Wigner quasi-probability distribution

See also Wigner distribution, a disambiguation page.

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 Schrodinger's equation with a probability distribution in phase space. It was independently derived by Hermann Weyl in 1931 as the symbol of the density matrix in representation theory in mathematics. It was once again derived 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. This is not true for a quantum particle due to the uncertainty principle. Instead, one can create a quasi-probability distribution, which necessarily does not satisfy all the properties of a normal probability distribution. For instance, the Wigner distribution can go negative for states which have no classical model (and hence, it can be used to identify non-classical states).

The Wigner distribution P(q, p) is defined as:

$P(x,p)=\frac{1}{\pi\hbar}\int_{-\infty}^{\infty}dy\, \psi^*(x+y)\psi(x-y)e^{2ipy}$

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:

$P(x,p)=\frac{1}{\pi\hbar}\int_{-\infty}^{\infty}dq\, \phi^*(p+q)\phi(p-q)e^{-2ixq}$

where φ is the Fourier transform of ψ.

In the case of a mixed state:

$P(x,p)=\frac{1}{\pi\hbar}\int_{-\infty}^{\infty}dy\, \langle x-y| \hat{\rho} |x+y \rangle e^{2ipy}$

where ρ is the density matrix.

Contents

1 Mathematical Properties

2 Uses of the Wigner function outside quantum mechanics

3 Measurements of the Wigner function

5 Historical Note

6 References

7 External links

Mathematical Properties

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Wigner_functions.jpg

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.
Note that 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:

$\psi(x) \rightarrow \psi(x+y) \Rightarrow P(x,p) \rightarrow P(x+y,p)$
Note that 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:

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

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

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Chirpedpulse.jpg

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 is replaced with ℏk=|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/ℏ 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/ℏ 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:

Glauber P representation
Husimi Q representation

Historical Note

As the introduction shows, 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. The latter 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, Th�orie et Applications de la Notion de Signal Analytique, Cables et Transmission, 2A: (1948) 61-74.
W. Heisenberg, �ber die inkoh�rente Streuung von R�ntgenstrahlen, 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).