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Frontiers of Physics

Evolution of a two-mode squeezed vacuum for amplitude decay via continuous-variable entangled state approach

Xiang‐Gao Meng, Ji‐Suo Wang, Bao‐Long Liang, Cheng-Xuan Han

Published Sep 29, 2018• 46 citations
Publisher

Abstract

Extending the recent work completed by Fan et al. [ Front. Phys. 9(1), 74 (2014)] to a two-mode case, we investigate how a two-mode squeezed vacuum evolves when it undergoes a two-mode amplitude dissipative channel, with the same decay rate κ, using the continuous-variable entangled state approach. Our analytical results show that the initial pure-squeezed vacuum state evolves into a definite mixed state with entanglement and squeezing, decaying over time as a result of amplitude decay. We also investigate the time evolutions of the photon number distribution, the Wigner function, and the optical tomogram in this channel. Our results indicate that the evolved photon number distribution is related to Jacobi polynomials, the Wigner function has a standard Gaussian distribution (corresponding to the vacuum) at long periods, losing its nonclassicality due to amplitude decay, and a larger squeezing leads to a longer decay time.

DOI
10.1007/s11467-018-0856-1

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Using the way of deriving infinitive sum representation of density operator as a solution to the master equation describing the amplitude dissipative channel by virtue of the entangled state representation, we show manifestly how the initial density operator of a single-mode squeezed vacuum state evolves into a definite mixed state which turns out to be a squeezed chaotic state with decreasing-squeezing and decoherence. We investigate average photon number, photon statistics distributions for this mixed state.
For the first time we derive the dissipating result of an initial two-mode squeezed pure vacuum state passing through a two-mode amplitude dissipative channel described by the direct product of two independent single-mode master equations. Although these two master equations do not mix the two modes (there is no coupling between them), since the two-mode squeezed state is simultaneously an entangled state, the final state which emerges from passing this channel is a two-mode mixed density operator. The compact expression of the outcoming state is obtained, which manifestly shows that as time evolves, the squeezing effect decreases.
We investigate the statistical properties of the photon-subtractions from the two-mode squeezed vacuum state and its decoherence in a thermal environment. It is found that the state can be considered as a squeezed two-variable Hermite polynomial excitation vacuum and the normalization of this state is the Jacobi polynomial of the squeezing parameter. The compact expression for Wigner function (WF) is also derived analytically by using the Weyl ordered operators' invariance under similar transformations. Especially, the nonclassicality is discussed in terms of the negativity of WF. The effect of decoherence on this state is then discussed by deriving the analytical time evolution results of WF. It is shown that the WF is always positive for any squeezing parameter and photon-subtraction number if the decay time exceeds an upper bound (}$κt>{1/2}\ln \frac{2\bar{n}+2}{2\bar{n}+1}).