1. Introduction

Magneto-optical effects are a class of physical effects that describe the interaction between magnetized materials and electromagnetic fields [1,2]. Aside from their importance in fundamental science, their applications in modern technologies have broadly impacted our daily lives. Optically accessed mass data storage media, geomagnetic studies of Earth and interstellar objects, detection of extremely weak magnetic fields, micro-Tesla magnetic resonance imaging, and even the search of cosmic particles are some of forefront fields in science and technology where magneto-optical effects play vital roles.

One of the well-known magneto-optical effects is the rotation of the polarization plane of a light field traversing a magnetized medium, an effect that was first observed by M. Faraday in 1845 and hence is referred to as the Faraday rotation. Atomic magnetometer (AM) is a class of magnetic field sensing devices that operates based on the principles of this Faraday magneto-optical rotation effect. The invention of the laser has fundamentally changed our knowledge of magneto-optical effects. For instance, it was found that the polarization rotation angle is dependent upon the intensity of the laser probing the magnetic field, a light power dependent effect for which the terminology of “nonlinear magneto-optical rotation" (NMOR) was given [3,4]. A survey of this AM research field has shown that ever since the first demonstration of this NMOR effect, AMs have primarily been a single-probe laser technique where a linearly polarized light field and an alkali atomic specie (usually, $F=1\rightarrow F^{'}=1$ transitions) form the core of the technique [5–7]. Since 1960, many pioneer studies and innovations [8–16] have contributed to the advancement of this vibrant field of physical science. However, to date, there has been no study that raises the question of why the NMOR effect, a nonlinear effect by name, does not have the characteristic nonlinear propagation dependency widely seen in nonlinear optics [17]. Here, we show that the root cause of this “nonlinear effect without nonlinear propagation characteristics" is a strong symmetry-enforced single-probe NMOR blockade. It is enacted by symmetries in atomic population distribution and transition rates, as well as cross-component energy conservation in a singe-probe excited $F=1\rightarrow F^{'}=1$ transition system. As a result, no significant energy circulation is permitted and the “nonlinear behavior" of NMOR effect is strongly suppressed. We further describe a novel colliding-probe atomic magnetometry scheme using a rigorous third-order nonlinear optics theoretical framework. We show that by adding a second probe field, this new magnetometry scheme can effectively lift the NMOR blockade, resulting in a significant directional energy circulation. Consequently, giant NMOR signal-to-noise ratio (SNR) enhancement and significant magnetic field sensitivity improvement can be achieved simultaneously. Remarkably, all experimental observations, including all single-probe AM based data known to date, can be analytically and qualitatively explained by this inelastic-wave-scattering-based [18,19] colliding-probe atomic magnetometry theory.

2. Summary of key experimental observations

The aim of this study is to develop a theoretical framework that can qualitatively and analytically predict or explain recent experimental observations of giant NMOR enhancements [20,21]. To facilitate readers we summarize key experimental observations in this short section.

Figures 1(a)–1(c) show three representative colliding-probe AM NMOR data. Figure 1(a) shows time-domain NMOR signals acquired using the colliding-probe AM (red trace) and a widely-studied single-probe AM (blue trace) under identical experimental conditions (i.e., the blue trace is acquired using identical operation parameters as used for the red trace except the second probe was turned off. For cases with small detunings, see discussions later). The nearly complete suppression of magnetic resonance in the blue trace indicates a strong and yet unknown signal suppression effect. Figure 1(b) is a re-plot of Fig. 1(a) using a significantly reduced vertical scale. Clearly, the NMOR signal from a single-probe AM (blue) is very small and noisy. The sharp vertical red trace with huge SNR is the NMOR signal from the colliding-probe AM. Typically, the ratio of the NMOR signal amplitude from the colliding-probe AM to that from a single-probe AM under the same operation conditions is larger than 20-dB [20–22]. Figure 1(c) shows Fast-Fourier Transform spectra of NMOR signals with (red) and without (blue) the second probe field, again indicating larger than 20-dB NMOR signal amplitude enhancement by the colliding-probe AM over the typical single-probe AM.

 

Fig. 1. Giant NMOR signal enhancement by the colliding-probe AM technique. (a) Time-domain NMOR spectra of the colliding-probe AM (red trace) and a single-probe AM (blue trace) under the same operation conditions (except the second probe is turned off). (b) Re-plot of (a) with a much smaller vertical scale showing very small and noisy single-probe AM NMOR signal. The nearly vertical red line is the signal from the colliding-probe AM. (c) NMOR signal FFT spectra: Colliding-probe AM (red, showing larger than 20-dB SNR enhancement) vs. single-probe AM (blue). Typically, the effective laser excitation rates are $<0.1\%$ of the corresponding one-photon on-resonance excitation rate (rubidium vapor temperature T=313 K).

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3. Theory of colliding-probe atomic magnetometry

Although experiments and corresponding numerical calculations have been published before [20,21] we believe that an analytical treatment is very desirable not only because it brings insights of underlying physics but also provides important guidance to future developments and applications of this new atomic magnetometry technique. Here, we focus on developing an analytical nonlinear optics inelastic-wave-scattering [19] theoretical framework. We show mathematically that all experimental observations reported with this colliding-probe AM to date can be predicted and explained even without complex numerical computations. Furthermore, we show that all experimental results known to date for single-probe AMs can also be analytically predicted and explained by this theory with high accuracy [23]. This is the first time that such high accuracy analytical solutions for single-probe NMOR and AM have ever been achieved since the demonstration of single-probe NMOR-based AMs more than 50 years ago.

We begin by considering a four-state atomic system [see Fig. 2(a)] where the atomic state $|j\rangle$ has energy $\hbar \omega _j$ ($j=1,\ldots,4$) and the lower three states form an $F=1\rightarrow F^{'}=1$ transition system. We assume that the probe field $\mathbf {E}_{p1}(\omega _{p1})$ is linearly-polarized along the $\hat {x}$-axis and it 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 $\delta _{p1}=\delta _2=\omega _{p1}-[\omega _2-(\omega _1+\omega _3)/2]$. Initially, the population is equally shared by the two ground states $|1\rangle =|F=1, m_F=-1\rangle$ and $|3\rangle =|F=1, m_F=+1\rangle$. 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 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 $|3\rangle =|m_F=+1\rangle$. A second probe field $\mathbf {E}_{p2}(\omega _{p2})$ propagates along the $-\hat {z}$-axis but has its linear polarization at an angle $\theta _0$ with respect to the $\hat {x}$-axis. Its $\sigma ^{(\pm )}$ components independently couple the $|3\rangle \Leftrightarrow |4\rangle$ and $|1\rangle \Leftrightarrow |4\rangle$ transitions with a large one-photon detuning $\delta _{p2}=\delta _4=\omega _{p2}-[\omega _4-(\omega _1+\omega _3)/2]$, again forming two opposite two-photon transitions between states $|1\rangle$ and $|3\rangle$ with the same two-photon detuning $2\delta _{B}$. More precisely, there are two ${\bf bi}$-${\bf directional}$ excitation channels via different upper excited state, i.e., $|1\rangle \stackrel {|2\rangle }{\Longleftrightarrow }|3\rangle$ and $|1\rangle \stackrel {|4\rangle }{\Longleftrightarrow }|3\rangle$. Each channel contains two competing two-photon transitions and the two channels share the ${\bf same}$ equally-populated states $|1\rangle$ and $|3\rangle$, creating a mutually influencing ground-state Zeeman coherence. We stress that the underlying physics of this colliding-probe AM technique is inelastic wave scattering [19], which is fundamentally different from the usual four-wave mixing or any electromagnetically-induced-transparency-based processes (see the Discussion Section). A close analogy of this inelastic wave scattering process is the $F=1\rightarrow F^{'}=1$ light scattering by an equally populated $F=1$ ground state spinors system of an atomic Bose-Einstein condensate.

 

Fig. 2. Schematic energy level diagrams for systems studied. (a) Energy level diagram of a colliding-probe AM showing energy circulation (large red and blue curved broken arrows). The probe $\mathbf {E}_{p1}$ couples $|5S_{1/2},F=1\rangle \rightarrow |5P_{1/2},F^{'}=1\rangle$ transition with a detuning of $-$5GHz. The probe $\mathbf {E}_{p2}$ couples $|5S_{1/2},F=1\rangle \rightarrow |5P_{3/2},F^{''}=2\rangle$ transition with a detuning of $-$3GHz. Horizontally displayed hollow arrows represent propagation directions of probe fields. The second probe (red color) enables an energy circulation that results in a significant NMOR SNR enhancement. (b) Energy level diagram of the conventional single-probe AM showing an NMOR blockade (the red X) due to symmetric restricted single-probe excitation. The reference point for Zeeman-shifted levels, i.e., the $|F=1,m_F=0\rangle$ state, is not shown in Fig. 2(a) since it does not contribute because of the transition selection rule. (c) Simplified experimental layout.

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Under the electric-dipole approximation, the interaction Hamiltonian describing the atom-light interaction scheme in Fig. 2(a) is given as

The third-order nonlinear optics calculation proceeds as follows [17]. First, using Eq. (1) in the Schrödinger equation and assuming large one-photon detunings (as used in experiments) we obtain adiabatic solutions of atomic state wave functions for states $|2\rangle$ and $|4\rangle$. We then seek third-order corrections to steady-state solutions of ground states $|1\rangle$ and $|3\rangle$ due to probe fields. Finally, the polarization source terms for Maxwell equations of circularly-polarized probe components $\Omega _{p1}^{(\pm )}$ and $\Omega _{p2}^{(\pm )}$ are constructed and properties of these differential equations are analyzed. These analysis give clear physical explanations to all experimental data shown in Figs. 1(a)–1(c), as well as all single-probe NMOR-based AM observations known to date.

Following the above prescribed methodology and under the slowly varying envelope approximation, we obtain the Maxwell equations describing the evolution of $\Omega _{p1}^{(\pm )}$ in the moving frame ($\xi =z-ct$, $\eta =z$) with linear absorption included

Writing $\Omega _{p1}^{(\pm )}=R_{\pm }\,e^{i\theta _{\pm }}$, $\Omega _{p2}^{(\pm )}=r_{\pm }\,e^{i\phi _{\pm }}$, where $R_{\pm }$, $\theta _{\pm }$, $r_{\pm }$ and $\phi _{\pm }$ are real quantities, expressing $S_{p1}^{(\pm )}=|R_{\pm }|^2/\gamma \Gamma$ and $S_{p2}^{(\pm )}=|r_{\pm }|^2/\gamma \Gamma$, then Eq. (2) gives

Letting $\Delta S_{p1}=S_{p1}^{(+)}-S_{p1}^{(-)}$ and $\Delta \theta =\theta _{+}-\theta _{-}$ we obtain (cp: colliding-probe; sp: single-probe)

4. Analysis and discussions

Remarkably, phenomena and underlying physics of all single-probe AMs known to date as well as the colliding-probe AM reported here can be qualitatively understood and well predicted by Eqs. (3a), (3b), (4a) and (4b) without carrying out complex numerical integration. In fact, Eqs. (4a) and (4b) separately give the colliding-probe AM heterodyne signal amplitude enhancement and theoretical magnetic field detection sensitivity limit.

We begin our analysis and discussions by first pointing out fundamental differences between the inelastic-wave-mixing process in colliding-probe AM and the usual optical four-wave mixing (FWM) [19] as well as electromagnetically-induced-transparency (EIT) [8] based processes.

A typical FWM process, such as third harmonic generation or double-$\Lambda$ frequency conversion schemes [19] routinely encountered in nonlinear optics, is an elastic process where the excitation starts from a single fully populated ground state and cycles back to the same ground state. Generally, in each cycle all intermediate states have negligible populations and there is no energy storage or population transfer in each cycle. The Zeeman coherence between any two states is negligibly small. In addition, all fields except the internally generated new field have no propagation effect other than linear absorption. In the case of parametric FWM generation where multiple fields are generated the growth of these new fields are highly nonlinear with strict phase matching requirement. The inelastic-wave-scattering based colliding-probe AM process does not have any of these behaviors.

The colliding-probe AM process is not related to any EIT-based processes. In fact, the cross-component optical power dependency known in all single-probe NMOR processes do not resemble any EIT-based power dependency. Generally, an EIT process requires a strong coupling field to produce a large ac Stark shift (i.e, an Autler-Townes splitting, sometimes referred to as a transparency window). Such strong coupling fields necessarily pre-empt any population in the corresponding two-photon terminal state. More importantly, for any argument attempting to interpret one probe component in the colliding-probe AM undergoes an EIT process there is an identical logical counter-argument against it, therefore invalidate any EIT argument.

Finally, due to large laser detunings (several GHz) and very weak probe fields this inelastic wave scattering process operating in $\sim nT$ regime cannot be attributed to any pump-probe process or light-shift-based alignment-orientation effect [24–26]. A typical pump-probe experiment, regardless of whether it employs saturated absorption spectroscopy or polarization spectroscopy, essentially relies on a two-level atomic system where the pump strongly saturates the transition while the probe is scanned across the absorption profile. In such a case only the linearized response of the probe field is considered. We note that SERF-based atomic magnetometers, aside from their spin-relaxation suppression mechanism, are examples of pump-probe schemes. In fact, the probe field induced angular momentum precession and projection representation widely used in SERF magnetometers is exactly the linearized response as in a pump-probe theory [27]. However, the colliding-probe AM is based on a third-order nonlinear optical process. We further note that the very weak laser excitation in $\sim nT$ magnetic field regime renders the pump-probe or alignment-orientation arguments (which is also basically a linear response argument) invalid. From nonlinear optics viewpoint neither single-probe NMOR-based AMs nor SERF-based pump-probe AMs are nonlinear atomic magnetometers. Usually, a nonlinear optics process should exhibit both nonlinear optical power as well as propagation dependencies. Yet, neither single-probe AM nor SERF AM exhibit such a nonlinear propagation behavior. The inelastic wave scattering based colliding-probe AM described in this work represents the first nonlinear-optical-atomic magnetometer known to date. In the following we discuss the underlying physics in Eqs. (4a) and (4b). We demonstrate a previously unknown NMOR blockade in single-probe AMs and explain how the colliding-probe AM removes this NMOR blockade.

1. NMOR blockade in single-probe AMs. When only $\mathbf {E}_{p1}$ is present [see Fig. 2(b)], $G=0$. Since the linearly-polarized $\mathbf {E}_{p1}$ is the only energy source in the system both of its circular components must add up, at any given propagation distance $\eta$, to satisfy the energy conservation relation $S_{p1}^{(-)}(\eta )=S_{p1}(\eta )-S_{p1}^{(+)}(\eta )$, where $S_{p1}(\eta )$ is the total energy of $\mathbf {E}_{p1}$ at $\eta$. Including linear absorption, Eq. (3a) can be expressed as

We immediately notice the probe gain clamping effect in the second term, i.e., $(S_{p1}-S_{p1}^{(+)})S_{p1}^{(+)}$. Indeed, the maximum value of this term is centered around $S_{p1}^{(+)}\approx \frac {1}{2}S_{p1}(0)$ which is precisely the input value of the $S_{p1}^{(+)}$ component. That is, no probe component can change appreciably (except the overall absorption). This is a single-probe energy-symmetry-based self-limiting effect, an intrinsic symmetry-enforced NMOR blockade that blocks any directional energy flow/circulation [see Fig. 2(b)]. Mathematically, by integrating Eq. (5) over the propagation distance, we obtain

 

Fig. 3. Very small probe field intensity changes for $S_{p1}^{(+)}$ (a) and $S_{p1}^{(-)}$ (b) by propagation. Notice that the ridges of the intensity changes only exhibit $\eta -$linear dependence. Due to the strong single-probe AM NMOR blockade only $<1\%$ field intensity change, with respect to the absorption base line, is allowed.

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Fig. 4. Single-probe AM NMOR blockade and the breaking of the NMOR blockade by the colliding-probe AM. (a) Single-probe AM NMOR signal $\Delta S_{p1}$ as functions of propagation distance and magnetic field. (b) Symmetry-forced NMOR blockade strongly suppresses the maximum nonlinear NMOR growth, resulting in a very weak $\eta -$linear growth characteristics in NMOR angle and reduced detection sensitivity. (c) Colliding-probe AM NMOR signal $\Delta S_{p1}$ as functions of propagation distance and magnetic field. Notice the more-than-two-order-of-magnitude difference in vertical scale in comparison with (a), indicating a strong enhancement to the NMOR signal. Experimentally, the NMOR signal amplitude enhancement ratio between (c) and (a) can reach to nearly 500 without hyper-fine state optical pumping. (d) The removal of NMOR blockade has led to a nonlinear growth of NMOR, resulting in about 9-dB (a factor of 8) enhancement in detection sensitivity. Parameters: $S_{p1}{(0)}=S_{p2}(0)=75$, $d_{p1}=-10$, $d_{p2}=-5$, and ${\cal N}_0=3\times 10^{10}$/cm$^3$.

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Inserting $G=0$ and $S_{p1}^{(\pm )}\approx \frac {1}{2} S_{p1}(0)$ into Eq. (3b) we obtain the analytical expression of polarization plane rotation angle for a single-probe AM with power broadening neglected, i.e., the prefactor $\Theta _{\rm sp}$ given in Eq. (4b). We note that Fig. 4(b) shows an extremely small NMOR angle $\Theta _{\rm sp}$ that exhibits only a weak $\eta -$linear dependency. In fact, all NMOR angle produced by single-probe AMs to date exhibit this similar $\eta -$linear dependency. However, in nonlinear optics a nonlinear propagation growth is a key characteristics for a process being associated with a nonlinear effect. The weak $\eta -$linear dependency is the direct consequence of the NMOR blockade.

2. Colliding-probe atomic magnetometry: breaking the single-probe NMOR blockade. Intuitively, the counter-propagating field $\mathbf {E}_{p2}$ injects energy to ground states shared by probe $\mathbf {E}_{p1}$. This breaks the energy-limited growth restriction (i.e., the NMOR blockade) imposed by the symmetry on $\mathbf {E}_{p1}$ since now $S_{p1}^{(-)}\!\ne \! S_{p1}\!-\!S_{p1}^{(+)}$. This enables a magnetic-field-dependent directional energy flow and circulation [see Fig. 2(a)]. Consequently, $\Omega _{p1}^{(\pm )}$ can be substantially different from their initial values by propagation. In addition to the removal of the NMOR blockade the gain $G$ also further contributes to enhancements of NMOR signal and magnetic field sensitivity.

Figure 4(c) shows the growth of the NMOR heterodyne signal $\Delta S_{p1}$ as the function of propagation distance. Notice the vertical scale is more than two orders of magnitude larger than the single-probe AM shown in Fig. 4(a), agreeing well with data shown in Figs. 1(a)–1(c). Figure 4(d) shows a nonlinear growth of the NMOR angle as a function of propagation. This $\eta$-nonlinear dependency in magnetic field sensitivity, which is nearly 9-dB larger comparing with Fig. 4(b), is the direct consequence of NMOR-blockade removal by the colliding-probe AM.

The magnetic field sensitivity of the colliding-probe AM can be estimated from Eq. (4b). Typically, it is at least 6-dB better than the theoretical sensitivity of a single-probe AM operating under the same conditions [28]. Experimentally, even at near resonant conditions (for instance, $d_{p1}\sim 1.5$ Doppler broadened linewidth) where a single-probe AM has reported $\sim fT/\sqrt {Hz}$ sensitivity [9], the colliding-probe technique consistently demonstrates better than 6-dB sensitivity improvement [29,30]. We also note that the $d_{p1}$ dependency of $G$ indicates that the detection sensitivity increases as $d_{p1}$ increases, a prediction that agrees with experimental observation [28].

The establishment of a strong directional energy flow resulted from breaking the NMOR blockade in colliding-probe AM can be further shown mathematically. The relevant nonlinear wave equations for $S_{p1}^{(+)}$ and $S_{p2}^{(+)}$, which are the two segments that form the energy circulation loop [see Fig. 2(a)], can be expressed as (for mathematics simplicity, let $\theta _0=0$),

Equations (7a) and (7b) clearly show that when the two probes are similarly detuned (experimentally, both $d_{p1}$ and $d_{p2}$ are red-detuned), “in-phase" gains for given $d_B$ occur simultaneously for both $\Omega _{p1}^{(+)}$ and $\Omega _{p2}^{(+)}$ [see Figs. 5(a) and 5(b)]. This indicates “synchronized increase" of stimulated emissions in $|2\rangle \rightarrow |1\rangle$ and $|4\rangle \rightarrow |3\rangle$ transitions, exactly as the energy circulation requires [see Fig. 2(a)]. Correspondingly, $S_{p1}^{(-)}$ and $S_{p2}^{(-)}$ decrease by propagation, just as the energy circulation requires. When the direction of the magnetic field is reversed the energy circulation is also reversed, as expected. Experimentally, by adjusting the ratio of two one-photon detunings we can achieve a large increase in NMOR signal interchangeably in either probe channel. The consequence of this directional energy flow, which is resulted from lifting the NMOR blockade, is a large polarization rotation signal [compare Figs. 4(d) and 4(b)]. To reach a similar NMOR signal amplitude without increasing the laser power in a single-probe AM, the one-photon detuning has to be reduced by a factor of five or more, which inevitably result in significantly linear absorption loss, much broadened magnetic resonance, as well as degradation of detection sensitivity [5,29]. We note that even though the function $G(\eta ;\delta _B)$ contains multiple fields and must be self-consistently evaluated with all fields, all above qualitative arguments and discussions are generally correct.

 

Fig. 5. Directional energy flow and NMOR cross-polarization dependency. Probe intensity (a) $S_{p1}^{(+)}$ and (b) $S_{p2}^{(+)}$ as functions of propagation distance and magnetic field. Notice $S_{p2}^{(+)}$ exhibits an“in-phase" growth as $S_{p1}^{(+)}$, indicating a “synchronized" growth of stimulated emission as required by energy circulation [see Fig. 2(a)]. (c) Colliding-probe AM cross-polarization angular dependency observed. (d) Cross-polarization angular dependency calculated at $\eta =5$ using Eq. (4b), showing a $\pi -$periodicity that agrees with (c).

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Finally, when the polarization cross angle $\theta _0$ is varied, the NMOR signal from Eq. (4b) has a $\pi -$periodicity [Fig. 5(d)]. This prediction agrees with experimental data shown in Fig. 5(c) [31].

5. Conclusion

In conclusion, we have developed a colliding-probe atomic magnetometry theory to explain significant NMOR signal SNR and magnetic field sensitivity enhancements in comparison to single-probe AMs under the same conditions. These enhancements, resulted from the combined action of removal of single-probe AM NMOR blockade and enact of a nonlinear gain, persist even when the probe is tuned close to one-photon resonance where a single-probe AM has reported $\sim fT/\sqrt {Hz}$ sensitivity. The inelastic wave-scattering based colliding-probe theoretical framework presented here can qualitatively and analytically explain all experimentally observed effects of this new atom magnetometry technique, as well as single-probe AMs known to date. We finally note that the multi-field-dependency of $G$ indicates a strong correlation between two probe channels and their NMOR SNRs. In a sense, the SNRs of NMOR from different probe channels may be viewed as a pair of “squeezing parameters" in this novel correlated colliding-probe nonlinear optical atomic magnetometer.