## A single-shot imaging magnetometer using cold atoms

Optics Express, Vol. 16, Issue 17, pp. 13062-13069 (2008)

http://dx.doi.org/10.1364/OE.16.013062

Acrobat PDF (1265 KB)

### Abstract

We demonstrate a technique for imaging magnetic fields using velocity-selective two-photon resonances in a cold atom cloud. Freely expanding ^{85}Rb 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 ≈250*µ*m spatial resolution.

© 2008 Optical Society of America

## 1. Introduction

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]

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]

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]

**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*=Δ

_{i}*m*=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

*h̄*

**k**=±2

*Mν*

_{r}**x̂**, where

*ν*is the recoil velocity and

_{r}*M*is the mass.

*δ*=0, which depends mainly on the magnetic field and momentum of the atom. In a small magnetic field,

*h̄ω*=

_{L}*g*, where

_{F}µ_{B}B*g*is the gyromagnetic ratio and

_{F}*µ*is the Bohr magneton. For

_{B}^{85}Rb,

*g*/

_{F}µ_{B}*h̄*=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]

*E*is the light shift. Raman transitions to a final state change the atom’s momentum by two photon momenta from

^{LS}_{i}**p**to

_{i}**p**±2

_{i}*h̄*

**k**. The final energy

*E*is then

_{f}**p**, so we symmetrize these equations by defining

_{avg}**p**=

_{avg}**pi**±

*h̄*

**k**. With this substitution, and setting

*E*=

_{i}*E*, we obtain

_{f}*E*≈

^{LS}_{i}*E*. Then resonance occurs for atoms with

^{LS}_{f}**v**=

**v**±

_{avg}*ν*

_{r}**x̂**such that

*T*=

*T*), 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

_{r}*T*=

*T*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.

_{i}## 2. Experiment

^{7}

^{85}Rb atoms. The 1/

*e*

^{2}MOT radius is ≈250

*µ*m and the temperature is ≈200

*µ*K. 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.

*T*=20 ms, the Raman pulse is switched on for 1 ms and at

_{r}*T*=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

_{i}*x*=±

*ω*/2

_{L}T_{r}*k*, 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.

*m*=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 Δ

*m*=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 Δ

*m*=0,±2 would be allowed. In between, all Δ

*m*values are allowed, but detailed calculations show that Δ

*m*=±2 transitions are weak under all polarization configurations. In Fig. 3, we see the third stripe for

**v**=0 due to Δ

_{avg}*m*=0 transitions, caused by slight deviations from the lin⊥lin polarization configuration.

**B**

*(*

_{AH}**r**) depends linearly on the spatial coordinates:

*B*′ is the magnetic field gradient along the

*z*-axis. The resonance condition for

*x*-directed Raman beams at the Raman pulse time

*T*is

_{r}*ω*′≡

_{L}*g*′/

_{F}µ_{B}B*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

*α*≡

*ω*′

*/4*

_{L}T_{r}*k*. For

*α*≪1, these slopes correspond to an angle from the vertical that changes linearly with the gradient:

*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 Δ

*m*=0 transitions. It is useful for identifying the origin along the

*x*-axis.

*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*) 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.

_{xz}*T*=

*T*, when atoms satisfying the resonance condition have two photon momenta added. The particles then continue freely expanding until

_{r}*T*=2

*T*=

_{r}*T*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

_{i}*mµ*′. 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.

_{B}B**B**=(

*B*+

_{x}*B*′

*x*,

*B*+

_{y}*B*′

*y*,

*B*-2

_{z}*B*′

*z*).

*B*and

_{y}*B*only shift the origin by

_{z}*y*=-

_{c}*B*/

_{y}*B*′ and

*z*=

_{c}*B*/2

_{z}*B*′, but for nonzero

*B*, the resonance condition becomes

_{x}*ω*

_{L0x}=

*g*/

_{F}µ_{B}B_{x}*h̄*. Eq. (9) is satisfied by a hyperbolic surface with center located at

*T*, not at the imaging time

_{r}*T*. Distances measured on the camera will all be larger by the factor

_{i}*T*/

_{i}*T*, but estimates of the slopes,

_{r}*m*,

_{xy}*m*,

_{xz}*θ*, and

_{xz}*θ*remain unchanged.

_{xy}*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

*x̂*. 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*. 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

_{x}*T*, is 5 mm. The spatial resolution is determined by the starting size of the atom cloud (250

_{r}*µ*m).

**B**and

**k**. In fact, for

**B**‖

**k**, Δ

*m*=±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.

*B*, the width and contrast of the left and right hyperbolic curves in Fig. 5 are unequal (e.g. at I=600 mA). For

_{x}*B*>0 and

_{x}*B*′>0, the feature width is larger for

*ν*>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}*x*=

*T*(

_{r}*ν*±

_{avg}*ν*). 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.

_{r}## 3. Feature visibility

*T*and

_{r}*T*result in better contrast.

_{i}*T*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/

_{r}*e*

^{2}radius ρ

_{0}, the radius at time

*T*is ρ≈(ρ

_{r}^{2}

_{0}+

*T*/

^{2}_{r}k_{B}T*M*)

^{1/2}where

*k*is the Boltzmann constant. With typical values in our experiment of ρ

_{B}_{0}=0.25mm,

*T*=20ms, and

_{r}*T*=100

*µ*K, 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. |

2. | V. Boyer, L. J. Lising, S. L. Rolston, and W. D. Phillips, “Deeply subrecoil two-dimensional Raman cooling,” Phys. Rev. A |

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 |

4. | J. Chabé, H. Lignier, P. Szriftgiser, and J. C. Garreau, “Improving Raman velocimetry of laser-cooled cesium atoms by spin-polarization,” Opt. Commun. |

5. | D. D. McGregor, “High-sensitivity helium resonance magnetometers,” Review of Scientific Instruments |

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 |

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 |

8. | M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, “Faraday spectroscopy in a dark optical trap,” Phys. Rev. A |

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. |

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 |

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 |

**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

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