King's graph

A quasiprobability distribution is a mathematical object similar to a probability distribution but which relaxes some of Kolmogorov's axioms of probability theory. Although quasiprobabilities share many of the same general features of ordinary probabilities such as the ability to take expectation values with respect to the weights of the distribution, they all violate the third probability axiom because regions integrated under them do not represent probabilities of mutually exclusive states. To compensate, some quasiprobability distributions also counterintuitively have regions of negative probability density, contradicting the first axiom. Quasiprobability distributions arise naturally in the study of quantum mechanics when treated in the phase space formulation, commonly used in quantum optics, time-frequency analysis, and elsewhere.

Introduction

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The coherent states, i.e. right eigenstates of the annihilation operator ${\hat {a}}$ serve as the overcomplete basis in the construction described above. By definition, the coherent states have the following property:

${\begin{aligned}{\hat {a}}|\alpha \rangle &=\alpha |\alpha \rangle \\\langle \alpha |{\hat {a}}^{\dagger }&=\langle \alpha |\alpha ^{*}.\end{aligned}}$

They also have some additional interesting properties. For example, no two coherent states are orthogonal. In fact, if |α 〉 and |β 〉 are a pair of coherent states, then

$\langle \beta |\alpha \rangle =e^{-{1 \over 2}(|\beta |^{2}+|\alpha |^{2}-2\beta ^{*}\alpha )}\neq \delta (\alpha -\beta ).$

Note that these states are, however, correctly normalized with 〈α|α〉 = 1. Owing to the completeness of the basis of Fock states, the choice of the basis of coherent states must be overcomplete. Click to show an informal proof.

In the coherent states basis, however, it is always possible to express the density operator in the diagonal form

${\hat {\rho }}=\int f(\alpha ,\alpha ^{*})|{\alpha }\rangle \langle {\alpha }|\,d^{2}\alpha$

where f is a representation of the phase space distribution. This function f is considered a quasiprobability density because it has the following properties:

• $\int f(\alpha ,\alpha ^{*})\,d^{2}\alpha =\mathrm {tr} ({\hat {\rho }})=1$ (normalization)
• If $g_{\Omega }({\hat {a}},{\hat {a}}^{\dagger })$ is an operator that can be expressed as a power series of the creation and annihilation operators in an ordering Ω, then its expectation value is

$\langle g_{\Omega }({\hat {a}},{\hat {a}}^{\dagger })\rangle =\int f(\alpha ,\alpha ^{*})g_{\Omega }(\alpha ,\alpha ^{*})\,d\alpha d\alpha ^{*}$ (optical equivalence theorem).

The function f is not unique. There exists a family of different representations, each connected to a different ordering Template:Mvar. The most popular in the general physics literature and historically first of these is the Wigner quasiprobability distribution, which is related to symmetric operator ordering. In quantum optics specifically, often the operators of interest, especially the particle number operator, is naturally expressed in normal order. In that case, the corresponding representation of the phase space distribution is the Glauber–Sudarshan P representation. The quasiprobabilistic nature of these phase space distributions is best understood in the Template:Mvar representation because of the following key statement:

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This sweeping statement is unavailable in other representations. For example, the Wigner function of the EPR state is positive definite but has no classical analog.

In addition to the representations defined above, there are many other quasiprobability distributions that arise in alternative representations of the phase space distribution. Another popular representation is the Husimi Q representation, which is useful when operators are in anti-normal order. More recently, the positive Template:Mvar representation and a wider class of generalized Template:Mvar representations have been used to solve complex problems in quantum optics. These are all equivalent and interconvertible to each other, viz. Cohen's class distribution function.

Characteristic functions

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

• $\chi _{W}(\mathbf {z} ,\mathbf {z} ^{*})=\operatorname {tr} (\rho e^{i\mathbf {z} \cdot {\widehat {\mathbf {a} }}+i\mathbf {z} ^{*}\cdot {\widehat {\mathbf {a} }}^{\dagger }})$
• $\chi _{P}(\mathbf {z} ,\mathbf {z} ^{*})=\operatorname {tr} (\rho e^{i\mathbf {z} \cdot {\widehat {\mathbf {a} }}}e^{i\mathbf {z} ^{*}\cdot {\widehat {\mathbf {a} }}^{\dagger }})$
• $\chi _{Q}(\mathbf {z} ,\mathbf {z} ^{*})=\operatorname {tr} (\rho e^{i\mathbf {z} ^{*}\cdot {\widehat {\mathbf {a} }}^{\dagger }}e^{i\mathbf {z} \cdot {\widehat {\mathbf {a} }}})$

Here ${\widehat {\mathbf {a} }}$ and ${\widehat {\mathbf {a} }}^{\dagger }$ are vectors containing the annihilation and creation operators for each mode of the system. These characteristic functions can be used to directly evaluate expectation values of operator moments. The ordering of the annihilation and creation operators in these moments is specific to the particular characteristic function. For instance, normally ordered (annihilation operators preceding creation operators) moments can be evaluated in the following way from $\chi _{P}\,$:

$\langle {\widehat {a}}_{j}^{\dagger m}{\widehat {a}}_{k}^{n}\rangle ={\frac {\partial ^{m+n}}{\partial (iz_{j}^{*})^{m}\partial (iz_{k}^{*})^{n}}}\chi _{P}(\mathbf {z} ,\mathbf {z} ^{*}){\Big |}_{\mathbf {z} =\mathbf {z} ^{*}=0}$

In the same way, expectation values of anti-normally ordered and symmetrically ordered combinations of annihilation and creation operators can be evaluated from the characteristic functions for the Q and Wigner distributions, respectively. The quasiprobability functions themselves are defined as Fourier transforms of the above characteristic functions. That is,

$\{W|P|Q\}(\mathbf {\alpha } ,\mathbf {\alpha } ^{*})={\frac {1}{\pi ^{2N}}}\int \chi _{\{W|P|Q\}}(\mathbf {z} ,\mathbf {z} ^{*})e^{-i\mathbf {z} ^{*}\cdot \mathbf {\alpha } ^{*}}e^{-i\mathbf {z} \cdot \mathbf {\alpha } }\,d^{2N}\mathbf {z} .$

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 space becomes multiplication in fourier space, moments can be calculated from these functions in the following way:

• $\langle {\widehat {\mathbf {a} }}_{j}^{\dagger m}{\widehat {\mathbf {a} }}_{k}^{n}\rangle =\int P(\mathbf {\alpha } ,\mathbf {\alpha } ^{*})\alpha _{j}^{n}\alpha _{k}^{*m}\,d^{2N}\mathbf {\alpha }$
• $\langle {\widehat {\mathbf {a} }}_{j}^{m}{\widehat {\mathbf {a} }}_{k}^{\dagger n}\rangle =\int Q(\mathbf {\alpha } ,\mathbf {\alpha } ^{*})\alpha _{j}^{m}\alpha _{k}^{*n}\,d^{2N}\mathbf {\alpha }$
• $\langle ({\widehat {\mathbf {a} }}_{j}^{\dagger m}{\widehat {\mathbf {a} }}_{k}^{n})_{S}\rangle =\int W(\mathbf {\alpha } ,\mathbf {\alpha } ^{*})\alpha _{j}^{m}\alpha _{k}^{*n}\,d^{2N}\mathbf {\alpha }$

Here $(\ldots )_{S}$ denotes symmetric ordering.

These representations are all interrelated through convolution by Gaussian functions:

• $W(\alpha ,\alpha ^{*})={\frac {2}{\pi }}\int P(\beta ,\beta ^{*})e^{-2|\alpha -\beta |^{2}}\,d^{2}\beta$
• $Q(\alpha ,\alpha ^{*})={\frac {2}{\pi }}\int W(\beta ,\beta ^{*})e^{-2|\alpha -\beta |^{2}}\,d^{2}\beta$

or using the property that convolution is associative

• $Q(\alpha ,\alpha ^{*})={\frac {1}{\pi }}\int P(\beta ,\beta ^{*})e^{-|\alpha -\beta |^{2}}\,d^{2}\beta .$

Time evolution and operator correspondences

Since each of the above transformations from Template:Mvar to the distribution functions is linear, the equation of motion for each distribution can be obtained by performing the same transformations to ${\dot {\rho }}$. Furthermore, as any master equation which can be expressed in Lindblad form is completely described by the action of combinations of annihilation and creation operators on the density operator, it is useful to consider the effect such operations have on each of the quasiprobability functions.

For instance, consider the annihilation operator ${\widehat {a}}_{j}\,$ acting on Template:Mvar. For the characteristic function of the P distribution we have

$\operatorname {tr} ({\widehat {a}}_{j}\rho e^{i\mathbf {z} \cdot {\widehat {\mathbf {a} }}}e^{i\mathbf {z} ^{*}\cdot {\widehat {\mathbf {a} }}^{\dagger }})={\frac {\partial }{\partial (iz_{j})}}\chi _{P}(\mathbf {z} ,\mathbf {z} ^{*}).$

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

${\widehat {a}}_{j}\rho \rightarrow \alpha _{j}P(\mathbf {\alpha } ,\mathbf {\alpha } ^{*}).$

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

• ${\widehat {a}}_{j}\rho \rightarrow \left(\alpha _{j}+\kappa {\frac {\partial }{\partial \alpha _{j}^{*}}}\right)\{W|P|Q\}(\mathbf {\alpha } ,\mathbf {\alpha } ^{*})$
• $\rho {\widehat {a}}_{j}^{\dagger }\rightarrow \left(\alpha _{j}^{*}+\kappa {\frac {\partial }{\partial \alpha _{j}}}\right)\{W|P|Q\}(\mathbf {\alpha } ,\mathbf {\alpha } ^{*})$
• ${\widehat {a}}_{j}^{\dagger }\rho \rightarrow \left(\alpha _{j}^{*}-(1-\kappa ){\frac {\partial }{\partial \alpha _{j}}}\right)\{W|P|Q\}(\mathbf {\alpha } ,\mathbf {\alpha } ^{*})$
• $\rho {\widehat {a}}_{j}\rightarrow \left(\alpha _{j}-(1-\kappa ){\frac {\partial }{\partial \alpha _{j}^{*}}}\right)\{W|P|Q\}(\mathbf {\alpha } ,\mathbf {\alpha } ^{*})$

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Examples

Coherent state

By construction, P for a coherent state $|\alpha _{0}\rangle$ is simply a delta function:

$P(\alpha ,\alpha ^{*})=\delta ^{2}(\alpha -\alpha _{0}).$

The Wigner and Q representations follows immediately from the Gaussian convolution formulas above:

$W(\alpha ,\alpha ^{*})={\frac {2}{\pi }}\int \delta ^{2}(\beta -\alpha _{0})e^{-2|\alpha -\beta |^{2}}\,d^{2}\beta ={\frac {2}{\pi }}e^{-2|\alpha -\alpha _{0}|^{2}}$

$Q(\alpha ,\alpha ^{*})={\frac {1}{\pi }}\int \delta ^{2}(\beta -\alpha _{0})e^{-|\alpha -\beta |^{2}}\,d^{2}\beta ={\frac {1}{\pi }}e^{-|\alpha -\alpha _{0}|^{2}}.$

The Husimi representation can also be found using the formula above for the inner product of two coherent states:

$Q(\alpha ,\alpha ^{*})={\frac {1}{\pi }}\langle \alpha |{\hat {\rho }}|\alpha \rangle ={\frac {1}{\pi }}|\langle \alpha _{0}|\alpha \rangle |^{2}={\frac {1}{\pi }}e^{-|\alpha -\alpha _{0}|^{2}}$

Fock state

The P representation of a Fock state $|n\rangle$ is

$P(\alpha ,\alpha ^{*})={\frac {e^{|\alpha |^{2}}}{n!}}{\frac {\partial ^{2n}}{\partial \alpha ^{*n}\partial \alpha ^{n}}}\delta ^{2}(\alpha ).$

Since for n>0 this is more singular than a delta function, a Fock state has no classical analog. The non-classicality is less transparent as one proceeds with the Gaussian convolutions. If L_n is the nth Laguerre polynomial, W is

$W(\alpha ,\alpha ^{*})=(-1)^{n}{\frac {2}{\pi }}e^{-2|\alpha |^{2}}L_{n}\left(4|\alpha |^{2}\right)~,$

which can go negative but is bounded. Q always remains positive and bounded:

$Q(\alpha ,\alpha ^{*})={\frac {1}{\pi }}\langle \alpha |{\hat {\rho }}|\alpha \rangle ={\frac {1}{\pi }}|\langle n|\alpha \rangle |^{2}={\frac {1}{\pi n!}}|\langle 0|{\hat {a}}^{n}|\alpha \rangle |^{2}={\frac {|\alpha |^{2n}}{\pi n!}}e^{-|\alpha |^{2}}$

Damped quantum harmonic oscillator

Consider the damped quantum harmonic oscillator with the following master equation:

${\frac {d{\hat {\rho }}}{dt}}=i\omega _{0}[{\hat {\rho }},{\hat {a}}^{\dagger }{\hat {a}}]+{\frac {\gamma }{2}}(2{\hat {a}}{\hat {\rho }}{\hat {a}}^{\dagger }-{\hat {a}}^{\dagger }{\hat {a}}{\hat {\rho }}-\rho {\hat {a}}^{\dagger }{\hat {a}})+\gamma \langle n\rangle ({\hat {a}}{\hat {\rho }}{\hat {a}}^{\dagger }+{\hat {a}}^{\dagger }{\hat {\rho }}{\hat {a}}-{\hat {a}}^{\dagger }{\hat {a}}{\hat {\rho }}-{\hat {\rho }}{\hat {a}}{\hat {a}}^{\dagger }).$

This results in the Fokker–Planck equation

${\frac {\partial }{\partial t}}\{W|P|Q\}(\alpha ,\alpha ^{*},t)=\left[(\gamma +i\omega _{0}){\frac {\partial }{\partial \alpha }}\alpha +(\gamma -i\omega _{0}){\frac {\partial }{\partial \alpha ^{*}}}\alpha ^{*}+{\frac {\gamma }{2}}(\langle n\rangle +\kappa ){\frac {\partial ^{2}}{\partial \alpha \partial \alpha ^{*}}}\right]\{W|P|Q\}(\alpha ,\alpha ^{*},t)$

where κ=0, 1/2, 1 for the P, W, and Q representations, respectively. If the system is initially in the coherent state $|\alpha _{0}\rangle$, then this has the solution

$\{W|P|Q\}(\alpha ,\alpha ^{*},t)={\frac {1}{\pi \left[\kappa +\langle n\rangle \left(1-e^{-2\gamma t}\right)\right]}}\exp {\left(-{\frac {\left|\alpha -\alpha _{0}e^{-(\gamma +i\omega _{0})t}\right|^{2}}{\kappa +\langle n\rangle \left(1-e^{-2\gamma t}\right)}}\right)}$