Quantum-enhanced metrology with large Fock states

Abstract

Quantum metrology uses non-classical states, such as Fock states with a specific number of photons, to achieve an advantage over classical sensing methods. Typically, quantum metrological performance can be enhanced by increasing the involved excitation numbers, for example, by using large photon-number Fock states. However, manipulating these states and demonstrating a quantum metrological advantage is experimentally challenging. Here we present an efficient method for generating large Fock states approaching 100 photons within a superconducting microwave cavity through the development of a programmable photon number filter. Using these states in displacement and phase measurements, we demonstrate quantum-enhanced metrology approaching the Heisenberg scaling for 40-photon Fock states and achieve a maximum metrological gain of up to 14.8 dB, highlighting the metrological advantages of large Fock states. Our study could be readily extended to mechanical and optical systems, promising potential applications in weak force detection and dark matter searches.

Figures (4)

• Figure 1: Efficient generation of large photon-number Fock states for quantum metrology.a Comparison between the conventional scheme with coherent state (left panel) and the Fock state scheme (right panel) for displacement and phase sensing in a probe cavity with an ancillary qubit. The conventional metrology scheme has a sensitivity limit of $1/2$ when measuring the displaced or rotated coherent states, while the Fock states provide an enhancement factor of $\sqrt{N}$ due to the fine structural features in their Wigner functions in the phase space. b Quantum circuit for the sinusoidal photon-number filter ($\mathcal{P}_\mathrm{S}$) on the bosonic state in the probe cavity (C) by projecting the ancilla qubit (Q) in the ground state. c Measured photon number populations of the cavity state at each stage of four successive sinusoidal PNFs for efficient generation of Fock state $|10\rangle$. Each PNF acts as a grating that periodically blocks certain photon numbers of the cavity state. Insets are simulated Wigner functions of the intermediate states at different stages and measured Wigner function of the final state.
• Figure 2: Characterization of large photon-number Fock states.a Measured qubit spectrum for various prepared Fock states $|N\rangle$, with $N=30$, 50, 70, and 100. The frequency detuning on the bottom horizontal axis is the drive frequency relative to the qubit frequency in the absence of any photons in the cavity. The solid lines are fits to a sum of Gaussian functions to extract the Fock state populations $P_n$. The total numbers of measurements (the numbers of postselected measurements) are approximately $4\times10^4$ ($2.2\times10^3$), $5.5\times10^4$ ($2.2\times 10^3$), $7.9\times 10^4$ ($2.5\times10^3$), and $1.0\times10^6$($2.1\times10^3$) with $N=30$, 50, 70, and 100, respectively. b Extracted photon number populations $P_n$ of the generated Fock states $|N\rangle$, with $N=30$, 50, 70, and 100. Error bars in the parentheses for the photon number populations $P_N$ are standard errors obtained from the fittings in a and others are not shown for clarity.
• Figure 3: Quantum metrology using Fock states.a Experimental circuit for displacement amplitude sensing. b The measured qubit ground state populations (dots) and corresponding fittings (solid lines) as a function of the displacement amplitude $\beta$ with the probe initially prepared in Fock states $|N\rangle$, with $N = 2,4,6$ as examples. c Fisher information extracted from the fittings in b. d The displacement measurement precision $\delta \beta$ against the number of photons $N$ of the initial Fock state, with a $14.8\pm 0.2\,\mathrm{dB}$ enhancement of the precision compared with SQL achieved at $N = 40$. A precision scaling of $N^{-0.35}$ is achieved, as determined from a linear fit in the logarithmic-logarithmic scale. e Experimental circuit for phase sensing. f The measured qubit ground state populations (dots) and corresponding fittings (solid lines) as a function of the phase $\phi$ with the probe initially prepared in various displaced Fock states $D(\gamma=\sqrt{N})|N\rangle$, with $N = 1,3,5$ as examples. g Fisher information extracted from the fittings in f. h The phase estimation precision $\delta\phi$ as a function of the average photon number $\bar{n} = 2N$ of the initial displaced Fock state. The precision scales with $\bar{n}^{-0.87}$ and a precision enhancement of $12.3\pm 0.5\,\mathrm{dB}$ surpassing the SQL is achieved at $\bar{n} = 60$. Error bars in d and h are standard errors obtained from error propagation of the fit parameter uncertainties in b and f, respectively. Error bars for other data are smaller than the marker sizes and not shown.
• Figure 4: Photon-number-resolved quantum metrology scheme for displacement amplitude sensing.a Quantum circuit illustrating the displacement amplitude sensing using coherent initial probe state ($|\alpha=\sqrt{3}\rangle$) and consequent photon-number-resolved measurements, which is implemented by three sinusoidal PNFs for simultaneously resolving 0-7 photons. b Measured qubit populations as a function of the displacement amplitude $\beta$ for various traces of the photon-number-resolved Fock states $|N\rangle$. Inset shows the probabilities of traces corresponding to various $|N\rangle$. c The Fisher information (dashed lines) extracted from b as a function of the displacement amplitude $\beta$ for different traces, as well as the total Fisher information (solid black line) by weighting all eight traces.