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Coherent state

In quantum mechanics, a coherent state is a specific quantum state of the harmonic oscillator that serves as an eigenstate of the annihilation operator a, satisfying a |\alpha\rangle = \alpha |\alpha\rangle where \alpha is a complex eigenvalue, and it minimizes the Heisenberg uncertainty product for position and momentum while exhibiting dynamics that closely mimic classical oscillations.^[1] These states were first derived by Erwin Schrödinger in 1926 as minimum-uncertainty wave packets for the harmonic oscillator and were later formalized in quantum optics by Roy J. Glauber in 1963 as part of his foundational work on the quantum theory of optical coherence, for which he received the Nobel Prize in Physics in 2005.^[2]^[3] Mathematically, a coherent state |\alpha\rangle can be expressed as a displaced vacuum state via the displacement operator D(\alpha) = \exp(\alpha a^\dagger - \alpha^* a), yielding |\alpha\rangle = D(\alpha) |0\rangle, or equivalently as an infinite superposition of number states: |\alpha\rangle = e^{-|\alpha|^2/2} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n\rangle.^[4] This form highlights their overcomplete nature, forming a basis for the Hilbert space through the resolution of identity \frac{1}{\pi} \int d^2\alpha \, |\alpha\rangle\langle\alpha| = \mathbb{1}, though the states are non-orthogonal with overlap \langle\alpha|\beta\rangle = \exp\left[-\frac{1}{2}(|\alpha|^2 + |\beta|^2) + \alpha^* \beta\right].^[4] Key properties of coherent states include a Poissonian photon number distribution with mean and variance both equal to |\alpha|^2, making them ideal for describing laser light fields where quantum noise is minimal and the electric field oscillates like a classical wave.^[5] The expectation values of position and momentum evolve as \langle x \rangle = \sqrt{\frac{\hbar}{2m\omega}} \operatorname{Re}(\alpha e^{-i\omega t}) and \langle p \rangle = \sqrt{\frac{\hbar m \omega}{2}} \operatorname{Im}(\alpha e^{-i\omega t}), respectively, demonstrating their classical-like time evolution under the harmonic oscillator Hamiltonian.^[1] In quantum optics, coherent states represent the output of ideal lasers operating well above threshold, bridging quantum and classical descriptions of light.^[5] Beyond optics, coherent states generalize to other bosonic systems, such as superconducting circuits and atomic ensembles, enabling applications in quantum information processing, including quantum computing gates and state preparation with reduced decoherence.^[4] Their minimal uncertainty and phase coherence make them essential for studying phenomena like squeezed states and non-classical light, where deviations from coherent behavior reveal quantum effects.^[5]

Introduction

Historical development

The concept of coherent light in classical optics, particularly in interferometry experiments dating back to Thomas Young's double-slit demonstration in 1801, laid foundational groundwork for understanding wave superposition and phase relationships, influencing later quantum interpretations. In 1926, Erwin Schrödinger introduced Gaussian wave packets as minimum-uncertainty states for the quantum harmonic oscillator, aiming to construct solutions to the Schrödinger equation that mimic classical oscillatory motion while preserving quantum features like minimal spreading. These wave packets, later recognized as coherent states, represented an early effort to bridge classical and quantum mechanics by ensuring the expectation values of position and momentum followed classical trajectories.^[6] During the 1950s and 1960s, mathematical developments advanced the framework of coherent states, notably through Valentine Bargmann's 1961 work establishing an overcomplete basis in Hilbert space using analytic functions, which facilitated representations beyond orthonormal bases. This period saw contributions from others, including E. C. G. Sudarshan, expanding the utility of coherent states in quantum field theory. The formalization of coherent states in quantum optics occurred in 1963 with Roy J. Glauber's and independently E. C. G. Sudarshan's seminal papers, where Glauber defined them as eigenstates of the annihilation operator for the electromagnetic field, providing a quantum description of optical coherence and photon statistics.^[3]^[7] This work explained the properties of laser light as coherent states and earned Glauber the 2005 Nobel Prize in Physics for contributions to quantum theory of optical coherence. Early experimental confirmations followed the 1960 invention of the laser by Theodore Maiman, with 1960s studies on photon correlation and statistics in laser beams verifying the coherent state model.

Overview and significance

Coherent states represent a class of quantum states for the harmonic oscillator that achieve the minimum possible uncertainty in position and momentum, as dictated by the Heisenberg uncertainty principle, while their wave packets evolve in time without spreading, mirroring the periodic motion of a classical oscillator.^[8] These states are particularly analogous to classical coherent radiation, such as that produced by lasers, where quantum fluctuations are minimized to closely approximate the deterministic behavior of electromagnetic waves.^[9] Originally introduced by Erwin Schrödinger in 1926 as minimum-uncertainty wave packets and later formalized by Roy J. Glauber and independently by E. C. G. Sudarshan in 1963 in the context of quantum optics, coherent states form an overcomplete basis in Hilbert space, enabling a resolution of unity that facilitates efficient representations of quantum operators and states.^[6]^[9]^[7]^[10] In quantum optics, coherent states provide the foundational description of light fields with classical-like properties, essential for modeling non-classical phenomena while bridging quantum and classical regimes.^[11] Their significance extends to quantum information science, where they serve as robust, error-corrected reference states for encoding and processing quantum data, and to many-body physics, where they underpin macroscopic quantum coherence in systems like superconductors and Bose-Einstein condensates.^[12]^[13] In modern quantum technologies, coherent states act as stable building blocks for applications in quantum computing, where they support fault-tolerant operations, and quantum sensing, enhancing precision in detecting weak signals through entangled coherent superpositions.^[14]

Fundamental Concepts

Quantum mechanical definition

In quantum mechanics, coherent states of the harmonic oscillator are formally defined as the right eigenstates of the annihilation operator a, satisfying the eigenvalue equation

a |\alpha\rangle = \alpha |\alpha\rangle,

where \alpha \in \mathbb{C} is the complex eigenvalue and a = \sqrt{\frac{m \omega}{2 \hbar}} \hat{x} + i \frac{\hat{p}}{\sqrt{2 m \omega \hbar}} is the lowering operator in terms of the position \hat{x} and momentum \hat{p} operators (with [\hat{x}, \hat{p}] = i \hbar). This definition, introduced in the context of quantum optics for radiation fields, identifies coherent states as those quantum states whose expectation values and fluctuations most closely mimic classical oscillatory behavior. The coherent state |\alpha\rangle admits an explicit expansion in the orthonormal Fock basis \{|n\rangle\}_{n=0}^\infty of number eigenstates, given by

|\alpha\rangle = e^{-|\alpha|^2 / 2} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n\rangle.

This series representation highlights the state's superposition nature, with coefficients that yield a Poissonian probability distribution for the photon (or excitation) number: P(n) = |\langle n | \alpha \rangle|^2 = e^{-|\alpha|^2} \frac{|\alpha|^{2n}}{n!}, where \langle n \rangle = |\alpha|^2 is the mean and the relative variance is unity, characteristic of classical-like statistics. The states are normalized, \langle \alpha | \alpha \rangle = 1, ensuring they form valid elements of the Hilbert space. Coherent states saturate the Heisenberg uncertainty relation, achieving the minimum \Delta x \Delta p = \hbar / 2 with variances (\Delta x)^2 = \hbar / (2 m \omega) and (\Delta p)^2 = m \omega \hbar / 2, displaced from the origin in phase space according to \mathrm{Re}(\alpha) and \mathrm{Im}(\alpha).^[15] Under time evolution governed by the harmonic oscillator Hamiltonian H = \hbar \omega (a^\dagger a + 1/2), the state transforms as |\alpha(t)\rangle = e^{-i \omega t / 2} |\alpha e^{-i \omega t}\rangle, preserving its coherent character while rotating the eigenvalue \alpha in the complex plane at the classical frequency \omega. In the Wigner phase-space quasiprobability representation, the coherent state appears as a Gaussian distribution centered at the classical position-momentum point (\sqrt{2 \hbar / m \omega} \mathrm{Re}(\alpha), \sqrt{2 m \omega \hbar} \mathrm{Im}(\alpha)), with no negative values, underscoring its classical-like localization.

Wavefunction representation

In the position representation, the coherent state of the quantum harmonic oscillator is described by a Gaussian wave packet that is displaced from the origin in both position and momentum. This form arises directly from the action of the displacement operator D(\alpha) = \exp(\alpha a^\dagger - \alpha^* a) on the ground state wavefunction, where a and a^\dagger are the lowering and raising operators, respectively. The resulting position-space wavefunction is

\psi_\alpha(x) = \left( \frac{1}{\sqrt{2\pi \sigma^2}} \right)^{1/2} \exp\left[ -\frac{(x - x_0)^2}{4\sigma^2} + i \frac{p_0 (x - x_0)}{\hbar} + i \phi \right],

with \sigma = \sqrt{\hbar / (2 m \omega)}, x_0 = \sqrt{2 \hbar / (m \omega)} \operatorname{Re}(\alpha), p_0 = \sqrt{2 m \omega \hbar} \operatorname{Im}(\alpha), and \phi a global phase.^[16]^[17] This expression is derived by solving the time-independent Schrödinger equation for a Gaussian ansatz displaced in position and boosted in momentum, ensuring it satisfies the minimum uncertainty relation \Delta x \Delta p = \hbar/2 while maintaining the harmonic oscillator potential V(x) = \frac{1}{2} m \omega^2 x^2. The displacement parameters x_0 and p_0 correspond to the expectation values \langle x \rangle and \langle p \rangle, linking the quantum state to classical-like motion.^[16] When \alpha = 0, the wavefunction reduces to the ground state \psi_0(x) = \left( \frac{1}{\sqrt{2\pi \sigma^2}} \right)^{1/2} \exp\left[ -\frac{x^2}{4\sigma^2} \right], a stationary Gaussian centered at the origin with no momentum offset. In contrast, nonzero \alpha shifts the center while preserving the width $2\sigma, highlighting the coherent state's role as a "displaced vacuum."^[17] The time-dependent evolution of the coherent state wave packet follows classical trajectories without spreading. Under the harmonic oscillator Hamiltonian, the center oscillates as x(t) = x_0 \cos([\omega](/page/Omega) t) + (p_0 / m \omega) \sin([\omega](/page/Omega) t) and p(t) = p_0 \cos([\omega](/page/Omega) t) - m \omega x_0 \sin([\omega](/page/Omega) t), with the wavefunction given by

\psi_\alpha(x, t) = \left( \frac{1}{\sqrt{2\pi \sigma^2}} \right)^{1/2} \exp\left[ -\frac{(x - x(t))^2}{4\sigma^2} + i \frac{p(t) x}{\hbar} + i \theta(t) \right],

where \theta(t) includes the dynamical phase \exp(-i \omega t / 2) and additional terms from the boost. This non-spreading behavior underscores the coherent state's classical correspondence.^[16]^[17]^[18] In phase space, the wave packet appears as a Gaussian blob of minimum uncertainty, centered at (x(t), p(t)) and rotating clockwise with angular frequency \omega, tracing an ellipse that matches the classical orbit for the given initial conditions. This visualization, often represented via the Wigner quasiprobability distribution, illustrates the state's localized, coherent propagation.^[17]

Key mathematical properties

Coherent states exhibit non-orthogonality, a fundamental property distinguishing them from the orthogonal Fock basis states. The inner product between two coherent states |\alpha\rangle and |\beta\rangle is given by

\langle \beta | \alpha \rangle = \exp\left( -\frac{|\alpha|^2}{2} - \frac{|\beta|^2}{2} + \beta^* \alpha \right),

resulting in a modulus squared overlap of |\langle \beta | \alpha \rangle|^2 = \exp(-|\alpha - \beta|^2). This non-zero overlap for \alpha \neq \beta implies that coherent states are not mutually orthogonal, yet they span the Hilbert space through overcompleteness, providing a redundant but continuous parameterization of quantum states. A hallmark of this overcompleteness is the resolution of the identity operator, expressed as

\frac{1}{\pi} \int d^2\alpha \, |\alpha\rangle\langle\alpha| = \hat{I},

where the integral covers the entire complex plane with measure d^2\alpha = d(\operatorname{Re} \alpha) d(\operatorname{Im} \alpha). This relation allows any operator or state to be expanded in the coherent state basis, simplifying calculations of traces, expectation values, and matrix elements in quantum mechanics and optics. It underpins applications in frame theory, where coherent states act as a frame for signal processing and quantum information tasks. In the Bargmann representation, coherent states are mapped to the space of entire analytic functions on the complex plane, offering a powerful holomorphic formulation of quantum mechanics for the harmonic oscillator. Here, an arbitrary state vector |\psi\rangle is represented by the analytic function f(\alpha) = \langle \alpha | \psi \rangle, which belongs to a Hilbert space equipped with the inner product \langle f | g \rangle = \int \frac{d^2\alpha}{\pi} f^*(\alpha) g(\alpha). This representation transforms differential operators into multiplication and differentiation in the complex domain, facilitating exact solutions for time evolution and symmetries. Expectation values in coherent states often involve generating functions tied to Laguerre polynomials, particularly for observables related to the number operator \hat{n} = \hat{a}^\dagger \hat{a}. The coherent state expansion |\alpha\rangle = e^{-|\alpha|^2/2} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n\rangle serves as a generating function, where higher-order moments \langle (\Delta \hat{n})^k \rangle for k \geq 2 incorporate associated Laguerre polynomials L_n^{(m)}(x) to capture deviations from the Poissonian statistics of \langle \hat{n} \rangle = |\alpha|^2. This structure is essential for analyzing intensity correlations and noise in quantum optical systems. In the semiclassical regime, as |\alpha| \to \infty, quantum expectation values in coherent states asymptotically approach classical predictions derived from Poisson bracket dynamics. The relative quantum fluctuations, such as \Delta n / \langle n \rangle \approx 1/\sqrt{\langle n \rangle}, diminish, allowing the state's behavior to mimic a classical oscillator with definite amplitude and phase. This limit bridges quantum and classical descriptions, validating Ehrenfest's theorem for large displacements.^[19] Coherent states achieve the minimal uncertainty \Delta x \Delta p = \hbar/2 inherent to their definition as displaced vacuum states, saturating the Heisenberg relation for non-commuting quadratures.

Role in Quantum Optics

Coherent states in electromagnetic fields

In quantum optics, coherent states provide a natural quantum description of electromagnetic fields, particularly for single-mode fields such as those in a laser cavity. A coherent state |\alpha\rangle for a single mode is defined as the eigenvector of the annihilation operator \hat{a} with eigenvalue \alpha, where \alpha is a complex number representing the field's amplitude and phase, satisfying \hat{a} |\alpha\rangle = \alpha |\alpha\rangle. Thus, the expectation value \langle \hat{a} \rangle = \alpha, and the state can be viewed as a displaced vacuum state obtained by applying the displacement operator \hat{D}(\alpha) = \exp(\alpha \hat{a}^\dagger - \alpha^* \hat{a}) to the vacuum |0\rangle.^[20] These states exhibit perfect coherence, as quantified by their correlation functions. The first-order coherence function for a coherent state is g^{(1)}(\tau) = \exp(-i \omega \tau), where \omega is the mode frequency, indicating phase stability over time \tau and no degradation in first-order correlations. The second-order coherence function at zero delay is g^{(2)}(0) = 1, corresponding to Poissonian photon statistics with variance equal to the mean photon number |\alpha|^2, implying no photon bunching or antibunching.^[20] Coherent states bridge quantum and classical descriptions of electromagnetic waves. The expectation value of the electric field operator \hat{E}(t) in the positive-frequency part is proportional to \langle \hat{E}(t) \rangle \propto \operatorname{Re}(\alpha e^{-i \omega t}), reproducing the oscillatory behavior of a classical monochromatic wave with amplitude |\alpha| and phase \arg(\alpha). This classical correspondence arises because normally ordered correlation functions factorize exactly as in classical theory.^[20] For multi-mode fields, such as those in beam propagation or interferometry, coherent states generalize to tensor products of single-mode coherent states across spatial or temporal modes, |\boldsymbol{\alpha}\rangle = \bigotimes_k |\alpha_k\rangle, where \boldsymbol{\alpha} = \{\alpha_k\} specifies the amplitudes for each mode k. This structure preserves coherence properties across modes, enabling descriptions of Gaussian beam evolution and phase-sensitive interference without mode entanglement in the coherent case.^[21] The phase-space representation via the Wigner function further highlights the Gaussian nature of noise in coherent states. For field quadratures \hat{X} = (\hat{a} + \hat{a}^\dagger)/\sqrt{2} and \hat{P} = -i(\hat{a} - \hat{a}^\dagger)/\sqrt{2}, the Wigner function is a Gaussian centered at ( \sqrt{2} \operatorname{Re} \alpha, \sqrt{2} \operatorname{Im} \alpha ) with equal variances of $1/2 in both quadratures, reflecting minimum uncertainty and symmetric quantum noise akin to the vacuum but displaced.

W(X, P) = \frac{1}{\pi} \exp\left( - (X - \sqrt{2} \operatorname{Re} \alpha)^2 - (P - \sqrt{2} \operatorname{Im} \alpha)^2 \right).

^[22]

Experimental realizations and laser physics

Coherent states are routinely generated in laser systems operating above threshold, where the output field approximates a coherent state with amplitude \alpha proportional to the square root of the pump power exceeding the threshold. In the quantum theory of the laser developed by Scully and Lamb, the steady-state solution for the intracavity field above threshold yields a large mean photon number, with the field's statistics closely matching those of a displaced vacuum state, characterized by Poissonian photon distribution and minimal phase diffusion for sufficiently high pump rates.^[23] This regime is achieved when the gain rate surpasses cavity losses, leading to stimulated emission dominance and suppression of spontaneous emission noise, as confirmed in experiments measuring higher-order coherence functions that align with theoretical predictions for coherent states.^[24] Detection of coherent states in quantum optics relies on homodyne and heterodyne methods, which enable reconstruction of the phase-space distribution through quadrature measurements. In balanced homodyne detection, the signal field interferes with a strong local oscillator coherent state at a beam splitter, allowing measurement of one quadrature (position or momentum) with high efficiency; for coherent states, this yields Gaussian distributions centered at \pm \operatorname{Re}(\alpha) or \pm \operatorname{Im}(\alpha). Heterodyne detection, by contrast, simultaneously measures both quadratures using an additional image-band local oscillator, albeit with added vacuum noise, facilitating full Wigner function tomography of the state. These techniques, formalized for arbitrary input states including coherent ones, have been pivotal in verifying the quantum properties of laser fields, with photocount moments matching theoretical expectations for coherent signals mixed with noise. The second-order correlation function g^{(2)}(\tau) for coherent states, measured via Hanbury Brown-Twiss (HBT) interferometry, confirms their Poissonian statistics with g^{(2)}(0) = 1, indicating no photon bunching or antibunching. In HBT setups, the laser beam is split and directed to two photodetectors, whose intensity correlations reveal the field's coherence; for coherent light from a stabilized laser, experiments yield g^{(2)}(0) \approx 1, distinguishing it from thermal sources (g^{(2)}(0) = 2) and single-photon states (g^{(2)}(0) < 1). This interferometric approach, originally proposed for stellar intensity measurements, has been adapted in quantum optics laboratories to characterize laser coherence, with precise agreement to unity for well-above-threshold operation.^[25] Coherent states maintain their coherence when propagating through linear optical elements such as beam splitters and interferometers, as these unitary transformations map coherent inputs to coherent outputs with transformed amplitudes. For a 50:50 beam splitter, an input coherent state |\alpha\rangle in one port and vacuum in the other produces entangled coherent states in the outputs, preserving the overall Gaussian phase-space structure and first-order coherence functions. This property underpins applications in quantum optics, where Mach-Zehnder interferometers demonstrate visibility close to unity for coherent inputs, reflecting the field's stable phase relationship across paths. Recent experiments up to 2025 have confirmed the generation and utilization of coherent state superpositions in nonlinear optical media, advancing their role in quantum gate implementations. In a 2025 study, intense infrared coherent superpositions—created via strong-field ionization and driven through beta-barium borate crystals—underwent efficient second-harmonic generation, producing non-classical UV output with measurable Wigner negativity.^[26] These results enable controlled superpositions for photonic quantum information, including proposals for entangling gates in nonlinear interferometers, where coherent superpositions serve as resources for deterministic two-qubit operations in integrated platforms.

Specialized Variants

Thermal coherent states

Thermal coherent states, also referred to as displaced thermal states, are mixed quantum states formed by applying the displacement operator D(\alpha) to a thermal equilibrium state of the quantum harmonic oscillator, where \alpha is a complex parameter representing the displacement in phase space.^[27] The thermal state is given by the diagonal density operator \rho_{\rm th} = \sum_{n=0}^{\infty} P(n) |n\rangle\langle n|, with the thermal occupation probabilities P(n) = (1 - e^{-\beta \hbar \omega}) e^{-n \beta \hbar \omega}, where \beta = 1/(k_B T) is the inverse temperature, \hbar is the reduced Planck's constant, \omega is the oscillator frequency, and k_B is Boltzmann's constant. Thus, the density operator for the thermal coherent state is \rho = D(\alpha) \rho_{\rm th} D^\dagger(\alpha).^[27] A key statistical property is the mean photon number, which combines the coherent displacement contribution with the thermal occupancy: \langle \hat{n} \rangle = |\alpha|^2 + \bar{n}_{\rm th}, where \bar{n}_{\rm th} = 1/(e^{\beta \hbar \omega} - 1) denotes the average photon number of the undisplaced thermal state.^[27] This additive structure highlights how the coherent amplitude superimposes on the thermal background noise, leading to super-Poissonian photon statistics with variance \Delta n^2 = |\alpha|^2 (2 \bar{n}_{\rm th} + 1) + \bar{n}_{\rm th} (\bar{n}_{\rm th} + 1).^[28] In phase space, the Wigner quasiprobability function of a thermal coherent state is a Gaussian centered at \alpha, with circular symmetry and a radial width broadened by thermal fluctuations:

W(\beta) = \frac{2}{\pi (2 \bar{n}_{\rm th} + 1)} \exp\left( -\frac{2 |\beta - \alpha|^2}{2 \bar{n}_{\rm th} + 1} \right),

where \beta is the phase-space coordinate.^[29] This form arises from convolving the narrow Gaussian Wigner function of a pure coherent state with the broader Gaussian of the thermal state, effectively smearing the distribution into a disk-like profile whose radius scales with \sqrt{\bar{n}_{\rm th}}.^[29] Due to the thermal admixture, these states deviate from the minimal uncertainty property of pure coherent states. Specifically, the variances in the quadrature operators \hat{X} and \hat{P} (scaled such that [\hat{X}, \hat{P}] = i/2) are equal and exceed the vacuum limit: \langle (\Delta \hat{X})^2 \rangle = \langle (\Delta \hat{P})^2 \rangle = (2 \bar{n}_{\rm th} + 1)/4 > 1/4 for \bar{n}_{\rm th} > 0, resulting in \Delta X \Delta P > 1/4. In cavity quantum electrodynamics, thermal coherent states model the intracavity field under coherent driving in the presence of a thermal reservoir, capturing realistic dissipative dynamics where bath photons introduce noise while the drive maintains displacement.^[30] Such states are pertinent for analyzing decoherence effects, quantum metrology in noisy environments, and the preparation of superpositions like hot Schrödinger cat states in microwave cavities.^[31]

Squeezed coherent states

Squeezed coherent states represent a class of quantum states derived from standard coherent states by applying a squeezing operator, which distorts the uncertainty ellipse in phase space to reduce noise in one quadrature at the expense of the orthogonal one. These states serve as a fundamental non-classical resource in quantum optics, enabling applications that surpass classical limits. The squeezing operator is defined as S(\zeta) = \exp\left[ \frac{\zeta^* \hat{a}^2 - \zeta \hat{a}^{\dagger 2}}{2} \right], where \hat{a} and \hat{a}^\dagger are the annihilation and creation operators, respectively, and \zeta = r e^{i\theta} is the complex squeezing parameter with magnitude r \geq 0 controlling the degree of squeezing and phase \theta determining the squeezing direction.^[32] A squeezed coherent state |\alpha, \zeta\rangle is then generated by acting this operator on a coherent state |\alpha\rangle, or equivalently as |\alpha, \zeta\rangle = \hat{D}(\alpha) \hat{S}(\zeta) |0\rangle, where \hat{D}(\alpha) is the displacement operator and |0\rangle is the vacuum state.^[33] The hallmark of these states is the unequal quadrature variances, which violate the symmetric uncertainty of coherent states. In natural units where \hbar = \omega = 1, the variance of the quadrature \hat{X}_\theta = \frac{1}{2} \left( \hat{a} e^{i\theta} + \hat{a}^\dagger e^{-i\theta} \right) is \Delta \hat{X}_\theta^2 = \frac{1}{4} e^{-2r}, while the orthogonal quadrature \hat{X}_{\theta + \pi/2} has \Delta \hat{X}_{\theta + \pi/2}^2 = \frac{1}{4} e^{2r}.^[32]^[33] This imbalance maintains the minimum uncertainty relation \Delta \hat{X}_\theta \Delta \hat{X}_{\theta + \pi/2} = \frac{1}{4} but allows one variance to fall below the vacuum level of $1/4, demonstrating non-classical behavior through sub-shot-noise fluctuations that cannot arise from classical fields.^[32] Squeezed coherent states thus exhibit non-classicality via this quadrature squeezing, as they cannot be expressed as statistical mixtures of classical coherent states.^[34] Generation of squeezed coherent states typically relies on nonlinear optical processes, such as degenerate parametric down-conversion in a nonlinear crystal within an optical cavity, where a pump photon splits into signal and idler photons correlated in a way that produces squeezing.^[35] Alternatively, quantum feedback schemes in cavities can stabilize and enhance squeezing by conditionally adjusting the field based on homodyne measurements, effectively reducing noise through active control.^[36] Recent advancements have expanded the utility of squeezed coherent states. In 2024, high-harmonic generation driven by bright squeezed vacuum in solids was demonstrated to produce non-perturbative harmonics with preserved quantum correlations, opening pathways for attosecond-scale quantum light sources.^[37] As of July 2025, squeezing-enhanced sensing at exceptional points in multimode systems achieved Heisenberg-limited precision, surpassing standard quantum limits in parameter estimation for applications like gravitational wave detection.^[38]

Applications in Condensed Matter Physics

In Bose-Einstein condensates

A Bose-Einstein condensate (BEC) represents a coherent state of matter in which a macroscopic number of bosonic atoms occupy the lowest quantum state, achieving near-complete ground state occupation of approximately 100% at absolute zero temperature. This coherence arises from the collective wavefunction of the condensate, which exhibits a well-defined phase, analogous to a classical wave but with quantum properties. The dynamics of such a system are described by the Gross-Pitaevskii equation (GPE), a mean-field model that treats the condensate as a nonlinear Schrödinger equation for the macroscopic wavefunction \psi(\mathbf{r}, t):

i \hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r}, t) + g |\psi|^2 \right] \psi,

where V(\mathbf{r}, t) is the external potential, g is the interaction strength proportional to the s-wave scattering length, and the normalization \int |\psi|^2 d\mathbf{r} = N gives the total atom number N. This equation captures the coherent evolution of the phase and density, enabling the BEC to behave as a quantum coherent source of matter waves. The experimental realization of BECs in dilute atomic gases occurred in 1995, first with rubidium-87 atoms by the JILA group and shortly thereafter with sodium-23 by the MIT group, using evaporative cooling in magnetic traps to reach nanokelvin temperatures. Phase coherence in these condensates was demonstrated through time-of-flight expansion, where released atoms expand ballistically, revealing interference patterns indicative of a coherent initial wavefunction with long-range phase order. This coherence extends to matter-wave interferometry, where BECs in double-well potentials—created by optical or magnetic fields—undergo coherent splitting analogous to beam splitters in optical interferometers, producing high-contrast interference fringes upon recombination. In interacting BECs, nonlinear effects from atomic collisions lead to dephasing, where the relative phase between components diffuses due to fluctuations in density or chemical potential, potentially disrupting coherence over time. However, revivals of coherence can occur periodically as interaction-induced phase shifts re-align, observed in experiments with time-dependent traps or multi-component systems, sustaining coherence for seconds in Ramsey-like interferometers.^[39] This phenomenon highlights the robust macroscopic quantum coherence inherent to BECs. The concept of BEC coherence also connects to superfluidity in liquid helium-4, where below the lambda transition at 2.17 K, a partial Bose-Einstein condensation occurs with a condensate fraction of approximately 6-8% at low temperatures, as measured by neutron scattering, enabling zero-viscosity flow and phase-coherent vortex dynamics.

In superconductivity

In the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity, the ground state of a superconductor is characterized as a coherent quantum state involving the condensation of Cooper pairs, which are bound pairs of electrons with opposite momenta and spins.^[40] This coherence arises from an attractive interaction mediated by phonons, leading to a macroscopic occupation of the paired state with a well-defined phase, distinguishing the superconducting phase from the normal state.^[40] The theory predicts that this collective pairing results in zero electrical resistance and other macroscopic quantum effects below the critical temperature.^[40] The superconducting order parameter, denoted as \psi = |\psi| e^{i\phi}, captures this macroscopic coherence, where |\psi| represents the density of the superconducting electron pairs and \phi is the global phase. This complex order parameter is analogous to the eigenvalue \alpha of a coherent state in quantum optics, encoding both the amplitude and phase of the condensed pairs, which enables long-range phase correlations across the material.^[41] In the BCS framework, the order parameter emerges self-consistently from the pairing gap \Delta, reflecting the energy scale of the coherent pairing.^[40] In Josephson junctions, which consist of two superconductors separated by a thin insulating barrier, the coherent tunneling of Cooper pairs occurs due to the phase difference \Delta\phi between the order parameters on either side. This results in a supercurrent I = I_c \sin(\Delta\phi), where I_c is the critical current, demonstrating macroscopic quantum interference and phase coherence across the barrier.^[42] The effect underscores the rigidity of the superconducting phase, allowing for applications in sensitive magnetometry and quantum computing.^[42] The global phase coherence in superconductors also underlies the Meissner effect, where magnetic fields are expelled from the interior, leading to perfect diamagnetism, and flux quantization in multiply connected geometries.^[43] In a superconducting ring, the magnetic flux threading the loop is quantized in units of \Phi_0 = h/(2e), arising from the single-valuedness of the wavefunction and the phase winding around the loop.^[43] This quantization, combined with the phase stiffness, enforces the expulsion of fields and the formation of vortices in type-II superconductors under applied fields.^[43] Links to the quantum Hall effect appear in hybrid systems where superconducting proximity induces pairing in the chiral edge states of quantum Hall insulators, forming chiral coherent modes that propagate unidirectionally.^[44] These edge modes, protected by topology, enable the transmission of superconducting correlations without backscattering, potentially realizing Majorana fermions for topological quantum computing.^[45]

Generalizations and Extensions

Angular momentum coherent states

Angular momentum coherent states, also known as SU(2) coherent states or spin coherent states, generalize the concept of coherent states from the harmonic oscillator to systems described by the SU(2) Lie algebra, which governs angular momentum in quantum mechanics. These states are defined for a given angular momentum quantum number j as |\theta, \phi \rangle = \exp(-i \phi J_z) \exp(-i \theta J_y) |j, j\rangle, where |j, j\rangle is the highest-weight state in the (2j+1)-dimensional representation of SU(2), and J_y, J_z are the corresponding angular momentum operators. This parametrization points the expectation value of the angular momentum vector \langle \mathbf{J} \rangle along a direction specified by the polar angle \theta and azimuthal angle \phi on the unit sphere, with |\langle \mathbf{J} \rangle| = j.^[46] The Perelomov construction provides a general framework for defining these coherent states as group displacements of the highest-weight state under the SU(2) group action: |g \cdot j, j \rangle = U(g) |j, j \rangle, where U(g) is the unitary representation of the group element g, often parametrized by Euler angles. For compact groups like SU(2), this yields states that are normalizable and resolve the identity via an overcomplete basis with a measure on the group manifold, specifically the Haar measure on the 2-sphere for SU(2). The overlap between two such states is \langle \theta, \phi | \theta', \phi' \rangle = \left[ \cos(\theta/2) \cos(\theta'/2) + \sin(\theta/2) \sin(\theta'/2) e^{i(\phi - \phi')} \right]^{2j}, which concentrates around unity when the directions are close, analogous to the Gaussian overlap in the oscillator case.^[46] These states minimize the uncertainty product for components perpendicular to the mean spin direction, satisfying \Delta J_1 \Delta J_2 \geq j/2, where J_1 and J_2 are orthogonal directions in the plane perpendicular to \langle \mathbf{J} \rangle, achieving equality for the coherent state.^[46] The resolution of the identity holds as (2j+1) \int d\Omega \, | \theta, \phi \rangle \langle \theta, \phi | = I, with d\Omega = \sin \theta \, d\theta \, d\phi / 4\pi the uniform measure on the sphere, mirroring the overcompleteness of harmonic oscillator coherent states but adapted to the finite-dimensional Hilbert space. In spin systems, SU(2) coherent states serve as trial states in variational methods for quantum many-body problems, such as approximating ground states in Heisenberg spin chains or facilitating path-integral formulations of spin dynamics.^[47] In nuclear physics, they model collective rotational excitations and isovector modes, aiding in the description of nuclear deformation and multipole responses within algebraic models like the interacting boson approximation restricted to SU(2) subspaces.^[48] Experimental realizations include preparing large atomic ensembles in coherent spin states via optical pumping or Ramsey interferometry, achieving high-fidelity superpositions of collective spin orientations in alkali vapors. For photons, SU(2) coherent states manifest as polarization states, generated using wave plates to rotate the Poincaré sphere representation of two orthogonal modes, as demonstrated in quantum optics setups with laser light.

Group-theoretic and modern generalizations

The generalization of coherent states to arbitrary Lie groups was independently developed by Perelomov and Gilmore in 1972, providing a unified framework beyond the Heisenberg-Weyl group associated with the quantum harmonic oscillator. In this approach, coherent states are constructed via the action of the group's unitary representation on a fiducial state, typically a highest-weight vector |μ⟩ in an irreducible representation space. Specifically, the Perelomov (or Gilmore-Perelomov) coherent states are defined as |g, μ⟩ = U(g) |μ⟩, where U(g) denotes the unitary operator corresponding to the Lie group element g, ensuring overcompleteness and resolution of the identity through integration over the group manifold.^[49] These group-theoretic coherent states find applications in quantum field theory, where they describe classical field configurations alongside quantum fluctuations, facilitate calculations involving infinite virtual particles, and aid in deriving functional integrals and effective theories.^[50] In relativistic systems, extensions to spinless particles incorporate approaches such as standard canonical coherent states, Lorentzian states, and those based on Newton-Wigner localization under the Salpeter equation, enabling analysis of expectation values for relativistic observables while satisfying Heisenberg uncertainty relations.^[51] Modern developments include photon-added coherent states, defined as â†^k |α⟩ normalized by the appropriate factor, which enhance entanglement when passed through beam splitters, yielding output states as superpositions of such states with Schmidt numbers up to three for single-photon additions, supporting quantum networks.^[52] In nonlinear optics, intense femtosecond infrared superpositions of coherent states, with mean photon numbers orders of magnitude beyond prior sources, drive processes like second-harmonic generation, imprinting non-classical features into the output and enabling quantum light engineering for information processing.^[53] Furthermore, superpositions of coherent states, such as cat states |α⟩ ± |-α⟩, serve as logical qubits in bosonic quantum error correction codes, achieving enhanced coherence times through exponential suppression of bit-flip errors and hardware-efficient concatenation with repetition codes, as demonstrated in superconducting implementations reaching below-threshold error rates.^[54]^[55]

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