Transformations In Phase Space Research Articles

Quantum de Finetti theorem in phase-space representation

The quantum versions of de Finetti's theorem derived so far express the convergence of n-partite symmetric states, i.e., states that are invariant under permutations of their n parties, towards probabilistic mixtures of independent and identically distributed (i.i.d.) states. Unfortunately, these theorems only hold in finite-dimensional Hilbert spaces, and their direct generalization to infinite-dimensional Hilbert spaces is known to fail. Here, we address this problem by considering invariance under orthogonal transformations in phase space instead of permutations in state space, which leads to a new type of quantum de Finetti's theorem that is particularly relevant to continuous-variable systems. Specifically, an n-mode bosonic state that is invariant with respect to this continuous symmetry in phase space is proven to converge towards a probabilistic mixture of i.i.d. Gaussian states (actually, n identical thermal states).

A New Kind of Two-Fold Integration Transformation in Phase Space and Its Uses in Weyl Ordering of Operators

We propose a new two-fold integration transformation in p-q phase spacewhich possesses some well-behaved transformation properties. We apply this transformation to the Weyl ordering of operators, especially those Q-P ordered and P-Q ordered operators.

A background suppression chopper for the High-Flux Backscattering Spectrometer at NIST

A characteristic of backscattering spectroscopy is the limited flux in achieving high resolution. This means that one requires a very high signal-to-noise ratio by keeping the background to a minimum. The high-flux backscattering spectrometer at the NIST Center for Neutron Research uses a phase-space transformation (PST) chopper to enhance the flux by a factor of 4.2. The chopper has six equal size sectors, alternating between open and pyrolytic graphite. The background in this spectrometer arises from neutrons from the main guide. Measurements show that neutrons which pass through an open sector of the PST chopper at the same time that the monochromatized beam passes through on its way to the sample, contribute to 60% of the background. In order to reduce this background while retaining 80% of the present neutron flux, we propose a second chopper placed between the local shutter and the entrance to the converging guide, synchronized with the PST chopper.

Generalized gauge transformations in phase space picture of quantum mechanics: Kirkwood representation

Here, the ordinary gauge transformations in configuration space are generalized to the phase space. The corresponding gauge functions are separable in terms of those of the configuration and the momentum spaces. Furthermore, we consider the gauge transformations in phase space for which their gauge functions are not, in general, separable. It is shown that, the assumption of the general global gauge symmetry in phase space immediately yields the quantization of the electric charge. We obtain this result without assuming the concept of the magnetic monopole that Dirac did in his approach to the problem of electric charge quantization and which is not detected yet.

Linear transformations and aberrations in continuous and finite systems

In geometric optics there is a natural distinction between the paraxial and aberration regimes, which contain respectively the linear and nonlinear canonical transformations of position and momentum in the phase space. In the Lie-theoretical presentation, linear inhomogeneous transformations are generated by linear and quadratic functions of the phase space, while aberrations of increasing order are generated by homogeneous functions with higher powers of these coordinates. In a way parallel but distinct from the Schrödinger quantization of continuous classical systems, we quantize the geometric optical model into discrete, finite-dimensional systems based on the Lie algebra , whose wavefunctions are N-point signals, phase space is a sphere and transformations are represented by the N × N unitary matrices that form the group . We factor this group into -linear and nonlinear unitary transformations of phase space and classify all its N2 − 4 aberrations. This offers a new parametrization of based on a chosen subgroup.

Primary spectrometer neutron optics simulations for a new cold neutron backscattering spectrometer

The future new cold neutron backscattering spectrometer IN16B at the Institut Laue-Langevin is being designed to maintain the extremely high energy resolution of the existing backscattering spectrometer IN16 (approximately 0.4 μeV full width at half maximum in a standard configuration). Simultaneously, a phase space transformation (PST) device will significantly increase the flux at the sample position at the expense of an acceptably more divergent incoming beam. A wide wavelength band (Δλ/λ ≈ 10%) has to be offered to the PST device in order to achieve a significant flux gain by a factor of at least 4. Thus, IN16B will have to be located at the end of a cold neutron guide, whilst the present IN16 is located at a side position along a guide. In order to optimize the layout of individual components and to estimate the instrument performance, the Monte Carlo simulation programs McStas and VITESS have been used. McStas and VITESS offer a general framework to compose virtual neutron scattering instruments and...

Monte-Carlo simulation of phase space transformation of ultra-cold neutrons

The very high phase space density of ultra-cold neutrons (UCN) originating from a superthermal UCN-source can be exploited for the production of intense cold neutron beams. UCN are accelerated by means of Doppler-shifter crystals. This method is called phase space transformation (PST). In the cold regime, gain factors of 100 are theoretically expected compared to standard beam generation. The Atominstitut in Vienna and the Paul Scherrer Institut have joined to design and construct a “proof of principle”-experiment for such a phase space transformer in the framework of the FP7-NMI3-JRA3 European project. The aims of this experiment are to explore its feasibility, its experimental limitations and to validate preceding MC-simulations. Employing a sophisticated mechanical system, stage-2 Potassium intercalated HOPG crystals ( d = 8.74 Å ) will be moved with velocities of up to 250 m/s. The experiment is planned to take place at the PF-2 UCN source at the Institut Laue-Langevin (ILL) in the second half of 2007. In this contribution recent results of preliminary Monte-Carlo simulations of the experiment are presented.

Reductions of self-dual Einstein equations

Transformations between the Pleba´nski equation and other reductions of the selfdual Einstein equations are given. A new reduction is presented. It consists of two equations describing canonical transformations in a 2-dimensional phase space. Symmetries and examples of linearizations of these equations are given.

On certain canonoid transformations and invariants for the parametric oscillator

In classical mechanics a transformation in phase space is said to be canonoid if it maps only some Hamiltonian systems into Hamiltonian systems. Once canonoid transformations are considered, these systems can be classically described by means of Lagrangians or Hamiltonians other than the conventional ones. In this context, a basic role is played by the dynamical invariants generated through the Poisson brackets of the new independent variables in phase space. Here we obtain the explicit form of special canonoid transformations of the polynomial type for systems which can be classically described by an equation of the parametric oscillator type and discuss some algebraic properties shown by the associated dynamical invariants.

Functional integral treatment of some quantum nondemolition systems

In the scheme of a quantum nondemolition (QND) measurement, an observable is measured without perturbing its evolution. In the context of studies of decoherence in quantum computing, we examine the ‘open’ quantum system of a two-level atom or, equivalently, a spin-1/2 system, in interaction with quantum reservoirs of either oscillators or spins, under the QND condition of the Hamiltonian of the system commuting with the system–reservoir interaction. For completeness, we also examine the well-known non-QND spin-Bose problem. For all these many-body systems, we use the methods of functional integration to work out the propagators. The propagators for the QND Hamiltonians are shown to be analogous to the squeezing and rotation operators, respectively, for the two kinds of baths considered. Squeezing and rotation being both phase-space area-preserving canonical transformations, this brings out an interesting connection between the energy-preserving QND Hamiltonians and the homogeneous linear canonical transformations.

Optical system design for orthosymplectic transformations in phase space

On the basis of a matrix formalism, we analyze the paraxial optical systems composed by generalized lenses and fixed free-space intervals, suitable for orthosymplectic transformations in phase space. Flexible configurations to perform the attractive operations for optical information processing such as image rotation, separable fractional Fourier transformation, and twisting for different parameters are proposed.

Monte Carlo simulations for instrumentation at SINQ

The Paul Scherrer Institut (PSI) operates a spallation source SINQ equipped with 11 different neutron scattering instruments. Beside the optimization of the existing instruments, the extension with new instruments and devices are continuously done at PSI. For design and performance studies different Monte Carlo packages are used. Presently two major projects are in an advanced stage of planning. These are the new thermal neutron triple-axis spectrometer Enhanced Intensity and Greater Energy Range (EIGER) and the ultra-cold neutron source (UCN-PSI). The EIGER instrument design is focused on an optimal signal-to-background ratio. A very important design part was to realize a monochromator shielding which covers best shielding characteristic, low background production and high instrument functionality. The Monte Carlo package MCNPX was used to find the best choice. Due to the sharp energy distribution of ultra-cold neutrons (UCN) which can be Doppler-shifted towards cold neutron energies, a UCN phase space transformation (PST) device could produce highly monochromatic cold and very cold neutrons (VCN). The UCN-PST instrumentation project running at PSI is very timely since a new-generation superthermal spallation source of UCN is under construction at PSI with a UCN density of 3000–4000 n cm −3. Detailed numerical simulations have been carried out to optimize the UCN density and flux. Recent results on numerical simulations of an UCN-PST-based source of highly monochromatic cold neutrons and VCN are presented.

How IN16 can maintain a world-leading position in neutron backscattering spectrometry

IN16 will be upgraded within the ILL Millenium program. We present briefly the concepts which will ensure that the extremely successful backscattering spectrometer IN16 maintains its leading position. The most important component is a phase space transformation (PST) chopper for which the spectrometer has to be placed on an end-of-guide position. We summarize first simulation results for the two guide positions in question, based on simulations of the source, guide, selector, focusing guide, PST and Doppler monochromator arrangement.

A space and energy focusing hybrid cold time-of-flight spectrometer based on a moving monochromator crystal

The concept of a time-of-flight spectrometer for cold neutrons is presented which makes use of the principle of phase–space transformation. A first curved monochromator is combined with a second moving crystal to a double monochromator. At the right velocity this crystal monochromatizes the neutron beam and reduces at the same time its horizontal divergence. With pyrolithic graphite as monochromator focusing can be realized for 5 Å ⩽ λ ⩽ 6.5 Å . At λ = 6 Å , an energy resolution δ E ∼ 60 μ eV is estimated. Compared to time-focusing instruments improved momentum transfer resolution and good energy resolution at all energy transfers is obtained. The instrument does not need an end position on a neutron guide.

ABCD rule for Gaussian beam propagation in the context of quantum optics derived by the IWOP technique

The development of technique of integration within an ordered product (IWOP) of operators extends the Newton–Leibniz integration rule, originally applying to permutable functions, to the non-commutative quantum mechanical operators composed of Dirac’s ket–bra, which enables us to obtain the images of directly mapping symplectic transformation in classical phase space parameterized by [ A, B; C, D] into quantum mechanical operator through the coherent state representation, we call them the generalized Fresnel operators (GFO) since they correspond to Fresnel transforms in Fourier optics. Based on GFO we find the ABCD rule for Gaussian beam propagation in the context of quantum optics (both in one-mode and two-mode cases) whose classical correspondence is just the ABCD rule in matrix optics. The entangled state representation is used in discussing the two-mode case.

Measurement of neutron flux and beam divergence at the cold neutron guide system of the new Munich research reactor FRM-II

A sophisticated neutron guide system has been installed at the new Munich neutron source FRM-II to transport neutrons from the D 2 cold neutron source to several instruments, which are situated in a separate neutron guide hall. The guide system takes advantage of supermirror coatings and includes a worldwide unique “twisted” guide for a desired phase space transformation of the neutron beam. During the initial reactor commissioning in summer 2004, the integral and differential neutron flux as well as the distribution of beam divergence at the exit of two representative and the twisted neutron guide were measured using time-of-flight spectroscopy and gold-foil activation. The experimental results can be compared to extensive simulation calculations based on MCNP and McStas. The investigated guides fulfill the expectations of providing high neutron fluxes and reveal good quality with respect to the reflective coatings and the installation precision.

Longitudinal phase space tomography and its implementation in energy recovery linacs

Applying tomographic techniques to study the longitudinal phase space of the electron beam in future energy recovery linacs presents a challenge as it requires nondestructive diagnostics and the manipulations of the phase space are restricted by the energy recovery. The different methods of tomographic reconstruction and previous experiments at the BNL DUV-FEL facility are presented. Different schemes to utilize tomography for an ERL are proposed, including non-destructive transverse and longitudinal beam profile measurements and the necessary phase space transformations to generate the tomographic projections.

Using phase–space transformation for an improved hybrid cold time-of-flight spectrometer

The concept and layout of a time-of-flight spectrometer for cold neutrons is presented which makes use of the principle of phase–space transformation. Close to the focal plane determined by a first-curved monochromator a second moving crystal is placed. If moving with the right velocity this crystal monochromatizes the neutron beam and reduces at the same time its horizontal divergence. With pyrolithic graphite as monochromator focusing can be realized for 5 Å ⩽ λ ⩽ 6.5 Å . At λ = 6 Å an energy resolution δ E ∼ 60 μ eV is calculated. Compared to time-focusing instruments improved momentum transfer resolution and good energy resolution at all energy transfers is obtained. A reduction of intensity by the second monochromator will be most likely balanced by improved background. The wavelength dependence of the energy focusing velocity restricts in practice the instrument to one or two fixed wavelengths. The instrument does not need an end position on a neutron guide.

Space charge lenses for particle beams

Space charge lenses provide strong cylinder symmetric focusing for low-energy high-perveance particle beams using a stable space charge cloud. They need drastically reduced magnetic and electrostatic field strength compared with conventional systems and are superior for a degree of lens filling above 17%. They can theoretically provide linear transformation in phase space and reduce beam aberrations and space charge forces. The density distribution of the enclosed space charge is given by the transverse and longitudinal enclosures of the cloud. By the use of self-consistent simulations of the space charge cloud, the focusing properties of space charge lenses in the presented design can be forecasted with sufficient quality. The presented simulations show that the theoretical values can be reached locally. The results of our investigations on the beam transport in a high-current test injector equipped with space charge lenses, including emittance measurements, will be presented and discussed. They show that significant improvements of lens operation have been reached by the reduction of the residual gas pressure and a careful design of the external fields using numerical simulation techniques to calculate the local density distributions. Comparisons of the experimental results with the beam transport simulations show good agreement concerning both focusing strength and linearity of phase space transformation. For the lens design used, the observed degree of lens filling is at least 38% of the theoretical value and more than twice the threshold value.

Sextupole correction of the longitudinal transport of relativistic beams in dispersionless translating sections

We examine the use of sextupole magnets to correct nonlinearities in the longitudinal phase space transformation of a relativistic beam of charged particles in a dispersionless translating section, or dogleg. Through heuristic analytical arguments and examples derived from recent experimental efforts, augmented by simulations using the particle tracking codes PARMELA and ELEGANT, sextupole corrections are found to be effective in optimizing the use of such structures for beam compression or for shaping the current profile of the beam, by manipulation of the second-order longitudinal dispersion. Recent experimental evidence of the use of sextupoles to manipulate second-order horizontal and longitudinal dispersion of the beam is presented. The theoretical and experimental results indicate that these manipulations can be used to create an electron bunch with a current profile having a long ramp followed by a sharp cutoff, which is optimal for driving large-amplitude wake fields in a plasma wake field accelerator.