ABSTRACT
Title of dissertation: BRIDGING QUANTUM, CLASSICAL
AND STOCHASTIC
SHORTCUTS TO ADIABATICITY.
Ayoti Patra, Doctor of Philosophy, 2017
Dissertation directed by: Professor Christopher Jarzynski
Department of Chemistry and Biochemistry;
Institute for Physical Science and Technology;
Department of Physics.
Adiabatic invariants – quantities that are preserved under the slow driving of
a system’s external parameters – are important in classical mechanics, quantum
mechanics and thermodynamics. Adiabatic processes allow a system to be guided
to evolve to a desired final state. However, the slow driving of a quantum system
makes it vulnerable to environmental decoherence, and for both quantum and clas-
sical systems, it is often desirable and time-efficient to speed up a process. Shortcuts
to adiabaticity are strategies for preserving adiabatic invariants under rapid driving,
typically by means of an auxiliary field that suppresses excitations, otherwise gen-
erated during rapid driving. Several theoretical approaches have been developed to
construct such shortcuts. In this dissertation we focus on two different approaches,
namely counterdiabatic driving and fast-forward driving, which were originally devel-
oped for quantum systems. The counterdiabatic approach introduced independently
by Dermirplak and Rice [J. Phys. Chem. A, 107:9937, 2003], and Berry [J. Phys.
A: Math. Theor., 42:365303, 2009] formally provides an exact expression for the
auxiliary Hamiltonian, which however is abstract and difficult to translate into an
experimentally implementable form. By contrast, the fast-forward approach devel-
oped by Masuda and Nakamura [Proc. R. Soc. A, 466(2116):1135, 2010] provides an
auxiliary potential that may be experimentally implementable but generally applies
only to ground states.
The central theme of this dissertation is that classical shortcuts to adiabatic-
ity can provide useful physical insights and lead to experimentally implementable
shortcuts for analogous quantum systems. We start by studying a model system of a
tilted piston to provide a proof of principle that quantum shortcuts can successfully
be constructed from their classical counterparts. In the remainder of the disser-
tation, we develop a general approach based on flow-fields which produces simple
expressions for auxiliary terms required for both counterdiabatic and fast-forward
driving. We demonstrate the applicability of this approach for classical, quantum as
well as stochastic systems. We establish strong connections between counterdiabatic
and fast-forward approaches, and also between shortcut protocols required for clas-
sical, quantum and stochastic systems. In particular, we show how the fast-forward
approach can be extended to highly excited states of quantum systems.
BRIDGING QUANTUM, CLASSICAL AND STOCHASTIC
SHORTCUTS TO ADIABATICITY
by
Ayoti Patra
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2017
Advisory Committee:
Professor Christopher Jarzynski, Chair/Advisor
Dr. Alexey V. Gorshkov
Professor Perinkulam S. Krishnaprasad, Dean’s Representative
Professor Rajarshi Roy
Professor Victor M. Yakovenko
© Copyright by
Ayoti Patra
2017
Dedication
To my father,
who envisoned a future scientist the first time he saw his infant daughter.
ii
Acknowledgments
First and foremost I’d like to thank my advisor, Prof. Christopher Jarzynski
for giving me an invaluable opportunity to work with him. Chris is an exemplary
scientist and an extremely humble person. Through his patient guidance and encour-
agement over the years, I have developed my scientific self, and learnt to persevere
through the challenges of research. He has shown me the importance of clarity in
research. His approach to solve difficult problems starting from simple text book
problems has left a deep impact on me. I will forever be grateful to Chris and will
strive to achieve the standards that he has set.
Next, I want to thank my first research advisor, Prof. Amit Dutta of IIT Kanpur.
I was not only fortunate to attend his courses during my entire time at IIT, but
also was priviledged to work with him for a year post M.Sc. My foundation as a
researcher was laid under the influence of his infectious enthusiasm for science. I
thank him for igniting my excitement and dedication for research. I also thank him
for introducing me to my life partner.
I thank all my physics teachers from UMD, IIT Kanpur and St. Stephen’s college
for building a strong foundation through rigourous training. I would specially like
to thank Dr. Paulo Bedaque and Dr. Zacharia Chacko from UMD, and Dr. Bikram
Phookun and Dr. Abhinav Gupta from St. Stephen’s. Late Dr. Swaminathan played
a pivotal role in starting my journey as a physicist and I am extremely thankful to
him for the chat we had during my initial days at St. Stephen’s.
My PhD experience has been enriched by the current and former members of
iii
the Jarzynski group. Thank you Sebastian, Yigit, Oren, Marcus, Shaon, Zhiyue,
Jeff and Andrew. A special thanks to Dibyendu for not only introducing me to the
group but also for all the scientific and moral support I got from him.
I thank the organizers of the annual workshops at Telluride, which have been
a rich hub for exchanging scientific ideas. I thank Prof. Krishnaprasad for the
stimulating discussions we had. I thank him and the other dissertation committee
members for volunteering their time and effort to help me improve this dissertation.
The past few years of my life would not have been as colorful and memorable
without my friends at College Park. I thank Wrick, Chana da (Anirban), Amit,
Swarnav, Manjistha, Sumalika, Mahashweta, Santanu, Arka, Apara, Jonathan,
Ranchu, Kim and my ex-roommates Shweta, Srimoyee, Vidya, Meera and Arthita
for all the good times we shared.
My deepest gratitude is reserved for my family – Patras and Dhabals – for
the myriad ways in which they have supported me. I consider myself extremely
fortunate to have a father and a husband who understand the technicalities of my
work, and therefore have supported me professionally as well as personally. Thank
you Baba for making science fun since an early age and for continuing to do so, and
Ma for always believing in my potential. Thanks Arnab for sharing your life with
me, and for making mine so grand as a result. Without you all, none of this would
have been possible.
Finally, I thank the U.S. Army Research Office (Grant number W911NF-13-1-
0390) and the University of Maryland for financial support.
iv
Table of Contents
Dedication ii
Acknowledgements iii
Table of Contents v
List of Tables vii
List of Figures viii
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Scope of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Shortcuts for a tilted piston 16
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Classical Dissipationless driving and the system under study . . . . . 18
2.3 Classical counterdiabatic terms . . . . . . . . . . . . . . . . . . . . . 21
2.3.1 Case (a): time-dependent slope at fixed length . . . . . . . . . 23
2.3.2 Case (b): time-dependent length at fixed slope . . . . . . . . . 25
2.4 Semiclassical counterdiabatic terms . . . . . . . . . . . . . . . . . . . 27
2.5 Solving the time-dependent Schro¨dinger Equation . . . . . . . . . . . 29
2.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3 Quantum shortcuts using flow-fields 45
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Review of quantum shortcuts . . . . . . . . . . . . . . . . . . . . . . 46
3.3 Setup and derivation of main results . . . . . . . . . . . . . . . . . . 48
3.4 Comparison with previous results . . . . . . . . . . . . . . . . . . . . 55
3.5 Divergences and a “no-flux” criterion . . . . . . . . . . . . . . . . . . 57
v
3.6 Scale-invariant dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.7 Numerical illustration of fast-forward driving . . . . . . . . . . . . . . 62
3.8 Extension to three degrees of freedom . . . . . . . . . . . . . . . . . . 69
3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4 Classical shortcuts using flow-fields 74
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2 Setup and definition of flow-field parameters . . . . . . . . . . . . . . 76
4.3 Counterdiabatic and fast-forward driving . . . . . . . . . . . . . . . . 80
4.4 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5 Quantum shortcuts for excited states 90
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2 Comparison of quantum and classical flow-fields . . . . . . . . . . . . 91
5.3 Auxiliary fields for excited states . . . . . . . . . . . . . . . . . . . . 94
5.4 Numerical illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.5 Semiclassical analysis of quantum peaks and final classical distribution107
5.6 Relating counterdiabatic and fast-forward driving . . . . . . . . . . . 111
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 Stochastic shortcuts using flow-fields 119
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2 Derivation of results . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.3 Comparision with engineered swift equilibration . . . . . . . . . . . . 125
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
A Derivation of Eq. 2.44 128
B Continuity conditions on H0(z, t) 132
C Flow under H0 +HCD preserves the adiabatic energy shell 135
D Local dynamical invariance of J(q, p, t) 137
E Evolution of the microcanonical measure under HFF 139
Bibliography 141
vi
List of Tables
2.1 The dependence of fidelity on the value of the reduced Planck’s con-
stant ~, keeping classical parameters fixed. The initial quantum num-
ber n is chosen such that the initial energy is En ≈ 80. Each simu-
lation is performed at fixed s = 3.0, while the box length is varied
from L = 25.0 to L = 15.0 at L˙ = −0.5. Fwcdmin is the minimum fi-
delity when the system evolves under Hˆ(t), and Fwocdmin is the minimum
fidelity when the system evolves under Hˆ0(t). . . . . . . . . . . . . . 42
vii
List of Figures
1.1 This figure shows a schematic representation of the adiabatic path
(blue dashed curve) and the path traversed by the system during
Hamiltonian evolution (red dotted curve). For counterdiabatic or
transitionless driving, the red and the blue curves coincide as the
system follows the adiabatic path at every instant. On the other hand,
for fast-forward driving, the system starts from a given state and
ends on the desired state at the final time τ , however at intermediate
times it is in general in a linear superposition of the instantaneous
eigenstates and does not follow the adiabatic path. . . . . . . . . . . 6
2.1 Three energy shells of H0 (Eq. 2.11) are shown for mass m = 1/2,
length L = 5 and slope s = 1.5. The green solid, red dashed and the
blue dotted curves correspond to E = 5.5(< sL), E = 7.5(= sL) and
E = 8.5(> sL) respectively. The brown dashed lines at q = 0 and
q = 5 denote the hard walls. . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Evolution of the probability density |ψ(q, t)|2 for a particle of mass
m = 1.0 in a box whose slope is fixed at s = 3.0 and whose length
is decreased from L = 25.0 to 15.0 at a rate L˙ = −0.5. Snapshots
of the wavefunction are taken at times t=0, 5.0, 10.0, 15.0 and 20.0.
The plots on the left depict evolution under the full Hamiltonian
Hˆ(t) = Hˆ0(t)+λ˙ · ξˆSC(λ(t)), while those on the right depict evolution
under Hˆ0(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 Same as Fig.2.2 except that the length of the box is fixed at L = 15.0,
while the slope is decreased from s = 13.0 to 3.0 at a rate s˙ = −0.5. . 37
2.4 Evolution of the fidelity F(t). The plot on the left is for the case
shown in Fig.2.2, whereas the plot on the right is for the same system
but subjected to the reverse process: the box length increases from
L = 15.0 to 25.0 at L˙ = 0.5. The dashed magenta curve depicts the
fidelity for evolution under Hˆ = Hˆ0 + λ˙ · ξˆSC , while the blue curve is
the fidelity upon evolution under Hˆ0. The inset is a magnified view
of the dashed magenta curve. . . . . . . . . . . . . . . . . . . . . . . 38
viii
2.5 Similar to Fig.2.4. The left plot is for the case shown in Fig.2.3,
whereas in the right plot the same system is subjected to the reverse
process: the slope s increases from s = 3.0 to 13.0 at s˙ = 0.5. . . . . . 38
2.6 Evolution of the fidelity under Hˆ0(t), for the simulations described
in Table 2.1. The lowermost (dashed magenta) curve corresponds to
~ = 1.0, the next one up (solid blue) corresponds to ~ = 2.0, and so
forth up to ~ = 7.0, which is the magenta curve that remains closest
to unity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1 The red curve φ2(q, t) depicts the probability distribution associated
with an energy eigenstate of Hˆ0(t). The blue vertical lines divide the
area under φ2(q, t) into K  1 strips of equal area. q(I, t) is the
right boundary of the shaded region, which has area I. The positions
of the vertical lines vary parametrically with t, and this “motion”
is described in terms of velocity and acceleration fields v(q, t) and
a(q, t), as given by Eq. 3.8. . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 U0(q, ξ) is plotted for five values of ξ. . . . . . . . . . . . . . . . . . . 63
3.3 Evolution under Hˆ0 (left panel) and Hˆ0 + UˆFF (right panel). The
solid magenta curves show Re(ψe−iα), and the dashed blue curves
show the eigenstate φ. Snapshots are shown at t = 0.05, at t = 0.1,
and at the end of the process, t = 0.2. . . . . . . . . . . . . . . . . . . 65
3.4 The variation of U0(q, t) and UFF (q, t) is plotted. The solid ma-
genta curves show U0(q, t) and the dashed blue curves show UFF (q, t)
at t = 0, 0.04, 0.055, 0.08, 0.12 and 0.2. U0(q, t) is initially a double
well potential, but as it evolves, the wells comes closer to the origin
and eventually U0(q, t) transforms to a single attractive well poten-
tial. UFF (q, t) smoothly increases from zero and quickly becomes
an attractive well, which then becomes a repulsive well that finally
transforms smoothly to zero at t = τ . . . . . . . . . . . . . . . . . . . 67
3.5 The blue dashed curve shows the fidelity |〈φ|ψ0〉|2, quantifying the
limited extent to which ψ0(q, t), evolving under Hˆ0, keeps pace with
the energy eigenstate φ(q, t). The solid red curve shows |〈φ|ψFF 〉|2,
which is the fidelity that is achieved when UˆFF is added to the Hamil-
tonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.1 Illustration of the classical adiabatic invariant. Fifty trajectories
evolving under a slowly varying Hamiltonian are shown at an ini-
tial time (on left) and a later time (on right). The closed curves are
instantaneous energy shells – level curves of H0 – with identical values
of the action I =
∮
p ·dq. Trajectories were generated using H(q, p, t)
given by Eq. 4.21, setting τ = 10.0 to achieve slow driving. . . . . . . 75
ix
4.2 The closed red curve, with upper and lower branches±p¯(q, t) (Eq. 4.7),
depicts the adiabatic energy shell E¯(t) in phase space. The blue ver-
tical lines divide E¯(t) into K  1 strips of equal phase space volume.
q(S, t) is the right boundary of the shaded region, of phase space
volume S. The parametric motion of the vertical lines defines the
velocity and acceleration fields v(q, t) and a(q, t). . . . . . . . . . . . 79
4.3 A snapshot, at t = τ/2, of 100 trajectories evolving under HFF(z, t)
using a rapid protocol, with τ = 0.2 (see text). The closed black
loop is the adiabatic energy shell E(t), and the red loop above it is
constructed by displacing each point on the lower loop by an amount
mv(q, t) along the p-axis. As predicted by Eq. 4.20, the trajectories
coincide with the red loop. . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4 Initial (a) and final (b,c) conditions for trajectories launched from a
single energy shell E(0). The trajectories in panel (b) evolved under
H(z, t) (Eq. 4.21), while those in panel (c) evolved under HFF =
H + UFF, with τ = 1.0. The solid black curves show the adiabatic
energy shell E(t) at initial and final times. . . . . . . . . . . . . . . . 86
4.5 A plot of U0(q, t) and UFF (q, t) is shown in sold magenta and dashed
blue curves respectively. UFF (q, t) is non-zero only in the interval
0 ≤ t ≤ τ . Shortly after t = 0, UFF (q, t) has a positive value to the
left of origin and a negative value to the right of origin, which ensures
that the particles from the left well are appropriately pushed towards
the right well. Thereafter, the value of UFF (q, t) to the right of the
origin begins to increase such that at t = τ/2 = 0.5, an attractive
well is formed. Beyond t = 0.5, UFF (q, t) starts to decrease to the
left of the origin, and finally it monotonically goes to zero at t = τ . . 87
5.1 A schematic plot of φ(q) vs. q is presented at times t and t + δt,
represented by dashed and solid curves respectively. The n’th node
q¯n is shown at times t and t + δt. The wavefunction ψ(q, t) evolving
under Hˆ0 +HˆCD should be guided by the auxiliary term in a way that
ψ(q¯n, t) = φ(q¯n, t) is satisfied for every node at every instant, i.e., the
nodes of φ(q, t) and ψ(q, t) should align at every instant. . . . . . . . 97
5.2 A plot of pn(t) = |〈φn(t)|ψ(t)〉|2 vs. n is presented at the initial time
t = 0. The system is initialized in the 35th energy eigenstate, which
is depicted by the single peak at n = 35 with p35(t = 0) = 1.0. . . . . 101
5.3 The plots above depict the overlap between the wavefunction |ψ(t)〉
as it evolves under Hˆ0(t) (defined in Eq. 5.9), and the instantaneous
energy eigenfunctions |φn(t)〉 for 0 ≤ n ≤ 70. The system is initial-
ized in the 35’th eigenstate, |ψ(0)〉 ≡ |φn(t)〉. The system is in a
superposition of instantaneous eigenstates at an intermediate time as
well as at the final time. The system has developed excitations dur-
ing the evolution and is unable to reach the final adiabatic state at
t = τ . This final state is analogous to the classical final state where
the trajectories do not end on the desired energy shell, see Fig. 4.4(b). 102
x
5.4 Same as Fig. 5.3, except that the system evolves under Hˆ0(t) +
UFF (qˆ, t), where UFF (qˆ, t) is the quantized counterpart of the clas-
sical fast-forward potential UFF (q, t) which is obtained numerically
from Eq. 4.11b. The system is in a superposition of instantaneous
eigenstates at an intermediate time, but it reaches the desired final
state at t = τ with high accuracy. At the final time, |ψ(τ)〉 has a
90% overlap with |φ35(τ)〉, i.e., p35 = 0.90. The combined probability
p34 + p35 + p36 = 0.98. Fig. 5.4(b) is analogous to the classical final
state where the trajectories end on the desired energy shell but the
initial uniform distribution is not preserved, see Fig. 4.4(c). . . . . . . 103
5.5 The final distribution of classical trajectories is non-uniform as de-
picted in the phase-space plot. This non-uniformity is reflected in the
quantum evolution as depicted in the plot of pn(τ) = |〈φn(τ)|ψ(τ)〉|2
vs. n (same as Fig. 5.4(b)). The peak value is p35(τ) = 0.90. . . . . . 104
5.6 Same as Fig. 5.6, except E = 25.08 and n = 28. The final classi-
cal distribution has a higher degree of non-uniformity compared to
Fig. 5.5(a), which is reflected in the quantum evolution. In Fig. 5.6(b),
the peak value is p28(τ) = 0.62, and sideband excitations are more
prominent compared to Fig. 5.5(b). . . . . . . . . . . . . . . . . . . . 105
5.7 A plot of pn(t) = |〈φn(t)|ψ(t)〉|2 vs. n is presented at the initial time
t = 0. The system is initialized in the 35th energy eigenstate, which
is depicted by the single peak at n = 35 with p35(t = 0) = 1.0. . . . . 111
5.8 The magenta curves depict the instantaneous probability density |ψ(q, t)|2
of the wavefunction ψ(q, t) evolving under Hˆ0+UˆFF . |ψ(q, t)|2 is plot-
ted with respect to q for times t = 0, τ/5, 2τ/5, 3τ/5, 4τ/5 and τ , for
τ = 1.0. The blue curves correspond to |φ18(q, t)|2, the probability
density of the instantaneous energy eigen state. The other parameters
for numerical evolution were chosen as m = 1, ~ = 2 and E = 53.76,
which corresponds to n = 18. As seen in the snapshots, the minima
of |ψ(q, t)|2 align with the nodes of |φ18(q, t)|2 at every instant, but
the amplitudes do not match. . . . . . . . . . . . . . . . . . . . . . . 115
6.1 The blue lines divide the equilibrium distribution into strips of equal
area. q(F , t) is the right boundary of the shaded region, which has
area F . The velocity field v(q, t) describes the motion of the vertical
lines with t (Eq. 6.6). . . . . . . . . . . . . . . . . . . . . . . . . . . 123
A.1 The function η0(t) plotted over one time period of oscillation is a
square wave (top figure). The function ητ (t) is obtained by shifting
this square wave leftward by an amount τ (middle figure). The auto-
correlation function C(τ) is the product of these square wave pulses,
integrated over one period, yielding a triangular wave (bottom figure). 129
xi
Chapter 1: Introduction
1.1 Background
1.1.1 Overview
In the last few decades, experimental advancements in quantum optics have led
to a surge of research in quantum information and computation. As a result, the need
to create and manipulate quantum states with high speed and accuracy has become
increasingly important. However, the quantum adiabatic theorem due to Max Born
and Vladimir Fock, which states that “a physical system remains in its instantaneous
eigenstate if a given perturbation is acting on it slowly enough and if there is a gap
between the eigenvalue and the rest of the Hamiltonian’s spectrum” [1,2], comes as
a hindrance. Adiabatic processes make a system robust against systematic errors
and various kinds of noise by enabling it to adapt its configuration and retain its
initial character. However, the requirement for slow driving is not only inefficient,
but also makes the system vulnerable to environmental decoherence thereby leading
to loss of important quantum mechanical features. Shortcuts to adiabaticity – a
term coined in 2010 by Chen et. al [3] – are strategies for achieving adiabatic results
with fast driving protocols. Shortcuts to adiabaticity have been applied to many
1
fields including adiabatic quantum computing [4–6], cold atom transport [3, 7, 8],
many-body state engineering [9–12], quantum sensing and metrology [13] quantum
simulation [14,15] and quantum thermodynamics [16–18].
Shortcut protocols achieve the desired evolution by means of an auxiliary Hamil-
tonian which supresses the excitations arising due to fast driving [19]. A variety
of shortcut protocols have been developed including invariant-based inverse engi-
neering [20, 21], transitionless counterdiabatic driving [22–25], fast-forward meth-
ods [11, 26–30] and methods based on unitary [31–35] or gauge [36] transforma-
tions. Shortcuts to adiabaticity have been extended to non-Hermitian Hamiltoni-
ans [37,38], open quantum systems [39,40] and Dirac-dynamics [41–43]. They have
been demonstrated experimentally [14,44–48], and their relationship with quantum
speed limits has been clarified [49,50].
Adiabatic processes are significant also for classical systems. They preserve
quantities known as adiabatic invariants. While the quantum number of an initial
energy eigenstate is the quantum adiabatic invariant, the classical action of a system,
equivalently the volume of phase space enclosed by a surface of constant energy, is
the classical adiabatic invariant in one degree of freedom [51]. Studies on shortcuts
to adiabaticity for classical dynamics, which aim at preserving the classical action of
a rapidly driven system, has also gained attention recently [25,34,52–56]. Analogous
problems for stochastic dynamics have also been studied [57–60].
2
1.1.2 Methods
In this dissertation, we focus on developing general methods to construct short-
cuts to adiabaticity whilst its direct applications on physical systems are not dis-
cussed in detail. Here, we review the transitionless quantum driving protocol, de-
veloped independently by Dermirplak and Rice [22, 23], and Berry [24]; and the
fast-forward method developed by Masuda and Nakamura [26], which are of partic-
ular relevance to this work. We use the terms ‘counterdiabatic driving’ and ‘transi-
tionless driving’ synonymously to indicate that during these processes, the system
tracks the adiabatic path at every instant, as shown in Fig. 1.1(a). In contrast,
fast-forward driving refers to those processes where the system starts from a given
state and ends up at the desired state at the final time, but need not follow the
adiabatic trajectory at intermediate times, see Fig. 1.1(b).
Consider a system initialized in the n’th energy eigenstate and evolving under
a time-dependent Hamiltonian Hˆ0(t). In the counterdiabatic approach, an auxiliary
Hamiltonian HˆCD(t) is constructed, such that when the system evolves under
Hˆ(t) = Hˆ0(t) + HˆCD(t), (1.1)
it follows the adiabatic trajectory of Hˆ0(t) at every instant even for rapid driving,
see Fig. 1.1(a). The auxiliary term HˆCD(t) ensures that the excitations to other
eigenstates are supressed at every instant [22,24]. The exact expression for HˆCD(t)
was derived by Berry [24] using reverse engineering. Berry’s derivation is straight-
3
forward and simple, and is presented below. An energy eigenstate |n(t)〉 of Hˆ0(t)
satisfies the eigenvalue equation
Hˆ0(t)|n(t)〉 = En(t)|n(t)〉, (1.2)
and the corresponding adiabatic path is given by [1]
|ψn(t)〉 = exp
[
− i
~
∫ t
0
dt′En(t′)−
∫ t
0
dt′〈n(t′)|∂t′n(t′)〉
]
|n(t)〉. (1.3)
The first term in the phase is dynamically generated while the second term arises
due to a geometric contribution where the integrand 〈n(t′)|∂t′n(t′)〉 acts as an ef-
fective vector potential. In the reverse engineering approach, a Hamiltonian Hˆ(t)
needs to be determined for which |ψn(t)〉 is an exact solution of the time-dependent
Schro¨dinger equation
i~∂t|ψn(t)〉 = Hˆ(t)|ψn(t)〉. (1.4)
It can be verified that any time-dependent unitary operator Uˆ(t) satisfies
i~∂tUˆ(t) = Hˆ(t)Uˆ(t), (1.5)
where
Hˆ(t) = i~(∂tUˆ(t))Uˆ †(t). (1.6)
4
By choosing
Uˆ(t) =
∑
n
|ψn(t)〉〈n(0)|
=
∑
n
exp
[
− i
~
∫ t
0
dt′En(t′)−
∫ t
0
dt′〈n(t′)|∂t′n(t′)〉
]
|n(t)〉〈n(0)|, (1.7)
and upon using Eq. 1.6, Berry [24] showed that the Hamiltonian driving the eigen-
states |n(t)〉 according to Eq. 1.4 is
Hˆ(t) =
∑
n
|n(t)〉En(t)〈n(t)|+ i~
∑
n
(|∂tn〉〈n| − 〈n|∂tn〉|n〉〈n|) = Hˆ0(t) + HˆCD(t).
(1.8)
Therefore the counterdiabatic Hamiltonian is given by
HˆCD(t) = i~
∑
n
(|∂tn〉〈n| − 〈n|∂tn〉|n〉〈n|), (1.9)
where |n〉 = |n(t)〉 denotes the instantaneous n’th eigenstate of Hˆ0(t), and |∂tn〉 ≡
∂t|n(t)〉. If a wavefunction evolves under Hˆ0 +HˆCD from an initial state |n(0)〉, then
the quantum number n is preserved at every instant during the evolution. Note
that HˆCD(t) in Eq. 1.9 does not depend on the choice of n. Transitionless quantum
driving is derived from basic principles of unitary evolution and is consequently
quite general: it applies both to spatially continuous systems such as a particle in a
time-dependent potential, and to discrete-state, e.g. spin, systems.
It was shown for a harmonic oscillator and a particle in a box by Chen, et al. [20]
and del Campo et al. [61] respectively, that Eq 1.9 can be recast in terms of position
5
| 0 
(a) Counterdiabatic driving
| 0 


(b) Fast-forward driving
Figure 1.1: This figure shows a schematic representation of the adiabatic path (blue
dashed curve) and the path traversed by the system during Hamiltonian evolution
(red dotted curve). For counterdiabatic or transitionless driving, the red and the
blue curves coincide as the system follows the adiabatic path at every instant. On
the other hand, for fast-forward driving, the system starts from a given state and
ends on the desired state at the final time τ , however at intermediate times it is
in general in a linear superposition of the instantaneous eigenstates and does not
follow the adiabatic path.
and momentum operators to yield
HˆCD ∝ pˆqˆ + qˆpˆ. (1.10)
Rewriting HˆCD in terms of operators corresponding to physical observables provide
physical insight into the underlying process and pave the way for experimental
implementation. Also, the knowledge of the entire eigenspectrum at every instant
will not be required if a systematic procedure is developed to obtain HˆCD in terms
of operators like pˆ and qˆ.
In Ref. [25], Jarzynski has shown that the original Hamiltonian written as a
function of time-dependent parameters λ leads to an auxiliary Hamiltonian HˆCD(t)
6
which can be cast in the form:
Hˆ(t) = Hˆ0(λ(t)) + HˆCD(t)
= Hˆ0(λ(t)) + λ˙ · ξˆ(λ(t)). (1.11)
Here, ξˆ(λ) is a vector of Hermitian operators: [24]
ξˆ(λ(t)) = i~
∑
m
(
|∂λm〉〈m| − 〈m|∂λm〉|m〉〈m|
)
, (1.12)
where the sum is taken over eigenstates |m(λ)〉 of Hˆ0(λ), and |∂λm〉 ≡ ∂λ|m(λ)〉.
The vector ξˆ(λ) can be viewed as a generator of adiabatic transformations which
associates infinitesimal displacements in parameter space, λ → λ + δλ, with dis-
placements in Hilbert space, |ψ〉 → |ψ〉+ |δψ〉, according to the rule
i~|δψ〉 =
[
δλ · ξˆ(λ)
]
|ψ〉. (1.13)
For an eigenstate |n(λ)〉, up to first order in δλ, Eq. 1.13 leads to the following
displacement in the Hilbert space [25]:
|n(λ)〉 →
(
1 +
1
i~
δλ · ξˆ
)
|n(λ)〉 = exp [−δλ · 〈n|∇n〉] |n(λ + δλ)〉 (1.14)
When Eq. 1.13 is applied stepwise along a curve λs in parameter space, the wave-
function gets transported along eiϕs|n(λs)〉, where the phase is the line integral of
i〈n|∇n〉. Thus the system is escorted along the eigenstates of Hˆ0(λ) as a result of
7
the unitary flow in Hilbert space generated by the operator ξˆ.
Alternatively, it can be stated that a small displacement in time, t → t + δt
leads to the following displacement in the Hilbert space:
e−iδt HˆCD(t)/~|n(t)〉 = |n(t+ δt)〉 , (1.15)
aside from an overall phase. Eq. 1.15 clarifies why adding HˆCD(t) to Hˆ0 produces
transitionless driving [25]: for each time step δt, the evolution operator under Hˆ0 +
HˆCD(t) is e
−iδt Hˆ0/~ e−iδt HˆCD/~, which first evolves the state |n(t)〉 to |n(t+ δt)〉, and
then contributes an increment in the dynamical phase, e−iδt En/~. Here, δt is taken
to be infinitesimal, and O(δt2) corrections have been ignored.
The generator ξˆ can alternatively be specified by the conditions: [25]
[
ξˆ, Hˆ0
]
= i~
(
∇Hˆ0 − diag(∇Hˆ0)
)
(1.16a)
〈n|ξ|n〉 = 0, (1.16b)
where diag(∇Hˆ0) =
∑
m |m〉〈m|∇Hˆ0|m〉〈m|. Eq. 1.16a determines the off-diagonal
elements of ξ, while Eq. 1.16b sets the diagonal elements. The identity 〈m|∇n〉 =
〈m|∇Hˆ0|n〉/(En − Em) [24] establishes the equivalence of the two definitions of ξˆ
(Eqs. 1.12 and 1.16).
Eqs. 1.16 have a classical counterpart described in Eqs. 1.18. This observation
leads to the following question: Can a classical generator ξ(λ(t)) solve a problem
on dissipationless classical driving – the classical analogue of transitionless quantum
8
driving? In one degree of freedom, the classical adiabatic invariant is the action
I0(q, p;λ) =
∮
p′ dq′, equivalently the volume of phase space enclosed by a surface of
constant energy [51]. The problem of dissipationless classical driving is formulated
as follows: For a classical time-dependent Hamiltonian H0(q, p;λ(t)), find the coun-
terdiabatic term HCD(q, p, t) = λ˙ · ξ(q, p;λ(t)) such that the action I0(q, p;λ(t))
(defined with respect to H0) remains constant along any trajectory evolving under
the Hamiltonian
H(t) = H0(λ(t)) + λ˙ · ξ(λ(t)). (1.17)
It was shown in by Jarzynski in Ref. [25] that the classical generator ξ(λ(t)) can be
determined by the following equations [62]:
{ξ, H0} = ∇H0 − 〈∇H0〉H0,λ ≡∇H˜0 (1.18a)
〈ξ〉E,λ = 0, (1.18b)
where {·, ·} denotes the Poisson bracket. Note that Eqs. 1.18 are the classical ana-
logues of Eqs. 1.16.
A natural question then arises: if we solve for the classical generator ξ(q, p;λ)
and then quantize it to obtain an operator ξˆ(λ) ≡ ξ(qˆ, pˆ;λ), will the term λ˙· ξˆ(λ(t))
suppress non-adiabatic transitions under quantum evolution? In other words, can
we construct HˆCD – either exactly or approximately – by first obtaining its classical
counterpart and then quantizing it? In Ref. [25], it was shown that for even-power-
law potentials in one degree of freedom, the classical HCD obtained in terms of
9
position and momentum, upon quantization produces the correct quantum auxiliary
term HˆCD. Deffner et al. extended the idea of solving for transitionless quantum
driving using an analogous problem on dissipationless classical driving for the general
class of scale-invariant dynamical processes which describe expansion and transport
[34]. In this dissertation, we investigate the example of a time-dependent tilted
piston, which does not follow scale invariance, and explore whether the aproach
from Ref. [25] is applicable to a more general system.
The fast-forward approach, due to Masuda et al. [26], solves for an auxiliary
potential UFF (qˆ, t) for a system evolving under a kinetic plus potential Hamiltonian
of the form Hˆ0(t) = pˆ
2/2m + U0(qˆ, t). This is in contrast with the counterdiabatic
approach which solves for a general auxiliary Hamiltonian HˆCD(t). As shown in
Fig. 1.1(b), the system need not follow the adiabatic path but the goal is to make
the system end in the desired state at the final time. The system may choose
any path from the initial to the final state and one such path is accessible by a
potential UˆFF (t). The experimental implementation of a fast-forward shortcut is
more feasible as compared to that of a CD shortcut. The derivation of UFF (qˆ, t)
derived in Refs. [26,28] is straightforward and briefly shown below.
It is assumed that an ansatz of the form
ψ¯n(q, t) = ψn(q, t) exp
[
i
S(q, t)
~
]
= φn(q, t) exp
[
− i
~
∫ t
0
En(t
′)dt′
]
exp
[
i
S(q, t)
~
]
(1.19)
describes the state of the system during its evolution, where ψn(q, t) is the position
10
space representation of the adiabatic path described in Eq. 1.3, φn(q, t) = 〈q|n〉
and En(t) are the corresponding instantaneous energy eigenfunction and eigen-
value respectively. The ansatz ψ¯n(q, t) therefore must satisfy the time-dependent
Schro¨dinger equation
i~ψ¯n(q, t) = (Hˆ0 + UFF (qˆ, t))ψ¯n(q, t). (1.20)
We have introduced two unknowns S(q, t) and UFF (q, t) which can be solved by
substituting Eq. 1.19 in Eq. 1.20. For a one-dimensional system, separating the
imaginary and real parts of the resulting equation gives rise to the following two
differential equations respectively [28]:
∂tφn +
1
m
∂qφn∂qS +
1
2m
φn∂
2
qS = 0, (1.21a)
1
2m
φn(∂qS)
2 + UFFφn = −φn∂tS. (1.21b)
Solving Eq. 1.21a determines the function S(q, t), which can then be substituted
in Eq. 1.21b to obtain UFF (q, t). These results can be easily generalized for three
dimensional systems as well.
The main differences between counterdiabatic and fast-forward driving is listed
below. The auxiliary fast-forward potential UFF (qˆ, t) in general depends on the ini-
tial state, i.e., on the initial quantum number n unlike the auxiliary counterdiabatic
Hamiltonian HˆCD(t) which is independent of n. Therefore, UFF (qˆ, t) is different for
every energy level while a single HˆCD(t) is valid for the entire eigenspectrum. Also
11
note that Eq. 1.21a can be recast as a continuity equation of the probability density:
∂tφ
2
n + ∂q(vφ
2
n) = 0, (1.22)
where v(q, t) = ∂qS/m. It can be seen that in general, the velocity diverges at nodes
of φn. Therefore, S(q, t) and consequently UFF (q, t) can in general be solved only for
the ground state where nodes are absent. Since the fast-forward potential typically
(though not always) becomes singular at nodes of the instantaneous eigenstate, the
applicability of the fast-forward method becomes restricted to the ground state only.
Such singularities do not occur in HˆCD(t). As pointed out earlier, UFF (qˆ, t) is easier
to implement experimentally compared to HˆCD(t).
A scale-invariant process is an exception for which UFF (qˆ, t) in independent
of n and does not suffer from singularities. This will be shown in Sec. 3.6. The
auxiliary potential UFF (qˆ, t) for a scale invariant process was obtained by Deffner
et al. in Ref. [34]. A canonical transformation was performed on the classical CD
term, HCD(t), required for dissipationless classical driving to obtain an auxiliary
classical potential UFF (q, t). The classical UFF (q, t) was then quantized to obtain
UFF (qˆ, t). In this dissertation, we reformulate Eqs. 1.21 by introducing ‘flow-fields’.
We extend the fast-forward method to classical systems evolving more generally, i.e.,
not following scale invariance, and use the classical solutions to solve for approximate
UFF (qˆ, t) beyond the ground state.
12
1.2 Scope of this work
The motivation behind this thesis comes from the need to obtain experimentally
implementable quantum shortcut protocols for generic systems which are subjected
to an arbitrary driving protocol. We study beyond the simple scale invariant or
self-similar driving which preserves the topology of a system and only accounts for
expansion and translation. In an attempt to solve for quantum shortcuts, we often
solve for the analogous classical problem first and gain useful physical insights. We
focus on obtaining auxiliary counterdiabatic and fast-forward terms which can be
applied to a generic system and are robust against the limitations specified in the
previous section. In the rest of the thesis, we study one-dimensional systems unless
mentioned otherwise. The thesis is structured as follows:
ˆ In the second chapter, we investigate the example of a tilted piston subjected
to non-scale-invariant driving. We solve exactly for the classical CD Hamil-
tonian HCD(q, p, t) following the method introduced in Ref. [25]. HCD(q, p, t)
is then quantized to obtain a Hermitian operator HˆCD(t). Using numerical
simulations, we find that HˆCD effectively suppresses non-adiabatic excitations
under rapid driving. This chapter offers a proof of principle – beyond the spe-
cial case of scale-invariant driving – that quantum shortcuts to adiabaticity
can successfully be constructed from their classical counterparts. However,
even for such a simple example, the expression for HCD(q, p, t) is very compli-
cated when the system is driven in a non scale invariant fashion as compared
13
to the HCD(q, p, t) required for scale invariant driving. We therefore follow a
different approach to shortcuts in the subsequent chapters.
ˆ In chapter three, we introduce a general framework for constructing shortcuts
to adiabaticity from flow-fields that describe the desired adiabatic evolution.
This approach provides surprisingly compact expressions for both counter-
diabatic Hamiltonians and fast-forward potentials. We illustrate our method
with numerical simulations of a model system, and we compare our shortcuts
with previously obtained results. Our method, like the fast-forward approach
developed previously [26], is susceptible to singularities when applied to ex-
cited states of quantum systems; we propose a simple, intuitive criterion for
determining whether these singularities will arise, for a given excited state.
ˆ We extend the flow-fields based framework to classical systems in chapter four,
and construct approximate counterdiabatic Hamiltonian and fast-forward po-
tential which preserves the classical action under non-adiabatic conditions.
We show that the fast-forward potential guides all trajectories with an ini-
tial action I0 to end with the same value of action. We also construct a
local dynamical invariant J(q, p, t) whose value remains constant along these
trajectories. We illustrate our results with numerical simulations of a model
double-well system. We sketch how these classical results may be used to
design approximate quantum shortcuts to adiabaticity.
ˆ In chapter five, we demonstrate how to construct counterdiabatic Hamilto-
nian and fast-forward potential for quantum excited states subjected to an
14
arbitrary driving protocol. In order to construct the shortcut, we will use
semiclassical analysis of a quantum wavefunction along with classical tools for
obtaining shortcuts. We illustrate our results using the quantum analogous
model double-well system from chapter four. We also carry out a semiclassical
analysis of the final distribution of trajectories on a classical energy shell to
quantitatively predict the accuracy of the quantal shortcut.
ˆ In chapter six, we demonstrate the universality of the flow-fields based method
by extending it to solve analogous shortcut protocols for stochastic systems.
We solve the counterdiabatic potential for a system of overdamped Browninan
particles subjected to a rapidly changing trapping potential. This counterdia-
batic potential enables the system to track its instantaneous equilibrium state
at every instant. We compare our method with previous theoretical and ex-
perimental work on swift equilibration [59] of Brownian particles.
This dissertation has elaborated the results presented by the author and col-
laborators in the following papers. Chap. 2 presents the contents from Ref. [54],
Chaps. 3 and 6 present contents from Ref. [63], and Chap. 4 presents content from
Ref. [56]. The contents of Chap. 5 are under preparation for submission to a journal.
15
Chapter 2: Shortcuts for a tilted piston
2.1 Overview
In this chapter, we study if Jarzynski’s method from Ref. [25] can be extended
to more general systems subjected to an arbitrary driving protocol. We study a
test case: a particle in a box with infinite walls and a slanted base, i.e. a tilted
piston. Recasting the counterdiabatic Hamiltonian HˆCD(t) (Eq. 1.9) in terms of
position and momentum operators may provide physical insights and pave way for
experimental implementation. As discussed in Chap. 1, for a harmonic oscillator
with a time-dependent stiffness k(t), Eq. 1.9 reduces to: [20]
HˆCD = − k˙
8k
(pˆqˆ + qˆpˆ). (2.1)
Similar expressions hold for a particle in a box [61], for attractive power law po-
tentials [25], and more generally for arbitrary potentials undergoing scale invariant
driving (see Eq. 2.10), characterized by simple expansion, contraction or translation
of the potential [32, 34]. However, for general Hˆ0(t) it is not clear how to rewrite
Eq. 1.9 in terms of operators such as qˆ and pˆ.
In Ref. [25] it was proposed that a problem on transitionless quantum driving
16
can usefully be approached by studying dissipationless classical driving – the classi-
cal counterpart of transitionless quantum driving. Consider a Hamiltonian Hˆ0(λ),
where λ denotes a vector of externally controlled parameters that are varied with
time according to a protocol λ(t). In the transitionless quantum driving approach,
the counterdiabatic Hamiltonian HˆCD(t) of Eq. 1.9 ensures that when the system
evolves under the full Hamiltonian Hˆ(t) given by Eq. 1.1, it follows the adiabatic
trajectory of Hˆ0(λ(t)) even for rapid driving, i.e. the term HˆCD(t) suppresses non-
adiabatic excitations. The counterdiabatic Hamiltonian can be expressed as [25]
HˆCD(t) = λ˙ · ξˆ(λ(t)). (2.2)
From Eq. 2.2, it is clear that the more rapidly the parameters λ are varied, the
greater the magnitude of the term HˆCD = λ˙·ξˆ needed to suppress excitations [10,64].
It was shown in Ref. [25] that the proposed strategy for constructing HˆCD yields
correct result for attractive power law potentials, including the harmonic oscillator
and the particle in a box as limiting cases. This encouraging result was generalized
to arbitrary potentials undergoing scale-invariant driving (Eq. 2.10) [32, 34]. In all
these cases the classical counterdiabatic term takes the form HCD = g(t)p+ h(t)qp,
and its quantized counterpart
HˆCD = g(t)pˆ+
h(t)
2
(pˆqˆ + qˆpˆ) (2.3)
can be shown to be equivalent to Eq. 1.9.
17
In this chapter, we investigate whether this strategy succeeds for non-scale-
invariant driving protocols. In Sec. 2.2, we briefly review dissipationless classical
driving, and we specify the tilted piston Hamiltonian and the driving protocols that
will be studied. We solve exactly for HCD(q, p, t) in Sec. 2.3, and we quantize it semi-
classically in Sec. 2.4. Finally we study numerically whether the resulting quantum
operator HˆCD(t) produces the desired transitionless quantum driving. Details of
the numerical approach are described in Sec. 2.5, and the results are presented in
Sec. 2.6. We summarize the results from this chapter in Sec. 2.7.
2.2 Classical Dissipationless driving and the system under study
Let z = (p, q) denote a point in the system’s two-dimensional phase space.
The counterdiabatic Hamiltonian HCD(z; t) = λ˙ · ξ(z;λ(t)), when added to the
unperturbed Hamiltonian H0(z;λ(t)), ensures that the classical system follows an
adiabatic trajectory – along which the action I0 is constant – even when the driving
is rapid. Below we briefly summarize how ξ(z;λ) is constructed [25].
Let the microcanonical average of a quantity be denoted by
〈. . .〉E,λ ≡ 1
∂EΩ
∫
dzδ(E −H0) . . . , (2.4)
and the volume of phase space enclosed by an energy shell E be denoted by
Ω(E,λ) ≡
∫
dzθ [E −H0(z;λ)] . (2.5)
18
Then the desired classical generator ξ satisfies [25]:
ξ(zb;λ)− ξ(za;λ) =
∫ b
a
dt∇H˜0(z(t);λ), (2.6)
where za and zb are two points on the energy shell E, z(t) is a trajectory that
evolves from za to zb under H0(z;λ) (with λ fixed), and the integrand is defined
as ∇H˜0 ≡ ∇H0 − 〈∇H0〉E,λ, with ∇ ≡ ∂/∂λ. By convention the microcanonical
average of ξ is set to zero,
〈ξ〉E,λ = 0. (2.7)
Eqs. 2.6 and 2.7 uniquely specify the generator ξ(z;λ).
For a system with one degree of freedom, the time average and the microcanon-
ical average of a quantity are equivalent, therefore we can compute 〈∇H0〉E,λ by
evaluating the time average of ∇H0 along a periodic trajectory of energy E. Al-
ternatively, this microcanonical average can be determined by defining the inverse
function E(Ω,λ) from Ω(E,λ), and by using the cyclic identity of partial derivatives:
∇E(Ω,λ) = −∇Ω(E,λ)
∂EΩ(E,λ)
= 〈∇H0〉E,λ. (2.8)
For a harmonic oscillator with a time-dependent stiffness k(t), the procedure
described above leads to the classical counterdiabatic Hamiltonian
HCD(z, t) = − k˙
4k
qp. (2.9)
19
Upon quantization, this result agrees with the quantum counterdiabatic Hamilto-
nian for the harmonic oscillator, Eq. 2.1, which was originally obtained by direct
evaluation of the Demirplak-Rice-Berry formula, Eq. 1.9 in Ref. [20]. More generally,
a time-dependent potential of the form
V (q; f, γ) =
1
γ2
V0
(
q − f
γ
)
(2.10)
where f = f(t) and γ = γ(t), is said to undergo scale-invariant driving. For scale-
invariant driving, Eqs. 2.6 and 2.7 lead to a simple expression for HCD(z; t) that,
upon quantization, give the exact quantum counterdiabatic Hamiltonian HˆCD(t) [34],
in the form given by Eq. 2.3.
To investigate how well these results extend to systems that are driven in non-
scale-invariant fashion, we will study a tilted piston: a particle of mass m confined
in a one-dimensional box with infinite walls and a slanted base. In terms of the
length of the box L and slope of its base s, the classical Hamiltonian is given by
H0 (q, p; s, L) =
p2
2m
+ sq + Θ(q; 0, L), (2.11)
where the function
Θ(q; qL, qR) =

0, qL < q < qR
∞ otherwise
(2.12)
describes hard walls at q = qL and q = qR.
20
We subject the system to two different driving protocols. In case (a), the slope
s is changed while the length L is held fixed, whereas in case (b), s is held fixed and
the box length is changed by moving the wall at q = L. Without loss of generality,
we will assume that the slope is positive, s > 0, and that the wall at q = 0 remains
fixed. Although the protocols (a) and (b) are both non-scale-invariant, when s and
L are varied simultaneously while holding sL3 fixed, the system undergoes scale-
invariant driving, as verified directly from Eqs. 2.10 and 2.11.
2.3 Classical counterdiabatic terms
Fig.2.1 illustrates the classical energy shells of the Hamiltonian H0(q, p; s, L)
defined by Eq. 2.11. There is a critial value of energy, Ec = sL, below which a
classical particle interacts only with the wall at q = 0, and above which the particle
interacts with both walls. When E ≤ Ec, the energy shell is a single curve that is
symmetric about the q-axis and has a discontinuity at q = 0, whereas when E > Ec,
the energy shell is a pair of curves symmetric about the q-axis with discontinuities
at q = 0 and q = L. The expression for the classical counterdiabatic generator
ξ(z,λ(t)) depends on Ec.
Let Ω< denote the volume enclosed by an energy shell of energy E < Ec and Ω>
denote the volume enclosed when E > Ec. From Eq. 2.11 using p =
√
2m(E − sq)
21
0 1 2 3 4 5
-3
-2
-1
0
1
2
3
q
p
Figure 2.1: Three energy shells of H0 (Eq. 2.11) are shown for mass m = 1/2, length
L = 5 and slope s = 1.5. The green solid, red dashed and the blue dotted curves
correspond to E = 5.5(< sL), E = 7.5(= sL) and E = 8.5(> sL) respectively. The
brown dashed lines at q = 0 and q = 5 denote the hard walls.
we obtain
Ω<(E, s, L) = 2
∫ E/s
0
p dq =
4
√
2mE
3
2
3s
, (2.13a)
Ω>(E, s, L) = 2
∫ L
0
p dq =
4
√
2m
3s
[
E
3
2 − (E − sL) 32
]
. (2.13b)
We now solve explicitly for the classical generator ξ(z;λ) specified by Eqs. 2.6
and 2.7. We analyze separately the two protocols (a) and (b) mentioned above.
22
2.3.1 Case (a): time-dependent slope at fixed length
The time-dependent parameter λ is the slope s in this case. Let us first solve
for ξ when E ≤ Ec. Using Eqs. 2.8 and 2.13a
〈∂sH0〉E,s = − ∂sΩ<(E, s, L)
∂EΩ<(E, s, L)
=
2E
3s
, (2.14)
and from Eq. 2.11
∇H0 = ∂sH0 = q. (2.15)
Hence, from Eq. 2.6, upon using dt = mdq/
√
2m(E − sq), we arrive at
ξ(zb; s)− ξ(za; s) =
∫ b
a
dt∇H˜0(z(t); s) = −qp
3s
∣∣∣∣b
a
. (2.16)
Together with the condition 〈ξ〉E,s = 0, this result gives us, for E < sL,
ξ<C (q, p; s) = −
qp
3s
, (2.17)
where the subscript C stands for classical. This simple form for Eq. 2.17 reflects
the fact that when E < sL, varying the slope of the base at fixed box length is
tantamount to scale invariant driving.
We now construct the generator ξ>C (q, p; s), for E ≥ Ec. For the upper and
the lower branches of the energy shell, let ξ(0, p(0); s) = ξ0, and ξ(L, p(L); s) = ξ1
23
denote two constants to be determined later. Eqs. 2.8 and 2.13b yield
〈∂sH0〉E,s = E + sL−
√
E(E − sL)
3s
. (2.18)
Using Eqs. 2.6, 2.15 and 2.18, we obtain the following two equations for the upper
and the lower branches respectively:
ξ(z; s)− ξ0 = − p
3s2
[
E − sL+
√
E(E − sL)
]
− pq
3s
+
√
2mE
3s2
[
E − sL+
√
E(E − sL)
]
, (2.19a)
ξ(z; s)− ξ1 = − p
3s2
[
E − sL+
√
E(E − sL)
]
− pq
3s
−
√
2m(E − sL)
3s2
[
E +
√
E(E − sL)
]
. (2.19b)
The constants ξ0 and ξ1 are now determined by demanding continuity of ξ at q = 0
and q = L, along with the condition that 〈ξ〉E,s = 0. A series of simple calculations
yield ξ0 = 0 = ξ1. Eqs. 2.19 can be rearranged to obtain the general expression for
the classical generator when E > sL
ξ>C (q, p; s) = −
p
3s2
[
E − sL+
√
E(E − sL)
]
− pq
3s
+ sign(p) ·
√
2m
3s2
[
E
√
E − sL+
√
E(E − sL)
]
, (2.20)
where sign(p) = +1 for the upper branch and −1 for the lower branch. As a
consistency check, we note that at the critical energy E = Ec = sL, Eq. 2.20
reduces to Eq. 2.17.
24
2.3.2 Case (b): time-dependent length at fixed slope
In this case, the length L plays the role of the parameter λ. A particle with
energy E ≤ Ec is not influenced by the motion of the wall at q = L. Hence we
expect
ξ<C (q, p;L) = 0. (2.21)
This is mathematically verified by noting that ∂LH0 = 0 for q < L, and 〈∂LH0〉E,L =
0 (from Eqs. 2.8 and 2.13a), therefore Eq. 2.6 gives us ξ<C = 0. When E > Ec,
Eqs. 2.8 and 2.13b yield 〈∂LH0〉E,L = −
[
E − sL+√E(E − sL)] /L. Hence at all
points except at q = L,
∂LH˜0(z(t);L) =
E − sL+√E(E − sL)
L
. (2.22)
Analogous to case (a), to-be-determined constants ξ(0, p(0);L) = ξ′0 and
ξ(L, p(L);L) = ξ′1 are introduced for the upper and lower branches respectively.
Using Eqs. 2.6 and 2.22 we obtain, for the upper and the lower branches respectively:
ξ(z;L)− ξ′0 =
E − sL+√E(E − sL)
sL
[√
2mE − p
]
, (2.23a)
ξ(z;L)− ξ′1 =
E − sL+√E(E − sL)
sL
×
[
−
√
2m(E − SL)− p
]
. (2.23b)
25
Setting 〈ξ〉E,L = 0 and demanding continuity of ξ at q = 0 and q = L, we get ξ′0 = 0
and
ξ′1 =
E − sL+√E(E − sL)
sL
×
[√
2m(E − SL)−
√
2mE
]
. (2.24)
Eqs. 2.23 and 2.24 can be combined to give the classical generator for E > Ec:
ξ>C (q, p;L) = −
p
sL
[
E − sL+
√
E(E − sL)
]
+ sign(p) ·
√
2m
sL
[
E
√
E − sL+
√
E(E − sL)
]
, (2.25)
which is consistent with Eq. 2.21 at E = Ec.
Eqs. 2.17 and 2.21 provide explicit expression for ξ at energies E < Ec, and
Eqs. 2.20 and 2.25 give ξ for E ≥ Ec. As mentioned earlier, below the critical
energy the system is effectively driven in a scale-invariant manner. We will focus
our attention on energies above the critical energy, where the driving is non-scale-
invariant.
Comparing Eqs. 2.17 and 2.20 with Eqs. 2.21 and 2.25 respectively, we note
that the classical generators for cases (a) and (b) are related to each other by the
following relation:
ξC(q, p; s) +
pq
3s
= ξC(q, p;L) · L
3s
. (2.26)
26
2.4 Semiclassical counterdiabatic terms
Having obtained exact classical expressions for the generator ξ(z;λ), we now
wish to utilize these results to construct its quantum counterpart ξˆ(λ), in terms
of position and momentum operators qˆ and pˆ. In later sections we will study,
numerically, the extent to which the operator constructed in this manner produces
transitionless quantum driving for the quantum tilted piston.
We seek a semiclassical approximation for the quantum generator, denoted by
ξˆSC . In cases (a) and (b) described above, HˆCD(t) is given by s˙ · ξˆSC(q, p; s) and
L˙ · ξˆSC(q, p;L) respectively, where ξˆSC is Hermitian. As the operators qˆ and pˆ do not
commute, merely putting ‘hats’ on the observables in Eqs. 2.17, 2.20 and 2.25 will
not ensure Hermiticity. Rather, the terms in ξC must be symmetrized. Complete
symmetrization as prescribed in Ref. [65] becomes unfeasible as ξC contains terms
with non-integer powers of q and p. We therefore implement the following procedure
to symmetrize the expressions.
Any term in ξC of the form f(p) · g(E), where f and g are arbitrary functions,
is symmetrized as
f(pˆ) · g(Hˆ0) + g(Hˆ0) · f(pˆ)
2
, (2.27)
where Hˆ0 is the quantized version of Eq. 2.11. The semiclassical operators for
27
E ≤ sL are given by
ξˆ<SC(qˆ, pˆ; s) = −
1
6s
(qˆpˆ+ pˆqˆ), (2.28a)
ξˆ<SC(qˆ, pˆ;L) = 0, (2.28b)
where Eqs. 2.17 and 2.21 have been used. Let ηˆ be the semiclassical operator for
sign(p) (to be determined later). For E > sL, from Eq. 2.20 and 2.25, we obtain
ξˆ>SC(qˆ, pˆ; s) = −
1
3s2
ξˆ1 − 1
3s
ξˆ2 +
1
3s2
ξˆ3, (2.29a)
ξˆ>SC(qˆ, pˆ;L) = −
1
sL
ξˆ1 +
1
sL
ξˆ3, (2.29b)
where
ξˆ1 =
pˆ · f(Hˆ0) + f(Hˆ0) · pˆ
2
(2.29c)
ξˆ2 =
qˆpˆ+ pˆqˆ
2
(2.29d)
ξˆ3 =
ηˆ · g(Hˆ0) + g(Hˆ0) · ηˆ
2
(2.29e)
f(Hˆ0) = Hˆ0 − sL+
√
Hˆ0(Hˆ0 − sL) (2.29f)
g(Hˆ0) =
√
2m
[
Hˆ0
√
Hˆ0 − sL+
√
Hˆ0(Hˆ0 − sL)
]
(2.29g)
The generators ξˆSC(qˆ, pˆ; s) and ξˆSC(qˆ, pˆ;L) defined by Eqs. 2.28-2.29 satisfy
ξˆSC(qˆ, pˆ; s) +
1
3s
· qˆpˆ+ pˆqˆ
2
= ξˆSC(qˆ, pˆ;L) · L
3s
, (2.30)
28
which is the semiclassical counterpart of Eq. 2.26.
2.5 Solving the time-dependent Schro¨dinger Equation
In the previous section, we obtained semiclassical expressions for the generators
ξˆSC(qˆ, pˆ;λ), where λ = s for case (a) and λ = L for case (b). We now aim to
simulate the evolution of the system under the time dependent Schro¨dinger equation,
to establish how well these generators produce transitionless quantum driving.
Let the wavefunction
un(q,λ) = 〈q|n(λ)〉 (2.31)
denote the nth eigenstate of the unperturbed Hamiltonian Hˆ0(λ), in the position
representation. For a given protocol λ(t), we will evolve a wavefunction ψ(q, t) under
the time dependent Schro¨dinger equation, Hψ = i~ ∂ψ/∂t, using the Hamiltonian
Hˆ(t) = Hˆ0(λ(t)) + λ˙ · ξˆSC(qˆ, pˆ;λ(t)), (2.32)
with initial condition ψ(q, 0) = un(q,λ(0)). We will compare the evolving wavefunc-
tion ψ(q, t) with the instantaneous nth energy eigenstate by evaluating the fidelity :
F(t) = |〈n(λ(t))|ψ(t)〉| =
∣∣∣∣∫ dq u∗n(q,λ(t))ψ(q, t)∣∣∣∣ . (2.33)
The fidelity provides a direct measure, between 0 and 1, of the degree to which the
term λ˙ · ξˆSC appearing in Eq. (2.32) suppresses transitions out of the nth energy
eigenstate.
29
In this section we describe our approach to solving the time dependent Schro¨dinger
equation numerically, and we develop the tools required to implement this proce-
dure. We expand the time-dependent wave function as
ψ(q, t) =
∑
n
an(t)un(q,λ(t)) exp
[
− i
~
∫ t
0
En(t
′)dt′
]
, (2.34)
where En(t) is the n
th eigenvalue of Hˆ0(λ(t)), and the expansion coefficients satisfy∑
n |an(t)|2 = 1. Upon substituting Eq. 2.34 in the time dependent Schro¨dinger
equation, using the Hamiltonian Hˆ(t) given by Eq. 2.32, we obtain
a˙m =
∑
n
Nmnan, (2.35)
where
Nmn = λ˙ exp
[
− i
~
∫ t
0
(En(t
′)− Em(t′))dt′
]
Mmn, (2.36)
and
Mmn = −〈m|∇n〉+ 1
i~
〈m|ξˆSC |n〉
= M0mn +M
CD
mn . (2.37)
The term M0mn ≡ −〈m|∇n〉 arises from the term Hˆ0 in Eq. 2.32, while MCDmn ≡
(i~)−1〈m|ξˆSC |n〉 is the contribution from the semiclassical counterdiabatic generator,
λ˙ · ξˆSC . Solving the Schro¨dinger equation is equivalent to solving the first order
matrix differential equation Eq. 2.35 for the expansion coefficients an(t).
30
In order to obtain explicit expressions for the matrices M0 and MCD appearing
in Eq. 2.37, it is convenient to make use of two different time-dependent basis sets
in Hilbert space. The first is the energy basis, {|n(λ)〉}, consisting of the eigenstates
of Hˆ0(λ). The second is the sine basis, {|α(L)〉}, by which we mean the orthogonal
sinusoidal functions of length L:
〈q|α(L)〉 =
√
2
L
sin
(αpiq
L
)
, α ≥ 1 (2.38)
where L = L(t). We will use Latin and Greek letters, respectively, to denote energy
and sine basis states. Given a Hermitian operator Oˆ, its representation in the energy
and sine bases will be denoted by the matrices
O¯mn = 〈m|Oˆ|n〉 (2.39a)
O˜αβ = 〈α|Oˆ|β〉 (2.39b)
The operators f(Hˆ0) and g(Hˆ0), defined by Eqs. 2.29f and 2.29g, are conve-
niently represented in the energy basis, in which they become diagonal matrices
with entries
f¯mm = Em − sL+
√
Em(Em − sL) (2.40)
g¯mm =
√
2m
[
Em
√
Em − sL+
√
Em(Em − sL)
]
(2.41)
The operators pˆ, ξˆ2 and ηˆ are more conveniently represented in the sine basis.
31
Using Eq. 2.38, we obtain
p˜αβ =

0 α− β = even
4i~αβ
L(β2−α2) α− β = odd
, (2.42)
and
(ξ˜2)αβ =

0 α = β
− 2i~αβ
β2−α2 α− β = even , α 6= β
2i~αβ
β2−α2 α− β = odd
(2.43)
A representation of ηˆ in the sine basis is obtained by semiclassical means in Ap-
pendix A, yielding the result:
η˜αβ =

0 α− β = even
2i
(β−α)pi α− β = odd
. (2.44)
In order to use Eqs. 2.40-2.44 to construct the matrix elements MCDmn , we re-
quired the similarity transformation
O¯mn =
∑
αβ
Z†mαO˜αβZβn, (2.45)
where Zβn = 〈β|n〉. Z is the matrix that diagonalizes H˜0 – the sine basis represen-
32
tation of Hˆ0 – which can be evaluated explicitly:
(H˜0)αβ =

0 α− β = even , α 6= β
− 8αβsL
(α2−β2)2pi2 α− β = odd
(αpi~)2
2mL2
+ sL
2
α = β
. (2.46)
We obtained Z from H˜0 numerically, and we used the result to transform p˜, ξ˜2 and η˜
(Eqs. 2.42 - 2.44) into p¯, ξ¯2 and η¯ via Eq. 2.45. We then combined these expressions
with f¯ and g¯ (Eqs. 2.40, 2.41) to construct ξ¯1, ξ¯2 and ξ¯3 (see Sec. 2.4). Finally, from
these we obtained ξ¯SC and therefore M
CD
mn (Eq. 2.37).
In addition to MCDmn , Eq. 2.37 contains the term
M0mn = −〈m|∇n〉 = −
〈m|∇Hˆ0|n〉
En − Em (2.47)
For case (a), ∇Hˆ0 = ∂sHˆ0 = qˆ. The elements of qˆ in the sine basis are
Q˜αβ =

0 α− β = even , α 6= β
− 8αβL
(α2−β2)2pi2 α− β = odd
L
2
α = β
. (2.48)
After obtaining Q¯ = ZT Q˜Z, we have
−〈m|∂sn〉 =

− Q¯mn
En−Em m 6= n
0 m = n
. (2.49)
33
For case (b) we have ∇Hˆ0 = ∂LHˆ0, whose classical counterpart ∂LH0 is singular
at q = L(t). We will determine 〈m|∂Ln〉 by relating it to 〈m|∂sn〉 using scale
invariance.
The potential V (q; s, L) = sq + Θ(q; 0, L) that appears in our Hamiltonian,
Eq. 2.11, depends parametrically on both the slope s and the length L. If these two
parameters are constrained to satisfy
s(L)L3 = constant (2.50)
(treating the slope s as a function of the length L) then the potential function
satisfies
V (q; s(L), L) =
1
L2
V
( q
L
; s(1), 1
)
(2.51)
which is the condition for scale invariance. In this situation the nth energy eigen-
function satisfies [34]
um(q; s(L), L) =
1√
L
um
( q
L
; s(1), 1
)
. (2.52)
Differentiating both sides of Eq. 2.52 with respect to L and equating the results, we
get
∂um
∂L
=
3s
L
∂um
∂s
− um
2L
− q
L
∂um
∂q
. (2.53)
Since the entire parameter space can be filled by a set of non-intersecting curves
defined by Eq. 2.50, Eq. 2.53 is valid for any slope s and any positive length L.
34
Now consider the expressions
〈n|∂Lm〉 =
∫ L
0
dq u∗n
∂um(q)
∂L
=
3s
L
〈n|∂sm〉 − 1
2L
δmn − 1
L
∫ L
0
dq u∗n q
∂um(q)
∂q
(2.54)
and
1
i~
〈n| qˆpˆ+ pˆqˆ
2
|m〉 = −1
2
∫ L
0
dq u∗n q
∂um(q)
∂q
− 1
2
∫ L
0
dq u∗n
∂(q um(q))
∂q
= −
∫ L
0
dq u∗n q
∂um(q)
∂q
− δmn
2
(2.55)
Substituting Eq. 2.55 into Eq. 2.54, we arrive at
〈n|∂Lm〉 = 3s
L
〈n|∂sm〉+ 1
L
1
i~
〈n| qˆpˆ+ pˆqˆ
2
|m〉, (2.56)
which can alternatively be obtained from Eq. 2.30. We can therefore compute the
matrix representation of 〈n|∂Lm〉 as we have already determined both the terms on
the right side of Eq. 2.56: Eq. 2.49 gives the first term while the second is obtained
after performing a similarity transformation on the matrix given in Eq. 2.43.
2.6 Numerical Results
Having determined the matrices M0 and MCD (Eq. 2.37), we solved the time de-
pendent Schro¨dinger equation by numerically integrating Eq. 2.35 using the (fourth-
order) Runge-Kutta-Gill method [66]. In each simulation the system was initialized
35
ÈΨHq
L2
5 10 15 20 25
0.001
0.002
0.003
0.004
0.005
t = 0.0
5 10 15 20 25
0.001
0.002
0.003
0.004
0.005
t = 0.0
0 5 10 15 20 25
0.001
0.002
0.003
0.004
0.005
t = 5.0
0 5 10 15 20 25
0.001
0.002
0.003
0.004
0.005
t = 5.0
0 5 10 15 20 25
0.001
0.002
0.003
0.004
0.005
t = 10.0
5 10 15 20 25
0.001
0.002
0.003
0.004
0.005
t = 10.0
0 5 10 15 20 25
0.001
0.002
0.003
0.004
0.005
t = 15.0
0 5 10 15 20 25
0.001
0.002
0.003
0.004
0.005
t = 15.0
0 5 10 15 20 25
0.001
0.002
0.003
0.004
0.005
t = 20.0
0 5 10 15 20 25
0.001
0.002
0.003
0.004
0.005
t = 20.0
q
Figure 2.2: Evolution of the probability density |ψ(q, t)|2 for a particle of mass
m = 1.0 in a box whose slope is fixed at s = 3.0 and whose length is decreased from
L = 25.0 to 15.0 at a rate L˙ = −0.5. Snapshots of the wavefunction are taken at
times t=0, 5.0, 10.0, 15.0 and 20.0. The plots on the left depict evolution under
the full Hamiltonian Hˆ(t) = Hˆ0(t) + λ˙ · ξˆSC(λ(t)), while those on the right depict
evolution under Hˆ0(t).
36
ÈΨHq
L2
2 4 6 8 10 12 14
0.001
0.002
0.003
0.004
0.005
t = 0.0
2 4 6 8 10 12 14
0.001
0.002
0.003
0.004
0.005
t = 0.0
0 2 4 6 8 10 12 14
0.001
0.002
0.003
0.004
0.005
t = 5.0
0 2 4 6 8 10 12 14
0.001
0.002
0.003
0.004
0.005
t = 5.0
0 2 4 6 8 10 12 14
0.001
0.002
0.003
0.004
0.005
t = 10.0
0 2 4 6 8 10 12 14
0.001
0.002
0.003
0.004
0.005
t = 10.0
0 2 4 6 8 10 12 14
0.001
0.002
0.003
0.004
0.005
t = 15.0
0 2 4 6 8 10 12 14
0.001
0.002
0.003
0.004
0.005
t = 15.0
0 2 4 6 8 10 12 14
0.001
0.002
0.003
0.004
0.005
t = 20.0
0 2 4 6 8 10 12 14
0.001
0.002
0.003
0.004
0.005
t = 20.0
q
Figure 2.3: Same as Fig.2.2 except that the length of the box is fixed at L = 15.0,
while the slope is decreased from s = 13.0 to 3.0 at a rate s˙ = −0.5.
37
F
id
e
li
ty
0 5 10 15 20
0.2
0.4
0.6
0.8
1.0
0 10 20
0.9998
0.9999
Time
F
id
e
li
ty
0 5 10 15 20
0.2
0.4
0.6
0.8
1.0
0 10 20
0.99975
0.99985
Time
Figure 2.4: Evolution of the fidelity F(t). The plot on the left is for the case shown
in Fig.2.2, whereas the plot on the right is for the same system but subjected to the
reverse process: the box length increases from L = 15.0 to 25.0 at L˙ = 0.5. The
dashed magenta curve depicts the fidelity for evolution under Hˆ = Hˆ0 + λ˙ · ξˆSC ,
while the blue curve is the fidelity upon evolution under Hˆ0. The inset is a magnified
view of the dashed magenta curve.
F
id
e
li
ty
0 5 10 15 20
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1.00
0 10 20
0.99935
0.99965
Time
F
id
e
li
ty
0 5 10 15 20
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1.00
0 10 20
0.9994
0.9998
Time
Figure 2.5: Similar to Fig.2.4. The left plot is for the case shown in Fig.2.3, whereas
in the right plot the same system is subjected to the reverse process: the slope s
increases from s = 3.0 to 13.0 at s˙ = 0.5.
38
in the nth energy eigenstate, ak(0) = δkn, then it was evolved in time as either the
slope s was varied at fixed length L (case (a)), or else the length was varied at
constant slope (case (b)). The rate of change, s˙ or L˙, was set to a constant value
sufficiently large to produce non-adiabatic evolution in the absence of the counter-
diabatic term.
Simulations were performed both under the original Hamiltonian Hˆ0(t), describ-
ing the time-dependent tilted piston (Eq. 2.11), and under the composite Hamilto-
nian
Hˆ(t) = Hˆ0(t) + λ˙ · ξˆSC(λ(t)) (2.57)
that includes the counterdiabatic term. In both cases the fidelity
F(t) = |〈n(λ(t))|ψ(t)〉| was computed. In these simulations the particle mass was
set to m = 1 and Planck’s reduced constant to ~ = 2, and the system was initialized
in the quantum number n = 35. The results, Figs. 2.2 - 2.5, are discussed in the
following paragraphs.
Fig. 2.2 shows the evolving probability distribution |ψ(q, t)|2 as the length of
the tilted piston is reduced from L = 25.0 to 15.0, at a rate L˙ = −0.5 and fixed
slope s = 3.0. The left column shows snapshots of |ψ|2 at five instants in time,
for evolution under the Hamiltonian Hˆ(t). The right column shows evolution under
Hˆ0(t). In these simulations the initial energy is En = 79.52. The plots on the
left are visually indistinguishable from the probability distribution of the adiabatic
energy eigenstate, |un(q,λ(t))|2, with n = 35. By contrast, in the plots on the
right the probability distribution develops noticeable shock waves, due to the rapid
39
compression of the piston length. Thus, with the addition of the counterdiabatic
term the system faithfully follows a fixed eigenstate of Hˆ0(t) (left plots), while in the
absence of this term it is unable to keep pace with the rapidly changing Hamiltonian
(right plots).
Fig.2.3 presents evolution in a tilted piston of fixed length L = 15.0, with a slope
that decreases from s = 13.0 to 3.0 at a rate s˙ = −0.5. As in Fig. 2.2, the plots
in the left and right columns depict evolution with and without the counterdiabatic
term λ˙ · ξˆSC . Once again, the plots on the left are indistinguishable from the
instantaneous energy eigenstate |un(q,λ(t))|2, while those on the right reveal (mild)
shock waves that are evidence of non-adiabatic evolution. The counterdiabatic term
again successfully guides the wavefunction along the desired adiabatic trajectory.
These claims are supported by analyses of the fidelity F(t). Fig.2.4 shows
fidelity plots for a tilted piston undergoing compression (left plot) and expansion
(right plot). The former corresponds to the evolution shown in Fig.2.2, while the
latter depicts the reverse process, in which the length increases from L = 15.0 to
25.0 at L˙ = 0.5. Similarly, Fig.2.5 shows a fidelity plot for the evolution depicted
in Fig.2.3 (left plot), and for the reverse process in which the slope is varied from
s = 3.0 to 13.0 at s˙ = 0.5 (right plot). In these figures, the solid blue curves depict
the fidelity for evolution under Hˆ0(t), while the dashed magenta curves correspond
to evolution under Hˆ(t). In all four plots the blue curves deviate significantly,
while the dashed magenta curves remain very close to unity, confirming that our
semiclassically obtained counterdiabatic term has the desired effect of enforcing
adiabatic evolution, with high accuracy.
40
As a side comment we observe that, in Fig.2.4, the oscillations in F(t) become
more rapid in time when the tilted piston is compressed (left plot), and less rapid
as it expands (right plot). These oscillations reflect the shock waves propagating
between the two walls of the box, hence it makes sense that the period of oscillation
diminishes or grows as the length L decreases or increases.
Because the counterdiabatic term λ˙· ξˆSC was obtained semiclassically, we expect
its efficacy to degrade as we approach the deep quantum regime. To test this
hypothesis, we performed simulations at fixed slope s = 3.0, with piston length
decreasing from L = 25.0 to 15.0 at L˙ = −0.5, and with particle mass m = 1, as in
Fig.2.2. We carried out seven such simulations, with the value of ~ ranging from 1.0
to 7.0, choosing the initial state n so that the particle starts with energy En ≈ 80 in
each simulation. Thus Planck’s constant was varied while the classical parameters
remained essentially fixed. As before, the system was subjected to evolution under
both Hˆ0(t) and Hˆ(t), and the fidelity F(t) was computed. Table 2.1 lists Fwcdmin,
which is the minimum fidelity (over the duration of the process) when the system
evolves under Hˆ(t), and Fwocdmin , the minimum fidelity when the system evolves under
Hˆ0(t). We see that as ~ increases and n decreases – that is, as we go deeper into the
quantum regime – Fwcdmin deviates further from unity. As expected, the semiclassical
counterdiabatic term λ˙ · ξˆSC works best in the semiclassical limit of small ~ / large
n.
Interestingly, Table 2.1 reveals that Fwocdmin increases with ~: in the absence of
the counterdiabatic term, the fidelity improves as we go deeper into the quantum
regime. We attribute this behavior to the fact that the spacing between adjacent
41
~ n Fwcdmin Fwocdmin
1.0 70 0.999 0.092
2.0 35 0.999 0.641
3.0 23 0.999 0.842
4.0 17 0.997 0.917
5.0 14 0.992 0.939
6.0 12 0.979 0.953
7.0 10 0.943 0.970
Table 2.1: The dependence of fidelity on the value of the reduced Planck’s constant
~, keeping classical parameters fixed. The initial quantum number n is chosen such
that the initial energy is En ≈ 80. Each simulation is performed at fixed s = 3.0,
while the box length is varied from L = 25.0 to L = 15.0 at L˙ = −0.5. Fwcdmin is the
minimum fidelity when the system evolves under Hˆ(t), and Fwocdmin is the minimum
fidelity when the system evolves under Hˆ0(t).
0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
Time
Fi
de
lit
y
Figure 2.6: Evolution of the fidelity under Hˆ0(t), for the simulations described in
Table 2.1. The lowermost (dashed magenta) curve corresponds to ~ = 1.0, the next
one up (solid blue) corresponds to ~ = 2.0, and so forth up to ~ = 7.0, which is the
magenta curve that remains closest to unity.
42
energy levels increases with ~. Let us picture a classical process in which initial
conditions are sampled from a single energy shell, and trajectories evolve from these
initial conditions under the HamiltonianH0(q, p, t). The final distribution of energies
then provides a crude estimate of the final energy distribution in the corresponding
quantum process, in which the system begins in an energy eigenstate. For a fixed
final distribution of energies, the distribution of final quantum numbers decreases
with increasing ~, simply because of the decreasing density of energy levels. As a
result, excitations out of the initial energy level are inhibited.
Fig.2.6 shows F(t) for the seven simulations of evolution under Hˆ0(t) that are
represented in the rightmost column of Table 2.1. These plots confirm that the
fidelity improves with increasing ~ (in the absence of the counterdiabatic term),
and they display similar oscillatory behavior, with peaks and valleys occurring at
nearly the same times for the seven curves. These observations are consistent with
the interpretation that the frequency of the oscillations reflect the corresponding
classical evolution, while the magnitude is governed by the quantum energy spacing.
2.7 Summary
We have studied a model system undergoing non-scale-invariant driving: the
one-dimensional tilted piston described by the Hamiltonian H0(q, p;λ(t)) (Eq. 2.11).
We derived exact, closed-form expressions for the classical counterdiabatic Hamil-
tonian HCD(q, p, t), which we quantized to obtain a Hermitian operator HˆCD(t). In
numerical simulations of the time dependent Schro¨dinger equation, we compared
43
evolution under Hˆ0 to that under Hˆ = Hˆ0 + HˆCD, with the system initialized in
an energy eigenstate. The simulations reveal that HˆCD very effectively suppresses
non-adiabatic transitions: when evolving under Hˆ, the system remains in an eigen-
state of Hˆ0 with nearly perfect fidelity. These results establish a proof of principle –
beyond the regime of scale-invariant driving [32,34] – that quantum counterdiabatic
Hamiltonians can successfully be constructed from their classical counterparts.
For most Hamiltonians H0(z;λ(t)) of interest, a closed-form expression for
HCD(z, t) will not be available. Even for the quite simple system we have studied,
the expression for HCD is somewhat involved, and the final result for the operator
HˆCD = λ˙ · ξˆSC – while given in terms of qˆ and pˆ (Eq. 2.29) rather than as a spectral
sum (Eq. 1.12) – would certainly be difficult to implement in a laboratory setting.
The difficulty in realizing HˆCD experimentally arises not just because it is given
by a complicated expression, but because it is non-local, i.e. because it depends on
the momentum operator, pˆ. This is also true for the much simpler counterdiabatic
Hamiltonians that have been derived for scale-invariant driving (Eq. 2.3), including
the harmonic oscillator (Eq. 2.1) as a particular example. In the scale-invariant
case, an appropriate canonical (or unitary) transformation of HCD(q, p, t) gives a
local counterdiabatic potential UFF (q, t); in effect, the non-locality can be “gauged
away” to construct a local shortcut to adiabaticity equivalent to the fast-forward
method, as described in Ref. [34]. Whether a transformation of this sort could be
applied to our counterdiabatic Hamiltonian HˆCD (Eq. 2.29) is an open question.
44
Chapter 3: Quantum shortcuts using flow-fields
3.1 Overview
In the previous chapter, we have seen that even for the simple example of a tilted
piston, the method prescribed in Ref. [25] yields a complicated expression for the
counterdiabatic Hamiltonian HCD, which is difficult to implement experimentally
not only due to its complicated nature but also due to its non-locality, i.e., the
momentum dependence of HCD. Note that the HCD from Chap. 2 is same regardless
of the choice of the initial state. In this chapter, we find that by trying to develop
HCD that depends on the choice of initial state, we get much simpler expressions.
We introduce a new framework for constructing the desired auxiliary Hamilto-
nian, which consists of three steps. First, we identify the adiabatic evolution as the
evolution that the system of interest would undergo if the process were performed
adiabatically. We then define velocity and acceleration flow fields v(q, t) and a(q, t)
that characterize the adiabatic evolution, as illustrated in Fig. 3.1 for a quantum
system. Finally, from these fields we immediately construct auxiliary terms that
provide both counterdiabatic (Eqs. 3.11) and fast-forward (Eqs. 3.14) shortcuts.
This chapter is structured as follows. In Sec. 3.2 we recapituate the salient
featues of counterdiabatic driving developed in Ref. [24] and the fast-forward method
45
developed in Ref. [26]. After deriving the main results of this chapter, Eqs. 3.11 and
3.14, in Sec. 3.3, we compare them with previously obtained shortcuts in Sec. 3.4. In
Sec. 3.5, we show how our approach provides insight into the singularities that may
arise in the fast-forward approach. We analyze the special case of scale invariant
dynamics in Sec. 3.6 and illustrate our approach with numerical simulations in
Sec. 3.7. We briefly discuss generalizations to three degrees of freedom in Sec. 3.8
and summarize the results of this chapter in Sec. 3.9.
3.2 Review of quantum shortcuts
Transitionless quantum driving, due to Demirplak and Rice [22] and Berry [24],
involves the counterdiabatic Hamiltonian in Eq. 1.9. If a wavefunction evolves under
Hˆ0 + HˆCD from an initial state |n(0)〉, then it remains in the n’th instantaneous
eigenstate of Hˆ0(t) at all times, as the term HˆCD(t) suppresses excitations to other
eigenstates [22, 24]. Note that the counterdiabatic term (Eq. 1.9) does not depend
on the choice of n.
The fast-forward approach, due to Masuda and Nakamura [26], pertains to a
Hamiltonian of the form
Hˆ0(t) =
pˆ2
2m
+ U0(qˆ, t) . (3.1)
For a given time interval 0 ≤ t ≤ τ , and a particular quantum number n, a fast-
forward potential UFF (qˆ, t) is constructed with the following property: if a wave-
function evolves under Hˆ0+UˆFF from the initial state |n(0)〉, then it will arrive in the
eigenstate |n(τ)〉 (up to an overall phase) at t = τ . For intermediate times 0 < t < τ ,
46
the wavefunction is in a superposition of eigenstates of Hˆ0(t), as illustrated in Fig. 1.1
and Fig. 3.3 below. The fast-forward potential depends on the chosen quantum num-
ber n. Moreover, UFF (q, t) typically (though not always) becomes singular at nodes
of the instantaneous eigenstate, that is, where φn(q, t) ≡ 〈q|n(t)〉 = 0. Hence the
applicability of the fast-forward method is generally restricted to the ground state,
n = 0, although there are exceptions to this statement. We will return to this point
later in our discussion.
Both HˆCD and UˆFF are auxiliary Hamiltonians that are tailored to achieve the
desired acceleration of adiabatic dynamics. We will use the term counterdiabatic
to refer to methods in which the auxiliary term causes the system to follow the
adiabatic evolution – at an accelerated pace – for the duration of the process. This
is the case with transitionless quantum driving: the wavefunction remains in a
given eigenstate of Hˆ0(t) at all times, when evolving under Hˆ0 + HˆCD. The term
fast-forward (or FF ) will refer to methods in which the system strays from the
adiabatic evolution at intermediate times, but returns to the adiabatic state at the
final time t = τ , as in the Masuda-Nakamura method. As illustrated by the previous
paragraphs, auxiliary terms in the fast-forward approach are local, in the sense that
they are explicit functions of qˆ and t. By contrast, counterdiabatic driving generally
requires non-local auxiliary terms, given either by spectral sums (as with Eq. 1.9)
or by expressions involving both coordinates and momenta (see Refs. [25,34,54], or
Eq. 3.11 below). Thus, fast-forward auxiliary terms may generically be easier to
implement experimentally, than counterdiabatic terms.
In what follows, we bridge the two approaches. We consider a Hamiltonian of
47
interest
Hˆ0(t) =
pˆ2
2m
+ U0(qˆ, t) (3.2)
in one degree of freedom. We assume that Hˆ0 varies with time only during the
interval 0 ≤ t ≤ τ , and that this time-dependence is turned on and off smoothly
– specifically, Hˆ0, ∂Hˆ0/∂t and ∂
2Hˆ0/∂t
2 are continuous functions of time for all t,
and ∂Hˆ0/∂t = 0 for t /∈ (0, τ). For a given choice of quantum number n, we will
define velocity and acceleration flow fields v(q, t) and a(q, t), that characterize how
the eigenstate probability distribution |〈q|n(t)〉|2 deforms with t. From these flow
fields we will construct simple expressions for both a counterdiabatic Hamiltonian
HˆCD(qˆ, pˆ, t) (Eq. 3.11), and a local fast-forward potential UˆFF (qˆ, t) (Eq. 3.14).
3.3 Setup and derivation of main results
Let the real-valued wavefunction φ(q, t) = 〈q|n(t)〉 denote the n’th eigenstate of
Hˆ0(t), with eigenenergy E(t):
Hˆ0(t)φ(q, t) =
[
− ~
2
2m
∂2
∂q2
+ U0(q, t)
]
φ(q, t) = E(t)φ(q, t). (3.3)
For convenience we omit the subscript n on φ and E. The adiabatic evolution is
identified as follows: 1
ψad(q, t) = φ(q, t) e
iα(t) , α(t) = −1
~
∫ t
0
E(t′) dt′ (3.4)
1The dynamical phase α is generically accompanied by a geometric phase [67], but the latter
vanishes for a kinetic-plus-potential Hamiltonian in one degree of freedom.
48
When the time-dependence of Hˆ0(t) is quasi-static, ψad is a solution of the Schro¨dinger
equation, i~ ∂tψad = Hˆ0ψad [1,2]. When the time-dependence is arbitrary, ψad obeys
i~
∂ψad
∂t
= (Hˆ0 + HˆCD)ψad (3.5)
for the counterdiabatic term given by Eq. 1.9 [22,24]. Thus the addition of the term
HˆCD causes the system to follow the adiabatic evolution, ψad, when Hˆ0 is varied
rapidly.
In what follows we construct a counterdiabatic term given as an explicit func-
tion of qˆ and pˆ (Eq. 3.11), which accomplishes the same result for a given choice of
n. Before proceeding further, let us first focus on the results from scale invariant
driving. Under scale invariant driving, if ψ0(q) is an eigenfunction of the Hamilto-
nian Hˆ0(γ = 1), then ψ(q, γ) = ψ
0(q/γ)/
√
γ is the eigenfunction of Hˆ0(γ), where γ
represents the expansion co-efficient [34]. Then, from Eqs. 1.10 and 1.15, it follows
that HˆCD ∝ pˆqˆ+ qˆpˆ drives the system along the desired adiabatic path by stretching
the wavefunction appropriately [20,25,34,61]. The Hamiltonian (pˆqˆ+ qˆpˆ) can there-
fore be viewed as a linear stretching operator, as it is linear in p and q. This result
suggests that the simplest form of HˆCD for an arbitrarily driven system should be
linear in p and non-linear in q so that HˆCD ∝ pˆvˆ+ vˆpˆ, where vˆ = v(qˆ, t). We choose
to retain the linear dependence on momentum instead of position because pˆ is a
non-local operator while qˆ is a local operator. Therefore, higher order terms in qˆ
are more practical and relatively easier to implement experimentally compared to
higher order terms in pˆ. We describe a method of obtaining v(qˆ, t) for a general
49
Figure 3.1: The red curve φ2(q, t) depicts the probability distribution associated
with an energy eigenstate of Hˆ0(t). The blue vertical lines divide the area under
φ2(q, t) into K  1 strips of equal area. q(I, t) is the right boundary of the shaded
region, which has area I. The positions of the vertical lines vary parametrically
with t, and this “motion” is described in terms of velocity and acceleration fields
v(q, t) and a(q, t), as given by Eq. 3.8.
system below.
Let us define the integrated probability density function,
I(q, t) ≡
∫ q
−∞
φ2(q′, t)dq′. (3.6)
and let us invert this function to obtain q(I, t). We then define a velocity flow field 2
v(q, t) =
∂
∂t
q(I, t) = −∂tI
∂qI , (3.7)
This flow field can be pictured by dividing the area under φ2(q, t) into K  1 strips
of equal area, delimited by vertical lines at locations q1(t), q2(t), . . . , qK−1(t), so that
2The quantity −v(q, t) was identified as a “hydrodynamic velocity” in Ref. [30], Eq. (6).
50
I(qk(t), t) = k/K; see Fig. 3.1. The locations {qk(t)} evolve parametrically with t,
according to
dqk
dt
= v(qk, t) (3.8a)
Note that Eq. 3.8a does not reflect the unitary dynamics generated by Hˆ0(t), but
rather the variation of the eigenstate probability density φ2(q, t) with t. We similarly
introduce an acceleration flow field,
d2qk
dt2
= a(qk, t) (3.8b)
By Eq. 3.8a this field satisfies
a(q, t) = v′v + v˙ (3.9)
where the prime and dot denote ∂q and ∂t, respectively. Both flow fields vanish
outside the interval 0 < t < τ :
v(q, t) = 0 = a(q, t) for t /∈ (0, τ) (3.10)
as follows from the assumptions spelled out after Eq. 3.2. We will now use these flow
fields to construct counterdiabatic and fast-forward shortcuts, given by Eqs. 3.11 and
3.14 below.
51
We begin by defining the counterdiabatic Hamiltonian,
HˆCD(t) =
pˆvˆ + vˆpˆ
2
, vˆ(t) = v(qˆ, t) (3.11)
We claim that the adiabatic wavefunction ψad (Eq. 3.4) satisfies Eq. 3.5 for arbitrary
time-dependence of Hˆ0(t), with HˆCD now given by Eq.3.11. To show this, we first
rearrange Eq. 3.7 as ∂tI + v∂qI = 0. Differentiating both sides with respect to q
leads to the continuity equation ∂tφ
2 + ∂q(vφ
2) = 0, equivalently,
φ˙+ vφ′ +
1
2
v′φ = 0. (3.12)
We now use Eqs. 3.3, 3.4 and 3.12 to evaluate the right side of Eq. 3.5:
(Hˆ0 + HˆCD)ψad =
[
Hˆ0φ− i~
2
∂q(vφ)− i~
2
v(∂qφ)
]
eiα
=
[
Eφ− i~
(
vφ′ +
1
2
v′φ
)]
eiα
=
(
Eφ+ i~ φ˙
)
eiα = i~
∂ψad
∂t
(3.13)
which establishes that ψad(q, t) is a solution of Eq. 3.5. Thus if a wavefunction
evolves under Hˆ0 + HˆCD from an initial state ψ(q, 0) = 〈q|n(0)〉, then it remains in
the n’th instantaneous eigenstate of Hˆ0(t) during the entire process, just as in the
case of transitionless quantum driving [22,24].
Turning our attention to the fast-forward approach, we construct a potential
52
UFF (q, t) and a companion function S(q, t) as follows:
−∂qUFF = ma(q, t) (3.14)
∂qS = mv(q, t) (3.15)
By Eq. 3.9, these functions satisfy
∂q
[
∂tS +
1
2m
(∂qS)
2 + UFF
]
= 0 (3.16)
Eqs. 3.14 and 3.15 specify UFF and S only up to arbitrary functions of time. We
use this freedom, along with Eqs. 3.10 and 3.16, to impose the conditions
UFF (q, t) = 0 for t /∈ (0, τ) (3.17)
and
∂tS +
1
2m
(∂qS)
2 + UFF = 0 (3.18)
Eq. 3.10 further implies that
S(q, 0) = S− , S(q, τ) = S+ (3.19)
where S± are constants (i.e. independent of q).
53
We now show that the ansatz
ψ¯(q, t) = ψad(q, t) exp
[
i
S(q, t)
~
]
= φ eiαeiS/~ (3.20)
is a solution of the Schro¨dinger equation
i~
∂ψ¯
∂t
= (Hˆ0 + UˆFF )ψ¯ (3.21)
Evaluating the right side with the help of Eqs. 3.3, 3.12, 3.15 and 3.18, we obtain
(
− ~
2
2m
∂2
∂q2
+ U0 + UFF
)
φ(q, t) eiα eiS/~
=
[
− ~
2
2m
φ′′ − i~
m
φ′S ′ − i~
2m
φS ′′ +
1
2m
φ(S ′)2 + U0φ+ UFFφ
]
eiα eiS/~
=
[
Eφ+
1
2m
(S ′)2φ+ UFFφ− i~
(
vφ′ +
1
2
v′φ
)]
eiα eiS/~
=
(
i~φ˙+ Eφ− S˙φ
)
eiα eiS/~ = i~
∂ψ¯
∂t
(3.22)
which is the desired result. By Eq. 3.19, the wave function ψ¯(q, t) begins in the n’th
energy eigenstate at t ≤ 0 and ends in the n’th energy eigenstate at t ≥ τ , which
establishes that UˆFF produces fast-forward evolution.
Note that we introduced the function S(q, t) (Eq. 3.15) only to facilitate the
derivation of our fast-forward approach. This function need not be evaluated if one
simply wishes to construct the potential UFF (q, t). That potential can be determined
directly from the acceleration field a(q, t), by Eq. 3.14.
Also, we imposed Eq. 3.17 so as to obtain an auxiliary potential that is turned
54
on at t = 0 and off at t = τ , but this condition is not necessary. Any UFF satisfying
Eq. 3.14 will provide a shortcut that transports the n’th eigenstate of Hˆ0(0) to the
n’th eigenstate of Hˆ0(τ). The addition of an arbitrary function f(t) to UFF (q, t)
affects only the overall phase of the evolving wavefunction.
3.4 Comparison with previous results
Eqs. 3.11 and 3.14 are recipes for constructing shortcuts directly from the flow
fields v(q, t) and a(q, t). Let us compare these results with previously published
counterdiabatic and fast-forward shortcuts.
Our result for HˆCD (Eq. 3.11) is given explicitly in terms of the operators qˆ and
pˆ. This appealing feature comes with a cost: in general, a different counterdiabatic
term is required for each eigenstate n, since v(q, t) depends on the choice of n.
By contrast the Demirplak-Rice-Berry [22, 24] counterdiabatic term (Eq. 1.9) is
independent of n, as noted earlier. We conclude that Eqs. 1.9 and 3.11 are not
equivalent, although the two counterdiabatic terms produce the same effect when
they act on the chosen adiabatic eigenstate:
HˆEq.1.9CD 6= HˆEq.3.11CD but HˆEq.1.9CD |n〉 = HˆEq.3.11CD |n〉 (3.23)
Using the identity 〈m|∂tm〉 = 0, which holds for Hˆ0 given by Eq. 3.2, we rewrite the
equality in Eq. 3.23 as follows:
i~
∂
∂t
|n〉 = pˆvˆ + vˆpˆ
2
|n〉 (3.24)
55
In other words, the operator Dˆ ≡ (pˆvˆ + vˆpˆ)/2 acts as the generator of adiabatic
transport (see Eq. 1.15) for the state |n〉 that was used to construct v(q, t):
e−iδtDˆ/~|n(t)〉 = |n(t+ δt)〉 (3.25)
for infinitesimal δt.
Substituting Eq. 3.15 into Eq. 3.12 yields
φ˙+
1
m
φ′S ′ +
1
2m
φS ′′ = 0. (3.26)
Eqs. 3.18 and 3.26 are essentially equivalent to Eqs. 5 and 6 of Torrontegui et al [28],
to Eqs. 17 and 15 of Takahashi [35], and to Eqs. 4 and 3 of Mart´ınez-Garaot et al [30].
In Refs. [28, 30, 35] these equations were used to provide streamlined derivations of
the fast-forward approach pioneered by Masuda and Nakamura [26]. (Our Eq. 3.26
is also equivalent to Eq. 2.18 of Ref. [26], and our Eq. 3.15 appears as Eq. 5 in
Ref. [30].) Thus our fast-forward potential UˆFF is equivalent to that derived by
previous authors [26, 28,30,35].
The observation that the quantum counterdiabatic and fast-forward approaches
are closely related is not surprising, as previous authors have argued that UˆFF can be
constructed from HˆCD by appropriate unitary [31–35] or gauge [36] transformations.
The novelty of our approach is that we obtain both HˆCD and UˆFF directly from
the velocity and acceleration fields that describe the time-dependence of φ2(q, t)
(Fig. 3.1). Our results are given by compact, intuitive expressions (Eqs. 3.11, 3.14).
56
This approach highlights the connection between counterdiabatic and fast-forward
shortcuts, and – as we shall see – it provides insight into the divergences that often
plague the fast-forward method when it is applied to excited states. Moreover, the
construction of HˆCD and UˆFF from v and a is mirrored in classical shortcuts to
adiabaticity, as will be discussed in Chap. 4.
Finally, we note that Eq. 3.18 is a Hamilton-Jacobi equation for the Hamiltonian
p2/2m+UFF . Okuyama and Takahashi [55] have recently used the Hamilton-Jacobi
formalism to explore the correspondence between quantum and classical shortcuts
to adiabaticity. It would be interesting to explore the relationship between their
approach and ours.
3.5 Divergences and a “no-flux” criterion
By Eq. 3.7, v(q, t) generically diverges at nodes of the wavefunction, where
∂qI = φ2 vanishes; this in turn leads to divergences in a(q, t), and in HˆCD and UˆFF .
These observations suggest that our method is in general restricted to ground state
wavefunctions (n = 0), which have no nodes.
While nodes in φ(q, t) typically spoil the applicability of our method, this need
not always be the case: the numerator and denominator in Eq. 3.7 might vanish
simultaneously in a way that prevents the ratio v = −∂tI/∂qI from blowing up at
a node. Here we propose a simple criterion for determining whether our approach
(and by extension the fast-forward approach [26, 28, 30, 35]) is applicable when an
eigenstate φ(q, t) has one or more nodes.
57
Let qν(t) denote the location, and uν(t) ≡ dqν/dt the velocity, of the ν’th node
of φ(q, t). We assume |uν | < ∞, as will generally be the case when the potential
U0(q, t) is well-behaved. As t varies parametrically, the flux of probability across
this node, from the region q < qν to the region q > qν , is given by
Φν(t) = − d
dt
I(qν , t) = [v(qν , t)− uν(t)]φ2(qν , t) (3.27)
using Eqs. 3.6 and 3.7. This result has the familiar interpretation of “flux equals
velocity times density”, in the node’s co-moving frame of reference. Eq. 3.27 implies
that if v(q, t) does not blow up at a given node, then the probability flux Φν(t) across
that node must be zero. This suggests a simple criterion: if the time-dependence
of φ2(q, t) is such that there is no flux of probability across any node, i.e. if Φν = 0
for all ν, then the velocity field v(q, t) will not diverge at the nodes and our method
will remain valid and applicable 3. Generalizing the term “nodes” to include the
boundaries at q = ±∞, the no-flux criterion can alternatively be stated as follows:
if the probability between all pairs of adjacent nodes remains independent of t [i.e. if
(d/dt)
∫ qν+1
qν
φ2dq = 0 for all ν], then v(q, t) will be free of divergences and UFF (q, t)
will be well-behaved.
This “no-flux” criterion is not generically satisfied for Hˆ0(t) given by Eq. 3.2.
However, in Sec. 3.6 we consider a particular class of time-dependent Hamiltoni-
ans for which this criterion is satisfied for every eigenstate, due to scale-invariance
(Eq. 3.28). In agreement with the arguments presented above, our method provides
3In fact, if Φν(t) = 0 then v(qν , t) = uν(t), although we will not make use of this result here.
58
non-singular counterdiabatic and fast-forward shortcuts for all energy eigenstates,
for this class of Hamiltonians. In Sec. 3.7 we present the results of numerical simula-
tions for a non-scale-invariant Hamiltonian, for which the no-flux criterion is satisfied
for the first excited state; again, our method successfully provides a non-singular
shortcut for this situation.
Divergences associated with eigenstate nodes are problematic not only for our
approach, but also for those of Refs. [26, 28, 30, 35], since all these approaches lead
to equivalent expressions for UFF . This problem has not received much attention in
the literature, although Mart´ınez-Garaot et al [30] consider it in a slightly different
context. In Sec. III.D of their paper, they develop a fast-forward strategy to drive a
wavefunction from a ground state φ0 to a first excited state φ1. In their approach the
fast-forward potential becomes singular due to the node in φ1, but they demonstrate
that ad hoc truncation of the singularity produces a well-behaved potential that
achieves near-perfect fidelity. It would be interesting to test whether such truncation
is also useful in the context of our method, when the no-flux criterion is not satisfied.
3.6 Scale-invariant dynamics
In the special case of scale-invariant driving, U0(q, t) undergoes expansions,
contractions and translations. As shown in Ref. [34] (and anticipated in Refs. [25,
32,52,61,68–70]), simple expressions for counterdiabatic and fast-forward shortcuts
can be obtained when a system is driven in a scale-invariant manner. In this section
we show that these shortcuts are obtained naturally within our framework.
59
The Hamiltonian for scale-invariant driving takes the form [34]
Hˆ0(t) = Hˆ0(γ, f) =
pˆ2
2m
+
1
γ2
U0
(
qˆ − f
γ
)
, (3.28)
where γ(t) and f(t) are parameters that describe expansions/contractions, and
translations, respectively. If we let φ˜(q) denote the n’th eigenstate of Hˆ0(γ = 1, f =
0), then the n’th eigenstate for a general choice of (γ, f) is given by [34]
φ(q) =
1√
γ
φ˜
(
q − f
γ
)
(3.29)
This scaling result immediately reveals how the “picket fence” of lines {qk} depicted
in Fig. 3.1 behaves when γ and f are varied with time. Variations in f result in
translations of the entire picket fence, and variations in γ cause the picket fence to
expand or contract linearly. These considerations give us
v(q, t) =
γ˙
γ
(q − f) + f˙ , (3.30)
and therefore (by Eq. 3.9)
a(q, t) =
γ¨
γ
(q − f) + f¨ . (3.31)
Eq. 3.30 also follows from Eq. 3.7, v = −∂tI/∂qI, since ∂qI = φ2 and (making use
60
of Eq. 3.29)
∂tI =
(
γ˙ ∂γ + f˙ ∂f
)∫ q
−∞
1
γ
φ˜2
(
q′ − f
γ
)
dq′
= γ˙
∫ q
−∞
− 1
γ2
[
φ˜2 + 2(q′ − f)φ˜(∂q′φ˜)
]
dq′ + f˙
∫ q
−∞
−2φ˜(∂q′φ˜)
γ
dq′
= − γ˙
γ
∫ q
−∞
[
φ2 + (q′ − f)∂q′(φ2)
]
dq′ − f˙
∫ q
−∞
∂q′(φ
2)dq′
=
[
− γ˙
γ
(q − f)− f˙
]
φ2 (3.32)
Combining Eqs. 3.30 and 3.31 with Eqs. 3.11 and 3.14, we obtain
HˆCD =
γ˙
2γ
[(qˆ − f)pˆ+ pˆ(qˆ − f)] + f˙ pˆ (3.33a)
UˆFF = −m
2
γ¨
γ
(qˆ − f)2 −mf¨qˆ (3.33b)
in agreement with Eqs. 9 and 30 of Ref. [34]. Thus the shortcuts obtained previously
for scale-invariant driving emerge naturally within our framework, from the flow
fields v and a (Eqs. 3.30, 3.31).
We end this section by highlighting two exceptional features of scale-invariant
Hamiltonians, both of which are due to the fact that all of the eigenstates of Hˆ0
satisfy the same scaling property, Eq. 3.29. First, although φ(q) = 〈q|n〉 denotes
a specific energy eigenstate in the above calculations, the resulting flow fields and
shortcuts (Eq. 3.33) are independent of the choice of n. This suggests that HˆEq.1.9CD =
HˆEq.3.11CD for scale-invariant driving, in contrast with the general situation discussed
in Sec. 3.4. Indeed, it has been shown elsewhere that Eq. 3.33a – which we derived
from Eq. 3.11 – follows directly from Eq. 1.9 [34]. Secondly, the shortcuts given by
61
Eq. 3.33 do not suffer from divergences at the nodes of excited energy eigenstates.
This is easy to understand in terms of the no-flux criterion of Sec. 3.5: because
variations in γ merely cause the eigenstate φ to expand or contract linearly, and
variations in f induce simple translations of φ, the probability between adjacent
nodes of the wavefunction is independent of t.
3.7 Numerical illustration of fast-forward driving
The parameter-dependent potential
U0(q; ξ) =
ξ2
16
[cosh(4q)− 1]− 3ξ
2
cosh(2q), (3.34)
belongs to a class of potentials studied by Razavy [71], for which convenient analyt-
ical expressions for low-lying eigenstates can be obtained. Here and below, we have
set the quantities β, m and ~ (appearing in Ref. [71]) to unity, so as to work with
an effectively dimensionless Hamiltonian. As illustrated in Fig. 3.2, U0(q; ξ) changes
from a broad double well to a narrow single well as ξ is increased from 0.1 to 6.0.
Now consider
Hˆ0(t) =
pˆ2
2
+ U0(qˆ; ξ(t)) (3.35)
where ξ(t) varies monotonically from 0.5 to 8.5 over the interval 0 ≤ t ≤ τ , according
to
ξ(t) = 4.5 + cos
(
pit
τ
)[
cos
(
2pit
τ
)
− 5
]
, (3.36)
62
Figure 3.2: U0(q, ξ) is plotted for five values of ξ.
and ξ(t) remains constant outside this interval. Note that ξ˙(0) = ξ˙(τ) = 0 and
ξ¨(0) = ξ¨(τ) = 0, hence Hˆ0(t) satisfies the continuity conditions described after
Eq. 3.2.
The wavefunction for the first excited state of Hˆ0(t) is given by [71]
φ(q, t) = κ(t) sinh(2q) exp
[
−1
4
ξ(t) cosh(2q)
]
(3.37)
where κ(t) is set by normalization. The corresponding eigenenergy is E(t) = −2.
Although this eigenstate has a node at the origin, the no-flux criterion of Sec. 3.5
is satisfied by the anti-symmetry of the wavefunction: φ(−q, t) = −φ(q, t), hence
I(0, t) = 1/2 for all t. Thus we expect our approach to apply despite the presence
of the node.
Using the above expressions and setting τ = 0.2, we numerically computed the
function I(q, t) (Eq. 3.6), from which we constructed the flow fields v and a and
63
the fast-forward potential UFF (Eq. 3.14). As ξ(t) increases from 0.5 to 8.5, U0(q, t)
becomes increasingly narrow (Fig. 3.2), as does the eigenstate φ(t); this is reflected in
the fields v and a, which describe the flow of probability toward the origin. UFF (q, t)
initially develops into a potential well that resembles a parabola (though it is not
precisely quadratic) – this brings about the acceleration of probability flow toward
the origin.
We performed two numerical simulations of evolution under the time-dependent
Schro¨dinger equation
i~
∂ψ
∂t
= Hˆ(t)ψ (3.38)
using Hˆ = Hˆ0 in the first simulation and Hˆ = Hˆ0 + UˆFF in the second; we will use
the notation ψ0 and ψFF to distinguish between the two simulations. In both cases
the wavefunction was initialized in the state ψ(q, 0) = φ(q, 0). The time evolution
was performed using the split-time propagation scheme [72, 73], which involves the
repeated application of the fast Fourier transform to toggle between the position
and momentum representations.
Fig. 3.3 shows snapshots of ψ(q, t)e−iα(t) (solid curves) and φ(q, t) (dashed
curves) for both simulations. Note that ψe−iα = φ in the adiabatic limit (Eq. 3.4)
– this is our motivation for plotting ψe−iα rather than ψ, though in the following
paragraphs we largely will stop writing the factor e−iα, for convenience.
The left panel of Fig. 3.3 shows the evolution of ψ0(q, t). Due to the nonadiabatic
time-dependence of Hˆ0, the wavefunction ψ
0 “lags” behind the instantaneous eigen-
state φ. This is particularly evident in Fig. 3.3(b), where the probability associated
64
ϕ,
R
e
[ψ
e
-
iα
]
-3 3
(a) t = 0.05
-3 3
(d) t = 0.05
-3 3
(b) t = 0.1
-3 3
(e) t = 0.1
-3 3
(c) t = 0.2
-3 3
(f) t = 0.2
q
Figure 3.3: Evolution under Hˆ0 (left panel) and Hˆ0 + UˆFF (right panel). The solid
magenta curves show Re(ψe−iα), and the dashed blue curves show the eigenstate φ.
Snapshots are shown at t = 0.05, at t = 0.1, and at the end of the process, t = 0.2.
65
with φ has shifted substantially toward the origin, while ψ0 remains somewhat be-
hind. This lag leads to shock waves, which are nascent in Fig. 3.3(b). These shocks
propagate inward, and ψ0 ends in a superposition of excited states [Fig. 3.3(c)].
The right panel shows the evolution of ψFF (q, t). Here the wavefunction devel-
ops excitations at short times [Fig. 3.3(d)], in response to large forces generated by
UˆFF (t). These forces eliminate the lag that is observed in the left panel, by “squeez-
ing” the wavefunction and causing probability to accelerate toward the origin. At
later times this flow is decelerated – again, due to UˆFF (t) – and the excitations sub-
side [Fig. 3.3(e)]. The wavefunction gently arrives at the desired energy eigenstate
at the final time [Fig. 3.3(f)].
In the present context Eq. 3.20 can be written as
ψFF e−iα = φ eiS/~ (3.39)
which implies that the probability densities |ψFF |2 = |φ|2 at all times, despite the
excitations that develop in ψFF (q, t). We have verified this result in our simulations
(data not shown). Eq. 3.39 further implies that Re(ψFF e−iα) = φ(q, t) cos[S(q, t)/~],
which is illustrated in Fig. 3.3(d), where the dashed line is manifestly the envelope
of the solid line.
The variation of the potential U0(q, t) and UFF (q, t) is plotted in Fig. 3.4. While
U0(q, t) follows Eq. 3.34 and ξ is varied according to Eq. 3.36, UFF (q, t) is nu-
merically evaluated using Eq. 3.14. For τ = 0.2, snapshots are presented at t =
0, 0.04, 0.055, 0.08, 0.12 and 0.2. The solid magenta curves represent U0(q, t) while
66
U
(q
,t
)
-2 2
-5
5
(a) t = 0
-2 2
-5
5
(b) t = 0.04
-2 2
-5
5
(c) t = 0.055
-1.5 1.5
-5
5
(d) t = 0.08
-1 1
-10
-5
5
(e) t=0.12
-1 1
-15
-10
-5
5
(f) t=0.2
q
Figure 3.4: The variation of U0(q, t) and UFF (q, t) is plotted. The solid ma-
genta curves show U0(q, t) and the dashed blue curves show UFF (q, t) at t =
0, 0.04, 0.055, 0.08, 0.12 and 0.2. U0(q, t) is initially a double well potential, but
as it evolves, the wells comes closer to the origin and eventually U0(q, t) transforms
to a single attractive well potential. UFF (q, t) smoothly increases from zero and
quickly becomes an attractive well, which then becomes a repulsive well that finally
transforms smoothly to zero at t = τ .
67
0.00 0.05 0.10 0.15 0.20
0.0
0.2
0.4
0.6
0.8
1.0
Time
Fi
de
lit
y
Figure 3.5: The blue dashed curve shows the fidelity |〈φ|ψ0〉|2, quantifying the
limited extent to which ψ0(q, t), evolving under Hˆ0, keeps pace with the energy
eigenstate φ(q, t). The solid red curve shows |〈φ|ψFF 〉|2, which is the fidelity that is
achieved when UˆFF is added to the Hamiltonian.
the blue dashed curves represent UFF (q, t). As shown, UFF (q, t) slowly increases
from zero and becomes an attractive well potential. This property of UFF (q, t)
squeezes the wavefunction appropriately such that its envelope aligns with the adi-
abatic path. As a result, excitations are developed in the evolving wavefunction.
After a while, the attractive nature of UFF (q, t) becomes repulsive and the well gets
inverted as shown. This ensures that the excitations start to die down. The inverted
well smoothly becomes zero at the final time τ and a perfect fast-forward driving is
achived by UFF (q, t) as supported by Fig. 3.3.
Finally, for both simulations we computed the fidelity F (t) = |〈φ|ψ〉|2, that is
the degree of overlap between the evolving wavefunction and the energy eigenstate.
Fig. 3.5 shows the results. In the absence of the auxiliary term, the fidelity |〈φ|ψ0〉|2
decays monotonically to F ≈ 0.3. When UˆFF is included in the Hamiltonian, the
fidelity |〈φ|ψFF 〉|2 at first drops rapidly to nearly zero – due to the excitations that
68
develop in ψFF – but then it claws its way back to unity, illustrating the effectiveness
of the fast-forward potential obtained from the acceleration flow field a(q, t).
3.8 Extension to three degrees of freedom
Although the focus in this paper is on systems with one degree of freedom, here
we briefly discuss how the results of Sec. 3.3 might be extended to three dimensions.
We will use boldface to denote vector quantities.
For a given choice of quantum number n, let φ(q, t) and E(t) denote the n’th
eigenstate and eigenenergy, respectively, of the Hamiltonian Hˆ0(t) given by Eq. 3.1.
Let us define a vector field v(q, t) by the equation
∂tφ
2 +∇ · (vφ2) = 0 (3.40)
which describes how the eigenstate probability density φ2(q, t) varies parametrically
with t. We assume that Hˆ0(t) and its first two time derivatives vanish outside the
interval 0 < t < τ (as in the one-dimensional case), therefore v can be constructed
to vanish outside this interval as well: v = 0 for t /∈ (0, τ). 4
Since Eq. 3.40 is a continuity equation, it can be interpreted as describing an
ensemble of independent trajectories, each evolving according to q˙ = v(q, t). The
acceleration of these trajectories is described by a field a(q, t) whose i’th component
4Eq. 3.40 defines v(q, t) only up to gauge-like transformations of the form v → v + (∇ ×
B)/φ2, where B(q, t) is an arbitrary, well-behaved vector field. Hence we have some freedom in
constructing v. This freedom was not present in Sec. 3.3, where the v(q, t) was defined using the
construction shown in Fig. 3.1, rather than from the continuity equation.
69
satisfies
ai = q¨i =
∂vi
∂t
+
∑
j
∂vi
∂qj
dqj
dt
(3.41)
We now define a counterdiabatic Hamiltonian
HˆCD(t) =
pˆ · vˆ + vˆ · pˆ
2
, vˆ(t) = v(qˆ, t) (3.42)
Using φ˙ + (1/2)(∇ · v)φ + v · ∇φ = 0 (which follows from Eq. 3.40), it is readily
verified that the wavefunction
ψad(q, t) = φ(q, t) e
iα(t) , α(t) = −1
~
∫ t
0
E(t′) dt′ (3.43)
is a solution of the Schro¨dinger equation i~ ∂tψad = (Hˆ0 + HˆCD)ψad.
Now let us suppose that the field v(q, t) can be chosen to be curl-free:
∇× v = 0 (3.44)
We can then introduce a function S(q, t) that satisfies
∇S = mv (3.45)
which allows us to rewrite Eq. 3.41 as
ai(q, t) =
1
m
∂
∂qi
[
∂S
∂t
+
(∇S)2
2m
]
(3.46)
70
We have used both Eqs. 3.44 and 3.45 in going from Eq. 3.41 to Eq. 3.46. If we now
define UFF (q, t) by the equation
∂tS +
(∇S)2
2m
+ UFF = 0 (3.47)
then Eq. 3.46 implies −∇UFF = ma (compare with Eq. 3.14). It is now a matter
of algebra to verify that ψ¯ ≡ φeiαeiS/~ obeys the Schro¨dinger equation i~ ∂tψ¯ =
(Hˆ0 + UˆFF )ψ¯. Since S(q, t) is a constant outside the interval 0 < t < τ , we see
that the addition of the fast-forward potential UFF causes the chosen eigenstate
of the initial Hamiltonian to evolve to the corresponding eigenstate of the final
Hamiltonian.
As in the one-dimensional case, divergences in v(q, t) may arise whenever φ(q, t) =
0, potentially causing the method to break down for excited eigenstates. This issue
deserves further exploration.
3.9 Summary
We have developed a framework for constructing counterdiabatic and fast-
forward shortcuts for quantum systems. This framework is organized around velocity
and acceleration flow fields v(q, t) and a(q, t), which describe the time-dependence
of the desired adiabatic evolution. Once the flow fields have been determined, the
shortcuts are given by simple expressions involving these fields (Eqs. 3.11, 3.14).
The flow-fields can be pictured in terms of the evolution of a “picket fence” of lines
(Figs. 3.1) that glide around as time is varied parametrically. The fields v and a are
71
constructed from integrated function I that define the picket fence.
As noted in Sec. 3.5, the nodes of excited energy eigenstates φ(q, t) generically
pose a problem for our method, as they do for the fast-forward approach in general.
The divergences in v(q, t) that result from these nodes can be understood intuitively
by considering Eq. 3.27, which gives the probability flux across the ν’th node: Φν =
(v − uν)φ2. The two factors on the right represent the flow velocity relative to the
motion of the node, v−uν , and the local density, φ2. If we momentarily imagine that
φ2 is very small but non-zero at qν , then we see that v−uν must be very large in order
to “push through” a fixed probability flux – an apt analogy is water flowing through
a pipe that becomes narrow at a certain point. Thus v− uν diverges as φ2 → 0: an
infinite velocity is required to achieve a finite flux, at vanishing probability density.
When the time-dependence of φ2(q, t) is such that there is no flux of probability
across nodes, i.e. when the probability between neighboring nodes remains constant
even as the eigenstate deforms, then the flow fields v and a are non-singular and we
expect our method (and more generally the fast-forward approach [26]) to work well.
This no-flux criterion is satisfied for scale-invariant driving, as well as for the model
system studied numerically in Sec. 3.7. In the latter case the criterion is satisfied
because the potential U0(q, t) (Eq. 3.34) is symmetric about the origin. It would be
useful to identify a more generic (i.e. non-symmetric) potential and eigenstate for
which the no-flux criterion is satisfied, and to test whether our method continues
to work in that situation. This would provide a more stringent test of the no-flux
criterion than the one studied in Sec. 3.7.
Our framework connects the counterdiabatic and fast-forward approaches for
72
quantum systems. The fields v(q, t) and a(q, t) provide two mathematical descrip-
tions the for same flow of probability. The former defines the counterdiabatic Hamil-
tonian HˆCD, while the latter (together with the mass, m) determines the fast-forward
potential UˆFF . It is remarkable that no other input is required to construct these
shortcuts. For the moment we lack a deeper or intuitive understanding of why this
should be the case.
This chapter presents an open problem in the development of fast-forward short-
cuts for excited states of quantum systems, when the no-flux criterion is not sat-
isfied. Our next goal is to probe into this problem. We first study an analogous
classical problem on shortcuts in the next chapter, and we extend the flow-fields
based method to classical systems. In a subsequent chapter, we will investigate if
the intuitions from classical shortcuts may help in constructing exact or approxi-
mate quantum shortcuts for excited states when the no-flux criterion is not satisfied,
at least in the semiclassical limit.
73
Chapter 4: Classical shortcuts using flow-fields
4.1 Overview
For a classical system in one degree of freedom, the action variable I =
∮
p ·dq is
an adiabatic invariant [51]. As an example, when the length of a pendulum is slowly
varied, both its energy E and frequency of oscillation ω change with time, but their
ratio E/ω, which is proportional to the action, remains constant. The adiabatic
invariant can be visualized in phase space by imagining a collection of trajectories
evolving under a slowly time-dependent Hamiltonian, H0. If all initial conditions
are sampled from a single energy shell (that is, a level curve) of H0(q, p, 0), then a
snapshot of these trajectories at a later time t will find them located on a single
energy shell of H0(q, p, t), with the same action as the initial shell, as shown in
Fig. 4.1.
In this chapter, we pose and answer the following question: How can the adia-
batic invariant be preserved under nonadiabatic driving conditions? We consider a
Hamiltonian H0(q, p, t) = p
2/2m + U0(q, t) that varies at an arbitrary rate. Under
the evolution generated by this Hamiltonian, the action I(q, p, t) does not remain
constant: If at time t = 0 we launch a collection of trajectories, each with the
same initial action I0, then at later times their actions will generally differ from one
74
t=0.0 t=6.0
Figure 4.1: Illustration of the classical adiabatic invariant. Fifty trajectories evolv-
ing under a slowly varying Hamiltonian are shown at an initial time (on left) and
a later time (on right). The closed curves are instantaneous energy shells – level
curves of H0 – with identical values of the action I =
∮
p · dq. Trajectories were
generated using H(q, p, t) given by Eq. 4.21, setting τ = 10.0 to achieve slow driving.
another and from the initial action. Thus under nonadiabatic driving, trajectories
wander away from the energy shell associated with the action I0. But suppose we
want these trajectories to “return home” at a specified later time τ , i.e., we demand
that the action of each trajectory be equal to I0 at t = τ , given that its action
had this value at t = 0. In this chapter we solve for the additional forces that are
required to steer the trajectories back to the action I0 at t = τ . More precisely,
we show how to construct an auxiliary fast-forward potential UFF (q, t) with the fol-
lowing property. Under the dynamics generated by the Hamiltonian H0 + UFF , all
trajectories that begin with action I0 at t = 0 will end with the same action, I0, at
t = τ . Throughout this chapter, the action I(q, p, t) is defined with respect to the
original Hamiltonian H0(q, p, t).
The motivation behind this work comes from the fact that quantum shortcuts
derived in Chap. 3 can not be applied to excited states in general. We investigate
75
a classical analogous problem in shortcuts hoping that it might provide useful in-
sights for designing quantum shortcuts for excited states in the semiclassical limit.
However, in this chapter, we focus on solving a self-contained problem of general
theoretical interest in elementary classical dynamics, for which we obtain a simple
and appealing solution (Eq. 4.11). In Sec. 4.2, we set up the classical problem and
define the flow-field velocity and acceleration. We state the expressions for coun-
terdiabatic Hamiltonian and fast-forward potential in Sec. 4.3, and show that these
auxiliary fields achieve counterdiabatic and fast-forward driving respectively. We
illustrate our results in Sec. 4.4 by solving a model double well Hamiltonian numer-
ically, and conclude in Sec. 4.5. We defer the discussion on the applicability of the
results from this chapter to quantum systems in the next chapter.
4.2 Setup and definition of flow-field parameters
Consider a classical system in one degree of freedom, described by a kinetic-
plus-potential Hamiltonian
H0(z, t) =
p2
2m
+ U0(q, t) , z = (q, p) (4.1)
H0 varies with time during the interval 0 ≤ t ≤ τ , but is constant outside this
interval. We assume that H0 is twice continuously differentiable with respect to
time [74], and hence both ∂H0/∂t and ∂
2H0/∂t
2 vanish at t = 0 and t = τ . In
Appendix B, we discuss how this assumption can be relaxed.
The term energy shell will denote a level curve of H0(z, t); that is, the set of all
76
points where H0 takes on a particular value, E, at time t. We will assume that each
energy shell forms a simple, closed loop in phase space. The function
Ω(E, t) =
∫
dz θ [E −H0(z, t)] =
∮
E
p · dq (4.2)
is the volume of phase space enclosed by the energy shell E of H0(z, t), and the
action,
I(z, t) = Ω(H0(z, t), t), (4.3)
is the volume enclosed by the energy shell that contains the point z. Eq. 4.3 implies
{I,H0} ≡ ∂I
∂q
∂H0
∂p
− ∂I
∂p
∂H0
∂q
= 0, (4.4)
which will prove useful.
Let us choose an arbitrary action value I0 > 0, and define the adiabatic energy
E¯(t) by the condition
Ω(E¯(t), t) =
∮
E¯(t)
p′ dq′ = I0. (4.5)
The adiabatic energy shell E(t) = {z|H0(z, t) = E¯(t)} is the level curve of H0(z, t)
with the value E¯(t), enclosing a phase space volume I0. Hence I(z, t) = I0 for all
z ∈ E(t). The action value I0 and the adiabatic energy E¯(t) are classical analogues
of the quantum number n and eigenenergy En(t).
At t = 0, the adiabatic energy shell E(0) defines a set of initial conditions that
form a closed loop in phase space. As trajectories evolve under H0(z, t) from these
77
initial conditions, this loop evolves in time,
L(t) = {z = zt(z0)|z0 ∈ E(0)}, (4.6)
where zt(z0) indicates the trajectory that evolves under H0(z, t) from initial condi-
tions z0. If H0 varies slowly with time, then these trajectories remain close to the
adiabatic energy shell, but under more general conditions the loop L(t) strays away
from E(t) for t > 0.
We now assume that H0 varies at an arbitrary – i.e. non-adiabatic – rate, but
we continue to use the term adiabatic energy to refer to E¯(t) defined by Eq. 4.5,
for chosen value of action, I0. For a trajectory with initial action I0, we wish to
construct a counterdiabatic Hamiltonian HCD(q, p, t) and a fast-forward potential
UFF (q, t) such that: (1) if the trajectory evolves under H0 + HCD, it remains on
the adiabatic energy shell at all times, that is, I(t) = I0; and (2) if the trajectory
evolves under H0 + UFF , it returns to the adiabatic energy shell at the final time:
I(τ) = I(0) = I0. Here and below, I(t) = I(q(t), p(t), t) denotes the value of the
action function along the trajectory.
To construct these shortcuts for a given choice of I0, let
p¯(q, t) = [2m(E¯ − U0)]1/2 (4.7)
78
adiabatic energy 
shell
Figure 4.2: The closed red curve, with upper and lower branches ±p¯(q, t) (Eq. 4.7),
depicts the adiabatic energy shell E¯(t) in phase space. The blue vertical lines divide
E¯(t) into K  1 strips of equal phase space volume. q(S, t) is the right boundary of
the shaded region, of phase space volume S. The parametric motion of the vertical
lines defines the velocity and acceleration fields v(q, t) and a(q, t).
specify the upper branch of the adiabatic energy shell, and let
S(q, t) = 2
∫ q
q0(t)
p¯(q′, t)dq′ (4.8)
denote the volume of phase space enclosed by the adiabatic energy shell E¯(t) between
the left turning point q0(t) and a point q. The function S(q, t) can be inverted to
obtain q(S, t), which in turn is used to define flow fields
v(q, t) =
∂
∂t
q(S, t) = −∂tS
∂qS (4.9a)
a(q, t) =
∂2
∂t2
q(S, t) = v′v + v˙ (4.9b)
These flow fields are pictured by dividing the adiabatic energy shell into K  1
79
strips enclosing equal phase space volume, delimited by lines drawn at locations
{qk(t)}; see Fig. 4.2. The fields v and a describe the motion of these lines as the
parameter t is varied: q˙k = v(qk, t) and q¨k = a(qk, t). Since ∂H0/∂t = ∂
2H0/∂t
2 = 0
at t = 0 and t = τ (see comments following Eq. 4.1) we have
v(q, 0) = v(q, τ) = 0 , a(q, 0) = a(q, τ) = 0. (4.10)
4.3 Counterdiabatic and fast-forward driving
Using the flow fields parameters defined in Eq. 4.9, we now define a counterdia-
batic Hamiltonian
HCD(q, p, t) = pv(q, t), (4.11a)
and a fast-forward potential UFF that satisfies
− ∂qUFF (q, t) = ma(q, t), (4.11b)
both of which vanish for t /∈ (0, τ). We will now demonstrate that Eqs. 4.11a and
4.11b achieves classical counterdiabatic and fast-forward driving.
Consider a point in phase space, (qn(t), pn(t)), attached to the top of the nth
line segment: pn = p¯(qn, t) (see Fig. 4.2). As the shape of the energy shell and
the locations of the line segments vary parametrically with time, this point (qn, pn)
moves in phase space, surfing the upper branch of the energy shell. This motion is
80
described by the equations
q˙n = v(qn, t) , p˙n = −pnv′(qn, t) (4.12)
where the equation for p˙n is obtained by demanding that the phase space volume of
the strip between neighboring vertical lines, δSn ≡ 2pn(qn+1− qn), remain constant.
In Eq. 4.12 and throughout this chapter, dots and primes denote derivatives with
respect to t and q respectively. Eq. 4.12 also describes the motion of a point attached
to the bottom of one of the vertical lines. We easily verify that Eq. 4.12 is generated
by the Hamiltonian HCD of Eq. 4.11a. If we start with initial conditions distributed
over the energy shell E(0), and we evolve trajectories from these initial conditions
under the Hamiltonian HCD(q, p, t), then these trajectories cling to the evolving
adiabatic energy shell, with each trajectory attached to the upper or lower end of
one of the vertical line segments. Hence the flow generated by HCD preserves the
adiabatic energy shell, in the following sense: for each time step δt, this flow maps
points on E(t) to points on E(t + δt). Equivalently, the action I(z, t) is conserved
under this flow, for those trajectories with action I0. Therefore we have
0 =
∂I
∂t
+
∂I
∂q
q˙ +
∂I
∂p
p˙ =
∂I
∂t
+ {I,HCD} ∀ z ∈ E(t) (4.13)
Next, we consider the full Hamiltonian H0 +HCD, which generates equations of
motion
q˙ =
p
m
+ v(q, t) , p˙ = −U ′0(q, t)− pv′(q, t) (4.14)
81
Along a trajectory z(t) obeying these dynamics,
I˙ =
d
dt
I(z(t), t) =
∂I
∂t
+ {I,H0}+ {I,HCD} (4.15)
Eqs. 4.4, 4.13 and 4.15 imply that I˙ = 0 for all z ∈ E(t). Thus the flow generated by
H0+HCD preserves the adiabatic energy shell and generates counterdiabatic driving.
This is easily understood: with each time step δt, the term HCD(z, t) generates a
flow that maps E(t) onto E(t + δt) while the term H0(z, t) generates flow parallel
to the adiabatic energy shell. As a consistency check, we can verify directly from
Hamilton’s equations that the flow generated by HCD preserves the adiabatic energy
shell (see Appendix C).
To this point, we have constructed a Hamiltonian H0 + HCD that generates
trajectories which cling to the adiabatic energy shell E(t). Along these trajectories,
I(z, t) remains constant. We now introduce a change of variables that effectively
transforms HCD(q, p, t) into the fast-forward potential UFF (q, t).
Consider the evolution of the observables
Q(q, p, t) = q , P (q, p, t) = p+mv(q, t) (4.16)
along a trajectory that evolves under Eq. 4.14. By direct substitution we get
dQ
dt
=
P
m
,
dP
dt
= −U ′0(Q, t) +ma(Q, t) (4.17)
82
using Eq. 4.9. Eq. 4.17 is generated by the Hamiltonian
HFF(Z, t) = H0(Z, t) + UFF (Q, t) (4.18)
where Z = (Q,P ) and UFF satisfies Eq. 4.11b. Thus Eq. 4.16 defines a time-
dependent transformation Mt : z → Z, which maps any trajectory z(t) evolving
under H0 + HCD to a counterpart trajectory Z(t) evolving under HFF(Z, t). Now
consider specifically a trajectory z(t) that evolves, under H0 + HCD, from initial
conditions on the adiabatic energy shell E(0). As we have already seen, this tra-
jectory remains on the adiabatic energy shell E(t) for all times t ∈ [0, τ ]. Under
the mapping Mt, its image Z(t) (which evolves under HFF) is displaced along the
momentum axis by an amount mv(q, t) (Eq. 4.16). By Eq. 4.10, Z(t) begins and
ends on the adiabatic energy shell: Z(0) ∈ E(0), Z(τ) ∈ E(τ). This is precisely the
fast-forward driving, which concludes our proof.
Consider the loop
LFF(t) = {z = zFFt (z0)|z0 ∈ E(0)} (4.19)
which evolves in phase space under HFF. The results of the previous paragraph can
be written compactly as follows:
Mt : E(t)→ LFF(t). (4.20)
At any time t, LFF(t) is the image of E(t) under the transformation defined by
83
Figure 4.3: A snapshot, at t = τ/2, of 100 trajectories evolving under HFF(z, t) using
a rapid protocol, with τ = 0.2 (see text). The closed black loop is the adiabatic
energy shell E(t), and the red loop above it is constructed by displacing each point
on the lower loop by an amount mv(q, t) along the p-axis. As predicted by Eq. 4.20,
the trajectories coincide with the red loop.
Eq. 4.16 (see Fig. 4.3). This result implies that the function J(q, p, t) ≡ I(q, p −
mv(q, t), t) is a local dynamical invariant. That is, if a trajectory z(t) is launched
from the energy shell E(0) and then evolves under HFF, then the value of J is
conserved along this trajectory: J(z(t), t) = I0. For consistency, we can verify
directly from Hamilton’s equations that dJ/dt = 0 for any point z ∈ LFF (see
Appendix D).
4.4 Numerical example
To illustrate our results, we chose the dimensionless Hamiltonian
H(z, t) =
p2
2
+ q4 − 16q2 + λ(t)q (4.21a)
84
with
λ(t) = 4 cos(pit/τ)[5− cos(2pit/τ)] (4.21b)
This Hamiltonian describes a particle in a double-well potential, with a linear contri-
bution whose slope λ(t) evolves from +16 at t = 0, to −16 at t = τ , with λ˙ = λ¨ = 0
at initial and final times. As illustrated in Fig. 4.1, when τ = 10.0 the driving
is sufficiently slow for the adiabatic invariant to be conserved with high accuracy.
For the simulations described in the following paragraph, we set τ = 1.0 to obtain
non-adiabatic driving.
We considered an initial adiabatic energy shell E(0) with energy E¯(0) = 50.0,
which corresponds to I0 = 214.035. We numerically determined the fields v(q, t) and
a(q, t) and constructed UFF(q, t) according to Eq. 4.11b. We then generated fifty
initial conditions on the energy shell E(0), shown in Fig. 4.4(a), and we performed
two sets of simulations. In the first set, trajectories were evolved from these initial
conditions under H(z, t). In the second set, trajectories were evolved from the same
initial conditions under the Hamiltonian HFF = H+UFF. In the absence of the fast-
forward potential UFF, the trajectories belonging to the first set have final actions
I(z, τ) that span a range of values, as seen in Fig. 4.4(b). By contrast, the addition
of UFF guides the second set of trajectories back to the adiabatic energy shell E(τ),
where each trajectory ends with I(z, τ) = I0; see Fig. 4.4(c). Note, however, that
while the initial conditions in Fig. 4.4(a) are spaced uniformly with respect to the
microcanonical measure, this is not the case for the final conditions in Fig. 4.4(c).
As discussed in the Appendix E, this non-uniformity is due to the fact that UFF(q, t)
85
Figure 4.4: Initial (a) and final (b,c) conditions for trajectories launched from a sin-
gle energy shell E(0). The trajectories in panel (b) evolved under H(z, t) (Eq. 4.21),
while those in panel (c) evolved under HFF = H + UFF, with τ = 1.0. The solid
black curves show the adiabatic energy shell E(t) at initial and final times.
depends on the choice of I0. The variation of U(q, t) and UFF (q, t) with q is shown
in Fig. 4.5 with solid magenta curves and dashed blue curves respectively for times
t = 0.2, 0.5 and 0.8.
We also performed simulations with a shorter duration, τ = 0.2. After con-
structing UFF(q, t) for this faster protocol, we simulated fifty trajectories evolving
under HFF = H + UFF, using the initial conditions in Fig. 4.4(a). Fig. 4.3 depicts a
snapshot of these trajectories at t = τ/2. The two closed curves show the adiabatic
energy shell E(t) and its image under the mapping p → p + v(q, t) (see Eq. 4.16).
86
U
(q
,t
)
-5 5
-120
-60
60
(a) t = 0.2
-5 5
-120
-60
60
(b) t = 0.5
-5 5
-120
-60
60
(c) t = 0.8
q
Figure 4.5: A plot of U0(q, t) and UFF (q, t) is shown in sold magenta and dashed
blue curves respectively. UFF (q, t) is non-zero only in the interval 0 ≤ t ≤ τ .
Shortly after t = 0, UFF (q, t) has a positive value to the left of origin and a negative
value to the right of origin, which ensures that the particles from the left well are
appropriately pushed towards the right well. Thereafter, the value of UFF (q, t) to
the right of the origin begins to increase such that at t = τ/2 = 0.5, an attractive
well is formed. Beyond t = 0.5, UFF (q, t) starts to decrease to the left of the origin,
and finally it monotonically goes to zero at t = τ .
This figure confirms Eq. 4.20: the trajectories evolving under HFF = H + UFF are
located on a loop LFF(t) that is obtained by “shearing” the instantaneous energy
shell E(t) along the momentum axis, by an amount mv(q, t).
For scale invariant driving [34], we consider the classical analogus Hamiltonian
from Eq. 3.28. The flow-field velocity is given by Eq. 3.30, and subsequently the
fast-forward potential is the classical analogue of Eq. 3.33b which does not depend
on I0 [34]. In this rather special case, every trajectory evolving under HFF returns
to its adiabatic energy shell at t = τ , J(z, t) is a global dynamical invariant – it is
the Lewis-Riesenfeld invariant [75,76] – and microcanonical initial distributions are
mapped to microcanonical final distributions.
87
4.5 Summary
Adiabatic invariants enjoy a distinguished history in quantum and classical me-
chanics [77], but the problem of how to achieve adiabatic invariance under non-
adiabatic conditions has gained attention only recently. Here we have shown how
to construct a potential UFF(q, t) that guides trajectories launched from a given
energy shell of an initial Hamiltonian to the corresponding energy shell of the final
Hamiltonian, so that the initial and final values of action are identical for every
trajectory.
The results presented in this chapter effectively extend the flow-fields based
method introduced in Chap. 3 to classical systems. The numerical illustrations show
that trajectories distributed uniformly over an initial energy shell, under fast-forward
driving, end up on the final energy shell which preserves the action. However the
initial uniform distribution is not preserved. These results may offer an alternative
approach to solve for quantum shortcuts to adiabaticity, which could overcome the
problem that arises due to nodes discussed in Sec. 3.5. The classical fast-forward
potential evaluated in this chapter, when quantized, may lead to the quantum fast-
forward potential. The classical potential is free from singularities, and for large
n the Correspondence Principle suggests that evolution under Hˆ + Uˆ
(n)
FF will cause
the initial eigenstate φn(q, 0) to evolve to the final eigenstate φn(q, τ). This will be
investigated in the next chapter.
The counterdiabatic Hamiltonian Eq. 4.11a generated by the flow-fields ap-
proach can be compared to the approaches in Refs. [25, 52, 54]. In both cases, the
88
classical action I(z, t) is preserved along the entire trajectory. However, a crucial
difference lies in the fact that the counterdiabatic Hamiltonians from the previous
approaches are independent of the initial energy of the system, but the counterdia-
batic Hamiltonian obtained from the flow-fields method depends on the choice of the
initial energy shell (except for scale invariant driving). The quantum counterpart of
the classical Hamiltonian obtained in this chapter can be compared to the previous
results from Refs. [22, 24], where the quantum eigenstate |n(t)〉 is preserved along
the entire trajectory. We will discuss this in the next chapter.
It is natural to ask whether our results can be applied to systems with d > 1
degrees of freedom. In certain situations of experimental relevance, such as ultra-
cold gases in optical lattices, a separation of variables reduces a three-dimensional
problem to an effectively one-dimensional one [11,78], providing a potential platform
to test our predictions. More generally, the distinction between integrable, chaotic,
and mixed phase space systems becomes crucial for d-dimensional systems [79]. For
integrable systems, the transformation to action-angle variables [51] may provide a
useful first step to extending our results, but for chaotic or mixed systems the task
is likely to be more challenging.
89
Chapter 5: Quantum shortcuts for excited states
5.1 Overview
We now focus on obtaining shortcuts – that is deriving a counterdiabatic Hamil-
tonian and a fast-forward potential – for a quantum system initialized in an excited
energy state. We particularly study the case of an arbitrary driving protocol where
the no-flux criterion is not satisfied, and therefore Eqs. 3.11 and 3.14 cannot yield ex-
act shortcuts due to the divergences associated with eigenstate, see Sec. 3.5. In this
chapter, we analyze the semiclassical limit of a quantum excited state, and investi-
gate whether classical shortcuts from Chap. 4 provide useful insights for obtaining
quantum shortcuts for excited states.
We start this chapter by making a comparison of the flow-fields method for
quantum and classical systems – as discussed in Chapters 3 and 4 – in Sec. 5.2. We
find that the classical and quantum flow-fields do not match in the semi-classical
limit. Therefore, the quantum counterpart of a classical shortcut is not the exact
quantum shortcut. In Sec. 5.3, we carry out a semiclassical analysis of an energy
eigenfunction. Following our analysis, we hypothesize that the classical auxiliary
fields – both the counterdiabatic Hamiltonian and the fast-forward potential – from
Chap. 4 should be able to produce a quantum shortcut to a very good approxi-
90
mation. We test this hypothesis numerically for a model system in Sec. 5.4. We
numerically solve the time-dependent Schro¨dinger equation for a quantum system
which is evolving under UˆFF – the quantum counterpart of Eq. 4.1, and test the
accuracy with which the quantum counterpart of the classical fast-forward potential
of Eq. 4.11b achieves quantum fast-forward driving. In Sec. 5.5, we analyze the final
distribution on the classical energy shell and combine it with the numerical results of
Sec. 5.4 to quantitatively establish the relationship between the classical and quan-
tum shortcut. In Sec. 5.6, we analytically argue that the accuracy of the quantum
fast-forward driving illustrated in Sec. 5.4 will be reflected in the counterdiabatic
driving as well, i.e., the quantum counterpart of the classical counterdiabatic Hamil-
tonian (Eq. 4.11a) will make an excited state track its adiabatic path to a very good
approximation. We present concluding remarks in Sec. 5.7.
5.2 Comparison of quantum and classical flow-fields
As proposed in Ref. [25] and illustrated in Chap. 2, a classical counterdiabatic
Hamiltonian emerges when the right side of Eq. 1.9 is evaluated in the semiclassical
limit. Similarly, it is natural to speculate that the classical shortcuts of Chap. 4
are the semiclassical limit of the quantum shortcuts of Chap. 3. In that case the
close similarity between Eqs. 4.11a, 4.11b and Eqs. 3.11, 3.14 would simply reflect
the Correspondence Principle. In the following paragraph we address this issue by
asking whether the flow fields v(q, t) and a(q, t) defined in Chap. 4 emerge from
those of Chap. 3 in the semiclassical limit (~ → 0). We will temporarily use the
91
superscript Q (for “quantum”) to denote certain quantities defined in Chap. 3, SC
to denote their semiclassical limits, and C to denote quantities defined in Chap. 4.
When we consider the semiclassical limit of the field vQ(q, t) (Eq.3.7), we imme-
diately run into a difficulty: the divergences discussed in Sec. 3.5 proliferate in this
limit, as the number of nodes of φ becomes large. This proliferation of divergences
(nodes) arises from the rapid spatial oscillations of high-lying eigenstates φ(q, t). To
obtain a non-singular velocity field, we replace the oscillatory probability density
φ2 (used to construct vQ) by a locally averaged counterpart, φ2, that smooths over
these oscillations. The semiclassical limit of φ2 is the microcanonical probability
distribution, projected from phase space onto the coordinate axis [2]:
lim
~→0
φ2(q, t) = µ(q, t) ∝
∫
dp δ(E¯ −H0) ∝ 1
p¯(q, t)
(5.1)
with p¯ given by Eq. 4.7. Using µ in place of φ¯2 in Eq. 3.6 we obtain
ISC(q, t) =
∫ q
−∞
µ(q′, t) dq′. (5.2)
We use this function to define vSC = −∂tISC/∂qISC , which is free of divergences and
can be viewed as the semiclassical limit of vQ = −∂tIQ/∂qIQ (Eq. 3.7). Comparing
vSC(q, t) with the field vC(q, t) defined by Eq. 4.9a, we see that while one is con-
structed from the integrated microcanonical distribution ISC = ∫ q µ dq′, the other
is constructed in terms of the phase space enclosed by the energy shell, S = ∫ q p¯ dq′.
Therefore, in general, the two fields differ: vSC 6= vC . We conclude that Eq. 4.11a
92
should not be viewed as the semiclassical limit of Eq. 3.11. Similar comments apply
to the acceleration field a(q, t).
We summarize the situation as follows: while the flow fields v and a are defined
similarly in the quantum and classical cases (compare Figs. 3.1 and 4.2), and while
the construction of counterdiabatic and fast-forward terms from the flow fields is
essentially identical in the two cases, the Correspondence Principle does not provide
an adequate explanation for this striking similarity. We also note that scale-invariant
driving (Eq. 3.28) provides an exception to this general conclusion: in that case the
quantum and classical flow fields are in fact identical [34,56].
As a final item of semiclassical comparison, let us consider trajectories evolving
under the classical Hamiltonian H0 +UFF , with initial conditions sampled from the
adiabatic energy shell (Fig. 4.4). It was shown in Ref. [56] that the function
J(q, p, t) = I(q, p−mv(q, t), t) (5.3)
remains constant along these trajectories: J(t) = I0 for all t. This is illustrated in
Fig. 4.3, where the thick red curve is obtained by “boosting” the thin black curve –
the adiabatic energy shell – by an amount mv(q, t) along the momentum direction.
Now consider the fast-forward wavefunction ψ¯ = φ eiα eiS/~ (Eq. 3.20) evolving under
Hˆ0 + UˆFF . Let us approximate the eigenstate φ(q, t) by the semiclassical form [2]
φ = A+e
+(i/~)
∫ q p¯ dq′ + A−e−(i/~) ∫ q p¯ dq′ (5.4)
93
where |A±(q, t)| ∝
√
1/p¯. The terms on the right side of Eq. 5.4 represent a right-
moving wave train and a left-moving wave train, with local momenta corresponding
to the upper and lower branches ±p¯ of the adiabatic energy shell (Fig. 4.2). Then
for the fast-forward wavefunction we get
ψ¯ = φ eiα eiS/~ = A+e
iα e(i/~)
∫ q(p¯+mv) dq′ + A−eiα e(i/~) ∫ q(−p¯+mv) dq′ (5.5)
since S =
∫ q
mv dq′ (Eq. 3.15). The terms in Eq. 5.5 are wave trains with local
momenta ±p¯ + mv. Thus the fast-forward wavefunction ψ¯ is represented, in the
semiclassical sense, by a “boosted” adiabatic energy shell similar to the one shown
as a thick red curve in Fig. 4.3. Although this interpretation provides a neat corre-
spondence between the quantum and classical fast-forward methods, it should not be
taken too literally, since the fast-forward method of Chap. 3 generally applies only
to the ground state (as discussed earlier), where the semiclassical approximation
(Eq. 5.4) is not generally accurate.
5.3 Auxiliary fields for excited states
Motivated by the need to obtain experimentally implementable shortcuts for
general, i.e., non scale-invariant systems, we proceed to find HˆCD(t) and UFF (qˆ, t)
for excited states. Note that UˆFF (t) due to its local nature is more practical for
experimental purposes compared to HˆCD(t) which has a non-local behaviour. In
Chaps. 3 and 4, we have shown that it is straightforward to obtain UˆFF (t) once a
counterdiabatic Hamiltonian of the form HˆCD(t) ∝ pˆvˆ + vˆpˆ is obtained. The coun-
94
terdiabatic Hamiltonian HˆCD(t) produces an appropriate non-linear stretching in an
energy eigenfunction in order to drive it through the adiabatic path, as described in
Sec. 3.3. However a quantum flow-field velocity v(qˆ, t) cannot be obtained for ex-
cited states due to the presence of nodes, see Sec. 3.5. Therefore we cannot obtain
a perfect counterdiabatic Hamiltonian of the form HˆCD(t) ∝ pˆvˆ + vˆpˆ for excited
states. Based on the quantum-classical correspondence principle, we anticipate that
in the semiclassical limit, the classical velocity v(q, t) may lead to an approximate
expression for a quantum velocity v(qˆ, t) which does not suffer from divergences. As
a result we might obtain approximate expressions for HˆCD(t) and UFF (qˆ, t) in the
semiclassical limit.
In the semiclassical limit, an energy eigenfunction of a Hamiltonian H0(t) be
expressed in terms of an amplitude and a phase as shown in Eq. 5.4. The semiclas-
sical eigenfunction has a a natural interpretation in terms of the classical energy
shell in phase space whose energy corresponds to the quantum energy eigenvalue as
discussed in Sec. 5.2. The phase
∫
p¯dq is half of S(q, t) defined in Eq. 4.8 (not to be
confused with S =
∫ q
mv dq′, (Eq. 3.15)), and the amplitude denotes a microcanon-
ical probability distribution on this energy shell as shown in Eq. 5.1.
Consider the eigenvalue equation, H0|φ〉 = E|φ〉, (Eq. 3.3), for a high lying state
|φ〉. The corresponding adiabatic wavefunction is ψad(q, t) = 〈q|φ〉 exp
[
−i/~(∫ t
0
E(t′)dt′)
]
,
(Eq. 3.4). For an ideal HˆCD(t) ∝ pˆvˆ+ vˆpˆ which would carry out perfect counterdia-
95
batic driving, ψad(q, t) satisfies Eq. 3.5. It follows that
[
Hˆ0 +
1
2
(pˆvˆ + vˆpˆ)
]
|ψad〉 = i~∂|ψad〉
∂t
=⇒ Hˆ0|φ〉+ 1
2
(pˆvˆ + vˆpˆ)|φ〉 = E|φ〉+ i~∂|φ〉
∂t
=⇒ 1
2
(pˆvˆ + vˆpˆ)|φ〉 = i~∂|φ〉
∂t
. (5.6)
Substituting Eq. 5.4 in Eq. 5.6 and separating the real and the imaginary parts of
the equation, we get
∂tS + v∂qS = 0, and (5.7a)
∂tA± + v∂qA± +
1
2
∂qvA± = 0
or equivalently ∂tρ+ ∂q(vρ) = 0 (5.7b)
respectively. Note that since we are considering a kinetic plus potential type of
Hamiltonian, the energy shell in phase space is symmetric about the q-axis. Defining
ρ as ρ = |A±(q, t)|2 ∝ 1/p¯, leads to Eq. 5.7b. A perfect velocity v(q, t) must satisfy
both the conditions of Eq. 5.7. While Eq. 5.7a describes the deformation of the
energy shell, Eq. 5.7b describes the deformation of the microcanonical distribution
in space. Except for the special case of scale-invariant driving where the topology
of the system is preserved throughout the evolution, there is no reason why a single
function v(q, t) should satisfy both the conditions in Eqs. 5.7 simultaneously. This
means that for a generic system, we cannot find a perfect v(q, t). This deduction
derived from a semiclassical analysis of |ψad〉 aligns with the deduction made in
96
Figure 5.1: A schematic plot of φ(q) vs. q is presented at times t and t + δt,
represented by dashed and solid curves respectively. The n’th node q¯n is shown at
times t and t + δt. The wavefunction ψ(q, t) evolving under Hˆ0 + HˆCD should be
guided by the auxiliary term in a way that ψ(q¯n, t) = φ(q¯n, t) is satisfied for every
node at every instant, i.e., the nodes of φ(q, t) and ψ(q, t) should align at every
instant.
Sec. 3.5, although the analysis carried out to reach the deduction are different. We
now focus on deriving a function v(q, t) which satisfies at least one of the conditions
of Eq. 5.7. Since the quantum number is the quantum adiabatic invariant, Eq. 5.7a,
which is an equation dependent on the energy shell (instead of the distribution on
the shell), gains priority and we focus on obtaining a v(q, t) that satisfies Eq. 5.7a.
We expect that v(q, t), should be able to move the nodes of the wavefunction ψ(q, t)
evolving under Hˆ0 + HˆCD appropriately such that they align with the nodes of the
instantaneous eigenfunction φ(q, t). If q¯n(t) denotes the position of the n’th node
of φ(q, t), then the relation ψ(q¯n, t) = φ(q¯n, t) is desired at every instant for all the
nodes. Since the v(q, t) we look for need not satisfy Eq. 5.7b, we anticipate that
the amplitude of ψ(q, t) need not overlap with the amplitude of ψad(q, t). A close
97
inspection of Eq. 5.7a indicates that it is identical to Eq. 4.9a which defines the
classical flow-field velocity v(q, t). We therefore hypothesize that the classical flow-
field velocity and the resulting classical auxiliary fields, upon quantization, will lead
to counterdiabatic and fast-forward driving to a very good approximation.
We can also arrive at the hypothesis stated above by carrying out a slightly
different analysis of the semiclassical eigenfunction. As the quantum number of
the semiclassical wavefunction is related more closely with the equivalent classical
energy shell than with the distribution on it, we look for HˆCD, or equivalently
a v(q, t) which drives the phase appropriately. Driving the phase of the evolving
wavefunction |ψ(t)〉 in conjunction with the phase of |φ(t)〉 is equivalent to driving
|ψ〉 in such a way that at every instant its nodes move appropriately and coincide
with the nodes of |φ〉. The auxiliary fields should therefore preserve the following
relation for every set of consecutive nodes, see Fig. 5.1:
p¯(qn¯)(qn+1 − qn) = p¯(qn¯)δq ' pi~. (5.8)
This condition is similar to the Bohr-Sommerfeld quantization condition. The quan-
tity pi~ on the right side was obtained from Eq. 5.4 upon using |A±| ∝
√
1/p¯, and
combining the two terms to obtain a cosine term. Eq. 5.8 equivalent to demand-
ing that the action of the classical energy shell, whose energy corresponds to the
energy of the quantum eigenstate should be preserved. The problem of preserv-
ing the classical action has been addressed in Chap. 4 and Eqs. 4.11 provide the
solution. We therefore arrive at the same hypothesis that the classical flow-field
98
velocity and the resulting classical auxiliary fields, upon quantization, should lead
to counterdiabatic and fast-forward driving to a very good approximation. We test
this hypothesis numerically in the next section.
5.4 Numerical illustration
We now test the hypothesis proposed in the previous section by using the ex-
ample of a model double-well Hamiltonian. We use the quantized counterpart Hˆ0(t)
of the classical model Hamiltonian H0(t), defined in Eq. 4.21. Unless specified oth-
erwise, the mass of the particle m and Planck’s reduced constant ~ are set to unity.
In this section, we initialize the system in an energy eigenstate of Hˆ0(t) and com-
pare the final states obtained numerically after subjectng it to the following two
evolutions governed by time-dependent Schro¨dinger equation – one under the bare
Hamiltonian Hˆ0(t), and the other under the full Hamiltonian Hˆ0(t) + UFF (qˆ, t).
The quantum fast-forward potential UFF (qˆ, t) was obtained by quantizing the clas-
sical fast-forward potential UFF (q, t), which was in turn numerically obtained upon
implementing Eq. 4.11b.
We start by describing how to obtain the n’th energy eigenstate |φn〉 numerically.
We first rewrite Hˆ0(t) as a sum of a harmonic-oscillator Hamiltonian HˆHO(t) and a
99
potential U(qˆ, t) as shown below:
Hˆ0(t) =
pˆ2
2
+ qˆ4 − 16qˆ2 + λ(t)qˆ
=
[
pˆ2
2
+
1
2
ω2qˆ2
]
+
[
qˆ4 −
(
16 +
1
2
ω2
)
qˆ2 + λ(t)qˆ
]
= HˆHO(t) + U(qˆ, t). (5.9)
We drop the time argument in the remainder of the paragraph as we are only
interested in the eigenfunction φn(q, t) at the initial time t = 0. We choose ω = 2
for our analysis. Let the eigenstates of HˆHO be represented by Greek letters, such as
|α〉, |β〉, etc. The matrix representation of Hˆ0 in the basis of HˆHO can be obtained
by making use of the following set of equations:
HSO,αβ =

(
β + 1
2
)
~ω α = β
0 otherwise
, (5.10)
λqαβ = λ
√
~
2mω

√
β α = β − 1
0 otherwise
, (5.11)
q4αβ =
(
~
2mω
)2

6β2 + 6β + 3 α = β√
β(β − 1)(β − 2)(β − 3) α = β − 2
(4β − 2)√β(β − 1) α = β − 4
0 otherwise
, (5.12)
100
t=0
Figure 5.2: A plot of pn(t) = |〈φn(t)|ψ(t)〉|2 vs. n is presented at the initial time
t = 0. The system is initialized in the 35th energy eigenstate, which is depicted by
the single peak at n = 35 with p35(t = 0) = 1.0.
ηq2αβ = η
(
~
2mω
)

2β + 1 α = β√
β(β − 1) α = β − 2
0 otherwise
, (5.13)
where η = 16 +ω2/2. In the equations above, we have assumed α ≤ β and obtained
the upper triangular elements. Since Hˆ0 is Hermitian, the relation H0,αβ = H0,βα
enables us to obtain the remaining elements. The Hamiltonian matrix can then be
diagonalized to obtain eigenvectors. The n’th energy eigenfunction |φn〉 of Hˆ0 can
be expressed as
|φn〉 =
∑
α
cn,α|α〉, (5.14)
where the coefficients cn,α are the elements of the n’th eigenvector.
The results below are presented for the 35’th energy eigenstate, corresponding
101
t=τ/2
(a) Plot of pn(t) = |〈φn(t)|ψ(t)〉|2 vs. n at time t = τ/2 for evolution
under Hˆ0(t).
t=τ
(b) Plot of pn(t) = |〈φn(t)|ψ(t)〉|2 vs. n at the final time t = τ for
evolution under Hˆ0(t).
Figure 5.3: The plots above depict the overlap between the wavefunction |ψ(t)〉
as it evolves under Hˆ0(t) (defined in Eq. 5.9), and the instantaneous energy eigen-
functions |φn(t)〉 for 0 ≤ n ≤ 70. The system is initialized in the 35’th eigenstate,
|ψ(0)〉 ≡ |φn(t)〉. The system is in a superposition of instantaneous eigenstates at an
intermediate time as well as at the final time. The system has developed excitations
during the evolution and is unable to reach the final adiabatic state at t = τ . This
final state is analogous to the classical final state where the trajectories do not end
on the desired energy shell, see Fig. 4.4(b).
102
t=τ/2
(a) Plot of pn(t) = |〈φn(t)|ψ(t)〉|2 vs. n at time t = τ/2 for evolution
under Hˆ0(t) + UFF (qˆ, t).
t=τ
(b) Plot of pn(t) = |〈φn(t)|ψ(t)〉|2 vs. n at the final time t = τ for
evolution under Hˆ0(t) + UFF (qˆ, t).
Figure 5.4: Same as Fig. 5.3, except that the system evolves under Hˆ0(t)+UFF (qˆ, t),
where UFF (qˆ, t) is the quantized counterpart of the classical fast-forward potential
UFF (q, t) which is obtained numerically from Eq. 4.11b. The system is in a superpo-
sition of instantaneous eigenstates at an intermediate time, but it reaches the desired
final state at t = τ with high accuracy. At the final time, |ψ(τ)〉 has a 90% overlap
with |φ35(τ)〉, i.e., p35 = 0.90. The combined probability p34 + p35 + p36 = 0.98.
Fig. 5.4(b) is analogous to the classical final state where the trajectories end on
the desired energy shell but the initial uniform distribution is not preserved, see
Fig. 4.4(c).
103
t=τ
E~50
(a) Plot of the final distribution of classical trajectories on the energy shell
for E = 51.76, when fifty uniformly distributed initial trajectories evolve
under the analogous classical Hamiltonian H0(t) + UFF (q, t).
t=τ
(b) Plot of pn(t) = |〈φn(t)|ψ(t)〉|2 vs. n at time t = τ . The system is
initialized in |ψ(0)〉 ≡ |φ35(0)〉 and evolves under Hˆ0(t) + UFF (qˆ, t).
Figure 5.5: The final distribution of classical trajectories is non-uniform as depicted
in the phase-space plot. This non-uniformity is reflected in the quantum evolution
as depicted in the plot of pn(τ) = |〈φn(τ)|ψ(τ)〉|2 vs. n (same as Fig. 5.4(b)). The
peak value is p35(τ) = 0.90.
104
t=τ
E~25
(a) Plot of the final distribution of classical trajectories on the energy shell
for E = 25.08, when fifty uniformly distributed initial trajectories evolve
under the analogous classical Hamiltonian H0(t) + UFF (q, t).
t=τ
(b) Plot of pn(t) = |〈φn(t)|ψ(t)〉|2 vs. n at time t = τ . The system is
initialized in |ψ(0)〉 ≡ |φ28(0)〉 and evolves under Hˆ0(t) + UFF (qˆ, t).
Figure 5.6: Same as Fig. 5.6, except E = 25.08 and n = 28. The final classical
distribution has a higher degree of non-uniformity compared to Fig. 5.5(a), which is
reflected in the quantum evolution. In Fig. 5.6(b), the peak value is p28(τ) = 0.62,
and sideband excitations are more prominent compared to Fig. 5.5(b).
105
to an initial eigenvalue E = 51.76. The time-dependent parameter λ(t) varied from
−16 to +16 over a time τ = 1.0 according to Eq. 4.21b, which resulted in a non-
adiabatic driving of the Hamiltonian. The initial energy eigenfunction was subjected
to two different evolutions – one under time-dependent Schro¨dinger equation with
the bare Hamiltonian Hˆ0(t), and the other with Hˆ0(t) + UFF (qˆ, t). The numerical
methods used to evolve the quantum and classical systems are specified in Secs. 4.4
and 3.7 respectively. Figs. 5.2 - 5.4 plot the probability pn(t) = |〈φn(t)|ψ(t)〉|2, which
quantifies the overlap between the evolving wavefunction |ψ(t)〉 and the instanta-
neous eigenfunction |φn(t)〉. At t = 0, ψ(0) ≡ φ35(0) which means pn(0) = 1 for
n = 35 and vanishes for all other values of n, as illustrated in Fig. 5.2. Figs. 5.3(a)
and 5.4(a) plots pn(t) at an intermediate time t = τ/2 = 0.5. It can be seen that
the evolving wavefunction |ψ(t)〉 is in a superposition of the instantaneous eigen-
functions |φn(t)〉 for both the evolution protocols. The plots pn(t) at the final time
t = τ = 1.0 are illustrated in Figs. 5.3(b) and 5.4(b). Fig. 5.3(b) indicates that the
final state |ψ(τ)〉 which evolves under Hˆ0(t) is in a superposition of eigenfunctions
having undergone excitations during the evolution. On the other hand, as seen in
Fig. 5.4(b), the final state |ψ(τ)〉 which has evolved under Hˆ0(t) + UFF (qˆ, t) is very
close to the 35’th eigenfunction φ35(τ). At the final time, UFF (qˆ, t) produces a peak
at n = 35 with p35(τ) = 0.90 indicating that it achieves fast-forward driving with a
high accuracy by supressing excitations. The combined probability of the peak and
the immediate sidebands yield p34(τ) + p35(τ) + p36(τ) = 0.98.
Several other initial conditions were tested and it was verified that the composite
Hamiltonian Hˆ0(t) + UFF (qˆ, t) always outperformed the bare Hamiltonian Hˆ0(t)
106
in supressing excitations. The numerical results support our hypothesis that the
quantized counterpart of a classical fast-forward potential can achieve quantum
fast-forward driving with a high accuracy. It was also observed that the degree of
accuracy achieved by the quantum fast-forward potential UFF (qˆ, t) is directly related
to the degree of uniformity in the final distribution of trajectories achieved by the
analogous classical fast-forward potential UFF (q, t). This is illustrated in Figs. 5.5
and 5.6, where the system is initialized in n = 35 and n = 28 with a corresponding
value of E = 51.76 and E = 25.08 respectively. The evolution protocol is same
as specified in the previous paragraph. The degree of non-uniformity of the final
distribution of classical trajectories is lower in Fig. 5.5(a) than in Fig. 5.6(a). This is
reflected in the quantum evolution as seen in Figs. 5.5(b) and 5.6(b). The sidebands
for n = 28 are significantly more prominent than for n = 35. For the quantum state
initialized in n = 28, p28(τ) = 0.62 and p27(τ) + p28(τ) + p29(τ) = 0.77.
5.5 Semiclassical analysis of quantum peaks and final classical dis-
tribution
In this section, we will show that the numerical peaks obtained in the previ-
ous section can be determined quantitatively by analyzing the final distribution on
the classical energy shell (from Sec. 4.4), and relating it to the semiclassical repre-
sentation ψSC,E(q, τ) of the final state |ψ(τ)〉 which has evolved under fast-forward
driving. We assume throughout this section that ψSC,E(q, t) is initialized in the n’th
energy eigenstate of Hˆ0, and evolves under Hˆ0 + UˆFF during the interval 0 ≤ t ≤ τ .
107
Here, Hˆ0 and UˆFF are obtained by quantizing Eqs. 4.1 and 4.11b respectively. Let
ψSC,k(q, t) denote the semiclassical wavefunction of the k’th energy eigenstate of
Hˆ0. Let ρMC,k(q, τ) denote the microcanonical distribution on the final energy shell
whose energy corresponds to the k’th quantum level. For a classical system starting
initialized in a microcanonical distribution on an energy shell and evolving under
H0 +UFF , let ρE(q, τ) denote the final distribution on the classical energy shell. For
an initial action Ik we define
S¯k(q, Ik) =
∫ q
p¯(q′, Ik)dq′, (5.15)
where the intergral is carried out on an energy shell whose energy is equal to the
k’th energy eigenvalue of Hˆ0. The wavefunctions ψSC,k and ψSC,E can be expressed
respectively as
ψSC,k(q, t) = AMC,k,+ exp
(
i
S¯k
~
)
+ AMC,k,− exp
(
i
S¯k
~
)
(5.16)
and
ψSC,E(q, t) = AE,+ exp
(
i
S¯n
~
)
+ AE,− exp
(
i
S¯n
~
)
, (5.17)
where the subscripts ‘+’ and ‘−’ corresponds to the upper and lower branch of the
energy shell, the functional dependences have been dropped. The amplitudes are
108
related to the distribution on the energyshell as
AE ≡ √ρE (5.18a)
AMC,k ≡ √ρMC,k (5.18b)
The overlap between ψSC,k(q, τ) and ψSC,E(q, τ) can be calculated as shown below.
We show the analysis only for the upper branch of a classical energy shell and drop
the functional dependences. The analysis for the lower branch follows the exact
same behaviour. We have for the uppper branch,
〈ψSC,k|ψSC,E〉 =
∫ qR
qL
dq′AE,+AMC,k,+ exp
[
i
~
(S¯n − S¯k)
]
, (5.19)
where qL and qR denote the left and right turning points respectively. The Taylor
expansion of S¯(q, In) is carried out to obtain
S¯(q, In) = S¯(q, Ik) + ∂S¯
∂I
(In − Ik) + . . .
= S¯(q, Ik) + (n− k)~θ +O(~2), (5.20)
where θ = ∂S¯/∂I is the angle variable and the Bohr-Sommerfeld quantization re-
lation, In = (n + 1/2)~, has been used for simplication. By applying a change of
variables from q to θ, Eq. 5.19 becomes
〈ψSC,k|ψSC,E〉 =
∫ pi
0
dθei(n−k)θ
A˜E,+(θ)√
2pi
(5.21)
109
where A˜E,+ is the representation of AE,+ in the angle coordinates and the following
transformations have been made:
dq → dθ, (5.22a)
qL → 0, (5.22b)
qR → pi, (5.22c)
A2MC,k,+dq →
1
2pi
dθ (5.22d)
and A2E,+dq → A˜2E,+dθ. (5.22e)
Note that Eqs. 5.22d and 5.22e are obtained by equating the fractional probability
of final trajectories in an infinitesimal region of the energy shell. Note that Eq. 5.21
indicates that the numerical peaks of Sec. 5.4 are related to the Fourier transform
of the final classical distribution expressed in the angle coordinates. The quantum
superposition pn(τ) = |〈φn(τ)|ψ(τ)〉|2, of the final state |ψ(τ)〉 which has evolved
under fast-forward driving, and the energy eigenfunctions |φn(τ)〉 is plotted as a bar-
graph, while |〈ψSC,n|ψSC,E〉|2 (Eq. 5.21) is depicted by blue dots in Fig. 5.7. The
blue dots, which are derived from the classical final distribution overlap perfectly
with the bars, which result from the quantum fast-forward evolution under time-
dependent Schro¨dinger equation. The semiclassical analysis clearly establishes a
relation between the classical and the quantum fast-forward driving.
110
Quantum simulation
Classical analysis
Figure 5.7: A plot of pn(t) = |〈φn(t)|ψ(t)〉|2 vs. n is presented at the initial time
t = 0. The system is initialized in the 35th energy eigenstate, which is depicted by
the single peak at n = 35 with p35(t = 0) = 1.0.
5.6 Relating counterdiabatic and fast-forward driving
We have numerically shown in Sec. 5.4 that the quantum counterpart of the
classical fast-forward potential (Eq. 4.11b) leads to quantum fast-forward driving
with a high accuracy. Given this result, we can analytically show that the quan-
tum counterpart of the classical counterdiabatic Hamiltonian (Eq. 4.11a) leads to
quantum counterdiabatic driving with the same level of accuracy. In this section we
prove the following statement: If a wavefunction ψFF satisfies the time-dependent
Schro¨dinger equation
[
Hˆ0(t) + UFF (qˆ, t)
]
ψFF (t) = i~
∂ψFF (t)
∂t
, (5.23)
111
then, the wavefunction ψCD related to ψFF as
ψCD(q, t) = ψFF (q, t) exp
[
−iS(q, t)
~
]
, (5.24)
will satisfy the time-dependent Schro¨dinger equation
[
Hˆ0(t) + HˆCD(t)
]
ψCD(t) = i~
∂ψCD(t)
∂t
. (5.25)
In the statement above UFF (qˆ, t) is obtained by quantizing Eq. 4.11b and HˆCD(t) is
obtained by quantizing Eq. 4.11a to obtain
HˆCD(t) =
1
2
(pˆvˆ + vˆpˆ) , (5.26)
where vˆ is the quantum counterpart of Eq. 4.9a. The function S(q, t) (not to be
confused with S of Eq. 4.8) is determined by solving the equation
∂qS(q, t) = mv(q, t). (5.27)
We also provide numerical evidence to show that the wavefunction evolving under[
Hˆ0(t) + HˆCD(t)
]
will follow the adiabatic path with a high accuracy.
To show that Eq. 5.23 implies Eq. 5.25, we start by substituting Eq. 5.24 in
Eq. 5.25 and then simplify the left and right sides of the equation. Henceforth, we
will drop the arguments of the functions in this section. Firstly note from Eq. 5.24
112
that
∂ψ2CD
∂q2
· exp
[
i
S(q, t)
~
]
=
[
∂ψ2FF
∂q2
− 1
~2
ψFF
(
∂S
∂q
)2]
− i
~
[
2
∂ψFF
∂q
∂S
∂q
+ ψFF
∂2S
∂q2
]
.
(5.28)
Upon using pˆ = −i~∂q and Eq. 3.1, the left side (LS) of Eq. 5.25 simplifies to
LS · exp
[
i
S(q, t)
~
]
=
[
− ~
2
2m
∂ψ2FF
∂q2
+ U0ψFF +
1
2m
ψFF
(
∂S
∂q
)2
− vψFF ∂S
∂q
]
+ i~
[
1
m
∂ψFF
∂q
∂S
∂q
+
1
2m
ψFF
∂2S
∂q2
− v∂ψFF
∂q
− 1
2
∂qvψFF
]
,
(5.29a)
while the right side (RS) simplifies to
RS · exp
[
i
S(q, t)
~
]
=
∂ψCD
∂t
· exp
[
i
S(q, t)
~
]
=
∂ψFF
∂t
+
1
i~
∂S
∂t
ψFF (5.29b)
After equating the left and right sides of Eq. 5.29, we separate the real and
imaginary parts to obtain
Re[LS-RS] exp
[
i
S(q, t)
~
]
=
[
1
2m
(
∂S
∂q
)2
− v∂S
∂q
− ∂S
∂t
]
ψFF (5.30)
and
Im[LS-RS] exp
[
i
S(q, t)
~
]
=
~
m
∂ψFF
∂q
[
∂S
∂q
−mv
]
+ ψFF
~
2m
[
∂S2
∂q2
−m∂qv
]
(5.31)
respectively. The term in the parenthesis of Eq. 5.30 is only a function of time as
113
shown below. We start from the definition of the flow-field acceleration, Eq. 4.9b
and use Eqs. 5.27 and 4.11b for simplication.
a = v∂qv + ∂tv
⇒ ma = mv∂qv +m∂tv
⇒ −∂qUFF = m
2
v∂qv +m∂tv
⇒ −∂qUFF = 1
2m
∂q(mv)
2 +m∂t(∂qS)
⇒ ∂q
[
1
2m
(
∂S
∂q
)2
− v∂S
∂q
− ∂S
∂t
]
= 0 (5.32)
Since Eq. 5.27 defines S upto an arbitrary function of time, we can exploit this
freedom to choose an S such that the the right side of Eq. 5.30 vanishes. Eq. 5.27
also implies that the terms in the parenthesis on the right side of Eq. 5.31 vanish.
This concludes our proof.
We have shown that if a wavefunction ψFF satisfies Eq. 5.23, then a wavefunc-
tion ψCD related to ψFF by Eq. 5.24 satisfies Eq. 5.25. In other words, once we
numerically establish that the fast-forward potential UFF of Eq. 4.11b carries out
quantum fast-forward driving with a high accuracy, we can deduce that the coun-
terdiabatic Hamiltonian HCD of Eq. 5.26 will carry out quantum counterdiabatic
driving with the same level of accuracy. This claim is supported by the numerical
result shown in Fig. 5.8 which plots the probability density |φ18(q, t)|2 of the instan-
taneous eigenfunction, and |ψ(q, t)|2, the probability density of the wavefunction
evolving under Hˆ0 + UˆFF . As seen in the figure, the minima of |ψ(q, t)|2 align with
114
|ϕ
2
,
|ψ
2
-6 -3 0 3 6
0.4
(a) t = 0
-6 -3 0 3 6
0.4
(b) t = τ / 5
-6 -3 0 3 6
0.4
(c) t = 2τ / 5
-6 -3 0 3 6
0.4
(d) t = 3τ / 5
-6 -3 0 3 6
0.4
(d) t = 4τ / 5
-6 -3 0 3 6
0.4
(d) t = τ
q
Figure 5.8: The magenta curves depict the instantaneous probability density
|ψ(q, t)|2 of the wavefunction ψ(q, t) evolving under Hˆ0 + UˆFF . |ψ(q, t)|2 is plot-
ted with respect to q for times t = 0, τ/5, 2τ/5, 3τ/5, 4τ/5 and τ , for τ = 1.0. The
blue curves correspond to |φ18(q, t)|2, the probability density of the instantaneous
energy eigen state. The other parameters for numerical evolution were chosen as
m = 1, ~ = 2 and E = 53.76, which corresponds to n = 18. As seen in the snap-
shots, the minima of |ψ(q, t)|2 align with the nodes of |φ18(q, t)|2 at every instant,
but the amplitudes do not match.
115
the nodes of |φ18(q, t)|2 at every instant, although the amplitudes do not align. For
the evolution depicted in Fig. 5.8, ~ = 2 and E = 53.76 so that n = 18. The
final time τ was set to unity. Combining this numerical result with the fact that a
wavefunction ψCD related to ψFF by Eq. 5.24 satisfies Eq. 5.25, we can deduce that
HˆCD obtained from the classical v(q, t) achieves counterdiabatic driving with a high
accuracy.
5.7 Summary
In this chapter, we have explored the problem of achieving shortcuts to adia-
baticity for excited states which are driven under an arbitrary protocol. We have
shown that the flow-fields method for quantum and classical systems, as discussed
in Chaps. 3 and 4 respectively, have many similarities but are not equivalent in
the semiclassical limit. This led to the study of the semiclassical limit of an en-
ergy eigenfunction. We determined that a perfect flow-field velocity must satisfy
Eqs. 5.7 in order to produce exact counterdiabatic and fast-forward fields. However,
as pointed out in Sec. 3.5, the nodes of the wavefunction make it impossible to
obtain a well defined flow-field velocity and therefore the set of Eqs. 5.7 can not be
satisfied simultaneously. We then attempt to find a velocity which satisfies at least
one of Eqs. 5.7. We prioritized Eq. 5.7a as it preserves the phase of the wavefunction
which is directly related to the quantum nummber. Eq. 5.7a is in fact equivalent to
the condition that the classical action of a system undergoing analogous classical dy-
namics be preserved. Following this, we hypothesized that the quantum counterpart
116
of a classical fast-forward potential should be able to achieve quantum fast-forward
driving to a good approximation.
We tested our hypothesis using a model double-well Hamiltonian subjected to
a non scale invariant driving. We first obtained the classical fast-forward poten-
tial UFF (q, t) for a chosen dynamics. We then quantized this potential to obtain
UFF (qˆ, t), and subjected a quantum system to the analogous quantum driving un-
der time-dependent Schro¨dinger equation. We compared the evolutions under the
bare Hamiltonian Hˆ0(t) and the composite Hamiltonian Hˆ0(t) + UFF (qˆ, t). We de-
duced from Figs. 5.2 - 5.4 that UFF (qˆ, t) carries out fast-forward driving with a very
high accuracy.
The accuracy of UFF (qˆ, t) was tested for different initial conditions by studying
pn(t) = |〈φn(t)|ψ(t)〉|2 which represents the overlap between the evolved state and
the energy eigenstate at time t. It was observed that the non-uniformity in the
distribution of the final trajectories on classical energy shell is directly related to
the accuracy with which UFF (qˆ, t) achieves fast-forward driving. Our analysis shows
that the higher the quantum number of the initial state, the more accurate is the
fast-forward driving. The quantum peaks obtained at the final time after fast-
forward driving was infact obtained quantitatively by comparing the semiclassical
analysis of the semiclassical wavefunction and the final classical distribution. It
was shown that the Fourier transform of the final classical distribution represented
in the angle coordinates matches perfectly with the quantum peaks obtained from
numerical evolution under time-dependent Schro¨dinger equation. It was also shown
analytically and supported numerically in Fig. 5.8 that the numerical accuracy of the
117
quantum fast-forward driving will be reflected in quantum counterdiabatic driving as
well, where the auxiliary counterdiabatic Hamiltonian is obtained from the quantum
counterpart of the classical flow-field velocity, see Eq. 5.26.
This chapter provides a novel approach to solve quantum shortcuts for a system
initialized in an excited state and subjected to an arbitrary driving protocol. The
importance of the classical auxiliary fields in obtaining quantum auxiliary fields for
analogous driving protocols is established. The problem arising due to nodes of an
excited wavefunction, as discussed in Sec.3.5, has been overcome to a great extent
and highly accurate auxiliary fields have been derived. A crucial point to note about
the analysis in this chapter is that the auxiliary fields depend on the choice of the
initial state. It remains an open problem to solve for a flow-field velocity which
satisfies Eq. 5.7b instead of Eq. 5.7a, and compare the auxiliary fields obtained as a
result. Once a solution is obtained, one may try to find an optimal way to combine
these auxiliary fields in order to obtain a resulting field which is closest to the ideal
auxiliary field which can achieve perfect shortcuts to adiabaticity.
118
Chapter 6: Stochastic shortcuts using flow-fields
6.1 Overview
In this chapter, we establish the broad applicability of the flow-fields method
by extending it to stochastic systems. This work has been motivated by a recent
experiment by Mart´ınez et al, where swift-equilibration was achieved in a system of
overdamped Browninan particles [59]. The experimental setup in Ref. [59] consisted
of a microsphere immersed in water, which was trapped by an optical harmonic
potential of the form U(q, t) = κ(t)q2/2. The power of the trapping laser controlled
the stiffness κ(t) of the trapping potential. The dynamics of the system was over-
damped and described by a Langevin equation. In the experiment, the stiffness κ(t)
was doubled over a time tf much shorter than the natural relaxation time τrelax of
the system. It was shown that when κ(t) was doubled according to the protocol
termed engineered swift equilibration, the system of Brownian particles reached final
equilibrium much faster than the natural relaxation time. In fact the system reached
equilibrium at the final time tf , i.e., as soon κ(t) reached its final value.
The experiment in Ref. [59] prompted us to frame a problem on stochastic
shortcuts as follows: Given a system of overdamped Browninan particles trapped
in a potential U0(q, t), is it possible to guide the system along the instantaneous
119
equilibrium state at all times? This problem is equivalent to counterdiabatic driving
discussed in previous chapters. We would like to construct an auxiliary trapping
potential UCD(q, t), such that when the system evolves under the composite trapping
potential U0(q, t) + UCD(q, t), it tracks the instantaneous equilibrium distribution
corresponding to U0(q, t) at all times. This is the problem for counterdiabatic driving
in a stochastic system. Unlike the quantum and classical Hamiltonian systems, it
will be shown that for an overdamped stochastic system, counterdiabatic driving
can be achieved using a potential (or a local field). This may be attributed to the
separation of time-scales between the particles in the system and its surroundings
for an overdamped system.
We derive the flow-fields method for stochastic systems in Sec. 6.2, compare our
results with previous results from Mart´ınez et al (Ref. [59]) in Sec. 6.3 and present
a brief summary in Sec. 6.4.
6.2 Derivation of results
Let us consider an overdamped Brownian particle in a time-dependent potential
U0(q, t), which varies smoothly for 0 ≤ t ≤ τ but is fixed outside this interval. The
particle is in contact with a thermal reservoir at temperature T . The interactions
with the degrees of freedom of the reservoir give rise to the random and dissipative
forces that characterize Brownian dynamics.
We will work in the ensemble picture, in which the dynamics are described by
a probability density function ρ(q, t) that evolves according to the Fokker-Planck
120
equation
∂tρ =
1
γ
∂q [(∂qU0)ρ] +D∂
2
qρ ≡ Lˆ0(t)ρ, (6.1)
The friction and diffusion coefficients, γ and D, obey the Einstein-Smoluchowski
relation, γD = kBT ≡ 1/β, where kB is the Boltzmann constant. The equilibrium
distribution associated with the potential U0(q, t) is given by
ρeq(q, t) =
1
Z(t)
exp [−βU0(q, t)] (6.2)
where Z(t) is the partition function. It is straightforward to verify the identity
Lˆ0(t)ρeq(q, t) = 0 (6.3)
which confirms that the equilibrium distribution is a stationary solution of the dy-
namics, when the potential does not vary with time. When U0(q, t) changes quasi-
statically, then the slowly varying ρeq(q, t) is a solution of Eq. 6.1 (both sides tend
toward zero in that limit), as expected for a reversible process. Therefore, the
adiabatic evolution is identified by the ensemble evolving through the continuous
sequence of equilibrium states ρeq(q, t). 1
We now consider the case in which the potential U0(q, t) is varied at an arbitrary
rate. To this potential we will add a counterdiabatic term UCD(q, t), so that the
1Here we use the term adiabatic consistently with its usage in the rest of the paper, namely to
denote a slow process. This differs from its usage in thermodynamics, where an adiabatic process
is one in which heat is not absorbed or released by a system. The dual meaning of this term is
unfortunate.
121
system evolves under
∂tρ =
1
γ
∂q {[∂q(U0 + UCD)] ρ}+D∂2qρ
= Lˆ0ρ+ 1
γ
∂q [(∂qUCD)ρ] (6.4)
We wish to design UCD(q, t) so as to achieve adiabatic evolution, i.e. so that ρ
eq(q, t)
(Eq. 6.2) is an exact solution of Eq. 6.4. We define an integrated distribution
F(q, t) =
∫ q
−∞
ρeq(q′, t)dq′ (6.5)
similar to I(q, t) (Eq. 3.6). Inverting this function to obtain q(F , t) (see Fig. 6.1),
we construct a velocity flow field
v(q, t) =
∂
∂t
q(F , t) = −∂tF
∂qF . (6.6)
Rearranging this result as ∂tF + v∂qF = 0 and differentiating with respect to q
produces the continuity equation
∂tρ
eq + ∂q(vρ
eq) = 0. (6.7)
Since we wish ρeq(q, t) to be a solution of Eq. 6.4, we use Eqs. 6.3 and 6.7 to
rewrite that equation as follows:
− ∂q(vρeq) = 1
γ
∂q [(∂qUCD)ρ
eq] (6.8)
122
Figure 6.1: The blue lines divide the equilibrium distribution into strips of equal
area. q(F , t) is the right boundary of the shaded region, which has area F . The
velocity field v(q, t) describes the motion of the vertical lines with t (Eq. 6.6).
Integrating both sides gives
− vρeq − J(t) = 1
γ
(∂qUCD)ρ
eq (6.9)
Here, J(t) is an arbitrary function of time that we set to zero, for convenience,
arriving at the result
− ∂qUCD(q, t) = γv(q, t) (6.10)
Eq. 6.10 defines UCD(q, t) up to an additive function of time that has no influ-
ence on the dynamics, and which can be adjusted so that UCD = 0 for t /∈ (0, τ).
The potential UCD(q, t) has the desired counterdiabatic property: when the system
evolves in the time-dependent potential U0+UCD, it remains in equilibrium (with re-
spect to U0) over the entire duration of the process. Our potential UCD is equivalent
to the auxiliary potential obtained by Li et al [60], as can be seen by differentiating
123
both sides of Eq. 12 of Ref. [60] with respect to x.
Eq. 6.10 has elements in common with both the counterdiabatic and fast-forward
shortcuts of previous chapters. It is counterdiabatic in that the system follows the
adiabatic evolution (it remains in the state ρeq) at all times. Moreover, UCD is given
in terms of the velocity field v (compare Eqs. 3.11, 4.11a, and 6.10), rather than
the acceleration field a. However, just as with the fast-forward shortcuts described
earlier, UCD is local, i.e. it is a time-dependent potential (compare Eqs. 3.14, 4.11b,
and 6.10). Also, as in Chaps. 3 and 4, Eq. 6.10 defines the auxiliary potential only
up to an arbitrary function of time that does not affect the dynamics.
When the Brownian particles evolve under U0(q, t) alone, the state of the en-
semble, ρ(q, t), lags behind the instantaneous equilibrium state, ρeq(q, t), as illus-
trated in Fig. 1 of Ref. [80]. The addition of the counterdiabatic potential UCD(q, t)
eliminates this lag. Lagging distributions are relevant for numerical free-energy es-
timation methods, where the lag gives rise to poor convergence of the free-energy
estimate. Vaikuntanathan and Jarzynski [57] have developed a method in which
this lag is reduced or eliminated by the addition of an artificial flow field to the dy-
namics, although in Ref. [57] this flow field was not related to an auxiliary potential
UCD. Comparing Eq. 6.7 above with Eq. 15 of Ref. [57], we see that our field v is
equivalent to the perfect flow field (u∗) that “escorts” the system faithfully along
the equilibrium path.
While our analysis has been restricted to overdamped Brownian motion, Le
Cununder and colleagues [81] have recently used a micromechanical cantilever to
implement engineered swift equilibration for an underdamped harmonic oscillator.
124
For underdamped motion in a general one-dimensional, time-dependent potential,
Li, Quan and Tu [60] have proposed a momentum-dependent counterdiabatic term
that achieves the desired adiabatic evolution. It remains to be seen whether this
progress will lead to expressions for a momentum-independent counterdiabatic po-
tential that extends the fast-forward method to underdamped Brownian dynamics
beyond the harmonic regime.
6.3 Comparision with engineered swift equilibration
As a simple example, which makes a connection to the engineered swift equili-
bration approach in Ref. [59], we consider a time-dependent harmonic potential,
U0(q, t) =
1
2
κ0(t)q
2 (6.11)
The instantaneous equilibrium distribution is
ρeq(q, t) =
√
σ
pi
exp(−σq2) , σ(t) ≡ βκ0(t)/2. (6.12)
Eq. 6.5 then gives
F(q, t) = 1
2
[
1 + erf(
√
σ(t)q)
]
, (6.13)
where erf(·) is the Gaussian error function. In turn, Eq. 6.6 yields v(q, t) = −σ˙/2σ,
and from there we use Eq. 6.10 to obtain
UCD(q, t) =
(
γσ˙
2σ
)
q2
2
. (6.14)
125
Therefore, under a harmonic trap of stiffness
κ(t) = κ0 +
γσ˙
2σ
= κ0 +
γκ˙0
2κ0
(6.15)
the ensemble remains in the equilibrium state ρeq (Eq. 6.12) at all times.
Rearranging Eq. 6.15 and using σ = βκ0/2, we get
σ˙
σ
=
2κ
γ
− 4kBTσ
γ
. (6.16)
This result is identical to Eq. 6 of Mart´ınez et al [59], where the goal was to bring the
system rapidly to the final equilibrium state, without concern for the intermediate
states visited along the way. Our approach achieves the same result by guiding
the system along the equilibrium path during the entire process. Eq. 6.16 was also
obtained by Schmiedl and Seifert, in the contexts of optimal finite-time control [82]
and stochastic heat engines [83].
6.4 Summary
In this chapter, we have extended the flow-fields method – developed in the
previous chapters for classical and quantum Hamiltonian systems – to a system
of overdamped Brownian particles. We have constructed an auxiliary potential
UCD(q, t) in Eq. 6.10 which guides an ensemble of Brownian particles along the
instantaneous equilibrium distribution of the trap potential of interest U0(q, t). We
have related UCD(q, t) with the shortcut protocols discussed in the previous chapters.
126
The results from Chaps. 3, 4 and 6 demonstrate that the flow-fields approach is
a unifying framework to obtain shortcuts for quantum, classical as well as stochastic
systems. The flow-fields approach also relates the the two distinct shortcut protocols
– the counterdiabatic driving and the fast-forward driving.
127
Appendix A: Derivation of Eq. 2.44
Because the classical function η(q, p) = sign(p) is non-analytic, the matrix rep-
resentation of its quantal counterpart ηˆ cannot be obtained by a procedure like the
one used in Sec. 2.5. Here we instead construct the matrix representation of η by
equating its classical and quantum auto-correlation functions.
Consider a quantum particle in a box with a flat base (s = 0) and hard walls at
q = 0 and q = L, described by the Hamiltonian Hˆ ′ = pˆ2/2m+ Θ(qˆ; 0, L). Following
Ref. [84], we write the quantum auto-correlation function of ηˆ, for the eigenstate
|α〉, as
Cα(τ) = 〈α|ηˆ exp
(
iHˆ ′τ
~
)
ηˆ exp
(
−iHˆ
′τ
~
)
|α〉
=
∑
β
|η˜αβ|2 exp
[
i(Eβ − Eα)τ
~
]
, (A.1)
where η˜αβ = 〈α|ηˆ|β〉, and Eα is the energy corresponding to the eigenstate |α〉. The
Fourier transform of the auto-correlation function is
Cα(ω) =
∑
β
|η˜αβ|2δ(ω − ωαβ), (A.2)
128


τ
()
 τ


0
()
t
τ
t
Figure A.1: The function η0(t) plotted over one time period of oscillation is a square
wave (top figure). The function ητ (t) is obtained by shifting this square wave left-
ward by an amount τ (middle figure). The autocorrelation function C(τ) is the
product of these square wave pulses, integrated over one period, yielding a triangu-
lar wave (bottom figure).
where
ωαβ ≡ Eβ − Eα~ . (A.3)
For a classical particle evolving under the equivalent Hamiltonian, η = sign(p) is
a square wave pulse with unit amplitude over a time period around the energy shell.
The functions ηE0 (t) and η
E
τ (t) describe the dependence of η on time for a particle of
energy E that starts from L = 0 at times t = 0 and t = −τ respectively, as depicted
in Fig.A.1. The classical auto-correlation function, CE(τ) = (1/T )
∫ T
0
dt ηE0 (t)η
E
τ (t),
is a triangular wave given by
CE(τ) =

T−4τ
T
, 0 ≤ τ ≤ T
2
4τ−3T
T
, T
2
≤ τ ≤ T
, (A.4)
129
shown in Fig.A.1. The Fourier transform of CE(τ) is
CE(ω) =
∞∑
odd γ=−∞
4
pi2γ2
δ(ω − ωγ), (A.5)
where
ωγ =
2piγ
T
. (A.6)
The correspondence principle suggests that the functions Cα(ω) and CE(ω)
ought to be equal, in the semiclassical limit, when Eα = E. To compare these
functions, we first note that for one dimensional systems, the classical action J(E) =∮
E
p · dq satisfies
dJ
dE
= T. (A.7)
For neighboring energy levels |α〉 and |α + 1〉, the energy spacing is
dE = Eα+1 − Eα = ~ωα,α+1, (A.8)
and the action spacing is given by the Bohr-Sommerfeld quantization condition:
dJ = 2pi~. (A.9)
From Eqs.(A.7) - (A.9) we obtain ωα,α+1 = 2pi/T , which generalizes to
ωαβ =
2pi(β − α)
T
, (A.10)
130
provided α and β are not too far apart.
Comparing Eqs.(A.6) and (A.10) we confirm that the delta-functions in Eqs.(A.2)
and (A.5) appear at the same frequencies, and by equating the coefficients of these
delta-functions we obtain
|η˜αβ| =

2
|α−β|pi α− β = odd
0 α− β = even
. (A.11)
To ensure that the operator ηˆ is Hermitian (as it represents a physical observable),
we impose the condition η˜αβ = η˜
∗
βα, which then implies
η˜αβ =

± 2i
(α−β)pi α− β = odd
0 α− β = even
(A.12)
Finally to determine the sign in Eq.(A.12), the ground state eigenfunction of Hˆ ′(t)
was boosted by a momentum p = pik/L, where k ∈ Z, which results in the wave
packet ψ(q) =
√
2
L
sin(piq
L
) exp( ipikq
L
). By demanding that 〈ψ|ηˆ|ψ〉 → 1 for k  1
and 〈ψ|ηˆ|ψ〉 → −1 for k  −1, a series of straightforward calculations yields
η˜αβ =

2i
(β−α)pi α− β = odd
0 α− β = even
(A.13)
131
Appendix B: Continuity conditions on H0(z, t)
In Sec. 4.2, we specified that H0(z, t) is constant in time for t < 0, then varies
between t = 0 and t = τ , then remains constant in time for t > τ . As a result,
H0 cannot be an entirely smooth function of time: for some n ≥ 0, the derivative
∂nH0/∂t
n must be discontinuous. We explicitly assumed that this discontinuity
occurs at n ≥ 3, giving us
∂H0
∂t
(z, 0) =
∂H0
∂t
(z, τ) = 0 (B.1a)
∂2H0
∂t2
(z, 0) =
∂2H0
∂t2
(z, τ) = 0 (B.1b)
leading to Eq. 4.10.
The assumption that H0 is twice continuously differentiable was made both for
clarity of presentation, and because it arises in proofs of the adiabatic invariance of
the action [74]. In our context, however, the assumption is not necessary, therefore
in the following we will discuss how Eq. B.1 can be relaxed. We will continue to
require that H0 itself is a continuous function of time. Without loss of generality,
we will assume that discontinuities in ∂H0/∂t and ∂
2H0/∂t
2 occur only at t = 0 and
t = τ , and not within the time interval 0 < t < τ .
132
We first consider the simpler case, in which the above-mentioned discontinuity
occurs at n = 2, i.e. Eq. B.1a holds but B.1b is violated. Then v(q, 0) = v(q, τ) = 0,
but a(q, t) changes abruptly at t = 0 and/or t = τ . In this situation the fast-forward
potential will also be discontinuous at these times (see Eq. 4.11b) but otherwise the
analysis in the main text remains valid. Thus the violation of Eq. B.1b simply
implies that VFF(q, t) is turned on and/or off suddenly rather than continuously.
Now consider the case in which the discontinuity occurs at n = 1, hence Eq. B.1a
is violated. Specifically, suppose the time-dependence of the Hamiltonian is turned
on abruptly: ∂H0/∂t 6= 0 at t = 0+, hence
v0(q) ≡ v(q, 0+) 6= 0 (B.2)
The velocity field changes suddenly from v(q, 0−) = 0 to v(q, 0+) = v0(q). The term
∂v/∂t in Eq. 4.10 then leads to a singular term v0(q)δ(t) in the acceleration field
a(q, t). By Eq. 4.11b, this term leads to a contribution to UFF that is proportional
to δ(t), which produces an impulsive force field at t = 0:
− ∂UFF
∂q
(q, t) = mv0(q)δ(t) + [other terms] (B.3)
The effect of this impulse is simple to state: a trajectory located at (q, p) at time
t = 0− is instantaneously “boosted” to (q, p + mv0(q)) at time t = 0+ as it evolves
under HFF.
133
Similar comments apply if ∂H0/∂t 6= 0 at t = τ−. Then
vτ (q) ≡ v(q, τ−) 6= 0 (B.4)
and we get a singular term in UFF that produces an impulsive force
−mvτ (q)δ(t− τ). (B.5)
Now consider a collection of trajectories that, for t < 0, are found on the
adiabatic energy shell E(0). As in the main text, let the loop LFF(t) describe the
evolution of these trajectories, under HFF(z, t). At t = 0, the impulsive force in
Eq. B.3 boosts these trajectories from LFF(0−) = E(0) to a loop LFF(0+) that is
displaced along the momentum axis by an amount mv0(q). Subsequently, this loop
evolves exactly as described in the main text: for 0 < t < τ , LFF(t) is displaced from
the adiabatic energy shell E(t) by an amount mv(q, t) (Eq. 4.20). In particular, at
t = τ− this loop is displaced from E(τ) by mvτ (q). The final impulse at t =
τ (Eq. B.5) instantaneously brings the collection of trajectories from LFF(τ−) to
LFF(τ+) = E(τ).
Thus, non-vanishing derivatives ∂H0/∂t at initial and final times can be accom-
modated by impulse-like terms in UFF(q, t). See Section III.A. of Ref. [34] for an
example that illustrates this point in the context of scale-invariant driving.
134
Appendix C: Flow under H0 + HCD preserves the adiabatic energy
shell
The Hamiltonian H0 +HCD generates the flow (Eq. 4.14)
q˙ =
p
m
+ v(q, t) , p˙ = −∂U0
∂q
(q, t)− p∂v
∂q
(q, t) (C.1)
Let H˙0(q, p, t) denote the instantaneous rate of change of H0, along a trajectory that
passes through the point (q, p) at time t as it evolves under these dynamics:
H˙0(q, p, t) ≡ ∂H0
∂q
q˙ +
∂H0
∂p
p˙+
∂H0
∂t
=
∂U0
∂q
v − p
2
m
∂v
∂q
+
∂U0
∂t
(C.2)
To establish that the flow given by Eq. C.1 preserves the adiabatic energy shell, we
must show that
H˙0(q, p, t) =
d
dt
E¯(t) when (q, p) ∈ E(t) (C.3)
135
We evaluate H˙0 at a point (q, p) ∈ E(t), by setting p = ±p¯(q, t) = ±
[
2m(E¯ − U0)
]1/2
:
H˙0(q,±p¯, t) = ∂U0
∂q
v − 2(E¯ − U0)∂v
∂q
+
∂U0
∂t
= −1
v
∂
∂q
[
(E¯ − U0)v2
]
+
∂U0
∂t
=
∂qS
∂tS
∂
∂q
[
p¯2
2m
(
∂tS
∂qS
)2]
+
∂U0
∂t
=
p¯
2m
∂q∂tS +
∂U0
∂t
=
p¯
m
∂p¯
∂t
+
∂U0
∂t
=
∂
∂t
[
p¯2(q, t)
2m
+ U0(q, t)
]
=
d
dt
E¯(t) (C.4)
which is the desired result. In obtaining Eq. C.4 we have made repeated use of the
identities ∂qS = 2p¯ and v = −∂tS/∂qS (Eqs. 4.8 and 4.9a).
136
Appendix D: Local dynamical invariance of J(q, p, t)
HFF(z, t) generates the equations of motion
q˙ =
p
m
, p˙ = −U ′0 +ma = −U ′0 +mv′v +m
∂v
∂t
(D.1)
Consider the quantity
J(q, p, t) = I(q, p−mv(q, t), t) (D.2)
and let J˙(z, t) denote the instantaneous rate of change of J along a trajectory that
passes through the point z = (q, p) at time t. We have, by direct substitution,
J˙(z, t) =
∂I
∂q
q˙ +
∂I
∂p
(
p˙−mv′q˙ −m∂v
∂t
)
+
∂I
∂t
=
∂I
∂q
p
m
− ∂I
∂p
U ′0 −
∂I
∂p
v′(p−mv) + ∂I
∂t
(D.3)
where the derivatives of I are evaluated at (q, p−mv(q, t), t).
In general J˙(z, t) 6= 0. However, let us now restrict our attention to a point z
137
that satisfies J(z, t) = I0 at a particular time t. At such a point, we have
p = ± p¯(q, t) +mv(q, t) (D.4)
with p¯ =
[
2m(E¯ − U0)
]1/2
. Taking p = p¯+mv for specificity (the case p = −p¯+mv
gives the same result) we get
J˙(z, t) =
∂I
∂q
( p¯
m
+ v
)
− ∂I
∂p
U ′0 −
∂I
∂p
v′p¯+
∂I
∂t
= {I,H0}+ {I,HCD}+ ∂I
∂t
(D.5)
where all quantities on the right side are evaluated at (q, p¯) ∈ E(t). From Eqs. 4.4
and 4.13 we conclude that the right side of the above equation is zero, hence
J(z, t) = I0 ⇒ J˙(z, t) = 0 (D.6)
where the symbol ⇒ is short for “implies that”.
Eq. D.6 establishes that J(z, t) is a local dynamical invariant, in the following
sense. Along trajectories zt evolving under HFF(z, t) from initial conditions z0 ∈
E(0), the value of J remains constant:
J(zt, t) = I0 (D.7)
138
Appendix E: Evolution of the microcanonical measure under HFF
As mentioned in the main text, initial conditions that are sampled from a mi-
crocanonical distribution on E(0) generally evolve (under HFF) to final conditions
that are not distributed microcanonically on E(τ), as illustrated in Fig. 4.4(c). To
understand this point, let
ΦFF : z0 → zτ (E.1)
denote evolution under HFF(z, t) from t = 0 to t = τ . ΦFF maps initial points
z0 ∈ E(0) to final points zτ ∈ E(τ). Now consider an initial phase space distribution
ρ(z, 0) that is uniform in the thin annular region R between the energy shells E(0)
and
EdE(0) ≡ {z|H(z, 0) = E¯(0) + dE} (E.2)
and zero elsewhere. In the limit dE → 0, this distribution converges to a micro-
canonical distribution on E(0).
For finite dE, ρ(z, 0) evolves to a distribution ρ(z, τ) that is uniform (by Liou-
ville’s theorem) in the region R′ = ΦFF(R) between the images of E(0) and EdE(0)
under ΦFF. Although ΦFF maps E(0) to E(τ), in general it does not map EdE(0) to
an energy shell of H(z, t). As a result, in the limit dE → 0, ρ(z, τ) converges to
139
a distribution on E(τ) that is not microcanonical. Thus the clustering of points in
Fig. 4.4(c) traces back to the fact that VFF(q, t) depends on the choice of I0.
140
Bibliography
[1] M. Born and V. A. Fock. Beweis des adiabatensatzes. Zeitschrift fu¨r Physik A,
51:165, 1928.
[2] D. J. Griffiths. Introduction to Quantum Mechanics. Pearson Prentice Hall,
2nd edition, 2004.
[3] X. Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Gue´ry-Odelin, and
J. G. Muga. Fast optimal frictionless atom cooling in harmonic traps: Shortcut
to adiabaticity. Phys. Rev. Lett., 104:063002, 2010.
[4] M. S. Sarandy, E. I. Duzzioni, and R. M. Serra. Quantum computation in
continuous time using dynamic invariants. Phys. Lett. A, 375:3343, 2011.
[5] A. C. Santos and M. S. Sarandy. Superadiabatic controlled evolutions and
universal quantum computation. Nat. Sci. Rep., 5:15775, 2015.
[6] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Infor-
mation. Cambridge University Press, Cambridge, UK, 2000.
[7] E. Torrontegui, S. Iba´n˜ez, X. Chen, A. Ruschhaupt, D. Gue´ry-Odelin, and
J. G. Muga. Fast atomic transport without vibrational heating. Phys. Rev. A,
83:013415, 2011.
[8] R. Bowler, J. Gaebler, Y. Lin, T. R. Tan, D. Hanneke, J. D. Jost, J. P. Home,
D. Leibfried, and D. J. Wineland. Coherent diabatic ion transport and separa-
tion in a multizone trap array. Phys. Rev. Lett., 109:080502, 2012.
[9] A. del Campo. Frictionless quantum quenches in ultracold gases: A quantum-
dynamical microscope. Phys. Rev. A, 84:031606(R), 2011.
[10] A. del Campo, M. M. Rams, and W. H. Zurek. Assisted finite-rate adiabatic
passage across a quantum critical point: Exact solution for the quantum ising
model. Phys. Rev. Lett., 109:115703, 2012.
141
[11] S. Masuda, K. Nakamura, and A. del Campo. High-fidelity rapid ground-state
loading of an ultracold gas into an optical lattice. Phys. Rev. Lett., 113:063003,
2014.
[12] V. Mukherjee, S. Montangero, and R. Fazio. Local shortcut to adiabaticity for
quantum many-body systems. Phys. Rev. A, 93:062108, 2016.
[13] V. Giovanetti, S. Lloyd, and L. Maccone. Quantum metrology. Phys. Rev.
Lett., 96:010401, 2006.
[14] M. G. Bason, M. Viteau, N. Malossi, P. Huillery, E. Arimondo, D. Ciampini,
R. Fazio, V. Giovannetti, R. Mannella, and O. Morsch. High-fidelity quantum
driving. Nature Physics, 8:147, 2011.
[15] A. Trabesinger. Quantum simulation. Nature Physics, 8:263, 2012.
[16] P. Salamon, K. H. Hoffmann, Y. Rezek, and R. Kosloff. Maximum work in
minimum time from a conservative quantum system. Phys. Chem. Chem. Phys.,
11:1027, 2009.
[17] Y. Rezek, P. Salamon, K. H. Hoffman, and R. Kosloff. The quantum refriger-
ator: The quest for absolute zero. Euro. Phys. Lett., 85:30008, 2009.
[18] P. Skrzypczyk, J. A. Short, and S. Popescu. Work extraction and thermody-
namics for individual quantum systems. Nat. Comm., 5:4185, 2014.
[19] E. Torrontegui, S. Iba´n˜ez, S. Mart´ınez-Garaot, M. Modugno, A. del Campo,
D. Gue´ry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga. Chapter 2 –
shortcuts to adiabaticity. Adv. At. Mol. Opt. Phys., 62:117, 2013.
[20] X. Chen, E. Torrontegui, and J. G. Muga. Lewis-riesenfeld invariants and
transitionless quantum driving. Phys. Rev. A, 83:062116, 2011.
[21] E. Torrontegui, S. Mart´ınez-Garaot, and J. G. Muga. Hamiltonian engineering
via invariants and dynamical algebra. Phys. Rev. A, 89:043408, 2014.
[22] M. Demirplak and S. A. Rice. Adiabatic population transfer with control fields.
J. Phys. Chem. A, 107:9937, 2003.
[23] M. Demirplak and S. A. Rice. Assisted adiabatic passage revisited. J. Phys.
Chem. B, 109:6838, 2005.
[24] M. V. Berry. Transitionless quantum driving. J. Phys. A: Math. Theor.,
42:365303, 2009.
[25] C. Jarzynski. Generating shortcuts to adiabaticity in quantum and classical
dynamics. Phys. Rev. A, 88:040101(R), 2013.
[26] S. Masuda and K. Nakamura. Fast-forward of adiabatic dynamics in quantum
mechanics. Proc. R. Soc. A, 466(2116):1135, 2010.
142
[27] S. Masuda and K. Nakamura. Acceleration of adiabatic quantum dynamics in
electromagnetic fields. Phys. Rev. A, 84:043434, 2011.
[28] E. Torrontegui, S. Mart´ınez-Garaot, A. Ruschhaupt, and J. G. Muga. Shortcuts
to adiabaticity: Fast-forward approach. Phys. Rev. A, 86:013601, 2012.
[29] A. Kiely, J. P. L. McGuinness, J. G. Muga, and A. Rauschhaupt. Fast and
stable manipulation of a charged particle in a penning trap. J. Phys. B: At.
Mol. Opt. Phys., 48:075503, 2015.
[30] S. Mart´ınez-Garaot, M. Palmero, J. G. Muga, and D. Gue´ry-Odelin. Fast
driving between arbitrary states of a quantum particle by trap deformation.
Phys. Rev. A, 94:063418, 2016.
[31] S. Iba´n˜ez, X. Chen, E. Torrontegui, J. G. Muga, and A. Ruschhaupt. Multiple
schro¨dinger pictures and dynamics in shortcuts to adiabaticity. Phys. Rev.
Lett., 109:100403, 2012.
[32] A. del Campo. Shortcuts to adiabaticity by counterdiabatic driving. Phys. Rev.
Lett., 111:100502, 2013.
[33] S. Mart´ınez-Garaot, E. Torrontegui, X. Chen, and J. G. Muga. Shortcuts to adi-
abaticity in three-level systems using lie transforms. Phys. Rev. A, 89:053408,
2014.
[34] S. Deffner, C. Jarzynski, and A. del Campo. Classical and quantum shortcuts
to adiabaticity for scale-invariant driving. Phys. Rev. X, 4:021013, 2014.
[35] K. Takahashi. Unitary deformations of counterdiabatic driving. Phys. Rev. A,
91:042115, 2015.
[36] D. Sels and A. Polkovnikov. Minimizing irreversible losses in quantum systems
by local counterdiabatic driving. Proc. Natl. Acad. Sci. USA, 114:3909, 2017.
[37] S. Iba´n˜ez, S. Mart´ınez-Garaot, X. Chen, E. Torrontegui, and J. G. Muga. Short-
cuts to adiabaticity for non-hermitian systems. Phys. Rev. A, 84:023415, 2011.
[38] B. T. Torosov, G. D. Valle, and S. Longhi. Non-hermitian shortcut to adia-
baticity. Phys. Rev. A, 87:052502, 2013.
[39] G. Vacanti, R. Fazio, S. Montangero, G. M. Palma, M. Paternostro, and V. Ve-
dral. Transitionless quantum driving in open quantum systems. New J. Phys.,
16:053017, 2014.
[40] J. Song, Z. Zhang, Y. Xia, X. Sun, and Y. Jiang. Fast coherent manipulation
of quantum states in open systems. Opt. Exp., 24:21674, 2016.
[41] S. Deffner. Shortcuts to adiabaticity: suppression of pair production in driven
dirac dynamics. New Journal of Physics, 18:012001, 2016.
143
[42] J. G. Muga, M. A. Simo´n, and A. Tobalina. How to drive a dirac system fast
and safe. New. J. Phys., 18:021005, 2016.
[43] X. Song, F. Deng, L. Lamata, and J. G. Muga. Robust state preparation in
quantum simulations of dirac dynamics. Phys. Rev. A, 95:022332, 2017.
[44] J.-F. Schaff, X.-L. Song, P. Vignolo, and G. Labeyrie. Fast optimal transition
between two equilibrium states. Phys. Rev. A, 82:033430, 2010.
[45] J.-F. Schaff, X.-L. Song, P. Capuzzi, P. Vignolo, and G. Labeyrie. Shortcut
to adiabaticity for an interacting Bose-Einstein condensate. Europhys. Lett.,
93:23001, 2011.
[46] A. Walther, F. Ziesel, T. Ruster, S. T. Dawkins, K. Ott, M. Hettrich, K. Singer,
F. Schmidt-Kaler, and U. Poschinger. Controlling fast transport of cold trapped
ions. Phys. Rev. Lett., 109:080501, 2012.
[47] J. Zhang, J. Hyun Shim, I. Niemeyer, T. Taniguchi, T. Teraji, H. Abe, S. On-
oda, T. Yamamoto, T. Ohshima, J. Isoya, and D. Suter. Experimental imple-
mentation of assisted quantum adiabatic passage in a single spin. Phys. Rev.
Lett., 110:240501, 2013.
[48] S. An, D. Lv, A. del Campo, and K. Kim. Shortcuts to adiabaticity by coun-
terdiabatic driving in trapped-ion transport. Nat. Comm., 7:12999, 2016.
[49] S. Deffner and S. Campbell. Trade-off between speed and cost in shortcuts to
adiabaticity. Phys. Rev. Lett., 118:100601, 2017.
[50] K. Funo, J.-N. Zhang, C. Chatou, K. Kim, M. Ueda, and A. del Campo. Uni-
versal work fluctuations during shortcuts to adiabaticity by counterdiabatic
driving. Phys. Rev. Lett., 118:100602, 2017.
[51] H. Goldstein. Classical Mechanics. Addison-Wesley, Reading, Massachusetts,
2nd edition, 1980.
[52] J. Deng, Q. Wang, Z. Liu, P. Ha¨nggi, and J. Gong. Boosting work charac-
teristics and overall heat engine performance via accelerated adiabatic control:
quantum and classical. Phys. Rev. E, 88:062122, 2013.
[53] M. Okuyama and K. Takahashi. From classical nonlinear integrable systems to
quantum shortcuts to adiabaticity. Phys. Rev. Lett., 117:070401, 2016.
[54] A. Patra and C. Jarzynski. Classical and quantum shortcuts to adiabaticity in
a tilted piston. J. Phys. Chem. B, 121:3403, 2017.
[55] M. Okuyama and K. Takahashi. Quantum-classical correspondence of shortcuts
to adiabaticity. J. Phys. Soc. Japan, 86:043002, 2017.
[56] C. Jarzynski, S. Deffner, A. Patra, and Y. Subasi. Fast forward to the classical
adiabatic invariant. Phys. Rev. E, 95:032122, 2017.
144
[57] S. Vaikuntanathan and C. Jarzynski. Escorted free energy simulations: Improv-
ing convergence by reducing dissipation. Phys. Rev. Lett., 100:190601, 2008.
[58] Z. C. Tu. Stochastic heat engine with the consideration of inertial effects and
shortcuts to adiabaticity. Phys. Rev. E, 89:052148, 2014.
[59] I. A. Mart´ınez, A. Petrosyan, D. Gue´ry-Odelin, E. Trizac, and S. Ciliberto.
Engineered swift equilibration of a brownian particle. Nat. Phys., 12:843, 2016.
[60] G. Li, H. T. Quan, and Z. C. Tu. Shortcuts to isothermality and nonequilibrium
work relations. Phys. Rev. E, 96:012144, 2017.
[61] A. del Campo and M. G. Boshier. Shortcuts to adiabaticity in a time-dependent
box. Sci. Rep., 2:648, 2012.
[62] C. Jarzynski. Geometric phases and anholonomy for a class of chaotic classical
systems. Phys. Rev. Lett., 74:1732, 1995.
[63] A. Patra and Jarzynski. C. Shortcuts to adiabaticity using flow fields.
arXiv:1707.01490, 2017.
[64] M. Demirplak and S. A. Rice. On the consistency, extremal, and global prop-
erties of counterdiabatic fields. J. Chem. Phys., 129:154111, 2008.
[65] A. Julg. The complete symmetrization of quantum operators: new thoughts on
an old problem. Theor. Chim. Acta, 74:323, 1988.
[66] S. Gill. A process for the step-by-step integration of differential equations in
an automatic digital computing machine. Cambridge Philos. Soc., 47:96, 1951.
[67] M. V. Berry. Quantal phase factors accompanying adiabatic changes. Proceed-
ings of the Royal Society A., 392:45, 1984.
[68] H. R. Lewis and P. G. Leach. A direct approach to finding exact invariants for
onedimensional timedependent classical hamiltonians. J. Math. Phys., 23:2371,
1982.
[69] J. G. Muga, X. Chen, S. Iba´n˜ez, I. Lizuain, and A. Ruschhaupt. Transitionless
quantum drivings for the harmonic oscillator. J. Phys. B: At. Mol. Opt. Phys.,
43:085509, 2010.
[70] S. Di Martino, F. Anza, P. Facchi, A. Kossakowski, G. Marmo, A. Messina,
B. Militello, and S. Pascazio. A quantum particle in a box with moving walls.
J. Phys. A: Math. Theor., 46(36):365301, 2013.
[71] M. Razavy. An exactly soluble schrdinger equation with a bistable potential.
Am. J. Phys., 48:285, 1980.
[72] M. D. Feit, J. A. Fleck, Jr., and A. Steiger. Solution of the schro¨dinger equation
by a spectral method. J. Comp. Phys., 47:412, 1982.
145
[73] R. Kosloff. Time-dependent quantum-mechanical methods for molecular dy-
namics. J. Phys. Chem., 92:2087, 1988.
[74] V. I. Arnold. Mathematical Methods of Classical Mechanics. Springer, New
York, Reading, Massachusetts, USA, 1978.
[75] H. R. Lewis. Class of exact invariants for classical and quantum timedependent
harmonic oscillators. J. Math. Phys., 9:1976, 1968.
[76] H. R. Lewis and W. B. Riesenfeld. An exact quantum theory of the time-
dependent harmonic oscillator and of a charged particle in a time-dependent
electromagnetic field. J. Math. Phys., 10:1458, 1969.
[77] P. Navarro and E. Pe´rez. Paul ehrenfest: The genesis of the adiabatic hypoth-
esis, 19111914. Arch. Hist. Exact Sci., 60:209, 2006.
[78] I. Bloch. Ultracold quantum gases in optical lattices. Nat. Phys., 1:23, 2005.
[79] M. Berry. Topics in nonlinear dynamics. AIP Conference Proceedings No. 46,
ed. S. Jorna, page 16, 1978.
[80] S. Vaikuntanathan and C. Jarzynski. Dissipation and lag in irreversible pro-
cesses. Europhys. Lett., 87:60005, 2009.
[81] Anne Le Cununder, Ignacio Mart´ınez, Artyom Petroysan, David Gue´ry-Odelin,
Emmanuel Trizac, and Sergio Ciliberto. Fast equilibrium switch of a micro
mechanical oscillator. Appl. Phys. Lett, 109:113502, 2016.
[82] T. Schmiedl and U. Seifert. Optimal finite-time processes in stochastic ther-
modynamics. Phys. Rev. Lett., 98:108301, 2007.
[83] T. Schmiedl and U. Seifert. Efficiency at maximum power: An analytically
solvable model for stochastic heat engines. Europhys. Lett., 81:20003, 2008.
[84] M. Feingold and A. Peres. Distribution of matrix elements of chaotic systems.
Phys. Rev. A, 34:591, 1986.
146