Draft:Optically pumped magnetometer

OPM

An OPM (optically pumped magnetometer) is a sensitive magnetometer used to measure extremely weak magnetic fields, by using properties of alkali vapors.

OPM's are often rated as the most sensitive magnetometers currently available reaching noise levels below 1 fTˑHz^-½. Their high sensitivity makes them good candidates for many biomedical applications such as Magnetoencephalography.

Theory

In it's most basic form an OPM consists of three components: A monochromatic light source, A vapor cell, and a detection system. In this simple setup the light source, which is most commonly a laser circularly polarized, emits light in order to optically pump the atoms in the cell to a magnetically sensitive spin state parallel to it's axis of propagation. As the atoms align their spins to the axis of propagation they become transparent to the light of the laser and the laser's intensity becomes essentially constant. When a magnetic field with components perpendicular to the axis of propagation of the laser is introduced to the system however, it applies a torque on the net magnetization of the system as the atoms begin to rotate around that field. The atoms deviate from their alignment which in turn reduces their transparency to the laser light, as those atoms begin to absorb light from the laser producing a drop of intensity of the laser after it passes through the cell which is measurable by a photodiode.

Alkali atom splitting

OPMs often use alkali atoms in their construction because these atoms have only one valence electron. This property of the atom makes energy levels produced by the atoms angular momentum easy to manipulate, and makes the number of possible states and absorptions limited. This is useful because OPMs rely on transfers of polarization between the laser light and the atoms in the cell, the fewer ways transfers of this kind can occur the more straight forward it generally is to interpret the physical result.

Fundamentally, referring to a transfer in polarization is the same as referring to a transfer in the angular momentum of the laser to the atom. The drop in laser intensity is due to a transfer of angular momentum from the beam of polarized photons to the atoms in the vapor cell, as they get absorbed to allow the atom to reach a certain energy level. For an OPM the most important part of this transfer happens with the electron of the atom. An electron's angular momentum (J) is most easily split into two components; an orbital momentum term, L; and a spin momentum term, S:

$J=L+S$

In an alkali atom this means that J can only take on two values for a given value of L because the atom only has one electron. The angular momentum levels defined through J are often referred to as fine structure levels, and it is these levels that the laser light directly effects. Each level requires a certain amount of energy to reach from the atoms ground state the atom is only capable of absorbing light of certain frequency. This is because the energy of a photon E is determined primarily by its frequency ν:

$E=\nu \cdot \hbar$

where h is planks constant.

For a specific fine structure level, the momentum carried by the atoms nucleus also defines a hyper fine level for the total angular momentum. This total momentum F is thusly split into the electron component J and the nuclear component I:

$F=J+I$

For an OPM the most important part of the energy level structure is the fine structure, as these transitions are what determine the frequency of laser light to be used in the magnetometer. It is also the fine structure coherence times that determine how well the magnetometer can make a measurement. Often the transition from the ground state to the firs and second fine structure states are referred to as the D1 and D2 transitions.

As the laser traverses the vapor cell, it imparts its angular momentum to the alkali vapor atoms polarizing them along the same axis that it is polarized along. It is for this reason that it is essential that the polarization of the laser be circular so that the direction of its polarization lies along the axis of propagation of the laser, allowing it to polarize the atoms along that axis.

Once an atom in some hyperfine state is subjected to a magnetic field, it begins to rotate in what is called larmor precession. The frequency of which is defined by the strength of the magnetic field and a proportionality constant γ:

$\omega =B\cdot \gamma$

This precession creates further energy level splitting which put the atom into what are often referred to as Zeeman sublevels. The number often used to differentiate these levels is m_f which is representative of the total amount of angular momentum along the beam axis.

In the absence of a magnetic field atoms are pumped up to a fine structure state and spontaneously emit light. Through this process the L states of the atoms change while, due to the spontaneously emitted light not being polarized in any particular direction the m_f state is on average unchanged. This means that eventually all the atoms find themselves pumped to the highest available Zeeman state leaving the cell in a highly polarized condition. In this state the atoms are transparent to the laser light as they cannot absorb anything more and so the power of the laser is stable. However, when a magnetic field with components perpendicular to the pump axis are introduced to the system the polarized atoms begin Larmor precession which places the atom in a lower Zeeman sublevel. This lower sublevel is no longer fully transparent to the laser light and so the subpopulation in the cell that has dropped to this sublevel creates a dip in laser intensity as they begin absorbing light again to return to the higher Zeeman level.

Vapor Cell considerations

Absorption and linewidth

The vapor cloud in a cell is often bigger than area of the laser beam, due to this atoms in the cell have a rate of absorption of the on resonance light:

$R_{abs}(\nu )=\sum _{res}\sigma (\nu )\Phi (\nu )$

defined using the total photon flux Φ(ν) and the photon absorption cross section σ(ν). The photon flux is often considered to be more or less constant due to the lasers highly monochromatic light. Similarly the photon absorption cross section can be viewed as a constant across all the frequencies:

$\int \limits _{0}^{\inf }\sigma (\nu )=\pi r_{e}cf_{res}$

where r_e is the classical electron radius, c is the speed of light, and f_res is the oscillator strength. It is good to note that f_res is the integral over the classical cross section for a given resonance, so one can relate the absorption cross section to the classical cross section of an effect. The value over all frequencies is somewhat irrelevant as what is of most interest is how well the atoms are absorbing within the linewidth of the laser used in the magnetometer. So for convenience sub sections of the photon absorption cross section owing to specific effects are considered.

The first common contribution that is considered is by convention is produced by the linewidth from the time-energy uncertainty and the linewidth from a process known as pressure broadening. The two fine structure states for the excited states in an alkali metal atom have natural lifetimes t_nat which serve as the time uncertainty Δt so the time energy uncertainty shifted to the time-frequency domain requires

$2\pi \Delta \nu \Delta t\geq 1$

Which produces a line width of:

$\Gamma _{nat}={\frac {1}{2}}\pi t_{nat}$

The absorption linewidth is also effected by the collisions that occur between the alkali atoms and other gases in the cell. The magnitude of this effect is largely determined by the number density of the non alkali atoms in the cell, increasing as they become more dense. Thus the line width associated with this effect is approximated as:

$\Gamma _{pres}\approx {\frac {1}{\pi }}t_{pres}$

where t_pres is the average time between the collisions. These linewidths are usually combined into one linewidth Γ_L. The frequency response to these effects is Lorentzian in nature with full width at half mast (FWHM) of Γ_L:

$\Gamma _{L}=\Gamma _{nat}+\Gamma _{pr}$

The absorption profile can also be broadened by the doppler shifts that occur due to parallel components of the atoms direction of travel to the laser beams propagation direction. Experiencing a light shift described by the doppler effect. In this scenario the atom absorbs light that is off resonance for the transition. The probability distribution for a certain magnitude of travel along the laser axis is Maxwellian and produces a frequency response that is gaussian with a line width of:

$\Gamma _{G}={\frac {2\nu _{res}}{c}}*{\sqrt {\frac {2*k_{b}T}{M}}}$

because the absorption cross section is a function of the classical cross section each of these frequency response curves can be plugged back into the absorption cross section equations to find the contribution of the effects on the overall absorption. For the natural and pressure broadening effects the Lorentzian frequency response produce an absorption cross section of:

$\sigma _{L}(\nu )=\pi r_{e}cfL(\nu -\nu _{res})$

which produces the absorption cross section at resonance:

$\sigma (\nu _{res})={\frac {2r_{e}cf}{\Gamma _{L}}}$

which demonstrates why increasing the gas pressure of a vapor cell also creates a requirement for more intense lasers.

Similarly the gaussian response produced by the doppler effects produces an absorption cross section of:

$\sigma _{G}=\pi r_{e}cfG(\nu -\nu _{res})$

which will decrease as temperature is increased. However, for the temperatures where the materials used to construct vapor cells are usable, this behavior is not observable.

Since the frequency response is dependent on all three of the described effects it is standard for these line shapes to be combined into the

Voigt profile:

$V(\nu -\nu _{res})=\int \limits _{0}^{\inf }L(\nu -\nu ')*G(\nu '-\nu _{res})d\nu '$

which similarly produces an absorption cross-section of:

$\sigma _{\nu }(\nu )=\pi r_{e}cfRe[V(\nu -\nu _{res})]$

the shape of which is largely decided by the magnitude of the pressure broadening effect. When pressure broadening is large the Lorenzian shape dominates and when it is small the doppler effects dominate causing the Voigt profile to be gaussian.

There also effects that are often explored relating to the hyperfine splitting sublevels. These can be found via the Wigner-Eckart theorem and add a summation over transitions from one hyperfine state to another.

Spontaneous emission

An OPM can also experience interference from the spontaneous emission of on resonance photons from pumped atoms. The photons emitted this way are unpolarized, so when they are absorbed by another alkali atom in the ground state, that atom does not become polarized. In the highly dense environment of the vapor cell this means that it can take several emission-absorption cycles for a single photon to escape the system, and as this happens it limits the capacity of the alkali atoms in the cell to polarize as an ensemble. This is referred to as radiation trapping.

There are several techniques that can be employed to reduce the effect of this drop in polarization, however the one most commonly employed in an OPM is to introduce a molecular gas such as nitrogen which can absorb the energy from excited atoms when it collides with an alkali atom. The rate that these collisions occur can be given as:

$R_{Q}=n_{Q}\sigma _{Q}{\bar {v}}$

where n_Q is the number density of the quenching gas, σ_Q is the quenching cross-section, and ṽ is the relative velocity between a molecule of the quenching gas and an atom of the alkali metal vapor. The greater the number of these collisions the smaller the effect of radiation trapping on the polarization attainable in the cell. Aside from the quenching gas, it is also common to include a buffer gas whose transitions are unlikely to absorb any photons emitted either by the quenching atoms, the alkali atoms, or the light source. This is done to limit the diffusion of the alkali vapor throughout the vapor cell, stabilizing the number density along the beam area for a reasonably short measurement period.

Spin Relaxation

Radiation trapping is just one example of how collisions between particles in the vapor cell can cause the total spin polarization in the vapor cell to degrade. The rate at which those collisions cause the spin to degrade has been presented as:

$R_{rt}=K(M-1)R_{op}{\frac {R_{sd}}{R_{sd}+R_{Q}}}(1-P)$

where K is a fit factor describing the depolarization by the absorption of a spontaneously emitted photon. M is a multiplicity factor, describing the amount of times the photon is emitted before it can escape. R_op is the rate of optical pumping, R_sd is the spontaneous decay rate of the alkali atoms in the cell, and P is the polarization of the ensemble. However, in a vapor cell with an appropriate amount of quenching gas this rate becomes negligible.

However there are many kinds of collisions that can still cause spin relaxation. The general form for collision rate is:

$R=n\sigma {\bar {v}}$

where n is the number density, σ is the collisional cross-section, and ṽ is the thermal velocity. The relative thermal velocity is defined in this scenario to be:

${\bar {v}}={\sqrt {\frac {8k_{b}T}{\pi M_{reduced}}}}$

where M_reduced is the reduced mass.

Collisions rates of this form will work to produce several different collision rates corresponding to collisions that effect different components of the polarization. These rates are related to their effect on the coherence of the system through the polarization lifetimes T1(the polarization lifetimes of the pumping parallel component denoted as z) and T2(the polarization lifetimes of the pumping perpendicular components, denoted as x and y).

The lifetime in the pump direction is defined by the rates of mechanisms affecting the momentum along the optical pumping axis:

${\frac {1}{T1}}={\frac {1}{q}}(R_{sd}+R_{op})+R_{wall}$

Namely the optical pumping rate R_op, the spin destruction collision rate R_sd, and the wall collision rate R_wall. The life time is also notably affected by the nuclear slowing down factor q which is defined as:

$q=(2I+1)$

in principle this factor also includes a ratio between the resonance frequency and the frequency of a time varying magnetic field, however for most application of OPM the field is so short lived that this ratio can be neglected. This factor q is the result of spin collisions and photon absorption changing the spin of the electron much faster than the nucleus can change its spin. This results in a lag time after the electron has experience spin destruction where the nuclear spin is keeping the atomic spin somewhat coherent with the rest of the ensemble.

The most important term in this formula is that of the spin destruction collision relaxation rate. Collisions of this type can happen between alkali atoms and any other atoms present in the cell, meaning that R_sd actually has three terms:

$R_{sd}=R_{sd}^{alkali-alkali}+R_{sd}^{alkali-Q}+R_{sd}^{alkali-buffer}$

one term for spin destruction collisions between alkali atoms and other alkali atoms, one modeling spin destruction collisions between the alkali atoms and the quenching gas, and one modeling the spin destruction collisions between alkali atoms and the buffer gas. These collisions are understood to be the result of spin-rotation interactions:

$V_{AA}={\frac {2}{3}}\lambda S(3{\widehat {R}}{\widehat {R}}-1)S$ (for Alkali-Alkali collisions)

where S is the total spin for the colliding alkali atoms, λ is a coupling coefficient, and Ř is the direction of the internuclear axis of the alkali collision pair. Similarly:

$V_{ang}=\gamma _{sr}NS_{alk}$ (for Alkali-Noble Gas collisions)

where γ_sr is a coupling coefficient, N is the angular momentum of the noble-gas and the alkali atoms about each other, and S_alk is the electron spin of the alkali atom. In general the contributions to the total relaxation process for these interactions is small. For the Alkali-Alkali collisions this is due to their contribution being so much smaller than the spin exchange interactions that their contribution is negligent. In the Alkali-noble atom case it is due to the fact that collisions between the alkali ground state and a noble gas can happen multiple time before the alkali atom begins to depolarize. There is also an interaction that can occur between the quenching gas and the alkali vapor atoms, but this is not covered as the quenching gas pressure is almost always chosen to sufficiently quench radiation trapping while ensuring that this contribution remain negligible.

The transverse lifetime T2 is also important in characterizing how the rates of spin relaxation effect the ability of the sensor to detect a magnetic field. In fact T2 is often a much more important metric as it is often much shorter than T1. T2 is defined mathematically as:

${\frac {1}{T2}}={\frac {1}{T1}}+{\frac {1}{q_{se}}}R_{SE}+R_{gr}$

where q_SE is a factor produced by spin-exchange broadening, R_SE is the rate of relaxation related to the spin exchange collisions, and R_gr is the rate due to magnetic field gradients.

Spin exchange collisions, are probably the most important source of spin relaxation in an OPM; because they often dominate in the high optical density environment of the vapor cell. Unlike spin destruction collisions, spin exchange collisions conserve the total spin of the system. However, as the nuclei of the atoms begin to realign they can change hyperfine zeeman states. When atoms occupy the two ground state zeeman levels they will rotate with the same larmor frequency but in opposite ways which will cause decoherence. This decoherence is causes a relaxation in the overall transverse spin.

It is because of this dominance of the spin exchange interactions that most OPM's operate in the Spin Exchange Relaxation Free (SERF) regime. Which is characterized by high optical density combined with near zero ambient magnetic field. In this regime the broadening factor grows large bringing the contribution of the Spin Exchange rate to zero. The high alkali vapor density also helps bring the broadening caused by spin exchange collisions to vanish. Allowing for the spin-exchange rate to both dominate all other sources of relaxation while allowing the atomic spin populations to come to a state similar to thermal equilibrium. Both of these effects combine to allow for a highly magnetically sensitive vapor state in a cell.

Two beam magnetometers

In practice the single beam OPM is limited in what it can actually measure. Namely it lacks the capability to distinguish the actual direction of the perpendicular field as the dip in intensity of the pumping beam can only distinguish the magnitude of the perpendicular field. It is for this reason that many OPM's employ both a pump laser to put the atoms in their sensitive state and a linearly polarized probe beam, set at an off resonant frequency and linewidth that will not contribute to the pumping of the vapor states.

There are several ways to produce measurements with a two beam setup but the most common was presented in its roughest form by Bell and Bloom in 1957 and has since been refined. In this setup the probe beam is placed to propagate perpendicular to to the direction of propagation for the pump laser. This means that the electric field of the linearly polarized probe beam is of a form:

$E(0)={\frac {E_{0}}{2}}e^{i\omega t}{\widehat {y}}+c$

where E₀ is the field amplitude and ω is the frequency. This field can then be separated into positive and negative circularly polarized parts. This field can then be adjusted as it travels some distance:

$l={\frac {tc}{n(\nu )}}$

where n(ν) is the index of refration of some medium. The field then becomes:

$E(l)={\frac {E_{0}}{4}}e^{\frac {i\omega n_{+}(\nu )l}{c}}({\widehat {y}}+i{\widehat {z}})+{\frac {E_{0}}{4}}e^{\frac {i\omega n_{-}(\nu )l}{c}}({\widehat {y}}-i{\widehat {z}})$

where n_- and n₊ are indexes of refraction that differ for positively and negatively polarized light. Then with some simplifications eventually the field can be produced as a function of some angle θ as:

$E(l)=E_{0}(cos(\theta ){\widehat {y}}-sin(\theta ){\widehat {z}})$

which will be accurate for any medium that is birefringent with differing indexes of refraction for positively and negatively polarized light. This angle θ corresponds to a difference in polarization of the light before and after it travels through the medium and is often referred to as the angle of faraday rotation.

It is understood that when subjected to an external magnetic field the vapor cell gas mixture is a medium that fulfills the requirements to create faraday rotation. Light traveling a distance l through the vapour cell ends up with a total optical rotation of:

$\theta ={\frac {\pi }{2}}lnr_{e}cP_{x}(-f_{D1}Im[V(\nu -\nu _{D1})]+{\frac {1}{2}}f_{D2}Im[V(\nu -\nu _{D2})]$

where P_x is the non-zero polarization in the x direction and ν_D1 and ν_D1 are the resonant frequencies for the D1 and D2 transitions. By measuring the difference in polarization one is thus able to find the magnitude of the applied field in the direction of either x or y depending on the orientation of the probe beam.

In this scenario, a single photo diode is not sufficient to measure the angle. Instead a device to non-destructively measure the polarization is setup in front of the cell, and a polarimeter is setup after the cell. There are a few variations on this, examples of which are included in the figure.

Sensitivity

An OPM's performance is measured primarily by its sensitivity, or its ability to detect small fast changes in the magnetic field. The typical units that this metric is measured in are fT/sqrt(hz). Primarily sensitivity is limited by the introduction of noise sources either inherent to the magnetometer or noise sources present in the environment around the magnetometer.

The first component of this sensitivity limiting noise comes from spin-projection produced by the uncertainty between the x and y components of the total angular momentum:

$\delta F_{x}\delta F_{y}\geq {\frac {|F_{z}|}{2}}$

this uncertainty means that there will always be some noise in the estimation of the value for the angular momentum in the direction for x. Due to the fact that we are mostly concerned about the value of this over the period of the T2 time this uncertainty is equal to:

$\delta (F_{x})_{rms}={\sqrt {\frac {2F_{z}T2}{N}}}$

Where N is the total number of independent spins interrogated by the probe beam.

There is also noise inherent to the sensor, because the photon density is limited which is comparable to the spatial resolution of the laser beam. If we take two photon total flux densities Φ₁ and Φ₂ which can be used to calculate an optical rotation angle:

$\theta ={\frac {\Theta _{1}-\Theta _{2}}{2(\Theta _{1}-\Theta _{2})}}$

We assume that these two total flux densities are roughly the same so we can calculate the uncertainty in this angle as:

$\delta \left\langle \theta \right\rangle _{rms}={\sqrt {{\frac {1}{2}}\Theta }}$

which will produce an uncertainty in the polarization from the x field of:

$\delta \left\langle P_{x}\right\rangle _{rms}={\frac {2}{\pi lnr_{e}cf{\sqrt {2\Theta }}Im[V(\nu -\nu _{0})]}}$

Because the polarization angle is the tool used in a two beam magnetometer to estimate the strength of the magnetic field this uncertainty produces noise in the estimation of the magnetic field estimate.

There is also uncertainty produced by the AC stark shift that occurs as the laser pumps the transition in the Alkali atoms. Which takes the form of:

$\delta B_{stark}={\frac {\pi r_{e}cf{\sqrt {2\Theta }}}{(2I+1)\gamma A_{beam}}}Im[V(\nu -\nu _{res})]$

An OPM can also be limited by various noise sources that are part of it's environment in a real world setting. Noise is often created and picked up by an OPM from the electronics around it, this often penetrates magnetic shields that an OPM is often placed in. OPM's will also experience noise from small fluxuations in the lasers and their polarization which are close but never fully consistent. Once identified the noise components are often combined into the total noise level through quadrature sum.