Extreme Tunability of Interactions in a ${}^7\mathrm{Li}$ Bose-Einstein Condensate

We use a Feshbach resonance to tune the scattering length $a$ of a Bose-Einstein condensate of ${}^7\mathrm{Li}$ in the $|F=1,m_F=1⟩$ state. Using the spatial extent of the trapped condensate, we extract $a$ over a range spanning 7 decades from small attractive interactions to extremely strong repulsive interactions. The shallow zero crossing in the wing of the Feshbach resonance enables the determination of $a$ as small as 0.01 Bohr radii. Evidence of the weak anisotropic magnetic dipole interaction is obtained by comparison with different trap geometries for small $a$.

Figure 1

Representative in situ polarization phase-contrast images of condensates with various interaction strengths. (a) $B=719.1\mathrm{G}$, $a=396a_0$, $N=1.7\times {10}^5$; (b) $B=597.4\mathrm{G}$, $a=8a_0$, $N=2.9\times {10}^5$; (c) $B=544.7\mathrm{G}$, $a=0.1a_0$, $N=2.0\times {10}^5$; (d) $B=542.4\mathrm{G}$, $a=-0.1a_0$, $N=1.2\times {10}^5$; (e) same as (d) but with a faster field ramp from 710 to 542.4 G, resulting in multiple solitons with $N\approx {10}^4$ per soliton. The probe laser detuning from resonance is adjusted to keep a nearly constant signal level, and varies between $20\gamma$ for large $a$ to $150\gamma$ for small $a$, where $\gamma /2\pi \approx 5.9\mathrm{MHz}$ is the excited state linewidth. The color map is adjusted to maximize contrast for each image.

Figure 2

Axial size of the condensate as a function of magnetic field. The axial size is defined as the $1/e$ radius of the axial density profile and is scaled by the axial harmonic oscillator size $l_z=\sqrt{\hslash /m{\omega }_z}\approx 22\mu \mathrm{m}$. The resolution of the optical imaging system is $\sim 3.3\mu \mathrm{m}$ (dotted line). The dashed line is the size of the condensate ($l_d\approx 0.62l_z$) found by solving Eq. (1) with $a=0$. The zero crossing in $a$ occurs when the size of the condensate equals $l_d$ (arrow). Neglecting dipolar effects results in a zero crossing about 0.5 G higher, where the axial size equals $l_z$. The inset shows the axial size corrected for number variation as described in the text. Individual data points and error bars are the average and standard error of approximately 10 shots taken at each field. Systematic uncertainty in the axial size is $\sim 3\%$ from uncertainty in temperature and the uncertainty in imaging magnification. The systematic uncertainty in the magnetic field due to calibration (via radio frequency transitions from the $|2,2⟩$ to the $|1,1⟩$ state) is $\sim 0.1\mathrm{G}$. We have binned the data into intervals of this size.

Figure 3

Mapping functions of axial size to $a$ using a Gaussian trial wave function in a variational solution to Eq. (1), including (solid line) and neglecting (dashed line) the MDI; also shown is the Thomas-Fermi approximation (dotted line). These mappings were computed for $N=3\times {10}^5$, ${\omega }_r/2\pi =193\mathrm{Hz}$, and ${\omega }_z/2\pi =3\mathrm{Hz}$. In practice, we compute the mapping individually for each imaged condensate to account for variations in $N$ and a field dependent variation in ${\omega }_z$ of $\sim 5\%$ over the relevant magnetic field range. The Gaussian solution neglecting the MDI asymptotically approaches $l_z$ at zero interactions, while their inclusion causes the solution to asymptotically approach a value smaller than $l_z$.

Figure 4

Axial size data of Fig. 2 mapped onto $a$. Results of a coupled-channels calculation are shown by the solid line. The Feshbach resonance fit is indicated by the dashed line. The inset shows the extracted values of $a$ near the zero crossing. The mean and standard error of approximately 10 shots taken at each field is shown. In addition, we estimate a systematic uncertainty of $\sim 20\%$ in $a$.

Figure 5

Extracted values of $a$ near the zero crossing for trapping potentials with ${\omega }_z/2\pi =3\mathrm{Hz}$ (filled squares) or ${\omega }_z/2\pi =16\mathrm{Hz}$ (unfilled squares), when (a) neglecting or (b) including the MDI in the mapping function. The MDI has a negligible effect on the extracted values of $a$ for the 16 Hz trap, but neglecting the MDI in analysis of the 3 Hz trap systematically lowers the mapped values of $a$, especially for $a\lesssim 0.15a_0$.