Fig. 1. (a) Lab-frame energy level diagram for velocity-selective resonances in ⁸⁵Rb. (b) Schematic picture of imaging principle in a uniform field. Resonant velocity classes, at v=v_avg±ν_r x̂ (with v_avg as defined in Eq. 4 for Δm=0,±1 transitions), are drawn in red for atoms emanating from a point source. The photon momentum kicks give these atoms average velocity v_avg, leading to increased fluorescence at specific locations.

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1. Introduction

An atom in a counterpropagating laser field can undergo a two-photon Raman transition only within a narrow velocity band [1]. This atomic velocity selection is useful for a variety of experiments in laser cooling, including subrecoil Raman cooling [2], atom interferometry [3], and atom velocimetry [4]. Depending on the experimental arrangement, these transitions may occur between different magnetic sublevels, in which case local magnetic fields will shift the resonant velocity classes. These magnetic fields may be nonuniform, which can lead to inefficiency in any experiment requiring velocity selectivity.

The local magnetic field in a cold atom sample can be measured by traditional alkali vapor magnetometry [5, 6, 7], in which the Larmor precession of the atom’s magnetic sublevels is detected. Faraday spectroscopy can be used to determine the scalar field at a single point by using a balanced polarimeter to detect the magneto-optic birefringence [7, 8]. However, to accumulate an image of the scalar field one must raster scan over the atomic sample, although phase-contrast imaging with a fast frame-transfer CCD camera allows spatially-resolved Larmor precession to a range determined by the CCD transfer rates [9].

In this paper, we demonstrate a simple, single-shot technique for directly imaging a nonuniform magnetic field with a sample of cold atoms. The technique relies on the sensitivity of velocity selection to magnetic fields, which we previously showed for constant, uniform fields [10], and it does not require any additional laser frequencies than are already employed in typical cold atom setups. Because the visual cues are striking, the technique works best as a qualitative measure of the magnetic field variation across the sample, but we show that one can use it to visually estimate linear magnetic field gradients to ≈30mG/cm.

The basic principle was described in Ref. [10] and will only briefly be described here. Figure 1(a) shows the relevant energy levels viewed from the lab frame. An atom with velocity v along the x-axis is exposed to a light field composed of two counterpropagating laser beams with wave vectors k and -k along the x-axis. The polarizations of the beams are lin ⊥ lin, and the one-photon detuning, Δ, is chosen to be much larger than the excited state hyperfine splittings. For an arbitrary B-field, this polarization configuration couples m_f-m_i=Δm=0,±1,±2 magnetic sublevels of a single hyperfine level, where we choose our quantization axis along the magnetic field. Photon absorption from one beam and emission into the other results in a linear momentum change of ±2h̄ k=±2Mν_r x̂, where ν_r is the recoil velocity and M is the mass.

Resonance occurs when the two-photon detuning δ=0, which depends mainly on the magnetic field and momentum of the atom. In a small magnetic field, h̄ω_L=g_Fµ_BB, where g_F is the gyromagnetic ratio and µ_B is the Bohr magneton. For ⁸⁵Rb, g_Fµ_B/h̄=466.74 kHz/Gauss [11]. We analyze the resonance condition in the lab frame as follows. An atom has initial energy

where E^LS_i is the light shift. Raman transitions to a final state change the atom’s momentum by two photon momenta from p_i to p_i±2h̄ k. The final energy E_f is then

The Raman transitions occur between two momentum states with average momentum p_avg, so we symmetrize these equations by defining p_avg=pi ±h̄ k. With this substitution, and setting E_i=E_f, we obtain

If Δ is much larger than the excited state hyperfine splittings, and the initial and final states are in the same hyperfine ground state, then E^LS_i≈E^LS_f. Then resonance occurs for atoms with v=v_avg±ν_r x̂ such that

From the atom center-of-mass frame, this is equivalent to the requirement that the Doppler shift offset the Zeeman shift.

To observe the velocity selection, the fluorescence from a ballistically expanding cloud of atoms is imaged onto a CCD camera from a direction orthogonal to the Raman beam axis. This process is shown schematically in Fig. 1(b). Midway through the expansion (T=T_r), the atoms are exposed to a 1 ms pulse from the Raman beam. The momentum of resonant atoms is altered by absorption of two photons, and because the image of the expanded cloud at T=T_i is a record of the average velocity distribution of the atoms, those within the narrow resonant velocity classes add distinct features to the images. In a uniform magnetic field, the resonant planes appear as vertical stripes through the expanded cloud. In a nonuniform magnetic field, the resonant planes can be distorted depending on the gradient, magnitude, and direction of the field. The characteristics of these distortions provide an intuitive, direct image of the magnetic field and are the subject of this paper.

2. Experiment

The layout of our apparatus is shown in Fig. 2. The experiment begins with a vapor cell magneto-optical trap (MOT) containing 10^{7 85}Rb atoms. The 1/e ² MOT radius is ≈250 µm and the temperature is ≈200 µK. Our Raman beam is spatially filtered by polarization maintaining (PM) fiber, and is collimated by a 60mm focal length gradient-index lens (1/e ² beam waist ω ₀=7.5 mm). The beam has up to 20 mW laser power, which is retroreflected in a lin ⊥ lin configuration. For B-field bias control, we use three orthogonal pairs of Helmholtz coils. The Raman beam travels horizontally along the axis of the x-directed coil pair. A benefit of this technique is that the repumper beam has a sufficiently large one-photon detuning to serve as the Raman beam, simplifying the implementation of this technique into existing cold atom setups.

At time T=0, the atoms are released from the MOT by extinguishing all laser beams and the MOT coils. The bias field coils remain on. We perform no molasses cooling, because the large velocity spread of the hotter sample of atoms provides greater range over which velocity selection can occur. At time T_r=20 ms, the Raman pulse is switched on for 1 ms and at T_i=40 ms, the MOT cooling and repump beams are switched on to image the expanded cloud onto the CCD camera. For uniform magnetic fields, the r.h.s. of Eq. 4 is constant, and the resonance condition reduces to the two planes at x=±ω_LT_r/2k, which is proportional to |B|. In Fig. 3, we show images taken as a function of bias current along the z-axis. The resonant planes show up as vertical stripes from this camera direction. In our system, the z-axis compensation current is 243 mA, at which setting the stripe separation is minimized. The coils produce a field of ≈1.5 G/A.

 

Fig. 2. Experimental setup for magnetic field imaging. A retroreflected Raman beam propagates along the x-axis to form a lin⊥lin field. A CCD along the y-axis images fluorescence. Gravity is out of the plane of the figure. Helmholtz bias coils are used along each axis.

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Fig. 3. a) Images of expanded atom cloud in the presence of a uniform magnetic field for different bias field values. dB_z/dI=1.5G/A and B_z=0 at I=243mA.

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Although one might expect five distinct stripes arising from Δm=0,±1,±2 transitions, this is not the case. If we choose the quantization axis along the B-field direction (ẑ), and the Raman beam polarizations are lin⊥lin (along ŷ and ẑ), then Δm=0,±2 transitions are forbidden. On the other hand, if the Raman polarizations are lin-lin (both along ẑ), or B is parallel to k̂, then only Δm=0,±2 would be allowed. In between, all Δm values are allowed, but detailed calculations show that Δm=±2 transitions are weak under all polarization configurations. In Fig. 3, we see the third stripe for v_avg=0 due to Δm=0 transitions, caused by slight deviations from the lin⊥lin polarization configuration.

In a nonuniform magnetic field, the resonance condition (Eq. 4) depends on all spatial coordinates. We first calculate the resonance condition for a linear quadrupole field, which is used in MOTs and is formed at the center of an anti-Helmholtz coil pair. Near the center of the coil pair, this field, B _AH(r) depends linearly on the spatial coordinates:

where B′ is the magnetic field gradient along the z-axis. The resonance condition for x-directed Raman beams at the Raman pulse time T_r is

 

Fig. 4. Top: Background-subtracted images along the y-axis for different magnetic field gradients. (a) Experiment (Media 1). (b) Simulations. Bottom: (c) Signal integrated along radial cuts through the experimental images as a function of angle from the vertical.

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where we have defined ω_L′≡g_Fµ_BB′/h̄. Equation 6 is satisfied by two conical surfaces whose axes are parallel to k (x-axis). The slopes in the x-z and x-y planes are

where we have defined the dimensionless quantity α≡ω′_LT_r/4k. For α≪1, these slopes correspond to an angle from the vertical that changes linearly with the gradient:

To demonstrate this scenario experimentally, we produce a test gradient by passing small current through the MOT coils. Each coil is a 5-cm-long solenoid with radius 3.7cm and 220 windings centered 5.2cm from the MOT, producing a gradient of ≈6.7G/cm per A. The results are shown in Fig. 4(a) for currents from 0–300mA. The images are taken along the y-axis so that the atom distribution is projected onto the x-z plane. For a linear quadrupole field, this projection of a conical surface causes the images to display a characteristic X-shape. For all images, there is a strong vertical stripe which corresponds to Δm=0 transitions. It is useful for identifying the origin along the x-axis.

We extract the slope of the conical shell in the x-z plane by integrating along radial slices, emanating from the intersection of the stripes, as a function of azimuthal angle. In principle, image processing techniques could be used to determine the intersection point, but to demonstrate the principle we have chosen this location manually. We plot these profiles as a function of angle from the vertical (θ_xz) in Fig. 4(c). Peaks in the analyzed data correspond to the edges of the cone where the integrated density is highest. The locations of these peaks are easily estimated visually to within ≈20mrad, or 30 mG/cm for this experiment.

 

Fig. 5. (a) Experimental images of the atom cloud for different values of the x-directed bias field, in the presence of a slight gradient (B′=0.5G/cm) (Media 2). B_x=0 at I=500mA, and dB_x/dI=-0.844G/A. (b) Simulations using the same conditions. The difference in visibility between simulation and experiment is explained in the text.

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Additionally, we perform simple Monte Carlo particle simulations of the experiment. The atoms are initially normally distributed in position and velocity to match the MOT size and temperature. They freely expand until T=T_r, when atoms satisfying the resonance condition have two photon momenta added. The particles then continue freely expanding until T=2T_r=T_i when the spatial positions are recorded. The simulations for this linear quadrupole case are shown in Fig. 4(b), using the above gradient estimate of 6.7G/cm per A. We have neglected forces on the atoms due to the magnetic field gradient, of magnitude mµ_BB′. These forces are dependent on the magnetic sublevel. Our sample is unpolarized, so this effect may blur the images slightly, but the average location of the features is unperturbed within the error of this technique.

Next, we consider the case in which a uniform bias field is added to the linear quadrupole field such that B=(B_x+B′x,B_y+B′y,B_z-2B′z). B_y and B_z only shift the origin by y_c=-B_y/B′ and z_c=B_z/2B′, but for nonzero B_x, the resonance condition becomes

where ω _L0x=g_Fµ_BB_x/h̄. Eq. (9) is satisfied by a hyperbolic surface with center located at

with m_xy and m_xz from Eq. 7 representing the asymptotic slopes. The major axis is

It is important to note that all formulas so far apply to the positions at T_r, not at the imaging time T_i. Distances measured on the camera will all be larger by the factor T_i/T_r, but estimates of the slopes, m_xy, m_xz, θ_xz, and θ_xy remain unchanged.

For the experimental demonstration, we add a bias field with a pair of Helmholtz coils along x. These 7-cm-diameter coils are 2.5 cm long and made of 171 windings separated by 28 cm, producing a field of -0.844 G/Ax̂. The results are shown in Fig. 5(a). For these cases, we chose a fixed gradient of 1.0G/cm in the MOT coils while varying B_x. Comparisons with simulation are shown. The images in Figs. 4–5 cover ≈10 mm, so the linear dimension over which the measurement is made, at T_r, is 5 mm. The spatial resolution is determined by the starting size of the atom cloud (250µm).

 

Fig. 6. Plot of the squares of the two sides of Eq. 9 showing the bands over which resonance occurs. For bias fields along k (x̂), the stripe features have asymmetric widths. The shaded parabola (square of l.h.s. of Eq. 9) is thickened because the resonance occurs over a small band of velocities.

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One striking difference between simulation and experiment is the visibility of the stripe features. The reason is that the simulations neglect Raman transition strengths, which vary as a function of angle between B and k. In fact, for B‖k, Δm=±1 transitions are forbidden. Therefore, along lines through the centers of the hyperbolas, where B is along the x-direction, the features are difficult to see. The features are much more apparent in general when B_x≪|B|. Detailed calculations of Raman transition strengths will be discussed elsewhere.

For nonzero B_x, the width and contrast of the left and right hyperbolic curves in Fig. 5 are unequal (e.g. at I=600 mA). For B_x>0 and B′>0, the feature width is larger for ν_x>0. This is understood by plotting the squares of the l.h.s. and r.h.s. of Eq. 9 in the presence of a bias field for y=z=0 (Fig. 6). The l.h.s of Eq. 9 is broadened because the resonance occurs between x=T_r(ν_avg ±ν_r). What remains are two offset parabolas, so that the curves intersect with different relative angle. Thus, a broader velocity class of atoms is resonant and the feature width is increased.

3. Feature visibility

Several geometric factors affect the visibility of the features in the images. We image the cloud with the MOT beams, which are large compared to the expanded atom cloud size, so the entire resonant shell of atoms is projected onto the camera plane. Contrast is only high for viewing along tangents to the shell, where the integrated atom density is largest. Thus, contrast is best for constant magnetic fields, when integration occurs through an entire plane of atoms, and decreases as the gradient increases. Contrast could be improved with thin sheet illumination.

The initialMOT size also controls the contrast of the stripe features and the spatial resolution. For a point trap, the fluorescence images provide a direct measure of the velocity spectrum, but in practice the image is a convolution of the initial MOT size and the velocity spectrum. For this reason, increasing T_r and T_i result in better contrast.

The MOT temperature changes the features in a few different ways. First, a hotter atom sample increases the range of measurable magnetic fields by increasing the range over which velocity selection can occur. Second, because the Raman pulse occurs after the atoms are released from the trap, a hotter sample is larger at T_r and therefore samples the magnetic field over a larger volume. This increased sampling range is compensated by the reduction in atom density during imaging. If we assume an initial Gaussian atom density with 1/e ² radius ρ₀, the radius at time T_r is ρ≈(ρ² ₀+T²_rk_BT/M)^1/2 where k_B is the Boltzmann constant. With typical values in our experiment of ρ₀=0.25mm, T_r=20ms, and T=100µK, the 1/e ² cloud radius is ρ≈2mm.

4. Conclusion

We have demonstrated a single-shot imaging magnetometer using two-photon velocity selection in cold atoms. Magnetic field variations manifest themselves as perturbations to the fluorescence images of expanded atom clouds. The technique is particularly suited to visual diagnostics of magnetic fields in cold atom experiments, and should work in any atomic species with hyperfine structure. This work was supported by the Office of Naval Research and the Defense Advanced Research Projects Agency.