Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Wigner function as the expectation value of a parity operator

Antoine Royer

Published Feb 1, 1977• 372 citations
Publisher

Abstract

It is pointed out that the Wigner function $f(r, p)$ is $\frac{2}{h}$ times the expectation value of the parity operator that performs reflections about the phase-space point $r$, $p$. Thus $f(r, p)$ is proportional to the overlap of the wave function $\ensuremath{\psi}$ with its mirror image about $r$, $p$; this is clearly a measure of how much $\ensuremath{\psi}$ is centered about $r$, $p$, and the Wigner distribution function now appears physically more meaningful and natural than it did previously.

DOI
10.1103/physreva.15.449

Related papers

The Wigner Distribution

Robert F. O’Connell

In contrast to classical physics, the language of quantum mechanics involves operators and wave functions (or, more generally, density operators). However, in 1932, Wigner formulated quantum mechanics in terms of a distribution function $W(q,p)$, the marginals of which yield the correct quantum probabilities for $q$ and $p$ separately \cite{wigner}. Its usefulness stems from the fact that it provides a re-expression of quantum mechanics in terms of classical concepts so that quantum mechanical expectation values are now expressed as averages over phase-space distribution functions. In other words, statistical information is transferred from the density operator to a quasi-classical (distribution) function.

arXiv:1009.4431•Sep 22, 2010•0 citations

Working in phase‐space with Wigner and Weyl

J. Ben-Benjamin, Moochan B. Kim, Wolfgang P. Schleich +2 more

Quantum phase‐space distributions offer a royal road into the fascinating quantum–classical interface; the Wigner function being the first and best example. However, the subject is frequent with subtleties and textbook‐level misinformation; e.g. “The Wigner distribution can give wrong answers for some operator expectation values” . Since the Wigner distribution is just another representation of the density matrix, it must yield correct answers. To that end, Marlan Scully has asked at several international conferences (the 2015 Prague conference being one of them) the following question: “Starting with the density matrix (not the Moyal characteristic function), could you give me a simple direct derivation of the Wigner distribution?” Section contains his answer. In Appendix D , we give a related treatment and make contact with other approaches. We hope that as a result of our studies, the Wigner distribution will become more deeply appreciated.

Fortschritte der Physik•Nov 15, 2016•11 citations

Direct measurement of the Wigner function of atoms in an optical trap

Falk-Richard Winkelmann, Carrie Ann Weidner, Gautam Ramola +3 more

Abstract We present a scheme to directly probe the Wigner function of the motional state of a neutral atom confined in an optical trap. The proposed scheme relies on the well-established fact that the Wigner function at a given point (x, p) in phase space is proportional to the expectation value of the parity operator relative to that point. In this work, we show that the expectation value of the parity operator can be directly measured using two auxiliary internal states of the atom: parity-even and parity-odd motional states are mapped to the two internal states of the atom through a Ramsey interferometry scheme. The Wigner function can thus be measured point-by-point in phase space with a single, direct measurement of the internal state population. Numerical simulations show that the scheme is robust in that it applies not only to deep, harmonic potentials but also to shallower, anharmonic traps.

arXiv:2205.15248•Journal of Physics B Atomic Molecular and Optical Physics•Aug 22, 2022•19 citations

Is the Wigner function directly measurable in quantum optics?

Ulf Leonhardt, Henryk Paul

Physics Letters A•Sep 1, 1994•9 citations

Generalized parity and quasi-probability density functions

M. G. Benedict, Attila Czirják

By shifting the parity operator in phase space, one obtains a class of operators, which we call the Wigner operator, because its expectation value is equal to the Wigner function. Calculating the commutator of Wigner operators with different arguments, we show that the corresponding observables cannot be measured simultaneously. Introducing a new parameter, we give a trace-class generalization of this operator. The displaced number states of the harmonic oscillator constitute the natural eigenstate basis of the corresponding generalized Wigner operator. Establishing the integral representation of this operator we show that its expectation value is proportional to the Q- and P- and Wigner functions at special values of the parameter. We illustrate this connection with the coherent, number and squeezed states of the harmonic oscillator.

Journal of Physics A Mathematical and General•Aug 21, 1995•9 citations

Relativistic Wigner function as the expectation value of the PT operator

Christopher Dewdney, Peter Holland, Anastasios Kyprianidis +1 more

Physics Letters A•Mar 1, 1986•12 citations
It is pointed out that the Wigner function f(r,p)f(r, p)f(r,p) is 2h\frac{2}{h}h2​ times the expectation value of the parity operator that performs reflections about the phase-space point rrr, ppp. Thus f(r,p)f(r, p)f(r,p) is proportional to the overlap of the wave function \ensuremathψ\ensuremath{\psi}\ensuremathψ with its mirror image about rrr, ppp; this is clearly a measure of how much \ensuremathψ\ensuremath{\psi}\ensuremathψ is centered about rrr, ppp, and the Wigner distribution function now appears physically more meaningful and natural than it did previously.
In contrast to classical physics, the language of quantum mechanics involves operators and wave functions (or, more generally, density operators). However, in 1932, Wigner formulated quantum mechanics in terms of a distribution function W(q,p)W(q,p)W(q,p), the marginals of which yield the correct quantum probabilities for qqq and ppp separately \cite{wigner}. Its usefulness stems from the fact that it provides a re-expression of quantum mechanics in terms of classical concepts so that quantum mechanical expectation values are now expressed as averages over phase-space distribution functions. In other words, statistical information is transferred from the density operator to a quasi-classical (distribution) function.
Quantum phase‐space distributions offer a royal road into the fascinating quantum–classical interface; the Wigner function being the first and best example. However, the subject is frequent with subtleties and textbook‐level misinformation; e.g. “The Wigner distribution can give wrong answers for some operator expectation values” . Since the Wigner distribution is just another representation of the density matrix, it must yield correct answers. To that end, Marlan Scully has asked at several international conferences (the 2015 Prague conference being one of them) the following question: “Starting with the density matrix (not the Moyal characteristic function), could you give me a simple direct derivation of the Wigner distribution?” Section contains his answer. In Appendix D , we give a related treatment and make contact with other approaches. We hope that as a result of our studies, the Wigner distribution will become more deeply appreciated.
Abstract We present a scheme to directly probe the Wigner function of the motional state of a neutral atom confined in an optical trap. The proposed scheme relies on the well-established fact that the Wigner function at a given point (x, p) in phase space is proportional to the expectation value of the parity operator relative to that point. In this work, we show that the expectation value of the parity operator can be directly measured using two auxiliary internal states of the atom: parity-even and parity-odd motional states are mapped to the two internal states of the atom through a Ramsey interferometry scheme. The Wigner function can thus be measured point-by-point in phase space with a single, direct measurement of the internal state population. Numerical simulations show that the scheme is robust in that it applies not only to deep, harmonic potentials but also to shallower, anharmonic traps.
By shifting the parity operator in phase space, one obtains a class of operators, which we call the Wigner operator, because its expectation value is equal to the Wigner function. Calculating the commutator of Wigner operators with different arguments, we show that the corresponding observables cannot be measured simultaneously. Introducing a new parameter, we give a trace-class generalization of this operator. The displaced number states of the harmonic oscillator constitute the natural eigenstate basis of the corresponding generalized Wigner operator. Establishing the integral representation of this operator we show that its expectation value is proportional to the Q- and P- and Wigner functions at special values of the parameter. We illustrate this connection with the coherent, number and squeezed states of the harmonic oscillator.