The Time-Dependent Jastrow Ansatz Exact Quantum Dynamics Shortcuts to Adiabaticity and Quantum Quenches in Strongly-Correlated Many-Body Systems Jing Yang1and Adolfo del Campo1 2

2025-05-06 0 0 580.61KB 23 页 10玖币

侵权投诉

The Time-Dependent Jastrow Ansatz: Exact Quantum Dynamics, Shortcuts to Adiabaticity, and

Quantum Quenches in Strongly-Correlated Many-Body Systems

Jing Yang 1, ∗and Adolfo del Campo 1, 2, †

1Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg

2Donostia International Physics Center, E-20018 San Sebasti´an, Spain

The description of strongly correlated quantum many-body systems far from equilibrium presents a funda-

mental challenge due to the vast amount of information it requires. We introduce a generalization of the Jastrow

ansatz for time-dependent wavefunctions that oﬀers an eﬃcient and exact description of the time evolution of

various strongly correlated systems. Previously known exact solutions are characterized by scale invariance,

enforcing self-similar evolution of local correlations, such as the spatial density. However, we demonstrate that

a complex-valued time-dependent Jastrow ansatz (TDJA) is not restricted to scale invariance and can describe a

broader class of dynamical processes lacking this symmetry. The associated time evolution is equivalent to the

implementation of a shortcut to adiabaticity (STA) via counterdiabatic driving along a continuous manifold of

quantum states described by a real-valued TDJA, providing a framework for engineering exact STA in strongly

correlated many-body quantum systems. We illustrate our ﬁndings in systems with inverse-square interactions,

such as the Calogero-Sutherland and hyperbolic models, supplemented with pairwise logarithmic interactions,

as well as in the long-range Lieb-Liniger model, where bosons experience both contact and Coulomb interac-

tions in one dimension. Our results enable the study of quench dynamics in all these models and serve as a

benchmark for numerical and quantum simulations of nonequilibrium strongly correlated systems with contin-

uous variables.

I. INTRODUCTION

The nonequilibrium dynamics of isolated quantum many-

body systems give rise to rich and intriguing physics, en-

compassing phenomena such as quantum chaos, many-body

localization, and thermalization [1,2]. Understanding these

phenomena becomes particularly challenging in the presence

of strong interactions, where perturbative approaches fail. In

such cases, numerical and analytical techniques are essen-

tial, with notable examples including exact diagonalization,

the density-matrix renormalization group [3], and quantum

Monte Carlo algorithms.

In one spatial dimension [4,5], additional methods are

available for studying strongly correlated systems. For in-

stance, the Bethe ansatz and the quantum inverse scatter-

ing method allow for the exact solution of certain integrable

many-body quantum systems [6–8]. However, even when the

exact eigenstates are known, evaluating correlation functions

remains a signiﬁcant challenge, both in thermal equilibrium

and, even more so, in nonequilibrium settings [9,10]. Further-

more, numerical techniques such as the density-matrix renor-

malization group and quantum Monte Carlo algorithms face

accuracy limitations when simulating long-time unitary dy-

namics. These diﬃculties become particularly pronounced in

systems with continuous variables.

Recent advancements in quantum simulation have made it

possible to experimentally realize strongly interacting mod-

els, fulﬁlling Feynman’s vision of quantum emulation [11]. A

paradigmatic platform for analog quantum simulation is pro-

vided by ultracold gases [12]. For example, Olshanii demon-

∗jing.yang@su.se; Present Address: Nordita, KTH Royal Institute of Tech-

nology and Stockholm University, Hannes Alfv´

ens v¨

ag 12, 106 91 Stock-

holm, Sweden.

†adolfo.delcampo@uni.lu

strated that ultracold bosons interacting via s-wave scattering

are eﬀectively described by the celebrated Lieb-Liniger (LL)

model [13,14] when conﬁned in a tight waveguide [15].

Moreover, the implementation of these models in ultracold

atomic systems enables precise control over interactions, al-

lowing them to be tuned from the strongly attractive to the

strongly repulsive regimes via Feshbach resonance. This tun-

ability has made it possible to explore the quench dynamics

of many-body systems in the laboratory, such as those in-

duced by an interaction quench or a change in conﬁnement

[4,10,16,17]—a regime where strong correlations pose sig-

niﬁcant theoretical challenges [18–24].

Further progress in quantum simulation has been driven by

the digital approach, where the dynamics of interest are ap-

proximated using a quantum circuit. This paradigm is un-

der extensive investigation, with applications spanning con-

densed matter physics [25,26], quantum ﬁeld theory [27],

and quantum chemistry [28]. Additionally, hybrid analog-

digital approaches oﬀer an alternative framework for ex-

ploring nonequilibrium quantum phenomena [29]. However,

progress in quantum simulation is constrained by the scarcity

of analytical and exact results for the nonequilibrium dynam-

ics of strongly correlated many-body systems [10,30]. These

results are not only valuable from a fundamental perspective

but also serve as crucial benchmarks for quantum simulation

algorithms and provide insights into complex experimental

observations.

Many interacting many-body systems of interest exhibit

strong correlations in their ground state [31–33]. One of the

simplest wavefunctions that captures these ground-state cor-

relations is the (Bijl-Dingle-) Jastrow ansatz [34–36], which

assumes that the ground-state wavefunction can be expressed

as a product of pair functions that depend only on the inter-

particle spacing. While generally considered an approximate

wavefunction for many-body quantum systems, the Jastrow

ansatz is particularly useful in describing perturbative expan-

arXiv:2210.14937v2 [quant-ph] 14 Mar 2025

sions at low densities [37] and is widely employed as a trial

wavefunction in quantum Monte Carlo techniques.

Moreover, the Jastrow ansatz provides an exact description

of the ground state for certain interacting many-body systems.

To identify such models, one can assume a Jastrow-form

wavefunction and determine the corresponding parent Hamil-

tonian by solving the time-independent Schr¨

odinger equation.

This approach was pioneered by Calogero [38] and Sutherland

[8,39], leading to the discovery of the well-known family of

one-dimensional integrable Calogero-Sutherland (CS) mod-

els. These models include particles interacting via inverse-

square potentials in unbounded space, as well as inverse-

square sinusoidal interactions under periodic boundary con-

ditions [8,40].

Similarly, the attractive Lieb-Liniger (LL) gas has long

been known to exhibit bright quantum soliton states with a

Jastrow-form wavefunction [41,42]. By now, the complete

family of models describing interacting identical particles

with a Jastrow ground state has been established in any spa-

tial dimension [43–46]. These parent Hamiltonians generally

include both two-body and three-body interactions. While the

latter are absent in the celebrated CS and LL models, renor-

malization group calculations have shown that three-body in-

teractions are irrelevant for long-wavelength, low-temperature

physics [47], further reinforcing the practical utility of the Jas-

trow ansatz in physical applications.

In contrast to its widespread use in equilibrium settings,

the application of the Jastrow ansatz to nonequilibrium sce-

narios remains relatively unexplored. Notably, the dynamics

of the rational Calogero-Sutherland (CS) model—describing

one-dimensional bosons with inverse-square interactions con-

ﬁned in a harmonic trap—are exactly captured by a time-

dependent Jastrow ansatz at all times [48–50]. This model

not only serves as a valuable testbed for studying nonequi-

librium phenomena but also includes hard-core bosons in the

Tonks-Girardeau regime as a limiting case [51–54], making

it highly relevant to ultracold atom experiments [55–57]. As

a result, this model has inspired a wide range of studies on

topics such as nonexponential decay, quantum speed limits,

Loschmidt echoes, and orthogonality catastrophe [58–62]. It

has also played a key role in advancing research in ﬁnite-

time quantum thermodynamics [63–67], quantum quenches

[17,68], and quantum control [69–72], among other areas.

While these applications help to illustrate the value of ex-

act solutions in quantum dynamics, they are all characterized

by scale-invariance. This dynamical symmetry is highly re-

strictive and results in self-similar time evolution, meaning

that the spatial probability densities (the absolute square of

the wavefunction in the coordinate representation) at any two

diﬀerent times are simply related by a rescaling of the coor-

dinate variables. For instance, in one spatial dimension, the

time dependence of a many-body quantum state Ψ(t) satisﬁes

|Ψ(x1,...,xN,t)|2=|Ψ(x1/b(t),...,xN/b(t),t=0)|2

b(t)N,(1)

where b(t)>0 is the scaling factor. In this case, the complex-

ity of many-body time evolution reduces signiﬁcantly to deter-

mining the scaling factor, which obeys an ordinary diﬀerential

equation known as the Ermakov equation. The latter was ﬁrst

introduced in the context of the time-dependent harmonic os-

cillator [73,74], and together with its generalizations, it plays

a fundamental role in the study and control of Bose-Einstein

condensates [75–77] and ultracold Fermi gases [78–83].

Beyond the realm of scale-invariant dynamics, results are

scarce, and time-dependent Jastrow ans¨

atze have only recently

been explored in numerical methods. Notably, integrating

the Jastrow ansatz with quantum Monte Carlo algorithms has

been applied to the study of the quench dynamics of the Lieb-

Liniger (LL) model [23].

A natural question arises: can the construction of the par-

ent Hamiltonian used for the stationary Jastrow ansatz be ex-

tended to the time-dependent setting for arbitrary processes?

However, a direct extension of this approach generally leads

to a parent Hamiltonian that is not necessarily Hermitian. The

underlying reason for this non-Hermiticity is that the dynam-

ics implicitly assumed by the time-dependent trial wavefunc-

tion may break unitarity. It is worth noting that recent progress

in ﬁnding parent Hamiltonians [84] for time-dependent quan-

tum states has primarily focused on discrete spin systems,

which are not directly applicable to the continuous-variable

many-body systems considered here.

In this work, we extend the program of constructing par-

ent Hamiltonians for Jastrow wavefunctions to the time-

dependent case in one-dimensional quantum many-body sys-

tems in the continuum, i.e., with continuous variables. We

derive consistency conditions for the one-body and two-body

pair functions that deﬁne the Jastrow ansatz and apply them to

systems embedded in a harmonic trap. These consistency con-

ditions lead to the parent Hamiltonian of the complex-valued

time-dependent Jastrow ansatz (TDJA), where the amplitude

of the two-body trial wavefunction retains a functional form

similar to that of the ground state of the Calogero-Sutherland

(CS), Hyperbolic, and attractive Lieb-Liniger models.

Interestingly, while the Ermakov equation—ubiquitous in

the study of scale-invariant dynamics with speciﬁc interac-

tions [58,77,78]— emerges in our approach, the dynamics

described by the complex-valued TDJA are not necessarily

scale-invariant. Applying our framework to the aforemen-

tioned trial wavefunctions leads to generalizations of the CS,

Hyperbolic, and LL models with additional interactions. In

these systems, we establish the relations between the initial

density distribution and its time evolution, as well as the long-

time momentum distribution.

As a further application of our results, we demonstrate

that the parent Hamiltonian of the complex-valued TDJA can

be utilized for the engineering of Shortcuts to Adiabaticity

(STA) [85,86], which enable the fast preparation of a tar-

get state from a given initial eigenstate without requiring the

long timescales necessary for traditional adiabatic evolution.

More speciﬁcally, we show that the parent Hamiltonian of the

complex-valued TDJA can be interpreted as an implementa-

tion of the adiabatic evolution of the real-valued TDJA. Fur-

thermore, we illustrate how our ﬁndings can be applied to

study the exact dynamics following a quantum quench of the

interparticle interactions. In particular, we demonstrate this

in the context of the long-range Calogero-Sutherland models

with logarithmic interactions and the long-range Lieb-Liniger

model with Coulomb interactions.

II. THE TIME-DEPENDENT JASTROW ANSATZ (TDJA)

The original Jastrow ansatz [34–36], constructed in terms

of products of a pair function and a one-body function, is

time-independent and real-valued. Its use turned out to be a

fruitful approach and led to the discovery of many integrable

models, including the family of Calogero-Sutherland models

[38,39,43–45]. More recently, other models with ground

state wave function supporting a real-valued time-independent

Jastrow ansatz (TIJA) have been discovered [42,44,46].

In this section, we focus on the generalization describing

the time-dependent case and introduce the TDJA,

Ψ(x,t)=1

exp[N(t)]Y

i<j

fi j(t)Y

gk(t),(2)

where x=(x1,x2,··· ,xN), exp[N(t)]is the normalization

of the Jastrow wave function, where N(t) is a real-valued

function that only depends on time. As a shorthand, we also

deﬁne dNx=QN

m=1dxmfor short. In addition, fi j(t)≡f(xi j,t)

and gk(t)≡g(xk,t) are functions of the particles’ coordi-

nates and time. Throughout this work, we focus on bosonic

wave functions exclusively so that f(x,t)=f(−x,t). The

TDJA (2) describes the exact solution to the time-dependent

Schr¨

odinger equation

iℏ˙

Ψ(x,t)=ˆ

H(t)Ψ(x,t),(3)

when the dynamics is generated by the many-body Hamilto-

nian

H(t)=X

ˆp2

2m+ℏ2

2mX

v(i)

1+ℏ2

i<j

v(i j)

+ℏ2

i<j<k

v(i jk)

3−iℏ˙

N(t),(4)

where

v(i)

1≡g′′

+2im

ℏ

˙gi

,(5)

v(i j)

2≡

f′′

i j

fi j

+



g′

gi−

g′j

gj





f′

i j

fi j

+im

ℏ

fi j

,(6)

v(i jk)

3≡ −





f′

i j f′

fi j fjk

f′

i j f′

fi j fki

f′

ki f′

fki fjk 



,(7)

are the normalized one-body, two-body, and three-body po-

tentials bearing the dimension of inverse length squared.

Throughout the work, the overdot denotes the time derivative,

while the derivative with respect to a spatial coordinate of a

function fis denoted by f′.

We note that the many-body time-dependent potential in

Eq. (4) is not Hermitian in general. Once the Hermiticity

is guaranteed, the unitarity of the dynamics dictates that the

norm of Ψmust be constant in time. This, in turn, determines

the time-dependence of a the multi-dimensional integral over

the spatial coordinates up to a constant, i.e.,

exp[2N(t)]∝ZdNxY

i<j|fi j(t)|2Y

k|gk(t)|2.(8)

However, we note that imposing a constant norm of the trial

wave function alone is not suﬃcient to yield a Hermitian

Hamiltonian. Additional constraints are required on on the

functional forms of fi j(t) and gk(t), as discussed in Sec. III.

The family of Hamiltonians (4) provides a generalization

to driven systems of the seminal result known in the station-

ary case, i.e., the family of parent Hamiltonians with station-

ary real-valued Jastrow ground state [8,43–45]. Naturally,

this family and the corresponding real-valued TIJA as the

ground state can be recovered by choosing fi j and gktime-

independent and real-valued in Eqs. (4) and (2), respectively.

We consider two options in the time-dependent case. The

most general option is to make fi j(t) and gk(t) both time-

dependent and complex-valued. Since any complex number

can be represented in the polar form by a non-negative am-

plitude multiplied by a phase, without loss of generality, we

take

fi j(t)=eΓi j(t)+iθi j(t),gk(t)=eΛk(t)+iϕk(t),(9)

where we use the compact notation Γi j(t)≡Γ(xi j,t), Λk(t)≡

Λ(xk,t), etc. Here, θi j(t) is a two-body phase, and ϕk(t) is a

one-body phase. Note that the zero-body phase τ(t) can be

incorporated into the phase factor eiϕk(t)by redeﬁning ϕk(t)→

ϕk(t)=ϕk(t)+τ(t)/Nand hence is omitted.

We propose to reverse engineer the parent Hamiltonian us-

ing the time-dependent Schr¨

odinger equation (3) with the Jas-

trow ansatz as an exact solution. Speciﬁcally, we shall solve

for H(t) and demonstrate that the solution admits a variety of

applications, including the use of STA and quench dynamics

for several one-dimensional many-body strongly correlated

quantum models.

Before doing so, we note that an alternative ansatz can be

constructed by considering fi j and gkto be time-dependent,

while keeping them real-valued, i.e.,

Φ(x,t)=1

exp[N(t)]Y

i<j

eΓi j (t)Y

eΛk(t),(10)

which we shall call real-valued TDJA. The parent Hamilto-

nian of the real-valued TDJA according to the time-dependent

Schr¨

odinger equation

iℏ˙

Φ(x,t)=ˆ

H′(t)Φ(x,t) (11)

can also be found analogously. Clearly,

Ψ(x,t)=UP(x,t)Φ(x,t),(12)

where the many-particle phase unitary operator is deﬁned as

UP(x,t)≡Y

i<j

eiθi j (t)Y

eiϕk(t).(13)

The corresponding Hamiltonians ˆ

H′(t) and ˆ

H(t) are unitar-

ily equivalent in the sense that

H′(t)=U†

P(x,t)ˆ

H(t)UP(x,t)−iℏU†

P(x,t)˙

UP(x,t).(14)

In what follows, we ﬁnd ˆ

H(t) ﬁrst and subsequently deter-

mine ˆ

H′(t) through Eq. (14). We will see that the many-body

potential in ˆ

H′(t) is non-local, i.e., it involves a term linear

in the particles’ momenta. Nevertheless, one can still deﬁne

the prime parent Hamiltonian according to the instantaneous

time-independent Schr¨

odinger equation

H′

0(t)Φ(x,t)=0,(15)

i.e., Φ(x,t) is the instantaneous eigenstate of ˆ

H′

0(t) with zero

eigenvalue. Note that if Φ(x,t) has no nodes, then it is gen-

erally the ground state of ˆ

H′

0(t) provided ˆ

H′

0(t) is bounded

from below. The procedure of ﬁnding ˆ

H′

0(t) leads to

H′

0(t)=X

ˆp2

2m+ℏ2

2mX

(Λ′′

i+ Λ′2

+ℏ2

i<j

[Γ′′

i j + Γ′2

i j +(Λi−Λj)Γ′

i j]

−ℏ2

i<j<k

(Γ′

i jΓ′jk + Γ′

i jΓ′

ki + Γ′

kiΓ′jk),(16)

which is the same as the parent Hamiltonian in the real-valued

TIJA, except that ˆ

H′

0(t) is now time-dependent.

We conclude this section by noting that the trial wave func-

tions (2) and (10) must be normalizable so that Ψand Φdo

not blow up when particles are far apart.

III. CONSISTENCY CONDITIONS BY IMPOSING

HERMICITY

As already advanced, the Hermicity of ˆ

H(t) may introduce

strong constraints on the functional forms of fi j(t) and gk(t).

To ﬁnd such constraints, in principle, one can rewrite ˆ

H(t) in

terms of Γi j(t), θi j(t), Λk(t) and ϕk(t) by substituting Eq. (9)

into Eq. (4) and then impose that the imaginary part of Vvan-

ishes. Speciﬁcally, we note that the two-body and three-body

terms can by no means be reduced to a one-body potential

unless they are independent of the particles’ positions. This

dictates that Imv(i)

1must be a function of time only. We shall

assume similar constraints on the two-body and three-body

interactions, i.e., that Imv(i j)

2and Imv(i jk)

3are functions of time

only. This assumption ignores the fact that in some cases the

three-body interaction v(i jk)

3can reduce to two-body interac-

tions [46,87] for the sake of simplicity. Based on the above

analysis, we introduce

Ns(t)=Imv(i1···is)

s,s=1,2,3,(17)

where e

Ns(t) is a real-valued function of time and again, the

over-dot denotes time derivation. Thus, the Hermiticity of

H(t) imposes

N(t)∼Nℏ

2me

N1(t)+ℏN(N−1)

2me

N2(t)+ℏN(N−1)(N−2)

6me

N3(t),

(18)

where ∼denotes equivalence up to a constant independent of

time and particles’ positions. This is our ﬁrst result for con-

sidering the complex-valued TDJA. We will exemplify these

results in several examples.

In general, Eq. (17) for s=3 can lead to complicated con-

ditions. In this work, we shall restrict our attention to the case

where the three-body interactions are either real-valued, de-

pending on both coordinates and time or complex-valued and

depending on time only. The former assumes a vanishing two-

body phase angle θi j(t), while the latter essentially boils the

trial function down to three types of functions

eΓi j (t)∝









|xi j|λ(t)CS

|sinh[c(t)xi j]|λ(t)Hyperbolic

exphc(t)|xi j|iLL

,(19)

corresponding to the celebrated CS, hyperbolic, and LL mod-

els, respectively in the case of TDJA [8,44]. In all these cases,

the following quantity

W3(x,t)≡ − X

i<j<k

(Γ′

i jΓ′jk + Γ′

i jΓ′

ki + Γ′

kiΓ′jk) (20)

is independent of the particles’ coordinates. Note that care

needs to be taken when calculating the derivatives of Γi j(t) ac-

cording to Eq. (19) since it involves absolute value of a func-

tion, see e.g., Chapter 5 of Ref. [8]and Supplemental Mate-

rial of Ref. [46] and for details. In particular, one can ﬁnd

that W3(t) reduces to a constant for the trial wave functions in

Eq. (19), i.e.,

W3(t)=









0 CS

N(N−1)(N−2)λ2(t)c2(t)/6 Hyperbolic

N(N−1)(N−2)c2(t)/6 LL

.(21)

In Appendix A, we show that the two-body phase angle can

be written in a uniﬁed expression,

θi j(t)=η(t)Γi j(t),fi j(t)=eΓi j(t)[1+η(t)],(22)

where η(t) is any given real-valued function of time. Then the

three-body term deﬁned in Eq. (7) becomes

i<j<k

v(i jk)

3=W3(t)[1 +iη(t)]2.(23)

Combining Eq. (23) with Eq. (17), one ﬁnds N(N−1)(N−

2) ˙

N3(t)/6=2η(t)W3(t).Note that when η(t)=0, ˙

N3(t)=0

but W3(t) may depend on the coordinates in general. We can

combine ˙

N2(t) and ˙

N3(t) and obtain

N(t)∼ N1(t)+N23(t),(24)

where

N1(t)≡Nℏ

2me

N1(t),(25)

N23(t)≡ℏN(N−1)

2me

N2(t)+2ℏ

mZt

η(τ)W3(τ)dτ. (26)

Next, we note that in general Eq. (17) with s=1,2

yields for these models consistency conditions between the

trial functions, which are discussed in detail in Appendix A.

Throughout this work, we shall focus on the case where the

particles are embedded in the harmonic trap, i.e., Rev(i)

1is

quadratic in xiand therefore Λk(t)=−mω(t)x2

k/(2ℏ). As ar-

gued in Appendix B, the one-body consistency condition, in

this case, simpliﬁes dramatically. It explicitly gives the func-

tional form of C2(t), i.e.,

C2(t)≡η(t)ω(t)+˙ω(t)

2ω(t),(27)

with

N1(t)=−m˙ω(t)

2ℏω(t),(28)

ϕk(t)=−m˙ω(t)

4ℏω(t)x2

k+τ(t).(29)

The two-body consistency condition corresponding to s=2

η(t)Γ′′

i j +2η(t)Γ′2

i j +m

ℏ

Γi j −m

ℏC2(t)Γ′

i j xi j =˙

N2(t).(30)

The Hamiltonian (4) then becomes

H(t)=1

2mX

i+1

2mΩ2(t)X

+ℏ2

i<jΓ′′

i j +[1 −η2(t)]Γ′2

i j 

−ℏϖ(t)X

i<j

Γ′

i j xi j −ℏX

i<j

dt [η(t)Γi j]+E(t),(31)

where the frequency of the trap Ω(t) is deﬁned as

Ω2(t)≡ω2(t)+d

dt "˙ω(t)

2ω(t)#−"˙ω(t)

2ω(t)#2

,(32)

and

E(t)≡ −1

2Nℏϑ(t)+ℏ2

m[1 −η2(t)]W3(t),(33)

ϖ(t)≡ω(t)"1−η(t) ˙ω(t)

2ω2(t)#,(34)

ϑ(t)≡2˙τ(t)+ω(t).(35)

Upon making the change of variables

ω(t)=ω0

b2(t), ω0≡ω(0),(36)

one immediately observes that ˙ω(t)/[2ω(t)] =−˙

b(t)/b(t) and

Eq. (32) becomes

b(t)+ Ω2(t)b(t)=ω2

0b−3(t),(37)

with the initial condition b(0) =1. Remarkably, Eq. (37) is

the celebrated Ermakov equation governing the dynamics of

the scaling factor in scale-invariant quantum many-body evo-

lution, emerging when the interactions of particles have given

scaling properties [58,77,78]. However, as we shall dis-

cuss in Sec. IX, although the eﬀective scale-invariant dynam-

ics also appears here, the interactions need not be restricted to

a given scaling dimension.

Let us remark on some diﬀerences with previous ap-

proaches in the literature [58,77,78]. (i) Works on scale in-

variance consider ˙

b(0) =0 when the initial state is stationary.

Here we only impose the condition b(0) =1 for the Ermakov

equation, and ˙

b(0) can be arbitrary. (ii) In the Ermakov equa-

tion discussed by the previous literature, it is assumed that

Ω0≡Ω(0) is always equal to ω0≡ω(0). Here, such a con-

straint does not necessarily hold. As we shall discuss in detail

in Sec. IX, if we further impose ˙

b(0) =0 and Ω0=ω0together

with the conditions for η(t) at t=0, we obtain ˆ

H(0)Ψ(0) =0,

i.e., at time t=0, the TDJA is also an eigenstate of the corre-

sponding parent Hamiltonian, which is always a requirement

for ﬁnding the counterdiabatic driving for scale-invariant dy-

namics, as discussed previously [88,89]. We emphasize that

in our discussion, the initial time t0is not necessarily zero. In

fact, both Eq. (38) and the interaction in Eq. (31) can break, in

general, scale-invariance. As we shall see subsequently, both

scale-invariant and non-scale-invariant dynamics follow natu-

rally from our results.

Finally, we shall refer to b(t) along with η(t) and γ(t) [de-

ﬁned in Sec. VII B], as the fundamental parameters that gov-

ern the exact many-body dynamics and regard other time-

dependent parameters as secondary given that they can be de-

rived from the former. Throughout this work, while the time-

dependent parent Hamiltonian (31) may contain both types of

parameters, we shall express the generic time-dependent Jas-

trow ansatz (2) mainly in terms of the fundamental parame-

ters. To this goal, Eq. (2) can be rewritten as

Ψ(x,t)=1

eN23(t)bN/2(t)Y

i<j

eΓi j(t)[1+iη(t)] Y

e−x2

2x2

0b2(t)+i

b(t)x2

2ω0x2

0b(t)+iτ(t),

(38)

where we have ignore the constant factor ωN/4

0and x0is the

length scale of the harmonic trap deﬁned as x0≡qℏ

mω0.

To summarize: given the functions Γi j(t) and η(t) satisfy the

consistency condition (30), the nonequilibrium dynamics of

one-dimensional interacting bosons in a harmonic trap de-

scribed by associated with the Hamiltonian (38) admits an

exact description in terms of Eq. (31), which bears the form

of TDJA. Such nonequilibrium dynamics are exempliﬁed by

specifying the functional form of Γi j(t) and η(t) in Sec. V-

VIII. Knowledge of the exact nonequilibrium dynamics of

strongly-correlated quantum systems is rare and precious, and

we shall explore its applications to counterdiabatic driving in

STA [85,86,90–93] and quench dynamics subsequently.

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TheTime-DependentJastrowAnsatz:ExactQuantumDynamics,ShortcutstoAdiabaticity,andQuantumQuenchesinStrongly-CorrelatedMany-BodySystemsJingYang1,∗andAdolfodelCampo1,2,†1DepartmentofPhysicsandMaterialsScience,UniversityofLuxembourg,L-1511Luxembourg,Luxembourg2DonostiaInternationalPhysicsCenter,E-20018San...

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The Time-Dependent Jastrow Ansatz Exact Quantum Dynamics Shortcuts to Adiabaticity and Quantum Quenches in Strongly-Correlated Many-Body Systems Jing Yang1and Adolfo del Campo1 2

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