Harmonic-oscillator Trapping Research Articles

Matter-wave soliton interferometer based on a nonlinear splitter

We elaborate a model of the interferometer which, unlike previously studied ones, uses a local (δ-functional) nonlinear repulsive potential, embedded into a harmonic-oscillator trapping potential, as the splitter for the incident soliton. An estimate demonstrates that this setting may be implemented by means of the localized Feshbach resonance controlled by a focused laser beam. The same system may be realized as a nonlinear waveguide in optics. Subsequent analysis produces an exact solution for scattering of a plane wave in the linear medium on the δ -functional nonlinear repulsive potential, and an approximate solution for splitting of the incident soliton when the ambient medium is nonlinear. The most essential result, obtained by means of systematic simulations, is that the use of the nonlinear splitter provides the sensitivity of the soliton-based interferometer to the target, inserted into one of its arms, which is much higher than the sensitivity provided by the usual linear splitter.

Quantum magnetism in strongly interacting one-dimensional spinor Bose systems.

Strongly interacting one-dimensional quantum systems often behave in a manner that is distinctly different from their higher-dimensional counterparts. When a particle attempts to move in a one-dimensional environment it will unavoidably have to interact and ‘push’ other particles in order to execute a pattern of motion, irrespective of whether the particles are fermions or bosons. A present frontier in both theory and experiment are mixed systems of different species and/or particles with multiple internal degrees of freedom. Here we consider trapped two-component bosons with short-range inter-species interactions much larger than their intra-species interactions and show that they have novel energetic and magnetic properties. In the strongly interacting regime, these systems have energies that are fractions of the basic harmonic oscillator trap quantum and have spatially separated ground states with manifestly ferromagnetic wave functions. Furthermore, we predict excited states that have perfect antiferromagnetic ordering. This holds for both balanced and imbalanced systems, and we show that it is a generic feature as one crosses from few- to many-body systems.

Ab initio study of neutron drops with chiral Hamiltonians

We report ab initio calculations for neutron drops in a 10 MeV external harmonic-oscillator trap using chiral nucleon–nucleon plus three-nucleon interactions. We present total binding energies, internal energies, radii and odd–even energy differences for neutron numbers N=2–18 using the no-core shell model with and without importance truncation. Furthermore, we present total binding energies for N=8,16,20,28,40,50 obtained in a coupled-cluster approach. Comparisons with quantum Monte Carlo results, where available, using Argonne v8′ with three-nucleon interactions reveal important dependences on the chosen Hamiltonian.

Creation of two-dimensional composite solitons in spin-orbit-coupled self-attractive Bose-Einstein condensates in free space.

It is commonly known that two-dimensional mean-field models of optical and matter waves with cubic self-attraction cannot produce stable solitons in free space because of the occurrence of collapse in the same setting. By means of numerical analysis and variational approximation, we demonstrate that the two-component model of the Bose-Einstein condensate with the spin-orbit Rashba coupling and cubic attractive interactions gives rise to solitary-vortex complexes of two types: semivortices (SVs, with a vortex in one component and a fundamental soliton in the other), and mixed modes (MMs, with topological charges 0 and ±1 mixed in both components). These two-dimensional composite modes can be created using the trapping harmonic-oscillator (HO) potential, but remain stable in free space, if the trap is gradually removed. The SVs and MMs realize the ground state of the system, provided that the self-attraction in the two components is, respectively, stronger or weaker than the cross attraction between them. The SVs and MMs which are not the ground states are subject to a drift instability. In free space (in the absence of the HO trap), modes of both types degenerate into unstable Townes solitons when their norms attain the respective critical values, while there is no lower existence threshold for the stable modes. Moving free-space stable solitons are also found in the present non-Galilean-invariant system, up to a critical velocity. Collisions between two moving solitons lead to their merger into a single one.

Numerical method for the stochastic projected Gross-Pitaevskii equation

We present a method for solving the stochastic projected Gross-Pitaevskii equation (SPGPE) for a three-dimensional weakly interacting Bose gas in a harmonic-oscillator trapping potential. The SPGPE contains the challenge of both accurately evolving all modes in the low-energy classical region of the system, and evaluating terms from the number-conserving scattering reservoir process. We give an accurate and efficient procedure for evaluating the scattering terms using a Hermite-polynomial based spectral-Galerkin representation, which allows us to precisely implement the low-energy mode restriction. Stochastic integration is performed using the weak semi-implicit Euler method. We extensively characterize the accuracy of our method, finding a faster-than-expected rate of stochastic convergence. Physical consistency of the algorithm is demonstrated by considering thermalization of initially random states.

Interaction for the trapped fermi gas from a unitary transformation of the exact two-body spectrum

We study systems of few two-component fermions interacting in a Harmonic Oscillator trap. The fermion-fermion interaction is generated in a finite basis with a unitary transformation of the exact two-body spectrum given by the Busch formula. The few-body Schrodinger equation is solved with the formalism of the No-Core Shell Model. We present results for a system of three fermions interacting at unitarity as well as for finite values of the S-wave scattering length a 2 and effective range r 2. Unitary systems with four and five fermions are also considered. We show that the many-body energies obtained in this approach are in excellent agreement with exact solutions for the three-body problem, and results obtained by other methods in the other cases.

Quenched dynamics of two-dimensional solitary waves and vortices in the Gross–Pitaevskii equation

We consider a two-dimensional (2D) counterpart of the experiment that led to the creation of quasi-1D bright solitons in Bose–Einstein condensates (BECs) (2002 Nature 417 150–3). We start by identifying the fundamental state of the 2D Gross–Pitaevskii equation for repulsive interactions, with a harmonic-oscillator (HO) trap, and with or without an optical lattice (OL). Subsequently, we switch the sign of the interaction to induce interatomic attraction and monitor the ensuing dynamics. Regions of stable self-trapping and a catastrophic collapse of 2D fundamental states are identified in the parameter plane of the OL strength and BEC norm. The increase of the OL strength expands the persistence domain for the solitary waves to larger norms. For single-charged solitary vortices, in addition to the survival and collapse regimes, an intermediate one is identified, where the vortex resists the collapse but loses its structure, transforming into a single-hump state. The same setting may also be implemented in the context of optical solitons and vortices, using photonic-crystal fibers.

Thermodynamic Properties of Ultracold Bose Gas: Transition Exponents and Universality

We report exact numerical calculation of chemical potential, condensate fraction and specific heat of $N$ non-interacting bosons confined in an isotropic harmonic oscillator trap in one, two and three dimensions, as also for interacting bosons in a 3D trap. Quasi phase transitions are observed in all these cases, including one-dimension, as shown by a rapid change of all the thermodynamic quantities at the transition point. The change becomes more rapid as $N$ increases in 2D and 3D cases. However with increase in $N$, the sudden change in the nature of specific heat, gets gradually wiped out in 1D, while it becomes more drastic in 2D and 3D. The sudden change in the nature of condensate fraction and chemical potential as $N$ increases becomes more drastic even in 1D. Defining transition exponents, which characterize the nature of a thermodynamic quantity at the transition point of a quasi phase transition, we evaluate them by careful numerical calculation very near the transition temperature. These exponents are found to be independent of the size of the system and whether the bosons are interacting or not, demonstrating their universality property.

Stabilizing single- and two-color vortex beams in quadratic media by a trapping potential

We consider two-dimensional (2D) localized modes in the second-harmonic-generating (χ(2)) system with the harmonic-oscillator (HO) trapping potential. In addition to its realization in optics, the system describes the mean-field dynamics of mixed atomic-molecular Bose–Einstein condensates (BECs). The existence and stability of various modes is determined by their total power, N, topological charge, m/2 [m is the intrinsic vorticity of the second-harmonic (SH) field], and χ(2) mismatch, q. The analysis is carried out in a numerical form and, in parallel, by means of the variational approximation (VA), which produces results that agree well with numerical findings. Below a certain power threshold, N≤Nc(m)(q), all trapped modes are of the single-color type, represented by the SH component only, while the fundamental frequency (FF) one is absent. In contrast with the usual situation, where such modes are always unstable, we demonstrate that they are stable, for m=0, 1, 2 (the mode with m=1 may be formally considered as a semivortex with topological charge m/2=1/2), at N≤Nc(m)(q), and unstable above this threshold. On the other hand, Nc(m)(q)≡0 at q≥qmax (in our notation, qmax=1); hence the single-color modes are unstable in the latter case. At N=Nc(m), the modes with m=0 and m=2 undergo a pitchfork bifurcation, which gives rise to two-color states, which remain completely stable for m=0. The two-color vortices with m=2 (topological charge 1) have an upper stability border, N=Nc2(q). Above the border, they exhibit periodic splittings and recombinations, while keeping their vorticity. The semivortex does not bifurcate; at N=Nc(m=1), it exhibits quasi-chaotic oscillations and a rotating “groove” resembling a screw-edge dislocation induced by the semi-integer vorticity.

Quasi-one-dimensional Bose–Einstein condensates in nonlinear lattices

We consider the three-dimensional (3D) mean-field model for the Bose–Einstein condensate, with a one-dimensional (1D) nonlinear lattice (NL), which periodically changes the sign of the nonlinearity along the axial direction, and the harmonic-oscillator trapping potential applied in the transverse plane. The lattice can be created as an optical or magnetic one, by means of available experimental techniques. The objective is to identify stable 3D solitons supported by the setting. Two methods are developed for this purpose: the variational approximation, formulated in the framework of the 3D Gross–Pitaevskii equation, and the 1D nonpolynomial Schrödinger equation (NPSE) in the axial direction, which allows one to predict the collapse in the framework of the 1D description. Results are summarized in the form of a stability region for the solitons in the plane of the NL strength and wavenumber. Both methods produce a similar form of the stability region. Unlike their counterparts supported by the NL in the 1D model with the cubic nonlinearity, kicked solitons of the NPSE cannot be set in motion, but the kick may help to stabilize them against the collapse, by causing the solitons to shed the excess norm. A dynamical effect specific to the NL is found in the form of freely propagating small-amplitude wave packets emitted by perturbed solitons.

Hidden vorticity in binary Bose-Einstein condensates

We consider a binary Bose-Einstein condensate (BEC) described by a system of two-dimensional (2D) Gross-Pitaevskii equations with the harmonic-oscillator trapping potential. The intraspecies interactions are attractive, while the interaction between the species may have either sign. The same model applies to the copropagation of bimodal beams in photonic-crystal fibers. We consider a family of trapped hidden-vorticity (HV) modes in the form of bound states of two components with opposite vorticities ${S}_{1,2}=\ifmmode\pm\else\textpm\fi{}1$, the total angular momentum being zero. A challenging problem is the stability of the HV modes. By means of a linear-stability analysis and direct simulations, stability domains are identified in a relevant parameter plane. In direct simulations, stable HV modes feature robustness against large perturbations, while unstable ones split into fragments whose number is identical to the azimuthal index of the fastest growing perturbation eigenmode. Conditions allowing for the creation of the HV modes in the experiment are discussed too. For comparison, a similar but simpler problem is studied in an analytical form, viz., the modulational instability of an HV state in a one-dimensional (1D) system with periodic boundary conditions (this system models a counterflow in a binary BEC mixture loaded into a toroidal trap or a bimodal optical beam coupled into a cylindrical shell). We demonstrate that the stabilization of the 1D HV modes is impossible, which stresses the significance of the stabilization of the HV modes in the 2D setting.

Three and four harmonically trapped particles in an effective-field-theory framework

We study systems of few two-component fermions interacting via short-range interactions within a harmonic-oscillator trap. The dominant interactions, which are two-body, are organized according to the number of derivatives and defined in a two-body truncated model space made from a bound-state basis. Leading-order (LO) interactions are solved for exactly using the formalism of the No-Core Shell Model, whereas corrections are treated as many-body perturbations. We show explicitly that next-to-LO and next-to-next-to-LO interactions improve convergence as the model space increases. We present results at unitarity for three- and four-fermion systems, which show excellent agreement with the exact solution (for the three-body problem) and results obtained by others methods (in the four-body case). We also present results for finite scattering lengths and non-zero range of the interaction, including (at positive scattering length) observation of a change in the structure of the three-body ground state and extraction of the atom-dimer scattering length.

Effective interactions for light nuclei: an effective (field theory) approach

One of the central open problems in nuclear physics is the construction of effective interactions suitable for many-body calculations. We discuss a recently developed approach to this problem, where one starts with an effective field theory containing only fermion fields and formulated directly in a no-core shell-model space. We present applications to light nuclei and to systems of a few atoms in a harmonic-oscillator trap. Future applications and extensions, as well as challenges, are also considered.

Hard-core Bose-Fermi mixture in one-dimensional split traps

We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a hard-wall trap or a harmonic oscillator trap with a tunable $\ensuremath{\delta}$-function barrier at the trap center. The mixture consists of a 1D Bose gas with repulsive interactions and of a 1D noninteracting spin-aligned Fermi gas, with both species interacting through hard-core interactions. Using a generalized Bose-Fermi mapping, we calculate the reduced single-particle density matrix and the momentum distribution of the gas as a function of barrier strength and the parity of particle number. The secondary peaks in the momentum distribution show remarkable correlation between particles on the two sides of the split.

An effective field theory approach to two trapped particles

We discuss the problem of two particles interacting via short-range interactions within a harmonic-oscillator trap. The interactions are organized according to their number of derivatives and defined in truncated model spaces made from a bound-state basis. Leading-order (LO) interactions are iterated to all orders, while corrections are treated in perturbation theory. We show explicitly that next-to-LO and next-to-next-to-LO interactions improve convergence as the model space increases. In the large-model-space limit we regain results from a pseudopotential. Arbitrary scattering lengths are considered, as well as a generalization to include the non-vanishing range of the interaction.

Effect of an impulsive force on vortices in a rotating Bose–Einstein condensate

The effects of a sudden increase and decrease of the interatomic interaction and harmonic-oscillator trapping potential on vortices in a quasi two-dimensional rotating Bose–Einstein condensate are investigated using the mean-field Gross–Pitaevskii equation. We also study the decay of vortices when the rotation of the condensate is suddenly stopped. Upon a free expansion of a rotating BEC with vortices the radius of the vortex core increases more rapidly than the radius of the condensate.

The Schrödinger cat state oftrapped ions in harmonic and anharmonic oscillator traps

We examine the time evolution of a two-level ion interactingwith a light field in a harmonic oscillator trap and in a trapwith anharmonicities. The anharmonicities of the trap arequantified in terms of the deformation parameter τcharacterizing the q-analogue of the harmonic oscillator trap.Initially the ion is prepared in a Schrödinger cat state.The entanglement of the centre-of-mass motional states and theinternal degrees of freedom of the ion results in a characteristiccollapse and revival pattern. We numerically calculate thepopulation inversion I(t), quasi-probabilities Q(t) andpartial mutual quantum entropy S(P) for the system as afunction of time. Interestingly, small deformations of the trapenhance the contrast between population inversion collapse andrevival peaks as compared to the zero-deformation case. Forβ = 3 and 4 (β determines the average number oftrap quanta linked to centre-of-mass motion) the best collapseand revival sequence is obtained for τ = 0.0047 and0.004, respectively. For large values of τ, decoherencesets are accompanied by loss of amplitude of population inversion,and for τ ~0.1 the collapse and revival phenomenondisappear. Each collapse or revival of population inversion ischaracterized by a peak in the S(P) versus t plot. During thetransition from collapse to revival and vice-versa, we have aminimum mutual entropy value that is S(P) = 0. Successiverevival peaks show a lowering of the local maximum pointindicating a dissipative irreversible change in the ionic state.An improved definition of the collapse and revival pattern as theanharmonicity of the trapping potential increases is alsoreflected in the quasi-probability versus t plots.

Collapse of attractive Bose-Einstein condensed vortex states in a cylindrical trap.

The quantized vortex states of a weakly interacting Bose-Einstein condensate of atoms with attractive interatomic interaction in an axially symmetric harmonic oscillator trap are investigated using the numerical solution of the time-dependent Gross-Pitaevskii equation obtained by the semi-implicit Crank-Nicholson method. The collapse of the condensate is studied in the presence of deformed traps with the larger frequency along either the radial or the axial direction. The critical number of atoms for collapse is calculated as a function of the vortex quantum number L. The critical number increases with increasing angular momentum L of the vortex state but tends to saturate for large L.

Self-similar BEC dynamics: accuracy and applications

We discuss a simplified description of the dynamics ofinhomogeneous dilute Bose-Einstein condensates (BECs), based on theassumption that the time-evolving BEC preserves its shape (i.e. thedensity profile remains similar to the initial profile). While thevalidity of the self-similarity assumption is not a priori obvious,weshow that this description becomes exact in the opposite limitsof large and small condensates contained by time-dependent sphericallysymmetric harmonic oscillator trap potentials. We discussapplications of this simple description and we show how the insightsderived from it suggest that phase-coherent BECs expanding fromoptical lattice sites can be used for the purpose of imaging.

Comment on “Single two-level ion in an anharmonic oscillator trap: Time evolution of theQfunction and population inversion”

We show here that the model Hamiltonian used in our paper for ion vibrating in a q-analog harmonic oscillator trap and interacting with a classical single-mode light field is indeed obtained by replacing the usual bosonic creation and annihilation operators of the harmonic trap model by their q-deformed counterparts. The approximations made in our paper amount to using for the ion-laser interaction in a q-analog harmonic oscillator trap, the operator $F_{q}=exp{-(|\epsilon|^2}/2)}exp{i\epsilon A^{\dagger}}exp{i\epsilon A}$, which is analogous to the corresponding operator for ion in a harmonic oscillator trap that is $F=exp{-(|\epsilon|^2 /2)}exp{i\epsilon a^{\dagger }}exp{i\epsilon a}$. In our article we do not claim to have diagonalized the operator, $F_q = exp{i \epsilon (A^{\dagger}+A)}$, for which the basis states |g,m> and |e,m> are not analytic vectors.