1. Introduction

The dynamics of atomic spin polarization in the presence of radio-frequency (rf) fields has been extensively studied (for example, see [1–4]). Recently, it has been demonstrated that nonlinear magneto-optical rotation (NMOR) is a sensitive probe of atomic spin dynamics [5–7], and can in fact be used to selectively create and probe different multipole moments of the atomic spin polarization [8–10]. There are numerous practical applications for the use of NMOR to detect the response of atomic spins to rf fields: for example, in nuclear magnetic resonance (NMR) [11], including nuclear quadruple resonance (NQR) [12], and magnetic resonance imaging (MRI) [13] experiments. One can also use such methods in tests of fundamental physics [14]. In our previous research, we built an alkali-vapor magnetometer for the detection of rf fields [15]. Recent work by researchers using a similar experimental setup is described in Refs. [16, 17]. The experiment [15] demonstrated a sensitivity to oscillating magnetic fields of $100\hspace{0.17em}\text{pG}/\sqrt{\text{Hz}}$. The line shapes observed in the magnetometry experiment in the limit of low rf power are well understood. However, when higher-strength rf fields are applied, nontrivial line shapes are seen, with similarities to those of the of the “Majorana-Brossel effect” in double-resonance spectroscopy [18]. Such signals have been subsequently studied in work such as that of Refs. [19, 20] and references therein. We have conducted experimental and theoretical investigations to explain the mechanism that produces these line shapes.

Experimental and theoretical analysis of double-resonance spectroscopy measured in transmission with a cesium paraffin-coated vapor cell was carried out in Refs. [21,22]. Good agreement was obtained between experimental data and the theoretical model.

This work employs the D₂ line of ⁸⁷Rb atoms contained in an paraffin-coated vapor cell, and optical rotation is detected. The signals are primarily due to interaction with the F_g = 2 → F_e = 1 transition (the subscripts 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, 22]. At higher rf powers, however, additional effects are observed which are described by a more general numerical model. We also present a qualitative explanation for the line shapes in terms of the underlying atomic spin dynamics of the system. An atomic polarization visualization technique [23] is used to illustrate the mechanisms and their relationship to the detected NMOR signals. The character of the observed line shapes is found to be different depending on the value of the Rabi frequency for the rf field, Ω_rf, relative to the ground-state atomic-polarization relaxation rate γ_t and the magnetic resonance frequency Ω_L. This work is motivated by an ongoing project aimed at measuring collisional transfer of alignment in collisions between different ground-state alkali atoms [24].

2. Description of the experiment and theory

2.1. Description of experiment

The experiment employs a spherical paraffin-coated glass vapor cell (diameter = 10 cm) filled with a natural isotopic mixture of rubidium. The cell coating allows polarization of ground-state alkali atoms to survive several thousand wall collisions [25, 26], thereby extending the lifetime of atomic polarization. The cell is heated by a constant air stream from a heat exchanger. During all measurements in this work the temperature of the cell is stabilized at ≈ 41°C, resulting in a Rb vapor density of 7.2(7) × 10¹⁰ atoms per cm³. The density was extracted from fitting a low-light power (≈ 8 μW) transmission spectrum for the Rb D₂ line to a calculated spectrum assuming linear absorption. Under these experimental conditions the longitudinal relaxation rate of ground-state alignment of the isotope ⁸⁷Rb 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]. A four-layer μ-metal magnetic shield (see Fig. 1) surrounds the vapor cell and reduces the external magnetic field by a factor of ∼10⁶ [26]. Additionally, three orthogonal solenoidal coils placed inside the innermost shield are used to compensate residual magnetic fields to below 1 μG and to create additional static and oscillating fields. For our measurements we apply a static magnetic field B₀ in the ẑ direction (typical strength B₀ = 0.8 mG) and an oscillating magnetic field B_rf = B_rfx̂cosω_rft along the x-axis (typical amplitude up to B_rf = 0.25 mG) using the sine wave output of a lock-in amplifier.

 

Fig. 1 A laser beam linearly polarized along the z axis propagates through a paraffin-coated vapor cell, producing ground-state alignment in the ⁸⁷Rb atoms. A constant magnetic field B₀ and an oscillating rf magnetic field B_rf are applied within the magnetic shield that surrounds the cell. A balanced polarimeter and lock-in amplifier are used to analyze polarization rotation of the light induced by the atoms. A resonance is observed when the rf frequency is equal to the ground-state Larmor frequency of the atoms.

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A laser beam, initially polarized along the 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_g = 2 → F_e = 1 transition in the 780 nm D₂ line of ⁸⁷Rb, 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, 29]. The light intensity is 10 μW throughout the measurements and the beam diameter is ≈ 2 mm. Linear dichroism of the atomic medium induces changes in the light polarization. (For high light power, the medium can also acquire circular birefringence [30].) After transmission through the vapor cell the polarization of the light beam is analyzed using a balanced polarimeter setup, consisting of a polarizing beam splitter (Rochon crystal) and two photodiodes detecting the intensities of the two beams exiting the crystal. The component of the difference signal that oscillates at the frequency of the rf field is then extracted by the lock-in amplifier.

Experimental signals as a function of rf frequency, along with predictions of the theory described in Sec. 2.2, are given in Fig. 2 for different amplitudes of the rf field. At the lowest amplitudes, the observed line shapes are Lorentzians, while for higher amplitudes, additional features are seen.

 

Fig. 2 Experimental spectra (dotted lines) and theoretical predictions (solid lines) for three different values of the rf field amplitude. The experimental values (obtained from calibrated measurements of the coil current) in terms of the Rabi frequency Ω_rf are (a) Ω_rf/(2π) = 3 Hz, (b) Ω_rf/(2π) = 63 Hz, (c) Ω_rf/(2π) = 158 Hz. These values are used in the theory, along with the parameters natural width Γ = 38 μs⁻¹ = 2π× (6.1 MHz) (chosen equal to the natural width of the Rb D₂ transition), light power 5 μW, ground-state relaxation rate γ_t = 50 s⁻¹ = 2π × (8 Hz), and bias-field Larmor frequency Ω_L/(2π) = 554.75 Hz. These last three parameters are chosen for optimal agreement with the data—they are reasonably close to the experimental values. The value for light power is applied to the theoretical model using the method described in Sec. 2.2. At low rf-field strengths, as in row (a), the spectra are Lorentzians. An additional central feature appears in the resonances when Ω_rf exceeds γ_t, as in rows (b) and (c). Boxes in row (c) indicate regions plotted on expanded scales in Fig. 3.

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2.2. Description of theory

The experimental signal is primarily due to interaction with the F_g = 2 → F_e = 1 transition of the D₂ line of ⁸⁷Rb, although there are also contributions from the F_g = 2 → F_e = 2 and F_g = 2 → F_e = 3 transitions. Theoretical modeling shows that signals produced on each of these transitions have similar line shapes, although the signal from a F_g = 2 → F_e = 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 F_g ≥ 1. For the theoretical treatment presented here, we therefore consider the simplest case of a F_g = 1 → F_e = 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.

The atoms are subject to a ẑ-directed field B₀ = B₀ẑ, corresponding to the Larmor frequency Ω_L = gμ_BB₀, where μ_B is the Bohr magneton and 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_rfx̂cosω_rft, corresponding to the rf Rabi frequency Ω_rf = gμ_BB_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_L and ω_rf. Numerical solutions are obtained in the general case.

The experiment is performed using a vapor cell with an antirelaxation coating. In such a cell, atoms can be optically pumped in the light beam and then exit and return to the beam after undergoing collisions with the cell walls, without the polarization relaxing. Thus a complete theoretical description must take into account the different conditions—and the different state of the atoms—inside and outside the beam. If the light power is low enough so that saturation effects do not occur in the beam, however, the system can be modeled by considering the average state of the atoms over the entire cell. This amounts to mapping the case of a coated cell onto that of an uncoated cell (i.e., one in which atomic polarization completely relaxes between exiting and reentering the light beam). The light beam in the uncoated cell is taken to have an intensity equal to the average intensity over the entire cross-section of the coated cell, and the atomic transit rate through the beam is taken to be equal to the ground-state polarization relaxation rate in the coated cell. (This “effective” uncoated cell must be very large, in order to account for the slow ground-state relaxation of the coated cell.) This is the case that we will consider.

Another complication arising in a complete model of an atomic vapor cell is the velocity dependence of the atomic state due to Doppler broadening and collisional velocity mixing effects. These effects tend to change the dependence of a signal on the light frequency. Since we hold the light frequency fixed in this experiment, the main effect is the introduction of an overall scaling factor. Therefore, we neglect the velocity dependence in our model.

In the Zeeman basis {|F_g = 1, m = 1〉, |F_g = 1, m = 0〉, |F_g = 1, m = −1〉, |F_e = 0, m = 0〉}, the total time-dependent Hamiltonian H of the Doppler-free system under the optical rotating-wave approximation is

Under the rotating-wave approximation for the rf field described above, the evolution equations for the rotating-frame density matrix contain no explicit time dependence. We can therefore set the time derivatives to zero and solve the resulting system of linear equations for the steady state. Taking the case of low light power, we solve the equations to third order in the optical Rabi frequency, which is the lowest order at which nonlinear optical rotation signals appear. Using the inverse transformation U⁻¹, we transform back to the laboratory frame to find the time-dependent density matrix.

The expectation value of the medium polarization is found from the laboratory-frame density matrix. By substituting this value into the wave equation, we can calculate the optical-rotation signal measured in the laboratory frame in terms of the rotating-frame density-matrix elements. After multiplying by the reference signal and averaging over time, we find the in-phase and quadrature (out-of-phase) signals per unit length dℓ of the medium to be

The expressions obtained after substituting in the solution for the density matrix are complicated; to simplify the presentation we assume that the light field is on resonance and that Γ is much greater than all other rates in the problem. This gives

When Ω_rf becomes of the same order as or exceeds Ω_L, the rotating-wave approximation for the rf field is no longer valid. In this case we use the Hamiltonian 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

3. Discussion

The predictions of the density-matrix calculation described in Sec. 2.2 are compared to the experimental data in Fig. 2. For each value of the rf field strength (characterized by the rf Rabi frequency Ω_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_t, the in-phase and quadrature resonances in rf frequency take the form of dispersive and absorptive Lorentzians of characteristic width γ_t [Fig. 2(a)]. At intermediate field strengths, γ_t < Ω_rf < |Ω_L| (we assume γ_t ≪ |Ω_L|), 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.

For higher fields, Ω_rf > |Ω_L|, 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 ω_rf = −Ω_L symmetric to the one at ω_rf = Ω_L. 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 ω_rf of the same order as Ω_L 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.

 

Fig. 3 Row (c) of Fig. 2 (boxed regions) plotted on expanded scales. (The vertical scale is different for each plot.) A resonance due to high-field effects can be seen near ${\omega }_{\text{rf}}/(2\pi )=\frac{1}{3}{\Omega }_L/(2\pi )\approx 185\hspace{0.17em}\text{Hz}.$.

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3.1. Low-field regime

We first consider the low-rf-field regime, Ω_rf < γ_t. This case was discussed in Ref. [15]. An example of experimental data taken in this regime compared to theoretical predictions is shown in Fig. 2(a). As described by Eqs. (12)–(14), the resonance observed in the in-phase component, quadrature component, and magnitude of the optical-rotation signal takes the form of the real (dispersive) part, imaginary (absorptive) part, and the magnitude of a complex Lorentzian, respectively. The absolute value of a complex Lorentzian has the line shape of the square root of the absorptive part. Near resonance the quadrature component is the primary contributor to the magnitude, while farther from resonance, the in-phase component provides the main contribution. The characteristic width of the observed resonances is determined by the ground-state relaxation rate.

Optical pumping by the light field removes atoms from the m_g = 0 sublevel, leaving an incoherent mixture of atoms in the m_g = ±1 sublevels. The atomic polarization can be illustrated using the angular-momentum-probability surface (AMPS) [23,31,32], whose radius in a given direction is determined by the probability of measuring the maximum possible angular-momentum projection in that direction. This provides the quantum-mechanical analog of the classical angular-momentum probability distribution. The optically pumped distribution corresponds to atomic alignment along the 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_g = 0 sublevel has been depleted, the atomic medium transmits 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]. This linear dichroism can induce rotation of the light polarization if the transmission axis is tilted away from the light polarization axis.

 

Fig. 4 Angular-momentum-probability surfaces illustrating the behavior of the system with a resonant (Δ_rf = 0), weak rf field. (a) The light, with electric field E, produces atomic alignment along the z-axis. (b) In the rotating frame, the atoms precess around the x-axis. Due to relaxation, a steady state is reached. (c) In the laboratory frame, the atoms precess around the z-axis, with the linear dichroism of the ensemble inducing oscillating optical rotation in the z-polarized, ŷ-propagating light field. ( Media 1 animates the precession and optical rotation as a function of time.) The transmission axis of the polarized ensemble, in this case parallel to the atomic alignment axis, is indicated with a line drawn on the surface.

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The observed signals in the low- and intermediate-field regimes can be most readily understood in terms of Larmor precession of the atomic alignment in the combined static and rf magnetic field. In the rotating frame, under the rotating-wave approximation for the rf field, the effect of the magnetic fields B₀ 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 B₀, 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_L. The rf field 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.

The atomic polarization evolves in the rotating frame under the action of 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_t, 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 z-axis in the yz plane [Fig. 4(b)].

In the laboratory frame, the alignment precesses about the 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.

As the rf field is tuned away from resonance, the total field in the rotating frame begins to point away from the 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_t, 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 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.

 

Fig. 5 As Fig. 4, but with an off-resonant rf field. (a) In the rotating frame, the atoms precess around the total effective field, which lies in the xz plane. As a result, the polarization is no longer entirely in the yz plane. (b) Far off resonance, the effective magnetic field is large enough that the polarization is completely averaged about the magnetic-field direction. (The arrow representing B′_tot is truncated, as it is too long to fit entirely on the plot.) Media 2 shows the polarization, along with the in-phase component of the optical-rotation signal, as the rf-field detuning is swept through resonant and off-resonant conditions

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The preceding description is seen to correspond to the signals shown in Fig. 2(a)—the dispersive (in-phase) and absorptive (quadrature) components and the magnitude of a Lorentzian with characteristic width γ_t —as well as to the lowest order terms of Eqs. (12)–(14).

3.2. Intermediate-field regime

When the rf field is large enough that γ_t < Ω_rf < |Ω_L|, 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_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_g = ±1 sublevels, and leaves them in the dark state made up of the opposite superposition, as well as in the m_g = 0 sublevel. However, due to the precession induced by the x̂ directed magnetic field, atoms oscillate between the bright and dark superpositions, so that the pump light removes atoms from the m_g = ±1 sublevels incoherently. The atoms are then left in the m_g = 0 sublevel, i.e., the atoms have no angular-momentum projection on the x-axis and are symmetric about the x-axis, as seen in Fig. 6(a).

 

Fig. 6 Rotating-frame AMPS for intermediate rf field strength, in the ideal case in which γ_t ≪ Ω_rf ≪ |Ω_L|. Optical rotation is indicated for the instant in which the rotating frame coincides with the laboratory frame—this means that the generation of the in-phase component of the signal is shown. (a) When the rf field is on resonance, the atomic polarization (created along the z-axis) is averaged about the x-axis. The resulting polarization is a “doughnut” aligned along the x-axis. The transmission axis, perpendicular to the alignment axis, is marked with a line on the surface. (b) For small detunings, the alignment axis follows the effective magnetic field direction. (c) When the effective magnetic field is at the magic angle ${\theta }_m=\text{arccos}(1/\sqrt{3})$ to the light polarization direction, the atomic polarization is completely averaged out due to precession. (d) For larger detunings, the polarization regains its original “peanut” shape, and the transmission axis is along the alignment axis. Media 3 shows the atomic polarization and in-phase optical rotation as the rf-field detuning is swept through resonance.

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Polarization along the 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.)

To understand the generation of optical rotation in this case, it is important to note the effect of the different character of the ensemble polarization. In the low-field case, the angular-momentum probability distribution has maxima along the ensemble polarization axis (a peanut), while in the current case, the distribution has minima (a doughnut). (This result depends on the particular type of transition considered—for a F_g → F_e = F_g 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 m_g = 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 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.)

As the rf field is tuned farther from resonance, 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) ∝ 3cos²θ − 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 ${\theta }_m=\text{arccos}(1/\sqrt{3})$, analogous to the magic angle observed in nuclear magnetic resonance experiments, the polarization moment goes to zero. This means that if 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.

 

Fig. 7 Cross section of the AMPS for pure alignment along the z-axis, described by Y_2,0 (θ, ϕ) ∝ 3cos²θ – 1. Positive function values are shown as solid lines, negative function values as dashed lines. The magic angle ${\theta }_m=\text{arccos}(1/\sqrt{3})$ is indicated.

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When the detuning is even larger, there is once again residual polarization after averaging about the magnetic-field direction. However, now the magnetic-field direction is close enough to the initial alignment axis that the polarization resulting from averaging resembles the peanut shape of the initially pumped polarization [Fig. 6(d)]. Thus, the optical-rotation signal in this case is opposite in sign to that for small detuning and has the same sign as that for the low-power case.

As the detuning continues to increase, B′_tot and the averaged atomic alignment point more toward the z-axis, reducing the optical-rotation signal.

The preceding discussion describes the in-phase signal shown in Fig. 2(b) and 2(c): a power-broadened Lorentzian with a narrower feature of the opposite sign in the center. If the discussion is strictly interpreted, there should be no quadrature signal in this regime, as the polarization in the rotating frame is always in the xz plane. Figure 2 does display (strongly suppressed) quadrature signals, which are a remnant of the low-field regime.

3.3. High-field regime

As Ω_rf becomes of the same order as Ω_L 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]. As the rf field strength increases, many additional resonances are predicted by the theory.

4. Conclusion

We have conducted a detailed experimental and theoretical investigation of magneto-optical rotation in the presence of a strong radio-frequency field. A model has been developed that allows both the reproduction of the nontrivial line shapes that appear as the rf field strength is increased, and also the qualitative understanding of the atomic polarization dynamics responsible for the signals. The results of this study will be useful for the analysis of the behavior of atomic magnetometers exposed to strong radio-frequency fields. The results may also have applications in the design of efficient methods for the preparation of specific polarization states of atoms, for example, atomic states with large alignment but no orientation, as are used in the work of Ref. [24].