Quantum sensing of the phase-space-displacement parameters using a single trapped ion

We introduce a quantum sensing protocol for detecting the parameters characterizing the phase-space displacement by using a single trapped ion as a quantum probe. We show that, thanks to the laser-induced coupling between the ion's internal states and the motion mode, the estimation of the two conjugated parameters describing the displacement can be efficiently performed by a set of measurements of the atomic state populations. Furthermore, we introduce a three-parameter protocol capable of detecting the magnitude, the transverse direction, and the phase of the displacement. We characterize the uncertainty of the two- and three-parameter problems in terms of the Fisher information and show that state projective measurement saturates the fundamental quantum Cramér-Rao bound.

Figure 1

(a) Probe system sensitive to the magnitude $F$ and the phase $\xi$ of the external applied force is represented by a three-level system. We assume that the atomic transitions $\left|0\right.〉\leftrightarrow \left|-1\right.〉$ and $\left|0\right.〉\leftrightarrow \left|1\right.〉$ are driven respectively by red- and blue-sideband laser fields. (b) For a time-varying force with oscillation frequency far from the resonance with respect to the harmonic trap frequency the information of the two parameters is encoded into the magnitude and the phase of the Rabi frequencies of the respective atomic transitions.

Figure 2

Probabilities $p_m\left(t\right)$ versus time. We compare the numerical solution of the time-dependent Schrödinger equation (circles) with the Hamiltonian (11) to the analytical solution for the state vector (16) (solid lines). We set the laser detuning $\mathrm{\Delta }=-g^2/\omega$, which compensates the ac-Stark shift. The parameters are set to $g=4$ kHz, $\omega =150$ kHz, $F=-35$ yN, and $\xi =1.7\pi$.

Figure 3

(a) Probe system capable of detecting the three parameters of the displacement operator, namely, the force components along the two orthogonal directions $F_x$ and $F_y$ and the phase $\xi$. The quantum system consists of one ground state $\left|0\right.〉$ and four excited states $\left|\pm 1\right.〉$ and $\left|\pm 2\right.〉$. The red- and blue-detuned laser fields couple the internal states with the motional states along $x$ and $y$ directions. (b) Adiabatic elimination of the phonon states leads to a closed set of states. The information of the magnitude and the transverse direction of the force and its phase is encoded onto the magnitude of the Rabi frequency $\mathrm{\Omega }$ and the phases ${\phi }_{\pm }=\xi \pm \varphi$.

Figure 4

Probabilities $p_m\left(t\right)$ versus time for the four-pod system. We compare the numerical solution of the time-dependent Schrödinger equation (red circles) with the Hamiltonian (24d) after applying two $\pi /2$ pulses with the analytical solution for the state vector (28) (solid lines). We set the laser detuning $\mathrm{\Delta }=-2g^2/\omega$, which compensates the ac-Stark shift. The parameters are set to $g=4$ kHz, $\omega =150$ kHz, $F_x=-35$ yN, $F_y=-30$ yN, and $\xi =0.51\pi$.