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Physical Review A

Bateman Hamiltonian and squeezing

A. L. De Brito, B. Baseia

Published Oct 1, 1989• 11 citations
Publisher

Abstract

We show in this report that the Bateman Hamiltonian belongs to the class of quadratic Hamiltonians, in terms of \^a and a${^}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$, that generates squeezed states of the electromagnetic field.

DOI
10.1103/physreva.40.4097

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Physical Review Letters•May 31, 1993•21 citations
We show in this report that the Bateman Hamiltonian belongs to the class of quadratic Hamiltonians, in terms of ^a and a{^}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}, that generates squeezed states of the electromagnetic field.
The phase operators P ± of the electromagnetic field, whose classical analogues are exp (∓ iθ), θ being the classical phase, and the number operator N associated with a harmonic oscillator mode are used to construct a class of operators T m = N(P −)m and T + m = (P +)m N, m = 1, 2,…. It is shown that T m, T + m and N satisfy a simple algebra. T m and T + m are used to construct model Hamiltonians H m describing the coupling of a two-level atom to a field that leads to decay and exact revivals in the atomic dynamics. In the second part of the paper, a family of unitary operators U m is constructed in terms of T m and T + m. For m = 1 and m = 2 the states generated by U m from the vacuum are non-Gaussian squeezed states. The variance properties of the squeezed states so constructed are analysed and compared with those of the Gaussian squeezed states. It is shown that the non-Gaussian squeezed states constructed in this paper have small values for the variance product that rise only logarithmically with for large values of , the mean number of quanta.
A number-phase interaction is suggested that is analogous to the usual quadrature squeezing Hamiltonian for a mode of the electromagnetic field. Applied to an initial coherent state, it produces number-phase squeezed states. For moderate squeezing in either direction the result is close to a minimum uncertainty state. Two different ${\mathrm{\ensuremath{\chi}}}^{(3)}$ approximations to the ideal interaction are found, appropriate for the different directions of squeezing. Even with the approximate interaction, the fluctuations in photon number can be reduced to less than half a photon for an arbitrarily intense field.