arXiv subjects

N. L. Harshman

Publications and source records attributed to N. L. Harshman.

At least 37 records · Page 2Linked to original sources

arXiv2015

One-Dimensional Traps, Two-Body Interactions, Few-Body Symmetries: I. One, Two, and Three Particles

This is the first in a pair of articles that classify the configuration space and kinematic symmetry groups for $N$ identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies. The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly-unitary interactions are universal, i.e.\ independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article sequentially analyzes the symmetries of one, two and three particles in asymmetric, symmetric, and harmonic traps; the sequel article treats the $N$ particle case.

quant-ph↗

arXiv2015

One-Dimensional Traps, Two-Body Interactions, Few-Body Symmetries: II. $N$ Particles

This is the second in a pair of articles that classify the configuration space and kinematic symmetry groups for $N$ identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies. The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly-unitary interactions are universal, i.e.\ independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article analyzes the symmetries of $N$ particles in asymmetric, symmetric, and harmonic traps; the prequel article treats the one, two and three particle cases.

quant-ph↗

arXiv2014

Spectroscopy for a Few Atoms Harmonically-Trapped in One Dimension

Spectroscopic labels for a few particles with spin that are harmonically trapped in one-dimension with effectively zero-range interactions are provided by quantum numbers that characterize the symmetries of the Hamiltonian: permutations of identical particles, parity inversion, and the separability of the center-of-mass. The exact solutions for the non-interacting and infinitely repulsive cases are reduced with respect to these symmetries. This reduction explains how states of single-component and multi-component fermions and bosons transform under adiabatic evolution from non-interacting to strong hard-core repulsion. These spectroscopic methods also clarify previous analytic and numerical results for intermediate values of interaction strength. Several examples, including adiabatic mapping for two-component fermionic states in the cases $N=3-5$, are provided.

quant-ph↗

arXiv2012

Symmetries of Three Harmonically-Trapped Particles in One Dimension

We present a method for solving trapped few-body problems and apply it to three equal-mass particles in a one-dimensional harmonic trap, interacting via a contact potential. By expressing the relative Hamiltonian in Jacobi cylindrical coordinates, i.e. the two-dimensional version of three-body hyperspherical coordinates, we discover an underlying ${\rm C}_{6v}$ symmetry. This symmetry simplifies the calculation of energy eigenstates of the full Hamiltonian in a truncated Hilbert space constructed from the trap Hamiltonian eigenstates. Particle superselection rules are implemented by choosing the relevant representations of ${\rm C}_{6v}$. We find that the one-dimensional system shows nearly the full richness of the three-dimensional system, and can be used to understand separability and reducibility in this system and in standard few-body approximation techniques.

quant-ph↗

arXiv2012

Observables and Entanglement in the Two-Body System

Using the quantum two-body system as a familiar model, this talk will describe how entanglement can be used to select preferred observables for interrogating a physical system. The symmetries and dynamics of the quantum two-body system provide a backdrop for testing the relativity of entanglement with respect to observable-induced tensor product structures. We believe this exploration leads us to a general statement: the physically-meaningful observable subalgebras are the ones that minimize entanglement in typical states.

quant-ph↗

arXiv2010

Entanglement in Massive Coupled Oscillators

This article investigates entanglement of the motional states of massive coupled oscillators. The specific realization of an idealized diatomic molecule in one-dimension is considered, but the techniques developed apply to any massive particles with two degrees of freedom and a quadratic Hamiltonian. We present two methods, one analytic and one approximate, to calculate the interatomic entanglement for Gaussian and non-Gaussian pure states as measured by the purity of the reduced density matrix. The cases of free and trapped molecules and hetero- and homonuclear molecules are treated. In general, when the trap frequency and the molecular frequency are very different, and when the atomic masses are equal, the atoms are highly-entangled for molecular coherent states and number states. Surprisingly, while the interatomic entanglement can be quite large even for molecular coherent states, the covariance of atomic position and momentum observables can be entirely explained by a classical model with appropriately chosen statistical uncertainty.

quant-ph↗

arXiv2010

Inertial and gravitational mass in quantum mechanics

We show that in complete agreement with classical mechanics, the dynamics of any quantum mechanical wave packet in a linear gravitational potential involves the gravitational and the inertial mass only as their ratio. In contrast, the spatial modulation of the corresponding energy wave function is determined by the third root of the product of the two masses. Moreover, the discrete energy spectrum of a particle constrained in its motion by a linear gravitational potential and an infinitely steep wall depends on the inertial as well as the gravitational mass with different fractional powers. This feature might open a new avenue in quantum tests of the universality of free fall.

quant-ph↗

arXiv2008

Entanglement Mechanisms in One-Dimensional Potential Scattering

When two non-relativistic particles scatter in one dimension, they can become entangled. This entanglement process is constrained by the symmetries of the scattering system and the boundary conditions on the incoming state. Applying these constraints, three different mechanisms of entanglement can be identified: the superposition of reflected and transmitted modes, momentum correlations of the reflected mode due to inversion of the relative momentum, and momentum correlations in the transmitted and reflected modes due to dependence of the scattering amplitude on the relative momentum. We consider three standard potentials, the hard core, Dirac delta, and double Dirac delta, and show that the relative importance of these mechanisms depends on the interaction and on the properties of the incoming wave function. We find that even when the momenta distributions of the incoming articles are sharply peaked, entanglement due to the momentum correlations generated by reflection can be quite large for particles with unequal mass.

quant-ph↗

arXiv2007

Entanglement Generation in the Scattering of One-Dimensional Particles

This article provides a convenient framework for quantitative evaluation of the entanglement generated when two structureless, distinguishable particles scatter non-relativistically in one dimension. It explores how three factors determine the amount of entanglement generated: the momentum distributions of the incoming particles, their masses, and the interaction potential. Two important scales emerge, one set by the kinematics and one set by the dynamics. This method also provides two approximate analytic formulas useful for numerical evaluation of entanglement and reveals an interesting connection between purity, linear coordinate transformations, and momentum uncertainties.

quant-ph↗

arXiv2007

Continuous-Discrete Entanglement: An Example with Non-Relativistic Particles

This article discusses entanglement between two subsystems, one with discrete degrees of freedom and the other with continuous degrees of freedom. The overlap integral between continuous variable wave functions emerges as an important parameter to characterize this kind entanglement. ``Beam-like'' entanglement and ``shape-like'' entanglement are contrasted. One example of this kind of entanglement is between between the spin degrees of freedom and the momentum degrees of freedom for a non-relativistic particle. This intraparticle entanglement is Galilean invariant.

quant-ph↗

arXiv2006

Tensor Product Structures, Entanglement, and Particle Scattering

Particle systems admit a variety of tensor product structures (TPSs) depending on the complete system of commuting observables chosen for the analysis. Different notions of entanglement are associated with these different TPSs. Global symmetry transformations and dynamical transformations factor into products of local unitary operators with respect to certain TPSs and not with respect to others. Symmetry-invariant and dynamical-invariant TPSs and corresponding measures of entanglement are defined for particle scattering systems.

quant-ph↗

arXiv2006

Galilean and Dynamical Invariance of Entanglement in Particle Scattering

Particle systems admit a variety of tensor product structures (TPSs) depending on the algebra of observables chosen for analysis. Global symmetry transformations and dynamical transformations may be resolved into local unitary operators with respect to certain TPSs and not with respect to others. Symmetry-invariant and dynamical-invariant TPSs are defined and various notions of entanglement are considered for scattering states.

quant-ph↗

arXiv2006

Dynamical Entanglement in Non-Relativistic, Elastic Scattering

This article considers dynamical entanglement in non-relativistic particle scattering. Three questions are explored: what kinds of entanglement occur in this system, how do global symmetries constrain entanglement, and how do the boundary conditions of scattering affect dynamical entanglement? First, a simple model of scattering spin systems is considered, then the full system is discussed.

quant-ph↗

arXiv2006

Limits on entanglement in rotationally-invariant scattering of spin systems

This paper investigates the dynamical generation of entanglement in scattering systems, in particular two spin systems that interact via rotationally-invariant scattering. The spin degrees of freedom of the in-states are assumed to be in unentangled, pure states, as defined by the entropy of entanglement. Because of the restriction of rotationally-symmetric interactions, perfectly-entangling S-matrices, i.e. those that lead to a maximally entangled out-state, only exist for a certain class of separable in-states. Using Clebsch-Gordan coefficients for the rotation group, the scattering phases that determine the S-matrix are determined for the case of spin systems with $σ= 1/2$, 1, and 3/2.

quant-ph↗

arXiv2005

Poincare Semigroup Symmetry as an Emergent Property of Unstable Systems

The notion that elementary systems correspond to irreducible representations of the Poincare group is the starting point for this paper, which then goes on to discuss how a semigroup for the time evolution of unstable states and resonances could emerge from the underlying Poincare symmetry. Important tools in this analysis are the Clebsch-Gordan coefficients for the Poincare group.

hep-ph↗

arXiv2005

Dynamical Entanglement in Particle Scattering

This paper explores the connections between particle scattering and quantum information theory in the context of the non-relativistic, elastic scattering of two spin-1/2 particles. An untangled, pure, two-particle in-state is evolved by an S-matrix that respects certain symmetries and the entanglement of the pure out-state is measured. The analysis is phrased in terms of unitary, irreducible representations (UIRs) of the symmetry group in question, either the rotation group for the spin degrees of freedom or the Galilean group for non-relativistic particles. Entanglement may occurs when multiple UIRs appear in the direct sum decomposition of the direct product in-state, but it also depends of the scattering phase shifts. \keywords{dynamical entanglement, scattering, Clebsch-Gordan methods}

quant-ph↗

arXiv2005

Basis States for Relativistic, Dynamically-Entangled Particles

In several recent papers on entanglement in relativistic quantum systems and relativistic Bell's inequalities, relativistic Bell-type two-particle states have been constructed in analogy to non-relativistic states. These constructions do not have the form suggested by relativistic invariance of the dynamics. Two relativistic formulations of Bell-type states are shown for massive particles, one using the standard Wigner spin basis and one using the helicity basis. The construction hinges on the use of Clebsch-Gordan coefficients of the Poincaré group to reduce the direct product of two unitary irreducible representations (UIRs) into a direct sum of UIRs.

quant-ph↗

arXiv2004

Clebsch-Gordan Coefficients for the Extended Quantum-Mechanical Poincaré Group and Angular Correlations of Decay Products

This paper describes Clebsch-Gordan coefficients (CGCs) for unitary irreducible representations (UIRs) of the extended quantum mechanical Poincaré group $\pt$. `Extended' refers to the extension of the 10 parameter Lie group that is the Poincaré group by the discrete symmetries $C$, $P$, and $T$; `quantum mechanical' refers to the fact that we consider projective representations of the group. The particular set of CGCs presented here are applicable to the problem of the reduction of the direct product of two massive, unitary irreducible representations (UIRs) of $\pt$ with positive energy to irreducible components. Of the sixteen inequivalent representations of the discrete symmetries, the two standard representations with $U_C U_P = \pm 1$ are considered. Also included in the analysis are additive internal quantum numbers specifying the superselection sector. As an example, these CGCs are applied to the decay process of the $Υ(4S)$ meson.

hep-ph↗