Abstract
We consider quantum many-body systems evolving under a time-independent Hamiltonian $H$ from a nonequilibrium initial state at time $t=0$ towards a close-to-equilibrium state at time $t=\tau$. Subsequently, this state is slightly perturbed and finally propagated for another time period $\tau$ under the inverted Hamiltonian $-H$. The entire procedure may also be viewed as an imperfect time inversion or "echo dynamics". We unravel a remarkable persistence of such dynamics with respect to the observable deviations of the time-dependent expectation values from the equilibrium expectation value: For most perturbations, the deviations in the final state are essentially independent of the inversion time point $\tau$. Our quantitative analytical predictions compare very well with exact numerical results.
Highlights
A trademark of classical chaos [1,2,3] is the sensitivity against small perturbations: Tiny changes in the initial conditions grow exponentially in time
II, we introduce the general setting of the considered echo dynamics and types of imperfections
The perturbation Hamiltonian V is of the “spin-glass” form (25) for one realization of the normally distributed random couplings Jiαjβ . (For other realizations, we found practically the same results.) Solid lines correspond to the numerical results using exact diagonalization, color coded as black for t ∈ [0, τ ], and brown-to-purple as indicated in the legend for t ∈ [τ, 2τ + δ] with the peak height decreasing as δ increases
Summary
A trademark of classical chaos [1,2,3] is the sensitivity against small perturbations: Tiny changes in the initial conditions grow exponentially in time. While this effect is readily observable in low-dimensional systems, things are less obvious in the case of many-body dynamics. The analytical result (21) predicts that the observable perturbation of the quantum echo due to tiny imperfections during the reversal procedure does not grow at all upon increasing the inversion time point τ , and this behavior is recovered numerically up to possible transient effects for very short values of τ.
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