SU(1,1) Displaced Coherent States

• SU(1,1) displaced coherent states, or Perelomov coherent states, are quantum states formed by applying the SU(1,1) displacement operator to the lowest-weight state.
• They exhibit distinctive features such as overcompleteness, minimum uncertainty, and unique phase-space structures that are crucial for quantum optics and metrology.
• The analytic construction using Baker-Campbell-Hausdorff formulas supports practical implementations like squeezing, sub-Planck structures, and enhanced phase sensitivity in experiments.

A SU(1,1) displaced coherent state—commonly called a Perelomov coherent state—refers to quantum states constructed by applying the SU(1,1) group displacement operator to the lowest-weight state of a positive discrete series irreducible representation. These states generalize canonical (Heisenberg–Weyl) coherent states to systems whose symmetry algebra is non-compact, specifically su(1,1). As such, they play a central role in quantum optics, quantum information, and the analysis of quantum dynamical systems exhibiting hyperbolic symmetry. Their algebraic construction, analytic properties, phase-space structures, and nonclassical correlations underpin their utility across a wide range of theoretical and experimentally relevant models.

1. Algebraic Framework and Displacement Operator

The SU(1,1) Lie algebra is generated by operators $K_0$, $K_+$, $K_-$ with the commutation relations: $$[K_0, K_\pm] = \pm K_\pm, \qquad [K_-, K_+] = 2K_0.$$ The Casimir is $C = K_0^2 - \tfrac{1}{2}(K_+K_- + K_-K_+)$, and in the positive discrete series representations, the basis $\{\lvert k,n\rangle\}_{n=0}^\infty$ satisfies $K_0\lvert k,n\rangle = (k+n)\lvert k,n\rangle$, $K_-\lvert k,0\rangle=0$ for $k>0$ (the "Bargmann index").

The SU(1,1) displacement operator, defining the Perelomov class of coherent states, is

$$D(\zeta) = \exp(\zeta K_+ - \zeta^* K_-), \quad \zeta \in \mathbb{C},\; |\zeta|<1.$$

A standard Weyl-Baker-Campbell-Hausdorff disentanglement yields: $K_+$0 This formula underpins all analytic and computational work regarding SU(1,1) displaced coherent states (Mojaveri et al., 2012, Guerrero, 2018).

2. Construction and Expansion in the Basis

The SU(1,1) Perelomov coherent state is defined by the action of the displacement operator on the lowest-weight state: $K_+$1 Its basis expansion is

$K_+$2

The expansion coefficients $K_+$3 satisfy

$K_+$4

These states are normalized for $K_+$5: $K_+$6 (Mojaveri et al., 2012, Salazar-Ramírez et al., 2016, Gazeau et al., 2020, Gazeau et al., 2023).

3. Completeness, Overcompleteness, and Resolution of Unity

The family $K_+$7 is (over)complete on the SU(1,1) positive-discrete series Hilbert space. The resolution of the identity is

$K_+$8

with the SU(1,1) invariant measure

$K_+$9

Insertion of the explicit expansion into this relation, with evaluation using Beta function integrals, verifies the identity (Mojaveri et al., 2012, Guerrero, 2018).

The overlap (reproducing kernel) is

$K_-$0

supporting reproducing-kernel Hilbert space structures.

4. Expectation Values, Quadrature Squeezing, and Minimum Uncertainty

Expectation values for the generators in $K_-$1 are

$K_-$2

For Hermitian quadratures $K_-$3, $K_-$4,

$K_-$5

so these states saturate the generalized Heisenberg–Robertson–Schrödinger lower bound (minimum-uncertainty states for the su(1,1) algebra) (Mojaveri et al., 2012, Guerrero, 2018, Gazeau et al., 2023).

Quadrature squeezing arises as the variances in $K_-$6 or $K_-$7 fall below the conventional coherent-state (vacuum) value for suitable loci in $K_-$8 (Gazeau et al., 2023).

5. Phase-Space Representations, Sub-Planck Structure, and Displacement Sensitivity

On the Poincaré disk parameterized by $K_-$9 with $$[K_0, K_\pm] = \pm K_\pm, \qquad [K_-, K_+] = 2K_0.$$0, a Perelomov coherent state is centered at $$[K_0, K_\pm] = \pm K_\pm, \qquad [K_-, K_+] = 2K_0.$$1. The SU(1,1) (hyperbolic) phase-space measure involves $$[K_0, K_\pm] = \pm K_\pm, \qquad [K_-, K_+] = 2K_0.$$2.

The SU(1,1) Wigner function $$[K_0, K_\pm] = \pm K_\pm, \qquad [K_-, K_+] = 2K_0.$$3—where $$[K_0, K_\pm] = \pm K_\pm, \qquad [K_-, K_+] = 2K_0.$$4 is a displaced parity—exhibits core features of coherent-state localization. Superpositions of two or more such states (cat, compass, circular states) exhibit quantum interference structures ("tiles") at sub-Planck scales in this phase space; specifically, the fundamental linear scale of such tiles is $$[K_0, K_\pm] = \pm K_\pm, \qquad [K_-, K_+] = 2K_0.$$5 for individual coherent states and $$[K_0, K_\pm] = \pm K_\pm, \qquad [K_-, K_+] = 2K_0.$$6 for certain multipartite superpositions (n-component compass states with $$[K_0, K_\pm] = \pm K_\pm, \qquad [K_-, K_+] = 2K_0.$$7) (Akhtar et al., 2022, Akhtar et al., 16 Feb 2026). This sub-Planck structure underpins metrological applications; the state’s distinguishability under (hyperbolic) phase-space displacements is set by the area of such tiles.

6. Physical Realizations and Quantum Optical Interpretation

Physical systems supporting SU(1,1) symmetry (and thus admitting displaced coherent states) include:

• Two-photon Hamiltonians and squeezing in quantum optics ($$[K_0, K_\pm] = \pm K_\pm, \qquad [K_-, K_+] = 2K_0.$$8, $$[K_0, K_\pm] = \pm K_\pm, \qquad [K_-, K_+] = 2K_0.$$9, $C = K_0^2 - \tfrac{1}{2}(K_+K_- + K_-K_+)$0), relevant for generating squeezed vacuum and even/odd photon-number subspaces,
• Radial modes of Laguerre-Gaussian beams (photon counting and squeezing in the radial index),
• Quantum oscillators with hyperbolic or radial symmetry (Dunkl oscillator, pseudo-harmonic/Calogero–Sutherland models, Dirac–Kepler–Coulomb systems) (Salazar-Ramírez et al., 2016, Ojeda-Guillen et al., 2013, Ojeda-Guillén et al., 2014, Karimi et al., 2012).

Experimentally, such states may be engineered via nonlinear parametric processes, optical fibers with SU(1,1) symmetry, or, more generally, in any system where the Hamiltonian and quantum numbers admit an embedding into the positive discrete series of su(1,1) (Gazeau et al., 2023).

The photon counting distribution is negative binomial: $C = K_0^2 - \tfrac{1}{2}(K_+K_- + K_-K_+)$1 The Mandel parameter $C = K_0^2 - \tfrac{1}{2}(K_+K_- + K_-K_+)$2 indicates Poissonian and sub/super-Poissonian regimes, controlled by $C = K_0^2 - \tfrac{1}{2}(K_+K_- + K_-K_+)$3 and the photon number.

7. Deformations, Nonlinear Generalizations, and Applications in Quantum Metrology

Generalized (nonlinear, "deformed") SU(1,1) displaced coherent states arise by modifying the ladder operator structure (in particular, their structure function $C = K_0^2 - \tfrac{1}{2}(K_+K_- + K_-K_+)$4 or a nonlinear f-deformation), introducing, for example, polynomial algebras or Pöschl–Teller–type potentials (Sadiq et al., 2012, Firouzabadi et al., 2014, Mojaveri et al., 2014, Abouelkhir et al., 29 Aug 2025). The resulting states interpolate between Barut–Girardello eigenstates ($C = K_0^2 - \tfrac{1}{2}(K_+K_- + K_-K_+)$5) and Perelomov displaced states, sometimes also approaching Heisenberg–Weyl (standard coherent) states in suitable limits.

In applications to quantum metrology, specifically phase sensitivity in interferometry, deformed SU(1,1) coherent states can achieve phase sensitivities approaching the Heisenberg limit. Quantum Fisher information analysis and achievable quantum Cramér–Rao bounds under realistic detection (difference intensity, single mode, or balanced homodyne) demonstrate that such states offer tunable precision enhancement in Mach–Zehnder setups, outperforming the shot-noise (standard quantum) limit in appropriate parameter regimes (Abouelkhir et al., 29 Aug 2025). Higher-order superpositions (such as compass or circular states) provide isotropic sub-Planck resolution, advantageous for quantum sensing tasks (Akhtar et al., 16 Feb 2026).

Table: Properties of SU(1,1) Displaced (Perelomov) Coherent States

Property	Formula/Description	Reference
Expansion	$C = K_0^2 - \tfrac{1}{2}(K_+K_- + K_-K_+)$6	(Mojaveri et al., 2012, Gazeau et al., 2023)
Overlap	$C = K_0^2 - \tfrac{1}{2}(K_+K_- + K_-K_+)$7	(Mojaveri et al., 2012)
Resolution of Unity	$C = K_0^2 - \tfrac{1}{2}(K_+K_- + K_-K_+)$8	(Mojaveri et al., 2012, Salazar-Ramírez et al., 2016)
Min-uncertainty	$C = K_0^2 - \tfrac{1}{2}(K_+K_- + K_-K_+)$9	(Mojaveri et al., 2012, Guerrero, 2018)
Negative binomial stats	$\{\lvert k,n\rangle\}_{n=0}^\infty$0	(Gazeau et al., 2023)
Sub-Planck structure	Superpositions lead to phase-space tiles of area $\{\lvert k,n\rangle\}_{n=0}^\infty$1 (compass state, $\{\lvert k,n\rangle\}_{n=0}^\infty$2)	(Akhtar et al., 2022, Akhtar et al., 16 Feb 2026)

• "SU(1,1) Nonlinear Coherent States" (Mojaveri et al., 2012)
• "SU(1,1)–displaced coherent states, photon counting and squeezing" (Gazeau et al., 2023)
• "Finer sub-Planck structures and displacement sensitivity of SU(1,1) circular states" (Akhtar et al., 16 Feb 2026)
• "Sub-Planck phase-space structure and sensitivity for SU(1,1) compass states" (Akhtar et al., 2022)
• "Quantum Phase Sensitivity with Generalized Coherent States Based on Deformed su(1,1) and Heisenberg Algebras" (Abouelkhir et al., 29 Aug 2025)
• "Coherent states for polynomial su(1,1) algebra and a conditionally solvable system" (Sadiq et al., 2012)
• "Algebraic and group treatments to nonlinear displaced number states and their nonclassicality features" (Firouzabadi et al., 2014)
• "Generalized su(1,1) coherent states for pseudo harmonic oscillator and their nonclassical properties" (Mojaveri et al., 2014)

These sources provide detailed mathematical and application-specific perspectives on the construction and utilization of SU(1,1) displaced coherent states in various physical and mathematical settings.