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Physica Scripta

Two particles in one-dimensional confinement: the role of mass imbalance and Pauli's exclusion principle within short and long range interactions

Daniel Vieira

Published Oct 15, 2024• 1 citation
Publisher

Abstract

Abstract We consider the problem of two interacting particles in one-dimensional harmonic confinement. By considering both particles are subject to the same harmonic curvature, we obtain exact numerical solutions in a straightforward way for any choice of interaction potential. The formulation is then applied in the situations of (i) non-identical particles with different masses m 1 and m 2; (ii) bosons and fermions; (iii) short and long range interactions; and (iv) combining all these ingredients within a time-dependent applied electric field. We analyze the role of the mass imbalance and Pauli's exclusion principle by investigating the formation of bound pairs, the expected value of the separation between the particles and the effects of interaction and exclusion principle on the density distributions.

DOI
10.1088/1402-4896/ad86fe

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Abstract We consider the problem of two interacting particles in one-dimensional harmonic confinement. By considering both particles are subject to the same harmonic curvature, we obtain exact numerical solutions in a straightforward way for any choice of interaction potential. The formulation is then applied in the situations of (i) non-identical particles with different masses m 1 and m 2; (ii) bosons and fermions; (iii) short and long range interactions; and (iv) combining all these ingredients within a time-dependent applied electric field. We analyze the role of the mass imbalance and Pauli's exclusion principle by investigating the formation of bound pairs, the expected value of the separation between the particles and the effects of interaction and exclusion principle on the density distributions.
Abstract Strongly interacting particles in one dimension subject to external confinement have become a topic of considerable interest due to recent experimental advances and the development of new theoretical methods to attack such systems. In the case of equal mass fermions or bosons with two or more internal degrees of freedom, one can map the problem onto the well-known Heisenberg spin models. However, many interesting physical systems contain mixtures of particles with different masses. Therefore, a generalization of the recent strong-coupling techniques would be highly desirable. This is particularly important since such problems are generally considered non-integrable and thus the hugely successful Bethe ansatz approach cannot be applied. Here we discuss some initial steps towards this goal by investigating small ensembles of one-dimensional harmonically trapped particles where pairwise interactions are either vanishing or infinitely strong with focus on the mass-imbalanced case. We discuss a (semi)-analytical approach to describe systems using hyperspherical coordinates where the interaction is effectively decoupled from the trapping potential. As an illustrative example we analyze mass-imbalanced four-particle two-species mixtures with strong interactions between the two species. For such systems we calculate the energies, densities and pair-correlation functions.
Abstract In this paper we study a mixed system of bosons and fermions with up to six particles in total. All particles are assumed to have the same mass. The two-body interactions are repulsive and are assumed to have equal strength in both the Bose–Bose and the Fermi–Boson channels. The particles are confined externally by a harmonic oscillator one-body potential. For the case of four particles, two identical fermions and two identical bosons, we focus on the strongly interacting regime and analyze the system using both an analytical approach and density matrix renormalization group calculations using a discrete version of the underlying continuum Hamiltonian. This provides us with insight into both the ground state and the manifold of excited states that are almost degenerate for large interaction strength. Our results show great variation in the density profiles for bosons and fermions in different states for strongly interacting mixtures. By moving to slightly larger systems, we find that the ground state of balanced mixtures of four to six particles tends to separate bosons and fermions for strong (repulsive) interactions. On the other hand, in imbalanced Bose–Fermi mixtures we find pronounced odd–even effects in systems of five particles. These few-body results suggest that question of phase separation in one-dimensional confined mixtures are very sensitive to system composition, both for the ground state and the excited states.
Quantum escapes of two particles with Coulomb interactions from a confined one-dimensional region to a semi-infinite lead are discussed by using the probability of finding all particles within the confined region, that is, the survival probability, in comparison with free particles. By taking into account the quantum effects of two identical particles, such as the Pauli exclusion principle, it is shown analytically that for two identical free fermions (bosons), the survival probability decays asymptotically in power ~t(-10) (~t(-6)) as a function of time t, although it decays in power ~t(-3) for one free particle. On the other hand, for two particles with attractive Coulomb interactions it is shown numerically that the survival probability decays in power ~t(-3) after a long time. Moreover, for two particles with repulsive Coulomb interactions it decays exponentially in time ~exp (-αt) with a constant α, which is almost independent of the initial energy of particles.
We examine the quantum field theoretical impact of the Pauli exclusion principle on the formation of the ground state of two fermions that are coupled with each other through the interaction with bosons. As expected, in case where the two fermions are indistinguishable, their binding strength due to the force-mediating bosons is reduced and their spatial distribution in the ground state is wider compared to the corresponding state for the distinguishable particles. Surprisingly, the spatial distribution of the bosons in the ground states is even fundamentally different for both systems. In fact, the Pauli exclusion principle leads to a strong accumulation of the bosons between the two fermions.
We present a systematic study of entanglement control in one-dimensional systems of two distinguishable interacting particles in harmonic confinement. Using density matrix renormalization group calculations, we examine how the von Neumann entropy evolves under variations of mass ratio, trap-frequency asymmetry, interaction range, and localized external fields. Our results reveal a consistent picture connecting the real-space density overlap, the redistribution of quantum fluctuations, and the resulting entanglement. Within the parameter regimes investigated, mass imbalance produces more pronounced changes in entanglement than trap-frequency asymmetry. Furthermore, the competition between interaction range and field-induced polarization determines whether short- or long-range correlations dominate. By connecting the von Neumann entropy with measurable quantities such as the density difference Δ n ( x ) and the phase-space products 〈 x i 2 〉 〈 p x i 2 〉 , we establish experimentally accessible fingerprints of quantum correlation. Our analysis demonstrates how external and geometrical parameters can tune entanglement in confined two-particle models, providing a basis for exploring related control protocols in suitable experimental settings.
We explore a few-fermion mixture consisting of two components that are repulsively interacting and confined in a one-dimensional harmonic trap. Different scenarios of population imbalance ranging from the completely imbalanced case where the physics of a single impurity in the Fermi sea is discussed to the partially imbalanced and equal population configurations are investigated. For the numerical calculations the multiconfigurational time-dependent Hartree method is employed, extending its application to few-fermion systems. Apart from numerical calculations we generalize our ansatz for a correlated pair wave function proposed recently [I. Brouzos and P. Schmelcher, Phys. Rev. Lett. 108, 045301 (2012)] for bosons to mixtures of fermions. From weak to strong coupling between the components the energies, the densities and the correlation properties of one-dimensional systems change vastly with an upper limit set by fermionization where for infinite repulsion all fermions can be mapped to identical ones. The numerical and analytical treatments are in good agreement with respect to the description of this crossover. We show that for equal populations each pair of different component atoms splits into two single peaks in the density while for partial imbalance additional peaks and plateaus arise for very strong interaction strengths. The case of a single-impurity atom shows rich behavior of the energy and density as we approach fermionization and is directly connected to recent experiments [G. Z\"urn et al., Phys. Rev. Lett. 108, 075303 (2012)].