Quantum phases from competing short- and long-range interactions in an optical lattice

Journal name:
Nature
Volume:
532,
Pages:
476–479
Date published:
DOI:
doi:10.1038/nature17409
Received
Accepted
Published online

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.

At a glance

Figures

  1. Illustration of the experimental scheme that realizes a lattice model with on-site and infinite-range interactions.
    Figure 1: Illustration of the experimental scheme that realizes a lattice model with on-site and infinite-range interactions.

    Left, a stack of 2D systems along the y axis is loaded into a 2D optical lattice (red arrows) between two mirrors (shown grey). The cavity induces atom–atom interactions of infinite range. Right, illustration of the competing energy scales: tunnelling t, on-site interactions Us and long-range interactions U1.

  2. Characterization of the phases.
    Figure 2: Characterization of the phases.

    ad, Characterization via spatial coherence (a, b) and via even–odd imbalance (c, d). a, Absorption images in the xz plane (upper panels), and the same signal integrated along the cavity axis (lower panels, red), taken after a ballistic expansion for lattice depths V2D of 2ER (I), 6.5ER (II), 11ER (III) and 16ER (IV) at Δc/2π = −22 MHz. Black lines show fits with a bimodal distribution including higher momentum peaks. Owing to the cavity mirrors, the field of view along the x direction is restricted. b, Extracted BEC fraction f as a function of V2D and Δc. White points mark the transition from a superfluid to an insulating phase and are obtained from a piecewise linear fit to the BEC fraction (see Extended Data Fig. 4). Error bars indicate fit uncertainties and contain contributions from the s.d. and from the stability of the fit (see Methods). c, Scattered photons nph of single repetitions as a function of V2D for pump–cavity detunings Δc/2π of −12 MHz (i), −22 MHz (ii) and −32 MHz (iii). d, Imbalance Θ mapped as a function of Δc and V2D. We assign the onset of a scattered cavity light field (black points) to the formation of a phase with even–odd imbalance. In the region indicated by the three dotted lines at values Δc/2π = {−47, −49.5, −52} MHz, the onset of the cavity light field showed a large variation. Error bars indicate the s.d. of the fit; an additional systematic error of 0.2ER stems from the data analysis. The detection background is growing with decreasing V2D and increasing detuning from cavity resonance (see Methods). Grey areas were not recorded.

  3. Phase diagram.
    Figure 3: Phase diagram.

    The four phases are indicated by different colours: SF (red), SS (violet), CDW (blue) and MI (yellow). Simplified density distributions are schematically illustrated for the homogeneous case with, on average, one atom per site. Data points (from Fig. 2b, d) show the experimentally obtained phase transition points recorded for increasing V2D: Black data points indicate the onset of an even–odd imbalance, white data points depict where spatial coherence is lost. Increasing the 2D lattice depth V2D simultaneously increases short- and long-range interactions. The detuning Δc changes only the strength of the long-range interactions. The slanted lines indicate the region where CDW and MI phases may coexist. At detuning Δc/2π = +8 MHz, U1 becomes negative and favours zero imbalance, thus only SF and MI phases appear. No data were taken at detunings indicated by the grey bar. A version of the phase diagram in Hamiltonian parameters is shown in Extended Data Fig. 2.

  4. Hysteretic behaviour of the CDW to MI transition.
    Figure 4: Hysteretic behaviour of the CDW to MI transition.

    Shown is imbalance Θ recorded by varying Δc/2π at a rate of 0.67 MHz ms−1, for fixed V2D = 14ER. The initial detunings Δc/2π, indicated by stars, are −32 MHz (a), −42 MHz (b) and −52 MHz (c). Arrows signify the ramp directions; dashed lines show the return to the starting point. Curves are rescaled to take atom loss into account and contain three to nine averages, binned at 400 μs (see Methods).

  5. Validity of the single-band approximation.
    Extended Data Fig. 1: Validity of the single-band approximation.

    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.

  6. Phase diagram plotted as a function of Hamiltonian parameters Us/t and Ul/t.
    Extended Data Fig. 2: Phase diagram plotted as a function of Hamiltonian parameters Us/t and Ul/t.

    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.

  7. Influence of the trapping potential on the long-range interacting system.
    Extended Data Fig. 3: Influence of the trapping potential on the long-range interacting system.

    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. bd, The state of the system for increasing but finite Ul, starting with small but finite Ul (see legend at right).

  8. Determination of the phase boundaries.
    Extended Data Fig. 4: Determination of the phase boundaries.

    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.

  9. Momentum distribution in the SS phase.
    Extended Data Fig. 5: Momentum distribution in the SS phase.

    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.

  10. Strength of the self-consistent chequerboard lattice.
    Extended Data Fig. 6: Strength of the self-consistent chequerboard lattice.

    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.

  11. Sensitivity to the ramp speed.
    Extended Data Fig. 7: Sensitivity to the ramp speed.

    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.

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Author information

Affiliations

  1. Institute for Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland

    • Renate Landig,
    • Lorenz Hruby,
    • Nishant Dogra,
    • Manuele Landini,
    • Rafael Mottl,
    • Tobias Donner &
    • Tilman Esslinger

Contributions

R.L., L.H., N.D. and M.L. took the data and analysed them together with T.D. Contributions to the design of the experiment were made by R.M. All work was supervised by T.E. All authors contributed to discussions and the preparation of the manuscript.

Competing financial interests

The authors declare no competing financial interests.

Corresponding author

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Author details

Extended data figures and tables

Extended Data Figures

  1. 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.

  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.

  3. 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. bd, The state of the system for increasing but finite Ul, starting with small but finite Ul (see legend at right).

  4. 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.

  5. 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.

  6. 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.

  7. 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.

Additional data