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Journal of Southwest Jiaotong University

Wigner Functions for Eigenstates of Arbitrary Power of Annihilation Operators

Lian-Fu Wei

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

Wigner functions for the eigenstates of arbitrary power of annihilation operators were reconstructed using their expressions in Fock presentations.The distribution of the reconstructed Wigner functions in phase spaces was analyzed,based on which the nonclassical properties of these eigenstates were discussed.The results show that,the distribution of the Wigner functions for these eigenstates depend on the eigenvalues of the annihilation operators;the eignestates of the annihilation operators of the first power(i.e.,coherent states) are quasiclassical(their Wigner functions are always non-negative),and those of higher powers obviously exhibit nonclassical properties(their Wigner functions may become negative).

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Wigner functions for the eigenstates of arbitrary power of annihilation operators were reconstructed using their expressions in Fock presentations.The distribution of the reconstructed Wigner functions in phase spaces was analyzed,based on which the nonclassical properties of these eigenstates were discussed.The results show that,the distribution of the Wigner functions for these eigenstates depend on the eigenvalues of the annihilation operators;the eignestates of the annihilation operators of the first power(i.e.,coherent states) are quasiclassical(their Wigner functions are always non-negative),and those of higher powers obviously exhibit nonclassical properties(their Wigner functions may become negative).
Non-classical properties for the eigen-states of k-th power of q-boson annihilation operators were calculated by their representations in Fock spaces.The non-classical properties of these eigen-states were further discussed by numerically discussing their quasiprobability distributions in phase spaces.It is shown that the certain non-classical quantum states could be generated from quasiclassical coherent states by q-deformations;these eigen-states are typically non-classical quantum states,as their relevant Wigner functions reveal negatives for certain parameters.
By using the expression of second-order coherence degree and Wigner function in Fock presentation,the second-order coherence degrees and the Wigner functions of the eigenstates of annihilation operator a3 were calculated.Based on the anti-bunching effect and the distributions of Wigner functions,the nonclassical properties of these eigenstates were analyzed.The numerical results show that these eigenstates produce anti-bunching effect,and their relevant Wigner functions can be negative in phase spaces,which indicates that the eigenstates of annihilation operator a3 are non-classical states.
Wigner functions in phase space are reconstructed for the excited coherent states，which are generated by applying the inverse of k-Boson operators repeatedly to the coherent states. The non-classicality of these states is discussed by calculating their Wigner functions in Fock-state space. Numerical results show that these excited coherent states always reveal non-classical characteristics，no matter whether the excited number k is even or odd，and also the corresponding non-classicality is more obvious with the number k increasing.
Abstract The Wigner function was introduced as an attempt to describe quantum-mechanical fields with the tools inherited from classical statistical mechanics. In particular, it is widely used to describe the properties of radiation fields. In fact, a useful way to distinguish between classical and nonclassical states of light is to ask whether their Wigner function has a Gaussian profile or not, respectively. In this paper, we use the basis of Fock states to provide the closed-form expression for the Wigner function of an arbitrary quantum state. Thus, we provide the general expression for the Wigner function of a squeezed Fock, coherent and thermal states, with an arbitrary squeezing parameter. Then, we consider the most fundamental quantum system, Resonance Fluorescence, and obtain closed-form expressions for its Wigner function under various excitation regimes. With them, we discuss the conditions for obtaining a negative-valued Wigner function and the relation it has with population inversion. Finally, we address the problem of the observation of the radiation field, introducing physical detectors into the description of the emission. Notably, we show how to expose the quantumness of a radiation field that has been observed with a detector with finite spectral resolution, even if the observed Wigner function is completely positive.
In this paper, we show that the Pegg–Barnett formalism accepts coherent states constructed as eigenstates of the annihilation operator, considering both the number and the phase. These operators are defined within a ( s + 1 ) -dimensional Hilbert space H s and with periodic conditions. The coherent states that we find are determined by the eigenvalue of the annihilation operator, which leads to a discrete spectrum. This approach allows calculation of the discrete-finite counterpart of the Wigner function in a phase space defined by the variables of number and phase.