General relation between quantum ergodicity and fidelity of quantum dynamics

Abstract

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 ∝ δ −2 , δ ∼ strength of perturbation, whereas faster, typically gaussian decay on shorter time scale ∝ δ −1 is predicted for integrable, or generally non-ergodic dynamics. This surprising result is demonstrated in quantum Ising spin-1/2 chain periodically kicked with a tilted magnetic field where we find finite parameter-space regions of non-ergodic and non-integrable motion in thermodynamic limit.PACS number: 03.65.Yz, 75.10.Jm The quantum signatures of various types of classical motion, ranging from integrable to ergodic, mixing and chaotic, are still lively debated issues (see e.g. [1]). Most controversial is the absence of exponential sensitivity to variation of initial condition in quantum mechanics which prevents direct definition of quantum chaos [2]. However, there is an alternative concept which can be used in classical as well as in quantum mechanics [3]: One can study stability of motion with respect to small variation in the Hamiltonian. Clearly, in classical mechanics this concept, when applied to individual trajectories, is equivalent to sensitivity to initial conditions. Integrable systems with regular orbits are stable against small variation in the hamiltonian (the statement of KAM theorem), wheres for chaotic orbits varying the hamiltonian has similar effect as varying the initial condition: exponential divergence of two orbits for two nearby chaotic hamiltonians.The quantity of the central interest here is the fidelity of quantum motion. Consider a unitary operator U being either (i) a short-time propagator, or (ii) a Floquet map U =T exp(−i p 0 dτ H(τ )/ ) of (periodically timedependent) Hamiltonian H (H(τ + p) = H(τ )), or (iii) a quantum Poincaré map. The influence of a small perturbation to the unitary evolution, which is generated by a hermitean operator A, U δ = U exp(−iAδ), δ being a small parameter, is described by the overlap ψ δ (t)|ψ(t) measuring the Hilbert space distance between exact and perturbed time evolution from the same initial pure state |ψ(t) = U t |ψ , |ψ δ (t) = U t δ |ψ , where integer t is a discrete time (in units of period p) [4]. This defines the fidelitywhere the average is performed either over a fixed pure state . = ψ|.|ψ , or, if convenient, as a uniform average over all possible initial states . = (1/N ) tr (.), N being the Hilbert space dimension. The quantity F (t) has already raised considerable interest, though under different names and interpretations: First, it has been proposed by Peres [3] as a measure of stability of quantum motion. Second, it is the Loschmidt echo measuring the dynamical irreversibility of quantum phases, used e.g. in spin-echo experiments [5] where one is interested in the ov...

Quantum Reversibility: Is there an Echo?

“…3 where we present our numerical results for t r and various T . The points corresponding to the same λ but different values of ε evol and ε prep fall onto the same smooth curve with a good accuracy, confirming the scaling hypothesis (6). Moreover, we see that for λ > λ * , with λ * ≈ 1.3, we have t r = T /2 (no echo) irrespective of the actual value of ε prep and ε evol .…”

Analyses of the transmission of the disorder from a disturbed environment to a spin chain

“…The dynamical processes on the spin chain are induced by ultra-short kicks on all spins. The subject concerning decoherence processes of regularly kicked spin chains has been studied by some authors [10,11,12,13,14,15,16]. In these previous studies, the kick processes are regular.…”

Pinning quantum phase transition in a Tonks-Girardeau gas: Diagnostics by ground-state fidelity and the Loschmidt echo

“…We find that the GSF is in agreement with recent experiments on the pinning quantum phase transition (QPT) for a Luttinger liquid of strongly interacting bosons [7] in the sense that it confirms that the particle number used in the experiment was sufficient to observe the pinning transition. All of the observed properties of ground-state fidelity are also reflected in the dynamical evolution of the system, i.e., in survival probability or the Loschmidt echo [11][12][13][14][15] (for a review see, e.g., [16]). The average value of the Loschmidt echo decreases for lower value of ground-state fidelity; that is a general observation.…”