## Nonlinear magneto-optical rotation in the presence of a radio-frequency field |

Optics Express, Vol. 18, Issue 25, pp. 25494-25508 (2010)

http://dx.doi.org/10.1364/OE.18.025494

Acrobat PDF (1383 KB)

### Abstract

We report measurements of nonlinear magneto-optical rotation (NMOR) for the D_{2} line of ^{87}Rb atoms in an antirelaxation-coated vapor cell in the presence of a radio-frequency (rf) field. The experimental NMOR signals as a function of rf field frequency for various rf field powers are compared to a theoretical model based on the density-matrix formalism. The comparison between experiment and theory enables understanding of the ground-state atomic spin polarization dynamics, illustrated using plots of the probability distribution of the atomic angular momentum.

© 2010 Optical Society of America

## 1. Introduction

1. A. Kastler, “Quelques suggestions concernant la production optique et la détection optique d’une inégalité de population des niveaux de quantifigation spatiale des atomes. Application à l’expérience de Stern et Gerlach et à la résonance magnétique,” J. Phys. Radium **11**, 255–265 (1950). [CrossRef]

4. A. Okunevich, “Laser pumping and magneto-optical rotation of the light polarization plane in a cell with an antirelaxation coating of the walls: I. Statement and solution of the problem,” Opt. Spectrosc. **97**, 834–841 (2004). [CrossRef]

5. D. F. Jackson Kimball, L. R. Jacome, S. Guttikonda, E. J. Bahr, and L. F. Chan, “Magnetometric sensitivity optimization for nonlinear optical rotation with frequency-modulated light: Rubidium D2 line,” J. Appl. Phys. **106**, 063113 (2009). [CrossRef]

7. D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. **74**, 1153 (2002). [CrossRef]

8. V. V. Yashchuk, D. Budker, W. Gawlik, D. F. Kimball, Y. P. Malakyan, and S. M. Rochester, “Selective addressing of high-rank atomic polarization moments,” Phys. Rev. Lett **90**, 253001 (2003). [CrossRef] [PubMed]

10. V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. Jackson Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker, “Electric-field-induced change of the alkali-metal vapor density in paraffin-coated cells,” Opt. Express **16**, 11423 (2008). [CrossRef] [PubMed]

11. S. Xu, S. M. Rochester, V. V. Yashchuk, M. H. Donaldson, and D. Budker, “Construction and applications of an atomic magnetic gradiometer based on nonlinear magneto-optical rotation,” Rev. Sci. Instrum. **77**, 083106 (2006). [CrossRef]

12. A. Garroway, M. Buess, J. Miller, B. Suits, A. Hibbs, G. Barrall, R. Matthews, and L. Burnett, “Remote sensing by nuclear quadrupole resonance,” IEEE Trans. Geosci. Remote Sens. **39**, 1108–1118 (2001). [CrossRef]

13. S. Xu, V. V. Yashchuk, M. H. Donaldson, S. M. Rochester, D. Budker, and A. Pines, “Magnetic resonance imaging with an optical atomic magnetometer,” Proceedings of the National Academy of Sciences **103**, 12668–71 (2006). [CrossRef]

14. R. Bradley, J. Clarke, D. Kinion, L. J. Rosenberg, K. Bibber, S. Matsuki, M. Muck, and P. Sikivie, “Microwave cavity searches for dark-matter axions,” Rev. Mod. Phys. **75**, 777 (2003). [CrossRef]

15. M. P. Ledbetter, V. M. Acosta, S. M. Rochester, D. Budker, S. Pustelny, and V. V. Yashchuk, “Detection of radio-frequency magnetic fields using nonlinear magneto-optical rotation,” Phys. Rev. A **75**, 023405 (2007). [CrossRef]

16. W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, “Quantum noise limited and entanglement-assisted magnetometry,” Phys. Rev. Lett. **104**, 133601 (2010). [CrossRef] [PubMed]

17. W. Chalupczak, P. Josephs-Franks, S. Pustelny, and W. Gawlik, “Optical–radio-frequency resonances free from power broadening,” Phys. Rev. A **81**, 013422 (2010). [CrossRef]

15. M. P. Ledbetter, V. M. Acosta, S. M. Rochester, D. Budker, S. Pustelny, and V. V. Yashchuk, “Detection of radio-frequency magnetic fields using nonlinear magneto-optical rotation,” Phys. Rev. A **75**, 023405 (2007). [CrossRef]

18. J. Brossel and F. Bitter, “A new “double resonance” method for investigating atomic energy levels. Application to Hg ^{3}*P*_{1}*,” Phys. Rev. **86**, 308–316 (1952). [CrossRef]

19. H. Friedmann and A. D. Wilson-Gordon, “Dispersion profiles of the absorptive response of a two-level system interacting with two intense fields,” Phys. Rev. A **36**, 1333–1341 (1987). [CrossRef] [PubMed]

20. S. Chakmakjian, K. Koch, and J. C. R. Stroud, “Observation of resonances at subharmonics of the rabi frequency in the saturated absorption of a 100% amplitude-modulated laser beam,” J. Opt. Soc. Am. B **5**, 2015–2020 (1988). [CrossRef]

21. G. Di Domenico, G. Bison, S. Groeger, P. Knowles, A. S. Pazgalev, M. Rebetez, H. Saudan, and A. Weis, “Experimental study of laser-detected magnetic resonance based on atomic alignment,” Phys. Rev. A **74**, 063415 (2006). [CrossRef]

22. A. Weis, G. Bison, and A. S. Pazgalev, “Theory of double resonance magnetometers based on atomic alignment,” Phys. Rev. A **74**, 033401 (2006). [CrossRef]

_{2}line of

^{87}Rb atoms contained in an paraffin-coated vapor cell, and optical rotation is detected. The signals are primarily due to interaction with the

*F*= 2 →

_{g}*F*= 1 transition (the subscripts

_{e}*g*and

*e*indicate the ground and excited states, respectively). At low rf powers, the signals agree with the theoretical model, of the same form as that obtained in Refs. [21

21. G. Di Domenico, G. Bison, S. Groeger, P. Knowles, A. S. Pazgalev, M. Rebetez, H. Saudan, and A. Weis, “Experimental study of laser-detected magnetic resonance based on atomic alignment,” Phys. Rev. A **74**, 063415 (2006). [CrossRef]

22. A. Weis, G. Bison, and A. S. Pazgalev, “Theory of double resonance magnetometers based on atomic alignment,” Phys. Rev. A **74**, 033401 (2006). [CrossRef]

23. S. M. Rochester and D. Budker, “Atomic polarization visualized,” Am. J. Phys. **69**, 450–4 (2001). [CrossRef]

_{rf}, relative to the ground-state atomic-polarization relaxation rate γ

*and the magnetic resonance frequency Ω*

_{t}*. This work is motivated by an ongoing project aimed at measuring collisional transfer of alignment in collisions between different ground-state alkali atoms [24].*

_{L}## 2. Description of the experiment and theory

### 2.1. Description of experiment

25. M. A. Bouchiat and J. Brossel, “Relaxation of optically pumped Rb atoms on paraffin-coated walls,” Phys. Rev. **147**, 41–54 (1966). [CrossRef]

26. D. Budker, V. Yashchuk, and M. Zolotorev, “Nonlinear magneto-optic effects with ultranarrow widths,” Phys. Rev. Lett. **81**, 5788–5791 (1998). [CrossRef]

^{10}atoms per cm

^{3}. The density was extracted from fitting a low-light power (≈ 8 μW) transmission spectrum for the Rb D

_{2}line to a calculated spectrum assuming linear absorption. Under these experimental conditions the longitudinal relaxation rate of ground-state alignment of the isotope

^{87}Rb was measured to be 31.7(7) s

^{−1}. The value represents the effective relaxation rate of alignment polarization due to diverse types of atomic collisions occurring in the vapor cell, e.g. electron-randomization collisions with the wall, uniform relaxation due to the reservoir effect and spin-exchange collisions among Rb atoms [27

27. M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D. Budker, E. B. Alexandrov, M. V. Balabas, and V. V. Yashchuk, “Relaxation of atomic polarization in paraffin-coated cesium vapor cells,” Phys. Rev. A **72**, 023401 (2005). [CrossRef]

^{6}[26

26. D. Budker, V. Yashchuk, and M. Zolotorev, “Nonlinear magneto-optic effects with ultranarrow widths,” Phys. Rev. Lett. **81**, 5788–5791 (1998). [CrossRef]

**B**

_{0}in the

**ẑ**direction (typical strength

*B*

_{0}= 0.8 mG) and an oscillating magnetic field

**B**

_{rf}=

*B*

_{rf}

**x̂**cos

*ω*

_{rf}

*t*along the

*x*-axis (typical amplitude up to

*B*

_{rf}= 0.25 mG) using the sine wave output of a lock-in amplifier.

*z*-axis, propagates through the vapor cell in the

**ŷ**direction. The beam is generated by a distributed feedback (DFB) laser, which is tuned near the

*F*= 2 →

_{g}*F*= 1 transition in the 780 nm D

_{e}_{2}line of

^{87}Rb, on the red-detuned slope of the Doppler-broadened absorption line, in order to maximize the optical rotation signal. The wavelength is locked using a dichroic atomic vapor laser lock [28

28. K. L. Corwin, Z.-T. Lu, C. F. Hand, R. J. Epstein, and C. E. Wieman, “Frequency-stabilized diode laser with the Zeeman shift in an atomic vapor,” Appl. Opt. **37**, 3295 (1998). [CrossRef]

29. V. Yashchuk, D. Budker, and J. Davis, “Laser frequency stabilization using linear magneto-optics,” Rev. Sci. Instr. **71**, 341 (2000). [CrossRef]

30. D. Budker, D. Kimball, S. Rochester, and V. Yashchuk, “Nonlinear magneto-optical rotation via alignment-to-orientation conversion,” Phys. Rev. Lett. **85**, 2088 (2000). [CrossRef] [PubMed]

### 2.2. Description of theory

*F*= 2 →

_{g}*F*= 1 transition of the D

_{e}_{2}line of

^{87}Rb, although there are also contributions from the

*F*= 2 →

_{g}*F*= 2 and

_{e}*F*= 2 →

_{g}*F*= 3 transitions. Theoretical modeling shows that signals produced on each of these transitions have similar line shapes, although the signal from a

_{e}*F*= 2 →

_{g}*F*= 3 transition is of the opposite sign. In fact, the effects that we describe here are present for any transition with ground-state angular momentum

_{e}*F*≥ 1. For the theoretical treatment presented here, we therefore consider the simplest case of a

_{g}*F*= 1 →

_{g}*F*= 0 transition, for which analytical solutions are readily obtained. The model for this system gives results in reasonable agreement with the experimental data, indicating that the effects are not strongly dependent on the angular momenta characterizing the transition.

_{e}**ẑ**-directed field

**B**

_{0}=

*B*

_{0}

**ẑ**, corresponding to the Larmor frequency Ω

*=*

_{L}*gμ*

_{B}B_{0}, where

*μ*is the Bohr magneton and

_{B}*g*is the Landé factor (we set

*h̄*= 1). Lin-early polarized light propagating in the

**ŷ**direction with polarization in the

**ẑ**direction optically pumps the system and creates an aligned state. An oscillating rf magnetic field is applied in the

**x̂**direction,

**B**

_{rf}=

*B*

_{rf}

**x̂**cos

*ω*

_{rf}

*t*, corresponding to the rf Rabi frequency Ω

_{rf}=

*gμ*

_{B}B_{rf}. The dependence on Ω

_{rf}of the rf line shape of the optical rotation signal is studied. An analytic solution can be obtained in case in which the rf power is low enough that the rf power-broadened line width is much smaller than Ω

*and*

_{L}*ω*

_{rf}. Numerical solutions are obtained in the general case.

*F*= 1,

_{g}*m*= 1〉, |

*F*= 1,

_{g}*m*= 0〉, |

*F*= 1,

_{g}*m*= −1〉, |

*F*= 0,

_{e}*m*= 0〉}, the total time-dependent Hamiltonian

*H*of the Doppler-free system under the optical rotating-wave approximation is

*is the Rabi frequency of the optical transition induced by the linearly polarized light and Δ =*

_{R}*ω*–

*ω*

_{0}is the optical detuning;

*ω*is the frequency of the light and

*ω*

_{0}is the frequency of the ground to excited state transition in the absence of a magnetic field. For the case in which the ground-state relaxation rate and Ω

_{rf}are both much smaller than Ω

*and*

_{L}*ω*

_{rf}, we can also perform the rotating-wave approximation on the rf field, in order to remove the Larmor-frequency time dependence from the Hamiltonian. In the rotating frame obtained using the unitary transformation the density-matrix evolution can be written in terms of an effective Hamiltonian

_{rf}=

*ω*

_{rf}– Ω

*is the rf detuning. The evolution of the density matrix ρ (normalized so that Tr*

_{L}*ρ*= 1) is described by the Liouville equation where [ ] denotes the commutator and { } the anticommutator. The relaxation of the system is given by the matrix where the excited state decays spontaneously with a rate Γ and the ground and excited states relax with a rate

*γ*due to the exit of atoms from the light beam. The matrix Λ describes re-population of the ground state due to atoms entering the beam and spontaneous decay from the upper state, and is given by where the Zeeman ground and excited sublevels are denoted as

_{t}*g*and

_{mg}*e*, and

_{me}*ρ*

_{e0e0}is the population in the excited state.

*U*

^{−1}, we transform back to the laboratory frame to find the time-dependent density matrix.

*ℓ*of the medium to be where

*λ*is the transition wavelength,

*N*is the atomic density, and

*ρ*

_{e0g−1}and

*ρ*

_{e0g1}are the optical coherences between the excited state and the ground

*g*

_{−1}and

*g*

_{1}Zeeman sublevels, respectively. The magnitude of the optical-rotation signal is given by

22. A. Weis, G. Bison, and A. S. Pazgalev, “Theory of double resonance magnetometers based on atomic alignment,” Phys. Rev. A **74**, 033401 (2006). [CrossRef]

_{rf}, we obtain

_{rf}centered at Δ

_{rf}= 0. To lowest order in Ω

_{rf}, they are proportional to the real part, imaginary part, and absolute value, respectively, of a complex Lorentzian. Additional features appear at higher orders, as discussed in the next section.

_{rf}becomes of the same order as or exceeds Ω

*, the rotating-wave approximation for the rf field is no longer valid. In this case we use the Hamiltonian*

_{L}*H*of Eq. (1) and proceed in the laboratory frame. The Liouville equation now has explicit time dependence, and the density matrix

*ρ*(

*t*) is a function of time. Considering

*ρ*(

*t*) as a vector of the density matrix elements, the Liouville equation can be written as where

*A*

_{1}and

*A*

_{2}are time-independent matrices and

*b*is a time-independent vector. Periodic solutions can be found by expanding the density matrix in a Fourier series in harmonics of the rf frequency, with constant coefficients

*a*, and substituting into Eq. (15). This gives or resulting in the recursion relation where δ is the Kronecker delta. Retaining a finite number of harmonics, this provides a linear system of time-independent equations that can be solved numerically for the Fourier coefficients. The observed optical rotation signals can then be found as before.

_{n}## 3. Discussion

_{rf}), the in-phase and quadrature components and the magnitude of the optical-rotation signal as a function of the rf frequency

*ω*

_{rf}are shown. The version of the theoretical treatment valid for arbitrary rf-field strength discussed in Sec. 2.2 is used to generate the theoretical predictions, although for the lowest rf power, the signal is well described by the lowest-order terms of the expansions (12)–(14). Three regimes in the dependence on Ω

_{rf}can be identified. At the lowest field strengths, Ω

_{rf}<

*γ*, the in-phase and quadrature resonances in rf frequency take the form of dispersive and absorptive Lorentzians of characteristic width

_{t}*γ*[Fig. 2(a)]. At intermediate field strengths,

_{t}*γ*< Ω

_{t}_{rf}< |Ω

*| (we assume*

_{L}*γ*≪ |

_{t}*Ω*|), the Lorentzians broaden and additional narrow features are seen at the center of the resonances [Fig. 2(b)], the result of polarization-averaging effects discussed below.

_{L}_{rf}> |Ω

*|, effects due to ac Zeeman shifts and far-off-resonant fields are predicted to become important. We did not perform measurements in this regime, but the beginning of these effects can be seen in the data [Fig. 2(c)]. The negative-frequency component of the rf field results in a resonance at*

_{L}*ω*

_{rf}= −Ω

*symmetric to the one at*

_{L}*ω*

_{rf}=

*Ω*. In Fig. 2(b) and 2(c), the off-resonant tail of this negative-frequency resonance produces an overall slope in the in-phase component of the positive-frequency signal. Calculations for values of

_{L}*ω*

_{rf}of the same order as Ω

*predict higher-order resonances at odd fractions (1/3, 1/5, etc.) of Ω*

_{L}*. When the experimental data presented in Fig. 2(c) are plotted on expanded scales, as in Fig. 3, a higher-order resonance can be observed at one third the frequency of the main resonance, in agreement with theoretical predictions.*

_{L}### 3.1. Low-field regime

_{rf}<

*γ*. This case was discussed in Ref. [15

_{t}15. M. P. Ledbetter, V. M. Acosta, S. M. Rochester, D. Budker, S. Pustelny, and V. V. Yashchuk, “Detection of radio-frequency magnetic fields using nonlinear magneto-optical rotation,” Phys. Rev. A **75**, 023405 (2007). [CrossRef]

*m*= 0 sublevel, leaving an incoherent mixture of atoms in the

_{g}*m*= ±1 sublevels. The atomic polarization can be illustrated using the angular-momentum-probability surface (AMPS) [23

_{g}23. S. M. Rochester and D. Budker, “Atomic polarization visualized,” Am. J. Phys. **69**, 450–4 (2001). [CrossRef]

31. M. Auzinsh, “Angular momenta dynamics in magnetic and electric field: Classical and quantum approach,” Can. J. Phys. **75**, 853–872 (1997). [CrossRef]

*z*-axis with a “peanut”-shaped probability distribution [Fig. 4(a)]. (All of the AMPS shown here are obtained directly from the density-matrix calculation. A quantity of the lowest-rank, isotropic polarization moment is subtracted from each figure so that the anisotropic polarization can be more clearly seen [32].) Because the

*m*= 0 sublevel has been depleted, the atomic medium transmits

_{g}*z*-polarized light, while tending to absorb orthogonally polarized light—i.e., the atoms function as a polarizing filter with transmission axis along the atomic alignment axis [33

33. S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hänsch, “Quantitative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping,” Phys. Rev. A **47**, 1220–6 (1993). [CrossRef] [PubMed]

**B**

_{0}and

**B**

_{rf}can be described in terms of fictitious static fields

**B**′

_{0}and

**B**′

_{rf}. These fields can be determined by examining the rotating-frame Hamiltonian (3). The bias field in the rotating frame,

**B**′

_{0}, points along

**B**

_{0}, while its field strength is such that it produces a Larmor frequency given by the detuning of the rf field from resonance: Ω′

_{rf}= −Δ

_{rf}+

*ω*

_{rf}+ Ω

*. The rf field*

_{L}**B**

_{rf}becomes a static field

**B**′

_{rf}in the

*xy*plane with associated Larmor frequency Ω′

_{rf}= Ω

_{rf}/2. The direction of this field in the

*xy*plane depends on the arbitrary phase chosen for the rotating frame; in our convention

**B**

_{rf}points along

**x̂**. Thus the resultant field

**B**′

_{tot}=

**B**′

_{0}+

**B**′

_{rf}lies in the

*xz*plane.

**B**′

_{tot}. For low field strengths, this evolution is Larmor precession, so that the polarization continues to correspond to alignment. Because all of the external fields are static in the rotating frame, the effect of relaxation leads to a steady state for the atomic polarization. First consider the case in which the rf field is exactly on resonance (Δ

_{rf}= 0). Then

**B**′

_{tot}is equal to

**B**′

_{rf}and points in the

**x̂**direction [Fig. 4(b)], and the atomic alignment precesses in the

*yz*plane. Under the assumption Ω

_{rf}≪

*γ*, the precession frequency is much less than the relaxation rate, so that each atom precesses through a small angle before relaxing. The rotating-frame steady-state ensemble polarization thus consists of alignment at a small angle to the

_{t}*z*-axis in the

*yz*plane [Fig. 4(b)].

*z*-axis [Fig. 4(c)]. At the instant that the alignment is in the

*yz*plane, it does not induce any polarization rotation in the

**ŷ**-propagating light field. On the other hand, whenever the alignment axis is tilted away from the initial light polarization axis (the

*z*-axis) in the plane transverse to the light propagation direction (the

*xz*plane), the atoms can induce optical rotation. Because of the precession of the alignment, the optical-rotation signal oscillates. The amplitude of the signal is determined by the amount of the alignment and the angle between the alignment axis and the

*z*-axis (zenith angle). The phase of the oscillating signal is determined by the angle of the alignment axis about the

*z*-axis in the rotating frame (azimuthal angle). The direction of the alignment axis in the rotating frame corresponds to its direction in the laboratory frame when the rf field is maximum. Only the component in the

*xz*plane will induce optical rotation in

*y*-propagating light, so an alignment axis in the

*xz*plane in the rotating frame produces a signal in phase with the rf field oscillation, while the component in the

*yz*plane produces a quadrature component in the signal. In the case of Fig. 4 the optical-rotation signal is entirely in the quadrature component.

*x*-axis and toward the

*z*-axis. Precession about this field then takes the alignment in the rotating frame out of the

*yz*plane [Fig. 5(a)]. This tends to reduce the angle that the steady-state alignment makes with the

*z*-axis, reducing the amplitude of the laboratory-frame optical rotation signal. On the other hand, because the alignment now has a component in the

*xy*plane in the rotating frame, the oscillating rotation signal gains an in-phase component. When the rf field is tuned far enough away from resonance so that |Δ

_{rf}| >

*γ*, the precession frequency in the rotating frame becomes large enough that the atoms undergo an entire precession cycle before relaxing. The ensemble polarization is then averaged about the direction of the total magnetic field [Fig. 5(b)]. Because

_{t}**B**′

_{tot}lies in the

*xz*plane, the signal is now predominately in phase with the rf field, and the quadrature component is strongly suppressed. As the detuning becomes large, the average polarization points more and more along the

*z*-axis, and the signal amplitude drops to zero.

### 3.2. Intermediate-field regime

*γ*< Ω

_{t}_{rf}< |Ω

*|, the rotating-frame precession frequency is high even at zero detuning. This causes averaging of the atomic polarization about the magnetic-field axis. For Δ*

_{L}_{rf}= 0 this is the

*x*-axis: polarization transverse to the

*x*-axis is averaged out. However, the polarization along the

*x*-axis is preserved, so that the

*x*-axis becomes the preferred axis for the polarization [Fig. 6(a)]. The “doughnut”-shaped probability distribution seen in Fig. 6(a) is obtained from the initially pumped “peanut”-shaped distribution [Fig. 4(a)] when copies of the peanut distribution rotated by arbitrary angles about the

*x*-axis are averaged together. Another way to explain the doughnut shape is to transform to the basis in which the quantization axis is along

**x̂**. In this basis, the excitation light is σ polarized, so that it pumps atoms out of the bright state consisting of a superposition of the

*m*= ±1 sublevels, and leaves them in the dark state made up of the opposite superposition, as well as in the

_{g}*m*= 0 sublevel. However, due to the precession induced by the

_{g}**x̂**directed magnetic field, atoms oscillate between the bright and dark superpositions, so that the pump light removes atoms from the

*m*= ±1 sublevels incoherently. The atoms are then left in the

_{g}*m*= 0 sublevel, i.e., the atoms have no angular-momentum projection on the

_{g}*x*-axis and are symmetric about the

*x*-axis, as seen in Fig. 6(a).

*x*-axis remains in the

*xy*plane as it precesses around the

*z*-axis in the laboratory frame and so does not induce any optical rotation. However, when the rf field is tuned slightly away from resonance,

**B**′

_{tot}points away from the

*x*-axis, and so also does the averaged atomic polarization [Fig. 6(b)]. The polarization then causes optical rotation. As the polarization in the rotating frame is in the

*xz*plane, the signal in the laboratory frame is in phase with the rf field. (The plots in Fig. 6 are shown for the instant at which the rotating frame coincides with the laboratory frame, so that the in-phase component of the optical rotation is shown.)

*F*→

_{g}*F*=

_{e}*F*transition the situation is reversed.) We can think of the doughnut shape as an unpolarized distribution (sphere) with a peanut shape removed. From this viewpoint, it is reasonable that this “negative polarization” produces rotation of the opposite sign. More concretely, we can note that, as described above, in a doughnut distribution atoms are concentrated in the

_{g}*m*= 0 sublevel with the quantization axis along the alignment axis. The state then preferentially absorbs light that is polarized along the alignment axis. In the analogy with a polarizing filter, the transmission axis of the doughnut-shaped probability distribution is transverse to, rather than along, the alignment axis. This explains the sign of the rotation shown in Fig. 6(b) when

_{g}**B**′

_{tot}points away from the

*x*-axis. (The transmission axis of each polarization state is marked with a line on the surfaces plotted in Fig. 6.)

**B**′

_{tot}points farther away from the

*x*-axis, bringing the averaged alignment axis with it. This larger angle produces a larger optical rotation signal. However, another trend eventually takes over: because of the shape of the initially pumped polarization distribution, the amount of polarization that lies along the magnetic-field direction decreases. As a result, the amount of averaged polarization is reduced, tending to reduce the signal. In order to analyze this, we can plot just the aligned part of the initially pumped density matrix (the rank κ = 2,

*q*= 0 polarization moment), neglecting the isotropic part that is included in Fig. 4(a). The surface corresponding to this moment is described by the spherical harmonic

*Y*

_{2,0}(θ, ϕ) ∝ 3cos

^{2}θ − 1, plotted in cross section in Fig. 7. Negative values of the function are indicated by dashed lines. There are maximum positive values along

*z*(θ= 0), and maximum negative values in the

*xy*plane (θ=

*π*/2). As θ moves away from either of these values, the magnitude of the polarization is reduced. At a particular angle

**B**′

_{tot}is at this angle to the

*z*-axis, the averaged polarization completely cancels, sending the optical rotation signal to zero [Fig. 6(c)]. This condition corresponds to the additional zero crossings seen in the in-phase component of Fig. 2(b) and 2(c) above and below the center of the resonance. The direction of the effective magnetic field in the rotating frame depends on the rf field strength and the detuning from resonance—as the field strength is increased, the detuning required to achieve the magic angle also increases.

**B**′

_{tot}and the averaged atomic alignment point more toward the

*z*-axis, reducing the optical-rotation signal.

*xz*plane. Figure 2 does display (strongly suppressed) quadrature signals, which are a remnant of the low-field regime.

### 3.3. High-field regime

_{rf}becomes of the same order as Ω

*or exceeds it, various higher-order effects appear in the data and the full theory that cannot be described under the rotating-wave approximation for the rf field. In particular, a resonance near Ω*

_{L}*/3 is seen, as shown in Fig. 3. This can be explained as due to ac Zeeman shifts, which produce evenly spaced sidebands that result in resonances at odd subharmonics of the lowest-order resonance. The additional features can also be interpreted as arising from higher-order resonances between the Larmor precession and the rf frequency, similar to those seen in nonlinear magneto-optical rotation with frequency-modulated light [34*

_{L}34. E. B. Alexandrov, M. Auzinsh, D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, “Dynamic effects in nonlinear magneto-optics of atoms and molecules: review,” J. Opt. Soc. Am. B **22**, 7 (2005). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | A. Kastler, “Quelques suggestions concernant la production optique et la détection optique d’une inégalité de population des niveaux de quantifigation spatiale des atomes. Application à l’expérience de Stern et Gerlach et à la résonance magnétique,” J. Phys. Radium |

2. | C. Cohen-Tannoudji, |

3. | T. G. Walker and W. Happer, “Spin-exchange optical pumping of noble-gas nuclei,” Rev. Mod. Phys. |

4. | A. Okunevich, “Laser pumping and magneto-optical rotation of the light polarization plane in a cell with an antirelaxation coating of the walls: I. Statement and solution of the problem,” Opt. Spectrosc. |

5. | D. F. Jackson Kimball, L. R. Jacome, S. Guttikonda, E. J. Bahr, and L. F. Chan, “Magnetometric sensitivity optimization for nonlinear optical rotation with frequency-modulated light: Rubidium D2 line,” J. Appl. Phys. |

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. | D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. |

8. | V. V. Yashchuk, D. Budker, W. Gawlik, D. F. Kimball, Y. P. Malakyan, and S. M. Rochester, “Selective addressing of high-rank atomic polarization moments,” Phys. Rev. Lett |

9. | S. Pustelny, D. F. J. Kimball, S. M. Rochester, V. V. Yashchuk, W. Gawlik, and D. Budker, “Pump-probe nonlinear magneto-optical rotation with frequency-modulated light,” Phys. Rev. A |

10. | V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. Jackson Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker, “Electric-field-induced change of the alkali-metal vapor density in paraffin-coated cells,” Opt. Express |

11. | S. Xu, S. M. Rochester, V. V. Yashchuk, M. H. Donaldson, and D. Budker, “Construction and applications of an atomic magnetic gradiometer based on nonlinear magneto-optical rotation,” Rev. Sci. Instrum. |

12. | A. Garroway, M. Buess, J. Miller, B. Suits, A. Hibbs, G. Barrall, R. Matthews, and L. Burnett, “Remote sensing by nuclear quadrupole resonance,” IEEE Trans. Geosci. Remote Sens. |

13. | S. Xu, V. V. Yashchuk, M. H. Donaldson, S. M. Rochester, D. Budker, and A. Pines, “Magnetic resonance imaging with an optical atomic magnetometer,” Proceedings of the National Academy of Sciences |

14. | R. Bradley, J. Clarke, D. Kinion, L. J. Rosenberg, K. Bibber, S. Matsuki, M. Muck, and P. Sikivie, “Microwave cavity searches for dark-matter axions,” Rev. Mod. Phys. |

15. | M. P. Ledbetter, V. M. Acosta, S. M. Rochester, D. Budker, S. Pustelny, and V. V. Yashchuk, “Detection of radio-frequency magnetic fields using nonlinear magneto-optical rotation,” Phys. Rev. A |

16. | W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, “Quantum noise limited and entanglement-assisted magnetometry,” Phys. Rev. Lett. |

17. | W. Chalupczak, P. Josephs-Franks, S. Pustelny, and W. Gawlik, “Optical–radio-frequency resonances free from power broadening,” Phys. Rev. A |

18. | J. Brossel and F. Bitter, “A new “double resonance” method for investigating atomic energy levels. Application to Hg |

19. | H. Friedmann and A. D. Wilson-Gordon, “Dispersion profiles of the absorptive response of a two-level system interacting with two intense fields,” Phys. Rev. A |

20. | S. Chakmakjian, K. Koch, and J. C. R. Stroud, “Observation of resonances at subharmonics of the rabi frequency in the saturated absorption of a 100% amplitude-modulated laser beam,” J. Opt. Soc. Am. B |

21. | G. Di Domenico, G. Bison, S. Groeger, P. Knowles, A. S. Pazgalev, M. Rebetez, H. Saudan, and A. Weis, “Experimental study of laser-detected magnetic resonance based on atomic alignment,” Phys. Rev. A |

22. | A. Weis, G. Bison, and A. S. Pazgalev, “Theory of double resonance magnetometers based on atomic alignment,” Phys. Rev. A |

23. | S. M. Rochester and D. Budker, “Atomic polarization visualized,” Am. J. Phys. |

24. | D. F. Jackson Kimball, O. Neitzke, E. J. Bahr, S. Guttikonda, S. M. Rochester, M. P. Ledbetter, I. Novikova, B. Coste, S. A. Rangwala, J. M. Higbie, A. I. Okunevich, V. V. Yashchuk, and D. Budker (in preparation). |

25. | M. A. Bouchiat and J. Brossel, “Relaxation of optically pumped Rb atoms on paraffin-coated walls,” Phys. Rev. |

26. | D. Budker, V. Yashchuk, and M. Zolotorev, “Nonlinear magneto-optic effects with ultranarrow widths,” Phys. Rev. Lett. |

27. | M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D. Budker, E. B. Alexandrov, M. V. Balabas, and V. V. Yashchuk, “Relaxation of atomic polarization in paraffin-coated cesium vapor cells,” Phys. Rev. A |

28. | K. L. Corwin, Z.-T. Lu, C. F. Hand, R. J. Epstein, and C. E. Wieman, “Frequency-stabilized diode laser with the Zeeman shift in an atomic vapor,” Appl. Opt. |

29. | V. Yashchuk, D. Budker, and J. Davis, “Laser frequency stabilization using linear magneto-optics,” Rev. Sci. Instr. |

30. | D. Budker, D. Kimball, S. Rochester, and V. Yashchuk, “Nonlinear magneto-optical rotation via alignment-to-orientation conversion,” Phys. Rev. Lett. |

31. | M. Auzinsh, “Angular momenta dynamics in magnetic and electric field: Classical and quantum approach,” Can. J. Phys. |

32. | M. Auzinsh, D. Budker, and S. M. Rochester, |

33. | S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hänsch, “Quantitative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping,” Phys. Rev. A |

34. | E. B. Alexandrov, M. Auzinsh, D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, “Dynamic effects in nonlinear magneto-optics of atoms and molecules: review,” J. Opt. Soc. Am. B |

**OCIS Codes**

(020.0020) Atomic and molecular physics : Atomic and molecular physics

(020.3690) Atomic and molecular physics : Line shapes and shifts

(020.7490) Atomic and molecular physics : Zeeman effect

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: August 18, 2010

Revised Manuscript: October 25, 2010

Manuscript Accepted: October 30, 2010

Published: November 22, 2010

**Citation**

T. Zigdon, A. D. Wilson-Gordon, S. Guttikonda, E. J. Bahr, O. Neitzke, S. M. Rochester, and D. Budker, "Nonlinear magneto-optical rotation in the presence of a radio-frequency field," Opt. Express **18**, 25494-25508 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-25494

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### References

- A. Kastler, "Quelques suggestions concernant la production optique et la d’etection optique dúne inégalité de population des niveaux de quantifigation spatiale des atomes. Application á l’expérience de Stern et Gerlach et á la résonance magnétique," J. Phys. Radium 11, 255-265 (1950). [CrossRef]
- C. Cohen-Tannoudji, Atoms in Electromagnetic Fields (World Scientific, Singapore, 1994).
- T. G. Walker, and W. Happer, "Spin-exchange optical pumping of noble-gas nuclei," Rev. Mod. Phys. 69, 629 (1997). [CrossRef]
- A. Okunevich, "Laser pumping and magneto-optical rotation of the light polarization plane in a cell with an antirelaxation coating of the walls: I. Statement and solution of the problem," Opt. Spectrosc. 97, 834-841 (2004). [CrossRef]
- D. F. Jackson Kimball, L. R. Jacome, S. Guttikonda, E. J. Bahr, and L. F. Chan, "Magnetometric sensitivity optimization for nonlinear optical rotation with frequency-modulated light: Rubidium D2 line," J. Appl. Phys. 106, 063113 (2009). [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, 043403 (2000). [CrossRef]
- D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153 (2002). [CrossRef]
- V. V. Yashchuk, D. Budker, W. Gawlik, D. F. Kimball, Y. P. Malakyan, and S. M. Rochester, "Selective addressing of high-rank atomic polarization moments," Phys. Rev. Lett. 90, 253001 (2003). [CrossRef] [PubMed]
- S. Pustelny, D. F. J. Kimball, S. M. Rochester, V. V. Yashchuk, W. Gawlik, and D. Budker, "Pump-probe nonlinear magneto-optical rotation with frequency-modulated light," Phys. Rev. A 73, 023817 (2006). [CrossRef]
- V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. Jackson Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker, "Electric-field-induced change of the alkali-metal vapor density in paraffin-coated cells," Opt. Express 16, 11423 (2008). [CrossRef] [PubMed]
- S. Xu, S. M. Rochester, V. V. Yashchuk, M. H. Donaldson, and D. Budker, "Construction and applications of an atomic magnetic gradiometer based on nonlinear magneto-optical rotation," Rev. Sci. Instrum. 77, 083106 (2006). [CrossRef]
- A. Garroway, M. Buess, J. Miller, B. Suits, A. Hibbs, G. Barrall, R. Matthews, and L. Burnett, "Remote sensing by nuclear quadrupole resonance," IEEE Trans. Geosci. Rem. Sens. 39, 1108-1118 (2001). [CrossRef]
- S. Xu, V. V. Yashchuk, M. H. Donaldson, S. M. Rochester, D. Budker, and A. Pines, "Magnetic resonance imaging with an optical atomic magnetometer," Proc. Natl. Acad. Sci. U.S.A. 103, 12668-12671 (2006). [CrossRef]
- R. Bradley, J. Clarke, D. Kinion, L. J. Rosenberg, K. Bibber, S. Matsuki, M. Muck, and P. Sikivie, "Microwave cavity searches for dark-matter axions," Rev. Mod. Phys. 75, 777 (2003). [CrossRef]
- M. P. Ledbetter, V. M. Acosta, S. M. Rochester, D. Budker, S. Pustelny, and V. V. Yashchuk, "Detection of radio-frequency magnetic fields using nonlinear magneto-optical rotation," Phys. Rev. A 75, 023405 (2007). [CrossRef]
- W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, "Quantum noise limited and entanglement-assisted magnetometry," Phys. Rev. Lett. 104, 133601 (2010). [CrossRef] [PubMed]
- W. Chalupczak, P. Josephs-Franks, S. Pustelny, and W. Gawlik, "Optical-radio-frequency resonances free from power broadening," Phys. Rev. A 81, 013422 (2010). [CrossRef]
- J. Brossel, and F. Bitter, "A new "double resonance" method for investigating atomic energy levels. Application to Hg 3P1*," Phys. Rev. 86, 308-316 (1952). [CrossRef]
- H. Friedmann, and A. D. Wilson-Gordon, "Dispersion profiles of the absorptive response of a two-level system interacting with two intense fields," Phys. Rev. A 36, 1333-1341 (1987). [CrossRef] [PubMed]
- S. Chakmakjian, K. Koch, and J. C. R. Stroud, "Observation of resonances at subharmonics of the rabi frequency in the saturated absorption of a 100%amplitude-modulated laser beam," J. Opt. Soc. Am. B 5, 2015-2020 (1988). [CrossRef]
- G. Di Domenico, G. Bison, S. Groeger, P. Knowles, A. S. Pazgalev, M. Rebetez, H. Saudan, and A. Weis, "Experimental study of laser-detected magnetic resonance based on atomic alignment," Phys. Rev. A 74, 063415 (2006). [CrossRef]
- A. Weis, G. Bison, and A. S. Pazgalev, "Theory of double resonance magnetometers based on atomic alignment," Phys. Rev. A 74, 033401 (2006). [CrossRef]
- S. M. Rochester, and D. Budker, "Atomic polarization visualized," Am. J. Phys. 69, 450-454 (2001). [CrossRef]
- D. F. Jackson Kimball, O. Neitzke, E. J. Bahr, S. Guttikonda, S. M. Rochester, M. P. Ledbetter, I. Novikova, B. Coste, S. A. Rangwala, J. M. Higbie, A. I. Okunevich, V. V. Yashchuk, and D. Budker (in preparation).
- M. A. Bouchiat, and J. Brossel, "Relaxation of optically pumped Rb atoms on paraffin-coated walls," Phys. Rev. 147, 41-54 (1966). [CrossRef]
- D. Budker, V. Yashchuk, and M. Zolotorev, "Nonlinear magneto-optic effects with ultranarrow widths," Phys. Rev. Lett. 81, 5788-5791 (1998). [CrossRef]
- M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D. Budker, E. B. Alexandrov, M. V. Balabas, and V. V. Yashchuk, "Relaxation of atomic polarization in paraffin-coated cesium vapor cells," Phys. Rev. A 72, 023401 (2005). [CrossRef]
- K. L. Corwin, Z.-T. Lu, C. F. Hand, R. J. Epstein, and C. E. Wieman, "Frequency-stabilized diode laser with the Zeeman shift in an atomic vapor," Appl. Opt. 37, 3295 (1998). [CrossRef]
- V. Yashchuk, D. Budker, and J. Davis, "Laser frequency stabilization using linear magneto-optics," Rev. Sci. Instrum. 71, 341 (2000). [CrossRef]
- D. Budker, D. Kimball, S. Rochester, and V. Yashchuk, "Nonlinear magneto-optical rotation via alignment-to-orientation conversion," Phys. Rev. Lett. 85, 2088 (2000). [CrossRef] [PubMed]
- M. Auzinsh, "Angular momenta dynamics in magnetic and electric field: Classical and quantum approach," Can. J. Phys. 75, 853-872 (1997). [CrossRef]
- M. Auzinsh, D. Budker, and S. M. Rochester, Optically Polarized Atoms: Understanding Light-Atom Interactions (Oxford University Press, Oxford, 2010).
- S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hänsch, "Quantitative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping," Phys. Rev. A 47, 1220-1226 (1993). [CrossRef] [PubMed]
- E. B. Alexandrov, M. Auzinsh, D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, "Dynamic effects in nonlinear magneto-optics of atoms and molecules: review," J. Opt. Soc. Am. B 22, 7 (2005). [CrossRef]

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