$P$-Band Induced Self-Organization and Dynamics with Repulsively Driven Ultracold Atoms in an Optical Cavity

We investigate a Bose-Einstein condensate strongly coupled to an optical cavity via a repulsive optical lattice. We detect a stable self-ordered phase in this regime, and show that the atoms order through an antisymmetric coupling to the $P$ band of the lattice, limiting the extent of the phase and changing the geometry of the emergent density modulation. Furthermore, we find a nonequilibrium phase with repeated intense bursts of the intracavity photon number, indicating nontrivial driven-dissipative dynamics.

Figure 1

(a),(b) We prepare a BEC of ${}^{87}\mathrm{Rb}$ atoms in the initially unpopulated mode of an optical resonator, and expose it to a repulsive optical lattice (blue) localizing the atoms (red) in the field minima. (c) Populating the cavity mode with photons can become energetically favorable for the system because interference between the pump and the cavity fields gives the atoms more space to expand close to the nodes of the total field. The atoms form a Bragg grating that allows scattering into the cavity mode. Photons leaking from the cavity are detected. (d) The atoms recoil and acquire momentum (red arrows) from scattering photons (blue arrows). These momenta are $\pm {\mathit{k}}_{-}=\pm ({\mathit{k}}_c-{\mathit{k}}_p)$ (upper processes) and $\pm {\mathit{k}}_{+}=\pm ({\mathit{k}}_c+{\mathit{k}}_p)$ (lower processes).

Figure 2

Self-organized phase with a repulsive pump. (a) Phase diagram showing the intracavity lattice depth as a function of pump lattice depth and cavity detuning. A self-ordered phase with limited extent, identified by a finite lattice in the cavity, can be identified. The dotted lines are the phase boundaries obtained from numerical mean-field calculations. The deviation from the data can be attributed to finite temperatures [29]. Insets display atomic momentum distributions for a pump lattice of $3.0(2)E_{\mathrm{rec}}$ recorded with absorption imaging after ballistic expansion. (b) For ${\mathrm{\Delta }}_c=-2\pi \times 4.0(1)\mathrm{MHz}$, the system is in the normal phase with momentum components at 0 and $\pm 2\hslash {\mathit{k}}_p$. In real space, this distribution corresponds to the density modulation of the pump lattice with $\lambda /2$ spacing. (c) At ${\mathrm{\Delta }}_c=0$, the system is self-organized, and the atoms assume a 1D density modulation with $\lambda$ spacing with momentum components $\pm \hslash ({\mathit{k}}_c-{\mathit{k}}_p)$. Both the pump lattice modulation at $\pm 2\hslash {\mathit{k}}_p$ and the recoil from the other scattering process at $\pm \hslash ({\mathit{k}}_c+{\mathit{k}}_p)$ [see Fig. 1] are suppressed.

Figure 3

Finite extent of the self-organized phase. (a) Cut through the phase diagram for ${\mathrm{\Delta }}_c=-2\pi \times 2\mathrm{MHz}$ showing the cavity lattice depth rising from and returning to 0. (b) The atoms (gray) get localized to the nodes of the repulsive standing wave field (black) along the $x$ direction. Taking the atomic density maxima as origin of the coordinate system, the coupling term with the double period (blue) turns into a sine. It thus couples to a state of opposite symmetry, as indicated. (c) This opposite-symmetry state lives in the $P$ band of the pump lattice. With increasing pump lattice depth, the coupling term leads to a mode softening at wave vector ${\mathit{k}}_{-}={\mathit{k}}_c-{\mathit{k}}_p$ (pale blue lines), and finally to self-organization. At the same time, the increasingly deep pump lattice increases the energy of the $P$ band (blue arrows). For deeper lattices, this effect becomes dominant and leads to the end of the self-ordered phase.

Figure 4

Superradiant dynamics. (a) Time trace of the cavity lattice depth after a far-detuned ramp-up of the pump to $4.1E_{\mathrm{rec}}$ and a subsequent jump of detuning from ${\mathrm{\Delta }}_c=2\pi \times -10\mathrm{MHz}$ to ${\mathrm{\Delta }}_c=2\pi \times 2\mathrm{MHz}$ at time 0. The difference in spike heights stems from the finite photon count rate at which the single photon counting module saturates [29]. The gray bar indicates the estimated uncertainty. (b),(c) Momentum distribution from absorption imaging after ballistic expansion after a photon spike in the cavity-pump-plane [$xy$, (b)] and an orthogonal plane [$xz$, (c)]. Very high momenta up to $\pm 20\hslash {\mathit{k}}_c$ are visible, possibly limited by the aperture of the imaging system. (d) Schematic phase diagram with stable and dynamic regions. (e)–(g) Numerical calculations of the free energy per atom as a function of the cavity photon field for ${\mathrm{\Delta }}_c=2\pi \times 2\mathrm{MHz}$ and pump lattice depths of $0.2E_{\mathrm{rec}}$ (normal phase with local minimum at $⟨\widehat{a}⟩=0$), $3.4E_{\mathrm{rec}}$ (self-ordered phase with local minima at a finite photon number) and $13E_{\mathrm{rec}}$ (spike region, no local minimum).