The Glauber-Sudarshan, Wigner and Husimi quasiprobability distributions are indispensable tools in quantum optics. However, although mathematical relations between them are well established, not much is known about their operational connection. In this paper, we prove that a single composition of finite-strength quantum limited amplifier and attenuator channels, known for their noise-adding properties, turns the Glauber-Sudarshan distribution of any input operator into its Wigner distribution, and its Wigner distribution into its Husimi distribution. As we dissect, the considered process, which can be performed in a quantum optical laboratory with relative ease, may be interpreted as realizing a quantum-to-classical transition.
Various phase-space distributions, from the celebrated Wigner function, to the Husimi
Q function and the Glauber–Sudarshan
P distribution, have played an interesting and important role in the phase-space formulation of quantum mechanics in general, and quantum optics in particular. A unified approach to all these distributions based on the notion of the
s -ordered phase-space distribution was introduced by Cahill and Glauber. With the intention of illuminating the physical meaning of the parameter
s , we interpret the
s -ordered phase-space distribution as the Wigner function of a state under the Gaussian noise channel, and thus reveal an intrinsic connection between the
s -ordered phase-space distribution and the Gaussian noise channel, which yields a physical insight into the
s -ordered phase-space distribution. In this connection, the parameter
−s/2 (rather than the original
s ) acquires the role of the noise occurring in the Gaussian noise channel. An alternative representation of the Gaussian noise channel as the scaling-measurement preparation in a coherent states is illuminated. Furthermore, by exploiting the freedom in the parameter
s , we introduce a computable and experimentally testable quantifier for optical nonclassicality, reveal its basic properties, and illustrate it by typical examples. A simple and convenient criterion for optical nonclassicality in terms of the
s -ordered phase-space distribution is derived.
The quasiprobability distributions (QPD) introduced by Cahill and Glauber much simplify calculating the density operator of the output of a linear quantum channel from the density operator of the input when the channel attenuates and is corrupted by thermal noise. This channel models, for instance, an attenuator viewed as a simple harmonic oscillator in contact with a heat bath, or the relation between the radiating field mode in the aperture of a transmitter and the field mode it excites at the aperture of a distant receiver. It is shown how the attenuation and the noise in the channel modify the dependence of the QPD on the ordering parameter s. The form of the QPD at the output indicates that the greater the attenuation, the more nearly the state of the output resembles a classical state. Indeed, the addition of thermal noise contributing on the average only one photon suffices to convert an arbitrary unimodal state into a classical state. The principal result of the paper is applied to determining the output of a quantum channel whose input mode is in a generalized coherent state.
Energy is a cornerstone concept in physics, embodying both conservation principles and the dynamics of systems. In quantum mechanics, energy manifests as the expectation value of the Hamiltonian operator, yet its intuitive understanding often remains elusive to students. This paper adopts a pedagogical approach to demystify quantum energy by employing three key phase-space representations: the Glauber-Sudarshan P-function, the Husimi Q-function, and the Wigner function. We show that, despite their distinct mathematical frameworks and interpretations, all three representations yield equivalent expressions for the mean energy of thermal states. This result provides an engaging platform for introducing concepts such as coherent states, operator ordering, and quasiprobability distributions. By bridging classical and quantum perspectives in an accessible format, we present a teaching model that not only conveys technical skills but also cultivates deeper conceptual insights, making it feasible to implement in a single lecture. This approach facilitates a richer understanding of the interplay between quantum mechanics and statistical physics, preparing students for advanced topics in quantum theory.
It is shown that, to any quasiprobability distribution corresponding to a given density operator, one can associate a class of positive distributions in an extended space. In situations where the quasiprobability distributions are singular, these distributions are shown to provide regularized versions thereof. The positive distribution introduced by Drummond and Gardiner (1981) in the context of the Glauber-Sudarshan P-representation is shown to arise as a special case. One can thus associate a positive distribution to the Wigner function such that its moments give the averages of Weyl-ordered operators with respect to the given density operator. It is also found that one can associate to the Glauber-Sudarshan P-function, a distribution in the extended space which, though not positive, interestingly, involves the Wigner function (1932). A measurement scheme which directly yields these positive distributions is presented.
We calculate exactly the quantum mechanical, temporal Wigner quasiprobability density for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz., a displaced-squeezed thermal state. The Wigner function allows us to calculate the fluctuations in photon number and the quadrature variance. We contrast the difference between the non-classicality criteria, which is independent of the displacement parameter α, based on the Glauber–Sudarshan quasiprobability distribution P(β) and the classical/non-classical behaviour of the Mandel QM(τ) parameter, which depends strongly on α. We find a phase transition as a function of α such that at the critical point αc, QM(τ), as a function of τ, goes from strictly classical, for |α|<|αc|, to a mixed classical/non-classical behaviour, for |α|>|αc|.
We show that there is a way to unify distribution functions that describe simultaneously a classical signal in space and (spatial) frequency and position and momentum for a quantum system. Probably the most well known of them is the Wigner distribution function. We show how to unify functions of the Cohen class, Rihaczek's complex energy function, and Husimi and Glauber-Sudarshan distribution functions. We do this by showing how they may be obtained from ordered forms of creation and annihilation operators and by obtaining them in terms of expectation values in different eigenbases.