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arXiv:quant-ph/0307230 · Physical Review A

Theory of output coupling for trapped fermionic atoms

P. A. S. Pires Filho, Claudio Lenz Cesar, L. Davidovich

Published Feb 24, 2004• 0 citations

Abstract

We develop a dynamic theory of output coupling for fermionic atoms initially confined in a magnetic trap. We consider an exactly soluble one-dimensional model, with a spatially localized \ensuremath{\delta}-type coupling between the atoms in the trap and a continuum of free-particle external modes. The transient dynamics of the atoms, as they leave the trap, is investigated in detail. Two important special cases are considered for the confinement potential: the infinite box and the harmonic oscillator. We establish that in both cases a bound state of the coupled system appears for any value of the coupling constant, implying that the trap population does not vanish in the infinite-time limit. For weak coupling, the infinite-time spectral distribution of the outgoing atoms exhibits peaks corresponding to the initially occupied energy levels in the trap; the heights of these peaks increase with the energy. As the coupling gets stronger, the infinite-time spectral distribution is displaced towards dressed energies of the fermions in the trap. The corresponding dressed states result from the coupling between the unperturbed fermionic states in the trap, mediated by the coupling between these states and the continuum. In the strong-coupling limit, there is a reinforcement of the lowest-energy dressed mode, which contributes to the spectral distribution of the outgoing beam more strongly than the other modes. This effect is especially pronounced for the one-dimensional box, which indicates that the efficiency of the mode-reinforcement mechanism depends on the steepness of the confinement potential. In this case, a quasimonochromatic antibunched atomic beam is obtained. Results for a bosonic sample are also shown for comparison.

DOI: 10.1103/physreva.69.023615

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We develop a dynamic theory of output coupling for fermionic atoms initially confined in a magnetic trap. We consider an exactly soluble one-dimensional model, with a spatially localized \ensuremath{\delta}-type coupling between the atoms in the trap and a continuum of free-particle external modes. The transient dynamics of the atoms, as they leave the trap, is investigated in detail. Two important special cases are considered for the confinement potential: the infinite box and the harmonic oscillator. We establish that in both cases a bound state of the coupled system appears for any value of the coupling constant, implying that the trap population does not vanish in the infinite-time limit. For weak coupling, the infinite-time spectral distribution of the outgoing atoms exhibits peaks corresponding to the initially occupied energy levels in the trap; the heights of these peaks increase with the energy. As the coupling gets stronger, the infinite-time spectral distribution is displaced towards dressed energies of the fermions in the trap. The corresponding dressed states result from the coupling between the unperturbed fermionic states in the trap, mediated by the coupling between these states and the continuum. In the strong-coupling limit, there is a reinforcement of the lowest-energy dressed mode, which contributes to the spectral distribution of the outgoing beam more strongly than the other modes. This effect is especially pronounced for the one-dimensional box, which indicates that the efficiency of the mode-reinforcement mechanism depends on the steepness of the confinement potential. In this case, a quasimonochromatic antibunched atomic beam is obtained. Results for a bosonic sample are also shown for comparison.

Analytical results on the correlation functions of strongly correlated many-body systems are rare in the literature and their importance cannot be overstated. We present determinant representations for the space-, time-, and temperature-dependent correlation functions of the strongly interacting one-dimensional Hubbard model in the presence of an external trapping potential. These representations are exact and valid in both equilibrium and nonequilibrium scenarios like the ones initiated by a sudden change of the confinement potential. In addition, they can be implemented numerically very easily, significantly outperforming other numerical approaches. As applications of our results we investigate the single-particle spectral functions of systems with harmonic trapping and show that dynamical quasicondensation occurs for both fermionic and bosonic spin-

1/2

systems released from a Mott insulator state.

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 noninteracting, 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.

We conduct a theoretical study of

\mathrm{SU}(N

) fermions confined by a one-dimensional harmonic potential. First, we introduce a numerical approach for solving the trapped interacting few-body problem, by which one may obtain accurate energy spectra across the full range of interaction strengths. In the strong-coupling limit, we map the

\mathrm{SU}(N

) Hamiltonian to a spin-chain model. We then show that an existing, extremely accurate ansatz---derived for a Heisenberg SU(2) spin chain---is extendable to these

N

-component systems. Lastly, we consider balanced

\mathrm{SU}(N

) Fermi gases that have an equal number of particles in each spin state for

N=2,\phantom{\rule{0.16em}{0ex}}3,\phantom{\rule{0.16em}{0ex}}4

. In the weak- and strong-coupling regimes, we find that the ground-state energies rapidly converge to their expected values in the thermodynamic limit with increasing atom number. This suggests that the many-body energetics of

N

-component fermions may be accurately inferred from the corresponding few-body systems of

N

distinguishable particles.

We address the effects of dissipative defects giving rise to a localized particle loss, in one-dimensional noninteracting lattice fermionic gases confined within a region of size $\ensuremath{\ell}$. We consider homogeneous systems within hard walls and inhomogeneous systems where the particles are trapped by space-dependent external potentials, such as harmonic traps. We model the dissipative particle-decay mechanism by Lindblad master equations governing the time evolution of the density matrix. The resulting quantum dynamics is analyzed in protocols starting from the ground state of the Hamiltonian for

{N}_{0}

particles, then evolving under the effect of one dissipative particle-loss defect, for example, at the center of the system. We study the interplay between time, size $\ensuremath{\ell}$, and the number

{N}_{0}

of initial particles, considering two different situations: (i) fixed number

{N}_{0}

of initial particles; and (ii) fixed ratio ${N}_{0}/\ensuremath{\ell}$, corresponding to the thermodynamic limit of the initial equilibrium state. We show that the quantum evolutions of the particle number and density develop various intermediate and asymptotic dynamic regimes, and nontrivial large-time states when the dissipative mechanism acts at the center of the system.

A theory for the coupling constants between spin-polarized degenerate fermionic atoms is developed for the case of atoms confined to a quasi-one-dimensional harmonic trap. Exploiting the presence of the Fermi edge and the large number of fermions, it is shown that the resulting effective one-dimensional interaction can be parametrized by three sets of interaction coefficients, two for forward scattering and one for backward scattering. In the case of identical fermions, backscattering dominates, because the contact part of the effective one-dimensional interaction must be subtracted. Analytic expressions for the interaction coefficients for the effective contact interaction, which is relevant for the inter-component interactions, and for the 'p-wave' interaction appearing in next order, are given. As an example, we calculate and discuss in detail the effective coupling constants for the marginally long ranged dipole–dipole interaction.

The exactly solvable model of two indistinguishable quantum particles (bosons or fermions) confined in a one-dimensional harmonic trap and interacting via finite-range soft-core interaction is presented and many properties of the system are examined. Particularly, it is shown that independently on the potential range, in the strong interaction limit bosonic and fermionic solutions become degenerate. For sufficiently large ranges a specific crystallization appears in the system. The results are compared to predictions of the celebrated Busch et al. model and those obtained in the Tonks-Girardeau limit. The assumed inter-particle potential is very similar to the potential between ultra-cold dressed Rydberg atoms. Therefore, the model can be examined experimentally.