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Journal of Mathematical Physics

Para-squeezed states

G. M. Saxena, C. L. Mehta

Published Mar 1, 1991• 4 citations
Publisher

Abstract

In this paper the squeezed states for a one mode para-bose system is considered and it will be shown that a recently published contradictory assertion that no such states exist is based on an erroneous consideration. It is found that the squeeze operator is identical for both the ordinary and para-bose systems. The uncertainties in the canonical variables q (position) and p (momentum) in the para-squeezed state are obtained. The normal ordered form of the para-squeeze operator is also obtained and is then used to write the number and the coordinate representation of the para-bose vacuum.

DOI
10.1063/1.529371

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In this paper the squeezed states for a one mode para-bose system is considered and it will be shown that a recently published contradictory assertion that no such states exist is based on an erroneous consideration. It is found that the squeeze operator is identical for both the ordinary and para-bose systems. The uncertainties in the canonical variables q (position) and p (momentum) in the para-squeezed state are obtained. The normal ordered form of the para-squeeze operator is also obtained and is then used to write the number and the coordinate representation of the para-bose vacuum.
We find that the exponential operators U=exp[-i(${\ensuremath{\lambda}}_{1}$Q1{\mathit{Q}}_{1}Q1​P2{\mathit{P}}_{2}P2​+${\ensuremath{\lambda}}^{2}$Q2{\mathit{Q}}_{2}Q2​P1{\mathit{P}}_{1}P1​] and W=exp[-i(${\ensuremath{\lambda}}_{1}$Q1{\mathit{Q}}_{1}Q1​Q2{\mathit{Q}}_{2}Q2​-${\ensuremath{\lambda}}_{2}$pP1{\mathrm{P}}_{1}P1​P2{\mathit{P}}_{2}P2​)] (where ${\ensuremath{\lambda}}_{1}$ and ${\ensuremath{\lambda}}_{2}$ are real; Pi{P}_{i}Pi​ and Qi{Q}_{i}Qi​ are coordinate and momentum operators, respectively; and i=1,2) are two types of generalized squeezing operators responsible for generating two types of one- and two-mode combination squeezed states, respectively. The coordinate and/or momentum representations of U and W are presented, and their normal product forms are first derived in terms of the newly developed technique of integration within an ordered product, which provides us with a direct approach to constructing these new squeezed states. The fluctuations in quadrature phases for these states are analyzed.
The one- and two-mode combination squeezed states are constructed by applying the operator U identical to exp(-i lambda (erQ1P2+e-rQ2P1)) or W identical to exp(-i lambda (erQ1Q2-e-rP1P2)) (where lambda , r are real; Qi, Pi, i=1, 2 are coordinate and momentum operators, respectively) to the vacuum state. When the condition tanh lambda <1/(1+cosh r) is satisfied, such states can have less fluctuations in one quadrature phase than the usual two-mode squeezed state. Here, the authors examine the higher-order squeezing properties for such states. The result shows that under the same conditions, these states can exhibit a stronger form of higher-order squeezing than the usual two-mode squeezed state.
We find that the exponential operator V ≡exp[ i λ( Q 1 P 2 + Q 2 P 3 +⋅⋅⋅+ Q n −1 P n + Q n P 1 )], Q i , P i are, respectively, the coordinate and momentum operators, is an n -mode squeezing operator which engenders standard squeezing. By virtue of the technique of integration within an ordered product of operators we derive V 's normally ordered expansion and obtain the n -mode squeezed vacuum states, its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.
The construction of eigenstates of the square of annihilation operators for a two-mode para-Bose system is reported. Bose coherent states can be deduced from these eigenstates as a special case. These states are termed para-Bose coherent states. These states are degenerate. The expansion of the coherent states in terms of two-mode para-Bose energy eigenstates has been obtained and their salient properties are discussed. Also discussed is the uncertainty product of the square of position and momentum operators in the para-Bose coherent states for a two-mode system.
It is known that exp [iλ ( Q 1 P 1 – i/2)] is a unitary single-mode squeezing operator, where Q 1 , P 1 are the coordinate and momentum operators, respectively. In this paper we employ Dirac's coordinate representation to prove that the exponential operator , is an n -mode squeezing operator which enhances the standard squeezing. By virtue of the technique of integration within an ordered product of operators we derive S n 's normally ordered expansion and obtain new n -mode squeezed vacuum states, its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.