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Dilute dipolar quantum droplets beyond the extended Gross-Pitaevskii equation

Fabian Böttcher, Matthias Wenzel, Jan-Niklas Schmidt, Mingyang Guo, Tim Langen, Igor Ferrier-Barbut, Tilman Pfau, Raúl Bombín, Joan Sánchez-Baena, Jordi Boronat, and Ferran Mazzanti
Phys. Rev. Research 1, 033088 – Published 8 November 2019
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  • INTRODUCTION
  • EXPERIMENT
  • THEORY
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Dipolar quantum droplets are exotic quantum objects that are self-bound due to the subtle balance of attraction, repulsion, and quantum correlations. Here we present a systematic study of the critical atom number of these self-bound droplets, comparing the experimental results with extended mean-field Gross-Pitaevskii equation and quantum Monte Carlo simulations of the dilute system. The respective theoretical predictions differ, questioning the validity of the current theoretical state-of-the-art description of quantum droplets within the extended Gross-Pitaevskii equation framework and indicating that correlations in the system are significant. Furthermore, we show that our system can serve as a sensitive testing ground for many-body theories in the near future.

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    • Received 23 April 2019

    DOI:https://doi.org/10.1103/PhysRevResearch.1.033088

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Bose-Einstein condensatesDipolar gases
    1. Techniques
    Quantum Monte Carlo
    Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Fabian Böttcher, Matthias Wenzel, Jan-Niklas Schmidt, Mingyang Guo, Tim Langen, Igor Ferrier-Barbut*, and Tilman Pfau†

    • 5. Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

    Raúl Bombín, Joan Sánchez-Baena, Jordi Boronat, and Ferran Mazzanti‡

    • Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4-B5, 08034 Barcelona, Spain

    • *Present address: Laboratoire Charles Fabry, Institut d'Optique Graduate School, CNRS, Universit Paris–Saclay, 91127 Palaiseau Cedex, France.
    • †t.pfau@physik.uni-stuttgart.de
    • ‡ferran.mazzanti@upc.edu

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    Vol. 1, Iss. 3 — November - December 2019

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    Images

    • Figure 1
      Figure 1

      Critical atom number of a self-bound dipolar quantum droplet for Dy162 (blue circles) and Dy164 [5] (black squares). We extract the critical atom number by analyzing the atom number decay curves. The theoretical boundary of the phases is obtained from numerical EGPE simulations. The red dashed and dash-dotted lines show the corresponding boundary as expected from an increased effective dipolar length due to a finite collision energy of 30–50 nK and 100 nK [30, 31]. The red triangles show the results obtained by QMC simulations, with the error bars chosen to cover the uncertainties of both the statistical error and the nonuniversality. See the text for more information.

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    • Figure 2
      Figure 2

      (a) Pair distribution function g(r) for the bulk system at a density of n=5.88×1021m−3, corresponding to the central density of a saturated quantum droplet at as=60a0. The red (blue) symbols correspond to a scattering length of as=60a0 (as=90a0), while the squares (circles) indicates the direction along (perpendicular to) the polarization direction. The solid lines act as a guide to the eye. (b) Condensate depletion as predicted by the PIGS calculations and the Bogoliubov theory, without and with dipolar interaction, for a scattering length of as=60a0.

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    • Figure 3
      Figure 3

      Combination of Feshbach resonances used to tune the scattering length. We measure (a) the atom number and (b) the temperature of a thermal cloud after forced evaporation at different magnetic fields. The extracted positions are B1=5.126(1)G and B2=5.209(1)G with widths of ΔB1=35(1)mG and ΔB2=12(1)mG, respectively. (c) Together with another resonance at B3=21.95(5)G with a width of ΔB3=2.4(8)G, we can calculate the dependence of the scattering length on the magnetic field. The dashed, vertical gray lines represent the positions of the zero crossing of the scattering length, while the blue area corresponds to the region where we observe self-bound quantum droplets. (d) Measured three-body loss coefficient L3 in a thermal cloud, which increases the closer we get to the resonance, explaining the shorter lifetime of the observed self-bound droplets.

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    • Figure 4
      Figure 4

      Phase diagram and schematic steps of the experiment. The phase diagram is calculated using the Gaussian ansatz to solve the EGPE for a cylindrically trapped Dy162 BEC, with mean trap frequency ω¯=ωxωyωz=30Hz and containing 2×104 atoms. For the trap aspect ratio λ below the critical point λ<λc, the stable BEC solution and the single quantum droplet state are connected through a continuous crossover. Above the critical point λc there is a multistable region where both solutions are stable (shown in gray).

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    • Figure 5
      Figure 5

      Expansion velocity across the crossover from a BEC to a quantum droplet. We extract the expansion velocity vexp from the evolution of the widths of the atomic cloud in time of flight for up to 20 ms, averaged over five realizations. This procedure can be applied in both the x and y directions, for which we find comparable results. We thus plot here the average over both directions. As error bars we show the quadratic sum of the uncertainty of the determination of vexp along the two directions and for the scattering length the uncertainty due to the experimental field stability, the knowledge of the Feshbach resonances, and the background scattering length. The two insets show example single-shot images for a self-bound droplet (top) and an expanding BEC (bottom) after a 20-ms time of flight.

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    • Figure 6
      Figure 6

      Exemplary atom number decay curve of a self-bound droplet. The measured decay of the atom number is plotted as a function of the levitation time, averaged over ten realizations, and the error bars denote the respective standard deviations. After a fast decay for short times (gray circles), we observe a constant atom number (blue circles). To extract the critical atom number Ncrit (horizontal blue line) we analyze the atom number distribution (histogram on the right-hand side) and fit our convolution model to the blue data (black line).

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    • Figure 7
      Figure 7

      Determination of the background scattering length of Dy162. (a) Difference res164 between measured critical atom numbers for Dy164 and the results obtained from numerical simulations shifted along the atom number axis. The vertical red line indicates the shift with the lowest residual from the shifted theory and the lighter red area marks the range of uncertainty in which the residual doubles. (b) Residual difference between the observed critical atom numbers of Dy162 and the shifted theory curve (black) with the minimum value shown by the vertical red line. The two dashed lines represent the residual using the boundaries of the uncertainty of the shifted theory. (c) Summary of measured critical atom number versus scattering length. The red line shows the shifted theory curve for a shift along the atom number axis and the green line similarly for a shift along the scattering length axis. The lighter areas show the respective uncertainty of the shift.

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    • Figure 8
      Figure 8

      Energy per particle in units of ℏ2/madd2 for the dipolar system with the three interactions of Eq. (I3) for the s-wave scattering length as=60a0. The lines represent a fit to the data and the intersection with the E=0 axis defines the critical number of the model at this scattering length value.

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    • Figure 9
      Figure 9

      Density profiles along the z direction in the EGPE (red solid line) and PIGS (blue circles) approximations for a scattering length a=60a0. The left and right panels show the EGPE results for N=1000 and N=2000 atoms, compared with the PIGS results for N=1024 and N=2048 atoms, respectively. Each profile has been properly normalized to its corresponding particle number.

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