Insights into complex phenomena in quantum matter can be gained from simulation experiments with ultracold atoms, especially in cases where theoretical characterization is challenging. However, these experiments are mostly limited to short-range collisional interactions; recently observed perturbative effects of long-range interactions were too weak to reach new quantum phases1, 2. Here we experimentally realize a bosonic lattice model with competing short- and long-range interactions, and observe the appearance of four distinct quantum phases—a superfluid, a supersolid, a Mott insulator and a charge density wave. Our system is based on an atomic quantum gas trapped in an optical lattice inside a high-finesse optical cavity. The strength of the short-range on-site interactions is controlled by means of the optical lattice depth. The long (infinite)-range interaction potential is mediated by a vacuum mode of the cavity3, 4 and is independently controlled by tuning the cavity resonance. When probing the phase transition between the Mott insulator and the charge density wave in real time, we observed a behaviour characteristic of a first-order phase transition. Our measurements have accessed a regime for quantum simulation of many-body systems where the physics is determined by the intricate competition between two different types of interactions and the zero point motion of the particles.
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Extended data figures and tables
Extended Data Figures
- Extended Data Figure 1: Validity of the single-band approximation. (123 KB)
The energy scales of the Hamiltonian are plotted in units of the minimum gap Δex between the lowest and the first excited Bloch band. The single-band approximation is assumed to be valid if all the energy scales (Us, Ul and t) are at least 5 times smaller than Δex, that is, if they lie below the black dashed line. This criterion is fulfilled for Δc/2π < −18.3 MHz and 18ER > V2D > 3ER. For detunings in the interval −18.3 MHz < Δc/2π < −10.9 MHz, the approximation is only partially valid, depending on V2D. We use this information to illustrate the region of validity in Extended Data Fig. 2.
- Extended Data Figure 2: Phase diagram plotted as a function of Hamiltonian parameters Us/t and Ul/t. (203 KB)
The experimental parameters in Fig. 3 have been converted to Hamiltonian parameters: the region of validity for this conversion lies to the right of the solid black line, grey areas were not recorded. The white data points indicate where spatial coherence is lost, and the black data points depict the onset of an even–odd imbalance. The white shaded regions around the data points represent the respective converted error bars. The dotted black lines show, as in Fig. 3, the region where the onset of the cavity light field showed a large variation.
- Extended Data Figure 3: Influence of the trapping potential on the long-range interacting system. (401 KB)
Shown are sketches of a 1D slice through a 2D layer, displaying the ground state configurations of 13 particles that depend on the relative influence of trapping potential and long-range interaction. a, Results for a homogeneous system with finite Ul. b–d, The state of the system for increasing but finite Ul, starting with small but finite Ul (see legend at right).
- Extended Data Figure 4: Determination of the phase boundaries. (214 KB)
Shown are the BEC fraction f (averaged into 100 equally spaced bins) and maximum photon number nph,max (filled and open symbols, respectively) as a function of Us/t for detunings Δc/2π of −12 MHz (a), −22 MHz (b) and −47 MHz (c). The red curve in each panel shows the result of a piecewise linear fit to f. We confirmed that the initial BEC fraction has no systematic dependence on Δc. The blue curve displays a power law fit to nph,max.
- Extended Data Figure 5: Momentum distribution in the SS phase. (73 KB)
Absorption image from a calibration measurement taken after a short ballistic expansion of 7 ms at a detuning of Δc/2π = −23 MHz and a lattice depth V2D 39% above the onset of an even–odd imbalance in the SS phase. We observe interference peaks at pz = ±2ħk. Additional interference peaks resulting from the emerging chequerboard lattice appear at (px, pz) = (±hk, ±hk). This observation indicates an SS phase. These additional momentum peaks lie outside the field of view for the longer ballistic expansion time of 15 ms.
- Extended Data Figure 6: Strength of the self-consistent chequerboard lattice. (148 KB)
The chequerboard (CB) lattice depth extracted from the measured mean intracavity photon number nph is shown as a function of the applied lattice depth V2D and detuning Δc. The CB lattice depth becomes comparable to the depth of the static lattices close to cavity resonance, but drops rapidly when moving away due to its detuning dependence. Exemplary equipotential lines at 0.05ER, 1ER and 3ER are shown.
- Extended Data Figure 7: Sensitivity to the ramp speed. (190 KB)
The hysteretic behaviour in the insulating regime at V2D = 18ER is shown. The detuning Δc/2π is ramped at two speeds, 0.67 MHz ms−1 (blue) and 0.33 MHz ms−1 (orange). Lines result from an average of two to five measurements, using 400 μs time bins. Stars signify starting points, arrows show the scan direction and dashed lines indicate the return to the starting point.