1. Introduction

In an optical magnetometer, the Larmor frequency or the Zeeman splitting, due to the presence of a magnetic field, is measured through the interaction of light and atoms [‎1]. The field of optical magnetometry is roughly divided into two detection techniques: non-linear magneto-optical rotation [‎2], based on measuring the polarization rotation of a probe laser [‎3, ‎4, ‎5], and intensity detection, which utilizes a narrow and steep change in the probe transmission due to electromagnetically induced transparency (EIT) [‎6, ‎7, ‎8]. In the latter, a sample of alkali atoms is probed by two laser beams or electromagnetic modes, with a frequency difference that corresponds to the splitting of two magnetic-sensitive levels [‎9]. At Raman resonance conditions, the atoms reach a coherent superposition in which the absorption decreases significantly. The magnetic field determines the frequency of the resonance, and the accuracy of the method is determined by the width and amplitude of the resonance.

In a simple magnetometry configuration, an EIT transition is created with a circularly polarized light between two hyperfine ground-state sub-levels with the same $m_F\ne 0$, where $m_F$ is the projection of angular momentum in the direction of the magnetic field. Figure 1a depicts this transition with $m_F=1$, for the D1 manifold of ⁸⁷Rb (dashed lines). The resonance frequency depends on the Zeeman shifts of the lower states involved. Two major trade-offs limit the performance of this scheme. First, increasing the laser power in order to improve the coherent pumping into the EIT state results in a power broadening of the resonance and also incoherently pumps the atoms away from the EIT system and towards the maximally polarized states (end states). Second, increasing the density in order to increase the line's amplitude results in an elevated spin-exchange decoherence rate, broadening the resonance. These factors impose an upper limit on both the laser power and the atomic density. Here, we introduce a new EIT magnetometry method which overcomes these disadvantages. The method exploits the beneficial characteristics of an EIT resonance with the maximally polarized states; it is enhanced due to both coherent and incoherent pumping, and it is substantially more resistant to spin-exchange [‎10].

 

Fig. 1 (a) Level scheme of the D1 transition in ⁸⁷Rb. The blue arrows represent the (0,0) resonance (the so-called clock resonance), the dashed arrows represent the ( + 1, + 1) resonance, and the red arrows represent the ( + 1, + 2) resonance. (b) The experimental system (see description in text).

Download Full Size | PDF

In many cases, spin-exchange collisions are the dominant decoherence mechanism in the system and the primary limitation to the magnetometer sensitivity. Around 50°C (atomic density of ~10¹¹ cm⁻³), for example, spin exchange determines the width of the resonance to the order of 100 Hz, compared with the Zeeman splitting, of the order of 10⁷ KHz/Tesla. In the current work, we perform the experiments at a low Rubidium density, in which spin-exchange relaxation is negligible relatively to the other decoherence mechanisms. Therefore, we do not examine the improved spin-exchange resistance and concentrate on the analysis of the pumping mechanism. In section 2, we introduce the principles of the suggested magnetometry method, alongside an analysis of its advantages. In section 3, we describe the experimental system and the main results achieved. The last section includes an in-depth comparison of the two EIT-based magnetometry methods, through detailed numerical simulations, which accurately reproduce the experimental results.

2. Principle of operation

We study the suggested magnetometry method within the D1 transition of ⁸⁷Rb. The energy-levels scheme is depicted in Fig. 1a, showing various possible EIT ground-state resonances. Specifically, we study the EIT resonance ( + 1, + 2), involving the two ground-state sub-levels $|F=1,m_F=+1〉$ and $|F=2,m_F=+2〉$ (solid and dashed red arrows in the figure), and compare it to the EIT resonance ( + 1, + 1), involving the $|F=1,m_F=+1〉$and $|F=2,m_F=+1〉$ (red and green dashed arrows). The end-state resonance can be observed when the propagation direction of a right-circularly polarized beam (${\sigma }^{+}$) is at a non-zero angle θ to the magnetic field direction, which defines the quantization axis. In that case, the electric field of the laser can be expressed in the form of all possible polarizations [‎7],

where,

The ratio of the different polarizations, and thus the intensities of different EIT resonances, is controlled by the angle between the beam and the magnetic field (see Fig. 1b). In a practical system, this magnetometer can exploit the advantages of the optimal-performance angle by tuning to this angle, either by a physical alignment of the system or by applying a DC magnetic field in the optimal direction.

We identify three distinct advantages of the ( + 1, + 2) magnetometer over the ( + 1, + 1) magnetometer. First, the Zeeman splitting of the ( + 1, + 2) resonance is 1.5 times that of the ( + 1, + 1) resonance. Second, for $\theta <90°$, the ${\sigma }^{+}$ light intensity exceeds the ${\sigma }^{-}$ intensity, resulting in an incoherent pumping of the alkali population towards the end state. This pumping increases the contrast of the ( + 1, + 2) resonance, which comprises the high $m_F$ states of the ground manifold, but decreases the contrast of all other resonances [‎11]. Third, when most of the atomic population is pumped to the end state, the spin-exchange decoherence rate is reduced, narrowing the EIT resonance. The latter is especially pronounced in miniature cells due to the high atomic density required.

In the following sections, we compare the experimental results with an elaborated numerical model. Our numerical model takes into account all 16 sub-levels of the D1 transition and the three laser polarizations. The system is modeled by means of solving the Master's equation,

where ρ is the density matrix of the Rubidium atom, $H=H_0+H_C$ is the full Hamiltonian of the system. Here $H_0$ is the free Hamiltonian, $H_C$is the interaction Hamiltonian, and L is the Lindblad super-operator which effectively describes the decay and decoherence processes at the ground and the excited states of the Rubidium atoms. In $H_C$, we take into account all the allowed transitions, with their respective dipole amplitudes (Clebsch-Gordan coefficients), including those that are far detuned from resonance (12 transitions for each polarization component). The ground-state decoherence rate (${\gamma }_{12}$) and the laser power (in terms of the power broadening ${\Gamma }_P$), were calibrated from the $\theta =0$ measurement, depicted in Fig. 2 (blue), to be approximately ${\gamma }_{12}=500\text{Hz}$and ${\Gamma }_P=1.5\text{KHz}$. A more elaborated description of the numerical model is given in ref. [‎12].

 

Fig. 2 Frequency scan of the RF modulation in 6 experiments (full lines) and the corresponding simulations (dashed lines). f₀ is the frequency of the clock transition. The EIT lines, from left to right, are the (0,0) clock line, (0, + 1) line, ( + 1, + 1) line, and ( + 1, + 2) line. The angle between the magnetic field and beam is 0 (blue), 5, 15, 25, 34 and 40 (brown) degrees. Inset: The ratio between the frequency of the EIT lines and ${\omega }_{Zeeman}^{\left(+1,+1\right)}$.

Download Full Size | PDF

3. Experimental setup & results

The experimental system is depicted in Fig. 1b. A Vertical Cavity Surface Emitting Laser (VCSEL) is tuned to the D1 transition of ⁸⁷Rb (~795nm). The VCSEL is current-modulated at $f_{RF}\approx 3.4\text{\hspace{0.17em} GHz}$, creating two main sidebands with $f=2\times f_{RF}\approx 6.8\text{\hspace{0.17em} GHz}$, matching the hyperfine splitting of the ⁸⁷Rb ground-state and carrying an intensity of approximately 0.1 mW/cm² each. The light that enters the vapor cell is ${\sigma }^{+}$ polarized by a linear polarizer and a quarter wave plate. The vapor cell is placed in a 4-layer magnetic shield, which attenuates the Earth magnetic field. Three pairs of Helmholtz coils located inside the shield, enable us to control the magnitude and the direction of the magnetic field. The vapor cell is about 25 mm long and contains isotopically pure ⁸⁷Rb with 10 torr of Nitrogen buffer gas. The experiments were conducted at temperatures of 40-50 °C providing a vapor density of ~5⋅10¹⁰ cm⁻³.

We performed several measurements in which we held constant the magnetic field on the z axis, while increasing the magnetic field on the y axis, thus controlling the angle between the beam and the magnetic field. Figure 2 depicts the results of six experiments (full lines) and the appropriate simulations (dashed lines), in which different magnetic fields on the y axis were applied. The measurements were conducted by scanning the RF frequency of the laser current modulation and measuring the laser power transmitted through the cell. The rate of the scan was slow enough to reach a steady-state at each point. In Fig. 2 we observe, from left to right, four groups of transparency lines, corresponding to four EIT resonances: (0,0), (0, + 1), ( + 1, + 1), and ( + 1, + 2). The observed Zeeman shift of the $m_F\ne 0$ states, given by ${\omega }_{Zeeman}=m_F\left(g_L{\mu }_B/\hslash \right)B$, depends only on the magnitude of the magnetic field, B. Here, $m_F$ is the projection of the angular momentum, $g_L$ is the Lande factor, ${\mu }_B$ is the Bohr magneton, and $2\pi \hslash$ is the Planck constant. The Lande factor for the ground states of ⁸⁷Rb is −1/2 for $F=1$, and 1/2 for $F=2$. We thus find for the ( + 1, + 1) resonance, ${\omega }_{Zeeman}^{\left(+1,+1\right)}=1.4\text{KHz}/\text{mG}$. In the inset of Fig. 2, we show the ratio of the experimental EIT resonance frequencies and ${\omega }_{Zeeman}^{\left(+1,+1\right)}$. The ratio is 1.5 for the ( + 1, + 2) resonance, 1 for the ( + 1, + 1) resonance and 0.5 for the (0, + 1) resonance. The measured amplitude of the different resonances varies with the angle of the magnetic field. It is determined by a complex dynamics of coherent and incoherent pumping, which is affected directly by the angular dependence of the three polarizations components, see Eq. (1). As described in the next section this dynamics results in a decrease in the amplitudes of the (0,0) and ( + 1, + 1) resonances with the increase of the magnetic field in the y axis, as well as an increase in the amplitudes of the (0, + 1) and ( + 1, + 2) resonances.

4. Discussion

In a typical magnetometer set-up, the magnetic field is deduced from a measurement of the magnetic resonance frequency. The possible accuracy is governed by two factors: the magnetic field sensitivity of the shift in the central frequency and the width and amplitude of the resonance. The magnetic field dependence of the central frequency ($\delta f$) is determined by the relative Zeeman splitting. For ⁸⁷Rb ground-state, the frequency dependence is 1.4 KHz/mG for the ( + 1, + 1) resonance and 2.1 KHz/mG for the ( + 1, + 2) resonance.

In order to compare the relative accuracy of the suggested magnetometer scheme to the ( + 1, + 1) scheme, we define a figure of merit (FOM), given by

Here, the contrast refers to the amplitude of the resonance, and the width of the resonance is given by the full width at half maximum (FWHM). The suggested FOM is related to the dispersion of the EIT line. A simple derivation from the susceptibility χ of the resonance gives the dispersion at zero detuning: ${\genfrac{}{}{0.1ex}{}{d\left(\mathrm{Im}\chi \right)}{d\Delta }|}_{\Delta =0}=\genfrac{}{}{0.1ex}{}{\text{Contrast}}{{\Gamma }_{EIT}}$, hence the FOM can be written as

Using our numerical model, we scanned two experimental parameters that influence the slope of the resonances, the laser power and the angle θ between the magnetic field and the beam, and examined the FOM of each resonance. Figure 3a and Fig. 3b depict the FOM of the ( + 1, + 1) and ( + 1, + 2) schemes respectively, as a function of the angle θ and the laser intensity (the color represents the calculated FOM). We observe several differences between the two schemes. First, the peak FOM-value of the ( + 1, + 2) resonance is approximately double the peak FOM-value of the ( + 1, + 1) resonance, thus twice more sensitive to magnetic field. This should further increase due to the expected reduction of spin-exchange decoherence in the ( + 1, + 2) scheme. Second, we see that the maximal FOM of the ( + 1, + 1) resonance is achieved at $\theta =0$ and at low laser power, while the maximal FOM of the ( + 1, + 2) resonance is found at $\theta \approx 65°$ and at higher laser power. This difference between the resonances is related to the angular dependence of the three polarization components and to the impact of the incoherent pumping on the atomic population. As seen in Fig. 1b, the ( + 1, + 1) resonance is mostly an interaction of ${\sigma }^{+}$ polarizations, while the ( + 1, + 2) resonance is a coherent superposition of the ${\sigma }^{+}$ and the π polarizations. Consequently, at $\theta >0$, the contrast of the ( + 1, + 1) resonance decreases, due to a decrease in the ${\sigma }^{+}$ power, while the ( + 1, + 2) contrast improves due to an increase in the π power. The effect of incoherent pumping, which drives the atomic population towards the maximally polarized state, grows stronger at higher laser powers. Therefore, while for the ( + 1, + 1) resonance, the incoherent pumping is a disadvantage, the ( + 1, + 2) resonance benefits from both coherent and incoherent pumping. Ultimately, we see in Fig. 3b that at sufficiently high laser powers, the power broadening of the ( + 1, + 2) resonance becomes dominant, resulting in a decrease in the FOM. We note that the higher optimal laser intensity of the ( + 1, + 2) resonance might point to another potential advantage due to higher signal-to-noise ratio of the measurement. This advantage is manifested when the noise is not intensity proportional, such as with a shot noise (not taken into account in the FOM). In Fig. 3c, we quantitatively compare the calculations with the experimental results by taking the ratio between the ( + 1, + 2) FOM for different θ and the ( + 1, + 1) FOM at $\theta =0$ (where it is maximal and usually employed). The model parameters used for Fig. 3 are the same as those in Fig. 2.

 

Fig. 3 (a) and (b), calculated FOM versus the logarithmic scale of laser power (in terms of power broadening in KHZ units) and the angular deviation of the magnetic field for (a) the ( + 1, + 1) resonance and (b) the ( + 1, + 2) resonance. The ratio between the maximal values is about 2. (c) The ratio between the ( + 1, + 2) FOM and the ( + 1, + 1) FOM versus the angular deviation at the experimental conditions. Note that the measurements were done at a laser power slightly higher than the optimal for the ( + 1, + 1) scheme, thus the maximum of 3.5 instead of 2.

Download Full Size | PDF

In conclusion, we have introduced an EIT magnetometry method that exploits the advantages of maximally-polarized states. The measurements and the simulations exhibit some of these advantages over the simple ( + 1, + 1) magnetometry. We have concentrated mainly on the role of the pumping mechanism in determining the sensitivity of the schemes. Two possible contributions to the sensitivity of the method were not analyzed here - decrease in the spin-exchange relaxation, which is especially relevant at miniature systems, and improvement in the signal-to-noise ratio. The behavior of the new scheme in a high-density regime dominated by spin-exchange relaxation and the noise properties of the method is left for future research.