Parametric pair production of collective excitations in a Bose–Einstein condensate

Summary form only given. Bose-Einstein condensation (BEC) of trapped atomic vapor has been realized in several atomic species. The static and dynamical properties of BEC crucially depend on the sign of the interatomic interaction. When the interaction is attractive, BEC in a spatially uniform 3D system is unstable to collapse into a denser phase. In a spatially confined system, however, the zero-point energy serves as a kinetic obstacle against collapse, allowing metastable BEC to be formed if the number of BEC atoms is below a certain critical number. Above the critical number of atoms, BEC collapses. The BEC with attractive interactions, therefore, has been restricted to a small number of atoms (/spl sim/1000) and the number has not been controllable.

Summary form only given. Bose-Einstein condensation (BEC) of trapped atomic vapor has been realized in several atomic species. The static and dynamical properties of BEC crucially depend on the sign of the interatomic interaction. When the interaction is attractive, BEC in a spatially uniform 3D system is unstable to collapse into a denser phase. In a spatially confined system, however, the zero-point energy serves as a kinetic obstacle against collapse, allowing metastable BEC to be formed if the number of BEC atoms is below a certain critical number. Above the critical number of atoms, BEC collapses. The BEC with attractive interactions, therefore, has been restricted to a small number of atoms (/spl sim/1000) and the number has not been controllable.

We have proposed an analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped, two-dimensional, and ideal atoms. It is found that the number of atoms in vapor is characterized by an analytical function, which involves a series of q-digamma functions in mathematics. We employ the analytical solution to calculate the internal energy E, the entropy S, the Helmholtz free energy F, and the heat capacity at constant number CN of ideal Bose atoms in a two-dimensional isotropic harmonic trap. The first main finding in this paper is that the variation with temperature of the internal energy of a finite number of ideal Bose atoms has an inflection point, which occurs at the transition temperature Tc. The second main finding is that the internal energy of a finite number of ideal Bose atoms has a classic limit as . The third main finding is that the variation with temperature of the heat capacity of a finite number of ideal Bose atoms has a maximal value, which occurs at Tc. The fourth main finding is that the heat capacity of a finite number of ideal Bose atoms has a classic limit as . The fifth main finding is that in the thermodynamic limit, at critical temperature Tc the heat capacity at constant number shows a cusp singularity, which is analogous to the λ-transition of liquid helium. Consequently, the transition between normal and condensed states is a second-order phase transition.

We studied plasmonlike resonances in one-dimensional (1D) atomic chain systems by using time-dependent density-functional theory (TDDFT) and local density functional theory. Recent TDDFT studies have shown the coexistence of longitudinal and transverses collective plasmonlike resonances in the atomic chains of simple and noble metals. Such atomic chains contain only a few atoms. The induced polarization occurs along the atomic chain in longitudinal mode and perpendicular to the atomic chain in transverse mode. To understand the emergence of plasmonlike resonance in 1D atomic chains better, we studied carbon chains in which plasmonic resonances are not expected to occur. We used TDDFT to study the emergence of collective resonances in various forms of carbon chains, cumulenes ${\mathrm{C}}_{n}{\mathrm{H}}_{4}$, polyynes ${\mathrm{C}}_{n}{\mathrm{H}}_{2}$, and alkenes ${\mathrm{C}}_{n}{\mathrm{H}}_{n+2}$. The excitation energy and dipole oscillation strengths of these systems were determined through TDDFT by using the turbomole package. We determined how collective plasmonlike resonances arise from single-electron excitations when the number of electrons increases as the carbon chain lengthens. The collective excitation behavior is then compared with that of metallic atomic chains. Our TDDFT results showed longitudinal collective modes for cumulenes and polyynes, as well as for finite-length chains. These collective excitations exhibit the same behavior as that of longitudinal ``plasmon'' previously identified in sodium and silver chains, although polyynes are gapped in the long chain limit. Such longitudinal excitations are absent in alkenes. However, unlike metal atomic chains, carbon chains exhibited no transverse collective mode. The band structure of periodic atomic chains was calculated by using the standard local density functional method. These structures were used to interpret the results and to relate the single-electron excitation to the collective plasmonlike response. Within the one-particle quantum-well picture, the longitudinal mode in the linear atomic chain arises from intraband transition with $\ensuremath{\Delta}q=1$, where $q$ is the quantum number of quantum wells. $\ensuremath{\Delta}q=1$ intraband transitions can be found in metallic (e.g., Na) chains and in carbon chains (cumulenes and polyynes), such longitudinal collective mode is rather ``generic''. Meanwhile, the transverse modes of the sodium chains are attributed to interband transitions with an even $\ensuremath{\Delta}q$ (dominated by $\ensuremath{\Delta}q=0$), and such transverse collective excitations only form if the allowed $\ensuremath{\Delta}q=0$ transitions occur between bands that are parallel to each other. Such bands can be found in simple metals, but not in carbon chains.

The Bose–Einstein condensation is a fascinating phenomenon, which results from quantum statistics for identical particles with an integer spin. Surprising properties, such as superfluidity, vortex quantization or Josephson effect, appear owing to the macroscopic quantum coherence, which spontaneously develops in Bose–Einstein condensates. Realization of Bose–Einstein condensation is not restricted in fluids like liquid helium, a superconducting phase of paired electrons in a metal and laser-cooled dilute alkali atoms. Bosonic quasi-particles like exciton-polariton and magnon in solids-state systems can also undergo Bose–Einstein condensation in certain conditions. Here, we report that the quantum coherence in Bose–Einstein condensate of the magnon quasi particles yields spontaneous electric polarization in the quantum magnet TlCuCl3, leading to remarkable magnetoelectric effect. Very soft ferroelectricity is realized as a consequence of the O(2) symmetry breaking by magnon Bose–Einstein condensation. The finding of this ferroelectricity will open a new window to explore multi-functionality of quantum magnets.

Using a corrected sum rule and a generalized virial identity, we study the analytical expression for entire modes of the collective elementary excitation spectrum in a trapped Bose–Einstein condensate at any atom number. Explicit analytical formulas for the spectrum are obtained for the harmonic traps with both spherical symmetry and axial symmetry using the gaussian approximation for the N-body ground-state wave function of the condensate. These formulas give the simple dependence of all energy levels on the atom numbers, their interaction strength and trap geometry parameters.

We examine the effect of the transverse breathing mode on the longitudinal sound propagation in a Bose-Einstein condensate in a one-dimensional optical lattice. In particular, we discuss how the coupling with the transverse breathing mode influences the sound velocity in an optical lattice. Using a variational approach we calculate the dispersion relations for the longitudinal sound mode and the transverse breathing mode analytically and find that the shift in the sound velocity from the uncoupled result can be large enough to be experimentally relevant. We also find that the effective mass of the transverse breathing mode is affected considerably by the coupling to longitudinal sound.

We study the dynamics of bright solitons in a Bose–Einstein condensate confined in a highly asymmetric trap. While working within the framework of a variational approach we carry out the stability analysis of the Bose– Einstein condensate solitons against collapse. When the number of atoms in the soliton exceeds a critical number Nc, it undergoes the so-called primary collapse. We find an analytical expression for Nc in terms of appropriate experimental quantities that are used to produce and confine the condensate. We further demonstrate that, in the geometry of the problem considered, the width of the soliton varies inversely as the number of constituent atoms.

Oscillations of a Bose–Einstein condensate in a rapidly contracting circular box

Quantum thermal engines have received much attention in recent years due to their potential applications. For a candidate group, harmonically trapped gases under Bose-Einstein condensation (BEC), we see little investigation on the energy transference around that transition. Therefore, we present an empirical study with rubidium-87 gas samples in a magnetic harmonic trap. We developed an empirical equation of state model to fit to our experimental dataset, expressing the pressure parameter in terms of temperature, and six technical coefficients, functions of the volume parameter and the number of atoms. By using standard thermodynamic relations, we determine the system's entropy, shown to be constant at the BEC transition, as expected. Being isentropic makes the BEC transition an energy source for non-mechanical work. Hence, we observed that the enthalpy at the BEC transition, at fixed values of the volume parameter, grows fairly linearly with the number of atoms. We fitted a linear function to that data, finding the specific enthalpy of the BEC transformation and the intrinsic enthalpic loss for BEC. We deem this study to be a step closer to practical quantum-based engines.

We study both static and dynamic properties of a weakly interacting Bose–Einstein condensate (BEC) in a quasi one-dimensional gravito-optical surface trap, where the downward pull of gravity is compensated by the exponentially decaying potential of an evanescent wave. First, we work out approximate solutions of the Gross–Pitaevskii equation for both a small number of atoms using a Gaussian ansatz and a larger number of atoms using the Thomas–Fermi limit. Then we confirm the accuracy of these analytical solutions by comparing them to numerical results. From there, we numerically analyze how the BEC cloud expands non-ballistically, when the confining evanescent laser beam is shut off, showing agreement between our theoretical and previous experimental results. Furthermore, we analyze how the BEC cloud expands non-ballistically due to gravity after switching off the evanescent laser field in the presence of a hard-wall mirror which we model by using a blue-detuned far-off-resonant sheet of light. There we find that the BEC shows significant self-interference patterns for a large number of atoms, whereas for a small number of atoms, a revival of the BEC wave packet with few matter-wave interference patterns is observed.

Quasiparticles in quantum many-body systems have essential roles in modern physical problems. Bose–Einstein condensation (BEC) of excitons in semiconductors is one of the unobserved quantum statistical phenomena predicted in the photoexcited quasiparticles in many-body electrons. In particular, para-excitons in cuprous oxide have been studied for decades because the decoupling from the radiation field makes the coherent ensemble a purely matter-like wave. However, BEC has turned out to be hard to realize at superfluid liquid helium-4 temperatures due to a two-body inelastic collision process. It is therefore essential to set a lower critical density by further lowering the exciton temperature. Here we cool excitons to sub-Kelvin temperature and spatially confine them to realize the critical number for BEC. We show that BEC manifests itself as a relaxation explosion as has been discussed in atomic hydrogen. The results indicate that dilute excitons are purely bosonic and BEC indeed occurs.

A Bose gas in a double-well potential, exhibiting a true Bose-Einstein condensate (BEC) amplitude and initially performing Josephson oscillations, is a prototype of an isolated, nonequilibrium many-body system. We investigate the quasiparticle (QP) creation and thermalization dynamics of this system by solving the time-dependent Keldysh-Bogoliubov equations. We find avalanchelike QP creation due to a parametric resonance between BEC and QP oscillations, followed by slow, exponential relaxation to a thermal state at an elevated temperature, controlled by the initial excitation energy of the oscillating BEC above its ground state. The crossover between the two regimes occurs because of an effective decoupling of the QP and BEC oscillations. This dynamics is analogous to elementary particle creation in models of the early universe. The thermalization in our setup occurs because the BEC acts as a grand canonical reservoir for the quasiparticle system.

Recent advances in the precise control of ultracold atomic systems have led to the realisation of Bose–Einstein condensates (BECs) and degenerate Fermi gases. An important challenge is to extend this level of control to more complicated molecular systems. One route for producing ultracold molecules is to form them from the atoms in a BEC. For example, a two-photon stimulated Raman transition in a 87Rb BEC has been used to produce 87Rb2 molecules in a single rotational-vibrational state1, and ultracold molecules have also been formed2 through photoassociation of a sodium BEC. Although the coherence properties of such systems have not hitherto been probed, the prospect of creating a superposition of atomic and molecular condensates has initiated much theoretical work3,4,5,6,7. Here we make use of a time-varying magnetic field near a Feshbach resonance8,9,10,11,12 to produce coherent coupling between atoms and molecules in a 85Rb BEC. A mixture of atomic and molecular states is created and probed by sudden changes in the magnetic field, which lead to oscillations in the number of atoms that remain in the condensate. The oscillation frequency, measured over a large range of magnetic fields, is in excellent agreement with the theoretical molecular binding energy, indicating that we have created a quantum superposition of atoms and diatomic molecules—two chemically different species.

The properties and stability of a trapped Bose-Einstein condensate are strongly influenced by attractive interactions between the particles. We describe the spatial distribution, stability, and collisional loss rates for a weakly interacting gas in the mean-field limit. We show how the condensate contracts and becomes unstable as the number of condensate atoms increases. We further show how the number of atoms is limited by the collisional loss rates associated with the contraction of the condensate; this loss is in addition to the particle ejection decay indentified by Kagan et al. \textcopyright{} 1996 The American Physical Society.