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Journal of Physics B Atomic Molecular and Optical Physics

Interplay of optical potential and condensate properties for bosons in different optical lattice geometries

Apurba Barman, Saurabh Basu

Published May 14, 2012• 1 citation
Publisher

Abstract

Motivated by the availability of optical lattices with tunable geometries in experiments, we compute different physical properties such as condensate fraction, fluctuation and depletion of the condensate density and quasi-particle velocity as a function of the interparticle interaction strength for bosons in different 2D optical potentials that correspond to square, triangular and honeycomb geometries. Our results demonstrate an interplay of these features with that of the coordination number and underlying lattice geometries. A triangular lattice, which has a coordination number z = 6, shows larger condensate fraction and fluctuation of the condensate density, along with a low depletion of the ground state occupancy. The other candidates, namely the square lattice (z = 4) and honeycomb lattice (z = 3) occupy second and third places, respectively, with regard to these properties. The quasi-particle velocities for these geometries follow a similar pattern as that of the condensate fraction data. The observations are explained by the behaviour of the low-energy quasi-particle density of states. Finally, to make the role of lattice geometry obvious, we include a discussion on a kagome lattice, which in spite of having the same coordination number as that of a square lattice, i.e. z = 4, yields a significantly low condensate fraction than that of the square lattice for all values of the interacting strength.

DOI
10.1088/0953-4075/45/10/105303

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Motivated by the availability of optical lattices with tunable geometries in experiments, we compute different physical properties such as condensate fraction, fluctuation and depletion of the condensate density and quasi-particle velocity as a function of the interparticle interaction strength for bosons in different 2D optical potentials that correspond to square, triangular and honeycomb geometries. Our results demonstrate an interplay of these features with that of the coordination number and underlying lattice geometries. A triangular lattice, which has a coordination number z = 6, shows larger condensate fraction and fluctuation of the condensate density, along with a low depletion of the ground state occupancy. The other candidates, namely the square lattice (z = 4) and honeycomb lattice (z = 3) occupy second and third places, respectively, with regard to these properties. The quasi-particle velocities for these geometries follow a similar pattern as that of the condensate fraction data. The observations are explained by the behaviour of the low-energy quasi-particle density of states. Finally, to make the role of lattice geometry obvious, we include a discussion on a kagome lattice, which in spite of having the same coordination number as that of a square lattice, i.e. z = 4, yields a significantly low condensate fraction than that of the square lattice for all values of the interacting strength.
We investigate the effects of disorder and lattice geometry against localisation phenomena in a weakly interacting ultracold bosonic gas confined in a 2D optical lattice. The behaviour of the quantum fluid is studied at the mean-field level performing computational experiments, as a function of disorder strength for lattices of sizes similar to current experiments. Quantification of localisation, away from the Bose glass phase, was obtained directly from the stationary density profiles through a robust statistical analysis of the condensate component, as a function of the disorder amplitude. Our results show a smooth transition, or crossover, to localisation induced by disorder in square and triangular lattices. In contrast, associated to its larger tunneling amplitude, honeycomb lattices show absence of localisation for the same range of disorder strengths and same lattice amplitude, while also exhibiting partial localisation for large disorder amplitudes. We also conclude that the coordination number z have a partial influence on how fast this smooth transition occurs as the system size increases. Signatures of disorder are also found in the ground state energy spectrum, where a continuous distribution emerges instead of a distribution of sharp peaks proper to the system in the absence of disorder.
In this work we study the properties of interacting bosons in a tilted triple well optical lattice potential. A tilted optical lattice is possible to create in an experiment with charged bosons in presence of an electric field. The ground state energy, occupation densities in each well and first order correlation function of occupation densities as a function of the tilt parameter for N = 90 bosons using recently developed Multiconfigurational time-dependent Hartree method for bosons (MCTDHB) are studied. The ground state energy falls off with the tilt parameter for a given value of the barrier height and interparticle interaction. Also the tilt parameter is observed to restrict the fragmentation of the occupation densities in each well and thus promote condensation. Further the correlation between the bosons in different wells enhances with the tilt parameter and at large values the bosons in the first and the third wells are correlated apart from the correlation among the bosons in the same well, while the occupancy in the middle one seems to be uncorrelated with the other two. Such correlation between different wells is not observed in absence of the tilt effect for the same barrier height of the potential and interparticle interaction strength.
The quantum geometry of Bloch states fundamentally affects a wide range of physical phenomena. The quantum Hall effect, for example, is governed by the Chern number, and flat-band superconductivity by the distance between the Bloch states: the quantum metric. While understanding quantum geometry phenomena in the context of fermions is well established, less is known about the role of quantum geometry in bosonic systems where particles can undergo Bose-Einstein condensation (BEC). In conventional single-band or continuum systems, excitations of a weakly interacting BEC are determined by the condensate density and the interparticle interaction energy. In contrast to this, we discover here fundamental connections between the properties of a weakly interacting BEC and the underlying quantum geometry of a multiband lattice system. We show that, in the flat-band limit, the defining physical quantities of BEC, namely, the speed of sound and the quantum depletion, are dictated solely by the quantum geometry. We find that the speed of sound becomes proportional to the quantum metric of the condensed state. Furthermore, the quantum distance between the Bloch functions forces the quantum depletion and the quantum fluctuations of the density-density correlation to obtain finite values for infinitesimally small interactions. This is in striking contrast to dispersive bands where these quantities vanish with the interaction strength. Additionally, we show how in the flat-band limit the supercurrent is carried by the quantum fluctuations and is determined by the Berry connections of the Bloch states. Our results reveal how nontrivial quantum geometry allows reaching strong quantum correlation regime of condensed bosons even with weak interactions. This is highly relevant, for example, for polariton and photon BECs where interparticle interactions are inherently small. Our predictions can be experimentally tested with flat-band lattices already implemented in ultracold gases and various photonic platforms.
The mean-field treatment of the Bose-Hubbard model predicts properties of lattice-trapped gases to be insensitive to the specific lattice geometry once system energies are scaled by the lattice coordination number z. We test this scaling directly by comparing coherence properties of ^{87}Rb gases that are driven across the superfluid to Mott insulator transition within optical lattices of either the kagome (z=4) or the triangular (z=6) geometries. The coherent fraction measured for atoms in the kagome lattice is lower than for those in a triangular lattice with the same interaction and tunneling energies. A comparison of measurements from both lattices agrees quantitatively with the scaling prediction. We also study the response of the gas to a change in lattice geometry, and observe the dynamics as a strongly interacting kagome-lattice gas is suddenly "hole doped" by introducing the additional sites of the triangular lattice.
We evaluate the Bose-Einstein condensate density and the superfluid fraction of bosons in a periodic external potential using path-integral Monte Carlo methods. The periodic lattice consists of a cubic cell containing a potential well that is replicated along one dimension using periodic boundary conditions. The aim is to describe bosons in a one-dimensional optical lattice or helium confined in a periodic porous medium. The one-body density matrix is evaluated and diagonalized numerically to obtain the single boson natural orbitals and the occupation of these orbitals. The condensate fraction is obtained as the fraction of bosons in the orbital that has the highest occupation. The superfluid density is obtained from the winding number. From the condensate orbital and superfluid fraction, we investigate (1) the impact of the periodic external potential on the spatial distribution of the condensate, and (2) the correlation of localizing the condensate into separated parts and the loss of superflow along the lattice. For high-density systems, as the well depth increases, the condensate becomes depleted in the wells and confined to the plateaus between successive wells, as in pores between necks in a porous medium. For low-density systems, as the well depth increases the BEC is localized at the center of the wells (tight binding) and depleted between the wells. In both cases, the localization of the condensate suppresses superflow leading to a superfluid-insulator crossover. The impact of the external potential on the temperature dependence of the superfluidity is also investigated. The external potential suppresses the superfluid fraction at all temperatures, with a superfluid fraction significantly less than one at low temperature. The addition of an external potential does not, however, significantly reduce the transition temperature.