Control methods for improved Fisher information with quantum sensing

Recently new approaches for sensing the frequency of time dependent Hamiltonians have been presented, and it was shown that the optimal Fisher information scales as $T^4.$ We present here our interpretation of this new scaling, where the relative phase is accumulated quadratically with time, and show that this can be produced by a variety of simple pulse sequences. Interestingly, this scaling has a limited duration, and we show that certain pulse sequences prolong the effect. The performance of these schemes is analyzed and we examine their relevance to state-of-the-art experiments. We analyze the $T^3$ scaling of the Fisher information which appears when multiple synchronized measurements are performed, and is the optimal scaling in the case of a finite coherence time.

Figure 1

Top: FI achieved with the first control method (see Sec. 2a), compared with the optimal FI. The pink curve corresponds to $\delta =0.08\omega$, the blue to $\delta =0.04\omega$, and the red to the optimal FI. The optimal FI coincides with this method only for $\delta t\ll 1,$ as can be seen from Eq. (14). Bottom: Comparison between the two different control methods. The blue curve [labeled as (1)] corresponds to the first method, the green [labeled as (2)] to the second (see Sec. 2b), and the red (upper) curve to the optimal FI. For short times ($\delta t\ll 1$) the first method is superior as it achieves the optimal FI. However it loses this advantage quite quickly as it suffers from a shorter lifetime. In both plots $\mathrm{\Omega }=50\omega .$

Figure 2

Illustration of the second control method (see Sec. 2b) in the Bloch sphere ($x\text{−}y$ plane). Top: Adiabatic evolution of the spin under the Hamiltonian of Eq. (13). The unitary is rotated with a frequency of $2\delta$ and the state evolves along it. The state undergoes a $\pi$ pulse with a different unitary every $\mathrm{\Delta }t$, as $\mathrm{\Delta }t=\frac{\pi }{2\mathrm{\Omega }}.$ However these different angles do not accumulate and it rotates with the same frequency as the unitary. Bottom: Now in addition to the unitary evolution, a $\pi$ pulse around the ${\sigma }_X$ axis is applied every $\mathrm{\Delta }t.$ Due to these $\pi$ pulses the different angles are accumulated, and the rotation angle of the state goes as $\delta \frac{t^2}{\mathrm{\Delta }t}.$

Figure 3

Numerical analysis of the FI behavior under different timings of $\pi$ pulses in the second control method (see Sec. 2b). Top: Dynamics of transition probability for different timings, where $\mathrm{\Delta }t=\frac{\pi }{2\mathrm{\Omega }}\text{[red,}\text{labeled}\text{(1)]},\frac{\pi }{1.96\mathrm{\Omega }}\text{[yellow,}\text{labeled}\text{(2)]},\text{and}\frac{\pi }{1.9\mathrm{\Omega }}$ [brown, labeled (3)]. It can be seen that for $\mathrm{\Delta }t=\frac{\pi }{2\mathrm{\Omega }}$, perfect $T^4$ oscillations are achieved, while for other values the amplitude drops significantly. Bottom: Comparison of the FI for different timings: $\mathrm{\Delta }t=\frac{\pi }{2\mathrm{\Omega }}\text{(red)},\frac{\pi }{1.96\mathrm{\Omega }}\text{(brown)},\frac{\pi }{1.8\mathrm{\Omega }}\text{(blue)}$, and no control (purple).

Figure 4

Analysisof the FI as a function of $\delta ,$ when multiple measurements, each one with a duration of $\tau ,$ are performed. Top: Behavior of the FI in the regime of $\delta \tau \ll 1$ ($\tau =1\left[\frac{1}{\mathrm{\Omega }}\right]$ in this plot). Observe that the FI (blue, lower curve) coincides with the ultimate limit (orange, upper curve) for $\delta T\ll 1$ and then oscillates and drops to half of this limit. Bottom: The FI for larger $\delta$, the different curves correspond to different values of $\tau :1\text{(blue)},0.8\text{(green,}\text{dotted)},\text{and}0.5\text{(purple,}\text{dashed)}\left[\frac{1}{\mathrm{\Omega }}\right].$ The FI goes as $I_{\text{tot}}=8{\left(\frac{\mathrm{\Omega }}{\delta }\right)}^2\frac{T^3}{3\tau }sin\left(\delta \tau \right)$; this serves as a good approximation except for $\delta =0,\frac{\pi }{2\tau }$ in which there are sharp oscillations. $\tau$ is the optimal measurement period only for $\delta <\frac{1.16}{\tau }.$