Allan I. Solomon

Complete controllability of quantum systems

Sufficient conditions for complete controllability of N-level quantum systems subject to a single control pulse that addresses multiple allowed transitions concurrently are established. The results are applied in particular to Morse and harmonic-oscillator systems, as well as some systems with degenerate energy levels. Controllability of these model systems is of special interest since they have many applications in physics, e.g., Morse and harmonic oscillators serve as models for molecular bonds, and the standard control approach of using a sequence of frequency-selective pulses to address a single transition at a time is either not applicable or only of limited utility for such systems.

Thermal entanglement in three-qubit Heisenberg models

We study pairwise thermal entanglement in three-qubit Heisenberg models and obtain analytic expressions for the concurrence. We find that thermal entanglement is absent from both the antiferromagnetic XXZ model, and the ferromagnetic XXZ model with anisotropy parameter Δ≥1. Conditions for the existence of thermal entanglement are discussed in detail, as is the role of degeneracy and the effects of magnetic fields on thermal entanglement and the quantum phase transition. Specifically, we find that the uniform magnetic field can induce entanglement in the antiferromagnetic XXX model, but cannot induce entanglement in the ferromagnetic XXX model.

New generalized coherent states

We construct a new family of boson coherent states using a specially designed function which is a solution of a functional equation de(q,x)/dx=e(q,qx) with 0⩽q⩽1 and e(q,0)=1. We use this function in place of the usual exponential to generate new coherent states |q,z〉 from the vacuum, which are normalized and continuous in their label z. These states allow the resolution of unity, and a corresponding weight function is furnished by the exact solution of the associated Stieltjes moment problem. They also permit exact evaluation of matrix elements of an arbitrary polynomial given as a normally-ordered function of boson operators. We exemplify this by showing that the photon number statistics for these states is sub-Poissonian. For any q<1 the states |q,z〉 are squeezed; we obtain and discuss their signal to quantum noise ratio. The function e(q,x) allows a natural generation of multiboson coherent states of arbitrary multiplicity, which is impossible for the usual coherent states. For q=1 all the above resul...

The general boson normal ordering problem

Abstract We solve the boson normal ordering problem for F[(a † ) r a s ] , with r , s positive integers, [a,a † ]=1 , i.e., we provide exact and explicit expressions for its normal form N {F[(a † ) r a s ]} , where in N (F) all a s are to the right. The solution involves integer sequences of numbers which are generalizations of the conventional Bell and Stirling numbers whose values they assume for r = s =1. A comprehensive theory of such generalized combinatorial numbers is given including closed-form expressions (extended Dobinski-type formulas) and generating functions. These last are special expectation values in boson coherent states.

GENERALIZED Q-BOSONS AND THEIR SQUEEZED STATES

Generalized q-boson operators associated with the simultaneous creation of several q-bosons are introduced. These operators give rise to a realization of the quantum Weyl-Heisenberg algebra and are applied to the construction of corresponding Holstein-Primakoff realizations of the quantum algebras SUq(2) and SUq(1,1). Coherent states of these algebras are defined in the various ways suggested by the equivalent definitions of the harmonic oscillator coherent states, and some of their properties are studied. Particular attention is devoted to the squeezing properties of the quadrature of the electromagnetic field in these states.

A characteristic functional for deformed photon phenomenology

A characteristic functional for the description of the dynamics of deformed photons is introduced. This parameter σ(n), which takes the value 1 for conventional photons, is shown to play a determining role in describing phenomena in deformed quantum optics. We exemplify its use with specific reference to the following applications: (i) photon counting statistics; (ii) squeezing; (iii) signal-to-quantum noise ratio.

Degrees of controllability for quantum systems and application to atomic systems

Precise definitions for different degrees of controllability for quantum systems are given, and necessary and sufficient conditions for each type of controllability are discussed. The results are applied to determine the degree of controllability for various atomic systems with degenerate energy levels and transition frequencies.

Complete controllability of finite-level quantum systems

Complete controllability is a fundamental issue in the field of control of quantum systems, not least because of its implications for dynamical realizability of the kinematical bounds on the optimization of observables. In this paper we investigate the question of complete controllability for finite-level quantum systems subject to a single control field, for which the interaction is of dipole form. Sufficient criteria for complete controllability of a wide range of finite-level quantum systems are established and the question of limits of complete controllability is addressed. Finally, the results are applied to give a classification of complete controllability for four-level systems.

Mittag-Leffler coherent states

We create a family of boson coherent states using the functions of Mittag-Leffler (ML) E(z), (>0) and their generalizations E,(z), (,>0) instead of exponentials. These states are shown to satisfy the usual requirements of normalizability, continuity in the label and the resolution of unity with a positive weight function. This last quantity is found for arbitrary ,>0 as a solution of an associated Stieltjes moment problem. In addition, for = m = 1,2,3 ... and = 1 (corresponding to Em(z)) we propose and analyse special q-deformations (0<q1) of the functions Em(z) which serve as a tool to define q-deformed coherent states of ML type. We provide the expressions for expectation values of physical quantities for all the above states. We discuss physical properties of these states, noting that they are squeezed. The ML coherent states are sub-Poissonian in nature, whereas the q-deformed ML states can be sub- and super-Poissonian depending on q. All these states are shown to be eigenstates of deformed boson operators whose commutation relations are given.

Combinatorics and Boson normal ordering: A gentle introduction

We discuss a general combinatorial framework for operator ordering problems by applying it to the normal ordering of the powers and exponential of the boson number operator. The solution of the problem is given in terms of Bell and Stirling numbers enumerating partitions of a set. This framework reveals several inherent relations between ordering problems and combinatorial objects, and displays the analytical background to Wick’s theorem. The methodology can be straightforwardly generalized from the simple example we discuss to a wide class of operators.