Fractals,coherent states and self-similarity induced noncommutative geometry

The self-similarity properties of fractals are studied in the framework of the theory of entire analytical functions and the q-deformed algebra of coherent states. Self-similar structures are related to dissipation and to noncommutative geometry in the plane. The examples of the Koch curve and logarithmic spiral are considered in detail. It is suggested that the dynamical formation of fractals originates from the coherent boson condensation induced by the generators of the squeezed coherent states, whose (fractal) geometrical properties thus become manifest. The macroscopic nature of fractals appears to emerge from microscopic coherent local deformation processes.     

Paraboson coherent states

It is known that the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) are equivalent to the defining (triple) relations of n pairs of paraboson operators b _i ^±. In particular, the “parabosons of order p” correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V(p) of osp(1|2n). Recently we constructed these representations V(p) giving the explicit actions of the osp(1|2n) generators. We apply these results for the n = 2 case in order to obtain “coherent state” representations of the paraboson operators.     

Hysteresis from dynamically pinned sliding states

We report a surprising hysteretic behavior in the dynamics of a simple one-dimensional nonlinear model inspired by the tribological problem of two sliding surfaces with a thin solid lubricant layer in between. In particular, we consider the frictional dynamics of a harmonic chain confined between two rigid incommensurate substrates which slide with a fixed relative velocity. This system was previously found, by explicit solution of the equations of motion, to possess plateaus in parameter space exhibiting a remarkable quantization of the chain center-of-mass velocity (dynamic pinning) solely determined by the interface incommensurability. Starting now from this quantized sliding state, in the underdamped regime of motion and in analogy to what ordinarily happens for static friction, the dynamics exhibits a large hysteresis under the action of an additional external driving force F_ext. A critical threshold value F_c of the adiabatically applied force F_ext is required in order to alter the robust dynamics of the plateau attractor. When the applied force is decreased and removed, the system can jump to intermediate sliding regimes (a sort of “dynamic” stick-slip motion) and eventually returns to the quantized sliding state at a much lower value of F_ext. Hysteretic behavior is also observed as a function of the external driving velocity.     

Multi-matrix vector coherent states

A class of vector coherent states is derived with multiple of matrices as vectors in a Hilbert space, where the Hilbert space is taken to be the tensor product of several other Hilbert spaces. As examples vector coherent states with multiple of quaternions and octonions are given. The resulting generalized oscillator algebra is briefly discussed. Further, vector coherent states for a tensored Hamiltonian system are obtained by the same method. As particular cases, coherent states are obtained for tensored Jaynes-Cummings type Hamiltonians and for a two-level two-mode generalization of the Jaynes-Cummings model.     

Quanta and coherent states

The quantum harmonic oscillator can be considered as a composite system of indistinguishable Bose-Einstein symmetric two-level-systems (quanta). In analogy to the classical Poisson limit theorem, we show that a coherent state is the limit of a sequence of homogeneous product states (coherent spin states) and discuss statistical properties of the quanta in classical and nonclassical states.     

Superposed graphene coherent states

Using an annihilation operator, coherent states related to the electron of graphene layer placed in a magnetic field, can be obtained. In this paper, we define even and odd superposed graphene coherent states and then, we consider their entanglement, squeezing and statistical properties. To study the entanglement, we use concurrence. The results show that odd superposed graphene coherent states are maximally entangled states for all values of coherence parameter. However, the entanglement of graphene coherent states and also even superposed depend on the coherence parameter. In addition, examining the Mandel parameter shows sub-Poissonian statistics for graphene coherent states and their odd superposition; while, even superposed states do not show sub-Poissonian statistics at all. Also, we find that graphene coherent states and even superposition may be squeezed while the odd states do not show squeezing.     

Cluster-type entangled coherent states

We present the cluster-type entangled coherent states (CTECS) and discuss their properties. A cavity QED generation scheme using suitable choices of atom-cavity interactions, obtained via detunings adjustments and the application of classical external fields, is also presented. After the realization of simple atomic measurements, CTECS representing nonlocal electromagnetic fields in separate cavities can be generated.     

Quantum coherent operators: A generalization of coherent states

We introduce a technique to compare different, but related, quantum systems, thereby generalizing the way that coherent states are used to compare quantum systems to classical systems in semiclassical analysis. We then use this technique to estimate the dependence of the free energy of the quantum Heisenberg model on the spin value, and to estimate the relation between the ferromagnetic and antiferromagnetic free energies.Work supported in part by the U.S. National Science Foundation grant PHY-9019433.Work supported in part by the U.S. National Science Foundation grant DMS-9002416.     

Nonclassical properties of coherent states

It is demonstrated that a weak measurement of the squared quadrature observable may yield negative values for coherent states. This result cannot be reproduced by a classical theory where quadratures are stochastic c-numbers. The nonclassicality of coherent states can be associated with negative values of the Terletsky–Margenau–Hill distribution.     

Decoupled bands and antialigned states: An illustrative example

We suggest that the properties of decoupled bands associated with high -j single-particle orbitals are similar to those of $\text{K}{}^{\mathrm{\pi }}\text{=}\text{1}\text{2}{}^{\mathrm{?}}$ bands associated with $\text{j =}\text{1}\text{2}$ orbitals. Special attention is focussed on the so-called antialigned states. As an illustrative example we have studied the $\text{K}{}^{\mathrm{\pi }}\text{=}\text{1}\text{2}{}^{\mathrm{?}}$ bands in ¹⁹F and ²³Na, where the level energies and electromagnetic transition rates are evaluated in a simple weak-coupling model.     

Painlevé IV coherent states

A simple way to find solutions of the Painlevé IV equation is by identifying Hamiltonian systems with third-order differential ladder operators. Some of these systems can be obtained by applying supersymmetric quantum mechanics (SUSY QM) to the harmonic oscillator. In this work, we will construct families of coherent states for such subset of SUSY partner Hamiltonians which are connected with the Painlevé IV equation. First, these coherent states are built up as eigenstates of the annihilation operator, then as displaced versions of the extremal states, both involving the related third-order ladder operators, and finally as extremal states which are also displaced but now using the so called linearized ladder operators. To each SUSY partner Hamiltonian corresponds two families of coherent states: one inside the infinite subspace associated with the isospectral part of the spectrum and another one in the finite subspace generated by the states created through the SUSY technique.     

A note on coherent states

Given a connected Lie groupG with an Abelian invariant Lie subgroup and a continuous unitary representation ofG on the Hilbert space ?, we investigate a relationship between the first cohomology groupH ¹(G, ?) and classes of sectors, determined by coherent states with a projectivelyG-covariant Weyl system. This result is applied to calculateH ¹(G, ?), if the groupG has in addition a compact subgroup with certain properties.     

Entangled coherent states under dissipation

We study the evolution of entangled coherent states of the two quantized electromagnetic fields under dissipation. Characteristic time scales for the decay of the negativity are found in the case of large values of the phase space distance among the states of each mode. We also study how the entanglement emerges among the reservoirs.     

Soliton equations and coherent states

The τ-functions, which represent the totality of solutions for hierarchies of equations in soliton theory, are identified with the coherent states of the infinite dimensional Lie algebra gl(∞). The associated quantum system can be realized by an infinite set of harmonically interacting fermionic modes. The soliton dynamical evolution is thus mapped into a quantum hamiltonian evolution, and the latter back into a classical hamiltonian flow corresponding to a succession of infinitesimal contact Bäcklund transformations.     

Realistic teleportation of coherent states using squeezed vacuum states

This paper presents a realistic scheme for the teleportation of coherent states in which a two-mode squeezed vacuum state serves as the quantum channel and the position-sum and momentum-difference of two local modes serve as the measuring observables. The average fidelity of the teleportation of coherent states is derived for finite squeezing parameters and it turns out that fidelity greater than 1/2 cannot be achieved by using a classical channel alone and the probability distribution of the measurement result is a Gaussian distribution around the unknown parameter of the input coherent state with a width given by the squeezing parameter.     

Geometric phases and coherent states

A cyclic evolution of a pure quantum state is characterized by a closed curve γ in the projective Hilbert space , equipped with the Fubini-Study geometry. It is known that the geometric phase for this evolution is given by the integral of the symplectic form of the Fubini-Study geometry over an arbitrary surface spanning γ. This result extends to an infinite-dimensional Hilbert space for a bosonic quantum field. We prove that is bounded above by the infimum area over all surfaces spanning γ, and that the bound is attained if γ can be spanned by a holomorphic curve. Using an earlier result concerning the intrinsic Euclidean geometry of the coherent state submanifold , we derive an expression for the geometric phase for a cyclic evolution amongst coherent states. We indicate how the intensity of a classical configuration can be inferred from the winding number of the exponential geometric phase about the origin in the complex plane. In the case of photon states we present group theoretic and 2-component spinor representations of . We derive an expression for in the case of a sequence of measurements such that the resulting states are coherent at each step, in terms of a sequence of projection operators. The situation in relation to some earlier experiments of Pancharatnam and Tomita–Chiao is explained.     

Communication with dynamically fluctuating states of light polarization

The fast, irregular, fluctuations of the state of polarization of light output from an erbium doper fiber ring laser are used to communicate digital information. The experiments illustrate the application of a dynamical encoding and information recovery scheme that is robust to perturbations of the communication channel, a standard single mode fiber. A fiber-optic polarization analyzer was used to measure and visualize the polarization dynamics at nanosecond time scales on the Poincaré sphere.