Geometric Phase using Diagonal Coherent State Representation

TL;DR

This paper introduces a method to calculate the geometric phase of non-classical light states using the diagonal coherent state representation, simplifying the process compared to traditional Fock basis approaches.

Contribution

The authors propose a novel approach employing the diagonal coherent state basis to determine geometric phases, avoiding complex calculations associated with Fock state representations.

Findings

Successfully computed geometric phases for non-classical light states.

Demonstrated the efficiency of the diagonal coherent state approach.

Connected geometric phase to symplectic area in phase space.

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.