Glauber–Sudarshan P representation

Definition

We wish to construct a function $P (α)$ with the property that the density matrix $\hat{ρ}$ is diagonal in the basis of coherent states ${| α ⟩}$ , i.e.,

\hat{ρ} = \int P (α) | α ⟩ ⟨ α | d^{2} α, d^{2} α \equiv d R e (α) d I m (α) .

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

{\hat{ρ}}_{A} = \sum_{j, k} c_{j, k} \cdot {\hat{a}}^{j} {\hat{a}}^{† k} .

Inserting the resolution of the identity

\hat{I} = \frac{1}{π} \int | α ⟩ ⟨ α | d^{2} α,

we see that

\begin{matrix} ρ_{A} (\hat{a}, {\hat{a}}^{†}) & = \frac{1}{π} \sum_{j, k} \int c_{j, k} \cdot {\hat{a}}^{j} | α ⟩ ⟨ α | {\hat{a}}^{† k} d^{2} α \\ = \frac{1}{π} \sum_{j, k} \int c_{j, k} \cdot α^{j} | α ⟩ ⟨ α | α^{* k} d^{2} α \\ = \frac{1}{π} \int \sum_{j, k} c_{j, k} \cdot α^{j} α^{* k} | α ⟩ ⟨ α | d^{2} α \\ = \frac{1}{π} \int ρ_{A} (α, α^{*}) | α ⟩ ⟨ α | d^{2} α, \end{matrix}

and thus we formally assign

P (α) = \frac{1}{π} ρ_{A} (α, α^{*}) .

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

χ_{N} (β) = tr (\hat{ρ} \cdot e^{i β \cdot {\hat{a}}^{†}} e^{i β^{*} \cdot \hat{a}})

and then take the Fourier transform

P (α) = \frac{1}{π^{2}} \int χ_{N} (β) e^{- β α^{*} + β^{*} α} d^{2} β .

Another useful integral formula for P is

P (α) = \frac{e^{| α |^{2}}}{π^{2}} \int ⟨ - β | \hat{ρ} | β ⟩ e^{| β |^{2} - β α^{*} + β^{*} α} d^{2} β .

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 $\hat{ρ}$ in the Fock basis ${| n ⟩}$ . The following formula shows that it is always possible to write the density matrix in this diagonal form without appealing to operator orderings using the inversion (given here for a single mode),

P (α) = \sum_{n} \sum_{k} ⟨ n | \hat{ρ} | k ⟩ \frac{\sqrt{n! k!}}{2 π r (n + k)!} e^{r^{2} - i (n - k) θ} [{(- \frac{\partial}{\partial r})}^{n + k} δ (r)],

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.

2 more chapters are free with a Lably account.

Discussion

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 state or entangled 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 $P (α)$ behaved like a true probability density [function], it would not describe probabilities of mutually exclusive states."

Examples

Fock states

Fock states, $| ψ ⟩ = | n ⟩$ for integer $n$ , correspond to a highly singular P distribution, which can be written as $P_{n} (α, α^{*}) = \frac{e^{| α |^{2}}}{n!} \frac{\partial^{2 n}}{\partial α^{* n} \partial α^{n}} δ^{2} (α) .$ While this is not a function, this expression corresponds to a tempered distribution. In particular for the vacuum state $| 0 ⟩$ the P distribution is a Dirac delta function at the origin, as $P_{0} (α, α^{*}) = e^{| α |^{2}} δ^{2} (α) = δ^{2} (α)$ . Similarly, the Fock state $| 1 ⟩$ gives $P_{1} (α, α^{*}) = e^{| α |^{2}} \frac{\partial^{2}}{\partial α \partial α^{*}} δ^{2} (α) = δ^{2} (α) + \frac{\partial^{2}}{\partial α \partial α^{*}} δ^{2} (α) .$ We can also easily verify that the above expression for $P_{n}$ works more generally observing that $\int d^{2} α (\frac{e^{| α |^{2}}}{n!} \frac{\partial^{2 n}}{\partial α^{n} \partial α^{* n}} δ^{2} (α)) | α ⟩ ⟨ α | = \sum_{j, k = 0}^{\infty} \frac{| j ⟩ ⟨ k |}{n! \sqrt{j! k!}} \int d^{2} α (\frac{\partial^{2 n}}{\partial α^{n} \partial α^{* n}} δ^{2} (α)) α^{j} α^{* k},$ together with the identity $\int d^{2} α (\frac{\partial^{2 n}}{\partial α^{n} \partial α^{* n}} δ^{2} (α)) α^{j} α^{* k} = {\frac{\partial^{2 n}}{\partial α^{n} \partial α^{* n}} (α^{j} α^{* k}) |}_{α = 0} = n!^{2} δ_{j, n} δ_{k, n} .$ The same reasoning can be used to show more generally that the P function of the operators $| n ⟩ ⟨ m |$ is given by $P_{n, m} (α, α^{*}) = \frac{e^{| α |^{2}}}{\sqrt{n! m!}} \frac{\partial^{n + m}}{\partial α^{n} \partial α^{* m}} δ^{2} (α) .$

Another concise formal expression for the P function of Fock states using the Laguerre polynomials is $P_{n} (α, α^{*}) = L_{n} (- \frac{1}{4} \partial_{α} \partial_{α^{*}}) δ^{2} (α) .$

Thermal radiation

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

⟨ {\hat{n}}_{k, s} ⟩ = \frac{1}{e^{ℏ ω / k_{B} T} - 1} .

The P representation of the black body is

P ({α_{k, s}}) = \prod_{k, s} \frac{1}{π ⟨ {\hat{n}}_{k, s} ⟩} e^{- | α |^{2} / ⟨ {\hat{n}}_{k, s} ⟩} .

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

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

| ψ ⟩ = c_{0} | α_{0} ⟩ + c_{1} | α_{1} ⟩

where c₀ , c₁ are constants subject to the normalizing constraint

1 = | c_{0} |^{2} + | c_{1} |^{2} + 2 e^{- (| α_{0} |^{2} + | α_{1} |^{2}) / 2} Re (c_{0}^{*} c_{1} e^{α_{0}^{*} α_{1}}) .

Note that this is quite different from a qubit because $| α_{0} ⟩$ and $| α_{1} ⟩$ are not orthogonal. As it is straightforward to calculate $⟨ - α | \hat{ρ} | α ⟩ = ⟨ - α | ψ ⟩ ⟨ ψ | α ⟩$ , we can use the Mehta formula above to compute P,

\begin{matrix} P (α) = & | c_{0} |^{2} δ^{2} (α - α_{0}) + | c_{1} |^{2} δ^{2} (α - α_{1}) \\ + 2 c_{0}^{*} c_{1} e^{| α |^{2} - \frac{1}{2} | α_{0} |^{2} - \frac{1}{2} | α_{1} |^{2}} e^{(α_{1}^{*} - α_{0}^{*}) \cdot \partial / \partial (2 α^{*} - α_{0}^{*} - α_{1}^{*})} e^{(α_{0} - α_{1}) \cdot \partial / \partial (2 α - α_{0} - α_{1})} \cdot δ^{2} (2 α - α_{0} - α_{1}) \\ + 2 c_{0} c_{1}^{*} e^{| α |^{2} - \frac{1}{2} | α_{0} |^{2} - \frac{1}{2} | α_{1} |^{2}} e^{(α_{0}^{*} - α_{1}^{*}) \cdot \partial / \partial (2 α^{*} - α_{0}^{*} - α_{1}^{*})} e^{(α_{1} - α_{0}) \cdot \partial / \partial (2 α - α_{0} - α_{1})} \cdot δ^{2} (2 α - α_{0} - α_{1}) . \end{matrix}

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:

\begin{matrix} ⟨ ψ | \hat{n} | ψ ⟩ & = \int P (α) | α |^{2} d^{2} α \\ = | c_{0} α_{0} |^{2} + | c_{1} α_{1} |^{2} + 2 e^{- (| α_{0} |^{2} + | α_{1} |^{2}) / 2} Re (c_{0}^{*} c_{1} α_{0}^{*} α_{1} e^{α_{0}^{*} α_{1}}) . \end{matrix}