Multiparticle Systems Research Articles

Quantum entanglement in multiparticle systems of two-level atoms

We propose the necessary and sufficient condition for the presence of quantum entanglement in arbitrary symmetric pure states of two-level atomic systems. We introduce a parameter to quantify quantum entanglement in such systems. We express the inherent quantum fluctuations of a composite system of two-level atoms as a sum of the quantum fluctuations of the individual constituent atoms and their correlation terms. This helps to separate out and study solely the quantum correlations among the atoms and obtain the criterion for the presence of entanglement in such multiatomic systems.

Three-dimensional ensemble averages for tensorial interactions in partially oriented, multi-particle systems

In a variety of three-dimensional, multi-particle systems, interactions of tensorial form occur between individual components and an applied stimulus that operates uniformly throughout the ensemble. When each material component has an identical, fixed orientation, its own response is replicated in the observed form of behaviour by the system as a whole, with respect to the angular disposition of the stimulus. The same complete correlation between microscopic and macroscopic response does not, however, operate in other systems where there is a degree of orientational freedom amongst the microscopic components. One limiting case is where the extent of such freedom allows random orientations, such conditions delivering a system that is macroscopically isotropic. When each particle has incomplete orientational freedom, the formulation is in general less mathematically tractable; here, the theory required to describe ensemble response yields analytically solvable equations only in cases where the distribution function takes a relatively simple form, such as a scalar product, and few explicit results are available. This paper addresses general systems in which there is an orientation-dependent complex exponential factor, weighting the response of each component. By means of irreducible tensor decomposition, results are determined for tensor interactions up to and including rank 2, where the weighting exponent is also of any rank up to the same order. Illustrative applications are drawn from the theory of laser particle orientation and photon absorption.

Simulating strongly correlated multiparticle systems in a truncated Hilbert space

Representing a strongly interacting multi-particle wave function in a finite product basis leads to errors. Simple rescaling of the contact interaction can preserve the low-lying energy spectrum and long-wavelength structure of wave functions in one-dimensional systems and thus correct for the basis set truncation error. The analytic form of the rescaling is found for a two-particle system where the rescaling is exact. Detailed comparison between finite Hilbert space calculations and exact results for up to 5 particles show that rescaling can significantly improve the accuracy of numerical calculations in various external potentials. In addition to ground state energies, the low-lying excitation spectrum, density profile and correlation functions are studied. The results give a promising outlook for numerical simulations of trapped ultracold atoms.

Voronoi-based discretizations for fracture analysis of particulate materials

Damage and mass transport behavior of particulate materials depend on the spatial distribution and orientation of the phase fractions and the properties of their interfaces. Discretization of this three-dimensional setting is the first critical step in the modeling of such behavior. In this paper, a general procedure is developed for discretizing particulate materials, such as particles embedded within a cementing matrix, in which the particle inclusions are assumed to be randomly generated polyhedra. Polyhedron discretization is based on the Voronoi tessellation of a set of points, which is obtained through a multistep process that relies on the concept of point saturation. The matrix–particle and/or particle–particle interfaces are explicitly constructed. To demonstrate the utility of the approach, uniaxial tensile tests (of a single particle, embedded in a matrix) are simulated using a lattice model. It is seen that interface debonding is strongly affected by inclusion shape and orientation with respect to the tension field. A procedure is developed for the discretization and analysis of multi-particle systems.

Experimental amplification of an entangled photon: what if the detection loophole is ignored?

The experimental verification of quantum features, such as entanglement, at large scales is extremely challenging because of environment-induced decoherence. Indeed, measurement techniques for demonstrating the quantumness of multiparticle systems in the presence of losses are difficult to define, and if they are not sufficiently accurate they can provide wrong conclusions. We present a Bell test where one photon of an entangled pair is amplified and then detected by threshold detectors, whose signals undergo postselection. The amplification is performed by a classical machine, which produces a fully separable micro–macro state. However, by adopting such a technique one can surprisingly observe a violation of the Clauser–Horne–Shimony–Holt inequality. This is due to the fact that ignoring the detection loophole opened by the postselection and the system losses can lead to misinterpretations, such as claiming micro–macro entanglement in a setup where evidently it is not present. By using threshold detectors and postselection, one can only infer the entanglement of the initial pair of photons, and so micro–micro entanglement, as is further confirmed by the violation of a nonseparability criterion for bipartite systems. How to detect photonic micro–macro entanglement in the presence of losses with the currently available technology remains an open question.

Symplectic Geometry of Entanglement

We present a description of entanglement in composite quantum systems in terms of symplectic geometry. We provide a symplectic characterization of sets of equally entangled states as orbits of group actions in the space of states. In particular, using Kostant-Sternberg theorem, we show that separable states form a unique Kaehler orbit, whereas orbits of entanglement states are characterized by different degrees of degeneracy of the canonical symplectic form on the complex projective space. The degree of degeneracy may be thus used as a new geometric measure of entanglement and we show how to calculate it for various multiparticle systems providing also simple criteria of separability. The presented method is general and can be applied also under different additional symmetry conditions stemming, eg. from the indistinguishability of particles.

Simulation of thermal ionization in a dense helium plasma by the Feynman path integral method

The region of equilibrium states is studied where the quantum nature of the electron component and a strong nonideality of a plasma play a key role. The problem of negative signs in the calculation of equilibrium averages a system of indistinguishable quantum particles with a spin is solved in the macroscopic limit. It is demonstrated that the calculation can be conducted up to a numerical result. The complete set of symmetrized basis wave functions is constructed based on the Young symmetry operators. The combinatorial weight coefficients of the states corresponding to different graphs of connected Feynman paths in multiparticle systems are calculated by the method of random walk over permutation classes. The kinetic energy is calculated using a viral estimator at a finite pressure in a statistical ensemble with flexible boundaries. Based on the methods developed in the paper, the computer simulation is performed for a dense helium plasma in the temperature range from 30000 to 40000 K. The equation of state, internal energy, ionization degree, and structural characteristic of the plasma are calculated in terms of spatial correlation functions. The parameters of a pseudopotential plasma model are estimated.

Drag force model corrections based on nonuniform particle distributions in multi-particle systems

The drag force is one major particle–fluid interaction force in multiphase flows. Conventional drag force models do not accurately consider the effect of nonuniform particle distributions which limits their applicability with considerable nonuniform particle distributions. A meshless method (Element-Free Gelerkin method) was used for direct numerical simulations to simulate real particle collisions and investigate the drag coefficient variations in various nonuniform particle distributions. The classical drag force model is corrected using nonuniformity coefficients based on various nonuniform particle distributions in various multiphase flow types. The results show that the nonuniform particle distribution greatly affects the drag coefficient with different compacting directions having different results. In most situations, only the effect of compacting in the direction perpendicular to the main flow direction needs to be considered and the effect of compacting in the main flow direction can be neglected. The particle nonuniformity in the direction perpendicular to the main flow direction reflects the overall dynamic effect of the drag coefficient while the nonuniformity in the main flow direction reflects local fluctuations. Thus, the modified drag force model considering the nonuniform distribution more accurately reflects the dynamic effect of the drag coefficient. It can be more widely used than conventional drag force models.

Creating and observing N-partite entanglement with atoms

The Mermin inequality provides a criterion for experimentally ruling out local-realistic descriptions of multiparticle systems. A violation of this inequality means that the particles must be entangled, but does not, in general, indicate whether N-partite entanglement is present. For this, a stricter bound is required. Here we discuss this bound and use it to propose two different schemes for demonstrating N-partite entanglement with atoms. The first scheme involves Bose–Einstein condensates trapped in an optical lattice and the second uses Rydberg atoms in microwave cavities.

Surface reconstruction using Delaunay triangulation for applications in life sciences

The Delaunay triangulation is an established method to define neighborhood relations in multi-particle systems. In particular, this method was employed for interacting multi-cellular systems in Biology. The extension of this method to a sub-cellular level that considers the membrane and the inner structure of cells is not straight forward and subject of this article. It is the objective to use a three-dimensional Delaunay-triangulation as a basis for the definition of a triangulation of a subset of particles that form a surface. An essential problem of this objective is the conservation of the number of particles belonging to the surface. This excludes established surface reconstruction algorithms. The presented algorithm allows the definition of a triangulation within a subset of particles attributed to a surface without the deletion of particles. A particular challenge is the deletion of connection that infer three-dimensional structures in the surface. The presented method is suitable for many configurations. Its performance and its limitations are analyzed and discussed. The developed algorithm for the reconstruction of connections in a surface is suitable to be used for simulations of biological cells because of the inherent conservation of the number of particles attributed to the membrane.

Mathematical analysis of the dimensional scaling technique for the Schrödinger equation with power-law potentials

The dimensional scaling (D-scaling) technique is an innovative asymptotic expansion approach to study the multiparticle systems in molecular quantum mechanics. It enables the calculation of ground and excited state energies of quantum systems without having to solve the Schrödinger equation. In this paper, we present a mathematical analysis of the D-scaling technique for the Schrödinger equation with power-law potentials. By casting the D-scaling technique in an appropriate variational setting and studying the corresponding minimization problem, the D-scaling technique is justified rigorously. A new asymptotic dimensional expansion scheme is introduced to compute asymptotic expansions for ground state energies.

Solved problems in classical mechanics: analytical and numerical solutions with comments

1. Introduction 2. Miscellanea 3. One-dimensional motion 4. Linear oscillations 5. Energy and potentials 6. Momentum and angular momentum 7. Motion in two and three dimensions 8. Spherically symmetric potentials 9. The Coulomb and oscillator problems 10. Two-body problems 11. Multi-particle systems 12. Rigid bodies 13. Nonlinear oscillations 14. Translation and Rotation of the reference frame 15. The relativity principle and some of its consequences

New Generalization of Supersymmetric Quantum Mechanics to Arbitrary Dimensionality or Number of Distinguishable Particles

We present here a new approach to generalize supersymmetric quantum mechanics to treat multiparticle and multidimensional systems. We do this by introducing a vector superpotential in an orthogonal hyperspace. In the case of N distinguishable particles in three dimensions this results in a vector superpotential with 3N orthogonal components. The original scalar Schrödinger operator can be factored using a 3N-component gradient operator and introducing vector "charge" operators: Q(1) and Q(1)(dagger). Using these operators, we can write the original (scalar) Hamiltonian as H(1) = Q(1)(dagger) x Q(1) + E(0)((1)), where E(0)((1)) is the ground-state energy. The second sector Hamiltonian is a tensor given by H(2) = Q(1)Q(1)(dagger) + E(0)((1)) and is isospectral with H(1). The vector ground state of sector 2, psi(0)((2)), canbe used with the charge operator Q(1)(dagger) to obtain the excited-state wave function of the first sector. In addition, we show that H(2) can also be factored in terms of a sector 2 vector superpotential with components W(2j) = -(partial partial differential ln psi(0j)((2)))/partial partial differentialx(j). Here psi(0j)((2)) is the jth component of psi(0)((2)). Then one obtains charge operators Q(2) and Q(2)(dagger) so that the second sector Hamiltonian can be written as H(2) = Q(2)(dagger)Q(2) + E(0)((2)). This allows us to define a third sector Hamiltonian which is a scalar, H(3) = Q(2) x Q(2)(dagger) + E(0)((2)). This prescription continues with the sector Hamiltonians alternating between scalar and tensor forms, both of which can be treated by the variational method to obtain approximate solutions to both scalar and tensor sectors. We demonstrate the approach with examples of a pair of separable 1D harmonic oscillators and the example of a nonseparable 2D anharmonic oscillator (or equivalently a pair of coupled 1D oscillators). We consider both degenerate and nondegenerate cases. We also present a generalization to arbitrary curvilinear coordinate systems in the Appendix.

Exact solvability of interacting many body lattice systems

We address the problem of exactly describing stochastic nonequilibrium systems that are widely used to model one-dimensional transport, in biology, traffic flow and others. We review the matrix product states ansatz to interacting multiparticle systems and its extension to a tridiagonal (generalized Onsager) algebra approach. The stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra defined by the dynamics of the process. The states involved in the matrix dements are determined by the boundary conditions. This reflects the intriguing feature of open systems that the bulk behaviour in the steady state strongly depends on the boundary rates. The importance of the boundary conditions manifests itself in the fact that the boundary operators are generators of a tridiagonal algebra whose irreducible modules are the Askey-Wilson polynomials. The matrices of the matrix product ansatz obey the tridiagonal algebraic relations for particular values of the structure, constants. Previously known representations, both infinite dimensional and finite dimensional ones, are recovered within the tridiagonal framework. The boundary Askey-Wilson and. tridiagonal symmetry is the deep algebraic property of driven diffusive systems allowing for the exact solvability in the steady state and the exact description of the stochastic dynamics.

Symmetry effects on the optical coupling between plasmonic nanoparticles with applications to hierarchical structures

We present theoretical studies on the collective optical response of symmetric configurations of metallic nanoparticles. We show that within the electrostatic approximation, the surface plasmon resonance of these symmetric multiparticle systems can be expressed as symmetry-adapted linear combinations of the plasmon modes of each particle of the ensemble, closely resembling the situation encountered in molecular systems. By making use of group theoretical arguments, we show that such linear combinations can be written down by simple geometrical considerations through the use of point group character tables, without using extensive numerical computations. Furthermore, we apply this formalism to study the coupling of hierarchical arrays containing a large number of nanoparticles. This theory thus provides an intuitive and formal approach for the rational design of plasmonic nanostructures.

Superconformal Calogero Models as a Gauged Matrix Mechanics

We present basics of the gauged superfield approach to constructing the N-superconformal multi-particle Calogero-type systems developed in arXiv:0812.4276, arXiv:0905.4951 and arXiv:0912.3508. This approach is illustrated by multi-particlesystems possessing SU(1, 1|1) and D(2, 1; α) supersymmetries, as well as by the model of new N = 4 superconformal quantum mechanics.

The QM‐MM interface for CHARMM‐deMon

We present a new QM/MM interface for fast and efficient simulations of organic and biological molecules. The CHARMM/deMon interface has been developed and tested to perform minimization and atomistic simulations for multi-particle systems. The current features of this QM/MM interface include readability for molecular dynamics, tested compatibility with Free Energy Perturbation simulations (FEP) using the dual topology/single coordinate method. The current coupling scheme uses link atoms, but further extensions of the code to incorporate other available schemes are planned. We report the performance of different levels of theory for the treatment of the QM region, while the MM region was represented by a classical force-field (CHARMM27) or a polarizable force-field based on a simple Drude model. The current QM/MM implementation can be coupled to the dual-thermostat method and the VV2 integrator to run molecular dynamics simulations.

Origins of Spin Noise

Spin systems, like any multiparticle systems, show fluctuations. Fluctuations are central to the discussion of spin relaxation, but when the spin system is in a magnetic field, these fluctuations are also measurable as spin noise. No pulse is required; we just need to collect a long continuous dataset. There is no phase coherence, so accumulation of amplitude data is counter-productive, but if the dataset is broken into blocks, the signal is relatively easy to see on standard spectrometers. The blocks should be roughly the length of T 2, the spin–spin relaxation time. These blocks are each Fourier transformed, and their power spectrum is calculated, to remove any phase effects. These power spectra can then be accumulated to provide a measurable signal. The dynamics of a spin system is usually calculated using the density matrix, following Abragam’s “The Principles of Nuclear Magnetism” (A. Abragam, The Principles of Nuclear Magnetism, Clarendon Press, Oxford, 1961), for instance. However, the usual formulation in terms of the density matrix predicts that the signal will decay exponentially to zero, and does not address the spin noise. In this article, we draw on related mathematical ideas in the theory of electromagnetic scattering from random media and apply the modern methods of stochastic calculus in combination with a detailed quantum mechanical description to include spin noise. The core of the work concerns the definition of a pure state for a spin, and how these pure states are combined when the density matrix is formulated. We retain the raw, as opposed to ensemble averaged, density matrix which is constructed in terms of the probability-weighted sum of projection operators corresponding to the constituent pure states of the system. We successfully describe spin noise and further discuss how this illuminates some of the processes of spin relaxation. The approximations that Abragam uses are justified, and we show how the standard density matrix fits within this more general formulation.

From real fields to complex Calogero particles

We provide a novel procedure to obtain complex -symmetric multi-particle Calogero systems. Instead of extending or deforming real Calogero systems, we explore here the possibilities for complex systems to arise from real nonlinear field equations. We exemplify this procedure for the Boussinesq equation and demonstrate how singularities in real-valued wave solutions can be interpreted as N complex particles scattering amongst each other. We analyse this phenomenon in more detail for the two- and three-particle cases. Particular attention is paid to the implementation of -symmetry for the complex multi-particle systems. New complex -symmetric Calogero systems together with their classical solutions are derived.

Entanglement in non-Hermitian quantum theory

Entanglement is one of the key features of quantum world that has no classical counterpart. This arises due to the linear superposition principle and the tensor product structure of the Hilbert space when we deal with multiparticle systems. In this paper, we will introduce the notion of entanglement for quantum systems that are governed by non-Hermitian yet PT-symmetric Hamiltonians. We will show that maximally entangled states in usual quantum theory behave like non-maximally entangled states in PT-symmetric quantum theory. Furthermore, we will show how to create entanglement between two PT qubits using non-Hermitian Hamiltonians and discuss the entangling capability of such interaction Hamiltonians that are non-Hermitian in nature.