Sudarshan–Glauber Distribution in Quantum Optics

• The Sudarshan–Glauber distribution is a quasiprobability function that represents quantum states as integrals over coherent state projectors, establishing a unique quantum-classical correspondence.
• It distinguishes classical from nonclassical states by exhibiting singularities or negativity, with Fock and squeezed states acting as prime examples of nonclassicality.
• The distribution transforms under physical processes like attenuation exactly as a classical probability, and filtering techniques enable experimental reconstruction of regularized, measurable versions.

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 (Kiesel, 2013, Moya-Cessa, 2013).

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 (Barnett et al., 2022).

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 (Kiesel, 2013, Przanowski et al., 2015): $$W^{(s)}(\alpha) = \frac{1}{\pi^2}\int d^2\beta\, \Phi^{(s)}(\beta) e^{\beta^*\alpha - \beta\alpha^*}\,, \quad \Phi^{(s)}(\beta) = \operatorname{Tr}[\hat\rho D(\beta)]\,e^{s|\beta|^2/2}.$$

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 (Kiesel, 2013). 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 (Sperling, 2016).

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 (Kiesel, 2013).

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 (Kiesel, 2013, Tyagi et al., 17 Oct 2025).

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) (Linowski et al., 2023).
• 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 (Zartab et al., 2022).

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 (Tyagi et al., 17 Oct 2025).

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 (Moya-Cessa, 2013). The number-phase Wigner function can also be explicitly related to P via intricate integral transforms involving photon-number and phase-variable marginals (Przanowski et al., 2015).

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 $N(H_n - E, H_n - E)$ (Werbos, 2013).

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 (Barnett et al., 2022).

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 (Yoneya et al., 4 Aug 2025).

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 (Kiesel, 2013, Sperling, 2016, Zartab et al., 2022, Yoneya et al., 4 Aug 2025, Tyagi et al., 17 Oct 2025, Linowski et al., 2023, Werbos, 2013, Przanowski et al., 2015, Barnett et al., 2022, Moya-Cessa, 2013).