| coherent states | |
|---|---|
| Name | Coherent state |
| Field | Quantum mechanics |
| Introduced | 1926 |
| Introduced by | Erwin Schrödinger |
| Notable examples | Glauber state, squeezed state |
coherent states
Coherent states are specific quantum states of harmonic-oscillator-like systems that most closely resemble classical oscillations. They are minimum-uncertainty wavepackets that follow classical equations of motion in phase space and play a central role in quantum optics and semiclassical analysis. Coherent states provide a bridge between classical mechanics and quantum mechanics and are widely used in both theoretical work and experimental implementations.
A coherent state is commonly defined as an eigenstate of the annihilation operator â of a quantum harmonic oscillator: â|α⟩ = α|α⟩, where α is a complex number labeling the point in phase space. The first explicit construction was proposed by Erwin Schrödinger in 1926 as localized wavepackets for the harmonic oscillator and later developed for electromagnetic fields by Roy J. Glauber in the 1960s. Coherent states minimize the Heisenberg uncertainty principle for canonical position and momentum operators, so that Δx·Δp = ħ/2 for the idealized oscillator. They form an overcomplete, non-orthogonal basis of the oscillator Hilbert space and admit a resolution of the identity via an integral over phase space.
Key properties include: temporal stability (the time evolution of a coherent state remains coherent under the harmonic oscillator Hamiltonian), displacement-operator generation (|α⟩ = D(α)|0⟩ with the displacement operator D(α)), Poissonian number statistics for the photon-number distribution in optical coherent states, and simple phase-space representations such as the Wigner quasi-probability distribution and the Glauber–Sudarshan P representation.
Coherent states can be constructed in several equivalent ways. The most common representations are: - Eigenstate definition: the annihilation-operator eigenstate â|α⟩ = α|α⟩. - Displacement operator: |α⟩ = D(α)|0⟩ where D(α) = exp(α↠− α*â) and |0⟩ is the vacuum state; this connects coherent states to the Weyl–Heisenberg group. - Expansion in Fock basis: |α⟩ = e^{-|α|^2/2} ∑_{n=0}^∞ α^n/√{n!} |n⟩ giving rise to Poissonian photon statistics.
Phase-space representations include the Wigner function, Husimi Q function, and Glauber–Sudarshan P representation, each useful for different analytic and experimental tasks. Coherent states are also characterized by the Bargmann representation (analytic functions on the complex plane) and appear naturally in the context of the Segal–Bargmann space. Their overcompleteness is expressed through the identity ∫ |α⟩⟨α| d^2α/π = I.
The paradigmatic physical realization of coherent states is in the electromagnetic field of a classical laser mode, where Glauber showed that laser light near threshold is well described by a coherent state (often called a Glauber state). Other realizations include vibrational modes of trapped ions in ion trap experiments, motional states of nanomechanical resonators, and coherent microwave fields in superconducting circuit QED devices.
Examples: - Optical coherent states: single-mode laser output approximated by |α⟩ with controllable amplitude and phase. - Atomic coherent states: large-spin analogues (spin coherent states) used in Bose–Einstein condensate experiments and quantum metrology. - Coherent phonon states in solid-state systems and coherent magnon states in spintronics.
Experimental generation techniques include stabilized lasers, coherent driving of resonators, displacement pulses in cavity QED, and reservoir engineering protocols in quantum optics and quantum information platforms.
Coherent states are central to the semiclassical description of light and matter. In quantum optics, they model classical electromagnetic waves with quantized excitations and provide a natural basis for describing optical coherence, correlation functions (Glauber coherence theory), homodyne and heterodyne detection, and quantum state tomography. Coherent states underpin the theory of laser operation and photon counting statistics.
In semiclassical analysis, coherent states enable phase-space path integrals and semiclassical propagation methods (e.g., the Herman–Kluk propagator). They serve as initial wavepackets in semiclassical approximations to study tunneling, molecular dynamics, and the quantum-classical correspondence addressed in works by Martin Gutzwiller and others. Coherent-state path integrals are also used in many-body physics and statistical mechanics.
Coherent states admit extensive mathematical generalization via group theory. The Perelomov construction defines coherent states as orbits of a reference state under a Lie group action: |g⟩ = U(g)|ψ_0⟩ with U a unitary representation; notable families include SU(2), SU(1,1), and the Heisenberg–Weyl group. These generalizations yield spin coherent states, squeezed states, and SU(1,1) coherent states relevant to parametric amplification.
Mathematical frameworks involve reproducing-kernel Hilbert spaces, geometric quantization, and the theory of coherent-state transforms (e.g., the Segal–Bargmann transform). Important contributors include Alexander Perelomov, John R. Klauder, and Emile B. Gilmore. Group-theoretic coherent states are widely used in representation theory, quantum information, and models of collective quantum dynamics.
Coherent states occupy a privileged role in quantum measurement theory and phase-space analysis. Because they saturate the Heisenberg uncertainty bound, they are often employed in metrology as near-optimal probe states for phase estimation under classical-noise-limited conditions; however, squeezed states can surpass coherent-state precision for certain tasks. Coherent states' phase-space distributions (Wigner, Q, P) provide intuitive pictures of quantum noise and nonclassicality: negative regions in the Wigner function or nonregular P distributions signal nonclassical states departing from coherent-like behavior.
In measurement contexts—homodyne detection, heterodyne detection, and quantum state tomography—coherent states serve as calibration standards and basis states for reconstructing unknown quantum states. Their robustness under loss and decoherence explains their prevalence in experiments spanning quantum communication, quantum cryptography, and optical implementations of quantum computing.