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QUANTUM COHERENCE AND LOSCHMIDT ECHO

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In a famous controversy with Ludwig Boltzmann at the dawn of modern statistical mechanics, Joseph Loschmidt pointed out that, if one reverses the velocities of all particles in a physical system, the latter would evolve back to its initial state, thus violating the second law of thermodynamics. The main objection to this line of reasoning is that velocity reversal is an extremely unstable operation and that tiny errors would quickly restore “normal” entropy increase. Nevertheless, time reversal is indeed possible, as was shown in spin echo experiments performed since the 1950s.

Loschmidt’s idea has recently experienced a resurgence of interest in the context of quantum information theory. Indeed, any attempt at coding information using quantum bits is prone to failure if a small coupling to the environment destroys the unitary evolution of the wave function (decoherence). In order to estimate the robustness of a physical system, the following procedure has been suggested: a single quantum particle evolves under the action of a chaotic Hamiltonian H0 until a time T; then, it is evolved backwards in time until 2T with the original Hamiltonian plus a small perturbation (the “environment”). The square of the scalar product of the initial and final states defines the Loschmidt echo or fidelity of the system, and is a reasonable measure of its quantum coherence. Theoretical and numerical studies showed that the Loschmidt echo decays exponentially with the time delay T.

What happens when one deals, not with a single particle, but with a large system of interacting particles, such as the electrons in a quantum well or a Bose-Einstien condensate (BEC) confined in a magnetic trap? We have simulated numerically the evolution of the quantum fidelity for such many-body systems and the results were intriguing: the fidelity does not decay exponentially, but rather stays close to unity until a critical time, after which it drops abruptly. This unusual behaviour is related to the fact that, for a many-body system in the mean-field approximation, the Hamiltonian depends on the wave function itself.

To test this behaviour on an atomic BEC, we proposed an experiment where two identical condensates are created in a double trap (see figure). The trap is suddenly shifted by a small distance Δz, and a random potential is switched on. The condensates evolve in each trap for some time, after which the confinement is removed, so that they expand freely and eventually overlap. The visibility of the observed interference fringes should display a behaviour similar to that of the quantum fidelity, characterized by a rapid drop after a critical time. The proposed experiment could be performed using standard optical techniques currently used in the manipulation of ultracold atoms.

This effect seems to be a generic feature of interacting many-particle systems. If confirmed, it would have an impact on the decoherence properties of solid-state quantum computation devices, which may then behave differently in the single-electron and many-electron regimes.

Publications

"Loschmidt echo for the many-electron dynamics in nonparabolic quantum wells",
G. Manfredi and P. -A. Hervieux, New Journal of Physics 11, 013050 (2009).

"Fidelity decay in trapped Bose-Einstein condensates",
G. Manfredi and P. -A. Hervieux, Phys. Rev. Lett. 100, 050405 (2008).

"Loschmidt Echo In a System of Interacting Electrons",
G. Manfredi and P. -A. Hervieux, Phys. Rev. Lett. 97, 190404 (2006).