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arXiv:nlin/0201038 · Journal of Physics A Mathematical and General

Decoherence of spin echoes

Tomaž Prosen, T. H. Seligman

Published May 24, 2002• 53 citations
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Abstract

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.

DOI
10.1088/0305-4470/35/22/304

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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.
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.
A general relation is derived, which expresses the fidelity of quantum dynamics, measuring the stability of time evolution to small static variation in the Hamiltonian, in terms of ergodicity of an observable generating the perturbation as defined by its time correlation function. Fidelity for ergodic dynamics is predicted to decay exponentially on time scale proportional to delta(-2), delta approximately strength of perturbation, whereas faster, typically Gaussian decay on shorter time scale proportional delta(-1) is predicted for integrable, or generally nonergodic dynamics. This result needs the perturbation delta to be sufficiently small such that the fidelity decay time scale is larger than any (quantum) relaxation time, e.g., mixing time for mixing dynamics, or averaging time for nonergodic dynamics (or Ehrenfest time for wave packets in systems with chaotic classical limit). Our surprising predictions are demonstrated in a quantum Ising spin-(1/2) chain periodically kicked with a tilted magnetic field where we find finite parameter-space regions of nonergodic and nonintegrable motion in the thermodynamic limit.
We derive a simple and general relation between the fidelity of quantum motion, characterizing the stability of quantum dynamics with respect to arbitrary static perturbation of the unitary evolution propagator, and the integrated time auto-correlation function of the generator of perturbation. Surprisingly, this relation predicts the slower decay of fidelity the faster the decay of correlations. In particular, 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 timescale ∝ 1/δ as opposed to mixing dynamics where the fidelity is found to decay exponentially on a timescale ∝ 1/δ 2 , where δ is the strength of perturbation. A detailed discussion of a semiclassical regime of small effective values of Planck constant is given where classical correlation functions can be used to predict quantum fidelity decay. Note that the correct and intuitively expected classical stability behaviour is recovered in the classical limit → 0, as the two limits δ → 0 and → 0 do not commute. In addition, we also discuss non-trivial dependence on the number of degrees of freedom. All the theoretical results are clearly demonstrated numerically on the celebrated example of a quantized kicked top.
We study the stability of unitary quantum dynamics of composite systems (for example: central system + environment) with respect to weak interaction between the two parts. Unified theoretical formalism is applied to study different physical situations: (i) coherence of a forward evolution as measured by the purity of the reduced density matrix, (ii) stability of time evolution with respect to small coupling between subsystems and (iii) Loschmidt echo measuring dynamical irreversibility. Stability has been measured either by fidelity of pure states of a composite system, or by the so-called reduced fidelity of reduced density matrices within a subsystem. Rigorous inequality among fidelity, reduced fidelity and purity is proved and a linear response theory is developed expressing these three quantities in terms of time correlation functions of the generator of interaction. The qualitatively different cases of regular (integrable) or mixing (chaotic in the classical limit) dynamics in each of the subsystems are discussed in detail. Theoretical results are demonstrated and confirmed in a numerical example of two coupled kicked tops.
We study the decay rate of the Loschmidt echo or fidelity in a chaotic system under a time-dependent perturbation V(q,t)V(\mathbf{q},t)V(q,t) with typical strength $\ensuremath{\hbar}∕{\ensuremath{\tau}}_{V}$. The perturbation represents the action of an uncontrolled environment interacting with the system, and is characterized by a correlation length ${\ensuremath{\xi}}_{0}$ and a correlation time ${\ensuremath{\tau}}_{0}$. For small perturbation strengths or rapid fluctuating perturbations, the Loschmidt echo decays exponentially with a rate predicted by the Fermi ``golden rule,'' $1∕\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}={\ensuremath{\tau}}_{c}∕{\ensuremath{\tau}}_{V}^{2}$, where ${\ensuremath{\tau}}_{c}\ensuremath{\sim}\mathrm{min}[{\ensuremath{\tau}}_{0},{\ensuremath{\xi}}_{0}∕v]$ and vvv is the typical particle velocity. Whenever the rate $1∕\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}$ 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.
The stability of quantum systems to perturbations of the Hamiltonian is studied. This stability is quantified by the fidelity. Dependence of fidelity on the initial state as well as on the dynamical properties of the system is considered. In particular, systems having a chaotic or regular classical limit are analysed. The fidelity decay rate is given by an integral of the correlation function of the perturbation and is thus smaller the faster correlation function decays. If the perturbation can be written as a time derivative of another operator, meaning that the time averaged perturbation vanishes, fidelity freezes at a constant value and starts to decay only after a long time inversely proportional to the perturbation strength. In composite systems stability of entanglement to perturbations of the Hamiltonian is analysed in terms of purity. For regular systems purity decay is shown to be independent of Planck's constant for coherent initial states in the semiclassical limit. The accelerated decoherence of macroscopic superpositions is also explained. The theory of fidelity decay is applied to the stability of quantum computation and an improved quantum Fourier transform algorithm is designed.