Force sensors with precision beyond the standard quantum limit

We propose force sensing protocols using a linear ion chain which can operate beyond the quantum standard limit. We show that oscillating forces that are off resonance with the motional trap frequency can be detected very efficiently by using quantum probes represented by various spin-boson models. We demonstrate that the temporal evolution of a quantum probe described by the Dicke model can be mapped on the nonlinear Ramsey interferometry which allows us to detect far-detuned forces simply by measuring the collective spin populations. Moreover, we show that the measurement uncertainty can reach the Heisenberg limit by using initial spin-correlated states, instead of motional entangled states. An important advantage of the sensing technique is its natural robustness against the thermally induced dephasing, which extends the coherence time of the measurement protocol. Furthermore, we introduce a sensing scheme that utilizes the strong spin-phonon coupling to improve the force estimation. We show that for a quantum probe represented by the quantum Rabi model the force sensitivity can overcome the one achieved by the simple harmonic oscillator force sensor.

Figure 1

Time evolution of the expectation value of ${\widehat{J}}_z$ operator for a system of $N=8$ spins. We assume an initial state $|\mathrm{\Psi }\left(0\right)\rangle =|j,j\rangle |0_x,0_y\rangle$. We compare the numerical solution of the time-dependent Schrödinger equation with Hamiltonian (1) (solid lines) with the solution using effective Hamiltonian (8) for $\mathrm{\Delta }={\chi }_x{\chi }_y$ (blue circles) and $\mathrm{\Delta }=2{\chi }_x{\chi }_y$ (red triangles). The parameters are set to $g_x=5$ kHz, $g_y=3$ kHz, ${\delta }_x=-85$ kHz, ${\delta }_y=-80$ kHz, $f_{d,x}=10$ yN, and $f_{d,y}=15$ yN.

Figure 2

Time evolution of the expectation value of ${\widehat{J}}_z$ operator for a system of $N=6$ spins. We assume an initial thermal vibrational state with average phonon number $\overline{n}=0.6$. We compare the numerical solution of the time-dependent Schrödinger equation with Hamiltonian (1) (solid lines) with the solution using effective Hamiltonian (10) (red circles). The parameters are set to $g_x=5$ kHz, ${\delta }_x=60$ kHz, $r_{0,x}=14.5$ nm, $f_{d,x}=1.5$ yN.

Figure 3

Time evolution of the expectation value of ${\widehat{a}}_x^{†}{\widehat{a}}_x$ operator for various number of ions. We assume an initial $|j,-j\rangle |0_x\rangle$. We compare the numerical solution of the time-dependent Schrödinger equation with Hamiltonian (1) (solid lines) with the solution using effective Hamiltonian (17). The parameters are set to $g_x=2.5$ kHz, $\mathrm{\Delta }=300$ kHz, ${\delta }_x=0.5$ kHz, $r_{0,x}=14.5$ nm, $f_{d,x}=3$ yN.

Figure 4

Force sensitivity as a function of the coupling $g_x$ for a single trapped ion. We compare the exact numerical solution for ${\delta }_x=0.14$ kHz (dots), ${\delta }_x=0.18$ kHz (triangles), and ${\delta }_x=0.25$ kHz (squares) with the analytical expression (20) (solid line). The other parameters are set to $\mathrm{\Delta }=270$ kHz and $r_{0,x}=14.5$ nm.

Figure 5

Signal-to-noise ratio vs heating rate for various ${\delta }_x$. We solve the master equation numerically (21) for single trapped assuming for $g_y=0$. We set $g_x=25$ kHz, $\mathrm{\Delta }=2.7$ MHz, and $f_{d,x}=1.05f_{d,x}^{\min }, {\delta }_x=1.4$ kHz (squares), $f_{d,x}=1.14f_{d,x}^{\min }, {\delta }_x=2.1$ kHz (circles), and ${\delta }_x=3.0$ kHz (triangles).