Abstract
Twomode interferometers lay the foundations for quantum metrology. Instead of exploring quantum entanglement in the twomode interferometers, a single bosonic mode also promises a measurement precision beyond the shotnoise limit (SNL) by taking advantage of the infinitedimensional Hilbert space of Fock states. Here, we demonstrate a singlemode phase estimation that approaches the Heisenberg limit (HL) unconditionally. Due to the strong dispersive nonlinearity and long coherence time of a microwave cavity, quantum states of the form \(\left( {\left 0 \right\rangle + \left N \right\rangle } \right)/\sqrt 2\) can be generated, manipulated and detected with high fidelities, leading to an experimental phase estimation precision scaling as ∼N^{−0.94}. A 9.1 dB enhancement of the precision over the SNL at N = 12 is achieved, which is only 1.7 dB away from the HL. Our experimental architecture is hardware efficient and can be combined with quantum error correction techniques to fight against decoherence, and thus promises quantumenhanced sensing in practical applications.
Introduction
High precision measurement is one of the main driving forces for science and technology, and the interferometer based on the coherent interference effect is one of the most extensively used tools^{1,2,3,4,5,6}. Twomode interferometers, in particular, are widely used to precisely measure the phase difference between the two modes induced by certain physical quantities. For example, the twomode atomic Ramsey interferometer that manipulates the superpositions of two internal states of an atomic ensemble^{7} has been used in various applications, such as clock^{8,9}, gravimeter^{10}, and gyro^{11}. Similarly, by separating photons into two spatial modes, twomode photonic Michelson interferometers have been extensively used in gravitationalwave observatory^{5}, optical coherence tomography^{12}, and spectrometry^{13}. Recently, quantum metrology^{14,15,16}, which makes use of quantum mechanical effects, such as entanglement, has gained a lot of attention in the twomode interferometers, as it can achieve measurement precisions beyond the classical limit. In the applications of quantum metrology, highly entangled states, such as the Greenberger–Horne–Zeilinger state of an atomic ensemble^{17,18} or the NOON state of optical interferometer^{19,20}, are essential. However, to prepare these exotic quantum states, nonlocal operations are required. In addition, the optimal measurements are also typically highly nonlocal. These pose significant challenges for practical applications of quantum metrology.
Instead of exploring quantum entanglement in the twomode interferometer, quantum sensors with a single mode are of great interest. For example, based on a single bosonic mode, quantum metrology schemes have been proposed^{21,22} by taking advantage of the infinitedimensional Hilbert space of Fock states. Such singlemode quantum sensors hold the advantages of hardware efficiency, compactness, and robustness against nonlocal perturbations. Similarly, by exploring the large angular momentum states, a high precision electrometer beating the shotnoise limit (SNL) is promising with a single atom, as demonstrated in refs. ^{23,24}. Alternatively, by utilizing coherence and implementing phase estimation algorithms, a quantumenhanced magnetometry was recently demonstrated with a single artificial atom^{25,26,27}, with a precision approaching the Heisenberg limit (HL).
In this paper, we implement the singlemode photonic quantum metrology with a superconducting qubitoscillator system^{28} and demonstrate an unconditional phase estimation approaching the HL. For a single mode, the phase can be measured based on the photon numberdependent phase accumulation. By preparing the superpositions of Fock states as \(\left {\Psi \left( N \right)} \right\rangle = \left( {\left 0 \right\rangle + \left N \right\rangle } \right)/\sqrt 2\) up to N = 12, we demonstrate a phase estimation precision that scales as \(\delta \tilde \theta \sim N^{  0.94}\) and approaches the HL. At N = 12, \(\delta \tilde \theta\) corresponds to an enhancement of \(20{\mathrm{log}}_{10}(\delta \tilde \theta _{{\mathrm{SNL}}}/\delta \tilde \theta )\,{\mathrm{dB}} = 9.1\,{\mathrm{dB}}\) over the SNL \(\delta \tilde \theta _{{\mathrm{SNL}}}\). Envisioning future applications in the optical regime with microwavetooptical transduction, we also realize a measurement scheme that is easy to implement in optics and only uses displacement operations and photon counting. Under this restricted measurement scheme, a subSNL precision, which scales as \(\delta \tilde \theta \sim N^{  0.62}\), is also achieved.
Results
Theory of optimal sensing and experimental architecture
According to the quantum Cramér–Rao bound^{29}, the estimation precision of parameter θ encoded in the state ψ(θ)〉 = e^{−iθH}ψ〉 is lower bounded as \(\delta \tilde \theta \ge \frac{1}{{2\Delta H}}\), where \(\delta \tilde \theta\) is the standard deviation of an unbiased estimator \(\tilde \theta\), and (ΔH)^{2} = 〈ψH^{2}ψ〉 − 〈ψHψ〉^{2} is the variance of the Hamiltonian H with the initial probe ψ〉. The quantum states with a maximum variance therefore are optimal for the singlemode sensing, i.e. the equal superpositions of the eigenstates of H corresponding to the extreme eigenvalues are the most preferable quantum states. For example, as illustrated in Fig. 1a, b, an atom prepared in the equal superposition of angular momentum states J, −J〉 and J, J〉 has maximal sensitivity to external field (H = J_{z} and J_{z} is the angular momentum operator). Recently, a high precision electrometer, using the Schröinger cat state of large angular momentum states to enhance ΔH, has also been demonstrated to beat SNL^{23,24,30}. Similarly, the phase precision with a single bosonic mode would be enhanced by using the state \(\left {\Psi \left( N \right)} \right\rangle = \left( {\left 0 \right\rangle + \left N \right\rangle } \right)/\sqrt 2\) (Fig. 1c, d), since it has the maximum variance for \(H = a^\dagger a\) (a is the bosonic operator of the sensing mode), given a mean photon number (average energy). Such a maximum variance state (MVS) can in principle achieve the HL precision \(\delta \tilde \theta = 1/N\) with \(\sqrt N\) times enhancement over the SNL.
As schematically illustrated in Fig. 1e, f, our experiment is carried out with a superconducting system consisting of a transmon qubit dispersively coupled to two threedimensional cavities^{31,32,33}. The longlived cavity serves as the sensing mode; the transmon qubit as an ancilla assists the preparation, manipulation, and detection of the photonic states in the sensing mode; the shortlived cavity is employed for a highfidelity readout of the qubit state. The Hamiltonian of the qubitoscillator system is \(H =  \hbar \chi _{{\mathrm{qs}}}e\rangle \langle ea^\dagger a\)^{28}, where e〉 is the excited state of the qubit (the ground state is g〉), and χ_{qs} reflects the dispersive interaction strength between the qubit and the mode (Supplementary Note 1). In our system, χ_{qs}/2π = 1.90 MHz is much stronger than the decoherence rates of the qubit and the sensing mode, thus allows full control of the photonic quantum state^{32,33,34,35,36}.
Preparation of the MVS states
In our experiment, the probe states of the sensing mode are deterministically created by implementing a qubitassisted unitary operation on the mode based on carefully calibrated experimental parameters. The unitary operation is realized through the socalled gradient ascent pulse engineering method^{37,38}, which is an optimization algorithm designed to numerically find pulses that most accurately realize a unitary operation and has been widely used for creating cat states^{32,35} and other superpositions of Fock states^{33}. With numerically optimized control pulses, the probe states Ψ(N)〉 with N = 1, 2, …, 12 are prepared faithfully. Compared with the previous scheme that prepares superpositions of Fock states by climbing the Jaynes–Cummings ladder stepbystep^{39}, our onestep approach allows arbitrary state preparation with higher fidelity and shorter operation time. In Fig. 2 the experimentally measured Wigner functions (bottom panels) of the typical MVSs are plotted, agreeing well with the ideal ones (top panels). In the phasespace, there are interesting periodic fringes in the polar direction with Nfold rotational symmetry for Ψ(N)〉. As the rotation of the Wigner function by θ corresponds to the phase operation \(U\left( \theta \right) = {e}^{i\theta a^\dagger a}\) on the oscillator, the enhanced measurement precision by the MVS can be intuitively explained: because of the fine fringe features, Ψ(N)〉 would be rotated to an orthogonal state when the phase θ = π/N, the measurement precision with the MVS is thus proportional to N, instead of \(\sqrt N\).
Optimal singlemode sensing scheme
Figure 3a depicts the experimental circuit for the optimal sensing scheme with a Ramseylike interference (Fig. 1c), which can attain the ultimate HL for the singlemode sensing. After an initialization process, the cavity is prepared in Ψ(N)〉 while the qubit ends up with e〉. The phase operation U(θ) on the sensing mode can then be generated after the system evolves for a time period τ, where θ = −χ_{qs}τ. Then, a unitary U_{H} is implemented to rotate \(\left {{\mathrm{\Phi }}_ + } \right\rangle = \left e \right\rangle \left( {\left 0 \right\rangle + \left N \right\rangle } \right)/\sqrt 2\) to g〉 0〉 and \(\left {{\mathrm{\Phi }}_  } \right\rangle = \left e \right\rangle \left( {\left 0 \right\rangle  \left N \right\rangle } \right)/\sqrt 2\) to e〉 0〉. Finally, the ancillary qubit is projectively measured on g〉, giving projection of U(θ)Ψ(N)〉 onto Ψ(N)〉 with the ideal probability oscillation (Supplementary Note 2)
The experimental results of the optimal scheme \(P_{{\mathrm{opt}}}^{\left( N \right)}\) are shown in Fig. 3b. As intuitively expected from Fig. 2, the period of the Ramsey interference fringes reduces with N and the contrast of the fringes are nearly ideal. By fitting the experimentally measured probability with P^{(N)}(θ) = A + B cos(Nθ), where A−B and B represent the detected background and the contrast of the Ramsey interference fringes, respectively, the phase estimation precision can be inferred as
Figure 3c shows the results of \(\delta \tilde \theta\) (blue dots) as a function of N in a logarithmic–logarithmic scale. Clearly, the optimal scheme beats the SNL, with the green region representing the experimental results that surpass the SNL with a maximum precision enhancement of 9.1 dB at N = 12, which is only 1.7 dB away from the ultimate HL. The results demonstrate the quantum advantage of our singlemode sensing unambiguously. The obtained precision scales as N^{−0.94}, approaching the Heisenberg scaling (N^{−1}). The slight deviation mainly attributes to the Ndependent imperfections including the larger operation errors for larger Hilbert space of Fock states (errors in the control pulse and parameter uncertainties) and higher probability of photon loss.
The demonstrated optimal scheme can be utilized in practical sensing applications. For example, if there is another microwave signal coupling with the qubit with unknown amplitude or frequency but largely offresonant with the qubitoscillator system, the frequency of the sensing mode would be shifted due to the crossKerr effect mediated by the qubit. This effect is equivalent to an AC Starkeffectinduced shift of the sensing mode. This frequency shift can cause an accumulated phase on N〉 with respect to 0〉, which can be detected by the presented optimal sensing scheme. In turn, we can estimate the frequency or amplitude of the unknown microwave signal.
Hybrid singlemode sensing scheme
By utilizing the recently developed highefficient bidirectional microwavetooptical quantum transduction^{40,41}, our scheme with the MVS can also be employed for the optical metrology. However, the Ramseylike measurement is very challenging in optical domain due to the limited capability of deterministic quantum state manipulation of optical photons. We thus propose a hybrid sensing scheme, as shown in Fig. 4a, by employing a measurement scheme that only uses easy operations in the optical domain, such as displacement operation and photon counting^{42}.
Envisioning the application of such a hybrid scheme, we simulate the scheme in our superconducting system with the restricted measurement. It is worth noting that different from photon counting in the real optical system^{42}, the measurement through the ancillary qubit in the superconducting system can only obtain a binary output, i.e. a result of whether the photon number is n or not. So, the outcome of the measurement has the probability \(P_n^{(N)} = \left {\langle nD\left( \alpha \right)U\left( \theta \right)\left {\Psi \left( N \right)} \right\rangle } \right^2\), where D(α) is the displacement operator. We optimize the parameters α and n for each MVS to maximize the measurement precision (Supplementary Note 2).
The experimental results for the simulated hybrid scheme are summarized in Fig. 4b. Although the fringe period reduces with N similar to that in the optimal scheme, the contrast for the hybrid scheme reduces with N. The reason is mainly that the probability of the binary photon number detection reduces for large N as the state spreads in the Fock space after a displacement operation. However, this hybrid scheme beats the SNL as well, as indicated by the green region in Fig. 4c, with a maximum precision enhancement of 0.7 dB at N = 12. The obtained scaling N^{−0.62} (N^{−0.69} for an ideal experiment) is lower than that for the optimal scheme because of the suboptimal detection process, but can still beat the standard scaling N^{−0.5} due to the initial MVS. Actually, by using a photon number resolving detector, which is available in optical domain, a better precision could be achieved by the hybrid scheme in future optical sensing applications (as shown by purple circles in Fig. 4c).
Discussion
Our singlemode quantum metrology architecture achieves a precision near the HL and holds the advantage of hardware efficiency, minimized sensing configuration, and compatibility with quantum error correction that can be employed for further enhancement of the precision^{43}. Our scheme can also be directly applied to other physical systems such as trapped ions^{44} and nitrogenvacancy centers^{45}. As demonstrated in the hybrid scheme, the precision still beats the SNL with the restricted detecting scheme consisting of only displacement operation and photon counting, which are easy to implement in optics. Additionally if we use microwavetooptical upconversion and downconversion twice, nearHL precisions with the optimaldetecting scheme can be achieved. Our scheme thus also adds a powerful new platform to optical quantum metrology, which is quantum resource saving and robust compared to the multiplepath optical interferometer.
Note added: When submitting this work, we became aware of a related work which demonstrates the single phonon mode sensing using the superposition of Fock states in the trapped ion system^{46}.
Methods
Device parameters
The superconducting system consists of two threedimensional cavities and one ancillary transmon qubit, where the qubit couples with the two cavities simultaneously. The ancillary qubit has a frequency ω_{q}/2π = 5.692 GHz, an energy relaxation time T_{1} = 30 μs, and a pure dephasing time T_{ϕ} = 120 μs. The shortlived cavity is at a frequency of ω_{r}/2π = 8.610 GHz, has a lifetime of 44 ns due to its strong coupling to the external microwave drive line, and assists the fast highfidelity readout of the ancillary qubit. The longlived cavity serves as the sensing mode and has a frequency ω_{s}/2π = 7.634 GHz, a singlephoton lifetime \(T_1^{\mathrm{s}} = 143\,\mu {\mathrm{s}}\), and a coherence time \(T_2^{\mathrm{s}} = 250\,\mu {\mathrm{s}}\).
Analysis of experimental imperfections
In practical quantum systems, there are inevitable imperfections, such as the decoherence of the ancillary qubit and the sensing mode, and the finite control precision. By including those imperfections in numerical simulations, we can estimate the phase estimation precision for practical experiments. It is worth noting that only the calibrated parameters of the system and the numerically optimized control pulses are used in the simulation, and neither further assumptions nor fitting parameters are introduced. The excellent agreement between the experimental and numerical results indicates that all essential experimental imperfections are captured in our numerical model and all experimental parameters are well calibrated. The numerical simulations further provide valuable information on the contribution of each imperfection to the loss of measurement precision. The detailed analysis of the errors in the three stages of the quantum metrology experiment: initial state preparation, evolution of the system during the sensing process, and final detection are described in Supplementary Note 3. Note that all the calculations and experiments are performed unconditionally with no postselection of the experimental and numerical outcomes.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
Code availability
The code used for simulations is available from the corresponding authors upon reasonable request.
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Acknowledgements
This work was supported by National Key Research and Development Program of China (Grant no. 2017YFA0304303) and the National Natural Science Foundation of China (Grant nos. 11474177 and 11874235). H.Y. was supported by RGC Hong Kong (Grant no. 14207717). C.L.Z. was supported by National Natural Science Foundation of China (Grant no. 11874342) and Anhui Initiative in Quantum Information Technologies (AHY130200). Y.W. acknowledges support from Shuimu Tsinghua Scholar Program and International Postdoctoral Exchange Fellowship Program (TalentIntroduction Program).
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W.W. performed the experiment and analyzed the data with the assistance of Y.M. Y.W., Z.J.C., H.Y., C.L.Z., and L.M.D. provided theoretical support. L.M.D., C.L.Z., and H.Y. proposed the experiment. L.S. directed the project. W.C. fabricated the JPA. L.H. and X.M. fabricated the devices with the assistance of Y.X., H.W., and Y.P.S. W.W., Y.W., H.Y., C.L.Z., L.M.D., and L.S. wrote the manuscript with feedback from all authors.
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Wang, W., Wu, Y., Ma, Y. et al. Heisenberglimited singlemode quantum metrology in a superconducting circuit. Nat Commun 10, 4382 (2019). https://doi.org/10.1038/s41467019122907
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