dorsal/arxiv
View SchemaA phase-space study of the quantum Loschmidt Echo in the semiclassical limit
| Authors | M. Combescure, D. Robert |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0510151 |
| URL | https://arxiv.org/abs/quant-ph/0510151 |
| DOI | 10.1007/s00023-006-0301-9 |
| Journal | Annales de l'Institut Henri Poincare Physique Theorique 8 (2007) 91-108 |
Abstract
The notion of Loschmidt echo (also called "quantum fidelity") has been introduced in order to study the (in)-stability of the quantum dynamics under perturbations of the Hamiltonian. It has been extensively studied in the past few years in the physics literature, in connection with the problems of "quantum chaos", quantum computation and decoherence. In this paper, we study this quantity semiclassically (as $\hbar \to 0$), taking as reference quantum states the usual coherent states. The latter are known to be well adapted to a semiclassical analysis, in particular with respect to semiclassical estimates of their time evolution. For times not larger than the so-called "Ehrenfest time" $C | \log \hbar |$, we are able to estimate semiclassically the Loschmidt Echo as a function of $t$ (time), $\hbar$ (Planck constant), and $\delta$ (the size of the perturbation). The way two classical trajectories merging from the same point in classical phase-space, fly apart or come close together along the evolutions governed by the perturbed and unperturbed Hamiltonians play a major role in this estimate. We also give estimates of the "return probability" (again on reference states being the coherent states) by the same method, as a function of $t$ and $\hbar$.
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"abstract": "The notion of Loschmidt echo (also called \"quantum fidelity\") has been\nintroduced in order to study the (in)-stability of the quantum dynamics under\nperturbations of the Hamiltonian. It has been extensively studied in the past\nfew years in the physics literature, in connection with the problems of\n\"quantum chaos\", quantum computation and decoherence. In this paper, we study\nthis quantity semiclassically (as $\\hbar \\to 0$), taking as reference quantum\nstates the usual coherent states. The latter are known to be well adapted to a\nsemiclassical analysis, in particular with respect to semiclassical estimates\nof their time evolution. For times not larger than the so-called \"Ehrenfest\ntime\" $C | \\log \\hbar |$, we are able to estimate semiclassically the Loschmidt\nEcho as a function of $t$ (time), $\\hbar$ (Planck constant), and $\\delta$ (the\nsize of the perturbation). The way two classical trajectories merging from the\nsame point in classical phase-space, fly apart or come close together along the\nevolutions governed by the perturbed and unperturbed Hamiltonians play a major\nrole in this estimate. We also give estimates of the \"return probability\"\n(again on reference states being the coherent states) by the same method, as a\nfunction of $t$ and $\\hbar$.",
"arxiv_id": "quant-ph/0510151",
"authors": [
"M. Combescure",
"D. Robert"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"doi": "10.1007/s00023-006-0301-9",
"journal_ref": "Annales de l\u0027Institut Henri Poincare Physique Theorique 8 (2007)\n 91-108",
"title": "A phase-space study of the quantum Loschmidt Echo in the semiclassical limit",
"url": "https://arxiv.org/abs/quant-ph/0510151"
},
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