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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 17 — Aug. 18, 2008
  • pp: 13062–13069
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A single-shot imaging magnetometer using cold atoms

Matthew L. Terraciano, Mark Bashkansky, and Fredrik K. Fatemi  »View Author Affiliations


Optics Express, Vol. 16, Issue 17, pp. 13062-13069 (2008)
http://dx.doi.org/10.1364/OE.16.013062


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Abstract

We demonstrate a technique for imaging magnetic fields using velocity-selective two-photon resonances in a cold atom cloud. Freely expanding 85Rb atoms released from a magneto-optical trap are exposed to a brief (≈1 ms), off-resonant, retro-reflected laser pulse in a lin-perp-lin configuration. Two-photon resonance between magnetic sublevels occurs only for atoms in narrow velocity classes dependent on the magnetic field strength. The momentum of resonant atoms is altered by the pulse, and this two-photon momentum change is easily visible after further ballistic expansion. When the momentum pulse is applied to an atom cloud with finite size, magnetic field variations across the sample result in position-dependent features in images of the expanded cloud. We demonstrate the technique by imaging magnetic field variations over ≈5 mm with ≈250µm spatial resolution.

© 2008 Optical Society of America

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=vavg±νr (with vavg 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 vavg, leading to increased fluorescence at specific locations.

1. Introduction

An atom in a counterpropagating laser field can undergo a two-photon Raman transition only within a narrow velocity band [1

1. M. Kasevich, D. S. Weiss, E. Riis, K. Moler, S. Kasapi, and S. Chu, “Atomic velocity selection using stimulated Raman transitions,” Phys. Rev. Lett. 66(18), 2297–2300 (1991). [CrossRef] [PubMed]

]. This atomic velocity selection is useful for a variety of experiments in laser cooling, including subrecoil Raman cooling [2

2. V. Boyer, L. J. Lising, S. L. Rolston, and W. D. Phillips, “Deeply subrecoil two-dimensional Raman cooling,” Phys. Rev. A 70(4), 043405 (pages 8) (2004). [CrossRef]

], atom interferometry [3

3. J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, “Sensitive absolute-gravity gradiometry using atom interferometry,” Phys. Rev. A 65(3), 033608 (2002). [CrossRef]

], and atom velocimetry [4

4. J. Chabé, H. Lignier, P. Szriftgiser, and J. C. Garreau, “Improving Raman velocimetry of laser-cooled cesium atoms by spin-polarization,” Opt. Commun. 274, 254–259 (2007). [CrossRef]

]. 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

5. D. D. McGregor, “High-sensitivity helium resonance magnetometers,” Review of Scientific Instruments 58(6), 1067–1076 (1987). URL http://link.aip.org/link/?RSI/58/1067/1. [CrossRef]

, 6

6. D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, “Sensitive magnetometry based on nonlinear magneto-optical rotation,” Phys. Rev. A 62(4), 043403 (2000). [CrossRef]

, 7

7. T. Isayama, Y. Takahashi, N. Tanaka, K. Toyoda, K. Ishikawa, and T. Yabuzaki, “Observation of Larmor spin precession of laser-cooled Rb atoms via paramagnetic Faraday rotation,” Phys. Rev. A 59(6), 4836–4839 (1999). [CrossRef]

], 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

7. T. Isayama, Y. Takahashi, N. Tanaka, K. Toyoda, K. Ishikawa, and T. Yabuzaki, “Observation of Larmor spin precession of laser-cooled Rb atoms via paramagnetic Faraday rotation,” Phys. Rev. A 59(6), 4836–4839 (1999). [CrossRef]

, 8

8. M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, “Faraday spectroscopy in a dark optical trap,” Phys. Rev. A 77, 063417 (2008). [CrossRef]

]. 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

9. M. Vengalattore, J. M. Higbie, S. R. Leslie, J. Guzman, L. E. Sadler, and D. M. Stamper-Kurn, “High-Resolution Magnetometry with a Spinor Bose-Einstein Condensate,” Phys. Rev. Lett. 98, 200801 (2007). [CrossRef] [PubMed]

].

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

10. M. L. Terraciano, S. E. Olson, M. Bashkansky, Z. Dutton, and F. K. Fatemi, “Magnetically controlled velocity selection in a cold-atom sample using stimulated Raman transitions,” Phys. Rev. A 76, 053421 (2007). [CrossRef]

], 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

10. M. L. Terraciano, S. E. Olson, M. Bashkansky, Z. Dutton, and F. K. Fatemi, “Magnetically controlled velocity selection in a cold-atom sample using stimulated Raman transitions,” Phys. Rev. A 76, 053421 (2007). [CrossRef]

] 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 mf-mim=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 ±2 k=±2r , where νr 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, h̄ωL=gFµBB, where gF is the gyromagnetic ratio and µB is the Bohr magneton. For 85Rb, gFµB/=466.74 kHz/Gauss [11

11. E. B. Alexandrov, M. V. Balabas, A. K. Vershovski, and A. S. Pazgalev, “Experimental demonstration of the sensitivity of an optically pumped quantum magnetometer,” Technical Physics 49, 779–783 (2004). [CrossRef]

]. We analyze the resonance condition in the lab frame as follows. An atom has initial energy

Ei=pi22M+mih¯ωL+EiLS
(1)

where ELSi is the light shift. Raman transitions to a final state change the atom’s momentum by two photon momenta from pi to pi±2 k. The final energy Ef is then

Ef=(pi±2h¯k)22M+mfh¯ωL+EfLS
(2)

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

(pavgh¯k)22M+mih¯ωL+EiLS=(pavg±h¯k)22M+mfh¯ωL+EfLS
(3)

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 ELSiELSf. Then resonance occurs for atoms with v=vavg±νr such that

2k·vavg=±ΔmωL
(4)

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=Tr), 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=Ti 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 107 85Rb 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 Tr=20 ms, the Raman pulse is switched on for 1 ms and at Ti=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 xωLTr/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.
Fig. 3. a) Images of expanded atom cloud in the presence of a uniform magnetic field for different bias field values. dBz/dI=1.5G/A and Bz=0 at I=243mA.

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 (), and the Raman beam polarizations are lin⊥lin (along ŷ and ), then Δm=0,±2 transitions are forbidden. On the other hand, if the Raman polarizations are lin-lin (both along ), or B is parallel to , 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 vavg=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:

BAH(r)=0.5B(xx̂+yŷ2zẑ)
(5)

where B′ is the magnetic field gradient along the z-axis. The resonance condition for x-directed Raman beams at the Raman pulse time Tr 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.
2kvx=2kxTr=±(0.5ωLx)2+(0.5ωLy)2+(ωLz)2
(6)

where we have defined ωL′≡gFµBB′/. 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

mxz=mxy2=±1α22α
(7)

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

θxz=2θxy=±2α=±ωLTr2k
(8)

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 (θxz) 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). Bx=0 at I=500mA, and dBx/dI=-0.844G/A. (b) Simulations using the same conditions. The difference in visibility between simulation and experiment is explained in the text.

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=Tr, when atoms satisfying the resonance condition have two photon momenta added. The particles then continue freely expanding until T=2Tr=Ti 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 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=(Bx+Bx,By+By,Bz-2Bz). By and Bz only shift the origin by yc=-By/B′ and zc=Bz/2B′, but for nonzero Bx, the resonance condition becomes

2kvx=2kxTr=(ωL0x+0.5ωLx)2+(0.5ωLy)2+(ωLz)2
(9)

where ω L0x=gFµBBx/. Eq. (9) is satisfied by a hyperbolic surface with center located at

xc=2ωL0xωLα21α2
(10)

with mxy and mxz from Eq. 7 representing the asymptotic slopes. The major axis is

D=2ωL0xωLα1α2=xcα
(11)

It is important to note that all formulas so far apply to the positions at Tr, not at the imaging time Ti. Distances measured on the camera will all be larger by the factor Ti/Tr, but estimates of the slopes, mxy, mxz, θxz, and θxy 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/A. 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 Bx. 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 Tr, 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 (), 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.

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 Bk, 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 Bx≪|B|. Detailed calculations of Raman transition strengths will be discussed elsewhere.

For nonzero Bx, the width and contrast of the left and right hyperbolic curves in Fig. 5 are unequal (e.g. at I=600 mA). For Bx>0 and B′>0, the feature width is larger for νx>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=Tr(νavg ±νr). 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 Tr and Ti 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 Tr 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 Tr is ρ≈(ρ2 0+T2rkBT/M)1/2 where kB is the Boltzmann constant. With typical values in our experiment of ρ0=0.25mm, Tr=20ms, and T=100µK, the 1/e 2 cloud radius is ρ≈2mm.

4. Conclusion

References and links

1.

M. Kasevich, D. S. Weiss, E. Riis, K. Moler, S. Kasapi, and S. Chu, “Atomic velocity selection using stimulated Raman transitions,” Phys. Rev. Lett. 66(18), 2297–2300 (1991). [CrossRef] [PubMed]

2.

V. Boyer, L. J. Lising, S. L. Rolston, and W. D. Phillips, “Deeply subrecoil two-dimensional Raman cooling,” Phys. Rev. A 70(4), 043405 (pages 8) (2004). [CrossRef]

3.

J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, “Sensitive absolute-gravity gradiometry using atom interferometry,” Phys. Rev. A 65(3), 033608 (2002). [CrossRef]

4.

J. Chabé, H. Lignier, P. Szriftgiser, and J. C. Garreau, “Improving Raman velocimetry of laser-cooled cesium atoms by spin-polarization,” Opt. Commun. 274, 254–259 (2007). [CrossRef]

5.

D. D. McGregor, “High-sensitivity helium resonance magnetometers,” Review of Scientific Instruments 58(6), 1067–1076 (1987). URL http://link.aip.org/link/?RSI/58/1067/1. [CrossRef]

6.

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, “Sensitive magnetometry based on nonlinear magneto-optical rotation,” Phys. Rev. A 62(4), 043403 (2000). [CrossRef]

7.

T. Isayama, Y. Takahashi, N. Tanaka, K. Toyoda, K. Ishikawa, and T. Yabuzaki, “Observation of Larmor spin precession of laser-cooled Rb atoms via paramagnetic Faraday rotation,” Phys. Rev. A 59(6), 4836–4839 (1999). [CrossRef]

8.

M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, “Faraday spectroscopy in a dark optical trap,” Phys. Rev. A 77, 063417 (2008). [CrossRef]

9.

M. Vengalattore, J. M. Higbie, S. R. Leslie, J. Guzman, L. E. Sadler, and D. M. Stamper-Kurn, “High-Resolution Magnetometry with a Spinor Bose-Einstein Condensate,” Phys. Rev. Lett. 98, 200801 (2007). [CrossRef] [PubMed]

10.

M. L. Terraciano, S. E. Olson, M. Bashkansky, Z. Dutton, and F. K. Fatemi, “Magnetically controlled velocity selection in a cold-atom sample using stimulated Raman transitions,” Phys. Rev. A 76, 053421 (2007). [CrossRef]

11.

E. B. Alexandrov, M. V. Balabas, A. K. Vershovski, and A. S. Pazgalev, “Experimental demonstration of the sensitivity of an optically pumped quantum magnetometer,” Technical Physics 49, 779–783 (2004). [CrossRef]

OCIS Codes
(020.7490) Atomic and molecular physics : Zeeman effect
(020.3320) Atomic and molecular physics : Laser cooling

ToC Category:
Atomic and Molecular Physics

History
Original Manuscript: June 30, 2008
Revised Manuscript: July 31, 2008
Manuscript Accepted: August 4, 2008
Published: August 11, 2008

Citation
Matthew L. Terraciano, Mark Bashkansky, and Fredrik K. Fatemi, "A single-shot imaging magnetometer using cold atoms," Opt. Express 16, 13062-13069 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13062


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References

  1. M. Kasevich, D. S. Weiss, E. Riis, K. Moler, S. Kasapi, and S. Chu, "Atomic velocity selection using stimulated Raman transitions," Phys. Rev. Lett. 66(18), 2297-2300 (1991). [CrossRef] [PubMed]
  2. V. Boyer, L. J. Lising, S. L. Rolston, and W. D. Phillips, "Deeply subrecoil two-dimensional Raman cooling," Phys. Rev. A 70(4), 043405 (pages 8) (2004). [CrossRef]
  3. J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, "Sensitive absolute-gravity gradiometry using atom interferometry," Phys. Rev. A 65(3), 033608 (2002). [CrossRef]
  4. J. Chab??e, H. Lignier, P. Szriftgiser, and J. C. Garreau, "Improving Raman velocimetry of laser-cooled cesium atoms by spin-polarization," Opt. Commun. 274, 254-259 (2007). [CrossRef]
  5. D. D. McGregor, "High-sensitivity helium resonance magnetometers," Review of Scientific Instruments 58(6), 1067-1076 (1987). URL http://link.aip.org/link/?RSI/58/1067/1. [CrossRef]
  6. D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62(4), 043403 (2000). [CrossRef]
  7. T. Isayama, Y. Takahashi, N. Tanaka, K. Toyoda, K. Ishikawa, and T. Yabuzaki, "Observation of Larmor spin precession of laser-cooled Rb atoms via paramagnetic Faraday rotation," Phys. Rev. A 59(6), 4836-4839 (1999). [CrossRef]
  8. M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, "Faraday spectroscopy in a dark optical trap," Phys. Rev. A 77, 063417 (2008). [CrossRef]
  9. M. Vengalattore, J. M. Higbie, S. R. Leslie, J. Guzman, L. E. Sadler, and D. M. Stamper-Kurn, "High-Resolution Magnetometry with a Spinor Bose-Einstein Condensate," Phys. Rev. Lett. 98, 200801 (2007). [CrossRef] [PubMed]
  10. M. L. Terraciano, S. E. Olson, M. Bashkansky, Z. Dutton, and F. K. Fatemi, "Magnetically controlled velocity selection in a cold-atom sample using stimulated Raman transitions," Phys. Rev. A 76, 053421 (2007). [CrossRef]
  11. E. B. Alexandrov, M. V. Balabas, A. K. Vershovski, and A. S. Pazgalev, "Experimental demonstration of the sensitivity of an optically pumped quantum magnetometer," Technical Physics 49, 779-783 (2004). [CrossRef]

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