Spin-echo-based magnetometry with spinor Bose-Einstein condensates

We demonstrate detection of a weak alternate-current magnetic field by application of the spin-echo technique to $F=2$ Bose-Einstein condensates. A magnetic field sensitivity of 12 pT$/\sqrt{\mathrm{Hz}}$ is attained for an atom number of $5\times {10}^3$ at a spatial resolution of 100 $\mu$m${}^2$. Our observations indicate magnetic field fluctuations synchronous with the power supply line frequency. We show that this noise is greatly suppressed by application of a reverse phase magnetic field. Our technique is useful in order to create a stable magnetic field environment, which is an important requirement for atomic experiments which require a weak bias magnetic field.

Figure 1

(a) Schematic illustration of the experimental setup. (b) The upper panel depicts the evolution of spin direction in the $S_x$-$S_y$ plane. The lower panel shows the rf Hahn-echo pulse sequence used for sensing a weak ac magnetic field.

Figure 2

Rabi-type oscillation of our spin-2 BEC. (a) and (b) show the population of each $m_F$ component ($N_{m_F}/{\Sigma }_{m_F=-2}^{+2}{N}_{m_F}$) and $\langle S_z\rangle$ as a function of the rf irradiation time, respectively. The rf pulse envelope is shaped by a Gaussian function and the irradiation time corresponds to twice the standard deviation. Each plotted point represents the average over the ten measurements. The solid curves in (a) and (b) are fitting curves obtained using the spin-2 rotation matrix [19] and the cosine function, respectively.

Figure 3

Operation of the ac magnetometer using the Hahn echo sequence. The $\langle S_z\rangle$ values were measured as a function of $b_{\mathrm{ac}}$, where a single frequency ac magnetic field $b\left(t\right)$ is applied. Each plotted point represents an average over multiple BEC measurements. The points at $\langle S_z\rangle \sim 0$ and other points are obtained from 60 and 30 times measurement, respectively. The error bars are sample standard deviation calculated from 60 $\langle S_z\rangle$ values.

Figure 4

(a) The upper panel shows a typical optical density distribution measured at $\langle S_z\rangle \sim 0$. The lower panel shows the column density, where the optical density along the $y$ direction is integrated. The optical density distribution of each $m_F$ component is divided into three regions. Taking into account the finite resolution of our imaging system ($\sim 7.5$ $\mu$m) [20], each region is separated by 9.4 $\mu$m. (b) $\delta \langle S_z\rangle$ calculated from a single BEC image as a function of the total atom number averaged over three regions, $N_{\mathrm{atom}}$, where the $N_{\mathrm{atom}}$ is changed by increasing the region along the $y$ direction, $dy$. The points and error bars indicate the $\delta \langle S_z\rangle$ averaged over the multiple BEC measurements and standard error, respectively.

Figure 5

$\tau$ dependence of $\langle S_z\rangle$ values after the Hahn-echo sequence. Each point represents the average over ten measurements with the error bars giving the standard deviation over those measurements. (a) $\langle S_z\rangle$ values measured without the artificial application of the ac magnetic field. (b) $\langle S_z\rangle$ values measured with application of an inverse phase magnetic field at 50 Hz (parameters given in main text). (c) $\langle S_z\rangle$ values measured with application of an inverse phase magnetic field at 50 and 100 Hz.