Geometric Phase using Diagonal Coherent State Representation

Abstract

Glauber-Sudarshan diagonal coherent state P-representation has been used to determine geometric phase for non-classical states of light. For a given density operator $\hat{\rho_1}$ of two mode optical beam, we evolve it in complex projective ray space $(\cal R)$ to $\hat{\rho_2}$ and to $\hat{\rho_3}$ by changing its state of polarisation using unitary operator $\mathrm{\hat{U}_p}(\theta)$. The diagonal coherent state basis has been utilized to represent the density operators instead of fock state basis as in the fock state basis the state vector in present work evolve under unitary operator produe infinitely numerous terms which make the density operators very messy to handle. This cumbersome situation can be easily avoided by using the approach proposed above. The trace of the product of $\hat{\rho_1}$, $\hat{\rho_2}$ and $\hat{\rho_3}$ is taken to get three vertex Bargmann invariant. Argument of this gives the geometric phase which is represented in terms of phase space variables containing a notion of symplectic area.