Emergent structure in a dipolar Bose gas in a one-dimensional lattice

We consider an ultracold dipolar Bose gas in a one-dimensional lattice. For a sufficiently large lattice recoil energy, such a system becomes a series of nonoverlapping Bose-Einstein condensates that interact via the long-range dipole-dipole interaction (ddi). We model this system via a coupled set of nonlocal Gross-Pitaevskii equations (GPEs) for lattices of both infinite and finite extent. We find significantly modified stability properties in the lattice due to the softening of a discrete roton-like mode, as well as “islands” in parameter space where biconcave densities are predicted to exist and that only exist in the presence of the other condensates on the lattice. We solve for the elementary excitations of the system to check the dynamical stability of these solutions and to uncover the nature of their collapse. By solving a coupled set of GPEs exactly on a full numeric grid, we show that this emergent biconcave structure can be realized in a finite lattice with atomic ${}^{52}\mathrm{Cr}$.

Figure 1

Differences in energy of a DBEC in a trap with aspect ratio $\lambda =7$ as a function of interaction strength between that calculated using the ansatz given in Eq. (10) and that calculated exactly on a full numeric grid. The blue dotted line shows the energy difference calculated using no second-order harmonic-oscillator wave function and a fixed axial width $l_z=a_z$, the red dashed line shows the energy difference using the same wave function but with $l_z$ treated variationally and the black solid line shows the energy difference including the second-order harmonic-oscillator wave function where the relative amplitude $A_2$ and $l_z$ are treated variationally.

Figure 2

The values of the axial wave-function parameters that, together with the radial wave function calculated on a grid, minimize the energy of a single DBEC in a trap with aspect ratio $\lambda =7$. The blue dotted line shows the results for $l_z=a_z$, the red dotted line shows $l_z$ when it is treated variationlly and $A_2=0$, and the black solid line shows $l_z$ when both it and $A_2$ are treated variationally. In this last case, the behavior of $A_2$ is given by the black dotted line.

Figure 3

Structure-stability diagram for a single DBEC. The colored regions indicate a dynamically stable condensate, and the pink (darker) regions indicate parameters for which the DBEC has biconcave density. (a) and (b) are calculated using the ansatz for the axial wave function given in Eq. (10) and (c) is calculated using a full numeric grid. For (a), $A_2=0$ and $l_z/a_z=1$ and, for (b), $A_2$ and $l_z$ are treated variationally.

Figure 4

Structure-stability diagram for an infinite lattice of DBECs in traps with aspect ratio $\lambda =10$ as a function of lattice spacing $d_{\mathrm{lat}}/a_z$ and interaction strength $(N-1)g_d/a_{\rho }$. The colored region indicates dynamic stability, while the pink (darker) regions indicate parameters where the DBECs have biconcave density. The inset shows an isodensity plot of a DBEC with biconcave density.

Figure 5

Structure-stability diagram for an infinite lattice of DBECs in traps with aspect ratio $\lambda =20$ as a function of lattice spacing $d_{\mathrm{lat}}/a_z$ and interaction strength $(N-1)g_d/a_{\rho }$. The inset shows a close-up of the diagram at the parameters indicated. The pink (darker) region in the inset indicates parameters where the DBECs have biconcave density. An isodensity plot of a DBEC with biconcave density is shown in this inset.

Figure 6

Stability lines for an infinite lattice of DBECs for aspect ratios $\lambda =50,\hspace{0.5em}100,\hspace{0.5em}150$ as a function of lattice spacing $d_{\mathrm{lat}}/a_z$ and interaction strength $(N-1)g_d/a_{\rho }$. The parameters beneath the lines are dynamically stable, while those above the lines are dynamically unstable.

Figure 7

Radial densities at $z=0$ of a DBEC with $(N-1)g_d/a_{\rho }=550$ in a trap with aspect ratio $\lambda =50$. The blue dashed line shows the density of a DBEC in a single harmonic trap, and the red solid line shows the density of a DBEC in the center site of a 1D lattice with nine occupied sites ($j_{\mathrm{lat}}=4$). This DBEC exhibits biconcave structure, while the single DBEC does not, demonstrating the emergence of this structure in the lattice system. These densities were calculated by solving the GPE (coupled GPEs) exactly on a full numeric grid.