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Glauber–Sudarshan P representation

We wish to construct a function ${\displaystyle P(\alpha )}$ with the property that the density matrix ${\displaystyle {\hat {\rho }}}$isdiagonal in the basis of coherent states ${\displaystyle \{|\alpha \rangle \}}$, i.e.,

${\displaystyle {\hat {\rho }}=\int P(\alpha )|{\alpha }\rangle \langle {\alpha }|\,d^{2}\alpha ,\qquad d^{2}\alpha \equiv d\,{\rm {Re}}(\alpha )\,d\,{\rm {Im}}(\alpha ).}$

We also wish to construct the function such that the expectation value of a normally ordered operator satisfies the optical equivalence theorem. This implies that the density matrix should be in anti-normal order so that we can express the density matrix as a power series

${\displaystyle {\hat {\rho }}_{A}=\sum _{j,k}c_{j,k}\cdot {\hat {a}}^{j}{\hat {a}}^{\dagger k}.}$

Inserting the resolution of the identity

${\displaystyle {\hat {I}}={\frac {1}{\pi }}\int |{\alpha }\rangle \langle {\alpha }|\,d^{2}\alpha ,}$

we see that

${\displaystyle {\begin{aligned}\rho _{A}({\hat {a}},{\hat {a}}^{\dagger })&={\frac {1}{\pi }}\sum _{j,k}\int c_{j,k}\cdot {\hat {a}}^{j}|{\alpha }\rangle \langle {\alpha }|{\hat {a}}^{\dagger k}\,d^{2}\alpha \\&={\frac {1}{\pi }}\sum _{j,k}\int c_{j,k}\cdot \alpha ^{j}|{\alpha }\rangle \langle {\alpha }|\alpha ^{*k}\,d^{2}\alpha \\&={\frac {1}{\pi }}\int \sum _{j,k}c_{j,k}\cdot \alpha ^{j}\alpha ^{*k}|{\alpha }\rangle \langle {\alpha }|\,d^{2}\alpha \\&={\frac {1}{\pi }}\int \rho _{A}(\alpha ,\alpha ^{*})|{\alpha }\rangle \langle {\alpha }|\,d^{2}\alpha ,\end{aligned}}}$

and thus we formally assign

${\displaystyle P(\alpha )={\frac {1}{\pi }}\rho _{A}(\alpha ,\alpha ^{*}).}$

More useful integral formulas for P are necessary for any practical calculation. One method^[6] is to define the characteristic function

${\displaystyle \chi _{N}(\beta )=\operatorname {tr} ({\hat {\rho }}\cdot e^{i\beta \cdot {\hat {a}}^{\dagger }}e^{i\beta ^{*}\cdot {\hat {a}}})}$

and then take the Fourier transform

${\displaystyle P(\alpha )={\frac {1}{\pi ^{2}}}\int \chi _{N}(\beta )e^{-\beta \alpha ^{*}+\beta ^{*}\alpha }\,d^{2}\beta .}$

Another useful integral formula for Pis^[7]

${\displaystyle P(\alpha )={\frac {e^{|\alpha |^{2}}}{\pi ^{2}}}\int \langle -\beta |{\hat {\rho }}|\beta \rangle e^{|\beta |^{2}-\beta \alpha ^{*}+\beta ^{*}\alpha }\,d^{2}\beta .}$

Note that both of these integral formulas do not converge in any usual sense for "typical" systems . We may also use the matrix elements of ${\displaystyle {\hat {\rho }}}$ in the Fock basis ${\displaystyle \{|n\rangle \}}$. The following formula shows that it is always possible^[3] to write the density matrix in this diagonal form without appealing to operator orderings using the inversion (given here for a single mode),

${\displaystyle P(\alpha )=\sum _{n}\sum _{k}\langle n|{\hat {\rho }}|k\rangle {\frac {\sqrt {n!k!}}{2\pi r(n+k)!}}e^{r^{2}-i(n-k)\theta }\left[\left(-{\frac {\partial }{\partial r}}\right)^{n+k}\delta ($r)\right],}

where r and θ are the amplitude and phase of α. Though this is a full formal solution of this possibility, it requires infinitely many derivatives of Dirac delta functions, far beyond the reach of any ordinary tempered distribution theory.

Discussion[edit]

If the quantum system has a classical analog, e.g. a coherent state or thermal radiation, then P is non-negative everywhere like an ordinary probability distribution. If, however, the quantum system has no classical analog, e.g. an incoherent Fock stateorentangled system, then P is negative somewhere or more singular than a Dirac delta function. (By a theorem of Schwartz, distributions that are more singular than the Dirac delta function are always negative somewhere.) Such "negative probability" or high degree of singularity is a feature inherent to the representation and does not diminish the meaningfulness of expectation values taken with respect to P. Even if P does behave like an ordinary probability distribution, however, the matter is not quite so simple. According to Mandel and Wolf: "The different coherent states are not [mutually] orthogonal, so that even if ${\displaystyle P(\alpha )}$ behaved like a true probability density [function], it would not describe probabilities of mutually exclusive states."^[8]

Examples[edit]

Thermal radiation[edit]

From statistical mechanics arguments in the Fock basis, the mean photon number of a mode with wavevector k and polarization state s for a black body at temperature T is known to be

${\displaystyle \langle {\hat {n}}_{\mathbf {k} ,s}\rangle ={\frac {1}{e^{\hbar \omega /k_{B}T}-1}}.}$

The P representation of the black body is

${\displaystyle P(\{\alpha _{\mathbf {k} ,s}\})=\prod _{\mathbf {k} ,s}{\frac {1}{\pi \langle {\hat {n}}_{\mathbf {k} ,s}\rangle }}e^{-|\alpha |^{2}/\langle {\hat {n}}_{\mathbf {k} ,s}\rangle }.}$

In other words, every mode of the black body is normally distributed in the basis of coherent states. Since P is positive and bounded, this system is essentially classical. This is actually quite a remarkable result because for thermal equilibrium the density matrix is also diagonal in the Fock basis, but Fock states are non-classical.

Highly singular example[edit]

Even very simple-looking states may exhibit highly non-classical behavior. Consider a superposition of two coherent states

${\displaystyle |\psi \rangle =c_{0}|\alpha _{0}\rangle +c_{1}|\alpha _{1}\rangle }$

where c₀ , c₁ are constants subject to the normalizing constraint

${\displaystyle 1=|c_{0}|^{2}+|c_{1}|^{2}+2e^{-(|\alpha _{0}|^{2}+|\alpha _{1}|^{2})/2}\operatorname {Re} \left(c_{0}^{*}c_{1}e^{\alpha _{0}^{*}\alpha _{1}}\right).}$

Note that this is quite different from a qubit because ${\displaystyle |\alpha _{0}\rangle }$ and ${\displaystyle |\alpha _{1}\rangle }$ are not orthogonal. As it is straightforward to calculate ${\displaystyle \langle -\alpha |{\hat {\rho }}|\alpha \rangle =\langle -\alpha |\psi \rangle \langle \psi |\alpha \rangle }$, we can use the Mehta formula above to compute P,

${\displaystyle {\begin{aligned}P(\alpha )={}&|c_{0}|^{2}\delta ^{2}(\alpha -\alpha _{0})+|c_{1}|^{2}\delta ^{2}(\alpha -\alpha _{1})\\[5pt]&{}+2c_{0}^{*}c_{1}e^{|\alpha |^{2}-{\frac {1}{2}}|\alpha _{0}|^{2}-{\frac {1}{2}}|\alpha _{1}|^{2}}e^{(\alpha _{1}^{*}-\alpha _{0}^{*})\cdot \partial /\partial (2\alpha ^{*}-\alpha _{0}^{*}-\alpha _{1}^{*})}e^{(\alpha _{0}-\alpha _{1})\cdot \partial /\partial (2\alpha -\alpha _{0}-\alpha _{1})}\cdot \delta ^{2}(2\alpha -\alpha _{0}-\alpha _{1})\\[5pt]&{}+2c_{0}c_{1}^{*}e^{|\alpha |^{2}-{\frac {1}{2}}|\alpha _{0}|^{2}-{\frac {1}{2}}|\alpha _{1}|^{2}}e^{(\alpha _{0}^{*}-\alpha _{1}^{*})\cdot \partial /\partial (2\alpha ^{*}-\alpha _{0}^{*}-\alpha _{1}^{*})}e^{(\alpha _{1}-\alpha _{0})\cdot \partial /\partial (2\alpha -\alpha _{0}-\alpha _{1})}\cdot \delta ^{2}(2\alpha -\alpha _{0}-\alpha _{1}).\end{aligned}}}$

Despite having infinitely many derivatives of delta functions, P still obeys the optical equivalence theorem. If the expectation value of the number operator, for example, is taken with respect to the state vector or as a phase space average with respect to P, the two expectation values match:

${\displaystyle {\begin{aligned}\langle \psi |{\hat {n}}|\psi \rangle &=\int P(\alpha )|\alpha |^{2}\,d^{2}\alpha \\&=|c_{0}\alpha _{0}|^{2}+|c_{1}\alpha _{1}|^{2}+2e^{-(|\alpha _{0}|^{2}+|\alpha _{1}|^{2})/2}\operatorname {Re} \left(c_{0}^{*}c_{1}\alpha _{0}^{*}\alpha _{1}e^{\alpha _{0}^{*}\alpha _{1}}\right).\end{aligned}}}$

See also[edit]

• Quasiprobability distribution § Characteristic functions
• Nonclassical light
• Wigner quasiprobability distribution
• Husimi Q representation
• Nobel Prize controversies

References[edit]

Citations[edit]

Bibliography[edit]