dorsal/arxiv
View SchemaHigher power squeezed states, Jacobi matrices, and the Hamburger moment problem
| Authors | Bengt Nagel |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9711028 |
| URL | https://arxiv.org/abs/quant-ph/9711028 |
Abstract
k:th power (amplitude-)squeezed states are defined as the normalized states giving equality in the Schroedinger-Robertson uncertainty relation for the real and imaginary parts of the k:th power of the one-mode annihilation operator. Equivalently they are the set of normalized eigenstates (for all possible complex eigenvalues) of the Bogolubov transformed "k:th power annihilation operators". Expressed in the number representation the eigenvalue equation leads to a three term recursion relation for the expansion coefficients, which can be explicitly solved in the cases k = 1, 2. The solutions are essentially Hermite and Pollaczek polynomials, respectively. k = 1 gives the ordinary squeezed states, i.e. displaced squeezed vacua. For k equal to or larger than three, where no explicit solution has been found, the recursion relation for the symmetric operator given by the real part of the k:th power of the annihilation operator defines a Jacobi matrix corresponding to a classical Hamburger moment problem, which is undetermined. This implies that the operator has an infinity of self-adjoint extensions, all with disjoint discrete spectra. The corresponding squeezed states are well-defined, however.
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"abstract": "k:th power (amplitude-)squeezed states are defined as the normalized states\ngiving equality in the Schroedinger-Robertson uncertainty relation for the real\nand imaginary parts of the k:th power of the one-mode annihilation operator.\nEquivalently they are the set of normalized eigenstates (for all possible\ncomplex eigenvalues) of the Bogolubov transformed \"k:th power annihilation\noperators\". Expressed in the number representation the eigenvalue equation\nleads to a three term recursion relation for the expansion coefficients, which\ncan be explicitly solved in the cases k = 1, 2. The solutions are essentially\nHermite and Pollaczek polynomials, respectively. k = 1 gives the ordinary\nsqueezed states, i.e. displaced squeezed vacua. For k equal to or larger than\nthree, where no explicit solution has been found, the recursion relation for\nthe symmetric operator given by the real part of the k:th power of the\nannihilation operator defines a Jacobi matrix corresponding to a classical\nHamburger moment problem, which is undetermined. This implies that the operator\nhas an infinity of self-adjoint extensions, all with disjoint discrete spectra.\nThe corresponding squeezed states are well-defined, however.",
"arxiv_id": "quant-ph/9711028",
"authors": [
"Bengt Nagel"
],
"categories": [
"quant-ph"
],
"title": "Higher power squeezed states, Jacobi matrices, and the Hamburger moment problem",
"url": "https://arxiv.org/abs/quant-ph/9711028"
},
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