Quasi-probability distribution

In the most general form, the dynamics of a quantum-mechanical system are determined by a master equation - an equationof motion for the density operator (usually written ρ) of thesystem. Although it is possible to directly integrate this equation forvery small systems (i.e., systems with few particles or degrees of freedom),this quickly becomes impossible for larger systems. For this reason, itis frequently useful to represent the density operator as a distributionover some (over-)complete operator basis. The evolution of the system isthen completely determined by the evolution of a quasi-probability distribution function. This general technique has a long history,especially in the context of quantum optics. The most common examplesof quasi-probability representations are the
Wigner,
P- and Q-functions.More recently, the positive P function and awider class of generalized P functions havebeen used to solve complex problems in both quantum optics and the newerfield of quantum atom optics.

Characteristic functions

Analogous to probability theory, quantum quasi-probability distributionscan be written in terms of characteristic functions,from which all operator expectation values can be derived. The characteristicfunctions for the Wigner, Glauber P and Q distributions of an "N" mode systemare as follows:

* $chi\_W(mathbf\{z\},mathbf\{z\}^*)=\; operatorname\{tr\}(\; ho\; e^\{imathbf\{z\}cdotwidehat\{mathbf\{a+imathbf\{z\}^*cdotwidehat\{mathbf\{a^\{dagger)$
* $chi\_P(mathbf\{z\},mathbf\{z\}^*)=operatorname\{tr\}(\; ho\; e^\{imathbf\{z\}cdotwidehat\{mathbf\{a\}e^\{imathbf\{z\}^*cdotwidehat\{mathbf\{a^\{dagger)$
* $chi\_Q(mathbf\{z\},mathbf\{z\}^*)=\; operatorname\{tr\}(\; ho\; e^\{imathbf\{z\}^*cdotwidehat\{mathbf\{a^\{daggere^\{imathbf\{z\}cdotwidehat\{mathbf\{a\})$

Here $widehat\{mathbf\{a$ and $widehat\{mathbf\{a^\{dagger\}$are vectors containing the annihilation and creation operators for each modeof the system. These characteristic functions can be used to directly evaluateexpectation values of operator moments. The ordering of the annihilation andcreation operators in these moments is specific to the particularcharacteristic function. For instance, normally ordered (annihilation operatorspreceding creation operators) moments can be evaluated in the following wayfrom $chi\_P,$:

: $langlewidehat\{a\}\_j^\{dagger\; m\}widehat\{a\}\_k^n\; angle\; =\; frac\{partial^\{m+n\{partial(iz\_j^*)^mpartial(iz\_k^*)^n\}chi\_P(mathbf\{z\},mathbf\{z\}^*)Big|\_\{mathbf\{z\}=mathbf\{z\}^*=0\}$

In the same way, expectation values of anti-normally ordered and symmetricallyordered combinations of annihilation and creation operators can be evaluated fromthe characteristic functions for the Q and Wigner distributions, respectively.

Quasi-probability functions

The quasi-probability functions themselves are defined as Fourier transformsof the above characteristic functions. That is,

: $\{W|P|Q\}(mathbf\{alpha\},mathbf\{alpha\}^*)=frac\{1\}\{pi^\{2Nint\; d^\{2N\}mathbf\{z\}chi\_\{\{W|P|Q(mathbf\{z\},mathbf\{z\}^*)e^\{-imathbf\{z\}^*cdotmathbf\{alpha\}^*\}e^\{-imathbf\{z\}cdotmathbf\{alpha.$

Here $alpha\_j,$ and $alpha^*\_k$ may be identified as
coherent state amplitudes in the case of the Glauber P and Q distributions,but simply c-numbers for the Wigner function. Since differentiation in normal spacebecomes multiplication in fourier space, moments can be calculated fromthese functions in the following way:

* $langlewidehat\{mathbf\{a\_j^\{dagger\; m\}widehat\{mathbf\{a\_k^n\; angle=int\; d^\{2N\}mathbf\{alpha\}P(mathbf\{alpha\},mathbf\{alpha\}^*)alpha\_j^nalpha\_k^\{*m\}$
* $langlewidehat\{mathbf\{a\_j^mwidehat\{mathbf\{a\_k^\{dagger\; n\}\; angle=int\; d^\{2N\}mathbf\{alpha\}Q(mathbf\{alpha\},mathbf\{alpha\}^*)alpha\_j^malpha\_k^\{*n\}$
* $langle(widehat\{mathbf\{a\_j^\{dagger\; m\}widehat\{mathbf\{a\_k^n)\_S\; angle=int\; d^\{2N\}mathbf\{alpha\}W(mathbf\{alpha\},mathbf\{alpha\}^*)alpha\_j^malpha\_k^\{*n\}$

Here $(ldots)\_S$ denotes symmetric ordering.

These relationships motivate comparisons between the distribution functions and classical
probability densities. The analogy - though strong - is notperfect, as the above functions are not necessarily positive for all$mathbf\{alpha\}$. Hence the term "quasi-"probability function.

Time evolution and operator correspondences

Since each of the above transformations from $ho,$ through to thedistribution function is linear, the equation of motion for each distributioncan be obtained by performing the same transformations to $dot\{\; ho\}$.Furthermore, as any master equation which can be expressed in
Lindblad form is completely described by the action of combinationsof annihilation and creation operators on the densityoperator, it is useful to consider the effect such operations have on each ofthe quasi-probability functions.

For instance, consider the annihilation operator $widehat\{a\}\_j,$acting on $ho,$. For the characteristic function of the P distributionwe have

: $operatorname\{tr\}(widehat\{a\}\_j\; ho\; e^\{imathbf\{z\}cdotwidehat\{mathbf\{a\}e^\{imathbf\{z\}^*cdotwidehat\{mathbf\{a^\{dagger)\; =\; frac\{partial\}\{partial(iz\_j)\}chi\_P(mathbf\{z\},mathbf\{z\}^*).$

Taking the Fourier transform with respect to $mathbf\{z\},$ to find theaction corresponding action on the Glauber P function, we find

$widehat\{a\}\_j\; ho\; ightarrow\; alpha\_j\; P(mathbf\{alpha\},mathbf\{alpha\}^*).$

By following this procedure for each of the above distributions, the following"operator correspondences" can be identified:

* $widehat\{a\}\_j\; ho\; ightarrow\; left(alpha\_j\; +\; sfrac\{partial\}\{partialalpha\_j^*\}\; ight)P\_s(mathbf\{alpha\},mathbf\{alpha\}^*)$
* $howidehat\{a\}^\{dagger\}\_j\; ightarrow\; left(alpha\_j^*\; +\; sfrac\{partial\}\{partialalpha\_j\}\; ight)P\_s(mathbf\{alpha\},mathbf\{alpha\}^*)$
* $widehat\{a\}^\{dagger\}\_j\; ho\; ightarrow\; left(alpha\_j^*\; -\; (1-s)frac\{partial\}\{partialalpha\_j\}\; ight)P\_s(mathbf\{alpha\},mathbf\{alpha\}^*)$
* $howidehat\{a\}\_j\; ightarrow\; left(alpha\_j\; -\; (1-s)frac\{partial\}\{partialalpha\_j^*\}\; ight)P\_s(mathbf\{alpha\},mathbf\{alpha\}^*)$

Here "s" = 0, 1/2 or 1 for P, Wigner and Q distributions, respectively.

In this way, master equations can be expressed as an equations ofmotion of quasi-probability functions.

Example — the anharmonic oscillator

Consider a single-mode system evolving under the following Hamiltonian operator:

: $widehat\{H\}\; =\; frac\{hbar\}\{2\}widehat\{a\}^\{dagger\; 2\}widehat\{a\}^2.$

References

* H. J. Carmichael, "Statistical Methods in Quantum Optics I: Master Equations and Fokker-Planck Equations", Springer-Verlag (2002).
* C. W. Gardiner, "Quantum Noise", Springer-Verlag (1991).