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arXiv:2304.08157 · International Journal of Geometric Methods in Modern Physics

Geometric phase for two-mode entangled squeezed-coherent states

S. Mohammadi Almas, G. Najarbashi, Ali Tavana

Published Feb 3, 2024• 1 citation
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Abstract

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.

DOI
10.1142/s0219887824501470

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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.
In this paper, we investigate the geometric phase (GP) acquired by two-mode mixed squeezed-coherent states (SCSs) during unitary cyclic evolution, examining the influence of squeezing parameter and classical weight. We analyze the GP for three distinct mixed states, each characterized by different configurations of the SCSs. Our results reveal that increasing the squeezing parameters of individual modes compresses the GP contours in different patterns: linearly, hyperbolically, and elliptically, depending on the mixed state configuration. This behavior demonstrates the enhancement of quantum measurement precision in squeezed states through uncertainty adjustment, consistent with the established theoretical predictions.
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.
Although the geometric phase for one-mode squeezed state had been studied in detail, the counterpart for two-mode squeezed state is vacant. In this paper, we aim at the special case, namely, two-mode squeezed vacuum state. Furthermore, the total phase factor is to be written in an elegant 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 the sum of the counterparts of two isolated one-mode squeezed vacuum state. Finally, the relationship between the cyclic geometric phase and entanglement of two-mode squeezed vacuum state is established. Key words: Quantum optics, Geometric phases, Entanglement
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.
Recently, we presented (Berrada K, Abdel-Khalek S and Raymond Ooi C H 2012 Phys. Rev. A 86 033823) an improved phase estimation scheme employing entangled spin coherent states (ESCSs) using the Holstein?Primakoff realization of angular momentum algebra. Here, we study the nonlinear phase enhancement from a generalized nonlinearity on ESCSs under perfect and lossy conditions with the same mean photon number and nonlinearity order. The results show that an increase in the spin number gives the smallest variance in the phase parameter in comparison to N00N states for different orders of nonlinearity. Finally, we study the physical properties of the input optical field and explore a connection between this quantity and the output state phase uncertainty. In particular, we show that the Mandel parameter may be used as an indicator of the phase estimation behavior in this interferometric setting.
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.