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Journal of Physics A Mathematical and Theoretical

Bose condensation in systems withp-particle tunneling and multi-body interactions

V. M. Travin, T. K. Kopeć

Published Aug 4, 2015• 4 citations
Publisher

Abstract

In this paper, we have studied a class of Hamiltonians that generalizes the Bose–Hubbard (BH) model of interacting bosons to include p -particle hopping as well as the multi-body interactions . While the case of two-body interaction and single-particle tunneling is well known, it is interesting to examine whether the resulting phase diagram for the general case has different features from that of the standard BH system. To this end we have applied a mean-field theory to locate critical lines, combined with the resolvent and Laplace transform method for the efficient calculation of the statistical sum for the quantum Hamiltonian. We have studied both ground state properties and temperature evolution of the phase diagram. Several new features, as an expansion of Mott insulator lobes for multi-body interaction and the emergence of a new insulating plateau for , have been demonstrated. These results may be of interest for experiments on cold atom systems in optical lattices.

DOI
10.1088/1751-8113/48/34/345001

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In this paper, we have studied a class of Hamiltonians that generalizes the Bose–Hubbard (BH) model of interacting bosons to include p -particle hopping as well as the multi-body interactions . While the case of two-body interaction and single-particle tunneling is well known, it is interesting to examine whether the resulting phase diagram for the general case has different features from that of the standard BH system. To this end we have applied a mean-field theory to locate critical lines, combined with the resolvent and Laplace transform method for the efficient calculation of the statistical sum for the quantum Hamiltonian. We have studied both ground state properties and temperature evolution of the phase diagram. Several new features, as an expansion of Mott insulator lobes for multi-body interaction and the emergence of a new insulating plateau for , have been demonstrated. These results may be of interest for experiments on cold atom systems in optical lattices.
Experiments with ultracold atoms in optical lattice have become a versatile testing ground to study diverse quantum many-body Hamiltonians. A single-band Bose-Hubbard (BH) Hamiltonian was first proposed to describe these systems in 1998 and its associated quantum phase-transition was subsequently observed in 2002. Over the years, there has been a rapid progress in experimental realizations of more complex lattice geometries, leading to more exotic BH Hamiltonians with contributions from excited bands, and modified tunneling and interaction energies. There has also been interesting theoretical insights and experimental studies on “un- conventional” Bose-Einstein condensates in optical lattices and predictions of rich orbital physics in higher bands. In this thesis, I present our results on several multi- band BH models and emergent quantum phenomena. In particular, I study optical lattices with two local minima per unit cell and show that the low energy states of a multi-band BH Hamiltonian with only pairwise interactions is equivalent to an effec- tive single-band Hamiltonian with strong three-body interactions. I also propose a second method to create three-body interactions in ultracold gases of bosonic atoms in a optical lattice. In this case, this is achieved by a careful cancellation of two contributions in the pair-wise interaction between the atoms, one proportional to the zero-energy scattering length and a second proportional to the effective range. I subsequently study the physics of Bose-Einstein condensation in the second band of a double-well 2D lattice and show that the collision aided decay rate of the con- densate to the ground band is smaller than the tunneling rate between neighboring unit cells. Finally, I propose a numerical method using the discrete variable repre- sentation for constructing real-valued Wannier functions localized in a unit cell for optical lattices. The developed numerical method is general and can be applied to a wide array of optical lattice geometries in one, two or three dimensions.
The zero temperature phase diagram for ultracold bosons in a random 1D potential is obtained through a site decoupling mean-field scheme performed over a Bose-Hubbard (BH) Hamiltonian, whose hopping term is considered as a random variable. As for the model with random on-site potential, the presence of disorder leads to the appearance of a Bose glass phase. The different phases—i.e., Mott insulator, superfluid, and Bose glass—are characterized in terms of condensate fraction and superfluid fraction. Furthermore, the boundary of the Mott lobes is related to an off-diagonal Anderson model featuring the same disorder distribution as the original BH Hamiltonian.
We study the system of multibody interacting bosons on a two-dimensional optical lattice and analyze the formation of bound bosonic pairs in the context of the Bose-Hubbard model. Assuming a repulsive two-body interaction we obtain the signatures of pair formation in the regions between the Mott insulator lobes of the phase diagram for different choices of higher-order local interactions. Considering the most general Bose-Hubbard model involving local multibody interactions we investigate the ground state properties utilizing the cluster mean-field theory approach and further confirm the results by means of sophisticated infinite projected entangled pair states calculations. By using various order parameters, we show that the choice of higher-order interaction can lead to pair superfluid phase in the system between two different Mott lobes. We also analyze the effect of temperature and density-dependent tunneling to establish the stability of the PSF phase.
In Chapter 1 we set up the physical and mathematical framework for the description of interacting bosons on a lattice. After a brief introduction to the basic concepts of Bose-Einstein condensation, and how the condensate can be manipulated using optical lattices, we derive the Bose-Hubbard Hamiltonian (BHH) which is the paradigm model as far as the quantum description is concerned. Then the semiclassical limit of the BHH – the discrete nonlinear Schroedinger equation (DNLS) – as well as the mean-field (classical) description, the so-called Gross-Pitaevskii equation (GPE), is discussed. We give an overview of the relevant literature and of the various physical systems that are captured by the BHH including, among others, arrays of Josephson junctions and bond vibrations in small molecules. The chapter ends with an outline of the remaining thesis. In Chapter 2, we look at some interesting experiments done in cold bosonic systems. Ostensibly, done to realize the dc and ac Josephson effect in a system of cold bosons, it was soon found that the system can show some rather interesting non-linear behavior. This includes features such as solitons and breathers , but most interestingly, it shows the damping of the center-of-mass motion. Theoretical explanations are given for both the experiments. The damping of center-of-mass of the system is seen as a result of modulational instability. We give a different interpretation to this opening up lines for further investigations. We outline these at the end of the chapter. In Chapter 3, we discuss various experiments which contradicted the idea of a sharp modulational instability. Rather, the loss of coherence is smeared out over a large domain. We then reproduce some calculations from the literature showing how the dissipation can be thought of as arising from quantum phase slips. We calculate the tunneling rates for quantum phase slips and show a phase diagram demarcating stable and unstable regions of current which have been have obtained from mean-field calculations via the Gutzwiller ansatz. In Chapter 4, we look at the problem in details analyzing it as a Tomonaga-Luttinger liquid. The framework of Tomonaga-Luttinger liquid is actually a non-perturbative tool through which we can investigate an one-dimensional interacting bosonic fluid. We look at two possible mechanisms of decay of current- We look at the effect of a weak link in a Josephson Junction array as well as local quantum phase slips on a bosonic superfluid. We find that quantum phase slips are actually irrelevant under renormalisation group in both cases and the supercurrent is revived. In chapter 5, we look at the non-linear dynamics of the structures in greater detail, thus elucidating chaos. This mechanism of chaos is different from all realizations of chaos known so far and is a result of phase fluctuations, which are encoded by quantum phase slips.
We present an extensive study of Mott insulator (MI) and superfluid (SF) shells in Bose-Hubbard (BH) models for bosons in optical lattices with harmonic traps. For this we apply the inhomogeneous mean-field theory developed by Sheshadri et al. [Phys. Rev. Lett. 75, 4075 (1995)]. Our results for the BH model with one type of spinless bosons agree quantitatively with quantum Monte Carlo simulations. Our approach is numerically less intensive than such simulations, so we are able to perform calculations on experimentally realistic, large three-dimensional systems, explore a wide range of parameter values, and make direct contact with a variety of experimental measurements. We also extend our inhomogeneous mean-field theory to study BH models with harmonic traps and (a) two species of bosons or (b) spin-1 bosons. With two species of bosons, we obtain rich phase diagrams with a variety of SF and MI phases and associated shells when we include a quadratic confining potential. For the spin-1 BH model, we show, in a representative case, that the system can display alternating shells of polar SF and MI phases, and we make interesting predictions for experiments in such systems.
Abstract We analyse the distinction between the three different ground states presented by a system of spinless bosons with short-range interactions submitted to a random potential using the disordered Bose–Hubbard model. The criteria for identifying the superfluid, the Mott-insulator, and the Bose-glass phases at finite temperatures are discussed for small values of the kinetic energy associated with the tunnelling of particles between potential wells. Field theoretical considerations are applied in order to construct a diagrammatic hopping expansion to the finite-temperature Green’s function. By performing a summation of subsets of diagrams we are able to find the condition to the long-range correlations which leads to the phase boundary between superfluid and insulating phases. The perturbative expression to the local correlations allows us to calculate an approximation to the single-particle density of states of low-energy excitations in the presence of small hopping, which characterizes unambiguously the distinction between the Mott-insulator and the Bose-glass phases. We obtain the phase diagram for bounded on-site disorder. It is demonstrated that our analysis is capable of going beyond the mean-field theory results for the classification of these different ground states.