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Physical Review A

Geometric phase with photon statistics and squeezed light for the dispersive fiber

Amitabh Joshi, Arun Kumar Pati, Arup Banerjee

Published Jun 1, 1994• 17 citations
Publisher

Abstract

The nonadiabatic geometric phase for an arbitrary cyclic evolution of the state vector for a system described by the quantum theory of the propagation of a single-mode off-resonant field through the dispersive fiber is discussed here. The sensitivity of the geometric phase on the initial field statistics has been explicitly shown in the calculations. We also derive the expressions of the geometric phase when the system is prepared initially in a squeezed coherent state.

DOI
10.1103/physreva.49.5131

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The nonadiabatic geometric phase for an arbitrary cyclic evolution of the state vector for a system described by the quantum theory of the propagation of a single-mode off-resonant field through the dispersive fiber is discussed here. The sensitivity of the geometric phase on the initial field statistics has been explicitly shown in the calculations. We also derive the expressions of the geometric phase when the system is prepared initially in a squeezed coherent state.
The geometric phases of the cyclic states of a generalized harmonic oscillator with nonadiabatic time-periodic parameters are discussed in the framework of squeezed state. It is shown that the cyclic and quasicyclic squeezed states correspond to the periodic and quasiperiodic solutions of an effective Hamiltonian defined on an extended phase space, respectively. The geometric phase of the cyclic squeezed state is found to be a phase-space area swept out by a periodic orbit. Furthermore, a class of cyclic states is expressed as a superposition of an infinite number of squeezed states. Their geometric phases are found to be independent of $\ensuremath{\Elzxh},$ and equal to $\ensuremath{-}(n+1/2)$ times the classical nonadiabatic Hannay angle.
A simple elegant expression of nonadiabatic light wave evolution is necessary in order to have a deeper insight for complicated optical phenomena in light science as well as in everyday life. Light wave propagation in linear media which have time-dependent electromagnetic parameters is investigated by utilizing a quadratic invariant of the system. The time behavior of the nonadiabatic geometric phase of the waves that yield a cyclic nonadiabatic evolution is analyzed in detail. Various quantum properties of light waves in this situation, such as variances of electric and magnetic fields, uncertainty product, coherent and squeezed states, and their classical limits, are developed. For better understanding of our research, we applied our analysis in a particular case. The variances of the fields D and B are illustrated and their time behaviors are addressed. Equivalent results for the corresponding classical systems are deduced from the study of the time evolution of the appropriate coherent and squeezed states.
The geometric phases of the cyclic states of a generalized harmonic oscillator with nonadiabatic time-periodic parameters are discussed in the framework of a squeezed state. A class of cyclic states is expressed as a superposition of an infinite number of squeezed states. Then their geometric phases are obtained explicitly and found to be $\ensuremath{-}(n+1/2)$ times the classical nonadiabatic Hannay angle. It is shown that the analysis based on the squeezed state approach provides a clear picture of the geometric meaning of the quantal phase.
Although the geometric phase for one-mode squeezed state had been studied in detail, the counterpart for two-mode squeezed state is vacant. It is be evaluated explicitly in this paper. Furthermore, the total phase factor is in an elegent form, which is just identical to one term of product of two squeezed operators. In addition, when this system undergoes cyclic evolutions, the corresponding geometric phase is obtained, which is just the sum of the counterparts of two isolated one-mode squeezed state. Finally, the relationship between the cyclic geomtric phase and entanglement of two-mode squeezed state is established.
Recently, geometric phases of nonlinear coherent and squeezed states have been investigated by Yang et al (2011 J. Phys. B: At. Mol. Opt. Phys. 44 075502). In this paper, after a modification of Yang et al's derivation procedure, we deduced the non-cyclic and non-unitary geometric phases of nonlinear coherent and nonlinear squeezed states with a new approach which is consistent with the nonlinear coherent states theory. Indeed, the most important distinguishable feature of our presentation lies in the fact that we employed the (nonlinear) Hamiltonian which appropriately generates the time evolution of nonlinear coherent and squeezed states. It is established that this modification directly affects the resultant geometric phases of the states under consideration. As a physical realization, we applied our formalism to a particular nonlinear physical system corresponding to the centre-of-mass motion of a trapped ion coherent state. Then, the variation of the geometric phases of the considered nonlinear states in terms of time, intensity of radiation field, squeezing and Lamb–Dicke parameters are investigated. Finally, we discuss our numerical results.