There are several distinct definitions and constructions of coherent states in the literature, each with its own mathematical formulation, physical interpretation, and domain of applicability. Below are some of the most prominent types of coherent states, along with their definitions, mathematical formulations, descriptions, applications, and foundational references.

Glauber–Sudarshan (Canonical) Coherent States

• Definition
  Canonical coherent states are quantum states of the harmonic oscillator that are defined as eigenstates of the annihilation operator. They exhibit the closest quantum analogue to classical harmonic motion.

• Mathematical definition
  They are defined by

  \[\hat{a}\,\lvert \alpha\rangle = \alpha\,\lvert \alpha\rangle,\qquad \alpha \in \mathbb{C},\]

  with explicit expansion in the Fock basis

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

• Description highlighting its essence
  Glauber–Sudarshan coherent states represent minimum-uncertainty states that saturate the Heisenberg uncertainty relation and possess equal uncertainties in position and momentum. Their time evolution preserves the functional form of the state, leading to classical-like oscillatory behavior of expectation values. These states form an overcomplete, nonorthogonal basis of the Hilbert space and provide a resolution of identity. In quantum optics, they describe ideal laser light and classical electromagnetic fields. The phase-space description through the Glauber–Sudarshan $P$-representation establishes a direct bridge between classical probability distributions and quantum states, making them central to semiclassical analysis.

• Where it can be used and its use in the context of rational extension of the harmonic oscillator
  Canonical coherent states are widely used in quantum optics, semiclassical physics, quantum information, and phase-space methods. In the context of rationally extended harmonic oscillators, they serve primarily as a reference or limiting case. While the standard annihilation operator does not close on the extended spectrum, Glauber coherent states help benchmark nonclassical effects such as squeezing and non-Gaussianity when compared with coherent states constructed from deformed or polynomial ladder operators of the rational extension.

• Fundamental and foundational papers

  • R. J. Glauber, Coherent and Incoherent States of the Radiation Field, Phys. Rev. 131, 2766 (1963).
  • E. C. G. Sudarshan, Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams, Phys. Rev. Lett. 10, 277 (1963).
  • J. R. Klauder and B.-S. Skagerstam, Coherent States, World Scientific (1985).
  • W.-M. Zhang, D. H. Feng, and R. Gilmore, Coherent States: Theory and Some Applications, Rev. Mod. Phys. 62, 867 (1990).

Perelomov Coherent States

• Definition
  Perelomov coherent states are defined as states obtained by the action of a continuous Lie group on a fixed reference (fiducial) state, usually chosen as a lowest- or highest-weight state of an irreducible representation.

• Mathematical definition
  Given a Lie group $G$ with generators ${T_i}$ and a reference state $\lvert \psi_0\rangle$, Perelomov coherent states are

  \[\lvert g\rangle = \hat{U}(g)\,\lvert \psi_0\rangle,\qquad g\in G,\]

  where $\hat{U}(g)=\exp!\left(\sum_i \xi_i T_i\right)$.
  For the $SU(1,1)$ algebra,

  \[\lvert z;k\rangle = (1-\lvert z\lvert ^2)^k \exp(z K_+)\,\lvert k,0\rangle,\qquad \lvert z\lvert <1 .\]

• Description highlighting its essence
  Perelomov coherent states provide a group-theoretical generalization of canonical coherent states. Their defining feature is covariance under the action of a symmetry group, making them naturally adapted to systems whose dynamics are governed by Lie algebras rather than the Heisenberg–Weyl algebra. These states inherit geometric and algebraic properties from the underlying group, such as natural phase-space manifolds and resolutions of identity. They minimize generalized uncertainty relations associated with the group generators and allow a transparent semiclassical interpretation. Perelomov coherent states are particularly powerful when the Hamiltonian admits an exact or dynamical symmetry, enabling a unified treatment of quantum and classical aspects.

• Where it can be used and its use in the context of rational extension of the harmonic oscillator
  Perelomov coherent states are used in quantum optics, atomic and molecular physics, quantum chaos, and semiclassical analysis of systems with $SU(2)$, $SU(1,1)$, or related symmetries. For rationally extended harmonic oscillators, where ladder operators often generate deformed or polynomial $SU(1,1)$-type algebras, Perelomov coherent states can be constructed using the corresponding group action. They provide a systematic way to explore classical limits, phase-space geometry, and uncertainty properties of the extended systems.

• Fundamental and foundational papers

  • A. M. Perelomov, Coherent States for Arbitrary Lie Groups, Commun. Math. Phys. 26, 222 (1972).
  • A. M. Perelomov, Generalized Coherent States and Their Applications, Springer (1986).
  • J. P. Gazeau, Coherent States in Quantum Physics, Wiley (2009).
  • W.-M. Zhang, D. H. Feng, and R. Gilmore, Coherent States: Theory and Some Applications, Rev. Mod. Phys. 62, 867 (1990).

Barut–Girardello Coherent States

• Definition
  Barut–Girardello coherent states are defined as eigenstates of the lowering operator of a non-compact Lie algebra, most notably the $SU(1,1)$ algebra, rather than being generated by a group displacement operator.

• Mathematical definition
  For the $SU(1,1)$ generators $(K_0, K_\pm)$ satisfying

  \[[K_0,K_\pm]=\pm K_\pm,\qquad [K_+,K_-]=-2K_0,\]

  the Barut–Girardello coherent states $\lvert z;k\rangle$ are defined by

  \[K_-\,\lvert z;k\rangle = z\,\lvert z;k\rangle,\qquad z\in\mathbb{C}.\]

  In the basis $\lvert k,n\rangle$,

  \[\lvert z;k\rangle = \mathcal{N}(\lvert z\lvert ^2)\sum_{n=0}^{\infty} \frac{z^n}{\sqrt{n!\,\Gamma(2k+n)}}\,\lvert k,n\rangle .\]

• Description highlighting its essence
  Barut–Girardello coherent states generalize the canonical coherent-state concept to systems governed by non-compact algebras. Their defining characteristic is the eigenvalue equation of the lowering operator, which parallels the Glauber construction but in a non-Heisenberg algebraic setting. These states are typically non-Gaussian and exhibit rich nonclassical features such as squeezing and sub-Poissonian statistics. They form an overcomplete set and admit a resolution of identity with a nontrivial measure. Unlike Perelomov coherent states, they are not generated by group action but directly encode the algebraic structure of the ladder operators, making them especially suited to systems with deformed or nonlinear spectra.

• Where it can be used and its use in the context of rational extension of the harmonic oscillator
  Barut–Girardello coherent states are used in quantum optics, relativistic quantum systems, and models with $SU(1,1)$ symmetry. In rationally extended harmonic oscillators, ladder operators often close a polynomial or deformed $SU(1,1)$ algebra. Barut–Girardello coherent states can be constructed as eigenstates of these generalized lowering operators, providing a natural framework to analyze nonclassicality, uncertainty relations, and spectral deformation effects induced by the rational extension.

• Fundamental and foundational papers

  • A. O. Barut and L. Girardello, New “Coherent” States Associated with Non-Compact Groups, Commun. Math. Phys. 21, 41 (1971).
  • A. M. Perelomov, Generalized Coherent States and Their Applications, Springer (1986).
  • J. P. Gazeau, Coherent States in Quantum Physics, Wiley (2009).
  • D. Popov and M. A. Shirokov, Coherent States for $SU(1,1)$ Systems, J. Math. Phys. 36, 1820 (1995).

Gazeau–Klauder Coherent States

• Definition
  Gazeau–Klauder coherent states are coherent states constructed directly from the energy spectrum of a quantum system, characterized by temporal stability, action identity, and continuity in their labeling parameters.

• Mathematical definition
  For a system with discrete, nondegenerate spectrum $E_n$ $(E_0=0)$, the Gazeau–Klauder coherent states are defined as

  \[\lvert J,\gamma\rangle = \mathcal{N}(J)^{-1/2} \sum_{n=0}^{\infty}\frac{J^{n/2}e^{-iE_n\gamma}}{\sqrt{\rho_n}}\,\lvert n\rangle,\]

  where $J\ge0$, $\gamma\in\mathbb{R}$, $\rho_n=E_1E_2\cdots E_n$, and $\mathcal{N}(J)$ is a normalization factor.

• Description highlighting its essence
  Gazeau–Klauder coherent states provide a unifying framework for constructing coherent states for systems with arbitrary discrete spectra, without requiring an underlying Lie group or simple ladder-operator structure. Their defining properties include exact temporal stability under time evolution, a direct correspondence between the classical action variable and the parameter $J$, and continuity in the state labels. These states reduce to canonical coherent states for the harmonic oscillator but naturally extend to systems with nonlinear or non-equidistant energy levels. As a result, they are particularly suited to exploring semiclassical limits, quantum dynamics, and nonclassical effects in exactly solvable models.

• Where it can be used and its use in the context of rational extension of the harmonic oscillator
  Gazeau–Klauder coherent states are widely used in semiclassical analysis, quantum chaos, and systems with nonlinear spectra. For rationally extended harmonic oscillators, where energy levels are modified and often non-equidistant due to SUSY or Darboux transformations, Gazeau–Klauder coherent states remain well-defined because they depend only on the known spectrum. They are therefore one of the most robust and physically meaningful coherent-state constructions for rational extensions, allowing direct investigation of temporal stability, uncertainty relations, and non-Gaussian behavior.

• Fundamental and foundational papers

  • J. P. Gazeau and J. R. Klauder, Coherent States for Systems with Discrete and Continuous Spectrum, J. Phys. A: Math. Gen. 32, 123 (1999).
  • J. R. Klauder, Continuous-Representation Theory II. Generalized Relation between Quantum and Classical Dynamics, J. Math. Phys. 4, 1058 (1963).
  • J. P. Gazeau, Coherent States in Quantum Physics, Wiley (2009).
  • S. Twareque Ali, J. P. Antoine, and J. P. Gazeau, Coherent States, Wavelets, and Their Generalizations, Springer (2014).

Nonlinear (f-Deformed) Coherent States

• Definition
  Nonlinear (or $f$-deformed) coherent states are generalized coherent states defined as eigenstates of a deformed annihilation operator, where the deformation is governed by a nonlinear function of the number operator.

• Mathematical definition
  Introducing the deformed operators

  \[\hat{A}=\hat{a}\,f(\hat{n}), \qquad \hat{A}^\dagger=f(\hat{n})\,\hat{a}^\dagger,\]

  the nonlinear coherent states $\lvert \alpha,f\rangle$ satisfy

  \[\hat{A}\,\lvert \alpha,f\rangle=\alpha\,\lvert \alpha,f\rangle .\]

  In the Fock basis,

  \[\lvert \alpha,f\rangle=\mathcal{N}(\lvert \alpha\lvert ^2)^{-1/2} \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}\,[f(n)]!}\,\lvert n\rangle ,\]

  where $[f(n)]!=f(n)f(n-1)\cdots f(1)$.

• Description highlighting its essence
  Nonlinear coherent states extend the Glauber coherent-state concept by incorporating spectral or dynamical nonlinearities directly into the ladder operators. The deformation function $f(n)$ encodes deviations from equally spaced energy levels and leads to states with intrinsically nonclassical features. These states are generally non-Gaussian, display squeezing, amplitude-dependent uncertainties, and sub- or super-Poissonian statistics. By choosing different deformation functions, a wide variety of coherent-state families can be generated, each adapted to a specific physical system. This flexibility makes nonlinear coherent states a powerful tool for modeling quantum systems with anharmonicity or effective interactions.

• Where it can be used and its use in the context of rational extension of the harmonic oscillator
  Nonlinear coherent states are used in quantum optics, nonlinear media, trapped ions, and models with intensity-dependent couplings. In rationally extended harmonic oscillators, SUSY-generated ladder operators naturally lead to polynomial or nonlinear commutation relations. The corresponding deformation function $f(n)$ can be derived from the extended spectrum, allowing coherent states to be constructed as eigenstates of the deformed annihilation operator. This approach is one of the most natural and direct ways to define coherent states for rational extensions and to analyze their nonclassical properties.

• Fundamental and foundational papers

  • V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, f-Oscillators and Nonlinear Coherent States, Phys. Scr. 55, 528 (1997).
  • A. J. Bracken, D. Ellinas, and I. Smyrnakis, Free-Particle Coherent States and the f-Oscillator, Phys. Rev. A 53, 2381 (1996).
  • S. Sivakumar, Nonlinear Coherent States, J. Phys. A: Math. Gen. 32, 3441 (1999).
  • J. R. Klauder and B.-S. Skagerstam, Coherent States, World Scientific (1985).

Photon-Added and Photon-Subtracted Coherent States

• Definition
  Photon-added and photon-subtracted coherent states are nonclassical states obtained by the successive application of creation or annihilation operators, respectively, on canonical coherent states, followed by normalization.

• Mathematical definition
  Starting from a Glauber coherent state $\lvert \alpha\rangle$, the $m$-photon-added coherent states are

  \[\lvert \alpha,m\rangle_{\text{add}}=\mathcal{N}_m^{(+)}\,(\hat{a}^\dagger)^m\lvert \alpha\rangle,\]

  while the $m$-photon-subtracted coherent states are

  \[\lvert \alpha,m\rangle_{\text{sub}}=\mathcal{N}_m^{(-)}\,\hat{a}^m\lvert \alpha\rangle,\]

  where $\mathcal{N}_m^{(\pm)}$ are normalization constants.

• Description highlighting its essence
  Photon-added and photon-subtracted coherent states represent controlled departures from classicality introduced through discrete quantum operations. Photon addition enhances higher Fock-state components, whereas photon subtraction reshapes the photon-number distribution by preferentially removing lower-excitation contributions. Both constructions lead to non-Gaussian states with strong nonclassical features, including enhanced squeezing, oscillatory photon-number distributions, and negativity of quasiprobability distributions. These states interpolate between coherent states and number states, providing a tunable platform to study the transition from classical to quantum behavior.

• Where it can be used and its use in the context of rational extension of the harmonic oscillator
  These states are widely used in quantum optics, quantum metrology, and quantum information processing, particularly in state engineering and entanglement generation. In the context of rationally extended harmonic oscillators, photon-added and photon-subtracted constructions can be generalized using the modified ladder operators of the extended system. This allows systematic exploration of excitation-dependent nonclassicality and provides a bridge between standard coherent states and coherent states adapted to the deformed spectrum.

• Fundamental and foundational papers

  • G. S. Agarwal and K. Tara, Nonclassical Properties of States Generated by the Excitation on a Coherent State, Phys. Rev. A 43, 492 (1991).
  • M. Dakna, T. Anhut, T. Opatrný, L. Knöll, and D.-G. Welsch, Generation of Arbitrary Quantum States of Traveling Fields, Phys. Rev. A 55, 3184 (1997).
  • A. Zavatta, S. Viciani, and M. Bellini, Quantum-to-Classical Transition with Single-Photon-Added Coherent States, Science 306, 660 (2004).
  • J. Fiurášek, R. García-Patrón, and N. J. Cerf, Conditional Generation of Nonclassical States, Phys. Rev. A 72, 033822 (2005).

Intelligent / Minimum-Uncertainty Coherent States

• Definition
  Intelligent or minimum-uncertainty coherent states are quantum states that saturate a given uncertainty relation, typically the Heisenberg or generalized Robertson–Schrödinger uncertainty relation, for a chosen pair of noncommuting observables.

• Mathematical definition
  For two operators $\hat{A}$ and $\hat{B}$ with $[\hat{A},\hat{B}]\neq0$, intelligent states satisfy

  \[(\hat{A}+i\lambda \hat{B})\lvert \psi\rangle = \beta \lvert \psi\rangle,\]

  where $\lambda$ is a real parameter.
  For position and momentum,

  \[\Delta x\,\Delta p = \frac{\hbar}{2},\]

  which is saturated by canonical coherent states and squeezed states.

• Description highlighting its essence
  Intelligent coherent states are defined by an extremal property rather than a specific algebraic construction. They represent states in which quantum fluctuations are optimally distributed between noncommuting observables. Depending on the choice of operators, these states may coincide with canonical coherent states, squeezed states, or more general non-Gaussian states. The intelligent-state approach is particularly useful when standard ladder operators are absent or deformed, as it relies only on operator uncertainty relations. This makes the concept broadly applicable across quantum systems with noncanonical commutation relations.

• Where it can be used and its use in the context of rational extension of the harmonic oscillator
  Intelligent states are used in quantum optics, precision measurements, and quantum metrology. For rationally extended harmonic oscillators, where ladder operators generate polynomial or deformed algebras, intelligent coherent states can be constructed using generalized position–momentum-like operators or deformed quadratures. They provide insight into how rational extensions modify uncertainty bounds and allow direct comparison with canonical coherent states regarding squeezing and nonclassicality.

• Fundamental and foundational papers

  • J. P. Robertson, The Uncertainty Principle, Phys. Rev. 34, 163 (1929).
  • E. Schrödinger, Zum Heisenbergschen Unschärfeprinzip, Sitzungsber. Preuss. Akad. Wiss. 19, 296 (1930).
  • H. P. Yuen, Two-Photon Coherent States of the Radiation Field, Phys. Rev. A 13, 2226 (1976).
  • V. Dodonov and V. Man’ko, Invariants and the Evolution of Nonstationary Quantum Systems, Proc. Lebedev Phys. Inst. 183, 3 (1989).