Glauber–Sudarshan P representation
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| Glauber–Sudarshan P representation | |
|---|---|
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| Name | Glauber–Sudarshan P representation |
| Field | Quantum optics |
| Introduced | 1963 |
| Introduced by | Roy J. Glauber; E. C. G. Sudarshan |
Glauber–Sudarshan P representation The Glauber–Sudarshan P representation is a quasiprobability distribution used to express density operators of single-mode bosonic fields as weighted mixtures of coherent states. It plays a central role in quantum optics and the theory of photodetection by connecting operator descriptions with phase-space methods developed in quantum mechanics. The representation elucidates distinctions between classical-like and nonclassical states and underpins techniques in state reconstruction and quantum information processing.
Introduction
The P representation was developed in the context of quantum optics by Roy J. Glauber and independently by E.C.G. Sudarshan to provide a diagonal expansion of the density operator in the overcomplete basis of coherent states related to the Glauber coherent state formalism. It complements other phase-space distributions such as the Husimi Q representation and the Wigner quasi-probability distribution introduced by Kurt Gödel contemporaries and later formalized in the work of Eugene Wigner. The formalism has influenced experimental programs at institutions like Bell Laboratories, Bell Labs, Jet Propulsion Laboratory, and research by groups at Harvard University and California Institute of Technology.
Mathematical Definition
For a single bosonic mode with annihilation operator a and coherent state |α⟩ labeled by complex α, the density operator ρ can be written as ρ = ∫ P(α) |α⟩⟨α| d^2α, where P(α) is the Glauber–Sudarshan P function. The representation relies on the resolution of the identity afforded by coherent states originally studied by John Klauder and connected to the optical coherence theory of Leonard Mandel and Emil Wolf. The transform relationships between P(α), the Wigner function W(α), and the Husimi Q(α) function are given by Gaussian convolution operations analogous to integral transforms used by Norbert Wiener and Joseph Fourier; inversion formulas can involve distribution theory as in the work of Laurent Schwartz.
Properties and Interpretation
When P(α) is a bona fide probability density (nonnegative and regular), the state admits a classical mixture interpretation akin to statistical optics models of Roy J. Glauber and E. C. G. Sudarshan. For many quantum states, P(α) is highly singular—containing derivatives of delta functions—or attains negative values, signaling nonclassical features analogous to negativity in the Wigner quasi-probability distribution encountered in studies by Eugene Wigner and Marcel Riesz. The representation preserves normal ordering of field operators: expectation values of normally ordered observables correspond to classical averages over P(α), a property exploited in treatments by Roy J. Glauber and Howard J. Carmichael.
Examples and Applications
For coherent states |β⟩ the P function is a delta function P(α)=δ^2(α−β), mirroring coherent-state properties emphasized in early experiments at Bell Laboratories. Thermal states yield Gaussian P(α) functions as in treatments by L. Mandel and E. Wolf of blackbody radiation. Squeezed states and Fock states produce P functions that are singular or highly nonclassical; such behaviors were analyzed in theoretical work by H. P. Yuen and in experimental squeezers developed at University of Tokyo and Institut d'Optique. Applications include photon-counting theory of Roy J. Glauber and quantum state engineering for platforms like Harvard University’s cavity QED, MIT’s trapped ions, and NIST’s superconducting circuits.
Nonclassicality and P-function Singularities
Nonclassicality criteria are often phrased in terms of the impossibility of representing a state by a nonnegative P(α). Tests for negativity or singularity relate to measurable quantities such as sub-Poissonian statistics discussed by Leonard Mandel and violations of classical inequalities akin to those used in John Bell experiments. The presence of derivatives of delta functions in P(α) for Fock states was highlighted in analyses by E.C.G. Sudarshan and later mathematical treatments leveraging generalized function theory by Laurent Schwartz and distribution analysts at University of Cambridge.
Measurement and Reconstruction
Reconstructing P(α) directly is challenging due to singularities; practical protocols reconstruct smoothed quasiprobabilities such as the Wigner function via homodyne tomography developed by groups at Universität Wien and University of Oxford. Techniques include balanced homodyne detection pioneered by experimentalists at California Institute of Technology and statistical inversion methods influenced by work at Max Planck Society and European Laboratory for Non-Linear Spectroscopy (LENS). Regularization schemes and pattern-function approaches from the theory of inverse problems by Andrei Tikhonov are applied to obtain approximations to P(α).
Extensions and Related Representations
Generalizations include multimode P representations used in studies of entanglement at Perimeter Institute and in continuous-variable quantum information at IQOQI Vienna and Institute for Quantum Computing (IQC). The s-parameterized family of quasiprobabilities introduced by H. W. Lee interpolates between P (s=1), Wigner (s=0), and Q (s=−1) functions, connecting to phase-space formulations by Roy J. Glauber and E.C.G. Sudarshan. Other related constructs include positive-P and generalized P representations developed by P. D. Drummond and C. W. Gardiner to handle open quantum systems and stochastic simulation of quantum dynamics, with applications across platforms at Imperial College London, Australian National University, and University of Sydney.