Bogoliubov depletion of the fragmented condensate in the bosonic flux ladder

We theoretically analyze the ground state of weakly interacting bosons in the flux ladder—the system that has been recently realized by means of ultracold atoms in the specially designed optical lattice [M. Atala, M. Aidelsburger, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Nat. Phys. 10, 588 (2014)]. It is argued that, for the system parameters corresponding to two degenerate minima in the Bloch dispersion relation, the ground state is a fragmented condensate. We study the Bogoliubov depletion of this condensate and discuss the role of boundary conditions.

Figure 1

Energy spectrum (dispersion relation) of a quantum particle in the flux ladder for $\alpha =1/3$ and $J_{\perp }=0.5$. The right-hand panel shows occupation of the ladder legs for a given Bloch state, where the dashed and solid lines refer to the lower and upper legs, respectively.

Figure 2

Blue curves are the exact low-energy spectrum of $N=4$ weakly interacting bosons in the flux ladder of length $L=12$. The other parameters are the same as in Fig. 1. The dimension of the Hilbert space is $\mathcal{N}=17550$. Dashed lines are the spectrum in the single-band approximation, where the dimension of the Hilbert space is reduced to $\mathcal{N}=1365$. The straight magenta lines are the diagonal approximation (10), which ignores population of the other quasimomentum states besides $\kappa =\pm q$.

Figure 3

Pictorial presentation of the Bogoliubov depletion. Two panels in the figure show two different kinds of the depletion processes which conserve the total quasimomentum of $N$ particles. The example is given for $L=12$ where discretization of the quasimomentum is $\mathrm{\Delta }\kappa =\pi /6$.

Figure 4

Low-energy spectrum of system (7) as the function of the macroscopic interaction constant $g$. The other parameters are $N=512, \alpha =1/2$, and $L=8$. The energy is measured with respect to mean-field energy $E_0$. Asterisks are the spectrum calculated by using ansatz (11).

Figure 5

Eigenvalues of the one-particle density matrix (19) (a) as the function of the hopping matrix element $J_{\perp }$ and (b) as the function of the interaction constant $U$. The other parameters are $N=4, L=12, \alpha =1/3$, and (a) $U=1$ and (b) $J_{\perp }=0.5$. Notice that altogether we have 24 eigenvalues where the two degenerate largest eigenvalues, which are associated with the fragmented condensate, are depicted by the thick red lines.

Figure 6

(a) The energy difference $\left|\mathrm{\Delta }E\right|$ between the two lowest eigenstates for $g=0$ (asterisk) and $g=1$ (open circles). The dashed line is the energy difference between the lowest duplet and the next eigenstate. The considered system size is $3\le L\le 300$ with step $\mathrm{\Delta }L=3$. (b) Symmetric and (c) antisymmetric ground states for $N=4, L=12$, and $U=0$ (dashed lines) and $U=1$ (solid lines). Shown are the squared amplitudes $|{\tilde{\psi }}_{l,m}{|}^2$ and $|{\varphi }_{l,m}{|}^2$, which are the same for $m=1$ and $m=2$.

Figure 7

(a) Eigenvalues of the one-particle density matrix (19) for the Dirichlet boundary condition. The two largest eigenvalues, which are associated with the symmetric and antisymmetric natural orbitals, are depicted by the thick red lines. (b) The overlap coefficients (24) as a function of the interaction constant $U$. The system parameters are the same as in Fig. 5 except for the different (Dirichlet) boundary condition.

Figure 8

Occupations $|{\varphi }_{l,m}{|}^2$ of the ladder sites for $L=120$ and $g=0$ (dashed lines) and $g=1$ (solid lines) for the (a) symmetric and (b) antisymmetric macroscopic wave functions.