| coherent states | |
|---|---|
| Name | Coherent state |
| Field | Quantum mechanics |
| Introduced | 1926–1960s |
| Introduced by | Erwin Schrödinger, Roy J. Glauber |
| Notable for | Minimal uncertainty, semiclassical behavior |
coherent states
Coherent states are specific quantum states of a dynamical system that most closely resemble classical motion while respecting the principles of Quantum mechanics. They play a central role in the study of the quantum–classical correspondence, semiclassical approximations, and the modern theory of quantum optics. Coherent states are important both for foundational understanding and for practical technologies such as lasers and protocols in quantum information.
Coherent states were first introduced by Erwin Schrödinger in 1926 as wavepackets for the harmonic oscillator that follow classical orbits with minimal spreading. The concept was later generalized and popularized in the 1960s by Roy J. Glauber in the context of quantum optics to describe the field state of an ideal laser. Subsequent mathematical formalization involved work by John R. Klauder, Jean-Pierre Gazeau, A. Perelomov, and J. R. von Neumann on group-theoretic constructions and representations, linking coherent states to notions of symmetry and phase space such as the Heisenberg group and Lie group representations. The development reflects a conservative emphasis on continuity between classical and quantum descriptions, preserving established physical intuition while extending applicability.
A coherent state is commonly defined as an eigenstate of the annihilation operator a for the quantum harmonic oscillator: a|α⟩ = α|α⟩, where α ∈ ℂ. Equivalently, one obtains |α⟩ by acting with the displacement operator D(α) = exp(α a† − α* a) on the vacuum |0⟩. This construction uses algebraic structures from the Weyl–Heisenberg group and yields states that are overcomplete and provide a resolution of the identity. Group-theoretic coherent states (also called generalized coherent states or Perelomov coherent states) arise from the action of a Lie group G on a reference vector in a Hilbert space, linking to representation theory and homogeneous spaces G/H.
Mathematically relevant properties include nonorthogonality ⟨α|β⟩ = exp(−|α|^2/2 − |β|^2/2 + α*β), displacement operator commutation relations, and a phase-space description via Wigner quasiprobability distribution and Glauber–Sudarshan P-representation.
Coherent states minimize the Heisenberg uncertainty principle for conjugate quadratures, exhibiting equal and minimal variances in position and momentum (or field amplitude and phase quadratures). Their expectation values follow classical equations of motion for linear systems, establishing them as optimal semiclassical states. They are not orthogonal and form an overcomplete basis, which facilitates path-integral semiclassical expansions and quantization schemes like Berezin–Toeplitz quantization. Coherent states often exhibit Poissonian number statistics, distinguishing them from thermal or Fock states, and their phase properties are captured by phase-space representations used in quantum measurement theory.
- Harmonic oscillator: The canonical example where |α⟩ are eigenstates of a. Relevant to molecular vibrations and quantum oscillators in condensed matter. - Electromagnetic field: Glauber coherent states model the output of an ideal laser and are central to quantum optics experiments; linked to the photodetection theory developed by Glauber. - Spin coherent states: Constructed via SU(2) group actions on highest-weight vectors, used in descriptions of large-spin systems, magnetic resonance, and semiclassical spin dynamics; related to Bloch sphere representations. - Atomic and molecular coherent states: Employed in descriptions of Bose–Einstein condensates via macroscopic coherent occupation and in models such as the Jaynes–Cummings model where quasi-classical field states interact with two-level atoms.
Coherent optical states are routinely generated by stabilized laser oscillators operating well above threshold; filtering and cavity techniques improve coherence. In cavity quantum electrodynamics experiments at institutions like Max Planck Institute for Quantum Optics and Harvard University groups, displacement operations, beam splitters, and coherent drives prepare coherent states in microwave and optical resonators. Trapped ions and superconducting circuits produce motional and microwave coherent states via drive pulses; notable experimental platforms include Paul trap systems and circuit QED experiments at Yale University and University of California, Berkeley research groups.
Coherent states underpin laser theory, interferometry, and precision measurement such as homodyne detection and heterodyne detection. In quantum information coherent states serve in continuous-variable protocols, coherent-state quantum key distribution, and optical communication schemes; firms like ID Quantique and research at IBM explore coherent-state methods for transmission and sensing. Semiclassical techniques using coherent-state path integrals enable approximations for transition amplitudes and quantum chaos studies; they are used in chemical physics for approximating molecular dynamics and reaction rates.
Coherent states embody the classical limit of quantum systems, providing a controlled route to the classical mechanics description via large quantum numbers or small effective Planck constant. Their construction is deeply tied to group symmetries and unitary representations (Weyl–Heisenberg, SU(2), SU(1,1)), connecting to topics in representation theory, geometric quantization, and phase-space methods. Coherent-state techniques also intersect with studies of decoherence and the emergence of classicality in open quantum systems treated by quantum master equations and Lindblad equation frameworks. Conservative scientific practice values coherent states for preserving continuity with classical intuition while supporting rigorous quantum analysis.
Category:Quantum states Category:Quantum optics Category:Semiclassical physics