Impenetrable mass-imbalanced particles in one-dimensional harmonic traps

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.

Doubly Excited Resonance States of Helium Atom: Complex Entropies

“…In particular, there has been a remarkable increase of interest in exploring properties of various quantum composite systems [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. In recent years, both the entanglement and Shannon Entropy in helium and helium-like ions have also accelerated a great research attention [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33].…”

One-dimensional mixtures of several ultracold atoms: a review

“…With a pair-correlated wave function approach, it has been shown that through the transition from non-interacting case (BEC-BEC, g B = g AB = 0) to the CP limit (g B = 0 and g AB → ∞), the impurity particle A tends to localize in the edges of the majority species B, when N B = 2, 3 atoms [182]. Then, it has been shown that one can force the impurity particle to again localize in the center of the trap for more massive impurities when N B = 2 to 4 atoms [183] (for the 3+1 system see also appendix in [174]). The limit when g B → ∞ has been studied in [184], showing that when also g AB → ∞ the particles cannot exchange their initial ordering, while when g AB is large but not infinity, the system maps into the spin chain Hamiltonian.…”

Experimentally Accessible Invariants Encoded in Interparticle Correlations of Harmonically Trapped Ultra-cold Few-Fermion Mixtures

“…The motion of the centre of mass does not depend on the interaction which influences only the motion of the remaining degrees of freedom. The two-body problem [11] is an example but the separation can also be done for larger number of particles [12][13][14]26,27] at the cost of using more sophisticated methods of describing the relative motion [15,22]. In practice, the mentioned transformation of variables is difficult to perform straightforwardly, however it is a standard tool in a theoretical analysis of few-body problems [30][31][32].…”