The quantum Loschmidt echo on flat tori. (English) Zbl 1459.35321
Summary: The quantum Loschmidt echo is a measurement of the sensitivity of a quantum system to perturbations of the Hamiltonian. In the case of the standard two-torus, we derive some explicit formulae for this quantity in the transition regime where it is expected to decay in the semiclassical limit. The expression involves both a two-microlocal defect measure of the initial data and the form of the perturbation. As an application, we exhibit a non-concentration criterium on the sequence of initial data under which one does not observe a macroscopic decay of the quantum Loschmidt echo. We also apply our results to several examples of physically relevant initial data such as coherent states and plane waves.
MSC:
| 35Q40 | PDEs in connection with quantum mechanics |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
Keywords:
semiclassical analysis; semiclassical measures; quantum fidelity; completely integrable systemsReferences:
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