Machine-Designed Sensor to Make Optimal Use of Entanglement-Generating Dynamics for Quantum Sensing

We use machine optimization to develop a quantum sensing scheme that achieves significantly better sensitivity than traditional schemes with the same quantum resources. Utilizing one-axis twisting dynamics to generate quantum entanglement, we find that, rather than dividing the temporal resources into separate “state-preparation” and “interrogation” stages, a complicated machine-designed sequence of rotations allows for the generation of metrologically useful entanglement while the parameter is interrogated. This provides much higher sensitivities for a given total time compared to states generated via traditional one-axis twisting schemes. This approach could be applied to other methods of generating quantum-enhanced states, allowing for atomic clocks, magnetometers, and inertial sensors with increased sensitivities.

Figure 1

(a) Traditional OAT scheme. With total available time $T$, time ${\tau }_{\text{prep}}$ is devoted to preparing the quantum state, while ${\tau }_{\mathrm{int}}=T-{\tau }_{\text{prep}}$ is devoted to interrogating the system. (b) Machine-designed scheme customized to maximize the sensitivity, which includes dynamics which can not be separately classified as state-preparation and interrogation.

Figure 2

Performance of the traditional OAT scheme. (a)–(d) The Husimi-$Q$ function at (a) $t=0$, (b) $t={\tau }_{\text{prep}}$ before the rotation is applied, (c) $t={\tau }_{\text{prep}}$ after the rotation $\exp (i{\theta }_0{\widehat{J}}_x)$ is applied, and (d) $t=T$. (e) $\mathrm{\Omega }(t)$. In this case, we chose $\mathrm{\Omega }(t)={\theta }_0\delta (t-{\tau }_{\text{prep}})$, resulting in an instantaneous rotation of magnitude ${\theta }_0$. (f) $F_Q(t)/T^2$ (blue solid line) and $f_0$ (red circles). $F_Q(t)=N{t}^2$ for an unentangled system is represented by the blue dashed line. The shot-noise limit ($f_0=N$) and Heisenberg limit ($f_0=N^2$) are represented by the lower and upper black dotted lines, respectively. Parameters: $N=100$, $\chi T=0.1$. The parameters ${\theta }_0=-1.35$ and ${\tau }_{\text{prep}}=0.48$ where chosen to maximize $F_Q(T)$. The Husimi-$Q$ function is defined by $Q(\theta ,\varphi )={|⟨\alpha (\theta ,\varphi )|\mathrm{\Psi }⟩|}^2$, where $|\alpha (\theta ,\varphi )⟩$ is a coherent spin state formed by rotating the maximal ${\widehat{J}}_z$ eigenstate: $|\alpha (\theta ,\varphi )⟩=\exp (i\varphi {\widehat{J}}_z)\exp (i\theta {\widehat{J}}_y)|N/2⟩$.

Figure 3

Performance of the machine-designed scheme. (a)–(d) The Husimi-$Q$ function at (a) $t=0.1T$, (b) $t=0.4T$, (c) $t=0.7T$, and (d) $t=T$. (e) $\mathrm{\Lambda }(t)$. (f) $F_Q(t)/T^2$ (blue solid line) and $f_0$ (red circles). $F_Q(t)=N{t}^2$ for an unentangled system is represented by the blue dashed line. The shot-noise limit ($f_0=N$) and Heisenberg limit ($f_0=N^2$) are represented by the lower and upper black dotted lines, respectively. Parameters: $N=100$, $\chi T=0.1$. $F_Q$ reaches a maximum of $28.95N{T}^2$, compared to $7.8N{T}^2$ for the OAT scheme.

Figure 4

Optimized scheme for different values of $\chi T$. (a)–(c) The optimum $\mathrm{\Lambda }(t)$. The value of ${\tau }_{\text{prep}}$ for the optimum OAT scheme for the same parameters is indicated by the vertical dashed line. (d)–(f) $F_Q(t)$ and $f_0$ for our generalized scheme (blue solid line and red circles, respectively) and the optimum OAT scheme (blue squares and red dot-dashed line, respectively). The shot-noise limit ($F_Q=N{T}^2$ and $f_0=N$) and Heisenberg limit ($f_0=N^2$) are indicated by the blue dashed line and the upper and lower black dotted lines, respectively. Parameters: (a),(d) $\chi T=0.02$, (b),(e) $\chi T=0.06$, (c),(f) $\chi T=0.2$. $N=100$ was used throughout.

Figure 5

Optimized scheme in the presence of detection noise. (a)–(d) The Husimi-$Q$ function at (a) $t=0.25T$, (b) $t=0.5T$, (c) $t=0.75T$, and (d) $t=T$. (e) $\mathrm{\Lambda }(t)$. (f) ${\tilde{F}}_C(t)/T^2$ (blue solid line), $F_Q(t)/T^2$ (blue dashed line), and $f_0$ (red circles). The SNL and ${\tilde{F}}_C(T)$ for a CSS are indicated by the upper and lower black dotted lines, respectively. Parameters: $N=100$, $\chi T=0.1$, $\sigma =10$. ${\tilde{F}}_C$ reaches a maximum of $\sim 1.4N{T}^2$.

Figure 6

Susceptibility of the final probability distribution to detection noise. ${\partial }_{\omega }{P}_m$ (narrow blue bars) and ${\partial }_{\omega }{\tilde{P}}_m$ (wide pink bars) for the scheme illustrated in Figs. 3 (top) and 5 (bottom). (Top) ${\partial }_{\omega }{\tilde{P}}_m$ has been multiplied by 500 in order for it to be displayed on the same scale.