Self-bound droplet clusters in laser-driven Bose-Einstein condensates

We investigate a two-dimensional Bose-Einstein condensate that is optically driven via a retro-reflecting mirror, forming a single optical feedback loop. This induces a peculiar type of long-range atomic interaction with highly oscillatory behavior, and we show here how the sign of the underlying interaction potential can be controlled by additional optical elements and external fields. This additional tunability enriches the behavior of the system substantially and gives rise to a surprising range of ground states of the condensate. In particular, we find the emergence of self-bound crystals of quantum droplets with various lattice structures, from simple and familiar triangular arrays to complex superlattice structures and crystals with entirely broken rotational symmetry. This includes mesoscopic clusters composed of small numbers of quantum droplets as well as extended crystalline structures. Importantly, such ordered states are entirely self-bound and stable without any external in-plane confinement, having no counterpart to other quantum-gas settings with long-range atomic interactions.

Figure 1

(a) Illustration of the considered setup in which a quasi-2D BEC is illuminated by laser light that is back-reflected by a mirror and thereby generates optical feedback in the condensate. The atoms are coupled to the incoming and back-reflected light fields, with amplitudes $E_{+}$ and $E_{-}$, respectively. These light fields drive separate atomic transitions with respective frequency detunings ${\mathrm{\Delta }}_{+}$ and ${\mathrm{\Delta }}_{-}$, as indicated in the depicted three-level scheme. Rescattering of the reflected photons generates an effective atomic interaction that is shown in panel (b). The blue and red lines depict the resulting potentials for ${\mathrm{\Delta }}_{+}{\mathrm{\Delta }}_{-}>0$ and ${\mathrm{\Delta }}_{+}{\mathrm{\Delta }}_{-}<0$, respectively.

Figure 2

The ground-state phase diagram of a finite BEC is displayed in panel (a), while panels (b) and (c) provide an enlarged view around the critical points. Examples of type I ($gN=5.6\times {10}^3$), II ($gN={10}^4$), and III ($gN=1.2\times {10}^4$) states are shown, respectively, in panels (d), (e), and (f) for a fixed long-range interaction strength $\gamma N=1.6\times {10}^3$. The color depth in panels (d), (e), and (f) represents the relative amplitude $\left|\psi \right|/{\left|\psi \right|}_{\max }$.

Figure 3

Different pentagon-type droplet-cluster states for (a) $\gamma N=20, gN=0$, (b) $\gamma N=60, gN=0$, and (c) $\gamma N=20, gN=10$. A droplet cluster with sixfold symmetry is shown in panel (d) at $\gamma N=20, gN=20$ and forms a precursor of the triangular lattice shown in Fig. 5.

Figure 4

Ground-state energy as a function of the contact-interaction strength, $gN$, at a fixed strength of the long-range interaction $\gamma N=2\times {10}^3$. Here the solid (dashed) lines show the energy of the ground (metastable) states, and the insets display the amplitude profiles of the corresponding states at the indicated vales of $gN$. The color depth represents the relative amplitude $\left|\psi \right|/{\left|\psi \right|}_{\max }$. The energy of the trivial flat state is shown by the black line.

Figure 5

Exemplary, density profiles of extended cluster crystals with (a) a fourfold rotational symmetry ($g\rho =28$), (b) with no rotational symmetry ($g\rho =31$), (c) with a twofold rotational symmetry ($g\rho =34$), and (d) with a sixfold rotational symmetry ($g\rho =36$). The strength of the long-range interaction is fixed at $\gamma \rho =3$.

Figure 6

Excitation spectrum of the condensate for different strengths, $g\rho$, of the contact interaction at a fixed long-range interaction strength $\gamma \rho =3$.