Position-momentum Correlations Research Articles

Light-field imaging from position-momentum correlations

Correlation plenoptic imaging (CPI) is a light-field imaging technique employing intensity correlation measurements to simultaneously detect the spatial distribution and the propagation direction of light. Compared to standard methods, in which light-field images are directly encoded in intensity, CPI provides a significant enhancement of the volumetric reconstruction performance in terms of both achievable depth of field and 3D resolution. In this article, we present a novel CPI configuration where light-field information is encoded in correlations between position and momentum measurements, namely, points on a given object plane and points of the Fourier plane of the imaging lens. Besides the fundamental interest in retrieving the properties of position-momentum correlation, the proposed scheme overcomes practical limitations of previously proposed setups, providing higher axial homogeneity and robustness with respect to the identification of reference planes.

Continuous-Variable Entanglement through Central Forces: Application to Gravity between Quantum Masses

We describe a complete method for a precise study of gravitational interaction between two nearby quantum masses. Since the displacements of these masses are much smaller than the initial separation between their centers, the displacement-to-separation ratio is a natural parameter in which the gravitational potential can be expanded. We show that entanglement in such experiments is sensitive to initial relative momentum only when the system evolves into non-Gaussian states, i.e., when the potential is expanded at least up to the cubic term. A pivotal role of force gradient as the dominant contributor to position-momentum correlations is demonstrated. We establish a closed-form expression for the entanglement gain, which shows that the contribution from the cubic term is proportional to momentum and from the quartic term is proportional to momentum squared. From a quantum information perspective, the results find applications as a momentum witness of non-Gaussian entanglement. Our methods are versatile and apply to any number of central interactions expanded to any order.

Phase-space quantum distributions and information theory

The local structure present in Wigner and Husimi phase-space distributions and their marginals are studied and quantified via information-theoretic quantities. Shannon, R\'enyi, and cumulative residual entropies of the Wigner and Husimi distributions are examined in the ground and excited states of a harmonic oscillator. The entropies of the Wigner function marginals are lower than the corresponding entropies of the Husimi function marginals, which illustrates how the nodal structure present in the Wigner function is lost upon consideration of the Husimi function. Shannon and cumulative residual entropies of the Wigner function yield entropies which are complex valued. Absolute values and real components of these quantities increase with quantum number, displaying a behavior which is consistent with their real-valued Husimi function counterparts. This comportment is also similar to the real-valued R\'enyi entropies of the Wigner function. The entropies of the Wigner function are observed to be lower than the corresponding Husimi function ones, in agreement with the results for the marginals. The real components of the Wigner function entropies are seen to be closer to the uncertainty relation bound compared to the corresponding Husimi function entropies. These real components are also closer to the bound when contrasted to the entropic sum of the marginal densities. The R\'enyi or collision entropy of the Wigner function sits exactly on the bound. Related statistical correlation measures show that the position-momentum correlation is larger in the Wigner function compared to the Husimi function and increases with quantum number.

The effect of in-doping on the quantum information entropy of hydrogenic impurity states in the InxGa1-xN semiconductor quantum dot

ABSTRACT The effect of In-doping on the quantum information entropy of the hydrogenic impurity states in the InxGa1-xN quantum dot has been studied for the first time. By exploring the Shannon entropy in the position space (Sr ), in the momentum-space (Sp ) and the Shannon entropy sum St , we find some novel and interesting phenomenon. First, although Sr increases and Sp decreases monotonically with the radius of InxGa1-xN quantum dot, the Shannon entropy sum St exhibits a series of peaks and valleys with the confinement radius. The physical origin of the minimum and maximum values in the Shannon entropy sum St is analysed in great detail. Second, the BBM entropy uncertainty principle () holds for the hydrogenic impurity state in the semiconductor quantum dot, which can be used to replace the Heisenberg uncertainty principle in the quantum mechanics. Third, the content of the doping In element in the InxGa1-xN quantum dot has a great influence on the Shannon entropy of the hydrogenic impurity state. Since the Shannon entropy sum St is usually used to measure the position-momentum correlation, therefore, we can explore the uncertainty of position and momentum of the hydrogenic impurity states by changing the content of the doping element in the semiconductor quantum dots. This study has some practical applications in semiconductor material and quantum information measurement, and may guide future experimental researches for the localisation and delocalisation characteristics of hydrogenic impurities states in the semiconductor quantum dots.

Uncertainty Relations in the Madelung Picture.

Madelung showed how the complex Schrödinger equation can be rewritten in terms of two real equations, one for the phase and one for the amplitude of the complex wave function, where both equations are not independent of each other, but coupled. Although these equations formally look like classical hydrodynamic equations, they contain all the information about the quantum system. Concerning the quantum mechanical uncertainties of position and momentum, however, this is not so obvious at first sight. We show how these uncertainties are related to the phase and amplitude of the wave function in position and momentum space and, particularly, that the contribution from the phase essentially depends on the position–momentum correlations. This will be illustrated explicitly using generalized coherent states as examples.

Analysis of slice transverse emittance evolution in a very-high-frequency gun photoinjector

Emittance is a key figure-of-merit for high brightness electron beams that enable the success of a broad range of modern accelerator-based instruments. With the successful implementation of the emittance compensation scheme and beam-shaping techniques to align each temporal beam slice in phase space, slice emittance starts to play a more dominating role in determining the beam quality. In this paper, we develop an approach, combining simulations and analytical calculations, to systematically study the evolution of the slice emittance in a very high frequency gun photoinjector, which is under intense R worldwide for driving the next generation continuous-wave accelerators. Our approach is capable of tracing various sources that introduce curvature in the transverse phase space, corresponding to the growth of the slice emittance. It is shown that nonlinear transverse space charge forces in the vicinity of cathodes and the spherical aberration of the focusing solenoid are the main sources of nonlinear forces, while in a long drift section space charge forces can actually compensate for nonlinear transverse position-momentum correlations and hence recover the emittance. We further demonstrate that the sources of nonlinearities can be controlled by tailoring the transverse beam density profile and the design of solenoid lenses.

Negative Probability Current in a Freely Falling Quantum Particle

A quantum particle (represented by a Gaussian wave packet) under uniform gravity, projected upward (the conventional positive direction) along the vertical degree of freedom x, may have a negative probability current on a set of points q. This imposes the condition that the position and momentum observables of the state are negatively correlated. We calculate the effect of the negative correlation on the individual contributions to the probability current by the positive and negative momentum amplitudes of the state using an integral representation of the probability current. Negative position-momentum correlation causes a transient attenuation of all contributions to the probability flow, and wave packet spreading is temporarily mitigated.

Critical properties of the Floquet time crystal within the Gaussian approximation

The periodically driven $O(N)$ model is studied near the critical line separating a disordered paramagnetic phase from a period doubled phase, the latter being an example of a Floquet time crystal. The time evolution of one-point and two-point correlation functions are obtained within the Gaussian approximation and perturbatively in the drive amplitude. The correlations are found to show not only period doubling, but also power-law decays at large spatial distances. These features are compared with the undriven $O(N)$ model, within the Gaussian approximation, in the vicinity of the paramagnetic-ferromagnetic critical point. The algebraic decays in space are found to be qualitatively different in the driven and the undriven cases. In particular, the spatiotemporal order of the Floquet time crystal leads to position-momentum and momentum-momentum correlation functions which are more long-ranged in the driven than in the undriven model. The light-cone dynamics associated with the correlation functions is also qualitatively different as the critical line of the Floquet time crystal shows a light cone with two distinct velocities, with the ratio of these two velocities scaling as the square-root of the dimensionless drive amplitude. The Floquet unitary, which describes the time evolution due to a complete cycle of the drive, is constructed for modes with small momenta compared to the drive frequency, but having a generic relationship with the square-root of the drive amplitude. At intermediate momenta, which are large compared to the square-root of the drive amplitude, the Floquet unitary is found to simply rotate the modes. On the other hand, at momenta which are small compared to the square-root of the drive amplitude, the Floquet unitary is found to primarily squeeze the modes, to an extent which increases upon increasing the wavelength of the modes, with a power-law dependence on it.

Probability backflow for correlated quantum states

In its original formulation, quantum backflow (QB) is an interference effect that manifests itself as a negative probability transfer for free-particle states comprised of plane waves with only positive momenta. Quantum reentry (QR) is another interference effect in which a wave packet expanding from a spatial region of its initial confinement partially returns to the region in the absence of any external forces. Here we show that both QB and QR are special cases of a more general classically-forbidden probability flow for quantum states with certain position-momentum correlations. We further demonstrate that it is possible to construct correlated quantum states for which the amount of probability transferred in the "wrong" (classically impossible) direction exceeds the least upper bound on the corresponding probability transfer in the QB and QR problems, known as the Bracken-Melloy constant.

Experimental demonstration of Einstein-Podolsky-Rosen entanglement in rotating coordinate space

Einstein-Podolsky-Rosen (EPR) entanglement involving a pair of particles entangled in their positions and momenta is of special interest in the field of quantum information. Previously, EPR entanglement has been studied in different physical systems but in fixed coordinate spaces. Here, we demonstrate an experiment of ghost imaging and ghost interference in rotated position-momentum spaces by using position-momentum entangled photons generated from a hot atomic ensemble. By using different image objects, the measured position-momentum correlations exhibit intriguing dynamics, including gradual decrease and axis-independent EPR entanglement. The reported results on manipulating the EPR entanglement in rotating coordinate spaces hold promise in quantum communication and distant quantum image processing.

Self-focusing quantum states

We discuss a class of solutions of the time-dependent Schrödinger equation such that the position uncertainty temporarily decreases. This self-focusing or contractive behavior is a consequence of the anti-correlation of the position and momentum observables. Since the associated position density satisfies a continuity equation, upon contraction the probability current at a given fixed point may flow in the opposite direction of the group velocity of the wave packet. For definiteness, we consider a free particle incident from the left of the origin, and establish a condition for the initial position-momentum correlation such that a negative probability current at the origin is possible. This implies a decrease in the particle's detection probability in the region x > 0, and we calculate how long this occurs. Analogous results are obtained for a particle subject to a uniform gravitational force if we consider the particle approaching the turning point. We show that position-momentum anti-correlation may cause a negative probability current at the turning point, leading to a temporary decrease in the particle's detection probability in the classically forbidden region.

Phase-Space Approach to the Position-Momentum Correlations of the Conduction Electron States in a Double-Barrier Resonant Nanosystem

Phase-Space Approach to the Position-Momentum Correlations of the Conduction Electron States in a Double-Barrier Resonant Nanosystem

Harmonic Chain with Velocity Flips: Thermalization and Kinetic Theory

We consider the detailed structure of correlations in harmonic chains with pinning and a bulk velocity flip noise during the heat relaxation phase which occurs on diffusive time scales, for $t=O(L^2)$ where $L$ is the chain length. It has been shown earlier that for non-degenerate harmonic interactions these systems thermalize, and the dominant part of the correlations is given by local thermal equilibrium determined by a temperature profile which satisfies a linear heat equation. Here we are concerned with two new aspects about the thermalization process: the first order corrections in $1/L$ to the local equilibrium correlations and the applicability of kinetic theory to study the relaxation process. Employing previously derived explicit uniform estimates for the temperature profile, we first derive an explicit form for the first order corrections to the particle position-momentum correlations. By suitably revising the definition of the Wigner transform and the kinetic scaling limit we derive a phonon Boltzmann equation whose predictions agree with the explicit computation. Comparing the two results, the corrections can be understood as arising from two different sources: a current-related term and a correction to the position-position correlations related to spatial changes in the phonon eigenbasis.

Wigner representation of ionization and scattering in strong laser fields

The interaction of single-electron atoms with a strong laser field is studied in the Wigner representation. The Wigner function is a quasiprobability function in phase space that allows one to study position-momentum correlations. These correlations give a physical interpretation of the emergence of the above-threshold-ionization (ATI) energy spectrum. Conversely, the quantum-mechanical interference between electrons from neighboring photon orders can explain the spatial bunching of the electron density by the laser field. Furthermore, the Wigner function offers one a rather accurate and relatively efficient quasiclassical estimate of the bound-state population. This method is applied to laser-induced electron-ion scattering and the stationary regime of the bound-state population can be determined. The present calculations are performed for a one-dimensional Rosen-Morse potential. Extensions to general spherically symmetric atomic potentials are indicated.

Position–momentum correlations in matter waves double-slit experiment

We present a treatment of the double-slit interference of matter-waves represented by Gaussian wavepackets. The interference pattern is modelled with Green's function propagator which emphasizes the coordinate correlations and phases. We explore the connection between phases and position–momentum correlations in the intensity, visibility and predictability of the wavepacket interference. This formulation will indicate some aspects that can be useful for theoretical and experimental treatment of particle, atom or molecule interferometry and can be discussed in introductory quantum mechanics courses.

Optimum phase space probabilities from quantum tomography

We determine a positive normalised phase space probability distribution P with minimum mean square fractional deviation from the Wigner distribution W. The minimum deviation, an invariant under phase space rotations, is a quantitative measure of the quantumness of the state. The positive distribution closest to W will be useful in quantum mechanics and in time frequency analysis. The position-momentum correlations given by the distribution can be tested experimentally in quantum optics.

The position–momentum correlation for quantum and classical probability distributions

We compare the definition and the interpretation of the position–momentum correlation in classical statistical physics and quantum mechanics. The introduction of a regional momentum distribution makes a rigorous discussion in both theories possible. We investigate the mutual dependence of position and momentum for one- and two-centred probability distributions.

Enhanced backscattering of a dilute Bose-Einstein condensate

We study experimentally and numerically the quasi-bidimensional transport of a 87Rb Bose-Einstein condensate launched with a velocity v0 inside a disordered optical potential created by a speckle pattern. A time-of-flight analysis reveals a pronounced enhanced density peak in the backscattering direction −v0, a feature reminiscent of coherent backscattering. Detailed numerical simulations indicate, however, that other effects also contribute to this enhancement, including a “backscattering echo” due to the position-momentum correlations of the initial wave packet.

Position–momentum correlations in electron–ion scattering in strong laser fields

The quantum Wigner function of an electron scattered by an ion in a strong laser field is considered in the framework of a one-dimensional scattering model with a soft-core Coulomb potential. The Wigner function contains much more information on the scattering process than the projected probability distributions in position and momentum space considered previously. The formation of the above-threshold ionization (ATI) energy spectrum, including ATI peaks, modulations and transients, can be easily explained by using the interference of phase-space trajectories.

On the Existence and Robustness of Steady Position-Momentum Correlations for Time-Dependent Quadratic Systems

We discuss conditions giving rise to stationary position-momentum correlations among quantum states in the Fock and coherent basis associated with the natural invariant for the one-dimensional time-dependent quadratic Hamiltonian operators such as the Kanai-Caldirola Hamiltonian. We also discuss some basic features such as quantum decoherence of the wave functions resulting from the corresponding quantum dynamics of these systems that exhibit no timedependence in their quantum correlations. In particular, steady statistical momentum averages are seen over well-defined time intervals in the evolution of a linear superposition of the basis states of modified exponentially damped mass systems.