Crystalline phases of laser-driven dipolar Bose-Einstein condensates

Although crystallization is a ubiquitous phenomenon in nature, crystal formation and melting still remain fascinating processes with several open questions yet to be addressed. In this work we study the emergent crystallization of a laser-driven dipolar Bose-Einstein condensate due to the interplay between long-range magnetic and effectively infinite-range light-induced interactions. The competition between these two interactions results in a collective excitation spectrum with two-roton minima that introduce two different length scales at which crystalline order can emerge. In addition to the formation of regular crystals with simple periodic patterns due to the softening of one of the rotons, we find that both rotons can also soften simultaneously, resulting in the formation of exotic, complex periodic, or aperiodic density patterns. We also demonstrate dynamic state-preparation schemes for achieving all the crystalline ground states found for experimentally relevant and feasible parameter regimes.

Figure 1

(a) Schematic of a dipolar BEC in the presence of two counterpropagating laser beams of orthogonal polarization. (b) Stability diagram of a homogeneous BEC as a function of dipole-interaction strength $g_d$ and light amplitude $E_0$ calculated from the excitation spectrum. The insets in the four regimes show examples of the typical spectrum that is stable (S) or consisting of either light roton instability (LRI) or magnetic roton instability (MRI), or biroton instability (LMRI). The gray dashed lines are a guide to the eye for the location of any roton instabilities. (c)–(f) Atomic density patterns in the crystalline ground-state phases. (c) Magnetic droplet crystal (MC) for $\{g_d/g,\sqrt{\alpha }{E}_0/\sqrt{E_{\text{rec}}}\}=\{1.52,0\}$ and (d) light crystal (LC) for $\{g_d/g,\sqrt{\alpha }{E}_0/\sqrt{E_{\text{rec}}}\}=\{0.92,12\}$. (e) and (f) Light-magnetic crystal (LMC) for $\{g_d/g,\sqrt{\alpha }{E}_0/\sqrt{E_{\text{rec}}}\}=\{1.42,4.2\}$ and $\{1.7,5\}$, respectively, for (e) droplets of supersolid and (f) an aperiodic crystal. Common parameters for (b)–(f) include $L=50{\lambda }_0, {\omega }_{\rho }=100E_{\mathrm{rec}}/\hslash , \zeta =0.1, a=70a_0$ ($a_0$ is the Bohr radius), and $N={10}^5$ atoms.

Figure 2

(a) Maximum energy difference between converged states of the eGPE for six different initial guesses for three $g_d/g$ values. (b)–(d) Mean-field phase diagram as a function of $\{g_d/g,\sqrt{\alpha }{E}_0/\sqrt{E_{\mathrm{rec}}}\}$ characterized by (b) reflection coefficient $r_c$, (c) density contrast $\mathrm{\Delta }n$, and (d) superfluid fraction $f_s$. White dashed lines are the stability diagram boundaries from the excitation spectrum. The yellow dotted curve demarcates the droplets of supersolid phase and the gray strip highlights the domain where density patterns are always periodic. All other parameters are the same as in Fig. 1.

Figure 3

Preparation of (a) MC, (b) LC, and (c) LMC states starting from an unordered state and (d) the LMC state starting from a MC for Dy atoms with their corresponding ramping schemes of $g_d/g$ and the light intensities $E_0$ shown in the middle row. (e)–(h) Corresponding densities in momentum space at the final time. For concreteness, the final-time density distributions in real space are shown in the insets. All other parameters are the same as in Fig. 1.

Figure 4

Reflection coefficients $r_c$ for converged states of eGPE obtained from different initial guesses for (a) $g_d/g=1.35$, (b) $g_d/g=1.7$, and (c) $g_d/g=1.88$, corresponding to the same parameters used in Fig. 1.

Figure 5

(a) and (b) Density profiles shown in Figs. 1 and 1, respectively. (c) Density profile corresponding to $\{g_d/g,\sqrt{\alpha }{E}_0/E_{\text{rec}}\}=\{1.7,10\}$ with all other parameters the same as in Figs. 1 and 1. (d)–(f) Light field intensity profiles corresponding to (a)–(c), respectively, obtained from the solution of the Helmholtz equation; the curves with the decreasing (increasing) amplitude of oscillations as a function of $z$ represent $|E_L{|}^2$ ($|E_R{|}^2$).

Figure 6

Preparation of (a) MC, (b) LC, and (c) LMC states starting from an unordered state with additional thermal noise and (d) a LMC starting from a MC with added thermal noise for Dy atoms with their corresponding ramping schemes of $g_d/g$ and the light intensities $E_0$ shown in the middle row, the same as in Fig. 3. (e)–(h) Corresponding incoherence plots with time, which is an average of at least three pairs of droplets where the droplets have length scales of either (e) MC or (f)–(h) LC. The inset in (h) is the incoherence plot when the chosen droplets are of the MC length scale.

Figure 7

Threshold value of the LC captured by the parameter $r_c$ as $N$ is varied for (a) $g_d/g=0.9$ and (b) $g_d/g=1.88$ with the other parameters similar to those in Fig. 1. (c)–(h) Two cases from the LMC phase where gradually reducing $N$ results in shifts of the phase boundaries and the LMC state either changing to an MC and eventually unordered state or changing into an LC state. (c)–(e) correspond to Fig. 1 except with (d) $N=80000$ and (e) $N=50000$. (f)–(h) correspond to Fig. 1 with (g) $N=50000$ and (h) $N=10000$.