Double-well Trap Research Articles

Multistability and Critical Fluctuation in a Split Bose-Einstein Condensate**The project supported in part by National Natural Science Foundation of China under Grant Nos 90 103 026, 60 078 023 and 10 125 419, and the Talents Project and KJCX2-W7-4 of the Chinese Academy of Sciences

By using a two-mode description, we show that there exist the multistability, phase transition and associated critical fluctuations in the macroscopic tunnelling process between the halves of a double-well trap containing a Bose–Einstein condensate. The phase transition that two of the triple stable states and an unstable state merge into one stable state or a reverse process takes place whenever the ratio of the mean field energy per particle to the tunnelling energy goes across a critical value of order one. The critical fluctuation phenomenon corresponds to squeezed states for the phase difference between the two wells accompanying with large fluctuations of atom numbers.

Chaotic dynamics of a Bose-Einsteincondensate in a double-well trap

The dynamics of Bose-Einstein condensates in a double-wellmagnetic trap is studied using the Gross-Pitaevskii equation (GPE)and a two-mode approximation. The self-trapping instabilityoccurs when a sufficiently high potential barrier is formed inthe centre of the trap. Close to the instability threshold, theGPE is reduced to a simple amplitudeequation. In the case where weakly dissipative effects aretaken into account, the condensate exhibits aperiodicoscillations which can be described by the Lorenz thermal convection model.

Tunneling of trapped-atom Bose condensates

We obtain the dynamics in number and phase difference, for Bose condensates that tunnel between two wells of a double-well atomic trap, using the (nonlinear) Gross-Pitaevskii equation.The dynamical equations are of the canonical form for the two conjugate variables, and the Hamiltonian corresponds to that of a momentum-shortened pendulum, supporting a richer set of tunneling oscillation modes than for a superconductor Josephson junction, that has a fixed-length pendulum as a mechanical model. Novel modes include "inverted pendulum" oscillations with an average angle of \pi; and oscillations about a self-maintained population imbalance that we term "macroscopic quantum self-trapping". Other systems with this phase-number nonlinear dynamics include twocomponent (interconverting) condensates in a single harmonic trap, and He^{3}B superfluids in two containers connected by micropores.

Possible realization of Josephson charge qubits in two coupled Bose-Einstein condensates

We demonstrate that two coupled Bose-Einstein condensates (BEC) at zero temperature can be used to realize a qubit which is the counterpart of Josephson charge qubits. The two BEC are weakly coupled and confined in an asymmetric double-well trap. When the "charging energy" of the system is much larger than the Josephson energy and the system is biased near a degeneracy point, the two BEC represent a qubit with two states differing only by one atom. The realization of the BEC qubits in realistic BEC experiments is briefly discussed.

Quantum fluctuations in one-dimensional arrays of condensates

The effects of quantum and thermal fluctuations upon the fringe structure predicted to be observable in the momentum distribution of coupled Bose-Einstein condensates are studied by the effective-potential method. For a double-well trap, the coherence factor recently introduced by Pitaevskii and Stringari [Phys. Rev. Lett. 87, 180402 (2001)] is calculated using the effective potential approach and is found in good agreement with their result. The calculations are extended to the case of a one-dimensional array of condensates, showing that quantum effects are essentially described through a simple renormalization of the energy scale in the classical analytical expression for the fringe structure. The consequences for the experimental observability are discussed.

Nonlinear tunneling and chaos between two Bose–Einstein condensates trapped in time-dependent potential

In this Letter, we discuss the macroscopic quantum self-trapping (MQST) and coherent atomic tunneling between two weakly coupled Bose–Einstein condensates confined in a time-dependent double-well trap. It is shown that the nonlinear interaction and the time-periodic external field can dramatically affect the MQST, lead to the high-order Bloch harmonics of the population imbalance, and periodic or chaotic behavior. We give out the onset point of chaos by Lyapunov exponent and the phase plots. It is found that the sudden change on the quasi-energy corresponds to the transition from libration to rotation.

Optimal conditions for observing Josephson oscillations in a double-well Bose-Einstein condensate

The Josephson oscillations between condensates in a double-well trap are known theoretically to be strongly effected by the mean field interaction in dilute atomic gases. The most important effect is that the amplitude of oscillation in the relative population of the two wells is greatly suppressed due to the mean field interaction, which can make it difficult to observe the Josephson effect. Starting from the work of Raghavan, Smerzi, Fantoni, and Shenoy, we calculate the maximum amplitude of oscillation in the relative population as a function of various physical parameters, such as the trap aspect ratio, the Gaussian barrier height and width, and the total number of atoms in the condensate. We also compare results for ${}^{23}$Na and ${}^{87}$Rb. Our main new result is that the maximum amplitude of oscillation depends strongly on the aspect ratio of the harmonic trap and can be maximized in a ``pancake'' trap, as used in the experiment of Anderson and Kasevich.

Thermally activated transitions in a bistable three-dimensional optical trap

Activated escape from a metastable state underlies many physical, chemical and biological processes: examples include diffusion in solids, switching in superconducting junctions1,2, chemical reactions3,4 and protein folding5,6. Kramers presented the first quantitative calculation7 of thermally driven transition rates in 1940. Despite widespread acceptance of Kramers’ theory8, there have been few opportunities to test it quantitatively as a comprehensive knowledge of the system dynamics is required. A trapped brownian particle (relevant to our understanding of the kinetics, transport and mechanics of biological matter9,10) represents an ideal test system. Here we report a detailed experimental analysis of the brownian dynamics of a sub-micrometre sized dielectric particle confined in a double-well optical trap. We show how these dynamics can be used to directly measure the full three-dimensional confining potential—a technique that can also be applied to other optically trapped objects11,12. Excellent agreement is obtained between the predictions of Kramers’ theory and the measured transition rates, with no adjustable or free parameters over a substantial range of barrier heights.

Bose condensate in a double-well trap: Ground state and elementary excitations

We study the Bose-Einstein condensate in the MIT double-well trap. We calculate the ground-state density profile of ${}^{23}$Na atoms and the Bogoliubov spectrum of the elementary excitations as a function of the strength of the double-well barrier. In particular, we analyze the behavior of quantum-mechanical collective excitations. Finally, we discuss the observability criteria for macroscopic quantum tunneling and macroscopic quantum self-trapping.

Splitting a trap containing a Bose-Einstein condensate: Atom number fluctuations

We theoretically study atom number fluctuations between the halves of a double-well trap containing a Bose-Einstein condensate. The basic tool is the two-mode approximation, which assumes that only two one-particle states are involved. An analytical harmonic-oscillator-like model is developed and verified numerically for both stationary fluctuations in the ground state of the system, and for the fluctuations resulting from splitting of a single trap by dynamically erecting a barrier in the middle. With increasing strength of the atom-atom interactions and/or increasing height of the potential barrier, the fluctuations tend to evolve from Poissonian to sub-Poissonian. Limits of validity of the two-mode model and its relations to the phase-atom-number approach of Leggett and Sols [A. J. Leggett and F. Sols, Found. Phys. 21, 353 (1991)] are discussed in detail.

Bose-condensate tunneling dynamics: Momentum-shortened pendulum with damping

Bose-Einstein condensates in a double-well trap, as well ${}^{3}\mathrm{H}\mathrm{e}\ensuremath{-}\mathrm{B}$ baths connected by micropores, have been shown to exhibit Josephson-like tunneling phenomena. Unlike the superconductor Josephson junction of phase difference $\ensuremath{\varphi}$ that maps onto a rigid pendulum of energy $\mathrm{cos}(\ensuremath{\varphi})$, these systems map onto a momentum-shortened pendulum of energy $\ensuremath{-}\sqrt{1\ensuremath{-}{p}_{\ensuremath{\varphi}}^{2}}\mathrm{cos}(\ensuremath{\varphi})$ and length $\sqrt{1\ensuremath{-}{p}_{\ensuremath{\varphi}}^{2}}$, where ${p}_{\ensuremath{\varphi}}$ is a population imbalance between the wells/baths. We study here the effect of damping on the four distinct modes of the nonrigid pendulum, characterized by distinct temporal mean values, $〈\ensuremath{\varphi}〉$ and $〈{p}_{\ensuremath{\varphi}}〉$. Damping is shown to produce different decay trajectories to the final equilibrium $\ensuremath{\varphi}{=0=p}_{\ensuremath{\varphi}}$ state that are characteristic dynamic signatures of the initial oscillation modes. In particular, damping causes $\ensuremath{\pi}$-state oscillations with $〈\ensuremath{\varphi}〉=\ensuremath{\pi}$ to increase in amplitude and pass through phase-slip states, before equilibrating. Similar behavior has been seen in ${}^{3}\mathrm{H}\mathrm{e}\ensuremath{-}\mathrm{B}$ experiments.

BOSON JOSEPHSON JUNCTION WITH TRAPPED ATOMS

We consider coherent atomic tunneling between two weakly coupled Bose-Einstein condensates at T =0 in a double-well trap. The condensate dynamics of the macroscopic amplitudes in the two wells is modeled by two Gross-Pitaevskii equations (GPE) coupled by a tunneling matrix element. Analytic elliptic function solutions are obtained for the time evolution of the inter-well fractional population imbalance z(t) (related to the condensate phase difference) of the Boson Josephson junction (BJJ). Surprisingly, the neutral-atom BJJ shows (non-sinusoidal generalizations of) effects seen in charged-electron superconductor Josephson junctions (SJJ). The BJJ elliptic-function behavior has a singular dependence on a GPE parameter ratio Λ at a critical ratio Λ=Λc, beyond which a novel 'macroscopic quantum self-trapping' effect sets in with a non-zero time-averaged imbalance <z>≠0.

Coherent oscillations between two weakly coupled Bose-Einstein condensates: Josephson effects,πoscillations, and macroscopic quantum self-trapping

We discuss the coherent atomic oscillations between two weakly coupled Bose-Einstein condensates. The weak link is provided by a laser barrier in a (possibly asymmetric) double-well trap or by Raman coupling between two condensates in different hyperfine levels. The boson Josephson junction (BJJ) dynamics is described by the two-mode nonlinear Gross-Pitaevskii equation that is solved analytically in terms of elliptic functions. The BJJ, being a neutral, isolated system, allows the investigations of dynamical regimes for the phase difference across the junction and for the population imbalance that are not accessible with superconductor Josephson junctions (SJJ's). These include oscillations with either or both of the following properties: (i) the time-averaged value of the phase is equal to $\ensuremath{\pi} (\ensuremath{\pi}$-phase oscillations); (ii) the average population imbalance is nonzero, in states with macroscopic quantum self-trapping. The (nonsinusoidal) generalization of the SJJ ac and plasma oscillations and the Shapiro resonance can also be observed. We predict the collapse of experimental data (corresponding to different trap geometries and the total number of condensate atoms) onto a single universal curve for the inverse period of oscillations. Analogies with Josephson oscillations between two weakly coupled reservoirs of ${}^{3}\mathrm{He}\ensuremath{-}B$ and the internal Josephson effect in ${}^{3}\mathrm{He}\ensuremath{-}A$ are also discussed.

Quantum Coherent Atomic Tunneling between Two Trapped Bose-Einstein Condensates

We study the coherent atomic tunneling between two zero-temperature Bose-Einstein condensates (BEC) confined in a double-well magnetic trap. Two Gross-Pitaevskii equations for the self-interacting BEC amplitudes, coupled by a transfer matrix element, describe the dynamics in terms of the inter-well phase-difference and population imbalance. In addition to the anharmonic generalization of the familiar $ac$ Josephson effect and plasma oscillations occurring in superconductor junctions, the non-linear BEC tunneling dynamics sustains a self-maintained population imbalance: a novel "macroscopic quantum self-trapping effect".

Electron radiative self-energy of highly stripped heavy atoms: Neal J. Snyderman. University of California, Lawrence Livermore National Laboratory, Livermore, California 94550

We study Floquet states of a nonlinear two-state model which is associated with a Bose–Einstein condensate (BEC) held in a driven double-well trap on the mean-field level. We reveal that the system exhibits pitchfork bifurcation and saddle-node bifurcation for resonance and off-resonance driven cases. Such a feature provides a new method for quasienergy band engineering and for precise control of the nonlinear Floquet states.