Non-invasive detection of animal nerve impulses with an atomic magnetometer operating near quantum limited sensitivity

Magnetic fields generated by human and animal organs, such as the heart, brain and nervous system carry information useful for biological and medical purposes. These magnetic fields are most commonly detected using cryogenically-cooled superconducting magnetometers. Here we present the first detection of action potentials from an animal nerve using an optical atomic magnetometer. Using an optimal design we are able to achieve the sensitivity dominated by the quantum shot noise of light and quantum projection noise of atomic spins. Such sensitivity allows us to measure the nerve impulse with a miniature room-temperature sensor which is a critical advantage for biomedical applications. Positioning the sensor at a distance of a few millimeters from the nerve, corresponding to the distance between the skin and nerves in biological studies, we detect the magnetic field generated by an action potential of a frog sciatic nerve. From the magnetic field measurements we determine the activity of the nerve and the temporal shape of the nerve impulse. This work opens new ways towards implementing optical magnetometers as practical devices for medical diagnostics.

: Pulse sequence for measuring the nerve impulse.

MAGNETOMETER PRINCIPLE
The total spin of the atomic ensemble is defined as J = (J x , J y , J z ). Here J is a quantum operator, and the components have the commutation relation [J y , J z ] = iJ x . J is here defined as being unitless and equals the total angular momentum divided by the reduced Planck constanth. The equations of motion for the spin vector can be derived using the Heisenberg equation of motionJ The dot denotes the time-derivative and the bracket denotes the commutator. The Hamiltonian describing the coupling between the spin and the magnetic field is where γ is the gyromagnetic ratio which for the cesium atom in the F = 4 ground state equals 2.2×10 10 rad/(sec·Tesla).
In vector form the equation of motion readsJ In the experiment, the atoms are spin-polarized in the x-direction and located in a static magnetic field B x pointing in the x-direction. In the presence of a small time-dependent magnetic field B y (t) or B z (t) pointing in the y-or z-direction, the spin vector will acquire a transverse component J ⊥ = (J y , J z ) = |J ⊥ | (cos θ, sin θ). We will assume that J x is large compared to J y and J z , and that J x is independent of time. We now introduce spin operators J y and In the rotating frame, the equations of motion reaḋ The transverse spin component will eventually decay, and we have therefore added decay terms in the above equations.
The decay rate is denoted by Γ and the associated decay time is T 2 = 1/Γ. We also added Langevin noise operators F y (t) and F z (t) with zero mean values and correlation functions F These equations can be integrated and the solutions are

Free precession
Assume that the transverse spin has some mean value at t = 0 and that it is then left free to precess. At a later time t the transverse spin component in the rotating frame is We see that the mean value decays in time. In the lab frame the transverse spin will perform a damped oscillation.
Atomic response to a pulse of magnetic field Consider the case where a magnetic field B y (t) is applied for a duration τ . We assume that τ T 2 such that any decay of the spin components can be neglegted. If initially the transverse spin component is zero J ⊥ (0) = 0, we find and Here we have defined the Fourier component of the magnetic field at the Larmor frequency as Similarly, if the magnetic field is applied along the z-direction instead, the transverse spin component will be We see that magnetic fields in y-and z-directions have similar effects on the spins: the fields create transverse spin components with lengths proportional to the Fourier components of the magnetic fields at the Larmor frequency.

Projection noise limited detection
The measurement of the transverse spin component is fundamentally limited by the spin-projection noise originating from the Heisenberg uncertainty principle. This uncertainty is ∆ |J ⊥ | = J x /2. By equating the created mean value given by Eq. (13) to the projection noise we find the uncertainty on the magnetic field Fourier component due to the projection noise: We can also calculate the uncertainty on the amplitude of an oscillating magnetic field due to the projection noise.
For a sinusoidal magnetic field with total duration τ equal to an integral multiple of the Larmor period, the amplitude B 0 is related to the Fourier component by |B(Ω)| = B 0 τ /2. From this and Eq. (14) we find the projection noise limited uncertainty on the amplitude: which is often called the minimal detectable field. The magnetic field sensitivity can be found by multiplying ∆B PN with the square-root of the total measurement time √ T tot and setting T tot = τ = T 2 : The Standard Quantum Limit Besides the projection noise, the magnetic field measurement will be limited by the quantum shot noise of the probing light, and the back-action noise imposed by the probe on the atomic spins. The total uncertainty is which can be written as 1,2 where κ = ∆B PN /∆B SN is a dimensionless light-atom coupling constant. For small coupling strengths, the measurement noise will be dominated by the shot noise, and for large coupling strengths, the measurement noise will be dominated by the back-action noise. By minimizing the uncertainty given by Eq. (18), we find the optimal coupling strength κ ≈ 1.3 and the standard quantum limit on the magnetic field measurement which is ≈ 1.5 times larger than the uncertainty due to the projection noise.

Measuring the atomic signal
The atomic spin can be measured optically. Assume that a linearly polarized pulse of light is propagating in the z-direction through the atomic ensemble. The polarization of the light will be rotated by an angle proportional to J z due to the Faraday paramagnetic effect. The polarization of the light is described using Stokes operators S x (t), S y (t), and S z (t) which have the unit of 1/time. Here S x (t) = [Φ x (t) − Φ y (t)] /2 equals one half the difference in photon flux of x-and y-polarized light. S y (t) refer to the differences of +45 • and −45 • polarized light, and S z (t) to the differences of right hand and left hand circular polarized light. Assuming that the input light before the atomic ensemble is either x or y-polarized (such that S x (t) is a large quantity) and that the rotation angle is small, the output light after the atomic ensemble can be described by the equation The parameter a describes the coupling strength between the atoms and the light 1 . The Stokes operator S out y (t) can be measured with polarization homodyning. There are several ways that one can extract information about the transverse spin components and therefore about the magnetic field from the measured signal. One can for instance measure the mean value S out y or the power spectral density of the signal. The power spectral density (P SD) for a function x(t) is defined as We will show below that the P SD of S out y (t) is proportional to the amplitude squared of the applied magnetic field.

Detection of a pulse of magnetic field
Assume that a pulse of magnetic field B y (t) of duration τ is applied from t = −τ to t = 0. After the pulse, the spins have acquired a non-zero transverse spin component J ⊥ (0) ∝ |B y (Ω)| as given by Eq. (11). At t = 0 the spin will continue to precess until it decays as described by Eq. (9). This spin vector can be measured using a pulse of light with duration T and starting at the time when the magnetic field pulse ends. The mean value of the measured signal is where θ is the polar angle of J ⊥ (0) . The amplitude of the transverse spin vector can be extracted from the measurement by, for instance, a fit of the experimental data to Eq. (22). Alternatively, one can calculate the P SD of the signal. For x(t) = A sin (Ωt + θ) e −Γt we calculate that the peak value of the P SD is where the second term (θ, Γ, Ω, T ) is much smaller than the first term for our experimental parameters. We see that

Continuous recording of the magnetic field
We will now discuss how one can measure the magnetic field as a function of time. Assume that a magnetic field B y (t) is applied and that light is continuously monitoring the atomic spin. If J ⊥ (0) = 0, then at a later time The mean value of the measured signal will be S out From this we see that the measured signal S out y (t) is proportional to the convolution of B y (t) with the function − cos (Ωt) e −Γt . Similarly, if the transverse magnetic field is pointing in the z-direction, the signal is proportional to the convolution of the magnetic field B z (t) with the function sin (Ωt) e −Γt . The magnetic field as a function of time can be extracted from the measured data using numerical deconvolution. The pulse sequence in Supplementary Fig. 3 is used for detection of the magnetic field from the nerve impulse B nerve . The atoms are first optically pumped using pump and repump light, then the magnetic field is present, and finally the atoms are measured using probe pulse A. The optically detected signal S A (t) will be a free induction decay as seen in Supplementary Fig. 3. Due to misalignment of the pump and repump laser beams with respect to the bias field B x (see Fig. 1

CONDUCTION VELOCITY
The nerve conduction velocity can be calculated by dividing the distance from the stimulation electrodes to the recording site [34(5) mm for optical recording and 53(10) mm for electrical recording] by the time interval between the stimulus artifact and the zero-crossing of the observed nerve signal (see Fig. 4(c) in the main text). The electrical recording measures the potential difference ∆V between two external electrodes separated by 5 mm. As the extent of the action potential Φ is much larger than electrode spacing, the electrical signal is proportional to the time-derivative of the action potential ∆V ∝ ∂Φ/∂t. The zero-crossing of the electrical signal therefore corresponds to when the peak of the action potential passes the electrodes. Also, when the peak of the action potential passes the magnetometer, the magnetic field is zero, as the magnetic field B(t) ∝ ∂Φ/∂t according to a simple model for the nerve 3 . The earlier arrival of the nerve impulse for optical recording compared to electrical recording [1.0(2) ms compared to 2.4(2) ms] is consistent with the magnetometer being positioned in between the stimulating and recording electrodes. From the measurements shown in Fig. 4(c) we calculate the conduction velocity of 34(8) m/s and 22(5) m/s for optical and electrical recording. We note that stating a single number for the conduction velocity is not entirely correct as the frog sciatic nerve contains thousands of axons of varying diameter (see Fig. S1). For a single myelinated axon, the conduction velocity is proportional to the axon's diameter, which leads to a distribution of velocities within the range 10-40 m/s for axons in the frog sciatic nerve 4 .

ESTIMATE OF THE AXIAL IONIC CURRENT
The detected magnetic field is created by axial ionic currents inside the nerve bundle. There is a forward current inside the axons and a return current outside the axons. The magnetic fields from the forward and return currents can cancel each other, the exact degree of cancellation depends on the anatomy of the nerve, and the geometry of the experiment, such as the size of the magnetic field sensor and the distance from the nerve to the sensor.
We can estimate the axial current in the nerve from our magnetic field measurements. We use a simple model, where we assume that the ionic current is concentrated at the center of the nerve, and that the nerve produces a magnetic field similar to that of an infinitely long conducting wire. The magnetic field from an infinitely long wire is |B| = µ 0 I/ (2πr), where µ 0 is the magnetic permeability, I is the current, and r is the radial distance from the wire.
Using r = 1.9 mm for the distance from the center of the nerve to the center of the vapor cell, we calculate that a current of 0.23 µA will produce a magnetic field of 24 pT.
Our estimate of 0.23 µA is smaller than the 0.4 µA which was estimated in previous work on the frog sciatic nerve 5 . This is expected as in that work, the nerve was put in a large container with saline solution and the magnetic field was measured by a coil with the nerve inside it, such that a large part of the return current could flow without being detected.