A Voigt effect based 3D vector magnetometer

A Voigt effect based 3D vector magnetometer

Tadas Pyragius    Hans Marin Florez    Thomas Fernholz thomas.fernholz@nottingham.ac.uk School of Physics & Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD, UK
Instituto de Física, Universidade de São Paulo, 05315-970 São Paulo, SP-Brazil
July 19, 2019
Abstract

We describe a method to dispersively detect all three vector components of an external magnetic field using alkali atoms based on the Voigt effect. Our method relies on measuring the linear birefringence of the radio frequency dressed atomic medium via polarization homodyning. This gives rise to modulated polarization signals at the first and second harmonic of the dressing frequency. The vector components of the external magnetic field are mapped onto the quadratures of these harmonics. We find that our scheme can be utilised in both cold and hot atomic gases to detect such external fields in shielded and unshielded environments. In the shielded hot vapour case we achieve field sensitivities well below 1 pT range for all 3 vector components, using pump-probe cycles with 125 Hz repetition rate, and limited by the short coherence time of the cell. Finally, our scheme has a simple single axis beam geometry making it advantageous for miniature magnetic field sensors.

pacs:
Valid PACS appear here
preprint: APS/123-QED

I Introduction

Optically pumped magnetometers, (OPMs), have increasingly been in the spotlight for their broad span of applications ranging from fundamental physics experiments to medical physics. Examples include measurements of the electric dipole moment (EDM) edm1 (); edm2 () and search for exotic physics cptviolation () as well as magneto-encephalography (MEG) meg1 (); meg2 () and magneto-cardiography mcg0 (); mcg1 (); mcg2 () where the detection of extremely small magnetic fields of the brain and the heart is required. A review can be found in  Budker2007 (). In recent years OPMs have become the state-of-the-art magnetic field sensors achieving sub fT sensitivity and surpassing the well established SQUID based sensors Romalis2002 (); Romalis2010 (); Romalis2013 (). In its simplest operation, an OPM uses a pump-probe laser to measure the atomic Larmor frequency, i.e. the frequency of precession, by interacting with polarized atomic spins, which in effect measures the strength of the external magnetic field. However, in a larger range of applications, a complete determination of the magnetic field is required. Some schemes employ a scalar magnetometer to run as a vector magnetometer by applying a rotating low frequency bias magnetic field Yakobson2004 (); Gao2016 (). Another possible approach uses multiple radio-frequency modulations to map the three vector components onto the harmonics of the signal Romalis2004 (); Budker2014 (). More recently, an all-optical scheme with crossed beams was demonstrated to extract the three field components Gao2015 (). However, in all of the described cases additional modulation or detection stages are required adding significant complexity towards realising a full vector magnetometer.

To date, most of the OPM schemes are based on pump-probe configurations that rely on the Faraday rotation, i.e. circular birefringence of the medium. As a result, the majority of the such schemes require an orthogonal pump-probe geometry for high efficiency of detection Romalis2002 (); ingleby (). However, this geometry is not convenient for developing miniature sensors, whilst a parallel configuration is compatible with chip-scale and compact atomic magnetometers.

In this paper we demonstrate a 3D vector magnetometer based on the measurement of the Voigt effect, i.e. linear birefringence of radio-frequency (rf) dressed states Jammi (). This method maps the three vector components of the external field, detected via demodulation of the probe beam’s ellipticity, onto orthogonal quadratures of the first and second harmonic of the dressing frequency. Therefore, vector magnetometer operation is achieved with a single radio-frequency modulation and without rotating bias fields. Furthermore, state preparation and detection are performed in a parallel pump-probe geometry. We analyzed sensitivity and vector capability of our magnetometer in a shielded environment in open loop operation, detecting fields in the range of 0.5 nT. An extension of dynamic range as well as operation in unshielded scenarios should be possible with active feedback on the external field.

The paper is organized as follows. In Section II, we briefly describe the linear birefringence induced by radio-frequency dressed states providing predictions of the field mapping onto the first and second harmonics of the rf oscillation. We report on the experimental realization using two different types of atomic ensemble. Section III demonstrates the detection principle with laser cooled atoms, prepared in a pure quantum state. Section IV, describes the extension to a magnetically shielded vapour cell by combining the Voigt effect with synchronous pumping. Experimental results on vector sensitivity are shown together with an analysis of noise performance. Section V presents our conclusions.

Ii Birefringence of radio-frequency dressed states

Optically pumped magnetometers utilise dispersive coupling of light to an atomic ensemble in the presence of an external magnetic field. The Larmor precession of spin-polarized atoms causes a modulation of the medium’s birefringence, which can be observed polarimetrically. In our scheme, we actively drive such precession with an additional radio-frequency field. In the following, we present a brief description of the driven medium and its interaction with the light field in terms of dressed states, as discussed in our previous work Jammi (). Our model includes the dependence on field orientation, which allows for the extraction of full vector information from measurements of either linear or circular birefringence.

Let us consider atoms interacting with a static field and a field oscillating at a radio-frequency in a transverse direction . The time-dependent interaction Hamiltonian of an atom with spin of constant magnitude can be approximated by where is the Bohr magneton, is the Landé factor, and is the reduced Planck constant. Depending on the sign of the factor, using positive , we transform the coordinates to a frame rotating about the -axis with a time-dependent rotation operator . Neglecting counter-rotating terms, the transformed, effective Hamiltonian takes the form

(1)

The effective magnetic field in this frame is given by , where corresponds to a fictitious magnetic field that defines a resonance condition for the Larmor precession. As depicted in Fig. 1 (a), the angle enclosed by the effective field and the -axis is

(2)

At resonance, i.e. , the effective field is pointing in the rotating -direction.

{overpic}

[scale=0.3]frame_rot.png a)b)

Figure 1: Geometrical description of the frame rotation due to the external field direction. a) Effective field oriented by angle . A small external field changes the angle . b) The presence of transverse external fields and changes the orientation of the effective field. Additional transverse external fields are mapped into the angles and .

The eigenstates of the effective, rotating-frame Hamiltonian, i.e. the dressed states can be written as in the rotating frame. The same state in the laboratory is then . These states can be prepared directly by synchronous optical pumping azunish (), or by adiabatic dressing of bare states , which are identical in both frames, apart from a time-dependent phase or different (quasi-)energy.

Given the geometric alignment of these states with the effective magnetic field direction, the atomic spin evolution and resulting vector magnetometer operation can be described in terms of frame rotations Sakurai (). Considering small angle rotations, the presence of transverse fields can be represented as small rotations of the laboratory frame with respect to and as it is shown in Fig. 1 b). The external transverse fields and generate rotations with angles and , respectively. Hence, applying a sequential rotation 111We use rotation matrices according to Rodrigues’s rotation formula ., the atomic spin operator in the new coordinate system is given by . In this approach, we make the false assumption that the applied rf field is co-rotated, while in fact there is a reduction of its effective amplitude in the rotating frame, given by .

Fig. 1 shows that from the geometrical relations the angles , and are given by

(3)
(4)

In the small angle approximation we can write

(5)

For the detection of the spin evolution we employ probe light which is described by propagating in the direction of the static magnetic field . The creation and annihilation operators are defined as density amplitudes in position space for circular polarization. Instead of using the Faraday rotation, we employ the Voigt rotation which depends on the second order AC Stark shift induced by the probe beam that is then detected on the output ellipticity of the light. After the light propagates through the medium in the off-resonant regime, where absorption of the fields can be neglected, the light matter interaction for the Faraday and Voigt rotation can be described by

(6)
(7)

where and represent the photon flux of elliptical and at polarized light; is the rank-k coupling strength and is the number of atoms. We have have neglected back-action and assumed that atoms are in the -manifold with the same state.

From the geometrical perspective in eqs.(5) one can determine the temporal atomic response for Faraday or Voigt rotation due to the presence of external magnetic fields. In particular, an adiabatic evolution of the state in the laboratory frame and rotated by , leads to a spectral decomposition of the Faraday rotation

(8)

where . From eqs.(22) and (23), in the low angle approximation meaning that external transverse fields are small in comparison to the static field, the spectral components are given by

(9)

The principal behaviour of these functions across rf resonance is depicted in Fig. 2 a). This spectral decomposition shows that the transverse components are mapped onto the quadratures of the first harmonic with maximum amplitude as it is shown with dashed-dotted line. Meanwhile, the zeroth harmonic presents a dispersive behaviour near the rf resonance (dashed line) mapping linearly the third component. In spite of that, this zeroth harmonic (or dc component) is quite vulnerable to electronic and technical noise, which in practice makes this strategy robust only to the two transverse field components.

In contrast, when the external transverse fields rotate the spin state by , the Voigt rotation in eq.(7) presents a spectral decomposition in the laboratory frame for the state given by

(10)

in which, according to eqs.(25)-(27), the harmonic components are

(11)

considering small angle approximation for and the light propagation parallel to the static field .

{overpic}

[scale=0.37]spectral_components.png a)b)

Figure 2: Spectral decomposition of a) Faraday rotation proportional to with harmonics and , and b) Voigt rotation proportional to with harmonics (solid black line), (dashed red line) and (dashed-dotted blue line).

Fig. 2 b) shows the principal behaviour of these spectral functions across the rf resonance. Again, the transverse field components are mapped onto the quadratures of the first harmonic but now with a maximum amplitude at (dashed red line). This angular condition is satisfied when the static field is . Unlike in the Faraday decomposition, now the the zeroth harmonic vanishes (solid black line) and the second harmonic is the one that carries the linear response with respect to at (dashed-dotted blue line). Hence, the Voigt rotation provides low-noise detection of the three magnetic components at first and second harmonics of the rf magnetic field.

From the geometrical definitions for , and in eq.(5), the magnetometer response maps linearly onto the externally applied fields

(12)
(13)
(14)

where .

Iii Experimental realization: Laser cooled atoms

iii.1 Laser cooled atoms setup

Our experimental cold atom setup was described in Jammi () and here we will present only a brief description. We measure the ellipticity of an ensemble of approximately Rb cold atoms at . The ensemble is initially prepared in the state after a previous sequence of optical pumping and state cleaning in . Atoms are then adiabatically dressed with a magnetic rf field in the -direction with frequency , generated by an external resonant coil. The rf field amplitude is ramped up to over while the static magnetic field is ramped down to a magnitude of , which tunes the atomic Larmor frequency near resonance.

Figure 3: Experimental setup. A laser cooled rubidium sample is prepared in a dressed state. After adiabatic dressing with a magnetic rf field along and optional rotation of the static field into the -direction, linear birefringence of the sample is probed polarimetrically by a laser pulse propagating along .

We employ a laser beam detuned by from the transition in the -line of Rb. A half-waveplate sets the polarization at with respect to the rf field axis. After interacting with the ensemble, a quarter-wave plate and a Wollaston prism allow us to measure the ellipticity induced by the medium. The light is detected on a balanced photodetector pair (Thorlabs PDB210A) and an optional high-pass filtering rf amplifier (Minicircuits Model ZFL-1000+). The output voltage is proportional to the observed ellipticity, i.e., with electronic gain . The output signal is acquired by an FPGA and is demodulated digitally.

iii.2 Field mapping in cooled atoms

Fig. 4 shows the demodulated signals at rf frequency and as a function of the static field . The dispersive response at is observed due to the presence of the transverse fields. In contrast, the mode amplitude at is maximum at the rf resonance, whereas the mode amplitude at is zeroed. As was shown in Fig. 2, when the static field reaches the G field, the amplitude of the mode becomes maximum in absolute value in both quadratures and the mode shows a linear response with respect to the static field.

{overpic}

[scale=0.37]cold_atoms_2f_1f_plots.png a)b)c)

Figure 4: Scan across RF resonance. (a) and (b) Quadratures of the mode amplitude at and (c) real part of the mode amplitude at when the field component is scanned through the rf resonance. The transverse static fields are kept constant.
{overpic}

[scale=0.31]Cold2DGrid.png a)c){overpic}[scale=0.31]Cold3DGrid.png b)d)

Figure 5: Atomic response mapping external magnetic field vector onto the rf harmonic components. a) Theoretical response of the first harmonic quadratures, i.e. at frequency , as a function of the transverse field and for constant . b) Full 3-dimensional response, including the real part of the mode amplitude at frequency , which allows for the measurement of the longitudinal field . c) Experimental realization showing how the orthogonal transverse fields map into the quadratures of the mode amplitude at . The separation between two vertical lines is about 9 mG. d) Full experimental response as a function of the scanned field.

The atomic ensemble operates then as a vector magnetometer at field in which the two frequency modes map the atomic response to the three components of the magnetic field. That is, at FWHM of the mode amplitude , the ellipticity induced by the atoms is sensitive to the longitudinal field, while the quadratures of mode amplitude map the transverse fields. To show the vector magnetometer operation, we scan the transverse fields and demodulate the and the quadratures, see Fig. 5. The separation between the two vertical lines is about 9mG for a 1ms probe pulse duration.

Fig. 5 shows that for magnetometry purposes the system should operate at where the deviation of the is minimum due to transverse fields. Since the mode signal maps the longitudinal field, the two frequency modes describe a complete basis for detecting the three field components from only one rf modulation.

We have shown the principle operation of the magnetometer based on the Voigt effect in cold atoms. However, given the limitations on the sampling rate due to the loading time of the MOT, we explored the Voigt rotation in a vapour cell with room temperature atoms to obtain faster detection rates, higher bandwidth and sensitivity.

Iv Experimental realisation: Room temperature vapour

iv.1 Room temperature vapour setup

Our setup consists of a paraffin coated Rb enriched vapour cell ( mm and mm) at room temperature inside a commercial 4–layer -metal shield (Twinleaf MS-2), see Fig. 6 a). The static magnetic fields as well as the radio frequency field inside the chamber are generated by a combination of a solenoid and cosine coils, respectively, which are driven by a lead-acid battery powered ultra low noise current sources based on the modified Hall-Librecht design hall (); dalin ().

The laser system consists of a combination commercial and home made lasers addressing the atomic transitions for state preparation and probing. The laser system is housed on a separate vibrationally damped table to the magnetometer. The light is coupled to SM-PM fibers and is sent to a separate non-magnetic and also vibrationally isolated table with non-magnetic optomechanics where the -metal chamber with the cell are housed.

The Voigt rotation is measured with a quarter wave-plate and polarizer cube and the light is detected with a balanced photodetector pair (Thorlabs PDB210A). The experimental sequence generation and data acquisition are performed using Labview FPGA (PCIe-7852). The magnetometer operation consists of two different stages, see Fig. 6 b). First, we perform the state preparation during the first 5ms pulse and then we probe the state for another 3ms after. This time is primarily limited by the coherence time of the cell i.e. this is the minimum pump/repump pulse duration required to prepare our state before probing it.

a)b)

Figure 6: Experimental realization. a) Layout of the experimental setup. The pump/repump comes at a slight angle to the probe and the static field. b) Single shot experimental pulse sequence with a typical 8 ms duration. During each cycle, the static fields are held at constant, scanned values after switching within 50 s at the start of the state preparation process.

iv.2 State preparation

The detection of magnetic fields is performed on an ensemble of approximately Rb atoms in a room temperature vapour state prepared in the two stretched states of equal population namely the , ground states in the . In order to prepare such a state, we first dress the atoms with a uniform radio frequency field of mG at 5 kHz in the -axis whilst simultaneously applying a uniform static field which is set on the magnetic resonance.

Note, that unlike in the cold atoms case, the adiabaticity condition here is not met. To pump the atoms in the chosen stretched states, we perform synchronous pumping with 9% duty cycle in phase with the rf field (see Fig. 6 b)). The pump is resonantly locked on the transition on the D1 line, which propagates along the -axis with a -polarization parallel to the rf field which corresponds to our quantization axis in the rotated frame.

To avoid atomic population loss by optical pumping, a co-propagating CW repump beam addressing of the D2 line of the same polarization is spatially overlapped with the pump which repopulates the atoms from the to the ground state in the . The pump and repump beams have a Gaussian intensity profile with intensities 1.1 mW/ and 0.8 mW/ respectively. To confirm our state preparation process we use stroboscopic microwave spectroscopy to probe the dressed atomic states. The details of this method are beyond the scope of this paper and will be discussed in detail in the follow up work.

iv.3 Signal Detection

Immediately after the state preparation process we couple a counter-propagating probe pulse along to measure the resultant Voigt rotation. The probe is detuned -550 MHz from the transition of the D1 manifold with a Gaussian profile of intensity 1.3 mW/ and linear polarization set to 45°  relative to the rf field quantization axis. The interaction between the probe beam and the atoms results in the probe being elliptically polarized where the ellipticity is proportional to the external magnetic field. The phase shift is measured by decomposing the elliptical polarization with the use of a quarter-wave plate, a polarizing beam splitter and a balanced photodetector. The quarter-wave plate is aligned such that the differential photo current measures the difference between right-hand left-hand circularly polarized components.

{overpic}

[scale=0.35]raw_atomic_and_fft_signal_5kHz_RF.png a)b)

Figure 7: Typical experimental signals. a) Raw balanced signal of the Voigt rotation. The state preparation occurs within the first 5ms of the cycle followed by a 3ms probing pulse. b) Single-sided, spectral power density of the amplified signal. Atomic signals arise at and , an additional harmonic can also be observed. The harmonic arises as a result of the spin evolution due to the nutation of the stretched state.
{overpic}

[scale=0.35]dedmoulated_profiles.png a)b)c){overpic}[scale=0.35]FOM_vs_det.png d)

Figure 8: Scan across the RF resonance. a), b) and c) and resonance profiles when the field is scanned, the transverse static fields are kept constant. The probe pulse is sent immediately after the state preparation process. d) Figure of merit as a function of probe detuning. The figure of merit is an effective slope of the resonance profile which essentially affects the overall sensitivity of the OPM. Maximizing this quantity, increases the overall magnetometer sensitivity.

Fig. 7 a) shows an example of the typical raw detector signals in time together with a signal spectrum, b), which shows that the main contribution to the rf signal is found at frequency 2 kHz and a lower signal at frequency kHz due to the presence of transverse fields. As can be seen in Fig. 7 a), the atomic signal decays in time. This is predominantly limited by the atom-wall and atom-atom collision rates which result in a reduction of state lifetime. In our case, the quality of the paraffin coating and the exchange of the atoms between the main cell body and the stem with the Rb reservoir are the major contributing factors to these fast relaxation times. This is a technical limitation, and in principle, a better quality coating with a lockable stem can be made to achieve coherence lifetimes in excess of 60 s balabas ().

The combination of detuning and power of the probe ensures that the prepared state is not appreciably affected. We check this by mapping the figure of merit of mode which we choose to be the amplitude of the mode over the FWHM of the mode, , as a function of the probe detuning from the on the D1 line and maximize this quantity, see Fig. 7 d).

The detected raw signals contain the field information originating from the 3 geometric axes , , and . These are digitally demodulated using the reference frequencies corresponding to the Larmor frequency, , for and field demodulation and Larmor frequency for demodulation. As it is shown in Fig. 8 a) and b) the quadratures of the mode amplitude at follow a dispersive profile, whilst figure c) shows the mode resonant response. This is consistent with the theoretical model described in section II and with the experimental results for cold atoms in section III.2.

In order to map the measured voltage signals of the magnetometer into real magnetic field signals, we initially apply a linear field ramp of a well known range to each of the fields independently and measure the corresponding demodulated voltage response which gives us a voltage to field conversion. For small fields, this has a linear relationship and as a result the voltage to field conversion factor is a constant. The field calibration is done by changing the Larmor resonance frequency and sweeping the external field (with no transverse fields present) and finding the maxima of the resonance, . The resonance field condition can be easily calculated and plotted against the applied voltage/current of the coils giving the field conversion. The presence of transverse fields changes the resonance condition to . Thus, to obtain the calibration for the transverse fields, we change one of the transverse fields whilst keeping the other one at zero and sweep the field to obtain a new location of the resonance. As before, the new resonance location is plotted against the control voltage/current of the coils. We additionally check the reliability of the field calibration with atoms using a commercial fluxgate magnetometer (Stefan Mayer Instruments, FLC3-70).

iv.4 3D Vector mapping

Figure 9: The 3D oval-shaped plot mapping of the OPM response at 5 kHz radio frequency dressing. The transverse fields are raster scanned during each cycle to produce the oval-shaped response. Each oval colour represents a different field point detuning from the point to demonstrate the full vector mapping (see inset showing resonance). For small external field perturbations in the transverse and longitudinal directions, the OPM produces a linear response (see 2D and 3D square grid insets in (a) and (b)). We attribute deviations from the ideal profiles to geometric misalignment between the probe beam and static and rf field coils. The asymmetric distortion increases for lower bias fields, consistent with cross talk from transverse coils to the field as defined by the direction of the probe beam.

Following the same procedure as in the cold atom case in section III.2, we can measure the three components of the field by setting the static field at which maximizes the mode amplitudes at . By linearly scanning the external transverse fields and demodulating the and quadratures, we are able to map the magnetometer response. The vector magnetometer operation can be visualized on a 3D plot shown in Fig. 9. It must be noted that the theoretical results in eqs. (12-14), which map the magnetometer response, do not take into account atomic collisions, various broadening effects (e.g. gradient fields, light power) and the fact that the atomic state does not satisfy the adiabaticity condition. As can be seen in Fig. 9 the 2D and 3D oval-shaped profiles contain some distortion which originates from the geometrical misalignment of the probe/pump beams relative to the static fields and/or the rf dressing field. Increasing the field scanning range increases the field leakage as can be seen in Fig. 9 where the small field approximation no longer applies and the OPM response is no longer linear. Despite these theoretical simplifications, the model is effective in qualitatively describing the vector magnetometer response to the external magnetic fields. The effects of distortion can be reduced by a better overlap between the probe/pump beams and the static fields, gradient field compensation and/or by increasing the Larmor frequency. However, the latter method reduces the sensitivity to the transverse fields. The full mathematical description of the oval-shaped plot can be found in the Appendix B.

iv.5 Noise performance

To perform the noise measurements we detune the static field to in order to maximize the mode amplitudes at which optimize the magnetometer sensitivity. We then adjust the transverse fields such that the modes become zero i.e. this is the point where atoms are efficiently sensitive to the magnitude and the direction of the transverse fields.

{overpic}

[scale=0.32]Bx_noise.png a){overpic}[scale=0.32]By_noise.png b){overpic}[scale=0.32]Bz_noise.png c)

Figure 10: OPM noise at 36C vapour cell temperature in a shielded environment in the frequency range of Hz at 5 kHz radio frequency dressing. a) and b) Quadrature noise performance of the two orthongonal transverse directions. c) Atomic noise along the longitudinal direction.
{overpic}

[scale=0.32]OPM_noise_vs_Temperature_w_error_bars.png a){overpic}[scale=0.32]OPM_noise_as_a_function_of_RF_freq_error_bars.png b)

Figure 11: Characterization of the noise performance. a) OPM noise as a function of cell temperature. b) OPM noise as a function of radio frequency dressing field in a shielded environment in the frequency range of Hz.

Based on the field calibration described above, Figure 10 shows the noise performance for each quadrature at and the in-phase quadrature at . At 5 kHz rf dressing frequency the magnetometer operates with an average noise level of 350 fT, with a bandwidth of 62.5 Hz. The dominant constraint on the magnetometer bandwidth and noise is down to the short coherence time of the cell ( ms) which is dominated by the quality of the paraffin coating and the exchange of the atoms between the main cell body and the stem with the Rb reservoir. Typically, paraffin or OTS coated cells have coherence times ranging from 30 ms to 300 ms seltzer (); coating (). Longer coherence time would improve the field sensitivity of the OPM due to a larger fraction of atoms remaining in the field sensitive state. In addition, higher quality paraffin coating would also shorten the pump/repump pulse time needed to prepare the stretched states and therefore allow us to increase the probe cycle rate thus increasing the bandwidth of the OPM. We have investigated the effects of heating the cell to increase the atomic density which should in principle improve the sensitivity, as it is shown in Fig. 11 a). As the temperature increase, the mode amplitude increased linearly by 33% for temperatures below C. The noise performance in the transverse direction improved by a factor of two and the longitudinal direction was not affected within that range of temperatures. Beyond C the atomic resonance at saturates and when reaches C its amplitude drops. This saturation effect was previously observed in ref. pustelny (), in which higher atomic concentrations lead to an increase of the collisional relaxation rate, depolarizing the prepared stretched states. An additional way of increasing the sensitivity of the OPM, at least in the transverse field directions, is to further decrease the dressing frequency of the rf, as it is shown in Fig. 11 b). This would result in a lower static field needed to achieve the Larmor resonance and as a result, the transverse field mapping angle would become larger making the OPM more sensitive to the transverse fields. In particular, from 40 kHz to 2.5 kHz the noise performance with respect to the transverse fields presents is reduced by a factor of four. The limitation of this method is the precise alignment that is needed between the probe/pump beams and the static field and the increased susceptibility and sensitivity to magnetic field gradients which would be distorting the oval-shaped mapping. Furthermore, reduction of the rf field amplitude increases the sensitivity of the and modes, however, additional mechanisms such as power, collisional and field gradient broadening are currently preventing the reduction of the rf field amplitude.

V Conclusions

oWe have presented and successfully demonstrated a full vector magnetometer based on the Voigt effect both in cold atom and hot vapour setups. As shown, our scheme has the advantage of requiring only a single optical axis for state preparation and detection making it ideal for compact magnetic field sensors. We have achieved fT/ sensitivity with a Hz bandwidth. Our current limitations in the sensitivity of the OPM stem from the coherence time of the cell and the low atom number. Future improvements will include a heated and buffer gas filled cell in order to increase the atom numbers and reduce the rate of atomic collisions that induce decoherence effects, respectively. These improvements should shorten the state preparation lifetime thus increasing the bandwidth and the field sensitivity of the OPM. Moreover, a cavity setup may be implemented in order to increase the interaction path between the light and the atoms thus further improving the sensitivity. The datasets generated for this paper are accessible at (Nottingham Research Data Management Repository).

Vi Acknowledgements

This work was funded by Engineering and Physical Sciences Research Council (Grant No. EP/M013294/1). We acknowledge the support from the School of Physics & Astronomy Engineering and Electronics workshops. We thank Sindhu Jammi for collecting the cold atoms data.

Appendix A Faraday and Voigt rotation for adiabatic evolution

In what follows we present a brief description of the atomic dynamics that leads into Faraday and Voig rotation, according to eq. (18) in ref. Jammi ().

Starting from our effective, rotating frame Hamiltonian in eq.(1), the adiabatic dressed state is

(15)

where is the angle of the effective magnetic field direction, given in eq.(2). In the laboratory frame, the same state is given by

(16)

where . Considering the atoms in a new coordinate system, any atomic observable is transformed according to given by Euler rotation as in Fig. 1. Therefore, the mean value of any atomic observable is given by

(17)
(18)

In particular, according to eq.(17), the longitudinal spin component is

(19)

where () stands for () with . It is worth noting that the average value with is evaluated for the state . Therefore, the contributions from operators to eq.(19) will be zero since in the z-basis they are formed of raising and lowering operators. Given the fact that the longitudinal spin polarization gives rise to Faraday rotation in eq.(6), we obtain

(20)

with . Now, the expression above can be written in terms of spectral components as

(21)

where the mode amplitudes are

(22)
(23)

Equivalently, for the Voigt Rotation, in which the ellipticity is given in eq.(7), the spectral decomposition of the mean value in eq.(17) for the bi-linear operator is

(24)

where the mode amplitudes are

(25)
(26)
(27)

Appendix B Field mapping in the linear regime

To measure the external magnetic field, we map the atomic signals and with the three components of the external magnetic field. To do that, we define the real and imaginary part of from eq.(26) as and respectively, and we define the real part of in eq.(27) as , such that

(28)
(29)
(30)

For a full vector field sensing, we set the magnetometer to a sensitive field point which gives . For small external field perturbations around the point e.g. small angle aproximation , the OPM response to external magnetic field is

(31)
(32)
(33)

where .

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