Wigner Transform Research Articles

Modeling Mixing in Stratified Heterogeneous Media: The Role of Water Velocity Discretization in Phase Space Formulation

Modeling solute transport in heterogeneous porous media faces two challenges: scale dependence of dispersion and reproducing mixing separately from spreading. Both are crucial since real applications may require km scales whereas reactions, often controlled by mixing, may occur at the pore scale. Methods have been developed in response to these challenges, but none has satisfactorily characterized both processes. In this paper, we propose a formulation based on the Water Mixing Approach extended to account for velocity variability. Velocity is taken as an independent variable, so that concentration depends on time, space and velocity. Therefore, we term the formulation the Multi-Advective Water Mixing Approach. A new mixing term between velocity classes emerges in this formulation. We test it on Poiseuille’s stratified flow using the Water Parcel method. Results show high accuracy of the formulation in both dispersion and mixing. Moreover, the mixing process exhibits Markovianity in space even though it is modeled in time.

Connes spectral distance and nonlocality of generalized noncommutative phase spaces

We study the Connes spectral distance of quantum states and analyse the nonlocality of a 4D generalized noncommutative phase space. By virtue of the Hilbert–Schmidt operatorial formulation, we obtain the Dirac operator and construct a spectral triple corresponding to the noncommutative phase space. Based on the ball condition, we obtain some constraint relations about the optimal elements, and then calculate the Connes spectral distance between two Fock states. Due to the noncommutativity, the spectral distances between Fock states in generalized noncommutative phase spaces are shorter than those in normal phase spaces. This shortening of distances implies some type of nonlocality caused by the noncommutativity. These spectral distances in the 4D generalized noncommutative phase space are additive and satisfy the normal Pythagoras theorem. When the noncommutative parameters go to zero, the results return to those in normal quantum phase spaces.

Algebras of variable coefficient quantized differential operators

In the framework of (vector valued) quantized holomorphic functions defined on non-commutative spaces, “quantized Hermitian symmetric spaces,” an obvious problem is to describe (quantum) holomorphically induced representations in terms of some manageable structures. An intimately related problem is to decide what the algebras of quantized differential operators with variable coefficients should be. It is an immediate point that even zeroth order operators, given as multiplications by polynomials, have to be specified as, e.g., left or right multiplication operators since the polynomial algebras are replaced by quadratic, non-commutative algebras. In the settings we are interested in, there are bilinear pairings that allow us to define differential operators as duals of multiplication operators. Indeed, there are different choices of pairings which lead to quite different results. We consider three different pairings while specializing to Uq(su(n,n)). The pairings are between quantized generalized Verma modules and quantized holomorphically induced modules. It is a natural demand that the corresponding representations can be expressed by (matrix valued) differential operators. We show that a quantum Weyl algebra Weylq(n,n) introduced by Hayashi [Commun. Math. Phys. 127(1), 129–144 (1990)] plays a fundamental role. In fact, for one pairing, the algebra of differential operators, though inherently depending on a choice of basis, is precisely matrices over Weylq(n,n). We determine explicitly the form of the (quantum) holomorphically induced representations and determine, for the different pairings, if they can be expressed by differential operators.

Variational quantum algorithm for Gaussian discrete solitons and their boson sampling

In the context of quantum information, highly nonlinear regimes, such as those supporting solitons, are marginally investigated. We miss general methods for quantum solitons, although they can act as entanglement generators or as self-organized quantum processors. We develop a computational approach that uses a neural network as a variational ansatz for quantum solitons in an array of waveguides. By training the resulting phase space quantum machine-slearning model, we find different soliton solutions, varying the number of particles and interaction strength. We consider Gaussian states that enable measuring the degree of entanglement and sampling the probability distribution of many-particle events. We also determine the probability of generating particle pairs and unveil that soliton bound states emit correlated pairs. These results may have a role in boson sampling with nonlinear systems and in quantum processors for entangled nonlinear waves.

On the Convergence of a Novel Time-Slicing Approximation Scheme for Feynman Path Integrals

Abstract In this note we study the properties of a sequence of approximate propagators for the Schrödinger equation, in the spirit of Feynman’s path integrals. Precisely, we consider Hamiltonian operators arising as the Weyl quantization of a quadratic form in phase space, plus a bounded potential perturbation in the form of a pseudodifferential operator with a rough symbol. The corresponding Schrödinger propagator belongs to the class of generalized metaplectic operators, a fact that naturally motivates the introduction of a manageable time-slicing approximation scheme consisting of operators of the same type. By leveraging on this design and techniques of wave packet analysis we are able to prove several convergence results with precise rates in terms of the mesh size of the time slicing subdivision, even stronger then those that can be achieved under the same assumptions using the standard Trotter approximation scheme. In particular, we prove convergence in the norm operator topology in $L^2$, as well as pointwise convergence of the corresponding integral kernels for non-exceptional times.

Angular momentum eigenstates of the isotropic 3-D harmonic oscillator: Phase-space distributions and coalescence probabilities

The isotropic 3-dimensional harmonic oscillator potential can serve as an approximate description of many systems in atomic, solid state, nuclear, and particle physics. In particular, the question of 2 particles binding (or coalescing) into angular momentum eigenstates in such a potential has interesting applications. We compute the probabilities for coalescence of two distinguishable, non-relativistic particles into such a bound state, where the initial particles are represented by generic wave packets of given average positions and momenta. We use a phase-space formulation and hence need the Wigner distribution functions of angular momentum eigenstates in isotropic 3-dimensional harmonic oscillators. These distribution functions have been discussed in the literature before but we utilize an alternative approach to obtain these functions. Along the way, we derive a general formula that expands angular momentum eigenstates in terms of products of 1-dimensional harmonic oscillator eigenstates.

Quantum Information Dimension and Geometric Entropy

A differential-geometric formulation of quantum mechanics and the continuity of quantum phase space are used to diagnose how the state of an open quantum system results from its correlations with a structured environment.

Quantum statistics of light emitted from a pillar microcavity

A quantum behavior of the light emitted by exciton polaritons excited in a pillar semiconductor microcavity with embedded quantum well is investigated. Considering the bare excitons and photon modes as coupled quantum oscillators allows for an accurate accounting of the nonlinear and dissipative effects. In particular, using the method of the quantum states representation in a quantum phase space via quasiprobability functions (namely, a P-function and a Wigner function), we study the impact of the laser and the exciton-photon detuning on the second order correlation function of the emitted photons. We determine the conditions under which the phenomena of bunching, giant bunching, and antibunching of the emitted light emerge. In particular, we predict the effect of a giant bunching for the case of a large exciton to photon population ratio. Within the domain of parameters supporting a bistability regime we demonstrate the effect of bunching of photons.

Semigroups for quadratic evolution equations acting on Shubin–Sobolev and Gelfand–Shilov spaces

We consider the initial value Cauchy problem for a class of evolution equations whose Hamiltonian is the Weyl quantization of a homogeneous quadratic form with non-negative definite real part. The solution semigroup is shown to be strongly continuous on several spaces: the Shubin-Sobolev spaces, the Schwartz space, the tempered distributions, the equal index Beurling type Gelfand-Shilov spaces and their dual ultradistribution spaces.

Propagation of global analytic singularities for Schrödinger equations with quadratic Hamiltonians

We study the propagation in time of 1/2-Gelfand-Shilov singularities, i.e. global analytic singularities, of tempered distributional solutions of the initial value problem{ut+qw(x,D)u=0u|t=0=u0, on Rn, where u0 is a tempered distribution on Rn, q=q(x,ξ) is a complex-valued quadratic form on R2n=Rxn×Rξn with nonnegative real part Req≥0, and qw(x,D) is the Weyl quantization of q. We prove that the 1/2-Gelfand-Shilov singularities of the initial data that are contained within a distinguished linear subspace of the phase space R2n, called the singular space of q, are transported by the Hamilton flow of Imq, while all other 1/2-Gelfand-Shilov singularities are instantaneously regularized. Our result extends the observation of Hitrik, Pravda-Starov, and Viola '18 that this evolution is instantaneously globally analytically regularizing when the singular space of q is trivial.

On the connection between the Wigner and the Bohm quantum formalism

In this paper, we discuss the connection between two alternative descriptions of the quantum motion proposed by Wigner in 1932 and by Bohm in 1952. Such formalism provides different representations of the quantum dynamics. The Wigner description is based on the definition of a non positive quasi-distribution function in the quantum phase space and the Bohm formalism on the introduction of the pilot wave field associated to the particle motion along trajectories. We discuss the representation of the Bohm dynamics in the phase space in terms of a suitable Wigner measure. We propose an extension of the Wigner transformation and we derive a family of models where the quantum dynamics is described by equivalent mathematical formulations. Similarly to the classical limit, we show the asymptotic convergence in distributional sense of the extended Wigner distribution to a measure. We prove that such a limit is the solution of a classical Liouville equation containing as a quantum correction the Bohm potential.

The transverse field XY model on the diamond chain

We consider the s=1/2 transverse field XY model on the frustrated diamond chain, considering anisotropic exchange parameters between nearest neighbor spins. To this end, we employ three different methodologies: mean-field approximations, and state-of-the-art exact diagonalizations (ED), and density matrix renormalization group (DMRG) simulations. Within a mean-field theory, the Hamiltonian is fermionized by introducing the Jordan–Wigner transformation, and the interacting (many-body) terms are approximated to single-particle ones by a Hartree–Fock approach. We analyze the behavior of the induced and spontaneous magnetization as functions of the external field, investigating the magnetic properties at the ground state, and at finite temperatures. Interestingly, the mean-field results are in reasonable agreement with the ED and DMRG ones, in particular for the distorted chain, or at an intermediate/large spin anisotropy parameter. As our key results, we present phase diagrams anisotropy × magnetic field at zero temperature, discussing the emergence of phases and its quantum critical points. Finally, our analysis at finite temperature provides a range of parameters in which an unusual behavior of the induced magnetization occurs — with it increasing as a function of temperature. This work presents a global picture of the XY model on the diamond chain, which may be useful to understand features of magnetism in more complex geometries.

Irreversible entropy production rate in a parametrically driven-dissipative system: The role of self-correlation between noncommuting observables

In this paper, we explore the emergence of the Wigner entropy production rate in the stationary state of a two-mode Gaussian system. The interacting modes dissipate into different local thermal baths. Also, one of the bosonic modes evolves into the squeezed-thermal state because of the parametric amplification process. Using the Heisenberg-Langevin approach, combined with quantum phase space formulation, we get an analytical expression for the steady-state Wigner entropy production rate. It contains two key terms. The first one is an Onsager-like expression that describes heat flow within the system. The second term resulted from vacuum fluctuations of the baths. Analyses show that self-correlation between the quadratures of the parametrically amplified mode pushes the mode towards the thermal squeezed state. It increases vacuum entropy production of the total system and reduces the heat current between the modes. The results imply that, unlike other previous proposals, squeezing can constrain the efficiency of actual non-equilibrium heat engines by irreversible flows.

New phase space formulations and quantum dynamics approaches

AbstractWe report recent progress on the phase space formulation of quantum mechanics with coordinate‐momentum variables, focusing more on new theory of (weighted) constraint coordinate‐momentum phase space for discrete‐variable quantum systems. This leads to a general coordinate‐momentum phase space formulation of composite quantum systems, where conventional representations on infinite phase space are employed for continuous variables. It is convenient to utilize (weighted) constraint coordinate‐momentum phase space for representing the quantum state and describing nonclassical features. Various numerical tests demonstrate that new trajectory‐based quantum dynamics approaches derived from the (weighted) constraint phase space representation are useful and practical for describing dynamical processes of composite quantum systems in the gas phase as well as in the condensed phase.This article is categorized under: Molecular and Statistical Mechanics > Molecular Dynamics and Monte‐Carlo Methods Theoretical and Physical Chemistry > Reaction Dynamics and Kinetics Theoretical and Physical Chemistry > Statistical Mechanics

Proof of the equivalence of the symplectic forms derived from the canonical and the covariant phase space formalisms

We prove that, for any theory defined over a space-time with boundary, the symplectic form derived in the covariant phase space is equivalent to the one derived from the canonical formalism.

Erratum to ‘Noncommutative coordinate picture of the quantum phase space’ Chinese Journal of Physics Volume 71, June 2021, Pages 418-434

Erratum to ‘Noncommutative coordinate picture of the quantum phase space’ Chinese Journal of Physics Volume 71, June 2021, Pages 418-434

Embeddings and Integrable Charges for Extended Corner Symmetry.

We revisit the problem of extending the phase space of diffeomorphism-invariant theories to account for embeddings associated with the boundary of subregions. We do so by emphasizing the importance of a careful treatment of embeddings in all aspects of the covariant phase space formalism. In so doing we introduce a new notion of the extension of field space associated with the embeddings which has the important feature that the Noether charges associated with all extended corner symmetries are in fact integrable, but not necessarily conserved. We give an intuitive understanding of this description. We then show that the charges give a representation of the extended corner symmetry via the Poisson bracket, without central extension.

Estimation of the Wigner distribution of single-mode Gaussian states: A comparative study

In this work, we consider the estimation of single mode Gaussian states using four different measurement schemes namely: i) homodyne measurement, ii) sequential measurement, iii) Arthurs-Kelly scheme, and iv) heterodyne measurement, with a view to compare their relative performance. To that end, we work in the phase space formalism, specifically at the covariance matrix level, which provides an elegant and intuitive way to explicitly carry out involved calculations. We show that the optimal performance of the Arthurs-Kelly scheme and the sequential measurement is equal to the heterodyne measurement. While the heterodyne measurement outperforms the homodyne measurement in the mean estimation of squeezed state ensemble, the homodyne measurement outperforms the heterodyne measurement for variance estimation of squeezed state ensemble up to a certain range of squeezing parameter. We then modify the Hamiltonian in the Arthurs-Kelly scheme, such that the two meters can have correlations and show that the optimal performance is achieved when the meters are uncorrelated. We expect that the results will be useful in analyzing various quantum information and quantum communication protocols.

Gravity from symmetry: duality and impulsive waves

We show that we can derive the asymptotic Einstein’s equations that arises at order 1/r in asymptotically flat gravity purely from symmetry considerations. This is achieved by studying the transformation properties of functionals of the metric and the stress-energy tensor under the action of the Weyl BMS group, a recently introduced asymptotic symmetry group that includes arbitrary diffeomorphisms and local conformal transformations of the metric on the 2-sphere. Our derivation, which encompasses the inclusion of matter sources, leads to the identification of covariant observables that provide a definition of conserved charges parametrizing the non-radiative corner phase space. These observables, related to the Weyl scalars, reveal a duality symmetry and a spin-2 generator which allow us to recast the asymptotic evolution equations in a simple and elegant form as conservation equations for a null fluid living at null infinity. Finally we identify non-linear gravitational impulse waves that describe transitions among gravitational vacua and are non-perturbative solutions of the asymptotic Einstein’s equations. This provides a new picture of quantization of the asymptotic phase space, where gravitational vacua are representations of the asymptotic symmetry group and impulsive waves are encoded in their couplings.

Hidden continuous quantum phase transition without gap closing in non-Hermitian transverse Ising model

Continuous phase transition in quantum matters is a significant issue in condensed matter physics. In general, the continuous quantum phase transitions in many-body systems occur with gap closing. On the other hand, non-Hermitian systems could display quite different properties as their Hermitian counterparts. In this paper, we show that a hidden, continuous quantum phase transition occurs without gap closing in non-Hermitian transverse Ising model. By using a projected Jordan–Wigner transformation, the one-dimensional (1D) non-Hermitian transverse Ising model with ferromagnetic order is mapped on to 1D non-Hermitian Kitaev model with topological superconducting order and becomes exactly solvable. A hidden, continuous quantum phase transition is really normal–abnormal transition for fermionic correlation in the 1D non-Hermitian Kitaev model. In addition, similar hidden, continuous quantum phase transition is discovered in two-dimensional non-Hermitian transverse Ising model and thus becomes a universal feature in certain non-Hermitian many-body systems.