Magneto-optical rotation: accurate approximated analytical solutions for single-probe atomic magnetometers

Lu Deng; Claire Deng

doi:10.1364/OE.456252

1. Introduction

Magneto-optical Faraday rotation describes the rotation of the polarization plane of a linearly polarized light field traversing through a magnetized medium [1]. When a laser was first used as the light source, it was found that the angle of polarization plane rotation was dependent upon the intensity of the laser, hence given the term “nonlinear magneto-optical rotation" effect (NMOR effect) [2]. For the past 60 years, atomic NMOR effect studies have primarily focused on a single probe laser interacting with $F=1$ atomic systems [3–5]. Many pioneering studies and innovations [6–14] have contributed to the advancement in this research field. However, to date there has been no analytic solution$-$or even an approximated analytic solution with sufficient accuracy$-$for NMOR phenomena. This is especially the case when the laser is tuned close to the relevant one-photon resonance in order to enhance the observability of NMOR signal [7]. Such a near resonance excitation inevitably introduces significant power broadening, rendering analytic solutions very difficult. Consequently, only numerical solutions are available for such near resonance excitation operations [3]. Recent experimental and theoretical progress [15,16], especially the demonstration of an NMOR blockade in a single-probe configuration, has for the first time opened the possibility for analytical or approximated analytical solutions with high accuracy for single-probe NMOR effects. In this work we show a high-accuracy approximate analytical solution for widely studied single-probe atomic systems with propagation effect, hyperfine levels as well as power broadening included and for wide range of laser powers and detunings.

2. Theory

The theoretical framework presented here encompasses both three and four-state systems for both near and far-detuned optical fields [see Fig. 1(a)].

Fig. 1. (a) Rubidium D-1 transitions relevant to single-probe atomic magnetometers. (b) A four-state model single-probe atomic magnetometer. Dashed blue arrows represent two opposite two-photon transitions. Population symmetry as well as transition symmetry lead to an NMOR blockade that strongly suppresses NMOR signal.

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We consider a four-state atomic system, depicted in Fig. 1(b), where the atomic state $|j\rangle$ has energy $\hbar \omega _j$ ($j=1,\ldots,4$) and the lower three states form an $F=1$ system. We assume that the probe field $\mathbf {E}_{p}$ (frequency $\omega _{p}$) is polarized along the $\hat {x}$-axis and propagates along the $\hat {z}$-axis. Its $\sigma ^{(\pm )}$ components independently couple the $|1\rangle \Leftrightarrow |2\rangle$ and $|3\rangle \Leftrightarrow |2\rangle$ transitions with a large one-photon detuning $\delta _2=\omega _{p}-[\omega _2-(\omega _1+\omega _3)/2]=\delta _p$. The second excited state $|4\rangle$ is assumed to be $\Delta _0$ above the state $|2\rangle$, enabling the second excitation channel by the same probe field with a detuning of $\delta _4=\omega _{p}-[\omega _4-(\omega _1+\omega _3)/2]=\delta _{p}+\Delta _0$ (in our notation $\delta _{p}<0$ and $\Delta _0=-\omega _4+\omega _2<0$). This atomic system describes, for example the Rubidium D-1 line with a hyper-fine splitting, i.e., $|2\rangle =|5P_{1/2}, F'=1\rangle$ and $|4\rangle =|5P_{1/2}, F'=2\rangle$ with $|\Delta _0|=820$ MHz [Fig. 1(a)]. Since initially the population is equally shared by the two ground states $|F=1, m_F=\pm 1\rangle$, therefore two opposite two-photon transitions between states $|1\rangle$ and $|3\rangle$ with a two-photon detuning $2\delta _B$ are simultaneously established. Here, the circular components of the probe field simultaneously access excited states $|2\rangle$ and $|4\rangle$ with different excitation rates. The magnetic field induced Zeeman frequency shift $\delta _B=g\mu _0 B$ in the axial magnetic field $B=B_z$ is defined with respect to the mid-point between the two equally but oppositely shifted Zeeman levels $|1\rangle =|m_F=-1\rangle$ and $|3\rangle =|m_F=+1\rangle$.

Under the electric-dipole approximation, the system interaction Hamiltonian reads

(1)$$\frac{\hat{H}}{\hbar}\!=\!\sum_{j=1}^{4}\delta_j|j\rangle\langle j|\!+\!\sum_{m=2,4}\!\left[\Omega_{m1}|m\rangle\langle 1|\!+\!\Omega_{m3}|m\rangle\langle 3|\!+\!{\rm c.c}\right],\:$$

where $\delta _j$ is the laser detuning from state $|j\rangle$. The total electric field is given by $\mathbf {E}\!=\!\left (\mathbf {\hat {e}}_{+}{\cal E}_{p}^{(+)}\!+\!\mathbf {\hat {e}}_{-}{\cal E}_{p}^{(-)}\right ){\rm e}^{i\theta _{p}}\!+\!{\rm c.c.}$ where $\theta _{p}\!=\!\mathbf {k}_{p}\!\cdot \mathbf {r}-\omega _{p}t$ with $k_{p}\!=\omega _{p}/c$ being the wavevector of the field. Expressing probe Rabi frequencies with respect to the state $|2\rangle$ as $\Omega _{21}\!=\!\Omega _{p}^{(+)}\!=\!D_{21}{\cal E}_{p}^{(+)}/2\hbar$ and $\Omega _{23}\!=\!\Omega _{p}^{(-)}\!=\!D_{23}{\cal E}_{p}^{(-)}/2\hbar$, we then have $\Omega _{41}\!=\!D_{41}{\cal E}_{p}^{(+)}/(2\hbar )\!=\!d_{42}\Omega _{p}^{(+)}$ and $\Omega _{43}\!=\!D_{43}{\cal E}_{p}^{(-)}/(2\hbar )\!=\!d_{42}\Omega _{p}^{(-)}$. Here, we define $d_{42}=D_{41}/D_{21}=D_{43}/D_{23}$ with $D_{nm}=\langle n|\hat {D}|m\rangle$ being the transition matrix element of the dipole operator $\hat {D}$ and $D_{21}=D_{23}$, $D_{41}=D_{43}$.

Under rotating wave approximation, the Schrodinger equations describing wave-function amplitudes are,

(2a)$$\dot{A}_{1}-i\delta_BA_1=i{\Omega_{p}^{(+)}}^{*}\left(A_2+d_{42}A_4\right)-\gamma A_1,$$

(2b)$$\dot{A}_{3}+i\delta_BA_3=i{\Omega_{p}^{(-)}}^{*}\left(A_2+d_{42}A_4\right)-\gamma A_3,$$

(2c)$$\dot{A}_{2}-i\delta_{p}A_2=i\Omega_{p}^{(+)}A_1+i\Omega_{p}^{(-)}A_3-\Gamma A_2,$$

(2d)$$\dot{A}_{4}\!-\!i\!\left(\delta_{p}\!+\!\Delta_0\right)\!A_4\!=\!id_{42}\!\left(\Omega_{p}^{(+)}A_1\!+\!\Omega_{p}^{(-)}A_3\right)\!-\!\Gamma\!A_4,$$

where for simplicity we have expressed the ground and excited states’ decay rates as $\gamma$ and $\Gamma$, respectively.

Applying the slowly varying envelope approximation and the third-order perturbation calculation [16,17], we obtain the Maxwell equations describing the evolution of both circular polarized components of the probe field $\mathbf {E}_{p}$ in the moving frame ($\xi =z-ct$, $\eta =z$),

(3)$$\frac{\partial\Omega_{p}^{({\pm})}}{\partial\eta}\approx{-}\sum_{n=2,4}\alpha_n\Omega_{p}^{({\pm})}\!+\!\sum_{n=2,4}\!\frac{\alpha_{n}}{2}(1+id_{n})S_{{\mp}}\Omega_{p}^{({\pm})}\!\!\left(\!\frac{\Gamma_{0}\!+\!i\beta_{0}}{\Gamma_{-}\!+\!i\beta_{-}}\!+\!\frac{\Gamma_{0}\!-\!i\beta_{0}}{\Gamma_{+}\!+\!i\beta_{+}}\!\right),$$

where $\alpha _{n}=\kappa _{n}/\Gamma (1+d_{n}^{2})$, $\kappa _{2}=\kappa _{23}$ and $\kappa _{4}=\kappa _{43}$ ($\kappa _2=2\pi |D_{21}|^{2}{\cal N}_0\omega _p/(\hbar c$) where ${\cal N}_0$ is the atom number density). The first summation on the right accounts for the linear absorption (we have neglected the linear propagation phase shift since it does not contribute to NMOR effects). The normalized detunings are defined as $d_{2}=\delta _{p}/\Gamma$, $d_{4}=(\delta _{p}+\Delta _0)/\Gamma =d_2+\Delta _0/\Gamma$, and $d_{B}=\delta _B/\gamma$. In addition, we have defined $\beta _{\pm }=-d_B\mp \beta _0S^{(\pm )}$, $\Gamma _{\pm }=1+\Gamma _0S^{(\pm )}$, $\beta _0=d_{2}/(1+d_2^{2})+d_4/(1+d_4^{2})$, and $\Gamma _0=1/(1+d_2^{2})+1/(1+d_4^{2})$ where $\beta _0$ and $\Gamma _0$ are the total light induced frequency shift and resonance broadening, respectively. The probe two-photon saturation parameters are defined as $S_{\pm }=|\Omega _{p}^{(\pm )}|^{2}/\gamma \Gamma$.

Letting $\Omega _{p}^{(\pm )}=R_{\pm }\,e^{i\theta _{\pm }}$ where $R_{\pm }$ and $\theta _{\pm }$ are real quantities, then Eq. (3) gives

(4a)$$\frac{\partial S_{{\pm}}}{\partial\eta}={-}\alpha S_{{\pm}}\!+\!S_{+}S_{-}{\cal P}_{{\pm}},$$

(4b)$$\frac{\partial\theta_{{\pm}}}{\partial\eta} =\frac{S_{{\mp}}}{2}{\cal Q}_{{\pm}}.$$

where ${\cal P}_{\pm }=\alpha {\cal A}\mp \alpha _{d}{\cal B}$, ${\cal Q}_{\pm }=\mp \alpha {\cal B}+\alpha _d{\cal A}$ with ${\cal A}\!=\!\Gamma _0(\Gamma _{-}/G_{-}\!+\!\Gamma _{+}/G_{+}\!)$, ${\cal B}\!=\!\Gamma _0(\beta _{-}/G_{-}\!+\!\beta _{+}/G_{+}\!)$, $\alpha =\alpha _2+\alpha _4$, $\alpha _d=d_2\alpha _2+d_4\alpha _4$, and $G_{\pm }=\Gamma _{\pm }^{2}+\beta _{\pm }^{2}$ (in the following calculations and without the loss of generality we neglect $\beta _0$ terms [18]). The advantage of this photon number representation is that intensity [i.e., Eq. (4a)] decouples from the phase [i.e., Eq. (4b)]. While both Eqs. (4a) and (4b) are highly nonlinear and complex, approximated analytical solutions can be obtained with excellent accuracy. As we show below, two key steps separately based on the physics of the single-probe system and mathematical considerations for approximation accuracy are necessary to achieve this.

The first key step is to realize the presence of a symmetry-enforced NMOR blockade in any single-probe system where population and transition symmetry are present [19]. We note that the probe field is the only energy source and therefore its two circular components must add up at any propagation distance to give $S_{-}(\eta )=S_0e^{-\alpha \eta }-S_{+}(\eta )$ where $S_0=\Omega _{p}(0)^{2}/\gamma \Gamma$ is the initial total energy of the single probe field. Taking the differential equation for the $S_{+}$ component in Eq. (4a) and inserting this energy restriction relation into intensity product in the second term on the right side of Eq. (4a), we immediately conclude a gain-clamping effect, i.e., $(S_0e^{-\alpha \eta }-S_{+})S_{+}$. This energy constraint locks the two probe components by enacting a self-restricted growth, resulting in $S_{\pm }(\eta )\approx S_{\pm }(0)=S_0(0)/2$ (i.e., the input value if linear absorption is neglected). That is, no appreciable magnetic field induced optical field change is allowed for either component (see numerical results and discussion later) [20]. It is this propagation growth restriction that limits the single-probe NMOR to be in characteristically linear. This is a single-probe blockade first postulated in Ref. [15] and then demonstrated mathematically in Ref. [16]. As we show below, it is this NMOR blockade and the linear absorption characteristics of both field components that lead to a high-accuracy approximate analytical solution that is well-applicable to both three and four-state atomic systems.

The second key step is based on the mathematics consideration of ${\cal P}_{\pm }$ and ${\cal Q}_{\pm }$ which contain $S_{\pm }(\eta )$. One of the consequences of the above described NMOR blockade is that the dominant propagation behavior is determined by linear absorption since the gain is clamped. The magnetic field is contained in functions ${\cal P}_{\pm }$ and ${\cal Q}_{\pm }$, and its effect should mostly be near magnetic resonance. Therefore, it is a reasonable expectation that errors by replacing $S_{\pm }(\eta )$ in ${\cal P}_{\pm }$ and ${\cal Q}_{\pm }$ with $S_{\pm }(\eta )\rightarrow (S_0/2)e^{-\alpha \eta }$ should be relatively small. We note that this replacement step has two benefits. One is the improved accuracy by propagation effect in comparison with simply using $S_{\pm }(\eta )\approx S_0/2$. The second benefit is that it also reveals the operation consideration between reduction of the NMOR effect signal strength, which reduces the detection ability, and the reduction of laser power broadening which improves the detection sensitivity. As the first trial we take $S_{\pm }(\eta )=(S_{0}/2)e^{-\alpha \eta }$ in denominators of ${\cal P}_{\pm }$ and ${\cal Q}_{\pm }$.

The major benefit of the second step is that now both ${\cal P}_{\pm }$ and ${\cal Q}_{\pm }$ contain $e^{-\alpha \eta }$ only, permitting analytical solutions to both field and NMOR angle. We get

(5)$$S_{{\pm}}(\eta; d_B)=\frac{S_0e^{S_0\Gamma_0{\cal L}_{{\pm}}}}{1+S_0e^{S_0\Gamma_0{\cal L}_{{\pm}}}}e^{-\alpha\eta},$$

where

{\cal L}_{{\pm}}(\eta; d_B)=2\Gamma_0\int_{0}^{\eta}\frac{\alpha(1+\Gamma_0S_0e^{-\alpha\eta}/2)\mp\alpha_dd_B}{d_B^{2}+(1+\Gamma_0S_0e^{-\alpha\eta}/2)^{2}}d\eta,

is analytically integrable. Inserting Eq. (5) into Eq. (4b), we obtain

\theta_{{\pm}}(\eta; d_B)\!=\!\Gamma_0\!\int_{0}^{\eta}\!\!S_{{\mp}}(\eta; d_B)\frac{\alpha_d(1\!+\!\Gamma_0S_0e^{-\alpha\eta}/2)\!\pm\!\alpha d_B}{d_B^{2}\!+\!(1\!+\!\Gamma_0S_0e^{-\alpha\eta}/2)^{2}}d\eta.

Using symbolic evaluation routines on MATLAB or MATHEMATICA we obtain an analytical expression for the NMOR angle [21,22]

(6)$$\begin{aligned} \Theta(\eta; d_B)_{_{\rm NMOR}}= & -C_0\left(\frac{8}{15}\right)\frac{e^{-\alpha\eta}\Gamma_0S_0}{2\alpha(1+d_B^{2})}\left\{i\left(\frac{\alpha+\alpha_dd_B}{1+{\cal M}_3{\cal M}_4^{i\frac{2\alpha_d}{\alpha}}}+\frac{\alpha-\alpha_dd_B}{1+{\cal M}_3{\cal M}_4^{{-}i\frac{2\alpha_d}{\alpha}}}\right){\rm ln}{\cal M}_1\right.\\ & \left.-\left(\frac{\alpha d_B-\alpha_d}{1+{\cal M}_3{\cal M}_4^{i\frac{2\alpha_d}{\alpha}}}+\frac{\alpha d_B+\alpha_d}{1+{\cal M}_3{\cal M}_4^{{-}i\frac{2\alpha_d}{\alpha}}}\right){\rm ln}{\cal M}_2\right\}, \end{aligned}$$

where the amplitude adjustment constant $C_0$ is very close to unity and has a narrow range (typically $<\pm 5\%$) depending on the choices of relative transition strength, laser power and detuning (see numerical results below). In addition,

(7a)$${\cal M}_1\!=\!\left[\frac{1\!+\!d_B^{2}\!+\!(1\!+\!id_B)\frac{\Gamma_0S_0}{2}}{1\!+\!d_B^{2}\!+\!(1\!-\!id_B)\frac{\Gamma_0S_0}{2}}\right]\!\left[ \frac{1\!+\!d_B^{2}\!+\!(1\!-\!id_B)e^{-\alpha\eta}\frac{\Gamma_0S_0}{2}}{1\!+\!d_B^{2}\!+\!(1\!+\!id_B)e^{-\alpha\eta}\frac{\Gamma_0S_0}{2}}\right],$$

(7b)$${\cal M}_2=\left[\frac{d_B^{2}+(1+\frac{\Gamma_0S_0}{2})^{2}}{d_B^{2}+(1+e^{-\alpha\eta}\frac{\Gamma_0S_0}{2})^{2}}\right]e^{{-}2\alpha\eta},$$

(7c)$${\cal M}_3=\frac{\left[d_B+i(1+\frac{\Gamma_0S_0}{2})\right]\left[d_B-i(1+e^{-\alpha\eta}\frac{\Gamma_0S_0}{2})\right]}{\left[d_B-i(1+\frac{\Gamma_0S_0}{2})\right]\left[d_B+i(1+e^{-\alpha\eta}\frac{\Gamma_0S_0}{2})\right]},$$

(7d)$${\cal M}_4=\left[\frac{d_B^{2}+(1+e^{-\alpha\eta}\frac{\Gamma_0S_0}{2})^{2}}{d_B^{2}+(1+\frac{\Gamma_0S_0}{2})^{2}}\right]^{2}.$$

When the power broadening terms $\left (\frac {\Gamma _0S_0}{2}\right )$ in ${\cal M}_m$ ($m=1,..,4$) are neglected, we have ${\cal M}_1\rightarrow 1$, ${\cal M}_2\rightarrow e^{-2\alpha \eta }$, ${\cal M}_3\rightarrow 1$, and ${\cal M}_4\rightarrow 1$. Consequently, Eq. (6) gives

(8)$$\Theta(\eta; d_B)_{_{\rm NMOR}}\approx{-}\frac{\alpha d_B}{2(1+d_B^{2})}\Gamma_0S_0e^{-\alpha\eta}\eta,$$

which is the NMOR angle for a four-level single-probe AM with power broadening neglected but linear absorption included [16]. When the state $|4\rangle$ is neglected, $\alpha =\alpha _2$, $\Gamma _0=1/(1+d_2^{2})$, and we recover the NMOR angle of a three-level single-probe AM without power broadening. Equations (5) and (6) are the most accurate approximated analytical solution to NMOR effect of three and four-state systems to date. The validity and accuracy of Eqs. (5) and (6) have been thoroughly verified using a 4th order Runge-Kutta numerical differential equation solver on MATLAB, as well as a MATHEMATICA symbolic evaluation routine for broad probe power and detuning ranges [23].

3. Comparison with numerical calculations

Figure 2(a) shows the NMOR angle of a four-state single-probe system obtained by a 4th-order Runge-Kutta code that simultaneously solves Eqs. (4a) and (4b). Figure 2(b) shows the NMOR angle evaluated using Eq. (6) with the same parameters. The small difference validates the proposal of replacing $S_{\pm }(\eta )\rightarrow (S_{0}/2)e^{-\alpha \eta }$ in ${\cal P}_{\pm }$ and ${\cal Q}_{\pm }$. When the numerical and the approximated analytical solutions are plotted at $\eta =10$ as a function of $d_B$ we have found that with $C_0=1.05\pm 0.02$ excellent agreement between the two methods in a broad probe detuning and power regions [also see Fig. 3], a testimonial of the accuracy of Eq. (6).

Fig. 2. (a) NMOR angle by numerically solving Eqs. (4a) and (4b) simultaneously. (b) NMOR angle evaluated using Eq. (6) by MATLAB. The two results are near identical (only differ by $<5\%$), indicating the excellent accuracy of the approximated analytical solution. Parameters: $S_0=5$, $d_2=-5$, $\Delta _0=-2$, $\kappa _4/\kappa _2=2$ ($\kappa _2=10^{5}$/cm.s), and $C_0=1.06$.

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Fig. 3. NMOR angle line profiles at $\eta =10$ by Eqs. (4a) and (4b) (black) and Eq. (6) (red). Figures 3(a) and 3(b): $d_2=-2$ and probe powers are $S_0$=2 and 10, respectively. Here, a single amplitude adjustment constant $C_0=1.17$ is used for Eq. (6). Figures 3(c) and 3(d): $S_0$=10 and probe detunings are $-5$ and $-10$, respectively, with $C_0=1.15\pm 0.02$ is used in Eq. (6).

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Figures 3(a)–3(d) show comparisons of numerical solutions with approximate analytical solutions for different probe powers and detunings. When $\kappa _4/\kappa _2=1$ the analytical solution achieves excellent agreement with the numerical solution in wide ranges of probe power and detuning for $C_0=1.00\pm 0.03$. When $\kappa _4/\kappa _2=2$, $d_2=-2$ we obtain remarkably accurate agreements in the probe power range of $S_0=$2 to 10 for a single amplitude adjustment constant of $C_0=1.17\pm 0.01$. For a fixed probe power of $S_0=10$, excellent agreements are obtained for a range of probe detuning $d_2$ from $-2$ to $-10$ with $C_0=1.15\pm 0.02$ [24]. Notice that the approximate analytical solution (red) show a slightly broader line shape than the numerical solution (black). This is precisely because the replacement slightly overestimates the power-broadened line width.

It is quite remarkable that accurate results from Eq. (6) can be obtained in such a tight range of amplitude adjustment constant $C_0$ for broad ranges of laser power and detuning. Indeed, with laser power varying from $S_0=$2 to 10 and in the probe detuning range of $d_2=-2$ to $-10$, Eq. (6) yields accurate results that are all within $\pm 5\%$ of that by the full numerical calculations. Figure 4 compares the numerical solution [Eqs. (4a) and (4b), solid black], analytical solution (Eq. (6), dashed red) and the simplified solution (Eq. (8), dash-dotted blue). For small probe detuning, which is the necessary operation condition for a single-probe scheme reported in [7], the simplified solution without power broadening does not agree with neither numerical solution nor the approximated analytical solution [Fig. (4a)]. However, when the probe detuning increases, all three methods approach the same result. We note that even though a small probe detuning [7] enhances NMOR signal amplitude, the power broadened line shape leads to a significant reduction in magnetic field detection sensitivity (i.e., much smaller slope near zero field). With a large probe detuning, the detection sensitivity is preserved but the NMOR signal amplitude is reduced significantly [Fig. 4(b)]. The recently reported colliding-probe bi-atomic magnetometer experiments and theory [15,16] overcome these issues, exhibiting excellent NMOR signal SNR, as well as increased field detection sensitivity at body temperature.

Fig. 4. NMOR angle line profiles obtained by numerical solving Eqs. (4a) and (4b) (black) and by evaluating the analytical solution Eq. (6) (red) as well as Eq. (8) (blue, power broadening is neglected). Here, $S_0=10$ and $\kappa _4/\kappa _2=2$ and probe detunings are (a): $d_2=-2$. (b): $d_2=-5$. As the probe detuning increases both numerical solution and the analytical solution approach that of Eq. (8). For large detunings, three results are indistinguishable.

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4. Conclusion

In conclusion, we have obtained the most accurate approximated analytical solutions to date for NMOR effects of three and four-state systems with both probe absorption and power broadening. When the probe power broadening is neglected, we recover the known single-probe three and four-state system NMOR effects. These general analytical solutions allow one to analyze multi-state magneto-optical rotation processes with excellent accuracy, revealing detailed effects and impacts of laser detuning, atomic hyper-fine splitting, as well as the probe power broadening on NMOR signal strength and magnetic field detection sensitivity.

Acknowledgments

Claire Deng thanks Dr. Changfeng Fang (SDU) for technical assistance on MATLAB coding. LD acknowledges the financial support from SDU.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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13. M. V. Balabas, T. Karaulanov, M. P. Ledbetter, and D. Budker, “Polarized Alkali-Metal Vapor with Minute-Long Transverse Spin-Relaxation Time,” Phys. Rev. Lett. 105(7), 070801 (2010). [CrossRef]

14. A. Korver, R. Wyllie, B. Lancor, and T. G. Walker, “Suppression of Spin-Exchange Relaxation Using Pulsed Parametric Resonance,” Phys. Rev. Lett. 111(4), 043002 (2013). [CrossRef]

15. F. Zhou, C. J. Zhu, E. W. Hagley, and L. Deng, “Symmetry-breaking inelastic wave-mixing atomic magnetometry,” Sci. Adv. 3(12), e1700422 (2017). [CrossRef]

16. L. Deng, “Colliding-probe bi-atomic magnetometry: Breaking symmetry-based NMOR blockade,” (submitted).

17. Y.R. Shen, The Principles of Nonlinear Optics, (John Wiley & Sons, Chapters 9 and 10, New York1984).

18. This is because frequency shifts cancels out in the calculations of NMOR angle which is the difference between the phase of two probe components. We verified this by comparing numerical calculations with and without β₀-dependent terms.

19. Most, if not all, warm vapor single-probe atomic magnetometers fall in this category.

20. For a 10-cm cell the probe field change is limited to only a few percent due to the single-probe-based NMOR blockade.

21. More rigorously, with S_± and θ_± obtained from Eq. (5) and the expression after the corresponding Stocks parameters for the probe field E(η;δ_B) can be constructed as usual and the final rotation angle of the probe polarization plane, i.e., $\Theta _(\eta ;\delta _B)$ can be obtained. Due to the NMOR blockade, the NMOR signal is very small (see numerical results in Figs. 2 and 3) the simplified Eq. (6) rather than the full expression using Stocks parameter is sufficiently accurate.

22. While NMOR angle Eq. (6) contains complex quantities we have verified that it indeed gives correct results. Inspections of real and imaginary parts of NMOR angle evaluated using Eq. (6) reveal extremely small residual imaginary parts on the order of 10⁻¹² ∼ 10⁻¹⁵. Therefore, we are justified to only plot the real part of Eq. (6) in Figs. 2–4.

23. For warm vapor experiments such as reported in Ref. [7] with sophisticated 5-layer magnetic shielding and RF modulation technique the laser is typically detuning by about δ_p=500 MHz (only about 1.5 Doppler linewidth) from F′ = 1 states. Since Rubidium hyper-fine splitting is |Δ₀|=820 MHz, which is about 2 Doppler linewidths, we therefore chose Δ₀=-2 in unit of Γ.

24. For instance, we first choose C₀ = 1.15 in Eq. (6) to obtained excellent agreement with numerical calculations for d₂ = −5. Then, with a very narrow range of C₀ = 1.15 ± 0.02 excellent agreements with numerical calculations can be obtained for the entire detuning range from d₂ = −2 to −15.

Magneto-optical rotation: accurate approximated analytical solutions for single-probe atomic magnetometers

Abstract

1. Introduction

2. Theory

3. Comparison with numerical calculations

4. Conclusion

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Equations (17)

Optics Express