Adiabatic Ramsey Interferometry for Measuring Weak Nonlinearities with Super-Heisenberg Precision

Abstract

We propose an adiabatic Ramsey interferometry technique for detecting weak nonlinearities with trapped ions. The method relies on using the quantum Rabi model as a probe, which is sensitive to nonlinear symmetry-breaking perturbations. We show that the couplings which arise either from anharmonic terms of the trapping potential or due to higher order terms in the Coulomb interaction expansion can be efficiently estimated by measuring the spin state probabilities alone. We show that the spin signal is amplified by the mean-phonon excitations, which results in the estimation precision reaching the super-Heisenberg limit. Notably, achieving such high-precision estimation does not require specific entangled state preparation and can be reached even for initial thermal motion state. Furthermore, we show that the super-Heisenberg scaling can be observed even in the presence of weak spin-dephasing.

Figures (5)

• Figure 1: Nonlinear adiabatic Ramsey interferometer consists of initial state preparation, adiabatic evolution and spin detection. (a) Without symmetry-breaking term the adiabatic evolution creates a Schrödinger cat state between the spin and the motion mode with equal probability. The Bloch vector, initially pointing along the $x$-axis, points in the same direction at the end of the adiabatic evolution, which leads to $\langle\sigma_{z}\rangle=0$. (b) The presence of nonlinear symmetry breaking term leads to non-equally probable Schrödinger cat state at the end of the adiabatic transition. Consequently, the Bloch vector is rotated and $\langle\sigma_{z}\rangle\neq 0$.
• Figure 2: (a) The first four eigenenergies $E_n$ of $\hat{H}(t)$ as a function of time $t$ for $\xi/2\pi$ = 1.0 kHz. (b) The gap between the energies of the degenerate ground state and the first degenerate excited state of $\hat{H}(t)$ at $t_{f}$ for different values of the motion squeezing strength $\xi$. The dashed line shows $\tilde{\omega}$. The parameters are set to $\omega/ 2 \pi= 5$ kHz, $\Omega/ 2 \pi= 100$ kHz, $g/ 2 \pi = 4$ kHz, $\tau = 3.5$ ms. (c) Adiabatic parameter $\varepsilon$ as a function of time for different values of $\xi$. Here $\Omega/ 2 \pi= 150$ kHz and $f_{3}/2\pi= 0.5$ Hz. (d) Adiabatic parameter $\varepsilon$ as a function of time for different values of the spin-phonon coupling $g$ and for $\xi=0$.
• Figure 3: (a) Parameter estimation precision for $f_3/2\pi$ = 0.5 Hz as a function of the average number of phonon excitations $\bar{n}(\xi)$ for $\tau=3.5$ ms. The dashed line shows the analytical formula (\ref{['errorH3']}). (b) Parameter estimation precision for $f_3$ with respect to $\bar{n}(g)$ for $\tau=3.0$ ms, $\xi= 0$ and $\Omega/2\pi= 150$ kHz. (c) Parameter estimation precision for $f_5/2\pi$ = 0.5 Hz as a function of the average number of phonon excitations $\bar{n}(\xi)$ for $\tau=3.5$ ms. The dashed line shows the analytical formula (\ref{['errorH5']}) (d) Parameter estimation precision for $f_5$ with respect to $\bar{n}(g)$ for $\tau=3.0$ ms, $\xi= 0$ and $\Omega/2\pi= 150$ kHz.
• Figure 4: (a) Parameter estimation precision for $f_{ab}/2\pi$ = 0.5 Hz as a function of the average number of phonon excitations $\bar{n}_b$ for $\tau=3.5$ ms and $\Omega/2\pi$ = 100 kHz. The dashed line shows the analytical formula (\ref{['errfab']}). (b) Parameter estimation precision for $f_{abc}/2\pi$ = 0.5 Hz with respect to $\bar{n}_b\equiv\bar{n}_c$. The dashed line shows the analytical formula (\ref{['errfabc']})
• Figure 5: (a) The spin observable $\langle\sigma_z\rangle$ as a function of $\bar{n}_{\rm th}$ at $t=t_f$. The numerical result is compared with the analytical formula (\ref{['signal_th']}). (b) Parameter estimation precision for initial thermal state for $f_{\rm th}/2\pi$ = 5 Hz as a function of the spin-phonon coupling $g$ for different $\bar{n}_{\rm th}$. The parameters are $\tau=3.0$ ms, $\Omega/2\pi$ = 150 kHz and $\xi/2\pi$ = 0. (inset) The ratio $\frac{\delta f_{\rm th}(\bar{n}_{\rm th} \neq0)}{\delta f_{\rm th}(\bar{n}_{\rm th} =0)}$ as a function of the spin-phonon coupling $g$. (c) Parameter estimation precision in the presence of dephasing for $f_{\rm deph}/2\pi$ = 1 Hz as a function of the spin-phonon coupling $g$ for different dephasing rates $\Gamma$. (inset) The ratio $\frac{\delta f_{\rm deph}(\Gamma \neq0)}{\delta f_{\rm deph}(\Gamma =0)}$ as a function of $g$. (d) Parameter estimation precision outside the Lamb-Dicke regime for $f_{\rm LD}/2\pi$ = 1 Hz as a function of the spin-boson coupling $g$ for different value of the Lamb-Dicke parameter $\eta$. (inset) The ratio $\frac{\delta f_{\rm LD}(\eta \neq0)}{\delta f_{\rm LDR}}$ as a function of the spin-phonon coupling $g$.