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| coherent states | |
|---|---|
| Name | Coherent states |
| Introduced | 1926 |
| Introduced by | Erwin Schrödinger |
| Applications | Quantum optics; Bose–Einstein condensates; superconducting circuits; signal processing |
| Related | Squeezed states; Fock states; Glauber–Sudarshan P representation |
coherent states Coherent states are specific quantum states of harmonic-oscillator-like systems that minimize uncertainty and exhibit dynamics closest to classical motion. Developed in the early 20th century and formalized across mid-century by figures associated with Erwin Schrödinger, Roy J. Glauber, and Alexander Perelomov, coherent states bridge formal developments in quantum mechanics with experimental domains such as quantum optics, condensed matter physics, and atomic physics. Their mathematical structure connects to representation theory of Lie groups and to functional analytic techniques used in operator theory and signal processing.
Coherent states were first identified in seminal work by Erwin Schrödinger as minimum-uncertainty wavepackets for the harmonic oscillator and later adopted in a quantum-optical framework by Roy J. Glauber, who related them to the quantum theory of laser light and the photoelectric effect. Subsequent formalizations by Alexander Perelomov and A. O. Barut with L. Girardello extended the concept using group theory and non-compact groups like SU(1,1), linking coherent states to the representation theory of Heisenberg group, Lie algebra methods, and techniques used by researchers in Mathematical Physics such as E. C. G. Sudarshan.
Mathematically, a coherent-state construction typically uses ladder operators associated with the harmonic oscillator algebra of the Heisenberg group or with generators of a chosen Lie group representation. For the canonical oscillator, coherent states |α⟩ are eigenstates of the annihilation operator a with complex eigenvalue α; they are expressible as displaced vacuum states via the Weyl operator (elements related to Hermann Weyl and the Stone–von Neumann theorem). Key properties include non-orthogonality, resolution of the identity (overcompleteness), Poissonian number statistics, and minimal Heisenberg uncertainty consistent with the Robertson–Schrödinger relation. These features have formal parallels in constructions by Alexander Perelomov and Barut–Girardello for groups like SU(2), SU(1,1), and other semisimple and solvable groups used in representation theory.
The Glauber construction originates in work by Roy J. Glauber and the Glauber–Sudarshan formalism pioneered with E. C. G. Sudarshan, defining coherent states via displacement operators in the Fock space of the oscillator and linking to the Glauber–Sudarshan P representation. The Perelomov method, due to Alexander Perelomov, builds coherent states by acting with group elements from a parent group such as Heisenberg group, SU(2), or SU(1,1) on a reference vector in an irreducible representation; this approach leverages coherent-state manifolds like projective spaces related to Élie Cartan's theory. The Barut–Girardello construction by A. O. Barut and L. Girardello defines coherent states as eigenstates of lowering operators for non-compact algebras such as su(1,1), producing states with distinct analytic and normalization properties used in studies by I. M. Gel'fand school and others.
Coherent states model semiclassical regimes across experimental platforms: in quantum optics they describe laser light and coherent emission in optical cavities and free-electron lasers; in atomic physics they approximate center-of-mass motion in ion trap experiments involving apparatus developed in labs associated with William D. Phillips and David J. Wineland; in condensed matter physics they underlie descriptions of Bose–Einstein condensate mean-field states investigated following work by Eric A. Cornell and Carl E. Wieman. Coherent-state techniques appear in superconducting qubit control in circuits influenced by research at institutions like Bell Labs and IBM Research, and in time-frequency analysis methods rooted in the Gabor transform associated with Dennis Gabor.
In quantum optics, Glauber coherent states provide the standard quantum description of classical-like electromagnetic fields in modes of an optical cavity and are central to the quantum theory of photodetection developed by Roy J. Glauber, which elucidated experiments by groups such as those at Bell Telephone Laboratories and research led by John L. Hall. They exhibit Poissonian photon-number distributions, classical coherence functions explored in the context of Hanbury Brown and Richard Q. Twiss experiments, and form the basis for semiclassical treatments used in laser theory and quantum-state tomography efforts by groups associated with Raymond G. Beausoleil and others.
Coherent states admit representations on classical phase space via quasiprobability distributions: the Wigner function (developed by Eugene Wigner), the Glauber–Sudarshan P representation, and the Husimi Q function (Husimi stemming from work connected to Kôdi Husimi). Overcompleteness yields a resolution of the identity enabling coherent-state path integrals used in semiclassical approximations related to methods by Richard Feynman and stationary-phase techniques from analysts such as Hermann Weyl and James Joseph Sylvester-style integral transforms. These tools connect to deformation quantization explored by researchers like Francois Bayen and others in mathematical physics.
Generalizations include squeezed states (prominent in the work of Carlton M. Caves and applied in LIGO interferometry studies led by Rainer Weiss), spin coherent states for SU(2) used in magnetic resonance contexts and quantum information experiments by groups at MIT and Caltech, nonlinear coherent states tied to deformed algebras studied by mathematical physicists from the Soviet Union school, and entangled coherent states explored in quantum-optical proposals by researchers at institutions like Institute for Quantum Optics branches and university groups across Europe and North America. Further developments intersect with wavelet coherent states in signal analysis associated with Yves Meyer and Ingrid Daubechies, and with coherent-state quantization approaches developed within geometric quantization traditions following Jean-Marie Souriau and Bertram Kostant.