Transverse optical pumping of spin states

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.

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.
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.
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 pumping [1]. For a light field with an electric field E (t) = E 0 e i(ωLt−kx)ê , the relevant transitions depend on the relative detuning of the light frequency ω L from the atomic transition frequency ω 0 , on the external electric and magnetic fields, and on the selection rules of the dipole interaction for polarizationê. For alkali-metal vapors, the latter dominates the pumping process of spin orientation at moderate magnetic fields, because the ground and excited magnetic sublevels |F g , m g , |F e , m e within each hyperfine manifold F g , F e are optically unresolved.
For constant polarizationê, one-photon absorption of light does not produce a considerable spin orientation transversely to the optical axis. Circular light polarizationê ± = (ŷ ± iẑ)/ √ 2 orients the spins along the optical axis ±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 lightê =ŷ,ẑ generates spin alignment alongx ×ê 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 moment [1], leaving both |m g = F g and |m g = −F g dark. It thus seems that no orientation is built perpendicularly to the optical axisx for any light polarization. Our scheme overcomes this limitation and allows for transverse optical pumping of the spins by temporally modulating the light polarization.

RESULTS
We employ the experimental setup shown schematically in Fig. 1(a), containing cesium vapor at room temperature. Setting a constant magnetic field Bẑ determines the quantization axisẑ and the Larmor frequency ω B = gB, where g = 0.35 (2π)MHz/G is the gyromagnetic ratio for cesium. For the transverse pumping, we use a pump beam, which frequency is tuned to the D 1 transition F g = 4 → F e = 3 and which polarization is modulated according tô e (t) = cos (θ)ẑ + ie iωt sin (θ)ŷ. (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 alongx andŷ, 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. A typical measurement during the pumping process is presented on the Bloch sphere in Fig. 2(a). We observe that the spin orientation follows a helical trajectory transversely to the optical axisx. In this experiment, the pump power is P 0 = 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. 3(a) 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 3(b) 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 S 2 x + S 2 y = 3.5%) as shown in Fig. 3(c).
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. 1(b). The magnetic field B = Bẑ (henceforth assume B > 0) breaks the isotropy in the transverse yz plane, setting our quantization axisẑ and splitting the Zeeman sublevels |0 , |±1 by ω 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 polarizationmodulated according to Eq. (1). We describe the effect of the polarization modulation on the spins dynamics by decomposing the polarization vectorê (t) into its Stokes componentsŝ = (s 1 , s 2 , s 3 ) [31]. The unmodulated linear polarizationẑ, represented by s 1 , aligns the atoms alonĝ z at a rate R a ∼ Γ cos 2 (θ), creating spin-alignment (see Supplementary Note). Here Γ = Ω 2 /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 (ŷ ±ẑ)/ √ 2, represented by s 2 , induces a tensor light shift of Γ sin (2θ) sin (ωt) along ±x, which acts like a magnetic field. The circular polarizationê ± , represented by s 3 , pumps the spins longitudinally along ±x at a rate Γ sin (2θ) cos (ωt). Therefore, the modulated polarization alternates between pumping (s 3 ) and lightshifting (s 2 ) the atomic spins alongx 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.
The toy model enables to reconstruct the main features of the measured trajectories as shown in Fig. 2(b), by solving the I = 1/2 model numerically and tuning its parameters (see Supplementary Note). 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, which may reduce the pumping efficiency.
The resonant nature of the pumping process can also be understood as originating from coherent population trapping (CPT) [33]. 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. 1(b) and decomposing the modulated pump into its two polarization components Eẑ and Eŷ, we identify two Λ systems: Λ + = {|1 , |e , |0 } and Λ − = {|−1 , |e , |0 }. System Λ + has the dark state |d + ∝ cos (θ) |1 − 1 √ 2 e iωt sin (θ) |0 at ω ≈ ω B , while system Λ − has the dark state |d − ∝ cos (θ) |−1 − 1 √ 2 e iωt sin (θ) |0 at ω ≈ −ω B . For θ 1, the dark states |d ± ≈ |±1 represent the polarized states perpendicular to the optical axis. The application of magnetic field Bẑ separates Figure 4. Transverse pumping mechanism. One period of the transverse pumping mechanism for synchronous modulation ω = ωB. The spins (green arrows) precess ωBt = π/2 radians between each subplot due to the magnetic field Bẑ. the CPT resonances of Λ + and Λ − by 2ω B , so that the states |d + and |d − cannot be simultaneously dark when ω B Γ. 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. |−1 for ω B > 0) absorbs photons, while the opposite state (|1 for ω B > 0) is transparent.
We associate the two resonances in Fig. 3(a) 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, the resonances would have seen like single peaks; it is the presence of F e = 4 that introduces the second peaks (with opposite sign) in each resonance, as well as the shift of the peak from ω = ω B and the asymmetry between positive and negative ω. As seen in Fig. 3(b), the pumping is most efficient at moderate modulation depths: For θ → π/2 the dark state is |d + → |0 , with zero net orientation. For θ → 0, the dark state is |d + ≈ |1 , but the depopulating rate of |d − , proportional to Γ sin 2 (θ), 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 msec according to θ (t) = arccos t/T . The spin state, initially pumped to |d + θ(t=0) ≈ |0 , adiabatically follows the varying dark-state |d + θ(t) to its final value |d + θ(t=T ) ≈ |1 , tracing a spherical-like trajectory as shown in Fig. 2(c) (experiment) and Fig. 2(d) (theory). This process is similar to stimulated Raman adiabatic passage [34] 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 Figs. 2(c,d).

DISCUSSION
It is relatively simple to implement the presented scheme in applications. Polarization modulation can be done using a single photo-elastic modulator [35] or with readily available, on chip, integrated photonics [36]. Various applications that rely on optical spin manipulation would 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 [25][26][27] 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 NMR-oscillators [27], where the size and complexity depend crucially on the beams configuration, especially if the light source, manipulation, and detection can be implemented on a single stack over a chip [26]. These sensors are used in various applications as well as in fundamental research, such as search for new physics [37,38]. 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 can now use our scheme to control and fine tune the final direction of the pumped spins (longitudinal, transversal, or combination thereof). Such applications include remote magnetometry of mesospheric sodium spins [25,39], steady-state entanglement generation during pumping [18], optical pumping of metastable 3 He for medical imaging [40], and coherent manipulation of the internal spin state of cold atoms without associate heating [41].
Third, transverse pumping may form the basis for an all-optical magnetometer using either alkali-metal atoms or metastable 4 He atoms designed for space applications [25,38]. This magnetometer would rely on measuring the resonant response to the modulated light, providing a dead-zone-free operation [31], or on measuring the Faraday-rotation of off-resonant probe light, thus reducing the photon shot-noise commonly limiting magnetometers based on electromagnetically induced transparency [42]. Moreover, the polarization-modulated pump generates the m = 1 Zeeman coherence [28], implying that these magnetometers could operate in the spin-exchange relaxation-free (SERF) regime [43].
Finally, our scheme does not rely on any process particular to vapor physics. It is thus readily application to any spin system having a non-degenerate Λ-system with a meta-stable ground manifold, such as those employed in diamond color centers [44,45], rare-earth doped crystals [46], and semiconductor quantum dots [47][48][49].
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.

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 ms [28]. We set a constant magnetic field Bẑ in the cell using Helmholtz coils and four layers of magnetic shields. For the transverse pumping, we use an 895 nm singlemode pump beam. We modulate the pump polarization by splitting it with a polarizing beam splitter (PBS) and sending each output arm to an acousto-optic modulator. The beams deflected by the modulators are recombined and mode-matched using a second PBS, resulting with the polarization given in Eq. 1. The laser frequency after passing the modulators is tuned to the D 1 transition F g = 4 → F e = 3. We control the modulation depth sin (θ) by rotating the linear polarization before the first PBS and control the modulation angular frequency ω by setting the relative RF frequencies of the two modulators. To keep the lower hyperfine manifold F g = 3 empty, we use 1 mW of auxiliary repump beam at 895 nm, resonant with the F g = 3 → F e = 4 transition and linearly polarized alongŷ. The pump and repump, both with a diameter of 8 mm, co-propagate alongx.
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 far-detuned monitor beams propagating alongx andŷ. 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 [2,50]. 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 B = Bẑ. Second, a measurement is conducted with B = Bŷ and θ changed by θ → π/2 − θ, keeping the other experimental parameters unchanged. As a result, spin is built alonĝ y and measured by theŷ 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 that confirming that the parameter change is appropriate.

Spin dynamics with polarization-modulated light:
For small modulation depths θ 1, the dynamics is governed by Bloch-like equations of the vector F = (F x , F y , F z ) (see Supplementary Note for the general treatment). The spin orientations F x (along the optical axis) and F y are subject tȯ which include a transverse decay rate γ ⊥ = γ + 2Γ cos 2 θ 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 s 3 . It describes a temporally-modulated optical pumping, which is maximal at ωt = 0 (and at all ωt = 2πn for any integer n) towards −x (Fig. 4a) and at ωt = π towards +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 s 2 . It describes a tensor light-shift, which acts as a magnetic field alongx 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 tȯ where the first term is a longitudinal decay at a rate γ = γ + 2Γ sin 2 θ, the second term is again light shift due to s 2 , 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 +ẑ, as it breaks the symmetry between the ±ẑ 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 +ẑ, with maximal tilting rate obtained at ωt = π/2 (Fig. 4b) and ωt = 3π/2 (Fig. 4d). The tensor term {F z , F x } ≡ {Fẑ, Fx} contributes similar spin buildup in amplitude but with a π/2 delay (see Supplementary Note). 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.

Supplementary Information
In this supplementary material, we derive the Bloch equations [Eqs. (2)(3)(4) in the main text] of I = 1/2 spins interacting with polarization-modulated light. The dynamics of the atoms can be described by the open quantum system Liouville equation where ρ is the density matrix of a single spin, and L is the Lindblad super-operator given by The first term in Eq. (S1) denotes the free Hamiltonian evolution; The second term denotes the excited state decay due to spontaneous emission (SE) at a rate γ e ; and the last term denotes the ground-state relaxation due to spin destruction (SD) at a rate γ, caused by interaction of the electronic spin operator S with some thermal bath. The free Hamiltonian term H 0 describes the interaction with external magnetic fields and laser beams. It is given by +Ω cos (θ) e iω L t |0 e| + (1/ √ 2) sin (θ) e i(ω L +ω)t (|−1 e| + |1 e|) + H.c. , where ω 0 is the optical transition to the excited state, ω B = gB/(2 ) is the Larmor frequency induced by the magnetic field Bẑ, ω L is the laser oscillation rate, ω is the polarization-modulation rate and sin (θ) is the modulation depth. Ω = E 0 d/ is pump laser Rabi frequency where d is the atomic dipole-moment and E 0 is the electric field amplitude of the pump beam. The matrix super-operator representation of the SE term in Eq. (S1) is given by In the derivation of this term, we assumed that the lower hyperfine level F g = 0 is not populated, due to the action of a strong repump beam. We can simplify the solution by transforming to a reference frame rotating at the laser frequency ω L . Assuming a resonant optical transition ω L = ω 0 , the Hamiltonian in the rotating frame is given by H 0 → ω B (|1 1| − |−1 −1|) + Ω cos (θ) |0 e| + (1/ √ 2) sin (θ) e iωt (|−1 e| + |1 e|) + H.c. , whereas the Liouville terms in Eqs. (S3)-(S4) are invariant under the transformation. Much below saturation (Ω γ e ), the excited state population is small (ρ ee 1), and we can adiabatically eliminate the excited state. This elimination results with the steady-state coherences