---
title: SU(1,1) Displaced Coherent States
url: https://www.emergentmind.com/topics/su-1-1-displaced-coherent-states
type: topic
---

# SU(1,1) Displaced Coherent States

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:
\[
D(\zeta) = \exp(\eta K_+) (1-|\eta|^2)^{K_0} \exp(-\eta^* K_-), \quad \text{with}\;\eta = e^{i\arg\zeta}\tanh|\zeta|.
\]
This formula underpins all analytic and computational work regarding SU(1,1) displaced coherent states [1212.6888], [1804.00051].

## 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:
\[
\lvert \zeta; k \rangle = D(\zeta)\lvert k,0\rangle.
\]
Its basis expansion is
\[
\lvert\zeta; k\rangle = (1-|\zeta|^2)^k \sum_{n=0}^\infty \sqrt{\frac{\Gamma(n+2k)}{n!\; \Gamma(2k)}}\,\zeta^n\,\lvert k,n\rangle.
\]
The expansion coefficients $c_n(\zeta)$ satisfy
\[
c_n(\zeta) = \langle k,n|\zeta;k\rangle = (1-|\zeta|^2)^k\sqrt{\frac{\Gamma(n+2k)}{n!\;\Gamma(2k)}}\,\zeta^n.\label{eq:su11-perelomov-expansion}
\]
These states are normalized for $|\zeta|<1$: $\langle \zeta;k | \zeta;k \rangle = 1$ [1212.6888], [1607.06169], [2011.10303], [2304.08031].

## 3. Completeness, Overcompleteness, and Resolution of Unity

The family $\{\lvert\zeta;k\rangle\}$ is (over)complete on the SU(1,1) positive-discrete series Hilbert space. The resolution of the identity is
\[
\int_{|\zeta|<1} \lvert\zeta; k\rangle\langle \zeta; k\rvert\, d\mu(\zeta, \zeta^*) = I,
\]
with the SU(1,1) invariant measure
\[
d\mu(\zeta,\zeta^*) = \frac{2k-1}{\pi} (1-|\zeta|^2)^{-2}\, d^2\zeta, \quad d^2\zeta = d\,\Re\zeta\; d\,\Im\zeta.
\]
Insertion of the explicit expansion into this relation, with evaluation using Beta function integrals, verifies the identity [1212.6888], [1804.00051].

The overlap (reproducing kernel) is
\[
\langle \zeta',k | \zeta,k \rangle = (1 - |\zeta'|^2)^k (1-|\zeta|^2)^k (1 - \zeta'^*\zeta)^{-2k},
\]
supporting reproducing-kernel Hilbert space structures.

## 4. Expectation Values, Quadrature Squeezing, and Minimum Uncertainty

Expectation values for the generators in $\lvert \zeta; k \rangle$ are
\[
\langle K_0 \rangle = k\, \frac{1 + |\zeta|^2}{1-|\zeta|^2},\qquad
\langle K_- \rangle = 2k\, \frac{\zeta}{1-|\zeta|^2},\qquad
\langle K_+ \rangle = 2k\, \frac{\zeta^*}{1-|\zeta|^2}.
\]
For Hermitian quadratures $K_1 = (K_+ + K_-)/2$, $K_2 = (K_+ - K_-)/(2i)$,
\[
\Delta K_1\;\Delta K_2 = \frac{1}{2}\,|\langle K_0\rangle|,
\]
so these states saturate the generalized Heisenberg–Robertson–Schrödinger lower bound (minimum-uncertainty states for the su(1,1) algebra) [1212.6888], [1804.00051], [2304.08031].

Quadrature squeezing arises as the variances in $K_1$ or $K_2$ fall below the conventional coherent-state (vacuum) value for suitable loci in $(|\zeta|,\arg\zeta)$ [2304.08031].

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

On the Poincaré disk parameterized by $\zeta = e^{i\theta}\tanh r$ with $|\zeta|<1$, a Perelomov coherent state is centered at $\zeta$. The SU(1,1) (hyperbolic) phase-space measure involves $d^2\zeta/(1-|\zeta|^2)^2$.

The SU(1,1) Wigner function $W_\rho(\zeta) = \mathrm{Tr}[\rho\,\Pi(\zeta)]$—where $\Pi(\zeta)$ 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 $1/\sqrt{k}$ for individual coherent states and $1/k$ for certain multipartite superpositions (n-component compass states with $n\geq 6$) [2207.12706], [2602.14752]. 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 ($K_- = a^2/2$, $K_+ = (a^\dagger)^2/2$, $K_0 = (a^\dagger a + 1/2)/2$), 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) [1607.06169], [1311.2843], [1411.1968], [1211.4162].

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) [2304.08031].

The photon counting distribution is negative binomial:
\[
P_n(\zeta) = | \langle k, n | \zeta; k \rangle |^2 = (1-|\zeta|^2)^{2k} \frac{\Gamma(n+2k)}{n!\, \Gamma(2k)} |\zeta|^{2n}.
\]
The Mandel parameter $Q$ indicates Poissonian and sub/super-Poissonian regimes, controlled by $k$ 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 $\Phi$ or a nonlinear f-deformation), introducing, for example, polynomial algebras or Pöschl–Teller–type potentials [1205.4401], [1407.8304], [1404.3277], [2508.21779]. The resulting states interpolate between Barut–Girardello eigenstates ($K_- |\xi;k\rangle_{\rm BG} = \xi |\xi;k\rangle_{\rm BG}$) 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 [2508.21779]. Higher-order superpositions (such as compass or circular states) provide isotropic sub-Planck resolution, advantageous for quantum sensing tasks [2602.14752].

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

| Property              | Formula/Description                                                                                           | Reference                  |
|-----------------------|--------------------------------------------------------------------------------------------------------------|----------------------------|
| Expansion             | $(1-|\zeta|^2)^k \sum_{n=0}^\infty \sqrt{\frac{\Gamma(n+2k)}{n! \Gamma(2k)}} \zeta^n |k,n\rangle$                  | [1212.6888], [2304.08031]  |
| Overlap               | $(1-|\zeta'|^2)^k(1-|\zeta|^2)^k(1-\zeta'^*\zeta)^{-2k}$                                                     | [1212.6888]                |
| Resolution of Unity   | $\int_{|\zeta|<1} |\zeta; k\rangle\langle \zeta; k|\, \frac{2k-1}{\pi}(1-|\zeta|^2)^{-2} d^2\zeta = I$        | [1212.6888], [1607.06169]  |
| Min-uncertainty       | $\Delta K_1 \Delta K_2 = \frac{1}{2} |\langle K_0 \rangle|$                                                  | [1212.6888], [1804.00051]  |
| Negative binomial stats| $P_n(\zeta) = (1-|\zeta|^2)^{2k} \frac{(2k)_n}{n!} |\zeta|^{2n}$                                            | [2304.08031]               |
| Sub-Planck structure  | Superpositions lead to phase-space tiles of area $1/k^2$ (compass state, $n \geq 6$)                        | [2207.12706], [2602.14752] |

## References

- "SU(1,1) Nonlinear Coherent States" [1212.6888]
- "SU(1,1)–displaced coherent states, photon counting and squeezing" [2304.08031]
- "Finer sub-Planck structures and displacement sensitivity of SU(1,1) circular states" [2602.14752]
- "Sub-Planck phase-space structure and sensitivity for SU(1,1) compass states" [2207.12706]
- "Quantum Phase Sensitivity with Generalized Coherent States Based on Deformed su(1,1) and Heisenberg Algebras" [2508.21779]
- "Coherent states for polynomial su(1,1) algebra and a conditionally solvable system" [1205.4401]
- "Algebraic and group treatments to nonlinear displaced number states and their nonclassicality features" [1407.8304]
- "Generalized su(1,1) coherent states for pseudo harmonic oscillator and their nonclassical properties" [1404.3277] 

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.

Source: https://www.emergentmind.com/topics/su-1-1-displaced-coherent-states