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Register a new userNon-Markovianity, Loschmidt echo and criticality: a unified picture
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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 is strongly affected by the environmental criticality; the qubit dynamics is shown to be Markovian exactly and only at the critical point. Therefore non-Markovianity is an indicator of criticality in the model considered here.
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Environment--induced decoherence causes entropy increase. It can be quantified using, e.g., the purity $varsigma={rm Tr}rho^2$. 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 $varsigma(t)$ 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 $varsigma$ and $bar M$ has a regime where it is dominated by the classical Lyapunov exponents
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 perturbations produce coherent oscillations in the width of LDOS that imprint clear signals of the perturbation in LE decay. We also show that if the perturbation acts in a small region of phase space (local perturbation) the effect is magnified and the decay is given by the width of the LDOS.
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 $hbar/tau_{V}$. The perturbation represents the action of an uncontrolled environment interacting with the system, and is characterized by a correlation length $xi_0$ and a correlation time $tau_0$. For small perturbation strengths or rapid fluctuating perturbations, the Loschmidt echo decays exponentially with a rate predicted by the Fermi Golden Rule, $1/tilde{tau}= tau_{c}/tau_{V}^2$, where typically $tau_{c} sim min[tau_{0},xi_0/v]$ with $v$ the particle velocity. Whenever the rate $1/tilde{tau}$ is larger than the Lyapunov exponent of the system, a perturbation independent Lyapunov decay regime arises. We also find that by speeding up the fluctuations (while keeping the perturbation strength fixed) the fidelity decay becomes slower, and hence, one can protect the system against decoherence.
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 turns to increase exponentially with the energy separation of the two components of the superposition.
We study the dynamics of an open quantum system interacting with a non-thermal bath. Here, non-thermal means that the bath modes do not need to have the same temperature, but they have an effective temperature distribution. We find that, when a quantum system is interacting with such a non-thermal bath far from thermal equilibrium, it is no longer proper to use any coarse-grained Markovian description for the system, even when their coupling strength is quite weak. Especially, when there is coherent transition with strong interference strength in the quantum system, the Markovian master equation would bring in a serious problem of negative probability. After we consider some proper non-Markovian corrections, the problem can be naturally resolved.
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