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arXiv:1604.01312 · Physical review. B./Physical review. B

Transient Loschmidt echo in quenched Ising chains

Carla Lupo, Marco Schirò

Published Jul 25, 2016• 16 citations
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Abstract

We study the response to sudden local perturbations of highly excited quantum Ising spin chains. The key quantity encoding this response is the overlap between time-dependent wave functions, which we write as a transient Loschmidt Echo. Its asymptotics at long time differences contain crucial information about the structure of the highly excited nonequilibrium environment induced by the quench. We compute the echo perturbatively for a weak local quench but for arbitrarily large global quench, using a cumulant expansion. Our perturbative results suggest that the echo decays exponentially, rather than power law as in the low-energy orthogonality catastrophe, a further example of quench-induced decoherence already found in the case of quenched Luttinger liquids. The emerging decoherence scale is set by the strength of the local potential and the bulk excitation energy.

DOI
10.1103/physrevb.94.014310

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We study the response to sudden local perturbations of highly excited quantum Ising spin chains. The key quantity encoding this response is the overlap between time-dependent wave functions, which we write as a transient Loschmidt Echo. Its asymptotics at long time differences contain crucial information about the structure of the highly excited nonequilibrium environment induced by the quench. We compute the echo perturbatively for a weak local quench but for arbitrarily large global quench, using a cumulant expansion. Our perturbative results suggest that the echo decays exponentially, rather than power law as in the low-energy orthogonality catastrophe, a further example of quench-induced decoherence already found in the case of quenched Luttinger liquids. The emerging decoherence scale is set by the strength of the local potential and the bulk excitation energy.
We study the response of a highly excited time-dependent quantum many-body state to a sudden local perturbation, a sort of orthogonality catastrophe problem in a transient nonequilibrium environment. To this extent we consider, as a key quantity, the overlap between time-dependent wave functions, which we write in terms of a novel two-time correlator generalizing the standard Loschmidt echo. We discuss its physical meaning, general properties, and its connection with experimentally measurable quantities probed through nonequilibrium Ramsey interferometry schemes. Then we present explicit calculations for a one-dimensional interacting Fermi system brought out of equilibrium by a sudden change of the interaction, and perturbed by the switching on of a local static potential. We show that different scattering processes give rise to remarkably different behaviors at long times, quite opposite from the equilibrium situation. In particular, while the forward scattering contribution retains its power-law structure even in the presence of a large nonequilibrium perturbation, with an exponent that is strongly affected by the transient nature of the bath, the backscattering term is a source of nonlinearity which generates an exponential decay in time of the Loschmidt Echo, reminiscent of an effective thermal behavior.
We study the Loschmidt echo (LE) in a central spin model in which a central spin is globally coupled to an environment (EEE) which is subjected to a small and sudden quench at t=0t=0t=0, so that its state at t=0+t={0}^{+}t=0+ remains the same as the ground state of the initial environmental Hamiltonian before the quench; this leads to a nonequilibrium situation. This state now evolves with two Hamiltonians, the final Hamiltonian following the quench and its modified version which incorporates an additional term arising due to the coupling of the central spin to the environment. Using a generic short-time scaling of the decay rate, we establish that in the early-time limit, the rate of decay of the LE close to the quantum critical point (QCP) of EEE is independent of the quenching. We also study the temporal evolution of the LE and establish the presence of a crossover to a situation where the quenching becomes irrelevant. In the limit of large quench amplitude the nonequilibrium initial condition is found to result in a drastic increase in decoherence at large times, even far away from a QCP. These generic results are verified analytically as well as numerically, choosing EEE to be a transverse Ising chain where the transverse field is suddenly quenched.
We analyze the dynamics of a Luttinger model following a quench in the electron-electron interaction strength, where the change in the interaction strength occurs over a finite time scale $\ensuremath{\tau}$. We study the Loschmidt echo (the overlap between the initial and final state) as a function of time, both numerically and within a perturbation scheme, treating the change in the interaction strength as a small parameter, for all $\ensuremath{\tau}$. We derive the corrections appearing in (a) the Loschmidt echo for a finite quench duration $\ensuremath{\tau}$, (b) the scaling of the echo following a sudden $(\ensuremath{\tau}\ensuremath{\rightarrow}0)$ quench, and (c) the scaling of the echo after an adiabatic $(\ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\infty})$ quench. We study in detail the limiting cases of the echo in the early-time and infinite-time limits, and provide scaling arguments to understand these in a general context. We also show that our perturbative results are in good agreement with the exact numerical ones.
The Loschmidt echo, defined as the overlap between quantum wave function evolved with different Hamiltonians, quantifies the sensitivity of quantum dynamics to perturbations and is often used as a probe of quantum chaos. In this work we consider the behavior of the Loschmidt echo in the many-body localized phase, which is characterized by emergent local integrals of motion and provides a generic example of nonergodic dynamics. We demonstrate that the fluctuations of the Loschmidt echo decay as a power law in time in the many-body localized phase, in contrast to the exponential decay in few-body ergodic systems. We consider the spin-echo generalization of the Loschmidt echo and argue that the corresponding correlation function saturates to a finite value in localized systems. Slow, power-law decay of fluctuations of such spin-echo-type overlap is related to the operator spreading and is present only in the many-body localized phase, but not in a noninteracting Anderson insulator. While most of the previously considered probes of dephasing dynamics could be understood by approximating physical spin operators with local integrals of motion, the Loschmidt echo and its generalizations crucially depend on the full expansion of the physical operators via local integrals of motion operators, as well as operators which flip local integrals of motion. Hence these probes allow one to get insights into the relation between physical operators and local integrals of motion and access the operator spreading in the many-body localized phase.
We investigate the Loschmidt echo, the overlap of the initial and final wave functions of Luttinger liquids after a spatially inhomogeneous interaction quench. In studying the Luttinger model, we obtain an analytic solution of the bosonic Bogoliubov-de Gennes equations after quenching the interactions within a finite spatial region. As opposed to the power-law temporal decay following a potential quench, the interaction quench in the Luttinger model leads to a finite, hardly time-dependent overlap; therefore, no orthogonality catastrophe occurs. The steady state value of the Loschmidt echo after a sudden inhomogeneous quench is the square of the respective adiabatic overlaps. Our results are checked and validated numerically on the XXZ Heisenberg chain.
We investigate the many-body generalization of the orthogonality catastrophe by studying the generalized Loschmidt echo of Luttinger liquids (LLs) after a global change of interaction. It decays exponentially with system size and exhibits universal behavior: the steady state exponent after quenching back and forth n times between 2 LLs (bang-bang protocol) is 2n times bigger than that of the adiabatic overlap and depends only on the initial and final LL parameters. These are corroborated numerically by matrix-product state based methods of the XXZ Heisenberg model. An experimental setup consisting of a hybrid system containing cold atoms and a flux qubit coupled to a Feshbach resonance is proposed to measure the Loschmidt echo using rf spectroscopy or Ramsey interferometry.