Striped states in a many-body system of tilted dipoles

We study theoretically and experimentally the behavior of a strongly confined dipolar Bose-Einstein condensate in the regime of quantum-mechanical stabilization by beyond-mean-field effects. Theoretically, we demonstrate that self-organized “striped” ground states are predicted in the framework of the extended Gross-Pitaevskii theory. Experimentally, by tilting the magnetic dipoles we show that self-organized striped states can be generated, likely in their metastable state. Matter-wave interference experiments with multiple stripes show that there is no long-range off-diagonal order (global phase coherence). We outline a parameter range where global phase coherence could be established, thus paving the way towards the observation of supersolid states in this system.

Figure 1

Striped ground states. (a) Total energy per atom $E_{\mathrm{tot}}/N$ for single (solid red) and double (solid green) droplet solutions obtained from a variational ansatz. For $f_y\gtrsim 200$ Hz the state with two droplets has energy lower than that of a state with a single droplet. The schematic shows a double-droplet configuration in the proposed harmonic trap. Numerical simulations of the extended Gross-Pitaevskii equation [see eq. (1)] predict higher numbers of droplets for increasing $f_y$ in the ground state. Insets show the integrated column density along $z$ for ground states with one (red dots), two (green squares), and four (blue diamonds) droplets. (b) Integrated column density of the ground state for $f_y=800$ Hz. There are several droplets with finite density between the droplets indicating overlap of the single-droplet wave functions. (c) For a similar system with periodic boundary conditions along $x$ ($f_x=0$) the ground state exhibits the same density modulation, thus breaking the continuous translational symmetry. Vertical lines represent the edge of the box. See text for further parameters.

Figure 2

Tunability of the mean-field dipolar interaction ${\mathrm{\Phi }}_{\mathrm{dip}}$. (a) Schematic of dipolar atoms strongly confined along the $z$ direction with a magnetic field tilted under an angle $\beta$ with respect to the confinement axis. (b) Determination of the critical angle ${\beta }_c$ via the Fourier anisotropy $A_{\mathrm{FT}}$ (see Appendix pp2). The gray bar marks the corresponding error. Insets show a BEC elongated by magnetostriction (left) and a double droplet state (right). Measured critical tilt angle ${\beta }_c$ over (c) $z$ trap frequency and (d) magnetic field close to a Feshbach resonance. In panel (c) the power of the light sheet is varied at fixed magnetic field $B=1240\left(5\right)$ mG. (d) The dependence on the scattering length is measured by changing the magnetic field for $f_z=950\left(10\right)$ Hz (red) and 300(10) Hz (green). The Feshbach resonance at $B_0=1326\left(3\right)$ mG with width $\mathrm{\Delta }=8\left(5\right)$ mG [3] is marked.

Figure 3

Striped states observed in an anisotropic trap. (a) Example single-shot in situ images for varying transversal trap aspect ratio ${\lambda }_{xy}=f_x/f_y$. (b) Critical tilt angle ${\beta }_c$ (red circles) and average number of droplets over ${\lambda }_{xy}$. We observe multiple droplets for ${\lambda }_{xy}\lesssim 1$ and single ones for ${\lambda }_{xy}\gtrsim 1$. Data are taken for $B=905\left(5\right)$ mG at a trap frequency of $f_z=945\left(5\right)$ Hz and averaged over 11 realizations. (c) Dynamic simulations of the eGPE confirm the creation of multiple droplets for conditions where a single droplet is the ground state. Simulation parameters are similar to those of the experiment; see main text and Appendix pp1.

Figure 4

Interference patterns after 8 ms of expansion. (a) Two example realizations with absorption image (top panels) and integrated density (bottom panels) showing fringes. We extract the phase $\theta$ with respect to the center of mass of the distribution. (b) Histograms and polar plot of relative phase $\theta$ and visibility $v$ for 650 atom distributions. There is no preferred phase visible, indicating that there is no phase coherence between the droplets.

Figure 5

Tuning the coherence properties. On-site interaction $E_C$ (red) and hopping term $E_J$ (green line) as defined in Appendix pp1 for varying confinement $f_x$ perpendicular to the magnetic field. This way the distance between droplets is varied and thus $E_J$ can be tuned over several orders of magnitude. The shape of the droplet wave function is not altered, keeping $E_C$ almost constant. The values are computed with the variational ansatz for a confinement where the double-droplet solution is the ground state ($f_y=300$ Hz). The temperature $T=30$–70 nK reflects the temperature range in the experiment.

Figure 6

Calculations with the variational ansatz. (a) Energy per particle $E_{\mathrm{tot}}/N$ over distance $d$ acquired with the variational ansatz. Curves represent calculations for different trap frequencies $f_y=50$–500 Hz. The solutions for a single-droplet state ($d=0$) and for a double-droplet state with distance $d>0$ are marked. (b) Phase diagram of atom number $N$ over confinement $f_x$ along the droplet axis. Solid lines mark the transition between the single-droplet ground state (left) and the double-droplet ground state (right) for different values of the scattering length $a_s$. See main text for parameters.

Figure 7

Measured critical angle ${\beta }_c$ over tilt speed $\dot{\beta }$.