Fig. 3

Optimal single-mode sensing scheme. a Quantum circuit. b The measured probability \(P_{{\mathrm{opt}}}^{(N)}\) of projecting to |Ψ(N)〉 as a function of the phase θ in the optimal scheme for different N. The outcomes are obtained with 10⁶ repetitions of the experiment. c Quantum advantage of the optimal scheme. Blue dots are experimental results. The blue solid line is a linear fit and gives \(\log _{10}\delta \tilde \theta = - 0.94\log _{10}N + 0.016\) with the precision scaling N^−0.94 approaching the Heisenberg scaling (N⁻¹). The small offset 0.016 in the log–log scale of the precision is dominantly due to the imperfections of the qubit readout process, where the spontaneous decay of the qubit gives wrong indication of the cavity state and lowers the contrast of the Ramsey interference. The red crosses are the results from numerical simulations including the decoherences of our system, in good agreement with the measured data. The error bars, obtained through the standard deviations of A and B in fitting the data in b, are smaller than marker sizes. The black and red dashed lines are the theoretically calculated SNL and HL, respectively. Green region represents the experimental results that surpass the standard limit by about 9.1 dB at N = 12