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arXiv:1011.0501 · Journal of Physics B Atomic Molecular and Optical Physics

Geometric phases for nonlinear coherent and squeezed states

Da-Bao Yang, Ying Chen, Fu-Lin Zhang, Jing-Ling Chen

Published Mar 16, 2011• 12 citations

Abstract

The geometric phases for standard coherent states which are widely used in quantum optics have attracted considerable attention. Nevertheless, few physicists consider the counterparts of nonlinear coherent states, which are useful in the description of the motion of a trapped ion. In this paper, the non-unitary and non-cyclic geometric phases for two nonlinear coherent and one squeezed states are formulated, respectively. Moreover, some of their common properties are discussed, such as gauge invariance, non-locality and nonlinear effects. The nonlinear functions have dramatic impacts on the evolution of the corresponding geometric phases. They speed the evolution up or down. So this property may have an application in controlling or measuring geometric phase. For the squeezed case, when the squeezed parameter r → ∞ , the limiting value of the geometric phase is also determined by a nonlinear function at a given time and angular velocity. In addition, the geometric phases for standard coherent and squeezed states are obtained under a particular condition. When the time evolution undergoes a period, their corresponding cyclic geometric phases are achieved as well. And the distinction between the geometric phases of the two coherent states may be regarded as a geometric criterion.

DOI: 10.1088/0953-4075/44/7/075502

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In this paper, we investigate the geometric phase (GP) of two-mode entangled squeezed-coherent states (ESCSs), undergoing unitary cyclic evolution. Results show that increasing the squeezing parameter of either mode of the balanced ESCS compresses the GP elliptically with respect to the coherence parameter of the corresponding mode. While in the case of unbalanced ESCS, the GP is compressed hyperbolically by increasing the squeezing parameters of the either mode. By generalizing the approach to higher constituting-state dimensions, it is found that the GPs of both balanced and unbalanced ESCSs, increase for a specific value of the coherence parameter. By analyzing the states through the Schmidt decomposition method, we find that, locally, the balanced and unbalanced ESCSs are unitarily equivalent. Finally, based on the interferometry technique, we suggest a theoretical scheme for the physical generation of ESCSs.

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.

The geometric phase, as well as the familiar dynamical phase, occurs in the evolution of a squeezed state in nano-optics as an extra phase. The outcome of the geometric phase in that state is somewhat intricate: its time behavior exhibits a combination of a linear increase and periodic oscillations. We focus in this work on the periodic oscillations of the geometric phase, which are novel and interesting. We confirm that such oscillations are due purely to the effects of squeezing in the quantum states, whereas the oscillation disappears when we remove the squeezing. As the degree of squeezing increases in q-quadrature, the amplitude of the geometric-phase oscillation becomes large. This implies that we can adjust the strength of such an oscillation by tuning the squeezing parameters. We also investigate geometric-phase oscillations for the case of a more general optical phenomenon where the squeezed state undergoes one-photon processes. It is shown that the geometric phase in this case exhibits additional intricate oscillations with small amplitudes, besides the principal oscillation. Such a sub-oscillation exhibits a beating-like behavior in time. The effects of geometric-phase oscillations are crucial in a wide range of wave interferences which are accompanied by rich physical phenomena such as Aharonov–Bohm oscillations, conductance fluctuations, antilocalizations, and nondissipative current flows.