## 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.

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## Main

Precision measurement and metrology are crucial for a broad range of research, as they enable the accurate measurement of physical quantities^{1}, the discovery of new phenomena^{2}, the validation of physics theories^{3} and the development of new technologies^{4,5}. Quantum metrology^{6,7,8}, which leverages quantum-mechanical principles, can achieve a measurement precision bounded by the Heisenberg limit (HL)^{9,10}, surpassing the standard quantum limit (SQL) in conventional measurements by a factor of \(1/\sqrt{N}\), with *N* being the number of particles or excitation number of probes. The quantum enhancement can be achieved by utilizing entanglement as a resource. This is usually realized by preparing the probe system into certain entangled states, such as the Greenberger–Horne–Zeilinger-type states^{11,12,13,14}, NOON states^{15,16,17}, spin squeezing states^{18,19,20} and other entangled states^{21,22,23} in multimode interferometry. However, manipulating these exotic quantum states in a large system remains a formidable challenge.

Alternatively, one may also achieve quantum-enhanced precision measurements in a hardware-efficient fashion without resorting to multipartite entanglement^{24}. One archetypal example employs a single bosonic mode^{25} as the probe system. Previous works have successfully demonstrated the advantages of quantum sensing by preparing the bosonic system into highly non-trivial states, including Schrödinger cat states^{26}, squeezing states^{27} and maximum variance states^{28,29}, as well as energy eigenstates or Fock states involving phonons^{30,31,32} and microwave photons^{33,34}. Nonetheless, the demonstrated metrological advantages have been confined to relatively small scales, with *N* typically on the order of 10, thereby not fully realizing the potential scaling advantages intrinsic to quantum metrology using a single bosonic mode.

In this work, we demonstrate quantum-enhanced metrology using Fock states with up to 100 photons in a superconducting microwave cavity, almost an order of magnitude improvement compared with previous works^{30,31,32,33,34}. To generate large Fock states, we develop a programmable photon number filter (PNF) by leveraging the photon-number-dependent response of an ancilla qubit coupled to the cavity. This parameterized state preparation method allows us to obtain arbitrary Fock states in logarithm steps. Based on these Fock states, we further demonstrate a close-to-Heisenberg scaling for both displacement and phase sensing and achieve a maximum metrological gain of 14.8 ± 0.2 dB and 12.3 ± 0.5 dB in each case, illustrating the metrological power of large Fock states in a single bosonic mode.

Employing quantum states with higher excitations in a single quantum system, a bosonic mode in a superconducting quantum circuit provides a hardware-efficient platform for realizing quantum metrology^{35,36}, as illustrated in Fig. 1a. In conventional metrology schemes, the bosonic mode serving as the probe is prepared to a quasiclassical coherent state \(\left\vert \alpha \right\rangle\). Its Wigner function distribution in the phase space is a Gaussian function, with an uncertainty of 1/2 and a mean photon number of *N* = ∣*α*^{2}∣. The left panel of Fig. 1a depicts that, by either external displacement operation \(D(\;\beta )={{\mathrm{e}}}^{\beta ({a}^{{\dagger} }-a)}\) (assuming *β* is a positive real number) or phase rotation \({{\mathrm{e}}}^{-i\phi {a}^{{\dagger} }a}\) of the probe state, the quantum state is displaced by a distance in the phase space. Here, *a*^{†}(*a*) is the creation (annihilation) operator of the bosonic mode. Therefore, the measurement sensitivity is limited by the width of the Gaussian function, which leads to the standard quantum limitations of displacement amplitude sensitivity and phase estimation sensitivity as

Recognized as the most fundamental quantum-mechanical states, Fock states possess an absolute certainty in the photon number of a quantized electromagnetic field, indicating an ultrasensitive response to field perturbations^{37}. To go beyond the SQL, non-classical Fock states were proposed to probe the displacement or phase rotation in ref. ^{31}. As shown by the right panel of Fig. 1a, Fock states exhibit sub-Planck phase-space structures of the Wigner functions^{38}, with their structural features inversely proportional to the photon number, indicating a higher resolution in phase space. By preparing the probe in the Fock state \(\left\vert N\;\right\rangle\) and only performing further displacement and parity measurement operations, the probe can achieve sensitivities

for displacement and phase sensing, respectively (see Methods for details). For both cases, the scheme based on the Fock state shows a \(\sqrt{N}\) enhancement of the measurement sensitivity compared with a coherent state with the same mean photon number, indicating the ability to approach the HL.

The generation of Fock states and their application in Heisenberg-limited quantum metrology are experimentally investigated with a probe mode in a high-*Q* three-dimensional (3D) microwave cavity, which is dispersively coupled to an ancillary superconducting qubit^{39}. With respect to the driving frequencies, the system Hamiltonian reads

Here, \({\sigma }_{x}=\left\vert e\right\rangle \langle g| +| g\rangle \left\langle e\right\vert\) and \(\left\vert e\right\rangle (\left\vert g\right\rangle )\) is the excited (ground) state of the superconducting qubit. Employing the dispersive coupling *χ* between the probe mode and qubit, and external coherent drives *ϵ*_{p,q} on them, efficient quantum control of the composite probe–qubit system can be realized^{26}. The displacements of the probe are trivial, and the parity measurement of the probe state can be realized by projectively measuring the qubit^{40,41}, while the preparation of the Fock state \(\left\vert N\;\right\rangle\) with *N* ≫ 1 remains elusive in this platform^{33,34}.

To tackle this challenge, a projection operation called sinusoidal PNF is developed. The quantum circuit is shown in Fig. 1b. By choosing the driving detuning on the qubit as *Δ* = *N**χ*, the output of the qubit on \(\left\vert g\right\rangle\) indicates a projection operation \({{{\mathcal{P}}}}_{{{\rm{S}}}}(\theta ,N\;)={\sum }_{n}{{\mathrm{e}}}^{i{\varphi }_{n}}\cos \frac{(n-N)\theta }{2}\left\vert n\right\rangle \left\langle n\right\vert\), with *φ*_{n} being an irrelevant phase for generating Fock states and *θ*/*χ* being the duration of dispersive interaction. By sequentially implementing the sinusoidal PNF with *θ* = π/2^{j−1} for the *j*th step (*j* = 1, 2, …, *m*), the probe is projected into a space of a general parity of 2^{m}, that is, \({\varPi }_{j}{{{\mathcal{P}}}}_{{\,{\rm{S}}}}({\theta }_{j},N\;)={\sum }_{k}\left\vert N+k{2}^{m}\right\rangle \left\langle N+k{2}^{m}\right\vert\), and the parity measurement is a special case of the sinusoidal PNF with *m* = 1 (ref. ^{41}). Such a PNF provides a unique and efficient approach for the preparation of Fock states using simple parameterized quantum circuits with a circuit depth of \(d={\log }_{2}\sqrt{N}\) for *N* ≫ 1, since the photon number uncertainty is \(\sqrt{N}\) for an initial coherent state \(\left\vert \alpha =\sqrt{N}\right\rangle\). This logarithmic scaling of the circuit depth is more advantageous for efficiently resolving all the photon number Fock states to achieve the quantum-enhanced metrology (Fig. 4) than the polynomial scaling using selective number-dependent arbitrary phase gates and displacement operations^{42}. In Fig. 1c, an example of the procedures for preparing Fock \(\left\vert 10\right\rangle\) by sinusoidal PNF is shown, with the probe initially prepared in a coherent state with \(\alpha =\sqrt{10}\). As expected, the measured photon number distributions evolve as they pass through a comb after each PNF. The simulated Wigner functions indicate the preservation of coherence in projected subspaces. The measured Wigner function of the final state verifies the successful preparation of the target Fock state, showing sub-Plank features and rotational symmetry with respect to the origin.

Figure 2 shows the experimental results for the prepared Fock states with *N* = 30, 50, 70 and 100. The states are prepared by two PNFs \({{{\mathcal{P}}}}_{{\,{\rm{S}}}}(\uppi ,N\;){{{\mathcal{P}}}}_{{\,{\rm{S}}}}(\uppi /2,N\;)\), with the assistance of a Gaussian PNF operation (Methods). In Fig. 2a, we characterize the generated Fock states by measuring their photon number distributions through a qubit spectroscopy experiment. By applying a selective Gaussian π pulse on the qubit while sweeping the drive frequency detuning, we measure the readout signal spectrum and fit it to a sum of Gaussian functions to extract the height of each peak (Methods). These peaks correspond to the photon number populations *P*_{n} of each Fock state \(\left\vert n\right\rangle\) after normalizing the Gaussian amplitudes such that ∑_{n}*P*_{n} = 1. The extracted photon number populations *P*_{n} for the generated large Fock state \(\left\vert N\right\rangle\) with *N* = 30, 50, 70 and 100 are shown in Fig. 2b. Comparing the detected photon number populations for different *N*, there are increasing probabilities for *N* − 1 and *N* − 2 with increasing *N* because the Fock state decay rate increases with *N*. These measured photon number amplitudes *P*_{N} for the Fock state \(\left\vert N\right\rangle\) are consistent with those estimated from our error analysis (Supplementary Fig. 9). The prepared Fock states are further characterized by Ramsey experiments on the ancilla qubit, with the Ramsey interference oscillating *N* times faster in the presence of *N* photons in the cavity (Supplementary Fig. 5). All these results confirm the successful generation of large-photon-number Fock states in the cavity.

Quantum-enhanced metrology using large-photon-number Fock states is investigated for both displacement and phase sensing. Using the quantum circuit shown in Fig. 3a, the displacement amplitude ∣*β*∣ is measured by directly applying a displacement operation to the prepared Fock state \(\left\vert N\;\right\rangle\) and detected by mapping the parity information to the ancilla qubit (\({{{\mathcal{P}}}}_{{\,{\rm{S}}}}(\uppi )\)). The measured qubit ground state populations *P*_{g} are plotted as a function of the displacement amplitude ∣*β*∣ for various initial Fock states, as shown in Fig. 3b. The data are fitted to the function \({P}_{{\mathrm{g}}}=A\exp \left(-2| \beta {| }^{2}\right){L}_{N}\left(4| \beta {| }^{2}\right)+B\), where *L*_{N} represents the *N*th order Laguerre polynomials, and *A* and *B* are the fitting parameters to account for the measurement imperfections (Supplementary Section II-A). From the population oscillations, we found that the oscillation period, corresponding to the range of the estimated parameter, scales as \(1/\sqrt{N}\) (Supplementary Section II-C). The Fisher information serves as a figure of merit for quantifying the average information learned about the parameter ∣*β*∣ and is calculated by \({F}_{N}(\;\beta )=\frac{1}{{P}_{{\mathrm{g}}}(1-{P}_{{\mathrm{g}}})}{\left(\frac{{\mathrm{d}}{P}_{{\mathrm{g}}}}{{\mathrm{d}}\beta }\right)}^{2}\) (refs. ^{6,9}), as shown in Fig. 3c. The maximal achievable sensitivity of displacement amplitude sensing can then be estimated using \(\updelta \beta =1/\sqrt{{F}_{{\mathrm{m}}}}\) with \({F}_{{\mathrm{m}}}={\max }_{\beta }({F}_{N})\) representing the maximal achieved Fisher information. We note that, although the Fisher information achieves its maximum at different displacement amplitudes ∣*β*∣ for different *N*, the maximal sensitivity to small displacement can be achieved experimentally by initially biasing to the specific ∣*β*∣ with a displacement operation. The corresponding extracted δ*β* as a function of the photon numbers *N* of the initial Fock state are shown in Fig. 3d on a logarithmic–logarithmic scale. The results demonstrate that the displacement amplitude sensing with Fock states surpasses the SQL with a maximum metrological gain of \(20\log (\updelta {\beta }_{{{\rm{SQL}}}}/\updelta \beta )=14.8\pm 0.2\) dB achieved at *N* = 40. Additionally, a linear fit in the logarithmic–logarithmic scale reveals that the enhancement of the sensitivity scales with the photon number as *N*^{−0.35} for *N* ≤ 40, approaching the Heisenberg scaling *N*^{−0.5}. To further enhance the metrological gain, improvements in the quality of the probe cavity and ancilla qubit are required, primarily focusing on suppressing decay and dephasing errors (Supplementary Section II-C).

For phase sensing, the direct application of the Fock state is hindered by the rotation-symmetry structures of the Wigner distributions in the phase space. However, this issue can be addressed by displacing the state away from the origin, as illustrated by the quantum circuit in Fig. 3e. By initially preparing the probe in a displaced Fock state \(D(\gamma )\left\vert N\right\rangle\) with \(\gamma =\sqrt{N}\), the phase rotation in the phase space is reminiscent of a displacement of *D*(−*i**γ**ϕ*). Here, the sensing resources are accounted for as the mean photon number of the initial state \(\bar{n}=2N\). Consequently, this displacement yields a similar response curve with respect to *ϕ*, as shown in Fig. 3f,g. The resulting phase estimation sensitivity exhibits a maximum metrological gain of 12.3 ± 0.5 dB for \(\bar{n}=60\), and a sensitivity scaling of \({\bar{n}}^{-0.87}\) for \(\bar{n}\le 60\), approaching the Heisenberg scaling \({\bar{n}}^{-1}\), as depicted in Fig. 3h.

The experimental results validate the quantum-enhanced metrology approaching the HL for large-photon-number Fock states. The PNFs generate a target Fock state from a coherent state in a probabilistic manner with the success probability scaling with the photon number as \(1/\sqrt{N}\) for *N* ≫ 1. However, we can still achieve a scaling enhancement of measurement precision with *N*^{1/4} compared with the classical limit (Methods). Therefore, it indicates a quantum metrology advantage even with the non-deterministic preparation of the initial probe state. Alternatively, it is unnecessary to select a specific Fock state of a given photon number *N* while discarding other Fock states with different photon numbers in practical metrology applications. Instead, by adaptively performing parity measurements, we can deterministically collapse the probe state to a photon number and obtain the corresponding Fock state \(\left\vert n\right\rangle\). Then, the sensing interrogation and the final measurement can be performed according to *n*, which promises an enhancement of precision scaling with *N*^{1/2} (\(N=\bar{n}\)) approaching the HL. Therefore, the advantages of the hardware-efficient quantum metrology platform based on bosonic modes can be further explored on the basis of an idea similar to the quantum jump tracking method^{43}. As a proof-of-principle demonstration, we implement an optimal displacement amplitude sensing scheme by recording all possible traces of the projection measurements on Fock \(\left\vert N\right\rangle\) for an initial coherent state of the probe and processing the data according to the measured *N*. The quantum circuit of the quantum jump tracking scheme is depicted in Fig. 4a, where a sequence of sinusoidal PNFs is applied and the outcomes are recorded to determine the *N* of the initial probe state. For an initial coherent state with \(\alpha =\sqrt{3}\), the experimental results for traces of various *N* are summarized in Fig. 4b, with the inset showing the probabilities of these traces. The corresponding Fisher information for each trace [\(F\left(\left\vert n\right\rangle \right)\)] is represented in Fig. 4c. The total Fisher information is upper bounded by the weighted Fisher information, given by \(\mathop{\sum }_{n = 0}^{7}{P}_{n}F\left(\left\vert n\right\rangle \right)\), due to its convexity^{44}. The experimental results indicate a maximum Fisher information gain of \(10\log ({F}_{{{\rm{weight}}}}/{F}_{{{\rm{SQL}}}})=5.04\pm 0.06\) dB for the initial coherent state with \(\alpha =\sqrt{3}\), where *F*_{SQL} = 4 (Fig. 4c, dashed line) is the Fisher information of the SQL for displacement amplitude sensing.

In conclusion, we have generated large Fock states with up to 100 photons in a high-quality superconducting microwave cavity and demonstrated quantum-enhanced displacement and phase sensing with these Fock states. The large-photon-number Fock states are efficiently generated with programmable parameterized quantum circuits, which provide an alternative way to inspire the future design of control protocols in bosonic systems. By utilizing these highly non-classical Fock states, we have demonstrated a hardware-efficient approach for quantum-enhanced metrology, with measurement sensitivities approaching the Heisenberg scaling for both displacement and phase sensing. Notably, a metrological gain of 14.8 dB has been achieved with *N* approaching 100. We anticipate that even higher metrological gain is possible with *N* exceeding 1,000 by further optimizing the device’s performance^{45}. In addition, we show that our approach—though depending on post-selection—still allows a *N*^{1/4} quantum-enhanced scaling and could further achieve a *N*^{1/2}-scaling enhancement through adaptive control.

Our hardware-efficient quantum metrology approach with large-photon-number Fock states can readily enable the detection of dark photons with a notably enhanced signal rate^{46}, since the action of the dark matter wave can be modelled as a classical displacement drive on the cavity states. One prominent advantage of utilizing Fock states is that generating these highly non-classical states with large photon numbers is less experimentally challenging compared with other non-classical states, such as the squeezing states^{27} and the Gottesman–Kitaev–Preskill states^{25}. Additionally, our control scheme for generating large Fock states can also be applied to trapped ion setups^{47} to achieve a high metrological gain, in which platform Fock states with 100 phonons have been achieved and their metrological advantages still await further exploration^{29}. Furthermore, our study can be extended to mechanical and optical sensors with the assistance of high-efficiency quantum transducers^{48,49,50}, opening up a promising avenue for practical quantum metrology with bosonic modes^{31,32,43,46}.

## Methods

### Device and setup

The experimental device, similar to that in ref. ^{51}, consists of a superconducting transmon qubit^{52}, a Purcell-filtered stripline readout resonator^{53} and a 3D coaxial stub cavity^{54} serving as the probe cavity for quantum metrology experiments. The probe cavity, machined from a single block of high purity (99.9995%) aluminium, is chemically etched to improve its quality factor^{55}. The transmon qubit and readout resonator are made by a thin tantalum film on a sapphire chip^{56,57}, which is fit with the small waveguide tunnel of the 3D cavity. The transmon qubit has a resonance frequency *ω*_{q}/2π = 4.878 GHz, an energy relaxation time *T*_{1} = 93 μs and a pure dephasing time *T*_{ϕ} = 445 μs. The high-*Q* probe cavity has a resonance frequency *ω*_{c}/2π = 6.597 GHz, a single-photon lifetime *T*_{1} = 1.2 ms and a pure dephasing time *T*_{ϕ} = 4.0 ms. The dispersive coupling strength between the qubit and the probe cavity is measured to be *χ*_{qc}/2π = 0.626 MHz. To achieve a fast high-fidelity qubit readout, the readout resonator is designed with a fast decay rate *κ*_{r}/2π = 1.92 MHz, matching the dispersive shift *χ*_{qr}/2π = 2.0 MHz. The control of the ancillary transmon qubit, probe cavity and readout resonator is achieved using Zurich Instruments HDAWG and UHFQA, employing single-sideband in-phase and quadrature modulations. Additionally, the UHFQA acquires the transmitted readout signal, performs demodulation and discrimination and sends the digitized results to the HDAWG in real time via a digital input/output link cable for fast feedback control (Supplementary Section I-A).

### Gaussian PNF

In principle, the sinusoidal PNF \({{{\mathcal{P}}}}_{{{\rm{S}}}}\) can be applied to the entire Hilbert space without any truncation. However, to concentrate on a subspace centred around the desired Fock state and, thus, realize a more efficient projection to a target Fock state with fewer operations, we employ an additional Gaussian PNF operation. This is accomplished by applying a qubit flip pulse with a Gaussian temporal envelope^{26,58} with a standard deviation of *σ*_{t} and a detuning of *N**χ*_{qc}, which leads to the projection operator \({{{\mathcal{P}}}}_{\,{\mathrm{G}}}(\sigma ,N\;)\approx {\sum }_{n}{{\mathrm{e}}}^{i{\varphi }_{n}}{{\mathrm{e}}}^{-{\left.(n-N)\right)}^{2}/4{\sigma }^{2}}\left\vert n\right\rangle \left\langle n\right\vert,\) with *φ*_{n} being an irrelevant phase similar to the sinusoidal filter. The Gaussian standard deviation of the filter function is approximated as \(\sigma \approx 1/\left(\sqrt{2}{\chi }_{{{\rm{qc}}}}{\sigma }_{{\mathrm{t}}}\right)\) for the Fock state distribution. Detailed experimental results regarding the Gaussian PNF can be found in Supplementary Fig. 3. Therefore, while \({{{\mathcal{P}}}}_{{\,{\rm{G}}}}\) selects a finite range of photon numbers, \({{{\mathcal{P}}}}_{{\,{\rm{S}}}}\) rules out photon numbers with certain parity. The combination of these two PNFs avoids the complicated optimization of pulses and is robust against decoherence (Supplementary Section I-B).

### Photon number population reconstruction

The generated Fock states with large photon numbers are characterized by measuring their photon number distribution through a qubit spectroscopy experiment. In this experiment, we apply a selective Gaussian π pulse on the ancillary qubit while sweeping the drive frequency. The measured readout signal as a function of the qubit drive frequency is fitted to a function of multicomponent Gaussian distributions denoted as \(S(\;f\;)=\mathop{\sum }_{n = N-{N}_{{\mathrm{c}}}}^{n = N+{N}_{{\mathrm{c}}}}{A}_{n}\exp \left(-\frac{{(f-{f}_{n})}^{2}}{2{\sigma }_{f}^{2}}\right)+B\), since each resolved peak corresponds to each photon number component with the peak height *A*_{n} proportional to the corresponding photon number population *P*_{n} (ref. ^{59}). Here, the fitting parameter *σ*_{f} is the frequency domain standard deviation of the selective Gaussian pulse and the background *B* accounts for the measurement imperfections due to residual crosstalk between the probe cavity and readout resonator. These two parameters are constrained to be the same for all *n*, while *f*_{n} is the pre-characterized qubit resonance frequency with *n* photons in the probe cavity. The cut-off number of Gaussian components *N*_{c} is chosen to be sufficiently large to include all the relevant Fock components. By normalizing the fitted Gaussian amplitudes *A*_{n}, the photon number populations by *P*_{n} = *A*_{n}/∑_{n}*A*_{n} can be extracted by ensuring the normalization of the photon number population ∑_{n}*P*_{n} = 1. More details can be found in Supplementary Section I-B.

### Precision limitations of quantum metrology

Quantum metrology involves the precise estimation of an unknown parameter *λ* by utilizing quantum resource states, and generally consists of three steps: (1) preparing the probe in a known quantum state; (2) interrogating the unknown parameter *λ*, that is, encoding the parameter into the probe state, usually using a unitary transformation \({U}_{\lambda }=\exp (-i\lambda h)\) with *h* being the interaction Hamiltonian; and (3) measuring the final state with the encoded parameter *λ*. By comparing the input and output states of the probe, the unknown parameter can be inferred, with a precision limitation of estimation \(\updelta \lambda \ge 1/\sqrt{Q}\) according to the Cramer–Rao bound^{9}. Here, *Q* = 4(*Δ**h*)^{2} = 4(〈*h*^{2}〉 − 〈*h*〉^{2}) represents the quantum Fisher information (QFI), which is the supremum of the Fisher information among all possible measurements, and Δ*h* is the standard deviation of the Hamiltonian *h* under the initial probe state.

For displacement amplitude sensing, where *λ* = *β* (assuming *β* is a real number here), the interrogation interaction leads to the unitary transformation \({U}_{\beta }=\exp (-i\beta {h}_{\beta })\) with *h*_{β} = *i*(*a*^{†} − *a*). The SQL is calculated by preparing the probe in a coherent state \(\left\vert \alpha \right\rangle\). The QFI is then determined as \({Q}_{{{\rm{SQL}}}}=4(\langle \alpha | {h}_{\beta }^{2}| \alpha \rangle -{\langle \alpha | {h}_{\beta }| \alpha \rangle }^{2})=4\), and the corresponding precision δ*β*_{SQL} = 1/2 is independent of the average photon number of the initial coherent state. By utilizing the Fock state \(\left\vert N\right\rangle\) as the initial probe state, the QFI is calculated as \({Q}_{{{\rm{Fock}}}}=4(\langle N| {h}_{\beta }^{2}| N\rangle -{\langle N| {h}_{\beta }| N\rangle }^{2})=4(2N+1)\), which leads to an estimation precision \(\updelta {\beta }_{{{\rm{Fock}}}}=1/2\sqrt{2N+1}\). Comparing the precision for coherent and Fock states, a \(\sqrt{N}\) enhancement of the measurement sensitivity is achievable considering the same input excitation numbers in the probe state, and the results indicate the scheme based on Fock states can approach the HL. It can be further demonstrated that, even when considering displacement in arbitrary directions, utilizing Fock states and parity measurements can still saturate the QFI. For more details, see Supplementary Section II-C.

For phase measurement, where *λ* = *ϕ*, the interrogation gives \({U}_{\phi }=\exp (-i\phi {h}_{\phi })\) with *h*_{ϕ} = *a*^{†}*a*. The SQL is calculated in a similar way with a coherent state \(\left\vert \alpha \right\rangle\) in the probe, as the QFI \({Q}_{{{\rm{SQL}}}}=4(\langle \alpha | {h}_{\phi }^{2}| \alpha \rangle -{\langle \alpha | {h}_{\phi }| \alpha \rangle }^{2})=4N\) is determined and the estimation precision \(\updelta {\phi }_{{{\rm{SQL}}}}=1/2\sqrt{N}\) is obtained, with *N* = ∣*α*∣^{2} representing the average photon number of the initial coherent state. By utilizing the displaced Fock state \(D(\sqrt{N})\left\vert N\right\rangle\) as the initial probe state, the QFI is calculated as \({Q}_{{{\rm{Fock}}}}=4(\langle N| D(-\sqrt{N}){h}_{\phi }^{2}D(\sqrt{N})| N\rangle -{\langle N| D(-\sqrt{N}){h}_{\phi }D(\sqrt{N})| N\rangle }^{2})=4N(2N+1)\). Thus, \(\updelta {\phi }_{{{\rm{Fock}}}}=1/2\sqrt{N(2N+1)}\), indicating a \(\sqrt{N}\) enhancement of the measurement sensitivity that approaches the HL. It can also be proven that the parity measurement combined with the displacement operation can saturate the QFI here (Supplementary Section II-D).

### Extracting the Fisher information from measured data

In the quantum metrology experiments, the classical Fisher information for estimating the parameter *λ* is extracted from the measured probability distributions *P*_{μ} using the equation \({F}_{\lambda }={\sum }_{\mu }\frac{1}{{P}_{\mu }}{\left(\frac{\partial {P}_{\mu }}{\partial \lambda }\right)}^{2}\) (refs. ^{6,9}), where a set of measurement projections {*M*_{μ}} is considered. In our experiment, the measurement performed on the ancillary qubit yields only two outcomes *μ* = *g*, *e*, corresponding to the populations with the qubit in the ground and excited states. Therefore, the Fisher information can be further expressed as \({F}_{\lambda }=\frac{1}{{P}_{{\mathrm{g}}}(1-{P}_{{\mathrm{g}}})}{\left(\frac{{\mathrm{d}}{P}_{{\mathrm{g}}}}{{\mathrm{d}}\lambda }\right)}^{2}\). Finally, the maximum achievable Fisher information \({F}_{{\mathrm{m}}}={\max }_{\lambda }({F}_{\lambda })\) is used to estimate the measurement precision of the parameter *λ* by \(\delta \lambda =1/\sqrt{{F}_{{\mathrm{m}}}}\).

For the displacement amplitude sensing experiment, the measured qubit ground state population *P*_{g} is plotted as a function of the displacement amplitude *β* in Fig. 3b. The population is fitted to a function of \({P}_{{\mathrm{g}}}=A{{\mathrm{e}}}^{-2| \beta {| }^{2}}{L}_{N}(4| \beta {| }^{2})+B\) (Supplementary Section II-A) with *L*_{N} being the *N*th order Laguerre polynomials and *A* and *B* being the fitting parameters to account for measurement imperfections. With the fitting result, the classical Fisher information *F*_{β} is calculated according to the equations described above, and plotted in Fig. 3c. The maximum achieved Fisher information is used to obtain the estimation precision δ*β* for the parameter *β*.

For the phase measurement, the measured qubit population *P*_{g} as a function of *ϕ* shown in Fig. 3f is fitted to a function of \({P}_{{\mathrm{g}}}=A{{\mathrm{e}}}^{-2N{(\phi +{\phi }_{0})}^{2}}{L}_{N}(4N{(\phi +{\phi }_{0})}^{2})+B\), with *A* and *B* being the fitting parameters to account for the measurement imperfections and *ϕ*_{0} being the fitting parameter to account for the phase rotation due to the self-Kerr interaction during the sensing process. In a similar way, the calculated Fisher information *F*_{ϕ} shown in Fig. 3g is employed to obtain the estimation precision δ*ϕ* for the parameter *ϕ*.

### Calculating the success probability and the scaling enhancement

In the quantum metrology experiment, the initial Fock states are generated using PNFs to project the coherent state into the target photon number *N* with a success probability of \({P}_{N}={{\mathrm{e}}}^{-N}{N}^{N}/N!\approx 1/\sqrt{2\uppi N}\) for *N* ≫ 1 according to the Stirling’s approximation.

With considering the post-selected preparation of the Fock states, the QFI for displacement sensing can be calculated as \({Q}_{{{\rm{postFock}}}}=4(2N+1){P}_{N}\approx \frac{8}{\sqrt{2\uppi }}{N}^{1/2}\) for *N* ≫ 1, resulting in a scaling factor of *N*^{1/2} better than *Q*_{SQL} = 4 for quasiclassical coherent states. Therefore, the measurement precision of the displacement amplitude \(\updelta {\beta }_{{{\rm{postFock}}}}=1/\sqrt{{Q}_{{{\rm{postFock}}}}}\approx \sqrt{\frac{\sqrt{2\uppi }}{8}}{N}^{-1/4}\) can still achieve a scaling enhancement factor of *N*^{1/4} over the classical limit, highlighting the quantum metrological advantage despite the probabilistic preparation of the initial Fock state.

Similarly, in the phase sensing experiment, the QFI is \({Q}_{{{\rm{postFock}}}}\approx \frac{2}{\sqrt{\uppi }}({\bar{n}}^{3/2}+{\bar{n}}^{1/2})\) when considering the post-selected preparation of the Fock states \(\left\vert N\right\rangle\) with \(N=\bar{n}/2\) in our scheme. The corresponding measurement precision \(\updelta {\phi }_{{{\rm{postFock}}}}=1/\sqrt{{Q}_{{{\rm{postFock}}}}}\approx \sqrt{\frac{\sqrt{\uppi }}{2}}{\bar{n}}^{-3/4}\) can also achieve a scaling enhancement factor of \({\bar{n}}^{1/4}\) over the classical limit.

## Data availability

Source data are provided with this paper. All other data relevant to this study are available from the corresponding authors upon reasonable request.

## Code availability

The code used for this study is available from the corresponding authors upon reasonable request.

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## Acknowledgements

This work was supported by the Key-Area Research and Development Program of Guangdong Province (grant no. 2018B030326001), the Guangdong Basic and Applied Basic Research Foundation (grant nos. 2024B1515020013 and 2022A1515010324), the National Natural Science Foundation of China (grant nos. 12274198, 92265210, 12061131011, 12322413 and 92365206), the Shenzhen Science and Technology Program (grant no. RCYX20210706092103021), the Guangdong Provincial Key Laboratory (grant no. 2019B121203002), the Shenzhen-Hong Kong cooperation zone for technology and innovation (contract no. HZQB-KCZYB-2020050), and the Innovation Program for Quantum Science and Technology (grant nos. 2021ZD0301703 and 2021ZD0301802). C.L.Z. also acknowledges supports from the Fundamental Research Funds for the Central Universities, the USTC Center for Micro and Nanoscale Research and Fabrication, and USTC Research Funds of the Double First-Class Initiative.

## Author information

### Authors and Affiliations

### Contributions

Y.X. and D.Y. supervised the project. Y.X. conceived and designed the experiment. X.D. and S. Li performed the experiment, analysed the data and carried out the numerical simulations under the supervision of Y.X. Z.N. and S. Li developed the feedback control technique. Z.-J.C. and C.-L.Z. provided theoretical support for the quantum metrology schemes and data analysis. F.Y. proposed the approach for efficiently generating Fock states with large photon numbers. L.Z., H.Y. and S. Liu support the device fabrication. Y.C., J.M. and P.Z. contributed to the experimental and theoretical support. X.D., S. Li, Z.-J.C., C.-L.Z., F.Y. and Y.X. wrote the paper with feedback from all authors.

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## Supplementary information

### Supplementary Information

Supplementary Figs. 1–12, Table 1, Sections I and II and references.

## Source data

### Source Data Fig. 1

Source data for Fig. 1c.

### Source Data Fig. 2

Source data for Fig. 2a,b.

### Source Data Fig. 3

Source data for Fig. 3b–d,f–h.

### Source Data Fig. 4

Source data for Fig. 4b,c.

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## About this article

### Cite this article

Deng, X., Li, S., Chen, ZJ. *et al.* Quantum-enhanced metrology with large Fock states.
*Nat. Phys.* (2024). https://doi.org/10.1038/s41567-024-02619-5

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DOI: https://doi.org/10.1038/s41567-024-02619-5