Phase space density representations in fluid dynamics

Phase space density representations of inviscid fluid dynamics were recently discussed by Abarbanel and Rouhi. Here it is shown that such representations may be simply derived and interpreted by means of the Liouville equation corresponding to the dynamical system of ordinary differential equations that describes fluid particle trajectories. The Hamiltonian and Poisson bracket for the phase space density then emerge as immediate consequences of the corresponding structure of the dynamics. For barotropic fluids, this approach leads by direct construction to the formulation presented by Abarbanel and Rouhi. Extensions of this formulation to inhomogeneous incompressible fluids and to fluids in which the state equation involves an additional transported scalar variable are constructed by augmenting the single-particle dynamics and phase space to include the relevant additional variable.     

Phase controllable dynamical localization of a quantum particle in a driven optical lattice

The Dunlap–Kenkre (DK) result states that dynamical localization of a driven quantum particle in a periodic lattice happens when the ratio of the field magnitude to the field frequency of the diagonal drive is a root of the ordinary Bessel function of order 0. This has been experimentally verified. A generalization of the DK result is presented here. The hitherto considered DK model contains only the diagonal forcing. In the present extended version of the DK model we consider both off-diagonal and diagonal driving fields with different frequencies and a definite relative phase between them. We analytically show that new dynamical localizations conditions exist where an important role is played by the relative phase. In appropriate limits our results reduce to DK result.     

Natural star-products on symplectic manifolds and related quantum mechanical operators

In this paper is considered a problem of defining natural star-products on symplectic manifolds, admissible for quantization of classical Hamiltonian systems. First, a construction of a star-product on a cotangent bundle to an Euclidean configuration space is given with the use of a sequence of pair-wise commuting vector fields. The connection with a covariant representation of such a star-product is also presented. Then, an extension of the construction to symplectic manifolds over flat and non-flat pseudo-Riemannian configuration spaces is discussed. Finally, a coordinate free construction of related quantum mechanical operators from Hilbert space over respective configuration space is presented.     

An atomtronics transistor for quantum gates

We present a mechanism for quantum gates where the qubits are encoded in the population distribution of two-component ultracold atoms trapped in a species-selective triple-well potential. The gate operation is a specific application of a different design for an atomtronics transistor where inter-species interaction is used to control transport, and can be realized with either individual atoms or aggregates like Bose–Einstein condensates (BEC). We demonstrate the operational principle with a static external potential, and show feasible implementation with a smooth dynamical potential.     

Low depth quantum circuits for Ising models

A scheme for measuring complex temperature partition functions of Ising models is introduced. Two applications of this scheme are presented. First, through appropriate Wick rotations, those amplitudes can be analytically continued to yield estimates for partition functions of Ising models. Bounds on the estimated error are provided through a central-limit theorem whose validity extends beyond the present context; it holds for example for estimations of the Jones polynomial. The kind of state preparations and measurements involved in this application can be made independent of the system size or the parameters of the system being simulated. Second, the scheme allows to accurately estimate non-trivial invariants of links. Another result concerns the computational power of estimations of partition functions for real temperature classical ferromagnetic Ising models. We provide conditions under which estimating such partition functions allows to reconstruct scattering amplitudes of quantum circuits, making the problem BQP-hard. We also show fidelity overlaps for ground states of quantum Hamiltonians, which serve as a witness to quantum phase transitions, can be estimated from classical Ising model partition functions. Finally, we discuss how accurate corner magnetisation measurements on thermal states of two-dimensional Ising models lead to fully polynomial random approximation schemes (FPRAS) for the partition function.     

Canonical transformations in quantum mechanics

This paper presents the general theory of canonical transformations of coordinates in quantum mechanics. First, the theory is developed in the formalism of phase space quantum mechanics. It is shown that by transforming a star-product, when passing to a new coordinate system, observables and states transform as in classical mechanics, i.e., by composing them with a transformation of coordinates. Then the developed formalism of coordinate transformations is transferred to a standard formulation of quantum mechanics. In addition, the developed theory is illustrated on examples of particular classes of quantum canonical transformations.     

Phase space analysis of metamaterial-based optical systems

Phase space analysis of light refraction in optical systems consisting of slabs or thin lenses from either metamaterials with negative refractive indices or common materials is performed with the aim of finding the conditions of perfect imaging for metamaterial-based optical systems. The analysis in the paraxial approximation uses ABCD matrices, whereas full ray tracing is employed in the non-paraxial case. The phase space analysis reveals that the ideality of planar metamaterial lenses only occurs when the absolute value of the refractive index in metamaterials is the same as in the surrounding medium.     

Numerical representation of quantum states in the positive-P and Wigner representations

Numerical stochastic integration is a powerful tool for the investigation of quantum dynamics in interacting many-body systems. As with all numerical integration of differential equations, the initial conditions of the system being investigated must be specified. With application to quantum optics in mind, we show how various commonly considered quantum states can be numerically simulated by the use of widely available Gaussian and uniform random number generators. We note that the same methods can also be applied to computational studies of Bose–Einstein condensates, and give some examples of how this can be done.     

Spin dynamics and quantum transport in magnetic semiconductor quantum structures

Quantum structures derived from magnetic semiconductors serve as a powerful arena within which to study the interplay between quantum electronics and thin film magnetism. In particular, the semiconductor aspects of these flexible systems allow direct access to the electronic spin degrees of freedom using both magneto-optical as well as magneto-transport probes. Here we provide an overview of recent developments in the experimental study of II–VI magnetic semiconductor quantum structures, with particular emphasis on the dynamical behavior of field-tunable electronic spin states and spin-dependent quantum transport.     

Distributed qutrit-qutrit entanglement via quantum Zeno dynamics

A scheme is proposed to generate a three-dimensional entangled state for two atoms trapped in two distant cavities via quantum Zeno dynamics. We study such system in the regime of resonant atom-field interactions, which makes the scheme work very fast. We also investigate the influence of dissipation, due to atomic spontaneous emission and photon leakage, on the qutrit-qutrit entanglement.     

Decoherence of quantum Brownian motion in noncommutative space

The decoherence of a harmonic oscillator under two-dimensional quantum Brownian motion on a noncommutative plane is investigated. The interaction with the environment is considered by two separate models so-called coupled and uncoupled. The two-dimensional master equation and its noncommutative counterpart are derived for both employed models. The rate of the linear entropy (predictability sieve) is chosen as a criterion to investigate the purity in the presence of the space noncommutativity. Besides, a two-dimensional charged harmonic oscillator on a plane which is imposed by a perpendicular magnetic field is introduced as a realization of our model. Therefore, our approach provides a formalism to investigate the influence of the magnetic field on the decoherence of the pure states. We show that in the high magnetic field limit the rate of the decoherence will be decreased.     

Picturing quantum phase transitions

We discuss the quantum phase transitions (QPT) in N-spin chains from the point of view of collective observables. We show that the measurement space representation is a convenient tool for the analysis of phase transitions, allowing the determination of an appropriate set of macroscopic order parameters (for a given Hamiltonian). Quantum correlations in the vicinity of the critical points are analyzed both in the ground states and low temperature thermal states.     

An alternative formulation of Hall effect and quantum phases in noncommutative space

A recent method of constructing quantum mechanics in noncommutative coordinates, alternative to implying noncommutativity by means of star product is discussed. Within this approach we study Hall effect as well as quantum phases in noncommutative coordinates. The θ-deformed phases which we obtain are velocity independent.     

On the relation between thermostatted ring-polymer molecular dynamics and exact quantum dynamics

Here we obtain the explicit difference in the propagator between thermostatted ring-polymer molecular dynamics (TRPMD) and Matsubara dynamics, a recently derived form of linearisation which conserves the quantum Boltzmann distribution. Examination of this approximation leads to the new results that the TRPMD force on the centroid is identical to the Matsubara force on the centroid, and that (in a harmonic potential) the friction matrix can be chosen to produce either the correct oscillation frequency of the higher ring-polymer normal modes or the correct maximum in their position spectrum. This is illustrated with the position-squared autocorrelation function where TRPMD improves upon other similar methods. However, no physical choice of friction resolves qualitatively incorrect fluctuation dynamics at barriers. These results are broadly consistent with previous numerical studies and advise the use of TRPMD for spectra.     

广义线性量子变换在相空间的新表示

用坐标和动量算子作为基本算子给出广义线性量子变换在相位空间的新表示,具体包括变换关系式、变换算子的幺正条件、变换算子的普通表达形式和正、反正规乘积形式.     

Generalized Weyl quantization on the cylinder and the quantum phase

Generalized Weyl quantization formalism for the cylindrical phase space S¹×R¹$S^1\times {\mathbb{R}}^1$ is developed. It is shown that the quantum observables relevant to the phase of the linear harmonic oscillator or electromagnetic field can be represented within this formalism by the self-adjoint operators on the Hilbert space L²(S¹)$L^2\left(S^1\right)$.