Environment--induced decoherence causes entropy increase. It can be quantified using, e.g., the purity $ς={\rm Tr}ρ^2$. When the Hamiltonian of a quantum system is perturbed, its sensitivity to such perturbation can be measured by the Loschmidt echo $\bar M(t)$. It is given by the average squared overlap between the perturbed and unperturbed state. We describe the relation between the temporal behavior of $ς(t)$ and $\bar M(t)$. In this way we show that the decay of the Loschmidt echo can be analyzed using tools developed in the study of decoherence. In particular, for systems with a classically chaotic Hamiltonian the decay of $ς$ and $\bar M$ has a regime where it is dominated by the classical Lyapunov exponents
The Loschmidt echo and the purity are two quantities that can provide invaluable information about the evolution of a quantum system. While the Loschmidt echo characterizes instability and sensitivity to perturbations, purity measures the loss of coherence produced by an environment coupled to the system. For classically chaotic systems both quantities display a number of -- supposedly universal -- regimes that can lead on to think of them as equivalent quantities. We study the decay of the Loschmidt echo and the purity for systems with finite dimensional Hilbert space and present numerical ev…
We investigate the sensitivity of the time evolution of semiclassical wave packets in two-dimensional chaotic billiards with respect to local perturbations of their boundaries. For this purpose, we address, analytically and numerically, the time decay of the Loschmidt echo (LE). We find the LE to decay exponentially in time, with the rate equal to the classical escape rate from an open billiard obtained from the original one by removing the perturbation-affected region of its boundary. Finally, we propose a principal scheme for the experimental observation of the LE decay.
The Loschmidt echo measures the sensitivity to perturbations of quantum evolutions. We study its short time decay in classically chaotic systems. Using perturbation theory and throwing out all correlation imposed by the initial state and the perturbation, we show that the characteristic time of this regime is well described by the inverse of the width of the local density of states. This result is illustrated and discussed in a numerical study in a 2-dimensional chaotic billiard system perturbed by various contour deformations and using different types of initial conditions. Moreover, the infl…
We study the decay rate of the Loschmidt echo or fidelity in a chaotic system under a time-dependent perturbation $V(q,t)$ with typical strength $\hbar/τ_{V}$. The perturbation represents the action of an uncontrolled environment interacting with the system, and is characterized by a correlation length $ξ_0$ and a correlation time $τ_0$. For small perturbation strengths or rapid fluctuating perturbations, the Loschmidt echo decays exponentially with a rate predicted by the Fermi Golden Rule, $1/\tildeτ= τ_{c}/τ_{V}^2$, where typically $τ_{c} \sim \min[τ_{0},ξ_0/v]$ with $v$ the particle veloci…
The Loschmidt echo is a measure of the stability and reversibility of quantum evolution under perturbations of the Hamiltonian. One of the expected and most relevant characteristics of this quantity for chaotic systems is an exponential decay with a perturbation independent decay rate given by the classical Lyapunov exponent. However, a non-uniform decay -- instead of the Lyapunov regime -- has been reported in several systems. In this work we find an analytical semiclassical expression for the averaged fidelity amplitude that can be related directly to the anomalous -- unexpected-- behaviour …
We investigate the Loschmidt echo in a one-dimensional spin chain having Kitaev-type interaction in constant and kicked magnetic fields. The Loschmidt echo for the initial states having different magnon excitations shows long-time revivals for smaller chains and has short-time revival peaks for the longer chains. The system near the critical point shows peculiarly long-time revival peaks of the Loschmidt echo for relatively larger chains. The presence of a magnon in the initial state affects the Loschmidt echo revival peaks. The momentum distribution function exhibits maxima for a few momenta …
Loschmidt echo (LE) is a measure of reversibility and sensitivity to perturbations of quantum evolutions. For weak perturbations its decay rate is given by the width of the local density of states (LDOS). When the perturbation is strong enough, it has been shown in chaotic systems that its decay is dictated by the classical Lyapunov exponent. However, several recent studies have shown an unexpected non-uniform decay rate as a function of the perturbation strength instead of that Lyapunov decay. Here we study the systematic behavior of this regime in perturbed cat maps. We show that some pertur…
We study the Loschmidt echo and Berry phase in a nonlinear system transported around a double and triple degeneracy. The nonlinearity of the system makes the Berry phase different from that in linear systems. We propose a witness of nonlinearity for the nonlinear system and show its dependence on the parameters of the system, taking the standard Landau-Zener model as an example. We calculate the Loschmidt echo(LE) or quantum fidelity in the quantum dynamics under perturbations around the degeneracy, and establish a connection between the LE and the witness of nonlinearity. These phases and Los…
We demonstrate analytically and verify numerically that the out-of-time order correlator is given by the thermal average of Loschmidt echo signals. This provides a direct link between the out-of-time-order correlator -- a recently suggested measure of information scrambling in quantum chaotic systems -- and the Loschmidt echo, a well-appreciated familiar diagnostic that captures the dynamical aspect of chaotic behavior in the time domain, and is accessible to experimental studies.
We study the Loschmidt echo for a system of electrons interacting through mean-field Coulomb forces. The electron gas is modeled by a self-consistent set of hydrodynamic equations. It is observed that the quantum fidelity drops abruptly after a time that is proportional to the logarithm of the perturbation amplitude. The fidelity drop is related to the breakdown of the symmetry properties of the wave function.
Time reversal is one of the most intriguing yet elusive wave phenomenon of major interest in different areas of classical and quantum physics. Time reversal requires in principle to flip the sign of the Hamiltonian of the system, leading to a revival of the initial state (Loschmidt echo). Here it is shown that Loschmidt echo of photons can be observed in an optical setting without resorting to reversal of the Hamiltonian. We consider photonic propagation in a binary waveguide lattice and show that, by exchanging the two sublattices after some propagation distance, a Loschmidt echo can be obser…
The Loschmidt echo is investigated to track the effect of the local QDP. It is also quite sensitive to whether the background dynamics is integrable or not. For the integrable case, viz. the Heisenberg model, the Loschmidt echo depends on the parameters operators corresponding to the QDP as well as the time of QDP. The probability of reviving the system to its initial state is higher for incoherent QDPs occurring at large time intervals. Whereas each time coherent QDP occurs certain probability of reviving the state is always lost. For For the non-integrable case, viz. a kicked Harper model, i…
We explore Loschmidt echo in two regimes of one-dimensional (1D) interacting Bose gases: the strongly interacting Tonks-Girardeau (TG) regime, and the weakly-interacting mean-field regime. We find that the Loschmidt echo of a TG gas decays as a Gaussian when small perturbations are added to the Hamiltonian (the exponent is proportional to the number of particles and the magnitude of a small perturbation squared). In the mean-field regime the Loschmidt echo decays faster for larger interparticle interactions (nonlinearity), and it shows richer behavior than the TG Loschmidt echo dynamics, with …
We study numerically the properties of entanglement of two interacting, or noninteracting, particles evolving in a regime of quantum chaos in the quantum Chirikov standard map. Such pairs can be viewed as interacting, on noninteracting, Einstein-Podolsky-Rosen pairs in a regime of quantum chaos. The analysis is done with such tools as the Loschmidt echo of entanglement and the Poincaré recurrences of entanglement in presence of absorption. The obtained results show unusual features of the entropy of entanglement and the spectrum of Schmidt decomposition with their dependence on interactions at…
Quantum states extended over a large volume in phase space have oscillations from quantum interferences in their Wigner distribution on scales smaller than $\hbar$ [W.H. Zurek, Nature {\bf 412}, 712 (2001)]. We investigate the influence of those sub-Planck scale structures on the sensitivity to an external perturbation of the state's time evolution. While we do find an accelerated decay of the Loschmidt Echo for an extended state in comparison to a localized wavepacket, the acceleration is described entirely by the classical Lyapunov exponent and hence cannot originate from quantum interferenc…
Identifying dynamical signatures of excited state quantum phase transitions (ESQPTs) in experimentally realizable quantum many-body systems is helpful for understanding the dynamical effects of ESQPTs. In such systems, the highly controllable spinor Bose-Einstein condensates (BECs) offer an exceptional platform to study ESQPTs. In this work, we investigate the dynamical characteristics of the ESQPT in spin-$1$ BEC by means of the Loschmidt echo spectrum. The Loschmidt echo spectrum is an extension of the well-known Loschmidt echo and definded as the overlaps between the evolved state and the e…
A critically enhanced decay of the Loschmidt echo is characteristic of sudden quench dynamics near a quantum phase transition. Here, we demonstrate that the decay and revival of the Loschmidt echo follows power-law scaling in the system size and the distance from a critical point with equilibrium critical exponents. We reveal such dynamical scaling laws by analyzing relevant perturbations to the Loschmidt echo cast in a scaling invariant form. We confirm the validity and the generality of the predicted dynamical scaling laws with a diverse range of critical models such as Ising spin models wit…
A quantum Loschmidt echo (also referred to as quantum time mirror) corresponds to an effective time inversion after which the quantum wave function reverses its previous time evolution and eventually reaches its initial distribution again. We propose a comparably simple protocol for such an effective time reversal for ultra-cold atoms in optical lattices which should be easier to realize experimentally than previous proposals.
We study the quantum critical behaviour of the Dicke Hamiltonian, with finite number of atoms and explore the signature of quantum chaos using measures like the ground state fidelity and the Loschmidt echo. We show that both these quantities clearly point to the chaotic nature of the system in the super-radiant phase.
We show that non-Hermitian biorthogonal many-body phase transitions can be characterized by the enhanced decay of Loschmidt echo. The quantum criticality is numerically investigated in a non-Hermitian transverse field Ising model by performing the finite-size dynamical scaling of Loschmidt echo. We determine the equilibrium correlation length critical exponents that are consistent with previous results from the exact diagonalization. More importantly, we introduce a simple method to detect quantum phase transitions with the short-time average of rate function motivated by the critically enhanc…
A simple relationship between recently proposed measures of non-Markovianity and the Loschmidt echo is established, holding for a two-level system (qubit) undergoing pure dephasing due to a coupling with a many-body environment. We show that the Loschmidt echo is intimately related to the information flowing out from and occasionally back into the system. This, in turn, determines the non-Markovianity of the reduced dynamics. In particular, we consider a central qubit coupled to a quantum Ising ring in the transverse field. In this context, the information flux between system and environment i…
Entanglement between a quantum system and its environment leads to loss of coherence in the former. In general, the temporal fate of coherences is complicated. Here, we establish the connection between decoherence of a central system and fidelity decay in the environment for a variety of situations, including both, energy conserving and dissipative couplings. We show how properties of unitary time evolution of the environment can be inferred from the non-unitary evolution of coherences in the central system. This opens up promising ways for measuring Loschmidt echoes in a variety of situations…
Non-Hermitian classical and open quantum systems near an exceptional point (EP) are known to undergo strong deviations in their dynamical behavior under small perturbations or slow cycling of parameters as compared to Hermitian systems. Such a strong sensitivity is at the heart of many interesting phenomena and applications, such as the asymmetric breakdown of the adiabatic theorem, enhanced sensing, non-Hermitian dynamical quantum phase transitions and photonic catastrophe. Like for Hermitian systems, the sensitivity to perturbations on the dynamical evolution can be captured by Loschmidt ech…
We study the crossover of the quantum Loschmidt echo (or fidelity) from the golden rule regime to the perturbation-independent exponential decay regime by using the kicked top model. It is shown that the deviation of the perturbation-independent decay of the averaged fidelity from the Lyapunov decay results from quantum fluctuations in individual fidelity, which are caused by the coherence in the initial coherent states. With an averaging procedure suppressing the quantum fluctuations effectively, the perturbation-independent decay is found to be close to the Lyapunov decay. We also show that …
We unveil the role of the long time average of 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 Loschmidt echo. The nonanalytic behaviours are illustrated by a sharp change in the long time ave…
We consider the degradation of the dynamics of a Gaussian wave packet in a harmonic oscillator under the presence of an environment. This last is given by a single non-degenerate two level system. We analyze how the binary degree of freedom perturbs the free evolution of the wave packet producing decoherence, which is quantified by the Loschmidt Echo. This magnitude measures the reversibility of a perturbed quantum evolution. In particular, we use it here to study the relative "fragility" of coherent superpositions (cat states) with respect to incoherent ones. This fragility or sensitivity tur…
We employ the Loschmidt Echo, i.e. the signal recovered after the reversal of an evolution, to identify and quantify the processes contributing to decoherence. This procedure, which has been extensively used in single particle physics, is here employed in a spin ladder. The isolated chains have 1/2 spins with XY interaction and their excitations would sustain a one-body like propagation. One of them constitutes the controlled system S whose reversible dynamics is degraded by the weak coupling with the uncontrolled second chain, i.e. the environment E. The perturbative SE coupling is swept thro…
A universal relation is established between the quantum work probability distribution of an isolated driven quantum system and the Loschmidt echo dynamics of a two-mode squeezed state. When the initial density matrix is canonical, the Loschmidt echo of the purified double thermofield state provides a direct measure of information scrambling and can be related to the analytic continuation of the partition function. Information scrambling is then described by the quantum work statistics associated with the time-reversal operation on a single copy, associated with the sudden negation of the syste…
I show how to perform a Loschmidt echo (time reversal) in the Bose-Hubbard model implemented with cold bosonic atoms in an optical lattice. The echo is obtained by applying a linear phase imprint on the lattice and a change in magnetic field to tune the boson-boson scattering length through a Feshbach resonance. I discuss how the echo can measure the fidelity of the quantum simulation, the intensity of an external potential (e.g. gravity), or the critical point of the superfluid-insulator quantum phase transition.
We propose a versatile Loschmidt echo protocol to detect and quantify multiparticle entanglement. It allows us to extract the quantum Fisher information for arbitrary pure states, and finds direct application in quantum metrology. In particular, the protocol applies to states that are generally difficult to characterize, as non-Gaussian states, and states that are not symmetric under particle exchange. We focus on atomic systems, including trapped ions, polar molecules, and Rydberg atoms, where entanglement is generated dynamically via long-range interaction, and show that the protocol is stab…
Coined discrete-time quantum walks are studied using simple deterministic dynamical systems as coins whose classical limit can range from being integrable to chaotic. It is shown that a Loschmidt echo like fidelity plays a central role and when the coin is chaotic this is approximately the characteristic function of a classical random walker. Thus the classical binomial distribution arises as a limit of the quantum walk and the walker exhibits diffusive growth before eventually becoming ballistic. The coin-walker entanglement growth is shown to be logarithmic in time as in the case of many-bod…
The equilibration dynamics of a closed quantum system is encoded in the long-time distribution function of generic observables. In this paper we consider the Loschmidt echo generalized to finite temperature, and show that we can obtain an exact expression for its long-time distribution for a closed system described by a quantum XY chain following a sudden quench. In the thermodynamic limit the logarithm of the Loschmidt echo becomes normally distributed, whereas for small quenches in the opposite, quasi-critical regime, the distribution function acquires a universal double-peaked form indicati…
A boundary transfer matrix formulation allows to calculate the Loschmidt echo for one-dimensional quantum systems in the thermodynamic limit. We show that non-analyticities in the Loschmidt echo and zeros for the Loschmidt amplitude in the complex plane (Fisher zeros) are caused by a crossing of eigenvalues in the spectrum of the transfer matrix. Using a density-matrix renormalization group algorithm applied to these transfer matrices we numerically investigate the Loschmidt echo and the Fisher zeros for quantum quenches in the XXZ model with a uniform and a staggered magnetic field. We give e…
We obtain an initial value representation for the quantum Loschmidt echo from the semiclassical theory of Wigner function evolution, together with classical first-order perturbation theory. In the limit of small actions, the amplitude of each trajectory reduces to unity, just as in the dephasing representation introduced by Vanicek, but these trajectories are here generated by the mean Hamiltonian for both the forward and the backward motion. This slight change of action may substantially alter the phase. The amplitude correction depends on the second derivative of the action. This improved de…
We study the short-time stability of quantum dynamics in quasi-one-dimensional systems with respect to small localized perturbations of the potential. To this end, we address, analytically and numerically, the decay of the Loschmidt echo (LE) during times short compared to the Ehrenfest time. We find that the LE is generally a non-monotonous function of time and exhibits strongly pronounced minima and maxima at the instants of time when the corresponding classical particle traverses the perturbation region. We also show that, under general conditions, the envelope decay of the LE is well appro…
A quantum phase transition is generally thought to imprint distinctive characteristics on the nonequilibrium dynamics of a closed quantum system. Specifically, the Loschmidt echo after a sudden quench to a quantum critical point $-$ measuring the time dependence of the overlap between initial and time-evolved states $-$ is expected to exhibit an accelerated relaxation followed by periodic revivals. We here introduce a new exactly solvable model, the extended Su-Schrieffer-Heeger model, the Loschmidt echo of which provides a counterexample. A parallell analysis of the quench dynamics of the thr…
In this paper we review our recent work on the theoretical approach to quantum Loschmidt echoes, i.e. various properties of the so called echo dynamics -- the composition of forward and backward time evolutions generated by two slightly different Hamiltonians, such as the state autocorrelation function (fidelity) and the purity of a reduced density matrix traced over a subsystem (purity fidelity). Our main theoretical result is a linear response formalism, expressing the fidelity and purity fidelity in terms of integrated time autocorrelation function of the generator of the perturbation. Surp…
In [PRE \textbf{90}, 022920 (2014), arXiv:1404.6738] a study of the ground state fidelity of the Dicke model as a function of the coupling parameter is presented. Abrupt jumps of the fidelity in the superradiant phase are observed and are assumed to be related to the transition to chaos. We show that this conclusion results from a misinterpretation of the numerics. In fact, if the parity symmetry is taken into account, the unexpected jumps disappear.
The out-of-time-ordered correlators (OTOC) and the Loschmidt echo are two measures that are now widely being explored to characterize sensitivity to perturbations and information scrambling in complex quantum systems. Studying few qubits systems collectively modelled as a kicked top, we solve exactly the three- and four- qubit cases, giving analytical results for the OTOC and the Loschmidt echo. While we may not expect such few-body systems to display semiclassical features, we find that there are clear signatures of the exponential growth of OTOC even in systems with as low as 4 qubits in app…
We study exact zeros of Loschmidt echo and quantum speed limit time for dynamical quantum phase transition in finite size systems. Our results illustrate that exact zeros of Loschmidt echo exist even in finite size quantum systems when the postquench parameter takes some discrete values in regions with the corresponding equilibrium phase different from the initial phase. As the system size increases and tends to infinity, the discrete parameters distribute continuously in the parameter regions. We further analyze the time for the appearance of the first exact zero of Loschmidt echo which is kn…
We study the Loschmidt echo (LE) of a coupled system consisting of a central spin and its surrounding environment described by a general XY spin-chain model. The quantum dynamics of the LE is shown to be remarkably influenced by the quantum criticality of the spin chain. In particular, the decaying behavior of the LE is found to be controlled by the anisotropy parameter of the spin chain. Furthermore, we show that due to the coupling to the spin chain, the ground-state Berry phase for the central spin becomes nonanalytical and its derivative with respect to the magnetic parameter $λ$ in spin c…
We study the effect of two simultaneous local quenches on the evolution of Loschmidt echo and entanglement entropy of a one dimensional transverse Ising model. In this work, one of the local quenches involves the connection of two spin-1/2 chains at a certain time and the other local quench corresponds to a sudden change in the magnitude of the transverse field at a given site in one of the spin chains. We numerically calculate the dynamics associated with the Loschmidt echo and the entanglement entropy as a result of such double quenches, and discuss various timescales involved in this proble…
We address the time decay of the Loschmidt echo, measuring sensitivity of quantum dynamics to small Hamiltonian perturbations, in one-dimensional integrable systems. Using semiclassical analysis, we show that the Loschmidt echo may exhibit a well-pronounced regime of exponential decay, alike the one typically observed in quantum systems whose dynamics is chaotic in the classical limit. We derive an explicit formula for the exponential decay rate in terms of the spectral properties of the unperturbed and perturbed Hamilton operators and the initial state. In particular, we show that the decay r…
Semiclassical results are usually expected to be valid in the semiclassical regime. An interesting question is, in models in which appropriate effective Planck constants can be introduced, to what extent will a semiclassical prediction stay valid when the effective Planck constant is increased? In this paper, we numerically study this problem, focusing on semiclassical predictions for the decay of the quantum Loschmidt echo in deep quantum regions. Our numerical simulations, carried out in the chaotic regime in the sawtooth model and in the kicked rotator model and also in the critical region …
In this work we develop the theory of the Loschmidt echo and dynamical phase transitions in non-interacting strongly disordered Fermi systems after a quench. In finite systems the Loschmidt echo displays zeros in the complex time plane that depend on the random potential realization. Remarkably, the zeros coalesce to form a 2D manifold in the thermodynamic limit, atypical for 1D systems, crossing the real axis at a sharply-defined critical time. We show that this dynamical phase transition can be understood as a transition in the distribution function of the smallest eigenvalue of the Loschmid…
Entropy production characterizes irreversibility. This viewpoint allows us to consider the thermodynamic uncertainty relation, which states that a higher precision can be achieved at the cost of higher entropy production, as a relation between precision and irreversibility. Considering the original and perturbed dynamics, we show that the precision of an arbitrary counting observable in continuous measurement of quantum Markov processes is bounded from below by Loschmidt echo between the two dynamics, representing the irreversibility of quantum dynamics. When considering particular perturbed d…
We explain and exploit the random matrix formulation of the Loschmidt echo for the XX spin chain, valid for multiple domain wall initial states and also for a XX spin chain generalized with additional interactions to more neighbours. For models with interactions decaying as $e^{-α\left\vert l-j\right\vert }/\left\vert l-j\right\vert ^{p+1}$, with $p$ integer or natural number and $α\geq 0$, we show that there are third order phase transitions in a double scaling limit of the complex-time Loschmidt echo amplitudes. For the long-range version of the chain, we use an exact result for Toeplitz det…
The Loschmidt echo (LE) is a measure of the sensitivity of quantum mechanics to perturbations in the evolution operator. It is defined as the overlap of two wave functions evolved from the same initial state but with slightly different Hamiltonians. Thus, it also serves as a quantification of irreversibility in quantum mechanics. In this thesis the LE is studied in systems that have a classical counterpart with dynamical instability, that is, classically chaotic. An analytical treatment that makes use of the semiclassical approximation is presented. It is shown that, under certain regime of th…
We address the computation of the Loschmidt echo in interacting integrable spin chains after a quantum quench. We focus on the massless regime of the XXZ spin-1/2 chain and present exact results for the dynamical free energy (Loschmidt echo per site) for a special class of integrable initial states. For the first time we are able to observe and describe points of non-analyticities using exact methods, by following the Loschmidt echo up to large real times. The dynamical free energy is computed as the leading eigenvalue of an appropriate Quantum Transfer Matrix, and the non-analyticities arise …
We investigate the time-dependent variance of the fidelity with which an initial narrow wavepacket is reconstructed after its dynamics is time-reversed with a perturbed Hamiltonian. In the semiclassical regime of perturbation, we show that the variance first rises algebraically up to a critical time $t_c$, after which it decays. To leading order in the effective Planck's constant $\hbar_{\rm eff}$, this decay is given by the sum of a classical term $\simeq \exp[-2 λt]$, a quantum term $\simeq 2 \hbar_{\rm eff} \exp[-Γt]$ and a mixed term $\simeq 2 \exp[-(Γ+λ)t]$. Compared to the behavior of th…
The Krylov subspace method is a standard approach to approximate quantum evolution, allowing to treat systems with large Hilbert spaces. Although its application is general, and suitable for many-body systems, estimation of the committed error is involved. This makes it difficult to automate its use. In this paper, we solve this problem by realizing that such error can be regarded as a Loschmidt echo in a tight-binding Hamiltonian. We show that the different time-regimes of the approximation can be understood using simple physical ideas. More importantly, we obtain computationally cheap error …
