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arXiv:2006.14440 · Physical review. B./Physical review. B

Dynamics of coherence: Maximal quantum Fisher information versus Loschmidt echo

Hadi Cheraghi, S. Mahdavifar

Published Jul 9, 2020• 15 citations
arXivHTMLPDF

Abstract

We consider the dynamics of maximal quantum Fisher information (MQFI) after sudden quenches for the one-dimensional transverse-field Ising model. Our results show, the same as Loschmidt echo, that there is a universality for the revival times, i.e., they do not depend on the initial state and the size of the quench and are given by integer multiples of ${T}_{\text{rev}}\ensuremath{\simeq}\frac{N}{2{v}_{\text{max}}}$, where $N$ is the system size and ${v}_{\text{max}}$ is the maximal group velocity of quasiparticles. Critically enhanced and decreased at revival and decay times as ${T}_{\text{rev}}\ensuremath{\equiv}{T}_{\text{dec}}$ are characterized by quenching from the order and disorder phases into the quantum phase transition, respectively, that can be utilized to detect the quantum critical point (QCP). In some quenches crossed from the QCP, nonanalytic behaviors appear at some times due to the turning of the local observable from one direction to another because of identifying the maximum value. We name this phenomenon the dynamical MQFI transitions, occurring at the critical times ${t}_{c}$. Interestingly, although no Fisher zero exists in the dynamics of MQFI, the first critical time emerged from the dynamical quantum phase transition is equal to the first time when the logarithm of MQFI is minimum. In addition, we unveil the long-time run of MQFI indicates a signature of a nonequilibrium quantum phase transition at the QCP. We also discuss the probability of the arising of macroscopic superpositions in the nonequilibrium dynamics of the system.

DOI
10.1103/physrevb.102.024304

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We consider the dynamics of maximal quantum Fisher information (MQFI) after sudden quenches for the one-dimensional transverse-field Ising model. Our results show, the same as Loschmidt echo, that there is a universality for the revival times, i.e., they do not depend on the initial state and the size of the quench and are given by integer multiples of Trev\ensuremath≃N2vmax{T}_{\text{rev}}\ensuremath{\simeq}\frac{N}{2{v}_{\text{max}}}Trev​\ensuremath≃2vmax​N​, where NNN is the system size and vmax{v}_{\text{max}}vmax​ is the maximal group velocity of quasiparticles. Critically enhanced and decreased at revival and decay times as Trev\ensuremath≡Tdec{T}_{\text{rev}}\ensuremath{\equiv}{T}_{\text{dec}}Trev​\ensuremath≡Tdec​ are characterized by quenching from the order and disorder phases into the quantum phase transition, respectively, that can be utilized to detect the quantum critical point (QCP). In some quenches crossed from the QCP, nonanalytic behaviors appear at some times due to the turning of the local observable from one direction to another because of identifying the maximum value. We name this phenomenon the dynamical MQFI transitions, occurring at the critical times tc{t}_{c}tc​. Interestingly, although no Fisher zero exists in the dynamics of MQFI, the first critical time emerged from the dynamical quantum phase transition is equal to the first time when the logarithm of MQFI is minimum. In addition, we unveil the long-time run of MQFI indicates a signature of a nonequilibrium quantum phase transition at the QCP. We also discuss the probability of the arising of macroscopic superpositions in the nonequilibrium dynamics of the system.
Dynamical quantum phase transitions (DQPTs) are temporal singularities marked by zeros of the Loschmidt echo, yet their underlying quantum-information structure remains elusive. Here, we introduce a momentum-resolved entanglement entropy as a direct probe of DQPTs in translation-invariant free systems. We analytically establish that every critical momentum mode k∗k^{*}k∗ associated with a DQPT saturates its entanglement to the maximal value ln⁡2\ln{2}ln2, coinciding with the vanishing of the Loschmidt echo. Crucially, we demonstrate that this maximal entanglement universally suppresses information scrambling: a momentum-resolved out-of-time-ordered correlator (OTOC) vanishes identically for all times at k∗k^{*}k∗. These three signatures -- Fisher zeros, maximal entanglement, and vanished OTOC -- are proved to be equivalent in both the transverse-field Ising and Su-Schrieffer-Heeger models, despite their distinct bipartitions (momentum-pair vs. sublattice). Our results establish a unified, information-theoretic framework for DQPTs, revealing them a points where quantum correlations saturate and information flow halts. This work elevates entanglement and scrambling to central dynamical order parameters, offering a universal perspective on nonequilibrium quantum critically.
We study the slow quenching dynamics (characterized by an inverse rate ${\ensuremath{\tau}}^{\ensuremath{-}1}$) of a one-dimensional transverse Ising chain with nearest neighbor ferromagentic interactions across the quantum critical point (QCP) and analyze the Loschmidt overlap measured using the subsequent temporal evolution of the final wave function (reached at the end of the quenching) with the final time-independent Hamiltonian. Studying the Fisher zeros of the corresponding generalized ``partition function,'' we probe nonanalyticities manifested in the rate function of the return probability known as dynamical phase transitions (DPTs). In contrast to the sudden quenching case, we show that DPTs survive in the subsequent temporal evolution following the quenching across two critical points of the model for a sufficiently slow rate; furthermore, an interesting ``lobe'' structure of Fisher zeros emerge. We have also made a connection to topological aspects studying the dynamical topological order parameter $[{\ensuremath{\nu}}_{D}(t)]$ as a function of time (t)(t)(t) measured from the instant when the quenching is complete. Remarkably, the time evolution of ${\ensuremath{\nu}}_{D}(t)$ exhibits drastically different behavior following quenches across a single QCP and two QCPs. In the former case, ${\ensuremath{\nu}}_{D}(t)$ increases stepwise by unity at every DPT (i.e., $\mathrm{\ensuremath{\Delta}}{\ensuremath{\nu}}_{D}=1$). In the latter case, on the other hand, ${\ensuremath{\nu}}_{D}(t)$ essentially oscillates between 0 and 1 (i.e., successive DPTs occur with $\mathrm{\ensuremath{\Delta}}{\ensuremath{\nu}}_{D}=1$ and $\mathrm{\ensuremath{\Delta}}{\ensuremath{\nu}}_{D}=\ensuremath{-}1$, respectively), except for instants where it shows a sudden jump by a factor of unity when two successive DPTs carry a topological charge of the same sign.
The dynamics of relative entropy and l1{l}_{1}l1​ norm of coherence, as well as, the Wigner-Yanase skew and quantum Fisher information are studied in the one-dimensional XYXYXY spin chain in the presence of a time-dependent transverse magnetic field. We show that, independent of the initial state of the system and while the relative entropy of coherence, the l1{l}_{1}l1​ norm of coherence, and quantum Fisher information are incapable, surprisingly, the Wigner-Yanase skew information dynamic can truly spotlight the equilibrium critical point. We also observe that, when the system is quenched to the critical point, these quantities show suppressions and revivals. Moreover, the first suppression (revival) time scales linearly with the system size and its scaling ratio is unique for all quenches independent of the initial phase of the system. This is the promised universality of the first suppression (revival) time.
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.
Abstract Quantum coherence will undoubtedly play a fundamental role in understanding the dynamics of quantum many-body systems; therefore, to be able to reveal its genuine contribution is of great importance. In this paper, we focus our discussions on the one-dimensional transverse field quantum Ising model initialized in the coherent Gibbs state, and investigate the effects of quantum coherence on dynamical quantum phase transition (DQPT). After quenching the strength of the transverse field, the effects of quantum coherence are studied using Fisher zeros and the rate function of the Loschmidt echo. We find that quantum coherence not only recovers DQPT destroyed by thermal fluctuations, but also generates some entirely new DQPTs, which are independent of the equilibrium quantum critical point. We also find that the Fisher zero cutting the imaginary axis is not sufficient to generate DQPT because it also requires the Fisher zeros to be tightly bound close enough to the neighborhood of the imaginary axis. It can be manifested that DQPTs are rooted in quantum fluctuations. This work reveals new information on the fundamental connection between quantum critical phenomena and quantum coherence.
We unveil the role of the long-time average of the Loschmidt echo in the characterization of nonequilibrium quantum phase transitions by studying sudden quench processes across quantum phase transitions in various quantum systems. While the dynamical quantum phase transitions are characterized by the emergence of a series of zero points at critical times during time evolution, we demonstrate that nonequilibrium quantum phase transitions can be identified by nonanalyticities in the long-time average of the Loschmidt echo. The nonanalytic behaviors are illustrated by a sharp change in the long-time average of the Loschmidt echo or the corresponding rate function or the emergence of divergence in the second derivative of the rate function when the driving quench parameter crosses the phase transition points. The connection between the second derivative of the rate function and fidelity susceptibility is also discussed.