Exploring the Berezinskii-Kosterlitz-Thouless Transition in a Two-dimensional Dipolar Bose Gas

Abstract

Long-range and anisotropic dipolar interactions induce complex order in quantum systems. It becomes particularly interesting in two-dimension (2D), where the superfluidity with quasi-long-range order emerges via Berezinskii-Kosterlitz-Thouless (BKT) mechanism, which still remains elusive with dipolar interactions. Here, we observe the BKT transition from a normal gas to the superfluid phase in a quasi-2D dipolar Bose gas of erbium atoms. Controlling the orientation of dipoles, we characterize the transition point by monitoring extended coherence and measuring the equation of state. This allows us to gain a systematic understanding of the BKT transition based on an effective short-range description of dipolar interaction in 2D. Additionally, we observe anisotropic density fluctuations and non-local effects in the superfluid regime, which establishes the dipolar nature of the 2D superfluid. Our results lay the ground for understanding the behavior of dipolar bosons in 2D and open up opportunities for examining complex orders in a dipolar superfluid.

Figures (10)

• Figure 1: Dipolar 2D Bose gas with tunable dipole angle(A) Schematics of the experiment. Atoms are loaded into a 2D trap and atomic dipoles are polarized by a bias magnetic field at angle $\theta$ from $z$ axis in the $y-z$ plane. (B) Probing the BKT transition of dipolar 2D sample in momentum space. When the sample crosses the BKT transition point a sharp zero momentum peak appears and the trap averaged first-order correlation $g_{1}(r)$ shows algebraic decay instead of exponential decay. The blue empty circles are $g_{1}(r)$ obtained by the Fourier transform of momentum distribution normalized by $g_{1}(0)=1$ and the orange curve on the left(right) is the power-law(exponential) fitting starting from the distance around $0.7\lambda_{db}$. (C)in situ density distribution of sample with different $\theta$, showing we can tune $\widetilde{g}_{eff}$ by tuning the dipole orientation. (D) Upper: Aspect ratio change when changing the dipole orientation due to the anisotropic nature of DDI. Lower: Theoretical $\widetilde{g}_{eff}$ as a function of angle $\theta$ in our experimental parameter. Filled squares above (below) are measured (theoretical) values for 5 specific angles($\theta=0^{\circ},30^{\circ},55^{\circ},70^{\circ},90^{\circ}$) where we conduct experiment.
• Figure 2: Measuring critical atom numbers of BKT transition with different dipole angles.(A) Examples of the time evolution of atom number(upper panel) and zero momentum peak(middle and bottom panels) of 2D samples which represent extended coherence. An empirical piecewise linear fit (orange solid line) is used to determine the transition point $t_c$ (vertical dash line). (B) Model-independent measurement of $N_c^{BKT}$ normalized by the critical number at $\theta=0^{\circ}$, blue solid curve is theoretical scaling factor from eq.(1) based on $\widetilde{g}_{eff}$. Inset: Critical atom number only considering atoms the ground state of axial harmonic oscillator. The dash-dotted curve is $N_c^{BKT}$ calculated by eq.(1) based on temperature fitted by critical samples and independently measured trap frequency. Blue shaded area denotes systematic uncertainty. Error bars are statistical errors.
• Figure 3: Equations of state measurement.(A)Upper panel: Scale-invariance behavior of quasi-2D dipolar samples at $\theta=0^{\circ}$ viewed by EoS constructed from different samples. The EoS follows the HFMF prediction (dash line) in normal regime (shaded orange) and classical field prediction (solid lind) in superfluid regime (shaded blue) like purely contact gas hung2011observation. Inset: Corresponding radial density profile with different atom number and temperature. Lower panel: Scaled-compressibility calculated from EoS. The orange line is the empirical fitting to determine $\widetilde{\mu}_c$ and $\widetilde{\kappa}_{sf}$. (B) Critical chemical potential $\mu_c$ and compressibility in superfluid regime $\widetilde{\kappa}_{sf}$ as a function of angle $\theta$. The solid line above is $\mu_c$ predicted by classical field theory with $\widetilde{g}_{eff}$. The solid line below is the compressibility estimated by TF approximation. (C) Rescaled EoS around BKT critical point of different angle $\theta$. The open circles are Monte Carlo calculations from prokof2002two. The slope of red dash line represents TF limit. Inset: Original EoS. Error bars denote statistical standard error and fitting error.
• Figure 4: Anisotropic atom number fluctuation.(A) Schematic of measuring anisotropic atom number fluctuation. We select two orthogonal rectangular cells within the central, nearly homogeneous region (the white box) of the 2D cloud. We then measure the fluctuation in the number of atoms inside these cells. To probe the number fluctuation, we take all possible configurations of blue (orange) cells inside the central box. Bottom plot is the center cut along the $x$ axis of the cloud for $\theta=90^\circ$. (B) Number fluctuation in the blue cell (blue squares) and orange cell (orange squares) under different tilting angles $\theta$ of the dipoles. Anisotropic number fluctuation emerges when tilting dipoles from $\theta=0^{\circ}$ to $\theta=90^{\circ}$. (C) Rotating two detection cells in a plane, it is clear from the anisotropic $\theta=90^{\circ}$ samples that the fluctuation is enhanced when the rectangular detection cells align with the dipoles. On the other hand, the isotropic $\theta=0^{\circ}$ samples do not exhibit significant variance when two detection cells are rotated. The shaded region serves as guidance for eyes. Error bars denote statistical standard error.
• Figure S1: Radial frequency $\omega_r$ of the vODT for different dipole angles. The polarizability of vODT shows no systematic difference when rotating B-field. Error bars denote 95% confidence interval of fitting.