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Abstract
Atomic vapour magnetometers sense the local magnetic field strength by measuring the resulting precession rate of a well-defined quantum state. An essential prerequisite for this approach is a requirement to drive the media into this quantum state, which is frequently achieved via optical pumping. In real-world alkali-metal atoms, with their multiplicity of ground states, the optical pumping process is necessarily lossy, with a large fraction of the atoms being lost to quantum states that do not contribute to the useful magnetically sensitive signal. This consequently reduces the sensitivity of all optically-pumped atomic sensors. Here we theoretically and experimentally study the population changes of the quantum ground states of 87Rb during optical pumping. We use this understanding to develop a repumping scheme that allows us to increase the number of atoms that are contributing to the useful magnetic sensing output. Unlike prior schemes, our approach delivers this improved sensitivity with significantly suppressed fictitious magnetic fields associated with the repumping, which would otherwise reduce the accuracy of the sensor. When operated at Earth’s field strength (∼50µT), the repumped sensor shows a magnetic sensitivity of 200 fT/$\sqrt {\text {Hz}}$, that is nearly three times higher than the non-repumped version.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
High-sensitivity magnetometry has a wide range of important applications, ranging from the tests of fundamental physics [1], explorations of geophysical fields [2], measurements of biomedical signals [3,4], nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) measurements [5,6], magnetonanoparticle detections [7], through to imaging of structural defects in materials [8]. Superconducting quantum interference devices (SQUIDs) were the original sensor of choice for these applications, demonstrating sensitivity better than $\sim$1 fT/$\sqrt {\text {Hz}}$ [9]. However, the SQUID detects a single component of magnetic field tensor and only operates at low temperatures [9]. In many circumstances, those characteristics are not the best choice for the application.
As an alternative, polarized atomic gases can be used to provide a high sensitivity measurement of magnetic fields. This work dates back 60 years [10], and in its most sensitive incarnation has shown a 0.16 fT/$\sqrt {\text {Hz}}$ sensitivity at 40 Hz in a spin-exchange relaxation-free (SERF) regime [11]. Although incredibly sensitive, SERF magnetometers have a very low bandwidth ($\sim$tens of Hz) and can only be used at near-zero fields ($\sim 0.01\,\mu$T) [11,12]. In addition, they require a calibration of the sensor, and have an inherently limited linearity and dynamic range [13,14]. Another type of magnetometer, the metastable Helium-4 magnetometer, shows a sensitivity of 50 fT/$\sqrt {\text {Hz}}$ at near-zero fields [15]. For measurements at magnetic field strengths similar to those of the Earth’s field ($\sim 50\,\mu$T) and with the potential for absolute accuracy, we find examples in the $M_x$, Bell-Bloom and Nonlinear Magneto-Optical Rotation (NMOR) schemes, which generally measure the resonant frequencies of Larmor precessions within the magnetic field [16–20]. The Bell-Bloom magnetometers can deliver a measurement of the absolute field at Earth’s field strength with femtoTesla-level sensitivities but need a cell heated up to 85-130$^\circ$C [14,21,22]. For the $M_x$ magnetometers, a sensitivity of 1.8 fT/$\sqrt {\text {Hz}}$ was also demonstrated at $50\,\mu$T by using a large cell (150 mm in diameter) at 42$^\circ$C [23].
Here we will use a repumping laser to construct a precise and accurate room-temperature atomic magnetometer with a dynamic range that allows high-sensitivity measurements in the Earth’s field. This will also enable the magnetometer to be free from the noise associated with the cell heaters [21,24,25]. For the biomedical applications, a magnetometer operated at room temperature could be placed closer to the human body for improved biomedical signal detection, compared to a magnetometer with vapor cells heated to 85-130$^\circ$C [14,21,22].
Let us first start with the fundamental sensitivity of an NMOR magnetometer, given by [26,27]:
where $\gamma$ is the gyromagnetic ratio, $N$ is the effective number of atoms that contribute to the magnetic sensing signal, $T_2$ is the transverse spin relaxation time, and $t$ is the measurement time. To enable the atomic ensemble to sense magnetic fields, it’s necessary to maximize the number of atoms, $N$, that are in the useful quantum state. The preparation of this quantum state is typically achieved by optical pumping the vapour with an optical beam with well chosen frequency and polarization [28,29]. This optical pump drives some fraction of the atoms into this well-defined quantum state that can be observed by a probe beam. Unfortunately, this optical pumping process unavoidably drives atoms into other quantum states which are not coupled to the field sensing transitions, leading to a reduction in the magnetometer sensitivity.To increase the effective number of atoms in the useful quantum state being probed, several techniques, including multi-passing the probe light [14], using large cells [23] and heating the cells [25], have been explored to enhance the magnetometer sensitivity. Alternatively, repumping lasers have been used to recover the lost atomic population into the original atomic states and thereby retrieve the lost magnetic sensitivity. A $300\,\%$ increase in the sensitivity of Cs magnetometer has been observed by exploiting a circularly-polarized laser with strong repumping effect [20,30]. A significant improvement in the sensitivity can also be realized by using a single intense circularly-polarized laser which simultaneously acted as the pump, probe and repump beams [31]. However, in both these cases the intense off-resonant repump light led to a strong light shift that limited the accuracy of the field measurements to the nanotesla level [32–34]. Recently, a Ramsey-style interrogation in a pulsed free-induction-decay modality was exploited to reduce the light shift inaccuracies to within 0.6 nT [35]. This paper proposes and demonstrates a scheme which significantly mitigates the production of fictitious magnetic fields associated with the repumping while preserving the increase in sensitivity that accrues from the repumping approach.
In particular, we analyze the fraction of the atomic population that is lost from the pump transition. To retrieve the lost atoms, and thereby improve the magnetic field sensitivity, we make use of a 1000 $\mu$W continuous-wave (cw) and linearly-polarized repump beam that is tuned between two atomic states that are not involved in the quantum state that is responding to the magnetic field. When we operate in the conventional mode (no repump), we observe a sensitivity of 570 fT/$\sqrt {\text {Hz}}$ at room temperature at a bias field of 50 $\mu$T. When the repump is implemented, we increase the number of atoms in the magnetically useful hyperfine state by more than an order of magnitude, and deliver a consequent increase in the magnetic field sensitity by about a factor of 3. We observe a shot-noise limited sensitivity of 200 fT/$\sqrt {\text {Hz}}$ in this repumped configuration. This is not competitive when compared to other types of magnetometers with an operating cell temperature of 42-130$^\circ$C mentioned above [14,21–23], but to the best of our knowledge this is among the best sensitivities a room-temperature atomic magnetometer has achieved at such a high field. For comparison, very recently Fourcault et al. reported a room-temperature Helium-4 magnetometer with a sensitivity of 50 fT/$\sqrt {\text {Hz}}$ but only at a near-zero field ($\sim 0.01\,\mu$T) [15]. In addition, unlike the circularly polarized repumper explored before [20,30–32], our approach produces no measurable light shift, consistent with the investigation in Ref. [36]. Our measurement-limited estimate of these light shifts is at a level of 3.6 pT, which is three orders of magnitude smaller than the repump-related light shift in prior work [32–34]. An NMOR magnetometer, with such a high performance at room temperature, opens a door to a wide range of new applications.
2. Analyzing atomic populations
A pump-probe configuration used in a conventional NMOR [37,38] is shown in the left panel of Fig. 1. The nonlinear magneto-optical rotation, also known as nonlinear Faraday rotation, is a light-power-dependent rotation of optical polarization due to resonant interaction with an atomic medium in the presence of a magnetic field $B$, and thus can be used to sense magnetic fields [17]. For simplicity, we focus on the situation where both beams are tuned to the $F=2 \rightarrow F'=1$ of $^{87}$Rb $D_1$ line $\vert 5^2S_{1/2}\rangle \rightarrow \vert 5^2P_{1/2}\rangle$. The right hand side of Fig. 1 shows the response of the atomic media in three situations: in grey is shown the initial thermal state, which has $1/8^\mathrm {th}$ of the population in each Zeeman states. In red, we show the optically pumped state in the presence of an intense linearly polarised pump pulse that propagates along the magnetic field direction ($z$ axis). This pumping produces non-uniform Zeeman populations in the $F=2$ ground state (seen on figure) as well as strong coherences between Zeeman levels with $\Delta m_F = 2$ (not shown), which can be detected by a counter-propagating pump-probe scheme. In addition, we note that much of the population has been pumped away to the other ground state, $F=1$. Finally, we also show the circumstances in which we use a strong repump beam (green) that connects the $F=1 \rightarrow F'=1$ transition. This pushes the population back into the original ground state, which we will show that can enhance the magnetic sensitivity.
Fig. 1. Illustration of the changing populations of two ground states due to the hyperfine pumping required in an $^{87}$Rb atomic magnetometer. The laser-induced transitions are shown in the left panel while the atomic population distribution is represented by the bars on the right. The heights of the bars represent the population states under different irradiation conditions according to Eq. (2), detailed in Supplemental Document. In the conventional scheme with a $50\mu$W pump ($F=2 \rightarrow F'=1$), $84\,\%$ of the atomic population will end up in the wrong probed state, $F=1$, due to the irradiation of the intense pump. However, by using a $50\mu$W repump ($F=1 \rightarrow F'$), it is possible to hold $93\,\%$ of the atomic population in the useful $F=2$ ground state.
A rate equation model, shown in the Supplemental Document, can quantitatively assess the population changes induced by optical pumping:
where $R_o$ is the optical pumping rate of the laser light, $R_r$ is the relaxation rate of the atoms in the cell, $\rho _F(0)$ is the initial population of the ground state $F$ ($37.5\%$ for $F=1$ and $62.5\%$ for $F=2$ of $^{87}$Rb), and $C_F$ is the decay branching ratio which shows how many atoms being pumped are decaying back to the original ground state $F$. Using Eq. (2) we calculate the probability of an atom being in a particular ground state $F$, which we assign to the symbol $\rho _F$. This can be validated against the experimental results by using the fact that measured optical density, $\text {OD}_{F,F'}$, on the $F \rightarrow F'$ transition is related to $\rho _F$. First, it’s well known that for a certain transition $F \rightarrow F'$ the measured OD depends on the absorption coefficient, $\alpha _{F,F'}$ [39]: where $l$ is the optical path length. The absorption can then be related to the effective number density for that transition: where $n_{\text {cell}}$ is the total atomic vapour density, $\rho _F$ is the probability of an atom being found in ground state $F$ under the measurement conditions, $\rho _F^0$ is the probability of the atom being in ground state $F$ under thermal conditions, and $\sigma _{F, F'}$ is the photon absorption cross-section when the probe is tuned onto the transition of $F \rightarrow F'$. From Eqs. (3) and (4), it can be seen that the OD is proportional to the atomic population giving a constant $n_{\text {cell}}$ and $\sigma _{F, F'}$. And below we will use the OD to indicate the changes in the atomic population, as it’s a direct measurable parameter in experiment.Since the hyperfine structure of the $D_1$ line is resolved in the low pressure cells used in this experiment, the OD for the $F \rightarrow F'$ transition is simply measured through the frequency dependent absorption of a very weak auxiliary laser:
where $I_0$ is the incident power, and $I(\nu _{F,F'})$ is the transmitted power when the laser is tuned to the $F \rightarrow F'$ transition.Table 1 shows the measured and predicted optical densities for different transitions of $^{87}$Rb $D_1$ when illuminated by a 50 $\mu$W pump light. In each row we consider the pump being tuned into resonance with the available transitions, as well as what happens when the pump is switched off (thermal equilibrium). The numbers shown in italics are predictions from Eq. (3), while the normal text values are derived from an experimental measurement using a weak laser (1$\,\mu$W) that can be tuned across the $D_1$ transition. The details of the experimental measurements are described below but first we consider the outcome of the measurements in comparison with the predictions.
Table 1. The measured and predicted Optical Density (OD) for the different transitions of $^{87}$Rb $D_1$. The first row of the table relates to thermal conditions, while the other rows consider illumination by a 50 $\mu$W pump light tuned onto resonance with the four available transitions. The numbers shown in italics are predictions from Eq. (3), while the bold text values are derived from an experimental measurement using a weak auxiliary laser (1$\,\mu$W) that is tuned over the $D_1$ manifold. The increases/decreases in the OD for different pumping schemes relative to the thermal equilibrium state (or pump-off case) are shown in the parentheses after the predicted numbers.
Due to symmetry considerations, in thermal equilibrium (i.e. the first row of Table 1, when the pump is off) one expects that the OD for the $F=1 \rightarrow F'=1$ transition should be one fifth of the other three hyperfine transitions, which should all be equal [40]. The experiment is in close agreement with this prediction. Examination of the other rows in the table shows that the pump light has the effect of pumping out the ground state to which it is tuned, while pushing the atoms into the other ground state. The conventional NMOR configuration has the pump and probe beams tuned to the $F=2 \rightarrow F'=1$ transition [37,38]. Table 1 shows that this induces a four fold reduction in the atomic population on the $F=2$ state compared to the thermal equilibrium state: that results in a significant reduction in the potential sensitivity of a magnetometer which ultimately depends on the number of atoms in the probed ground state. The table, however, also provides a hint of a pathway forward for an increased sensitivity: when pumped on the $F=1 \rightarrow F'=2$ transition it is possible to increase the atomic population of $F=2$ ground state to a value even greater than the thermal equilibrium value.
This observation motivated us to implement an additional repumping laser to retrieve the lost populations. The repump couples the other ground state ($F=1$) to the upper states as displayed in Fig. 1(a), to push atomic population that had been lost into the $F=1$ ground state back into the magnetically sensitive $F=2$ transition. Table 2 calculates the expected optical density for this configuration where we now consider the presence of three optical beams: a $50\,\mu$W pump and $10\,\mu$W probe beam that are tuned to the $F=2 \rightarrow F'=1$ transition as well as a 1000 $\mu$W repump that is tuned to either the $F=1 \rightarrow$ 1 $\mathrm {or}$ 2 transitions. The repump results in an increase in the $F=2$ state population by more than an order of magnitude. This larger population can potentially increase the signal-to-noise ratio (SNR) and thus improve the NMOR sensitivity, which will be experimentally demonstrated in Sec. 4. Note that the data in row 1 of Table 2 (repump, off) are different from row 4 of Table 1 (pump, $F = 2 \rightarrow F' = 1$), as the optical pumping effect of the probe is not included in Table 1.
Table 2. Theoretically predicted OD under the combined illumination of three beams (pump, probe and repump). The power of the beams is set to be similar to those of a real-world experiment. The pump (50 $\mu$W) and probe (10 $\mu$W) are tuned to the $F=2 \rightarrow F'=1$ transition. The first row of the table considers the situation where the 1 mW repump is absent, while the other rows consider tuning the repump to the two available transitions from the $F=1$ level.
3. Experimental details
A schematic of the experimental apparatus is shown in Fig. 2. A drop of isotopically enriched $^{87}$Rb was contained in a buffer-gas-free, paraffin-coated vapour cell. The cell had a spherical profile with a diameter of 40 mm while the wall thickness was estimated to be 0.1 mm, and the coating reduced the spin relaxation due to collisions with the wall to that the intrinsic relaxation rate, $2\pi \times 4$ (1/s). The room-temperature cell was placed in a 3-layer cylindrical magnetic shield made of $\mu$-metal, which was used to passively suppress the noisy quasi-static environmental magnetic fields. Two pairs of coils were mounted inside the innermost layer of the shield to create a quiet and homogeneous bias field: one was used to produce a 50$\,\mu$T bias field along $z$ axis, while the other suppressed any residual magnetic gradients from the bias field [41].
Fig. 2. Experimental setup. PBS, polarization beam splitter; HW, half-wave plate; P, linear polarizer; M, reflection mirror; WL, Wollaston prism; D1-D3, silicon photodetectors; AOM, acousto-optic modulator. The photodetector D1 is used for OD test when the optical frequency is swept across four hyperfine transitions. The intensity of pump beam is modulated by an AOM with a modulation depth of 100%, and the modulation frequency is scanned around twice the Larmor frequency, 2$\Omega _L$. The probe is a continuous-wave (CW) light and its power is stabilized by the AOM in probe path. For the repump, it’s a CW beam and power-controlled by a third AOM in magnetic field measurement but its intensity is modulated at 3 Hz by the AOM to facilitate the assessment of light shift it might introduce. Note that the AOMs also red-shift all three laser beams by 80 MHz, which is determined by the RF driver frequency and is a fixed parameter in our experiment.
Three laser beams were transmitted through the vapour cell without any overlaps: a conventional pump and probe beam along with a repumping beam. The laser path lengths of three beams were measured to be around 34 mm, as three spatially separated laser beams were not propagating through the cell center. They were all linearly polarized along the $y$ axis at the cell, and any changes in the pump polarization plane with respect to probe polarization plane can affect the phase of the lock-in amplifier required to get the right-shape magnetic resonance signal. The probe and repump propagated along the $z$ direction, opposite to the pump.
A distributed Bragg reflector (DBR) laser diode emitting radiation at 795 nm was split into a pump and a probe. The pump was red-shifted by 80 MHz by an acousto-optic modulator (AOM), which was used to both modulate and stabilize the laser intensity with the assistance of an electronic control system. The pump power was modulated at $\Omega _m$ by a square wave with a 20 $\%$ duty cycle and 100$\,\%$ modulation depth in order to periodically align the atomic spins (within the pump volume) along the direction of the light’s polarization. Note that the atomic alignment studied here is distinct from the atomic orientation generated by a circularly polarized light used in other types of magnetometers mentioned above [13,14,23]. The pump light had an $1/e^2$ intensity diameter of 10 mm and an averaged power of 50$\,\mu$W, which yields a pumping rate of 1200 (1/s). To facilitate the comparison between the cases with and without the repumping, the pump power was kept the same throughout the experiment. The direction of the aligned atoms will precess at the Larmor frequency $\Omega _L=\gamma B$, where the gyromagnetic ratio $\gamma$ is approximately $2\pi \times 7$ Hz/nT for $^{87}$Rb [40], around the magnetic field. Because of the symmetry of the aligned state [18,42], one observes that the aligned state completes a full rotation in half a Larmor cycle. And we will see a resonance in the alignment being built up by the modulated pump if its modulation frequency, $\Omega _m$, is equal to $2\,\Omega _L$. A precise measurement of $\Omega _m$ at which this resonance occurs will define the sensitivity for NMOR magnetometry.
The NMOR resonance can be observed via the polarization rotation of a probe beam caused by the nonlinear Faraday rotation [42]. This experiment used a probe beam with an $1/e^2$ intensity diameter of 1.6 mm and a power of 10$\,\mu \text {W}$. A Wollaston-based balanced polarimeter was used to detect the polarization rotation of the probe, expressed as $\theta = \text {arcsin}[(P_1-P_2)/(P_1+P_2)]/2$ where $P_1$ and $P_2$ are the optical powers received by two photodetectors (D2 and D3 in Fig. 2) [43]. A dual-phase lock-in amplifier (not shown), referenced to $\Omega _m$, was used to demodulate the polarization rotation signal. The NMOR resonance can be observed by scanning the modulation frequency, $\Omega _m$, around twice the Larmor frequency, $2\,\Omega _L$.
A second external-cavity diode laser (ECDL), also emitting at 795 nm, was used for two additional purposes. It provided the repumping light, in which case it was tuned to the transition of $F=1 \rightarrow 1$ $\mathrm {or}$ $2$. In that case it was expanded to a diameter of 10$\,\text {mm}$ before entering the cell and used a power of up to 1000$\,\mu$W. We also used this source to provide the measurements for the OD tests listed in Table 1. In this case, its output power was adjusted down to 1$\,\mu$W so as not to perturb the atomic populations. The laser frequencies of both lasers can be stabilized to any hyperfine transition of the $^{87}$Rb $D_1$ line through saturated absorption spectroscopy (SAS) systems (not shown). All the measurements reported here were performed in an applied magnetic field that is approximately equal to the Earth’s field ($\sim 50\,\mu$T inside the shields). This corresponded to a resonance (or pump modulation) frequency of around $2\pi \times$700 kHz.
4. Results and discussions
4.1 Optical density measurement
A weak laser with a power of 1$\,\mu$W, sufficiently low that it had no noticeable effect on the pumping process, was used to measure the OD so as to estimate the atomic populations of two ground states. The OD can be derived by measuring the incident and transmitted powers of this weak beam according to Eq. (5). A photodetector, D1 in Fig. 2, was used to detect the transmitted power. The detuning of the ECDL was assessed using a signal derived from an auxilliary saturation absorption spectroscopy cell as indicated by the black curve in Fig. 3.
Fig. 3. OD as a function of the frequency of a weak laser through the cell. We plot a family of curves which relate to the tuning of a pump onto the various available hyperfine transitions as well as being absent (see text for details). The auxiliary probe light was split into two branches: one transmitted through a reference cell filled with Rb in natural abundance, which gave the black curve to calibrate the frequency axis of the other five colorful traces, while the other one propagated through the coated magnetometry cell (shown in Fig. 2) with enhanced $^{87}$Rb and generated the colorful traces. A small mismatch in the exact location of the peaks in absorption signal and the OD measurement is a result of the AOM frequency shift that is used as part of the laser power stabilization. We note that the coating material inside the coated cell can also change the absorption frequency [44].
In Fig. 3, we present five curves where we have scanned the auxilliary laser across the $D_1$ manifold. These curves measure the frequency dependent absorption under conditions in which there is no pumping, or where the pump laser has been tuned to each of the hyperfine transitions in turn. The pump power was chosen to match that used in a typical NMOR measurement, 50 $\mu$W. The red and magenta curves represent the frequency dependent absorption for the pump tuned to the $F=1 \rightarrow F'=1,2$ respectively, while the cyan and green represent the frequency dependent absorption as the pump was tuned to the transitions of $F=2 \rightarrow F'=1,2$. The four peak absorptions, measured where the weak probe coincides with the transitions, are converted to an OD, and then tabulated in Table 1 where they are compared to the theoretical predictions. We see exceedingly good agreement for most of the numbers therein. Some observable discrepancies are found for the cases where the pump is interacting with a same ground or excited state as the weak auxiliary laser implemented for the OD measurement, as in these cases some atoms disturbed by the strong pump can affect the OD result.
The very significant modifications of the OD seen in the various pumping conditions on Fig. 3 show that the typical pumping conditions used in a magnetometry experiment almost completely deplete the atoms from the initial quantum state. Obviously, this is undesirable since the magnetic sensitivity is closely related to the number of atoms in this quantum state. On the other hand, Table 1 implies that, under the right conditions, it should be possible to implement a repump laser to significantly restore the population in that state.
4.2 Magnetic fields measurement
On Fig. 4 we show the measured NMOR resonance signal that is seen if we scan the pump modulation frequency, $\Omega _m$, around $2\,\Omega _L$. The grey and black curves show the output of the two lock-in amplifier channels when operated without repumping using a 50 $\mu$W average pump power, a 10 $\mu$W cw probe power, and where both lasers are locked to the $F=2 \rightarrow F'=1$ hyperfine transition. At the $50\,\mu$T bias field the in-phase channel shows a resonance linewidth of 60 Hz, broader than the 19 Hz linewidth observed at low fields mainly because of the magnetic field gradient, while the quadrature-phase channel shows a slope of 0.23 mrad/nT. We denote this slope the “scale factor” since it is closely related to the sensitivity of the magnetometer. When this scale factor is combined with detection noise of the apparatus it provides the estimate of the potential magnetic sensitivity.
Fig. 4. Comparison of the in-phase and quadrature signals recorded at 49$\,\mu$T in the absence (black and grey) and presence (blue and light blue) of a 1 mW repump centred onto the transition of $F=1 \rightarrow F'=1$.
As a demonstration of the potential for improved magnetic field sensitivity we tuned a 1 mW cw repump laser to the $F=1 \rightarrow F'=1$ transition. The in-phase and quadrature-phase signals from the lock-in amplifer are now very much larger (light and dark blue traces on Fig. 4) with a measured scale factor of 0.93 mrad/nT. We note that the linewidth is unchanged, which implies that repumping has not induced any additional relaxation processes, because the repump is off-resonant with the atoms being probed by the probe light.
In Fig. 5 we display the scale factor as a function of repump power where the repump is tuned to both $F=1 \rightarrow F'=1$ and 2 respectively. At low repump powers (<300$\mu$W), we see more effective repumping on the transition of $F=1 \rightarrow F'=2$ rather than $F=1 \rightarrow F'=1$. This is reasonably consistent with the higher strength of the $F=1 \rightarrow F'=2$ transition. When the repump power exceeds 300$\,\mu$W, the scale factor for both cases saturates at around 0.93 mrad/nT, indicating that approximately same amount of atoms are found in the state $F=2$ and this atomic population has been almost maximized under the illumination of such an intense repump.
Fig. 5. Scale factor of NMOR magnetometry signal versus the repump power. The optical frequency of the pump/probe is fixed at the transition of $F=2 \rightarrow F'=1$. The horizontal green line indicates the scale factor without the repump.
A closer examination of Fig. 5 shows that the highest scale factor increase ($\times$4) is not consistent with the increase in the population ($\times$20) seen in OD$_{2,1'}$. This is not that surprising: under these conditions, the OD is proportional to the total population in the $F=2$ ground state whereas the NMOR resonance is proportional to that fraction of the $F=2$ ground state population that is in the particular Zeeman state superposition that gives rise to the NMOR signal. In the absence of the repump laser, and with the low rates of relaxation in our cells, we expect that the entire $F=2$ is polarized into the correct quantum state by our strong pumping. However, once the repump laser is switched on it will contribute an incoherent component to the $F=2$ ground state which contributes to the OD$_{2,1'}$ signal but not to the NMOR signal. Nevertheless, the continuing action of the pump results in a significant overall increase to the coherent population. The overall strength of NMOR resonance is a three-way competition between the loss of population because of pumping, plus the gain of incoherent population from repumping, and the coherent population created by pumping.
We characterise the sensitivity of the device as a magnetic field sensor by measuring the power spectral density (PSD) of the output of the quadrature channel of the lock-in amplifier when we have tuned the pump modulation frequency to twice the Larmor frequency ($\Omega _m = 2 \Omega _L$) [27]. On Fig. 6 we display the value of the PSD at 20 Hz as a function of the repump power on both $F=1 \rightarrow F'=1$ and 2 transitions, respectively. We chose 20 Hz for this characterisation because the noise of the magnetic field environment around the sensor is below 100 fT/$\sqrt {\text {Hz}}$ at this frequency. In the absence of the repump, the noise floor is marked by the green line at the top of the plot: 570 fT/$\sqrt {\text {Hz}}$. When the repump is applied, the sensitivity floor is improved by a factor of 2-3. As with the scale factor, we see a saturation for repump above $100\,\mu$W. In this range the sensitivity of the magnetometer is improved to around 220 fT/$\sqrt {\text {Hz}}$ with the benefit of the repump laser.
Fig. 6. Magnetic noise floor of NMOR magnetometry (at 20 Hz) versus the repump power. The noise floor achieved without the repump is indicated by the green line.
The complete noise spectrum of the magnetometer, in the absence (black) and presence (red) of a 1 mW repump, is displayed in Fig. 7. At a bias field of 50$\,\mu$T, the conventional pump-probe scheme achieves a sensitivity of 570 fT/$\sqrt {\text {Hz}}$ at 20 Hz at room temperature. Using a 1 mW repump, the sensitivity can be improved by nearly a factor of 3, reaching 200 fT/$\sqrt {\text {Hz}}$. We note the worsening of the low frequency performance of the magnetic sensitivity at low frequencies (<20 Hz) of the repumped sensor. It is important to note that this is associated with the low-frequency drift of the 50 $\mu$T bias field created for the magnetometer sensitivity evaluation and not because of some intrinsic drift of the sensor itself. We note that this is only revealed because of the extremely good performance of the sensor in this configuration.
Fig. 7. The magnetic sensitivity of the NMOR device at 50$\,\mu$T with (red) and without (blue) the 1000 $\mu$W repump, which is driving the $F=1 \rightarrow F'=1$ transition.
We also display on Fig. 7 the detection noise limits of the magnetometer which we estimate by blocking the pump light. In the absence of the repump, this detection limit at low frequencies is around 570 fT/$\sqrt {\text {Hz}}$, which is consistent with photon shot noise on the detector and also consistent with the measured magnetic sensitivity. We thus note that the performance is as good as it can be given this effective atomic density.
For the repumped configuration we plot both the detection noise limit and the photon shot-noise limit, which in this case are slightly different. We estimate the detection limit by blocking the pump laser and obtain a limit of 200 fT/$\sqrt {\text {Hz}}$ at low frequencies. This improvement in the noise floor results from a higher population in the ground state of $F=2$ and hence a higher contrast NMOR spectra, shown in Fig. 4, thanks to the irradiation of the repump. For the photon shot-noise measurement we tune the probe frequency away from the centre of resonance to measure a low-frequency floor of 160 fT/$\sqrt {\text {Hz}}$. An independent calculation of the photon shot noise contribution at the power level on the photodiode is consistent with this level. Thus, interestingly, we find an additional contribution of around 50 fT/$\sqrt {\text {Hz}}$ in the absence of the pump, but in the presence of the repump. We believe that the high density of atoms in the $F=2$ state is generating spin noise similar to that observed in Refs. [45,46]. This observation deserves additional investigation and will be the subject of a future publication. It is the appearance of this additional noise that results in the improvement in magnetic sensitivity not being quite as good as the improved scale factor that we demonstrated in Fig. 5.
To assess whether the implementation of the repump will affect the accuracy of the NMOR magnetometry, we set the NMOR in the field-tracking mode [27] and look for any modulation of the resonant frequency as the repump is switched on/off at 3 Hz, shown in Fig. 8. The 3 Hz modulation rate was chosen to minimize the effects of magnetic field drifts, whilst allowing the cell to come to equilibrium. It is clear by observation that there are no observed changes in the apparent resonant frequency synchronous with the modulation of the repump power. We also note that the fluctuations observed here are of the same magnitude with or without the pump since it is limited by the surrounding magnetic environment as well as the noise of the closed-loop system. We analyzed the data in both the time domain (fitting a square wave signal to the frequency data) and frequency domain (Fourier transform) to look for a putative modulation. For the fitting process, the frequency data were fitted to a square wave with a fixed frequency of 3 Hz while the offset and the amplitude are the free parameters therein. The upper bound of the fitted amplitude gives us an estimation on the influence related to the 3 Hz-modulated repump. In both cases, we obtain an amplitude of this signal of below 0.05 Hz (or 3.6 pT) which is limited by electronic noise introduced in the field-tracking mode elevating the noise floor above that presented in Fig. 7. This is 1000-fold improvement over the repump perturbations that were noted in prior work [32–34]. Pleasingly, this would appear to indicate that we are able to benefit from the repumping in the sensitivity of our device without needing to suffer from a degraded accuracy associated with fictitious magnetic fields [36].
Fig. 8. Changes of the NMOR resonant frequency when the repump is switched on and off. The repump power is plotted against the left axis, while the changes in the resonant frequency is illustrated against the right axis. When the repump modulated between 0 mW and 1 mW at 3 Hz (blue trace) by the AOM is applied, no observed changes in the resonant frequency (red circles) synchronous with the modulation of the repump power. The changes in the resonant frequency is estimated to be below 0.05 Hz or 3.6 pT from a fitting curve or a Fourier transform of the response.
5. Conclusions
In summary, the atomic population under the irradiation of the laser lights used in NMOR magnetometer is investigated theoretically and experimentally. It is shown that the pump laser, necessary to prepare the atomic media in the right quantum state, has the unwanted effect of driving most of the atomic population into trapped states that can not participate in the magnetic sensing. The employment of an additional repump laser can bring these atoms back into the quantum system and thus allow a larger number of effective sensing atoms. In optimal configuration, we see more than an order of magnitude greater atoms in the right hyperfine level, along with a 4-fold increase in the number of atoms in the right quantum state. The greater scale factor enables us to improve the sensitivity of the NMOR by a factor of nearly 3, from 570 fT/$\sqrt {\text {Hz}}$ to 200 fT/$\sqrt {\text {Hz}}$ at 20 Hz, with a room-temperature cell at a geophysically useful magnetic field magnitudes (50$\mu$T). In addition, our particular repump configuration allows us to demonstrate this sensitivity increase with significantly suppressed unwanted shifts in the NMOR resonant frequency.
Funding
Science and Industry Endowment Fund; Commonwealth Scientific and Industrial Research Organisation.
Acknowledgements
The authors are grateful to M. V. Balabas for the manufacturing of the Rb cells. R. L. is a recipient of a SIEF Ross Metcalf STEM+ Business Fellowship in Australia.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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