Navigation and positioning technologies play a key role in modern human activities. Currently, navigation devices are mostly dependent on the global navigation satellite systems, for example, the global positioning system (GPS). However, it is known that GPS signals can be jammed or spoofed or otherwise fail. As the need for GPS independent navigation increases in some practical application scenarios, novel navigation techniques based on the geomagnetic field have been developed. A main task of geomagnetic navigation is to obtain the amplitude and direction of the geomagnetic field accurately. Here, we introduce an alternative scheme for vectorial measurements of the local geomagnetic field for magnetic positioning based on the biological ferric sulfide cluster, which exists in the magnetoreceptor protein/cryptochrome complex in certain avian species. We find that by observing the number of peaks and the proportional rate of spectrum on resonance, both the direction and intensity of the magnetic field can be determined. Therefore, our findings may provide a fresh insight into magnetic field measurement and also suggest further guidelines for the design and operation of satellite-free navigation systems based on the electrically tunable inorganic biological molecules.

## I. INTRODUCTION

The geomagnetic field is an inherent resource of Earth, which has a wide variety of applications, such as in earth science, navigation, mineral exploration, earthquake prediction, and other fields.^{1,2} Earth’s magnetic field is an omnipresent three-dimensional vector field, which has one-to-one correspondence with the latitude and longitude, providing a natural coordinate system for navigation in GPS-denied environments. Therefore, in principle, as long as the geomagnetic field vector of the geographical point on Earth is measured, global positioning can be realized.^{3}

Remarkably, long-distance migratory animals appear to have such a natural global positioning capability.^{4,5} Compelling behavioral evidence demonstrated that migratory birds can potentially use the information from Earth’s magnetic field for orientational and navigational purposes.^{6–9} The birds utilize the direction of the geomagnetic field as a magnetic compass^{10} to maintain a directed heading while sensing the intensity of the geomagnetic field as a component of the navigational magnetic map^{11,12} to assess their approximate geographic position. However, the underlying mechanism of how birds use the geomagnetic field to navigate remains an enduring mystery.^{13,14} One leading hypothesis assumes that the radical pair mechanism^{15–24} might account for the avian detection of the inclination of the geomagnetic field. Moreover, the magnetic map sensitivity of birds is likely to rely on biogenic magnetite^{25–28} associated with trigeminal nerves. A novel biocompass model^{29} based on the magnetoreceptor protein (MagR)/cryptochrome (Cry) complex has been suggested to explain the avian magnetoreception. Recently, we also proposed a compass-free migratory navigation hypothesis^{30} based on the spectrum of resonance fluorescence of a four-level model derived from the ferric sulfide cluster, which exists in the MagR/Cry protein complex. We have shown that the feature of the spectrum can be a cue for animal navigation. We also showed that the weak radio frequency magnetic field can disrupt avian magnetic orientation.

Meanwhile, for engineered systems, the geomagnetic field can be used in global positioning and navigation that do not rely on GPS satellites. The main objective of making geomagnetic navigation possible is achieving an accurate measurement of geomagnetic field information. Several different styles of magnetic sensors^{31} have been developed to detect the presence, strength, or direction of magnetic fields. One well-established magnetic sensor is the superconducting quantum interference device (SQUID),^{32–34} which is a scalar-type magnetic sensor. SQUIDs have been proved to be reliable mechanical geomagnetic surveying and mapping equipment. This device has the advantages of a wide measurement range and high sensitivity on the order of 1 fT$/Hz$, but the disadvantage is that it requires cryogenic cooling. The insulation and refrigeration equipment leads to the magnetometer’s large volume, high cost, and inflexible installation.^{35–37} Another class of sensitive magnetic sensor is the optically pumped magnetometer (OPM),^{38–40} which is based on the Zeeman effect in alkali atoms. As for OPMs, the atomic vapor is encapsulated in the glass bubble, and the magnetic resonance signal is extracted by the interaction between atoms and the environmental magnetic field so as to obtain the intensity of the magnetic field. OPMs are known as precise sensors for scalar measurement of Earth’s magnetic field without the hurdle of cryogenic operation, but they suffer from so-called dead zones^{41} and heading errors^{42,43} when used for modern geomagnetic surveys.

The aforementioned studies motivate us to perceive an alternative method to measure Earth’s magnetic field for global positioning, which is also inspired by our four-level system derived from the ferric sulfide cluster.^{29,30} By studying the resonance fluorescence spectrum of the system, the direction of the geomagnetic field can be determined by tuning the applied artificial electric field and observing the characteristics of the spectrum. The resonance fluorescence spectrum always exhibits three peaks as long as the direction of the geomagnetic field is parallel to the electric field. Moreover, the magnitude of the geomagnetic field can be evaluated by the proportional rate of observing the spectrum on resonance. From the viewpoint of spin dynamics, we numerically inspect the spin excitations of the cluster system and clarify that the feature of the spin resonance spectra also provides the precise information of the direction of the magnetic field.

## II. RESULTS AND DISCUSSION

### A. The model Hamiltonian and method

We shall specifically refer to the MagR/Cry complex,^{29} a potential magnetoreceptor that consists of ferric sulfide cluster protein. As schematically depicted in Fig. 1(a), we thus devised the ferric sulfide cluster as a three-site cluster minimal model consisting of two magnetic-ion sites (M) and a ligand site (L). Our simplified M-L-M cluster system under both an external (geo-)magnetic field ** B** and an (bio-)electric field

**is described by the following Hamiltonian:**

*E*^{30}

The first term in Eq. (1) refers to the contribution of the Zeeman energy of two spins *S*_{1} and *S*_{2} in the applied magnetic field, and *g* and *μ*_{B}, respectively, denote the *g*-factor and the Bohr magneton, while the second term arises from the magnetoelectric coupling between the spin-induced electric polarization and the applied electric field. The non-collinearity of two neighboring spins in the M-L-M cluster brings in a local electric polarization ** P** via the spin-current mechanism

^{44}of the following form:

where *g*_{ME} is the magnetoelectric coupling that depends on the spin–orbit interaction and $e12=sin\theta cos\varphi ,sin\theta sin\varphi ,cos\theta $ denotes a unit vector connecting the neighboring spins *S*_{1} and *S*_{2} in an arbitrary direction (*θ*, *ϕ*) relative to the coordinates (*ξ*, *η*, *ζ*), as shown in Fig. 1(b). For a Cartesian coordinate *ξ*-*η*-*ζ*, any vector is expressed as ** V** = (

*V*

_{ξ},

*V*

_{η},

*V*

_{ζ}). We choose the direction of the magnetic field

**as the**

*B**ζ*-axis and assume that there is a tilt angle

*ϑ*between

**and the electric field**

*B***, which can be written as**

*E***=**

*E**E*(sin

*ϑ*, 0, cos

*ϑ*) without loss of generality.

To inspect the spin dynamics of the M-L-M cluster, it is convenient to use the phenomenological Landau–Lifshitz–Gilbert (LLG) equation, which is given by^{45}

where *α*_{G} (=0.01) is the phenomenological Gilbert-damping coefficient and $Hieff=\u2212\u2202H/\u2202Si$ is the effective magnetic field that acts on the *i*th spin *S*_{i}. Each spin is normalized to be of unit length (i.e., *S* = 1), and the LLG equation is solved numerically with the fourth-order Runge–Kutta method.

We calculate the spin excitation modes and their resonance frequencies through the dynamical magnetic susceptibility,^{45} which is defined as

where the subscript *μ* stands for the orthogonal axes *ξ*, *η*, or *ζ*, and the *ζ*-axis is defined to be parallel to the static external magnetic field ** B**. Here,

*B*

_{μ}(

*ω*) and Δ

*S*

_{μ}(

*ω*) are Fourier transform of the magnetic field pulse

*B*

_{μ}(

*t*) and that of the simulated time-profile of the spatially averaged spin Δ

**(**

*S**t*) =

**(**

*S**t*) −

**(0), with**

*S***(**

*S**t*) = (1/

*N*)

*∑*

_{i}

*S*_{i}(

*t*), which is the transient response of the system under a magnetic field pulse. In our numerical simulation, we adopt an intensive short pulse with a

*δ*-function shape, i.e., $B(t)=\delta (t)e\u20d7\xi $ applied at

*t*= 0. We initialize the cluster system with a random spin configuration and relax it for a sufficient time to achieve a ground state by using the LLG equation simulation. After stabilizing the system, we then simulate the spin dynamics by applying an intensive pulse of magnetic field to obtain the spin excitation spectra.

### B. Spin excitation modes

We first calculate the spin excitation spectra of the model system and discuss how the tilt angle of the applied electric field ** E** and the orientation of the unit vector

*e*_{12}affect the characteristics of the spectral peaks. We show in Figs. 2(a)–2(d) the calculated imaginary parts of the dynamical magnetic susceptibility Im

*χ*

_{ξ}(

*ω*) of the M-L-M cluster under tilted electric fields (i.e.,

*ϑ*=

*π*/4) for various selected orientations of the unit vector

*e*_{12}with (

*θ*,

*ϕ*) = (

*π*/2,

*π*/2), (

*π*/4,

*π*/2), (

*π*/3, 0.1843

*π*), and (2

*π*/3, 0.6959

*π*), which correspond to the relative angle between

**and**

*E*

*e*_{12}as

*π*/2,

*π*/3,

*π*/6, and 3

*π*/4, respectively. Here, we fix

*E*= 0.06 and

*B*= 0.1 for the calculations. Note that the spin excitation modes are seen as peaks in the spectrum of Im

*χ*

_{ξ}(

*ω*). We find that each of the spectra has two distinct peaks, indicating the existence of two spin resonance modes for all orientations of

*e*_{12}. For comparison, Figs. 2(e)–2(h) show the calculated spectra Im

*χ*

_{ξ}(

*ω*) of the M-L-M cluster under a parallel electric field (i.e.,

*ϑ*= 0 with

**‖**

*E***) for various selected orientations**

*B*

*e*_{12}. The spectra of Im

*χ*

_{ξ}(

*ω*) clearly exhibit a single resonance peak for all

*e*_{12}orientations. These facts indicate that the peak numbers of the spectra Im

*χ*

_{ξ}(

*ω*) involve information of the relative direction of the magnetic field

**and the electric field**

*B***. When the applied electric field**

*E***has a tilt angle with respect to the applied magnetic field**

*E***, the spectra of Im**

*B**χ*

_{ξ}(

*ω*) exhibit two distinct resonance peaks for a randomly oriented

*e*_{12}; otherwise, the spectra exhibit a single resonance peak, which indicates the magnetic field

**and electric field**

*B***are parallel to each other. This property may allow a navigational signal for the birds to perform precise geomagnetic field direction sensing by adjusting the bioelectric field direction.**

*E*To further elucidate the influence of the tilted angle *ϑ* of ** E** on the spin excitation of the M-L-M cluster, we show in Fig. 3(a) the spectra Im

*χ*

_{ξ}(

*ω*) for various cases

*ϑ*. In the simulation, the strength of the magnetic and electric field is fixed at

*B*= 0.1 and

*E*= 0.06, while the direction of the unit vector

*e*_{12}= (0, 1, 0) is along the

*η*-axis. We find that as

*ϑ*decreases from

*π*/2 to 0, the resonance frequency of the lower-lying peak (

*ω*

_{1}) shifts to high values while that of the higher-lying peak (

*ω*

_{2}) barely changes. However, the intensity of the higher-lying peak

*ω*

_{2}decreases with decreasing

*ϑ*, and it finally disappears at

*ϑ*= 0. The plot in the inset of Fig. 3(a) illustrates how the difference in the resonance frequencies Δ

*ω*(=

*ω*

_{2}−

*ω*

_{1}) between the higher-lying peak and lower-lying peak varies as a function of the tilted angle

*ϑ*. It shows that as

*ϑ*decreases, Δ

*ω*monotonically decreases. Therefore, the peak’s difference Δ

*ω*would be sensitive to the relative direction of

**and**

*B***, which can be a cue for birds to adjust the angle of**

*E***and then sense the direction of the magnetic field**

*E***. In Fig. 3(b), we show Im**

*B**χ*

_{ξ}(

*ω*) when the magnetic field and the electric field are parallel to each other, i.e., $B\Vert E\Vert e\u20d7\zeta $, for several values of

*B*with fixed

*E*= 0.06. It can be found from Fig. 3(b) that the resonance frequencies are significantly influenced by the amplitude of

**. We can find that the positions of resonance peaks increase whereas the intensities of the peaks are monotonically decreased as**

*B**B*increases. The frequencies of the resonance modes (

*ω*

_{R}) increase nearly linearly with the magnitude of

*B*, as illustrated in the inset of Fig. 3(b).

### C. Four-level system under an SU(2) × SU(2) symmetry

As a step to calculate the resonance fluorescence spectra, we should solve the eigenenergies and eigenstates of the model system. Following our previous paper,^{30} we restate the derivation of a four-level system from the model Hamiltonian (1). Three operators are conveniently defined, namely, the total spin operator ** S** = (

*S*_{1}+

*S*_{2}), a new operator

**= 2**

*Q*

*S*_{1}×

*S*_{2}, and a vector field

**= (**

*C***×**

*E*

*e*_{12})/2, which satisfy the commutation relations as follows:

where the subscripts *i*, *j*, *k* = *ξ*, *η*, *ζ*. Their further combinations ** J** = (

**+**

*S***)/2 and**

*Q***= (**

*K***−**

*S***)/2 give rise to two sets of decoupled SU(2) generators. Then one can rewrite the model Hamiltonian (1) as**

*Q*Clearly, it indicates an SU(2) × SU(2) model characterized by two vector fields given by ** α** = (

**+**

*B***) and**

*C***= (**

*β***−**

*B***), which can be tuned by both the magnetic and electric fields.**

*C*If one chooses the direction of ** α** as the

*ξ*-axis of our coordinate frame, the SU(2) × SU(2) Hamiltonian (5) can be simplified as

which is solved by four eigenvectors: the ground sate |g⟩ and three excited states |a⟩, |b⟩, and |c⟩, with the corresponding eigenenergies $Eg=\u2212(\alpha +\beta )/2$, $Ea=(\alpha \u2212\beta )/2$, $Eb=(\beta \u2212\alpha )/2$, and $Ec=(\alpha +\beta )/2$, respectively. Therefore, one can construct a four-level system parameterized by *α* and *β*, and the corresponding Hamiltonian (5) can be written in Fock space as

The energy-level scheme of the four-level system is shown in Fig. 4(d), which is employed as the model system to calculate the resonance fluorescence spectrum.

Following the standard steps,^{46,47} we can derive the resonance fluorescence spectrum function of the four-level system as follows:^{30}

where $P(+)(t)=(P(\u2212)(t))\u2020$ is the effective polarization operator with *P*^{(+)} (*t*) = *d*_{1}*γ*_{1}(*t*) + *d*_{2}*γ*_{2}(*t*). Here, *d*_{1} and *d*_{2} denote the dipole moments, *γ*_{1} = |g⟩⟨a| + |b⟩⟨c| and *γ*_{2} = |g⟩⟨b| + |a⟩⟨c| are the energy-lowering operators related to the four-level systems. Generally, it is difficult to derive an exact analytical expression for the spectrum *S*(*ω*). Nevertheless, we found that it is possible to solve an analytical spectrum for our four-level system in the regime *α* = *β* under the double-resonance conditions.^{30} Now, we take a close look at ** B** and

**, assuming that the angle between**

*C***and**

*B***is**

*C**φ*. Then the expressions for

*α*and

*β*can be expressed concretely as

With Eq. (9) in mind, we then investigate the double resonance condition *α* = *β* = *ν*. From the above-mentioned equation, we can see that *α* and *β* are equal to each other when cos *φ* = 0, i.e., *φ* = *π*/2. Furthermore, we recall that

where *ψ* denotes the angle between ** E** and

*e*_{12}. Thus, the final formula for the double resonance condition

*α*=

*β*=

*ν*in terms of

**and**

*B***can be written as**

*E*Here, *ψ*_{0} specifies the angle when the double resonance condition is fulfilled. In the following, we would calculate the spectrum numerically for various parameters in more general situations.

### D. The resonance fluorescence spectrum

Figure 4 shows the numerical results for the three-dimensional plots of resonance fluorescence spectra of our four-level system under various situations. First, Fig. 4(a) shows the spectrum vs *ω* and the difference between *β* and *α*, i.e., *β* − *α*, which varies from −10Γ to 10Γ (with Γ being the natural width) by keeping *α*/Γ = 20. Here, we choose Ω/Γ = 10 and *ν*/Γ = 30, where Ω stands for the Rabi frequency associated with the driving field of frequency *ν*. It is seen that when *α* ≠ *β*, the spectrum generally exhibits a characteristic five-peaked resonance structure, where there is a central peak and two pairs of side-peaks symmetrically located at the left- and right-hand side, respectively. When *β* − *α* approaches zero, the magnitude of these side-peaks grows, and the separation between the inner and outer side-peak is getting smaller. It is clearly shown that the five-peaked spectrum collapses into a three-peaked structure at *β* − *α* = 0, which actually serves as a three-level system.^{30}

In Fig. 4(b), we present the resonance fluorescence spectra as a function of both *ω* and *φ*, where *φ* is the angle between ** B** and

**. In this situation, the magnitude of**

*C**α*and

*β*is no longer addition and subtraction of |

**| and |**

*B***|, as we simplified previously.**

*C*^{30}Instead, they are replaced by Eq. (9). We choose

*φ*ranging from

*π*/8 to 7

*π*/8, while the magnitude of |

**| and |**

*B***| is set to 3Γ and 10Γ, respectively. One finds that for the given amplitude of**

*C***and**

*B***, the spectrum is triple peaked for**

*C**φ*=

*π*/2, in which

**is perpendicular to**

*B***. In this case, one central peak is located at**

*C**ω*=

*ν*while two side-peaks are symmetrically distributed to both sides. However, as

*φ*deviates from

*π*/2, the two side-peaks will split into two peaks, and a five-peak feature appears. We can see that the more difference there is between the angle

*φ*and

*π*/2, the bigger the splitting of the two side peaks in the spectrum is.

Finally, we consider a special situation, *φ* = *π*/2. In Fig. 4(c), we show the resonance fluorescence spectra as a function of *ω* and the magnitude of ** B** under the condition that

**is perpendicular to**

*B***. The Larmor frequency caused by**

*C***is kept at 4Γ, and the other parameters are set to Ω/Γ = 6 and**

*C**ν*/Γ = 30. One finds that the spectrum generally exhibits a three-peak structure when

**is perpendicular to**

*B***. The intensity of the central and side peaks grows when the magnetic field**

*C***increases. Furthermore, the spacing of the side-peaks from the central peak decreases as**

*B***increases.**

*B*### E. Determination of the geomagnetic field direction

We can clearly determine the direction of the geomagnetic field from the characteristics of the resonance fluorescence spectra described in Fig. 4. As shown in Figs. 4(b) and 4(c), the spectrum will always exhibit a three-peak structure when ** B** is perpendicular to

**, i.e.,**

*C**φ*=

*π*/2; otherwise, the spectrum exhibits a five-peak structure. As we know, parameter

**is defined as**

*C***= (**

*C***×**

*E*

*e*_{12})/2, and the direction of

*e*_{12}is chosen randomly in our model; thus, all possible values of

**lie in a plane perpendicular to**

*C***. Therefore, if the spectrum generally exhibits a three-peak structure, it means that**

*E***is perpendicular to**

*B***; thus,**

*C***is exactly parallel to the direction of**

*B***in this case. This implies that the spectrum hosting a three-peak structure is a characteristic signature of the parallel nature of the electric field**

*E***and magnetic field**

*E***.**

*B*### F. Determination of geomagnetic field magnitude

In Subsection II E, we have determined the direction of the magnetic field by the direction of ** E**, and we will get the magnitude of the magnetic field by the proportional rate of resonance here.

We assume the angle between ** E** and

*e*_{12}is

*ψ*; thus, all possible values of

*e*_{12}constitute a sphere. According to the resonant condition illustrated in Eq. (10), it is obvious to see that the spectrum will be resonant only in the case of

*ψ*=

*ψ*

_{0}or

*ψ*=

*π*−

*ψ*

_{0}, and this corresponds to two circular lines on the sphere. In order to get a finite area, we relax the resonant condition to

*ψ*

_{0}∼

*ψ*

_{0}+

*dψ*and

*π*− (

*ψ*

_{0}+

*dψ*) ∼

*π*−

*ψ*

_{0}. The area of two bands is

*dS*= 4

*π*sin

*ψ*

_{0}

*dψ*. The proportional rate of observing the resonance spectrum is

In the following, we will make an estimation of *dψ*. We assume that the spectrum for *ψ*_{0} + *dψ* is slightly different from that of *ψ*_{0} by nearly about Γ, where Γ is the half width of the spectrum. According to the Rayleigh criterion, one cannot distinguish two peaks if their distance is within the half width. Moreover, we consider that the spectrum for *ψ*_{0} + *dψ* is resonant with *ν* + Γ. Recalling the resonant condition in Eq. (10), we have

In deriving the above-mentioned formula, we have made the assumption that *dψ* is a small quantity.

Substituting Eq. (13) into Eq. (11) and moving cos *ψ*_{0} to the left hand of the equation, we will finally get the expression

Once we know cos *ψ*_{0}, we can substitute Eq. (14) into Eq. (12a) and get the explicit expression of |** B**| caused by the magnetic field, which is given by

where Γ is the half width of the resonance fluorescence spectrum. The above-mentioned explicit expression indicates that the magnitude of the magnetic field is determined by the parameter $P$, the proportional rate of observing the spectrum of resonance fluorescence, which is related to the angle *ψ*. Therefore, it allows one to deduce the magnetic field intensity by measuring the proportional rate for detection of the resonance fluorescence in the experiments.

## III. CONCLUSION

In summary, we have investigated a four-level system derived from the biological ferric sulfide cluster existing in the MagR/Cry protein complex. By making Laplace transform of the double-time correlation function of the electric field and utilizing the quantum regression theorem, the resonance fluorescence spectrum of this four-level system is obtained. Further analysis reveals that the spectrum generally exhibits a five-peak structure, with one central peak located at the driving frequency *ν* and four side peaks. However, when *α* = *β*, the spectrum will contain only three peaks, with one central peak and two side peaks symmetrically distributed at the left- and right-hand side, respectively. Moreover, the separation between the side peaks and the central peak grows when *α* or *β* is farther away from resonance, with the driving frequency *ν* increasing. Thus, a new method for measuring Earth’s magnetic field is proposed based on the characteristics of the spectrum. Explicitly, the direction of the geomagnetic field can be determined by tuning the applied electric field. When the spectrum is always observed as three peaks, the direction of the magnetic field is the same with the applied electric field. In addition, the magnitude of the geomagnetic field is related to the proportional rate of observing the spectrum on resonance. Our study provides an entirely new insight into magnetic field measurement and may be helpful for potential manmade practical devices.

We should note that the iron-sulfur cluster in the intact MagR protein binds either the 2Fe–2S or 3Fe–4S form.^{48–50} It is reasonable to extend our model calculations to the 2Fe–2S cluster since the spin-induced polarization in the 2M-2L cluster exhibits an analogous form to Eq. (2) with an anisotropic amplitude.^{51} Extensions of this work to more realistic 2Fe–2S or 3Fe–4S cluster situations are future works of interest.

## ACKNOWLEDGMENTS

This work was supported by the National Key R&D Program of China, Grant No. 2017YFA0304304, and the NSFC, Grant No. 11935012.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Xin Zhao**: Investigation (equal); Methodology (equal); Writing – original draft (equal). **Hong-Bo Chen**: Investigation (equal); Methodology (equal); Writing – original draft (equal). **Li-Hua Lu**: Formal analysis (lead); Project administration (lead); Writing – review & editing (equal). **You-Quan Li**: Conceptualization (lead); Supervision (lead); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.