Analytic representations based on SU(1,1) coherent states and their applications

We consider two analytic representations of the SU(1,1) Lie group: the representation in the unit disk based on the SU(1,1) Perelomov coherent states and the Barut-Girardello representation based on the eigenstates of the SU(1,1) lowering generator. We show that these representations are related through a Laplace transform. A ``weak'' resolution of the identity in terms of the Perelomov SU(1,1) coherent states is presented which is valid even when the Bargmann index $k$ is smaller than one half. Various applications of these results in the context of the two-photon realization of SU(1,1) in quantum optics are also discussed.Comment: LaTeX, 15 pages, no figures, to appear in J. Phys. A. More information on http://www.technion.ac.il/~brif/science.htm

Two-Photon Algebra Eigenstates: A Unified Approach to Squeezing

We use the concept of the algebra eigenstates that provides a unified description of the generalized coherent states (belonging to different sets) and of the intelligent states associated with a dynamical symmetry group. The formalism is applied to the two-photon algebra and the corresponding algebra eigenstates are studied by using the Fock-Bargmann analytic representation. This formalism yields a unified analytic approach to various types of single-mode photon states generated by squeezing and displacing transformations.Comment: To appear in Annals of Physics, REVTeX with AMSsymbols, 27 pages, no figures. More information on http://www.technion.ac.il/~brif/science.htm

A Quantum Group Approach to some Exotic States in Quantum Optics

This subject of this thesis is the physical application of deformations of Lie algebras and their use in generalising some exotic quantum optical states. We begin by examining the theory of quantum groups and the q-boson algebras used in their representation theory. Following a review of the properties of conventional coherent states, we describe the extension of the theory to various deformed Heisenberg-Weyl algebras, as well as the q-deformations of su(2) and su(1,1). Using the Deformed Oscillator Algebra of Bonatsos and Daskaloyannis, we construct generalised deformed coherent states and investigate some of their quantum optical properties. We then demonstrate a resolution of unity for such states and suggest a way of investigating the geometric effects of the deformation. The formalism devised by Rembielinski et al is used to consider coherent states of the q-boson algebra over the quantum complex plane. We propose a new unitary operator which is a q-analogue of the displacement operator of conventional coherent state theory: This is used to construct q-displaced vacuum states which are eigenstates of the annihilation operator. Some quantum mechanical properties of these states are investigated and it is shown that they formally satisfy a Heisenberg-type minimum uncertainty relation. After briefly reviewing the theory of conventional squeezed states, we examine the various q-generalisations. We propose a q-analogue of the squeezed vacuum state, and use this in conjunction with the unitary q-displacement operator to construct a general q-squeezed state, parameterised by noncommuting variables.. It is shown that, like their conventional counterparts, such states satisfy the Robertson-Schrodinger Uncertainty Relation. We conclude with a brief discussion about the appearance of noncommuting variables in the states that have been considered

Low dimensional manifolds for exact representation of open quantum systems

Weakly nonlinear degrees of freedom in dissipative quantum systems tend to localize near manifolds of quasi-classical states. We present a family of analytical and computational methods for deriving optimal unitary model transformations based on representations of finite dimensional Lie groups. The transformations are optimal in that they minimize the quantum relative entropy distance between a given state and the quasi-classical manifold. This naturally splits the description of quantum states into quasi-classical coordinates that specify the nearest quasi-classical state and a transformed quantum state that can be represented in fewer basis levels. We derive coupled equations of motion for the coordinates and the transformed state and demonstrate how this can be exploited for efficient numerical simulation. Our optimization objective naturally quantifies the non-classicality of states occurring in some given open system dynamics. This allows us to compare the intrinsic complexity of different open quantum systems.Comment: Added section on semi-classical SR-latch, added summary of method, revised structure of manuscrip

Sub-Planck phase-space structure and sensitivity for SU(1,1) compass states

We investigate the sub-Planck-scale structures associated with the SU(1,1) group by establishing that the Planck scale on the hyperbolic plane can be considered as the inverse of the Bargmann index $k$. Our discussion involves SU(1,1) versions of Wigner functions, and the quantum-interference effect is easily visualized through plots of these Wigner functions. Specifically, the superpositions of four Perelomov SU(1,1) coherent states (compass state) yield nearly isotropic sub-Planck structures in phase space scaling as $\frac1{k}$ compared with $\frac1{\sqrt{k}}$ scaling for individual SU(1,1) coherent states and anisotropic quadratically improved scaling for superpositions of two SU(1,1) coherent states (cat state). We show that displacement sensitivity exhibits the same quadratic improvement to scaling.Comment: 15 pages, 7 figure

On the squeezed states for n observables

Three basic properties (eigenstate, orbit and intelligence) of the canonical squeezed states (SS) are extended to the case of arbitrary n observables. The SS for n observables X_i can be constructed as eigenstates of their linear complex combinations or as states which minimize the Robertson uncertainty relation. When X_i close a Lie algebra L the generalized SS could also be introduced as orbit of Aut(L^C). It is shown that for the nilpotent algebra h_N the three generalizations are equivalent. For the simple su(1,1) the family of eigenstates of uK_- + vK_+ (K_\pm being lowering and raising operators) is a family of ideal K_1-K_2 SS, but it cannot be represented as an Aut(su^C(1,1)) orbit although the SU(1,1) group related coherent states (CS) with symmetry are contained in it. Eigenstates |z,u,v,w;k> of general combination uK_- + vK_+ + wK_3 of the three generators K_j of SU(1,1) in the representations with Bargman index k = 1/2,1, ..., and k = 1/4,3/4 are constructed and discussed in greater detail. These are ideal SS for K_{1,2,3}. In the case of the one mode realization of su(1,1) the nonclassical properties (sub-Poissonian statistics, quadrature squeezing) of the generalized even CS |z,u,v;+> are demonstrated. The states |z,u,v,w;k=1/4,3/4> can exhibit strong both linear and quadratic squeezing.Comment: 25 pages, LaTex, 4 .pic and .ps figures. Improvements in text, discussion on generation scheme added. To appear in Phys. Script

Multiphoton Quantum Optics and Quantum State Engineering

We present a review of theoretical and experimental aspects of multiphoton quantum optics. Multiphoton processes occur and are important for many aspects of matter-radiation interactions that include the efficient ionization of atoms and molecules, and, more generally, atomic transition mechanisms; system-environment couplings and dissipative quantum dynamics; laser physics, optical parametric processes, and interferometry. A single review cannot account for all aspects of such an enormously vast subject. Here we choose to concentrate our attention on parametric processes in nonlinear media, with special emphasis on the engineering of nonclassical states of photons and atoms. We present a detailed analysis of the methods and techniques for the production of genuinely quantum multiphoton processes in nonlinear media, and the corresponding models of multiphoton effective interactions. We review existing proposals for the classification, engineering, and manipulation of nonclassical states, including Fock states, macroscopic superposition states, and multiphoton generalized coherent states. We introduce and discuss the structure of canonical multiphoton quantum optics and the associated one- and two-mode canonical multiphoton squeezed states. This framework provides a consistent multiphoton generalization of two-photon quantum optics and a consistent Hamiltonian description of multiphoton processes associated to higher-order nonlinearities. Finally, we discuss very recent advances that by combining linear and nonlinear optical devices allow to realize multiphoton entangled states of the electromnagnetic field, that are relevant for applications to efficient quantum computation, quantum teleportation, and related problems in quantum communication and information.Comment: 198 pages, 36 eps figure

Topics in Modern Quantum Optics

This is the written version of lectures presented at "The 17th Symposium on Theoretical Physics - Applied Field Theory", 29 June - 1 July, 1998, the Sangsan Mathematical Science Building, Seoul National University, Seoul, Korea.Comment: 97 pages, 23 figures, 187 references. Misprints corrected, most figures redrawn and references update