Dissipative transport of a Bose-Einstein condensate

We investigate the effects of impurities, either correlated disorder or a single Gaussian defect, on the collective dipole motion of a Bose-Einstein condensate of ${}^7\mathrm{Li}$ in an optical trap. We find that this motion is damped at a rate dependent on the impurity strength, condensate center-of-mass velocity, and interatomic interactions. Damping in the Thomas-Fermi regime depends universally on the disordered potential strength scaled to the condensate chemical potential and the condensate velocity scaled to the speed of sound. The damping rate is comparatively small in the weakly interacting regime, and, in this case, is accompanied by strong condensate fragmentation. In situ and time-of-flight images of the atomic cloud provide evidence that this fragmentation is driven by dark soliton formation.

Figure 1

Disordered potential created from laser speckle. (a) Cut through an image of the speckle potential. The disorder strength $V_D$ is proportional to the average value of the intensity $\langle I\rangle$ (dashed line). (b) The autocorrelation of the intensity distribution is well fit by a Gaussian with $1/e^2$ radius ${\sigma }_D=5.5\hspace{0.5em}\mu \text{m}$. For some of the data in this paper (Figs. 6, 8, and 10), ${\sigma }_D=3.4\hspace{0.5em}\mu \text{m}$.

Figure 2

Damping of a condensate initially traveling supersonically through a disordered potential with $V_D/h=280\hspace{0.5em}\mathrm{Hz}$. The center of the BEC (circles) is extracted from a Thomas-Fermi fit to the radially integrated column density (the axial density). The thick lines that trace the amplitude are phenomenological guides to the eye. The initial amplitude is $A=0.6\hspace{0.5em}\mathrm{mm}$, which yields an initial peak velocity of $v_0=20\hspace{0.5em}\mathrm{mm}/\mathrm{s}$. For these data, ${\omega }_z=2\pi \times 5.5\hspace{0.5em}\mathrm{Hz}$, ${\omega }_r=2\pi \times 260\hspace{0.5em}\mathrm{Hz}$, $a=25\hspace{0.5em}{a}_0$, and $\mu =\frac{1}{2}m{\omega }_z^2{R}_{\mathrm{TF}}^2=h\times 1.1\hspace{0.5em}\mathrm{kHz}$, where $R_{\mathrm{TF}}$ is the axial Thomas-Fermi radius. In addition, $c_0=5.6\hspace{0.5em}\mathrm{mm}/\mathrm{s}$, $\xi =0.8\hspace{0.5em}\mu \text{m}$, and $\xi /{\sigma }_D=0.2$. The insets show details of the oscillation at early and late times.

Figure 3

Velocity-dependent damping. Results of fitting the data of Fig. 2 to Eq. (5) by using a traveling four-period window. The peak velocity $v_0$ is obtained from $v_0=A{\omega }_z$. The solid line is a square-root function convolved with an exponential decay and is meant as a guide to the eye. The inset shows the same data on a semilogarithmic plot, which emphasizes the nearly exponential decay of $\beta /{\omega }_z$ for large $v_0/c_0$. Vertical error bars correspond to the range in $\beta$ for which $\Delta {\chi }^2=1$ for the fit to Eq. (5) while simultaneously adjusting $A$, $\beta$, and $\varphi$ to minimize ${\chi }^2$. Horizontal error bars are determined by using an identical process for $A$ in Eq. (5) and are typically $~15\%$. The effects of systematic uncertainty in imaging magnification and variations in $N$ are $~10\%$ in the horizontal axis and $~5\%$ in the vertical axis; these are not included in the displayed error bars.

Figure 4

Characteristic in situ polarization phase-contrast images of the data shown in Fig. 2 at various times. The images are nearly equally spaced in time between the time labels.

Figure 5

Generation of a noncondensed component. (a) Squares show the center of the Thomas-Fermi (condensed) component, and circles show the center of the Gaussian (noncondensed) component. The Gaussian center trails behind the Thomas-Fermi center and has a smaller amplitude of oscillation. Within experimental uncertainty, ${\omega }_z=2\pi \times 5.1(2)\hspace{0.5em}\mathrm{Hz}$ for both components. For these data, $a=200\hspace{0.5em}{a}_0$, $N=3\times {10}^5$, $\mu /h=1.8\hspace{0.5em}\mathrm{kHz}$, $V_D/\mu =0.22$, $v_0=28\hspace{0.5em}\mathrm{mm}/\mathrm{s}$, $c_0=7.2\hspace{0.5em}\mathrm{mm}/\mathrm{s}$, and ${\omega }_r=2\pi \times 220\hspace{0.5em}\mathrm{Hz}$. (b)–(d) Axial density distributions with bimodal fits (solid lines) and a single-component Thomas-Fermi fit (dashed lines) at various times during the oscillation: (b) 28 ms, (c) 100 ms, and (d) 190 ms. The condensates in (b) and (d) are traveling in the positive direction, whereas the condensate in (c) is traveling in the negative direction.

Figure 6

Damping vs $V_D$. Open circles correspond to the data shown in Fig. 7 ($a=200\hspace{0.5em}{a}_0$) in the range $0.7<v_0/c_0<0.9$; filled circles correspond to $a=200\hspace{0.5em}{a}_0$, $v_0/c_0=1.2$, $\mu /h=2.2\hspace{0.5em}\mathrm{kHz}$; squares correspond to $a=25\hspace{0.5em}{a}_0$, $v_0/c_0=1.2$, $\mu /h=750\hspace{0.5em}\mathrm{Hz}$. The damping parameter $\beta$ follows a power law with $p~2$ (solid and dashed lines), independent of $\mu$ or $a$. To minimize systematic effects associated with the velocity dependence of $\beta$ (e.g., Figs. 2 and 3), we fit a four-period window for which the data are described well by Eq. (5). Vertical error bars are as defined in Fig. 3. The inset shows the fit values of $p$ as a function of $v_0/c_0$ for a collection of data sets at $a=200\hspace{0.5em}{a}_0$. The dashed lines indicate the plus-and-minus one standard deviation extent for the collection of measured velocities. Vertical error bars for $p$ are determined as in Fig. 3 by using a fit to Eq. (6) for each oscillation at a given $v_0/c_0$. Data that correspond to filled circles and squares were taken by using an optical trap setup different from that described in Sec. 2 with $\lambda =1064\hspace{0.5em}\mathrm{nm}$ and a beam waist of $24\hspace{0.5em}\mu \text{m}$, which results in ${\omega }_z=2\pi \times 4.9\hspace{0.5em}\mathrm{Hz}$, ${\omega }_r=2\pi \times 460\hspace{0.5em}\mathrm{Hz}$, and $N=3\times {10}^5$. Also, for these data sets, ${\sigma }_D=3.4\hspace{0.5em}\mu \text{m}$.

Figure 7

Transport regimes of a BEC that travels through a disordered harmonic potential. Black squares show the values of disorder strength $V_D/\mu$ and initial peak c.m. velocity $v_0/c_0$ for the data used to extract $\beta$ from a fit to Eq. (5) by using four to six periods of oscillation. The interpolated color map and contour lines for $\beta /{\omega }_z$ are derived from the measured results. These measurements have $a=200\hspace{0.5em}{a}_0$, $N=2\times {10}^5\hspace{0.5em}\mathrm{atoms}$, $\mu /h=1.5\hspace{0.5em}\mathrm{kHz}$, $c_0=6.5\hspace{0.5em}\mathrm{mm}/\mathrm{s}$, ${\omega }_r=2\pi \times 260\hspace{0.5em}\mathrm{Hz}$, and ${\omega }_z=2\pi \times 5.5\hspace{0.5em}\mathrm{Hz}$. The variable experimental quantities are $A$ and $V_D$. Due to small shot-to-shot fluctuations in the position of the c.m. of the cloud, measurements with $v_0<0.2c_0$ are not reliable. Data with $\beta ⩽2\times {10}^{-3}$ are consistent with undamped motion.

Figure 8

Universal damping vs $v_0/c_0$. The disorder strength was adjusted to keep $0.30<V_D/\mu <0.35$ for all of the data. Squares correspond to $a=28\hspace{0.5em}{a}_0$, $N=2.5\times {10}^5$, $\mu /h=550\hspace{0.5em}\mathrm{Hz}$, $c_0=4.0\hspace{0.5em}\mathrm{mm}/\mathrm{s}$, ${\omega }_z=2\pi \times 5.5\hspace{0.5em}\mathrm{Hz}$, and ${\omega }_r=2\pi \times 260\hspace{0.5em}\mathrm{Hz}$; open circles correspond to $a=200\hspace{0.5em}{a}_0$, $N=3\times {10}^5$, $\mu /h=2.4\hspace{0.5em}\mathrm{kHz}$, $c_0=8.3\hspace{0.5em}\mathrm{mm}/\mathrm{s}$, ${\omega }_z=2\pi \times 4.5\hspace{0.5em}\mathrm{Hz}$, and ${\omega }_r=2\pi \times 460\hspace{0.5em}\mathrm{Hz}$; filled circles correspond to the same parameters as in Fig. 7. Error bars are as defined in Fig. 3.

Figure 9

Peak damping vs $a$ with fixed $V_D/\mu$ and $v_0/c_0$. For these data, $V_D$ and $v_0$ were adjusted to keep $0.3<V_D/\mu <0.4$ and $0.6<v_0/c_0<1.4$ with all other parameters as in Fig. 7. The upper horizontal axis shows values for $\mu$ obtained from a variational solution of the GPE [21] (note that the upper tick marks are not strictly logarithmically spaced). The linear fit has a slope $0.002a_0^{-1}$. Vertical error bars are as defined in Fig. 3.

Figure 10

Damping vs $\mu$ with fixed $V_D$ and $v_0$. Squares and filled circles correspond to $V_D/h=370\hspace{0.5em}\mathrm{Hz}$ with $v_0=11$ and 6 mm/s, respectively. Open circles correspond to $V_D/h=140\hspace{0.5em}\mathrm{Hz}$ and $v_0=6\hspace{0.5em}\mathrm{mm}/\mathrm{s}$. The vertical dashed line denotes $\mu =\hslash {\omega }_r$, at which point, $v_0/c_0=1.7$ for the open and filled circles and $v_0/c_0=3$ for the squares. We varied $\mu$ by adjusting $a$, shown on the upper horizontal axis (note that the upper tick marks are not strictly logarithmically spaced). Values for $\mu$ are obtained from a variational solution of the GPE [21] using the following measured experimental parameters: ${\omega }_r=2\pi \times 460\hspace{0.5em}\mathrm{Hz}$, ${\omega }_z=2\pi \times 4.5\hspace{0.5em}\mathrm{Hz}$, and $N=4\times {10}^5\hspace{0.5em}\mathrm{atoms}$. For this data, ${\sigma }_D=3.4\hspace{0.5em}\mu \text{m}$. Vertical error bars are as defined in Fig. 3.

Figure 11

Damping of a nearly noninteracting gas. For this data, $a=0.4\hspace{0.5em}{a}_0$, $N=2\times {10}^5$, $\mu /h=26\hspace{0.5em}\mathrm{Hz}$, $c=1.2\hspace{0.5em}\mathrm{mm}/\mathrm{s}$, $V_D=4\mu$, ${\omega }_r=2\pi \times 240\hspace{0.5em}\mathrm{Hz}$, and ${\omega }_z=2\pi \times 5.3\hspace{0.5em}\mathrm{Hz}$. (a) C.m. position (circles) and radius (squares) of the condensate as a function of time. Here, we use statistically determined values for the c.m. $z_{\mathrm{cm}}=\int zn(z)\mathit{dz}/N$ and radius $R$, given by $R^2=4\int (z-z_{\mathrm{cm}}){}^{2}n(z)\mathit{dz}/N$. (b)–(d) Axial density traces at various times in the oscillation: (b) 40 ms (c) 100 ms, (d) 960 ms. After two full oscillations, the cloud has fragmented and has spread to a size comparable with the initial oscillation amplitude.

Figure 12

BEC in a harmonic trap with a single Gaussian defect. (a)–(c) correspond to a repulsive defect, while (d)–(f) correspond to an attractive one. (a) and (d) In situ polarization phase-contrast images, (b) and (e) axial densities, which correspond to the images, and (c) and (f) numerical solutions to the GPE with the dashed lines that show the solution in the absence of a defect. The inset trace shows the characteristic shape of the potential. For all panels, $a=200\hspace{0.5em}{a}_0$, $N=4\times {10}^5$, ${\omega }_z=2\pi \times 5.0\hspace{0.5em}\mathrm{Hz}$, and ${\omega }_r=2\pi \times 360\hspace{0.5em}\mathrm{Hz}$.

Figure 13

Axial densities at various times during a supersonic oscillation in the presence of (a) an attractive or (b) a repulsive defect. The arrows are proportional to the instantaneous velocity of the condensate. The vertical dashed lines denote the location of the defect. For these data, $a=200\hspace{0.5em}{a}_0$ and $v_0=13\hspace{0.5em}\mathrm{mm}/\mathrm{s}$; (a) corresponds to $N=4\times {10}^5$, $\mu =1.5\hspace{0.5em}\mathrm{kHz}$, $v_0/c_0=2$, $V_D=-0.8\mu$, ${\omega }_z=2\pi \times 4.7\hspace{0.5em}\mathrm{Hz}$, and ${\omega }_r=2\pi \times 360\hspace{0.5em}\mathrm{Hz}$; (b) corresponds to $N=1\times {10}^6$, $\mu =3\hspace{0.5em}\mathrm{kHz}$, $v_0/c_0=1.4$, $V_D=0.4\mu$, ${\omega }_z=2\pi \times 5.0\hspace{0.5em}\mathrm{Hz}$, and ${\omega }_r=2\pi \times 360\hspace{0.5em}\mathrm{Hz}$.

Figure 14

Density fluctuations produced, which cross a repulsive defect. Absorption images after 4-ms time of flight. The vertical dashed line denotes the location of the defect. The experimental parameters are as stated in Fig. 13b.

Figure 15

Velocity dependence of $\beta$ induced by a single Gaussian defect. (a) Attractive defect with (squares) $V_D/\mu =-0.8$ or (circles) $V_D/\mu =-0.3$ and other parameters as stated in Fig. 13a except with $N=8\times {10}^5$ and $\mu /h=2\hspace{0.5em}\mathrm{kHz}$. (b) Repulsive defect with (squares) $V_D/\mu =0.4$ or (circles) $V_D/\mu =0.2$ and other parameters as stated in Fig. 13b. Both types of impurities show critical behavior at low velocities as well as undamped motion at large $v_0/c_0$. Note the difference in scale between damping induced by an attractive vs a repulsive impurity. Vertical and horizontal error bars are as described in Fig. 3.

Figure 16

Transport regimes of a BEC that travels through a harmonic potential with a central Gaussian defect. Coordinates of the black squares are the values of $V_D/\mu$ and $v_0/c_0$ for the data sets used to extract $\beta /{\omega }_z$ from a fit to Eq. (5). The interpolated color map and contour lines for $\beta /{\omega }_z$ are derived from the measured results. Dashed white lines show the local Landau critical velocity as given by Eqs. (7) and (9). The attractive and repulsive cases are qualitatively similar: superfluidity for $v_0/c_0\ll 1$, increased damping as $v_0/c_0\to 1$, and reduced damping for $v_0/c_0\gg 1$. Damping induced by an attractive impurity is an order of magnitude weaker than for a repulsive one. Data with $V_D<0$ and $V_D>0$ correspond to parameters in Figs. 13a and 13(b), respectively.

Figure 17

In situ density distributions of a condensate that passes through a Gaussian defect. These images are taken at the instant the center of the BEC first crosses the defect. Rows correspond to the three flow regimes: subsonic superfluid ($v_0/c_0<1$), dissipative ($v_0/c_0~1$), and supersonic nondissipative ($v_0/c_0>1$). (a)–(c) Attractive defect with $V_D/\mu =-0.85$ and $v_0/c_0=0.31$, $v_0/c_0=1.0$, and $v_0/c_0=3.0$, respectively. (d)–(f) Repulsive defect with $V_D/\mu =0.65$ and $v_0/c_0=0.15$, $v_0/c_0=0.90$, and $v_0/c_0=2.0$, respectively. The arrows indicate the direction and the relative speed of the condensate. For these data, all other parameters are as described in Fig. 13.

Figure 18

Oscillation of a weakly interacting BEC in the presence of a repulsive defect, with $a=0.6\hspace{0.5em}{a}_0$, $N=2.5\times {10}^5$, $\mu /h=44\hspace{0.5em}\mathrm{Hz}$, $V_D/\mu =0.8$, ${\omega }_z=2\pi \times 4.7\hspace{0.5em}\mathrm{Hz}$, ${\omega }_r=2\pi \times 300\hspace{0.5em}\mathrm{Hz}$, and $v_0/c_0=1.6$. (a) C.m. position as a function of time (computed as in Fig. 11), we find $\beta /{\omega }_z=0.03$, (b) root-mean-square density deviation from a Gaussian fit to the axial density distribution (see text). The solid line shows the decay of the oscillation energy (in arbitrary units) found from the fit in (a). (c)–(e) In situ axial density traces and Gaussian fits at various oscillation times: (c) 0 ms, (d) 140 ms, at the second crossing of the defect, and (e) 1260 ms, after several crossings of the defect. At increasingly large times, we find that the large density modulations are accompanied by only a slight increase of the axial size of the condensate.

Figure 19

Velocity dependence of damping with a repulsive defect. Circles correspond to a nearly noninteracting BEC with $a=0.6\hspace{0.5em}{a}_0$, $\mu /h=31\hspace{0.5em}\mathrm{Hz}$, $c_0=0.9\hspace{0.5em}\mathrm{mm}/\mathrm{s}$, and $V_D=0.9\mu$. Shot-to-shot variations in the position of the cloud limit the extraction of $\beta$ to $v_0>1\hspace{0.5em}\mathrm{mm}/\mathrm{s}$, which corresponds to $v_0/c_0>1.1$. Squares correspond to data from Fig. 15b for comparison, $a=200\hspace{0.5em}{a}_0$, $\mu /h=3\hspace{0.5em}\mathrm{kHz}$, $c_0=9.2\hspace{0.5em}\mathrm{mm}/\mathrm{s}$, and $V_D=0.4\mu$. Error bars are as defined in Fig. 3.

Figure 20

Dark soliton formation. In situ axial densities of BECs during the first (a)–(d) and fourth (e)–(h) passes through a semipermeable defect. The defect is located at $z=0$, and its strength was adjusted to keep $V_D/\mu ~0.7$. Oscillation amplitudes were adjusted to keep $v_0~c_0$. (a), (e): $a=0.1\hspace{0.5em}{a}_0$, $N=1.0\times {10}^5$, $\mu /h=5\hspace{0.5em}\mathrm{Hz}$, $\xi =12.5\hspace{0.5em}\mu \text{m}$, ${\xi }_s=16(6)\hspace{0.5em}\mu \text{m}$; (b), (f): $a=0.5\hspace{0.5em}{a}_0$, $N=2.2\times {10}^5$, $\mu /h=30\hspace{0.5em}\mathrm{Hz}$, $\xi =4.9\hspace{0.5em}\mu \text{m}$, ${\xi }_s=6.6(2)\hspace{0.5em}\mu \text{m}$; (c), (g): $a=1.7\hspace{0.5em}{a}_0$, $N=2.6\times {10}^5$, $\mu /h=77\hspace{0.5em}\mathrm{Hz}$, $\xi =3.06\hspace{0.5em}\mu \text{m}$, ${\xi }_s=2.8(4)\hspace{0.5em}\mu \text{m}$; (d), (h): $a=5.4\hspace{0.5em}{a}_0$, $N=2.2\times {10}^5$, $\mu /h=144\hspace{0.5em}\mathrm{Hz}$, $\xi =2.24\hspace{0.5em}\mu \text{m}$, ${\xi }_s=2.5(3)\hspace{0.5em}\mu \text{m}$. The trap frequencies for these data are ${\omega }_r=2\pi \times 240\hspace{0.5em}\mathrm{Hz}$ and ${\omega }_z=2\pi \times 4.75\hspace{0.5em}\mathrm{Hz}$. The dashed lines show fits to Eq. (11). We omit the fit in (d) for clarity. For comparison, the thin dashed line in (a) is only the Gaussian portion of the fit. Error bars for ${\xi }_s$ are given by the standard deviation of a collection of images.