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Register a new userDecay of the Loschmidt echo in a time-dependent environment
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Added by
Fernando M. Cucchietti
Publication date
2006
fields
Physics
and research's language is
English
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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.
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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.
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.
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.
Evaluating the role of perturbations versus the intrinsic coherent dynamics in driving to equilibrium is of fundamental interest to understand quantum many-body thermalization, in the quest to build ever complex quantum devices. Here we introduce a protocol that scales down the coupling strength in a quantum simulator based on a solid-state nuclear spin system, leading to a longer decay time T2, while keeping perturbations associated to control error constant. We can monitor quantum information scrambling by measuring two powerful metrics, out-of-time-ordered correlators (OTOCs) and Loschmidt Echoes (LEs). While OTOCs reveal quantum information scrambling involving hundreds of spins, the LE decay quantifies, via the time scale T3, how well the scrambled information can be recovered through time reversal. We find that when the interactions dominate the perturbation, the LE decay rate only depends on the interactions themselves, T3 ~ T2, and not on the perturbation. Then, in an unbounded many-spin system, decoherence can achieve a perturbation-independent regime, with a rate only related to the local second moment of the Hamiltonian.
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