Entanglement Involved in Time Evolution of Two-Mode Squeezed State in Single-Mode Diffusion Channel

We derive the evolution law of an initial two-mode squeezed vacuum state ( text {sech}^{2}lambda e^{a^{dag }b^{dagger }tanh lambda }left vert 00right rangle left langle 00right vert e^{abtanh lambda }) (a pure state) passing through an a-mode diffusion channel described by the master equation

$$frac{drho left( tright) }{dt}=-kappa left[ a^{dagger}arho left( tright) -a^{dagger}rho left( tright) a-arho left( tright) a^{dagger}+rho left( tright) aa^{dagger}right] , $$

since the two-mode squeezed state is simultaneously an entangled state, the final state which emerges from this channel is a two-mode mixed state. Performing partial trace over the b-mode of ρ(t) yields a new chaotic field, (rho _{a}left (tright ) =frac {text {sech}^{2}lambda }{1+kappa t text {sech}^{2}lambda }:exp left [ frac {- text {sech}^{2}lambda }{1+kappa ttext {sech}^{2}lambda }a^{dagger }a right ] :,) which exhibits higher temperature and more photon numbers, showing the diffusion effect. Besides, measuring a-mode of ρ(t) to find n photons will result in the collapse of the two-mode system to a new Laguerre polynomial-weighted chaotic state in b-mode, which also exhibits entanglement.