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Physical Review A

Higher-order squeezing properties and correlation functions for squeezed number states

Paulina Marian

Published Sep 1, 1991• 58 citations
Publisher

Abstract

Higher-order squeezing conditions for squeezed number states are derived using a normal-ordering technique for calculating the moments of the field. Intrinsic higher-order squeezing is also investigated. It is found that the normally ordered moments of the quadrature operators are oscillating functions of the squeeze parameter. Also calculated is the exact nth-order correlation function for an arbitrary squeezed number state. Finally, its behavior for large and weak squeezing is analyzed.

DOI
10.1103/physreva.44.3325

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Higher-order squeezing conditions for squeezed number states are derived using a normal-ordering technique for calculating the moments of the field. Intrinsic higher-order squeezing is also investigated. It is found that the normally ordered moments of the quadrature operators are oscillating functions of the squeeze parameter. Also calculated is the exact nth-order correlation function for an arbitrary squeezed number state. Finally, its behavior for large and weak squeezing is analyzed.
By use of the squeezed-number-state basis, we study several features of the squeezed thermal states. Higher-order squeezing properties are found to be entirely determined by the second-order ones. The photon-number distribution is expressed in closed form in terms of a Legendre polynomial. We point out that at a critical value of the squeeze parameter, pairwise oscillations of the photon-number distribution set in. Finally, a compact analytical expression of the exact lth-order correlation function is derived and examined.
We examine the influence of thermal noise on several non-classical properties of squeezed number states. In order to evaluate arbitrary-order moments of both amplitude and quadrature operators of the superposition, we use a normal-ordering technique based on McCoy's theorem. Our analytical results are compact formulae involving Gauss hypergeometric functions. We find that the thermal mean occupancy sufficient to destroy squeezing to any order is less than . Other non-classical features of a squeezed number state, such as pairwise oscillations in the photon number distribution, sub-Poissonian statistics and amplitude-squared squeezing disappear completely above this threshold.
The concept of squeezing, which generally implies that one quadrature component of an optical field has measured fluctuations below the vacuum limit, is generalized to higher moments of even order. Higher-order squeezing is encountered in many of the nonlinear processes that generate second-order squeezing. It is shown that the field produced in degenerate parametric downconversion is squeezed to all even orders and is intrinsically squeezed to orders 2,6,10…. Higher-order squeezing also shows up in second harmonic generation and in resonance fluorescence from an atom. However, in those cases there is no intrinsic squeezing beyond the second order, although a strong second-order squeezing effect is manifest in the higher-order moments of the field.
We have investigated the properties of two-mode squeezed states of various kinds: squeezed coherent states, squeezed number states, squeezed thermal states, and squeezed number-thermal states. The properties studied include characteristic functions, photon number distributions, and high-order moments. We show that a two-mode squeezed coherent state appears in either of its modes as a Glauber-Lachs superposition of a coherent signal with thermal noise; this generalizes the well-known thermofield result that a two-mode squeezed vacuum state behaves like a pure thermal state in either of its modes. We investigate the higher-order squeezing (in the sense of Hong and Mandel) of squeezed coherent-thermal and squeezed number-thermal states. Although the squeezing is found to increase with order in both cases, there are nevertheless interesting differences that can be traced to the fact that the Wigner function has a Gaussian form for the former but not for the latter.