WIGNER QUASI-PROBABILITY DISTRIBUTION : Comprehensive Encyclopedia

The Wigner quasi-probability distribution was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. The goal was to replace the wavefunction that appears in Schrödinger's equation with a probability distribution in phase space. It corresponds to the density matrix in the map between phase space functions and Hermitean operators introduced by Hermann Weyl in 1931, in representation theory in mathematics. It was rederived by J. Ville in 1948 as a quadratic (in signal) representation of the local time-frequency energy of a signal. It is also known as the "Wigner function," "Wigner-Weyl transformation" or the "Wigner-Ville distribution". It has applications in statistical mechanics, quantum chemistry, quantum optics, classical optics and signal analysis in diverse fields such as electrical engineering seismology, biology, and engine design.

A classical particle has a definite position and momentum and hence, is represented by a point in phase space. When one has a collection (ensemble) of particles, the probability of finding a particle at a certain position in phase space is given by a probability distribution, the Liouville density. This is not true for a quantum particle, due to the uncertainty principle. Instead, the above quasi-probability Wigner distribution, plays an analogous role, but does not satisfy all the properties of a normal probability distribution, and satisfies boundedness properties unavailable to the classical distributions. For instance, the Wigner distribution can and normally does go negative for states which have no classical model --- and is a convenient indicator of quantum mechanical interference. Regions of such negative value are provable (by convolving them with a small Gaussian) to be "small": they cannot extend to compact regions over a few $\hbar$ , and hence disappear in the classical limit. They are shielded by the uncertainty principle, which does not allow precise location within phase-space regions smaller than $\hbar$ , and thus renders such "negative probabilities" less paradoxical.

where ψ is the wavefunction and x and p are position and momentum but could be any conjugate variable pair. (ie. real and imaginary parts of the electric field or frequency and time of a signal). It is symmetric in x and p:

Mathematical properties

Figure 1: The Wigner quasi-probability distribution for a) the vacuuum b) An n = 1 Fock state (e.g. a single photon) c) An n = 5 Fock state.

1. P(x, p) is real

2. The x and p probability distributions are given by the marginals:

$\int_{-\infty}^{\infty}dp\,P(x,p)=|\psi(x)|^2=\langle x|\hat{\rho}|x \rangle$
$\int_{-\infty}^{\infty}dx\,P(x,p)=|\phi(p)|^2=\langle p|\hat{\rho}|p \rangle$
$\int_{-\infty}^{\infty}dx\int_{-\infty}^{\infty}dp\,P(x,p)=Tr(\hat{\rho})$
Typically the trace of ρ is equal to 1.
1. and 2. imply the P(x,p) is negative somewhere, with the exception of the coherent state (and mixtures of coherent states) and the squeezed vacuum state.

3. P(x, p) has the following reflection symmetries:

Time symmetry: $\psi(x) \rightarrow \psi(x)^* \Rightarrow P(x,p) \rightarrow P(x,-p)$
Space symmetry: $\psi(x) \rightarrow \psi(-x) \Rightarrow P(x,p) \rightarrow P(-x,-p)$

4. P(x, p) is Galilei-invariant:

$\psi(x) \rightarrow \psi(x+y) \Rightarrow P(x,p) \rightarrow P(x+y,p)$
It is not Lorentz invariant.

5. The equation of motion for each point in the phase space is classical in the absence of forces:

$\frac{\partial P(x,p)}{\partial t}=\frac{-p}{m}\frac{\partial P(x,p)}{\partial x}$

6. State overlap is calculated as:

$|\langle \psi|\theta \rangle|^2=2\pi\hbar\int_{-\infty}^{\infty}dx\,\int_{-\infty}^{\infty}dp\,P_{\psi}(x,p)P_{\theta}(x,p)$

7. Operators and expectation values (averages) are calculated as follows:

$A(x,p)=\int_{-\infty}^{\infty}dy\, \langle x-y/2| \hat{A} |x+y/2 \rangle e^{ipy/\hbar}$
$\langle \psi|\hat{A}|\psi\rangle=Tr(\hat{\rho}\hat{A})=\int_{-\infty}^{\infty}dx\, \int_{-\infty}^{\infty}dp P(x,p)A(x,p)$

8. In order that P(x, p) represent physical (positive) density matrices:

$\int_{-\infty}^{\infty}dx\, \int_{-\infty}^{\infty}dp\, P(x,p)P_{\theta}(x,p)\ge 0$

where |θ> is a pure state.

Uses of the Wigner function outside quantum mechanics

Figure 2: A contour plot of the Wigner-Ville distribution for a chirped pulse of light. The plot makes it obvious that the frequency is a linear function of time.

In the modelling of optical systems such as telescopes or fibre telecommunications devices, the Wigner function is used to bridge the gap between simple ray tracing and the full wave analysis of the system. Here $p / \hbar$ is replaced with k=|k|sinθ≈|k|θ in the small angle (paraxial) approximation. In this context, the Wigner function is the closest one can get to describing the system in terms of rays at position x and angle θ while still including the effects of interference. If it becomes negative at any point then simple ray-tracing will not suffice to model the system.

In signal analysis, a time-varying electrical signal, mechanical vibration, or sound wave are represented by a Wigner function. Here, x is replaced with the time and $p / \hbar$ is replaced with the angular frequency ω=2πf, where f is the regular frequency.

In ultrafast optics, short laser pulses are characterized with the Wigner function using the same f and t substitutions as above. Pulse defects such as chirp (the change in frequency with time) can be visualized with the Wigner function. See Figure 2.

In quantum optics, x and $p / \hbar$ are replaced with the X and P quadratures, the real and imaginary components of the electric field (see coherent state). The plots in Figure 1 are of quantum states of light.

Measurements of the Wigner function

Tomography
Homodyne detection
FROG Frequency-resolved optical gating

Other related quasi-probability distributions

The Wigner distribution was the first quasi-probability distribution but many more followed with various advantages:

Historical note

As the introduction shows, the formula for the Wigner function was independently derived many times in different contexts. In fact, apparently Wigner was unaware that even within the context of quantum theory, it had been introduced previously by Heisenberg and Dirac. However, they missed its significance as they believed it was only an approximation to the true quantum description of a system such as the atom. Incidentally, Dirac would later become Wigner's brother-in-law. See references.

References

E.P. Wigner, "On the quantum correction for thermodynamic equilibrium", Phys. Rev. 40 (June 1932) 749-759.
H. Weyl, Z. Phys. 46, 1 (1927).
H. Weyl, Gruppentheorie und Quantenmechanik (Leipzig: Hirzel)(1928).
H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1931).
J. Ville, "Théorie et Applications de la Notion de Signal Analytique", Cables et Transmission, 2A: (1948) 61-74.
W. Heisenberg, "Über die inkohärente Streuung von Röntgenstrahlen", Physik. Zeitschr. 32, 737-740 (1931).
P.A.M. Dirac, "Note on exchange phenomena in the Thomas atom", Proc. Camb. Phil. Soc. 26, 376-395 (1930).
C. Zachos, D. Fairlie, and T. Curtright, Quantum Mechanics in Phase Space ( World Scientific, Singapore, 2005).

WIGNER QUASI-PROBABILITY DISTRIBUTION

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