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Exact Quantum Dynamics, Shortcuts to Adiabaticity, and Quantum Quenches in Strongly-Correlated Many-Body Systems: The Time-Dependent Jastrow Ansatz

Jing Yang, Adolfo del Campo

TL;DR

The paper introduces a general time-dependent Jastrow ansatz (TDJA) for continuous-variable quantum many-body systems and derives Hermiticity-consistency conditions that enable an exact, unitary description of nonequilibrium dynamics beyond scale invariance. By reverse-engineering a parent Hamiltonian, the authors connect the TDJA to shortcuts to adiabaticity (STA) and explicitly construct counterdiabatic terms, including nonlocal one- and two-body contributions, for one-dimensional models such as the Calogero-Sutherland (CS), hyperbolic, and Lieb-Liniger (LL) systems in a harmonic trap. They show that complex TDJA dynamics can be mapped to STA on the real-valued TDJA manifold, yielding exact quench dynamics for interactions and trap parameters, with analytic survival-probability expressions and asymptotic scaling tied to a scaling factor $b(t)$. The framework produces tractable exact results for scale-invariant and non-scale-invariant regimes, and is validated through analytical expressions and numerical quasi-Monte Carlo simulations, offering a robust benchmark for quantum simulations of nonequilibrium strongly correlated systems. Overall, the TDJA provides a powerful analytic route to study work statistics, Loschmidt echoes, and fidelity bounds in 1D continuous-variable quantum matter, with potential experimental relevance for ultracold gases and quantum technologies.

Abstract

The description of strongly correlated quantum many-body systems far from equilibrium presents a fundamental 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 interactions 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 continuous variables.

Exact Quantum Dynamics, Shortcuts to Adiabaticity, and Quantum Quenches in Strongly-Correlated Many-Body Systems: The Time-Dependent Jastrow Ansatz

TL;DR

The paper introduces a general time-dependent Jastrow ansatz (TDJA) for continuous-variable quantum many-body systems and derives Hermiticity-consistency conditions that enable an exact, unitary description of nonequilibrium dynamics beyond scale invariance. By reverse-engineering a parent Hamiltonian, the authors connect the TDJA to shortcuts to adiabaticity (STA) and explicitly construct counterdiabatic terms, including nonlocal one- and two-body contributions, for one-dimensional models such as the Calogero-Sutherland (CS), hyperbolic, and Lieb-Liniger (LL) systems in a harmonic trap. They show that complex TDJA dynamics can be mapped to STA on the real-valued TDJA manifold, yielding exact quench dynamics for interactions and trap parameters, with analytic survival-probability expressions and asymptotic scaling tied to a scaling factor . The framework produces tractable exact results for scale-invariant and non-scale-invariant regimes, and is validated through analytical expressions and numerical quasi-Monte Carlo simulations, offering a robust benchmark for quantum simulations of nonequilibrium strongly correlated systems. Overall, the TDJA provides a powerful analytic route to study work statistics, Loschmidt echoes, and fidelity bounds in 1D continuous-variable quantum matter, with potential experimental relevance for ultracold gases and quantum technologies.

Abstract

The description of strongly correlated quantum many-body systems far from equilibrium presents a fundamental 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 interactions 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 continuous variables.
Paper Structure (27 sections, 1 theorem, 162 equations, 4 figures, 1 table)

This paper contains 27 sections, 1 theorem, 162 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Time-dependent Jastrow ansatz (TDJA)
  3. Consistency conditions by imposing Hermicity
  4. Shortcuts to adiabaticity for the Jastrow ansatz
  5. The case of vanishing $\eta(t)$
  6. The time-dependent Calogero-Sutherland models
  7. The Calogero-Sutherland model: Recovering the scaling-invariant dynamics
  8. The logarithmic Calogero-Sutherland model
  9. The time-dependent Hyperbolic models
  10. The hyperbolic model
  11. The logarithmic hyperbolic model
  12. The time-dependent long-range Lieb-Liniger model
  13. Effective dynamics and the one-body reduced density matrix (OBRDM)
  14. Quench dynamics
  15. Interaction quench of the logarithmic Calogero-Sutherland gases
  16. ...and 12 more sections

Key Result

Lemma 1

For a differentiable function $F(x)$, the only possibility for $F(x)-F(y)$ to be only dependent on $x-y$ is that $F(x)$ is a linear function of $x$.

Figures (4)

  • Figure 1: Engineered time-dependence of the control parameters as a function of $\kappa t$. For the plot of $\Omega^{2}(t)/\omega_{0}^{2}$, $\kappa/\omega_{0}=4$. The initial time is $t_{0}=-50\kappa^{-1}$ and $a(t)$ is the strength of the long-range Coulomb interactions defined as $a(t)\equiv\hbar\omega_{0}c_{0}\beta^{3}(t)$. For $\kappa t\gg1$, the time dependence of all these parameters is well approximated by the Heaviside function and, therefore, leads to a sudden-quench protocol.
  • Figure 2: The numerical calculation versus the analytic prediction of the survival probability for $4$ particles for the quench of the interactions in the long-range LL model. The initial time $t_{0}=-50\kappa^{-1}$. The values of parameters: $\hbar=1$, $N=4$, $\omega_{0}=1$, $\kappa=5$ and $c_{0}=-1$. For this case, the survival probability is perfectly predicted by Eq. (\ref{['eq:SP-LL']}).
  • Figure 3: Engineered time-dependence of the control parameters as a function of $\omega_{0}t$. The initial time $t_{0}=0$ and $a(t)$ is the strength of the long-range Coulomb interactions defined as $a(t)\equiv\hbar\omega_{0}c_{0}/b^{3}(t)$.
  • Figure 4: The numerical calculation versus the analytic prediction of the survival probability after a quench of the harmonic trap in the long-range LL model. The values of parameters: $\hbar=1$, $N=4$, $\omega_{0}=1$ and $c_{0}=-1$. The long-time decay is characterized by asymptotic expression in Eq. (\ref{['eq:SP-LL']}).

Theorems & Definitions (2)

  • Lemma 1
  • proof