Kaleidoscope of Classical Vortex Images and Quantum Coherent States (original) (raw)
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Vortex Images, q-Calculus and Entangled Coherent States
Journal of Physics: Conference Series, 2012
The two circles theorem for hydrodynamic flow in annular domain bounded by two concentric circles is derived. Complex potential and velocity of the flow are represented as q-periodic functions and rewritten in terms of the Jackson q-integral. This theorem generalizes the Milne-Thomson one circle theorem and reduces to the last on in the limit q → ∞. By this theorem problem of vortex images in annular domain between coaxial cylinders is solved in terms of q-elementary functions. An infinite set of images, as symmetric points under two circles, is determined completely by poles of the q-logarithmic function, where dimensionless parameter q = r 2 2 /r 2 1 is given by square ratio of the cylinder radii. Motivated by Möbius transformation for symmetrical points under generalized circle in complex plain, the system of symmetric spin coherent states corresponding to antipodal qubit states is introduced. By these states we construct the maximally entangled orthonormal two qubit spin coherent state basis, in the limiting case reducible to the Bell basis. Average energy of XYZ model in these states, describing finite localized structure with characteristic extremum points, appears as an energy surface in maximally entangled two qubit space. Generalizations to three and higher multiple qubits are found. We show that our entangled N qubit states are determined by set of complex Fibonacci and Lucas polynomials and corresponding Binet-Fibonacci q-calculus.
Quantum calculus of classical vortex images, integrable models and quantum states
Journal of Physics: Conference Series, 2016
From two circle theorem described in terms of q-periodic functions, in the limit q → 1 we have derived the strip theorem and the stream function for N vortex problem. For regular N-vortex polygon we find compact expression for the velocity of uniform rotation and show that it represents a nonlinear oscillator. We describe q-dispersive extensions of the linear and nonlinear Schrödinger equations, as well as the q-semiclassical expansions in terms of Bernoulli and Euler polynomials. Different kind of q-analytic functions are introduced, including the pq-analytic and the golden analytic functions.
Special functions with mod n symmetry and kaleidoscope of quantum coherent states
Journal of Physics: Conference Series
The set of mod n functions associated with primitive roots of unity and discrete Fourier transform is introduced. These functions naturally appear in description of superposition of coherent states related with regular polygon, which we call kaleidoscope of quantum coherent states. Displacement operators for kaleidoscope states are obtained by mod n exponential functions with operator argument and non-commutative addition formulas. Normalization constants, average number of photons, Heinsenberg uncertainty relations and coordinate representation of wave functions with mod n symmetry are expressed in a compact form by these functions.
A quantum group approach to some exotic states in quantum optics
1995
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...
Journal of Physics A: Mathematical and General, 2002
Following the discussion -in state space language -presented in a preceding paper, we work on the passage from the phase space description of a degree of freedom described by a finite number of states (without classical counterpart) to one described by an infinite (and continuously labeled) number of states. With that it is possible to relate an original Schwinger idea to the Pegg and Barnett approach to the phase problem. In phase space language, this discussion shows that one can obtain the Weyl-Wigner formalism, for both Cartesian and angular coordinates, as limiting elements of the discrete phase space formalism. PACS: 03.65.-w, 03.65.Bz, 03.65.Ca
Action-angle coherent states for quantum
2012
Quantum versions of cylindric phase space, like for the motion of a particle on the circle, are obtained through different families of coherent states. The latter are built from various probability distributions of the action variable. The method is illustrated with Gaussian distributions and uniform distributions on intervals, and resulting quantizations are explored.
Journal of Geometry and Physics, 2001
The manifold of pure quantum states can be regarded as a complex projective space endowed with the unitary-invariant Riemannian geometry of Fubini and Study. According to the principles of geometric quantum mechanics, the detailed physical characteristics of a given quantum system can be represented by specific geometrical features that are selected and preferentially identified in this complex manifold. In particular, any specific feature of projective geometry gives rise to a physically realisable characteristic in quantum mechanics. Here we construct a number of examples of such geometrical features as they arise in the state spaces for spin-1 2 , spin-1, and spin-3 2 systems, and for pairs of spin-1 2 systems. A study is undertaken on the geometry of entangled states, and a natural measure is assigned to the degree of entanglement of a given state for a general multi-particle system. The properties of this measure are analysed in detail for the entangled states of a pair of spin-1 2 particles, thus enabling us to determine the structure of the space of maximally entangled states. With the specification of a quantum Hamiltonian, the resulting Schrödinger trajectories induce an isometry of the Fubini-Study manifold. For a generic quantum evolution, the corresponding Killing trajectory is quasiergodic on a toroidal subspace of the energy surface. When the dynamical trajectory is lifted orthogonally to Hilbert space, it induces a geometric phase shift on the wave function. The uncertainty of an observable in a given state is the length of the gradient vector of the level surface of the expectation of the observable in that state, a fact that allows us to calculate higher order corrections to the Heisenberg relations. A general mixed state is determined by a probability density function on the state space, for which the associated first moment is the density matrix. The advantage of the idea of a general state is in its applicability in various attempts to go beyond the standard quantum theory, some of which admit a natural phase-space characterisation.
Two-circles theorem, q-periodic functions and entangled qubit states
Journal of Physics: Conference Series, 2014
For arbitrary hydrodynamic flow in circular annulus we introduce the two circle theorem, allowing us to construct the flow from a given one in infinite plane. Our construction is based on q-periodic analytic functions for complex potential, leading to fixed scale-invariant complex velocity, where q is determined by geometry of the region. Self-similar fractal structure of the flow with q-periodic modulation as solution of q-difference equation is studied. For one point vortex problem in circular annulus by fixing singular points we find solution in terms of q-elementary functions. Considering image points in complex plane as a phase space for qubit coherent states we construct Fibonacci and Lucas type entangled N-qubit states. Complex Fibonacci curve related to this construction shows reach set of geometric patterns. Γ 2πi 1 z−z 0. Residue at this singularity determines the vortex strength Γ = ud s.
Coherent States and Their Generalizations: A Mathematical Overview
Reviews in Mathematical Physics, 1995
We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.
Coherent States in Quantum Physics
Coherent States in Quantum Physics, 2009
Part One Coherent States 1 1 Introduction 3 1.1 The Motivations 3 2 The Standard Coherent States: the Basics 13 2.1 Schrödinger Definition 13 2.2 Four Representations of Quantum States 13 2.2.1 Position Representation 14 2.2.2 Momentum Representation 14 2.2.3 Number or Fock Representation 15 2.2.4 A Little (Lie) Algebraic Observation 16 2.2.5 Analytical or Fock-Bargmann Representation 16 2.2.6 Operators in Fock-Bargmann Representation 17 2.3 Schrödinger Coherent States 18 2.3.1 Bergman Kerne' as a Coherent State 18 2.3.2 A First Fundamental Property 19 2.3.3 Schrödinger Coherent States in the Two Other Representations 19 2.4 Glauber-Klauder-Sudarshan or Standard Coherent States 20 2.5 Why the Adjective Coherent? 20 3 The Standard Coherent States: the (Elementary) Mathematics 25 3.1 Introduction 25 3.2 Properties in the Hilbertian Framework 26 3.2.1 A "Continuity" from the Classical Complex Plane to Quantum States 26 3.2.2 "Coherent" Resolution of the Unity 26 3.2.3 The Interplay Between the Circle (as a Set of Parameters) and the Plane (as a Euclidean Space) 27 3.2.4 Analytical Bridge 28 3.2.5 Overcompleteness and Reproducing Properties 29 3.3 Coherent States in the Quantum Mechanical Context 30 3.3.1 Symbols 30 3.3.2 Lower Symbols 30