Abstract We investigate the effect of spin-motion coupling on the spectral properties of Rabi-coupled noninteracting fermions in a spin-dependent harmonic trapping potential plus an anharmonic term. It is shown that when the spin-motion coupling becomes strong, fermions tend to stay in one of the two components. In the limit of the strong spin-motion coupling, the entire energy spectrum exhibits a sequence of near degeneracy. In particular, in the case of the sextic anharmonic term, the system admits the exact analytical energies and wave functions of the bound states for an infinite number of the specific parameter conditions. The properties of the energy spectrum have also been discussed on basis of these obtained exact analytical solutions.
Abstract We investigate the effect of spin-motion coupling on the spectral properties of Rabi-coupled noninteracting fermions in a spin-dependent harmonic trapping potential plus an anharmonic term. It is shown that when the spin-motion coupling becomes strong, fermions tend to stay in one of the two components. In the limit of the strong spin-motion coupling, the entire energy spectrum exhibits a sequence of near degeneracy. In particular, in the case of the sextic anharmonic term, the system admits the exact analytical energies and wave functions of the bound states for an infinite number of the specific parameter conditions. The properties of the energy spectrum have also been discussed on basis of these obtained exact analytical solutions.
Abstract We investigate the spectral properties of coupled noninteracting trapped fermions in a spin-dependent harmonic trapping potential. It is shown that for certain specific parameter conditions, there exist an infinite number of the exact energies and wave functions of the bound states. Two different types of the exact energies have been identified, and they are associated with the level-crossings and avoided level-crossings, respectively. In addition, the effect of the energy bias on the energy spectrum is also discussed. In particular, it is found that under suitably chosen parameter conditions, the exact wave functions of the bound states provide an analytical demonstration of the existence of the spin segregation.
The exact bound-state wavefunctions and the corresponding energy equation are calculated for a new generalized harmonic oscillator problem describing a spin-1/2 fermion in 3+1-dimensions, involving scalar, vector, and tensor couplings acting simultaneously within a particular plane of motion. For the scalar and vector coupling, singular harmonic oscillator shapes are considered, such that the singular term is needed to allow analytical solutions for the wavefunctions while preserving binding under adequate conditions. For the tensor sector,the Dirac oscillator potential is employed, which adds another independent binding mechanism to the problem. The exact bound-state solutions are computed by specifically tuning coefficients for an appropriate pair of \textit{Ansätze} for the radial functions, which leads to wavefunctions in terms of generalized Laguerre polynomials. Although the energy equation cannot provide a general expression for the energy spectrum, specific constraints on the quantum numbers as simple functions of the external potential parameters can be derived for it, which determines the conditions for bound solutions to exist, and of what type: particle, antiparticle or both. It is shown that the results can be simply mapped to the spherically symmetric analogue problem, and this is used to show that the general result encompass several previous particular cases of spherically symmetric harmonic oscillator problems in the Dirac equation available in the literature.
We study the problem of the relativistic motion of a 1/2-spin particle in an exactly solvable potential, which consists of the harmonic oscillator potential plus a novel angle-dependent potential, The analytic bound state solutions of the Dirac equation for this potential are obtained by using the Nikiforov–Uvarov method. The wave functions of the radial and angle-dependent parts of the Dirac equation are derived in the form of the Laguerre and Jacobi polynomials. The contribution of the angle-dependent potential to the relativistic energy spectra is discussed under the condition that the scalar potential is equal to or minus the vector potential.
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.
We provide a detailed study of the properties of a few interacting spin 1 / 2 fermions trapped in a one-dimensional harmonic oscillator potential. The interaction is assumed to be well represented by a contact delta potential. Numerical results obtained by means of direct diagonalization techniques are combined with analytical expressions for both the non-interacting and strongly interacting regime. The N = 2 case is used to benchmark our numerical techniques with the known exact solution of the problem. After a detailed description of the numerical methods, in a tutorial-like manner, we present the static properties of the system for N = 2 , 3 , 4 and 5 particles, e.g., low-energy spectrum, one-body density matrix, ground-state densities. Then, we consider dynamical properties of the system exploring first the excitation of the breathing mode, using the dynamical structure function and corresponding sum-rules, and then a sudden quench of the interaction strength.
We provide an analytically tractable toy model of an infinitely heavy impurity interacting with the spin-polarized gas of bath fermions via a contact potential in 1D and placed in a harmonic trap. The solution to this problem requires knowledge of the single-particle fermionic eigenstates in the presence of a harmonic trap and a contact potential. We show how the spectrum can be understood from a perturbative solution of the exact transcendental equation in the weak and strong-coupling regimes. We additionally provide expressions for normalized wavefunctions and different overlaps between the states in the presence and absence of the impurity. Using these exact results, we analyze the energy of the polaron, derive Tan's contact-like relation for the density at the center of the harmonic trap, and compute the quasiparticle residue.