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Canadian Journal of Physics

Evolution of the photon-added coherent state in a laser channel

Jun Zhou, Jun Song, Hao Yuan, Hong-Yi Fan

Published Jun 1, 2012• 1 citation
Publisher

Abstract

We investigate the time evolution of a quantum state in the laser channel with cavity gain coefficient g and loss coefficient κ by virtue of the thermo entangled state representation. This is a new approach for solving master equations. We exactly calculate the evolving result of the photon-added coherent state (PACS) governed by the master equation describing the laser process, and the normal ordered form of the density operator in the laser channel is derived, based on which we calculate the Wigner function analytically and graphically. The corresponding law of the PACS evolution in the laser channel is evaluated.

DOI
10.1139/p2012-060

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We investigate the time evolution of a quantum state in the laser channel with cavity gain coefficient g and loss coefficient κ by virtue of the thermo entangled state representation. This is a new approach for solving master equations. We exactly calculate the evolving result of the photon-added coherent state (PACS) governed by the master equation describing the laser process, and the normal ordered form of the density operator in the laser channel is derived, based on which we calculate the Wigner function analytically and graphically. The corresponding law of the PACS evolution in the laser channel is evaluated.
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.
We adopt a new approach (using the entangled state representation) to study time evolution of the mixed state in a dissipative channel (DC) with decay rate κ. We then investigate the statistical properties of the m -photon-added coherent state (PACS) in the DC, by deriving the analytical expressions of the Wigner function (WF), which turns out to be the Laguerre–Gaussian function. The criterion time for the definite positivity of WF: , valid for arbitrary m , is revealed. Time evolution of the photocount distribution is also derived analytically, which is related to two-variable Hermite polynomials. The tomogram of PACS is also calculated through the intermediate coordinate–momentum representation.
Master equations of density operators can be concisely solved by virtue of thermo entangled state representation and the technique of integration within an ordered product of operators.The solution of master equations of density operators is given.We get the Wigner function evolution formula in phase space and photon number distribution formula of the state,which is a superposition of vacuum state and single-photon state initially in laser process.By numerical calculation we plot its Wigner function and photon number distribution evolution curves,when cavity gain constant and cavity loss constant are both equal to certain values.The effect of decoherence on nonclassical property of the state is discussed by comparing negative area size and the depth of negativity of Wigner function.Numerical results show that the negativity area size and the depth of negativity of its Wigner function decrease and then vanish with dissipation.In addition,we calculate the variation of negative volume of Wigner function with dissipation time.We find that it decreases with dissipation time increasing.These results show that the nonclassical properties of the state are weakened and then disappear with dissipation time increasing.
Based on the density operator’s operator-sum representation recently obtained by Fan and Hu for a laser process (Opt. Commun., 2008, 281: 5571; Opt. Commun., 2009, 282: 932; Phys. Lett. B, 2008, 22: 2435), we derive the evolution law of Wigner operator, the law is concisely expressed in the normally ordered form $\Delta (\alpha ,\alpha ^* ,t) = \tfrac{T} {\pi }:\exp [ - 2T(a^\dag e^{ - (\kappa - g)t} - \alpha ^* )(ae^{ - (\kappa - g)t} - \alpha )] $ :, where g and κ are the cavity gain and the loss, respectively, and T ≡ (κ − g)(κ + g − 2ge−2(κ−g)t )−1:, When $t = 0,\Delta (\alpha ,\alpha ^* ,t) \to \tfrac{1} {\pi }:\exp [ - 2(a^\dag - \alpha ^* )(a - \alpha )] $ ;, which is the initial Wigner operator. Using this formalism the evolution law of Wigner functions in laser process can be directly obtained.
Abstract Using the operator correspondence of the real and fictious modes in the thermo entangled state representation, we solve the quantum master equation describing the diffusion channel and obtain the Kraus operator-sum representation of its analytical solution. we find that the pure coherent states evolve into the new mixed thermal superposed states in the diffusion channel. Also, we investigate the statistical properties of the initial coherent states and their entropy evolutions in the diffusion channel, and find that the entropy evolutions are only related to the decay time and without the amplitudes of the initial coherent states.