Weyl Ordering Symbol Method for Studying Wigner Function of the Damping Field
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We discuss nonclassicality of a superposition of coherent states in terms of sub-Poissonian photon statistics as well as the negativity of the Wigner function. We derive an analytic expression for the Wigner function from which we find that the function has some negative region in phase space. We obtain a compact form of the Wigner function when decoherence occurs and study the effect of decoherence on the state. We demonstrate the behavior of the nonclassicality indicator.PACS Nos.: 42.50.Dv, 03.65.Yz
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In 1983, Wigner outlined a modified Schrödinger--von-Neumann equation of motion for macroobjects, to describe their typical coupling to the environment. This equation has become a principal model of environmental decoherence which is beleived responsible for the emergence ofclassicality in macroscopic quantum systems. Typically, this happens gradually and asymptotically after a certain characteristic decoherence time. For the Wigner-function, however, one can prove that it evolves perfectly into a classical (non-negative) phase space distribution after a finite time of decoherence.
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Much of the discussion of decoherence has been in terms of a particle moving in one dimension that is placed in an initial superposition state (a Schr\{o}dinger cat state) corresponding to two widely separated wave packets. Decoherence refers to the destruction of the interference term in the quantum probability function. Here, we stress that a quantitative measure of decoherence depends not only on the specific system being studied but also on whether one is considering coordinate, momentum or phase space. We show that this is best illustrated by considering Wigner phase space where the measure is again different. Analytic results for the time development of the Wigner distribution function for a two-Gaussian Schrodinger cat state have been obtained in the high-temperature limit (where decoherence can occur even for negligible dissipation) which facilitates a simple demonstration of our remarks.
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In this paper we study the effect of decoherence on the photon-added even/odd coherent states. For this purpose, the time-dependent Wigner function of the considered states during their interaction with the environment in zero temperature (that is called photon-loss channel) is obtained in the framework of standard master equation. Then, by analyzing the evolution of Wigner distribution of the mentioned states, the loss of nonclassicality behavior as a result of decoherence is clearly shown.
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We analyse the coherence properties of neutron wavepackets, after they have interacted with a phase shifter undergoing different kinds of statistical fluctuation. We give a quantitative (and operational) definition of decoherence and compare it to the standard deviation of the distribution of the phase shifts. We find that in some cases the neutron ensemble is more coherent, even though it has interacted with a wider (i.e. more disordered) distribution of shifts. This feature is independent of the particular definition of decoherence: this is shown by proposing and discussing an alternative definition, based on the Wigner function, that displays a similar behaviour. We briefly discuss the notion of entropy of the shifts and find that, in general, it does not correspond to that of decoherence of the neutron.
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