Noninteracting Identical Particles Research Articles

Many body density of states of a system of non interacting spinless fermions

The modeling of out-of-equilibrium many-body quantum systems requires to go beyond low-energy physics and single or few bodies densities of states. Many-body localization, presence or lack of thermalization and quantum chaos are examples of phenomena in which states at different energy scales, including highly excited ones, contribute to dynamics and therefore affect the system’s properties. Quantifying these contributions requires the many-body density of states (MBDoS), a function whose calculation becomes challenging even for non-interacting identical particles due to the difficulty to enumerate accessible states while enforcing the exchange symmetry. In the present work, we introduce a new approach to evaluate the MBDoS in the general case of non-interacting systems of identical quantum particles. The starting point of our method is the principal component analysis of a filling matrix F describing how N particles can be distributed into L single-particle energy levels. We show that the many body spectrum can be expanded as a weighted sum of singular vectors of the filling matrix. The weighting coefficients only involve renormalized energies obtained from the single body spectrum. We illustrate our method in two classes of problems that are mapped into spinless fermions : (i) non-interacting electrons in a homogeneous tight-binding model in 1D and 2D, and (ii) interacting spins in a chain under a transverse field.

Investigating and improving student understanding of the basics for a system of identical particles

We discuss an investigation of students' difficulties with systems of non-interacting identical particles. The investigation was carried out in advanced quantum mechanics courses at both the undergraduate and graduate levels by administering free-response and multiple-choice questions and conducting individual interviews with students. This research on student difficulties was used as a guide to develop and evaluate a Quantum Interactive Learning Tutorial (QuILT) that helps students develop a functional understanding of the concepts necessary for determining the many-particle Hamiltonian and stationary state wavefunctions for a system of non-interacting identical particles.

The Shannon-McMillan Theorem Proves Convergence to Equiprobability of Boltzmann's Microstates.

This paper shows that, for a large number of particles and for distinguishable and non-interacting identical particles, convergence to equiprobability of the W microstates of the famous Boltzmann–Planck entropy formula S = k log(W) is proved by the Shannon–McMillan theorem, a cornerstone of information theory. This result further strengthens the link between information theory and statistical mechanics.

Nonequilibrium Green's Function's Approach to the Calculation of Work Statistics.

The calculation of work distributions in a quantum many-body system is of significant importance and also of formidable difficulty in the field of nonequilibrium quantum statistical mechanics. To solve this problem, inspired by the Schwinger-Keldysh formalism, we propose the contour-integral formulation for work statistics. Based on this contour integral, we show how to do the perturbation expansion of the characteristic function of work (CFW) and obtain the approximate expression of the CFW to the second order of the work parameter for an arbitrary system under a perturbative protocol. We also demonstrate the validity of fluctuation theorems by utilizing the Kubo-Martin-Schwinger condition. Finally, we use noninteracting identical particles in a forced harmonic potential as an example to demonstrate the powerfulness of our approach.

Effect of the resonance spectra in the propagation of two decaying entangled particles

An exact analytical solution of the decaying wave function of two identical noninteracting particles, which are entangled by spatial symmetry, is used to analyze the effect of the resonance spectra in the propagation of the decaying probability density outside the interaction potential region. We find, using exactly solvable problems, that a usual approximation that considers the two resonance levels associated with the initial states, is affected substantially in the case of sharp high energy resonances by disrupting the pure exponential decaying regime exhibited by the two resonance level approximation, whereas for broad high energy resonances, we find that the probability density profile is well described by the two resonance approximation.

Ruling out the class of statistical processes involving two noninteracting identical particles in two modes

In the framework of Generalized probabilistic theories (GPT), we illustrate a class of statistical processes in case of two noninteracting identical particles in two modes that satisfies a well motivated notion of physicality conditions namely the double stochasticity and the no-interaction condition proposed by Karczewski et. al. (Phys. Rev. Lett. 120, 080401 (2018)), which can not be realized through a quantum mechanical process. This class of statistical process is ruled out by an additional requirement called the evolution condition imposed on two particle evolution. We also show that any statistical process of two noninteracting identical particles in two modes that satisfies all of the three physicality conditions can be realized within quantum mechanics using the beam splitter operation.

Improving student understanding of a system of identical particles with a fixed total energy

We discuss an investigation of student difficulties with whether a given total energy is possible for the system under given constraints, the possible outcomes of an energy measurement, and the probability of obtaining a particular energy if we randomly measured the energy of one particle and concepts involved in determining the number of distinct many-particle states in a system of noninteracting identical particles with a fixed total energy. The investigation was carried out in advanced quantum mechanics courses by administering free-response and multiple-choice questions and conducting individual interviews with students. We also discuss how the research was used as a guide to develop, validate, and evaluate a Quantum Interactive Learning Tutorial (QuILT), which strives to help students develop a functional understanding of the relevant concepts for a system of non-interacting identical particles with a fixed total energy. We discuss the development and validation of the identical particles QuILT and its evaluation in the undergraduate and graduate courses that focused on these issues.

Three-body JP=0+ , 1+ , 2+ B*B*K¯ bound states

Three body systems with short-range interactions display universal features that have been extensively explored in atomic physics, but apply to hadron physics as well. Systems composed of two non-interacting identical particles (species H) of mass $M$ and a third particle (species P) of mass $m$ that interacts attractively with the other two have the property that they are more likely to bind for larger values of the mass ratio $M/m$. This is particularly striking if the HHP system is in P-wave (while the interacting pair is in S-wave), in which case one would not normally expect the formation of a three body state. If we assume that the $B^* \bar{K}$ binds to form the $B^*_{s1}$ heavy meson and notice that the mass ratio of the $B^*$ to $\bar{K}$ is $M /m = 10.8$, concrete calculations indicate that there should be a three body $B^* B^* \bar{K}$ bound state between $30-40\,{\rm MeV}$ below the $B^*_{s1} B^*$ threshold. For the $\Xi_{bb} \Xi_{bb} \bar{K}$ system the mass imbalance is about $M /m = 20.5$ and two bound states are expected to appear, a fundamental and an excited one located at $50-90$ and $5-15\,{\rm MeV}$ below the $\Xi_{bb} \Omega^*_{bb\frac{1}{2}}$ threshold (where $\Omega^*_{bb\frac{1}{2}}$ denotes the $\Xi_{bb} \bar{K}$ bound state). We indicate the possibility of analogous P-wave three body bound states composed of two heavy baryons and a kaon or antikaon and investigate the conditions under which the Efimov effect could appear in these systems.

Generalized Probabilistic Description of Noninteracting Identical Particles

We investigate an operational description of identical noninteracting particles in multiports. In particular, we look for physically motivated restrictions that explain their bunching probabilities. We focus on a symmetric 3-port in which a triple of superquantum particles admitted by our generalized probabilistic framework would bunch with a probability of 3/4. The bosonic bound of 2/3 can then be restored by imposing the additional requirement of product evolution of certain input states. These states are characterized by the fact that, much like product states, their entropy equals the sum of entropies of their one-particle substates. This principle is, however, not enough to exclude the possibility of superquantum particles in higher-order multiports.

Buildup of symmetrization entanglement for the nonescape probability of two identical particles

We study the nonescape probability for tunneling decay of two identical noninteracting particles which are entangled due to symmetrization. As a consequence, the two-particle nonescape probability, which gives the probability that the particles remain within the potential region, may be written as the sum of two terms: a direct term and an exchange term which is added or subtracted according to symmetrization. At the initial time the direct term decays exponentially in the usual way, but the exchange term vanishes due to the orthonormality of the initial states. As time evolves the exchange term builds up until it becomes comparable to the direct term at time ${t}_{0}$. Thereafter, the symmetric and antisymmetric nonescape probabilities decay in a distinctive way. The buildup time typically lasts several lifetimes and implies that during the relevant exponentially decaying regime the probability of finding the particles within the interaction region is insensitive to the symmetry of the decaying particles. That is, the particles decay as if they were distinguishable. This might be experimentally verified in the tunneling decay of atoms in ultracold gases.

Quantum statistical effects in multichannel wave-packet scattering of noninteracting identical particles

For a number of non-interacting identical particles entering a multi-channel scatterer in various wave packet states, we construct a generating function for the probabilities of various scattering outcomes. This is used to evaluate the mean numbers of particles $\overline{n}_m$ scattered into a given ($m$-th) channel, single-channel statistics, and inter-channel correlations. We show that for initially uncorrelated particles, indistinguishability changes single channel statistics without altering the the value of $\overline{n}_m$. For uncorrelated bosons and fermions, bunching and anti-bunching behaviour can be detected in the extreme-case probabilities, to have all particles scattered into the same channel, or none of particles scattered into a channel, or channels. As an example, we consider a cavity with a single long-lived resonance accessible to the particles, which allows them to "pile up" inside the scatterer.

Symmetry-assisted resonance transmission of noninteracting identical particles

We report novel interference effects in wave packet scattering of identical particles incident on the same side of a resonant barrier, different from those observed in Hong-Ou-Mandel experiments. These include significant changes in the mean number of transmissions and full counting statistics, as well as "bunching" and "anti-bunching" effects in the all-particles transmission channel. With several resonances involved, pseudo-resonant driving of the two-level system in the barrier, may result in sharp enhancement of scattering probabilities for certain values of temporal delay between the particles.

Exact analytical non-Hermitian formulation of the time evolution of decay of one and two identical quantum particles

An analytical solution to the time evolution of decay of one and two identical noninteracting particles is presented using the formalism of resonant states. It is shown that the time-dependent wave function and hence the survival and nonescape probabilities for the initial state of a single particle and entangled symmetric and antisymmetric initial states of two identical particles evolve in a distinctive form along the exponential and long-time nonexponential decaying regimes. In particular, for the last regime, they exhibit different inverse power of time behaviors.

Contact parameters in two dimensions for general three-body systems

We study the two-dimensional three-body problem in the general case of three distinguishable particles interacting through zero-range potentials. The Faddeev decomposition is used to write the momentum-space wave function. We show that the large-momentum asymptotic spectator function has the same functional form as derived previously for three identical particles. We derive the analytic relations between the three different Faddeev components for the three distinguishable particles. We investigate the one-body momentum distributions both analytically and numerically and analyze the tail of the distributions to obtain two- and three-body contact parameters. We specialize from the general cases to the examples of two identical, interacting or non-interacting, particles. We find that the two-body contact parameter is not a universal constant in the general case and show that the universality is recovered when a subsystem is composed of two identical non-interacting particles. We also show that the three-body contact parameter is negligible in the case of one non-interacting subsystem compared to the situation where all the subsystems are bound. As an example, we present the results for mixtures of lithium with two cesium or two potassium atoms, which are systems of current experimental interest.

DETERMINING SOURCE CUMULANTS IN FEMTOSCOPY WITH GRAM–CHARLIER AND EDGEWORTH SERIES

Lowest-order cumulants provide important information on the shape of the emission source in femtoscopy. For the simple case of noninteracting identical particles, we show how the fourth-order source cumulant can be determined from measured cumulants in momentum space. The textbook Gram–Charlier series is found to be highly inaccurate, while the related Edgeworth series provides increasingly accurate estimates. Ordering of terms compatible with the Central Limit Theorem appears to play a crucial role even for non-Gaussian distributions.

High temperature magnetic properties of magnesium ferrite nanoparticles

Magnetic properties such as Curie temperature (TC), saturation magnetization (Ms), remanent magnetization (Mr), and coercivity (Hc) of nanoparticles of magnesium ferrites (MgFe2O4) were studied in a broad range of temperatures varying from room temperature to 800 K. The magnetization decreases with increasing temperature, approaching 0 at ∼ 750 K. The Curie temperature, determined by means of the inverse susceptibility versus temperature, was ∼738 K. The saturation magnetization, coercivity, and remanence decreased with increasing temperature, being close to 0 at temperatures near TC. However, for temperatures 100 K above room temperature, these magnetic properties were still the same as those at room temperature. The coercivity temperature dependence could be expressed in terms of T3/4, indicating that MgFe2O4 nanoparticles may form a system of random and noninteracting identical particles. The results are discussed in terms of interparticle interactions induced by the thermal fluctuations, cation distribution, and other imperfections that exert fields on Mg2+ ions that could increase with temperature.

Extinction and the optical theorem Part II Multiple particles

The extinction due to a group of identical particles in a fixed orientation is studied. An analytical description based on the Maxwell volume integral equation is presented, describing the interference energy flow due to a group of noninteracting identical particles. As compared to a single particle, the effect of multiple particles is shown to narrow the angular region around the forward direction over which the dominant contribution to the group's extinction cross section occurs. The extinction due to a group of fully interacting nonspherical particles is studied using the discrete dipole approximation. Interparticle interactions are found not to change the essential character of the interference-based energy flow mechanism that describes extinction.

Diffusion in tight confinement: A lattice-gas cellular automaton approach. I. Structural equilibrium properties

The thermodynamic and transport properties of diffusing species in microporous materials are strongly influenced by their interactions with the confining framework, which provide the energy landscape for the transport process. The simple topology and the cellular nature of the alpha cages of a ZK4 zeolite suggest that it is appropriate to apply to the study of the problem of diffusion in tight confinement a time-space discrete model such as a lattice-gas cellular automaton (LGCA). In this paper we investigate the properties of an equilibrium LGCA constituted by a constant number of noninteracting identical particles, distributed among a fixed number of identical cells arranged in a three-dimensional cubic network and performing a synchronous random walk at constant temperature. Each cell of this network is characterized by a finite number of two types of adsorption sites: the exit sites available to particle transfer and the inner sites not available to such transfers. We represent the particle-framework interactions by assuming a differentiation in binding energy of the two types of sites. This leads to a strong dependence of equilibrium and transport properties on loading and temperature. The evolution rule of our LGCA model is constituted by two operations (randomization, in which the number of particles which will be able to try a jump to neighboring cells is determined, and propagation, in which the allowed jumps are performed), each one applied synchronously to all of the cells. The authors study the equilibrium distribution of states and the adsorption isotherm of the model under various conditions of loading and temperature. In connection with the differentiation in energy between exit and inner sites, the adsorption isotherm is described by a conventional Langmuir isotherm at high temperature and by a dual-site Langmuir isotherm at low temperature, while a first order diffuse phase transition takes place at very low temperature.

Steepest-Entropy-Ascent Irreversible Relaxation Towards Thermodynamic Equilibrium: The Dynamical Ansatz that Completes the Gyftopoulos-Hatsopoulos Unified Theory with a General Quantal Law of Causal Evolution

We overview the main features of the general equation of motion that completes the Gyftopoulos-Hatsopoulos unified theory of mechanics and thermodynamics with a quantal law of causal evolution that entails relaxation towards stable equilibrium for any non-equilibrium state, no matter how far from thermodynamic equilibrium. We illustrate with numerical examples the behavior of the equation of motion by discussing spontaneous energy redistribution within an isolated, closed system composed of non-interacting identical particles with energy levels ei and i = 1, 2,…, N. For this system the time-dependent occupation probabilities pi(t) obey the nonlinear rate equations which include functions of the pi(t)’s that maintain invariant the mean energy and the normalization condition. The entropy is a non-decreasing function of time until the initially nonzero occupation probabilities reach a Boltzmann-like canonical distribution over the occupied energy eigenstates. Initially zero occupation probabilities, instead, remain zero at all times. The solutions of the rate equations are unique and well-defined for arbitrary initial conditions pi(0) and for all times, -∞<t<+∞. Existence and uniqueness both forward and backward in time allows the reconstruction of the ancestral or primordial lowest entropy state. We also illustrate the structure and main properties of the nonlinear dynamics for a composite system. An initial version of this paper was published in July of 2006 in the proceedings of ECOS’06, Aghia Pelagia, Crete, Greece.

Spectral statistics in noninteracting many-particle systems

It is widely accepted that the statistical properties of energy level spectra provide an essential characterization of quantum chaos. Indeed, the spectral fluctuations of many different systems like quantum billiards, atoms, or atomic nuclei have been studied. However, noninteracting many-body systems have received little attention, since it is assumed that they must exhibit Poisson-like fluctuations. Apart from a heuristic argument of Bloch, there are neither systematic numerical calculations nor a rigorous derivation of this fact. Here we present a rigorous study of the spectral fluctuations of noninteracting identical particles moving freely in a mean field emphasizing the evolution with the number of particles N as well as with the energy. Our results are conclusive. For N > or =2 the spectra of these systems exhibit Poisson fluctuations provided that we consider sufficiently high excitation energies. Nevertheless, when the mean field is chaotic there exists a critical energy scale L(c); beyond this scale, the fluctuations deviate from the Poisson statistics as a reminiscence of the statistical properties of the mean field.