Linear and nonlinear coherent coupling in a Bell-Bloom magnetometer

Spin-exchange collisions in hot vapors are generally regarded as a decoherence mechanism. In contrast, we show that linear and nonlinear spin-exchange coupling can lead to the generation of atomic coherence in a Bell-Bloom magnetometer. In particular, we theoretically and experimentally demonstrate that nonlinear spin-exchange coupling, acting in an analogous way to a wave-mixing mechanism, can create additional modes of coherent excitation which inherit the magnetic properties of the natural Larmor coherence. The generated coherences further couple via linear spin-exchange interaction, leading to an increase of the natural coherence lifetime of the system. Notably, the measurements are performed in a low-density caesium vapor and for nonzero magnetic field, outside the standard conditions for collisional coherence transfer. The strategies discussed are important for the development of spin-exchange coupling into a resource for an improved measurement platform based on room-temperature alkali-metal vapors.

Figure 1

Schematics of the experiment: a circularly polarized pump with periodically modulated amplitude creates a collective atomic spin (green arrow) orthogonal to the applied dc magnetic field (thin black arrow). The atomic spin precession is then probed by detecting the polarization rotation undergone by a linearly polarized off-resonant probe, propagating through the atomic sample orthogonally with respect to the pump beam and the applied magnetic field. The relevant transitions for pumping and probing on the ${}^{133}\mathrm{Cs}$ $D_2$ line are shown.

Figure 2

FFT spectrum of the magneto-optical rotation signal detected for atoms in (a) $F=3$ and (b) $F=4$ for different frequency of modulation of the pump beam power. This portion of the spectrum shows the evolution around the Larmor frequency ${\omega }_L/2\pi \sim 270$ Hz. Together with the peak at ${\omega }_L$ and the first three harmonics of ${\omega }_M$, additional peaks (and dips) depending on ${\omega }_M$ are visible around integer multiples of ${\omega }_M/2$ (highlighted by black arrows in the figure). The pump and probe beam powers have been set to $80\mu \mathrm{W}$ and $600\mu \mathrm{W}$ and for each modulation frequency the spectrum is the result of over 100 different acquisitions. Insets: calculated FFT spectrum of the average macroscopic spin component along the direction of the probe beam ($\widehat{y}$ direction) for atoms in $F=3$ and $F=4$. The pump power has been set to ${\mathrm{\Omega }}_R=100{\omega }_M$, which corresponds to $20\mu \mathrm{W}$ power for our experimental settings. Due to computational issues we cannot simulate higher pump powers, for which we have compensated by using a stretched state to initiate the $F=3$ ground state, mimicking saturation by a strong pump.

Figure 3

(a) Positions of the FFT secondary coherence profile crossing ${\omega }_L$ (see text) as a function of the pump modulation frequency (black dots, scale on the left) and position of the natural coherence profile (blue triangles, scale on the right) for atoms in $F=4$. The red solid line is a linear fit to the secondary coherence data providing the functional dependence ${\omega }_{SC}=(3.06\pm 0.02){\omega }_M-(272\pm 2)\text{Hz}\sim (3{\omega }_M-{\omega }_L)$. The inset shows the fitted amplitude of the secondary coherence profile as a function of the modulation frequency. (b) Linewidth of the natural coherence for atoms in $F=4$ (blue triangles), secondary coherence for atoms in $F=4$ (black squares), and Larmor coherence for atoms in $F=3$ (red dots) as a function of the modulation frequency. The green solid line shows a quadratic fit to the secondary coherence data with function $\mathrm{\Gamma }=(0.16\pm 0.01){(3{\omega }_M-2{\omega }_L)}^2+(5.1\pm 0.4)\mathrm{Hz}\sim {({\omega }_{SC}-{\omega }_L)}^2+{\mathrm{\Gamma }}_0$, where we have used the linear fit above for substituting ${\omega }_M$. The inset of the figure is a zoom into the natural coherence linewidth for $F=4$, showing a reduction of the peak size at resonance ${\omega }_M={\omega }_0=180$ Hz. (c) Linewidth as a function of the pump beam power for the natural coherence of $F=4$ atoms (red dots), the secondary coherence for $F=4$ atoms (purple squares), and the natural coherence for $F=3$ atoms (green triangles). For the secondary coherence profile in $F=4$, measurements are taken on one side of the resonance with $160<{\omega }_M<170$ Hz. The green solid line is a linear fit to the $F=3$ data. Each point in the above figures is obtained from the average of 100 acquisitions.