Trapped Bose Gas Research Articles

The dynamics of three-mode coupling in a trapped Bose gas

ABSTRACTMultiple mode couplings in topological coherent modes of Bose–Einstein condensate are considered, by introducing an external alternating (resonating) field in the system. This analysis is based on the analytical solutions of nonlinear Gross–Pitaevskii equation for a trapped Bose gas at nearly absolute zero temperature. The dynamics of fractional populations of the generated coherent modes are analysed, particularly for a three-level system in the limit of small to large detuning of the intermediate state. These coupled topological modes, though nonlinear, are analogous to a resonant atom and exhibit a variety of significant non-trivial phenomena (effects), like: dynamic phase transitions, interference patterns, critical phenomena, mode-locking and chaotic motion.

Effective approach to impurity dynamics in one-dimensional trapped Bose gases

We investigate a temporal evolution of an impurity atom in a one-dimensional trapped Bose gas following a sudden change of the boson-impurity interaction strength. Our focus is on the effects of inhomogeneity due to the harmonic confinement. These effects can be described by an effective one-body model where both the mass and the spring constant are renormalized. This is in contrast to the classic renormalization, which addresses only the mass. We propose an effective single-particle Hamiltonian and apply the multilayer multiconfiguration time-dependent Hartree method for bosons to explore its validity. Numerical results suggest that the effective mass is smaller than the impurity mass, which means that it cannot straightforwardly be extracted from translationally invariant models.

Effects of Quantum Fluctuations on -Symmetric Solitons of a Trapped Bose Gas**Supported by the National Natural Science Foundation of China under Grant Nos. 11647017, 11805116, 21703166, and supported by Science Research Fund of Shaanxi University of Science and Technology under Grant No. BJ16-03

Considering the quantum fluctuation effects, the existence and stability of solitons in a Bose-Einstein condensate subjected in a -symmetric potential are discussed. Using the variational approach, we investigate how the quantum fluctuation affects the self-localization and stability of the condensate with attractive two-body interactions. The results show that the quantum fluctuation dramatically influences the shape, width, and chemical potential of the condensate. Analytical variational computation also predicts there exists a positive critical quantum fluctuation strength qc with each fixed attractive two-body interaction g0, if the quantum fluctuation strength q0 is bigger than qc, there is no bright soliton solution existence. We also study the effects of the quantum fluctuations on the stability of solitons using the Vakhitov-Kolokolov (VK) stability criterion. A robust stable bright soliton will always exist when the quantum fluctuation strength q0 belongs to the parameter regimes qc ≥ q0 > 0.

Anomalous frequency shifts in a one-dimensional trapped Bose gas

We consider a system of interacting bosons in one dimension at a two-body resonance. This system, which is weakly interacting, is known to give rise to effective three-particle interactions, whose dynamics is similar to that of a two-dimensional Bose gas with two-body interactions, and exhibits an identical scale anomaly. We consider the experimentally relevant scenario of a harmonically trapped system. We solve the three-body problem exactly and evaluate the shifts in the frequency of the lowest compressional mode with respect to the dipole mode, and find that the effect of the anomaly is to increase the mode's frequency. We also consider the weak-coupling regime of the trapped many-boson problem and find, within the local density approximation, that the frequency of the lowest compressional mode is also shifted upwards in this limit. Moreover, the anomalous frequency shifts are enhanced by the higher particle number to values that should be observable experimentally.

Coalescence of Two Impurities in a Trapped One-dimensional Bose Gas.

We study the ground state of a one-dimensional (1D) trapped Bose gas with two mobile impurity particles. To investigate this setup, we develop a variational procedure in which the coordinates of the impurity particles are slowlike variables. We validate our method using the exact results obtained for small systems. Then, we discuss energies and pair densities for systems that contain of the order of 100 atoms. We show that bosonic noninteracting impurities cluster. To explain this clustering, we calculate and discuss induced impurity-impurity potentials in a harmonic trap. Further, we compute the force between static impurities in a ring (in the manner of the Casimir force), and contrast the two effective potentials: the one obtained from the mean-field approximation, and the one due to the one-phonon exchange. Our formalism and findings are important for understanding (beyond the polaron model) the physics of modern 1D cold-atom systems with more than one impurity.

Quantum hydrodynamic approximations to the finite temperature trapped Bose gases

For the quantum kinetic system modeling the Bose–Einstein Condensate that accounts for interactions between condensate and excited atoms, we use the Chapman–Enskog expansion to derive its hydrodynamic approximations, include both Euler and Navier–Stokes approximations. The hydrodynamic approximations describe not only the macroscopic behavior of the BEC but also its coupling with the non-condensates, which agrees with the Landau two-fluid theory.

The Inhomogeneous Gaussian Free Field, with application to ground state correlations of trapped 1d Bose gases

This formalism is then applied to the study of ground state correlations of the Lieb-Liniger gas trapped in an external potential V(x)V(x). Relations with previous works on inhomogeneous Luttinger liquids are discussed. The main innovation here is in the identification of local observables \hat{O} (x)Ô(x) in the microscopic model with their field theory counterparts \partial_x h, e^{i h(x)}, e^{-i h(x)}∂xh,eih(x),e−ih(x), etc., which involve non-universal coefficients that themselves depend on position — a fact that, to the best of our knowledge, was overlooked in previous works on correlation functions of inhomogeneous Luttinger liquids —, and that can be calculated thanks to Bethe Ansatz form factors formulae available for the homogeneous Lieb-Liniger model. Combining those position-dependent coefficients with the correlation functions of the IGFF, ground state correlation functions of the trapped gas are obtained. Numerical checks from DMRG are provided for density-density correlations and for the one-particle density matrix, showing excellent agreement.

On the dynamics of finite temperature trapped Bose gases

The system that describes the dynamics of a Bose–Einstein Condensate (BEC) and the thermal cloud at finite temperature consists of a nonlinear Schrodinger (NLS) and a quantum Boltzmann (QB) equations. In such a system of trapped Bose gases at finite temperature, the QB equation corresponds to the evolution of the density distribution function of the thermal cloud and the NLS is the equation of the condensate. The quantum Boltzmann collision operator in this temperature regime is the sum of two operators C12 and C22, which describe collisions of the condensate and the non-condensate atoms and collisions between non-condensate atoms. Above the BEC critical temperature, the system is reduced to an equation containing only a collision operator similar to C22, which possesses a blow-up positive radial solution with respect to the L∞ norm (cf. [29]). On the other hand, at the very low temperature regime (only a portion of the transition temperature TBEC), the system can be simplified into an equation of C12, with a different (much higher order) transition probability, which has a unique global classical positive radial solution with weighted L1 norm (cf. [3]). In our model, we first decouple the QB, which contains C12+C22, and the NLS equations, then show a global existence and uniqueness result for classical positive radial solutions to the spatially homogeneous kinetic system. Different from the case considered in [29], due to the presence of the BEC, the collision integrals are associated to sophisticated energy manifolds rather than spheres, since the particle energy is approximated by the Bogoliubov dispersion law. Moreover, the mass of the full system is not conserved while it is conserved for the case considered in [29]. A new theory is then supplied.

Application of the Lagrangian variational method to a one-dimensional Bose gas in a dimple trap

We present an application of the Lagrangian variational method (LVM) to study the small-oscillation dynamics and ground-state properties of an interacting Bose gas in a one-dimensional dimple trap, for the first time. For this purpose, a Gaussian wavefunction ansatz—rich in variational parameters—is used for the analytical solutions of the equations of motion resulting from the LVM. The results of LVM are found to agree well with those due to the numerical Crank–Nicolson method. The strength of this method is shown to reveal details hidden in the wavefunction and its dynamics that are otherwise hard to obtain numerically or even experimentally. In this regard, the important role that the wavefunction-width plays in determining the properties of a trapped Bose gas is manifested. Via this method, the dependence of the breathing-mode frequency on the interactions, width of wavefunction, and dimple trap parameters, is demonstrated. A significant result is that in the ‘width-space’ of the wavefunction, the LVM yields a differential equation that describes the motion of a fictitious particle in an effective potential whose shape is by analogy similar to that of an interatomic interaction potential.

Quantum-fluctuation-induced time-of-flight correlations of an interacting trapped Bose gas

We investigate numerically the momentum correlations in a two dimensional, harmonically trapped interacting Bose system at $T=0$ temperature, by using a particle number preserving Bogoliubov approximation. Interaction induced quantum fluctuations of the quasi-condensate lead to a large anti-correlation dip between particles of wave numbers $\mathbf{k}$ and $-\mathbf{k}$ for $|\mathbf{k}|\sim 1/R_c$, with $R_c$ typical size of the condensate. The anti-correlation dip found is a clear fingerprint of coherent quantum fluctuations of the condensate. In contrast, for larger wave numbers, $|\mathbf{k}| >> 1/R_c$, a weak positive correlation is found between particles of wave numbers $\mathbf{k}$ and $-\mathbf{k}$, in accordance with the Bogoliubov result for homogeneous interacting systems.

Use of Two-Body Correlated Basis Functions with van der Waals Interaction to Study the Shape-Independent Approximation for a Large Number of Trapped Interacting Bosons

We study the ground state and the low-lying excitations of a trapped Bose gas in an isotropic harmonic potential for very small ($\sim 3$) to very large ($\sim 10^7$) particle numbers. We use the correlated two-body basis functions and the shape-dependent van der Waals interaction in our many-body calculations. We present an exhaustive study of the effect of inter-atomic correlations and the accuracy of the mean-field equations considering a wide range of particle numbers. We calculate the ground state energy and the one-body density for different values of the van der Waals parameter $C_{6}$. We compare our results with those of the modified Gross-Pitaevskii results, the correlated Hartree hypernetted-chain equations (which also utilize the two-body correlated basis functions), as well as of the Diffusion Monte Carlo for hard sphere interactions. We observe the effect of the attractive tail of the van der Waals potential in the calculations of the one-body density over the truly repulsive zero-range potential as used in the Gross-Pitaevskii equation and discuss the finite-size effects. We also present the low-lying collective excitations which are well described by a hydrodynamic model in the large particle limit.

Hydrodynamic versus collisionless dynamics of a one-dimensional harmonically trapped Bose gas

By using a sum rule approach we investigate the transition between the hydrodynamic and the collisionless regime of the collective modes in a 1D harmonically trapped Bose gas. Both the weakly interacting gas and the Tonks-Girardeau limits are considered. We predict that the excitation of the dipole compression mode is characterized, in the high temperature collisionless regime, by a beating signal of two different frequencies ($\omega_z$ and $3\omega_z$) while, in the high temperature collisional regime, the excitation consists of a single frequency ($\sqrt{7}\omega_z$). This behaviour differs from the case of the lowest breathing mode whose excitation consists of a single frequency ($2\omega_z$) in both regimes. Our predictions for the dipole compression mode open promising perspectives for the experimental investigation of collisional effects in 1D configurations.

Collective modes of a one-dimensional trapped Bose gas in the presence of the anomalous density

We study the collective modes of a one-dimensional harmonically trapped Bose-Einstein condensate in the presence of the anomalous density using the time-dependent Hartree-Fock-Bogoliubov theory. Within the hydrodynamic equations, we derive analytical expressions for the mode frequencies and the density fluctuations of the anomalous density which constitutes the minority component at very low temperature and feels an effective external potential exerted by the majority component, i.e., the condensate. On the other hand, we numerically examine the temperature dependence of the breathing mode oscillations of the condensate at finite temperature in the weak-coupling regime. At zero temperature, we compare our predictions with available experimental data, theoretical treatments, and Monte carlo simulations in all interaction regimes and the remaining hindrances are emphasized. We show that the anomalous correlations have a non-negligible role on the collective modes at both zero and finite temperatures.

Study of weakly interacting trapped Bose gas

We have obtained expressions for single particle density and two particle density of weakly interacting trapped quantum gases. These are valid for all temperature and in any dimension. These expressions have been simplified and expressed in terms of non-interacting single particle density. The ground fluctuations for T<Tc in grand canonical ensemble has been treated with care using the method of Kocharovsky et al. [Phys. Rev. A 61, 053606 (2000)]. Some numerical results are presented in one and three dimension for isotropic harmonically trapped Bose gas with contact interactions. It is seen that boson density decreases with increasing repulsive interactions. The expression for critical temperature is also shown to agree with earlier result and is in accordance with experiments.

Signature of Critical Point in Momentum Profile of Trapped Ultracold Bose Gases**Supported by the National Natural Science Foundation of China under Grant No 11104322, and the National Key Basic Research and Development Program of China under Grant No 2011CB921503.

We present a new method to identify the critical point for the Bose-Einstein condensation (BEC) of a trapped Bose gas. We calculate the momentum distribution of an interacting Bose gas near the critical temperature, and find that it deviates significantly from the Gaussian profile as the temperature approaches the critical point. More importantly, the standard deviation between the calculated momentum spectrum and the Gaussian profile at the same temperature shows a turning point at the critical point, which can be used to determine the critical temperature. These predictions are also confirmed by our BEC experiment for magnetically trapped 87Rb gases.

Energy fluctuation of a finite number of interacting bosons: A correlated many-body approach

We calculate the energy fluctuation of a truly finite number of interacting bosons and study the role of interaction. Although the ideal Bose gas in thermodynamic limit is an exactly solvable problem and analytic expression of various fluctuation measures exists, the experimental Bose-Einstein condensation (BEC) is a nontrivial many-body problem. We employ a two-body correlated basis function and utilize the realistic van der Waals interaction. We calculate the energy fluctuation $(△{E}^{2})$ of the interacting trapped bosons and plot $\frac{△{E}^{2}}{{k}_{B}^{2}{T}^{2}}$ as a function of $\frac{T}{{T}_{c}}$. In the classical limit $△{E}^{2}$ is related to the specific heat per particle ${c}_{v}$ through the relation $△{E}^{2}={k}_{B}{T}^{2}{c}_{v}$. We have obtained a distinct hump in $\frac{△{E}^{2}}{{k}_{B}^{2}{T}^{2}}$ around the condensation point for three-dimesional harmonically trapped Bose gas when the particle number $N\ensuremath{\simeq}5000$ and above which corresponds to the second-order phase transition. However for finite-size interacting bosons ($N\ensuremath{\simeq}$ a few hundred) the hump is not sharp, and the maximum in $\frac{△{E}^{2}}{{k}_{B}^{2}{T}^{2}}$ can be interpreted as a smooth increase in the scaled fluctuation below ${T}_{c}$ and then a decrease above ${T}_{c}$. To illustrate the justification we also calculate ${c}_{v}$, which exhibits the same feature, which leads to the conjecture that for finite-sized interacting bosons phase transition is ruled out.

Ground-state phase diagram of a spin-orbit-coupled bosonic superfluid in an optical lattice

In recent experiments, spin-orbit-coupled (SOC) bosonic gases in an optical lattice have been successfully prepared into any Bloch band [Hamner et al., Phys. Rev. Lett. 114, 070401 (2015)], which promises a viable contender in the competitive field of simulating gauge-related phenomena. However, the ground-state phase diagram of such systems in the superfluid regime is still lacking. Here we present a detailed study of the phase diagram in an optically trapped Bose gas with equal-weight Rashba and Dresselhaus SO coupling. We identify four different quantum phases, which include three normal phases and a mixed phase, by considering the wave vector ${k}_{1}$, the longitudinal $\ensuremath{\langle}{\ensuremath{\sigma}}_{z}\ensuremath{\rangle}$, and the transverse $\ensuremath{\langle}{\ensuremath{\sigma}}_{x}\ensuremath{\rangle}$ spin polarizations as three order parameters. The ground state of normal phases is a Bloch wave with a single wave vector ${k}_{1}$, which can position in arbitrary regions in the Brillouin zone. By contrast, the ground state of the mixed phase is a superposition of two Bloch waves with opposite ${k}_{1}$, which, remarkably, may lack periodicity even though the system's Hamiltonian is periodic. This mixed phase in the lattice setting can be seen as the counterpart of the stripe phase associated with the uniform SOC gas. Furthermore, due to the lattice-renormalized SOC, the phase diagram of the model system becomes significantly different from the uniform case when the lattice strength grows. Finally, a scheme for experimentally probing the mixed phase using Bragg spectroscopy is proposed.

Calorimetry of a harmonically trapped Bose gas

We experimentally study the energy-temperature relationship of a harmonically trapped Bose-Einstein condensate by transferring a known quantity of energy to the condensate and measuring the resulting temperature change. We consider two methods of heat transfer, the first using a free expansion under gravity and the second using an optical standing wave to diffract the atoms in the potential. We investigate the effect of interactions on the thermodynamics and compare our results to various finite temperature theories.

Quantum recurrences in a one-dimensional gas of impenetrable bosons.

It is well-known that a dilute one-dimensional (1D) gas of bosons with infinitely strong repulsive interactions behaves like a gas of free fermions. Just as with conduction electrons in metals, we consider a single-particle picture of the resulting dynamics, when the gas is isolated by enclosing it into a box with hard walls and preparing it in a special initial state. We show, by solving the nonstationary problem of a free particle in a 1D hard-wall box, that the single-particle state recurs in time, signaling the intuitively expected back-and-forth motion of a free particle moving in a confined space. Under suitable conditions, the state of the whole gas can then be made to recur if all the particles are put in the same initial momentum superposition. We introduce this problem here as a modern instance of the discussions giving rise to the famous recurrence paradox in statistical mechanics: on one hand, our results may be used to develop a poor man's interpretation of the recurrence of the initial state observed [T. Kinoshita et al., Nature 440, 900 (2006)] in trapped 1D Bose gases of cold atoms, for which our estimated recurrence time is in fair agreement with the period of the oscillations observed; but this experiment, on the other hand, has been substantially influential on the belief that an isolated quantum many-body system can equilibrate as a consequence of its own unitary nonequilibrium dynamics. Some ideas regarding the latter are discussed.

The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases

We study the ground state of a trapped Bose gas, starting from the full many-body Schrodinger Hamiltonian, and derive the nonlinear Schrodinger energy functional in the limit of large particle number, when the interaction potential converges slowly to a Dirac delta function. Our method is based on quantitative estimates on the discrepancy between the full many-body energy and its mean-field approximation using Hartree states. These are proved using finite dimensional localization and a quantitative version of the quantum de Finetti theorem. Our approach covers the case of attractive interactions in the regime of stability. In particular, our main new result is a derivation of the 2D attractive nonlinear Schrodinger ground state.