Geometric phases for two-mode squeezed vacuum state

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.


INTRODUCTION
Squeezed light plays an important role in the development of quantum optics (Walls, 1983). It preserves the minimum uncertainty and exhibits non-classical nature of light, such as sub-Poissonian statistics which can be observed as photon antibunching effect. It also has many applications in optical communications and detection of gravitational radiation. It was be generalized to nonlinear case (Kwek and Kiang, 2003). But their studies were just confined to one mode case. Moreover, two mode squeezed state was systematically studied Schumaker and Caves, 1985).
Since geometric phase had been discovered (Berry, 1984) in the quantum system which underwent adiabatic and unitary evolution, its research exploded. Subsequently, It was extended to non-Abelian case (Wilczek and Zee, 1984). Its non-adiabatic and cyclic counterpart was studied (Aharonov and Anandan, 1987;Anandan, 1988). Soon, by getting rid of the condition of cyclic evolution, it was generalized to a more general case (Samuel and Bhandari, 1988), which depended on the earlier study (Pancharatnam, 1956). Subsequently, using kinematic approach, geometric phase was derived as well (Mukunda and Simon, 1993).
Meanwhile, the interdisciplinary study between quantum optics and geometric phase has also emerged. Berry phase for coherent and squeezed states was researched (Chaturvedi et al., 1987); the non-adiabatic geometric phase for squeezed state was studied by Liu et al. (1998); the geometric phase for nonlinear coherent and squeezed state in kinematic approach was discussed (Yang et al., 2011). However, the above studies are all confined to one-mode case. In seeking for theoretical progress, the two-mode case will be researched in this paper. Moreover, the degree of entanglement between the two-mode state is to be evaluated. This paper is organized as follows. First is a presentation of the features of two-modes squeezed states followed by a review of the kinematic approach to geometric phase. Next, the geometric phase for two-mode squeezed state was calculated. From the above outcome, when the system undergoes cyclic evolution, the corresponding result was also obtained. Moreover, the Von Neumann entropy was calculated, and its relation with geometric phase established. Finally, the research was concluded.

REVIEW OF TWO-MODE SQUEEZED VACUUM STATES AND GEOMETRIC PHASES
The Hamiltonian for two-mode of electromagnetic field where   are the frequencies for the two-mode; also, we take 1  for simplicity. Furthermore,  and  can be regarded as a carrier frequency and a modulation frequency respectively. And the electromagnetic field are quantized by the following commutation relations † The squeezed operator where the real number r is called the squeeze factor and  is a real phase angle. Moreover, the above operator (2) Under the Hamiltonian (1), it evolves as which uses the following formulas

a a a a S r i a a a a S r
The geometric phases  (Mukunda and Simon, 1993) for arbitrary time t takes the form It is a physical reality, due to it is invariant under gauge transformation. And it can be explained as an outcome of parallel transportation in the framework of fiber bundle, that is, holonomy, hence it is fittingly called geometric phase.

EVALUATIONS OF THE GEOMETRIC PHASE FACTOR
For convenience, instead of calculating the geometric phase, we evaluate the geometric phase factor, which is identical to negative the dynamical phase.
At first, let us calculate the inner product which uses Equation (4). In order to work out the total phase, the following formula  is very useful † ( ) ( where the above parameters satisfy the matrix equation By use of the explicit decomposition of squeezed operator : which is identical to the result in Van Enk (1999). Finally,