Excitations of a Bose-Einstein condensate and the quantum geometry of a flat band

Abstract

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.

Quantum Geometry and Flat Band Bose-Einstein Condensation

“…In this way, the importance of quantum fluctuations and correlations can be significantly enhanced. We demonstrate this in our companion paper [35], where we calculate the density-density correlation function to show that the quantum geometry can provide access to a regime dominated by interaction effects even with infinitesimally small U. This is highly relevant in systems where interactions are inherently small, such as photon and polariton condensates.…”

“…While bosonic flat band geometries have been studied experimentally [26][27][28][29][30][31][32][33][34], and quantum geometry is experimentally accessible in bosonic systems [13], understanding how the quantum geometry affects the physical properties of a BEC is still lacking. In this Letter and in our more detailed companion paper [35], by using multiband Bogoliubov formalism, we theoretically unravel fundamental connections between a weakly interacting BEC taking place in a multiband lattice system and the quantum geometric properties of the underlying Bloch states. Our focus is on the systems where the condensation takes place within a flat band.…”

“…In this Letter, we consider a two-dimensional kagome lattice geometry that supports a flat band. In our companion paper [35], we provide the details of the calculations for a generic flat band system and, furthermore, show the results for density-density correlations and superfluid density. These two works together thus establish fundamental connections between quantum geometry and various physical properties of weakly interacting flat band condensates.…”

“…To analyze the stability and excitation properties of BEC, we utilize the multiband Bogoliubov approximation (details are provided in Ref. [35]), where the bosonic operators for the condensate are treated as complex numbers; i.e., we write c k c α ¼ ffiffiffiffiffiffiffiffi ffi Nn 0 p hαjϕ 0 i, where n 0 is the number of condensed bosons per unit cell and hαjϕ 0 i is the projection of jϕ 0 i to the αth sublattice. In the Bogoliubov theory, one considers only the interaction terms that are quadratic in fluctuations c kα and c † kα with k ≠ k c .…”

“…For anisotropic systems (for derivation see [35]), c s ðθ q Þ¼ð2Un 0 =MÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi êT q g 1 ðk c Þê q q , where êq ¼q=jqj, tan θ q ¼ q y =q x , ½g 1 μν ¼ g 1 μν , and M is the number of orbitals.…”