Technical limits of sensitivity for EIT magnetometry

J. A. McKelvy; M. A. Maldonado; I. Novikova; E. E. Mikhailov; A. B. Matsko

doi:10.1364/AO.497368

1. INTRODUCTION

Optical atomic magnetometry is among the most sensitive techniques for precision measurements of magnetic fields [1–12]. The first all-optical atomic magnetometer based on synchronous optical pumping was demonstrated over 60 years ago [13]. This device utilized incoherent lamp light modulated at Larmor frequency to generate spin procession in either Cs or Rb atomic vapor, thereby increasing optical absorption. Recently, the maturation of semiconductor lasers has led to further development of all-optical magnetometers based on a variety of magneto-optical effects, resulting in new devices with increased sensitivity [4,6,7,12].

Shortly after its discovery, the effect of electromagnetically induced transparency (EIT) was identified as a promising tool for sensitive detection of a magnetic field over a large dynamic range [14,15] while also being capable of measuring the magnetic field direction [16–19]. In a simplified model, an EIT magnetometer relies on the interaction of bichromatic coherent light with an atomic medium supporting the phenomenon of coherent population trapping (CPT). This phenomenon occurs when the frequency difference between the optical harmonics becomes equal to the frequency splitting of two ground-state atomic sublevels. In this case, atoms are prepared in a so-called “dark state,” decoupled from the laser light. Such atoms do not attenuate light, and a transmission resonance with subnatural linewidth emerges. The exact two photon frequency of EIT resonances involving magneto-sensitive ground sates provides information about the absolute value of the magnetic field [14,15], while the relative amplitudes of multiple EIT peaks can provide information about the direction of the field [17,18].

The quantum limited sensitivity of an ideal EIT magnetometer can reach ${10^{- 16}}\; {\rm T}$ [14]. However, there is a large discrepancy between these early theoretical estimates and the experimentally observed performance. At the time of writing, the reported sensitivity of EIT magnetometers does not exceed ${10^{- 11}}\; {\rm T}$ [20–23]. This inferior performance stems from various systematic and fundamental effects involving asymmetry and fluctuations in the contrast of the EIT resonances, which depend on the magnetometer configuration [24–26] and various frequency shifts associated with the measurement apparatus [27–29]. In this paper, we identify the leading causes for these statistical and systematic errors and estimate their impacts on the sensitivity of a realistic EIT magnetometer. We also discuss possible ways to mitigate these problems, and show that sensitivity better than 0.1 pT at 1 s measurement time is achievable under realistic conditions.

We consider the application of well known frequency modulation laser spectroscopy techniques [30–33] to enable accurate measurements of the frequencies and shapes of spectral lines and determine the magnetic field. In the case of measurements involving bichromatic light, we must scan the two photon detuning, which is defined by the frequency difference between the light harmonics and the frequency of the magnetically sensitive ground state transition of the atoms. The two photon detuning can be modified by changing the frequency of either a single optical harmonic or the relative frequency of both harmonics. The transmission spectrum of each optical harmonic reveals a set of transparency resonances observed on top of both the homogeneously and inhomogeneously broadened optical absorption line. The relative frequency separation of the resonances carries information about the absolute value of the magnetic field.

We can measure the frequency values of the resonances in several ways. The most common methods involve downconversion of the optical spectrum to either the radio frequency (RF) domain using the heterodyne technique or to the base band using the homodyne technique. The heterodyne method may involve direct measurement of the beat note of the optical harmonics, which is detected with a fast photodiode capturing light exiting the atomic sample. The beat note has a carrier frequency approximately equal to the hyperfine splitting of the atoms. The two photon detuning modulation results in modulation of the measured RF. The resultant RF spectrum resembles the optical transmission spectrum for each of the optical harmonics. In this case, the magnetically insensitive transition has the RF clock transition frequency of the atomic sample.

Alternatively, it is possible to perform heterodyne measurement by splitting the light into two parts. One part interrogates the atoms, while the other part is sent through a frequency modulator (frequency shifter) to introduce a frequency offset. The measurement signal is generated using one of the optical harmonics leaving the atomic cell and beating it on a fast photodiode against the same optical harmonic as the frequency offset. In this case, information of the clock frequency from the atomic transition is lost. The RF value in the heterodyne case is arbitrary. It is selected to be high enough to reduce the impact of the DC drifts associated with base band measurements. The sensitivity of the heterodyne technique is impacted by the stability of the frequency reference used to detect the position of the magnetically sensitive resonance.

With the homodyne technique, the amplitude and phase of each optical harmonic are detected separately without the introduction of a frequency offset. The spectrum spread of the measurement scales with the modulation frequency of the two photon detuning. In this case, the magnetically insensitive transition resonance corresponds to the zero spectral frequency. The modulation frequency is low, so the measurement can be considered to be performed in the base band.

The sensitivity of these measurements is limited due to the shot noise, relative intensity noise (RIN), and residual amplitude modulation (RAM) of light. It also depends on the symmetry of the spectral shape of the resonance under study. Shot noise and RIN reduce the signal to noise ratio (SNR), while RAM results in a nonzero background that shifts the visible center of the spectral line and limits measurement sensitivity. The resonance asymmetry results in the systematic shift of the error signal, but this shift can be compensated for.

In what follows, we analyze the sensitivity of the homodyne technique applied to an EIT resonance observed in a rubidium atomic vapor cell and identify measurement optimization strategies. We show, for example, that it is possible to reduce the impact of the EIT resonance asymmetry by adjusting the parameters of the amplitude modulation (AM) and phase modulation (PM) (controllable RAM). Furthermore, we demonstrate that simultaneous tracking of several EIT resonances enables removing their common frequency drifts as well as the effects of the quadratic Zeeman shift. Taking the majority of the technical effects into account, we show that the optimized measurement technique enables magnetic field detection with 0.1 pT sensitivity. Further improvements are possible with state-of-the-art low noise equipment.

The paper is organized as follows: in Section 2, we describe the physical model studied; in Section 3, we describe both fundamental and systematic effects limiting the sensitivity of the EIT magnetometer; in Section 4, we discuss results of the study in terms of the frequency stabilization of an EIT atomic system; Section 5 concludes the paper.

2. MODELING MAGNETOMETER RESPONSE UNDER EIT CONDITIONS

Early theoretical studies of an EIT magnetometer optimistically predicted high achievable sensitivity, expecting that the signal can be improved by simultaneously increasing the atomic vapor density and power of the pump light [14,15]. However, these studies neglect to take into account deterioration of spin coherence (due to phenomena such as radiation trapping [34]) or growing dynamic instabilities of the system. Practically, the best EIT magnetometers and clocks involve relatively low optical intensity and atomic density. Consequently, the linear attenuation of the atomic medium is relatively small, and the contrast of the EIT resonance is less than 10% [20,21,35]. In this regime, the linewidth of the resonance scales with the amplitude of the optical pump field instead of the power. Note that this statement describes only a system in which the Doppler broadened atomic transition does not contain a buffer gas [36]. There is no known unique power dependence of the EIT resonance for the case of a cell with an arbitrary amount of an arbitrary buffer gas.

To model the magnetometer response, we need to construct a transfer function that adequately captures the properties of EIT resonances in an atomic cell. At a basic level, it is possible to estimate the sensitivity of an EIT magnetometer using a steady state analytical solution of the density matrix equations for a three level $\Lambda$ level configuration [14,15]. This approach can be extended to the case of a Doppler broadened atomic transition in an atomic cell [36]. However, a realistic multilevel system becomes too cumbersome to obtain a meaningful analytical solution, and small variations in experimental parameters (e.g., buffer gas content and pressure, laser beam geometry, atomic diffusion) may require significant adjustments of the model. Instead, it is convenient to introduce an ansatz response function with experimentally determined parameters [25]. In our study, we consider a modified EIT model that also allows for incorporating the variation of the contrast of the EIT resonances.

Let us consider a toy model of an EIT resonance based on our general understanding of the physics of the system. This model is exact in the case of a naturally broadened atomic transition [15] and also in the case of a strong pump [36]. For a weaker pump, the model becomes approximate, but still valid in the vicinity of the EIT resonance.

In our model, the EIT resonance position is measured by scanning the probe laser frequency through the resonance, while the pump laser frequency is fixed (see Fig. 1). This is the case of the homodyne base-band measurement approach discussed in the Section 1. The normalized transmission for the probe light amplitude is given by the formula

(1)$$F({\omega _{p0}} + \Delta \omega) = 1 - \frac{\kappa}{{1 + i\frac{{{\Delta _0} + {\delta _B} - \Delta \omega}}{{{W_D}}} + \frac{{{g_s}}}{{1 + i(2{\delta _B} - \Delta \omega)/{\gamma _0}}}}}.$$

The linear attenuation coefficient $\kappa$ can be defined as

(2)$$\kappa = \frac{3}{{8\pi}}\eta {\cal N}\lambda _0^2L\frac{{{\gamma _r}}}{{{W_D}}}$$

in the case of inhomogeneously broadened atomic vapor. Here ${\cal N}$ is the particle density of the atomic vapor, ${\lambda _0} = 2\pi c/{\omega _{p0}}$ is the wavelength of the atomic probe transition, $L$ is the length of the optical path in the cell, ${\gamma _r}$ is the naturally broadened spectral width of the atomic optical transition at the probe frequency, ${\gamma _0}$ is the ground-state spin coherence decay rate, ${W_D}$ is the width of the Doppler resonance, and $\eta$ is a numeric coefficient dependent on the buffer gas utilized in the system as well as other technical experimental parameters (collisional broadening is expected to be much less than inhomogeneous broadening). We introduced single photon frequency detuning from the corresponding optical transition so that ${\Delta _0} = {\omega _{\textit{ac}}} - {\omega _{p0}} = {\omega _{\textit{ab}}} - {\omega _{d0}}$ when the magnetic field is zero, where ${\omega _{\textit{ab}}}$ and ${\omega _{\textit{ac}}}$ are the frequencies of the unperturbed drive and probe transitions, respectively. The probe field frequency changes during measurements. This frequency is defined by ${\omega _p} = {\omega _{p0}} + \Delta \omega$, where ${\omega _{p0}}$ is the time averaged probe frequency, and $\Delta \omega$ is the probe frequency deviation used to describe frequency modulation of the probe field. The two photon frequency detuning associated with the magnetic field is presented by the coefficient ${\delta _B}$. This detuning also adds to the single photon frequency detuning of the probe field, since we are working with high magnetic fields. This addition can be significant when considering measurements in an Earth magnetic field environment. The decoherence rate of the EIT resonance, ${\gamma _0}$, depends on the geometrical parameters and buffer gas content of the cell. A power-dependent parameter, ${g_s}$, defines the saturation of the EIT resonance and its spectral width. Depending on the pump power and atomic density, ${g_s}$ can vary. Usually it is proportional to optical intensity at higher power levels and to the square root of the optical intensity at lower power levels [36]. The dependence may change depending on the presence and amount of buffer gas in the cell.

Fig. 1. Three level scheme used to describe the electromagnetically-induced-transparency-based magnetometer.

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In the presence of a magnetic field, the frequency detunings become

(3)$${\omega _{\textit{ab}}} - {\omega _d} = {\Delta _0} - {\delta _B},$$

(4)$${\omega _{\textit{ac}}} - {\omega _p} = {\Delta _0} + {\delta _B} - \Delta \omega ,$$

(5)$${\omega _{\textit{bc}}} - {\omega _p} + {\omega _d} = 2{\delta _B} - \Delta \omega .$$

Utilizing these detuning values, we write Eq. (1) that captures the major features of a generic EIT resonance in a simplified form. The advantage of the ansatz Eq. (1) is that it allows for explaining EIT spectra in a vacuum cell. The addition of a buffer gas changes the shape of the EIT resonance significantly; however, the model still adequately describes the contrast and phase response of the system in the vicinity of the two photon resonance.

We assume that the medium is optically thin, $\kappa \lt 1$, and that the EIT resonance contrast is small (${g_s} \le 1$), since this appears to be the optimal operation regime for EIT magnetometers. In this case, it is useful to introduce the following experimentally measurable parameters:

(6)$$R = (1 - \kappa {)^2},$$

(7)$$T = {\left({1 - \frac{\kappa}{{1 + {g_s}}}} \right)^2},$$

(8)$$C = 2\kappa (1 - \kappa)\frac{{{g_s}}}{{1 + {g_s}}},$$

(9)$${\delta _{\rm{EIT}}} = (1 + {g_s}){\gamma _0},$$

where $R$ is the contrast of the single photon resonance, $T$ is transmission at the EIT point (for the low EIT contrast $T \approx R$), $C$ is the contrast of the EIT resonance (Fig. 2), and ${\delta _{\rm{EIT}}}$ is the half width at half maximum of the EIT resonance. The use of such phenomenological parameters also makes it easier to generalize the sensitivity analysis presented here to other EIT-like systems [37,38] by using proper dependencies of the EIT resonance characteristics on the specific experimental parameters.

Fig. 2. Illustration of an EIT resonance and its basic parameters measurable in an experiment. The inset shows a Lorentzian fit of the EIT resonance.

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A. Phase-Sensitive Measurements

Let us consider magnetic field measurements performed with a phase detection technique that evaluates only the probe field. In this method, the drive field is assumed to be filtered out. While this approach does not reflect a typical experimental setup, the study of this model is valuable for identifying the highest possible measurement sensitivity. The phase measurement is performed by first detecting the frequency of the probe field at which the measurement error signal is zero, and then measuring changes in the error signal to quantify variations of the magnetic field.

Phase spectroscopy allows for precision measurement of a magnetically sensitive center of the spectral line. This is accomplished by transforming the PM of the probe light to the AM of the frequency-dependent spectral profile of the response of a magnetically sensitive atomic cell. The phase modulated light sent directly to a fast photodiode does not produce any signal on the photodiode at the modulation frequency. An atomic spectral line modifies the phase relationship between the modulation spectral sidebands for the off-resonant tuning of the probe light with respect to a symmetrical spectral line. A signal characterizing the frequency detuning from the line center is produced on the photodiode at the modulation frequency. Minimizing the signal by tuning the probe laser frequency allows for locking the laser to the atomic transition. With this approach, either the error signal or the feedback magnitude becomes an indicator of the variation of the atomic resonance frequency.

In the case of EIT spectroscopy, the frequencies of the probe and drive fields should approximately coincide with the frequencies of the corresponding optical atomic transitions. The frequency detuning should be small if compared with the inhomogeneously broadened (Doppler) bandwidth of the transitions. The frequency difference of the optical frequencies should coincide with the ground state splitting of the $\Lambda$ level configuration used in the experiment. In one possible realization of the experiment, the frequency of the drive field stays constant, while the frequency of the probe field is phase modulated in the vicinity of the two photon resonance. In so doing, changes of the magnetic field result in a shift of the two photon resonance, which can be detected to determine the absolute value of the magnetic field.

B. Phase-Sensitive Spectroscopy

The typical magnetometer feedback signal is produced using the phase-sensitive detection of a modulated optical probe. In what follows, we assume that the probe field modulation frequency is ${\omega _m}$, and both small AM and PM are present:

(10)$$\begin{split}{E_{{p_{\rm{in}}}}} &= {E_{0p}}({1 + a\cos {\omega _m}t} ){e^{- i({\omega _{p0}}t + b\sin {\omega _m}t)}} \\ &\approx {E_{0p}}{e^{- i{\omega _{p0}}t}}\left[{1 + \left({\frac{{a - b}}{2}} \right){e^{i{\omega _m}t}} + \left({\frac{{a + b}}{2}} \right){e^{- i{\omega _m}t}}} \right],\end{split}$$

where coefficients $a$ and $b$ are AM and PM coefficients, respectively. For the complex amplitude of light exiting the cell, we find

(11)$$\begin{split}{E_{{p_{\rm{out}}}}} &= {E_{{0_p}}}{e^{- i{\omega _{p0}}t}}\left[{F({\omega _{p0}}) + \frac{{a - b}}{2}F({\omega _{p0}} - {\omega _m}){e^{i{\omega _m}t}} }\right.\\&\quad+ \left.{\frac{{a + b}}{2}F({\omega _{p0}} + {\omega _m}){e^{- i{\omega _m}t}}} \right],\end{split}$$

where we replaced $\Delta \omega$ in Eq. (1) with the frequency of the corresponding harmonics, ${\pm}{\omega _m}$.

The modulated light is sent to a photodiode with responsivity ${\cal R}$ (Fig. 3) to obtain the electric current error signal. In the first approximation, the drive field ${E_{{d_{\rm{out}}}}}$ is not modulated and thus does not contribute to the detected AC photocurrent ${i_{\rm{PD}}}$:

(12)$${i_{\rm{PD}}} = {\cal R}{P_{{p_{\rm{out}}}}} \propto E_{{p_{\rm{out}}}}^*{E_{{p_{\rm{out}}}}},$$

where ${P_{\rm{out}}}$ is the optical power at the photodiode (PD).

Fig. 3. Setup for the phase-sensitive spectroscopy of the EIT resonance.

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Only the components of the photocurrent oscillating at ${\omega _m}$ need to be considered. This simplification leads to the expression

(13)$$\begin{split}\frac{{{P_{{p_{\rm{out}}}}}}}{{{P_0}}}{|_{\exp (\pm i{\omega _m})}} &= \frac{1}{2}\left[{(a + b)F({\omega _{p0}}){F^*}({\omega _{p0}} + {\omega _m}) }\right.\\&\quad+ \left.{(a - b){F^*}({\omega _{p0}})F({\omega _{p0}} - {\omega _m})} \right]{e^{i{\omega _m}t}} + {\rm c.c.}\end{split}$$

The photocurrent is electronically mixed with an electronic local oscillator signal modulated at the same frequency,

(14)$${i_{\rm{LO}}} \sim {e^{i({\omega _m}t + {\phi _{\rm{LO}}})}} + {e^{- i({\omega _m}t + {\phi _{\rm{LO}}})}},$$

and filtered out with a low pass filter of bandwidth $\Delta f$, to produce a DC error signal with power ${P_{\rm{error}}}$:

(15)$$\begin{split}{P_{\rm{error}}} &= \rho i_{\rm{PD}}^2 \\ &= \rho {{\cal R}^2}P_0^2\frac{\mu}{4}{\left\{{\left[{(a + b)F({\omega _{p0}}){F^*}({\omega _{p0}} + {\omega _m}) }\right.}\right.}\\&\quad+ {\left.{\left.{(a - b){F^*}({\omega _{p0}})F({\omega _{p0}} - {\omega _m})} \right]{e^{- i{\phi _{\rm{LO}}}}} + {\rm c.c.},} \right\}^2},\end{split}$$

where ${\rm\mu}$ is the mixer efficiency, and $\rho$ is the photodetector resistance. Equation (15) allows us to find the maximum measurement sensitivity for the phase-sensitive detection and also to identify the systematic frequency shifts and associated noise observed in the system. In what follows, we discuss these effects.

3. LIMITATIONS OF EIT MAGNETOMETER SENSITIVITY

A. Fundamental Noises

First, we identify the fundamental sensitivity limit, which is determined by the laser shot noise and the relative intensity noise (RIN) of the laser. Strictly speaking, the latter falls under the category of technical noise, since it is determined by the technical characteristics of the laser. However, it is external to the EIT system, and is thus included in this section. We also assume pure PM of the probe field $a = 0$, and save RAM to be considered later. To ensure that the expression in Eq. (15) is real, we select ${\phi _{\rm{LO}}} = 0$ and obtain the following expression for the magnetometer error signal:

(16)$${P_{\rm{error}}} =\mu\rho {\left[{4C{\cal R}{P_0}b\frac{{\delta _{\rm{EIT}}^2{\omega _m}{\delta _B}}}{{{{(\delta _{\rm{EIT}}^2 + \omega _m^2)}^2}}}} \right]^2}.$$

The power of the first PM sideband is approximately given by $4{P_1}{P_0} = P_0^2{b^2}$, so that

(17)$${P_{\rm{error}}} =\mu\rho {\left[{8C{\cal R}\sqrt {{P_0}{P_1}} \frac{{\delta _{\rm{EIT}}^2{\omega _m}{\delta _B}}}{{{{(\delta _{\rm{EIT}}^2 + \omega _m^2)}^2}}}} \right]^2}.$$

As the expression shows, the signal magnitude maximizes when ${\omega _m} = {\delta _{\rm{EIT}}}/\sqrt 3$.

To find the SNR, we must consider the broadband shot noise as well as RIN of the probe light that reaches the photodiode. The noise power can be described as

(18)$${P_{\rm{RIN}}} =\mu\rho {{\cal R}^2}P_0^2\left({{\rm RIN}({\omega _m}) + \frac{{2\hbar \omega}}{{{P_0}}}} \right)\Delta f,$$

where $\Delta f \ll {\omega _m}$ is the bandwidth of the measurement, and ${\rm RIN}({\omega _m})$ indicates that the RIN at the modulation frequency must be taken into account.

The SNR is defined as ${P_{\rm{error}}}/{P_{\rm{RIN}}}$. Assuming that the accuracy of the lock is given by ${\rm SNR} = 1$, we find

(19)$$\frac{{{\delta _B}}}{{{\delta _{\rm{EIT}}}}} \approx \sqrt {\frac{{{P_0}}}{{{P_1}}}} \frac{{{\delta _{\rm{EIT}}}}}{{{\omega _m}}}\frac{{{{(\delta _{\rm{EIT}}^2 + \omega _m^2)}^2}}}{{\delta _{\rm{EIT}}^4}}\frac{{\sqrt {({\rm RIN}({\omega _m}) + 2\hbar \omega /{P_0})\Delta f}}}{{8C}}.$$

Let us make some numerical estimations for typical experimental parameters [18]. For an EIT magnetometer $2{\delta _{\rm{EIT}}} \,\sim\def\LDeqbreak{} 2\pi \times 5\;{\rm kHz} $, the optical power is 500 µW, $\Delta f = 1\;{\rm Hz} $, $C \sim 0.1$, ${\rm RIN} = {-}140 \;{\rm dB/Hz}$ at 2.5 kHz frequency offset (this is the RIN level of a relatively low noise laser). From Eq. (19), it is easy to see that the maximum demodulated signal occurs for ${\omega _m} \approx {\delta _{\rm{EIT}}}$. To avoid over-modulation, it is reasonable to assume ${P_0}/{P_1} \approx 3$. Finally, for $^{87}{\rm Rb}$ atoms, the relations between the Zeeman frequency and applied magnetic field is ${\delta _B} = \delta B \times 2\pi \times 0.7 \;{\rm MHz/G}$. Under these conditions, we can expect the magnetometer sensitivity to reach $\delta B \simeq 3\; {{\rm nG/Hz}^{1/2}}$ or $0.3 \;{{\rm pT/Hz}^{1/2}}$, as shown in Fig. 4.

Fig. 4. (a) Example of the typical relative intensity noise (RIN) of the probe laser resulting directly from a laser chip as well as converted technical noises. (b) Sensitivity of the magnetometer calculated using Eq. (19) for three different EIT widths and RIN values estimated in the top panel. (c) Comparison of the shot noise and RIN limited sensitivity.

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This model is valid for near DC magnetic fields measured in a band narrower than the EIT resonance, where we assume linear dependence of the error signal in Eq. (17) on the Zeeman shift ${\delta _B}$. While the position of the EIT resonance reacts rather quickly to changes of the magnetic field, the speed of detection is limited by the resonance bandwidth.

The sensitivity can be improved in multiple ways. The most obvious approach is reduction of the RIN below the shot noise level. The shot noise for the configuration identified above is ${-}{150}\;{\rm dB/Hz}$. Reaching the shot noise level results in achieving $\delta B = 1 \;{{\rm nG/Hz}^{1/2}}$ (0.1 pT) measurement sensitivity. Further improvements can become possible by increasing the overall power and contrast as well as decreasing ${\delta _{\rm{EIT}}}$. However, all steps towards improvement must be taken systematically. For instance, decreasing the bandwidth of the EIT resonance will call for reduction of the modulation frequency and reduction of the frequency offset frequency from the optical carrier, which usually leads to an increase of RIN.

The atomic projection noise does not impact measurement sensitivity directly, given that the number of atoms is not contained in Eq. (19). However, the noise becomes important when the undefined initial state of the atoms participating in the measurement process contributes to the SNR. In the measurement described here, the initial uncertainty of the atoms entering the interaction region contributes to the noise associated with the decoherence of the EIT transition, which directly relates to the attenuation of light in the EIT medium. The SNR decrease due to the attenuation associated with the atomic state preparation accounts for the random processes occurring in the medium. In other words, the medium does not change the coherent state of light regardless of the processes present during the interaction. The variations associated with the atomic spins introduce uncertainty of the signal magnitude, but this variation is a second order effect that can be omitted. This omission is common for continuous wave measurements.

B. Systematic Shifts

While it may be difficult to increase the fundamental sensitivity limit of an EIT magnetometer without significant modifications to the experimental apparatus, approaching this limit requires managing various parasitic effects that limit its accuracy and sensitivity. In this section, we discuss a selection of these effects, and distinguish between the systematic shifts that primarily impact accuracy of the measurement (e.g., the device shows a nonzero magnetic field when the actual ambient field is zero), and technical errors limiting the measurement sensitivity (discussed in the next section).

Let us identify the most prominent systematic frequency shifts. When only the variation of the magnetic field is measured, the overall contribution of these shifts can often be excluded (assuming they remain constant during the duration of measurement). However, these shifts can be detrimental to the absolute precision of the device. For example, in an EIT system, systematic shifts of the error signal appear due to the RAM of laser emission, elliptical polarization of light, and nonzero single photon detuning.

1. Shifts Due to Residual Amplitude Modulation

AM shifts the zero of the error signal in Eq. (15) by ${\delta _{\rm{AM}}}$, which can be written in the form of the frequency shift ${\delta _{\rm{AM}}}$ mimicking the true resonance shift ${\delta _B}$:

(20)$$\frac{{{\delta _{\rm{AM}}}}}{{{\delta _{\rm{EIT}}}}} \approx - \frac{a}{b}\frac{R}{{2C}}\frac{{{{(\delta _{\rm{EIT}}^2 + \omega _m^2)}^2}}}{{\delta _{\rm{EIT}}^4}}\frac{{{\delta _{\rm{EIT}}}}}{{{\omega _m}}}.$$

Even a small admixture of AM results in a significant shift if (i) ${\omega _m}$ is smaller than ${\delta _{\rm{EIT}}}$ and (ii) the contrast is small, $C \lt 1$. The frequency shift is power dependent since it depends on the bandwidth of the EIT resonance. This power dependence results in long term drifts in the system. The direct RAM is $a/b \ge {10^{- 4}}$ for a relatively good laser, resulting in the shift of the error signal (systematic error) of 3.4 µG (or 340 pT) for $R = 0.5$. Since a typical relative drift of standard laser power is worse than ${10^{- 3}}/{\rm s}$, this effect limits the sensitivity by one nG (0.1 pT).

2. Shifts Due to Nonzero One Photon Laser Detuning

Nonzero single photon detuning ${\Delta _0} \ne 0$ also results in a systematic shift of the error signal. In the most interesting case (${g_s}{W_D} \gg {\gamma _0}$ and $1 \gt {g_s}$), the frequency shift depends on the power through the bandwidth of the EIT resonance:

(21)$$\frac{{{\delta _{\Delta 0}}}}{{{\delta _{\rm{EIT}}}}} \approx - \frac{{{\Delta _0}}}{{{W_D}}}\frac{{\omega _m^2 + \delta _{\rm{EIT}}^2}}{{2\delta _{\rm{EIT}}^2}}\left({1 - \frac{{\Delta _0^2 + \gamma _0^2}}{{{\gamma _0}{g_s}{W_D}}}} \right).$$

The physical meaning of this shift is the impact of the dispersion of the inhomogeneously broadened resonance on the dispersion of the two photon EIT resonance. Variation of laser frequency as well as variations of the bandwidth of the EIT resonance result in the measurement error associated with this systematic shift.

The shift Eq. (21) might be significant if the atomic vapor is placed in a large magnetic field. For instance, the Earth level magnetic field 0.5 G ($5 \times {10^{- 5}}{\rm T}$) results in 0.35 MHz frequency shift. Taking into account the Doppler width ${W_D} = 800\;{\rm MHz} $, we find that the systematic error signal shift can be as large as 10 µG (1 nT) in the case of circularly polarized light. However, there are configurations of electric field polarization that minimize the impact of the single photon shift on the two photon shift. This is the case for the magnetometer involving both pump and probe characterized with the same linear polarization [17].

Equation (21) also suggests that the frequency shift is power dependent and might be minimized by optimizing the beam diameter as well as the optical power. It is possible to show that there is a turning point at which this effect becomes negligible.

We consider here a simplified three level $\Lambda$ configuration. The presence of the off-resonant hyperfine levels in the excited state also results in the effects similar to single photon detuning. These effects should be taken into account when the calculations are performed for a particular atomic system.

3. Effect of Residual Laser Ellipticity

Another laser-power-dependent systematic frequency shift is caused by the elliptical polarization of light. The relative frequency shift, not being a part of our simplified model, results from the far detuned atomic levels [39], and can be estimated as

(22)$$\frac{{{\delta _{\rm{AC}}}}}{{{\delta _{\rm{EIT}}}}} \approx \epsilon \frac{{{{\tilde g}_s}}}{2}\frac{{{W_D}}}{{{\Delta _{\rm{HF}}}}},$$

where $\epsilon$ is the residual ellipticity of light, ${\tilde g_s} \approx {g_s}$ is the saturation parameter associated with the far detuned transitions, and ${\Delta _{\rm{HF}}}$ is the frequency detuning from the transitions. Practically speaking, the residual ellipticity should be of the same order of magnitude as $a/b$ to ensure that the contribution of the effect to measurement accuracy is minimized. A systematic error also can occur due to the dynamic dependence of the AC-Stark on the scan frequency in the case of fast scanning and finite time redistribution of the population in the medium.

The aforementioned effects can be reduced by measuring the splitting between two resonance EIT peaks experiencing the same systematic shift. When measuring the frequency splitting between two resonances of slightly different shapes, a significant systematic error is observed. Therefore, one must compare the resonances with identical absolute values of the magnetic number $|m|$ to reduce the asymmetry.

4. Other Systematic Shifts

Systematic errors are also caused by changes of the shape of the EIT resonance stemming from attenuation, the optical beam profile, and buffer gas contents [24–26,40,41]. In measurement configurations that involve the detection of strong magnetic fields, the presence of nonzero nuclear spin leads to nonlinear Zeeman splitting caused by mixing the ground Zeeman states and by differences in Larmor frequencies for hyperfine manifolds [26]. These errors can be partially corrected by optimizing the measurement procedure for each particular measurement configuration. However, variation of the parameters of the measurement setup along with the systematic shifts results in unavoidable drifts in the systems.

Variations of temperature of the cell as well as the density and type of the buffer and residual gases also result in systematic frequency shifts in the magnetometer [42–44]. These effects are too complex to be evaluated analytically, and experimental study is needed to quantify the shifts. Fortunately, the expected differential shift mimicking the magnetic field is expected to be small if compared with the sensitivity level estimated above.

C. Technical Noises

While the systematic effects result in a constant shift and long term drift of the measured value of the magnetic field, noise is the primary restriction on both the short term sensitivity and accuracy of the measurements. In what follows, we discuss various noise sources impacting the EIT magnetometer.

1. Electronic Noise

Because optical signals are ultimately converted into photoelectric currents, the sensitivity of the measurement is restricted by the technical noise levels. This noise can arise from sources such as the dark current of the photodiode (usually negligible at high optical power) or input noise of the pre-amplifiers, which we denote as $\delta {i_{\rm{LO}}}$. It is possible to estimate the associated sensitivity value as follows:

(23)$$\frac{{{\delta _B}}}{{{\delta _{\rm{EIT}}}}} \approx \frac{{\delta {i_{\rm{LO}}}}}{{16C{\cal R}\sqrt {{P_0}{P_1}}}}\frac{{{\delta _{\rm{EIT}}}}}{{{\omega _m}}}\frac{{{{(\delta _{\rm{EIT}}^2 + \omega _m^2)}^2}}}{{\delta _{\rm{EIT}}^4}}.$$

Assuming that ${P_0} = 500\;{\unicode{x00B5}}{\rm W}$, ${P_1} = 170\;{\unicode{x00B5}}{\rm W}$, $\delta {i_{\rm{LO}}} \sim 5 \;{{\rm fA/Hz}^{1/2}}$ (a low noise current pre-amplifier SRS SR570; see also [45] for high end performance), and ${\cal R} = 0.8\;{\rm A/W}$, we get $\delta B \sim 0.4 \;{{\rm nG/Hz}^{1/2}}$ ($40\;{{\rm fT/Hz}^{1/2}}$), which is approximately at the level of the fundamental sensitivity of the measurement with an optimized measurement setup.

Electronic noise can be introduced by other sources in the measurement system. For instance, in experiments that use electro-optical modulators to produce probe and pump EIT fields by phase-modulating a monochromatic laser output, the fluctuations in amplitude and phase noise of RF modulation can result in amplitude noise of the pump and probe light as well as additional phase noise of the modulation signal.

2. Phase to Amplitude Noise Conversion

As discussed above, RIN represents one of the major noise sources of an optical magnetometer. This noise can be directly related to the laser source since white RIN acts similarly to shot noise. This contribution is considered in Eq. (19). Furthermore, RIN can be caused by the conversion of laser phase noise into amplitude noise PM-to-AM conversion due to the dispersion of the EIT resonance [46]. The transmitted beam can exhibit excess intensity noise (AM) when laser light propagates through a resonant medium. In a semiclassical description of the phenomenon, laser phase noise (PM) induces fluctuations in the medium’s electric susceptibility, which in turn cause fluctuations in the transmitted intensity. Intuition suggests that large linewidth lasers should exhibit much greater PM-to-AM conversion than narrow linewidth lasers [46]. However, the dependence on laser linewidth can be rather nontrivial in the general case, and needs careful study for each realization of the measurement setup [47–50].

It is possible to estimate the RIN due to PM-to-AM conversion as

(24)$${{\rm RIN}_{\rm{EIT}}} \approx \frac{{{C^2}\Delta {S_\nu}}}{{\delta _{\rm{EIT}}^2}} + \frac{{R{S_\nu}}}{{W_D^2}}\frac{{\Delta _0^2}}{{W_D^2}},$$

where $\Delta {S_\nu}$ is the relative frequency noise of the pump and probe light, and ${S_\nu}$ is the frequency noise of the laser. The relative frequency noise originates from the frequency fluctuations of two photon detuning. Strictly speaking, two photon detuning fluctuations are transferred to intensity fluctuations when ${\omega _m}$ is tuned to the slope of the EIT resonance. There is no conversion for the resonant tuning of the pump–probe light. Since the two photon frequency is modulated in a real system to identify the center of the EIT line, and since the measurement bandwidth is finite, the EIT-based PM-to-AM conversion impacts the measurement.

It is worth noting that Eq. (24), which shows how two photon and single photon resonances discriminate the frequency noise of the laser and the beat note between the probe and drive light, is not directly related to Eq. (21), which shows the systematic frequency shift of the magnetically sensitive resonance due to presence of single photon detuning.

Two photon residual noise depends on the quality of the phase locking of the pump and probe lasers or the laser modulation that produces both pump and probe light. It is also impacted by the RF amplifier noise that regulates the magnitude of the RF signal feeding the modulator, as well as by the fluctuations of the RF delay in the RF circuitry. It is reasonable to expect that the residual frequency noise is similar to a frequency noise of a regular dielectric resonator oscillator (DRO) used to modulate the driving light to produce the probe. For a standard 6.8 GHz DRO, the frequency noise at 10 kHz is ${10^{- 2}} \;{{\rm Hz}^2}/{\rm Hz}$, and the corresponding RIN is ${-}{129}\;{\rm dB/Hz}$. The estimated RIN level from the conversion of frequency noise to amplitude noise is of the order of the RIN of the laser. This number can be improved if a better microwave source is used to generate the dive–probe pair. To reduce the RIN contribution, it may also be beneficial to optimize the frequency modulation magnitude.

An important feature of ${{\rm RIN}_{\rm{EIT}}}$ is related to its dependence on the EIT bandwidth and contrast. The noise level increases for the case of a narrower EIT resonance. The fundamental white RIN and shot noise contribution does not depend on the EIT parameters.

The frequency noise of the optical signal is transferred to intensity fluctuations if the laser frequency is tuned away from the line center (${\Delta _0} \ne 0$). The transfer occurs in the frequency band of the broadened optical transition that exceeds the frequency at which the measurement occurs (${\omega _m}$). The frequency noise of a distributed-feedback (DFB) laser reaches ${10^6}\;{{\rm Hz}^2}/{\rm Hz}$ at 10 kHz offset (and even greater at smaller offsets). Single photon detuning is of the order of a MHz and results from the uncompensated for Earth magnetic field. The RIN associated with this noise is approximately ${-}{192}\;{\rm dB/Hz}$, which is an insignificant value. Therefore, the frequency noise of a laser is of low importance in these measurements. This noise can become more significant if the Doppler resonance acquires a saturated absorption structure due to residual reflections in the system [51].

3. Amplitude to Phase Noise Conversion

Amplitude to phase noise conversion (AM-to-PM) is the opposite of the PM-to-AM noise conversion effect. The dynamic (AC-)Stark shift observed in an atomic coherent medium [52–55] motivates the transformation of the optical AM-to-PM, which ultimately leads to magnetometer sensitivity that is restricted with a standard quantum limit (SQL) [39]. This is fundamental noise, but it is usually small for a real system. The AC-Stark shift also results in systematic effects hindering accuracy of the measurement. The so called vector AC-Stark shift directly mimics the magnetic field action [52–55]. It can be reduced or eliminated in some measurement schemes by proper selection of single photon optical detuning [56] and polarization of probe and pump light [17].

The magnitude of the AC-Stark shift is a function of the local intensity of light. The intensity changes because of the variations of the optical beam profile as well as because of the attenuation of the medium. These effects are hard to either predict or control, so identification of AC-Stark free configurations is desirable.

The AC-Stark shift might have a scalar, vector, or tensor nature [52–55]. The scalar shift is applied to all atomic levels and can be neglected in an EIT magnetometer. The tensor AC-stark shift vanishes if the excited-state hyperfine structure is not resolved, since the net quadrupole polarizability moment for the two unresolved transitions sums to zero. This condition can be fulfilled in an atomic cell with a buffer gas. The vector light shift mimics the magnetic field action and is absent in an EIT system interacting with linearly polarized light [17]. The residual elliptical polarization resulting from the circular dichroism of the windows of the cell can result in a systematic error and additional measurement noise coming from the conversion of the amplitude noise of the light to the phase of the EIT signal.

In addition to the AC-Stark shift, frequency noise can be introduced by the variation of the EIT bandwidth due to variations of optical power. This noise contribution nullifies when two photon detuning coincides with the EIT resonance. However, given that two photon detuning is modulated for locking purposes, the contribution of frequency noise might be significant.

To conclude this subsection, it is worth noting that AM-to-PM conversion usually is optimized experimentally. This process involves measuring the response of the system to artificially introduced fluctuations and estimating the contribution of unavoidable technical noise.

4. Variations of Ambient Temperature

Variations of the ambient temperature result in changes of the fractional clock and magnetometer frequencies to a limit of ${10^{- 10}}/{\rm K}$ [57–59]. This effect comes from the dependence of the atomic density on temperature. The variations of the atomic density lead to variations of the EIT parameters as well as the optical power impacting the signal.

The impact of thermal dependence can be significant. Variations of the frequency at the level of 0.1 ppb lead to 0.67 Hz frequency variation of the EIT transition. This corresponds to a magnetic field magnitude of approximately 0.5 µG (or 50 pT) for $^{87}{\rm Rb}$. To improve the sensitivity by three orders of magnitude, one must detect the frequency difference between EIT resonances in the Zeeman manifold. Reduction of the absolute frequency difference by three orders of magnitude leads to improvement of the thermal insensitivity by three orders of magnitude or more, assuming we admit possible correlations among the transitions. Regardless, the measurement sensitivity limitations due to variations of temperature can be less than a nG if a proper measurement configuration is selected.

4. DISCUSSION

The results of our study are qualitatively supported by the general observation that, in most situations, the accuracy of determining the resonance center frequency does not exceed 1 ppm. For a magnetically sensitive EIT resonance with 1 kHz bandwidth and 1.4 MHz/G magnetic sensitivity (for $^{87}{\rm Rb}$ atoms), this estimate puts the expected measurement accuracy at a level of a few nanoGauss. A balanced measurement, or a measurement of the relative frequencies of Zeeman resonances, may reduce common fluctuations and improve the sensitivity by an order of magnitude or more.

This prediction is quite plausible if we take into account that the EIT magnetometer is quite similar to CPT atomic clocks. A good continuous wave Rb EIT atomic clock has 1 s stability of the order of ${10^{- 11}}$ [35,60], which can be translated to 70 ppm splitting of the 1 kHz wide microwave clock transition at 6.8 GHz frequency. The inferred magnetometer sensitivity at 1 s averaging, corresponding to this frequency stability and line splitting, is at the level of 5 pT. This value is nearly two orders of magnitude worse than our prediction for the magnetometer discussed in this paper. We see the discrepancy in the not optimal interrogation optical technique utilized in the demonstrated clocks. Usage of low noise bichromatic light with optimized intensities of the frequency harmonics may result in the improvement predicted in this study. Moreover, advanced Rb clocks based on periodic optical pumping may improve their 1 s stability by two orders of magnitude [61]. Similar improvement was observed for a Cs CPT clock [59]. Thus, we can expect similar improvements in magnetometer performance if periodical pumping is used there. This configuration should be studied theoretically in the future.

While these estimates still put the expected sensitivity of EIT magnetometers behind some of the most precise optical atomic magnetometers [4,5,7–12], there are still applications in which such devices can offer clear advantages. Those include large dynamic range measurements enabled by the EIT magnetometers. In such situations, the presented work outlines a clear optimization strategy for achieving their best performance.

5. CONCLUSION

We have studied both fundamental and technical limitations of a magnetometer based on EIT. Our model is rather generic and can be applied to various EIT realizations and detection techniques. Our calculations applied to a Rb atomic system show that proper optimization of the measurement setup can lead to measurement sensitivity of the magnetic field magnitude better than one nanoGauss (${\lt}{100}\;{\rm fT}$) for 1 s measurement time under realistic experimental conditions. While better sensitivity is fundamentally possible, it becomes increasingly more difficult to reach with commercially available lasers and electronics.

Funding

National Aeronautics and Space Administration (80NM0018D0004); Defense Advanced Research Projects Agency (81-110107); Army Research Office (W911NF-21-2-0094).

Acknowledgment

The authors acknowledge stimulating discussions with Dr. Sheng-Wey Chiow at the Jet Propulsion Laboratory in Pasadena, CA, as well as Dr. John Kitching, Dr. Ying-Ju Wang, Dr. Isaac Fan, and Dr. Yang Li at the National Institute of Standards and Technology in Boulder, CO. The views, opinions, and/or findings expressed are those of the authors and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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Technical limits of sensitivity for EIT magnetometry

Abstract

1. INTRODUCTION

2. MODELING MAGNETOMETER RESPONSE UNDER EIT CONDITIONS

A. Phase-Sensitive Measurements

B. Phase-Sensitive Spectroscopy

3. LIMITATIONS OF EIT MAGNETOMETER SENSITIVITY

A. Fundamental Noises

B. Systematic Shifts

1. Shifts Due to Residual Amplitude Modulation

2. Shifts Due to Nonzero One Photon Laser Detuning

3. Effect of Residual Laser Ellipticity

4. Other Systematic Shifts

C. Technical Noises

1. Electronic Noise

2. Phase to Amplitude Noise Conversion

3. Amplitude to Phase Noise Conversion

4. Variations of Ambient Temperature

4. DISCUSSION

5. CONCLUSION

Funding

Acknowledgment

Disclosures

Data availability

REFERENCES

Data availability

Cited By

Figures (4)

Equations (24)

Applied Optics