Occupation numbers of the harmonically trapped few-boson system

We consider a harmonically trapped dilute $N$-boson system described by a low-energy Hamiltonian with pairwise interactions. We determine the condensate fraction, defined in terms of the largest occupation number, of the weakly-interacting $N$-boson system ($N \ge 2$) by employing a perturbative treatment within the framework of second quantization. The one-body density matrix and the corresponding occupation numbers are compared with those obtained by solving the two-body problem with zero-range interactions exactly. Our expressions are also compared with high precision {\em{ab initio}} calculations for Bose gases with $N=2-4$ that interact through finite-range two-body model potentials. Non-universal corrections are identified to enter at subleading order, confirming that different low-energy Hamiltonians, constructed to yield the same energy, may yield different occupation numbers. Lastly, we consider the strongly-interacting three-boson system under spherically symmetric harmonic confinement and determine its occupation numbers as a function of the three-body "Efimov parameter".Comment: 16 pages, 7 figure

Dynamics of small trapped one-dimensional Fermi gas under oscillating magnetic fields

Deterministic preparation of an ultracold harmonically trapped one-dimensional Fermi gas consisting of a few fermions has been realized by the Heidelberg group. Using Floquet formalism, we study the time dynamics of two- and three-fermion systems in a harmonic trap under an oscillating magnetic field. The oscillating magnetic field produces a time-dependent interaction strength through a Feshbach resonance. We explore the dependence of these dynamics on the frequency of the oscillating magnetic field for non-interacting, weakly interacting, and strongly interacting systems. We identify the regimes where the system can be described by an effective two-state model and an effective three-state model. We find an unbounded coupling to all excited states at the infinitely strong interaction limit and several simple relations that characterize the dynamics. Based on our findings, we propose a technique for driving transition from the ground state to the excited states using an oscillating magnetic field.Comment: 11 pages, 7 figure

Universal and non-universal effective NN-body interactions for ultracold harmonically-trapped few-atom systems

We derive the ground-state energy for a small number of ultracold atoms in an isotropic harmonic trap using effective quantum field theory (EFT). Atoms are assumed to interact through pairwise energy-independent and energy-dependent delta-function potentials with strengths proportional to the scattering length $a$ and effective range volume $V$, respectively. The calculations are performed systematically up to order $l^{-4}$, where $l$ denotes the harmonic oscillator length. The effective three-body interaction contains a logarithmic divergence in the cutoff energy, giving rise to a non-universal three-body interaction in the EFT. Our EFT results are confirmed by nonperturbative numerical calculations for a Hamiltonian with finite-range two-body Gaussian interactions. For this model Hamiltonian, we explicitly calculate the non-universal effective three-body contribution to the energy.Comment: 7 pages, 4 figure

Effective renormalized multi-body interactions of harmonically confined ultracold neutral bosons

We calculate the renormalized effective 2-, 3-, and 4-body interactions for N neutral ultracold bosons in the ground state of an isotropic harmonic trap, assuming 2-body interactions modeled with the combination of a zero-range and energy-dependent pseudopotential. We work to third-order in the scattering length a defined at zero collision energy, which is necessary to obtain both the leading-order effective 4-body interaction and consistently include finite-range corrections for realistic 2-body interactions. The leading-order, effective 3- and 4-body interaction energies are U3 = -(0.85576...)(a/l)^2 + 2.7921(1)(a/l)^3 + O[(a/l)^4] and U4 = +(2.43317...)(a/l)^3 + O[(a\l)^4], where w and l are the harmonic oscillator frequency and length, respectively, and energies are in units of hbar*w. The one-standard deviation error 0.0001 for the third-order coefficient in U3 is due to numerical uncertainty in estimating a slowly converging sum; the other two coefficients are either analytically or numerically exact. The effective 3- and 4-body interactions can play an important role in the dynamics of tightly confined and strongly correlated systems. We also performed numerical simulations for a finite-range boson-boson potential, and it was comparison to the zero-range predictions which revealed that finite-range effects must be taken into account for a realistic third-order treatment. In particular, we show that the energy-dependent pseudopotential accurately captures, through third order, the finite-range physics, and in combination with the multi-body effective interactions gives excellent agreement with the numerical simulations, validating our theoretical analysis and predictions.Comment: Updated introduction, correction of a few typos and sign error