Coherence of Rabi oscillations with spin exchange

Rabi measurements in atomic vapor cells are of current interest in a range of microwave imaging and sensing experiments, but are increasingly in a parameter space outside of theoretical studies of coherence defined by spin-exchange collisions. Here, we study the coherence of Rabi oscillations in vapor cells by employing continuous nondestructive readout of the hyperfine manifold of ${}^{87}\mathrm{Rb}$ using Faraday rotation. We develop a full model for spin-exchange coherence for hyperfine transitions that takes into account a nonstatic population distribution. In this regime, Rabi oscillations exhibit nontrivial time-domain signals that allow verification of vapor-cell parameters. We find excellent agreement between theory and experiment, which will aid in benchmarking sensitivities of Rabi measurement applications.

Figure 1

(a) Energy-level diagram for ${}^{87}\mathrm{Rb}$ showing the relevant microwave transitions and Rabi oscillations driving atomic populations (${\rho }_{ii}$) measured in our apparatus (green insets). A microwave sweep performs adiabatic rapid passage (ARP) to switch the ${\sigma }^{+}$ populations prior to driving the ${\sigma }^{-}$ and $\pi$ transitions (magenta). (b) The envelopes of simulated atomic population dynamics due purely to SE collisions during Larmor precession (left) which decays slower than during a ${\sigma }^{+}$ Rabi oscillation (right). Comparison of the envelopes for each state population to the full-simulated Faraday rotation signal (insets) shows nuanced population dynamics that are easier to individually observe with Rabi oscillations that couple a pair of states. The initial populations and SE rate here are set by a ST distribution with polarization $p=0.7$ and vapor temperature ${\mathcal{T}}_v=110{}^{\circ }\mathrm{C}$.

Figure 2

(a) Apparatus for driving Rabi oscillations. We drive and measure Rabi oscillations between hyperfine states $|1,1\rangle$ to $|2,0\rangle , |2,1\rangle$, and $|2,2\rangle$ at multiple detunings in real time using Faraday rotation readout of the macroscopic atomic spin state. (b) Energy-level diagram for adiabatic rapid passage (ARP). (c) Using ARP to maximize the $|1,1\rangle$ population and enhance the ${\mathrm{\Omega }}_{\pi }$ signal. Overlaying with ${\mathrm{\Omega }}_{+}$ (light green) denotes ${\theta }_F$ values for $|2,2\rangle$ and $|1,1\rangle$ occupation, where we observe ARP transfer (purple) after the microwave frequency sweep. Extracting the pure $\pi$ oscillation (dark green) requires filtering off-resonance driving of adjacent transitions.

Figure 3

(a) A mirror comparison of the simulated (black dashed) and measured (green) ${\sigma }^{+}, \pi$, and ${\sigma }^{-}$ Rabi oscillations. Here, ${\overline{\mathrm{\Delta }}}_{\pm ,\pi }={\mathrm{\Delta }}_{\pm ,\pi }/{\mathrm{\Omega }}_{\pm ,\pi }$ is the normalized microwave detuning. (b) A plot of the ${\sigma }^{+}$ Rabi oscillation alongside other relevant decay rates. Importantly, the weak-driving approximation predicts much higher coherence compared to our measurements. The inset shows the RMSE between the simulated and measured Rabi oscillations for different vapor temperatures ${\mathcal{T}}_v$ and buffer gas pressures $P_{{\text{N}}_2}$ at a fixed spin polarization $p=0.79$. By finding these optimal values where the theory and measured Rabi oscillations highly overlap, our model estimates these parameters for our cell. (c) Intracell thermometry with SE coherence measurements. The intracell temperature ${\mathcal{T}}_v$ is inferred from measured dephasing times $T_2^{m,+}$ using the calibrated model. The inset displays $T_2^{m,+}$ fluctuations over a 0.5 s period.