Transformations In Phase Space Research Articles

Implications of Coupling in Quantum Thermodynamic Machines

We study coupled quantum systems as the working media of thermodynamic machines. Under a suitable phase-space transformation, the coupled systems can be expressed as a composition of independent subsystems. We find that for the coupled systems, the figures of merit, that is the efficiency for engine and the coefficient of performance for refrigerator, are bounded (both from above and from below) by the corresponding figures of merit of the independent subsystems. We also show that the optimum work extractable from a coupled system is upper bounded by the optimum work obtained from the uncoupled system, thereby showing that the quantum correlations do not help in optimal work extraction. Further, we study two explicit examples; coupled spin- 1 / 2 systems and coupled quantum oscillators with analogous interactions. Interestingly, for particular kind of interactions, the efficiency of the coupled oscillators outperforms that of the coupled spin- 1 / 2 systems when they work as heat engines. However, for the same interaction, the coefficient of performance behaves in a reverse manner, while the systems work as the refrigerator. Thus, the same coupling can cause opposite effects in the figures of merit of heat engine and refrigerator.

Scale transformations in phase space and stretched states of a harmonic oscillator

Рассматриваются преобразования масштаба $(q,p)\to(\lambda q,\lambda p)$ в фазовом пространстве. Они индуцируют преобразования функций Хусими $H(q,p)$, заданных на этом пространстве. Рассматриваются функции Хусими состояний, которые являются произвольной суперпозицией $n$-частичных состояний гармонического осциллятора. Развит метод, позволяющий находить так называемые растянутые состояния, в которые преобразуются эти суперпозиции при таком преобразовании масштаба. Изучены их свойства. Вычислены в явном виде матрицы плотности этих растянутых состояний. Установлено, что структуру матриц плотности можно описать, используя отрицательные биномиальные распределения. Для растянутых состояний найдены выражения для их энергии и энтропии и вычислены средние значения оператора числа состояний. Показано, какой вид принимают для растянутых состояний соотношения неопределенностей Гейзенберга и Робертсона-Шредингера.

Scale transformations in phase space and stretched states of a harmonic oscillator

We consider scale transformations (q, p) → (λq, λp) in phase space. They induce transformations of the Husimi functions H(q, p) defined in this space. We consider the Husimi functions for states that are arbitrary superpositions of n-particle states of a harmonic oscillator. We develop a method that allows finding so-called stretched states to which these superpositions transform under such a scale transformation. We study the properties of the stretched states and calculate their density matrices in explicit form. We establish that the density matrix structure can be described using negative binomial distributions. We find expressions for the energy and entropy of stretched states and calculate the means of the number-ofstates operator. We give the form of the Heisenberg and Robertson–Schrodinger uncertainty relations for stretched states.

Nonlinear dynamics and band transport in a superlattice driven by a plane wave

A quantum particle transport induced in a spatially periodic potential by a propagating plane wave has a number of important implications in a range of topical physical systems. Examples include acoustically driven semiconductor superlattices and cold atoms in an optical crystal. Here we apply a kinetic description of the directed transport in a superlattice beyond standard linear approximation, and utilize exact path-integral solutions of the semiclassical transport equation. We show that the particle drift and average velocities have nonmonotonic dependence on the wave amplitude with several prominent extrema. Such nontrivial kinetic behavior is related to global bifurcations developing with an increase of the wave amplitude. They cause dramatic transformations of the system phase space and lead to changes of the transport regime. We describe different types of phase trajectories contributing to the directed transport and analyze their spectral content.

Maximally symmetric stabilizer MUBs in even prime-power dimensions

One way to construct a maximal set of mutually unbiased bases (MUBs) in a prime-power dimensional Hilbert space is by means of finite phase-space methods. MUBs obtained in this way are covariant with respect to some subgroup of the group of all affine symplectic phase-space transformations. However, this construction is not canonical: as a consequence, many different choices of covariance subgroups are possible. In particular, when the Hilbert space is 2n dimensional, it is known that covariance with respect to the full group of affine symplectic phase-space transformations can never be achieved. Here we show that in this case there exist two essentially different choices of maximal subgroups admitting covariant MUBs. For both of them, we explicitly construct a family of 2n covariant MUBs. We thus prove that, contrary to the odd dimensional case, maximally covariant MUBs are very far from being unique in even prime-power dimensions.

Canonical algorithms for numerical integration of charged particle motion equations

A technique for numerically integrating the equation of charged particle motion in a magnetic field is considered. It is based on the canonical transformations of the phase space in Hamiltonian mechanics. The canonical transformations make the integration process stable against counting error accumulation. The integration algorithms contain a minimum possible amount of arithmetics and can be used to design accelerators and devices of electron and ion optics.

Gyrokinetic Model beyond the Standard Ordering

AbstractAn energy‐conserving reduced kinetic model is formulated which is valid for perturbations with large amplitude and long wavelength beyond the standard gyrokinetic ordering. The standard gyrokinetic model is elegantly formulated by the two‐step phase space transformation consisting of the guiding‐center and gyro‐center transformations, but the amplitude of perturbations is assumed to be small in return for use of the two‐step transformation. Besides, the wavelength of perturbations is assumed to be short in deriving the gyrokinetic Poisson equation. Both limitations are relaxed by using the modified guiding‐center 1‐form for flowing plasmas [Miyato et al., J. Phys. Soc. Jpn. 78, 104501 (2009)]. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Entropy generation in Gaussian quantum transformations: applying the replica method to continuous-variable quantum information theory

AbstractIn spite of their simple description in terms of rotations or symplectic transformations in phase space, quadratic Hamiltonians such as those modelling the most common Gaussian operations on bosonic modes remain poorly understood in terms of entropy production. For instance, determining the quantum entropy generated by a Bogoliubov transformation is notably a hard problem, with generally no known analytical solution, while it is vital to the characterisation of quantum communication via bosonic channels. Here we overcome this difficulty by adapting the replica method, a tool borrowed from statistical physics and quantum field theory. We exhibit a first application of this method to continuous-variable quantum information theory, where it enables accessing entropies in an optical parametric amplifier. As an illustration, we determine the entropy generated by amplifying a binary superposition of the vacuum and a Fock state, which yields a surprisingly simple, yet unknown analytical expression.

Demonstrating nonclassicality and non-Gaussianity of single-mode fields: Bell-type tests using generalized phase-space distributions

We present Bell-type tests of nonclassicality and non-Gaussianity for single-mode fields employing a generalized quasiprobability function. Our nonclassicality tests are based on the observation that two orthogonal quadratures in phase space (position and momentum) behave as independent realistic variables for a coherent state. Taking four (three) points at the vertices of a rectangle (right triangle) in phase space, our tests detect every pure nonclassical Gaussian state and a range of mixed Gaussian states. These tests also set an upper bound for all Gaussian states and their mixtures, which thereby provide criteria for genuine quantum non-Gaussianity. We optimize the non-Gaussianity tests by employing a squeezing transformation in phase space that converts a rectangle (right triangle) to a parallelogram (triangle), which enlarges the set of non-Gaussian states detectable in our formulation. We address fundamental and practical limits of our generalized phase-space tests by looking into their relation with decoherence under a lossy Gaussian channel and their robustness against finite data and nonoptimal choice of phase-space points. Furthermore, we demonstrate that our parallelogram test can identify useful resources for nonlocality testing in phase space.

Noether theorem for non-conservative systems with time delay in phase space based on fractional model

In this paper, we focus on discussing the fractional Noether symmetries and fractional conserved quantities for non-conservative Hamilton system with time delay. Firstly, the fractional Hamilton canonical equations of non-conservative system with time delay are established; secondly, based upon the invariance of the fractional Hamilton action with time delay under the infinitesimal transformations of group, we obtain the definitions and criterion of fractional Noether symmetric transformations, fractional Noether quasi-symmetric transformations and fractional generalized Noether quasi-symmetric transformations in phase space; finally, the relationship between the fractional Noether symmetries and fractional conserved quantities with time delay in phase space is established. At the end of the paper, some examples are given to illustrate the application of the results.

Coherence conversion for optimized resolution in optical measurements – example of femtosecond time resolution using the transverse coherence of 100-picosecond X-rays

A way is proposed to obtain a femtosecond time resolution over a picosecond range in laser-pump, X-ray probe spectroscopic measurements where the light source and the detector are much slower than that. It is based on a phase-space transformation from the time/bandwidth to the spatial/wavenumber domain to match the coherence properties of synchrotron radiation to the requirements of femtosecond experiments. In a first step, the geometry of the laser incidence maps time, t, of laser-induced femtosecond dynamics to a spatial coordinate, x. Then, a far-field X-ray diffraction pattern, i.e. the optical Fourier transform, is obtained from the laser-induced modifications of the sample properties, including shifts of X-ray absorption edges and changes in crystallographic unit-cell form factors. Whereas the first step is similar to previously used schemes for femtosecond time resolution, the second one is substantially different with specific advantages discussed in the text. Key to this technique is that the modulus of the Fourier transform is invariant with respect to translations along x, which are due to the correlation. It can, therefore, be acquired in a simple intensity measurement with a slow detector. The phase, which does vary strongly with , is missing in the intensity data, but can be recovered through a heterodyning technique. Data from a demonstration experiment are presented. The same concept can be used to obtain attosecond time resolution with an X-ray free-electron laser.

Effective-one-body Hamiltonian with next-to-leading order spin-spin coupling

We propose a way of including the next-to-leading (NLO) order spin-spin coupling into an effective-one-body (EOB) Hamiltonian. This work extends [S. Balmelli and P. Jetzer, Phys. Rev. D 87, 124036 (2013)], which is restricted to the case of equatorial orbits and aligned spins, to general orbits with arbitrary spin orientations. This is done applying appropriate canonical phase-space transformations to the NLO spin-spin Hamiltonian in Arnowitt-Deser-Misner (ADM) coordinates, and systematically adding "effectiv" quantities at NLO to all spin-squared terms appearing in the EOB Hamiltonian. As required by consistency, the introduced quantities reduce to zero in the test- mass limit. We expose the result both in a general gauge and in a gauge-fixed form. The last is chosen such as to minimize the number of new coefficients that have to be inserted into the effective spin squared. As a result, the 25 parameters that describe the ADM NLO spin-spin dynamics get condensed into only 12 EOB terms.

Bubble chambers for experiments in nuclear astrophysics

A bubble chamber has been developed to be used as an active target system for low energy nuclear astrophysics experiments. Adopting ideas from dark matter detection with superheated liquids, a detector system compatible with γ-ray beams has been developed. This detector alleviates some of the limitations encountered in standard measurements of the minute cross-sections of interest to stellar environments. While the astrophysically relevant nuclear reaction processes at hydrostatic burning temperatures are dominated by radiative captures, in this experimental scheme we measure the time-reversed processes. Such photodisintegrations allow us to compute the radiative capture cross-sections when transitions to excited states of the reaction products are negligible. Due to the transformation of phase space, the photodisintegration cross-sections are up to two orders of magnitude higher. The main advantage of the new target-detector system is a density several orders of magnitude higher than conventional gas targets. Also, the detector is virtually insensitive to the γ-ray beam itself, thus allowing us to detect only the products of the nuclear reaction of interest. The development and the operation as well as the advantages and disadvantages of the bubble chamber are discussed.

Globally stabilizing state feedback control design for Lotka-Volterra systems based on underlying linear dynamics

Using translated X-factorable phase space transformations and nonlinear variable transformations a dynamically similar linear ODE model is associated to the Lotka-Volterra system models with a positive equilibrium point. This enables to use a linear full state feedback controller to stabilize the system in controllable cases, that leaves the open loop equilibrium point unchanged. The linear state feedback controller design problem in the general case has also been formulated to ensure the compartmental property of the closed loop system from which the existence of a diagonal Lyapunov function follows. Further extension has been obtained by using the time re-parametrization transformation defined for quasi-polynomial models.

Entropy-inspired Lyapunov Functions and Linear First Integrals for Positive Polynomial Systems

Two partially overlapping classes of positive polynomial systems, chemical reaction networks with mass action law (MAL-CRNs) and quasi-polynomial systems (QP systems) are considered. Both of them have an entropy-like Lyapunov function associated to them which are similar but not the same. Inspired by the work of Prof. Gorban [12] on the entropy-functionals for Markov chains, and using results on MAL-CRN and QP-systems theory we characterize MAL-CRNs and QP systems that enable both types of entropy-like Lyapunov functions. The starting point of the analysis is the class of linear weakly reversible MAL-CRNs that are mathematically equivalent to Markov chains with an equilibrium point where various entropy level set equivalent Lyapunov functions are available. We show that non-degenerate linear kinetic systems with a linear first integral (that corresponds to conservation) can be transformed to linear weakly reversible MAL-CRNs using linear diagonal transformation, and the coefficient matrix of this system is diagonally stable. This implies the existence of the weighted version of the various entropy level set equivalent Lyapunov functions for non-degenerate linear kinetic systems with a linear first integral. Using translated X-factorable phase space transformations and nonlinear variable transformations a dynamically similar linear ODE model is associated to the QP system models with a positive equilibrium point. The non-degenerate kinetic property together with the existence of positive equilibrium point form a sufficient condition of the existence of the weighted version of the various entropy level set equivalent Lyapunov functions in this case. Further extension has been obtained by using the time re-parametrization transformation defined for QP models.

Application of the phase space action principle to finite-size particle plasma simulations in the drift-kinetic approximation

We formulate a finite-size particle numerical model of strongly magnetized plasmas in the drift-kinetic approximation. We use the phase space action as an alternative to previous variational formulations based on Low’s Lagrangian or on a Hamiltonian with a non-canonical Poisson bracket. The useful property of this variational principle is that it allows independent transformations of particle coordinates and velocities, i.e., transformations in particle phase space. With such transformations, a finite degree-of-freedom drift-kinetic action is obtained through time-averaging of the finite degree-of-freedom fully-kinetic action. Variation of the drift-kinetic Lagrangian density leads to a self-consistent, macro-particles and fields numerical model. Since the computational particles utilize only guiding center coordinates and velocities, there is a large computational advantage in the time integration part of the algorithm. Numerical comparison between the time-averaged fully-kinetic and drift-kinetic charge and current, deposited on a computational grid, offers insight into the range of validity of the model. Being based on a variational principle, the algorithm respects the energy conserving property of the underlying continuous system. The development in this paper serves to further emphasize the advantages of using variational approaches in plasma particle simulations.

Дифузiї, породженi вiнеровим процесом

У роботi розглядається питання можливостi побудови дифузiйного процесу в $\mathbb R^m$ шляхом перетворення вiнерового процесу. Застосовуються гладке перетворення фазового простору та випадкова замiна часу з допомогою адитивного функцiоналу iнтегрального типу.

Phase space optics: an alternate approach to freeform optical systems

Freeform optical surfaces are of increasing interest in the design of modern imaging and illumination systems. However for the optical designer there are still methods missing in order to illustrate and quantify the effect of freeform surfaces in terms of aberrations or their impact on illumination performance. Phase space optics might offer an alternative way to understand freeform optics and their impact within optical systems. To prove this, phase space in geometrical optics is generally introduced, and some example systems are presented which allow us to relate phase space transformations to geometrical aberrations. In this way, we show that aberrations can be understood as nonlinear deformations of phase space, which need to be minimized in imaging designs or to be employed for homogenizing radiation in illumination designs.

Linear dispersion relation for the mirror instability in context of the gyrokinetic theory

The linear dispersion relation for the mirror instability is discussed in context of the gyrokinetic theory. The objective is to provide a coherent view of different kinetic approaches used to derive the dispersion relation. The method based on gyrocenter phase space transformations is adopted in order to display the origin and ordering of various terms.

On the form invariant volume transformation in phase space by focusing neutron guides: An analytic treatment

Tapered guides with supermirror coating are frequently used to focus neutron beams on specimens. The divergence distribution in the focused beam is of a great importance for the quality of neutron instrumentation. Using an analytic approach we derive the tapering which is needed to achieve a form invariant phase space transformation of a rectangular phase volume. In addition we consider the effect of beam attenuation by the finite reflectivity of supermirrors.