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arXiv:1006.0339 · Physical Review A

Long-time coherence in echo spectroscopy with π/2 - π - π/2 pulse sequence

Arseni Goussev, Philippe Jacquod

Published Nov 18, 2010• 2 citations
arXivHTMLPDF

Abstract

Motivated by atomic optics experiments, we investigate a class of fidelity functions describing the reconstruction of quantum states by time-reversal operations as ${M}_{\mathrm{Da}}(t)=|\ensuremath{\langle}\ensuremath{\psi}|{e}^{i{H}_{2}t/2}{e}^{i{H}_{1}t/2}{e}^{\ensuremath{-}i{H}_{2}t/2}{e}^{\ensuremath{-}i{H}_{1}t/2}|\ensuremath{\psi}\ensuremath{\rangle}|{}^{2}$. We show that the decay of ${M}_{\mathrm{Da}}$ is quartic in time at short times and that it freezes well above the ergodic value at long times, when ${H}_{2}\ensuremath{-}{H}_{1}$ is not too large. The long-time saturation value of ${M}_{\mathrm{Da}}$ contains easily extractable information on the strength of decoherence in these systems.

DOI
10.1103/physreva.82.052114

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In echo experiments, imperfect time-reversal operations are performed on a subset of the total number of degrees of freedom. To capture the physics of these experiments, we introduce a partial fidelity M(B)(t), the Boltzmann echo, where only part of the system's degrees of freedom can be time reversed. We present a semiclassical calculation of M(B)(t). We show that, as the time-reversal operation is performed more and more accurately, the decay rate of M(B)(t) saturates at a value given by the decoherence rate of the controlled degrees of freedom due to their coupling to uncontrolled ones. We connect these results with NMR spin echo experiments.
Motivated by neutron scattering experiments, we investigate the decay of the fidelity with which a wave packet is reconstructed by a perfect time-reversal operation performed after a phase-space displacement. In the semiclassical limit, we show that the decay rate is generically given by the Lyapunov exponent of the classical dynamics. For small displacements, we additionally show that, following a short-time Lyapunov decay, the decay freezes well above the ergodic value because of quantum effects. Our analytical results are corroborated by numerical simulations.
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We investigate the dephasing of ultra cold 85Rb atoms trapped in an optical dipole trap and prepared in a coherent superposition of their two hyperfine ground states by interaction with a microwave pulse. We demonstrate that the dephasing, measured as the Ramsey fringe contrast, can be reversed by stimulating a coherence echo with a pi pulse between the two pi / 2 pulses, in analogy to the photon echo. We also demonstrate that "echo spectroscopy" can be used to study the quantum dynamics in the trap even when more than 10(6) states are thermally populated and to study the crossover from quantum to classical dynamics.
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.