Rotation of atoms in a lattice is studied using a Hubbard model. It is found that the atoms are still contained in the trap even when the rotation frequency is larger than the trapping frequency. This is very different from the behavior in continuum. Bragg scattering and coupling between angular and radial motion are believed to make this stability possible. In this regime, density depletion at the center of the trap can be developed for spin polarized fermions.
We consider the system of two interacting atoms confined in axially symmetric harmonic trap. Within the pseudopotential approximation, we solve the Schroedinger equation exactly, discussing the limits of quasi-one and quasi-two-dimensional geometries. Finally, we discuss the application of an energy-dependent pseudopotential, which allows to extend the validity of our results to the case of tight traps and large scattering lengths.
We study the influence of the nonlinearity in the Schrodinger equation on the motion of quantum particles in a harmonic trap. In order to obtain exact analytic solutions, we have chosen the logarithmic nonlinearity. The unexpected result of our study is the existence in the presence of nonlinearity of two or even three coexisting Gaussian solutions.
A method is proposed to cool down atoms in a harmonic trap without phase-space compression as in a perfectly slow adiabatic expansion, i.e., keeping the populations of the instantaneous initial and final levels invariant, but in a much shorter time. This may require that the harmonic trap becomes an expulsive parabolic potential in some time interval. The cooling times achieved are also shorter than previous minimal times using optimal-control bang-bang methods and real frequencies.
We analyze the quantum entanglement between two interacting atoms trapped in a spherical harmonic potential. At ultra-cold temperature, ground state entanglement is generated by the dominated s-wave interaction. Based on a regularized pseudo-potential Hamiltonian, we examine the quantum entanglement by performing the Schmidt decomposition of low-energy eigenfunctions. We indicate how the atoms are paired and quantify the entanglement as a function of a modified s-wave scattering length inside the trap.
We propose an experimental configuration, within an ion trap, by which a quantum mechanical delta-kicked harmonic oscillator could be realized, and investigated. We show how to directly measure the sensitivity of the ion motion to small variations in the external parameters.
We have studied the atomic density of a cloud confined in an isotropic harmonic trap at the vicinity of the Bose-Einstein transition temperature. We show that, for a non-interacting gas and near this temperature, the ground-state density has the same order of magnitude as the excited states density at the centre of the trap. This holds in a range of temperatures where the ground-state population is negligible compared to the total atom number. We compare the exact calculations, available in a harmonic trap, to semi-classical approximations. We show that these latter should include the ground-s…
We investigate quantum entanglement between two (spin-1/2) fermions inside a cylindrical harmonic trap, making use of the von Neumann entropy for the reduced single particle density matrix as the pure state entanglement measure. We explore the dependence of pair entanglement on the geometry and strength of the trap and on the strength of the pairing interaction over the complete range of the effective BCS to BEC crossover. Our result elucidates an interesting connection between our model system of two fermions and that of two interacting bosons.
Self-organization effects related to light amplification in the collective atomic recoil laser system with the driven atoms confined in a harmonic trap are investigated further. In the dispersive parametric region, our study reveals that the spontaneously formed structures in the phase space contributes an important role to the light amplification of the probe field under the atomic motion being modified by the trap.
We study the absorption spectrum for a strongly degenerate Fermi gas confined in a harmonic trap. The spectrum is calculated using both the exact summation and also the Thomas-Fermi (TF) approximation. In the latter case, relatively simple analytical expressions are obtained for the absorption lineshape at large number of trapped atoms. At zero temperature, the approximated lineshape is characterized by a $(1-z^2)^{5/2}$ dependence which agrees well with the exact numerical calculations. At non-zero temperature, the spectrum becomes broader, although remains non-Gaussian as long as the fermion…
In this article we formulate frictionless atom cooling in harmonic traps as a time-optimal control problem, permitting imaginary values of the trap frequency for trasient time intervals during which the trap becomes an expulsive parabolic potential. We show that the minimum time solution has "bang-bang" form, where the frequency jumps suddenly at certain instants and then remains constant, and calculate estimates of the minimum cooling time for various numbers of such jumps. A numerical optimization method based on pseudospectral approximations is used to obtain suboptimal realistic solutions …
Complete description of the classical and quantum dynamics of a particle in an anisotropic, rotating, harmonic trap is given. The problem is studied in three dimensions and no restrictions on the geometry are imposed. In the generic case, for an arbitrary orientation of the rotation axis, there are two regions of instability with different characteristics. The analysis of the quantum-mechanical problem is made simple due to a direct connection between the classical mode vectors and the quantum-mechanical wave functions. This connection is obtained via the matrix Riccati equation that governs t…
We investigate laser-driven vibronic transitions of a single two-level atomic ion in harmonic and hard wall traps. In the Lamb-Dicke regime, for tuned or detuned lasers with respect to the internal frequency of the ion, and weak or strong laser intensities, the vibronic transitions occur at well isolated "Rabi Resonances", where the detuning-adapted Rabi frequency coincides with the level spacing of the vibrational modes. These vibronic resonances are characterized as avoided crossings of the dressed levels (eigenvalues of the full Hamiltonian). Their peculiarities due to symmetry constraints …
We propose a scheme for measuring the Berry phase in the vibrational degree of freedom of a trapped ion. Starting from the ion in a vibrational coherent state we show how to reverse the sign of the coherent state amplitude by using a purely geometric phase. This can then be detected through the internal degrees of freedom of the ion. Our method can be applied to preparation of Schrödinger cat states.
The influence of a tight isotropic harmonic trap on photoassociation of two ultracold alkali atoms forming a homonuclear diatomic is investigated using realistic atomic interaction potentials. Confinement of the initial atom pair due to the trap leads to a uniform strong enhancement of the photoassociation rate to most, but also to a strongly suppressed rate for some final states. Thus tighter traps do not necessarily enhance the photoassociation rate. A further massive enhancement of the rate is found for strong interatomic interaction potentials. The details of this interaction play a minor …
The problem of the Kohn mode in bosonized theories of one-dimensional interacting fermions in the harmonic trap is investigated and a suitable modification of the interaction is proposed which preserves the Kohn mode. The modified theory is used to calculate exactly the inhomogeneous linear mobility at position z in response to a spatial force pulse at another position. It is found the inhomogeneous particle mobility exhibits resonances not only at the trap frequency but also at multiples of a new renormalized collective mode frequency which depends on the strength of the interaction. In contr…
For both cases with and without interactions, bipartite entanglement of two-fermions from a Fermi gas in a trap is investigated. We show how the entanglement depends on the locations of the two fermions and the total number of particles. Fermions at the verge of trap have longer entanglement distance (beyond it, the entanglement disappears) than those in the center. We derive a lower limitation to the average overlapping for two entangled fermions in the BCS ground state, it is shown to be $\sqrt{Q/2M}$, a function of Cooper pair number $Q$ and total number of occupied energy levels $M$.
We study the exactly solvable quantum system of two particles confined in a three-dimensional harmonic trap and interacting via finite-range soft-core interaction by means of the separation of variables and ansatz method. Supposing the solution in the form of ansatz $Ψ(r)=r^{a}e^{-λr^2} ψ(r)$ we transform the time independent Schrödinger equation into Kummer's differential equation whose solution are given in the form of confluent hypergeometric function. We also discuss that in the absence of central force potential, the quantum system map into the problem of two quantum particle trapped in o…
The correct form of the Schmidt decomposition of the stationary wave functions for a system of two interacting particles trapped in a two-dimensional harmonic potential is given
A new strategy for trapping quantum particles is presented, which behaves like an effective harmonic oscillator potential trap wherever is desired. The approach is based on harmonic contraction and expansion of the system around a fixed point (trapping point) at high frequencies. Analytical results are presented for an arbitrary potential and contrasted with numerical calculations for a quantum particle between two impenetrable walls. Similarly, making use of the analogy between quantum mechanics and optics, it is shown that an harmonically breathing waveguide lattice can be used to trap optic…
We investigate the fast transport of an atom or a packet of atoms by different kinds of non-harmonic traps including power-law traps. The study is based on the reverse engineering method. Exact results are obtained and applied to design robust transport protocols. The optimization of the transport trajectory is performed with classical trajectories, and remains valid for the transport of a wave packet.
Recently predicted and observed Pauli crystals are structures formed by trapped ultracold non-interacting particles obeying Fermi statistics. The relative positions of the particles are determined by the trapping potential and the Pauli exclusion principle. Measuring all particles at once, their positions tend to be close to vertices of non-trivial polyhedrons. Similar shapes are found for systems of particles bound to the surface of a sphere.
We study Bose-Einstein condensation in a harmonic trap with a dimple potential. We specifically consider the case of a tight and deep dimple potential which is modelled by a Dirac delta function. This allows for simpler, explicit numerical and analytical investigations of noninteracting gases. Thus, the Schrodinger equation is used instead of the Gross-Pitaevski equation. Calculating the atomic density, chemical potential, critical temperature and condensate fraction, the role of the relative depth of the dimple potential with respect to the harmonic trap in large condensate formation at enhan…
We obtain second and higher order corrections to the shift of the Bose-Einstein critical temperature due to finite-size effects. The confinement is that of a harmonic trap with general anisotropy. Numerical work shows the high accuracy of our expressions. We draw attention to a subtlety involved in the consideration of experimental values of the critical temperature in connection with analytical expressions for the finite-size corrections.
We compute the pair entanglement between two interacting bosons in a two dimensional (2D)isotropic harmonic trap. The interaction potential is modeled by a 2D regularized pseudo-potential. By analytically decomposing the wave function into the single particle basis, we show the dependency of the pair entanglement on the scattering length. Our results turn out to be in good agreements with earlier results using a quasi-2D geometry.
We study the fast transport of a particle or a Bose-Einstein condensate in a harmonic potential. An exact expression for the final excitation energy in terms of the Fourier transform of the trap acceleration is used to engineer optimal transport protocols (with no final excitation) that are robust with respect to spring-constant errors. The same technique provides a way to design the simultaneous and robust transport of a few non-interacting species that experience different harmonic trap frequencies in the same trap.
We study spin-1/2 fermions, interacting via a two-body contact potential, in a one-dimensional harmonic trap. Applying exact diagonalization, we investigate their behavior at finite interaction strength, and discuss the role of the ground-state degeneracy which occurs for sufficiently strong repulsive interaction. Even low temperature or a completely depolarizing channel may then dramatically influence the system's behavior. We calculate level occupation numbers as signatures of thermalization, and we discuss the mechanisms to break the degeneracy.
We consider controlled collisions between two ultracold atoms guided by external harmonic potentials. We derive analytical solutions of the Schroedinger equation for this system, and investigate the properties of eigenergies and eigenstates for different trap geometries as a function of a trap separation and of the scattering length. When varying the trap separation the energy spectrum exhibits avoided crossings, corresponding to trap-induced shape resonances. Introducing an energy-dependent scattering length we investigate the behavior of the system in the vicinity of a magnetic Feshbach reso…
The time evolution of a single particle in a harmonic trap with time dependent frequency omega(t) is well studied. Nevertheless here we show that, when the harmonic trap is opened (or closed) as function of time while keeping the adiabatic parameter mu = [d omega(t)/dt]/omega(t)^2 fixed, a sharp transition from an oscillatory to a monotonic exponential dynamics occurs at mu = 2. At this transition point the time evolution has a third-order exceptional point (EP) at all instants. This situation, where an EP of a time-dependent Hermitian Hamiltonian is obtained at any given time, is very differe…
Employing a short-range two-channel description we derive an analytic model of atoms in isotropic and anisotropic harmonic traps at a Feshbach resonance. On this basis we obtain a new parameterization of the energy-dependent scattering length which differs from the one previously employed. We validate the model by comparison to full numerical calculations for Li-Rb and explain quantitatively the experimental observation of a resonance shift and trap-induced molecules in exited bands. Finally, we analyze the bound state admixture and Landau-Zener transition probabilities.
We consider a two-dimensional system of harmonically trapped particles with pseudo-spin-$\frac{1}{2}$ degree of freedom. This degree of freedom is coupled to the particle's momentum via the so-called Rashba spin-orbit interaction. We present our numerical results for a single-particle and few-particle systems, assuming the repulsive interparticle interaction to be of zero range.
System of a two-flavor mixture of ultra-cold fermions confined in a one-dimensional harmonic trap is studied in the frame of the center of mass. We present a numerical method of obtaining energetic spectra in this frame for an arbitrary mass ratio of fermionic species. We identify a specific invariant encoded in many-body correlations which enable one to determine an eigenstate of the Hamiltonian and to label excitations of the center of mass. The tool presented may be particularly useful in experimental analysis of the interparticle interactions which do not affect the center of mass excitati…
We investigate fast transport and spin manipulation of tunable spin-orbit-coupled Bose-Einstein condensates in a moving harmonic trap. Motivated by the concept of "shortcuts to adiabaticity", we design inversely the time-dependent trap position and spin-orbit coupling strength. By choosing appropriate boundary conditions we obtain fast transport and spin flip simultaneously. The non-adiabatic transport and relevant spin dynamics are illustrated with numerical examples, and compared with the adiabatic transport with constant spin-orbit-coupling strength and velocity. Moreover, the influence of …
Exactly solvable model of two Rydberg-dressed atoms moving in a quasi-two-dimensional harmonic trap is introduced and its properties are investigated. Depending on the strength of inter-particle interactions and the critical range of the potential, the two-particle eigenstates are classified with respect to the excitations of the center-of-mass motion, relative angular momentum, and relative distance variable. Having these solutions in hand, we discuss inter-particle correlations as functions of interaction parameters. We also present a straightforward prescription of how to generalize obtaine…
We investigate Bose-Einstein condensation of noninteracting gases in a harmonic trap with an off-center dimple potential. We specifically consider the case of a tight and deep dimple potential which is modelled by a point interaction. This point interaction is represented by a Dirac delta function. The atomic density, chemical potential, critical temperature and condensate fraction, the role of the relative depth and the position of the dimple potential are analyzed by performing numerical calculations.
In our previous paper, we have investigated the thermodynamics of the quantum ultra-cold neutron gas trapped in the two-dimensional square-well potential on the Earth's gravitational field. We have pointed out that they have strong dependence on the shape of the trapping potential. In the present paper, we estimate the Fermi energy and the density distribution of the Fermi gas with two-dimensional harmonic oscillator potential, instead of the square-well potential. Then we discuss them in terms of the Fermi energy and the density distribution as compared with both of cases.
We study energy levels of two heteronuclear molecules moving in a spherically symmetric harmonic trap. A role of electric dipole interactions is compared and contrasted with our earlier results (https://arxiv.org/abs/1512.00631) for two magnetic dipolar atoms. We stress importance of a rotational energy with its value which is very high compared to the energy of dipolar interaction. We show that dipolar forces do not play a significant role in the ground state of the system under typical experimental conditions. However, there exist excited states that exhibit anticrossings similar to the ones…
We develop an analytical many-body wave function to accurately describe the crossover of a one-dimensional bosonic system from weak to strong interactions in a harmonic trap. The explicit wave function, which is based on the exact two-body states, consists of symmetric multiple products of the corresponding parabolic cylinder functions, and respects the analytically known limits of zero and infinite repulsion for arbitrary number of particles. For intermediate interaction strengths we demonstrate, that the energies, as well as the reduced densities of first and second order, are in excellent a…
The ground-state properties of the two-flavored mixture of a few attractive fermions confined in a one-dimensional harmonic trap is studied. It is shown that for slightly imbalanced system the pairing between fermions of opposite spins has completely different features that in the balanced case. The fraction of correlated pairs is suppressed by the presence of additional particle and another uncorrelated two-body orbital dominates in the ground-state of the system.
Two identical dipolar atoms moving in a harmonic trap without an external magnetic field are investigated. Using the algebra of angular momentum a semi - analytical solutions are found. We show that the internal spin - spin interactions between the atoms couple to the orbital angular momentum causing an analogue of Einstein - de Haas effect. We show a possibility of adiabatically pumping our system from the s-wave to the d-wave relative motion. The effective spin-orbit coupling occurs at anti-crossings of the energy levels.
We investigate the properties of two interacting ultracold polar molecules described as distinguishable quantum rigid rotors, trapped in a one-dimensional harmonic potential. The molecules interact via a multichannel two-body contact potential, incorporating the short-range anisotropy of intermolecular interactions including dipole-dipole interaction. The impact of external electric and magnetic fields resulting in Stark and Zeeman shifts of molecular rovibrational states is also investigated. Energy spectra and eigenstates are calculated by means of the exact diagonalization. The importance a…
We study systems of two identical dipolar particles confined in quasi one-dimensional harmonic traps. Numerical results for the dependencies of the entanglement on the control parameters of the systems are provided and discussed in detail. In the limit of a strong interaction between the particles, the occupancies and the von Neumann entropies of the bosonic and fermionic ground states are derived in closed analytic forms by applying the harmonic approximation. The strong correlation regimes of the system with the dipolar bosons and the system with the charged ones are compared with each other…
We study harmonically trapped one-dimensional atoms subjected to an equal combination of Rashba and Dresselhaus spin-orbit coupling induced by Raman transition. We first examine the wave function and the degeneracy of the single-particle ground state, followed by a study of two weakly interacting bosons or fermions. For the two-particle ground state, we focus on the effects of the interaction on the degeneracy, the spin density profiles, and the density-density correlation functions. Finally we show how these studies help us to understand the many-body properties of the system.
We analyze the dynamics of two atoms with a short-ranged pair interaction in a one-dimensional harmonic trap with time-dependent frequency. Our analysis is focused on two representative cases: (i) a sudden change of the trapping frequency from one value to another, and (ii) a periodic trapping frequency. In case (i), the dynamics of the interacting and the corresponding non-interacting systems turn out to be similar. In the second case, however, the interacting system can behave quite differently, especially close to parametric resonance. For instance, in the regions where such resonance occur…
We show that the presence of a harmonic trap may in itself lead to many-body localization for cold atoms confined in that trap in a quasi-one-dimensional geometry. Specifically, the coexistence of delocalized phase in the center of the trap with localized region closer to the edges is predicted with the borderline dependent on the curvature of the trap. The phenomenon, similar in its origin to Stark localization, should be directly observed with cold atomic species. We discuss both the spinless and the spinful fermions, for the latter we address Stark localization at the same time as it has no…
We theoretically investigate the properties of two interacting ultracold highly magnetic atoms trapped in a one-dimensional harmonic potential. The atoms interact via an anisotropic long-range dipole-dipole interaction, which in one dimension effectively can be modeled by the contact interaction. We investigate the interplay of the external magnetic field, the spin-spin interaction, and the trapping potential and how they affect the magnetization of the system. We show the role of indistinguishability and symmetries in the dynamics by studying the time evolution of the observables that could b…
