Control-Enhanced Sequential Scheme for General Quantum Parameter Estimation at the Heisenberg Limit

The advantage of quantum metrology has been experimentally demonstrated for phase estimations where the dynamics are commuting. General noncommuting dynamics, however, can have distinct features. For example, the direct sequential scheme, which can achieve the Heisenberg scaling for the phase estimation under commuting dynamics, can have even worse performances than the classical scheme when the dynamics are noncommuting. Here we realize a scalable optimally controlled sequential scheme, which can achieve the Heisenberg precision under general noncommuting dynamics. We also present an intuitive geometrical framework for the controlled scheme and identify sweet spots in time at which the optimal controls used in the scheme can be prefixed without adaptation, which simplifies the experimental protocols significantly. We successfully implement the scheme up to eight controls in an optical platform and demonstrate a precision near the Heisenberg limit. Our work opens the avenue for harvesting the power of quantum control in quantum metrology, and provides a control-enhanced recipe to achieve the Heisenberg precision under general noncommuting dynamics.

Figure 1

Generators in the direct and control-enhanced sequential schemes. (a) Direct sequential scheme. (b) Control-enhanced sequential scheme with $N$ controls. Both schemes consist of three steps: state preparation (green module), evolution (blue), and measurement (purple). (c) The evolution of the Bloch vectors ${\mathit{s}}_T^{(N)}$ for the generators $S_T^{(N)}$, here $N$ denotes the number of controls. Without the controls the path of ${\mathit{s}}_T$ is a helical line. The controls change the velocity and increase the length of ${\mathit{s}}_T^{(N)}$. (d) The evolution of the generators at the plane orthogonal to ${\mathit{n}}_h$. Without the controls ${\mathit{s}}_T$ has a uniform circular motion at the plane orthogonal to ${\mathit{n}}_h$. The controls change the motion and increase the length, as shown by the paths of ${\mathit{s}}_T^{(N)}$.

Figure 2

Experimental scheme and setup. (a) Dynamics-controlled feedback scheme. (b) Experimental setup. The module of preparation prepares the probe state using the polarization degree of a heralded single photon from the SPDC process. The probe state then undergoes the evolution and the control in the module of evolution. The state is then measured in the module of measurement. Key devices in the setup are defined in the text.

Figure 3

Precision with the optimal and adaptive controls. (a) QFI. (b) The standard deviation. The performances with $N=1$, 2, and 4 controls are demonstrated, which are denoted by blue, purple, and red, respectively. Experimental results for ideal controls (dots) and adaptive controls (circles) are close to optimal theoretical values (solid lines). The error bars are discussed in the Supplementarl Material [31].

Figure 4

Experimental results at the sweet spots in time. (a) Probability distribution with respect to $x$. Red and black dots show frequencies measured experimentally with 50 000 measurements at sweet time $t$ and nonsweet time $0.5t$, respectively. The solid lines show the theoretical probability distribution. From upper to bottom, the four subplots correspond to control numbers $N=1$, 2, 4, and 8. Error bars are calculated from measurement statistics and too small to be visible. (b) The QFI for the case of $N=8$ is plotted, at both the sweet spot in time $T_8=4\pi$ and the nonsweet spot in time $0.5T_8=2\pi$. The solid lines are the theoretical value and dots are the experiment results.