Chapter 2. The dc bias is represented
as depicted in the figure as rji — I i / I c (i = 1,2).
At this point we have two goals: one is to recast the equations in terms of the flux
in the loop; the other is to explore the possibility of a formulation that eliminates a degree
of freedom. We start by re-defining the two phase difference terms p — | (pi + p 2) and
ip = | (pi —P2 ) and then we can express Eqs. (3.1) and (3.2) as
p + a p + cos ip sin p
=
ip + aip + cos p sin ip =
where
77
=
i
(rji
+
7 7 2 ) ,
r]' = A
(7 7 1
-
7 7 2 ) ,
n =
77
+
n(t)
(3-3)
r}' + n'(i),
(ni + n 2), and n’ —
(3.4)
(n\ - n2) (note the
dissipation-fluctuation relationships still hold).
Next, continuity in the total phase difference within the current loop is used to
establish the relation between ip and the total flux through the loop $ (see Ref. [42])
4>
2ip = 2ttp — 2tt— = 2np ~ 2tv—
4*0
d’o
fiiprj
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(3.5)
where p is an integer and 4>e is the externally imposed flux. The normalized loop inductance
is P i = 2 ttLIc/To, where L is the loop inductance in physical units. We use M = $ e/$o
to denote the fraction of external flux relative to the flux quantum To. Using these substi­
tutions, Eq. (3.4) can be more conveniently recast
2
ip + a i p + cosy? sin = —— Up + n (M ( t ) —p)} + n'{t).
(3.6)
PL
Here we consider only cases for p = 0 and 0 < M < | , which (due to periodicity), do
not restrict the generality of the results. Thus we have met the first goal to express the
equations as driven by an external flux through the superconducting loop.
To achieve a single degree of freedom for the coupled Eqs. (3.3) and (3.6), we
follow Ref. [49] which improves upon a suggestion by Refs. [50,55] for the case of small loop
inductance Pi. This analysis begins by expressing the static part of Eq. (3.6)
^ cos (p sin ip + ip =
=>•
=
tM
(3.7)
—7
ip
—ttM — ~ cospsmipo
ipo - ^-cospsin-ipo,
(3-8)
(3.9)
where ipo is the time-averaged value of ip for small oscillations. This expression is correct to
first order in Pi. Inserting this into Eq. (3.3) yields an equation for the two-junction loop
with a single degree of freedom
(p + atp
+
cosipo sin tp
+
^ sin2 ipo sin(2y>) —rj
+
n(t).
(3.10)
The linear resonance frequency can be found using a similar approach as shown
in Chapter 2. Using the ansatz p = <po + <5, the characteristic equation for Eq. (3.10) is
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written
5 4~ cos ipo cos (po +
Z
sin 2 ipo cos(2 ipo)
where £ is a constant (which is arbitrary to the homogeneous solution.) It can be seen that
in the limits |^o|
1
, a -C 1 , and
0
i <C 1 , the linear resonance is equivalent to that for
the single junction circuit
cui2 = y/1 - rj2.
(3.12)
Using these same limits, a driving term can be added to the standard pendulum
equation
(p + costpo sin 9? = 7? 4- £s smu>st.
(3.13)
The bias current at linear resonance can then be expressed
Vres = V cOS2 Ipo - UJS4.
(3.14)
Similarly, a nonlinear resonance analysis which includes contributions from 0 l can
be performed starting with Eq. (26) in Ref. [49]
u>r 2 — cos ipo cos(ip±) + -j3i sin 2 ipo cos(2y?±),
(3.15)
where ip± represents the phase-difference at the fixpoints in the potential well. From this,
cos <p_ (the fixpoint at the local minimum) can be found
■cos ipo + y jcos2 ipo + 4/3l Sin2 ipo (ujs2 + \(3l sin2 ipo)
cos <p_ = ----------------------------—— 2 f------------------------------ .
2f3L sin 2 ip0
(3.16)
Now, a nonlinear version of rjres can be found from substitution of the fixpoint phasedifference into Eq. (3.10) with the requisite driving term
rjres
= ^cos ipo +
sin 2 Ipo COS ( p _ ^ a /1 - cos2 (p_.
In the limit /3l —> 0, Eq. (3.17) becomes Eq. (3.14).
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(3-17)
30
aii
es
O^O.SO
• b r)a= 0 .O O 1 8
fl* = 0 .5 5
iWV r)d—0 0 0 4 0
n4=0.60
..;.ht.I\ rla= o .o o a o
j
f -
na=Q.65
'a'
1) 4 = 0
.a t
U A- 0 . 7 0
:j..... !...
i
r)4=0.0ZOO
na=a rs
rj^OOZBQ
n4=0, bo
a
1 )4 = 0 .0 3 7 0
ii.................. h i
OS
0 .7
0 .8
0 .9
30
1
V
Figure 3.2: Multi-peak switching distribution as a function of driving frequency. Markers
represent numerical integration results; dashed curve indicates Eq. (3.14); solid line from
Ref. [2]. Parameters were: I c = 11.05 /J.A, u a — 40 n s-1 , Pl = 0.07, a = 0.01, r] ~
10-8 , and T = 370 m K , or 0 = 1.4 x 10-3 . [34]
3 .3
R e s u lt s
Two sets of results are portrayed below. The first set of data presented repre­
sents a set of simulations which were conducted by initiating the system in a zero-voltage
state for bias current rj well below bias values of significant switching-probability for the
given temperature. In addition, a driving current was applied with constant amplitude
rjd and frequency
W ith no external flux present, the bias current r/ was then contin­
uously increased at a very low rate r/ until the system switched into a non-zero voltage
state. The bias current, for which the system switched, was recorded as a contribution
to a switching-probability distribution P(rj). For the simulations, we used a characteristic
damping parameter of a = 0.01. The temperature shown was 370 m K or © — 1.4 x 10-3
in normalized units. The markers in the top panel of Fig. 3.2 demonstrate the multi-peak
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switching distribution as the result of numerical integration of the system of equations (3.3)
and (3.6). Here rjd and Qrj represent the driving amplitude and frequency of the driving flux
respectively, flp is used to denote the linear resonance frequency w;, and p is the count of
escape events. The dashed curve indicates the relationship in Eq. (3.14) where the driving
frequency matches the linear resonance frequency. The solid line reflects the anharmonic
analysis outlined in Ref. [2]. The rightmost or primary peak in the lower, stacked plots
indicates the inevitable escape due to the flattening of the well for very high p. The left or
secondary peak follows the resonance predicted by Eq. (3.12).
In Ref. [34] additional figures of the type shown by Fig. 3.2 are presented for
comparison of temperature effects on the peaks in the distribution. The key effect is a
widening of the peaks. This widening can be attributed to the fact th at greater temperatureinduced fluctuations have the effect of increasing the probability of escape in the resonant
region.
The second set of data, shown in Fig. 3.3, pertain to a series of numerical simula­
tions combined with experimental results. The parameters used were similar to those used
in the first set, differing in that, instead of a driving current, a varying external flux was
applied with amplitude rjd and frequency Qd- The temperature used was the same as the
first data set and it should be noted that, in the experimental context, it was 70m K higher
than the device quantum transition temperature T*.
Two different methods were used in conducting the simulations corresponding
to the two types of open markers, circles and squares.
The open circles in the figure
correspond to simulations performed using the two-equation system. The open squares
represent simulations of the single-equation system Eq. (3.10). The overlap of the open
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32
0.5
Ud=0.56
(°) 0.5
0.4
0.4
■
6
£
0.3
&
0.2
0.4
V
0.8
0.8
•
N.
■
0.2
■
•K
■\
•J
0.1
0.2
0.3
\
•N .
0.0
0.0
0 .605
_____
0.0
0 .0
1.0
0.2
0.4
)
\
0.8
• G>\
■ \
,• J
0.8
1.0
0.6
0.8
0.1
1
(b) 0.5
(d) 0.5
0.4
0.4
0.3
&
a
0.2
0.1
0.0
0.0
0.2
0.4
0.8
1.0
0.0
0 .2
0.4
1 .0
V
Figure 3.3: Comparison of simulations to experiments. Open markers correspond to sim­
ulations; filled markers represent experimental results; dashed curves represent Eq. (3.12);
solid curves reflect Eq. 3.17. [34]
circles and open squares verifies the accuracy of the approximations used in developing the
single-degree-of- freedom equation.
The filled markers represent the experimental results with filled squares having
been obtained for negative flux and filled circles for positive flux. The close alignment of
these symbols indicates the symmetry of fabrication of the apparatus. The dashed curves
represent the solution of Eq. (3.12) for the maximum critical current of the interferometer
from the static solution of the system of equations (3.3) and (3.6). The solid curves reflect
the analysis in Eq. 3.17.
Prom Fig. 3.3, we note th at for increasing values of driving frequency the exper­
imental data tend to move apart from the numerical simulations and from the analytical
curves. The explanation of this effect may be related to sub harmonic generation in the
interferometer. However, it is also possible that systematic configurational irregularities
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were responsible for generating spurious resonances.
3 .4
C o n c lu s io n s
The simulations and the data reported in this chapter have shown that, well above
the classical to quantum transition temperature, a two junction interferometer can display
evidence of multiple states on the top of the Josephson super-current induced by external
ac signals. In zero-applied flux, we have shown that the frequency response of the inter­
ferometer is very similar th at expected from single junction measurements. Also, the data
indicate that the numerical simulations of the full interferometer equations provide results
that are close to the observed experimental behavior and are in good agreement with the
low-/?/, approximation of the potential.
We have shown that, for the interferometer, the spectrum of harmonic and sub­
harmonic excitations generated by resonant modes must be considered when investigating
the existence of quantum levels. These aspects of our investigations could be relevant for
studies of quantum coherence and computing and we have made an effort to relate our work
to experiments and theoretical modelling in the literature.
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C hapter 4
R ab i-typ e oscillations
The following chapter is an edited version of our work which appeared in Ref. [36].
The original introduction has been reworked and adapted into the thesis introduction.
Some of the theoretical developments have been included in Chapter 2. The figure captions
have been folded into the text. Portions of the conclusion have been incorporated into the
concluding chapter. Also note that a more basic and detailed derivation of the Rabi-type
modulation frequency has been included in Appendix B.
4 .1
O v e r v ie w
The research described in this chapter examined the phenomena of Rabi oscilla­
tions as reported by several experimental groups in Refs. [3,11,15,17-23,27]. In general,
the “programme” for these reports encompasses stimulation of Josephson junction system
in a zero-voltage steady state with an ac (or microwave) signal ({<£>) = 0 ) . At a given time, a
probe of the system is conducted with either another ac signal or a dc probe pulse. Escapes
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to a non-zero-voltage state (or to another well) are recorded and plotted as a percentage of
the total number of trials. Oscillation in the escape statistics as a function of the duration
of the ac signal are observed and denoted Rabi oscillations - according to the notion that
this mimics a type of atomic-transition effect.
We selected the work done in Ref. [3] to examine from a classical viewpoint. The
primary reasons for using this work included a simple functional apparatus, clear step-bystep instructions for reproducing the work with full description of all the parameters, and
clearly presented results including a quantitative depiction of the variations imposed on the
microwave signal amplitude ea. The following section details the theoretical model used,
including a perturbation theory which encompasses the modulation frequency responsible
for generating the Rabi oscillations in the classical mode - denoted Rabi-type oscillations.
The results of our simulations are then presented, followed by some conclusions.
4 .2
C la s s ic a l m o d e l
Prom Chapter 2, the normalized classical equation for a Josephson junction can
be written
<p + aip + sin ip =
t? + £s(t)sin(ujst + 9S) + ep(t) + n(t) ,
(4.1)
where we have added another signaling term ep as a system probe.
Using the model described in Chapter 2, we develop analysis and conduct sim­
ulations according to experiments reported in Ref. [3]. System parameters, such as
,
77
a, u>s, £s(t), and £p(t) are matched as closely as possible to reported values, and Rabitype oscillations are observed statistically through simulations of the distribution of probe
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36
I/IC
microwave pulse width, Ms
t
Figure 4.1: Signaling protocol schematic.
field induced switching from the zero-voltage state as a function of applied microwave field
es(t) sin(u;s£ + 8 S). Microwave and probe perturbations have the form sketched in Fig. 4.1.
We assume the starting phase 0S of the microwave field is random for each switching event.
Notice the difference in probe field between this presentation (along with Ref. [3])
compared to Refs. [18,23,33], where the probe was a microwave field with frequency slightly
lower than cos. In the experiments which use a microwave probe, the varying nature of the
probe can stimulate oscillating modes through the multi-valued nature of the response
shown in Fig. 2.6. But, this stimulation only occurs if the signal frequency is not equal to
the linear resonance frequency.
4 .3
P e r t u r b a t io n a n a ly s is
For T = sp{t) = 0 we propose an ansatz which uses a modified form of Eq. (2.31)
P — (po + Stpo) + (A + <5A) sin (ust +
6
+ 56) ,
(4.2)
which separates the phase-difference response into steady state and small time dependent
deviations, where |&4| <C 1, |<50| <C 1, and |<5y?o| <C 1.
Inserting Eq. (4.2) into (4.1) for ep{t) = T = 0 and es(t) — @(t)£s (0 being
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Heaviside’s step function), yields the steady state phase-locked relationships [33,35]
(4.3)
tan(d —9S) =
aui.
(4.4)
and the linearized expressions for the small deviations from steady state:
SipQ+ aSip0 + Jo (A) cos<po S(po — J\ {A) sin (po SA
(4.5)
SA + a{SA —(jjsA59) —2ujsAS9 + [Jq(A) —J 2 (A) —
cos <poSA =
2
J\{A) sin <po Spo
(4.6)
AS9 + a{AS9 A u s5A)+ 2 u sSA+ \_ J o { A )J i{A ) —
cos (poA S9 = 0 .
(4.7)
Equation (4.5) represents the small amplitude, slowly varying terms in (4.1), while (4.6)
and (4.7) represent small amplitude terms oscillating with
uj s
Evaluating a solution to
.
Eqs. (4.5)-(4.7) is not simple, but making the assumption th a t all three variables oscillate
slowly with the same frequency
Q, r
and decay with the same attenuation
we can write
0.
the simple relationships: SA = exp(iflt), AS9 = kSA, and Spo = 7&4, with 12 = 12# -f i0.
This results in the following polynomial
126 - 3 ia l2 5
15
-a —a 4
ia [wsa + rx]
12
124
—ia
— a 2 urs cos p 0 +
-a T 2a4
^2
15 4 3 2
— a — —a a 4 + a 2 n2
16
2
,
(4.8)
where the parameters a* and E, are given in Eqs. (4.10)-(4.14) below. The polynomial
can be simplified considerably by inserting
12
=
12
j? + if.3, and realizing th at
imaginary part of the roots (for all roots) with a non-zero real part
12
0
=
is the
#; i.e., for underdamped
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modulations. The notion of a dissipative system (a > 0) naturally implies
that
-0 +
for a non-zero threshold value of the microwave amplitude £s > 0 [31,32],
W ith 0 —
the (real) modulation frequency fIr is determined by the following real polynomial:
+ Cl4 fiR + fX2^H T
— o,
(4-9)
where
a4
—
a2
=
3
— 2Wj —(2 Jo{A) + 1) cos (fo
(4-10)
+ o?{^aA + J s2) + Ti
(4.11)
ao = ~ ^ a6 IT
+ a 2 ( (j - u J s2 cos<p0) - T2
(4.12)
= (Wj —H>r) (ujg + U>2 — 2Jq(A) COS <Po) + 2( Jo(A) COS ipo+ Wg) cos <po
—2 J 2 {A) sin2<po
(4-13)
r 2 = (u>2 - (D2) [(«j + ui2 — 2Jq(A) costpo) costpo + 2J 2 (,4) sin2ipo] •
(4.14)
The steady-state resonance frequency ojr is given by (2.35)
(4 ,5 )
Generally speaking, the three roots flR are real and positive. One is located near (2uj3)2, one
near lo2, and one much smaller than uo2. It is the latter we are interested in when studying
transients and modulations to phase-locking, and it can be very well approximated by either
of the following approximations, since the normalized frequency is small:
n2
R
*
nR
2 «
( 4 . 16 )
a2
02
2a4
4a°^ i.
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(4.i7)
A further simplification to the coefficients in the polynomial can be produced by
noticing that the damping parameter a in relevant Josephson experiments is very small,
allowing for omission of several terms ofhighorder in a in Eqs. (4.10)-(4.12).
limit
of the above solution can bederived for smalloscillation
amplitudes for a = 0 and
A specific
(A) and microwave (es)
= 1 —rj1
nR a
J E
(4.18)
for 4 ^ 0
=
=
4
16ul
(2 -
, for €. -
(4.19)
0
.
(4 .20 )
o; 4)3
The resonant choice w, s=s 1 —rj2 is consistent with the experimental studies [3,18]. However,
if ujg -f- 1 — rj2, then A <x es, and the relationship between Q2R and ej. becomes linear for
small values of A and ss. Further, if w] < 1
rj2, A becomes a multi-valued function of
es, corresponding to several energy states (but not quantized states) of the system for the
same parameter values as seen in Fig. 2.6.
The solution to Eq. (4.9) (which can be given in closed-form as a solution to a
third order polynomial) is both a complementary and more convenient approach to the one
presented in Ref. [35]. It provides the important additional information regarding the atten­
uation (3 of the transient modulations and it provides explicit expressions for the modulation
frequency as a function of all system parameters. The two different perturbation methods
agree qualitatively, and are quantitatively very similar. The main difference between the
two approaches is th at we have here considered phase 0 and amplitude A as harmonically
varying variables, instead of using the energy balance approach to transients [31], which
treats the total energy of the system as the linearized dynamical variable. Notice th at the
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linearized Eqs. (4.5)-(4.7) provide information about frequency, attenuation, and internal
phase-relationships, but not about the magnitude of the deviations. However, the mag­
nitude can be quantitatively estimated from the following reasoning: For a simulation, in
which a Josephson junction is described by Eq. (4.1) with ep(t) —T = 0 (or at least very low
temperature) and es(t) = &(t)es, the system is at rest <p — ipo = sin-1 rj for t < 0 (A = 0).
The onset of the microwave field at t = 0 therefore, within the harmonic approximation,
produces an envelope function of oscillation given by
(
0
, t <0
(4.21)
A = A + SA
A (l —e~Pf cos
, t > 0
The two other modulated variables, 5(po = 'ySA and A50 = k 5A, can then be found from the
coefficients
=
( 4 ,2 2 )
—ta il —cos<po
K =
02 — ta
■ il — J q(A) costpo + u>g2sV
(4-23)
The modulated system energy (for sp(t) = 0) can be calculated from Eq. (2.41) as
(H) 2n
= H + SH,
(4.24)
where H is the steady state energy of the phase-locked state in Eqs. (4.3) and (4.4), and SH
is the transient modulation. The average phase <po can be calculated either as a function
of A = A + SA as given by Eqs. (2.32) and (4.21), or it can be extracted from the theory
through Eq.(4.22). The
two results are very similar, but sincethe latter
approximation inthe perturbation theory, it
has a second-ordererror
is a first order
at t — 0. Using
Eq. (2.32), ipo(t) satisfies the true value both at t = 0 and for t —* oo.
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(a) --- %W±4(r)
.... %(t)
1.6
A
.n
14
>
^
u
W
/V V V V W '
,, ( /V V v w ^
W
L
. _
i.i
(h)
A
l\ A
tt?
3
0.00
iM /y v
w
1/ iV ■ v. 1 I I . j
0
1000
W
1
v w
i it
(d)
,
,
.
A
A /V W '
/ \-IvA. . 1 . I ■ 1 ■
I I I/—
2000
t
(C)
0
1000
2000
t
Figure 4.2: Amplitude and energy response. Panels (a,b) show results of the perturbation
theory Eqs. (2.32), (2.35), (2.41), and (2.42). Panels (c,d) show results of direct simulations
of Eq. (4.1) for the same parameters as (a and b). Parameters were: a = 0.00151477,
7] = y / l - v j = 0.904706, es = 0.00108, and T - £p(t) = 0.
4 .4
S im u la tio n o f t r a n s ie n t s
Numerical validation of the expressions for the transient modulation can be di­
rectly acquired from simulating Eq. (4.1) for £p{t) = T = 0 and £s(t) = es©(t) for different
values of w, = 1 —r f and a. Simulations were conducted by choosing rj < 1 and a > 0, then
initiating the system in the static state (y>(0),</?(0)) = (sin"1'/], 0). At t = 0, the microwave
field switches on, and we measure the phase
the average phase <po(t) = ( ^ ( t ) ) ^ ,
u> s
and the system energy H(t) as given by Eq. (2.30). Figure 4.2 shows a typical compar­
ison between the analysis of the previous section and direct numerical simulations. We
have chosen system parameters a = 0.00151477, r] = 0.904706, es = 0.00108 ■&(t), and
u>s = \ / l —r f = 0.6527147 (inspired by Ref. [3]). We can clearly see that the transient
in phase and amplitude is a very slow modulation, and th at this modulation provides a
slow transient oscillation in the system energy. Notice that the phase-locked frequency
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42
0.2
0.0
1000
2000
1000
2000
1000
2000
- 0.2
0.02
-
0.02
0.002
0.000
Figure 4.3: Linear transient responses to microwave onset as calculated from Eqs. (4.21)(4.23) for the parameter values of Fig. 4.2.
is much higher than the depicted modulation, Hr -C ujs. We have shown results of the
theory on the left (Figs. 4.2(a,b)) and the corresponding simulation results on the right
(Figs. 4.2(c,d)). The quantitative agreement between the results is obvious. We do see a
slightly larger predicted oscillation amplitude compared to the simulated. We also notice
a slightly higher predicted modulation frequency than what is observed in the simulations.
Despite these slight discrepancies, which are usually about 5% and no larger than about
10%, we consider this remarkable agreement given the simple monochromatic trial function
and the linearization in the perturbation treatment. Figure 4.3 shows the details of the per­
turbation variables as calculated from the theory, and they reveal th at the actual oscillation
frequency uj — ujs + 56 is only insignificantly different from ujs. Direct comparisons between
the predicted modulation H and the simulated values as a function of microwave amplitude
for different values of rj and a are summarized in Figs. 4.4 and 4.5. Comparisons for H r is
provided in Fig. 4.4 and the attenuation (3 is validated in Fig. 4.5. The general agreement
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it
asymptotic form
iff
Iff■2
Iff■3
Iff■4
ti - 0.94259
it - 0.10000
£
Figure 4.4: Normalized modulation frequency fin (at T = 0 ) as a function of sig­
nal amplitude es. Panel (a) shows a range of damping a values, with r) = 0.94259
(los = \ / l —r/ 2 = 0.577886). Gray line with “star” markers indicates the asymptotic form
given in Eq. (4.20). Arrows indicate the frequency for which small oscillations would be
overdamped for respective values of a. Panel (b) shows a range of dc bias r] values, with
a = 0.001. Curves represent solutions to Eq. (4.9). Markers represent data from numerical
simulations of Eq. (4.1) with ep(t) = T — 0 .
is very good for all parameters tested. We observe the simple relationship /3 = a /2 for
all oscillating solutions and the general trend of the polynomial (Eq. (4.8) or (4.9)) is to
provide a modulation frequency slightly larger than what is observed in simulations. This
is quantitatively similar to the values obtained from the energy balance perturbation the­
ory developed in Ref. [35], where the true modulation frequency consistently was slightly
under-estimated. We finally observe the broad applicability of the asymptotic value of the
modulation frequency given in Eq. (4.20).
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0.005
o " j 7 - 0.942590, a
0.004
o
7 - 0.9 4 2 5 9 0 a -- 0.00400000
7 - 0.904706, a = 0.00151477
7 = 0.942590, a = 0.00100000
0.003
0.002
0.001
mx> o
0.000
0.00
0.01
0.02
Figure 4.5: Characteristic attenuation P as a function of damping a and signal amplitude
es for T = 0. Lines represent the derived relationship p = a / 2 and markers represent data
from the simulations of Fig. 4.4.
4 .5
R a b i- t y p e o s c illa t io n s
As was argued in Ref. [35], the Rabi oscillations observed in Refs. [3,18,23] may
be closely related to the classical transient modulations described above. The numerical
simulations in Ref. [35] were conducted in close agreement with the procedures described
in Ref. [18], by applying the temperature T ~ 50m K (k T / H j ~ 10- 2 < 1) to a Josephson
junction, which is perturbed by a microwave pulse starting at t —
0
with a frequency
UJS = V T — r/2. Varying either the microwave amplitude or duration may yield Rabi-type
oscillations in switching probability when a subsequent microwave pulse with a smaller
frequency top < los is applied to probe the energy state of the junction as it is left by the
first microwave pulse. The frequency of the second (probe) field was chosen to match an
anharmonic amplitude th at could excite the junction beyond the energy saddle point and
lead to escape from the well. The measurements were conducted for varying microwave
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amplitude and fixed microwave duration.
The experiments described in Ref. [3] were conducted slightly differently. First, the
Josephson system was a small inductance
interferometer, which we can approximate with
a single Josephson junction, since the dynamics for small /3l is known to be well represented
by a single degree of freedom [49]. Second, the probe pulse was in this case a dc pulse as
sketched in Fig. 4.1, and the measurements are presented for fixed microwave amplitude and
variable duration. We provide simulations of the classical system described in Eq. (4.1) and
parameterized by information provided in Ref. [3]. Characteristic current and frequency
were Ic = 6.056 fj,A and ljq « 110 • lO9.?^1, which lead to a normalized temperature of
k T /H j ~
2
• 1 0 - 4 (T = 30 m K ), and a normalized microwave frequency uis — 0.6527147,
which is close to the resonance such that rj ~ yj 1 —to4. We have estimated the dissipation
parameter a from the reported decay of Rabi-oscillations, a = 0.00151477. The normalized
applied microwave amplitude es is varied between
0
and 0 .0 1 , and the normalized duration
is in the range [0;3000]. Simulations were conducted with a probe pulse £p(t) as shown
in Fig. 4.1, followed by a short time in which we determine whether or not the junction
has switched from the zero-voltage state. For every specific set of parameters, this type
of simulation is conducted 25,000 times, each with a randomly chosen value of microwave
phase 6 S, and the switching probability is recorded, before a new microwave duration is
chosen with the system reset in the zero-voltage state. Typical results are displayed in
Fig. 4.6(a), where the switching probability P, reported as population occupancy of excited
quantum state in Ref. [3], is shown as a function of microwave duration. We clearly observe
the Rabi-type oscillations of this classical system with a distinct frequency. Moreover, the
oscillation frequency is in near perfect agreement with the reported comparable figure of the
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46
^ 0.010
V 0.000
O
^
25.2 25.4
0 .0 1 0
0.005
25.2 25.4
0 .0 0 0
o 0.010
0.007
I
0.006
(4
0.005
25.2 25.4
0 5 1015202530354045
time (ns)
Figure 4.6: Simulation study of Rabi-type oscillations. Panel (a) depicts escape probability
P as a function of signal duration Ats. Numerically simulated quantities are represented by
dots (each of which represents 25,000 escape events.) Panel (b) depicts ensemble average of
normalized energy, simulated through Eq. (4.1), as a function of signal duration (ensemble
size N = 50,000, randomly chosen 6 S.) Parameters were: T = 30 mK, a = 0.00151477,
r] = V 1 - <4 = 0.904706, u>s = 2irv0 i/uj 0 = 0.652714, es = 0.00217, ev = 0.08474. These
parameters were inspired by those reported in Ref. [3]. Panel (c) Single trajectory of energy
versus time for parameters listed in (b). Panel (d) Single trajectory of energy versus time
for T = 0 mK. Panels (e-g) are expanded views of the time-averaged, zero-temperature,
and 30 mK energy curves, respectively.
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experimental measurements as well as the theoretical value
Hr
provided by the analysis in
the above section. Figure 4.6(b) shows the simulated system energy as an average of many
thermal realizations, each with a randomly chosen 6 S, of the trajectory (at T ~ 30m K ).
The different choices of 9S £ [—7r; vr] yield slightly different trajectories of SA(t), 59(t), and
5ipo(t), which in turn make possible switching a function of 9S. The importance of this
ensemble average can be viewed in Fig. 4.6(c), where a single trajectory of energy is shown
for the applied temperature of T = 30mK. After a short transient, the thermal effects
are dominating the individual trajectory, exciting the modulation frequency
Hr
at random
phase. Only after an appropriate average over trajectories do we observe the smoothly
decaying envelope of the modulated energy curve in Fig. 4.6(b), which in turn is very similar
to the (single) zero-temperature trajectory seen in Fig. 4.6(d). Parameters for this figure
group were as follows: a = 0.00151477, r] = 0.904706, u s — 2iti'oi/u)q = 0.652714, s s =
0.00217, and ep — 0.08474. Note the expanded view in panel (f) indicates the phase-locked
signal present in the energy response. Similarly, panel (g) shows the effect of temperature
fluctuations on the response.
The close connection between the transient energy modulation and the Rabi-type
modulation is obvious, and the close relationship between classical numerical simulations,
classical theory, and the experimental measurements is further emphasized in Fig. 4.7, where
the experimental data (open markers) of Ref. [3] are shown alongside theory (solid curve)
and simulations (closed markers). The three-way close agreement indicates that much about
these measurements can be understood from classical theory of driven, damped Josephson
junctions. Parameters a, rj, and T were the same as those for Fig. 4.6.
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woo
A n aly tical re su lt
750
O
S im u latio n d a ta
♦
E x p e rim en tal d a ta
500
0.002
0.004
0.006
0.008
0.010
Figure 4.7: Modulation frequency Qr as a function of signal amplitude £s for a =
0.00151477, rj = 0.904706, T = 30 mK. Lines represent calculations using Eq. (4.9), the
open markers represent statistical data from simulations, and the filled markers are the
measurements copied from Ref. [3].
4 .6
C o n c lu s io n s
We have presented a perturbation analysis of the classical, nonlinear model de­
scribing the microwave-driven Josephson junction. Our results show direct quantitative
analogy between the experimentally reported Rabi-oscillations in Ref. [3] and the classical
transients to phase-locking, and results are presented as specific functions of experimental
system parameters.
Qualitatively, the oscillating switching distributions we have simulated bear strong
resemblance to those shown in other experiments. One interesting facet of our results is the
nonlinear relationship between Rabi-type frequency and signal amplitude shown in our sim­
ulations and in several experiments [3,14,23,29], Several other studies have demonstrated
a linear relationship in these variables, including Refs. [17,21,24,27]. It is not entirely clear
why there is a difference (either from the quantum or classical viewpoint) - as it extends
across various types of apparatus. However, one must keep in mind that the model used
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here is that of a simple pendulum. The actual details of a real Josephson junction system
and all of the experimental components involved may play important roles in these effects.
Another consideration in this regard is that our selection of Ref. [3] was (partially) based
on their clear identification of the experimental methods and clear labelling of measured
quantities. This generally facilitated our work and specifically simplified the overlay process
in generating Fig. 4.7. The topic of linear vs. nonlinear Rabi-type frequency and driving
frequency relationships is considered further in Chapter 7.
The analysis in this chapter is both a complement and an extension to the analysis
of Ref. [35], where an energy balance perturbation approach was applied to produce a sim­
ilar connection between reports of Rabi-oscillations and the nonlinear classical Josephson
system. The extension provides direct closed-form solutions of a (Rabi-type) modulation
frequency as well as attenuation, and the resulting frequency agrees closely with the previ­
ously obtained result.
This consistency lends credibility to the value of a classical interpretation of the
reported Rabi-oscillations, since the quantum mechanical and classical intuition turns out
to be very similar (as is also the case in laser physics [59]).
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C hapter 5
Tem perature dependence in
R ab i-typ e oscillations
5 .1
O v e r v ie w
This chapter reports on the results described in [60], which was an important op­
portunity for further comparison of our results with experiment. In that work, we presented
experimental and computational results from a study of temperature dependence on the
frequency and amplitude of Rabi oscillations. Experiment and simulation show reasonable
agreement and indicate an exponential dependence for the decay of the oscillation ampli­
tude as temperature is increased. We also show the oscillations persisting up to 800 m K ,
although in the experimental case, the quantum crossover temperature is ~ 200 m K . Also,
the data obtained through the corresponding efforts are remarkably similar across the full
temperature range considered.
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5 .2
C la s s ic a l m o d e l
Figure 5.1: Single-junction loop schematic.
The normalized classical equation for the flux-driven single-junction loop (as shown
in Fig. 5.1) is written
Cp + a<p + sin
=
—1
— (<p - 2irM(t)) + n(t),
(5.1)
PL
where the normalization described in Chapter 2 applies.
The term M (t) represents the external magnetic flux in the loop and contains
components
M (t) = M dc + es(t) sin(o;sf + 9S) + ep(t),
(5.2)
where M dcrepresents the dc-bias flux and £s{t) and los represent the microwave signal ampli­
tude and frequency, respectively. A pulse for probing the state of the system is represented
by £p(t). These quantities are normalized to ToWe define the energy as
H
=
\ip 2 + 1 - cosiy? +
2
zpL
- 2irM)2.
A profile of the potential energy Hp is depicted in Fig. 5.2.
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(5.3)
Figure 5.2: Single-junction loop potential energy for: (a) Mdc = 0.32; (b) M ric = 0.50; (c)
Mdc = 0.68. Note that here p in Eq. (5.3) has been rescaled to <p* — (p —ir).
5 .3
S im u la tio n d e t a ils
We based the method and parameters for this study on documents received from
the requesting laboratory. The parameters used in integrating Eq. (5.1) were I c = 2.1 /i/1,
C = 0.4 pF, L = 580 pH, M dc = 0.780, es = 0.00135, u s = 0.706409, and ep = 0.0675.
The remaining parameter, characteristic damping a, was treated as a tuneable parameter.
By examining and comparing the time-decay depicted in the experiment, we chose a =
.
0 0 0 1
. Also note th at while the characteristic damping a is a function of temperature, the
experimentalists agreed that treating this as a constant would be a reasonable strategy.
The signalling used to generate the Rabi-type oscillations for this effort is the same
as that shown in Chapter 4, Fig. 4.1. Similar to the methods in that chapter, statistics
are gathered for the number of escape events triggered by this signalling at each given
temperature.
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1.0
800 mk
640 mk
400 mk
240 mk
.3
80 mk
0.6
40 mk
a,
?
X
I
a
0.4
0.01
0.0
0.2
0.4
0.6
0.8
Microwave pulse width (ns)
Figure 5.3: Simulation results: Temperature effects on escape probability. Panel (a) shows
our Rabi-type oscillations for I c = 2.1 p.A, C = 0.4 pF, L = 580 pH, M^c = 0.780, es =
0.00135, u s = 0.706409, and sp = 0.0675. Panel (b) indicates the relationship between
oscillation amplitude and temperature as measured from the first complete period in panel
(a). Line added as an aid to the eye. Note log scale on left axis.
5 .4
S im u la tio n r e s u lts
Figure 5.3(a) shows the observed Rabi-type oscillations at each temperature. Statis­
tics for each data point are based on 30,000 switch events. The creation of Fig. 5.3(b)
required a determination of the magnitude of the simulated oscillations from a succession
of plots of the form in panel (a). As a convention, we measured the difference in proba­
bility amplitude between the first complete peak and subsequent trough. The exponential
dependence of oscillation amplitude on temperature of our simulations is clearly observed.
5 .5
E x p e r im e n ta l r e s u lt s
Figure 5.4 depicts the temperature dependence as observed in the laboratory (note
the temperature of the traces is in reverse order of the legend.) Note that although the
quantum transition temperature for the device T* is 200 mK, Rabi oscillations are observed
at temperatures up to 800 m K .
While this is not necessarily antithetic to the notion
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2
1
2
■.. 1
4
*
6
i i 11
i
1
IS
Microwave pulse width (ns)
,i...... i
«
W
Figure 5.4: Experiment results: Temperature effects on escape probability. In this figure
the notation for the vertical axis, P (|l)), refers to the tunneling probability from the | 1 )
state. Reprinted with permission of Ref. [4].
of a quantum crossover temperature, the match between these experimental results and
our classical simulations, vanishing oscillations at a temperature of 800 m K astounded all
involved.
5 .6
C o n c lu s io n s
In agreement with the experiment, we see large Rabi-type oscillations at 40 mK
and vanishing oscillations at around 800 mK. This is significant as it reinforces the notions
from the previous chapters th at the oscillatory phenomena generated by Josephson junction
systems in the presence of microwave signals can be classically described. Also significant is
the fact th at the experiments show Rabi-type oscillations far above the quantum transition
temperature.
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C hapter 6
R am sey-typ e fringes and
spin-echo-type oscillations
6 .1
O v e r v ie w
The purpose of this chapter is to demonstrate the reproduction of Ramsey-type
fringe and spin-echo oscillations using the classical model when the recipe for experimental
measurements is followed. The analysis and results shown reflect work done for Ref. [38].
Here we illustrate the results with the simplest possible system, namely a single classical
Josephson junction, excited by a bias-current, microwave fields, probe fields, and thermal
noise. We develop a simple analytical expression for the observed frequency of oscillation
characteristic in these phenomena. Then we present a description of our methods and
describe the computational results from our model. Finally, we offer some conclusions from
our work.
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I/I
S p(t)
At
t
Figure 6.1: Signalling diagrams for simulating (a) Ramsey-type fringes and (b) spin-echotype oscillations.
6 .2
C la s s ic a l m o d e l
We start by recalling the equation of motion from Chapter 4
(p + aip + sin <p =
r} + £s(t)sin(Lost + Os) + £p(t) + n(t) ,
(6 . 1 )
To achieve Ramsey-type fringes we apply signalling as shown in Fig. 6.1. Starting from
an equilibrium condition, two microwave signals are applied in separate intervals, the first
from t = to =
0
to t = fi and another from
£4
to £5 . We denote the first interval A t a-
The second interval is labelled A te- These signals are defined to have the same duration
At,4 = A t e =
t r /4 .
Where t r is the period of Rabi-type oscillations 2 ir/ Qr from Chapter 4.
Also, note that the phase progression of the signal is continuous even while it is absent (i.e.,
in between the two pulses).
Spin-echo-type oscillations are generated through a similar method with the addi­
tion of an intervening pulse twice the duration of the others - i.e., a “7r-pulse” - noted as
AtR, between £ 2 and £3 , in Fig. 6 .1 . Again, the phase progression of the signal is continuous.
The quantum mechanical notion of a a7r / 2 -pulse” (and “7r-pulse”) is based on
nuclear magnetic resonance principles of rotating a precessing “atom” specific incrementally
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along prescribed axes, and parameterized to the Rabi frequency [19]. In the classical model,
the slowly modulating response of the system to the microwave signal pulse (as characterized
by fin, the progenitor of Rabi-type oscillations described in Chapter 4) over the period 7r/ 2
drives the system to the mid-point of the energy curve in its oscillation. (Obviously, a
r-pulse takes the system to the high point of the transient energy.) However none of this
7
is qualitatively significant in the classical view of the production of Ramsey-type fringes or
spin-echo-oscillations, since it is the phase-relationship between the signal pulses and the
evolution of the system during the signal’s absence which is the essential component in the
process.
In the case of Ramsey-type fringes, if we assume th at the oscillator is well described
by a phase-locked state with a transient modulation, we have (at t — ti) a sudden termination of the driving field, leaving the oscillator in a state with amplitude A(ti) = A + 5A(ti),
where A is the steady-state phase-locked amplitude. Since no driving is present, this am­
plitude decays due to dissipation, such that
A{ 1 - exp(—j t ) cos ClRt)
;t
G
[0;ti]
(6 .2)
A exp(—§ (t —fi))
; f e [ f i ; f 4]-
During this time, the [absent] signal advances its phase at a constant rate. We also recall
Eq. (2.35), the natural, anharmonic resonance frequency of the undriven, lightly damped
Josephson system
(6.3)
A simple reasoning of the origin of classical Ramsey-type fringes can now be for­
mulated.
We can show they are the result of the accumulated phase-slip between the
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microwave signal and the oscillator during the ballistic time interval t G ] 1 1 ; £4 [ . The
phase-slip is denoted 9p'
rt 4
9p =
Jti
(ws — tor)dt.
(6.4)
The Ramsey frequency for increasing At,i = (t 4 — t{) is therefore
nF =
6f
H ~ *1
(6 .5)
Notice th at the integral for evaluating 9p is not as simple as it looks at first glance.
The natural frequency is a nonlinear function of time through Eqs. (6.2) and (6.3). We
may be able to make some reasonable approximations, but the integral is straightforward to
calculate numerically. Also notice how the dissipation parameter a enters into the Ramseytype fringes as an important parameter for the change in amplitude, which in turn changes
a (low) anharmonic resonance frequency into a (high) linear resonance frequency. This is
in contrast to the expressions for Rabi-type oscillations in which (for the high-Q case) the
oscillation frequency is independent of a.
For extremely low (vanishing) dissipation, we may write
9f = {h - h ) (ws - tor (A (t4)))
(6 .6 )
= ws —uir (A (f4) ) .
(6-7)
A similar expression for spin-echo-type oscillation frequency is more problematic,
as the intervening 7r-pulse disturbs the ballistic path of the nonlinear oscillator in a complex
fashion. We have found through our simulations, however, that Eq. (6.5) still provides a
satisfactory description. We interpret this agreement to indicate that if A t# is small relative
to the interval A t j this disturbance is of minimal consequence. More work is needed to
determine the validity of this interpretation.
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6 .3
S im u la tio n d e ta ils
Following the procedure of reported experiments, we record switching from the
zero-voltage state ((tp) — 0 ) as a result of applying the external current sketched in 6 .1 .
We first equilibrate the system at a chosen temperature for a given value of rj. For
a randomly chosen, but temporally constant, phase 9S, a microwave pulse with frequency
ujs, which is associated with the natural resonance of the junction at bias point rj, is applied
for a duration of A t a- This duration is chosen to be A t a = Tr/4, where t r = 2 ir/Q r is
the modulation frequency period [35,36] for the microwave amplitude ss, as measured from
observed Rabi-type oscillations.
In the case of the Ramsey-type fringe experiment, another 7r / 2 -pulse, in phase
0
S with the first pulse, is applied at a later time with identical amplitude, frequency, and
duration (A t e = A t a)- Subsequently, a short pulse £p(t) is applied to probe whether the
system is in a high or low energy state after the sequential microwave pulse application. The
probe pulse is parameterized such that a state of relatively high system energy at t& results
in a likely escape event when £p(t) is activated, while a state of relatively low system energy
results in a low probability of escape when £p(t) is activated. This procedure is repeated
many times, each time with a new realization of 9S, in order to generate information about
the probability of switching for a given set of parameters, such as the time delay between
the two 7r/2-pulses.
Figure 6.2 illustrates typical examples of these events for 0 = 0. 7r / 2 -pulses are
applied in the intervals t £ [0 ; t\ —
and t e [£4 ; <5 = t4+ Ate] with A t a = A t e = 90.75
time units. This value is obtained from the corresponding Rabi-type oscillations reported
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n/ 2
n/ 2
n/ 2 n/ 2
Probe pulse
0
250 500 750 1000
0
t
250
I
Figure 6.2: Direct simulation of the Ramsey-type fringe switching response. Josephson
junction response to the sequential application of two n / 2 microwave pulses followed by a
probe field. Panels (a,b) show phase-difference and energy for a non-switching sequence in
the case of A t r{ = t^—ti = 400. Panels (c,d) indicate a switching event for A t ({ = t^—ti = 70.
Parameters were a = 1 0 ~4, tj — 0.904706, es = 2.17 x 1 0 ~3, uis — = \ / l — r f = 0.652714,
£p = 8 . 2 x 1 0 —2, and 0 = 0 .
in [36], After the second 7r / 2 -pulse, a short delay of A t =
20
is allowed before the probe
pulse sp(t) is initiated at t4. The probe has linear rise and fall times of 160 normalized time
units with a constant value interval of 275 time units.
Figures 6.2(a,b) show the response for A t j = 400. It is evident th a t the first n/2pulse elevates the system energy by leaving the system in a phase-locked state at t = t\ with
energy E(t{). At the onset of the second 7r / 2 -pulse, a significant phase-slip between the
oscillation of tp and the microwave has developed during the interval [fi; t^\, and the second
microwave pulse therefore decreases the energy of the Josephson system. The system is left
at t =
£5
with energy E{t§) < E (t\), which, when the probe field is applied shortly after <5 ,
results in only temporary energy increase while the probe is applied; this is not enough to
make the system switch into a non-zero voltage state, which can be seen from the fact that
E(t) —» 0 for large t. For comparison we show Figs. 6.2(c,d) for Aid = 70. Here we observe
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61
n/2
n/2
(a)
Probe pulse
Probe pub e
1.5
1.0
0.04 r
(c)
0.02
0.00
1000
2000
3000
0
1000
t
2000
3000
t
Figure 6.3: Direct simulation of the spin-echo-type oscillation switching response. Joseph­
son response to the application of two 7t/ 2 microwave pulses, with an intervening 7r-pulse,
followed by a probe field. The delay between the 7t / 2 pulses is At.rj = f 4 —t\ = 2182. The
7r-pulse offset is for (a,b) (non-switching) t 2 —ti = 1000. For (c,d) (switching) t 2 —C = 900.
Parameters were a — 10~4, r) = 0.904706, es = 2.17 x 10“ 3, ep = 7.5 x 10~2, and 0 = 0.
the same initial behavior, but the application of the second 7r/2-pulse leaves the system
in a relatively high energy state E(t$) > E (t\). since the phase-slip between the junction
and the microwave field is minor in this case. Consequently, the junction phase (p switches
out of the bound state when the probe pulse is applied after
15
. This is recognized by the
diverging energy for t > t$.
In the case of the spin-echo-type oscillation simulation, a 7r-pulse is applied prior to
the second 7r / 2 -pulse. Escape events resulting from the probe pulse are recorded in similar
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fashion to the Ramsey-type fringe simulation; however, the results are tabulated according
to keeping the delay At j constant and varying the interval (t>2 —t\). i.e., the 7r-pulse offset.
Examples of the evolution of the Josephson phase and the energy are seen in Fig. 6.3.
6 .4
S im u la tio n r e s u lts
(b)
^
(d)
so
0.4
(f)
-'N /V V j
2 000
............
1
2000
4000
At
Figure 6.4: Switching probabilities for the Ramsey-type fringe (a-d) and the Spin-echotype oscillation (e,f) simulations. Each point represents statistics of ~ 2,500 events at
0 = 2 . 0 0 x 1 0 4. The horizontal axis in (a-d) represents the 7r / 2 -pulse separation (Ata =
t,i —f]). Panels (a,b) have a = 10~ 3 and e = 0.085. In (a) the driving frequency uis = an —
\ J l —rj2 = 0.652714. Panel (b) shows nearly vanishing Ramsey-type fringe frequency at
ujs = 0.997a;;. Panels (c,d) have a = 1 0 ~ 4 and e — 0.082. Panel (c) uses the same driving
frequency as (a), (d) shows vanishing Ramsey-type fringe frequency near lus — 0.990a;/.
Other parameters not noted in (a-d) are the same as used in Fig. 6.2. Panels (e,f) reflect
data from spin-echo-type simulations for A t a — 2750 time units. The 7r-pulse offset (i 2 —ti)
is shown on the horizontal axis. Parameters used in (e,f) are the same as in Fig. 6.3 with
(e) having los = loi and (f) having tos — 0.990m/. The lines are an aid to the eye.
Ramsey-type fringes in the switching probability as a function of 7r / 2 -pulse sepa­
ration can be directly observed in Figs. 6.4(a-d), where we have shown the variation of the
switching probability for normalized temperature
0
=
2
- 1 0 ~4, calculated as averages of
2,500 events for different values of a and u>s. Panel (c) clearly shows a distinct frequency,
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63
0 .0 3 0
0.020
0.015
g» 0.010
5
§
0 .0 0 5
8 0.000
6§
aa*
k
£
0.020
0 .0 1 5
0.010
0 .0 0 5
0.000
0 .9 7
0 .9 9
1.00
Signalfrequency (&/(&])
0 .9 8
1.01
1.02
Figure 6.5: Frequency response as a function of applied microwave frequency for two dif­
ferent dissipation parameters: (a) a = 10~4, and (b) a = 1CT3. Filled diamonds depict
Ramsey-type fringe frequencies, Q f, with parameters corresponding to Fig. 6.2. The open
circles in (a) represent spin-echo-type frequencies with parameters as in Fig. 6.3. The gray
lines represent integration of Eq. (6.5) to £ 4 = 1 ; dashed lines to t.4 = 1 0 ,000; solid black to
t4 = 1 0 0 , 0 0 0 .
which we name the Ramsey-type fringe frequency tip , for tds = u>i =
—r/ 2 = 0.652714
and a = 10“ 4. For the same value of a, panel (d) shows that the fringe frequency depends
on the applied microwave frequency such th at when u>a = 0.990a.'/, Qp ~ 0. Similar behavior
is observed for a larger dissipation parameter a = 10- 3 shown in panels (a,b). In panel (a)
ljs
— uji = 0.652714. Panel (b) demonstrates th at for this value of a a driving frequency of
u a = 0.997a>/ results in
« 0.
A number of simulations of the Ramsey-type fringe frequency for different mi­
crowave frequencies were conducted. The results are summarized in Fig. 6.5, where the
fringe frequency flp is shown as a function of microwave frequency a , for the two different
dissipation parameters mentioned in Fig. 6.4. The simulation data are represented by filled
diamond markers with error bars. The frequencies are read from figures like Fig. 6.4. In
the region of low flp, less than one wavelength of fringe oscillation may be visible and the
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oscillation may be somewhat distorted [e.g., Fig. 6.4(b)]. In such cases we measured the
time from the first trough (peak) to the next peak (trough) and used the half period to
determine the frequency. In cases of saturation or serious distortion such as Figs. 6.4(c,e,f),
we measured the period of the first regular waveform. These issues also account for some
of the distortion seen in Fig. 6.5.
We observe the distinct “V-shape” signature in our data of the Ramsey-type fre­
quency Qp(u>s), the same as that seen in the experiments of Ref. [27], with a slope near
±1. The lines in Fig. 6.5 have been drawn which correspond to integration of Eq. (6.5)
for specific values of t$. The gray lines represent integration to
— 1 ; dashed lines to
t 4 = 10,000; solid black to
£4
frequency for which Q f =
increases with a, and we further notice th at this characteristic
0
= 100,000. We observe that the characteristic microwave
frequency approaches loi for f.4 —> 0 0 . This effect has not been studied experimentally, but
would be interesting for future comparison.
Our spin-echo-type oscillation data are shown in Figs. 6.4(e,f) and 6.5(a). Several
important observations can be made. The most critical is the fact that, outside a given
range of (tos/u>{) near which fIp —> 0 , the frequency of spin-echo-type oscillations very nearly
matches the Ramsey-type fringe frequency ftp- Also, the time-dependent nature of u>r (A(t))
presents difficulty with precise frequency measurement with both types of oscillations, but
seems to be even more pronounced with spin-echo-type oscillations (as seen in the degraded
signal in Fig. 6.4(e)). From Fig. 6.4(f) it is clear that the frequency of oscillation is changing
fairly rapidly with 7r-pulse offset. One may interpret from the initial 800 time units a zero
frequency, but in keeping with our methodology of reporting the first coherent waveform, we
recorded a frequency higher than that observed in the comparable oscillation in Fig. 6.4(d).
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Finally, obtaining data for higher values of characteristic damping a is somewhat more
difficult for spin-echo-type oscillations than for Ramsey-type fringes phenomena. Since we
intended to clearly identify conditions for either zero or variable frequency, we were unable to
examine ranges of larger A t ci for which low-frequency oscillations are more severely damped
by higher values of a.
6 .5
C o n c lu s io n s
Like Rabi-type oscillations and resonant multi-peak switching distributions, Ramsey-
type fringes and spin-echo-type oscillations have been studied to identify and characterize
expected macroscopic quantum behavior of Josephson systems. We have identified the
phase-relationship between external signaling and system evolution as the key factor in
classically generating these types of phenomena. The transient, modulating response of the
system to the imposed microwave signals determines the oscillatory nature of the switching
response. Thus, the same basic mechanism responsible for production of Rabi-type oscilla­
tions, Qr , is linked to Ramsey-type fringes and spin-echo-type oscillations in our classical
model.
In this chapter we have addressed the comparable classical system through direct
simulations of the well-established nonlinear classical model equation, which is driven ac­
cording to the recipe prescribed by the experimental reports of Ramsey-fringes. Our results
show that even the simplest possible classical Josephson model, the single RCSJ model,
clearly exhibits the quantitative and qualitative features of these types of oscillations when
the model is used to simulate a low temperature, low dissipation system.
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C hapter 7
R esonances in Josephson ju n ction
system s
This chapter extends the classical ideas and implementations we have developed for
the single-junction circuit to the more complex system of the flux-biased three-junction loop.
The three-junction loop has been extensively utilized by many of the research groups exam­
ining macroscopic quantum tunneling and other phenomena [14,21,22,25,27,29,39,61,62].
This extensive use can be attributed to nature of the potential well, as shaped through a
flux bias, which allows settings which are suitably insensitive to flux noise, particularly in
the case of small values of self-inductance /?£.
Following Ref. [40], we first develop an analytical description of resonant modes
for this system. Next, we demonstrate computer simulations which display these modes.
Simulations are also shown which depict the oscillatory phenomena of previous chapters.
Also, off-resonant modes are explored and compared to similar effects in the single-junction
circuit.
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67
R/k ______ ±1I
—Wv—
>N2
L
Figure 7.1: Three-junction loop schematic. Junction 3 is smaller than the other two and is
indicated by the factor of k applied to the capacitance, resistance, and critical current.
7 .1
C la s s ic a l m o d e l for t h e t h r e e - j u n c t io n lo o p
Figure 7.1 depicts the three-junction system. The normalized classical equations
can be written [61,62]
(7.1)
where <pi is the difference between the phases of the quantum mechanical wave functions
defining the ith junction, k is the scale factor relating the one smaller junction, represented
by ip3 , to the other two equivalent junctions, Pi is the normalized loop inductance, P i —
2ttL I c/ $ o- M (t) is a composite of the various external magnetic fluxes to which the system
is exposed
M (t) = M dc + £s(t) sin(u3t + 9S) + £p(t)
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(7.2)
where Mdc represents the dc bias flux, and £s(t) and u>s represent the ac flux (signal)
amplitude and frequency, respectively. A flux pulse for probing the state of the system is
represented by ep(t). The normalization is in keeping with the previous chapters.
We may identify a Hamiltonian for the undamped system as
H
Hk + H p
(7.3)
Hk
\ ( v l + V 2 + k <pI)
(7-4)
Hr,
2 - cos
- cos<p2 + k(1 - cos<p3) + vi-(<^i +
2PL
¥2
+
7>3
+ 2ttM)2.
(7.5)
For small /3l in Eq. (7.5), we can see that energy from the loop current (last term) dictates
the sum of the phases. We may therefore use the following orthogonal (and therefore
separable) transformation to analyze the system:
Vh =
<pi +
V>2
=
¥>1 -
^3
=
- i H k t o + V^ + i ^ P a
+ T’s
<P2
vi
=
V2
=
V3 =
+
5^2 -
5^3
(7.6)
T ^ ^ I + V’S-
The energy now reads
Hk
p
—
4" 2 ^ + 2 ^ 4 -
2
k)i/’3)
V>2 cos .
—2 cos —
=
2
+
K 1 — COS
V l +^ 2/c V 'l
2
1
~b 2 ft
V’l + V>3
-
(7.7)
^ 3
+ ^
(V'i + 2vrM)2
Under the condition /?l <C 1, this system can be simplified to a single degree of
freedom by examining the constraints which frame ipi and V'2 - First, it can be seen from
Eq. (7.7) that energy considerations prevent the deviation of ipi far from —27rM.
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A second constraint, on fa , is obtained by considering the equation of motion
fa + a fa +
2
cos ( j~
-
^ 3
J Sin y
=
(7.8)
n iOO ~ n 2 ^)-
Prom this it can seen that the static equilibrium value of fa is given by
1
^2
cosy
>
0
for c o s ( j ^ i l )i - \ f a ) <
0
for cos(T^ V ’l ”
(7.9)
=
-1
.
Note this is a weak constraint as it does not require much energy to break the equilibrium.
Equation (7.7) can now be written
Hn
1
2
1
1
T 2cos(— — V>: - - f a ) + «(1 - cos(- ■ —
K1 -f- 2k
+ fa )) + — (1&1 + 2irM) .
Wl
(7.10)
Now the constraint on fa can be further defined by minimizing Hp with respect to fa
dHp
d fa
± 2 tc
■sm
1 -{- 2K
K
•sm
+
1 +
2 k
27tM —(3l 5\
=> fa
k/3l
-27tM ■
1
-I- 2k
n . ( 2irM k
1
\
. ( ,
^ S,nIT T ^ + 2 * ) + sm f a
2ttM
- TT2k
which is correct to first order in y .
The energy can now be written as a function of fa only
TT
Hp =
2
( 27rM k
1 \
(
27xM \
/3l ro
2 cos ( - + ^ + - f a ) + k - K c o s [ fa - ■■■■| ^ ) - — 51;
1 + 2k
where the choice between =F should be such that the energy is minimized.
,
.
(7.11)
Figure 7.2
depicts this single-degree-of-freedom representation of the potential energy. Note that while
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70
2.0
(b)
1.4
1.2
1.0
0
n
2n
0
2n
7t
v3
0
jt
27t
V3
V3
Figure 7.2: Three-junction loop potential energy for several values of M^c: (a) Mdc = 0.5,
(b) Mdc — 0.52, and (c) Mdc — 0.542. Other parameters were k = 0.68 and (3 — 0.09994.
in general this potential consists of pairs of wells separated by higher energy “cusps” , for
purposes of this work we focus on a single well-pair, as the energies are kept below these
cusps.
The following definitions allow the energy to be expressed in a reduced form
( 2irM k
1
\
(I
C1 = COS1lT ^ + 2*7 = cos ( f 3+ M *
.
( 2irM k
Sl = sm
C2
cos ( (;':i
S2
sin
^
where
03
=
1 , \
.
(I n
,,
+ 2 * y = sm U ”3 + Mn
\ _ cosq3
2k
27xM
1 -f~ 2K
1
=
(1
+ 2 k) 2
= sin 0 3
Pl
+ 12wM
+ 2k '
This gives
1 =p cos ( ^ 0 3 + Mir) + k ( 1 —cos 0 3 ) — (3l
Hn
=
2 (1
4=2 sin ( - 0 3 + Mir) + sin 0 3
=f ci) + k ( 1 - c2) - % ( t 2 si + s2)2-
(7.12)
Calculating the linear resonance frequency for a given well, the equation of motion
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for the friction-less and constant-M system is
» (1 +
=
( 7 . 13 )
For ip3 = ipl + S, small oscillations around tpp are then given by
-( 1
+ 2k )5 +
± \c x +
7 .2
7 .2 .1
(7.14)
kc2
- PL {{C2 T Cl)2 - (g 2 T S2 )(S 2 =f 2 s i ) )
(7.15)
S im u la tio n r e s u lts
D isp e r sio n re sp o n se for th e th r e e -ju n c tio n lo o p
A convention we have adopted in our work is to assign values {M ric < 0.5} to a
configuration which initially places the inter-well barrier at a value of
-0 3
which is higher
than the minimum of the lower-energy well. Also, an applied force is defined as positive if it
acts to push the reduced phase difference ipz in a positive direction. To verify Eq. (7.15), we
developed a computer code for the three-junction loop Eqs. (7.1). The results are shown in
the frequency response chart (or d isp ersio n d ia g ra m ), Fig. 7.3. Each open circle represents
the minimum flux pulse ep required to obtain a statistical response of 50% escape probability.
Each trial in the simulation is conducted in the following manner: W ith initial conditions
of i/ls in the minimum energy configuration [according to the above convention] and
7 /3
=
0
,
the system is driven at frequency lus and amplitude es for a period of (2.0 /a ). At the end of
the driving phase and after a slight pause (50 time units), a pulse of magnitude ep is applied
(in similar fashion to Chapter 4) and the subsequent escape or non-escape to the alternate
well is recorded. Statistics are then gathered to determine £p0% while varying the driving
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.52
-
5
0.50
-
0
0.48
-
S
1
^
/
046- /
0.44 — J
0.48
/
>-
0.50
0.52
0.54
M dc
Figure 7.3: Resonant frequency response in the three-junction loop. Lines represent the
predicted frequency response from Eq. (7.15). The open circles indicate the resonances
from the results of our simulations. The inset is a switching distribution indicating the
resonance at the box marked ’A’. This provides an example for determining the placement
of open circles in the larger plot. Parameters for the simulation were Pl — 0.09994, k = 0.68,
a = 0.00004, e* = 0.0004, and 0 (T ) = 0.00912.
frequency. A resonance is determined for each value of Mdc, as indicated by a minimum
value of £p0% as a function of frequency.
The solid line in Fig. 7.3 represents the resonant frequency response as predicted
by Eq. (7.15). The open circles indicate the simulation results and depict the minimum
amplitude probe pulse, ep for which the escape rate equals 50%. The inset shows the
relationship between probe pulse (at 50% escape probability) and signal frequency for Mdc =
0.48. The minimum energy value in each well is marked with a boxed symbol to establish
the correspondence with the potential energy plot. The dashed line indicates the symmetric
relationship in linear resonance frequency (about Mdc = 0.5) for a well placement convention
which is opposite to our convention. The data presented is based on 2500-7000 escape events.
The simulation results show close agreement with our theory, though the agreement
diminishes slightly for increasing values of Mdc- This can be understood by noting that as
the flux bias is increased, the higher of the two wells becomes broader, allowing larger
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oscillations, which corresponds to a greater degree of anharmonic behavior.
7.2.2
Oscillatory phenom ena for the three-junction loop
10 L
0
,
L.
,---------------- 1---------------- ,---------------- L _ -------------,---------------- 1---------------- ,---------------- ,---------------- ,---------------- 1---------
2000
4000
6000
t
Figure 7.4: Direct simulation of Ramsey-type fringe switch events. Panels (a,c) represent
the reduced phase-difference for a trial event which does not switch and one th at switches,
respectively. Panels (b,d) display the resulting potential energy for panels (a,c), respectively.
Squares labelled “lo” and “hi” indicate the low and high potential energy wells, respectively.
Parameters were: uis = 0.42528, ev = 0.0178 and © = 0 .0 .
Next we adapted these simulations to generate Rabi-type oscillations, Ramsey-type
fringes, and spin-echo-type oscillations. Similar recipes are used as described in chapters 4
and 6 . A summary of the signalling used in the Ramsey-type fringe simulation of the threejunction loop is shown in Fig. 7.4, where the wells are marked with square boxes. Here we
display the phase and energy plots illustrating the dynamics of a Ramsey-type fringe switch
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event, where the signal timing is the same as th at in Fig. 6.2.
1.0
0.04
0.8
0.6
0.4
C f0.02
1
0.2
0.0
0
0
1000
200
400
600
nil pulse separation
Figure 7.5: Simulated switching phenomena for the three-junction loop: Rabi-type oscil­
lations and Ramsey-type fringes. Panel (a) shows Rabi-type oscillations for M rp. = 0.52,
(3l = 0.09559, k = 0.68, a = 0.00015, ws = 0.443, es = 0.00182, ep = 0.0149, and
0 (T ) = 0.003. Statistics were gathered for ~ 20,000 escape events. Panel (b) provides the
Rabi-type frequency as a function of microwave amplitude, with es for uis = 0.45629 ~ ojt .
Panel (c) is the resulting switching distribution for Ramsey-type fringes with u>s = 0.42528
and sp = 0.0195. The inset indicates the driving frequency which achieves a fringe frequency
of zero; here ujs = 0.45629 ~ u>r , sp = 0.0148. Panel (d) provides the relationship between
fringe frequency and driving frequency. The arrow indicates measurement of lji — 0.46557
by direct simulation. Unless indicated, all other parameters were the same as those for
panel (a).
The resulting Rabi-type oscillation and Ramsey-type fringe switching distribu­
tions are shown in Fig. 7.5. The Rabi-type frequency in panel (a) is found to be 0.01690.
In panel (b) the Rabi-type frequency is displayed as a function of microwave amplitude.
Panel (c) illustrates the results of our simulations for producing Ramsey-type fringes. The
Ramsey-type fringe frequency is flp — 0.03502. The inset indicates the driving frequency
which achieves a fringe frequency of zero; here u>s = 0.45629 ~ uir, sp = 0.0148. In panel (d)
the relationship between fringe frequency and driving frequency, u>s is shown. The arrow
indicates measurement of uii = 0.46557 by direct simulation. For comparison, Eq. (7.15)
predicts u>i = 0.46625. We observe again the ± unity slope relationship seen in experiments
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75
0.6
0.4
0.2
n-pulse offset
Figure 7.6: Oscillation phenomena for three-junction loop, spin-echo-type oscillations. Pa­
rameters were: Mdc = 0.52, (3l = 0.09559, k = 0.68, a = 0.00015, u>s — 0.419787,
£s — 0.00182, ep = 0.0195, and 0 (T ) = 0.003. Each dot represents 22,000 escape events.
such as Ref. [27]. In Fig. 7.6 the simulated spin-echo-type oscillation response is shown for
the three-j unction loop.
7 .2 .3
O ff-reso n a n ce p h en o m en a
In the conclusion to Chapter 4, we mentioned that our Rabi-type oscillations
display a nonlinear relationship with signal amplitude (Fig. 4.7). We indicated that, in this
specific instance, the agreement between our results and published experiments is somewhat
mixed. It is possible th at this inconsistency is related to the effect noted in Chapter 2, where
we noted in reference to Fig. 2.6(d) that the effect an of off-resonant signal frequency is
to generate a multi-valued response. This should be especially signifcant at small signal
amplitudes - where the curvature in Fig. 4.7 is greatest.
In an effort to explore this phenomenon further, we performed simulations using
an off-resonance driving signal on both the simple Josephson junction circuit and the threejunction. We show these effects on the Rabi-type frequency in Fig. 7.7. One important
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76
(a)
(b)
Cf
& 0.03
§
sO’
I
0.02
R
O
.3
0.01
0.00
0.0
0.002
0.004 0.0
0.002
0.004
Microwave amplitude, eg
Figure 7.7: Simulations of off-resonance Rabi-type oscillation frequencies. Panel (a) depicts
the off-resonance modes for the three-j unction loop. Parameters were: Mdc = 0.52, (3l =
0.09559, k = 0.68, cos = 0.443, £s = 0.00182, ep = 0.063, and @(T) = 0.003. Two
values of characteristic damping are given. The filled squares correspond to a = 0.00015,
empty circles:® = 0.0. The resonance frequency, cor (as shown in Fig. 7.5(d)) is 0.45629.
Panel (b) shows the off-resonance modes for the single-junction circuit. Parameters were
a = 0.00151477, r? = y jl - uif, u>s = 0.99u/j = 0.646188, and © - 0.0002.
aspect of these plots is that low-amplitude signals result in modulation frequencies which do
not lie on the “main sequence” curve to which higher-amplitude responses conform. While
this would seem to confirm the effect of multi-valued nature of the amplitude response
for off-resonant driving, it is not conclusive as to how this may contribute to the linear
relationships seen in Refs. [17,21,24,27],
7 .3
C o n c lu s io n s
The analysis we have developed reducing the three-junction loop to a single degree
of freedom provides a significant reduction in the complexity of the system. The resulting
equation provides direct agreement with our simulations in modelling the well resonances.
Our studies of off-resonant driving of Rabi-type oscillations reinforce earlier work
concerning multi-valued functions in signal amplitude. Although no distinct conclusions
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can be made with regard to a connection between off-resonant driving signals and linear
relationships between Rabi-type frequency and signal amplitude, we have shown evidence
that driving the system near (but not exactly at) resonance does extract nearly linear
behavior in the response for larger amplitudes. Again, the simple model we are using may
not completely account for this aspect of the phenomena.
We have used classical devices to demonstrate various phenomena heretofore at­
tributed to macroscopic quantum tunneling. W ith regard to Rabi-type oscillations, Ramseytype fringes, and spin-echo-type oscillations, the three-junction loop differs little, qualita­
tively, from the single-junction circuit. Given the relative insensitivity to noise inherent in
the three-junction loop, it is no surprise from a classical viewpoint th at research groups are
favoring its use.
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C hapter 8
C onclusions and future work
8 .1
T w o v ie w s
In this dissertation we have attem pted to present an alternative to the quantum
view of Josephson junction phenomena:
The macroscopic quantum picture (currently prevalent) interprets the observed
phenomena as a result of intrinsic states which, when stimulated with radiation, display
switching patterns which oscillate due to inherent incongruity in the tunneling probabilities
at each prescribed energy level. There are stochastic components to the system which are
resident in the absorption processes of the microwave photons, in the presence of quasipar­
ticles, and in the measurement and subsequent immediate collapse of the system wavefunction.
The classical picture offered here (and in our related references) presents a system
with a microwave induced temporally modulated energy. The slow variation is indirectly
observed through the associated variation in escape from the energy well when the probe is
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applied, with high probability for passing the energetic saddle point during times of elevated
energy and relatively little escape probability during times of depressed energy content. The
stochastic component in this system is due to the random phase of the microwave signal as
well as thermal fluctuations.
Hopefully, the analogies we have developed between 1) intrinsic quantum mechan­
ical energy levels, and
2
) the multi-valued resonances of the classical nonlinear system,
further bridge the gap between what can be expected from the two interpretations of the
microwave induced measurements. Since the experimental measurements are concerned
only with detecting the escape event, and since the classical and quantum mechanical in­
terpretations seem to provide the same (or very similar) signatures for that escape, we are
faced with a fundamental ambiguity of how to read this information.
Ultimately, potential applications of Josephson technology for quantum informa­
tion processing therefore benefit from the development of new unambiguous measurements,
which must present signatures of macroscopic quantum behavior that cannot be explained
classically.
8 .2
F u tu r e w o r k
References [16,21,27,29] have presented the details concerning an “energy gap”
seen in multi-well Josephson systems. This gap represents the frequency of tunneling as­
sociated with wavefunctions of mixed states. As the bias of the system is brought into
a degenerate state [Mdc = 0.5 in the three-j unction loop), the energy difference between
the two [conjugate] mixed states does not go to zero as would be expected classically. The
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classical description and subsequent simulation of this phenomenon is the subject of current
research.
In the previous chapters we have purposely based our examinations on simple
devices and systems.
In the future it may be intellectually profitable to explore more
complex systems. In addition to the systems we have considered, other configurations have
been utilized to elicit these phenomena, including charge-biased two-junction SQUIDs [24],
charge-biased three-j unction SQUIDs [17], and hybrid systems which often include inductorbased systems to create multiple potential wells [11,13,20,28,63].
The complexities intrinsic to these systems present significant challenges for nu­
merical simulation and are almost impervious to analytical methods. While the hybrid
systems are complex in terms of the number of devices and their interactions, finding clas­
sical descriptions for charge-biased systems is even more daunting. Charge-biased systems
use capacitances so small th at individual electrons are “trapped” and manipulated.
However, it is possible th at a classical depiction of the energy gap mentioned above
may require such treatment. It is also possible th at interactions between the system being
measured and the measuring apparatus may be a factor in the process and, thus, require
modelling as well.
8 .3
In c lo s in g ...
We have presented in this thesis a consistent picture of theory and simulations
which confirm the major features of the published experiments. Given the very close quan­
titative agreement between our classical theory and the experimental measurements, the
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intuition and the closed-form expressions presented here can be directly applied to guide
future experiments and microwave manipulations of Josephson systems.
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A p p en d ix A
N um erical m eth od s
The simulations performed for this thesis utilized the velocity Stormer-Verlet
method for integrating the equation of motion. This can be derived in the following manner.
Consider the second-order differential equation
y(t) = f{ t,y ,v ) ,
v — y(t),
Given the adjacent time coordinates
v = y = f.
(A.l)
separated by a time dt, the current point
yn = y(tn) can be related to the previous point through a second-order Taylor expansion
dt 2
yn = yn- \ + dtv n - 1 + — / n_i.
(A.2)
Next, consider the mixed Euler method for determining the current point from the deriva­
tives of current and previous points
dt
.
Zn ~ Zn—l “i“ ™ \^n "b Zn—l) •
(A.3)
This expression can be used to describe the velocity
Vn
—
Un_ l + —
(fn
+
f n - l )
■
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(A .4)
Together, Eqs. (A.2, A.4) - along with known initial conditions for y(n=o), ^(n=o)>
can be used to integrate the considered second-order differential equation. If f( t, y, v) takes
the form f ( t , y , v ) = f ( y , t ) —av (i.e., a linearly damped system), these equations become
(A.5)
where a represents the damping in the system and the undamped portion of the force is
only a function of the position:
fn {V n )-
Also, it should be noted th at the methods described above were consistently stable
in our application. Also, no trouble was encountered with regard to truncation or round-off
error.
A customized C + + program was developed to integrate the equation (or equations)
of motion using Eq. A.5 using a time-step of
dt
= 0.01. This gave reasonable energy
conservation (in the undamped mode) while allowing for acceptable run-times.
In generating the switching distributions, each trial only consumed 15 —60 seconds
of CPU time. Thus, to generate 10,000 trials for 50 data points required 90 —350 days of
CPU time. The stochastic nature of these simulations allowed parallel trials to be executed
on 30-80 CPUs; with each process using different seed values in generating the requisite
random numbers.
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A p p en d ix B
R abi-m odulation frequency
detailed derivation
Start the expansion of Eq. (4.1) with some definitions,
4/,, = sin(a;st + 9), 4>c = cos(a;sf + 9).
The phase difference and its derivatives can be written
= <po + 6<po + (A +
—A
+ A60VC
f
—SfQ ~f~ ((5vl
u ) s ~i~ i^AoJs 4“ cosSA
(p
= 5fo + (5A - A J l - 2A ujs69 - u)2s5 A )^ s + (A59 + 2ujsSA - Au%60)'f!c,
-t-
A69^\I/c
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where terms of second order and higher are discarded. Next the sine term in Eq. (4.1) can
be expanded, considering only monochromatic terms
sm<^ =
sin(<^o + j4^s)
4- cos(<^o 4-
cos
(5<po 4- 5 A ^ S + A 5 6 ^ c)
sin (5<po + S A 9 s + AdO^c)
— sin <po [cos A\l/S — (6(po + M $ s + A89^Aj sin(J4^rs)]
+ cos<^o [ s i n ^ ^ g ) + (J 970 + S A V s + A 8 9 ^ ^ ) cosj4\I/s]
= sin<po [(Jo(i) + 2J2(i)'£2c) - (6<po + SAW, + a89Vc) 2J1(A)'^S\
4- cos tpo [2vfi(^4)vE,s + (SipQ + <E4\I/s + a59'^fc) («^o(-'4) 4- 2«72 (.A)1!lr2c)]
= siny?0 [(Jo(i4) + 2J2(A)$2c) - (5<po*a + 6AVS2 + M M a* c) 2J X(I)]
4- cos ipo 2 Ji(A )'ks + (dtpo + SAVs 4- ASOVc) J0(A)
4- {S<po$2c + SA V teV , + A W M c ) 2J2(A)
sin^o Jo(A) - (V o 'I's +
2
J X{A)
4- cos ip0 2J i {A)'$!s + (<Vo + 5A V a + A56VC) J0(A) + ( S A V a 4 A 59*c) J2(A)
where the following identities are used
OO
sin(asinx)
=
2
J 2 n+i(a) sin[(2n 4 - l)x]
n
cost asm a?)
=0
J 0 (a) + 2 ^ 2
n~
n(a) cos [2-nx\
1
cos 2x cos x
=
—(cos 3x + cos x)
cos 2x sin x
=
- (sin 3x —sin x)
1
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Substituting the above into the original differential equation gives
7) + es sinu st
=
5^pq + { 5 A ~ A uj2s—2Au)s56 — uj25s A ) ^ s + {A56+ 2u)s5A —A lo25s 6 )^ c
-f-Q;
“ t~
{S-A
—
AcusdQ^^ s
[Alus
tusSA
-f-
c
+ sin (p0 J0(A) - ( S ip ^ s + \sA ^j 2J\(A )
+ COS
(fio
2J1(A )$s + (6<po + 6A V , + A 5 6 ^c) J0(A)
+ ( - 8 A ^ s + A 5 9 ^ c) J 2{A)
This equation can be separated into static (dc) components and dynamic (ac) components.
The dc equation is written
rj — d(po + aSipo + Jo(A) cos tpoStpo + Jo(A) sin (po — Ji(A ) sin poSA
For the ac equation yet another definition helps to keep the expression clear,
if>s = sin u^i, tfjc = cos uist,
so that
= V’s cos 9 + 4>c sin 9, \I/C= V’c cos 9 — rj;s sin I
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87
Then
^ si’s — {AA — Aug — 2 Au)sS8 —ui2 8 A)(ipcsin 9 + ips cos 9)
+(Ad9 + 2 ujs8 A —u)2A89)(ipc cos 9 — i/>8 sin 9)
+ a (8 A —ujsA 8 8 ) (tpc sin 9 + ips cos 9)
+(Au)s + ujs8 A + A89){ipc cos8 — ips sin#)
+ [cos<^o (2Ji(A) + Jo(A) 8 A — J 2 (A) 8 A) — 2 Ji(A ) sin<£o<fyo] {ipc sm 9 + ips cos 9)
+ cosifo (Jq(A)AS9 + J 2 (A)A89) (ipc cos 9 — ips sin#).
Collect on ipc and xps
0 =
sin 8
{8
A —Auig — 2AujsS9 —u>2s8 A) + a{ 8 A —u>sA89)
(B.l)
+ cos (po ( 2Ji(^4) + Jq(A) 8 A — J 2 (A) 8 A j —2J\(A) sm(poS<po
+ cos 9 (A89 + 2ojsSA —u)gAS9) + a.(Aus + cas8 A + A89)
+ cos 930 (Jo(A)A89 + J 2 (A)AS 9 )
cos
(■8A — Auj2 —2Alos89 —ajgSA) + a(8A —losA89)
+ cos <fo (2 J\{A) + Jq{A) 8 A — J 2 {A) 8 A )} — 2 J\{A) sin^o^^o
—smf (.A89 + 2 u>s8 A —uj2A89) + a(Auis + tos 8 A + A89)
+ cos<^o (J o(A)A89 + J2(A)A89)
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(B-2)
Equation (B.2) has the solution
£ s COS0
=
(8A —Au>2 —2 Auis89 —W jM ) + ct(8A — uisAS9)
+ cos<^o ( 2 Ji(A ) + Jq{A)8A —J2{A)5A) —2J\{A) sin <^0
-e, smt
(.ASd +
2
^ 0
(jjs8A —oj2A89) + a{Aus + uis8A + A89)
+ costpo {Jq(A)A89 + J2(A)A89)
Now, if the variations are small compared to the quantities A and rj, the dc equation gives
.
_
sirnpo
v
M AY
and the dc equation can then be rewritten
8<po + ot8(po + Jo(A) cos tpoStpo
Ji(A )
r]8A.
M A)
Similarly, if the ac variations are small with respect to A, es, and u>s
cost
-£s sin#
-AuYs + 2Ji(A ) cos (po
pa aAujs-
So that
e\
=
(2
Ji(A ) cos ipo — A los2) 2 + (aAu) s ) 2
tan#
=
aAuq
A uj^ — 2 Ji (A) cos ipo
Using these approximations simplifies the ac equations
8A —2ujsA89 —u>28A + a(8A —lusAS9) + [Jo(A) —/ 2 (A)] cos</?o<5A =
AS9 + 2lus8A —lu2AS9 + a(A89 +
ujs8A) + [Jo(A) + / 2 (A)] cos (poA89
=
2
Ji(A ) sint^o^^o
0
.
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If one assumes that 5A is periodic and varies with frequency Q. then aS6 must vary in the
same manner
6A = em t, A50 = Keiat.
Since 5A is the forcing term in the dc equation, the long-time response will be
Sipo = 7 elQt.
Substitution into the dc equation gives an expression for
7
—7 O2 + ijaCl + jJo (A ) cos po = Ji(A ) sin po­
or
7 -
Ji(A ) siny?o
-Q2 + iaQ. 4 - J q(A) cos tpo
Next define
b = cosipo,
D = J i(A )2 sin2 po-
Z \ = [Jo(A) + Jv{A)\ cospo ~ w2,
W
Zo = [Jo(A) - h {A )\ cospo - u 2s.
= 4w2 + Zi + Z 0 + Jo{A)b,R = \ w - Jo(A)b\J0{A)b + Z XZ 0 - 2D
The equations can then be written
[—Cl2 — 2 los ( i Q, k ) + a ( i f l ) — olujs k + Zo]
(—
k
-t~
Z x) k -I- 2iLosQ -1- ootjg ==
2D
—Q2 + iaQ + Jo(A)b
0
.
can be solved for
k
=
2icosCl + au>s
fl 2 —iaQ, —Z\
Solving for fl numerically results in three pairs of complex roots which are ± in the real
part. The real part of one pair corresponds to the modulation frequency. The other two
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correspond to frequencies very near the driving frequency and the next harmonic (second
harmonic).
Each of the six roots has an imaginary part which effectively equals a /2 for the
entire parameter space. This leads us to investigate the analytical properties of this equa­
tion.
Start by substituting
ft =
flR
+ i/3, where f l R , (3 are both real. Further substitution
of the expression /3 = a /2 reveals th a t each of the imaginary coefficients goes to zero (note
that all of the parameters in the above expression are real.) The real coefficients can then
be reduced to a third order equation in flR2 with the resulting form
( flR 2)3
+ C2(Qr2 ) 2 +
C iflR 2
+ Co = 0,
the coefficients cn can be expressed
c2
=
-a 2- W
4
00
=
^
+ (4ojs - W ) j q + (R - 4Jo(A)bu,2) ^
+ Z\2D - Z0Z 1J0(A)b.
The frequencies then can be expressed
12
n 2
4 'R
R .M
.M oodulation
d u la tio n
ci —4 c2 2 —4 c2 _B3 —j9 3 + iy/3 (l2 c\ —4c22 + B I ^
______________________________
1
l2 B s
12
ci —4 c2 2 —4c2s t —B i — iy/ 3 ^ 1 2 ci —4 c2 2 + B s'j
‘R .D r iv in g
'R Jln d H arm o n ic
where
B = —8 c 23 +
36ciC 2
—1 0 8 cq
+ 12 a / 12 c 23cq + 1 2 c i 3 — 3 c 22c i 2 + 8 1 c q 2 — 54 c 2CiCq.
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(B.3)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
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