Loschmidt echoes are a well-established method to characterize dynamical properties such as quantum chaos and sensitivity to perturbations. Here we present a straightforward generalization that identifies non-classical dynamical behavior. The generalized echoes involve four forward-backward time propagators. If these propagators do not commute, one can readily identify signatures of this property in the relations between echoes. As an example, detection of intrinsic non-commuting quantum features is analyzed in critical dynamics of a spin chain. Interestingly, this phenomenon is also observed for Gaussian dynamics and high temperature limits which are generally regarded as classical.
Loschmidt echoes are a well-established method to characterize dynamical properties such as quantum chaos and sensitivity to perturbations. Here we present a straightforward generalization that identifies non-classical dynamical behavior. The generalized echoes involve four forward-backward time propagators. If these propagators do not commute, one can readily identify signatures of this property in the relations between echoes. As an example, detection of intrinsic non-commuting quantum features is analyzed in critical dynamics of a spin chain. Interestingly, this phenomenon is also observed for Gaussian dynamics and high temperature limits which are generally regarded as classical.
Non-Markovianity can be characterized by performing a series of successive measurements and analyzing departures of the corresponding outcome statistics from a Markovian probabilistic structure. Considering a system coupled with its environment via a dephasing interaction, we show that joint outcome probabilities can be written in terms of a set of two-time environment correlations. Their definition involves forward and backward propagators with different Hamiltonians, associated with a recently introduced generalization of standard Loschmidt echoes [Cormick and Budini, Phys. Lett. A 593, 132011 (2026)]. This result establishes a solid connection between quantum non-Markovianity defined in an operational (measurement-based) way and complex quantum dynamics studied through their sensitivity to dynamical perturbations. We find conditions that guarantee a Markovian (system) behavior and also determine how the generalized echoes can identify information exchanges between the system and its environment. We illustrate our ideas considering examples of spin environments that realize these different dynamical regimes.
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. Surprisingly, this relation predicts that the decay of fidelity is the slower the faster the decay of correlations. In particular for a static (time-independent) perturbation, and for non-ergodic and non-mixing dynamics where asymptotic decay of correlations is absent, a qualitatively different and faster decay of fidelity is predicted on a time scale ∝1/δ as opposed to mixing dynamics where the fidelity is found to decay exponentially on a time-scale ∝1/δ2, where δ is a strength of perturbation. A detailed discussion of a semi-classical regime of small effective values of Planck constant ℏ is given where classical correlation functions can be used to predict quantum fidelity decay. Note that the correct and intuitively expected classical stability behavior is recovered in the classical limit ℏ→0, as the two limits δ→0 and ℏ→0 do not commute. The theoretical results are demonstrated numerically for two models, the quantized kicked top and the multi-level Jaynes Cummings model. Our method can for example be applied to the stability analysis of quantum computation and quantum information processing.
In the long quest to identify and compensate the sources of decoherence in many-body systems far from the ground state, the varied family of Loschmidt echoes (LEs) became an invaluable tool in several experimental techniques. A LE involves a time-reversal procedure to assess the effect of perturbations in a quantum excitation dynamics. However, when addressing macroscopic systems one is repeatedly confronted with limitations that seem insurmountable. This led to the formulation of the central hypothesis of irreversibility stating that the timescale of decoherence, T3, is proportional to the timescale of the many-body interactions we reversed, T2. We test this by implementing two experimental schemes based on Floquet Hamiltonians where the effective strength of the dipolar spin-spin coupling, i.e., 1/T2, is reduced by a variable scale factor k. This extends the perturbation timescale, ${T}_{\mathrm{\ensuremath{\Sigma}}}$, in relation to T2. Strikingly, we observe the superposition of the normalized Loschmidt echoes for the bigger values of k. This manifests the dominance of the intrinsic dynamics over the perturbation factors, even when the Loschmidt echo is devised to reverse that intrinsic dynamics. Thus, in the limit where the reversible interactions dominate over perturbations, the LE decays within a timescale, ${T}_{3}\ensuremath{\approx}{T}_{2}/R$ with $R=(0.15\ifmmode\pm\else\textpm\fi{}0.01)$, confirming the emergence of a perturbation independent regime. These results support the central hypothesis of irreversibility.
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.
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 the parameters, the LE decays exponentially. Furthermore, for strong enough perturbations, the decay rate is given by the Lyapunov exponent of the classical system. Some particularly interesting examples are given. The analytical results are supported by thorough numerical studies. In addition, some regimes not accessible to the theory are explored, showing that the LE and its Lyapunov regime present the same form of universality ascribed to classical chaos. In a sense, this is evidence that the LE is a robust temporal signature of chaos in the quantum realm. Finally, the relation between the LE and the quantum to classical transition is explored, in particular with the theory of decoherence. Using two different approaches, a semiclassical approximation to Wigner functions and a master equation for the LE, it is shown that the decoherence rate and the decay rate of the LE are equal. The relationship between these quantities results mutually beneficial, in terms of the broader resources of decoherence theory and of the possible experimental realization of the LE.
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 employed here 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 through arbitrary combinations of XY and Ising-like interactions, that contain the standard Heisenberg and dipolar ones. Different time regimes are identified for the Loschmidt echo dynamics in this perturbative configuration. In particular, the exponential decay scales as a Fermi golden rule, where the contributions of the different SE terms are individually evaluated and analyzed. Comparisons with previous analytical and numerical evaluations of decoherence based on the attenuation of specific interferences show that the Loschmidt echo is an advantageous decoherence quantifier at any time, regardless of the S internal dynamics.