Trapped Bose-Einstein Condensate Research Articles

Elements of Dynamics of a One-Dimensional Trapped Bose–Einstein Condensate Excited by a Time-Dependent Dimple: A Lagrangian Variational Approach

We examine the dynamics of a one-dimensional harmonically trapped Bose–Einstein condensate (BEC), induced by the addition of a dimple trap whose depth oscillates with time. For this purpose, the Lagrangian variational method (LVM) is applied to provide the required analytical equations. The goal is to provide an analytical explanation for the quasiperiodic oscillations of the BEC size at resonance, that is additional to the one given by Adhikari (J Phys B At Mol Opt Phys 36:1109, 2003). It is shown that LVM is able to reproduce instabilities in the dynamics along the same lines outlined by Lellouch et al. (Phys Rev X 7:021015, 2017). Moreover, it is found that at resonance the energy dynamics display ordered oscillations, whereas at off-resonance they tend to be chaotic. Further, by using the Poincare–Lindstedt method to solve the LVM equation of motion, the resulting solution is able to reproduce the quasiperiodic oscillations of the BEC.

Vinen turbulence via the decay of multicharged vortices in trapped atomic Bose-Einstein condensates

We investigate a procedure to generate turbulence in a trapped Bose-Einstein condensate which takes advantage of the decay of multicharged vortices to reduce surface oscillations. We show that the resulting singly charged vortices twist around each other, intertwined in the shape of helical Kelvin waves, which collide and undergo vortex reconnections, creating a disordered vortex state. By examining the velocity statistics, the energy spectrum, the correlation functions, and the temporal decay and comparing these properties with the properties of classical turbulence and observations in superfluid helium, we conclude that this disordered vortex state can be identified with the Vinen regime of turbulence which has been discovered in the context of superfluid helium.

Dynamics of straight vortex filaments in a Bose–Einstein condensate with the Gaussian density profile

The dynamics of interacting quantized vortex filaments in a rotating trapped Bose-Einstein condensate, which is in the Thomas-Fermi regime at zero temperature and described by the Gross-Pitaevskii equation, is considered in the hydrodynamic "anelastic" approximation. In the presence of a smoothly inhomogeneous array of filaments (vortex lattice), a non-canonical Hamiltonian equation of motion is derived for the macroscopically averaged vorticity, with taking into account the spatial non-uniformity of the equilibrium condensate density determined by the trap potential. A minimum of the corresponding Hamiltonian describes a static configuration of deformed vortex lattice against a given density background. The minimum condition is reduced to a vector nonlinear partial differential equation of the second order, for which some approximate and exact solutions are found. It is shown that if the condensate density has an anisotropic Gaussian profile then equation of motion for the averaged vorticity admits solutions in the form of a spatially uniform vector with a nontrivial time dependence. An integral representation is obtained for the matrix Green function determining the non-local Hamiltonian of a system of arbitrary shaped vortex filaments in a condensate with Gaussian density. ... A simple approximate expression for the two-dimensional Green function is suggested at rather arbitrary density profile, and its successful comparison to the exact result in the Gaussian case is done. Approximate equations of motion are derived which describe a long-wave dynamics of interacting vortex filaments in condensates with the density depending on the transverse coordinates only.

Collective excitation of a trapped Bose-Einstein condensate with spin-orbit coupling

We investigate the collective excitations of a Raman-induced spin-orbit coupled Bose-Einstein condensate confined in a quasi one-dimension harmonic trap using the Bogoliubov method. By tuning the Raman coupling strength, three phases of the system can be identified. By calculating the transition strength, we are able to classify various excitation modes that are experimentally relevant. We show that the three quantum phases possess distinct features in their collective excitation properties. In particular, the spin dipole and the spin breathing modes can be used to clearly map out the phase boundaries. We confirm these predictions by direct numerical simulations of the quench dynamics that excites the relevant collective modes.

Scalar field as a Bose–Einstein condensate in a Schwarzschild–de Sitter spacetime

In this paper, we analyze some properties of a scalar field configuration, where it is considered as a trapped Bose–Einstein condensate in a Schwarzschild–de Sitter background spacetime. In a natural way, the geometry of the curved spacetime provides an effective trapping potential for the scalar field configuration. This allows us to explore some thermodynamical properties of the system. Additionally, the curved geometry of the spacetime also induces a position-dependent self-interaction parameter, which can be interpreted as a kind of gravitational Feshbach resonance, that could affect the stability of the cloud and could be used to obtain information about the interactions among the components of the system.

Vortex dynamics in coherently coupled Bose-Einstein condensates

In classical hydrodynamics with uniform density, vortices move with the local fluid velocity. This description is rewritten in terms of forces arising from the interaction with other vortices. Two such positive straight vortices experience a repulsive interaction and precess in a positive (anticlockwise) sense around their common centroid. A similar picture applies to vortices in a two-component two-dimensional uniform Bose-Einstein condensate (BEC) coherently coupled through rf Rabi fields. Unlike the classical case, however, the rf Rabi coupling induces an attractive interaction and two such vortices with positive signs now rotate in the negative (clockwise) sense. Pairs of counter-rotating vortices are instead found to translate with uniform velocity perpendicular to the line joining their cores. This picture is extended to a single vortex in a two-component trapped BEC. Although two uniform vortex-free components experience familiar Rabi oscillations of particle-number difference, such behavior is absent for a vortex in one component because of the nonuniform vortex phase. Instead the coherent Rabi coupling induces a periodic vorticity transfer between the two components.

Bose polaronic soliton-molecule and vector solitons in [formula omitted]-symmetric potential

We study analytically and numerically the properties of polaronic soliton molecules and vector solitons of a trapped Bose–Einstein condensate (BEC)-impurity mixture subjected to a PT-symmetric potential in a quasi one-dimensional geometry employing our time-dependent Hartree-Fock-Bogoliubov equations. Analytical results, based on a variational approach and checked with direct numerical simulations reveal that the width, chirp, the vibration frequency and the profile of impurity solitons are enhanced by varying the strengths of real and imaginary parts of PT-symmetric potential as well as the boson-boson and boson-impurity interaction. We address the impact of the imaginary part of the potential, which represents a gain-loss mechanism, on the dynamics and on the stability of the impurity soliton-molecule. We show that for sufficiently strong complex part of the potential, the single soliton exhibits a snake instability and the molecule destroys analogous to the dissociation of a diatomic molecule. We discuss, on the other hand, the formation of several unusual families of three-component vector solitons in the BEC-impurity mixture. An unconventional dark (D)-bright (B) soliton conversion is found.

Quantifying the mesoscopic quantum coherence of approximate NOON states and spin-squeezed two-mode Bose-Einstein condensates

We examine how to signify and quantify the mesoscopic quantum coherence of approximate two-mode NOON states and spin-squeezed two-mode Bose-Einstein condensates (BEC). We identify two criteria that verify a nonzero quantum coherence between states with quantum number different by $n$. These criteria negate certain mixtures of quantum states, thereby signifying a generalised $n$-scopic Schrodinger cat-type paradox. The first criterion is the correlation $\langle\hat{a}^{\dagger n}\hat{b}^{n}\rangle\neq0$ (here $\hat{a}$ and $\hat{b}$ are the boson operators for each mode). The correlation manifests as interference fringes in $n$-particle detection probabilities and is also measurable via quadrature phase amplitude and spin squeezing measurements. Measurement of $\langle\hat{a}^{\dagger n}\hat{b}^{n}\rangle$ enables a quantification of the overall $n$-th order quantum coherence, thus providing an avenue for high efficiency verification of a high-fidelity photonic NOON states. The second criterion is based on a quantification of the measurable spin-squeezing parameter $\xi_{N}$. We apply the criteria to theoretical models of NOON states in lossy interferometers and double-well trapped BECs. By analysing existing BEC experiments, we demonstrate generalised atomic "kitten" states and atomic quantum coherence with $n\gtrapprox10$ atoms.

Some exact solutions of the local induction equation for the motion of a vortex in a Bose–Einstein condensate with a Gaussian density profile

The dynamics of a vortex filament in a trapped Bose-Einstein condensate is considered when the equilibrium density of the condensate, in rotating with angular velocity ${\bf\Omega}$ coordinate system, is Gaussian with a quadratic form ${\bf r}\cdot\hat D{\bf r}$. It is shown that equation of motion of the filament in the local induction approximation admits a class of exact solutions in the form of a straight moving vortex, ${\bf R}(\beta,t)=\beta {\bf M}(t) +{\bf N}(t)$, where $\beta$ is a longitudinal parameter, and $t$ is the time. The vortex is in touch with an ellipsoid, as it follows from the conservation laws ${\bf N}\cdot \hat D {\bf N}=C_1$ and ${\bf M}\cdot \hat D {\bf N}=C_0=0$. Equation of motion for the tangent vector ${\bf M}(t)$ turns out to be closed, and it has the integrals ${\bf M}\cdot \hat D {\bf M}=C_2$, $(|{\bf M}| -{\bf M}\cdot\hat G{\bf \Omega})=C$, where the matrix $\hat G=2(\hat I \mbox{Tr\,} \hat D -\hat D)^{-1}$. Intersection of the corresponding level surfaces determines trajectories in the phase space.

A Path Integral Monte Carlo Study of Anderson Localization in Cold Gases in the Presence of Disorder

We revisit the problem of Anderson localization in a trapped Bose–Einstein condensate in 1D and 3D in a disordered potential, applying Quantum Monte Carlo technique because the disorder cannot be treated accurately in a perturbative way as even a small amount of disorder can produce dramatic changes in the physical properties of the system under investigation. Till date no unambiguous evidence of localization has been observed for matter waves in 3D. Matter waves made up of cold atoms are good candidates for such investigations. Simulations are performed for Rb gas in continuous space using canonical ensemble in the case of random and quasi-periodic potentials. To realize random and quasiperiodic potentials numerically we use speckle and bichromatic potentials, respectively. Owing to the high degree of control over the system parameters we specifically study the interplay of disorder and interaction in the system. A dilute Bose gas placed in a random environment falls into a fragmented localized state and the ergodicity (the repetitiveness of the wave function) is lost. An arbitrary Interaction can slowly overcome the effect of disorder and restore the ergodicity again. We observe that as the interaction strength increases, the wave functions become more and more delocalized. Since vanishing of Lyapunov exponent is only a necessary but not a sufficient condition for delocalization for probing the localization we calculate the mean square displacements as an alternative measure of localization. The path integral Monte Carlo technique in this paper numerically establishes the existing predictions of the scaling theory so far and paves a clear path for the further investigation of scaling theory to calculate more complicated properties like ‘critical exponents’ etc. in disordered quantum gases.

OpenMP, OpenMP/MPI, and CUDA/MPI C programs for solving the time-dependent dipolar Gross–Pitaevskii equation

We present new versions of the previously published C and CUDA programs for solving the dipolar Gross-Pitaevskii equation in one, two, and three spatial dimensions, which calculate stationary and non-stationary solutions by propagation in imaginary or real time. Presented programs are improved and parallelized versions of previous programs, divided into three packages according to the type of parallelization. First package contains improved and threaded version of sequential C programs using OpenMP. Second package additionally parallelizes three-dimensional variants of the OpenMP programs using MPI, allowing them to be run on distributed-memory systems. Finally, previous three-dimensional CUDA-parallelized programs are further parallelized using MPI, similarly as the OpenMP programs. We also present speedup test results obtained using new versions of programs in comparison with the previous sequential C and parallel CUDA programs. The improvements to the sequential version yield a speedup of 1.1 to 1.9, depending on the program. OpenMP parallelization yields further speedup of 2 to 12 on a 16-core workstation, while OpenMP/MPI version demonstrates a speedup of 11.5 to 16.5 on a computer cluster with 32 nodes used. CUDA/MPI version shows a speedup of 9 to 10 on a computer cluster with 32 nodes.

Homogeneous one-dimensional Bose–Einstein condensate in the Bogoliubov’s regime

We analyze the corrections caused by finite size effects upon the ground state properties of a homogeneous one-dimensional (1D) Bose–Einstein condensate. We assume from the very beginning that the Bogoliubov’s formalism is valid and consequently, we show that in order to obtain a well-defined ground state properties, finite size effects of the system must be taken into account. Indeed, the formalism described in the present paper allows to recover the usual properties related to the ground state of a homogeneous 1D Bose–Einstein condensate but corrected by finite size effects of the system. Finally, this scenario allows us to analyze the sensitivity of the system when the Bogoliubov’s regime is valid and when finite size effects are present. These facts open the possibility to apply these ideas to more realistic scenarios, e.g. low-dimensional trapped Bose–Einstein condensates.

Solvable model of a trapped mixture of Bose–Einstein condensates

A mixture of two kinds of identical bosons held in a harmonic potential and interacting by harmonic particle–particle interactions is discussed. This is an exactly-solvable model of a mixture of two trapped Bose–Einstein condensates which allows us to examine analytically various properties. Generalizing the treatments in Cohen and Lee (1985) and Osadchii and Muraktanov (1991), closed form expressions for the mixture’s frequencies and ground-state energy and wave-function, and the lowest-order densities are obtained and analyzed for attractive and repulsive intra-species and inter-species particle–particle interactions. A particular mean-field solution of the corresponding Gross–Pitaevskii theory is also found analytically. This allows us to compare properties of the mixture at the exact, many-body and mean-field levels, both for finite systems and at the limit of an infinite number of particles. We discuss the renormalization of the mixture’s frequencies at the mean-field level. Mainly, we hereby prove that the exact ground-state energy per particle and lowest-order intra-species and inter-species densities per particle converge at the infinite-particle limit (when the products of the number of particles times the intra-species and inter-species interaction strengths are held fixed) to the results of the Gross–Pitaevskii theory for the mixture. Finally and on the other end, we use the mixture’s and each species’ center-of-mass operators to show that the Gross–Pitaevskii theory for mixtures is unable to describe the variance of many-particle operators in the mixture, even in the infinite-particle limit. The variances are computed both in position and momentum space and the respective uncertainty products compared and discussed. The role of the center-of-mass separability and, for generically trapped mixtures, inseparability is elucidated when contrasting the variance at the many-body and mean-field levels in a mixture. Our analytical results show that many-body correlations exist in a trapped mixture of Bose–Einstein condensates made of any number of particles. Implications are briefly discussed.

Hamilton’s equations of motion of a vortex filament in the rotating Bose–Einstein condensate and their “soliton” solutions

The equation of motion of a quantized vortex filament in a trapped Bose-Einstein condensate [A. A. Svidzinsky and A. L. Fetter, Phys. Rev. A {\bf 62}, 063617 (2000)] has been generalized to the case of an arbitrary anharmonic anisotropic rotating trap and presented in a variational form. For condensate density profiles of the form $\rho=f(x^2+y^2+\mbox{Re\,}\Psi(x+iy))$ in the presence of the plane of symmetry $y=0$, the solutions $x(z)$ describing stationary vortices of U and S types coming to the surface and solitary waves have been found in quadratures. Analogous three-dimensional configurations of the vortex filament uniformly moving along the $z$ axis have also been found in strictly cylindrical geometry. The dependence of solutions on the form of the function $f(q)$ has been analyzed.

Mean-field regime and Thomas–Fermi approximations of trapped Bose–Einstein condensates with higher-order interactions in one and two dimensions

We derive rigorously one- and two-dimensional mean-field equations for cigar- and pancake-shaped Bose–Einstein condensates (BECs) with higher-order interactions (HOIs), which originate from shape-dependent confinement corrections to the effective two-body atomic interaction potential. We show how the higher-order interaction modifies the contact interaction of the strongly confined particles. Surprisingly, we find that the usual Gaussian profile assumption for the strongly confining direction is inappropriate for the cigar-shaped BEC case, and a Thomas–Fermi-type profile should be adopted instead. Based on the derived mean-field equations, the Thomas–Fermi densities are analyzed in the presence of the contact interaction and HOI, and considering the limit of large contact interaction and HOI. For both box and harmonic traps in one, two and three dimensions, we identify the analytical Thomas–Fermi densities, which depend on the competition between the contact interaction and the HOI.

Formation of Vortex Lattices in Superfluid Bose Gases at Finite Temperatures

We study the dynamics of a rotating trapped Bose–Einstein condensate (BEC) at finite temperatures. Using the Zaremba–Nikuni–Griffin formalism, based on a generalized Gross–Pitaevskii equation for the condensate coupled to a semiclassical kinetic equation for a thermal cloud, we numerically simulate vortex lattice formation in the presence of a time-dependent rotating trap potential. At low rotation frequency, the thermal cloud undergoes rigid body rotation, while the condensate exhibits irrotational flow. Above a certain threshold rotation frequency, vortices penetrate into the condensate and form a vortex lattice. Our simulation result clearly indicates a crucial role for the thermal cloud, which triggers vortex lattice formation in the rotating BEC.

Conditions for order and chaos in the dynamics of a trapped Bose-Einstein condensate in coordinate and energy space

We investigate numerically conditions for order and chaos in the dynamics of an interacting Bose- Einstein condensate (BEC) confined by an external trap cut off by a hard-wall box potential. The BEC is stirred by a laser to induce excitations manifesting as irregular spatial and energy oscillations of the trapped cloud. Adding laser stirring to the external trap results in an effective time-varying trapping frequency in connection with the dynamically changing combined external+laser potential trap. The resulting dynamics are analyzed by plotting their trajectories in coordinate phase space and in energy space. The Lyapunov exponents are computed to confirm the existence of chaos in the latter space. Quantum effects and trap anharmonicity are demonstrated to generate chaos in energy space, thus confirming its presence and implicating either quantum effects or trap anharmonicity as its generator. The presence of chaos in energy space does not necessarily translate into chaos in coordinate space. In general, a dynamic trapping frequency is found to promote chaos in a trapped BEC. An apparent means to suppress chaos in a trapped BEC is achieved by increasing the characteristic scale of the external trap with respect to the condensate size.

Equation of state for a trapped quantum gas: remnant of zero-point energy effects

The study of the thermodynamic properties of trapped gases has attracted great attention during the last few years and can be used as a tool to characterize such clouds in the presence of other phenomena. Here, we obtain an equation of state for a harmonically trapped Bose–Einstein condensate taking the limit of by means of global themodynamic variables. These variables allow us to explore limits in which the standard thermodynamics are not defined. Our results are taken in the high density limit, and the extrapolation for is done later. Even in this situation, we qualitatively observe the well known existence of a zero-point energy for harmonic potentials in which the determination of conjugated variables is limited by the quantum nature of the system.

Stability and dispersion relations of three-dimensional solitary waves in trapped Bose–Einstein condensates

We analyse the dynamical properties of three-dimensional solitary waves in cylindrically trapped Bose–Einstein condensates. Families of solitary waves bifurcate from the planar dark soliton and include the solitonic vortex, the vortex ring and more complex structures of intersecting vortex lines known collectively as Chladni solitons. The particle-like dynamics of these guided solitary waves provides potentially profitable features for their implementation in atomtronic circuits, and play a key role in the generation of metastable loop currents. Based on the time-dependent Gross–Pitaevskii equation we calculate the dispersion relations of moving solitary waves and their modes of dynamical instability. The dispersion relations reveal a complex crossing and bifurcation scenario. For stationary structures we find that for the solitonic vortex is the only stable solitary wave. More complex Chladni solitons still have weaker instabilities than planar dark solitons and may be seen as transient structures in experiments. Fully time-dependent simulations illustrate typical decay scenarios, which may result in the generation of multiple separated solitonic vortices.

Stable multiple vortices in collisionally inhomogeneous attractive Bose-Einstein condensates

We study the stability of solitary vortices in a two-dimensional trapped Bose-Einstein condensate (BEC) with a spatially localized region of self-attraction. Solving the respective Bogoliubov--de Gennes equations and running direct simulations of the underlying Gross-Pitaevskii equation reveals that vortices with a topological charge up to $S=6$ (at least) are stable above a critical value of the chemical potential (i.e., below a critical number of atoms, which sharply increases with $S$). The largest nonlinearity-localization radius admitting stabilization of higher-order vortices is estimated analytically and accurately identified in numerical form. To the best of our knowledge, this is the first example of a setting which gives rise to stable higher-order vortices, $Sg1$, in a trapped self-attractive BEC. The same setting may be realized in nonlinear optics too.