Transverse optical pumping of spin states

Abstract

Optical pumping is an efficient method for initializing and maintaining atomic spin ensembles in a well-defined quantum spin state. Standard optical pumping methods orient the spins by transferring photonic angular momentum to spin polarization. Generally the spins are oriented along the propagation direction of the light due to selection rules of the dipole interaction. Here we present and experimentally demonstrate that by modulating the light polarization, angular momentum perpendicular to the optical axis can be transferred efficiently to cesium vapor. The transverse pumping scheme employs transversely oriented dark states, allowing for control of the trajectory of the spins on the Bloch sphere. This new mechanism is suitable and potentially beneficial for diverse applications, particularly in quantum metrology.

Introduction

Optical pumping is the prevailing technique for orienting atomic spins, conveying order from polarized light onto the state of spins^1,2,3. Many applications in precision metrology^4,5,6,7, quantum information^8,9,10, noble gas hyper-polarization^11,12,13, and searches for new physics beyond the standard model^14,15 employ optical pumping for initializing the orientation moment of the spins, that is, for pointing the spins towards a preferred direction. The required degree of polarization depends on the specific application, where optimized performance in quantum metrology is often practically achieved around 50% polarization^16,17,18. Standard optical pumping schemes generate polarization along the propagation direction of the laser beam. These schemes include depopulation pumping¹, synchronous pumping^19,20,21, spin-exchange indirect pumping^22,23, alignment-to-orientation conversion^24,25, and hybrid spin-exchange pumping¹⁶. However, in various applications, it is often desired to polarize the spins along an applied magnetic field, perpendicular to the optical axis^{18,26,27,28,29,30}. While at extreme magnetic fields, it is possible to polarize the spins transversely³¹, at moderate magnetic fields, typical to alkali-metal spin experiments for example, the pumping efficiency is rather low.

In standard optical pumping schemes, the atomic ground state is polarized via repeated cycles of absorption and spontaneous emission. Ideally, the atoms cease to absorb the pump photons when they reach a ‘dark state’, which is determined by the excited transitions during pumping¹. For a light field with an electric field ${\mathbf{E}}\left( t \right) = E_0e^{i(\omega _{\mathrm{L}}t - kx)}\hat e$, the relevant transitions depend on the relative detuning of the light frequency ω_L from the atomic transition frequency ω₀, on the external electric and magnetic fields, and on the selection rules of the dipole interaction for polarization $\hat e$. For alkali-metal vapors, the latter enables the pumping process of spin orientation at moderate magnetic fields, when the ground and excited magnetic sublevels $|m_g\rangle ,{\mkern 1mu} |m_e\rangle$ within each hyperfine manifold F_g, F_e are optically unresolved.

In the absence of magnetic field and for constant polarization $\hat e$, one-photon absorption of light does not produce spin orientation transversely to the optical axis. Circular light polarization $\hat e_ \pm = (\hat y \pm i\hat z)/\sqrt 2$ orients the spins along the optical axis $\pm \hat x_{}^{}$ via the allowed transitions m_e = m_g ± 1; For F_e ≤ F_g, the maximally polarized state |m_g = ±F_g〉 is dark. Linearly polarized light $\hat e = \hat y,\hat z$ generates spin alignment along $\hat x \times \hat e$ and zero net orientation with the selection rules m_e = m_g when tuned to the transition F_g → F_e = F_g − 1. This generates a quadrupole magnetic moment¹, leaving both |m_g = F_g〉 and |m_g = −F_g〉 dark. It thus seems that no orientation is built perpendicularly to the optical axis $\hat x$ for any light polarization. In the presence of a constant magnetic field, precession around it may orient the spins along the transverse direction, but this is never achieved with considerable orientation. Our scheme overcomes this limitation and allows for transverse optical pumping of the spins by temporally modulating the light polarization.

Here we propose and demonstrate an optical pumping scheme for efficient spin polarization transversely to the propagation direction of the laser beam. The scheme incorporates a polarization-modulated light beam, which steers the spins in helical-like trajectories on the Bloch sphere around and along a transverse magnetic field, while gradually increasing their polarization. The scheme exhibits sharp resonances, reaching maximum efficiency when the optical modulation is resonant with the Larmor precession of the spins. We develop a simple analytical model for analyzing the experimental results and discuss the applicability of the scheme for various applications.

Results

Experimental pumping of cesium spins

We employ the experimental setup shown schematically in Fig. 1a, containing cesium vapor at room temperature. The energy level for a I = 1/2 model is shown in 1b. Setting a constant magnetic field $B\hat z$ determines the quantization axis $\hat z$ and the Larmor frequency ω_B = gB, where g = 0.35(2π) MHz/G is the gyro-magnetic ratio for cesium. For the transverse pumping, we use a pump beam, whose frequency is tuned to the D₁ transition F_g = 4 → F_e = 3 and whose polarization is modulated according to

$$\hat e\left( t \right) = \cos \left( \theta \right)\hat z + ie^{i\omega t}\sin \left( \theta \right)\hat y.$$

(1)

Here, sin(θ) is the modulation depth and ω is the modulation angular frequency. For the sake of analysis and presentation, we introduce two far-detuned monitor beams propagating along $\hat x$ and $\hat y$, measuring the three-dimensional orientation state of the spins (2S_x, 2S_y, 2S_z) on the Bloch sphere during the pumping process. See Methods for additional experimental details.

Fig. 1

Experimental system and toy model. a Schematics of the experimental setup and the spiral motion of the atomic spins (green) towards $+ \hat z$. The polarization of the pump beam (red) alternates between linear and circular (blue arrows). The spin orientation is monitored using balanced polarimetry of a far-detuned monitor beam (yellow). Not shown are the repump beam and a second monitor beam, which co-propagate with the pump. b Toy model for alkali atoms with nuclear spin I = 1/2. The repump laser (blue arrow) empties the lower hyperfine state (with quantum number F_g = 0), while the pump laser (red and green arrows) drives the atoms into the dark state $|d_ + \rangle \propto \cos \theta \left| 1 \right\rangle - {\textstyle{1 \over {\sqrt 2 }}}e^{i\omega t}\sin \theta \left| 0 \right\rangle$ (gray shading), eventually oriented perpendicular to the beam direction

In Fig. 2a–d, we present measurement and theoretical calculation of the spin dynamics on the Bloch sphere. Figure 2a shows a typical measurement of the pumping process during continuous pumping operation. We observe that the spin orientation follows a helical trajectory transversely to the optical axis $\hat x$. In this experiment, the pump power is P₀ = 250 μW and the modulation frequency is tuned to resonate with the Larmor frequency ω_B ≈ ω = 1.5 (2π) kHz. The final value of 2S_z quantifies the pumping efficiency. Its dependence on the modulation parameters ω and θ is shown in Fig. 3. We identify two resonant features of 2S_z(ω) at ω ≈ ± ω_B as shown in Fig. 3a for θ = 0.24 and two laser powers. The laser power governs the width of the resonance, as well as, the shift of the peak from the actual Larmor frequency. Figure 3b presents 2S_z(θ) on one of the resonances [ω = 10.3 (2π) kHz]. We achieved an overall maximal polarization of 2S_z = 65% (with small residual transverse polarization 2$\sqrt {S_x^2 + S_y^2}$ = 3.5%) as shown in Fig. 3c.

Fig. 2

Spin trajectories on the Bloch sphere. The optical axis is $\hat x_{}^{}$, within the equatorial plane. a, c Measurements of the pumping process from t = 0 to t = 100 ms. b, d Theoretical toy model with nuclear spins I = 1/2. Red (green) circles mark the initial (final) states of the spin. a When pumping with a constant modulation depth θ = 0.2 rad, the measured cesium spins follow a spiral-helical trajectory around the $+ \hat z$ direction. c Adiabatically varying $\theta \left( t \right) = \arccos \sqrt {t/T}$ over T = 100 ms allows for driving the spins in a spherical-helical trajectory that ends along the $\hat z$ axis

Fig. 3

Dependence of the pumping degree on the modulation parameters. a Measured spin S_z(ω) at t = 200 ms for cesium atoms with a modulation depth θ = 0.24 rad and Larmor precession rate ω_B = 10.2 (2π) kHz. The resonance peaks at modulation frequency ω ≈ ± ω_B are associated with the two coherent population trapping (CPT) dark states |d_±〉. b Dependence on the modulation depth θ: measured S_z(θ) at t = 200 ms with ω = ±10.3 (2π) kHz and pump power P₀ = 120 μW. Pumping is optimal at moderate modulation depths, such that |d₊〉 is oriented towards $+ \hat z$, but the dark state pumping $\Gamma \sin ^2\left( \theta \right)$ competes the relaxation rate γ. Vertical error bars indicate standard deviation of the measured value. c Measured spin trajectory, corresponding to the point marked by an arrow in b. The final polarization (green circle) is along $\hat z$, i.e. perpendicular to the optical axis, with a small residual polarization along $\hat x$ and $\hat y$

Dynamics of pumped spins

To explain the transverse pumping mechanism we utilize a simple model of an alkali-like level structure with nuclear spin I = 1/2, as shown in Fig. 1b. The magnetic field ${\mathbf{B}} = B\hat z$ (henceforth, assume B > 0) breaks the isotropy in the transverse yz plane, setting our quantization axis $\hat z$ and splitting the Zeeman sublevels $\left| 0 \right\rangle ,\left| { \pm 1} \right\rangle$ by ħω_B. The F_g = 0 level is emptied by a repump field or by spin–exchange collisions. The pump field, resonant with the F_g = 1 → F_e = 0 transition, is polarization-modulated according to Eq. (1). We describe the effect of the polarization modulation on the spin dynamics by decomposing the polarization vector $\hat e\left( t \right)$ into its Stokes components $\widehat {\boldsymbol{s}} = (s_1,s_2,s_3)$^32,33. The unmodulated linear polarization $\hat z$, represented by s₁, aligns the atoms along $\hat z$ at a rate $R_a\sim \Gamma \cos ^2\left( \theta \right)$, creating spin alignment (see Supplementary Note 1). Here Γ = Ω²/γ_e is the characteristic pumping rate, with γ_e the spontaneous emission rate and Ω the Rabi frequency of the pump beam. The linear polarization $(\hat y \pm \hat z)/\sqrt 2$, represented by s₂, induces a tensor light shift of Γ sin(2θ) sin(ωt) along $\pm \hat x$, which acts like a magnetic field. This light shift appears when the linearly polarized light is neither parallel nor perpendicular to the spin alignment³⁴. The circular polarization $\hat e_ \pm$, represented by s₃, pumps the spins longitudinally along $\pm \hat x$ at a rate Γ sin(2θ) cos (ωt), while vector light shift is absent for the resonant optical transition. Therefore, the modulated polarization alternates between pumping (s₃) and light shifting (s₂) the atomic spins along $\hat x$ at a rate ω. For ω = ω_B, the pumping and light shifts are synchronous, efficiently driving the precessing spins away from the xy plane, transversely to the optical axis. The resulting evolution of the Bloch vector (2S_x, 2S_y, 2S_z) is shown graphically in Fig. 4 and further detailed in Methods.

Fig. 4

Transverse pumping mechanism. One period of the transverse pumping mechanism for ω = ω_B, where the modulation frequency ω is synchronous with the Larmor precession rate ω_B. The spins (green arrows) precess ω_Bt = π/2 radians between each subplot due to the magnetic field $B\hat z$. a The spins are optically pumped towards $- \hat x$. b The spins are tilted towards $+ \hat z$ due to tensor light shift. c The spins are optically pumped towards $+ \hat x$. d The spins are again tilted towards $+ \hat z$ due to tensor light shift. Black arrows indicate the pumping/tilting directions. The eventual spin orientation is along $+ \hat z$, perpendicular to the propagation direction of the pumping beam

The toy model enables one to reconstruct the main features of the measured trajectories as shown in Fig. 2b, by solving the I = 1/2 model numerically and tuning its parameters (see Supplementary Note 1). We note however that this model only aims at explaining the qualitative features of the process, while disregarding effects arising from the multilevel structure of cesium and from level mixing due to line broadening, which would reduce the pumping efficiency. We attribute these effects to the observed maximum of 65% polarization, rather than the 100% polarization expected for an I = 1/2 system (see Methods).

Coherent population trapping

The resonant nature of the pumping process can also be understood using the following supplementary picture, as originating from coherent population trapping (CPT)³⁵. In CPT, a dark state is formed within a Λ level-system via destructive interference of two excitation pathways. Considering the level structure in Fig. 1b and decomposing the modulated pump into its two polarization components $E\hat z$ and $E\hat y$, we identify two Λ systems: $\Lambda _ + = \{ \left| 1 \right\rangle ,\left| e \right\rangle ,\left| 0 \right\rangle \}$ and $\Lambda _ - = \{ \left| { - 1} \right\rangle ,\left| e \right\rangle ,\left| 0 \right\rangle \}$. System Λ₊ has the dark state $|d_ + \rangle \propto \cos \left( \theta \right)\left| 1 \right\rangle - {\textstyle{1 \over {\sqrt 2 }}}e^{i\omega t}\sin \left( \theta \right)\left| 0 \right\rangle$ at ω ≈ ω_B, while system Λ₋ has the dark state $|d_ - \rangle \propto \cos \left( \theta \right)\left| { - 1} \right\rangle - {\textstyle{1 \over {\sqrt 2 }}}e^{i\omega t}\sin \left( \theta \right)\left| 0 \right\rangle$ at ω ≈ −ω_B. For $\theta \ll 1$, the dark states $|d_ \pm \rangle \approx \left| { \pm 1} \right\rangle$ represent the polarized states perpendicular to the optical axis. The application of magnetic field $B\hat z$ separates the CPT resonances of Λ₊ and Λ₋ by 2ω_B, so that the states |d₊〉 and |d₋〉 cannot be simultaneously dark when $\omega _{\mathrm{B}} \gg \Gamma$. Consequently, setting ω = ω_B depopulates the Λ₋ system while pumping the Λ₊ system towards the transversely oriented dark state |d₊〉. We conclude that destructive interference of two excitation pathways effectively modifies the absorption selection rules, such that one polarized state (e.g. $\left| { - 1} \right\rangle$ for ω_B > 0) absorbs photons, while the opposite state ($\left| 1 \right\rangle$ for ω_B > 0) is transparent.

We associate the two resonances in Fig. 3a with the CPT dark states of Λ₊ at ω > 0 and Λ₋ at ω < 0. In the absence of the upper hyperfine level F_e = 4, we expect to find distinct resonance peaks at ± ω_B; it is the presence of F_e = 4 that breaks the symmetry between positive and negative ω [S_z(ω) ≠ − S_z(−ω)], generates the small Fano-like features, and shifts the peak from ω = ω_B. As seen in Fig. 3b, the pumping is most efficient at moderate modulation depths: For θ → π/2 the dark state is $|d_ + \rangle \to \left| 0 \right\rangle$, with zero net orientation. For θ → 0, the dark state is $|d_ + \rangle \approx \left| 1 \right\rangle$, but the depopulating rate of |d₋〉, proportional to $\Gamma \sin ^2\left( \theta \right)$, is too small compared to the overall depolarization rate γ.

A benefit of the CPT resonant operation is the ability to temporally vary the system state in a controlled, adiabatic manner. To demonstrate this, we monitor the pumping process on resonance [ω = ω_B = 1.5 (2π) kHz], while temporally varying θ over a duration T = 100 ms according to $\theta \left( t \right) = \arccos \sqrt {t/T}$. The spin state, initially pumped to $|d_ + \rangle _{\theta \left( {t = 0} \right)} \approx \left| 0 \right\rangle$, adiabatically follows the varying dark state |d₊〉_θ(t) to its final value $|d_ + \rangle _{\theta \left( {t = T} \right)} \approx \left| 1 \right\rangle$, tracing a spherical-like trajectory as shown in Fig. 2c (experiment) and Fig. 2d (theory). This process is similar to stimulated Raman adiabatic passage³⁶ and can therefore be used to tailor desired trajectories and final states. Notably, it enables the zeroing of the transverse spin components S_x and S_y at the end of the process, as shown in Fig. 2c, d.

Discussion

It is relatively simple to implement the presented scheme in applications. Polarization modulation can be done using a single photo-elastic modulator (Photoelastic Modulators, www.hindsinstruments.com/products/photoelastic-modulators) or with readily available, on-chip, integrated photonics³⁷. Various applications that rely on optical spin manipulation and feature resolved hyperfine spectra could potentially benefit from utilizing the scheme. Here we briefly consider some directions with spin vapors.

First, devices currently employing perpendicular beams in a pump-probe configuration^26,27,28 could be realized with a simpler, co-propagating arrangement, with the spins oriented transversely to the optical axis. Such arrangement is most beneficial for miniaturized sensors, such as nuclear magnetic resonance (NMR) oscillators^28,29, where the size and complexity depend crucially on the beam's configuration, especially if the light source, manipulation, and detection can be implemented on a single stack over a chip²⁷. These sensors are used in various applications as well as in fundamental research, such as search for new physics^29,38. Particularly for NMR oscillators, the projection of the alkali spin along the magnetic field will be unmodulated, thus sustaining the spin-exchange optical pumping of the noble gas spins.

Second, any application that is restricted to a single laser direction and requires moderate alkali polarization (tens of percents) can now use our scheme to control and fine-tune the final direction of the pumped spins (longitudinal, transversal, or combination thereof). One example includes remote magnetometry of mesospheric sodium spins^26,39. Third, transverse pumping may form the basis for an all-optical magnetometer using either alkali-metal atoms or metastable ⁴He atoms designed for space applications^26,38. This magnetometer would rely on measuring the resonant response to the modulated light, providing a dead-zone-free operation³², or on measuring the Faraday rotation of off-resonant probe light, thus reducing the photon shot-noise commonly limiting magnetometers based on electromagnetically induced transparency⁴⁰. Moreover, the polarization-modulated pump generates the m = 1 Zeeman coherence³⁰, implying that these magnetometers could operate in the spin-exchange relaxation-free (SERF) regime⁴¹, where spin-exchange collisions may also assist in repumping the lower hyperfine manifold⁴².

Finally, our scheme does not rely on any process particular to vapor physics. It is thus readily applicable to any spin system having a non-degenerate Λ-system with a meta-stable ground manifold, such as those employed in diamond color centers^43,44, rare-earth doped crystals⁴⁵, and semiconductor quantum dots^46,47,48.

In conclusion, we have demonstrated a new optical pumping technique, generating significant spin orientation transversely to the propagation direction of the pump beam. The spins are oriented along the external transverse magnetic field via alternating actions of pumping and tensor light shifts, which are resonant with the Larmor precession. The resonance features, associated with transversely orientated dark states, allow one to control the spin trajectory on the Bloch sphere by varying the modulation parameters. This scheme could be highly suitable for quantum-metrology applications.

Methods

Additional experimental details

We use a 10-mm-diameter, 30-mm-long cylindrical glass cell containing cesium vapor (I = 7/2, S = 1/2) at room temperature. The cell is paraffin coated and free of buffer gas, exhibiting spin coherence time of 150 ms³⁰. We set a constant magnetic field $B\hat z$ in the cell using Helmholtz coils and four layers of magnetic shields. For the transverse pumping, we use an 895-nm single-mode pump beam using a free-running distributed Bragg reflector (DBR) diode laser (see optical schematics in Supplementary Fig. 1). We modulate the pump polarization by splitting it with a polarizing beam splitter (PBS) and sending each output arm to an acousto-optic modulator (AOM), operating at 200 MHz. The two beams deflected by the modulators are recombined and mode-matched using a second PBS, resulting with the polarization given in Eq. (1). The pump frequency after passing the modulators is tuned to resonance with the D₁ transition F_g = 4 → F_e = 3, within the Doppler width. We control the modulation angular frequency ω by setting the relative radio frequencies of the two modulators. We control the modulation depth sin(θ) either by rotating the linear polarization using a half-wavelength wave-plate before the first PBS for constant θ (e.g., Figs. 2a, 3c), or by varying the relative RF amplitudes of the two AOMs for a time-varying θ(t) (e.g., Fig. 2d). We sample the pumping beam before the entrance to the cell and measure the Stokes component S₂ to determine the polarization state of the light. To keep the lower hyperfine manifold F_g = 3 empty, we use 1 mW of auxiliary repump beam at 895 nm, using a second free-running DBR laser. The repump is resonant with the F_g = 3 → F_e = 4 transition, within the Doppler width, and linearly polarized along $\hat y$. The pump and repump, both with a diameter of 8 mm, counter-propagate along the $\hat x$ axis.

Reconstruction of the spin state on the Bloch sphere

The spin state is reconstructed by evaluating the electronic spin orientations (2S_x, 2S_y, 2S_z) = 〈F〉/4, where 〈F〉 is the orientation moment of the total spin operator F = I + S. The spin orientations are measured by using balanced polarimetry of two linearly polarized monitor light beams propagating along $\hat x$ and $\hat y$. The monitor light is 30 GHz blue-detuned from the optical transition F_g = 4 → F_e = 3. Polarization rotation of far-detuned light is sensitive to spin orientation along the optical axis and insensitive to higher magnetic moments. At low atomic densities and depopulated F_g = 3 hyperfine manifold, the detected Faraday-rotation angles are proportional to the spin orientation along the direction of the beam^2,49. We calibrate the proportionality constants of each monitor beam by measuring its maximal polarization rotation when the ground state is fully pumped using two circularly polarized beams resonant with the two ground-state hyperfine manifolds. We reconstruct the three spin components by making two consecutive measurements: First, S_x and S_y are measured when ${\mathbf{B}} = B\hat z$. Second, a measurement is conducted with ${\mathbf{B}} = B\hat y$ and θ changed by θ → π/2 − θ, keeping the other experimental parameters unchanged. As a result, spin is built along $\hat y$ and measured by the $\hat y$ monitor. This provides the S_z component of the first configuration. We verify that S_x is unaffected by the change of θ, B, by confirming that the parameter change is appropriate.

Spin dynamics with polarization-modulated light

For small modulation depths $\theta \ll 1$, the dynamics is governed by Bloch-like equations of the vector 〈F〉 = (F_x, F_y, F_z) (see Supplementary Note 1 for the general treatment). This dynamics is qualitatively shown in Fig. 4 at four parts of the pumping period: at ωt = 0 [Fig. 4a], at ωt = π/2 [Fig. 4b], at ωt = π [Fig. 4c] and at ωt = 3π/2 [Fig. 4d]. The spin orientations F_x (along the optical axis) and F_y are subject to

$$\dot F_x = - \gamma _ \bot F_x - \omega _{\mathrm{B}}F_y - \Gamma \sin \left( {2\theta } \right)\cos \left( {\omega t} \right),$$

(2)

$$\dot F_y = - \gamma _ \bot F_y + \omega _{\mathrm{B}}F_x + \Gamma \sin \left( {2\theta } \right)\sin \left( {\omega t} \right)F_z,$$

(3)

which include a transverse decay rate $\gamma _ \bot = \gamma + (1 + \cos ^2\theta )\Gamma$ and Larmor precession at the rate ω_B. Here γ denotes a slow ground-state depolarization rate (e.g., due to wall collisions). The third term in Eq. (2) is due to s₃. It describes a temporally modulated optical pumping, which is maximal at ωt = 0 (and at all ωt = 2πn for any integer n) towards $- \hat x$ (Fig. 4a) and at ωt = π towards $+ \hat x$ (Fig. 4c). The pumping of F_x is thus most efficient when the optical modulation is synchronous with the Larmor precession ω = ω_B. The third term in Eq. (3) is due to the modulated linear polarization component s₂. It describes a tensor light shift, which acts as a magnetic field along $\hat x$ that rotates the spins in the yz plane at a modulated rate Γ sin (2θ) sin (ωt). The orientation F_z along the magnetic field, which we aim to generate, is subject to

$$\dot F_z = - \gamma _\parallel F_z - \Gamma \sin \left( {2\theta } \right)\left[ {\sin \left( {\omega t} \right)F_y - \cos \left( {\omega t} \right)\left\{ {F_z,F_x} \right\}} \right],$$

(4)

where the first term is a longitudinal decay at a rate $\gamma _\parallel = \gamma + 2\Gamma \sin ^2\theta$, the second term is again light shift due to s₂, and the third term is an alignment-induced shift. The temporal modulation sin(ωt) of the light shift is a key ingredient in pumping F towards $+ \hat z$, as it breaks the symmetry between the $\pm \hat z$ directions: The sign of the light shift changes together with the sign of F_y, thus acting as an alternating magnetic field that always tilts the spins -towards $+ \hat z$, with maximal tilting rate obtained at ωt = π/2 (Fig. 4b) and ωt = 3π/2 (Fig. 4d). The tensor term $\{ F_z,F_x\} \equiv \langle \left\{ {{\mathbf{F}}\hat z,{\mathbf{F}}\hat x} \right\}\rangle$, resonant at ω_B, contributes similar spin buildup in amplitude, but with a π/2 delay (see Supplementary Note 1). Contribution of other tensor terms resonant at 2ω_B is negligible for $\omega \approx \omega _{\mathrm{B}} \gg \Gamma$. For ω = ω_B, both the synchronous pumping and the light shift are most efficient, driving the precessing spins away from the xy plane, transversely to the optical axis. For large magnetic fields and strong control beam $\omega _{\mathrm{B}} \gg \Gamma \gg \Gamma \sin ^2\theta ,\gamma$, the steady-state polarization on resonance (ω = ω_B) is given by

$$F_z = \frac{{\Gamma \sin ^2\theta }}{{\Gamma \sin ^2\theta + \gamma }},$$

where $\Gamma \sin ^2\theta$ can be interpreted as the effective optical pumping rate for depopulating the bright state |−1〉 (see derivation in Supplementary Note 2). The transverse spin components are then given by F_x = tan θ cos (Δt + π)(1 + F_z)/2, and F_y = tan θ sin(Δt)(1 + F_z)/2, where Δ = ω − ω_B denotes the frequency mismatch from resonance. We thus conclude that high polarization along the magnetic field and transverse to the optical axis is achievable for I = 1/2 spins.