Measurement of $s$-wave scattering lengths in a two-component Bose-Einstein condensate

We use collective oscillations of a two-component Bose-Einstein condensate (2CBEC) of ${}^{87}$Rb atoms prepared in the internal states $|1\rangle \equiv |F=1,m_F=-1\rangle$ and $|2\rangle \equiv |F=2,m_F=1\rangle$ for the precision measurement of the interspecies scattering length $a_{12}$ with a relative uncertainty of $1.6\times {10}^{-4}$. We show that in a cigar-shaped trap the three-dimensional (3D) dynamics of a component with a small relative population can be conveniently described by a one-dimensional (1D) Schrödinger equation for an effective harmonic oscillator. The frequency of the collective oscillations is defined by the axial trap frequency and the ratio $a_{12}/a_{11}$, where $a_{11}$ is the intraspecies scattering length of a highly populated component 1 and is largely decoupled from the scattering length $a_{22}$, the total atom number and loss terms. By fitting numerical simulations of the coupled Gross-Pitaevskii equations to the recorded temporal evolution of the axial width we obtain the value $a_{12}=98.006\left(16\right)a_0$, where $a_0$ is the Bohr radius. Our reported value is in reasonable agreement with the theoretical prediction $a_{12}=98.13\left(10\right)a_0$ but deviates significantly from the previously measured value $a_{12}=97.66a_0$ [Phys. Rev. Lett. 99, 190402 (2007)] which is commonly used in the characterization of spin dynamics in degenerate ${}^{87}$Rb atoms. Using Ramsey interferometry of the 2CBEC we measure the scattering length $a_{22}=95.44\left(7\right)a_0$ which also deviates from the previously reported value $a_{22}=95.0a_0$ [Phys. Rev. Lett. 99, 190402 (2007)]. We characterize two-body losses for component 2 and obtain the loss coefficients ${\gamma }_{12}=1.51\left(18\right)\times {10}^{-14}{\mathrm{cm}}^3/\mathrm{s}$ and ${\gamma }_{22}=8.1\left(3\right)\times {10}^{-14}{\mathrm{cm}}^3/\mathrm{s}$.

Figure 1

(a) Oscillations of one-dimensional density $|f_2{(z,t)|}^2$ with frequency $2.91$ Hz evaluated from Eq. (18) and 1b the linear density with frequency $3.00$ Hz simulated with the coupled 3D GPE [Eqs. (3)].

Figure 2

Dependence of collective oscillations frequency of component $|2\rangle$ on the scattering lengths 2a $a_{12}$ and 2b $a_{22}$ and 2c on the total atom number $N$. Dotted lines are the analytical predictions of Eq. (20). The three-dimensional GPE simulations are represented by the solid lines ($\theta =\pi /10$) and dashed lines ($\theta =\pi /5$). $N={10}^5$ for panels 2a and 2b.

Figure 3

Dipole oscillations of state $|1\rangle$ BEC along the axial direction of magnetic trap. The measured value of the axial trap frequency is $f_z=11.507\left(7\right)$ Hz.

Figure 4

(a) Iterative convergence of interspecies scattering length $a_{12}$ and (b) intraspecies scattering length $a_{22}$. The solid lines represent the results obtained with a preparation-pulse area $\theta =\pi /10$, dashed lines are for $\theta =\pi /5$ in panel (a). Different colors in panel (a) represent the data for different sets of measurements of $a_{12}$.

Figure 5

Measurement of (a) intraspecies ${\gamma }_{22}$ and (b) interspecies ${\gamma }_{12}$ two-body loss coefficients. The plot of $N^{-2/5}$ versus hold time $t$ is almost linear in the ${\gamma }_{22}$ measurement. The measurement of ${\gamma }_{12}$ is plotted on a semilog scale, the increase in loss rate is observed at the point of maximum density of component 2 ($0.17$ s) due to the influence of ${\gamma }_{22}$. Black points are experimental results; blue solid lines represent fits with GPE simulations.

Figure 6

(a) Absorption images of two BEC components in the time-of-flight expansion for different evolution times after a $\pi /5$ preparation pulse. (b)–(d) Central cross sections of the column density of component 2 at evolution times of (b) 0, (c) 100, and (d) 170 ms. Black dots represent the measured optical density from the CCD pixels. Blue solid lines are the central cuts of the 2D Gaussian functions with axial widths of (b) $21.7$, (c) $11.1$, and (d) $7.6$ $\mu$m.

Figure 7

Temporal evolution of axial width of component 2 in a superposition of two states prepared with (a) $\pi /10$ or (b) $\pi /5$ pulse. The expansion times are (a) $6.6$ ms and (b) $20.1$ ms, and the initial total atom numbers are (a) $N=1.1\times {10}^5$ and (b) $N=7.6\times {10}^4$. Black dots are the data extracted with the 2D Gaussian functions and error bars represent statistical errors of the fits. Blue solid lines are the results of the GPE simulations [Eq. (23)] with (a) $a_{12}=98.025a_0$ and (b) $97.986a_0$, and $a_{22}=95.44a_0$.

Figure 8

Ramsey interferometry with variable-area $\theta$ for preparation pulse used in measurements of scattering length $a_{22}$. (a) The normalized number difference $P_z$ oscillates with the evolution time $t$. Dots are data points, blue solid line is fit GPE simulations for $\theta =\pi /10$, red solid line is fit GPE simulations for $\theta =\pi /2$. (a) Two-photon detuning $\Delta /\left(2\pi \right)=6.82\left(10\right)$ Hz was measured in Ramsey interferometry of noncondensed atoms. Dots are data points, solid line is a sinusoidal fit.