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Physical Review A

Tunneling of ultracold Bose gases in multiple wells

R. Paredes

Published Mar 17, 2006• 14 citations
Publisher

Abstract

We study the dynamics of an ultracold gas of interacting bosons confined in a one-dimensional potential composed of $n$ wells. We derive an effective $N$-particle Hamiltonian for this system that describes the tunneling of particles between adjacent wells and that includes the interactions among the particles within the same well. The problem is discussed in the light of current research on optical lattices. We demonstrate that the effective Hamiltonian gives the possibility of studying the transition from an ideal insulator to a real conductor when the parameters that modulate the interaction and the tunneling of the particles are varied. To show this we numerically solve Schr\"odinger equation for the case of $n=3$ wells. We find that for a given number of particles it is possible to establish a weak- and a strong-interaction regime. In the limit of strong interaction the tunneling term can be neglected. One important result in our analysis is that in the weak-interaction regime the system reaches a stationary state as a result of the loss of coherence in the transport of particles through the wells.

DOI
10.1103/physreva.73.033616

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•Jan 1, 2010•0 citations
We study the dynamics of an ultracold gas of interacting bosons confined in a one-dimensional potential composed of nnn wells. We derive an effective NNN-particle Hamiltonian for this system that describes the tunneling of particles between adjacent wells and that includes the interactions among the particles within the same well. The problem is discussed in the light of current research on optical lattices. We demonstrate that the effective Hamiltonian gives the possibility of studying the transition from an ideal insulator to a real conductor when the parameters that modulate the interaction and the tunneling of the particles are varied. To show this we numerically solve Schr"odinger equation for the case of n=3n=3n=3 wells. We find that for a given number of particles it is possible to establish a weak- and a strong-interaction regime. In the limit of strong interaction the tunneling term can be neglected. One important result in our analysis is that in the weak-interaction regime the system reaches a stationary state as a result of the loss of coherence in the transport of particles through the wells.
In this paper we study an interacting Bose gas at low temperatures, confined in a one-dimensional potential composed of four wells. In order to derive and validate the effective Hamiltonian that describes this system, we study the stationary states of a particle confined in the four-well potential. In particular, we calculate the energies and the corresponding wave functions for the ground state and for the three lowest excited states. It was established that the effective Hamiltonian of a four-well optical lattice is composed of tunneling terms among all the wells, and interaction terms between pairs of particles within the same well.
We study the dynamics of an ultracold interacting Bose-Einstein gas confined in a one-dimensional potential composed of three symmetrical wells. We numerically solve the time-dependent Schrödinger equation of the N -particle Hamiltonian for N = 50, 150, 500, 1000. We demonstrate that the quantum phase transition from a superfluid (SF) to a Mott insulator (MI) phase in the three-well potential depends on the strength of the interactions among the particles, the total number of particles, and the confining potential in which the particles move. We discuss the appearance of population revivals as a function of time and find that, even in the case when the interaction strength among the particles is very small, its effect has the consequence that the system never returns to the initial condition. A stationary state for long times is observed in the SF phase, while the particle population in each well remains almost equal to the initial condition in the MI phase.
The system under study consists of an interacting ultracold Bose gas confined by a finite n -well potential in one dimension ( n = 2, 3 and 4). By numerically solving the time-dependent Schrödinger equation for the effective Hamiltonian that describes the gas confined in each potential, we determine the mean population of particles in each well as a function of time. From this analysis, we obtain a continuous transition from a coherent state to a self-trapped state as a function of the parameter Λ ≈ Ng , which takes into account the number of particles N and the interaction strength g in each system. Three different behaviours as a function of Λ are found: a coherent oscillation regime, a partial-trapped state and a self-trapped state. The partial trapping regime appears to be a novel phase for finite potentials in one dimension. A systematic quantitative study allows us to conclude that the relaxation time observed in the coherent oscillation regime for a two-well potential scales as N 1/2 . Finally, a comparison with an experimental realization of a Bose–Einstein condensate in a two-well potential (Albiez et al 2005 Phys. Rev. Lett. 95 010402) is presented, exhibiting good agreement.
Ultracold Bose gases as precise quantum simulators have emerged amongst the leaders in the current race  – a second quantum revolution  – toward the promise of unfathomable sensitivity, scope, and speed of a new era of technology. The only path leading to these advancements is through many-body descriptions, especially in regimes where interatomic interactions render many approximations incompatible with the physics. The research in this dissertation unlocks a piece of this mystery through investigations of many-body quantum tunneling of ultracold bosons in optical lattices. A key mechanism to characterizing the dynamics, matrix product state simulations equip the many-body toolbox with efficient optimization of large Hilbert spaces together with correlation and entropy measures that help clarify when semiclassical or mean-field approximations fail. In this thesis we obtain the following results. Macroscopic quantum tunneling escape of a Bose-Einstein condensate  – comprised of tens of thousands of 87Rb atoms  – from a single well, as portrayed through both experiment and theory, highlights non-exponential decay in the number of atoms trapped behind the barrier. Additionally, explicit symmetry breaking of a biperiodic optical lattice ring trap to a uniform one illustrates a critical symmetry-breaking strength at which the system transitions from a state of remembering to a state of forgetting its initial symmetry state; this symmetry memory indicates a new form of quantum dynamics that requires the quantum advantage for scaling up particle number and lattice sites. Extensive characterization of dynamical regimes in a bosonic Josephson junction  – underlaid by an optical lattice as described by the Bose-Hubbard model  – reveal substantial deviation of many body dynamics from the mean field or semiclassical predictions underlying the usual operating regime of Josephson junctions. Initial state population and phase yield insight into dynamic tunneling regimes of a quasi one-dimensional double well potential, through the Z2 spontaneous symmetry-breaking phase transition from Josephson oscillations to macroscopic self-trapping. We demonstrate that with increasing repulsive interaction strength, the U(1) superfluid-Mott insulator phase transition induces localized dynamics in the number density, such as particle-hole and soliton formation. Moreover, g^(2) correlations unveil a new dynamic regime  – the Fock flashlight. The new field of atomtronics  – or atomic circuits  – has emerged with many studies of weakly-interacting condensates. In contrast, and building on our work on many-body effects in the double wells underlying Josephson junctions, we present a proof-of-principle of a Mott-insulating atomtronic switch that requires strongly-interacting bosons. Through transient analysis, we reveal that hole conductance can be modulated via tuning interaction strength through a critical value, thus propelling the state to the next ``wedding cake'' layer of the Mott insulating structure. This critical interaction strength is a…
The selective coherent destruction of tunneling (CDT) of ultracold Bose gas with three-particle interactions in a modulated double-well potential is discussed. It is shown that the effects of two- and three-particle interactions on the dynamics of the selective CDT are strongly coupled and the three-particle interactions significantly modify the selective CDT. For weak three-particle interactions, an upper bound of the boson number for realizing the selective CDT exists and the region of boson number for realizing the selective CDT is enlarged (reduced) with repulsive (attractive) three-particle interactions. For strong three-particle interactions, the boson number in the system for realizing the selective CDT not only has an upper bound, but also has a lower bound. The results are confirmed by numerical simulations.
Ultracold bosonic gases in optical lattices are strongly correlated quantum systems similar to solids. The strong correlation between the electrons in a solid on the one hand, and the bosonic atoms in optical lattice on the other, exhibit various quantum phenomena like insulation, conductivity, localization of electrons and atoms, respectively. Controlled by the intensity of the lattice laser, the ultracold bosonic gas can be transferred from a regime with superfluid character for shallow lattices into a regime of strong correlations, the Mott insulator. As an additional external parameter besides the lattice depth, one can generate spatial inhomogeneities by superimposing an additional standing wave (so-called two-color superlattices), which gives rise to localization effects or the formation of a Bose-glass phase. In the present work, numerical simulations are employed in order to investigate characteristic signatures of the quantum phases in the low-energy excitation spectrum of one-dimensional systems. We simulate temporal small amplitude modulations of the optical lattice in analogy to experiments, and evaluate the response of the system from the time-evolved initial state. The lattice systems are described in the framework of the Bose-Hubbard model. For the evaluation of the time-evolved state, we employ several numerical methods. We analyze systems of small size (6 particles on 6 sites) using an exact time-evolution by integration of the time-dependent Schrödinger equation. The formulation of an importance truncation scheme enables us to retain only the relevant components of the model space in the strongly correlated regime and, thus, allows for the investigation of systems with 10 particles on 10 sites using exact time-evolution. Based on this method, we present results of the Mott-insulating regime as well as for the Bose-glass phase. Furthermore, we employ particle-hole methods, which allow for the treatment of systems with experimental lattice sizes and particle numbers. Starting from the equation of motion method we adapt the Tamm-Dancoff approximation as well as the random-phase approximation for the occupation number representation of the Bose-Hubbard model. We present results of simulations of up to 50 particles on 50 sites and discuss the impact of the lattice depth on the low-energy excitations (U-resonance). Moreover, the impact of a two-color superlattice and the variation of its amplitude is investigated.