We present an all-optical method to measure and compensate for residual magnetic fields present in a cloud of ultracold atoms trapped in an optical dipole trap. Our approach leverages the increased loss from the trapped atomic sample through electromagnetically induced absorption. Modulating the excitation laser provides coherent sidebands, resulting in a Λ-type pump–probe scheme. Scanning an additional magnetic offset field leads to pairs of sub-natural linewidth resonances, whose positions encode the magnetic field in all three spatial directions. Our measurement scheme is readily implemented in typical quantum gas experiments and has no particular hardware requirements.

Magnetic field measurement and calibration are cornerstones in quantum gas experiments. It allows for precision control of the atomic Zeeman levels, which is needed in a large variety of applications, like Raman transitions, spinor gases,1 spin dynamics and entanglement,2 Feshbach resonances,3 or metrology.4,5 Traditional methods such as fluxgate magnetometers6,7 or high-end methods such as NV centers8 or SQUID sensors9,10 can only measure the field outside the vacuum chamber, and microwave equipment, which allows for the precise in situ measurement of magnetic fields, is not always available. Atomic magnetometers11–13 use magneto-optical effects to measure the spin precession in a magnetic field. Several approaches leverage coherent population trapping and electromagnetically induced transparency (EIT)14 to implement vector magnetometers.15–18 Recently, magnetometry via Ramsey-type measurements was demonstrated in an array of single Rubidium atoms.19 They are close to the applications discussed here; however, they are often optimized for vapor cell applications. In a quantum gas experiment, atomic magnetometry is straightforward to implement since the necessary components such as lasers, atom detection, and magnetic field management are readily available. It is, therefore, possible to adapt the core principles of atomic magnetometers to the requirements and constraints of ultracold atom experiments.

Here, we demonstrate an all-optical technique to measure a magnetic field vector in an ultracold atom experiment without prior calibration. Our method is easily applicable to most quantum gas and optical tweezer setups. The method relies on electromagnetically induced absorption (EIA)20–24 on the optical cooling transition in combination with fast chopping of the excitation laser. The latter creates coherent sidebands at a well-defined frequency. Spectroscopy is performed by scanning an additional magnetic field in all three spatial dimensions. Thereby, the carrier and the sidebands can be brought into two-photon resonance with neighboring Zeeman sublevels in a Λ type setting. The resonance condition is signaled by increased absorption due to EIA and corresponding peaks in the spectra. The measurement signal can be any experimentally accessible observable, which depends on the optical absorption, e.g., the fluorescence or the sample lifetime. Our method does not require any microwave field, and a single laser beam is, in principle, sufficient to measure the field.

In the following, we first introduce the optical tweezer setup, which we use in our experiment. We then introduce the general measurement scheme and the underlying theory and discuss the obtained spectra. We close with a discussion of possible applications, extensions, and further developments.

To demonstrate the method, we use mesoscopic atomic ensembles of rubidium atoms trapped in optical tweezers. In brief, we load 87Rb atoms from the background gas into a 6-axis magneto-optical-trap (MOT), located in an ultra-high vacuum glass cell. The optical tweezers (waist w0 ≈ 2 μm at 1064 nm) are directly loaded from the MOT. We typically trap 10–20 atoms in a single optical tweezer. An NA = 0.4 objective, which is used for the creation of the optical tweezers, is also used to collect fluorescence light from the atomic sample both on the D1 and D2 lines. Our setup has an overall photon detection efficiency of ∼2%. The fluorescence imaging is essentially background free (<0.01 photons/second/pixel). For the spectroscopy, we use a separate set of co-axial counter-propagating beams, which are resonant to the D2 cooling transition F = 2 → F′ = 3 of 87Rb. Both beams are switched by an acousto-optical modulator (AOM). See Fig. 1 for a sketch of the laser beam geometry and the relevant level scheme. To exclusively collect the fluorescence from trapped atoms, we only consider the camera counts from a 3 × 3 μm2 region centered around the optical tweezer. Additionally, we perform the fluorescence imaging after a time twait following the spectroscopy to allow any lost atoms to leave the imaging region. We choose twait = 15 ms, a timescale much larger than the inverse of trap frequencies, which we expect to be >10 kHz.

FIG. 1.

We probe the atomic sample with two counter-propagating beams (dashed black arrows) having circular polarization of opposite handedness. Three pairs of coils in Helmholtz configuration provide a constant magnetic offset field B=(Bx,By,Bz). The probing beams are not aligned with any of the coil axes. The field |B| splits the Zeeman states of the involved 5S1/2F = 2 and 5P3/2F′ = 3 levels.

FIG. 1.

We probe the atomic sample with two counter-propagating beams (dashed black arrows) having circular polarization of opposite handedness. Three pairs of coils in Helmholtz configuration provide a constant magnetic offset field B=(Bx,By,Bz). The probing beams are not aligned with any of the coil axes. The field |B| splits the Zeeman states of the involved 5S1/2F = 2 and 5P3/2F′ = 3 levels.

Close modal
For the spectroscopy, we chop the probing beams and the optical tweezers with a frequency of ν = 500 kHz and a phase shift of π with respect to each other [see Fig. 2(a)]. A duty-cycle of d = τonν = 0.5 for the probe beams and 0.4 for the optical tweezer ensures that the probe beams interact with free atoms without any residual trapping light and resulting lightshifts present. The modulation of the probing beams leads to the formation of coherent sidebands at frequencies ±(2n+1)νnN0, as shown in Fig. 2(b). We typically apply 250 pulses of the probe beams before we measure the fluorescence of the remaining atoms on the D1 line. As the probe beams are resonant to the atomic transition, photon scattering leads to the heating of the atomic sample and a loss of atoms. We model the time evolution of the atom number N̄(τ) with an exponential decay,
N̄(τ)=N0̄exp(γτ),
(1)
where N̄ denotes the mean number of trapped atoms and γ is the loss rate. Figure 3 shows typical decay curves of the atom number truncated to the first 90 µs. The setup is completed by three orthogonal pairs of coils in Helmholtz configuration around the science cell, which can be used to apply external homogeneous magnetic offset fields during the probing sequence.
FIG. 2.

(a) Alternating sequence of trapping (top) and probing (bottom) the sample. Modulating the probe beam with a modulation frequency ν and a duty cycle d = 0.5 leads to the formation of AM sidebands. Using a smaller duty cycle for the optical dipole trap (d = 0.4) omits trap-induced lightshifts during probing. (b) Measured frequency spectrum of the probing beam for ν = 500 kHz and d = 0.5, revealing sidebands with frequency offsets (2n+1)νnN0. S is normalized to the spectral power of the carrier.

FIG. 2.

(a) Alternating sequence of trapping (top) and probing (bottom) the sample. Modulating the probe beam with a modulation frequency ν and a duty cycle d = 0.5 leads to the formation of AM sidebands. Using a smaller duty cycle for the optical dipole trap (d = 0.4) omits trap-induced lightshifts during probing. (b) Measured frequency spectrum of the probing beam for ν = 500 kHz and d = 0.5, revealing sidebands with frequency offsets (2n+1)νnN0. S is normalized to the spectral power of the carrier.

Close modal
FIG. 3.

Mean D1 fluorescence signal count plotted against the probing duration τ. The dashed lines show an exponential fit to the data. The three curves correspond to the three vertical lines in Fig. 5(a).

FIG. 3.

Mean D1 fluorescence signal count plotted against the probing duration τ. The dashed lines show an exponential fit to the data. The three curves correspond to the three vertical lines in Fig. 5(a).

Close modal
The atomic sample is initially prepared in the F = 2 hyperfine ground state, for which the Zeeman levels split according to ΔE=mFgFμB|B| with gF = 1/2 in a constant magnetic field B=(Bx,By,Bz). For typical ambient fields, this splitting is well below the linewidth of the optical probing transition, and direct spectroscopic measurement of the magnetic field is impossible. However, in three-level systems, interferences between excitation pathways can suppress the admixture of the short-lived excited state and thus allow for a spectroscopic resolution below the linewidth of the optical transition. Here, we use EIA in a Λ scheme, as depicted in Fig. 4, where the pump and probe lasers are created by the carrier and the sidebands of the amplitude-modulated excitation laser. Since atoms are illuminated by two counter-propagating beams with circular polarization, every magnetic transition between the Zeeman levels in the upper F′ = 3 and the lower F = 2 manifold is possible as long as the quantization axis is along one of the three coil axes. While the two-photon transition in principle allows for (σ±, σ) transitions with ΔmF = ±2 and (π, σ±) transitions with ΔmF = ±1, a summation over the relevant Clebsch–Gordon coefficients reveals that the latter is up to a factor of eight stronger than the former. Therefore, it is possible to observe increased photon absorption due to EIA if the two-photon detuning vanishes, δ=gFμB|B|hν=0, i.e., if the magnetic energy splitting between adjacent Zeeman states matches the modulation frequency creating the sidebands,
hν=±gFμB|B|.
(2)
FIG. 4.

Schematics of the Λ scheme used in the experiment (top). Exemplarily, the carrier of the probe beam drives the π transition to the upper hyperfine manifold, whereas the sidebands couple to a neighboring mF state with σ± transitions. Additional EIA excitation pathways with the same initial and final state are possible but not shown for clarity. The bottom half shows the expected absorption spectrum. The left peak depicts the situation where the external field Bx brings the ΔmF = ±1 level in resonance with the ±ν sideband according to Eq. (2). On the right, the ΔmF = ±1 level is in resonance with the ∓ν sideband. Both lead to enhanced absorption via EIA. The center peak only occurs if the orthogonal field components B vanish, such that all Zeeman levels are degenerate.

FIG. 4.

Schematics of the Λ scheme used in the experiment (top). Exemplarily, the carrier of the probe beam drives the π transition to the upper hyperfine manifold, whereas the sidebands couple to a neighboring mF state with σ± transitions. Additional EIA excitation pathways with the same initial and final state are possible but not shown for clarity. The bottom half shows the expected absorption spectrum. The left peak depicts the situation where the external field Bx brings the ΔmF = ±1 level in resonance with the ±ν sideband according to Eq. (2). On the right, the ΔmF = ±1 level is in resonance with the ∓ν sideband. Both lead to enhanced absorption via EIA. The center peak only occurs if the orthogonal field components B vanish, such that all Zeeman levels are degenerate.

Close modal

As depicted in Fig. 4, different pathways generated by the sidebands come into resonance depending on the magnetic splitting of the Zeeman levels. Since the EIA also stays effective for small one-photon detunings (Δ < Γ5P), the magnetic splitting of the excited state Zeeman levels can be neglected and only becomes relevant if a detailed understanding of the observed line strength is necessary. The increased absorption of photons at δ = 0 directly leads to an increased heating-induced loss rate γ from the trap and comprises an easily accessible measurement signal that allows us to characterize the external magnetic field.

For the magnetic field, we consider a system of three pairs of coils that create a magnetic field with mutually perpendicular components
Bn=αnIn,
(3)
at the position of the atoms. Here, In is the current through the respective coil pair, and αn is the system specific conversion factor between the current and magnetic field. We further introduce an unknown offset field, Bres=(Bxres,Byres,Bzres). For convenience, we express the offset field with the help of the corresponding current in the coils, Bnres=αnIn0. The modulus of the magnetic field is then given by
|B|=αx2(Ix+Ix0)2+αy2(Iy+Iy0)2+αz2(Iz+Iz0)2.
(4)
Combining Eqs. (2) and (4), we obtain the following relation for the appearance of the EIA peaks:
hνgFμB2=αx2(Ix+Ix0)2+αy2(Iy+Iy0)2+αz2(Iz+Iz0)2.
(5)

To experimentally probe these field-induced EIA resonances, we create a varying magnetic field in one of the Cartesian directions, e.g., the x-direction, while setting Iy = Iz = 0. For each applied magnetic field, we measure the atom loss from the sample as a function of the probing duration and extract the decay constant γ (Fig. 3). Plotting the decay rates as a function of the applied magnetic field reveals the spectrum shown in Fig. 5(a). Except for an overall slope, the measured spectrum shows the expected peak structure from Fig. 4. The observed peaks correspond to the (π, σ±) transitions, while the strongly suppressed (σ±, σ) resonances remain hidden in the noise floor of the measurement. The central peak from Fig. 4 only appears if the orthogonal field components By = Bz ≈ 0 approximately vanish. This is only the case if the offset field is small in these two directions or already compensated. Still, the symmetry of the spectrum allows us to identify the current required to compensate for the offset field, i.e., Ix=Ix0 in the x direction, by measuring the center between the two first side peaks. Note that the width of the observed features is considerably smaller than the natural linewidth while being limited by the width of the amplitude sideband itself. In addition, drifting magnetic fields during the measurement period of several hours for one spectrum and potential remanent fields caused by apparatus undergoing repeated field changes further broaden the features.

FIG. 5.

(a) Measured sample loss rate γ while scanning the magnetic field in the x-direction and keeping the other respective coil currents fixed. We determine the positions of the peaks by fitting the spectrum with a multi-Lorentzian model (gray line). The dotted red line indicates Ix0 whereas the solid red lines mark the peak positions used to determine Ix. The sample decays for +, −, and 0 are shown in Fig. 3. (b) and (c) are the remaining measurements for the field directions y and z.

FIG. 5.

(a) Measured sample loss rate γ while scanning the magnetic field in the x-direction and keeping the other respective coil currents fixed. We determine the positions of the peaks by fitting the spectrum with a multi-Lorentzian model (gray line). The dotted red line indicates Ix0 whereas the solid red lines mark the peak positions used to determine Ix. The sample decays for +, −, and 0 are shown in Fig. 3. (b) and (c) are the remaining measurements for the field directions y and z.

Close modal

Compensating the field in the x-direction and conducting an analog measurement for the y-direction yields the peak structure shown in Fig. 5(b). After compensating the field in the y-direction, the spectrum for the z-direction shown in Fig. 5(c) differs in that a central peak emerges between the pair of symmetric EIA resonances. Note that the position of the middle peak is not fully centered. The origin of this effect is not clear at present and requires further investigation. This feature corresponds to a situation where the external fields are zeroed and all Zeeman sublevels are degenerate, as previously observed in Refs. 20, 25, and 26. In this situation, the carrier alone is able to drive all allowed transitions between F = 2 → F′ = 3, greatly increasing the sample absorption and loss rate. Note that this feature is only visible in the measurement in the z-direction since we deliberately compensated the field in the orthogonal directions by setting Ix=Ix0 and Iy=Iy0 through the values extracted from the measurements in the x- and y-directions.

With the information on the offset currents In0, we can apply Eq. (5) to the peaks In± appearing in the three measurements and obtain a system of three linear equations, which we then solve for the conversion coefficients αx, αy, and αz. Finally, we determine the residual magnetic fields in the experiment as
|Bnres|=|αnIn0|,
(6)
from the centers of the peak structures.

As an example, we show the conversion factors αn and residual fields Bnres of our setup in Table  I. While the residual field in the z-direction is given by the earth’s magnetic field, the deviation in the x- and y-direction can be qualitatively explained by a nearby ion-getter pump located 0.5 m beside the science chamber that creates a magnetic field mainly in the horizontal plane. For comparison of the extracted conversion coefficients, we calculated the conversion factors αsim by numerically integrating Biot–Savart’s law for our coil geometry. The measured values show an overall good agreement with the simulated values and deviate by less than 20%. The mismatch can be attributed to position uncertainties, small remanent magnetic fields of the setup, and imperfections in the coil manufacturing.

TABLE I.

Measured values of the current-to-field conversion factor α and residual magnetic field Bres for each coil pair. For comparison of the conversion factors, we list the conversion factors αsim simulated for our coil geometry. To compare the measured residual magnetic field, we list the earth magnetic field Bearth at the position of our lab according to the world magnetic model.27 The uncertainties are calculated from the covariance of the fitted peak positions.

Axis|α| (mGA−1)|αsim| (mGA−1)|Bres| (mG)|Bearth| (mG)
x 562 (17) 680 498 (16) 208 
y 530 (17) 580 91 (3) 
z 10 061 (17) 9060 488 (5) 482 
Axis|α| (mGA−1)|αsim| (mGA−1)|Bres| (mG)|Bearth| (mG)
x 562 (17) 680 498 (16) 208 
y 530 (17) 580 91 (3) 
z 10 061 (17) 9060 488 (5) 482 

This work demonstrates a technique to measure magnetic fields in a quantum gas experiment without prior calibration of the coil system. The possibility to determine the conversion factors Bn = αnIn on the fly distinguishes our method from magnetometers based on the Hanle effect while providing a comparable measurement accuracy of ≈±8 mG.12,13 While competing methods that work without prior coil calibration, for example, microwave spectroscopy, show higher accuracy, they typically require additional equipment. In contrast, our approach needs, in essence, only resonant laser light and beam chopping. Preliminary investigations show that the symmetric structure of the EIA peaks in Fig. 5(a) can even be produced with only one modulated beam. We employ two counter-propagating probe beams to omit excess atom loss by pushing the atoms out of the optical tweezer by one-sided radiation pressure. The prolonged interaction time within the spectroscopy thus decreases the Fourier-limit as one main influence on the width of the sidebands and improves the field resolution. Consequently, any further reduction of the atom loss rate for the baseline and the features themselves will result in a higher resolution.

Our method is also flexible with respect to the detection scheme. In principle, any signal that is dependent on the photon scattering rate, such as the sample loss rate, the fluorescence, or the transparency of the sample, can serve as an observable. The simplicity and independence of a prior calibration make the technique readily available in typical quantum gas experiments. Finally, since it is intrinsically Doppler-free in a single beam, our scheme might also find applications in vapor cell magnetometers, using a single probe beam while directly monitoring its transmitted intensity. The measurement speed is then only limited by the time it takes to perform the magnetic field sweeps.

We acknowledge financial support by the DFG within the collaborative research center TR185 OSCAR (Grant No. 277625399) and within the Major Research Instrumentation Programme (Grant No. INST 248/270-1).

The authors have no conflicts to disclose.

S.P., S.S., and A.T. performed the experiments. S.P. and S.S. analyzed the data. S.P. prepared the paper. H.O. and T.N. supervised the experiment. All authors contributed to the data interpretation and paper preparation.

Suthep Pomjaksilp: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Sven Schmidt: Data curation (equal); Formal analysis (equal); Investigation (equal). Aaron Thielmann: Data curation (equal); Formal analysis (equal); Investigation (equal). Thomas Niederprüm: Funding acquisition (lead); Methodology (lead); Project administration (lead); Supervision (lead); Writing – review & editing (equal). Herwig Ott: Conceptualization (equal); Funding acquisition (lead); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon request.

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