Abstract
We demonstrate detection of a weak alternate-current magnetic field by application of the spinecho technique to F = 2 Bose-Einstein condensates. A magnetic field sensitivity of 12 pT/ √ Hz is attained with the atom number of 5 × 10 3 at spatial resolution of 99 µm 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.

Characterization of the inhomogeneity of the ambient magnetic field is an important task in many experiments which utilize atomic systems with spin degrees of freedom. Such experiments include fundamental physics such as tests of symmetries [1] and identification of the ground state in spinor condensate systems [2, 3] as well as important applications such as optical lattice clocks [4] and long-lived coherence [5] . In all of these cases, the experimental accuracy is directly related to the inhomogeneity of the magnetic field. Recently, high sensitivity magnetometers have been made using both superconducting quantum interference devices and atomic systems. Although the former are known to provide ultrasensitive magnetic field sensing, the latter do not require cryogenic cooling and are particularly useful when measuring the ambient magnetic field inside a vacuum chamber. The most sensitive atomic magnetometer so far realized used a spin-exchange relaxation-free technique to achieve a magnetic field sensitivity of 0.5 fT/ √ Hz for a measurement volume of 0.3 cm 3 [6] . However, in such atomic vapour magnetometers the spatial resolution is limited by the diffusion of atoms. This problem can be largely overcome by using Bose-Einstein condensates (BECs) in an optical trap. For example, it was demonstrated that a sensitivity of 8 pT/ √ Hz at spatial resolution of 120 µm 2 could be achieved using an F = 1 spinor BEC in Ref. [7] .

All of the atomic systems mentioned above detect constant magnetic fields. However, specialized magnetometers also exist for detecting alternating current (AC) magnetic fields. Such devices have high sensitivity for an AC field within an certain frequency band and are useful for the characterization of temporal fluctuations of the magnetic field. Such magnetometers can be constructed using the spin-echo technique, as originally demonstrated in experiments on single nitrogen vacancy centers [8] [9] [10] . In this Letter, we perform a similar type of spin-echo AC magnetometry using 87 Rb F = 2 BECs. We attain a magnetic field sensitivity of 12 pT/ √ Hz with the spatial resolution of about 99 µm 2 . To the best of our knowledge this is the first realization of spin-echo based magnetometer using an atomic gas. In addition, we observe magnetic field noise synchronous with the power supply line at frequencies of 50 and 100 Hz using our AC magnetometer, and reconstruct the amplitude and phase of the AC magnetic field noise. By applying a magnetic field with opposite phase, we can suppress this noise to 1 nT order. We anticipate that the clean magnetic field environment created by this technique will be useful in applications requiring low ambient magnetic fields such as the search for magnetic ground states in systems with spin > 1/2 [2, 3] since a temporally fluctuating magnetic fields can cause undesired spin rotations, seriously affecting such experiments.

Figure 1(a) shows the outline of the experimental setup. A BEC of 87 Rb is created using radio frequency (RF) evaporative cooling in a magnetic trap [11, 12] . The BEC is then loaded into a crossed far-off-resonant optical trap (FORT) with axial and radial frequencies of 30 Hz and 100 Hz, respectively. After 300 ms hold time in the crossed FORT, typically 3×10 5 atoms remain in the |F = 2, m F = −2 state. A bias magnetic field along the z axis (B z ) shown in Fig. 1 and Sz 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 plot represents the average over the ten measurements. The solid curves in (a) and (b) are fitting curves obtained using the spin-2 rotation matrix [13] and the cosine function, respectively. the quantization axis. To generate the stable B z field, a laser diode source with a low ripple noise of less than 2 µA (Newport 505) is used as the current source for our z-axis Helmholtz coils and the whole experimental setup is installed inside a magnetic shield room whose walls consist of permalloy plates. The magnetic field along the x-and y-direction is carefully compensated by using two Helmholtz coils with the similar laser current sources. The strength of the B z field is calibrated from the Larmor frequency as measured using a Ramsey interferometric method [13] and is found to be B z = 9.26 µT for a Helmholtz coil current of 120 mA.

The time sequence used for sensing the weak magnetic field is shown in the bottom panel of Fig. 1(b) . In the detection of the weak AC magnetic field using the spin echo technique [8] , the RF Hahn-echo pulse sequence (π/2-π-π/2) is applied to the initial |2, −2 state in the crossed FORT. The first π/2 pulse rotates the spin vector from the S z direction to the S x direction and induces Larmor precession in the S x − S y plane as shown in Fig. 1(b) . We synchronize the RF source with the AC power line so that the timing of the first π/2 pulse is synchronous with the 50 Hz supply. The upper panel in Fig. 1(b) shows the evolution of the spin vector in the S x -S y plane between two π/2 pulses, where the frame is rotating at Larmor frequency of f 0 = g F µ B B z /h with g F and µ B being g-factor and the Bohr magneton. When we apply a time-varying magnetic filed b(t) along the zdirection, the spin direction in the S x -S y plane is changed

b(t)dt] relative to the −S x direction due to the presence of the AC magnetic field b(t) (shown by solid arrows in the top panel of Fig. 1(b) ). Note that the maximum θ variation for a single frequency field, b(t) = b AC sin (2πt/τ AC ), is reached when τ , the total precession time between two π/2 pulses, is equal to τ AC . The angle of θ yielded by b(t) is converted to S z , by the application of a second π/2 pulse, and the relationship between the expectation value of S z , S z , and b(t) can be expressed by

In the case that b AC = 0 (shown by dotted arrows in the top panel of Fig. 1(b) ), the spin direction returns to the |2, −2 state ( S z = −2). One can thus detect the weak AC magnetic field by measuring the S z . In addition, the effect of undesirable inhomogeneities such as magnetic field gradients and slowly fluctuating magnetic fields is reduced due to spin-echo [14] .

The techniques of the Stern-Gerlach separation and time-of-flight absorption imaging are used to obtain S z . The atomic density distributions of each m F component are measured by shining the imaging beam from the xdirection after a time of flight of 15 ms. The atom number in each m F component, N mF , is calculated over a small region in the center of the BEC (66 µm and 47 µm in the y-and z-direction) in order to extract the peak atomic number. Using N mF , we calculate S z from the following equation:

In the first experiment, we observed Rabi-type oscillations of the spin-2 BEC in order to determine the RF pulse durations for π/2 and π pulses. Instead of the Hahn echo sequence, a single RF pulse with various irradiation time was applied to |2, −2 state. Figure 2 In order to confirm that our system operates as an AC magnetometer, S z was measured in the presence of a purposely introduced AC magnetic field of amplitude b AC . Figure 3 shows the measured value of S z versus b AC at τ = 5 ms (filled circles) and 15 ms (empty circles), where b(t) = b AC sin (2πt/τ ) and the Hahn echo sequence is also applied. The solid curves in Fig. 3 are cosine functions fitted to the data where the amplitude, period and initial phase were the fitting parameters. The cosinusoidal variation shown in Fig. 3 means that the phase variation in the S x -S y plane accumulated by the AC magnetic field was successfully observed. The period at τ =15 ms is three times shorter than that at 5 ms due to the longer phase accumulation time.

The magnetometer sensitivity for a single measurement is given by δb min = δ S z /(d S z /db AC ), where δ S z is the standard deviation of S z in a single measurement and d S z /db AC is the slope with b AC . When we calculate the δ S z from the measured S z values at S z ∼ 0, δ S z is found to be 0.28 and 0.48 at τ = 5 and 15 ms, respectively. From these values, we find the sensitivity to be δb min = 0.97 and 0.66 nT. Note that the spin echo technique can only remove the effect of slowly fluctuating magnetic fields whose period is longer than τ . However, our experiment is subject to the influence of faster fluctuations that vary for each measurement. For example, the magnetic field noise caused by the ripple of the Helmholtz coil current is of order 0.1 nT. In order to evaluate the intrinsic sensitivity, which is unaffected by the temporal fluctuation of the magnetic field, we divide the optical density distribution of each m F component for a single measurement into the three regions as shown in Fig. 4(a) . We infer the value of δ S z from three expectation values of S z , S z i = Σ +2 mF =−2 m F N i,mF /Σ +2 mF =−2 N i,mF , where the subscript i = 1 − 3 indicates which of the three regions is being considered. In our previous work [14] , we theoretically and experimentally confirmed that the shape of the atomic distribution of each m F component in the optical trap is almost unchanged after a time-of-flight of 15 ms, although the distributions become more spread out. Each value of S z i thus reflects the value of S z in a different spatial region of the trapped BEC. A field sensitivity of 94 ± 9 pT for τ = 15 ms is attained with corresponding δ S z of 0.069 ± 0.006, when we select the size of each region as dy × dz = 15.7 µm × 6.3 µm = 99 µm 2 , where dy and dz represent the length along the y-and z-direction of each region. The field sensitivity for N times measurement per 1 second is to be δb atom is changed by increasing dy. The dotted curve represents the atom shot noise limited δ S z . The deviation from the the atom shot noise limited values has multiple origins which induce spatial distortion of the atomic distributions and the optical image: Interference fringes caused by unclean regions on the imaging optics is one well known origin of distortion in the image. Additionally, the effect of the magnetic field gradient (∼ 1.5 µT/cm), which cannot be completely removed by spinecho [15] , and spontaneous pattern formation [16] produce effective distortion of the atomic distribution. Another possible cause of the deviation is the effect of thermal atoms, whose Gaussian tails reduce the accuracy of discrimination between the m F components in the SternGerlach separation. Thus the sensitivity of our AC magnetometer will be improved by reduction of the magnetic field gradient and optimization of imaging system.

As a practical application of our magnetometer, we performed detection of the stray AC magnetic field present in our apparatus. exhibit a random distribution. Instead, we see that S z exhibits oscillatory behavior. Such behavior indicates the existence of a stable AC stray magnetic field in our apparatus. Note that the first π/2 pulse in the Hahn echo sequence is synchronous with the power supply line of 50 Hz. It is therefore reasonable to expect that the AC stray magnetic field is mainly induced by the magnetic field arising from the electronic devices surrounding the BEC apparatus which should be synchronous with the 50 Hz supply line.

In order to quantitatively investigate the effect of AC stray magnetic field, we simulate the τ dependence of S z by inserting a 50 Hz magnetic field, b(t) = b 20 sin [2πt/(20 × 10 −3 ) + θ 20 ], into the Eq. 1. The solid curve in Fig. 5(a) represents a fit of Eq. 1 to the data assuming such a 50 Hz magnetic field, where b 20 and θ 20 are used as the fitting parameters. The fitted curve reproduces the oscillating behavior of the data in Fig. 5(a) , and the parameter values are found to be b 20 = 9.5 nT and θ 20 = −0.76.

Assuming that the fitted parameter values are accurate, we should be able to remove the effects of the 50 Hz stray magnetic field along z-direction by application of a magnetic field with the same frequency and amplitude but inverse phase. Fig. 5(a) , and this suppression suggests that the 50 Hz magnetic field noise is reduced.

Nonetheless, the small remaining oscillation in Fig.  5(b) indicates that a synchronous magnetic field still remains along the z-direction.

We assumed the existence of a 100 Hz magnetic field, b 10 (t) = b 10 sin [2πt/(10 × 10 −3 ) + θ 10 ], and by fitting to the data in Fig. 5(b) we obtained the parameters b 10 = 3.4 nT and θ 10 = 1.50. Based on the parameter values, we further applied an inverse phase magnetic field at 100 Hz in addition to the application of that at 50 Hz [ Fig. 5(c) ]. As shown in Fig. 5(c) , S z is close to −2 for most values of τ , particularly when τ ≤ 10 ms. This result indicates that magnetic field noise which is synchronous with the power supply line is strongly suppressed by the application of an inverse phase field with frequency components at 50 and 100 Hz. In particular, we note that at τ = 20 ms, the value of S z is −1.7±0. −0.4 nT, implying that the magnetic field has been suppressed by almost one order of magnitude when compared with the uncompensated case shown in Fig. 5(a) .

In conclusion we have reported the demonstration of a spin-echo based magnetometer using 87 Rb F = 2 BoseEinstein condensates, with which weak AC magnetic field noise is detected. We attained a field sensitivity of 12 pT/ √ Hz at a spatial resolution of 99 µm 2 . In addition we observed magnetic field noise synchronous with the power supply line. By artificial application of an inverse phase AC magnetic filed, the synchronous noise at 50 Hz is suppressed down to 1 nT order. The techniques demonstrated here are use for characterizing and controlling the magnetic field environment, with particular applicability to ultracold atom experiments. By allowing the detection and creation of a stable, weak bias magnetic field we anticipate that our magnetometer will facilitate the development of fundamental research areas in atomic physics which require a weak magnetic field regime [1] [2] [3] [4] .

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