dorsal/arxiv
View SchemaWigner function statistics in classically chaotic systems
| Authors | Martin Horvat, Tomaz Prosen |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0601165 |
| URL | https://arxiv.org/abs/quant-ph/0601165 |
| DOI | 10.1088/0305-4470/36/14/307 |
| Journal | J. Phys. A: Math. Gen. 36 (2003) 4015-4034 |
Abstract
We have studied statistical properties of the values of the Wigner function W(x) of 1D quantum maps on compact 2D phase space of finite area V. For this purpose we have defined a Wigner function probability distribution P(w) = (1/V) int delta(w-W(x)) dx, which has, by definition, fixed first and second moment. In particular, we concentrate on relaxation of time evolving quantum state in terms of W(x), starting from a coherent state. We have shown that for a classically chaotic quantum counterpart the distribution P(w) in the semi-classical limit becomes a Gaussian distribution that is fully determined by the first two moments. Numerical simulations have been performed for the quantum sawtooth map and the quantized kicked top. In a quantum system with Hilbert space dimension N (similar 1/hbar) the transition of P(w) to a Gaussian distribution was observed at times t proportional to log N. In addition, it has been shown that the statistics of Wigner functions of propagator eigenstates is Gaussian as well in the classically fully chaotic regime. We have also studied the structure of the nodal cells of the Wigner function, in particular the distribution of intersection points between the zero manifold and arbitrary straight lines.
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"abstract": "We have studied statistical properties of the values of the Wigner function\nW(x) of 1D quantum maps on compact 2D phase space of finite area V. For this\npurpose we have defined a Wigner function probability distribution P(w) = (1/V)\nint delta(w-W(x)) dx, which has, by definition, fixed first and second moment.\nIn particular, we concentrate on relaxation of time evolving quantum state in\nterms of W(x), starting from a coherent state. We have shown that for a\nclassically chaotic quantum counterpart the distribution P(w) in the\nsemi-classical limit becomes a Gaussian distribution that is fully determined\nby the first two moments. Numerical simulations have been performed for the\nquantum sawtooth map and the quantized kicked top. In a quantum system with\nHilbert space dimension N (similar 1/hbar) the transition of P(w) to a Gaussian\ndistribution was observed at times t proportional to log N. In addition, it has\nbeen shown that the statistics of Wigner functions of propagator eigenstates is\nGaussian as well in the classically fully chaotic regime. We have also studied\nthe structure of the nodal cells of the Wigner function, in particular the\ndistribution of intersection points between the zero manifold and arbitrary\nstraight lines.",
"arxiv_id": "quant-ph/0601165",
"authors": [
"Martin Horvat",
"Tomaz Prosen"
],
"categories": [
"quant-ph",
"nlin.CD"
],
"doi": "10.1088/0305-4470/36/14/307",
"journal_ref": "J. Phys. A: Math. Gen. 36 (2003) 4015-4034",
"title": "Wigner function statistics in classically chaotic systems",
"url": "https://arxiv.org/abs/quant-ph/0601165"
},
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