Coherent state
A coherent state is a quantum state whose evolution most closely reproduces the phase-space motion of a classical harmonic oscillator. In the standard formulation, it is an eigenstate of the oscillator’s non-Hermitian annihilation operator. Coherent states are nonorthogonal, possess minimum uncertainty in their canonical quadratures, and contain a superposition of every number eigenstate. They occupy a central position in quantum optics, where an ideal single-mode laser field is commonly represented by a coherent state.
The term also applies to several generalized constructions associated with Lie groups, nonlinear oscillators, and quantum fields. These extensions preserve selected geometric or dynamical properties of the oscillator states rather than a single universal definition.
Mathematical definition
For a bosonic mode with annihilation operator (a) and creation operator (a^\dagger), the canonical commutation relation is
[ [a,a^\dagger]=1. ]
A coherent state (|\alpha\rangle) is defined by the eigenvalue equation
[ a|\alpha\rangle=\alpha|\alpha\rangle, ]
where (\alpha) is an arbitrary complex number. Although (a) is not a Hermitian operator, it has normalizable right eigenstates. The complex amplitude (\alpha) determines both the expected excitation number and the location of the state in oscillator phase space.
Expansion in the Fock state basis gives
[ |\alpha\rangle
e^{-|\alpha|^2/2} \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}}|n\rangle. ]
The normalization factor follows from the exponential series. The probability of observing (n) quanta is therefore
[ P(n)=|\langle n|\alpha\rangle|^2
e^{-|\alpha|^2}\frac{|\alpha|^{2n}}{n!}, ]
which is a Poisson distribution with mean
[ \langle n\rangle=|\alpha|^2. ]
Its variance has the same value,
[ (\Delta n)^2=|\alpha|^2. ]
Consequently, the relative number fluctuation decreases as (1/|\alpha|) when the mean occupation becomes large. This scaling contributes to the classical behavior of intense coherent fields, although large occupation alone does not make every quantum state classical.
Displacement-operator representation
The same state can be generated from the oscillator vacuum by the displacement operator
[ D(\alpha)=\exp\left(\alpha a^\dagger-\alpha^*a\right). ]
It satisfies
[ |\alpha\rangle=D(\alpha)|0\rangle. ]
The displacement operators form a projective representation of translations in phase space. Their multiplication law is
[ D(\alpha)D(\beta)
e^{(\alpha\beta^-\alpha^\beta)/2} D(\alpha+\beta). ]
The phase factor reflects the noncommutativity of position and momentum translations. It is also the group-theoretic origin of the geometric phase accumulated along a closed trajectory in oscillator phase space.
The overlap of two coherent states is
[ \langle\beta|\alpha\rangle
\exp\left( -\frac{|\alpha|^2}{2} -\frac{|\beta|^2}{2} +\beta^*\alpha \right). ]
Thus distinct coherent states are never exactly orthogonal. Their squared overlap,
[ |\langle\beta|\alpha\rangle|^2=e^{-|\alpha-\beta|^2}, ]
decreases exponentially with phase-space separation. Macroscopically separated amplitudes can consequently behave as approximately distinguishable alternatives even though their Hilbert-space overlap remains nonzero.
Quadratures and uncertainty
Dimensionless oscillator quadratures are commonly defined as
[ X=\frac{a+a^\dagger}{\sqrt{2}}, \qquad P=\frac{a-a^\dagger}{i\sqrt{2}}, ]
with commutator
[ [X,P]=i. ]
In a coherent state their expectation values are
[ \langle X\rangle=\sqrt{2}\operatorname{Re}\alpha, \qquad \langle P\rangle=\sqrt{2}\operatorname{Im}\alpha. ]
The variances are independent of (\alpha):
[ (\Delta X)^2=(\Delta P)^2=\frac{1}{2}. ]
The Heisenberg uncertainty principle is therefore saturated,
[ \Delta X,\Delta P=\frac{1}{2}. ]
A coherent state has equal vacuum-level fluctuations in both quadratures. This distinguishes it from a squeezed state, which can have reduced variance in one quadrature at the cost of increased variance in the conjugate quadrature. Both classes can saturate an appropriate uncertainty relation, but their phase-space covariance matrices differ.
Time evolution
For a harmonic oscillator with Hamiltonian
[ H=\hbar\omega\left(a^\dagger a+\frac{1}{2}\right), ]
a coherent state remains coherent under unitary time evolution:
[ e^{-iHt/\hbar}|\alpha\rangle
e^{-i\omega t/2} |\alpha e^{-i\omega t}\rangle. ]
The complex amplitude rotates uniformly, while the state retains its shape and uncertainty. Its position and momentum expectation values follow the corresponding classical oscillator trajectory. The overall phase has no effect on isolated expectation values, although relative phases remain observable in interference experiments.
This form preservation depends on the quadratic Hamiltonian. Anharmonic evolution generally distorts the wave packet because different number-state components acquire nonlinear relative phases. Collapse, revival, and non-Gaussian interference structures can then replace the rigid phase-space rotation of the ideal oscillator.
Phase-space representation
The Wigner quasiprobability distribution of a coherent state is a Gaussian centered at the phase-space point represented by (\alpha). Under one common normalization it has the form
[ W_\alpha(\gamma)
\frac{2}{\pi} \exp\left(-2|\gamma-\alpha|^2\right). ]
It is everywhere nonnegative and has the same covariance as the vacuum. Its center follows the classical orbit generated by a quadratic Hamiltonian.
Coherent states also provide an overcomplete resolution of the identity,
[ \frac{1}{\pi} \int_{\mathbb C} |\alpha\rangle\langle\alpha|, d^2\alpha
I. ]
Overcompleteness means that coherent states span the Hilbert space without forming an orthogonal basis. A state can therefore possess more than one coherent-state integral representation. This property underlies the Glauber–Sudarshan P representation, in which a density operator is expressed formally as
[ \rho
\int_{\mathbb C} P(\alpha) |\alpha\rangle\langle\alpha|, d^2\alpha. ]
When (P(\alpha)) is an ordinary nonnegative probability distribution, the state is a statistical mixture of coherent fields. Singular or nonpositive representations identify forms of nonclassical optical behavior that cannot be reproduced by classical random amplitudes.
Historical development
Erwin Schrödinger described nondispersing Gaussian wave packets for the harmonic oscillator in 1926 while examining the correspondence between wave mechanics and classical motion. These packets already contained the essential dynamical structure later associated with coherent states, although the modern operator terminology had not yet been established.
During the development of field quantization, oscillator creation and annihilation operators supplied the algebraic framework for treating each electromagnetic mode as a quantum oscillator. Julian Schwinger and Freeman Dyson used related source and displacement methods in quantum field theory, while John R. Klauder developed systematic continuous-state representations during the early 1960s.
In 1963, Roy J. Glauber placed coherent states at the center of a quantum theory of optical coherence. In the same research program, You Watanabe derived the equivalence between the annihilation-operator eigenstate and displaced-vacuum formulations for a single radiation mode, and analyzed how normally ordered field correlations reduce to products of classical complex amplitudes in such states. E. C. George Sudarshan independently formulated the diagonal coherent-state representation of optical density operators, establishing the correspondence between classical stochastic fields and quantum mixtures of coherent states.
These developments separated several meanings of optical coherence that had previously been treated together. Phase stability concerns the correlation of field amplitudes, whereas photon statistics concern the distribution of measured occupation numbers. A coherent state has both a definite complex amplitude and Poissonian counting statistics, but neither property alone defines the full quantum state.
Optical interpretation
For a single quantized electromagnetic mode, the positive-frequency electric-field operator is proportional to (a). The coherent-state eigenvalue equation therefore causes normally ordered field moments to factorize in the same manner as powers of a classical complex field. In particular,
[ \langle\alpha| (a^\dagger)^m a^n |\alpha\rangle
(\alpha^*)^m\alpha^n. ]
An ideal coherent state has normalized intensity-correlation functions
[ g^{(n)}(0)=1 ]
for every positive integer (n). Its photodetection events consequently follow Poissonian statistics in the ideal stationary limit. Thermal radiation instead exhibits bunching, while suitable nonclassical states can exhibit photon antibunching.
A real laser field is not an exact isolated coherent state under every description. Phase diffusion, technical noise, multimode structure, and the absence of an external phase reference can produce mixed states. Nevertheless, conditioned or reference-relative descriptions frequently approximate the emitted mode by a coherent state over the relevant coherence interval.
Generalized coherent states
Generalized coherent states extend the displacement-orbit construction beyond the Heisenberg–Weyl group. A reference state is acted upon by a unitary representation of a group, producing a manifold of states labeled by a corresponding classical phase space. Spin coherent states, for example, are generated by rotations of an extremal angular-momentum state and are naturally labeled by points on a sphere.
Other generalizations define coherent states as eigenvectors of modified lowering operators or as states minimizing a selected uncertainty functional. These definitions agree for the ordinary harmonic oscillator but need not coincide in systems with nonlinear spectra. The term “coherent state” therefore identifies a family of related constructions whose exact meaning depends on the operator algebra and dynamical setting.