Faraday waves in Bose-Einstein condensates

Motivated by recent experiments on Faraday waves in Bose-Einstein condensates we investigate both analytically and numerically the dynamics of cigar-shaped Bose-condensed gases subject to periodic modulation of the strength of the transverse confinement. We offer a fully analytical explanation of the observed parametric resonance, based on a Mathieu-type analysis of the non-polynomial Schrödinger equation. The theoretical prediction for the pattern periodicity versus the driving frequency is directly compared to the experimental data, yielding good qualitative and quantitative agreement between the two. These results are corroborated by direct numerical simulations of both the one-dimensional non-polynomial Schrödinger equation and of the fully three-dimensional Gross-Pitaevskii equation.

Figure 1

Imaginary part of the critical exponent $\mu$ as a function of $a$ for $b=0.25$. Notice the main lobe centered around $a=1$. For the first lobe, the difference between the approximation (13) and the numerics is smaller than the thickness of the line. The second lobe, shown in the inset, is so much smaller than the first one that they can hardly be seen on the same scale.

Figure 2

(Color online) Average spacing of adjacent maxima of the longitudinal patterns as a function of the transverse driving frequency. Blue diamonds depict the experimental data of Ref. [12] (the experimental error bar depicted here only takes into account the error bar in the pixel size of the recorded experimental image). The dashed black line, solid red line, and blue dashed-dotted line correspond, respectively, to spacings computed using $S_0$, $S_1$, and $S_2$. Green empty circles correspond to spacings extracted from the 1D NPSE numerics by the averaged spacing method. Pink squares correspond to the spacings extracted from the full 3D numerics using the FFT method and its associated error bars (see the text for details). Notice that for $\omega /2\pi$ close to 160.5 Hz the theoretical prediction is far from the experimental data. The radial breathing mode excited at these frequencies cannot be captured by the NPSE but is well captured by the 3D numerics.

Figure 3

(Color online) Faraday pattern formation for the 1D NPSE. The top panel depicts the growth rate $G$ ($L^2$ norm of the deviation from the initial density) of the pattern. The middle two rows display the Faraday patterns at the times indicated (see the vertical lines in the top panel) where the left subpanels show the density profile while the right subpanels show the deviation from the initial density profile. The bottom panel depicts the space-time evolution of the density. This case corresponds to the experiment in Ref. [12], namely, a cloud of $N=5\times {10}^5$ ${}^{87}\text{R}\text{b}$ atoms, trapped by $\left\{{\omega }_{r,0},{\omega }_z\right\}/\left(2\pi \right)=\left\{160.5,7\right\}\text{Hz}$ with a $20\%$ modulation of the radial confinement at a frequency $\omega /\left(2\pi \right)=321\text{Hz}$.

Figure 4

(Color online) Same as in Fig. 3 for a (stronger) $40\%$ modulation of the radial confinement at a (slower) driving frequency $\omega /\left(2\pi \right)=150\text{Hz}$.

Figure 5

(Color online) Method of averaging spacings in the inhomogeneous cloud for the NPSE model. The left and right columns of plots correspond, respectively, to $\omega /\left(2\pi \right)=200\text{Hz}$ and $\omega /\left(2\pi \right)=320\text{Hz}$. The top row of panels shows the density deviation $\Delta f$ from the initial density profile. The middle row of panels depicts with dots the measured spacing between maxima of $\Delta f$ inside the cloud. The averaged spacing is shown with the red horizontal line. The bottom row of panels shows the FFT of the density ${\left|f\right|}^2$ as a function of the wave number. The vertical lines depict the estimated window of the frequencies contained in the Faraday pattern due to the nonhomogeneity in the spacings (see the middle row).

Figure 6

(Color online) Faraday pattern from the $\left(r,z\right)$ 3D simulations. This case corresponds to the experiment in Ref. [12], namely, a cloud of $N=5\times {10}^5$ ${}^{87}\text{R}\text{b}$ atoms, trapped by $\left\{{\omega }_{r,0},{\omega }_z\right\}/\left(2\pi \right)=\left\{160.5,7\right\}\text{Hz}$ with a 20% modulation of the radial confinement at a driving frequency $\omega /\left(2\pi \right)=321\text{Hz}$. The top panel depicts the transverse radius of the cloud displaying impact-oscillator behavior (thin vertical lines depict the times of the snapshots shown in Fig. 7). The second panel depicts the growth of the Faraday pattern while the bottom two rows depict the $r$-integrated density profiles (left subpanels) and their deviation from the initial profile (right subpanels).

Figure 7

Development of the Faraday pattern in our 3D numerical simulations. Shown are the snapshots of the $y$-integrated profile density (i.e., the observable in the experiments) for the same data as in Fig. 6 at the times indicated.

Figure 8

(Color online) FFT analysis for the data from the experiments of Ref. [12]. The left and right columns correspond, respectively, to a driving frequency of $\omega /\left(2\pi \right)=140\text{Hz}$ and $\omega /\left(2\pi \right)=300\text{Hz}$. The top row depicts the images from the experiment. The second row depicts the corresponding FFTs and the third row the FFTs integrated over the $y$ direction.

Figure 9

(Color online) FFT analysis from the 3D numerics for $\omega /\left(2\pi \right)=321\text{Hz}$. The top two panels depict, respectively, the Faraday pattern and its FFT. The third panel depicts the FFT integrated over the $y$ direction. The bottom panel depicts the same as the third panel but the FFT is computed after adding a $2.5\%$ noise to the density to simulate the experimental noise in the picture (notice also the difference in the depicted windows of wave numbers). The vertical lines in the last panel represent our estimated window of possible frequencies for the Faraday pattern (see the error bars in the pink squares in Fig. 2).