Semiclassical evaluation of quantum fidelity

Abstract

We present a numerically feasible semiclassical ͑SC͒ method to evaluate quantum fidelity decay ͑Loschmidt echo͒ in a classically chaotic system. It was thought that such evaluation would be intractable, but instead we show that a uniform SC expression not only is tractable but it also gives remarkably accurate numerical results for the standard map in both the Fermi-golden-rule and Lyapunov regimes. Because it allows Monte Carlo evaluation, the uniform expression is accurate at times when there are 10 70 semiclassical contributions. Remarkably, it also explicitly contains the ''building blocks'' of analytical theories of recent literature, and thus permits a direct test of the approximations made by other authors in these regimes, rather than an a posteriori comparison with numerical results. We explain in more detail the extended validity of the classical perturbation approximation and show that within this approximation, the so-called ''diagonal approximation'' is automatic and does not require ensemble averaging. The question of stability of quantum motion, originally formulated by Peres ͓1͔, has recently attracted much interest, due to its relevance to quantum computation and decoherence in complex systems. Peres defined stability in terms of quantum fidelity M (t), the overlap at time t of two states, which were identical at time tϭ0, but afterwards propagated in slightly different dynamical systems, described by Hamiltonians H 0 and H V ϭH 0 ϩV,This quantity is also called Loschmidt echo, because it can be interpreted as an overlap of a state propagated forward for time t with H 0 and then backward for time t with H V , with the initial state. We consider H 0 to be strongly chaotic, although our method is not limited to this case. Even with this restriction, the decay of fidelity has a surprisingly rich behavior: Most surprising recently was the derivation in Ref.͓2͔ that for a certain range of perturbations the decay rate is independent of the perturbation strength.The Loschmidt echo is physically realizable, for example, in NMR spin echo experiments, where back propagation under a slightly different Hamiltonian is feasible ͓3-5͔. There are other examples, which often go unnoticed. An example is neutron scattering, where the scattering kernel can be written as in Eq. ͑1͒, with H V a momentum boosted version of H 0 . Many numerical investigations of fidelity decay ͑FD͒ have been undertaken in various systems ͓6-32͔. Depending on the strength of perturbation, there exist at least four qualitatively different regimes of the decay in chaotic systems ͓6͔: As the perturbation increases, these regimes are perturbative ͑PT͒, Fermi-golden-rule ͑FGR͒, Lyapunov ͑L͒, and the strong semiclassical ͑SC͒ regimes.In the PT regime, in which the characteristic matrix element of the perturbation is smaller than the mean level spacing ⌬, the decay can be described by a combination of perturbation theory and random-matrix theory ͑RMT͒, and is Gaussian ͓6,7͔,For intermediate perturbation strengths, the decay follows the Fermi g...

Quantum chaos, dynamical stability and decoherence

“…Confining ourselves to classically chaotic systems, the emerging picture which results from analytical and numerical investigations [7,[10][11][12][13][14][15][16][17][18] is that both exponential and Gaussian decays are present in the time behavior of fidelity. The strength of the perturbation determines which of the two regimes prevails.…”

Efficient evaluation of the accuracy of molecular quantum dynamics on an approximate analytical or interpolated ab initio potential energy surface

“…By using only the trajectories of the approximate PES, the dephasing representation is related to the semiclassical perturbation approximation [15]. However, in our case, due to the shadowing property, it is not necessary to assume that the trajectory of V acc remains near the trajectory of V appr with exactly the same initial condition [12,13]. Note that we suggest using trajectories of V appr for computing f DR .…”

“…Li, Mollica, and Vaníček have circumvented the necessity to compute ψ acc (t) by using the dephasing representation (DR) of quantum fidelity [10,11]. The DR is a semiclassical approximation of fidelity [12,13] that has been shown to be accurate in chaotic, integrable, and mixed systems even in nonuniversal regimes sensitive to the choice of the initial state and details of dynamics [10,11]. To evaluate the accuracy of V appr , a specific type of "perturbation" is considered, namely the difference V = V appr −V acc between the approximate and accurate PESs.…”

Efficient on-the-fly ab initio semiclassical method for computing time-resolved nonadiabatic electronic spectra with surface hopping or Ehrenfest dynamics

“…The MSDR is a generalization to nonadiabatic dynamics of the dephasing representation (DR) [7][8][9] of quantum fidelity. 10 In the context of spectroscopy, the DR has been known as phase-averaging 11 and used to compute electronic spectra within the Born-Oppenheimer approximation.…”