Spectroscopy for a few atoms harmonically trapped in one dimension

Abstract

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.PACS numbers: 03.75. Mn, 05.30.Fk The quantum system of a few particles in one dimension, confined by an external field and interacting via a short-range potential, has been a touchstone model for nuclear, atomic, statistical, and mathematical physics since the 1930's. This system possesses enough symmetry and regularity to be analytically tractable in many cases, but also manifests a rich enough phenomenological structure to serve as physically significant model for realistic systems. This class of models has been a playground for studying the interplay of integrability and solvability, the emergence of universality in few-body phenomena, and the transition from few-body to many-body physics.This article focuses specifically on the quantum system of N particles of mass m confined in a one-dimensional harmonic trap with frequency ω and interacting via an effectively zero-range potential. The Hamiltonian for the model iŝwith the particle coordinates x i expressed in units of /mω. This Hamiltonian has received intense attention recently because it serves as a model for neutral cold atoms with interactions tunable through the manipulation of Feshbach and confinement-induced resonances trapped in extremely prolate, approximately harmonic optical wells [1][2][3][4]. The model has been used to investigate few-body and many-body properties of one-dimensional trapped ultracold gases of bosons [5][6][7][8][9][10][11], fermions [12][13][14][15][16][17][18][19][20][21][22][23], and mixtures of bosons and fermions [24][25][26]. The so-called unitary limit g → ∞ of hard-core repulsive interactions is known as the TonksGirardeau gas and has been realized in the laboratory with bosons [27,28], as has the metastable super-TonksGirardeau limit g → −∞ [29]. Recent experiments in * Electronic address: harshman@american.edu which the state of a few trapped, interacting fermions can be controlled and measured [30][31][32][33] have generated excitement that tunable fermi gases will allow simulation of solid-state phenomena such as the transition to ferromagnetism [18,21,23,34].The solutions of Hamiltonian (1) form a complete basis for the spatial states of N distinguishabl...

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