We use Faraday spectroscopy of atoms confined to crossed hollow beam tweezers to map magnetic fields over 3 millimeters with 200 micron resolution in a single trap loading cycle. The hollow beams are formed using spatial light modulation, and the trap location is scanned using acousto-optic deflectors. We demonstrate the technique by mapping a linear quadrupole magnetic field with 10 nT sensitivity.
©2010 Optical Society of America
Dark optical traps for cold atoms, which use blue-detuned light to confine atoms to low intensity, are useful for precision measurements by providing perturbation-reduced environments and long interrogation times . By keeping atoms in the dark, these traps can operate at small detunings, resulting in deep potentials, so that many measurements can be made without reloading the trap . In particular, these traps are advantageous for magnetometry because they minimally perturb Zeeman sublevels of the trapped atoms .
High spatial resolution magnetometry with excellent sensitivity has been achieved using Larmor precession in cold atom samples, because dense, sub-millimeter trap sizes can readily be obtained. Isayama et al. demonstrated 18 pT sensitivity using Faraday spectroscopy in a falling cloud of 85Rb atoms . Vengalattore et al. used a Bose-Einstein condensate (BEC) to achieve 10 μm spatial resolution with pT/Hz1/2 sensitivity . Smith et al. used localized Faraday spectroscopy measurements in an optical lattice, but not for magnetometry purposes [6,7]. We recently demonstrated high-duty-cycle measurements of time-varying magnetic fields in a 500 μm spot size through Larmor precession . High resolution using cold atoms has also been achieved by imaging density fluctuations in BECs, which reveal the underlying potential . These magnetic probes could be of use for imaging fields near surfaces such as integrated circuits  and atom chips designed for atom interferometry , and for determining gradient fields in cold atom trap environments.
In this paper, we demonstrate mapping a magnetic field over a region with 150 μm resolution using blue-detuned optical tweezers for cold atoms . To measure the field in different locations, a crossed hollow beam trap is dynamically controlled using an acousto-optic deflector (AOD). The position can be precisely controlled through the rf drive frequency to the AOD. A probe laser beam independently tracks the location of the atoms using a second AOD. We demonstrate the technique by mapping a linear quadrupole field with 150 μm spatial resolution over 3 mm. Furthermore, we make these measurements within a single trap loading cycle by sampling the magnetic field at a 100 Hz rate. The technique is capable of higher spatial resolution, spatial extent, and sampling rate, and can be extended to two and three dimensional maps, which would be impractical if each measurement were taken in separate atom loading cycles. We also discuss some technical challenges that arise when moving the probe and trap beams. We previously imaged magnetic fields with submilliGauss sensitivity and 250 μm resolution using stimulated Raman transitions in cold atoms , but Faraday spectroscopy and optical tweezers can offer higher spatial resolution and sensitivity.
2. Experimental setup
The layout for the optical tweezers is shown in Fig. 1 . A hollow laser beam is formed by imparting an azimuthal phase exp(inθ) to a Gaussian beam, where n = 8 is the charge number . The phase modification is done by a 512 x 512 pixel reflective spatial light modulator (SLM, Boulder Nonlinear Systems). The pixel size is 15μm, and the incident Gaussian has a 1/e 2 radius of 1.71 mm. The modified beam intensity and phase profile are relayed to a plane near the vacuum chamber, where the beam is focused by a 200 mm focal length lens.
The focused hollow beam passes through the MOT, and is aligned so that the MOT is approximately located at the plane of peak intensity , a few centimeters away from the focal plane, where the beam diameter is ~200 μm. The beam is relayed back onto the atom sample at right angles through an 8f imaging system. The beam passes through AOD1 (CTI 4050-5) placed between the final two lenses. The intersection of the these beams forms a dark, box-like, bounded volume of order 10−2 mm3 that traps the atoms for blue-detuned light. For these experiments we used a detuning of 0.05nm and ~200mW total power, giving a peak intensity of ~105 mW/cm2, and trap depth of ~2ħΓ, where Γ is 2π × 6 MHz. To mitigate interference effects, we have used in-plane polarization, so the input and re-entrant beams at the sample are orthogonally polarized, and shifted in frequency by ~50 MHz by AOD1. The AOD deflection efficiency is ~65% over the scan range. We define x- and y-axes along the propagation directions of the hollow beam, and the MOT coils are along the z-axis.
To position the atomic sample, we use the following procedure. A MOT containing 107 87Rb atoms is cooled during a 10 ms molasses stage to a temperature of 10 μK and density ~1011/cm3. The hollow beams are first centered on the MOT, and remain on during the MOT loading phase. The hollow beam trap diameter is ~200 microns, so we load ~105 atoms. The alignment of AOD1 is set to the maximum deflection efficiency at the MOT location, so that the maximum number of atoms is loaded into the trap. After the molasses stage, only the hollow beams remain on; the MOT magnetic field, and the cooling and repumper beams are extinguished. The atoms are then moved over 30 ms to the starting measurement location. By moving the atoms over this time scale, motion-induced losses are reduced. In Fig. 1b, we show images of the atom cloud over a range of 2 mm. For our lens focal lengths, the maximum deflection is ~4mm at the MOT (150μm/MHz). Very high positioning resolution is possible, since the AOD frequency can easily be controlled at the kilohertz level. The use of AODs eliminates mechanical vibrations that might occur from beam-steering mirrors.
The π-polarized probe beam travels along the bisector of the hollow beam path. This beam is derived from an extended cavity diode laser (New Focus StableWave), and has a 1/e 2 waist of ~100 microns, a detuning of ~2π × 4 GHz, and peak intensity of ~30 mW/cm2. To maintain alignment with the atom trap, it passes through AOD2 (CTI 4080-4). A polarizer (LP) with 106 extinction ratio eliminates minor frequency-dependent polarization effects of the AOD. The AODs are calibrated so that the probe is aligned with the atoms at all locations. The probe propagates along the bisector of the hollow beams so that it only samples atoms in the trap. If it copropagates along one of the hollow beams, there are enough atoms contained within each individual beam (see Fig. 1b) that spatial resolution is lost.
To initiate Larmor precession, the sample is optically pumped to F = 2, mF = 2 by a brief (20 μs) σ + pulse connecting F = 2 to F’ = 2. The probe beam remains on at constant power throughout the experiment. After passing through the precessing atomic sample, the probe beam enters a polarimeter consisting of a half waveplate, a Wollaston polarizing beamsplitter, and a balanced photodetector . The photodetector output is a measure of the average spin projection along the optical pumping axis, and the Larmor precession frequency, ωL, is related to the scalar magnetic field B through ωL = gFμBB/ħ, where gF is the gyromagnetic ratio, μB is the Bohr magneton, and the quantity gFμB/ħ = 700 kHz/Gauss for 87Rb.
3.1 Spatially- varying magnetic field
In Figs. 2(a,b) , we show Faraday signals at different locations in a linear quadrupole field, created by passing a small current through the MOT coils; the figure shows these data for two gradients. In this section, the signals recorded for each position are obtained in different loading cycles of the MOT; the next section will demonstrate measurements in a single loading cycle. There is also a small bias field of ~10 mG orthogonal to the probe and optical pumping beam directions; the small bias allows the field variations to be visually apparent in the Faraday signals for this demonstration . The total field generated isFigs. 2(a,b), the line plots are Faraday signals at two locations separated by 1.5mm, where the blue trace is recorded with the atoms on the axis of the MOT coils, and the red trace is recorded 1.5 mm away. For Fig. 2a, the coil current is zero, so ωL is position independent. Figure 2b shows Faraday traces with a small coil current to generate a gradient of ~150 mG/cm along the x- and y-directions.
To visualize the measurements over a range of positions, the data can be presented as grayscale montages of the Faraday photodetector signals on the horizontal axis, and atom position along the vertical axis. These images are shown below each line plot in Figs. 2(a,b), resulting from 21 locations, spaced 150 μm apart, along x over 3 mm. The Larmor oscillations are shown over 1 ms in Fig. 2a, and 500 μs in Fig. 2b. With no gradient present the images show a series of parallel lines. As the gradient is increased, ωL increases away from the origin. The magnetic field curvature is apparent in the magnetic field maps, withFig. 2(c). The shot-noise-limited sensitivity for our parameters is approximately 20 μG, but we achieve sensitivities of ~100 μG (10 nT) in each sampling window with no averaging, due to systematic fluctuations and the simple photodetection circuit we used. A fit to Eq. (2) is also shown, giving a bias field of 24.4 ± 0.2 mG, and a gradient of 15.1 ± 0.4 mG/mm. We also retrieve x0 = −51 ± 19 μm.
In addition to an increasing ωL away from the axis, the Faraday signals are biased to a positive value that decays over a few milliseconds, as seen in the red trace of Fig. 2(b). This is due to the orientation of B with respect to the optical pumping axis. As the atoms are moved along x, B also tips along x (Eq. (1). Because the atoms are optically pumped along x – y, and precession occurs around B, the average spin polarization is also biased along the optical pumping direction. The gradient off axis causes oscillations to dephase after ~200 microseconds, followed by the net spin polarization a few milliseconds later. In situations with reduced projection of the spin precession along one axis, one could consider using sequential measurements along two orthogonal directions. This could be accomplished with beams that copropagate with the hollow beams, as discussed in Sec. 3.2. We also note that gradient information is contained within the Faraday signals: Variations of Larmor frequency across the probe beam can be determined by Fourier transform of the Faraday signal. This is useful for environments where the length scale of B-field variations is significantly smaller than the probe size.
3.2 Single-shot measurements
We have also made these measurements within a single loading cycle of the MOT. For this demonstration, we move the atoms to the initial measurement location 1.5 mm away from the MOT coil axis. At time T = 0, the probe and re-entrant hollow beam are scanned over 3 mm in 150 ms (20mm/sec). We note that we also relay the probe AOD plane onto the photodetector so that the detector alignment is fixed during the scan. Faraday measurements are recorded at 15 locations in one scan. A typical trace is shown in Fig. 3(a) . This measurement was also done for a linear quadrupole field. The retrieved frequencies are plotted in Fig. 3(b), and the measurements at x = −1.5mm and x = 0 are shown in Figs. 3(c) and 3(d) respectively.
As before, we fit B0 and B’ according to Eq. (2) [Fig. 3(b)]. Our values are B0 = 15.7 ± 0.4 mG and B’ = 15.6 ± mG/mm. We have found that the fits are not quite as good as the prior results for two main reasons. The first is that the atom number decreases during the scan, reducing the signal to noise. In principle this can be corrected by using a deeper trap. The second reason is that time-varying fields are also detected. The experiment is locked to the power line, so the dominant term will be a 60 Hz modulation of the magnetic field. This modulation does not occur in Fig. 2, where each measurement was taken at the same time in the power line cycle. We previously found that the amplitude of this oscillation in our lab is a few mG . If we add a 60 Hz sinusoidal term to the fit, the source of the scatter in the data becomes more apparent. These synchronous time-varying bias fields can be determined beforehand through a measurement without scanning the atoms because they should be only weakly position dependent. They can also be determined by sampling at a higher rate; we previously performed Faraday spectroscopy at 1 kHz sampling rate in a static trap . In this work, we only required 100 Hz sampling rate to achieve 200 μm resolution. Our measurement error (100 μG) and shot noise limit (20 μG) at T = 0 are the same as in Sec. 3.1.In the single-loading-cycle experiments, there is a gradual decay of the Faraday signal envelope. This is because atoms are lost during the scanning and occurs for a few reasons. First, each time the sample is optically pumped, the atom cloud heats up and gradually boils out of the trap . Second, the AOD loses diffraction efficiency across the scan range. At the extremal locations, the diffraction efficiency is approximately 75% of the peak diffraction efficiency, so the trap depth is lower. Third, we ramped the AOD drive frequency linearly, instead of using a smooth acceleration. These loss mechanisms can be reduced by using larger trap depths or by slowing the scan speed. We determine a rough estimate of the maximum scan acceleration, as, that should be used under the assumption of a harmonic potential with trap frequency, ω, as follows. For a stationary potential of radius R, the acceleration of an atom at the turning points is Rω 2. This is also an upper bound on the scan acceleration, as; if it is exceeded, the atom will have to reach a larger radius to obtain the necessary acceleration, and it will leave the trap. Similarly, if we assume that the trap scanning motion is sinusoidal with frequency ωs << ω, and amplitude As, the peak scan acceleration as = Asωs2. Setting as = Rω 2, the maximum scan frequency is roughly ωs ~ (Rω 2/As)0.5. For a trap depth U = 2ħΓ, amplitude As = 1.5 mm, and R = 100 μm, ωs < ~800 rad/sec. We note, however, that an exact expression depends on the specific trap geometry.
For low fields, these measurements suffer from systematic noise from the AOD at kHz frequencies. This noise comes from scanning-induced noise from the AOD, imparting 0.2% frequency-dependent power fluctuations onto the probe beam. In a typical balanced photodetector layout, power fluctuations can be eliminated to the shot noise limit. However, in the polarimeter, the beam passes through a half waveplate and Wollaston polarizing beamsplitter to equalize the powers hitting the photodetectors. Thus, both the signal and noise are on each split beam; if the signal occurs near ωL, the noise cannot be cancelled in this setup. The noise is suppressed by the orientation of the polarizing elements in the polarimeter. Figure 4(a) shows representative time domain signals when the probe is not scanned (black), when the polarimeter optics are optimized (red), and when they are slightly misaligned (blue). In Fig. 4(b), we show the spectral noise on the laser intensity (green), and for the scanned (red) and static (black) probes. The laser intensity noise in the balanced mode can be reduced by ~200x for optimum optical alignment. This agrees with the expected orientation performance of polarizing elements (~π/500). Because the intensity fluctuations are frequency-dependent, they can be subtracted with a subsequent scan with no atoms present, but the power fluctuations are not well subtracted due to noise on the voltage-controlled oscillator driving the AOD. Performance near the shot-noise limit could be achieved with a second polarimeter that records signals simultaneously from a beam that bypasses the MOT, but for this work we have only used one polarimeter, and suppressed noise by monitoring the frequency spectrum and by a subsequent background subtraction. From Fig. 4, we see that this noise would not affect B-field measurements above a few milliGauss at our scan rate. Also, other AODs and drive electronics may not suffer from these fluctuations.
Another way to eliminate scan-induced noise is to counterpropagate the probe beam with the static hollow beam. Because the trapping beam is hollow, it can be combined with the probe in a lossless manner. We attempted this approach, combining the probe with the hollow beam after the first lens shown in Fig. 1. We used a 1mm diameter silvered circle coated onto a glass plate oriented at Brewster's angle (for transparent transmission of the hollow beam). At this location the hollow beam diameter is a few millimeters, and the silvered spot does not perturb it. The probe is then extracted on the input side of the chamber in the same manner. However, due to the extent of the atoms confined in each independent hollow beam [Fig. 1(b)], the total Faraday signal is washed out and spatial resolution is lost. In principle, these atoms could be eliminated, but this was not done for this demonstration. Furthermore, if multidimensional optical tweezers are used , a dynamic probe may be required anyway.
We have demonstrated Faraday spectroscopy in moving dark optical traps with spatial resolution of 150 microns and sampling rate of 100 Hz over a 3 mm linear extent, and applied the measurement to a linear quadrupole field in a single MOT loading cycle. This approach could be extended to 2- and 3-D with additional AODs .
This work was supported by the Office of Naval Research and by the Defense Advanced Research Projects Agency.
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