Photon-recoil spectroscopy: Systematic shifts and nonclassical enhancements

In photon-recoil spectroscopy, signals are extracted from recoils imparted by the spectroscopy light on the motion of trapped ions as demonstrated by Hempel et al. [C. Hempel et al., Nat. Photon. 7, 630 (2013)] and Wan et al. [Y. Wan et al., Nat. Commun. 5, 3096 (2014)]. The method exploits the exquisite efficiency in the detection of phonons achievable in ion crystals and is thus particularly suitable for species with broad noncycling transitions where detection of fluorescence photons is impractical. Here we develop a theoretical model for the description of photon-recoil spectroscopy based on a Fokker-Planck equation for the Wigner function of the phonon mode. Our model correctly explains systematic shifts due to Doppler heating and cooling as observed in the experiment. Furthermore, we investigate quantum metrological schemes for enhancing the spectroscopic sensitivity based on the preparation and detection of nonclassical states of the phonon mode.

Figure 1

(a) Schematic overview of PRS consisting of initial-state preparation on the logic ion, pulsed laser-ion interaction during spectroscopy, and measurement $\mathcal{M}$ (columns from left to right, respectively). (b) Spectroscopy with the ground state of motion. The spectroscopy dynamics gives a coherent displacement and incoherent diffusion of the Wigner function in phase space. The measured signal is the overlap $\langle 0\left|{\rho }_{t_s}\right|0\rangle$. (c) Enhancement of the recoil sensitivity for individual photons. The schematic is for a momentum squeezed state where the overlap with the initial state changes faster under the spectroscopy dynamics.

Figure 2

(a) Schematic representation of two-point sampling. (b) Systematic Doppler-induced shift. The black solid line depicts the symmetric resonance profile $P_{\mathrm{sym}}$ obtained for $g=0$ and the red dashed line depicts the true overlap $P$ which is shifted by $\delta P$ and therefore asymmetric around $\mathrm{\Delta }=0$. The frequency shift $\delta \omega \left(P\right)$ is illustrated for two working points. The inset shows the linearization of $P$ used to determine the shift on one side of the resonance.

Figure 3

Wigner function overlap $P$ for a momentum squeezed state (blue dashed line) and the ground state (black solid line) against displacement $\alpha \left({\mathrm{\Delta }}_0\right)t_s$. The signal-to-noise ratio depends on the quantum projection noise $\sqrt{P(1-P)}$ and the slope of $P$, which changes with the initial state ${\rho }_0$. The interaction times ${\tau }_{sq}$ and ${\tau }_{vac}$ are defined to result in $P_0=1/2$ for both states.

Figure 4

(a) Recoil sensitivity under diffusive noise characterized by the diffusion-to-drift ratio $\epsilon =D/\alpha$. The comparison shows the ground state $|0\rangle$ (thick black line) along with a momentum squeezed state $|r,\pi /2\rangle$ (blue dashed line) and cat state proportional to $|\beta \rangle +|-\beta \rangle$ (red solid line), both constrained to $\overline{n}=4$. Additionally, optimized linear combinations of the first two even Fock states are considered (green dash-dotted line). (b) Energy scaling of the sensitivity improvement over the ground state for the cat state, squeezed state, and Fock states for fixed $\epsilon =D/\alpha =0.1$ as indicated in (a). The inset shows the approximate linear scaling at low $\overline{n}$.

Figure 5

(a) Recoil sensitivity for momentum squeezed states with $\overline{n}=4,6,8$ (blue dashed, green dash-dotted, and red solid lines, respectively) over the range of $\epsilon$. The ground state (black thick line) is added as a reference. The red dotted line corresponds to the QFI sensitivity limit according to Eq. (41) for the state with $15.3-\mathrm{dB}$ squeezing. (b) Sensitivity as a function of mean occupation $\overline{n}$ for a realistic ratio $\epsilon =0.1$ of the diffusion coefficient to the drift coefficient, as marked in (a). The black dotted line is again the QFI limit for squeezed states with the respective $\overline{n}$.

Figure 6

Reduction of FI for unitary displacement with inefficient state detection. (a) Ground state $\sqrt{{\tilde{F}}_{vac}}$ as a function of probability $\tilde{P}$ for various detection efficiency $\eta$ (the curves are for decreasing values of $\eta$ from top to bottom). The dotted line at the top is $\sqrt{2}$, the QFI for the ground state. (b) Fisher information $\sqrt{{\tilde{F}}_{\mathrm{cat}}}$ on the first oscillation of ${\tilde{P}}_{\mathrm{cat}}$.

Figure 7

Recoil sensitivity for squeezed states with a phase mismatch $\mathrm{\Delta }\varphi$ (in rad), compared to (a) the ratio $\epsilon =D/\alpha$ and (b) the squeezing strength $r$ in dB. The displayed dotted lines are identical sections within both diagrams: In (a) this corresponds to a squeezing of $12.5\mathrm{dB}$ used for all ratios and in (b) to the ratio $\epsilon =0.1$ used for all values of $r$.