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Physical Review A

Glauber-SudarshanPrepresentation of negative binomial states

Kuniaki Matsuo

Published Jan 1, 1990• 36 citations
Publisher

Abstract

The Glauber-Sudarshan P representation of negative binomial states is presented and its properties are discussed. It is shown that for the limiting cases the states approach the random-phase coherent state and the thermal noise state. Thus these states may allow a unified description of coherent and thermal states of the field. The two-dimensional representation of the P function is plotted on the complex-\ensuremath{\alpha} plane. The probability density function of waiting-time intervals between the arrivals of successive photons on a photodetector irradiated by a beam of the negative binomial light is also investigated.

DOI
10.1103/physreva.41.519

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The Glauber-Sudarshan P representation of negative binomial states is presented and its properties are discussed. It is shown that for the limiting cases the states approach the random-phase coherent state and the thermal noise state. Thus these states may allow a unified description of coherent and thermal states of the field. The two-dimensional representation of the P function is plotted on the complex-\ensuremath{\alpha} plane. The probability density function of waiting-time intervals between the arrivals of successive photons on a photodetector irradiated by a beam of the negative binomial light is also investigated.
We present the s-parametrized quasiprobability distributions for the negative binomial states. Marked changes in the quasiprobability distributions W(\ensuremath{\alpha},\ensuremath{\epsilon},s) are exhibited by states that are close to the random-phase coherent state (\ensuremath{\epsilon}=0), as the parameter s is varied continuously from s=-1, corresponding to the Q function, to s=1, corresponding to the P function. However, states close to the thermal-noise state (\ensuremath{\epsilon}=1) exhibit the usual Gaussian-like distributions for all values of s.
A class of normal ordering representations of quantum operators is introduced, that generalises the Glauber-Sudarshan P-representation by using nondiagonal coherent state projection operators. These are shown to have practical application to the solution of quantum mechanical master equations. Different representations have different domains of integration, on a complex extension of the usual canonical phase-space. The 'complex P-representation' is the case in which analytic P-functions are defined and normalised on contours in the complex plane. In this case, exact steady-state solutions can often be obtained, even when this is not possible using the Glauber-Sudarshan P-representation. The 'positive P-representation' is the case in which the domain is the whole complex phase-space. In this case the P-function may always be chosen positive, and any Fokker-Planck equation arising can be chosen to have a positive-semidefinite diffusion array. Thus the 'positive P-representation' is a genuine probability distribution. The new representations are especially useful in cases of nonclassical statistics.
We introduce excited binomial states and excited negative binomial states of the radiation field by repeated application of the photon creation operator on binomial states and negative binomial states. They reduce to Fock states and excited coherent states in certain limits and can be viewed as intermediate states between Fock states and coherent states. We find that both the excited binomial states and excited negative binomial states can be exactly normalized in terms of hypergeometric functions. Base on this interesting character, some of the statistical properties are discussed.
New states of electromagnetic field, i.e. even and odd negative binomial states are introduced here. These states interpolate between the even (odd) coherent states and the even (odd) quasi-thermal states. Various statistical properties of these states such as the mean photon number, second-order coherence function, normal squeezing are calculated. The Wigner function for these states are also discussed. These results may be useful for carrying out systematic study of the non-classical properties of `Schrödinger cat states' as one moves from even (odd) coherent state to even (odd) quasi-thermal state.