Model Of Quantum Particle Research Articles

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.

On free fall of fermions and antifermions

We propose a model describing spin-half quantum particles in curved spacetime in the framework of quantum field theory. Our model is based on embodying Einstein’s equivalence principle and general covariance in the definition of quantum-particle states. With this model at hand, we compute several observables which characterise spin-half quantum particles in a gravitational field. In particular, we find that spin precesses in a normal Fermi frame, even in the absence of torsion. The effect appears to be complementary to free-fall non-universality we have recently reported about for spinless quantum particles. Furthermore, we find that quantum-particle gravitational-potential energy is insensitive to wave-packet spreading in the Earth’s gravitational field, that is responsible for the non-universality of free fall in quantum theory. This theoretical result provides another channel for the experimental study of our quantum-particle model by using gravitational spectrometers. Finally, we also find that (elementary) fermions and antifermions are indistinguishable in gravity.

Combinatorics of the quantum symmetric simple exclusion process, associahedra and free cumulants

The quantum symmetric simple exclusion process (QSSEP) is a model of quantum particles hopping on a finite interval and satisfying the exclusion principle. Recently, Bernard and Jin have studied the fluctuations of the invariant measure for this process, when the number of sites goes to infinity. These fluctuations are encoded into polynomials, for which they have given equations and proved that these equations determine the polynomials completely. In this paper, we give an explicit combinatorial formula for these polynomials, in terms of Schröder trees. We also show that, quite surprisingly, these polynomials can be interpreted as free cumulants of a family of commuting random variables.

Q-Deformation and Free Statistics for Interaction of a Field and a Particle

Emerging of free (or quantum Boltzmann) statistics for a model of quantum particle interacting with quantum field is described in the stochastic limit without dipole approximation. The quantum field is considered in a Gaussian (for example temperature) state. Entangled operators which describe interaction of the field and the particle satisfy the q-deformed relations which in the stochastic limit generate free statistics.

Distribution of Brownian Coincidences

We study the probability distribution, $P_N(T)$, of the coincidence time $T$, i.e. the total local time of all pairwise coincidences of $N$ independent Brownian walkers. We consider in details two geometries: Brownian motions all starting from $0$, and Brownian bridges. Using a Feynman-Kac representation for the moment generating function of this coincidence time, we map this problem onto some observables in three related models (i) the propagator of the Lieb Liniger model of quantum particles with pairwise delta function interactions (ii) the moments of the partition function of a directed polymer in a random medium (iii) the exponential moments of the solution of the Kardar-Parisi-Zhang equation. Using these mappings, we obtain closed formulae for the probability distribution of the coincidence time, its tails and some of its moments. Its asymptotics at large and small coincidence time are also obtained for arbitrary fixed endpoints. The universal large $T$ tail, $P_N(T) \sim \exp(- 3 T^2/(N^3-N))$ is obtained, and is independent of the geometry. We investigate the large deviations in the limit of a large number of walkers through a Coulomb gas approach. Some of our analytical results are compared with numerical simulations.

The Quantum Group and Harper Equation on a Honeycomb Lattice

The tight-binding model of quantum particle on a honeycomb lattice is investigated in the presence of homogeneous magnetic field. The one-particle Hamiltonian is expressed in terms of the generators of the quantum group Uq(sl2). The corresponding Harper equation is rewritten as a system of two coupled functional equations in the complex plane. The system is shown to exhibit certain symmetry that allows one to resolve the entanglement, and the basic single equation determining the eigenvalues and eigenstates is obtained. Equations specifying the roots of eigenstates in the complex plane are found.

Decoherence induced by long wavelength gravitons

We discuss how a background bath of gravitons can induce decoherence of quantum systems. The mechanism is dephasing, the loss of phase coherence due to quantum geometry fluctuations caused by the gravitons. This effect is illustrated in a simple analog model of quantum particles in a cavity whose walls undergo position fluctuations, and create the same effect expected from spacetime geometry fluctuations. We obtain an explicit result for the decoherence rate in the limit where the graviton wavelength is large compared to the size of the quantum system, and make some estimates for this rate.

Directed-polymer systems explored via their quantum analogs: General polymer interactions and their consequences

The impact of polymer-polymer interactions of various types on the thermodynamics, structure, and accommodation of topological constraints is addressed for systems comprising many directed polymers in two spatial dimensions. The approach is predicated on the well-known equivalence between the classical equilibrium statistical mechanics of directed polymers in two spatial dimensions and the imaginary-time quantum dynamics of particles in one spatial dimension, originally exploited by P.-G. de Gennes [P.-G. de Gennes, J. Chem. Phys. 48, 2257 (1968)]. Known results concerning two exactly solvable microscopic models of quantum particles moving in one spatial dimension---the Lieb-Liniger model of contact interactions and the Calogero-Sutherland model of long-range interactions---are used to shed light on the behavior of the corresponding polymeric systems. In addition, the technique of bosonization is used to reveal how generic polymer interactions give rise to an emergent polymer fluid that has universal collective excitations. Additionally, the response of the system to topological constraints such as pins though which polymers cannot pass is explored. Immediately on the compressed side of a pin there is a divergent pile-up in polymer density, while on the other side there is a gap of finite area in which polymer density is negligible. Comparison of this response to that of a system of simply noncrossing (i.e., noncrossing but otherwise noninteracting) directed polymers, explored in a companion paper, reveals that generic interactions leave the structure quantitatively unchanged on the line transverse to the pin, and leave it qualitatively unchanged throughout the two dimensions of the system's extent. Furthermore, the free-energy cost associated with a pin that partitions a system having generic interactions is found to be proportional to the pin-partitioning cost for a system of simply noncrossing polymers.

The quantum group, Harper equation and structure of Bloch eigenstates on a honeycomb lattice

The tight-binding model of quantum particles on a honeycomb lattice is investigated in the presence of a homogeneous magnetic field. Provided the magnetic flux per unit hexagon is a rational of the elementary flux, the one-particle Hamiltonian is expressed in terms of the generators of the quantum group Uq(sl2). Employing the functional representation of the quantum group Uq(sl2), the Harper equation is rewritten as a system of two coupled functional equations in the complex plane. For the special values of quasi-momentum, the entangled system admits solutions in terms of polynomials. The system is shown to exhibit a certain symmetry allowing us to resolve the entanglement, and a basic single equation determining the eigenvalues and eigenstates (polynomials) is obtained. Equations specifying the locations of the roots of polynomials in the complex plane are found. Employing numerical analysis, the roots of polynomials corresponding to different eigenstates are solved and diagrams exhibiting the ordered structure of one-particle eigenstates are depicted.

SELF-SIMILAR BEHAVIOR IN SEMICONDUCTOR SUPERLATTICES

The scattering of particles in fractal superlattices has been analyzed by means of the transfer matrix method. The fractal superlattice consists of alternating layers of semiconductor materials following the rule of a Cantor set. This problem can be represented by a model of quantum particles scattering in piecewise constant potential wells. Fractal properties of the reflection coefficient versus the particle energy curves are examined comparatively to the curves when using the corresponding periodic potentials. The degree of self-similarity of the output variables has been quantified by means of the correlation function.

Observation and description of phonon interactions in molecular dynamics simulations

Three-phonon interactions are identified and analyzed in nonequilibrium molecular dynamics simulations of the Lennard-Jones face-centered cubic crystal. The small simulation cells studied and the classical nature of the simulations lead to a different description of the phonon transport than the quantum-particle model. The selection process is strictly governed by the mode wave vectors. In determining the strength of an allowed interaction, one must consider the mode polarizations and frequencies---the latter in the context of internal resonance. The results are consistent with intrinsic scattering rates calculated directly from the Lennard-Jones potential.

GROUND STATE OF THE MASSLESS NELSON MODEL WITHOUT INFRARED CUTOFF IN A NON-FOCK REPRESENTATION

We consider a model of quantum particles coupled to a massless quantum scalar field, called the massless Nelson model, in a non-Fock representation of the time-zero fields which satisfy the canonical commutation relations. We show that the model has a ground state for all values of the coupling constant even in the case where no infrared cutoff is made. The non-Fock representation used is inequivalent to the Fock one if no infrared cutoff is made.

Essential Spectrum of a Self-Adjoint Operator on an Abstract Hilbert Space of Fock Type and Applications to Quantum Field Hamiltonians

We establish general theorems on locating the essential spectrum of a self-adjoint operator of the form A⊗I+I⊗dΓ(S)+HI on the tensor product H⊗Fb(K) of a Hilbert space H and the abstract Boson Fock space Fb(K) over a Hilbert space K, where A is a self-adjoint operator on H bounded from below, dΓ(S) is the second quantization of a nonnegative self-adjoint operator S on K, and HI is a symmetric operator on H⊗Fb(K). We then apply the theorems to the generalized spin-boson model by A. Arai and M. Hirokawa (1997, J. Funct. Anal.151, 455–503) and a general class of models of quantum particles coupled to a Bose field including the Pauli–Fierz model in nonrelativistic quantum electrodynamics.