Continuously observing a dynamically decoupled spin-1 quantum gas

We continuously observe dynamical decoupling in a spin-1 quantum gas using a weak optical measurement of spin precession. Continuous dynamical decoupling modifies the character and energy spectrum of spin states to render them insensitive to parasitic fluctuations. Continuous observation measures this new spectrum in a single preparation of the quantum gas. The measured time series contains seven tones, which spectrogram analysis parses as splittings, coherences, and coupling strengths between the decoupled states in real time. With this we locate a regime where a transition between two states is decoupled from magnetic-field instabilities up to fourth order, complementary to a parallel work at higher fields [D. Trypogeorgos et al., preceding paper, Phys. Rev. A 97, 013407 (2018)]. The decoupled microscale quantum gas offers magnetic sensitivity in a tunable band, persistent over many milliseconds: the length scales, frequencies, and durations relevant to many applications, including sensing biomagnetic phenomena such as neural spike trains.

Figure 1

Energy spectrum and splittings of a radio-frequency coupled spin-1 for several $q_R=q/\mathrm{\Omega }$ between 0 and 1. The boldest curves have $q_R=q_{R,\text{magic}}$. Shown on the left are energies ${\omega }_n$ of dressed states $|n\rangle =|1\rangle$ (red), $|2\rangle$ (blue), and $|3\rangle$ (green) normalized to the Rabi frequency. Dashed lines indicate the energies of uncoupled states ($\mathrm{\Omega }=0$) in a frame rotating at ${\omega }_{\text{rf}}$. Shown on the right are splittings ${\omega }_{ij}={\omega }_j-{\omega }_i$ of dressed states $|i\rangle$ and $|j\rangle$ as a function of detuning. When $q_R=q_{R,\text{magic}}$ (bold curves), ${\omega }_1$ and ${\omega }_2$ share the same curvature and their difference ${\omega }_{12}$ is minimally sensitive to detuning and thus magnetic-field variations.

Figure 2

Continuous measurement of the dressed energy spectrum for $q_R=0.402\left(3\right), f_{\text{rf}}=3.521\text{MHz}$, and $B_0=5.013\text{G}$: spectrogram (left) and power spectral density (PSD) (normalized) (right) of the 90-ms-long signal. The inset shows the dressed-state energy diagram; the mean and difference of transition frequencies ${\omega }_{12}$ and ${\omega }_{23}$ are the dressed Larmor frequency ${\omega }_D$ and quadratic shift $q_D$, respectively. The sidebands about the carrier at $f_{\text{rf}}$ are associated with the ${\omega }_{13}$ (gold), ${\omega }_{23}$ (turquoise), and ${\omega }_{12}$ (lavender) transitions. Magnetic-field fluctuations are manifest as asymmetric frequency modulation of the ${\omega }_{13}$ and ${\omega }_{23}$ sidebands, while the ${\omega }_{12}$ transition remains relatively unaffected. The corresponding peaks in the PSD have linewidths 102, 97, and 24 Hz (near transform limited), respectively. The ${\omega }_{23}$ peaks are significantly skewed (the mean magnitude of the Pearson skew coefficient is 0.88), while the ${\omega }_{12}$ peaks are unskewed (Pearson skew coefficient 0.08).

Figure 3

Real-time observation of continuous dynamical decoupling for $q_R=0.402\left(3\right)$. (a) and (b) Spectrograms of a continuous weak measurement of $\langle {\widehat{F}}_x\rangle$. (a) Magnetometry of the bare Zeeman states is used to calibrate $B_z\left(t\right)=B_0+\delta B_z\left(t\right)$ during the measurement interval, in which the detuning varies over a range $\sim 2\mathrm{\Omega }$. We numerically track the Zeeman splittings (gold and orange lines) to determine the instantaneous Larmor frequency ${\omega }_L\left(t\right)$ and quadratic shift $q\left(t\right)$. (b) The field is swept over the same range but the rf dressing is applied $[\mathrm{\Omega }/2\pi =4.520(2\left)\text{kHz}\right]$. Three sidebands above (shown) and below the carrier at $f_{\text{rf}}=3.521\text{MHz}$ (dashed orange line) reveal the dressed-state splittings $f_{ij}={\omega }_{ij}/2\pi$. (c) Parametric plot of $f_{12}\left(t\right)$ and $f_{23}\left(t\right)$ versus $\delta B_z\left(t\right)$ by combining the analysis of (a) and (b). Solid curves in (b) and (c) are from an eigenspectrum calculation, provided only $f_{\text{rf}}, B_z\left(t\right)$, and $\mathrm{\Omega }$. The inset in (c) shows the variation of the hyperdecoupled transition $f_{12}$ for $0\le \delta B_z\le \mathrm{\Omega }/2\gamma =3.2\text{mG}$.

Figure 4

Curvature of the hyperdecoupled $|1\rangle \leftrightarrow |2\rangle$ transition for normalized quadratic shifts $q_R$ from 0.2 to 0.5. Measured curvature (black points) is determined from polynomial fits to the $(\delta B_z,f_{12})$ data, e.g., Fig. 3. Vertical and horizontal error bars correspond to the standard error of the regression and uncertainty in $q_R$, respectively. A linear fit (black dashed line) with a $1\sigma$ confidence band (gray shaded region) is shown; the intercept gives $q_{R,\text{magic}}\text{(expt)}=0.350\left(6\right)$. The analytic expression for the curvature (red) (see footnote 1) is consistent. The left axis is the curvature ${\partial }^2{f}_{12}/\partial B_z^2$ in $\mathrm{kHz}/{\mathrm{G}}^2$. The right axis is $\mathrm{\Omega }{\partial }^2{\omega }_{12}/\partial {\mathrm{\Delta }}^2$, i.e., normalized to the curvature of quadratic decoupling ($q_R=0$).