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Chinese Physics B

Analytical and numerical investigations of displaced thermal state evolutions in a laser process

Chuan-Xun Du, Xiang‐Gao Meng, Ran Zhang, Ji‐Suo Wang

Published Dec 1, 2017• 8 citations
Publisher

Abstract

We investigate how displaced thermal states (DTSs) evolve in a laser channel. Remarkably, the initial DTS, an example of a mixed state, still remains mixed and thermal. At long times, they finally decay to a highly classical thermal field only related to the laser parameters κ and g . The normal ordering product of density operator of the DTS in the laser channel leads to obtaining the analytical time-evolution expressions of the photon number, Wigner function, and von Neumann entropy. Also, some interesting results are presented via numerically investigating these explicit time-dependent expressions.

DOI
10.1088/1674-1056/26/12/120301

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We investigate how displaced thermal states (DTSs) evolve in a laser channel. Remarkably, the initial DTS, an example of a mixed state, still remains mixed and thermal. At long times, they finally decay to a highly classical thermal field only related to the laser parameters κ and g . The normal ordering product of density operator of the DTS in the laser channel leads to obtaining the analytical time-evolution expressions of the photon number, Wigner function, and von Neumann entropy. Also, some interesting results are presented via numerically investigating these explicit time-dependent expressions.
We investigate how the squeezed thermal state (STS) and squeezed number state (SNS) evolve undergoing decoherence in the laser process. Remarkably, the initial SNS, an example of a pure state, evolves into a mixed state, which turns out to be a Laguerre polynomial of combination of creation and annihilation operators within normal ordering; however, the STS, a mixed state, still keeps squeezed and thermal. At long times, these fields lose their nonclassical nature and decay to a highly classical thermal field. The normally ordered density operators of such states in the laser channel lead to deriving the analytical time-evolution expressions of the Wigner functions (WFs). Their nonclassicality is investigated in reference to the time-evolution WFs, which indicates that both of the WFs decay to the same thermal WF as a result of decoherence when the decay time t or the cavity loss κ of the laser is large enough.
A displaced thermal state (DTS) is first introduced by Jeong et al. [Phys. Rev. Lett. 97, 100401 (2006)] and its superpositions can be used in transfer of nonclassicality. In this paper, the analytical evolution law of the DTS exposed to dissipative environments is obtained according to the Kraus operator for amplitude damping. The results manifestly show that, after undergoing the amplitude damping, the initial DTSs still keep mixed and thermal with the exponential decay due to amplitude damping. An appealing feature is that the entropy evolution in this process completely depends on the mixedness.
We investigate systematically the evolution of the number state in a laser process by deriving the analytic expression of the density operator and putting it into a normal ordered form. The eigenvalue of the density operator is related to Jacobi polynomials. Then we derive the expression for the mean photon number, the second degree of coherence, the entropy, Wigner function and the photoncount distribution. The nonclassicality is discussed by virtue of the negativity of Wigner function. It is found that the Wigner function is always negative for t < t0, which is independent on the parameter m. On the other hand, the condition for the second degree of coherence larger than 1 is dependent on the parameter m.
Based on the Kraus operator of laser theory which we derived in [2] and by virtue of the technique of integration within ordered product of operators we exactly calculate the evolving result of a chaotic field, governed by the master equation describing the laser process, we find that the initial density operator evolves into a new chaotic field, with where g and κ represent the cavity gain and loss of the laser, respectively. When g = 0, the system reduces to an amplitude dissipative channel, and f' becomes −ln [(e−f − 1) e2κt + 1]. The law of photon number decay in this process is evaluated, and the corresponding Wigner function's evolution is also demonstrated.
Using the correspondence of the operators between the real and fictitious modes, we find the Kraus operator-sum form of the analytical solution of the diffusion equation and obtain the evolution of a displaced thermal state in the diffusion channel. Also, we investigate the quantum statistical properties of the initial displaced thermal state and its von Neumann entropy evolution in the diffusion channel, especially find that the entropy evolution is affected by diffusion and mixedness, but without the displacement. These results provide theoretical basis for quantifying the evolved entropy in quantum information.