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\begin{document}

\title{3-axis Vector SERF Magnetometer with Pulsed Pumping and a Single Beam Path}



\author[1]{Morgan Hedges}%
\affil[1]{Affiliation not available}%


\vspace{-1em}



  
  \date{May 26, 2024}


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\selectlanguage{english}
\begin{abstract}
We propose and demonstrate a single beam-axis design for a highly stable and sensitive 3-axis vector atomic magnetometer compatible with the SERF regime. The scheme uses short, intense optical pulses for pumping, while a co-linear probe measures the spins evolution under the influence of signal fields and applied magnetic pulses. Due to evolution in the absence of a strong pump field, the scheme is inherently insensitive to pump intensity and wavelength fluctuations. Further, drifts in the pump direction or the applied magnetic pulses can be distinguished from signal fields due to their differing temporal signature. Our demonstration uses a small $\sim$ 1mm $^3$ measurement volume of hot $^{87}$Rb vapor in the SERF regime, and achieves sub 100 \ftrhz precision on all three axes. Good low frequency noise of $<$ 30 \ptrhz at 1 mHz is also achieved despite long, $\sim$ 1 m in-air beam paths which can cause orders of magnitude more drift in similar schemes. The residual low-frequency noise is though to be limited by environmental magnetic noise. 
%The general method of measuring spin-precession in the absence of pump light is promising. Not only does it allow for eliminating pump noise, but can be used to read out of atom density, laser power, on-line self-calibration, and the possibility of spin-squeezing. 



%advantages of the design include low laser power (~1 mW), small meausrement volume  ($sim$ 1 mm$^3$ measurement volume), high sensitiity (80 \ftrhz precision on all axes). Measuremnets sensitivity to many slow noise sources such as pump or probe fluctuations to enable low drift (30 \ptrhz at 1 mHz). Disadvantages include the need for relatively sophisitcated analyisis and operation in low field. The scheme is highly suitable to applications in the field due to low power and volume requirements, and the fact that it can be made self-calibrating. It is also of interest for applications requiring very low drift as common noise sources such as pump beam parameters either cause no signal or can be distinguished based on their temporal signature.

%The demonstration uses a single beam axis for both pumping and probing a vapor cell filled with Rb87 atoms in the SERF regime. Short, intense pulses of pump light polarize the atoms, which then precess under the influence of signal fields and applied magnetic pulses while their polarization along a single axis is measured. Low drift is realised as the fluctuations in the pump laser direction or magnetic pulse areas can be distinguised from signal fields due to their differing temporal signatures. This enabled us to measure a stability of 10s of \ptrhz at the 1000s scale that is most likely limited by environmental noise. Due to the simple geometry, robustness to imperfections, and low laser power required, the scheme is well suited for applications in the field. The scheme is suitable for extension. For shot-noise limited situations, the scheme is amenable to squeezing, while the expense of a more complex pulse-sequence and analysis, the system can be made self-calibrating.
%
\end{abstract}%



\sloppy


\section{Experiment}

The scheme is implemented as shown schematically in figure 2(?). A pulsed laser diode resonant with the Rb87 D1 line and tuned by an external grating reflection is used as the pump beam. The probe beam is produced by a DFB laser tuned 150 GHz below resonance. The beams are combined at a shallow angle and incident on the a 3 mm deep (10 mm diameter) pancake cell heated to 165 C and housed within a 3 layer mu-metal shield. The pump beam is $\approx$ 4 mm in diameter with a peak power of $\approx 500 mW$, while the probe was (0.7 mm FWHM).

The reflection of the probe beam leaving the cell assembly is incident on a balanced polarimeter measuring polarization rotation. Most of the light from the pump pulse is filtered out by a 1 mm$^2$ aperture.

The pulse sequence used was shown in Figure 1. Measurement of all 3 axes is made using 4 pump/evolution periods, each period of which contains two magnetic $\pi/2$ pulses, one immediately after the pump pulse, and one at a time $\tone$ later. To describe the scheme we use the notation $(X,Y), (-X,Y), (Y, X), (-Y, X)$, where the letters denote the axis of the $\pi/2$ pulse and the position within the brackets denotes when it occurs. For the experiment an extra pulse is added at the end of each period which is the inverse of the initial magnetic pulse. This is to revert the remaining polarization to point back along the $z$-axis to start the next period, and reduces requirements on optical pumping.

The length, $\tau_2$ of each period was chosen to be 4.15 ms, such that the entire 4-pulse sequence lasted 16.6 ms. The duration of each optical pump pulse was 20$\mu $s, and magnetic pulses were all 4$\mu$s. The first magnetic pulse occurs 10$\mu$s after the end of each optical pulse, and the final reverse occurs 50$\mu$s before the next.

The pulse sequence was implemented via two arbitrary wave-form generator channels, each of which generates the full 4-pulse sequence on a trigger. To drive the coils (2-turn helmholtz arrangement, with 10 cm diameter), their outputs are amplified. Crossed diodes shield the coils from amplifier noise while off.

\subsection{Data Recording and Processing}

The balanced polarimetery signal is recorded continuously and processed in real-time. The signal is digitized at 6 MS/s and downsampled to 333 kS/s by a Picoscope 5000 USB oscilloscope. As the data arrives it is split into time segments between the trigger signal for the wave-from generators. Each segment contains a full sequence (i.e. the transient signal after 4 optical pump pulses).

To determine the signal fields, each segment is split into it's corresponding evolution periods. The 4 transient signals are reduced to two by subtracting adjacent signal windows, i.e. $S_{yz}= (X,Y) - (-X,Y)$, and $S_{xz}= (Y,X) - (-Y,X)$ where the subscripts on the $S$ denote the axes to which the trace is nominally sensitive to magnetic fields.  The subtracted signals are clipped to remove the first 100 $\mu$s after the pump pulse and the last $50\mu$s was also removed to ensure that no signal from the following pump pulse is included due to trigger jitter. A final combined trace $S$ is formed by concatenating the traces $S_yz$ and $S_xz$. This trace contains all information about magnetic interaction with atoms, with most slowly-changing measurement noise elminated.

To optimally estimate the values of $B_x, B_y, B_z$ contributing to each $S$, we need to accurately know how these fields affect it. This is done by simply measuring the change in $S$ when a small known field is applied along each of the axes. The recorded changes are shown in Figure 2 (a, c). A measurement is also made of the change in signal induced by a small change in pulse area, by making a small change to the height of the initial $\pi/2$ pulses. Only two pulse heights are varied: one inital X and one initial Y. This is because changing the -X (-Y) has an essentialy identical, but opposite, effect on the subtrated signal as changing the +X (+Y) pulse.  The measured deviations are shown in Figure 2 (b,d).

Each $S$ is fit to a linear combination of these 5 traces by linear least-squares. The entire processing routine, performed at 60 Hz and fit using an openly available Python library\cite{Statsmodels} without optimization, utilized less than 50\% of a single core of a desktop CPU. %Fitting can of course also be done without the pulse-height variation, but it was found

\section{Results}

Figure 3 shows results obtained with this method. Figure 3 a) dispays the spectrum recorded continously over $\approx$ 24 hours. Spectra are given for all 3 axes. Noise levels of a few 10s of pT$/\sqrt{\mathrm{Hz}}$ are found at long times, while $\approx 100$ \ftrhz is seen at short times.

The $x$ axis is the quietest axis at long times, but exhibits a large noise hump between about 10 mHz and 10 Hz. This hump is consistent with the spectrum of noise in the lab, shown in Fig 3 (x). The $x$-axis is the most susceptible as it corresponds with the long-axis of the cylindrical magnetic shields with a shielding factor (~1500) reduced by ~ 5 compared to Y/Z (10000).

The noise source causing the hump is not clear, but suspected to be associated with building cooling. Fig 3 c) shows readings from a flux-gate magnetometer mounted near the apparatus, showing noisier and quieter periods every few minutes. To obtain Fig (3b), the first 30 quieter periods recorded by the flux-gate, each of 100 s, were selected manually from the data set. The corresponding data from the atomic magnetometer were averaged to obtain the spectrum. Here we see that the noise on the $y$ axis is now consistent with $z$, down to below 1 Hz, while the $x$-axis reaches the level of the other axes at around $10 Hz$.

The high frequency noise level of all axes reaches a minimum of $\approx 75~ \mathrm{fT}/ \sqrt{\mathrm{Hz}}$. The similar spectra taken with the pump, and with both lasers blocked show minimium levels of below $ 50~ \mathrm{fT}/ \sqrt{\mathrm{Hz}}$, indicating that probe and measurement noise are contributing but not yet dominating the measurement.

We note that if pulse-height variation is not included in the fits, the high-frequency noise increases to ~500 fT for X and Y, while the Z-axis stays approximately the same.

We find that calibration of the apparatus (by applying small fields and measuring response as described above) is quite stable and varies only at the few percent level over the course of a day. Nonetheless, small modulating fields were also applied during the long measurement to ensure no drastic changes. These modulations were at 1, 3, and 5 Hz on the x,y,z axes respectively and had an amplitude 50 pT. These can be seen as narrow peaks in Figure 3.

These modulations can also be used to determine the frequency response of the system, shown in Figure 3(d). Here it can be seen that in this configuration the sensitivity of the \ux and \uy axes decreases by about 10\% in the given frequency range, while the \uz-axis decreases by about 25\%. This may be expected as at these frequencies the field is no longer constant over the measurement period and our fitting assumptions are no longer valid. This is not taken in the spectra shown in figure 3.

To quantify cross-talk and sensitivity to other potential noise sources of the magnetometer, we recorded the output of the fit procedure whith several paramters changed, shown in Table 1. The size of the deviations were chosen to give ~pT level effects. We find that cross-talk between field measurement directions is at the 1\% level. The source of this is not clear, but may be simply due to imperfect alignment of the coils used to define the field directions.

To obtain pT deviations by changing pulse-areas, deviations on the order of 1 degree are required. This level of cross-talk is about 1000 times lower than the case when the pulse-heights are not included in the fit. At this level, the pulse-area variations should have negligble effect on the observed noise floor. %The level of pulse-area noise was also checked independantly by monitoring voltage dropped accross a resistor in the coil path and confirmed to be at the 0.01\% level.

Changing the probe laser affects the $z$ axis primarily, presumably because any probe light shift is directed along the z axis. A movement of 10 GHz toward resonance by changing the diode temperature caused a ~10pT shift in the measured value of Bz. Thus if the probe's affect on the atoms were dominating high frequncy noise, we would expect the $z$ axis to be much noiser than the others. Finally it was necessary to reduce the pump pulse width by 25\% (from 20$\mu$s to $15\mu$s) to result in changes of ~1 pT. To simulate other sources of pump noise, the pump beam was purposely misaligned in the x-z plane by adjusting a mirror roughly 50 cm away. The spatial translation was ~0.5mm giving an angular deviation of $\sim$ 0.05 deg.

\section{More results discussion}
Noise of the shields
Larger optical path length to improve high frequency noise. Fluctiations in probe polarization could be limiting z-axis, but not clear.
A large amount of low frequency noise is pobably due to the environment and/or the shields themselves. Gradiometry/better or quieter shielding will be needed to be certain.

\section{Further Work}

\textbf{SERF Regime Comments}

For high alkali density and low fields, spins precess according to the Bloch equations but with a gyromagnetic ratio that depends on total polarization, varying between the non-SERF value of $g_L$ at full polarization (i.e. $P_z=1$) to $g_L/Q$ where $Q$ is the so-called 'slowing down' factor associated with the nuclear spin (Q=XXX for 87Rb). The reason for this is that as the spins relax there is redistribtion of spins to F states that have opposite gyromagnetic ratios. The slowing-down factor represents the average gyromagnetic ratio at equilibrium.

There are two effects of the SERF regime then. In the first case, the effect preecssion in the period up until $\tau$ is affected by the time varying $g$ factor as the net polarization decays. Second, after the second $\pi/2$ pulse, nearly $1/3$ of the polarization built up due to the $B_z$ field is lost. This is because at this point, the polarization distribution is roughly at the equilibrium limit, such that 1/6 of the atoms have the opposite gyormagnetic ratio. As the magnetic pulse is much faster than spin-relaxation, these 1/6 of atoms see a $\pi/2$ pulse with opposite sign, ending up facing the oppsoite direction to the majority of the population.

\textbf{Potential for fitting more parameters}

We note that the raw-recorded trace contains a lot of information, not just about the fields, which we haven't used here. For instance, the period immediately after the optical pulse but before the first magnetic pulse is largely independent of any magnetic field, but reflects the signal at maximum polarization which may fluctuate with probe intensity/wavelength, atom density (temperature), or pump polarization. Second, consider the signal following the final reversing $\pi/2$ pulse. The height of the signal here relative to the first period depends on fields only in second order, and is measure of the atoms decay time. Finally there is a small, $\approx 50\mu$s transient immediately after this pulse (as well as the pulse at $\tau$), which may be associated with spin-exchange relaxation. The size of this transient is a measure of misalignment as it would not exist with a perfect $\pi/2$ pulse, while the width is a measure of spin-exchange rate and thus alkali density.

\textbf{Possibility for high-field or scalar variations.}

The present scheme is promising for low-field magnetometry, however any applications in higher fields would be limited by noise in the bias fields required to maintain linearity (and operation in the SERF regime). We note that high-field modifications of the scheme are interesting, as using a high-intensity pump allows high polarization to be achieved even in high fields, and magnetic pulses can also be easily made shorter. Signifcant changes to the fitting procedure would have to be made however, and operation out of the SERF regime in a multi-pass cell would be preferred.

\textbf{Spin-squeezing.}

We also note that if the noise-level can be reduced to the spin shot-noise limit, this scheme will be a good candidate for the use of spin squeezing to go beyond. In this case seperate period would be taken to measure each axis, and the probe would only be pulsed on immediately before and after each pump pulse.

\textbf{ On-line zeroing/Calibration, Addressing Bias}

If it's necessary to have fields in different orientations, to correct poor alignment, to know what zero Bz is, or for effective scalar operation, it may be good to go through the zeroing procedure, which can be discussed here.

\section{Conclusions}
A vector magnetometer with small volume ($\approx 1 mm ^3 $) with performance of sub 100 fT$/\sqrt{\mathrm{Hz}}$ on all 3 axes for frequencies above a few Hz was demonstrated.
Performance at low frequncies is good, reaching a few 10s of pT $/\sqrt{Hz}$ without any effort to optimize pump or probe stability. Some of the remaining low frequency noise is likely to be ambient magnetic noise or from the shields. It's unclear what the major limiting factor for higher frequencies, but better performance should be possible with improved elecronics or greater optical path lengths. Indeed while the present focus is on low frequency noise the possibility for high sensitvitiy at higher frequencies, perhaps in a multi-pass cell, is intersting as the standard quantum limit is slightly lower than for traditional techniques, the ease of optical pumping makes it suitable for very high number densities, and with small modifcations could take advantage of spin-squeezing. From the point of view of commercial devices, the use of pulsed beams can drastically reduce power usage. In the present experiment each of the lasers used an average of ~1 mW, and could in principle be much lower.

We note some drawbacks of the scheme for some applications.  The technique is that it is not magnetically quiet, although the average applied field is zero. Further, without significant changes to the processing step, it can only be used for high fields without cancelling coils which will be the limiting factor on it's sensitivity. Finally, in order to suppress the effect of pulse-noise, relatively complex electronics and processing is needed than for some other schemes. On the plus side, we note that that it should be possible to extract extra information from the free decay that is not available in other schemes.

It is also interesting to compare this to standard pump-probe orthogonal continuously pumped magnetometer, where maximum sensitivity occurs at total polarization,  $|P|=50\%$, when the pumping rate is equal to the decay rate due to other sources, i.e.  it doubles the total decay rate. The sensitivity in this case is $\frac{d P_z}{dB}=0.25\gamma$.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/pm-experiment/default-figure}
\caption{{The beam from a pulsed laser diode (pump) is overlapped with that from a DFB laser (probe) at a narrow angle on the vapour cell. The polarization rotation of the probe laser due to spins in the cell is monitored via balanced polarimetry. The cell is 3 mm $\times$ deep and 10 mm in diameter.%
}}
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\includegraphics[width=0.70\columnwidth]{figures/pm-measurement-summary/default-figure}

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\includegraphics[width=0.70\columnwidth]{figures/pm-result-spectra/default-figure}

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%\begin{acknowledgements}
This work was supported by an ONR grant...
%\end{acknowledgements}

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