dorsal/arxiv
View SchemaThe Loschmidt echo in classically chaotic systems: Quantum chaos, irreversibility and decoherence
| Authors | Fernando M. Cucchietti |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0410121 |
| URL | https://arxiv.org/abs/quant-ph/0410121 |
Abstract
The Loschmidt echo (LE) is a measure of the sensitivity of quantum mechanics to perturbations in the evolution operator. It is defined as the overlap of two wave functions evolved from the same initial state but with slightly different Hamiltonians. Thus, it also serves as a quantification of irreversibility in quantum mechanics. In this thesis the LE is studied in systems that have a classical counterpart with dynamical instability, that is, classically chaotic. An analytical treatment that makes use of the semiclassical approximation is presented. It is shown that, under certain regime of the parameters, the LE decays exponentially. Furthermore, for strong enough perturbations, the decay rate is given by the Lyapunov exponent of the classical system. Some particularly interesting examples are given. The analytical results are supported by thorough numerical studies. In addition, some regimes not accessible to the theory are explored, showing that the LE and its Lyapunov regime present the same form of universality ascribed to classical chaos. In a sense, this is evidence that the LE is a robust temporal signature of chaos in the quantum realm. Finally, the relation between the LE and the quantum to classical transition is explored, in particular with the theory of decoherence. Using two different approaches, a semiclassical approximation to Wigner functions and a master equation for the LE, it is shown that the decoherence rate and the decay rate of the LE are equal. The relationship between these quantities results mutually beneficial, in terms of the broader resources of decoherence theory and of the possible experimental realization of the LE.
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"abstract": "The Loschmidt echo (LE) is a measure of the sensitivity of quantum mechanics\nto perturbations in the evolution operator. It is defined as the overlap of two\nwave functions evolved from the same initial state but with slightly different\nHamiltonians. Thus, it also serves as a quantification of irreversibility in\nquantum mechanics.\n In this thesis the LE is studied in systems that have a classical counterpart\nwith dynamical instability, that is, classically chaotic. An analytical\ntreatment that makes use of the semiclassical approximation is presented. It is\nshown that, under certain regime of the parameters, the LE decays\nexponentially. Furthermore, for strong enough perturbations, the decay rate is\ngiven by the Lyapunov exponent of the classical system. Some particularly\ninteresting examples are given.\n The analytical results are supported by thorough numerical studies. In\naddition, some regimes not accessible to the theory are explored, showing that\nthe LE and its Lyapunov regime present the same form of universality ascribed\nto classical chaos. In a sense, this is evidence that the LE is a robust\ntemporal signature of chaos in the quantum realm.\n Finally, the relation between the LE and the quantum to classical transition\nis explored, in particular with the theory of decoherence. Using two different\napproaches, a semiclassical approximation to Wigner functions and a master\nequation for the LE, it is shown that the decoherence rate and the decay rate\nof the LE are equal. The relationship between these quantities results mutually\nbeneficial, in terms of the broader resources of decoherence theory and of the\npossible experimental realization of the LE.",
"arxiv_id": "quant-ph/0410121",
"authors": [
"Fernando M. Cucchietti"
],
"categories": [
"quant-ph",
"nlin.CD"
],
"title": "The Loschmidt echo in classically chaotic systems: Quantum chaos, irreversibility and decoherence",
"url": "https://arxiv.org/abs/quant-ph/0410121"
},
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