---
title: Sudarshan–Glauber Distribution in Quantum Optics
url: https://www.emergentmind.com/topics/sudarshan-glauber-distribution
type: topic
---

# Sudarshan–Glauber Distribution in Quantum Optics

The Sudarshan–Glauber distribution, commonly referred to as the Glauber–Sudarshan P function, is a foundational quasiprobability distribution in quantum optics and quantum phase-space theory. It provides a diagonal representation of quantum states in terms of coherent-state projectors, establishing the most direct quantum-classical correspondence for bosonic systems. The distribution's ability to characterize nonclassicality, its singularity structure, its operational transformation under physical channels, and its extension to field theory and open-system dynamics position it as a central object in both theoretical analysis and experimental diagnostics.

## 1. Definition and Representations

Let $\hat\rho$ denote the density operator for a single-mode bosonic system, and let $\{|\alpha\rangle\}$ be the overcomplete set of coherent states defined by $|\alpha\rangle=D(\alpha)|0\rangle$, with $D(\alpha)=\exp(\alpha \hat a^\dagger-\alpha^* \hat a)$. The Sudarshan–Glauber P function $P(\alpha)$ is uniquely determined by the operator expansion
\[
\hat\rho = \int d^2\alpha\, P(\alpha) |\alpha\rangle\langle\alpha|\,,
\]
where $d^2\alpha=d\Re\alpha\, d\Im\alpha$, and the normalization $\int d^2\alpha\, P(\alpha) = 1$ holds for any physical state [1303.4718], [1311.5598].

Equivalently, $P(\alpha)$ is the inverse Fourier transform of the normally ordered characteristic function $\Phi(\beta)=\operatorname{Tr}[\hat\rho D(\beta)]$:
\[
P(\alpha) = \frac{1}{\pi^2}\int d^2 \beta\, \Phi(\beta) e^{\beta^*\alpha-\beta\alpha^*}\,.
\]
This formalism generalizes immediately to multimode bosonic systems and extends, with crucial Grassmannian modifications, to fermionic systems [2210.11980].

The "s-parameterized" family of quasiprobabilities $W^{(s)}(\alpha)$ (Cahill–Glauber) interpolates between the P function ($s=1$), the Wigner function ($s=0$), and the Husimi Q function ($s=-1$), via characteristic function filtering [1303.4718], [1512.01836]:
\[
W^{(s)}(\alpha) = \frac{1}{\pi^2}\int d^2\beta\, \Phi^{(s)}(\beta) e^{\beta^*\alpha - \beta\alpha^*}\,, \quad \Phi^{(s)}(\beta) = \operatorname{Tr}[\hat\rho D(\beta)]\,e^{s|\beta|^2/2}.
\]

## 2. Singularities and Nonclassicality Criteria

The most important property distinguishing the P function is its singularity and negativity structure. For genuinely classical states (i.e., mixtures of coherent states in the sense of Titulaer and Glauber), $P(\alpha)$ is a bona fide nonnegative function [1303.4718]. For nonclassical states, $P(\alpha)$ is either negative in some regions or more singular than the Dirac delta—in the distributional sense.

Explicitly, number (Fock) states yield
\[
P_{n,n}(\alpha) = \frac{(-1)^n}{n!}\, \partial_\alpha^n\,\partial_{\alpha^*}^n\,\delta^{(2)}(\alpha)\,, 
\]
and squeezed vacuum, photon-added, or cat states produce similar derivatives or nonpositive distributions [1605.04425].

The critical operational nonclassicality criterion is that a quantum state is classical **if and only if** its P function can be interpreted as a regular, nonnegative probability density. This equivalence does **not** hold for the Wigner function, which can become everywhere nonnegative under attenuation without eliminating quantum correlations or violations of classical inequalities [1303.4718].

## 3. Transformations under Physical Channels

The uniqueness of the P function is sharply manifest in its transformation laws under linear optical devices. Under attenuation (e.g., a loss channel with efficiency $\eta$ modeled as a beam splitter), $P(\alpha)$ transforms **exactly** as a classical probability density:
\[
P_{out}(\alpha) = \frac{1}{\eta}\, P_{in}\left(\frac{\alpha}{\sqrt{\eta}}\right)\,.
\]
No other $s$-parameterized quasiprobability enjoys this property; for instance, Gaussian convolution present in the Wigner or Q representations mixes quantum and classical features [1303.4718], [2510.15628].

Operationally, one may realize the quantum-to-classical smoothing via physical processes:
- Quantum-limited amplifier (squeezing transformation) maps a singular P distribution into the Wigner function (Gaussian convolution) [2312.08216].
- Pure-loss channel (attenuator) maps the Wigner function into the Husimi Q distribution.
This chain elucidates the Weierstrass transform structure of the three principal distributions, and provides a direct laboratory pathway to probe nonclassicality and singular features of $P(\alpha)$.

## 4. Filter-Regularized P Functions and Experimental Reconstruction

Because many nonclassical states have P functions that are not regular functions but generalized distributions, regularization is required for experimental procedures. A filter function $\Omega(\xi)$ acts via characteristic function multiplication and Fourier transform:
\[
P_\Omega (\alpha) = (P * \tilde{\Omega})(\alpha)\,, \quad \tilde{\Omega}(\beta) = \frac{1}{\pi^2}\int d^2\xi\, e^{\beta\xi^* - \beta^*\xi} \Omega(\xi)\,,
\]
so $P_\Omega$ is a smooth, directly measurable quasiprobability. The associated quantum map $\rho_\Omega$ is completely positive and trace preserving (CPTP) **if and only if** $\tilde{\Omega}(\beta)$ is a probability density [2206.05594].

The fidelity between the filtered and original states is lower-bounded by the overlap of their characteristic functions. As a result, any quantum state can be arbitrarily well approximated in trace-norm by a state with a regular P function. This facilitates process tomography, measurement statistics estimation, and visualization of nonclassicality in experiments.

## 5. Mathematical Properties and Connections to Other Quasiprobabilities

The P function underpins the star-product calculus in phase-space quantum mechanics. Any operator $\hat A$ admits both Wick (P-symbol) and Anti-Wick (Q-symbol) representations, related by Berezin transforms:
\[
A_W(\alpha,\alpha^*) = \exp\left(\frac{\partial}{\partial\alpha}\frac{\partial}{\partial\alpha^*}\right) A_{aW}(\alpha,\alpha^*)\,.
\]
Time evolution in the P representation yields a hierarchy of derivatives:
\[
\frac{\partial}{\partial t} P(\alpha, t) = \frac{i}{\hbar} \sum_{n=1}^\infty \frac{(-1)^n}{n!}\Big[ \partial_{\alpha^*}^n (\partial_\alpha^n H_W P ) - \partial_\alpha^n (\partial_{\alpha^*}^n H_W P ) \Big]\,,
\]
which truncates at quadratic order (giving a Fokker–Planck equation) for polynomial Hamiltonians of degree up to four [2510.15628].

Gaussian convolution links P to both Wigner and Husimi Q distributions. For instance, the Kirkwood–Riháczec function is the Gaussian smoothed P function:
\[
K(\beta, \beta^*) = \int d^2\alpha\, P(\alpha)\, \exp(-2|\alpha - \beta|^2)\,,
\]
highlighting that sharp features in K signal singularities in P [1311.5598]. The number-phase Wigner function can also be explicitly related to P via intricate integral transforms involving photon-number and phase-variable marginals [1512.01836].

## 6. Extensions to Field Theory and Fermionic Systems

The P mapping generalizes to ensembles in bosonic field theory. For arbitrary classical ensembles, the quantum density matrix constructed via the P mapping preserves the mean classical energy:
\[
\mathrm{Tr}(H_n \rho) = \langle E(S) \rangle_{\mathrm{ensemble}}\,,
\]
where $H_n$ is the normal-ordered quantization of the classical Hamiltonian. For microcanonical ensembles (definite classical energy), the mapped quantum state is supported exactly on the zero-eigenvalue subspace of the spectral operator $N(H_n - E, H_n - E)$ [1402.5116].

For fermionic systems, coherent states are parameterized by Grassmann variables, leading to two distinct P-type representations due to two inequivalent resolutions of identity. Singularities and nonclassicality criteria are formally analogous: a regular, even Grassmann P function signals classicality; derivatives or oscillations indicate quantum features [2210.11980].

## 7. Stochastic Trajectories and Open Quantum Dynamics

In open-system dynamics described by Lindblad equations, the P function is central to path-integral and stochastic differential equation (SDE) formulations. The s-parameterized path integral framework yields SDEs for quantum trajectories, valid only when the corresponding diffusion matrix is positive semidefinite. In typical situations (free Hamiltonian, linear losses), the P distribution corresponds to deterministic trajectories (i.e., classical motion in phase space recovers exact quantum evolution for observables). In interacting or multi-body open systems, negative-definite diffusion obstructs the P-function SDE, requiring transition to Wigner or Husimi representations [2508.01991].

These methodologies facilitate efficient simulation, benchmarking, and prediction of non-equal-time correlations in many-body quantum optics.

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The Sudarshan–Glauber distribution thus provides a rigorous, operationally unique, and structurally central description of quantum states in phase space. It sharply distinguishes classical from nonclassical states, elucidates their transformation under physical processes, supports high-fidelity experimental approximation via filtering, connects to alternative quasiprobabilities through explicit convolutions, and generalizes throughout quantum field theory and open-system dynamics [1303.4718], [1605.04425], [2206.05594], [2508.01991], [2510.15628], [2312.08216], [1402.5116], [1512.01836], [2210.11980], [1311.5598].

Source: https://www.emergentmind.com/topics/sudarshan-glauber-distribution