Heisenberg limited single-mode quantum metrology

TL;DR

This paper demonstrates a single-mode quantum phase estimation method that approaches the Heisenberg limit, achieving significant precision enhancement over the shot-noise limit using microwave cavity states.

Contribution

It provides the first experimental demonstration of near-Heisenberg-limited phase estimation with a single bosonic mode, utilizing high-fidelity state manipulation in a microwave cavity.

Findings

Achieved phase estimation precision scaling as ~N^{-0.94}

Realized a 9.1 dB enhancement over the shot-noise limit at N=12

Approached the Heisenberg limit within 1.7 dB in a microwave cavity system

Abstract

Two-mode interferometers, such as Michelson interferometer based on two spatial optical modes, lay the foundations for quantum metrology. Instead of exploring quantum entanglement in the two-mode interferometers, a single bosonic mode also promises a measurement precision beyond the shot-noise limit (SNL) by taking advantage of the infinite-dimensional Hilbert space of Fock states. However, the experimental demonstration still remains elusive. Here, we demonstrate a single-mode 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 $\left(\left|0\right\rangle +\left|N\right\rangle \right)/\sqrt{2}$ are generated, manipulated and detected with high fidelities, leading to an experimental phase estimation precision scaling as $\sim N^{-0.94}$. A $9.1$~$\mathrm{dB}$ enhancement of the precision over the SNL at $N=12$, which is only $1.7$~$\mathrm{dB}$ away from the HL, is achieved. Our experimental architecture is hardware efficient and can be combined with the quantum error correction techniques to fight against decoherence, thus promises the quantum enhanced sensing in practical applications.

Full text

Based on the coherent interference effects, interferometers have been extensively used in precision measurements. For example, the two-mode atomic Ramsey interferometer that manipulates the superpositions of two internal states of an atomic ensemble has been used in various applications, such as clock, gravimeter, and gyro Chu (2002). Similarly, by separating photons into two spatial modes, two-mode photonic Michelson interferometers have been extensively used in LIGO Adhikari (2014), optical coherence tomography and spectrometer. Recently, quantum metrology, which makes use of quantum mechanical effects, such as entanglement, has gained a lot of attention in the two-mode interferometers, as it can achieve measurement precisions beyond the classical limit Giovannetti et al. (2004); Schnabel et al. (2010); Giovannetti et al. (2011); Pezzè et al. (2018); Braun et al. (2018). In the applications of quantum metrology, highly entangled states, such as the Greenberger–Horne–Zeilinger state of an atomic ensemble Pezzè et al. (2018); Braun et al. (2018) or the NOON state of optical interferometer Nagata et al. (2007); Slussarenko et al. (2017), are essential. To prepare these exotic quantum states, non-local operations are required. In addition, the optimal measurements are also typically highly non-local. This poses significant challenges for practical applications of quantum metrology.

In this Letter, instead of exploring quantum entanglement in the two-mode interferometer we implement the single-mode photonic quantum metrology with a superconducting qubit-oscillator system Devoret and Schoelkopf (2013) and demonstrate an unconditional phase estimation with the precision approaching the HL. A quantum sensor with a single mode is of great interest Duivenvoorden et al. (2017); Jacobs et al. (2017) for its hardware efficiency, compactness, and robustness against non-local perturbations. For a single mode, the phase can be measured based on the photon number dependent phase accumulation. Using the state $\left|\Psi\left(N\right)\right\rangle=\left(\left|0\right\rangle+\left|N\right\rangle\right)/\sqrt{2}$, superpositions of Fock states, up to $N=12$, we demonstrate a phase estimation precision which 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 microwave-to-optical 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, an SNL-beating precision, which scales as $\delta\tilde{\theta}\sim N^{-0.62}$, is also achieved.

According to the quantum Cramer-Rao bound Braunstein and Caves (1994), the estimation precision of parameter $\theta$ encoded in the state $|\psi(\theta)\rangle=e^{-i\theta H}\left|\psi\right\rangle$ is lower bounded as $\delta\tilde{\theta}\geq\frac{1}{2\Delta H}$, where $\delta\tilde{\theta}$ is the standard deviation of an unbiased estimator $\tilde{\theta}$, and $(\Delta H)^{2}=\left\langle\psi\right|H^{2}\left|\psi\right\rangle-\left\langle\psi\right|H\left|\psi\right\rangle^{2}$ is the variance of the Hamiltonian $H$ under the initial probe $|\psi\rangle$. The quantum states with a maximum variance therefore are optimal for the single-mode 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 Figs. 1a and 1b, an atom prepared in the equal superposition of angular momentum states $\left|J,-J\right\rangle$ and $\left|J,J\right\rangle$ 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 $\Delta H$, has also been demonstrated to beat SNL Facon et al. (2016); Chalopin et al. (2018); Dietsche et al. (2019). Similarly, the phase precision with a single bosonic mode would be enhanced by using the state $\left|\Psi\left(N\right)\right\rangle=\left(\left|0\right\rangle+\left|N\right\rangle\right)/\sqrt{2}$, since it has the maximum variance for $H=a^{\dagger}a$ ($a$ is the bosonic operator of the sensing mode) (Figs. 1c and 1d). 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 Figs. 1e and 1f, our experiment is carried out with a superconducting system consisting of a transmon qubit dispersively coupled to two three-dimensional cavities Paik et al. (2011); Ofek et al. (2016); Hu et al. (2018). One long-lived 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 short-lived cavity is employed for a high-fidelity readout of the qubit state. The Hamiltonian of the qubit-oscillator system is $H=-\hbar\chi_{\mathrm{qs}}a^{\dagger}a|e\rangle\langle e|$ Devoret and Schoelkopf (2013), where $|e\rangle$ is the excited state of the qubit (the ground state is $\left|g\right\rangle$), and $\chi_{\mathrm{qs}}$ reflects the dispersive interaction strength between the qubit and the mode. In our system, $\chi_{\mathrm{qs}}/2\pi=1.90\,\mathrm{MHz}$ is much stronger than the decoherence rates of the qubit and the sensing mode, thus allows full control of the photonic quantum state Heeres et al. (2015); Ofek et al. (2016); Heeres et al. (2017); Wang et al. (2017); Hu et al. (2018).

In our experiment, the probe states of the sensing mode are deterministically created by implementing a qubit-assisted unitary operation on the mode. With numerically optimized control pulses Khaneja et al. (2005), the probe states $\left|\Psi\left(N\right)\right\rangle=\left(\left|0\right\rangle+\left|N\right\rangle\right)/\sqrt{2}$ with $N=1,2,\dots,12$ are prepared faithfully. 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 phase-space, there are interesting periodic fringes in the polar direction with $N$-fold rotational symmetry for $\left|\Psi\left(N\right)\right\rangle$. As the rotation of the Wigner function by $\theta$ 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, $\left|\Psi\left(N\right)\right\rangle$ would be rotated to an orthogonal state when the phase $\theta=\pi/N$, the measurement precision with the MVS is thus proportional to $N$, instead of $\sqrt{N}$.

Figure 3a depicts the experimental circuit for the optimal sensing scheme by a Ramsey-like interference (Fig. 1c), which is potential for attaining the ultimate precision HL for the single-mode sensing. After an initialization process, the cavity is prepared in $\left|\Psi\left(N\right)\right\rangle$ while the qubit ends up with $\left|e\right\rangle$, resulting in a phase operation $U\left(\theta\right)$ on the sensing mode with $\theta=-\chi_{\mathrm{qs}}\tau$ and $\tau$ being the waiting time. Then, a unitary $U_{H}$ is implemented to rotate $\left|\Phi_{+}\right\rangle=\left|e\right\rangle(\left|0\right\rangle+\left|N\right\rangle)/\sqrt{2}$ to $\left|g\right\rangle\left|0\right\rangle$ and $\left|\Phi_{-}\right\rangle=\left|e\right\rangle(\left|0\right\rangle-\left|N\right\rangle)/\sqrt{2}$ to $\left|e\right\rangle\left|0\right\rangle$. Finally, the ancillary qubit is projectively measured on $\left|g\right\rangle$, giving projection of $U\left(\theta\right)\left|\Psi\left(N\right)\right\rangle$ onto $\left|\Psi\left(N\right)\right\rangle$ with the ideal probability oscillation $P_{\mathrm{opt}}^{(N)}=\left(1+\mathrm{cos}N\theta\right)/2$.

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 $P^{(N)}\left(\theta\right)=A+B\cos\left(N\theta\right)$, 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 $\delta\tilde{\theta}=\sqrt{P^{(N)}\left(\theta\right)\left(1-P^{(N)}\left(\theta\right)\right)}/\frac{\partial P^{(N)}\left(\theta\right)}{\partial\theta}=\frac{\sqrt{A\left(1-A\right)}}{NB}$.

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 $1.7$ $\mathrm{dB}$ away from the ultimate HL. The results demonstrate the quantum advantage of our single-mode sensing unambiguously. The obtained precision scales as $N^{-0.94}$, approaching the Heisenberg scaling ($N^{-1}$). The slight deviation mainly attributes to the $N$-dependent 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 schemes can be utilized in practical sensing applications, for example, the detection of the frequency and power of a microwave signal based on the Stark-effect-induced phase shift of the sensing mode. By utilizing the recently developed high-efficient bidirectional microwave-to-optical quantum transduction Fan et al. (2018); Higginbotham et al. (2018), our scheme with the MVS can also be employed for the optical metrology. However, the Ramsey-like 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 Matthews et al. (2016).

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 Matthews et al. (2016), 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 $P_{n}^{(N)}=\left|\left\langle n\right|D\left(\alpha\right)U\left(\varphi\right)\left|\Psi\left(N\right)\right\rangle\right|^{2}$, where $D(\alpha)$ is the displacement operator and $U(\varphi)=\mathrm{exp}(-i\varphi a^{\dagger}a)$ is the phase operator. We optimize the parameters $\alpha$ and $n$ for each MVS to maximize the measurement precision (see Supplementary Materials).

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 sub-optimal 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).

Our single-mode 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 Zhou et al. (2018). Our scheme can also be directly applied to other physical systems such as trapped ions Zhang et al. (2018) and nitrogen-vacancy centers Golter et al. (2016). 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 microwave-to-optical up- and down-conversion twice, near-HL precisions with the optimal detecting 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 multiple-path optical interferometer.