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Higher power squeezed states, Jacobi matrices, and the Hamburger moment problem
Quantum Physics · Submitted 17 Nov 1997
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The corresponding squeezed states are well-defined, however.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 0.59 | 0.37 | 0.06 | asserted 1.00 |
| s1 | Equivalently they are the set of normalized eigenstates (for all possible complex eigenvalues) of the Bogolubov transformed "k:th power annihilation operators". | result 0.50 | 0.45 | 0.07 | asserted 1.00 |
| s2 | 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. | method 0.53 | 0.73 | 0.05 | asserted 0.99 |
| s3 | The solutions are essentially Hermite and Pollaczek polynomials, respectively. k = 1 gives the ordinary squeezed states, i.e. displaced squeezed vacua. | result 0.93 | 0.64 | 0.07 | asserted 1.00 |
| s4 | 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. | result 0.66 | 0.63 | 0.44 | asserted 1.00 |
| s5 | This implies that the operator has an infinity of self-adjoint extensions, all with disjoint discrete spectra. | interpretation 0.83 | 0.65 | 0.22 | asserted 1.00 |
| s6 | The corresponding squeezed states are well-defined, however. | result 0.61 | 0.60 | 0.20 | asserted 1.00 |
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Original 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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Among the diverse generalizations discussed in the literature, multiple authors considered $k$-photon generalizations of the squeezing operator in <ref> in the form [equation] with $k≥3$, together with similar generalizations of well-known models of light–matter interaction such as the Rabi and Jaynes–Cummings model [ref] .
Self-adjoint realizations of higher-order squeezing operators · Introduction · 0.86
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