# Shortcuts to adiabaticity in harmonic traps: a quantum-classical analog

Vincent Hardel<EMAIL_ADDRESS>Université de Strasbourg,
CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, UMR 7504,
F-67000 Strasbourg, France  Giovanni Manfredi
<EMAIL_ADDRESS>Université de Strasbourg, CNRS, Institut de
Physique et Chimie des Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg,
France  Paul-Antoine Hervieux<EMAIL_ADDRESS>Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux
de Strasbourg, UMR 7504, F-67000 Strasbourg, France  Rémi Goerlich  Raymond &
Beverly Sackler School of Chemistry, Tel Aviv University, Tel Aviv 6997801,
Israel

###### Abstract

We present a new technique for efficiently transitioning a quantum system from
an initial to a final stationary state in less time than is required by an
adiabatic (quasi-static) process. Our approach makes use of Nelson’s
stochastic quantization, which represents the quantum system as a classical
Brownian process. Thanks to this mathematical analogy, known protocols for
classical overdamped systems can be translated into quantum protocols. In
particular, one can use classical methods to find optimal quantum protocols
that minimize both the time duration and some other cost function to be freely
specified. We have applied this method to the time-dependent harmonic
oscillator and tested it on two different cost functions: (i) the cumulative
energy of the system over time and (ii) the dynamical phase of the
wavefunction. In the latter case, it is possible to construct protocols that
are “adiabatically optimal”, i.e., they minimize their distance from an
adiabatic process for a given duration.

††preprint: APS/123-QED

##  I Introduction

Optimization problems play an important role in both classical and quantum
physics, providing an elegant framework for describing natural phenomena.
Concepts such as Fermat’s principle and the stationary-action principle, which
serve as foundational pillars of physics, are deeply rooted in optimization
principles. Optimization techniques are also crucial for efficient resource
utilization, improving theoretical models, and controlling and manipulating
the state of a physical system.

The central question in optimization research revolves around bringing a
system from an initial state to a desired final state while minimizing a
certain quantity, known as the cost function. A very diverse array of systems
can be explored through the lens of optimization, ranging from Kramers-like
problems with double well potentials [1] to trapped ions [2], cavity quantum
electrodynamics [3], superconducting circuits [4], spin-orbit coupling [5],
nitrogen-vacancy centers [6], many-body and spin-chain models [7], and even
metrology applications [8]. Various methods can be employed depending on the
specific circumstances. Optimization techniques based on variational
principles are often utilized, such as in the quantum brachistochrone problem
[9, 10] or in the context of Bose-Einstein condensates trapped in harmonic
potentials [11, 12]. Additionally, optimization methods based on optimal
control theory have gained prominence in recent years [13, 14, 15, 16, 17].

In the present work, we focus on the case of the time-dependent quantum
harmonic oscillator, a system of paramount importance across multiple fields
of physics. Specifically, we will examine protocols for manipulating the
stiffness of the potential in order to efficiently transition the system from
one steady-state to another in a time shorter than that required by an
adiabatic (i.e., quasi-static) process. This concept is known as _shortcut to
adiabaticity_ (STA) (for recent reviews, see [18, 19]). While many of the
basic ideas were put forward about two decades ago, with various approaches
such as counterdiabatic driving [20, 21, 22], inverse engineering [23, 24],
scaling laws [25], and others [26, 27, 28, 29, 30, 31, 32, 33] being
introduced, the development of STA methods has gained much momentum in recent
years [34, 35, 36, 37].

Here, we will employ an approach recently proposed for the classical
overdamped dynamics of a Brownian particle confined in a harmonic trap and in
contact with a heat bath at given temperature [38]. This method, based on a
variational principle, allowed us to optimize the transfer from one thermal
equilibrium to another, by minimizing both the duration of the transfer and
the expended work. The trade-off between duration and work could be modulated
at will by tuning a single Lagrange multiplier.

The main purpose of the present work is to develop a quantum-classical analogy
that allows us to exploit the aforementioned method in order to control and
optimize the dynamics of a _quantum_ harmonic oscillator. Indeed, analogies
may serve as powerful tools in physics. For instance, the experiments
conducted by Couder in 2005 [39] established a hydrodynamic analogy to the
behavior of quantum particles, as described by the pilot wave theory
introduced by de Broglie [40] and later developed by Bohm [41]. With this
approach, it was possible to demonstrate the existence of quantum-like
diffraction using a fully classical experimental setup [42].

The analogy to be presented here is based on an alternative formulation of
quantum mechanics originally due to Nelson [43, 44, 45]. In Nelson’s
representation, the quantum evolution is governed by a first-order stochastic
equation, supplemented by Schrödinger’s equation for the wave guiding the
trajectory in a manner similar to the Bohm-de Broglie theory. Nelson’s
stochastic equation bears a close resemblance to the Langevin equation that
governs the overdamped motion of a classical Brownian particle, thus
suggesting the potential for a fruitful quantum-classical analogy. Such an
analogy has already been considered in recent years for an open quantum system
[46]. In the present work, we will demonstrate that this approach is
particularly well-suited for solving optimization problems for closed quantum
systems.

The objective of this paper is to devise a protocol that transitions the
system from a given stationary state to another in the shortest possible time,
while minimizing a specified cost function throughout the temporal evolution.
In classical stochastic thermodynamics, it is common to minimize the work done
on the system, which also corresponds to minimizing the dissipated heat [47,
23, 48, 38]. However, for closed Hamiltonian systems, such work is simply
equal to the difference between the final and the initial energies, and
therefore it is not a pertinent quantity to minimize [34].

In the ensuing sections, we will outline a method – based on Nelson’s dynamics
and the quantum-classical analogy mentioned above – which allows us to
minimize a generic cost function, usually written as the sum of the duration
of the protocol plus a functional $F$ of the control parameters. As relevant
examples, we will choose for $F$ either the cumulative energy of the system
over time or the dynamical phase of the wavefunction. The latter case allows
us to construct protocols that are “adiabatically optimal”, i.e. protocols
that, for a given duration, minimize their distance (in a precise mathematical
sense) from an adiabatic process.

In Section II, we will detail the basic features of the classical analog of
the time-dependent quantum harmonic oscillator using Nelson’s stochastic
formulation, illustrating the analogy with a simple numerical example. In
Section III, we will show how this analogy can be applied to the quantum
harmonic oscillator. We will also describe a general method to obtain a
protocol that is optimal in regards of both its duration and another cost
functional to be specified at will. In Sec IV we will present numerical
results for optimal protocols obtained using two different cost functions as
illustrative examples. Conclusions and perspectives for future work will be
detailed in Sec. V.

##  II Quantum-classical analogy

###  II.1 Fundamentals of the analogy

A one-dimensional particle of mass $m$ trapped in a time-dependent harmonic
potential obeys the Schrödinger equation

$i\hbar\frac{\partial}{\partial
t}\psi(x,t)=\hat{H}(t)\psi\equiv\left(-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
x^{2}}+\frac{1}{2}\kappa(t)x^{2}\right)\psi(x,t),$ (1)

where $\hbar$ is Planck’s constant, $\kappa(t)$ is the time-dependent
stiffness of the potential, and $\psi(x,t)$ is the wavefunction of the system
at time $t$ and position $x$. The optimization procedure developed in this
work consists in designing a protocol $\kappa(t)$ which brings the system from
an initial stationary state $\psi_{\mathrm{i}}(x)$ at time $t_{\mathrm{i}}$ to
a final (also stationary) state $\psi_{\mathrm{f}}(x)$ at time
$t_{\mathrm{f}}$, in the shortest possible time $\Delta
t=t_{\mathrm{f}}-t_{\mathrm{i}}$, while minimizing a given cost function [49,
50, 34].

The derivation of these optimal protocols will be based on Nelson’s
formulation of quantum mechanics. In this approach, similarly to the Bohm-de
Broglie formalism, quantum particles are supposed to have a well-defined
position $x(t)$ evolving in time. Unlike the deterministic trajectories
followed by quantum objects in the Bohm-de Broglie theory, Nelson’s theory
postulates that each trajectory obeys the stochastic differential equation

$\mathrm{d}x(t)=b(x(t),t)\mathrm{d}t+\sqrt{2D}\,\mathrm{d}W(t),$ (2)

where $b(x,t)$ is the deterministic drift velocity, $D=\hbar/2m$ is the
diffusion coefficient, and $\mathrm{d}W(t)$ is the Wiener increment for a
Markov process: $\langle\mathrm{d}W(t)\rangle=0$,
$\langle\mathrm{d}W(t)\,\mathrm{d}W(t^{\prime})\rangle=t-t^{\prime}$. The
origin of the stochastic nature of the dynamic of quantum particles is
postulated but not explained by Nelson’s theory. The key point in Nelson’s
approach is to define the drift coefficient as

$b(x,t)=\frac{\hbar}{m}\frac{\partial S(x,t)}{\partial
x}+\frac{\hbar}{2m}\frac{\partial\ln\rho(x,t)}{\partial x},$ (3)

where $S(x,t)$ and $\rho(x,t)$ are respectively the phase and squared modulus
of the wavefunction $\psi(x,t)$, expressed in polar form as
$\psi=\sqrt{\rho}\,\exp(iS)$. This definition of $b(x,t)$ ensures that the
probability distribution of a large ensemble of trajectories $x(t)$ obeying
Nelson’s equation (2) converges to the square-modulus of the wavefunction,
following Born’s rule [51]. If the initial probabilistic distribution of the
trajectories follows Born’s rule at $t=0$, it will do so for all successive
times $t>0$. Hence, Nelson’s theory reproduces the same results as the
standard quantum mechanics based on the Schrödinger equation.

For the ground state of the harmonic oscillator, the Gaussian form of the
wavefunction yields a simple form for the drift term of Eq. (3), as we now
show. Due to the quadratic nature of the Hamiltonian in Eq. (1), if the
initial wavefunction $\psi_{\mathrm{i}}(x,t=0)$ is Gaussian, then it remains
Gaussian for all times $t>0$, and can be written as

$\psi(x,t)=\frac{1}{\sqrt[4]{2\pi
s(t)}}\exp\left[-\frac{x^{2}}{4s(t)}+i\alpha(t)x^{2}+i\beta(t)\right],$ (4)

where $s(t)=\langle x(t)^{2}\rangle$ is the time-dependent variance of the
density, while $\alpha(t)$ and $\beta(t)$ are the dynamical and geometrical
phases of the wavefunction, respectively. For the wavefunction to be a
solution of Eq. (1), the time-dependent coefficients $\alpha(t)$ and
$\beta(t)$ must satisfy the following relations:

$\alpha(t)=\frac{m}{4\hbar}\frac{\dot{s}(t)}{s(t)},\quad\dot{\beta}(t)=-\frac{\hbar}{4ms(t)},$
(5)

where the dot denotes differentiation with respect to the time $t$. The
variance $s(t)$ is related to the standard deviation $\sigma=\sqrt{2s}$, which
must obey the following Ermakov equation [52]:

$\ddot{\sigma}(t)+\frac{\kappa(t)}{m}\sigma(t)=\frac{4D^{2}}{\sigma^{3}(t)}.$
(6)

The coupled equations alpha-ermakov provide a full, exact solution of the
Schrödinger equation in the Gaussian form psi_gaussian.

From Eqs. b_nelson and psi_gaussian, one obtains immediately Nelson’s drift
velocity:

$b(x,t)=\frac{\hbar}{m}\left(2\alpha(t)-\frac{1}{2s(t)}\right)x,$ (7)

which, importantly, is linear in $x$. Therefore, we can rewrite Nelson’s
stochastic equation nelson_equation as

$\mathrm{d}x(t)=\frac{\hbar}{m}\left(2\alpha(t)-\frac{1}{2s(t)}\right)x(t)\,\mathrm{d}t+\sqrt{2D}\,\mathrm{d}W(t).$
(8)

The above Nelson equation nelson_equation2 bears a striking resemblance with
the Langevin equation for a classical overdamped Brownian particle in a
harmonic potential of stiffness $\bar{\kappa}(t)$ and same diffusion
coefficient $D$, which we write here as:

$\mathrm{d}x(t)=-\frac{\bar{\kappa}(t)}{\gamma}x(t)\mathrm{d}t+\sqrt{2D}\,\mathrm{d}W(t),$
(9)

where $\gamma$ is the usual Stokes drag coefficient, which we keep for
dimensional reasons in the classical equation, but will disappear in the
quantum results. The equations nelson_equation2 and langevin_equation are
identical if we define the classical stiffness as:

$\bar{\kappa}(t)=\gamma\frac{\hbar}{m}\left(2\alpha(t)-\frac{1}{2s(t)}\right).$
(10)

Hence, our physical analogy is based on the mathematical equivalence between
Nelson’s equation nelson_equation2 for a quantum particle in a harmonic
oscillator and the Langevin equation langevin_equation for a classical
Brownian particle. In addition, for the classical Langevin equation, the
variance $s(t)$ obeys the following closed evolution equation [53, 54]

$\frac{\mathrm{d}s(t)}{\mathrm{d}t}=\frac{2}{\gamma}[D\gamma-\bar{\kappa}(t)s(t)].$
(11)

To complete the analogy, we need to specify the relationship between the
stiffness $\kappa(t)$ of the quantum oscillator and the the stiffness
$\bar{\kappa}(t)$ appearing in the classical stochastic process. Taking the
time derivative of Eq. kappabar and using Eqs. alpha and sdot, we arrive,
after some algebra, at the following expression for the quantum stiffness

$\kappa(t)=\frac{\hbar^{2}}{2ms^{2}(t)}+\frac{m}{\gamma}\dot{\bar{\kappa}}(t)-\frac{m}{\gamma^{2}}\bar{\kappa}^{2}(t),$
(12)

written in terms of the classical stiffness $\bar{\kappa}(t)$ and its time
derivative. Equation quantum_protocol serves as a crucial link in establishing
the quantum-classical analogy, acting as a bridge between the quantum system
and its classical analog.

Finally, from Eq. sdot it results that, at equilibrium:
$\bar{\kappa}_{\mathrm{eq}}=D\gamma/s_{\mathrm{eq}}$ for the classical case.
For the quantum case, from Eq. quantum_protocol we obtain

$\kappa_{\mathrm{eq}}=\frac{m}{\gamma^{2}}\bar{\kappa}_{\mathrm{eq}}^{2}=\frac{D^{2}m}{s_{\mathrm{eq}}^{2}},$
(13)

which does not depend on the classical parameter $\gamma$, as expected.

Our strategy will be to suggest a classical protocol $\bar{\kappa}(t)$ and use
Eq. quantum_protocol to obtain the corresponding quantum protocol. By
construction, the evolution of the variance $s(t)$ will be identical for both
cases and given by Eq. sdot. Therefore, if we can devise a classical protocol
that brings the variance from an initial equilibrium state with
$s(t_{\mathrm{i}})=s_{\mathrm{i}}$ to a final equilibrium state with $s(t_{\rm
f})=s_{\rm f}$, then the corresponding quantum protocol will do the same.

###  II.2 Example: STEP protocol

To illustrate the quantum-classical analogy, we examine a sudden protocol
(frequently referred to as STEP) which consists in an abrupt change of the
classical stiffness from $\bar{\kappa}_{\mathrm{i}}$ to
$\bar{\kappa}_{\mathrm{f}}$. However, because of the presence of a first
derivative in Eq. quantum_protocol, it is necessary to smooth out such STEP
protocol. The smoothed STEP is then defined as follows [55]

$\bar{\kappa}(t)=\frac{\bar{\kappa}_{\mathrm{f}}+\bar{\kappa_{\mathrm{i}}}}{2}+\frac{\bar{\kappa}_{\mathrm{f}}-\bar{\kappa_{\mathrm{i}}}}{2}\tanh\left(\frac{t-\tau}{\epsilon}\right).$
(14)

The classical stiffness is centered at $t=\tau$ and becomes steeper and
steeper as $\epsilon\to 0$. For simplicity, we have used units for which:
$\kappa_{\mathrm{i}}=2$, $\hbar=\gamma=1$, $m=1/2$, so that the quantum
diffusion coefficient is $D=\hbar/2m=1$. In these units, time is measured in
units of $2/\omega_{\mathrm{i}}$ and the variance in units of
$\hbar/m\omega_{i}$, where $\omega_{\mathrm{i}}=\sqrt{\kappa_{\mathrm{i}}/m}$
is the initial angular frequency of the harmonic potential. In the examples
below, we have used $\bar{\kappa}_{\mathrm{i}}=2$ and
$\bar{\kappa}_{\mathrm{f}}=4$, which in virtue of Eq. equilibrium yields:
$\kappa_{\mathrm{i}}=2$ and $\kappa_{\mathrm{f}}=8$, and variances
$s_{\mathrm{i}}=0.5$ and $s_{\mathrm{f}}=0.25$.

In Fig. 1 (bottom panel), we show the classical and quantum protocols for two
values of the width $\epsilon=0.1$ and $\epsilon=1$. For the smoother
classical protocol ($\epsilon=1$, dashed black curve), the quantum protocol
(shown in the inset) has a similar shape as its classical counterpart,
although its initial and final values are different, in accordance with Eq.
equilibrium. In contrast, the steeper classical protocol ($\epsilon=0.1$,
solid orange curve) yields an oscillating quantum protocol (inset). These
oscillations become stronger as $\epsilon\to 0$. The time evolution of the
variance $s(t)$ (top panel of Fig. 1) – which, as stated above, is by
construction the same for the classical and quantum cases – shows that the
variance of the system is smoothly brought from its initial value to its final
value, even for the case ($\epsilon=0.1$) where the quantum protocol is
strongly oscillating.

It is clear from Fig. 1 that the steeper protocol achieves the transition more
quickly, but let us try to quantify this speed-up more accurately.
Classically, the relaxation time for a STEP protocol is given by the final
stiffness $\bar{\kappa}_{\mathrm{f}}$, and reads as: $\tau_{\rm
rel}=\gamma/\kappa_{\mathrm{f}}$. Rewriting this in terms of the quantum
quantities, we get: $\tau_{\rm
rel}=\sqrt{m/\kappa_{\mathrm{f}}}=\omega_{\mathrm{f}}^{-1}$, where $\gamma$,
being a purely classical parameter, has naturally disappeared. Hence, the
relaxation time is the inverse of the final oscillator frequency. In the
present case, we have, in our units, $\tau_{\rm
rel}=\omega_{\mathrm{f}}^{-1}=0.25\,(2\omega_{\mathrm{i}}^{-1})$.

Now, it is important to understand that, for the quantum oscillator (which is
conservative), this is not really a relaxation time. If we apply a STEP
protocol directly on the quantum stiffness $\kappa$, the quantum system will
oscillate indefinitely, with no damping. The standard way to implement the
transition without oscillations would be to proceed adiabatically, which takes
an infinite time. Hence, any quantum protocols, like those of Fig. 1, that
take a finite time to complete, already do much better than the adiabatic one.
In the figure, the slower protocol takes about $\approx
6\,(2\omega_{\mathrm{i}}^{-1})$ to achieve the transition, while the faster
protocol takes $\approx 0.75\,(2\omega_{\mathrm{i}}^{-1})$. This speeding up
is achieved through the special temporal profile of the quantum protocol
$\kappa(t)$, which was obtained thanks to the quantum-classical analogy.

But the classical STEP protocol will never be able to go faster than the
relaxation time $\tau_{\rm rel}=\omega_{\mathrm{f}}^{-1}$, which therefore
constitutes a fundamental limit also for the quantum protocol, as the
evolution of the variance is by construction the same for both. In the next
section, we will develop a method to construct optimal protocols that break
this limit, and allow relaxation on a timescale shorter than
$\omega_{\mathrm{f}}^{-1}$. Therefore, these optimal protocols not only
outperform the adiabatic process ($\tau_{\rm rel}=\infty$), but also do better
than “naive” protocols such as STEP, for which $\tau_{\rm
rel}=\omega_{\mathrm{f}}^{-1}$. In addition, they also minimize some other
quantity of physical interest, such as the cumulative energy over time.

Figure 1: Top Panel: Evolution of the variance $s(t)$ as a function of time
(measured in units of $2\omega_{\mathrm{i}}^{-1}$), for $\epsilon=1$ (black
dashed lines) and $\epsilon=0.1$ (orange solid lines). The variance decreases
from $s_{\mathrm{i}}=0.5$ to $s_{\mathrm{f}}=0.25$, in units of
$\hbar/m\omega_{i}$ as detailed in the main text. The larger value of
$\epsilon$ correspond to the smoother protocol. Bottom panel: Time evolution
of the quantum stiffness $\kappa(t)$ (main plot) associated to the classical
STEP protocol $\bar{\kappa}(t)$ (inset), for the same values of $\epsilon$.
When the classical STEP is smooth ($\epsilon=1$, black dashed lines), the
quantum protocol is also smooth and follows the same behavior as the classical
one. In contrast, when the classical STEP protocol varies abruptly
($\epsilon=0.1$, orange solid lines), an oscillation appears at mid-time in
the quantum protocol.

##  III Optimal quantum protocols

As mentioned in the preceding sections, our objective is to transition a
quantum system from an initial state, $\psi_{\mathrm{i}}$, to a final state,
$\psi_{\mathrm{f}}$, in the shortest possible time, $\Delta t$, while
minimizing a certain cost function. For the time being, we keep this cost
function as general as possible. The initial and final states are supposed to
be the ground states in the respective harmonic potentials at $t_{\mathrm{i}}$
and $t_{\mathrm{f}}$. For a classical Brownian particle in a thermal bath,
several methods already exist for finding the optimal protocol, especially
when the cost function is the work done on the system [38, 56, 23]. In the
following, we will explore various cost functions and, for each of them, find
the optimal protocol using the method based on the classical analogy developed
in Sec. II. Our approach is similar to the one used in our earlier work [38]
and based on a variational principle.

In order to find the optimal protocol, we first need to define the functional
to be minimized $J[\kappa,\dot{\kappa}]$, which is a function of the control
parameter (stiffness of the oscillator) and its time derivative. This
functional will be written as the sum of the total duration of the protocol,
denoted $\Delta t[\kappa]$, plus another functional to be minimized, denoted
$F[\kappa,\dot{\kappa}]$ (our cost function), the latter associated with a
Lagrange multiplier $\lambda$. Then, the optimal protocol $\kappa_{\rm opt}$
is found as the solution of the Euler-Lagrange equation derived from the total
functional $J$. In practice, it will be necessary to add a third functional
$G[\dot{\kappa}]$, and its Lagrange multiplier $\mu$, in order to ensure that
the boundary conditions on $\kappa(t)$ are satisfied.

One has the choice to express the above functionals either in terms of the
classical stiffness $\kappa$ or the quantum stiffness $\bar{\kappa}$, and then
derive the corresponding Euler-Lagrange equations accordingly. In practice, as
we shall see, it will be easier to express all functionals as a function of
$\bar{\kappa}$ (and its time derivative) using Eq. quantum_protocol. For
simplicity of notation, we will use the same symbols ($J$, $F$ and $G$) for
the functionals, irrespective of their arguments.

Let us first express the time duration as a functional. We adopt the method
developed in Ref. [38], which consists in using the variance $s$ as an
independent variable, instead of the time $t$ 111This is possible only if
$s(t)$ is a monotonic function of $t$.. Each protocol can then be
characterized by a trajectory in the stiffness-variance space $(\kappa,s)$.
This allows us to express the protocols as a function of $s$ instead of $t$,
so that we write: $\kappa(s)={\kappa}(t(s))$ and
$\bar{\kappa}(s)={\bar{\kappa}}(t(s))$ (note that, for simplicity of notation,
we use the same variable name for a function of $t$ and the corresponding
function of $s$). Using Eq. sdot one obtains for the time duration functional

$\Delta
t[{\bar{\kappa}}]=\frac{1}{2}\int_{s_{\mathrm{i}}}^{s_{\mathrm{f}}}\mathrm{d}s~{}\frac{\gamma}{D\gamma-s{\bar{\kappa}}(s)}.$
(15)

Note that the above functional depends on the classical protocol
$\bar{\kappa}(s)$, and not the quantum one $\kappa(s)$.

Since we are interested in the dynamics of the quantum system, the yet-
unspecified cost functional $F$ is usually expressed in terms of $\kappa$, the
stiffness of the quantum harmonic oscillator. However it is easy to express it
in terms of $\bar{\kappa}$, by using Eq. (12) rewritten in the $s$-domain:

${\kappa}(s)=\frac{\hbar^{2}}{2ms^{2}}+\frac{2m}{\gamma^{2}}[D\gamma-s{\bar{\kappa}}(s)]{\bar{\kappa}}^{\prime}(s)-\frac{m}{\gamma^{2}}{\bar{\kappa}}^{2}(s),$
(16)

where the prime denotes the derivative with respect to $s$. Then, we can write
$F$ in integral form as:
$F[{\bar{\kappa}},{\bar{\kappa}}^{\prime}]=\int_{s_{\mathrm{i}}}^{s_{\mathrm{f}}}\mathrm{d}s~{}f(s,{\bar{\kappa}}(s),{\bar{\kappa}}^{\prime}(s))$,
where $f$ is a function obtained by transforming the quantum stiffness to the
classical stiffness using Eq. (16).

The need for another functional $G[\kappa^{\prime}]$ results from the
requirement to avoid strong gradients in $\kappa(s)$. As was noted in Ref.
[38], without this term the resulting optimal protocol displays jumps, i.e.
infinite gradients, at the initial and final times. Mathematically, this is
because, in the absence of this term, the Euler-Lagrange equation is an
algebraic one, so that one cannot fix the boundary conditions on the solution,
and instead has to “stitch” them artificially as jumps. That was not an issue
for the overdamped dynamics studied in Ref. [38], because a system with
vanishing inertia remains at equilibrium when the stiffness is suddenly
changed. But here the situation is different, as the underlying problem is the
standard Schrödinger equation, which does include inertia. Hence, if the
boundary conditions are not satisfied at $t=t_{\mathrm{f}}$, then the system
will continue to evolve and deviate from the target stationary state. A way to
ensure that boundary conditions are indeed satisfied is to render the Euler-
Lagrange equation a second-order differential equation [58]. This can be
achieved by adding the following functional

$G[{\bar{\kappa}}^{\prime}]=\int_{s_{\mathrm{i}}}^{s_{\mathrm{f}}}\mathrm{d}s~{}\left|{\bar{\kappa}}^{\prime}(s)\right|^{2},$
(17)

with the corresponding Lagrange multiplier $\mu$. Note that we expressed
$G[{\bar{\kappa}}^{\prime}]$ in terms of the classical stiffness, to be
consistent with the other functionals $\Delta t$ and $F$. But limiting the
gradient of $\bar{\kappa}(s)$ also limits the gradient of $\kappa(s)$, in
virtue of Eq. quantum_protocol_s. We also recall that the boundary conditions
are those for which the variance is stationary at the boundaries, i.e.
$\dot{s}_{\mathrm{i,f}}=0$. From Eq. (11), this is equivalent to imposing that
$\bar{\kappa}_{\mathrm{i,f}}\,s_{\mathrm{i,f}}=D\gamma$.

In summary, the total functional to be minimized is $J=\Delta t+\lambda F+\mu
G$, or explicitly:

$J[{\bar{\kappa}},{\bar{\kappa}}^{\prime}]=\frac{1}{2}\int_{s_{\mathrm{i}}}^{s_{\mathrm{f}}}\mathrm{d}s~{}\frac{\gamma}{D\gamma-s{\bar{\kappa}}(s)}+\lambda\int_{s_{\mathrm{i}}}^{s_{\mathrm{f}}}\mathrm{d}s~{}f(s,{\bar{\kappa}}(s),{\bar{\kappa}}^{\prime}(s))+\mu\int_{s_{\mathrm{i}}}^{s_{\mathrm{f}}}\mathrm{d}s~{}\left|{\bar{\kappa}}^{\prime}(s)\right|^{2}.$
(18)

It is important to understand that $\lambda$ and $\mu$ play very different
roles. While $\lambda$ corresponds to the weight given to the cost functional
we want to minimize on physical grounds, $\mu$ is present only to ensure that
the equilibrium conditions are satisfied. Nevertheless, both Lagrange
multipliers have an impact on the resulting optimal solution. The Lagrangian
associated to Eq. (18) is

$L[s,{\bar{\kappa}},{\bar{\kappa}}^{\prime}]=\frac{\gamma}{D\gamma-s{\bar{\kappa}}}+\lambda
f+\mu\left|{\bar{\kappa}}^{\prime}\right|^{2},$ (19)

where we removed the factor $1/2$ from the first term, because it can be
included in the Lagrange multipliers $\lambda$ and $\mu$. The optimal protocol
is obtained by solving the Euler-Lagrange equation associated with $L$, which
reads as

$2\mu{\bar{\kappa}}^{\prime\prime}=\frac{\gamma
s}{[D\gamma-s{\bar{\kappa}}(s)]^{2}}+\lambda\frac{\partial
f}{\partial{\bar{\kappa}}}-\lambda\frac{\mathrm{d}}{\mathrm{d}s}\frac{\partial
f}{\partial{\bar{\kappa}}^{\prime}}.$ (20)

This equation is, as expected, a second-order differential equation for
$\bar{\kappa}(s)$ and the two boundary conditions
$\bar{\kappa}_{\mathrm{i,f}}=D\gamma/s_{\mathrm{i,f}}$ can thus be imposed.

##  IV Optimization results

Some protocols have already been studied in the past for the case where the
cost function is the work done on the system [47], i.e.: $F\equiv
W=\int_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}\langle\psi|\partial_{t}\hat{H}(t)|\psi\rangle\mathrm{d}t$.
It was shown that the optimal protocol is highly degenerate [23], with the
minimal work simply corresponding to the difference between the final and
initial energies of the system [34]:
$W_{\mathrm{opt}}=E_{\mathrm{f}}-E_{\mathrm{i}}$. In particular, for any
protocol that satisfies the right boundary conditions – so that the initial
and final states are both stationary – the work done on the system will be the
optimal one, irrespective of the duration of the protocol [34]. This is not
too surprising, as the quantum oscillator system is conservative, and
therefore the energy expended to go from one stationary state to another
should only depend on these states, and not on the path connecting them. For
these reasons, the work $W$ does not appear to be the pertinent cost
functional to be minimized. The advantage of the method described in the
preceding section is that it allows us to select any cost functional, and find
the corresponding protocol that minimizes it for a given duration $\Delta t$.
Here, we will consider two different functional forms of $F$ and find the
optimal protocol for each of them.

In all the forthcoming simulations, we employ units in which $\hbar=\gamma=1$,
$m=0.5$, and then $D=1$. The variance increases in time from
$s_{\mathrm{i}}=1$ to $s_{\mathrm{f}}=2$, so that the classical equilibria
correspond to $\bar{\kappa}_{\mathrm{i}}=D\gamma/s_{\mathrm{i}}=1$ and
$\bar{\kappa}_{\mathrm{f}}=D\gamma/s_{\mathrm{f}}=0.5$. The quantum
equilibrium conditions are such that:
$\kappa_{\mathrm{i}}=mD^{2}/s_{\mathrm{i}}^{2}=0.5$ and
$\kappa_{\mathrm{f}}=mD^{2}/s_{\mathrm{f}}^{2}=0.125$. The classical
relaxation time is then: $\tau_{\rm
rel}=\sqrt{m/\kappa_{\mathrm{f}}}=\omega_{\mathrm{f}}^{-1}=2$. All optimal
protocols considered in the next two subsections are such that $\Delta
t<\tau_{\rm rel}$ (except for one case where $\Delta t=2.1$), confirming that
they can outperform the STEP protocols, as discussed in Sec. II.2.

Finally, in the Appendix A, we will also treat the case of the classical
optimal protocol developed in Ref. [38], and discuss how it can be translated
into an analog quantum protocol.

###  IV.1 Cumulative energy as cost function

First, we take the integral of the energy as the cost function, which, when
divided by the total duration, corresponds to the time-averaged energy
furnished to the system, a quantity of clear physical interest, both
theoretically and for experimental applications. As the wavefunction is
supposed to be Gaussian at all times, see Eq. psi_gaussian, it is
straightforward to write the energy as

$E(t)=\langle\psi|\hat{H}(t)|\psi\rangle=\frac{m}{4s(t)}\left(\frac{1}{2}\dot{s}^{2}(t)+\frac{2s^{2}(t)\kappa(t)}{m}+2D^{2}\right),$
(21)

where $\hat{H}(t)$ is defined in Eq. schrodinger_equation. We define the
functional $F\equiv F_{E}$ as the integral of the energy. After changing
variable from $t$ to $s$ and writing $\kappa(s)$ in terms of
${\bar{\kappa}}(s)$ using Eq. quantum_protocol_s, we obtain

$F_{E}[{\bar{\kappa}},{\bar{\kappa}}^{\prime}]=\frac{m}{4\gamma}\int_{s_{\mathrm{i}}}^{s_{\mathrm{f}}}\mathrm{d}s~{}\left[\frac{D\gamma-s{\bar{\kappa}}(s)}{s}+\frac{3D^{2}\gamma^{2}-s^{2}{\bar{\kappa}}^{2}(s)}{s(D\gamma-s{\bar{\kappa}}(s))}+2s{\bar{\kappa}}^{\prime}(s)\right].$
(22)

The corresponding Lagrangian is then

$L[s,{\bar{\kappa}},{\bar{\kappa}}^{\prime}]=\frac{\gamma}{D\gamma-s{\bar{\kappa}}}+\frac{\lambda}{\gamma}\left[\frac{D\gamma-s{\bar{\kappa}}}{s}+\frac{3D^{2}\gamma^{2}-s^{2}{\bar{\kappa}}^{2}}{s(D\gamma-s{\bar{\kappa}})}+2s{\bar{\kappa}}^{\prime}\right]+\mu|{\bar{\kappa}}^{\prime}|^{2},$
(23)

where the factor $m/4$ was absorbed in the Lagrange multiplier $\lambda$. The
resulting Euler-Lagrange equation is a second-order nonlinear differential
equation for $\bar{\kappa}(s)$, and the initial and final values of the
classical protocol can be imposed according to Eq. equilibrium. The Euler-
Lagrange equation is

$2\mu\gamma\,{\bar{\kappa}}^{\prime\prime}=\frac{\gamma^{2}s+3D^{2}\gamma^{2}\lambda-s^{2}{\bar{\kappa}}^{2}\lambda}{(D\gamma-s{\bar{\kappa}})^{2}}-\frac{2s\bar{\kappa}\lambda}{D\gamma-s{\bar{\kappa}}}-3\lambda,$
(24)

and it can be solved numerically using an iterative method such as Thomas’s
algorithm [59]. Note that, had we not added the Lagrange multiplier $\mu$, Eq.
euler_lagrange_energy would be an algebraic equation, and it would be possible
to fix the correct boundary conditions.

The solutions of Eq. (24) are represented in Fig. 2, for two distinct
situations: a case where $\mu$ is kept fixed (left panels) and a case where
$\lambda$ is kept fixed (right panels). For the first case ($\mu=0.1$ fixed),
three solutions, corresponding to $\lambda=0.01$, $0.10$, and $1.00$ are
represented, thus showing the impact of varying the Lagrange multiplier
associated with the cost function $F_{E}$. The classical protocols are
depicted in the upper panels of the figure, and the associated quantum
protocols in the lower panels. The equilibrium conditions at the initial and
final times are fulfilled, as expected, and all protocols are continuous
functions of the variance $s$. Moreover, as $\mu$ is constant, the derivatives
of each protocol at the boundaries of the $s$ domain are the same. Each
classical protocol exhibits a minimal value, which decreases as $\lambda$
increases while its quantum associated protocol has both minimum and maximum
values, whose amplitude increases as $\lambda$ increases. Hence, both
protocols can become negative, corresponding to a repulsive harmonic
potential, when $\lambda$ is large enough. This feature, while notable, is not
necessarily problematic and has been previously reported in the literature
[34, 60, 23].

The right panels of Fig. 2 show the solutions when $\lambda=1$ is kept
constant while $\mu$ varies, taking the values $0.1$, $0.5$ and $1.00$. The
shapes of the curves are similar to the preceding case, but it is notable that
the larger $\mu$, the smaller the gradient of the protocol around
$s_{\mathrm{i}}$ and $s_{\mathrm{f}}$, as expected because the functional $G$
limits the derivative of $\bar{\kappa}(s)$. Moreover, the range of values
taken by the protocols increases as $\mu$ decreases, and they can become
negative for small values of $\mu$. Caution should be taken for even smaller
values of $\mu$, which lead to large values of the quantum protocol, both
positive and negative. The same remark can also be made for very large values
of $\lambda$.

Figure 2: Energy-optimal protocols in the $s$-domain, where the variance
increases from $s_{\mathrm{i}}=1$ to $s_{\mathrm{f}}=2$. Left panels:
Classical protocols ${\bar{\kappa}}(s)$ (a) and quantum protocols $\kappa(s)$
(c), for a fixed value of $\mu=0.10$, and $\lambda=0.01$ (solid orange lines),
$0.10$ (dashed grey lines) and $1.00$ (dot-dashed black lines). Right panels:
Classical protocols ${\bar{\kappa}}(s)$ (b) and quantum protocols $\kappa(s)$
(d) for a fixed value of $\lambda=1$, and $\mu=0.1$ (solid orange lines),
$0.50$ (dashed grey lines) and $1.00$ (dot-dashed black lines). Note that the
quantum protocol can become negative for sufficiently large values of
$\lambda$ or sufficiently small values of $\mu$.

All the above results were given in the $s$-domain, i.e. as a function of the
variance. In order to go back to the time representation, one needs to solve
Eq. (11) for $t$:

$t(s)=t_{\mathrm{i}}+\frac{\gamma}{2}\int_{s_{\mathrm{i}}}^{s}\frac{\mathrm{d}y}{D\gamma-s\bar{\kappa}(y)},$
(25)

which can also be used to obtain the total duration $\Delta
t=t_{\mathrm{f}}-t_{\mathrm{i}}$. The time-dependent solutions
$\bar{\kappa}(t)={\bar{\kappa}}(s(t))$ and $\kappa(t)={\kappa}(s(t))$ are
depicted in Fig. 3, together with the time evolution of the energy of the
system, whose integral is the cost functional $F_{E}$ associated to $\lambda$.
The curves corresponds to the same values of $\lambda$ and $\mu$ as in Fig. 2,
with total durations $\Delta t=0.95$, $0.87$ and $0.82$ (left panels, $\mu$
fixed) and $\Delta t=0.82$, $1.27$ and $1.58$ (right panels, $\lambda$ fixed).
We recall that here the classical relaxation time is equal to $\tau_{\rm
rel}=\sqrt{m/\kappa_{\mathrm{f}}}=\omega_{\mathrm{f}}^{-1}=2$, and therefore
$\Delta t<\tau_{\rm rel}$ everywhere.

When the Lagrange multiplier $\lambda$, corresponding to the functional
$F_{E}=\int_{t_{\mathrm{i}}}^{t_{\mathrm{i}}}\mathrm{d}t\,E(t)$, is increased,
then the energy integral decreases, as seen in the inset of Fig. 3(e). This is
natural, as the role of the $\lambda$-term is precisely to limit the value of
the cumulative energy over time. However, we also observe that the time
duration $\Delta t$ decreases for increasing $\lambda$, while the opposite
occurred for classical systems [38]. This may seem surprising, as increasing
$\lambda$ should reduce the value of the functional $F_{E}$ while increasing
$\Delta t$ (energy-time trade-off). But, in contrast to the classical
overdamped case, here there are two Lagrange multipliers, so the trade-off is
actually among _three_ functionals, $F_{E}$, $\Delta t$, and $G$, which makes
the whole situation more complex.

Conversely, the time duration increases as $\mu$ increases, which is more in
line with a trade-off between $G$ and $\Delta t$. At the same time, the
cumulative energy over time increases with increasing $\mu$, as seen in the
inset of Fig. 3(f).

For all cases, the behavior of the variance appears very similar when plotted
against time normalized to the duration $\Delta t$. As expected, $s(t)$ is
strictly increasing in time and has vanishing derivatives close to the initial
and final times. The energy $E(t)$ follows closely the evolution of
$\kappa(t)$, which indicates that the potential energy is the preponderant
contribution to the total energy of the system. It is important to note that,
when $\mu$ becomes small enough, the energy might become negative, as seen in
Fig. 3(f), although its time integral remains strictly positive, ensuring that
the functional $F_{E}$ is positive.

Figure 3: Energy-optimal protocols in the time domain, where the variance
increases from $s_{\mathrm{i}}=1$ to $s_{\mathrm{f}}=2$. The time is
normalized to the total duration of the protocol $\Delta t$, which is
different for each pair of Lagrange multipliers $(\lambda,\mu)$. Left panels:
variance $s(t)$ (a), quantum protocols $\kappa(t)$ (c), and instantaneous
energy $E(t)$ (e), for a fixed value of $\mu=0.10$, and $\lambda=0.01$ (solid
orange lines), $0.50$ (dashed grey lines) and $1.00$ (dot-dashed black lines).
These values correspond to time durations: $\Delta t=0.95$, $0.87$ and $0.82$.
Right panels: variance $s(t)$ (b), quantum protocols $\kappa(t)$ (d), and
instantaneous energy $E(t)$ (f), for a fixed value of $\lambda=1$, and
$\mu=0.1$ (solid orange lines), $0.50$ (dashed grey lines) and $1.00$ (dot-
dashed black lines). These values correspond to time durations: $\Delta
t=0.82$, $1.27$ and $1.58$. Note that, although the various protocols are
quite different, the behavior of the variance is almost identical when time is
scaled to the total duration $\Delta t$. The boundary values of the quantum
protocols correspond to the conditions:
$s_{\mathrm{i,f}}^{2}\kappa_{\mathrm{i,f}}=mD^{2}$. The energy $E(t)$ is quite
similar to the quantum stiffness $\kappa(t)$, indicating that the potential
energy is the predominant contribution to the total energy of the system. The
insets in (e) and (f) show the cumulative energy over time for the three
protocols represented in the same panels.

It is interesting to compare our results with other protocols, such as the
ones that minimizes the work done on the system, proposed by Chen et al. [34].
This protocol is obtained from the Ermakov equation ermakov rewritten in terms
of the scale factor $q(t)=\sqrt{s(t)/s_{\mathrm{i}}}$, which reads as:
$\ddot{q}+\frac{\kappa(t)}{m}q=\kappa_{\mathrm{i}}/q^{3}$. This equation can
be inverted to obtain the stiffness as a function of the scale factor:
$\kappa=\kappa_{\mathrm{i}}/q^{4}-m\ddot{q}/q$. The protocol is then obtained
by imposing a suitable temporal profile for $q(t)$. The authors of [34] chose
a polynomial of fifth degree: $q(t)=(a-1)(6T^{5}-15T^{4}+10T^{3}+1)$, where
$T=t/t_{\mathrm{f}}$ and $a=(\kappa_{\mathrm{i}}/\kappa_{\mathrm{f}})^{1/4}$,
so that $q(t_{\mathrm{i}})=1$, $q(t_{\mathrm{f}})=a$, and its first and second
derivatives are zero at the initial and final times. We recall that imposing
such boundary conditions is sufficient to minimize the external work, which
becomes equal to the difference between the final and initial energies.

To compare Chen’s polynomial protocol to our optimal protocol, we can fix a
value of $\lambda$, for instance $\lambda=10$, and vary the value of $\mu$.
For each protocol, we compute the energy integral
$F_{E}=\int_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}E(t)\mathrm{d}t$, as in Eq.
Fintegral, and the time duration $\Delta t$, Eq. tf. The results are shown in
Fig. 4 for both our protocol (black squares) and Chen’s (triangles). It
appears that our protocol performs better, in terms of the integral of the
energy, than the polynomial protocol for small values of $\Delta t$ (gray area
in Fig. 4). For larger $\Delta t$, the opposite is true, which can be
explained by noticing that the long-duration regime corresponds to large
values of $\mu$. Indeed, the total functional $J=\Delta t+\lambda F+\mu G$,
see Eq. J, can be minimized by minimizing either $F$ or $G$. If the Lagrange
multiplier $\mu$ is large, then $G$ dominates, and the variational procedure
will end up minimizing $J$ by essentially minimizing $G$ instead of $F$.
However, this is not a significant constraint for our purposes, since our aim
is to minimize the energy integral for short durations, which correspond to
small values of $\mu$. In the inset of the same figure 4, we also plot the
cumulative energy over time divided by $\Delta t$, which represents the
_average_ energy of the system. Again, our optimal protocol is the one that
minimizes the averaged energy for short durations.

Figure 4: Cumulative energy of the system over time
$\int_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}\mathrm{d}t\,E(t)$ as a function of the
total time duration $\Delta t$ of the protocol. The black squares corresponds
to the optimal protocol, solution of the Euler-Lagrange equation
euler_lagrange_energy, while the orange triangles correspond to the polynomial
protocol of Ref. [34]. In the gray shaded region, the optimal protocols
display lower cumulative energy than the polynomial ones. The different times
durations are obtained by varying the Lagrange multiplier $\mu$, while keeping
$\lambda=10$ fixed. The dashed lines are empirical fits to the numerical data.
The inset shows the same plots for the time-averaged energy, given by the
integral of the energy divided by the time duration.

###  IV.2 Dynamical phase as cost function

As a second example, we consider the following functional to be optimized:
$F_{\alpha}=\int_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}\mathrm{d}t~{}\alpha^{2}(t)$,
where $\alpha(t)$ is the dynamical phase given in Eq. (4). Before carrying out
the optimization procedure, we provide some justification about the importance
and meaning of such functional. According to Eq. (5), $\alpha(t)$ is
proportional to the time derivative of the variance. During an adiabatic
process, ${\dot{s}}(t)\approx 0$, because the process is infinitely slow, and
therefore $\alpha(t)\approx 0$. Hence, the integral
$F_{\alpha}=\int_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}\mathrm{d}t~{}\alpha^{2}(t)$
represents the departure from adiabaticity, and by minimizing such quantity we
therefore minimize the “distance” of the optimal protocol from an adiabatic
one.

A visual way to represent the dynamical phase is to consider the Wigner
function $W(x,p,t)$ [61] corresponding to the Gaussian wave packet of Eq. (4).
The Wigner function is a quantum pseudo-probability density in the phase space
$(x,p)$. As the wavefunction is the exponential of a quadratic polynomial, its
Wigner function is non-negative, and can be written as [62]

$W(x,p,t)=A\exp\left(-\frac{x^{2}}{2s(t)}-\frac{2s(t)}{\hbar^{2}}\left[p-2\alpha(t)\hbar
x\right]^{2}\right),$ (26)

where $A$ is a normalization constant ensuring that
$\int_{\mathbb{R}^{2}}W\,dx\,dp=1$. Note that $W(x,p,t)$ peaks around the
straight line in phase space $p_{0}(x,t)=2\alpha(t)\hbar
x=\hbar\partial_{x}S(x,t)$, where $S(x,t)$ is the total phase of the Gaussian
wavefunction (4). When $\alpha=0$, the Wigner function is symmetric with
respect to both the position and momentum axes; when $\alpha\neq 0$, it is
tilted of an angle $\theta$ such that
$\tan\theta=2\alpha\hbar/(m\omega_{\mathrm{i}})$ (obtained by expressing $p$
and $x$ in our normalized units). During an adiabatic protocol, $W(x,p,t)$
remains symmetric and only changes its aspect ratio. For instance, during an
expansion ($s_{\mathrm{f}}>s_{\mathrm{i}}$), it becomes wider in $x$ and
narrower in $p$. Instead, during a faster-than-adiabatic process, $W$ first
gets tilted of an angle $\theta$, then expands by increasing its spatial
variance, and finally recovers a symmetric shape with $\alpha=0$. This is
illustrated in Fig. 5, where we show the phase space portraits for the optimal
and adiabatic protocols at several instants in time. It is clear that the
acceleration in the optimal protocol is achieved by tilting the Wigner
function of a certain angle before coming back to a symmetric configuration at
the end of the protocol.

Figure 5: Wigner functions $W(x,p,t)$ at several instants indicated on the
vertical time axis. The left panels refer to the phase-optimal protocol (with
$\lambda=\mu=1$), while the right panels correspond to an adiabatic protocol.
The variance grows from $s_{\mathrm{i}}=1$ to $s_{\mathrm{f}}=2$. The Wigner
function of the adiabatic protocol remains symmetric with respect to both the
$x$ and $p$ axes, and slowly reduces its width in $p$ while increasing its
width in $x$. In contrast, the phase-optimal Wigner function first becomes
elongated along the straight line $p=2\alpha\hbar x$, before reaching the same
final state in a time much shorter than the adiabatic protocol.

Turning back to the minimization procedure, the $F_{\alpha}$ functional is
given by, after changing variable from $t$ to $s$ and expressing the integrand
in terms of ${\bar{\kappa}}(s)$,

$F_{\alpha}[s,{\bar{\kappa}}]=\int_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}\mathrm{d}t~{}\alpha^{2}(t)=\frac{m^{2}}{8\gamma\hbar^{2}}\int_{s_{\mathrm{i}}}^{s_{\mathrm{f}}}\mathrm{d}s~{}\frac{D\gamma-s{\bar{\kappa}}(s)}{s^{2}},$
(27)

where we have used Eq. sdot. The total Lagrangian is

$L[s,{\bar{\kappa}},{\bar{\kappa}}^{\prime}]=\frac{\gamma}{D\gamma-s{\bar{\kappa}}(s)}+\lambda\frac{m^{2}}{8\gamma\hbar^{2}}\frac{D\gamma-s{\bar{\kappa}}(s)}{s^{2}}+\mu{\bar{\kappa}}^{\prime
2},$ (28)

leading to the following Euler-Lagrange equation:

$2\mu\,{\bar{\kappa}}^{\prime\prime}=\frac{\gamma
s}{(D\gamma-s{\bar{\kappa}})^{2}}-\frac{m^{2}\lambda}{8\gamma\hbar^{2}s}.$
(29)

Again, this is a second-order non-linear ordinary differential equation, whose
initial and final conditions ${\bar{\kappa}}_{\mathrm{i,f}}$ can be imposed to
ensure the equilibrium conditions.

We use the same parameters as in Sec. IV.1, notably $s_{\mathrm{i}}=1$ and
$s_{\mathrm{f}}=2$. In Fig. 6 we represent the classical and quantum protocols
for two cases: (i) a fixed value of $\mu=0.10$ and $\lambda$ varying in the
range $\lambda\in[0.01,0.10,1.00]$ (left panels), and (ii) a fixed value of
$\lambda=1.00$ and $\mu$ varying in the range $\mu\in[0.10,0.50,1.00]$ (right
panels). These correspond to the following time durations: $\Delta t=0.95$,
$0.97$, and $1.00$ (fixed $\mu$, left) and $\Delta t=1.00$, $1.67$, and $2.10$
(fixed $\lambda$, right). We recall that the classical relaxation time is
equal to $\tau_{\rm
rel}=\sqrt{m/\kappa_{\mathrm{f}}}=\omega_{\mathrm{f}}^{-1}=2$. The general
behaviors of the protocols look similar to those obtained in Sec. IV.1 for the
energy cost function, probably because the solutions are somewhat dominated by
the $\mu$ term. A notable difference is that the time duration is less
sensitive to the values of the Lagrange multipliers. Also, the duration
increases as $\lambda$ increases, in contrast to what was observed for the
energy-optimal protocols of Sec. IV.1.

Figure 6: Phase-optimal protocols in the $s$-domain, where the variance
increases from $s_{\mathrm{i}}=1$ to $s_{\mathrm{f}}=2$. Left panels:
Classical protocols ${\bar{\kappa}}(s)$ (a) and quantum protocols $\kappa(s)$
(c), for a fixed value of $\mu=0.10$ and $\lambda=0.01$ (solid orange lines),
$0.50$ (dashed grey lines) and $1.00$ (dot-dashed black lines). Right panels:
Classical protocols ${\bar{\kappa}}(s)$ (b) and quantum protocols $\kappa(s)$
(d) for a fixed value of $\lambda=1$ and for $\mu=0.1$ (solid orange lines),
$0.50$ (dashed grey lines) and $1.00$ (dot-dashed black lines). Note that the
quantum protocol can become negative for sufficiently large values of
$\lambda$ or sufficiently small values of $\mu$.

The temporal evolution of the variance, the quantum stiffness, and the
dynamical phase are depicted in Fig. 7, where the time has been normalized to
the total duration of each protocol. In accordance with the above discussion
on the Wigner functions, the dynamical phase vanishes at the initial and final
times and is maximal around $\Delta t/2$. We also note that the maximum of
$\alpha^{2}(t)$ and its time integral are smaller for the longer time
durations of the protocols, i.e., those protocols that are closer to
adiabaticity, in agreement with our earlier interpretation of the functional
$F_{\alpha}$ as quantifying the departure from adiabaticity.

Figure 7: Phase-optimal protocols in the time domain, where the variance
increases from $s_{\mathrm{i}}=1$ to $s_{\mathrm{f}}=2$. The time is
normalized to the total duration of the protocol $\Delta t$, which is
different for each pair of Lagrange multipliers $(\lambda,\mu)$. Left panels:
variance $s(t)$ (a), quantum protocols $\kappa(t)$ (c), and instantaneous
energy $E(t)$ (e), for a fixed value of $\mu=0.10$ and $\lambda=0.01$ (solid
orange lines), $0.50$ (dashed grey lines) and $1.00$ (dot-dashed black lines).
These values correspond to time durations: $\Delta t=0.95$, $0.97$ and $1.00$.
Right panels: variance $s(t)$ (b), quantum protocols $\kappa(t)$ (d), and
instantaneous energy $E(t)$ (f), for a fixed value of $\lambda=1$ and
$\mu=0.1$ (solid orange lines), $0.50$ (dashed grey lines) and $1.00$ (dot-
dashed black lines). These values correspond to time durations: $\Delta
t=1.00$, $1.67$ and $2.10$. The boundary values of the quantum protocols
correspond to the conditions:
$s_{\mathrm{i,f}}^{2}\kappa_{\mathrm{i,f}}=mD^{2}$. The insets in (e) and (f)
show the integral of $\alpha^{2}(t)$ over time for the three protocols
represented in the same panels. Note that the shortest durations correspond to
the largest values of the phase $\alpha(t)$.

In Fig. 8, we compare our results to the same polynomial protocol described in
Sec. IV.1, by representing the values of the functional $F_{\alpha}$ for
different time durations $\Delta t$ for both protocols. Here, we fix
$\lambda=10$ and vary $\mu$ from $0.001$ to $2.50$, in order to obtain
different time durations for the optimal protocol. The optimal protocol
displays significantly lower values of the cost function $F_{\alpha}$ for
short durations, which is the regime of interest. For longer durations, the
two protocols behave very similarly in this respect.

The above result is important, inasmuch as it shows that, for the same time
duration $\Delta t$, our optimal protocol is closer to adiabaticity than the
polynomial protocol of Ref. [34]. Both protocols constitute shortcuts to
adiabaticity, but the one we propose here is, in a precise sense,
“adiabatically optimal”.

Figure 8: Integral of the square of the dynamical phase
$\int_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}\mathrm{d}t\,\alpha^{2}(t)$ as a
function of the total time duration $\Delta t$ of the protocol. The black
squares corresponds to the optimal protocol, solution of the Euler-Lagrange
equation euler-lagrange_alpha, while the orange triangles correspond to the
polynomial protocol of Ref. [34]. In the grey shaded region the optimal
protocols perform better than the polynomial ones. The different times
durations are obtained by varying the Lagrange multiplier $\mu$ (from $0.001$
to $2.5$), while keeping $\lambda=10$ fixed. The dashed lines are empirical
fits to the numerical data.

##  V Conclusion

To bring a classical or quantum system from a stationary state to another, the
simplest strategy is to vary an external parameter very slowly, i.e.
adiabatically. By doing so, the system will be at steady state at each instant
of the evolution, but the transition will take an infinite time. The growing
field of research known as “shortcuts to adiabaticity” tries to accomplish the
same transition within a finite duration.

In the present work, we proposed a new strategy to achieve faster-than-
adiabatic transitions. The main idea is based on Nelson’s representation of
quantum mechanics as a classical stochastic process. In the case of a time-
dependent harmonic oscillator this quantum-classical analogy is particularly
simple and fruitful. Using Nelson’s procedure, the Schrödinger equation is
rewritten as an overdamped Langevin equation with a linear harmonic force of
stiffness $\bar{\kappa}(t)$, which is related to the stiffness of the quantum
oscillator $\kappa(t)$ through Eq. quantum_protocol. Thanks to this
mathematical analogy, it is possible to translate the classical protocols
developed for an overdamped oscillator into quantum protocols for a system
with finite inertia.

In particular, we utilized our experience on optimal classical protocols to
devise quantum protocols that minimize both the time duration and some other
arbitrary cost function. For instance, the cost function can be the cumulative
energy over time. Even more interestingly, we showed that minimizing the
dynamical phase of the wavefunction (again, together with the time duration)
amounts to minimizing the distance of the protocol from an adiabatic one.
Hence, we could devise a family of protocol that are “adiabatically optimal”:
for a given finite duration $\Delta t$, they are as close as possible to an
adiabatic (i.e., infinitely slow) process.

The proposed method is rather versatile, inasmuch as the cost functional to be
minimized can be chosen at will. Nevertheless, some functionals may lead to
complicated Euler-Lagrange equations, which are difficult to solve
numerically. This occurs because the cost functional $F$ that has a physical
relevance is expressed in terms of the quantum stiffness $\kappa$ and its time
derivative, but must be rewritten in terms of the classical stiffness
$\bar{\kappa}$ before performing the minimization procedure. This can
transform relatively a simple functional $F[\kappa,\dot{\kappa}]$ into a
rather complicated functional of $\bar{\kappa}$ and $\dot{\bar{\kappa}}$.

The original method proposed in this work, although limited here to the ground
state of the harmonic potential, opens up many possible avenues for future
research. For instance, as shown in [34], the same procedure also works for
transitioning an excited state of the harmonic oscillator $\psi_{n}(x)$, which
transforms according to the scale factor defined earlier as:
$q(t)=\sqrt{s(t)/s_{\mathrm{i}}}$. In contrast, for anharmonic systems, the
Gaussian wavefunction is no longer an exact solution. However, a Gaussian
ansatz can be used as an approximate solution, leading to a modified Ermakov
equation, similar to Eq. ermakov, which can be used as the basis for a
generalization of the present theory. Similarly, one may consider many-body
problems in the mean field approximation, either with contact interactions
(Gross-Pitaevskii equation for Bose-Einstein condensates) [63] or Coulomb
interactions (Schrödinger-Poisson equations for a quantum electron gas) [64].
These are nonlinear Schrödinger equations that are amenable to the Nelson
representation utilized in the present work. For weak coupling, the exact
solution can be approximated by a Gaussian wavefunction, leading again to a
modified Ermakov equation. For the Gross-Pitaevskii case, an interesting goal
would be to control the system by modulating the scattering length, which can
be done experimentally by varying an external magnetic field. A further avenue
for future research is to extend the present method to the case of an open
quantum system in contact with a bath at finite temperature. To do this, it
would be necessary to extend Nelson’s formalism, for instance by adding a
thermal noise to Eq. nelson_equation, for which several attempts have already
been proposed [65, 66].

###### Acknowledgements.

We thank Dr. Cyriaque Genet for many useful discussions. R. G. acknowledges
support from the Mark Ratner Institute for Single Molecule Chemistry at Tel
Aviv University.

##  Appendix A Classical work-optimal protocol

In order to illustrate the quantum-classical analogy, and to show the
importance of adding the functional $G[{\bar{\kappa}}^{\prime}]$ to the total
functional to be minimized, we propose to study a simple and well-documented
case: the classical work-optimal protocol developed in [38]. A Brownian
particle is trapped in a harmonic potential whose stiffness $\bar{\kappa}(t)$
can vary in time. The particle is immersed in a fluid of damping coefficient
$\gamma$ and thermal diffusion coefficient $D=k_{B}T/\gamma$, where $T$ is the
temperature of the fluid. In Ref. [38], the objective was to find the optimal
manner to vary $\bar{\kappa}(t)$ so that both the duration of the transition
and the work done on the system are minimal. The position of the Brownian
particle follows a Gaussian probability distribution of variance $s(t)$, which
obeys Eq. (11). Changing the independent variable from the time $t$ to the
variance $s$, we can write the time duration $\Delta t$ as in Eq. (15) and the
work done on the system as [67, 68, 38]: $W={1\over
2}\int_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}\mathrm{d}t\,\dot{\bar{\kappa}}(t)\langle
x^{2}\rangle=-{1\over
2}\int_{s_{\mathrm{i}}}^{s_{\mathrm{f}}}\mathrm{d}s~{}{\bar{\kappa}}(s)+{1\over
2}(s_{\mathrm{f}}\bar{\kappa}_{\mathrm{f}}-s_{\mathrm{i}}\bar{\kappa}_{\mathrm{i}})$.
Hence, the functional to be minimized is

$J[{\bar{\kappa}}]=\int_{s_{\mathrm{i}}}^{s_{\mathrm{f}}}\mathrm{d}s~{}\frac{\gamma}{D\gamma-s{\bar{\kappa}}(s)}-\lambda\int_{s_{\mathrm{i}}}^{s_{\mathrm{f}}}\mathrm{d}s~{}{\bar{\kappa}}(s),$
(30)

with $\lambda$ a Lagrange multiplier. It is straightforward to find the
solution of the associated Euler-Lagrange equation [38]:

$s{\bar{\kappa}}(s)=D\gamma\mp\sqrt{\frac{\gamma s}{\lambda}},$ (31)

where the upper and lower signs correspond to the cases of the compression or
expansion, respectively. Note that, as the Euler-Lagrange equation is purely
algebraic, the boundary conditions cannot be fixed at will. Hence, in the
classical case, the solution class-overdamped must be supplemented by “jumps”
at the initial and final times [38].

The associated quantum protocol is obtained using Eq. (16), yielding:

${\kappa}(s)=mD^{2}/s^{2},$ (32)

which is independent of $\lambda$. Surprisingly, this solution coincides with
the equilibrium solution equilibrium, which means that it represents an
adiabatic process for the quantum oscillator, albeit with a finite duration
that can be obtained from Eq. (15): $\Delta
t=\sqrt{\gamma\lambda}(\sqrt{s_{\mathrm{i}}}\pm\sqrt{s_{\mathrm{f}}})$. The
variance can be computed solving Eq. (11), yielding

$s(t)=(\sqrt{s_{\mathrm{i}}}\pm t/\sqrt{\gamma\lambda})^{2},$ (33)

and the classical and quantum protocols are, respectively,

$\bar{\kappa}(t)=\frac{D\gamma+\sqrt{\gamma s_{\mathrm{i}}/\lambda}\pm
t/\lambda}{(\sqrt{s_{\mathrm{i}}}\pm
t/\sqrt{\gamma\lambda})^{2}}\,;\quad\kappa(t)=\frac{mD^{2}}{(\sqrt{s_{\mathrm{i}}}\pm
t/\sqrt{\gamma\lambda})^{4}}.$ (34)

Note that, if the time is expressed in units of total time duration $\Delta
t$, then $\kappa(t/\Delta t)$ is indeed independent of the Lagrange multiplier
$\lambda$.

As mentioned above, the quantum solution is at equilibrium at each instant.
However, since the classical equilibrium conditions do not hold,
$s_{\mathrm{i,f}}\bar{\kappa}_{\mathrm{i,f}}\neq D\gamma$, the time derivative
of the variance at the initial and final times is not zero. From the point of
view of the classical system this is not a problem, because the overdamped
dynamics displays no inertia, so that one can change the stiffness abruptly to
bring it to the equilibrium value compatible with $s_{\mathrm{f}}$ [38]. But
for the (inertial) Schrödinger equation, if $\dot{s}\neq 0$ at
$t=t_{\mathrm{f}}$, then the system will continue to evolve in time after
$t_{\mathrm{f}}$.

It is therefore necessary to ensure that $D\gamma-s\bar{\kappa}=0$, both at
$t=t_{\mathrm{i}}$ and $t=t_{\mathrm{f}}$. In order to do that, the Euler-
Lagrangian equation should be a second-order differential equation, instead of
an algebraic one as was the case for the functional of Eq. Jclass. This is the
reason why one needs to add a second functional of the form
$G[{\bar{\kappa}}^{\prime}]=\int_{s_{\mathrm{i}}}^{s_{\mathrm{f}}}\mathrm{d}s~{}\left|{\bar{\kappa}}^{\prime}(s)\right|^{2}$,
associated with the Lagrange multiplier $\mu$, which leads to the following
Euler-Lagrange equations

$2\mu\,{\bar{\kappa}}^{\prime\prime}(s)=\frac{\gamma
s}{[D\gamma-s{\bar{\kappa}}(s)]^{2}}-\lambda.$ (35)

This being a second-order differential equation, the boundary conditions at
$t_{\mathrm{i}}$ and $t_{\mathrm{f}}$ can be imposed consistently with the
requirement that: $s_{\mathrm{i,f}}\bar{\kappa}_{\mathrm{i,f}}=D\gamma$.

The various results, both for the analytical solution class-overdamped-kappa-
classical (with jumps) and the smooth numerical solution of Eq. EL-classical,
are presented in Fig. 9 for the variance $s(t)$ (top panels), the classical
protocols $\bar{\kappa}(t)$ (middle panels), and the quantum protocols
$\kappa(t)$ (bottom panels). In the left panels, we take $\mu=0$ (no
smoothing) and vary $\lambda$ from 0.1 to 10, while the right panels keep
$\lambda=1$ fixed, while $\mu$ varies from 0.01 to 0.1. It is clear (top left
and bottom left panels) that the variance $s(t)$ and the quantum protocol
$\kappa(t)$ do not depend on $\lambda$, as suggested by Eqs. ks-classical and
kappa-classical. Instead, the classical protocol depends on $\lambda$, in
accordance with Eq. class-overdamped. It is also evident that the classical
protocol displays discontinuities at the initial and final times (which
disappear in the adiabatic limit $\lambda\to\infty$), while the quantum
protocol does not.

In the protocols with $\mu>0$ (right panels of Fig. 9), the variance varies
smoothly at the initial and final times, as expected, and the classical
protocols display no discontinuities at the boundaries. Hence, the equilibrium
conditions are fulfilled and the system’s variance will remain at its final
value at the end of the transition. Finally, we note that the quantum protocol
develops large spikes near $t_{\mathrm{i}}$ and $t_{\mathrm{f}}$ for small
values of $\mu$. Hence, although it must converge to the non-smoothed one for
$\mu\to 0$, it appears to do so in a singular way, displaying large positive
and negative spikes at the boundaries.

Figure 9: Classical work-optimal protocols. The left panels represent, from
top to bottom, the time evolution of the variance (a), the classical protocol
(c), and the quantum protocol (e), for the case $\mu=0$, which displays jumps
at the initial and final times (the initial and final conditions are
represented by red dots). The different curves are obtained with $\lambda=0.1$
(orange solid lines), $1.00$ (grey dashed lines) and $10.00$ (black dash-
dotted lines). Note that the variance (a) and the quantum protocol (e) do not
depend on $\lambda$. The right panels represent, from top to bottom, the time
evolution of the variance (b), the classical protocol (d), and the quantum
protocol (f), for fixed $\lambda=1$, and finite values of $\mu$: $\mu=0.10$
(orange solid lines), $0.05$ (grey solid lines) and $0.01$ (black solid
lines). Note that these finite-$\mu$ protocols are continuous and smooth at
the initial and final times. For comparison, the dashed grey line represents
the classical (discontinuous) protocol with $\mu=0$.

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