Wigner Function Research Articles

Preparation and properties of a non-Gaussian state by quantum catalysis with thermal state input

We theoretically introduce a new kind of non-Gaussian state, namely, the mixture of multiphoton added thermal states, achieved through multiphoton measurements at one output port of the beam splitter (termed multiphoton catalysis) for thermal state input. We discuss the success probabilities of detection (the normalization factors). Furthermore, we investigate their nonclassical properties based on the nonclassical depth, photon number distribution, and photocount distribution. It is shown that the multiphoton catalysis presents a better nonclassical performance by controlling the thermal parameter, catalyzed photon number and the transmissivity of the beam splitter. In addition, the nonclassicality is further examined through the partial negativity of the Wigner function, which reveals that multiphoton catalysis operation can prepare highly nonclassical quantum states. These results indicate that the preparation of a non-Gaussian states may have potential applications in the realms of quantum information processing and quantum metrology.

The Mth nonlinear coherent states

Since radiation field states are very important in the interaction with the various atomic structures, in this article we introduce a new type of quantum states, which we call ‘Mth nonlinear coherent states’. We define these states based on acting of the number operator M times nˆM on a nonlinear coherent state. By analyzing the photon distribution function and its overlap with the nonlinear coherent states, it becomes clear that these states are very similar to the corresponding nonlinear coherent states. We also investigate the nonclassical features of such states for different values of M, and it is determined that the occurrence of sub-Poissonian photon statistics, normal squeezing, and amplitude squared squeezing are considerable nonclassical properties of the Mth nonlinear coherent states. Also, examining the Husimi function shows a local behavior for the introduced states. Furthermore, by calculating the Wigner function, we find out that in some points of the phase space, the value of this function is negative which could be considered as a nonclassical feature for these states.

The entanglement and nonclassical properties of multi-photon amplified added based two-mode entangled coherent states

We explore into the non-classical characteristics of a new entangled coherent state (CS), also known as a multi-photon addition amplified coherent state (MPAACS) gn̂â†m, which is produced by adding any quantity of photons to two-mode coherent states with various gain factor values. A superposition of photon-added amplified two-mode CSs (MPAACS) also exists in this novel entangled state. It is proven that the gain factor and Laguerre polynomials are related to the MPAACS normalization factor. Analytical discussion is also had regarding entanglement and quantum statistical properties. Regarding concurrence, our findings further suggest that by selecting suitable values for the gain factor g, photon added (m,n), and coherent amplitude, entanglement for the output state can occur. Specifically, the enhancement of the entanglement can be found at large g values at fixed (m,n) in comparison to (m,n) values at fixed g and small amplitude with ϕ=0 or pi. Besides that, the analyses of nonclassicality reveal that, when comparing the results for several values (m,n) at fixed g and with several values g at fixed (m,n), the low-order antibunching effect, the cross-correlation function, and the Wigner function all support the sub-Poissonian statistics. When analyses of the nonclassicality and the negative value of the WF for ϕ=0 are larger at fixed parameter (m,n), but larger at g fixed for ϕ=π, the TMAA-TMC states perform better.

Chiral kinetic theory with self-energy corrections and neutrino spin Hall effect

We systematically derive the chiral kinetic theory for chiral fermions with collisions, including the self-energy corrections, from quantum field theories. We find that the Wigner functions and chiral kinetic equations receive both the classical and quantum corrections from the self-energies and their spacetime gradients. We also apply this formalism to study nonequilibrium neutrino transport due to the interaction with thermalized electrons and nucleons, as realized in core-collapse supernovae. We derive neutrino currents along magnetic fields and neutrino spin Hall effect induced by the density gradient at first order in the Fermi constant GF for anisotropic neutrino distributions. Published by the American Physical Society 2024

Properties of new even and odd nonlinear coherent states with different parameters

We construct a class of nonlinear coherent states (NLCSs) by introducing a more general nonlinear function and study their nonclassical properties, specifically the second-order correlation function [Formula: see text], Mandel parameter Q, squeezing, amplitude-squared squeezing and Wigner function of the optical field. The results indicate that the nonclassical properties of the new types of even and odd NLCSs crucially depend on the nonlinear functions. More concretely, we find that the new even NLCSs could exhibit the photon-bunching effect, whereas the new odd NLCSs could show the photon-antibunching effect. The degree of squeezing is also significantly affected by the parameter selection of these NLCSs. By employing various forms of nonlinear functions, it becomes possible to construct the NLCSs with diverse properties, thereby providing a theoretical foundation for the corresponding experimental investigations.

Numerical simulation of large-scale nonlinear open quantum mechanics

We introduce a method to solve nonlinear open quantum dynamics of a particle in situations where its state undergoes significant expansion in phase space while generating small quantum features at the phase-space Planck scale. Our approach involves simulating two steps. First, we transform the Wigner function into a time-dependent frame that leverages information from the classical trajectory to efficiently represent the quantum state in phase space. Next, we simulate the dynamics in this frame using a numerical method that implements this time-dependent nonlinear change of variables. To demonstrate the capabilities of our method, we examine the open quantum dynamics of a particle evolving in a one-dimensional weak quartic potential after initially being ground-state cooled in a tight harmonic potential. This approach is particularly relevant to ongoing efforts to design, optimize, and understand experiments targeting the preparation of macroscopic quantum superposition states of massive particles through nonlinear quantum dynamics. Published by the American Physical Society 2024

Unconditional Wigner-negative mechanical entanglement with linear-and-quadratic optomechanical interactions

The generation of entangled states that display negative values of the Wigner function in the quantum phase space is a challenging task, particularly elusive for massive, and possibly macroscopic, systems such as mechanical resonators. In this work, we propose two schemes based on reservoir engineering for generating Wigner-negative entangled states unconditionally. We consider two noninteracting mechanical resonators that are radiation-pressure coupled to either one or two common cavity fields; the optomechanical coupling with the field(s) features both a linear and quadratic part in the mechanical displacement and the cavity is driven at multiple frequencies. We show analytically that both schemes stabilize a Wigner-negative entangled state that combines the entanglement of a two-mode squeezed vacuum with a cubic nonlinearity, which we dub cubic-phase entangled (CPE) state. We then perform extensive numerical simulations to test the robustness of Wigner-negative entanglement attained by approximate CPE states stabilized in the presence of thermal decoherence. Published by the American Physical Society 2024

Wigner transport in linear electromagnetic fields

Applying a Weyl–Stratonovich transform to the evolution equation of the Wigner function in an electromagnetic field yields a multidimensional gauge-invariant equation which is numerically very challenging to solve. In this work, we apply simplifying assumptions for linear electromagnetic fields and the evolution of an electron in a plane (two-dimensional transport), which reduces the complexity and enables to gain first experiences with a gauge-invariant Wigner equation. We present an equation analysis and show that a finite difference approach for solving the high-order derivatives allows for reformulation into a Fredholm integral equation. The resolvent expansion of the latter contains consecutive integrals, which is favorable for Monte Carlo solution approaches. To that end, we present two stochastic (Monte Carlo) algorithms that evaluate averages of generic physical quantities or directly the Wigner function. The algorithms give rise to a quantum particle model, which interprets quantum transport in heuristic terms.

Wigner function of optical cumulant operator and its dissipation in thermo-entangled state representation

To conveniently calculate the Wigner function of the optical cumulant operator and its dissipation evolution in a thermal environment, in this paper, the thermo-entangled state representation is introduced to derive the general evolution formula of the Wigner function, and its relation to Weyl correspondence is also discussed. The method of integration within the ordered product of operators is essential to our discussion.

Phase-space evolution of quasiparticle excitations in electron gas

In this research, we use the dual lengthscale quasiparticle model for collective quantum excitations in electron gas to study the time evolution of the Wigner function. The linearized time-dependent Schrödinger–Poisson system for quasiparticles is used to study the dynamics of initial known stationary and damped solutions in an electron gas with arbitrary degree of degeneracy. The self-consistent potential in the Schrödinger–Poisson model is treated in a quite different manner in this analysis due to the effective coupling of the electrostatic field to the electron density, which leads to a modified Wigner function. It is shown that the modified Wigner function in the absence of external potential evolves similar to the system of free particles, a feature of collective quantum excitations which is quite analogous to freely evolving classical system of particles in the center of mass frame in the absence of external forces. The time evolution of the modified Wigner function reveals a grinding effect on large-amplitude density structures present at initial states, which is a characteristic feature of the Landau damping in plasmas. It is further shown that linear phase-space dynamics of spill-out electrons (damped quasiparticles) can be described similar to free quasiparticles with imaginary momentum. The later predicts the surface electron tunneling via the collective excitations of spill-out electrons at the half-space boundary, which is closely related to the Heisenberg's uncertainty principle. Current research can have applications in plasmonics and related fields.

Spin polarization and spin alignment from quantum kinetic theory with self-energy corrections

We derive the quantum kinetic theory for massive fermions with collision terms and self-energy corrections based on quantum field theory. We adopt an effective power-counting scheme with ℏ expansion to obtain the leading-order perturbative solutions of the vector and axial Wigner functions and the corresponding kinetic equations. We observe that both the on-shell relation and the structure of Wigner functions, along with the kinetic equations, are modified due to the presence of self-energies and their spacetime gradients. We further apply our formalism to investigate the spin polarization phenomena in relativistic heavy ion collisions and derive the modification to the spin polarization spectrum of massive quarks. We find that the gradient of vector self-energy plays a similar role to the background electromagnetic fields, which induces a more dominant contribution than the collisional effects by a naive power counting in the gradient expansion and weak coupling. Our findings could further modify the spin polarization of strange quarks and the spin alignment of ϕ mesons beyond local thermal equilibrium. Published by the American Physical Society 2024

Matrix formulation of the Gaussian expansion of coherent multiple beams in arbitrary dimensions.

Modeling the propagation of beams along laser beamlines is very challenging due to the multidimensional and multiscale configuration of the problem. Spatio-temporal couplings are particularly difficult to address with conventional numerical methods. Here we derive the Wigner function of a sum of Gaussian beams by calculating the multidimensional Fourier transform of the intercorrelation function of the fields. The matrix formulation allows for a simple propagation of the Wigner function in the framework of matrix optics. The relevancy of this approach is assessed by applying this model to one-dimensional and multidimensional configurations and by studying the influence of spatio-temporal couplings when considering propagation and dispersion by a diffraction grating.

Phase-space path-integral representation of the quantum density of states: Monte Carlo simulation of strongly correlated soft-sphere fermions.

The Wigner formulation of quantum mechanics is used to derive a path-integral representation of the quantum density of states (DOS) of strongly correlated fermions in the canonical ensemble. A path-integral Monte Carlo approach for the simulation of DOS and other thermodynamic functions is suggested. The derived Wigner function in the phase space resembles the Maxwell-Boltzmann distribution but allows for quantum effects. We consider a three-dimensional quantum system of strongly correlated soft-sphere fermions at different densities and temperatures. The calculated properties include the DOS, momentum distribution functions, spin-resolved radial distribution functions, potentials of mean force, and related energy levels obtained from the Bohr-Sommerfeld condition. We observe sharp peaks on DOS and momentum distribution curves, which are explained by the appearance of fermionic bound states.

Protecting quantum correlations of negative quantum states using weak measurement under non-Markovian noise

The weak measurement (WM) and quantum measurement reversal (QMR) are crucial in protecting the collapse of quantum states. The idea of WM and QMR has recently been used to protect and enhance quantum correlations and universal quantum teleportation (UQT) protocols. Here, we study the quantum correlations, maximal fidelity, and fidelity deviation of the two-qubit negative quantum states developed using discrete Wigner functions with(without) WM and QMR. To take into account the effect of a noisy environment, we evolve the states via non-Markovian amplitude damping and random telegraph noise quantum channels. To benchmark the performance of negative quantum states, we calculate their success probability. We compare our results with the two-qubit maximally entangled Bell state. Interestingly, we observe that some negative quantum states perform better with WM and QMR than the Bell state for different cases under evolution via noisy quantum channels.

Not All Probability Density Functions Are Tomograms.

This paper delves into the significance of the tomographic probability density function (pdf) representation of quantum states, shedding light on the special classes of pdfs that can be tomograms. Instead of using wave functions or density operators on Hilbert spaces, tomograms, which are the true pdfs, are used to completely describe the states of quantum systems. Unlike quasi-pdfs, like the Wigner function, tomograms can be analysed using all the tools of classical probability theory for pdf estimation, which can allow a better quality of state reconstruction. This is particularly useful when dealing with non-Gaussian states where the pdfs are multi-mode. The knowledge of the family of distributions plays an important role in the application of both parametric and nonparametric density estimation methods. We show that not all pdfs can play the role of tomograms of quantum states and introduce the conditions that must be fulfilled by pdfs to be "quantum".

Non-Gaussian quantum states generated via quantum catalysis and their statistical properties

Abstract A new kind of non-Gaussian quantum catalyzed state is proposed via multiphoton measurements and two-mode squeezing as an input of thermal state. The characteristics of generating multiphoton catalysis output state depends on the thermal parameter, catalyzed photon number and squeezing parameter. We then analyze the nonclassical properties by examining the photon number distribution, photocount distribution and partial negativity of the Wigner function. Our findings indicate that nonclassicality can be achieved through the implementation of multiphoton catalysis operations, and modulated by the thermal parameter, catalyzed photon number and squeezing parameter.

Local Structures in Proteins from Microsecond Molecular Dynamics Simulations: 2. The Role of Symmetry in GTPase Binding and Dimer Formation.

The Rho GTPase binding domain of Plexin-B1 (RBD) prevails in solution as dimer. Under appropriate circumstances, it binds the small GTPase Rac1 to yield the complex RBD-Rac1. Here, we study RBD dimerization and complex formation from a symmetry-based perspective using data derived from 1 μs long MD simulations. The quantities investigated are the local potentials, u(MD), prevailing at the N-H sites of the protein. These potentials are statistical in character providing an empirical description of the local structure. To establish more methodical description, a method for approximating them by explicit functions, u(simulated), was developed in the preceding article in this journal issue. These functions are combinations of analytical Wigner functions, DL,K, belonging to the D2h point group. The D2h subgroups Ag and B2u are found to dominate u(simulated); the B1u subgroup contributes in some cases. The Ag (B2u) functions have axial or rhombic symmetry. For the first time, local potentials in proteins can be quantitatively characterized in terms of their strength (rhombicity) evaluated by axial Ag (rhombic Ag and B2u) contributions. Until now, the chain-segment [β3-L3-β4] and to some extent the α2-helix have been associated with GTPase binding. Here, we find that this process causes an increase (decrease) in the potential strength of β3 and β4 (the preceding L2 loop and the remote chain-segment [(α2-helix)-(α2/β5-turn)-(β5-strand)]), suggesting effects of counterbalancing and allostery. There is evidence for the L2 loop being associated with RBD-GTPase binding. Until now only the L4 loop has been associated with RBD dimerization. The latter process is found to cause an increase (decrease) in the potential strength and rhombicity of the L4 loop (the adjacent chain-segment [(α2-helix)-(α2/β5-turn)-(β5-strand)]), suggesting counterbalancing activity. On average, the RBD dimer features stronger local potentials than RBD-Rac1. The novel information inherent in these findings is mesoscopic in character. Prospects of interest include exploring relation to atomistic force-field parameters.

Connecting Continuous and Discrete Wigner Functions Via GKP Encoding

Connecting Continuous and Discrete Wigner Functions Via GKP Encoding

On the fractional quark–antiquark confinement and symplectic quantum mechanics

Using the formalism of generalized fractional derivatives, a two-dimensional nonrelativistic meson system is studied. The mesons are interacting by a Cornell potential. The system is formulated in the domain of the symplectic quantum mechanics by means of the generalized fractional Nikiforov–Uvarov method. The corresponding Wigner function and the energy eigenvalues are then derived. The effect of fractional parameters [Formula: see text] and [Formula: see text] with the ground state solution is analyzed through the Wigner function for the charm–anticharm, bottom–antibottom and [Formula: see text] mesons. One of the fundamental achievements of such Cornell model is the determination of heavy quarkonia mass spectra. We have computed these masses and the present results are in agreement with the experimental data, improving previous theoretical results.

Non-Equilibrium Wigner Function and Application to Model of Catalyzed Polymerization.

The quantum Wigner function and non-equilibrium equation for a microscopic particle in one spatial dimension (1D) subject to a potential and a heat bath at thermal equilibrium are considered by non-trivially extending a previous analysis. The non-equilibrium equation yields a general hierarchy for suitable non-equilibrium moments. A new non-trivial solution of the hierarchy combining the continued fractions and infinite series thereof is obtained and analyzed. In a short thermal wavelength regime (keeping quantum features adequate for chemical reactions), the hierarchy is approximated by a three-term one. For long times, in turn, the three-term hierarchy is replaced by a Smoluchovski equation. By extending that 1D analysis, a new model of the growth (polymerization) of a molecular chain (template or te) by binding an individual unit (an atom) and activation by a catalyst is developed in three spatial dimensions (3D). The atom, te, and catalyst move randomly as solutions in a fluid at rest in thermal equilibrium. Classical statistical mechanics describe the te and catalyst approximately. Atoms and bindings are treated quantum-mechanically. A mixed non-equilibrium quantum-classical Wigner-Liouville function and dynamical equations for the atom and for the te and catalyst, respectively, are employed. By integrating over the degrees of freedom of te and with the catalyst assumed to be near equilibrium, an approximate Smoluchowski equation is obtained for the unit. The mean first passage time (MFPT) for the atom to become bound to the te, facilitated by the catalyst, is considered. The resulting MFPT is consistent with the Arrhenius formula for rate constants in chemical reactions.