Class Of Quantum Systems Research Articles

Systematic construction of non‐autonomous Hamiltonian equations of Painlevé‐type. III. Quantization

AbstractThis is the third article in our series of articles exploring connections between dynamical systems of Stäckel‐type and of Painlevé‐type. In this article, we present a method of deforming of minimally quantized quasi‐Stäckel Hamiltonians, considered in Part I to self‐adjoint operators satisfying the quantum Frobenius condition, thus guaranteeing that the corresponding Schrödinger equations possess common, multitime solutions. As in the classical case, we obtain here both magnetic and nonmagnetic families of systems. We also show the existence of multitime‐dependent quantum canonical maps between both classes of quantum systems.

A quantum prediction as a collection of epistemically restricted classical predictions

Spekkens has introduced an epistemically restricted classical theory of discrete systems, based on discrete phase space. The theory manifests a number of quantum-like properties but cannot fully imitate quantum theory because it is noncontextual. In this paper we show how, for a certain class of quantum systems, the quantum description of an experiment can be decomposed into classical descriptions that are epistemically restricted, though in a different sense than in Spekkens' work. For each aspect of the experiment—the preparation, the transformations, and the measurement—the epistemic restriction limits the form of the probability distribution an imagined classical observer may use. There are also global constraints that the whole collection of classical descriptions must satisfy. Each classical description generates its own prediction regarding the outcome of the experiment. One recovers the quantum prediction via a simple but highly nonclassical rule: the "nonrandom part" of the predicted quantum probabilities is obtained by summing the nonrandom parts of the classically predicted probabilities. By "nonrandom part" we mean the deviation from complete randomness, that is, from what one would expect upon measuring the fully mixed state.

Theory of dipole insulators

Insulating systems are characterized by their insensitivity to twisted boundary conditions as quantified by the charge stiffness and charge localization length. The latter quantity was shown to be related to the expectation value of the many-body position operator and serves as a universal criterion to distinguish between metals and insulators. In this work we extend these concepts to a new class of quantum systems having conserved charge and dipole moments. We refine the concept of a charge insulator by introducing notions of multipolar insulators, e.g., a charge insulator could be a dipole insulator or dipole metal. We develop a universal criterion to distinguish between these phases by extending the concept of charge stiffness and localization to analogous versions for multipole moments but with our focus on dipoles. We are able to relate the dipole localization scale to the expectation value of a recently introduced many-body quadrupole operator. This refined structure allows for the identification of phase transitions where charge remains localized but, e.g., dipoles delocalize. We illustrate the proposed criterion using several exactly solvable models that exemplify these concepts and discuss a possible realization in cold-atom systems.

On Con-Numbers and Con-Matrices with Applications to Control Systems

In this paper, the concepts of con-numbers and con-matrices are proposed by the introduction of conjugate operators into the field of complex numbers, and some properties of these two concepts are derived. In addition, two potential applications of these two concepts are addressed in details. Specifically, a class of quantum systems is expressed in terms of the proposed con-numbers, and the ℝ-linear mapping is represented by con-matrices. These two applications imply the importance of the con-numbers and con-matrices. In addition, some methods are discussed on the construction of the conjugate-operator-induced numbers. Also, some further research directions are provided on these extended numbers and matrices with their applications to systems control.

String-net models withZNfusion algebra

We study the Levin-Wen string-net model with a $Z_N$ type fusion algebra. Solutions of the local constraints of this model correspond to $Z_N$ gauge theory and double Chern-simons theories with quantum groups. For the first time, we explicitly construct a spin-$(N-1)/2$ model with $Z_N$ gauge symmetry on a triangular lattice as an exact dual model of the string-net model with a $Z_N$ type fusion algebra on a honeycomb lattice. This exact duality exists only when the spins are coupled to a $Z_N$ gauge field living on the links of the triangular lattice. The ungauged $Z_N$ lattice spin models are a class of quantum systems that bear symmetry-protected topological phases that may be classified by the third cohomology group $H^3(Z_N,U(1))$ of $Z_N$. Our results apply also to any case where the fusion algebra is identified with a finite group algebra or a quantusm group algebra.

Gamow Vectors and Borel Summability in a Class of Quantum Systems

We analyze the detailed time dependence of the wave function ψ(x,t) for one dimensional Hamiltonians \(H=-\partial_{x}^{2}+V(x)\) where V (for example modeling barriers or wells) and ψ(x,0) are compactly supported.

Controllability of quantum systems with switching control

This article focuses on wavefunction controllability of a class of quantum systems with switching control. First, we present a specific quantum system which is not wavefunction controllable under the action of any one of two controllers with different fixed structures. However, we prove that the system is wavefunction controllable if the control input is allowed to switch between the two controllers. Then we present a sufficient condition for wavefunction controllability of a class of quantum systems with switching control and briefly discuss the problem of control design for such quantum control systems. Finally, two illustrative examples are presented to demonstrate a possible control design process for the proposed method.

Quantum discord in matrix product systems

We consider a class of quantum systems with spin-flip symmetry and derive the quantum correlation measured by the quantum discord (QD). As an illustration, we investigate the QD in a three-body interaction model and an XYZ interaction model, whose ground states can be expressed as matrix product states, and the QD is exactly soluble. We show that the QD behaves differently than the quantum entanglement (QE) in many ways; for example, they may show opposite monotonicity and completely different finite-size effects. Furthermore, we compare the capability of the QD and the QE to detect quantum phase transitions (QPTs) and find that the QD is more reliable than the QE for signaling QPTs in these models: In the three-body interaction model, the QE is singular at the quantum critical point, however, it exhibits an additional singularity in the noncritical region, while the analyticity of the QD can be used to identify the quantum critical point perfectly; and in the XYZ interaction model, the QE vanishes in the thermodynamic limit, thus losing its ability to detect QPTs, while the QD still functions very well.

On the notion of ergodicity for finite quantum systems

We show that the orginal definition of ergodicity of Boltzmann can be directly applied to finite quantum systems, such as those arising from the quantization of classical systems on a compact phase space. It yields a notion of quantum ergodicity strictly stronger than the Von Neumann one. As an example, we remark that the quantized hyperbolic symplectomorphisms (a particular case is the quantized Arnold cat) are ergodic in this sense.

Incoherent control of locally controllable quantum systems

An incoherent control scheme for state control of locally controllable quantum systems is proposed. This scheme includes three steps: (1) amplitude amplification of the initial state by a suitable unitary transformation, (2) projective measurement of the amplified state, and (3) final optimization by a unitary controlled transformation. The first step increases the amplitudes of some desired eigenstates and the corresponding probability of observing these eigenstates, the second step projects, with high probability, the amplified state into a desired eigenstate, and the last step steers this eigenstate into the target state. Within this scheme, two control algorithms are presented for two classes of quantum systems. As an example, the incoherent control scheme is applied to the control of a hydrogen atom by an external field. The results support the suggestion that projective measurements can serve as an effective control and local controllability information can be used to design control laws for quantum systems. Thus, this scheme establishes a subtle connection between control design and controllability analysis of quantum systems and provides an effective engineering approach in controlling quantum systems with partial controllability information.

Decomposition of scalar potentials of natural Hamiltonians into integrable and perturbative terms: a naive approach

The geometry of Stäckel systems is employed to find, for a given (classical) natural Hamiltonian, another natural Hamiltonian which is integrable by separation of variables and in some sense close to the first one. The study is intended as a first step for a perturbative analysis of a given Hamiltonian system. The method proposed here, still in development, is in large part coordinate independent and effective mainly on manifolds of constant curvature. Examples are given for the quadrupole field and Hénon-Heiles systems. Stäckel systems are associated with quadratic in the momenta first integrals which play a fundamental role in quantization of classical systems.

Topological order and topological entropy in classical systems

We show that the concept of topological order, introduced to describe ordered quantum systems which cannot be classified by broken symmetries, also applies to classical systems. Starting from a specific example, we show how to use pure state density matrices to construct corresponding thermally mixed ones that retain precisely half the original topological entropy, a result that we generalize to a whole class of quantum systems. Finally, we suggest that topological order and topological entropy may be useful in characterizing classical glassy systems.

Equivalent circuits for two‐state quantum systems

AbstractThe development of nanocircuits designed to ultimately interact with classical electromagnetic environments demands a description of their performance in circuit terms, i.e. in terms of voltages and currents for incident, reflected and transmitted waves. A class of quantum systems which are particularly promising for applications (e.g. in spintronics) and at the same time easy to analyse and to understand physically are the two‐state systems. Here, we present equivalent circuits for a generic two‐state quantum system imbedded in a one‐port or two‐port structure guiding a linearly polarized electromagnetic wave. For the sake of concreteness we explicitly refer to a system of spin ½ particles whose magnetic moments are aligned with the axis of the guiding structure by a strong constant magnetic field; but the formalism can be easily extended to particles endowed with electric dipole moments or any other system with two degrees of freedom. Copyright © 2007 John Wiley & Sons, Ltd.

Star products, duality and double Lie algebras

Quantization of classical systems using the star-product of symbols of observables is discussed. In the star-product scheme an analysis of dual structures is performed and a physical interpretation is proposed. At the Lie algebra level duality is shown to be connected to double Lie algebras. The analysis is specified to quantum tomography. The classical tomographic Poisson bracket is found.

Quantization of natural systems

New rules for the quantization of classical systems are proposed which generalize the traditional rules and reduce to these rules, provided that transition to the Cartesian coordinates is possible. An equation is suggested that offers generalization of the Schrodinger equation to arbitrary natural systems. Using the minimum coupling (strong equivalence) principle, this equation can be extended to arbitrary spaces with curvature.

Quantum electrodynamics of the dispersive dielectric medium–a path integral approach

The Feynman path integration method of quantization of classical systems provides a useful alternative to the canonical quantization scheme. In the present paper this method is applied for the first time to be quantization of the electromagnetic field in a linear, inhomogeneous, dispersive and absorptive dielectric medium, described in the framework of the Hopfield model. The medium is assumed to be linear, so that the corresponding functional integrations are of the Gaussian type and can be explicitly performed. With the use of local microscopic action as the starting point, the method allows us to obtain in a simple and natural way the (non-local in time) effective action describing properties of the electromagnetic field in the medium on the macroscopic level. Effective action can be obtained directly by elimination of the matter degrees of freedom, facilitated by functional integration over the matter fields, and leaving the electromagnetic field on the classical level. The macroscopic effective action of the electromagnetic field was further used to obtain an explicit expression for the frequency dependent dielectric constant, fulfilling the Kramers–Kronig relations. The relation between electric and displacement fields is on this level classical, that is it does not contain Langevin quantum noise term, characteristic for the quantum electromagnetic field in a dispersive and absorptive medium. Complete quantization of the model is performed by functional integration over both the matter and the electromagnetic fields. This gives an expression for the generating functional, and for the propagators (Green functions) of the quantized electromagnetic and matter polarization fields. The propagators fulfil Dyson-type equations, which in some cases (e.g. homogeneous medium) can be solved exactly. In the case of a homogeneous dielectric, expressions for the field operators are obtained from the propagators.

QUANTIZATION OF CLASSICAL CONSTANTS OF MOTION AND EQUILIBRIUM DISTRIBUTIONS IN QUANTUM MECHANICS

We distinguish a class of quantum systems (trigonometric and hyperbolic potentials) with an additional constant of motion unrelated to any symmetry. The various quantum constants of motion result from non-uniqueness of the quantization procedure. We discuss equilibrium distributions in quantum statistical mechanics determined by the constants of motion.

Central extensions and realizations of one-dimensional Galilean systems and quantization

The unitary irreducible realizations (representations up to a factor) of the maximal non-trivial central extension of the (1 + 1) Galilei group, (1 + 1), are obtained via the linear unitary irreducible representations of its maximal non-trivial central extension, (1 + 1). As an application we construct the Stratonovich - Weyl correspondence, which allows Moyal quantization of classical systems, for two cases of great physical interest: a system in a external variable force field and a variable-mass system.

Equilibrium in quantum systems of particles with magnetic interaction

We compute density matrices determining equilibrium states in the thermodynamic limit for a class of quantum systems. Interaction in these systems is described as a collective electromagnetic field whose scalar potential is equal to zero.

Fractal versus quasiclassical diffusive transport in a class of quantum systems.

We compare the properties of transmission across one-dimensional finite samples which are associated with two types of quantum diffusion, one related to a classical chaotic dynamics, the other to a multifractal energy spectrum. We numerically investigate models exhibiting one or both of these features, and we find in all cases an inverse power-law dependence of the average transmission on the sample length. Although in all the considered cases the quadratic spread of wave packets increases linearly (or very close to linearly) in time for both types of dynamics, a proper Ohmic dependence is always observed only in the case of quasiclassical diffusion. The analysis of the statistics of transmission fluctuations in the case of a fractal spectrum exposes some new features, which mark further differences from ordinary diffusion, and enforce the conclusion that the two types of transmission are intrinsically different.