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AIP Advances

Time evolution of coherent state’s photon number statistics in laser channel gained via cumulant calculation

Ke Zhang, Lan-Lan Li, Hong-Yi Fan

Published Mar 1, 2024• 1 citation
Publisher

Abstract

In this paper, by using the method of integration within the ordered product of operators and entangled state representation, the degree of second-order coherence of coherent state evolving in a laser channel is derived for the first time, which exhibits its anti-bunching property. The master equation of the laser channel is solved by entangled state representation, and the cumulant for deriving mean and mean-square photon numbers is evaluated.

DOI
10.1063/5.0202405

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In this paper, by using the method of integration within the ordered product of operators and entangled state representation, the degree of second-order coherence of coherent state evolving in a laser channel is derived for the first time, which exhibits its anti-bunching property. The master equation of the laser channel is solved by entangled state representation, and the cumulant for deriving mean and mean-square photon numbers is evaluated.
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.
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Single-photon sources are critical optical components in quantum communication, in particular, for security applications. One of the essential parameters that define these sources is the magnitude of the second-order coherence function, whose investigation reveals the state of the emitted photon. In this study, we indicate that the second-order coherence function varies over time when using two lasers and preparing coherent population trapping. The calculation is based on solving the master equation to find the density matrix corresponding to the emission dynamics and provide the second-order coherence function. The changes of the second-order coherence function can be estimated and the system behavior regarding photon emission can be predicted by solving the master equation based on the parameters obtained from the experimental results of a nitrogen vacancy (NV) in a diamond. Here we report, for the first time to the best of our knowledge, that the state of the emitted photons persists in the strong interaction of the aforementioned process. As using two lasers is a familiar method for controlling the single-photon source and the stability of the source is an essential point in a quantum network, this study can be considered to develop quantum network components such as memory and on-demand single-photon sources. Also, it suggests a method for tuning photon statistics while controlling the photon states.
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.
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.