In echo experiments, imperfect time-reversal operations are performed on a subset of the total number of degrees of freedom. To capture the physics of these experiments, we introduce a partial fidelity M(B)(t), the Boltzmann echo, where only part of the system's degrees of freedom can be time reversed. We present a semiclassical calculation of M(B)(t). We show that, as the time-reversal operation is performed more and more accurately, the decay rate of M(B)(t) saturates at a value given by the decoherence rate of the controlled degrees of freedom due to their coupling to uncontrolled ones. We connect these results with NMR spin echo experiments.
We define a quantity, the so-called purity fidelity , which measures the rate of dynamical irreversibility due to decoherence, observed e.g. in echo experiments, in the presence of an arbitrary small perturbation of the total (system + environment) Hamiltonian. We derive a linear response formula for the purity fidelity in terms of integrated time correlation functions of the perturbation. Our relation predicts, similar to the case of fidelity decay, that the faster the decay of purity fidelity the slower is the decay of time correlations. In particular, we find exponential decay in quantum mixing regime and faster, initially quadratic and later typically Gaussian decay in the regime of non-ergodic, e.g. integrable quantum dynamics. We illustrate our approach by an analytical calculation and numerical experiments in the Ising spin 1/2 chain kicked with tilted homogeneous magnetic field where part of the chain is interpreted as a system under observation and part as an environment.
Motivated by atomic optics experiments, we investigate a class of fidelity functions describing the reconstruction of quantum states by time-reversal operations as ${M}_{\mathrm{Da}}(t)=|\ensuremath{\langle}\ensuremath{\psi}|{e}^{i{H}_{2}t/2}{e}^{i{H}_{1}t/2}{e}^{\ensuremath{-}i{H}_{2}t/2}{e}^{\ensuremath{-}i{H}_{1}t/2}|\ensuremath{\psi}\ensuremath{\rangle}|{}^{2}$. We show that the decay of
MDa is quartic in time at short times and that it freezes well above the ergodic value at long times, when ${H}_{2}\ensuremath{-}{H}_{1}$ is not too large. The long-time saturation value of
MDa contains easily extractable information on the strength of decoherence in these systems.
We examine an exactly solvable model of decoherence -- a spin-system interacting with a collection of environment spins. We show that in this simple model (introduced some time ago to illustrate environment--induced superselection) generic assumptions about the coupling strengths lead to a universal (Gaussian) suppression of coherence between pointer states. We explore the regime of validity of this result and discuss its relation to spectral features of the environment. We also consider its relevance to the experiments on the so-called Loschmidt echo (which measures, in effect, the fidelity between the initial and time-reversed or "echo" signal). In particular, we show that for partial reversals (e.g., when of only a part of the total Hamiltonian changes sign) fidelity will exhibit a Gaussian dependence on the time of reversal. In such cases echo may become independent of the details of the reversal procedure or the specifics of the coupling to the environment. This puzzling behavior was observed in several NMR experiments. Natural candidates for such two environments (one of which is easily reversed, while the other is ``irreversible'') are suggested for the experiment involving ferrocene.
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.
Abstract Echo protocols provide a means to investigate the arrow of time in macroscopic processes. Starting from a nonequilibrium state, the many-body quantum system under study is evolved for a certain period of time τ. Thereafter, an (effective) time reversal is performed that would – if implemented perfectly – take the system back to the initial state after another time period τ. Typical examples are nuclear magnetic resonance imaging and polarisation echo experiments. The presence of small, uncontrolled inaccuracies during the backward propagation results in deviations of the “echo signal” from the original evolution and can be exploited to quantify the instability of nonequilibrium states and the irreversibility of the dynamics. We derive an analytic prediction for the typical dependence of this echo signal for macroscopic observables on the magnitude of the inaccuracies and on the duration τ of the process, and verify it in numerical examples.
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.