Coherent states and geodesics: cut locus and conjugate locus

The intimate relationship between coherent states and geodesics is pointed out. For homogenous manifolds on which the exponential from the Lie algebra to the Lie group equals the geodesic exponential, and in particular for symmetric spaces, it is proved that the cut locus of the point $0$ is equal to the set of coherent vectors orthogonal to $\vert 0>$. A simple method to calculate the conjugate locus in Hermitian symmetric spaces with significance in the coherent state approach is presented. The results are illustrated on the complex Grassmann manifold.Comment: 19 pages, enlarged version, 14 pages, Latex + some macros from Revtex + some AMS font

A holomorphic representation of the Jacobi algebra

A representation of the Jacobi algebra $\mathfrak{h}_1\rtimes \mathfrak{su}(1,1)$ by first order differential operators with polynomial coefficients on the manifold $\mathbb{C}\times \mathcal{D}_1$ is presented. The Hilbert space of holomorphic functions on which the holomorphic first order differential operators with polynomials coefficients act is constructed.Comment: 34 pages, corrected typos in accord with the printed version and the Errata in Rev. Math. Phys. Vol. 24, No. 10 (2012) 1292001 (2 pages) DOI: 10.1142/S0129055X12920018, references update

The Jacobi group and the squeezed states - some comments

The generalized coherent states attached to the Jacobi group realize the squeezed states. Imposing hermitian conjugacy to the generators of the Jacobi algebra, we find out the form of the weight function appearing in the scalar product. We show effectively the orthonormality of the base functions with respect to the scalar product. From the explicit form of the reproducing kernel, we find out the expression of the multiplier in a holomorphic representation of the Jacobi group.Comment: 9 pages, to appear in AIP Conference Proceedings, Geometric Methods In Physics, Bialowieza (Poland), June 28 -July 4 2008, Editor(s): P. Kielanowski, S. T. Ali, A. Odzijewicz, M. Schlichenmaier, Th. Vorono

A convenient coordinatization of Siegel-Jacobi domains

We determine the homogeneous K\"ahler diffeomorphism $FC$ which expresses the K\"ahler two-form on the Siegel-Jacobi ball \mc{D}^J_n=\C^n\times \mc{D}_n as the sum of the K\"ahler two-form on \C^n and the one on the Siegel ball \mc{D}_n. The classical motion and quantum evolution on \mc{D}^J_n determined by a hermitian linear Hamiltonian in the generators of the Jacobi group G^J_n=H_n\rtimes\text{Sp}(n,\R)_{\C} are described by a matrix Riccati equation on \mc{D}_n and a linear first order differential equation in z\in\C^n, with coefficients depending also on W\in\mc{D}_n. $H_n$ denotes the $(2n+1)$-dimensional Heisenberg group. The system of linear differential equations attached to the matrix Riccati equation is a linear Hamiltonian system on \mc{D}_n. When the transform $FC:(\eta,W)\rightarrow (z,W)$ is applied, the first order differential equation in the variable \eta=(\un-W\bar{W})^{-1}(z+W\bar{z})\in\C^n becomes decoupled from the motion on the Siegel ball. Similar considerations are presented for the Siegel-Jacobi upper half plane \mc{X}^J_n=\C^n\times\mc{X}_n, where \mc{X}_n denotes the Siegel upper half plane.Comment: 32 pages, corrected typos, Latex, amsart, AMS font

Drag in a resonantly driven polariton fluid

We study the linear response of a coherently driven polariton fluid in the pump-only configuration scattering against a point-like defect and evaluate analytically the drag force exerted by the fluid on the defect. When the system is excited near the bottom of the lower polariton dispersion, the sign of the interaction-renormalised pump detuning classifies the collective excitation spectra into three different categories (Ciuti and Carusotto 2005 Phys. Status Solidi b 242 2224): linear for zero, diffusive-like for positive and gapped for negative detuning. We show that both cases of zero and positive detuning share a qualitatively similar crossover of the drag force from the subsonic to the supersonic regime as a function of the fluid velocity, with a critical velocity given by the speed of sound found for the linear regime. In contrast, for gapped spectra, we find that the critical velocity exceeds the speed of sound. In all cases, the residual drag force in the subcritical regime depends on the polariton lifetime only. Also, well below the critical velocity, the drag force varies linearly with the polariton lifetime, in agreement with previous work (Cancellieri et al 2010 Phys. Rev. B 82 224512), where the drag was determined numerically for a finite-size defect