1. Introduction

Nonlinear magneto-optic rotation (NMOR) has been extensively studied, particularly as a tool for sensitive magnetometry [1, 2]. In standard NMOR, a linearly polarized optical beam simultaneously pumps (i.e. polarizes) and probes the atoms, and as the strength of the magnetic field (collinear with the optical beam) is swept through zero, a dispersive-shaped resonance is observed in the rotation of the light’s polarization. For small fields, the degree of polarization rotation is proportional to the magnetic field. These resonances can be much narrower (≤ 1 Hz) than typical optical atomic resonances as they rely on the spin-coherence lifetime of the atoms’ ground state. If there are small fields transverse to the optical beam, additional features can arise within the dispersive-shaped polarization rotation signal, as shown in Fig. 1. This “twist” feature within the NMOR curve, associated with transverse magnetic fields, is the focus of this work, where we have used cold atoms to isolate various effects and characterize its dependencies.

 

Fig. 1 Typical NMOR data showing a “twist” as the axial magnetic field is swept through zero. This data was taken with a 10.6 μW probe beam of 1 mm diameter that is detuned 27 MHz from the zero-field atomic resonance. A static transverse field of 55 mG is applied parallel to the polarization vector.

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A similar nested coherence was observed in warm vapor [3], where it was termed a “twist.” It appears when a transverse magnetic field is present with a magnitude comparable to the width of the NMOR feature. It was ascribed to couplings among multiple hyperfine levels, which are unresolved due to Doppler broadening in the warm vapor, causing multiple atomic polarization subsystems. These subsystems derive from independent velocity classes of atoms that were optically pumped to different magnetic sublevels depending on the excited hyperfine state that Doppler-shifted into resonance with the light. These polarization subsystems can be orthogonal to each other resulting in optical polarization rotation towards opposite directions depending on which subsystem dominates.

Another similar twist feature in warm vapor was reported in double-resonance magneto-optic experiments [4–6], where a combination of resonant optical and radio-frequency (RF) fields were used. Zigdon et al. thoroughly explained the mechanism behind their twist through the evolution of the atomic alignment tensor (the rank-two tensor components in the polarization moment series expansion [7]). They described how the atomic alignment transforms between two distinct polarization moments in the series expansion, which have perpendicular axes of symmetry, due to the averaging effect of the off-axis magnetic field. These two different axes of symmetry rotate the light’s polarization in opposite directions resulting in the observed twist.

Here, by working with cold atoms in which the excited hyperfine levels are well resolved and individually addressable, we suppress the complication of multiple excited state hyperfine levels. This excludes the twist mechanism described in [3], where competition among excited hyperfine levels was responsible. The mechanism behind our twist is also distinct from that of the double-resonance experiments in that, aside from there being no RF fields present, ours is dominated by atomic orientation (the rank-one tensor components in the polarization moment series expansion). Acting alone, linearly polarized light can only induce alignment because the electric field has a preferred axis but not a preferred direction or orientation; in other words, averaged over a wavelength the light’s electric field points equally in opposite directions along its polarization axis, thus no orientation. But when a magnetic field is present and directed at some non-perpendicular angle to the optical polarization, then alignment-to-orientation conversion (AOC) occurs [8]. The AOC mechanism relies on AC Stark shifts from the probe light, and rotation due to AOC will dominate over that of alignment when the Stark shifts $(~{\mathrm{\Omega }}_R^2/\mathrm{\Delta })$ exceed the optical pumping rate $(~\mathrm{\Gamma }{\mathrm{\Omega }}_R^2/{\mathrm{\Delta }}^2)$. Using a detuned probe with optical intensities near saturation ensures we are in the AOC-dominated regime.

We found that the twist width is linearly proportional to the magnetic field component directed transverse to the laser beam. This suggests an application whereby the transverse and longitudinal components of a magnetic field can be simultaneously and independently measured (see Sec. 4.1). The sensitivity with which the twist can resolve fields that are present scales directly with the atoms’ spin coherence lifetime (just as it does with NMOR), and therefore the present methods can in principle perform at the same level as other vapor-based magnetometers. Furthermore, the twist signal provides a clear indication of the light’s polarization purity (see Sec. 4.2). Theoretical predictions based on the Lindblad density matrix master equation are in agreement with experimental results, and we characterize the impact of various parameters on the twist feature, including polarization angle and optical power. We have found these signals and associated techniques to be useful diagnostics for fields affecting our atomic experiments, and believe they could find use in a variety of other contexts, particularly for example spinor BECs [9], spin-orbit coupled systems [10], and, more broadly, qubit systems that require stringent quantum control [11–13].

2. Theory

Our model closely follows that developed in [7, 14, 15] and is implemented using the Atomic Density Matrix (ADM) Mathematica package developed by S. Rochester and D. Budker [16]. The Hamiltonian, H, is given by

After invoking the rotating wave approximation, we insert this Hamiltonian into the Lindblad master equation,

We are concerned with timescales long compared to the Rabi frequency, so we solve these coupled Lindblad equations for the steady-state density matrix by setting the time-derivative to zero,$\dot{\rho }=0$. Experimentally, we use the F = 2 → F′ = 3 hyperfine transition of the D2 manifold of ⁸⁷Rb, which has a total of 12 Zeeman sublevels, and the 12 × 12 matrix equations are solved numerically. With the steady state density matrix solution in hand, the polarization of the medium (P) can be calculated using$\mathit{P}=n\text{Tr}\left(\rho \overrightarrow{\mathit{d}}\right)$, where n is atomic density. The effects on the light due to passing through the polarized atoms, such as absorption, phase shifts, or polarization rotation, are then calculated from Maxwell’s equations. The numerically calculated results are in good agreement with our experimental data, and allow us to discriminate individual effects of various factors such as polarization impurity or optical power.

3. Experimental methods

The apparatus is set up in a Hanle configuration, where a single linearly-polarized probe beam propagates coaxially with an applied magnetic field through an atomic sample. As a practical matter, we took care to direct the beam coaxially with the relevant coil pair to better than one degree, as this keeps the spatial degrees-of-freedom independent. A schematic of our experimental configuration is shown in Fig. 2. The atomic sample is a cold cloud of roughly 10^{7 87}Rb atoms, collected from background vapor in a 1 x 1 x 3 inch glass vacuum chamber, and further cooled in an optical molasses to approximately 50 μK. We control the magnetic field using three perpendicular sets of Helmholtz coils (only 1 pair is shown in Fig. 2). We calibrated these coils using Faraday spectroscopy of the atoms [17]. The probe beam is detuned 27 MHz to the red of the F = 2 → F′ = 3 transition of the D₂ line with typical intensity of 11 μW/mm². The beam size is chosen such that the size (the 1/e² width) of the beam approximately matches that of the atomic cloud. We control the polarization with a Wollaston prism followed by a quarter-wave plate to compensate for induced ellipticity from the glass vacuum cell’s birefringence.

 

Fig. 2 Experimental configuration. λ/2: half-wave plate, PBS: polarizing beam splitter, PD: photodetectors.

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An NMOR experiment is performed by loading the atomic sample into the MOT for five seconds followed by brief cooling in the optical molasses with the MOT fields off. We then release the optical molasses and shine the probe light through the atomic sample. The polarization rotation of the probe light, induced by interaction with the atoms, is measured with a balanced polarimeter as the coaxial $(\mathrm{i}.\mathrm{e}.\widehat{\mathbf{z}})$) magnetic field is swept through zero. The $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{y}}$ magnetic fields in the perpendicular plane, which we call “transverse”, are held fixed.

While the data can be collected with a continuous sweep of the axial magnetic field, our data are recorded using discrete steps. A continuous sweep complicates the analysis in two small but measurable ways. One is the atom loss during the course of the sweep, resulting in fewer atoms at the end of the sweep than at the beginning. The second is the induction of the coils causing a delay in achieving the magnetic field for a given current. Stepping incrementally through the axial magnetic field values and reloading the MOT prior to each step eliminates these effects. We controlled optical power to better than 1% throughout the experiment. In each discrete step, the light-atom interaction time is 1.5 ms implying a ground state coherence rate of γ = 0.7 kHz.

4. Analysis

We experimentally and theoretically characterize the twist-feature’s dependence on magnetic field magnitude, polarization angle, and optical power. We start by showing the linear relationship between the twist width and the magnitude of the transverse magnetic field, because this is likely the most useful aspect of this work. In the following subsections we show the impact of the light’s polarization and intensity, which conveniently have distinct qualitative effects that are easily discernible. In fact in practice, we use the twist signal to first optimize the light’s polarization and intensity by adjusting these parameters until the signal is symmetric and has maximum contrast.

4.1. Twist dependence on magnitude of transverse magnetic field

The raw data from typical B_Z scans, shown in Fig. 3(a), reveals that the width of the twist changes with the magnitude of B_Tr. The error bars represent the standard deviation statistical uncertainty. The results of our numerical simulation, using no free parameters, is plotted on top of the experimental data. In Fig. 3(b) the black points show the measured twist width (i.e. the valley-to-peak width) as a function of B_Tr, and we see that it scales linearly. This is a satisfying result as it shows the twist feature can be a useful indicator of non-zero transverse field. It is the collinear components of the optical polarization and transverse magnetic field that give rise to the twist, and hence the twist is strongest when these are aligned - as will be seen the subsequent section. The data in Fig. 3(a) is taken under this collinear condition. The linear relationship seen in Fig. 3(b) holds for transverse magnetic field strengths whose Larmor frequencies (Ω_L−Tr) are on the order of the spin coherence lifetime (γ). If the transverse magnetic field is negligible (Ω_L−Tr ≪ γ) then the twist is not visible. If the transverse magnetic field is too strong (Ω_L−Tr ≫ γ), such that the atoms precess many times within the decoherence time then the feature is essentially washed out.

 

Fig. 3 (a) Twist’s dependence on the transverse magnetic field strength, lines are numerical simulation and points are experimental data. (b) Summary of twist widths & amplitudes versus the transverse field magnitude. In both cases the light is polarized along the direction of the transverse magnetic field with a fixed optical intensity of 1.03 mW/cm².

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4.2. Twist dependence on optical polarization

In Fig. 4, we show experimental and simulated data showing the dependence of the twist and NMOR features on the direction of the linear polarization vector relative to the transverse magnetic field direction. When the polarization is aligned with the transverse magnetic field, the twist is centered within the NMOR (both vertically and horizontally) and its contrast is maximized. Although not shown in the Figs., we note that when the polarization is perpendicular to the transverse magnetic field the twist is not observed, rather the NMOR is simply broadened. In between the two extreme cases, i.e. when there is some intermediate angle between the optical polarization and transverse B-field, the twist is pulled vertically off-center relative to the NMOR feature. The twist feature itself remains centered horizontally as the polarization angle is changed; this is distinct from the effect of elliptical polarization.

 

Fig. 4 Impact of the angle between polarization and B_Tr (a) Points are Experimental data, while lines are guides for clarity; (b) Numerical simulation. Both have a fixed transverse field of B_Tr = 48 mG, and a Rabi frequency of 9 MHz.

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For elliptical polarization, the twist becomes asymmetric, as shown in Fig. 5. We parameterize elliptical polarization using the angle of “ellipticity,” defined as the arctangent of the ratio of the minor to major axes of the polarization ellipse. As the degree of ellipticity is adjusted, we see that although the twist remains positioned about zero rotation (i.e. horizontal 0 mrad axis), it is not centered relative to the NMOR feature, and the central zero-crossing no longer occurs at B_Z = 0. These effects are due to the vector component of the AC stark shift, associated with the circular polarization component of the light, which behaves as a fictitious magnetic field along the axial direction [18]. This breaks the symmetry of the real magnetic fields. As mentioned in the experimental methods, to get pure linear polarization it was necessary to correct for the ellipticity induced by the birefringence of the glass vacuum cell by using a quarter-wave plate. The asymmetrically-shaped nature of the signal provided a clear qualitative indication of linear polarization, which made for simple optimization.

 

Fig. 5 Impact of elliptical polarization. (a) Points are Experimental data, while lines are guides for clarity; (b) Numerical simulation. Both have B_Tr = 55 mG, and a Rabi frequency of 9 MHz.

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4.3. Twist dependence on optical intensity

In Fig. 6(a), we show the twist feature’s dependence on optical intensity. At intensities much lower than our experimental parameters where the saturation parameter is small, ${\mathrm{\Omega }}_R^2/\mathrm{\Gamma }\gamma <1$, the twist vanishes. Despite being detuned, intensities this small do not allow for significant AOC and therefore the rank-two polarization moment (alignment) dominates and does not result in the sign reversal behavior associated with the twist. As the intensity is increased, AOC becomes more prominent and orientation begins to dominate in the observed rotation signals, which allows the twist to become visible. As the intensity is increased further, the twist contrast saturates and ultimately diminishes until it is no longer visible. This occurs approximately when Ω > Γ.

 

Fig. 6 Experimental data (dots) of the twist’s dependence on optical power with numerical simulation (lines) overlaid. A fixed transverse field of 70 mG is applied parallel to the light polarization. (a) Raw data of axial field sweep at three different optical intensities. (b) The dependence of twist widths and amplitudes on the Rabi frequency.

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The range of field strengths over which the twist is visible can be parameterized by ratio of the |E| to |B| fields. We use the ratio of the DC equivalent Stark shift, Ω_S, which is second order in the electric field, to the Larmor frequency, Ω_L. When the light is significantly detuned from resonance (∆ ≫ Γ), this stark shift becomes ${\mathrm{\Omega }}_S\simeq {\mathrm{\Omega }}_R^2/\mathrm{\Delta }$. In terms of this ratio, Ω_S/Ω_L, the twist is visible over roughly two orders of magnitude from 0.01 to 1.0, revealing a wide range of applicability for these methods.

5. Conclusion

In conclusion we have observed an NMOR twist feature in cold atoms, and analyzed it in terms various critical parameters. To the best of our knowledge this has previously only been observed in warm Doppler-broadened vapor. Having well resolved hyperfine states simplifies the theoretical modeling, and reveals that the present twist relies on polarization moments distinct from those in previous work. Stated simply, the twist is due to the change in the magnetic field direction relative to the light’s electric field direction as the axial magnetic field is swept through zero. We identified a favorable linear scaling with applied transverse magnetic field that makes this a suitable technique for simple and convenient in-situ diagnostic of background magnetic fields in an existing apparatus. We characterized systematic effects on the twist from probe light polarization angle and ellipticity. In the present case our spin coherence lifetime was limited by the light-atom interaction time (i.e. transit broadening), but with greater spin coherence times the NMOR and twist features get narrower and hence smaller fields could be measured.