Generalized Coherent States

• Generalized coherent states are quantum states that extend canonical coherent states through nonlinear and group-theoretic modifications to capture complex quantum-classical correspondence.
• They are constructed explicitly via generalized Heisenberg algebras, polynomial algebras, and supersymmetric methods, offering control over photon statistics and squeezing properties.
• Applications span quantum optics, condensed matter, and quantum information, providing a robust framework for modeling nonclassical phenomena and quantum measurements.

Generalized coherent states are quantum states that extend the classical notion of canonical (Glauber) coherent states to a broad spectrum of quantum systems, including those with nonlinear spectra, algebraic deformations, non-equidistant energy levels, or underlying group-theoretic structures beyond the Heisenberg-Weyl algebra. These states form an essential framework for describing the quantum-classical correspondence in complex systems and provide powerful tools for investigating nonclassical features, generalized statistics, and applications ranging from quantum optics to condensed matter and quantum information.

1. Algebraic Foundations and Generalization Schemes

The core principle underlying generalized coherent states (GCS) is the relaxation or extension of the canonical algebraic structure. In the canonical case, the creation and annihilation operators $a^\dagger,a$ satisfy $[a,a^\dagger] = 1$, and the energy spectrum is linear. GCS constructions generalize this in multiple directions:

• By adopting nonlinear algebraic structures, such as the generalized Heisenberg algebra (GHA) defined by operators $(H, A, A^\dagger)$ satisfying

$$H A^\dagger = A^\dagger f(H),\quad A H = f(H) A,\quad [A, A^\dagger] = f(H) - H$$

where $f(H)$ is an analytic characteristic function determined by the physical system's spectrum (Berrada et al., 2010).

• Through group-theoretic approaches—constructing Perelomov- and Barut–Girardello-type states for non-Heisenberg algebras, e.g., $su(2)$, $su(1,1)$, polynomial Weyl–Heisenberg algebras (Kibler et al., 2012), or more exotic Lie groups (Grigolo et al., 2015).
• By using system-adapted ladder operators obtained from supersymmetric quantum mechanics, shape invariance, or dynamical symmetries (Amir et al., 2016, C. et al., 2022, Myo et al., 26 Jan 2024).
• By considering quantum systems with degenerate spectra, leading to new forms of coherence across degenerate subspaces (Honarasa et al., 2011).

These approaches establish the existence of GCS as (i) eigenstates of generalized annihilation operators, (ii) displacements of a fiducial state under a suitable group action, or (iii) minimal uncertainty states with respect to generalized or time-dependent quadratures.

2. Explicit Constructions and Representative Examples

Several canonical methods have emerged for constructing GCS, with explicit forms dependent on the system:

• Generalized Heisenberg Algebra for power-law potentials: With $V(x,k) = V_0|x/a|^k$ and WKB-derived spectrum $E_{n,k}$, the associated GCS are

$$|z,k\rangle = \mathcal{N}(|z|^2, k) \sum_{n=0}^\infty \frac{z^n}{\sqrt{g(n,k)}} |n\rangle,\quad g(n,k) = \prod_{i=1}^n\left[ (i+\gamma/4)^{2k/(k+2)} - (\gamma/4)^{2k/(k+2)} \right]$$

where the normalization function $\mathcal{N}$ and the weight function $W(x)$ (for the resolution of unity) are determined by the Stieltjes moment problem, often solvable via Mellin transforms (Berrada et al., 2010).

• Polynomial Weyl–Heisenberg Algebra: For $r$ parameters $\{K_i\}$,

$F(N) = N \prod_{i=1}^r \left[1 + K_i (N-1)\right]$

Perelomov-type GCS are built via a Fock–Bargmann analytic approach:

$$|z,\varphi\rangle = \mathcal{N}^{-1} \sum_{n=0}^\infty \frac{z^n e^{-iF(n)\varphi}}{\sqrt{F(n)!}} |n\rangle$$

This structure unifies GCS for $su(2)$, $su(1,1)$, and oscillator algebras as special cases (Kibler et al., 2012).

• Systems with Position-Dependent Effective Mass: Employing SUSY QM and shape invariance, the system-specific ladder operators $L_-, L_+$ satisfy a modified commutation relation, and the coherent states are expanded as

$$|z\rangle = \frac{1}{\sqrt{\mathcal{N}(|z|^2)}} \sum_{n=0}^\infty \frac{z^n}{\sqrt{\rho_n}} |\varphi_n\rangle$$

where $\rho_n$ generalizes $n!$ and is set by spectral data (Amir et al., 2016).

• Graphene and Dirac Materials: Ladder operators are constructed from intertwining relations adapted to the Dirac structure, and the coherent states are shown to unify multiple standard definitions (Barut–Girardello, Gilmore–Perelomov, minimum uncertainty) when the nonlinearity $f(n)=1$ (C. et al., 2022).
• Truncated and Non-Poissonian GCS: By generalizing the factorial in the expansion to special functions (Mittag-Leffler, Wright), one obtains finite-dimensional/truncated GCS with photon-number statistics showing stretched exponential or power-law behavior, enabling modeling of a wide range of non-classical fields (Giraldi et al., 2022).

3. Statistical and Nonclassical Properties

A central aspect of GCS is the departure from canonical Poissonian statistics and the emergence of nonclassical features:

• Mandel's $Q$ Parameter and Photon Statistics: The $Q$ parameter,

$$Q = \frac{\langle (\Delta N)^2 \rangle - \langle N \rangle}{\langle N \rangle}$$

quantifies the deviation from Poissonian photon-counting statistics. For GCS related to loosely binding power-law potentials ($k<2$), $Q$ can be positive (super-Poissonian) at low $|z|$ and negative (sub-Poissonian) for larger $|z|$. Tightly binding cases ($k>2$) yield persistent sub-Poissonian statistics (Berrada et al., 2010). Families based on Mittag-Leffler or Wright generalizations exhibit super- or uniquely sub-Poissonian behavior, directly controlled by parameter choices (Giraldi et al., 2022).

• Squeezing and Entropic Uncertainty: Many GCS obey minimum uncertainty relations for generalized quadratures, but the squeezing—i.e., reduction of variance below the standard quantum limit—is sensitive to the deformation or nonlinearity function and the underlying group. For instance, generalized $su(1,1)$ and $su(2)$ GCS exhibit adjustable squeezing, antibunching, and anti-correlation effects via the group deformation parameter (Mojaveri et al., 2014, Dehghani et al., 2014). In degenerate systems, squeezing properties can switch between number and phase representations depending on the state parameter (Honarasa et al., 2011).
• Nonclassicality Signatures: Sub-Poissonian statistics, single-photon-like nodal structures, and entangled outputs in beamsplitter experiments distinguish GCS from standard coherent and number states (see, e.g., displaced Fock states $|n,\alpha\rangle$ for the oscillator (Philbin, 2013)).

4. Resolution of Identity and Operator-Theoretic Structure

For a set of states $|z\rangle$ to qualify as GCS, they must satisfy Klauder’s minimal set of conditions: normalizability, continuity in label $z$, and the existence of a resolution of identity: $$\int_\mathbb{C} d^2z\, |z\rangle\langle z|\, W(|z|^2) = 1$$ The weight function $W(|z|^2)$ is not universal but system-dependent, commonly determined by inversion of the moment problem associated with the GCS expansion coefficients. In complex cases, Meijer G-functions or Mellin transform techniques are used to solve for $W$.

Building on this, the framework of generalized coherent states provides a one-to-one correspondence with positive operator-valued measures (POVMs) (Heinosaari et al., 2011). Any POVM can be represented as an integral over rank-one operators constructed from (possibly unnormalized) generalized coherent states $d_k(x)$: $$M(X) = \sum_{k} \int_{x\in X} |d_k(x)\rangle\langle d_k(x)| \, d\mu(x)$$ This allows for an intrinsic diagonalization ("minimal diagonalization") of quantum observables and plays a key role in the theory of quantum measurements, including characterization of extremal POVMs and informational completeness.

5. Time-Dependent and Nonlinear Quantum Dynamics

GCS are crucial for encoding quantum dynamics in terms of classical or nonlinear analogues:

• Riccati Equation Formalism: Generalized Gaussian wavepackets with time-dependent width are described by complex Riccati equations. The evolution of the width parameter (governing uncertainties) obeys

$$\frac{2\hbar}{m}\dot{y}(t) + \left(\frac{2\hbar}{m}\right)^2 y^2(t) + \omega^2(t) = 0$$

where $y(t)$ is complex-valued and sensitive to initial conditions, with implications for quantum uncertainty, bifurcations in dissipative systems, and links to isospectrality in supersymmetric quantum mechanics (Castaños et al., 2012, Cruz-Prado et al., 2020).

• Geometric and Symplectic Structures: The dynamics of GCS, especially for quadratic Hamiltonians, can be realized as nonlinear (Hamiltonian) flows on lower-dimensional symplectic manifolds (such as the Poincaré disk or upper half-plane), revealing deep connections between quantum and classical phase spaces and recasting quantum evolution as complex Riccati dynamics (Cruz-Prado et al., 2020).

6. Physical Applications and Systems

The versatility of GCS makes them applicable in diverse quantum systems:

• Quantum Optics and Lasing Theory: GCS for power-law potentials are used to model statistical properties of both ideal (Poissonian) and real (sub/super-Poissonian) lasers, capturing nonideal photon statistics and squeezing observed in experiments (Berrada et al., 2010).
• Multi-Well and Many-Body Systems: For Bose–Einstein condensate dynamics in double- and triple-well traps, GCS built from $SU(2)$ and $SU(3)$ coherent states yield efficient representations for quantum propagation, respecting conservation laws, system symmetries, and particle indistinguishability. They enable accurate semiclassical simulations without exponential scaling (Grigolo et al., 2015).
• Graphene and Dirac Materials: Generalized coherent states provide a unified formalism for quasi-relativistic systems, ensuring mutual equivalence of several standard definitions, and facilitating the analysis of probability/current densities, time-evolution, and underlying group symmetries (C. et al., 2022).
• Nuclear Cluster Models: By using GCS constructed via exponentials of raising operators (including dilation parameters), basis states that respect the Pauli exclusion principle and optimize radial behavior for both bound and continuum states are constructed, with direct applications in the description of nuclear clustering and scattering (Myo et al., 26 Jan 2024).
• Systems with Unbounded Spectrum: Even for particles with purely continuous spectra and unbounded motion—such as the accelerated particle—GCS can be constructed by non-commutative integration and by identifying the proper algebra of constants of motion (Breev et al., 5 May 2025).

7. Unified Perspectives and Future Directions

Generalized coherent states provide a comprehensive, algebraically and analytically tractable platform for quantum state construction, quantum measurement theory, and modeling of nonclassical phenomena across a wide range of systems. Core unifying features include:

• Flexibility in accommodating both finite- and infinite-dimensional Hilbert spaces, via analytic or Grassmann-variable-based constructions.
• The existence of dualities (e.g., between Barut–Girardello and Klauder–Perelomov states), illuminating connections between eigenstate- and displacement-generated coherent states (Popov, 2023).
• Systematically controlled departures from classicality, enabling detailed tuning of squeezing, statistics, and entropic uncertainty.
• Accessibility of explicit analytic expressions for normalization, expansion coefficients, and operator expectation values, supporting both analytical and computational studies.
• Potential for application in quantum information (state tomography, circuit synthesis (Somma, 2018)), quantum simulation, and basic studies of the quantum-to-classical transition.

Future research continues to deepen the understanding of the interplay between nonlinearity, group-theoretical structures, geometry, and quantum measurement, with generalized coherent states serving as a central organizing concept.