The Minimum-Uncertainty Squeezed States for for Atoms and Photons in a Cavity

Abstract

We describe a six-parameter family of the minimum-uncertainty squeezed states for the harmonic oscillator in nonrelativistic quantum mechanics. They are derived by the action of corresponding maximal kinematical invariance group on the standard ground state solution. We show that the product of the variances attains the required minimum value 1/4 only at the instances that one variance is a minimum and the other is a maximum, when the squeezing of one of the variances occurs. The generalized coherent states are explicitly constructed and their Wigner function is studied. The overlap coefficients between the squeezed, or generalized harmonic, and the Fock states are explicitly evaluated in terms of hypergeometric functions. The corresponding photons statistics are discussed and some applications to quantum optics, cavity quantum electrodynamics, and superfocusing in channeling scattering are mentioned. Explicit solutions of the Heisenberg equations for radiation field operators with squeezing are found.

27 pages, no figures, 174 references J. Phys. B: At. Mol. Opt. Phys., Special Issue celebrating the 20th anniversary of quantum state engineering (R. Blatt, A. Lvovsky, and G. Milburn, Guest Editors), May 2013

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• Exceptional point in a coupled Swanson system
• Liouvillian Propagators, Riccati Equation and Differential Galois Theory
• On the Problem of Electromagnetic-Field Quantization
• Degenerate Parametric Amplification of Squeezed Photons: Explicit Solutions, Statistics, Means and Variances
• Comment on `The minimum-uncertainty squeezed states for atoms and photons in cavity'
• Swanson Hamiltonian: non-PT-symmetry phase
• On the $\mathcal{P}\mathcal{T}$-symmetric parametric amplifier
• Wave Packets and Coherent Structures for Nonlinear Schroedinger equations in Variable Nonuniform Media