Enhancement of damping in a turbulent atomic Bose-Einstein condensate

Abstract

Turbulence enhances momentum transport in classical fluids, effectively increasing their viscosity. We investigate an analogous effect in a superfluid by measuring the damping of collective oscillations in an atomic Bose-Einstein condensate (BEC) containing stationary spin-superflow turbulence. Using continuous spin driving to maintain turbulence in a spin-1 $^{23}$Na BEC, we excite its quadrupole mode and measure the damping rate over a range of temperatures. The damping consistently exceeds the Landau-damping rate expected for an equilibrium, non-turbulent BEC. The enhancement likely originates from two complementary processes: direct energy transfer from the mode to turbulent condensate fluctuations and turbulence-induced modification of the thermal cloud that amplifies Landau damping. These results establish collective-mode damping as a sensitive probe of momentum transport in superfluid turbulence.

Figures (7)

• Figure 1: Collective oscillations of a turbulent Bose-Einstein condensate (BEC). (a) Schematic of the experiment. A BEC with internal turbulent flow (gray arrows) is confined in a harmonic potential and undergoes shape oscillations. The turbulence is sustained by resonant RF spin driving, which steadily generates an irregular spin texture (color pattern). (b) Energy flow diagram illustrating two pathways for dissipation of collective excitation energy: direct interaction with thermal components and energy transfer into the internal turbulence. (c) Image of a turbulent BEC after an 18-ms time-of-flight.
• Figure 2: Observation of damped oscillations of a turbulent BEC. (a) Experimental sequence. The sample temperature is controlled by adjusting the final trap depth $U_f$ during evaporation cooling. After cooling, sample is held for a few seconds to thermalize and damp out residual motion. A resonant RF magnetic field is represented as a green solid line, continuously applied to generate and sustain turbulence. The hatched area indicates the period of transition to steady turbulence. After a steady turbulent state is prepared, the trap is perturbatively modulated during a time $t_0$, to excite the quadrupole mode. After a variable hold time $t_h$, an absorption image is taken after time-of-flight of 18-ms. (b) Time evolution of the normalized condensate width $\widetilde{W}$ along the $y$ direction as a function of the hold time $t_h$. Images at top were taken at $t_h=36$-ms, $168$-ms, and $299$-ms, from left to right, respectively. Each data point represents a single measurement and the solid red line is a damped sinusoidal function fit to the mean values of the data. This fit yields an oscillation frequency $\omega_\nu=18.8(9)$ Hz and damping rate $\Gamma=0.60(9)$ Hz. The thermal fraction of the sample was $0.44(1)$.
• Figure 3: Temperature dependence of the damping of collective oscillations of BECs. (a) Images of the three different types of BEC samples under investigation: single-component ($\mathcal{D}_{s}=1$) and two-component ($\mathcal{D}_{s}=2$) BECs without turbulence, and a three-component turbulent BEC ($\mathcal{D}_{s}=3$). To visualize the spin composition, the images were taken after Stern-Gerlach spin separation during free expansion footnote_SG. (b) Relative damping rates $\Gamma/\omega_\nu$ as functions of the reduced temperature $\widetilde{T}$ for $\mathcal{D}_{s}=1,2,3$. The solid lines represent weighted linear fits of $\Gamma/\omega_\nu=A_\nu \widetilde{T}$ [Eq. (1)] to the data, and the shaded regions indicate the 1$\sigma$ uncertainties of the fits including the individual uncertainties of the data points. The dashed lines show the predictions of the thermal damping model for $\mathcal{D}_{s}=2,3$, respectively, based on the measurement data for $\mathcal{D}_{s}=1$. The inset shows the effective kinematic viscosity $\nu_\text{T}$ calculated from the excess damping $\Gamma_\text{T}$ in the turbulent BECs [Eq. (2)] SM with $\kappa=h/m$. The horizontal dashed line marks the average of the $\nu_\text{T}$ values and the shaded area indicates its $1\sigma$ standard error.
• Figure S1: (a) Quadrupole oscillation modes for an oblate BEC with trapping frequencies $\omega_x<\omega_y<\omega_z$. The oscillatory motion is characterized by $(b_x, b_y, b_z)$, as shown in Eq. (\ref{['velocity']}), and the three modes are denoted by $\text{X, Y,}$ and $\text{Z}$, respectively, corresponding to the dominant oscillation axis. The variation in the density profile in the $xy$ and $xz$ planes for each mode is presented. (b),(c) Experimental observation of the quadrupole oscillations of BECs. BEC samples with $\mathcal{D}_s=1$ were prepared at our lowest temperatures and the $\text{X}$ or $\text{Y}$ mode was selectively excited by short trap modulations at a frequency slightly red-detuned from resonance. The upper panels in (b) and (c) display time-of-flight images of BECs oscillating in the $\text{X}$ and $\text{Y}$ modes, respectively, at different hold times. The lower panels show the time evolution of the normalized condensate widths in the $x$ and $y$ directions, respectively. Data points represent single measurements and the solid lines indicate damped sinusoidal fits to the experimental data.
• Figure S2: (a) Time evolution of the normalized condensate width $\widetilde{W}$ (blue diamond) during forced driving for different driving frequencies $\omega_d/2\pi=22$ Hz, 25.5 Hz and 26 Hz from left. The sample was an ordinary BEC sample with thermal fraction $\zeta_{th}= 0.3$. Each data point represents a single measurement. Blue solid lines are sinusoidal functions fitted to the data. Red lines denote guides for oscillations in phase with the trap modulations. Responses of BECs to the periodic modulations of the trapping potential. (b) Amplitude magnification factor $\widetilde{A}$ and (c) relative phases $\theta$ of the shape oscillations of the driven BECs as functions of the driving frequency $\omega_{d}$ for three different samples: ordinary single-component samples with thermal fractions $\zeta_{th}= 0.3$ (blue) and $= 0.6$ (cyan), and a turbulent BEC sample with $\zeta_{th}= 0.6$ (orange). Error bars indicate 1$\sigma$ uncertainties including fitting error in (a). Solid lines show Lorentzian curves in (b) and arctangent functions with offsets in (c), fitted to the corresponding data sets. The dashed vertical lines indicate the resonant frequency of the quadrupole oscillation, estimated from the trapping frequencies.