The three-step approach is a useful method for describing the evolution of optically pumped systems. This approach is however known to be limited to low light intensities [26–28]. For depopulation pumping of a spin \(1/2\) system with circularly polarized light, it was shown long ago that this approach remains valid for higher light intensities if an additional relaxation term \(\gamma _{p}\) due to the pumping, and proportional to light intensity *I* is added to the natural relaxation \(\varGamma _{e}\).

Some authors have discussed if a similar approach may be valid for a \(\mbox{spin} >1/2\) pumped by linearly polarized light towards an aligned state [38]. We study here if adding well-chosen relaxation terms to the three-step approach model could yield accurate predictions for the photodetection signals of helium-4 interacting with linearly polarized light. We first study the simplest scheme which is the Hanle effect. Then we consider the case of parametric resonance magnetometers.

### 3.1 Hanle effect

We will consider here the situation sketched in Fig. 1(a). Let us recall the multipole decomposition of the density matrix on the irreducible tensor operators \(\hat{T}_{q}^{(k)}\) [39]:

$$ \hat{\rho }=\sum_{k,q}m_{q}^{(k)} \hat{T}_{q}^{(k)\dagger } $$

(1)

with \(m_{q}^{(k)}=\langle \hat{T}_{q}^{(k)}\rangle \). The order \(k=1\) corresponds to orientation and \(k=2\) to alignment. The latter can be described by a column matrix *M* with five components \(m_{-2}^{(2)}\), \(m_{-1}^{(2)}\), \(m_{0}^{(2)}\), \(m_{1}^{(2)}\), \(m_{2}^{(2)}\).

Within the three-step approach, the evolution of *M* is given by [25, 35]:

$$ \frac{\mathrm{d}}{\mathrm{d}t}M=\mathbb{H} (B_{0} ) \cdot M- \varGamma (M-M_{ss}) $$

(2)

with \(\mathbb{H} (B_{0} )\) the matrix describing the magnetic evolution, the total relaxation rate \(\varGamma =\varGamma _{e}+\gamma _{p}\) and the pumping steady-state is \(M_{ss}=\gamma _{p}/\varGamma (0,0,1/\sqrt{6},0,0)^{t}\). The optical pumping rate \(\gamma _{p}=\eta I\) with *I* the light intensity and *η* a constant. The evolution of *M* causes the light absorption to vary. For an input intensity \(I_{0}\), the intensity at the output of the cell will be \(I_{0}-\Delta I\) with [40]:

$$ \Delta I=I_{0} \alpha _{r} \biggl[\frac{m_{0}^{(0)}}{\sqrt{3}}+ \frac{m_{0}^{(2)}}{\sqrt{6}}-\Re \bigl(m_{2}^{(2)} \bigr) \biggr], $$

(3)

where \(\alpha _{r}\) a constant which depends on the optical transition, and the quantization axis is set along the propagation axis *z*.

In order to probe this approach we have recorded a set of Hanle-effect resonance curves for growing light intensities from 35 *μ*W/cm^{2} up to 1.96 mW/cm^{2}. For each light power, a quasi-static ramp of the \(B_{z}\) component of the magnetic field is applied to the cell, and the photodetection signals are recorded with a National Instruments DAQmx board at the output of the transimpedance amplifier. These Hanle resonance curves are shown in Fig. 2.

This data is compared to the theoretical predictions resulting from Eqs. (2) and (3). The only free parameters of this model are the proportionality constants *η* and \(\alpha _{r}\) which are common for the whole data set: we have fitted them on the resonance curve corresponding to the lowest light intensity. The resulting theoretical curves show a fair agreement fort the lowest light intensities, which is progressively degraded for larger light intensities. A qualitative explanation for these disagreements is the following. In this model the effective relaxation \(\gamma _{p}\) does only depend on the light intensity, and not on the system alignment *M*. This is only accurate as far as the system is far from a fully aligned state, i.e. as far as optical power remains very low.

For larger optical powers, the complete equations describing optical pumping [41, 42], suggest that it may be possible to refine the previous model by considering the anisotropic nature of the effective relaxation brought by the pumping. The equation of evolution thus becomes:

$$ \frac{\mathrm{d}}{\mathrm{d}t}M = \mathbb{H} (B_{0} ) \cdot M -\mathbb{R}\cdot M + \gamma _{p} M_{ss}, $$

(4)

where \(\mathbb{R}=\mathbb{R}^{(2)}+\varGamma _{e}\mathbb{I}\) with \(\mathbb{I}\) the identity matrix and \(\mathbb{R}^{(2)}\) is a \(5 \times 5\) matrix which accounts for relaxation due to the optical pumping. From the complete equations of optical pumping [42], this latter is found to be for \(D_{0}\) line of helium-4

$$ \mathbb{R}^{(2)}= \begin{pmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{3}{2}\gamma _{p} & 0 & 0 & 0 \\ 0 & 0 & 2\gamma _{p} & 0 & 0 \\ 0 & 0 & 0 & \frac{3}{2}\gamma _{p} & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix} $$

(5)

and the pumping steady-state \(M_{ss}=(0,0,\sqrt{2/3},0,0)^{t}\), both with quantization axis parallel to the light polarization. The pumping rate is:

$$ \gamma _{p}=I_{0}\frac{2\pi ^{2}r_{e}cf}{\hbar \omega }\Re \bigl[ \mathcal{V} ( \omega -\omega _{0} ) \bigr] $$

(6)

with \(r_{e}\) the electron classical radius, *f* the oscillator strength of the transition, *ω* and \(\omega _{0}\) the angular frequencies of the light and of the transition respectively, ℜ is the real part and \(\mathcal{V}\) the complex Voigt profile function. Since the same beam is used as pump and probe, it is possible to write \(\alpha _{r}\) in terms of \(\gamma _{p}\): \(\alpha _{r}=3\alpha \gamma _{p}\) with \(\alpha =\omega l n \hbar /I_{0}\), where *l* is the inner cell length and \(n_{m}\) the density of atoms in the metastable state [40].

A first measurement at 9 *μ*W optical power allows to deduce the metastable density \(n_{m}\) and a second one at 37 *μ*W, the ratio between the effective pumping rate in the double-pass configuration and the \(\gamma _{p}\) given by Eq. (6). Except for these independently-measured parameters, the model contains no free parameters at all. We have calculated the resonance curves from Eq. (4) for the optical powers used in the experiments described above (Fig. 2). The agreement with experimental data is much better than with the previous model. Figure 2 also shows the widths of the Hanle resonances. For growing powers the experimentally measured widths become narrower than the three step approach prediction. Including an anisotropic relaxation term as explained above allows a very good agreement between the predicted widths and those observed.

### 3.2 Parametric resonance

We now consider the case of practical interest which is the PRM. Thanks to the dressed atom formalism, PRM equations of motion can be mapped to the ones of the Hanle effect of the atom dressed by the two RF fields [16, 25].

For optical powers larger than those described by the three step approach, this equivalence remains valid if, in addition to the usual conditions, all the elements of the \(\mathbb{R}\) matrix to remain much lower than the RF frequencies *ω* and *Ω*. If this condition is fulfilled, the dressed anisotropic relaxation matrix \(\mathscr{R}\) can be found to be [43]:

$$ \mathscr{R}_{q,q'}=\sum_{r=-\infty }^{{+\infty }}J_{r,q} \mathbb{R}_{q,q'}J_{r,q'}=J_{0, \vert q-q' \vert }\mathbb{R}_{q,q'}, $$

(7)

where the second equality comes from Graf’s theorem [44]. The PRM equations can then be solved as usual.

For single-RF parametric resonance (the scheme of Fig. 1(b) with \(B_{2}=0\)), the absorption signal at odd harmonics of the RF frequency *ω* is sensitive to the \(B_{z}\) component of the field. From Eq. (5), (7) and [25] one can find around null field:

$$ \Delta I=\frac{48 I_{0}\alpha }{2\varGamma _{e}+3\gamma _{p}}\times \frac{\gamma B_{z}\varGamma _{e}\gamma _{p}^{2}J_{0,2}J_{p,2}}{ (2\varGamma _{e}+\gamma _{p} ) (2\varGamma _{e}+3\gamma _{p} )-3\gamma _{p}^{2}J_{0,2}^{2}} \sin (p\omega t ), $$

(8)

where *p* is an odd integer and \(J_{q,q'}=J_{q}(q'\gamma B_{1}/\omega )\) with \(J_{q}(x)\) the Bessel function of first kind and order *q*.

To probe if this description of PRM is accurate we have acquired a set of parametric resonance dispersive response curves at different optical powers, in the presence of a single RF field. To do so we slowly varied the \(B_{z}\) component of the magnetic field in the presence of a RF field along the *z* axis, of 20 kHz frequency and \(B_{1}=\omega /(2\gamma )\). The photodetection signal at the output of the transimpedance amplifier was sent to the input of a Zurich MFLI lock-in amplifier. The output of this lock-in is then acquired with the same National DAQmx board used before.

The resulting resonance curves are plotted with crosses in Fig. 3(a). The theoretical predictions with no free parameters are superposed to this data set, and show an excellent agreement with the experimental data in the central null-field region. The agreement becomes slightly worse for high light intensities and larger magnetic field. We believe this may be due to slight imperfections on the cancellation of the transverse components of the magnetic field.

The Fig. 3(b) displays the slopes of the linear low-field portion of the Fig. 3(a). The agreement is very good for light power up to 1 mW (i.e. an intensity of 3.9 mW/cm^{2}), which is well above the light power which optimizes the magnetometer signal [14]. For larger powers a disagreement is clearly visible, probably due to the incipient saturation of the optical transition. Indeed the saturation intensity in our system is 7.42 mW/cm^{2}, corresponding to an integrated power of 2.1 mW.

The two RF case can be addressed in the very same way, dressing the atom with the faster RF field first, then rotating the quantization axis from *z* to *y* and finally dressing the atom with the slower RF field. The resulting relaxation matrix is \(\mathscr{R}=\varGamma _{e} \mathbb{I}+\gamma _{p} \mathbb{D}\), where \(\mathbb{D}\) is

$$ \begin{pmatrix} \frac{3}{4} (1+J_{0,2} ) & 0 & \sqrt{\frac{3}{32}} (1+J_{0,2} )\mathcal{J}_{0,2} & 0 & 0 \\ 0 & \frac{3}{8} (3-J_{0,2} ) & 0 & \frac{3}{8} (1+J_{0,2} )\mathcal{J}_{0,2} & 0 \\ \sqrt{\frac{3}{32}} (1+J_{0,2} )\mathcal{J}_{0,2} & 0 & \frac{1}{4} (5-3J_{0,2} ) & 0 & \sqrt{\frac{3}{32}} (1+J_{0,2} )\mathcal{J}_{0,2} \\ 0 & \frac{3}{8} (1+J_{0,2} )\mathcal{J}_{0,2} & 0 & \frac{3}{8} (3-J_{0,2} ) & 0 \\ 0 & 0 & \sqrt{\frac{3}{32}} (1+J_{0,2} )\mathcal{J}_{0,2} & 0 & \frac{3}{4} (1+J_{0,2} ) \end{pmatrix}. $$

(9)