Precise control of magnetic fields is a frequent challenge encountered in experiments with atomic quantum gases. Here we present a simple method for performing in situ monitoring of magnetic fields that can readily be implemented in any quantum-gas apparatus in which a dedicated field-stabilization approach is not feasible. The method, which works by sampling several Rabi resonances between magnetically field sensitive internal states that are not otherwise used in a given experiment, can be integrated with standard measurement sequences at arbitrary fields. For a condensate of 87Rb atoms, we demonstrate the reconstruction of Gauss-level bias fields with an accuracy of tens of microgauss and with millisecond time resolution. We test the performance of the method using measurements of slow resonant Rabi oscillations on a magnetic-field sensitive transition and give an example for its use in experiments with state-selective optical potentials.

Experiments with ultracold atomic quantum gases1 often call for the manipulation and control of the atoms’ spin degree of freedom, including work with spinor Bose-Einstein condensates (BECs)2 or homonuclear atomic mixtures in state-selective optical potentials3–10 where a control of the Zeeman energies to a fraction of the chemical potential (typically on the order of 1 kHz or 1 mG), may be required. With fluctuations and slow drifts of ambient laboratory magnetic fields on the order of several to tens of milligauss, achieving such a degree of control over an extended amount of time requires dedicated field-stabilization techniques. However, in a multi-purpose BEC machine, this may be challenging given geometric constraints that can interfere with shielding or with placing magnetic-field probes sufficiently close to an atomic cloud, which are often subject to short-range, drifting stray fields from nearby vacuum hardware or optomechanical mounts. To address this problem, we have developed a simple method for direct monitoring of the magnetic field at the exact position of the atomic cloud, by employing the cloud as its own field probe, in a way that does not interfere with its originally intended use. The idea is that hyperfine ground-state Zeeman sublevels that are not used in an experimental run can be employed for a rapid, concurrent sampling of Rabi resonances, in the same run, thus making it possible to record and “tag on” field information to standard absorption images, which can be used both for slow feedback control and for stable-field postselection. We emphasize that our pulsed, single-shot method, which features an accuracy of tens of microgauss and has an effective bandwidth of 1 kHz, is not meant to compete with state-of-the-art atomic magnetometers;11–16 rather, its distinguishing feature is that it can be implemented without additional hardware and independently of geometric constraints, while featuring a performance that is competitive with that of advanced techniques for field stabilization in a dedicated apparatus.17,18 It can, at least in principle, be used over a wide range of magnetic fields, starting in the tens of milligauss range.

This paper is structured as follows. Section II presents the principle and implementation of our method. Section III discusses the expected measurement accuracy as well as an experimental test based on a tagged measurement of slow Rabi oscillations on a magnetic-field sensitive transition. Section IV describes an application featuring the precise characterization of a state-selective optical lattice potential via microwave spectroscopy.8 

The principle of the method is illustrated in Fig. 1 for the S1/2(F = 1, 2) hyperfine ground states of 87Rb, which are split by 6.8 GHz. The atomic sample is located in an externally applied bias field B0 along z leading to a differential Zeeman shift δz/2π = 0.7 MHz/G × B0 between neighboring |F,mF states. Starting with all atoms in the state |a|F=1,mF=1, a sequence of microwave pulses i distributes population to |2,0,|2,1,|2,2 (i = 1, 2, 3), and then via |2,0 to |1,1 (i = 4) and further to |2,1 and |2,2 (i = 5, 6). To ensure isolated addressing of each transition, the detunings δi and Rabi couplings Ωi are chosen to be small compared to δz (by three orders of magnitude in the example discussed later), and the ordering of the individual pulses is chosen to avoid spurious addressing of near-degenerate single-photon transitions: |2,1|1,0|2,0|1,1 and |2,1|1,0|2,0|1,1. Other transitions are near-degenerate, but magnetic-dipole forbidden, ΔmF>1.

FIG. 1.

Magnetometry scheme. (a) A Bose-Einstein condensate of 87Rb atoms in a bias field B0 is subjected to a series of microwave pulses that distributes population over the |F,mF ground state manifold, depending on the exact value of the field. (b) Relevant states for the 6-pulse sequence, the |1,1 (red) and |2,2 (blue) states are used for the measurement of Fig. 3. (c) Outline of a typical experimental run (gray) with the magnetic field tagging added in (white). (d) Rabi resonance for Ωiτi = 0.94π, choice of detunings δi = 0.82Ωi (at B0) and effects of magnetic-field changes on the transfer probabilities pi (for identical Ωi) from which the field is then reconstructed.

FIG. 1.

Magnetometry scheme. (a) A Bose-Einstein condensate of 87Rb atoms in a bias field B0 is subjected to a series of microwave pulses that distributes population over the |F,mF ground state manifold, depending on the exact value of the field. (b) Relevant states for the 6-pulse sequence, the |1,1 (red) and |2,2 (blue) states are used for the measurement of Fig. 3. (c) Outline of a typical experimental run (gray) with the magnetic field tagging added in (white). (d) Rabi resonance for Ωiτi = 0.94π, choice of detunings δi = 0.82Ωi (at B0) and effects of magnetic-field changes on the transfer probabilities pi (for identical Ωi) from which the field is then reconstructed.

Close modal

The pulse parameters are adjusted such that the final populations PF=2,mF expected at B0 are comparable, and that their sensitivity to small deviations19δB from B0 is maximal, cf. Fig. 1(d). Assuming all δi > 0 at B0, the populations’ change away from the nominal field B0 is negative for i = 1, 2, 3 and positive for i = 4, 5, 6, with each transition shifted by a different amount. The change in the set of final populations then allows for an unequivocal and precise reconstruction of B = B0 + δB. In quantitative terms, the transition probabilities pi for the individual pulses can be calculated from the relative final-state populations PF,mF=N(F,mF)/N as

p1=P1,1+P2,0+P2,1+P2,2p2=P2,1/(P1,1+P2,1+P2,2)p3=P2,2/(P1,1+P2,2)p4=(P1,1+P2,1+P2,2)/(P1,1+P2,0+P2,1+P2,2)p5=P2,1/(P1,1+P2,1+P2,2)p6=P2,2/(P1,1+P2,2).

Each pi is related to the magnitude of the magnetic field B via

pi=(Ωi/Ω̃i)2sin2(Ω̃iτi/2),
(1)

where Ω̃i=(Ωi2+δi2)1/2 and δi = δi(B) is the modified detuning of the ith pulse from the ith addressed resonance. Assuming that the Rabi couplings Ωi are known from an independent calibration, the magnetic field B can then be extracted by fitting (δi+ωi)=E(Fi,mFi;B)E(Fi,mFi;B), where ωi is the microwave frequency for the ith pulse and

E(F,mF;B)=Δ8±Δ21+mFx+x2+gIμBmFB
(2)

is the Breit-Rabi energy of the levels involved in the transition, where the +(−) sign holds for F = 2(1), x = (gIgs)μBB/Δ, with gs the g-factor of the electron, Δ = 2π × 6.834 GHz and gI = −9.951 × 10−4 for 87Rb.20 

Our experiments are performed in a magnetic transporter apparatus,21 with an optically trapped condensate of N ∼ 1 × 105 atoms in the |a|1,1 ground state. At the end of an experimental run (which usually contains steps for the manipulation of the motional and/or internal state of the atoms), the atoms are released, given about 1 ms to expand (to avoid interaction effects), then subjected to the magnetometry pulse sequence described earlier and subsequently detected using absorption imaging. For the determination of the state populations PF,mF, we use Stern-Gerlach separation. In addition, to distinguish the F = 1, 2 states with |mF| = 1 (note that the gF factors in 87Rb have the same magnitude), absorption imaging of the F = 2 states is first performed using resonant F = 2 → F′ = 3 light, which disperses the F = 2 atoms while the F = 1 atoms continue their free fall. After optical pumping of the F = 1 atoms to F = 2 (using F = 1 → F′ = 2 light), these atoms are then imaged as well.

Several considerations determine the optimum choice of parameters for the magnetometry pulse sequence. Maximizing the magnetic-field sensitivity of the pi [see Eq. (1)] for a fixed coupling Ωi yields optimum detunings δi ≈ 0.58Ωi (at B0) and pulse durations τi1.24πΩi1 (the pulse area should be kept below 3π/2 in order to avoid side lobes as high as the main lobe in the Rabi spectrum). Additional minimization of the sensitivity to possible fluctuations of Ωi (with microwave amplifiers typically specified only to within 1 dB) modifies these conditions to δi ≈ 0.82Ωi and τi0.94πΩi1, respectively. Ideally, the chosen coupling strength Ωi of each transition should be proportional to the differential magnetic moment Δμi=BE(Fi,mFi;B)BE(Fi,mFi;B) of the two states involved in the transition. Furthermore, the expected range δB of fluctuations around B0 sets the optimum choice of Ωi through δBΩi/μB, and in turn the accuracy of the measurement goes down with increasing Ωi. In our experiment, we can comfortably realize kHz-range microwave couplings on all transitions (which are independently calibrated from sampling single Rabi resonances).

To demonstrate our method, we applied a bias field of 5.9 G using a pair of Helmholtz coils with 10 ppm current stabilization. Figure 2 shows the results of a typical short-time measurement of the magnetic field along the bias field direction, using an AC-line trigger to start the pulse sequence. The dominant contribution to field fluctuations around B0 is seen to be ambient AC-line noise with an amplitude around 1 mG, containing the first few harmonics of 60 Hz. From here on, we compensate for this by feeding forward the sign-reversed fit function onto an identical secondary coil of a single winding. The subtraction of the fit results in residual fluctuations up to ±0.4 mG, without apparent phase relationship with the AC-line.

FIG. 2.

Measurement of magnetic-field fluctuations (at B0 = 5.9 G), referenced to an AC-line trigger with a variable delay. (a) Reconstructed field noise, as a function of time after an AC-line trigger. The solid line is a fit function a cos(ωact + ϕ1) + b cos(3ωact + ϕ3) + c cos(5ωact + ϕ5)t, with ωac = 2π × 60 Hz. (b) Residual field variation after subtracting the fit function.

FIG. 2.

Measurement of magnetic-field fluctuations (at B0 = 5.9 G), referenced to an AC-line trigger with a variable delay. (a) Reconstructed field noise, as a function of time after an AC-line trigger. The solid line is a fit function a cos(ωact + ϕ1) + b cos(3ωact + ϕ3) + c cos(5ωact + ϕ5)t, with ωac = 2π × 60 Hz. (b) Residual field variation after subtracting the fit function.

Close modal

We characterize the remaining fluctuations further and, in particular, determine whether they represent the actual magnetic field in a time interval close to the measurement. For this purpose, we implement slow Rabi cycling (at B0 = 9.045 G) on the maximally magnetic-field sensitive transition |a|1,1|b|2,2, with a differential magnetic moment of Δμ3 = 2π × 2.1 kHz/mG. This measurement is performed by varying the coupling time of the oscillation and then recording the number of atoms in |b. To accommodate the Rabi cycling measurement, we choose a truncated pulse sequence in which the population in |a is subsequently distributed over five transitions instead of six. We note that this experiment is an example for the mode of operation depicted in Fig. 1(a), in which a “measurement” (of the Rabi cycling) is followed by a magnetic field “tag.” Magnetic-field fluctuations will lead to a rapid dephasing of the Rabi oscillation. However, using the field tag, the effect of (slow) magnetic-field fluctuations on the oscillation can be eliminated.

For a well-resolved, single-cycle oscillation, the instability of the detuning should not exceed about one tenth of the Rabi frequency. Here we choose Ω = 2π × 0.61(3) kHz, at an average detuning of δ = 2π × 0.44(3) kHz.

We see that the raw data resulting from multiple repetitions of the Rabi oscillation experiment have large associated scatter due to the long term drifts and shot-to-shot jitter of the magnetic field. To demonstrate the effect of the field tag, we plot the oscillation both as a function of inferred detuning (at a fixed duration) and time (at a fixed detuning). The results are shown in Figs. 3(b) and 3(c). In addition, we also plot all data, as scaled population pΩ̃2/Ω2 vs. scaled time tΩ̃. Clearly, the field tagging leads to a marked improvement of the oscillation contrast.

FIG. 3.

Slow Rabi cycling between |a=|1,1 and |b=|2,2, with magnetic-field reconstruction based on a 5-pulse sequence. (a) Observed time dependence of the transferred population after eliminating AC-line fluctuations as demonstrated in Fig. 2. The large shot-to-shot scatter is due to residual field fluctuations. (b) Data points post-selected to be within a 100 μG-window. A clear oscillation is recovered that only dephases after the first cycle. The line is the expected oscillation. (c) Population at a constant time of 400 μs vs. the measured field tag. The solid line is a Rabi resonance fit with the pulse time and Rabi frequency fixed to expectation. (d) Scaled population vs. scaled time. Gray points are original data scaled by average detuning. The shaded line is a simulation, with B0 known to within 55 μG and Ω known to within 1 dB (see text).

FIG. 3.

Slow Rabi cycling between |a=|1,1 and |b=|2,2, with magnetic-field reconstruction based on a 5-pulse sequence. (a) Observed time dependence of the transferred population after eliminating AC-line fluctuations as demonstrated in Fig. 2. The large shot-to-shot scatter is due to residual field fluctuations. (b) Data points post-selected to be within a 100 μG-window. A clear oscillation is recovered that only dephases after the first cycle. The line is the expected oscillation. (c) Population at a constant time of 400 μs vs. the measured field tag. The solid line is a Rabi resonance fit with the pulse time and Rabi frequency fixed to expectation. (d) Scaled population vs. scaled time. Gray points are original data scaled by average detuning. The shaded line is a simulation, with B0 known to within 55 μG and Ω known to within 1 dB (see text).

Close modal

Section III B, will give the details of a simulation of the exact behavior of the field reconstruction. For the given example and for the parameters of the five-pulse sequence used, we expect the reconstructed fields to scatter around the true magnetic field value with a 55 μG standard-deviation. The simulation and data agree very well, with a slight deviation at late times, potentially due to imperfect cancellation of the AC-line or higher-frequency noise that is uncorrelated with the AC-line.

In our measurements, the high degree of correlation between the transferred population and the detected magnetic field further confirms that the residual fluctuations occur on a scale that is long compared to the duration of the Rabi cycle preceding the field measurement (cf. Fig. 3). We note that on long time scales, the observed magnetic field drifts are typically on the order of one to several milligauss, over the course of one hour.

For the Rabi-oscillation measurements described in Sec. III A, the parameters of the magnetometry pulse sequence i = (1, 2, 4, 5, 6) were Ωi/2π = (2.3, 1.6, 2.6, 2.0, 2.7) kHz, τi = (150, 150, 150, 200, 120) μs, and δi/2π = (1.8, 2.8, 2.0, 2.1, 3.4) kHz, which yielded an inferred accuracy of 55 μG. To estimate the ultimate resolution and limits of our magnetometer for optimal parameters (see Sec. II B), we perform a Monte-Carlo simulation, using a six-transition sequence. We start with a set of fixed (true) fields Btr drawn from a Gaussian distribution around B0 that are supposed to be reconstructed. The number of atoms transferred in the ith pulse at fixed pi is drawn from a binomial distribution, while the transfer probabilities pi themselves are subject to uniformly distributed fluctuations of τi (±2 μs), Ωi (±1 dB), δi (±2π × 7 Hz) and the instantaneous magnetic field during each individual pulse due to uncanceled residual fluctuations (±100 μG). The Rabi frequencies are Ωi/2π = (0.9, 1.9, 3.1, 1.2, 1.9, 3.1) kHz, and the optimized detunings and pulse areas are δi = 0.82Ωi and τiΩi = 0.94π as mentioned earlier.

Results of the simulation are shown in Fig. 4. For the optimum pulse parameters, the reconstruction of Btr is accurate to within a standard deviation of 25 μG. A reconstruction is consistently possible within a ±500 μG window around B0, if outliers with large fit uncertainties are removed. For larger distances from B0, the default detunings δi can be readjusted, or alternatively larger Rabi couplings can be used, at the (inversely proportional) expense of the accuracy of the field reconstruction. The results of the simulation confirm that most of the apparent remaining fluctuations in Fig. 2 are actual fluctuations of the ambient magnetic field, at least to within the reconstruction uncertainty (±100 μG for the pulse parameters chosen in that experiment).

FIG. 4.

Simulated field reconstruction (over 104 runs). (a) Reconstruction error vs. fit uncertainty Be, with convergence in the shaded area Be < 80 μG. (b) Reconstruction error vs. distance of Brc from B0, after discarding fits with Be > 80 μG. Proper convergence is obtained in a ±500 μG window. (c) Histogram of reconstruction errors for the data in the gray shaded areas (a) and (b). The solid curve is a Gaussian with a σ of 25 μG.

FIG. 4.

Simulated field reconstruction (over 104 runs). (a) Reconstruction error vs. fit uncertainty Be, with convergence in the shaded area Be < 80 μG. (b) Reconstruction error vs. distance of Brc from B0, after discarding fits with Be > 80 μG. Proper convergence is obtained in a ±500 μG window. (c) Histogram of reconstruction errors for the data in the gray shaded areas (a) and (b). The solid curve is a Gaussian with a σ of 25 μG.

Close modal

A number of experimental applications involve the use of homonuclear mixtures of alkali atoms in state-selective optical lattice potentials,3–9,22 which rely on the existence of a differential Zeeman shift between the states involved. In certain cases, a highly stable separation between a deeply lattice-bound state and a less deeply bound or free state may be desired, such as when the states are subject to coherent coupling,23–25 requiring precise control of both the lattice depth and the magnetic field.

Figure 5(a) shows an experimental configuration in which we prepared an “untrapped” ensemble of atoms in state |b=|2,0, coupled to a state |r=|1,1 that is confined to the sites of a deep, blue-detuned lattice potential with a zero-point energy shift ho/2 = h × 20(1) kHz, generated with circularly polarized light from a titanium-sapphire laser.

FIG. 5.

Microwave spectroscopy of a free-to-bound transition in a state-selective optical lattice potential [wavelength 790.10(2) nm, σ polarization]. (a) Population is transferred from the untrapped state |b=|2,0 to the confined state |r=|1,1. The gray lines indicate the magnetometry sequence following the transfer. (b) Bound-state population after a 400 μs long pulse with Ω = 2π × 450(1) Hz and variable detuning and after accounting for magnetic-field fluctuations. The sequence of spectra was taken at regular intervals over the course of 1 h.

FIG. 5.

Microwave spectroscopy of a free-to-bound transition in a state-selective optical lattice potential [wavelength 790.10(2) nm, σ polarization]. (a) Population is transferred from the untrapped state |b=|2,0 to the confined state |r=|1,1. The gray lines indicate the magnetometry sequence following the transfer. (b) Bound-state population after a 400 μs long pulse with Ω = 2π × 450(1) Hz and variable detuning and after accounting for magnetic-field fluctuations. The sequence of spectra was taken at regular intervals over the course of 1 h.

Close modal

To stabilize the magnetic field, we utilize post-selection down to the 100 Hz level based on the magnetic-field tagging described earlier, using parameters similar to those in Fig. 4. The optical intensity I is stabilized to ∼1% using a photodiode and a proportional integral differential (PID) regulation circuit, yielding a transition frequency that should be stable to within about 100 Hz (since ωhoI). However, this does not eliminate the possibility of slow drifts of the lattice depth (such as due to temperature-induced birefringence or small wavelength changes of the laser) in the course of an experiment, as can be seen in Fig. 5(b). To address these issues, the precise resonance condition can now be monitored throughout data taking, using our method. The range of the drift is several hundreds of Hz. We emphasize that the spectroscopic precision necessary for this kind of experiment would not be attainable without canceling AC-line induced magnetic field noise and compensating the shot-to-shot fluctuations using the magnetic field tag.

In conclusion, we have demonstrated a simple method for in situ monitoring of magnetic fields in quantum gas experiments with alkali atoms, with a demonstrated accuracy of 55 μG, an inferred accuracy of 25 μG for optimized parameters, and a time resolution of 1 ms. As already seen for the examples earlier, the magnetometry pulse sequence can be tagged onto experiments that potentially involve several hyperfine states. In principle, the number of transitions used for magnetometry can be reduced down to two, as long as they move differently for a change of the magnetic field (this can be achieved by having two detunings of opposite sign or differential magnetic moments of opposite sign). For example, the method could work using only transitions 1 and 4 of Fig. 1(b). Using a smaller number of transitions generally degrades the accuracy (here by a factor 3 when all pulse parameters are left constant, compared to using six transitions), but it increases the measurement bandwidth (here by a factor of 3), which could be an important independent consideration for certain applications.

Thus far, we have only described the use of this method as a scalar magnetometer (in order to be able to ignore fluctuations in perpendicular directions). It should also be possible to access fluctuations of the ambient field in more than one spatial direction, if the bias field is rotated during the magnetometry pulse sequence (with two transitions used per direction). This can become important if one wants to use this method for stable field post-selection at low fields.

Finally, for comparison to other magnetometry techniques, a sensitivity may be specified as26η=ΔBminT, i.e., as the minimum detectable change in field ΔBmin=2ln2σ 60 μG multiplied by the square root of the cycle time. Since typical field fluctuations in laboratories usually stem from AC-mains or are very low frequency (such as fluctuations of Earth’s magnetic field), synchronizing the experiment to the AC-line can yield one measurement in the effective integration time of 1 ms. In this case, an effective sensitivity η300pT/Hz (in a measurement volume of 10 μm3) can be reached.

We thank M. G. Cohen for discussions and a critical reading of the manuscript. This work was supported by NSF PHY-1205894 and PHY-1607633. M.S. was supported from a GAANN fellowship by the DoEd. A.P. acknowledges partial support from ESPOL-SENESCYT.

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