Introduction

A closed quantum many-body system is highly sensitive to even small variations in the Hamiltonian parameters near quantum critical points (CPs). This sensitivity is quantified by the divergent susceptibilities of its ground state and directly leads to a divergent quantum Fisher information (QFI), a key metric that characterizes the ultimately achievable precision of a many-body sensor. This observation has motivated manifold criticality-enhanced sensing proposals1,2,3,4,5,6,7,8,9,10,11,12,13, offering promising alternatives towards quantum-enhanced sensitivity beyond conventional schemes that rely on the direct preparation of entangled states14,15,16.

However, typical quantum sensors are intrinsically open systems both because they need to be interrogated by measurements and because they are naturally coupled to environmental noise. Hence, a natural next step for criticality-enhanced sensing is to look at open critical system undergoing dissipative phase transitions17,18,19,20,21,22,23. In analogy to the case of a closed system, at a dissipative CP the steady-state susceptibilities of an open system exhibit universally divergent scaling behavior, enabling the potential of achieving nonclassical precision scaling. Moreover, in sharp contrast to closed systems, open critical systems continuously exchange radiation quanta with their environment, making it possible to sense unknown parameters through continuous measurement of the emission field. This approach naturally avoids the detrimental effect of critical slowing down in steady state preparation1,2,3,4,5,6,7,8,9,10,11,12,13, which needs to be factored into the resource budget of the metrology scheme thereby leading to a considerable reduction in the achievable precision24.

In the innovative framework of sensing via continuous measurement25,26,27,28,29,30,31,32,33,34,35,36,37,38,39, the ultimate precision limit of an open sensor is characterized by the global QFI of the joint sensor and its environment. In a previous work10, we have established a general scaling theory of the global QFI at dissipative CPs, by relating its scaling exponents to the critical exponents of the underlying CP, assuming perfect photon detection and the absence of environmental noise. It remains, however, an important open question whether such scaling is robust against imperfect photon detection and various sources of experimental noise. Obtaining a definitive and positive answer to this question is of utmost importance for the successful implementation of our sensing protocol in real-world applications—particularly in view of the vulnerability of highly entangled states and their nonclassical precision scaling in the presence of experimental noise sources. Moreover, a direct implementation of the sensing protocol of ref. 10 requires highly efficient detection of the emission field emanating from engineered critical open systems. This can typically be accomplished through the use of high finesse optical cavities, which is, however, challenging experimentally15.

Here, we provide a confirmative answer to the first question and we overcome the second challenge with the proposal of a general cavity-free scheme for the highly efficient monitoring of driven-dissipative open quantum sensors, and demonstrate the persistence of criticality enhanced precision scaling for imperfect detection. Our scheme is inspired by entanglement-based spectroscopy40,41,42 and is illustrated in Fig. 1 at the example of detecting the emission quanta of a bosonic mode: to detect a single emission event we correlate the phonon states with an auxiliary spin−\(\frac{1}{2}\) via sideband pulses that achieve the transition \(\left\vert n-1\right\rangle \left\langle n\right\vert \otimes \left\vert e\right\rangle \left\langle g\right\vert\) (\(\left\vert n\right\rangle\) is the bosonic Fock state) such that the event \(\left\vert n\right\rangle \to \left\vert n-1\right\rangle\) is correlated with population transfer \(\left\vert g\right\rangle \to \left\vert e\right\rangle\) of the auxiliary spin. The latter can be measured via standard electron shelving techniques with nearly unit efficiency43. The dissipative channel \(\left\vert e\right\rangle \to \left\vert g\right\rangle\) of the spin restores its internal state thus enabling continuous monitoring of the emission quanta of the bosonic mode with high efficiency. Natural extension of this scheme presents a versatile method for the continuous measurement of complex and interacting synthetic many-body quantum systems, which we demonstrate below using the critical open Rabi model realized with spin and phonons of trapped ions in a Paul trap. We will show that this system functions as a precise quantum sensor by harnessing driven-dissipative criticality and continuous monitoring, allowing for nonclassical precision scaling beyond the standard quantum limit. We showcase the criticality-enhanced precision scaling in terms of sensing the trapping frequency of the Paul trap, and demonstrate its robustness against realistic noise and imperfections. Since the trapping frequency directly reflects the electric field gradient at the ions’ location, our proposed setup can serve as a promising precise sensor device suitable for electric-field imaging and diagnostics.

Fig. 1: Schematic illustration of the proposed criticality-enhanced sensor.
figure 1

The sensor consists of a coupled bosonic mode and qubit system that realizes the critical open Rabi model. The quanta emitted into the bosonic mode can be monitored via a phonon detector module with high efficiency. In particular, via sideband coupling to an auxiliary spin, a single boson emission \(\left\vert n\right\rangle \to \left\vert n-1\right\rangle\) switches the spin from the dark state \(\left\vert g\right\rangle\) to the bright state \(\left\vert e\right\rangle\) measured with standard electron shelving techniques. An additional decay channel \(\left\vert e\right\rangle \to \left\vert g\right\rangle\) restores the internal state of the spin, allowing for continuous phonon counting at nearly perfect effective efficiency. Our measurement scheme can be viewed as an extension of the principle of quantum-logic spectroscopy for the implementation of continuous weak measurements.

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Results

The Open Rabi Model as an Open Critical Sensor

We consider a bosonic mode coupled to a qubit according to the quantum Rabi Hamiltonian (throughout this article ℏ = 1),

$${\hat{H}}_{{{{\rm{R}}}}}=\omega {\hat{c}}^{{\dagger} }\hat{c}+\frac{\Omega }{2}{\hat{\sigma }}_{z}-\lambda (\hat{c}+{\hat{c}}^{{\dagger} }){\hat{\sigma }}_{x},$$
(1)

where \(\hat{c}\)\(({\hat{c}}^{{\dagger} })\) denotes the annihilation (creation) operator of the mode, \({\hat{\sigma }}_{z,x}\) are the Pauli matrices of the qubit, ω is the mode’s frequency, Ω is the qubit transition frequency and λ is the coupling strength of the qubit-mode interaction. We assume that the bosonic mode is damped by a Markovian reservoir at a dissipation rate κ, and thus the dynamics of the open boson-qubit system can be described by a Lindblad master equation5,10,44

$$\dot{\rho }=-i[{\hat{H}}_{{{{\rm{R}}}}},\rho ]+\kappa \left(2\hat{c}\rho {\hat{c}}^{{\dagger} }-\{{\hat{c}}^{{\dagger} }\hat{c},\rho \}\right).$$
(2)

For such a zero-dimensional model, we can introduce the frequency ratio β = Ω/ω as the effective system size, with β → ∞ corresponding to the thermodynamic limit44. In the limit β → ∞, when the dimensionless coupling strength \(g=2\lambda /\sqrt{\omega \Omega }\) is tuned across the critical point (CP) \({g}_{c}=\sqrt{1+{(\kappa /\omega )}^{2}}\), the steady state of Eq. (2) breaks spontaneously the \({{\mathbb{Z}}}_{2}\) parity symmetry (\(\hat{c}\to -\hat{c},{\hat{\sigma }}_{x}\to -{\hat{\sigma }}_{x}\)) of the model, therefore undergoing a continuous dissipative phase transition44: from a normal phase for g < gc, as characterized by the order parameter \({\langle \hat{c}\rangle }_{{{{\rm{st}}}}}=0\), with \({\langle \cdot \rangle }_{{{{\rm{st}}}}}\equiv {{{\rm{tr}}}}[{\rho }_{{{{\rm{st}}}}}(\cdot )]\); to a superradiant phase for g > gc, where \({\langle \hat{c}\rangle }_{{{{\rm{st}}}}}\ne 0\).

The CP is characterized by a few critical exponents which have been extracted via numerical finite-size scaling10,44, in particular z = 1 is the dynamic and ν = 2 is the correlation length critical exponent. Moreover, at the CP the mean excitation of the bosonic mode, \(\hat{n}:= {\hat{c}}^{{\dagger} }\hat{c}\), diverges as \({\langle \hat{n}\rangle }_{{{{\rm{st}}}}} \sim {\beta }^{1/2}\) which identifies the scaling dimension of \(\hat{n}\) to be \({\Delta }_{\hat{n}}=-1/2\). Consequently, in such model the standard quantum limit (SQL) corresponds to SQL \(\sim \sqrt{\beta }\) while the Heisenberg limit (HL) is proportional to HL ~ β.

In ref. 10 we had demonstrated the metrological potential of open critical systems subjected to continuous monitoring at the hand of an illustrative example of sensing the bosonic mode frequency, ω, of the open Rabi model via photon counting. There, it was assumed that an annihilation of a phonon in the bosonic mode is associated with the emission of scattered photons all of which are directed to and counted by a photon detector with detection efficiency ϵ = 1. As a result, the dynamics of the joint boson-qubit system is subjected to measurement backaction conditioned on a series of photon detection events. Specifically, at any infinitesimal time step dτ there is probability \({p}_{1}=2\kappa {\langle {\hat{c}}^{{\dagger} }\hat{c}\rangle }_{c}d\tau\), with \({\langle \cdots \rangle }_{c}:= \langle {\tilde{\psi }}_{c}| \cdots | {\tilde{\psi }}_{c}\rangle /\langle {\tilde{\psi }}_{c}| {\tilde{\psi }}_{c}\rangle\), of photon detection, leading to the collapse of the conditional (unnormalized) state of the joint system, \(\left\vert {\tilde{\psi }}_{c}\right\rangle \to \sqrt{\kappa d\tau }\hat{c}\left\vert {\tilde{\psi }}_{c}\right\rangle\). On the other hand, with probability p0 = 1 − p1 we detect no photon and the system evolves according to the non-unitary evolution \(\left\vert {\tilde{\psi }}_{c}\right\rangle \to \left(\hat{1}-d\tau (i{\hat{H}}_{R}+\kappa {\hat{c}}^{{\dagger} }\hat{c})\right)\left\vert {\tilde{\psi }}_{c}\right\rangle\)45,46,47,48,49,50. Repeating such a stochastic evolution defines a specific quantum trajectory, D(t, 0), consisting of the accumulated photon detection signal up to time t, with probability \(P[D(t,0)]=\langle {\tilde{\psi }}_{c}(t)| {\tilde{\psi }}_{c}(t)\rangle\).

Processing the continuous signal D(t, 0) provides an estimator of the bosonic mode frequency whose achievable precision is represented by the Fisher information (FI) of the detected signal

$${F}_{\omega }(t)=\mathop{\sum}\limits_{D(t,0)}P[D(t,0)]{\left\{{\partial }_{\omega }{{{\rm{\ln }}}}P[D(t,0)]\right\}}^{2}.$$
(3)

where the sum is over all, practically sufficient many, trajectories. Importantly, at the critical point g = gc, the FI in Eq. (3) has been shown to exhibit a universal transient and long-time scaling behavior which surpasses the standard quantum limit10

$${F}_{\omega }(t,\beta )={(\kappa t)}^{2}{f}_{F}(\kappa t/\beta ),\quad \kappa t\,\lesssim\, \beta ,$$
(4)
$${F}_{\omega }(t,\beta )={{{\rm{const.}}}}\times \kappa t\beta ,\quad \kappa t\,\gg\, \beta .$$
(5)

where fF(κt/β) is a universal scaling function reflecting the finite-size correction.

The criticality-enhanced scaling, Eqs. (4) and (5), demonstrates the potential of the critical open Rabi model as a quantum sensor when combined with highly efficient continuous measurements of the emitted quanta. Below we will describe a concrete realization of such a sensor model with trapped ions, where the model parameter ω translates directly to the actual trapping frequency of the ion, thus ultimately revealing the amplitudes and frequencies of unknown electric field gradients at ions’ location. As mentioned in the introduction, a key achievement of our implementation scheme is the highly efficient detection of the emission quanta of the open sensor—corresponding to individual phonon excitation in our trapped-ion implementation—despite a low photon detection efficiency.

Highly efficient continuous monitoring of the open sensor

A promising platform for implementing our critical sensing scheme is a trapped-ion setup depicted in Fig. 2a. We consider two trapped ions in a linear Paul trap sharing a quantized vibrational motion (phonon mode), which we assume is cooled down to the ground state. The two ions can be individually manipulated by lasers via single-ion addressability with focused laser beams51 or via frequency space addressing with a crystal of mixed species52. As described in refs. 53,54,55,55,56, the Rabi Hamiltonian, \({\hat{H}}_{{{{\rm{R}}}}}\), can be implemented by driving the spin transition \(\left\vert \downarrow \right\rangle \to \left\vert \uparrow \right\rangle\) of the system ion with two traveling-wave laser beams with the same Rabi frequency Ω0, and with frequencies slightly detuned from the blue and red-sideband transitions,

$$\begin{array}{rcl}{\omega }_{1}&=&{\omega }_{{{{\rm{I}}}}}+{\omega }_{{{{\rm{ph}}}}}-{\delta }_{{{{\rm{b}}}}},\\ {\omega }_{2}&=&{\omega }_{{{{\rm{I}}}}}-{\omega }_{{{{\rm{ph}}}}}-{\delta }_{{{{\rm{r}}}}}.\end{array}$$
(6)

Here, ωI is the frequency of the ionic spin transition, ωph the center-of-mass (COM) phonon frequency and δb(r) ≪ ωph is a small frequency offset. The transition \(\left\vert \downarrow \right\rangle \to \left\vert \uparrow \right\rangle\) can represent a narrow optical transition subjected to laser driving as in 40Ca+; or a stimulated Raman transition between a pair of hyperfine ground states as in 171Yb+56, which is illustrated in Fig. 2a. After moving to a suitably rotating frame and performing an optical and a vibrational rotating-wave approximation (RWA), the Hamiltonian of the system ion reads (see Supplementary Note 2 for the detailed derivation)

$${\hat{H}}_{{{{\rm{R}}}}}=\frac{{\delta }_{{{{\rm{b}}}}}-{\delta }_{{{{\rm{r}}}}}}{2}{\hat{c}}^{{\dagger} }\hat{c}+\frac{{\delta }_{{{{\rm{r}}}}}+{\delta }_{{{{\rm{b}}}}}}{4}{\hat{\sigma }}_{z}-\frac{{\eta }_{{{{\rm{LD}}}}}{\Omega }_{0}}{2}{\hat{\sigma }}_{x}(\hat{c}+{\hat{c}}^{{\dagger} }),$$
(7)

where \({\eta }_{{{{\rm{LD}}}}}=k/\sqrt{2m{\omega }_{{{{\rm{ph}}}}}}\) is the Lamb-Dicke parameter, with m the ion mass and k the magnitude of the laser wavevector along the direction of the quantized oscillation. Eq. (7) has exactly the same form as the Rabi Hamiltonian in Eq. (1), with the new set of parameters ω = (δb − δr)/2, Ω = (δb + δr)/2 and λ = ηLDΩ0/2. The last term of Eq. (7) formally corresponds to a Mølmer-Sørensen interaction widely used for generating effective spin-spin interaction in an ion crystal57,58. In contrast to refs. 57,58, here we focus on individual trapped ions coupled with selected phonon modes. In our scenario, remarkably, Eq. (7) generates highly squeezed phonon states which ultimately manifests as a phase transition in the soft-phonon limit β ≡ Ω/ω → ∞59. Importantly, all the aforementioned parameters can be adjusted experimentally and thus allowing for tuning the spin-phonon system to the CP experimentally.

Fig. 2: Implementation of the critical quantum sensor using two co-trapped 171Yb+ ions in a linear Paul trap which share a common vibrational (phonon) mode.
figure 2

a The relevant energy levels are shown in the dashed boxes (not to scale). The system ion is driven by two laser beams slightly detuned by δb and δr from the blue and red sideband transition respectively, realizing the Rabi Hamitonian, \({\hat{H}}_{R}\). The ancilla ion is also driven by two laser beams of strength Ωs and Ωw which are tuned to the carrier resonance and slightly detuned from the red sideband transition respectively. The annihilation of the phonon mode triggers the emission of multiple photons that are collected and counted by a photon detector. Finally, an additional laser of strength \({\widetilde{\Omega }}_{s}\), detuned by Δ from the auxiliary transition \(\left\vert e\right\rangle \to \left\vert m\right\rangle\), restores the internal state of the ancilla ion with a controllable rate and thus accomplishing continuous phonon detection with near unit efficiency. b The most detrimental noise sources in the trapped ion setup are: (i) motional diffusion which results in effective phonon heating and cooling with rate Γh and Γc respectively, (ii) motional decoherence with rate Γm and (iii) spin dephasing of the system ion at a rate Γdph.

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Hence, our implementation realizes the parameter ω of the Rabi model as the differential detuning of the laser beams, which is directly related to the trapping frequency via Eq. (6),

$$\omega =\frac{{\delta }_{{{{\rm{b}}}}}-{\delta }_{{{{\rm{r}}}}}}{2}\equiv \frac{{\omega }_{2}-{\omega }_{1}}{2}+{\omega }_{{{{\rm{ph}}}}}.$$
(8)

As the frequency difference between the two lasers can be accurately controlled, estimating ω is equivalent to estimating the trapping frequency ωph. The access to ωph provides us with a versatile tool for sensing unknown electric field gradients, as detailed in later sections.

Our sensing proposal further requires controlled dissipation of the phonon mode and efficient continuous detection of the phonons, both of which can be achieved via an ancilla ion, cf. Fig. 1. In particular, we assume that the ancilla ion is driven by two additional laser beams: one slightly detuned from the red phonon sideband \(\left\vert g\right\rangle \left\vert n\right\rangle \leftrightarrow \left\vert e\right\rangle \left\vert n-1\right\rangle\), with a detuning Δw = − ωph − ω; and a second one on resonance with the carrier transition \(\left\vert e\right\rangle \left\vert n-1\right\rangle \leftrightarrow \left\vert d\right\rangle \left\vert n-1\right\rangle\), i.e., Δs = 0. The first laser drives the cooling transition with strength \(\sqrt{n}{\eta }_{{{{\rm{LD}}}}}^{(2)}{\Omega }_{{{{\rm{w}}}}}\), where \({\eta }_{{{{\rm{LD}}}}}^{(2)}\) is the Lamb-Dicke parameter of the ancilla ion; the second laser realizes the strong carrier transition with Rabi frequency Ωs. In the regime Ωs, Γs ≫ Ωw, Γw, where Γs(w) is the spontaneous emission rate of the strong (weak) transition \(\left\vert d\right\rangle \to \left\vert e\right\rangle \,\)\((\left\vert e\right\rangle \to \left\vert g\right\rangle )\), a controllable dissipation rate \(\kappa \,\approx\, {\Gamma }_{{{{\rm{s}}}}}{\Omega }_{{{{\rm{w}}}}}^{2}/{\Omega }_{{{{\rm{s}}}}}^{2}\) is realized (see Supplementary Note 1), where each phonon annihilation is accompanied with the emission of a significant number of photons from the strongly driven transition which in turn are collected and counted by a photon detector with efficiency ϵ. We note that although in this section we assume that the state \(\left\vert e\right\rangle\) relaxes directly to \(\left\vert g\right\rangle\), in practice the weak \(\left\vert e\right\rangle \to \left\vert g\right\rangle\) transition, necessary for the restoration of the internal state of the ancilla ion, may be accomplished through an auxiliary excited state \(\left\vert m\right\rangle\), as illustrated in Fig. 2a and analyzed below. This results in the efficient continuous measurement of the phonon mode, featuring an enhancement factor (see Supplementary Note 3)

$${N}_{{{{\rm{ph}}}}}\equiv \frac{\,{{\mbox{detected photon \#}}}}{{{\mbox{annihilated phonon \#}}}\,}\,\approx\, \epsilon \frac{{\Gamma }_{{{{\rm{s}}}}}}{{\Gamma }_{{{{\rm{w}}}}}}.$$
(9)

Quantitatively, the conditional dynamics of the system and ancilla ions interacting with the phonon mode can be described by a stochastic master equation (SME)45,46,47,48,49,50 of the unnormalized joint state (see Supplementary Note 3)

$$\begin{array}{lll}d{\tilde{\rho }}_{{{{\rm{c}}}}}\,=\,\Big(-i[{\hat{H}}_{{{{\rm{R}}}}}+{\hat{H}}_{{{{\rm{a}}}}},{\tilde{\rho }}_{{{{\rm{c}}}}}]\\\qquad\quad\; -\frac{{\Gamma }_{{{{\rm{w}}}}}}{2}\{\left\vert e\right\rangle \left\langle e\right\vert ,{\tilde{\rho }}_{{{{\rm{c}}}}}\}+{\Gamma }_{{{{\rm{w}}}}}\left\vert g\right\rangle \left\langle e\right\vert {\rho }_{{{{\rm{c}}}}}^{{\prime} }\left\vert e\right\rangle \left\langle g\right\vert \\\qquad\quad\;-\frac{{\Gamma }_{{{{\rm{s}}}}}}{2}\{\left\vert d\right\rangle \left\langle d\right\vert ,{\tilde{\rho }}_{{{{\rm{c}}}}}\}+(1-\epsilon ){\Gamma }_{{{{\rm{s}}}}}\left\vert e\right\rangle \left\langle d\right\vert {\rho }_{{{{\rm{c}}}}}^{{\prime} }\left\vert d\right\rangle \left\langle e\right\vert \Big)dt\\\qquad\quad\;+(\epsilon {\Gamma }_{{{{\rm{s}}}}}\left\vert e\right\rangle \left\langle d\right\vert {\rho }_{{{{\rm{c}}}}}^{{\prime} }\left\vert d\right\rangle \left\langle e\right\vert -{\tilde{\rho }}_{{{{\rm{c}}}}})dN(t),\end{array}$$
(10)

where \({\hat{H}}_{{{{\rm{a}}}}}={\Omega }_{{{{\rm{s}}}}}(\left\vert d\right\rangle \langle e| +| e\rangle \left\langle d\right\vert )+{\eta }_{{{{\rm{LD}}}}}^{(2)}{\Omega }_{{{{\rm{w}}}}}(\left\vert e\right\rangle \langle g| \hat{c}+| g\rangle \left\langle e\right\vert {\hat{c}}^{{\dagger} })\) is the Hamiltonian of the ancilla ion in the appropriate rotating-frame. Here, we have introduced the notation \({\rho }_{{{{\rm{c}}}}}^{{\prime} }=\int\nolimits_{-1}^{1}duN(u){e}^{-i{\eta }_{{{{\rm{LD}}}}}^{(2)}(\hat{c}+{\hat{c}}^{{\dagger} })u}{\tilde{\rho }}_{{{{\rm{c}}}}}{e}^{i{\eta }_{{{{\rm{LD}}}}}^{(2)}(\hat{c}+{\hat{c}}^{{\dagger} })u}\) with N(u) being the angular distribution of the emitted photons. Such term accounts for the momentum transfer by spontaneous emission to the center of mass motion of the ions when the excitation returns from the \(\left\vert d\right\rangle\) to the \(\left\vert e\right\rangle\) state or due to the transition \(\left\vert e\right\rangle \to \left\vert g\right\rangle\). Although the effect of such process is explicitly analyzed later as one of the dominant noise sources of our detection scheme, in the remaining part of this section let us temporarily neglect it and study the ideal performance of our measurement scheme in the absence of noise. Moreover, we note that in Eq. (10)dN(t) is a stochastic Poisson increment which, similar to the ideal case of perfect detection, can take two values: if there is a photon detected, dN(t) = 1 with a modified probability \({\tilde{p}}_{1}=\epsilon {\Gamma }_{{{{\rm{s}}}}}{{{\rm{tr}}}}\{\left\vert d\right\rangle \left\langle d\right\vert {\rho }_{{{{\rm{c}}}}}\}dt\), where \({\rho }_{{{{\rm{c}}}}}\equiv {\tilde{\rho }}_{{{{\rm{c}}}}}/{{{\rm{tr}}}}({\tilde{\rho }}_{{{{\rm{c}}}}})\) is the normalized conditional density matrix, while if there is no photon detection dN(t) = 0 with probability \({\tilde{p}}_{0}=1-{\tilde{p}}_{1}\). Taking the ensemble average over all conditional states and after elimination of the ancilla ion leads to the definition of the joint system ion–phonon state which evolves according to Eq. (2).

The enhanced metrological precision via our measurement scheme is demonstrated in Fig. 3a, which shows the FI for the estimation of the frequency of the phonon mode, Fω(t), at various enhancement factors, Nph. As illustrated, increasing Nph leads to larger FI and thus enhanced precision, which can in principle achieve the ideal precision for perfect photon detection. In the experimental relevant case of photon detector efficiency ϵ = 0.01 and Nph ≈ 80ϵ, we perform a numerical finite-size scaling analysis for different experimentally accessible system sizes β (see below), as shown in Fig. 3c where the perfect data collapse indicates that the Fω(t) for ϵph < 1 follows a similar transient and long time behavior as in Eqs. (4) and (5). In the following we analyze a concrete experimental realization of our measurement scheme, shown in Fig. 1, and we examine its robustness against various experimental noise sources.

Fig. 3: Numerical estimation of Fisher information for photon counting signals with imperfect photon detectors.
figure 3

a The behavior of the FI, Fω(t), in comparison with the ideal case of ϵ = 1 (black dashed line) for ϵ = 0.1, Γs/Γw = 80 → Nph ≈ 8 (red dashed-dotted line); ϵ = 0.01, Γs/Γw = 80 → Nph ≈ 0.8 (blue solid line); ϵ = 0.01, Γs/Γw = 10 → Nph ≈ 0.1 (orange dotted line). In all curves Γw = 40, Ωw = 14.3Γw, Ωs = 2Γs and \({\eta }_{{{{\rm{LD}}}}}^{(2)}=0.07\). The above set of parameters for the ancilla ion gives rise to a phonon dissipation rate κ = 1.5 × 10−3Γw. The system ion is tuned to the dissipative CP and we choose ω = 10κ, β = 50. b Long time behavior of the achieved FI with respect to the effective system size β (blue line). For comparison we plot the scaling of FI that corresponds to SQL10 (orange line) obtained with the use of a coherent state of the bosonic model (see main text). c Finite size scaling analysis of Fω(t) averaged over 104 trajectories for the case of ϵ = 0.01 and Γs/Γw = 80. The remaining parameters are the same as in Fig. 3a.

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Parameters for experimental implementation

Various ion species can be used for implementing our measurement scheme, and here, as a concrete example, we consider 171Yb+ as the ancilla ion, whose internal level structure is shown in Fig. 2a. We choose from the ground 2S1/2 manifold the hyperfine states \(\left\vert g\right\rangle \equiv \left\vert F=0,{m}_{F}=0\right\rangle\) and \(\left\vert e\right\rangle \equiv \left\vert F=1,{m}_{F}=0\right\rangle\) to implement the phonon sideband cooling, cf. Fig. 1, with a frequency difference ΔHFS ≈ 2π × 12.6 GHz, while the readout state is chosen as \(\left\vert d\right\rangle \equiv \left\vert F=0,{m}_{F}=0\right\rangle\) from the excited 2P1/2 manifold. The cooling transition \(\left\vert g\right\rangle \left\vert n\right\rangle \to \left\vert e\right\rangle \left\vert n-1\right\rangle\) can be accomplished either by exploiting techniques utilizing microwave fields on resonance with the red phonon sideband60,61, or more straightforwardly by stimulated Raman transition at a two-photon detuning ΔHFS − ωph62. The strong cycling transition \(\left\vert e\right\rangle \leftrightarrow \,\)\(\left\vert d\right\rangle\), with a spontaneous emission rate Γs = 2π × 19.6 MHz of the 2P1/2 manifold, can be driven by a laser beam at 369.5 nm with adjustable Rabi frequency Ωs, cf. Fig. 2a. Since the \(\left\vert d\right\rangle \to \left\vert g\right\rangle\) emission channel is forbidden by dipole selection rules, an additional laser of strength \({\widetilde{\Omega }}_{{{{\rm{s}}}}}\), detuned by Δ from the \(\left\vert e\right\rangle \to \left\vert m\right\rangle \equiv\)\({}^{2}{P}_{1/2}\left\vert F=1,{m}_{F}=0\right\rangle\) transition, induces an effective decay \(\left\vert e\right\rangle \to \left\vert g\right\rangle\) at an adjustable rate63,

$${\Gamma }_{{{{\rm{w}}}}}\approx \frac{{\Gamma }_{{{{\rm{s}}}}}}{2}\frac{{(2{\widetilde{\Omega }}_{{{{\rm{s}}}}}/{\Gamma }_{{{{\rm{s}}}}})}^{2}}{1+{\left(\frac{2{\widetilde{\Omega }}_{{{{\rm{s}}}}}}{{\Gamma }_{{{{\rm{s}}}}}}\right)}^{2}+{\left(\frac{2\Delta }{{\Gamma }_{{{{\rm{s}}}}}}\right)}^{2}}.$$
(11)

As described in refs. 55,56, the two states of the ground state manifold of a second 171Yb+ ion can be chosen as the qubit states, which together with the spatial motion of the ion along one of its principle axis, with frequency ωph = 2π × 2.35 MHz, provide the two degrees of freedom for implementing the Rabi Hamiltonian. A typical set of experimental parameters is ω ≈ 2π × 2 kHz and \({({\eta }_{{{{\rm{LD}}}}}{\Omega }_{0})}_{\max }\approx 2\pi \times 20{{{\rm{kHz}}}}\) with ηLD = 0.07. Therefore, effective system size of β ≲ 100 can be easily achieved. To achieve even larger β > 100, a stronger Rabi frequency Ω0 is required to tune the system to the CP, which may ultimately break the vibrational RWA and modifies the resulting Hamiltonian of the system ion. This can be overcome by suppressing the corresponding carrier transition, e.g., by using standing wave configuration54 or exploiting the ac Stark shift of traveling waves64 to implement the sideband transitions, which allows for exploring the critical physics close to the thermodynamic limit. As a possible implementation of the ancilla-enhanced continuous readout, we consider that \({\eta }_{{{{\rm{LD}}}}}^{(2)}{\Omega }_{{{{\rm{w}}}}}\,\approx\, 2\pi \times 150{{{\rm{kHz}}}}\) with \({\eta }_{{{{\rm{LD}}}}}^{(2)}=0.07\) and Γw can be adjusted to Γw = Γs/80 = 2π × 245 kHz. These realistic parameters result in a phonon dissipation rate κ = 2π × 200 Hz with Nph ≈ 80ϵ (see Supplementary Note 3) which, remarkably, allow for a clear demonstration of the criticality enhanced precision scaling as shown in Fig. 3b.

Robustness against experimental noise

Noise can be detrimental to any quantum enhanced sensing protocol and certainly our proposal would be incomplete without a systematic study of the impact of realistic noise sources, found in the experiments, in the performance of our proposed sensor setup. As illustrated in Fig. 2b, the main relevant experimental noise sources are (i) spin decoherence of the system ion at an effective rate Γdph, including also the effect of laser frequency noise, (ii) motional decoherence of the phonon mode at a rate Γm, and (iii) motional diffusion of the phonon mode mainly caused by photon recoil in the strong transition \(\left\vert d\right\rangle \left\vert n\right\rangle \to \left\vert e\right\rangle \left\vert n-1\right\rangle\) of the ancilla ion (at a rate Γs, cf. Fig. 2). In the sideband resolved regime, such photon recoil results in effective phonon heating and cooling, \(\left\vert d\right\rangle \left\vert n\right\rangle \to \left\vert e\right\rangle \left\vert n\pm 1\right\rangle\), at a rate Γh and Γc respectively. The rates are given by \({\Gamma }_{h}\approx {\Gamma }_{c}\approx \frac{2}{5}R{({\eta }_{{{{\rm{LD}}}}}^{(2)})}^{2}\) where \(R=\frac{{N}_{{{{\rm{ph}}}}}}{\epsilon }\kappa\) is the rate of the scattered photons and \(\frac{2}{5}{\eta }_{{{{\rm{LD}}}}}^{(2)}\) is the probability to decay in a sideband65. Note that all the above imperfections do not affect either the effective dissipation rate of the phonon mode κ or the highly efficient ancilla-assisted continuous phonon counting. Consequently, for the sake of numerical efficiency, we examine the effect of noise under the assumption of perfect photon detection efficiency ϵ = 1. In this case the SME of the unnormalised system ion-phonon state reads

$$\begin{array}{lll}d{\tilde{\rho }}_{{{{\rm{c}}}}}\,=\,\Big(-i[{\hat{H}}_{R},{\tilde{\rho }}_{{{{\rm{c}}}}}]+{\Gamma }_{{{{\rm{dph}}}}}{{{\mathcal{L}}}}[{\hat{\sigma }}_{z}]+{\Gamma }_{{{{\rm{m}}}}}{{{\mathcal{L}}}}[{\hat{c}}^{{\dagger} }\hat{c}]\\\qquad\quad\;+\,{\Gamma }_{{{{\rm{h}}}}}{{{\mathcal{L}}}}[{\hat{c}}^{{\dagger} }]+{\Gamma }_{{{{\rm{c}}}}}{{{\mathcal{L}}}}[\hat{c}]-\kappa \{{\hat{c}}^{{\dagger} }\hat{c},{\tilde{\rho }}_{{{{\rm{c}}}}}\}\Big)dt\\\qquad\quad\;+\,(2\kappa \hat{c}{\tilde{\rho }}_{{{{\rm{c}}}}}{\hat{c}}^{{\dagger} }-{\tilde{\rho }}_{{{{\rm{c}}}}})dN(t)\end{array}$$
(12)

where \({{{\mathcal{L}}}}[\hat{x}]\equiv \hat{x}{\tilde{\rho }}_{{{{\rm{c}}}}}{\hat{x}}^{{\dagger} }-\frac{1}{2}\{{\hat{x}}^{{\dagger} }\hat{x},{\tilde{\rho }}_{{{{\rm{c}}}}}\}\). We take typical experimental numbers Γdph = 2π × 1Hz and Γm = Γh = Γc = Γ = 2π × 20 Hz56,66. Consequently, fixing Γdph = κ/200 and Γ = κ/10 while keeping the rest of the parameters the same as Fig. 3, we show the resulting FI Fω(t) in Fig. 4a. The results indicate that although the various noise sources degrades the perfect data collapse, a scaling behavior similar to the noiseless case persists. The quality of the data collapse can be further quantified by the dimensionless quality factor Q67 that captures the mean relative spread among different sets of data (see Methods). We identify that for the current state-of-the-art experimental setups the main impact emanates from the motional decoherence and diffusion. As shown in Fig. 4b, Q ~ 1 persists for small motional decoherence rate, indicating that the associated Fisher information follows the same scaling behavior as the noiseless case. This demonstrates the feasibility of our sensing scheme under realistic experimental imperfections.

Fig. 4: Performancce under noise.
figure 4

a Finite size scaling analysis of Fω(t) for ϵ = 1 including noise with Γdph = κ/200 and Γm = Γh = Γc = κ/10. The rest of the parameters are the same as in Fig. 3. Each data point represents an average over 5 × 103 independent trajectories. b The quality of data collapse, Q, in comparison with the noiseless scenario. For Γ ≤ κ/40, the Fisher information, Fω(t), follows the same scaling behavior as predicted for the noiseless case.

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Performance for sensing external electric field gradients

The proposed sensor utilizes the COM phonon mode of the co-trapped ions along the radial x-direction to implement the bosonic degrees of freedom of the Rabi Hamiltonian, Eq. (7), and the associated dissipation and continuous readout. In a linear Paul trap, the radial trapping frequency ωph is determined via the trap parameters as43

$${\omega }_{{{{\rm{ph}}}}}=\sqrt{\frac{4q}{m{\Omega }_{{{{\rm{rf}}}}}^{2}}\left(\frac{{U}_{r}}{{R}^{2}}-\gamma {U}_{0}\right)+\frac{1}{2}{\left(\frac{2q{V}_{0}}{{\Omega }_{{{{\rm{rf}}}}}^{2}m{R}^{2}}\right)}^{2}}\frac{{\Omega }_{{{{\rm{rf}}}}}}{2}.$$
(13)

Here, q and m is the charge and the mass of the ions, R is the distance between the trap center and the surface of the electrodes and γ is a geometric factor. As in standard linear Paul traps, we consider the application of a radio-frequency potential \(V={U}_{r}+{V}_{0}\cos ({\Omega }_{{{{\rm{rf}}}}}t)\) between diagonally opposite electrodes of the trap, with Ur(V0) being its dc(ac) component. This results in a quadrupolar potential near the trap center \({\Phi }_{{{{\rm{rf}}}}}=\frac{1}{2}\left[{U}_{r}+{V}_{0}\cos ({\Omega }_{{{{\rm{rf}}}}}t)\right](1+\frac{{x}^{2}-{y}^{2}}{{R}^{2}})\). Meanwhile, a dc voltage U0 in the outer segments of the electrodes realizes a static trapping potential \({\Phi }_{{{{\rm{st}}}}}=\gamma {U}_{0}({z}^{2}-\frac{{x}^{2}+{y}^{2}}{2})\). Consequently, our scheme is capable of sensing variations in the pseudopotential’s curvature, ωph, resulted either by small drifts of the frequency Ωrf due to different technical noise sources, or external electric field gradients (static or oscillating at frequency Ωrf) at the ions’ location.

To verify the high sensitivity of the proposed sensor, let us analyze its performance under typical parameter choices as in Fig. 3 and an experimentally achievable effective system size β ≃ 50. From Fig. 3a, it is clear that for the experimentally reachable system size β = 50 we obtain Fωω2 ≈ 36 × 102 for a realistic evolution time κt ≈ 200. As a result, the sensitivity for measuring the model parameter ω is \(\frac{\Delta \omega }{\omega }=\frac{1}{{F}_{\omega }\omega }\approx 1.6\times 1{0}^{-2}\). This directly translates to the sensitivity of measurement of the trap frequency

$$\frac{\Delta {\omega }_{{{{\rm{ph}}}}}}{{\omega }_{{{{\rm{ph}}}}}}=\frac{\Delta \omega }{\omega }\frac{\omega }{\frac{{\omega }_{1}-{\omega }_{2}}{2}+\omega }\approx 1.6\times 1{0}^{-5},$$
(14)

where ω1(2) ~ 2π × 12.6 GHz for 171Yb+, and we have assumed an accurate control of the two laser beams in implementing the Rabi Hamiltonian. We note that such sensitivity is comparable to the one achieved using excited motional Fock states in a single run of experiments as described in ref. 68. In such approach the trap’s frequency, ωz, can be estimated by detecting a residual displacement \(\tilde{\alpha }\) where the interrogation time of each experimental run lasts tf ≈ 7.8 ms. Consequently, using the reported experimental numbers and taking into account that in our main text we have assumed a scheme lasting t ≈ 0.16 s, leads to sensitivity \(| \frac{\Delta {\omega }_{{{{\rm{z}}}}}}{{\omega }_{{{{\rm{z}}}}}}| \approx 6\times 1{0}^{-6}\) which is similar to the one achieved with our protocol as derived in Eq. (14).

Furthermore, our scheme, besides its use for estimating the trap frequency and its direct connection with electric field gradients, can be particularIy useful for measuring other physical parameters of interest characterizing the potential applied in the electrodes. For example, if we are interested in estimating U0 and assuming Ur = 0, using Eq. (13) we straightforwardly obtain

$$\left| \frac{\Delta {U}_{0}}{{U}_{0}}\right| =8\frac{\Delta {\omega }_{{{{\rm{ph}}}}}}{{\omega }_{{{{\rm{ph}}}}}}\left| \frac{{\omega }_{{{{\rm{ph}}}}}}{A-4{\omega }_{{{{\rm{ph}}}}}^{2}}\right| ,$$
(15)

where \(A:= \frac{{\Omega }_{{{{\rm{rf}}}}}^{2}}{2}{(\frac{2q{V}_{0}}{{\Omega }_{{{{\rm{rf}}}}}^{2}m{R}^{2}})}^{2}=4({\omega }_{{{{\rm{z}}}}}^{2}-{\omega }_{{{{\rm{ph}}}}}^{2})\) with \({\omega }_{{{{\rm{z}}}}}=\sqrt{2q\gamma {U}_{0}/m}\) being the frequency of the COM mode in the axial direction. Using typical experimental numbers where ωz/ωph ≈ 0.169 we obtain

$$\left| \frac{\Delta {U}_{0}}{{U}_{0}}\right| \approx 3.2\times 1{0}^{-3}.$$
(16)

Importantly, in our proposal the Fisher information and thus the precision of estimating the unknown parameter ω, or equivalently ωph, scales as Fω(t → ∞) ~ βt ~ N2t, i.e., quadratically with respect to the emitted photons as clearly shown in Fig. 3b. Such a Heisenberg scaling outperforms significantly the one that can be achieved for sensing the trap frequency via the preparation of a coherent state with N average excitations of the phonon mode, where the QFI scales as \({I}_{{{{\rm{coh}}}}} \sim N \sim \sqrt{\beta }\) corresponding to the SQL; cf Fig. 3b. We emphasize that while such a Heisenberg scaling can be achieved by using the highly non-classical state \({\left\vert \psi \right\rangle }_{{{{\rm{nc}}}}}=\frac{1}{\sqrt{2}}(\left\vert 0\right\rangle +\left\vert N\right\rangle )\) of the phonon mode, the preparation. of such state for large enough N is however very challenging in practice.

Finally, in the proposed continuous-monitoring setting, the processing the continuously detected signal D(t, 0) in a single experimental run via Eq. (10) allows us to construct the likelihood function Lω[D(t, 0)] = Pω[D(t, 0)]/∫dωPω[D(t, 0)] which in turn provides us with an estimate of the unknown parameter ω via a simple maximum-likelihood strategy as illustrated in Fig. 5.

Fig. 5: Extracting the information of the frequency of the bosonic mode via continuous measurement.
figure 5

We consider sensing the bosonic mode’s frequency ω via the detection of the emitted photons from the strong transition of the ancilla’s ion as described in the main text. Processing the simulated continous signal D(t, 0) (upper panel) via Eq. (10) allow us to construct the likelihood function Lω[D(t, 0)] shown for {κt1, κt2} = {424, 1197}. In turn, a maximum-likelihood strategy provides us with an estimate value ωest(t) shown here for t = t2. The rest of the parameters are the same as in Fig. 3 with ϵ = 0.01.

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Discussion

Our study has proposed and analyzed the implementation of criticality-enhanced quantum sensing with single trapped ions via continuous measurement. A key innovation of our scheme is the use of a co-trapped ancillary ion as a detection module, allowing for highly efficient continuous monitoring (e.g., phonon counting) of the critical sensor. We have shown that the criticality enhanced precision scaling persists for imperfect detection efficiency, and is robust against various sources of noise in realistic experimental setups. Our proposal allows for a clear demonstration of criticality-enhanced precision scaling well beyond the standard quantum limit, paving the way for harnessing driven-dissipative criticality to build next generation ultra-precise sensor devices, which may play a key role in electric-field imaging70 and noise diagnosing near metallic surfaces71.

Methods

Numerical calculation of the classical Fisher Information

As explained in the Results, in our protocol the measured signal up to time T = ndt, D(t, 0), consists of a collection of the detected photons emitted from the strong transition \(\left\vert d\right\rangle \to \left\vert e\right\rangle\) i.e., D(t, 0) ≔ {dN(ndt), ⋯ , dN(dt), dN(0)}, where dN(t) is a stochastic increment which for every time step dt can take two values: dN(t) = 1 with a modified probability \({\tilde{p}}_{1}=\epsilon {\Gamma }_{{{{\rm{s}}}}}{{{\rm{tr}}}}\{\left\vert d\right\rangle \left\langle d\right\vert {\rho }_{{{{\rm{c}}}}}\}dt\), where \({\rho }_{{{{\rm{c}}}}}\equiv {\tilde{\rho }}_{{{{\rm{c}}}}}/{{{\rm{tr}}}}({\tilde{\rho }}_{{{{\rm{c}}}}})\) is the normalized conditional density matrix, while if there is no photon detection dN(t) = 0 with probability \({\tilde{p}}_{0}=1-{\tilde{p}}_{1}\). Such a conditional evolution gives rise to the SME of the unnormalized-state written in Eq. (10). The process of such signal gives rise to an estimator value of the bosonic mode frequency, ω, whose precision in quantified by the classical Fisher Information (FI) as written in Eq. (3):

$${F}_{\omega }(t)=\mathop{\sum}\limits_{D(t,0)}P[D(t,0)]{\left\{{\partial }_{\omega }{{{\rm{\ln }}}}P[D(t,0)]\right\}}^{2}.$$
(17)

Crucially, the FI can be calculated numerically by approximating the ensemble average ∑D(t, 0)P[D(t, 0)] by a statistical average over sufficient (but finite) number of trajectories which are generated by typical Monte Carlo techniques following the numerical propagation of the normalized version of Eq. (10). For each generated trajectory we can in turn calculate the respective probability \(P[D(t,0)]={{{\rm{tr}}}}\{{\tilde{\rho }}_{{{{\rm{c}}}}}\}\) for slightly different values of ω up to an arbitrary multiplication factor, C, of our choice. Consequently, the \({\left\{{\partial }_{\omega }{{{\rm{\ln }}}}P[D(t,0)]\right\}}^{2}\) can be extracted by a subsequent numerical differentiation (which is independent of C) and thus the FI can be estimated. We note that although straightforward, such approach requires the propagation of a SME which can be numerically very demanding especially for large system sizes.

As an alternative, we follow an equivalent strategy which gives rise to the same FI but requires only the propagation of a stochastic Schrödinger equation (SSE) and thus is numerically more efficient. In particular, we unravel the detected strong transition \(\left\vert d\right\rangle \to \left\vert e\right\rangle\) with detection efficiency ϵ and rate Γs, into two possible channels: (i) one which is associated to the collapse of the conditional unnormalized state of the system into a"bright" channel \(\left\vert {\tilde{\Psi }}_{{{{\rm{c}}}}}\right\rangle \to \sqrt{\epsilon {\Gamma }_{{{{\rm{s}}}}}dt}\left\vert e\right\rangle \langle d| {\tilde{\Psi }}_{{{{\rm{c}}}}}\rangle\) with probability pb = ϵΓsdt〈Ψc∣d〉〈d∣Ψc〉 where \(\left\vert {\Psi }_{c}\right\rangle := \left\vert {\tilde{\Psi }}_{c}\right\rangle /\sqrt{\langle {\tilde{\Psi }}_{{{{\rm{c}}}}}| {\tilde{\Psi }}_{{{{\rm{c}}}}}\rangle }\) and (ii) a second which is associated to the collapse of the conditional unnormalized state of the system into a “dark" channel \(\left\vert {\tilde{\Psi }}_{{{{\rm{c}}}}}\right\rangle \to \sqrt{(1-\epsilon ){\Gamma }_{{{{\rm{s}}}}}dt}\left\vert e\right\rangle \langle d| {\tilde{\Psi }}_{c}\rangle\) with probability pd = (1 − ϵ)Γsdt〈Ψc∣d〉〈d∣Ψc〉. Following a quantum jump approach45 we further unravel the undetected weak transition \(\left\vert e\right\rangle \to \left\vert g\right\rangle\) into individual quantum trajectories where at each time step there is probability pun = Γwdt〈Ψc∣e〉〈e∣Ψc〉 that the state of the system collapses into the ground state i.e., \(\left\vert {\tilde{\Psi }}_{{{{\rm{c}}}}}\right\rangle \to \sqrt{{\Gamma }_{{{{\rm{w}}}}}dt}\left\vert g\right\rangle \langle e| {\tilde{\Psi }}_{{{{\rm{c}}}}}\rangle\). Consequently, at any infinitesimal time step dt there is probability pb that the systems collapses into the “bright" channel, probability pd that it collapses into the “dark" channel, probability pun that it collapses to the ground state and finally probability \({p}_{0}^{{\prime} }=1-{p}_{{{{\rm{b}}}}}-{p}_{{{{\rm{d}}}}}-{p}_{{{{\rm{un}}}}}\) to evolve according to the non-Hermitian evolution \(\left\vert {\tilde{\Psi }}_{{{{\rm{c}}}}}\right\rangle \to \left(\hat{1}-idt{\hat{H}}_{{{{\rm{eff}}}}}\right)\left\vert {\tilde{\Psi }}_{{{{\rm{c}}}}}\right\rangle\) where \({\hat{H}}_{{{{\rm{eff}}}}}={\hat{H}}_{{{{\rm{R}}}}}+{\hat{H}}_{{{{\rm{a}}}}}-i{\Gamma }_{{{{\rm{w}}}}}\left\vert e\right\rangle \langle e| -i\epsilon {\Gamma }_{{{{\rm{s}}}}}| d\rangle \langle d| -i(1-\epsilon ){\Gamma }_{{{{\rm{s}}}}}| d\rangle \left\langle d\right\vert\). Quantitatively, the dynamics of the conditional unnormalized state, \(\left\vert {\tilde{\Psi }}_{{{{\rm{c}}}}}\right\rangle\), can described in terms of a SSE

$$\begin{array}{lll}d\left\vert {\tilde{\Psi }}_{{{{\rm{c}}}}}\right\rangle \,=\,-idt{\hat{H}}_{{{{\rm{eff}}}}}\left\vert {\tilde{\Psi }}_{{{{\rm{c}}}}}\right\rangle +d{N}_{{{{\rm{b}}}}}(t)\left(\sqrt{\epsilon {\Gamma }_{{{{\rm{s}}}}}dt}\left\vert e\right\rangle \left\langle d\right\vert -\hat{1}\right)\left\vert {\tilde{\Psi }}_{{{{\rm{c}}}}}\right\rangle \\\qquad\qquad\;+\,d{N}_{d}(t)\left(\sqrt{(1-\epsilon ){\Gamma }_{{{{\rm{s}}}}}dt}\left\vert e\right\rangle \left\langle d\right\vert -\hat{1}\right)\left\vert {\tilde{\Psi }}_{{{{\rm{c}}}}}\right\rangle \\\qquad\qquad\;+\,d{N}_{{{{\rm{un}}}}}(t)\left(\sqrt{{\Gamma }_{{{{\rm{w}}}}}dt}\left\vert g\right\rangle \left\langle e\right\vert -\hat{1}\right)\left\vert {\tilde{\Psi }}_{{{{\rm{c}}}}}\right\rangle ,\end{array}$$
(18)

where dNj(t) with j ≔ {b, d, un}is a stochastic increment associated with the respective collapse of the system and can take two values: dNj(t) = 1 with probability pj and dNj(t) = 0 with probability 1 − pj. Notice that the probability of two counts in a time interval dt vanishes faster than dt and thus at any time interval we have at most one count in of the possible channels. In such approach the measured signal, \({D}^{{\prime} }(t,0)\), and thus the single trajectory generated via Monte Carlo methods is a collection of which “jump" occurred at each time step i.e., \({D}^{{\prime} }(t,0):= \{\left(d{N}_{{{{\rm{b}}}}}(ndt),d{N}_{{{{\rm{d}}}}}(ndt),d{N}_{{{{\rm{un}}}}}(ndt)\right),\cdots \left(d{N}_{{{{\rm{b}}}}}(dt),d{N}_{{{{\rm{d}}}}}(dt),d{N}_{{{{\rm{un}}}}}(dt)\right),\left(d{N}_{{{{\rm{b}}}}}(0),d{N}_{{{{\rm{d}}}}}(0),d{N}_{{{{\rm{un}}}}}(0)\right)\}\) and requires only the numerical propagation of the normalized version of the SSE in Eq. (18). Fixing the trajectory we can in turn neglect the “jumps" happened in the “dark" or the undetected channel, construct the trajectory \({D}_{{{{\rm{b}}}}}^{{\prime} }(t,0):= \{d{N}_{{{{\rm{b}}}}}(ndt)\cdots \,,d{N}_{{{{\rm{b}}}}}(0)\}\subseteq {D}^{{\prime} }(t,0)\) corresponding only to the collapse of the system in the “bright" channel and finally calculate numerically \({\left\{{\partial }_{\omega }{{{\rm{\ln }}}}P[{D}_{b}^{{\prime} }(t,0)]\right\}}^{2}\). Therefore, we construct the corresponding FI

$${F}_{\omega }^{{\prime} }(t)=\mathop{\sum}\limits_{{D}^{{\prime} }(t,0)}P[{D}^{{\prime} }(t,0)]{\left\{{\partial }_{\omega }{{{\rm{\ln }}}}P[{D}_{{{{\rm{b}}}}}^{{\prime} }(t,0)]\right\}}^{2},$$
(19)

which is equal to Eq. (17), i.e., \({F}_{\omega }^{{\prime} }(t)={F}_{\omega }(t)\), for sufficiently enough numbers of generated trajectories.

Quantifying the quality of the scaling collapse

The finite-size scaling analysis is a powerful numerical method to extract the relevant exponents of continuous phase transitions. Consider a general quantity \({{{\mathcal{A}}}}\) dependent on two parameters h and L according to

$${{{\mathcal{A}}}}(L,h)={h}^{a}f(h/{L}^{b}).$$
(20)

Depending on the nature of the model system, \({{{\mathcal{A}}}}\), L and h can refer to different quantities. For example, \({{{\mathcal{A}}}}\) might refer to the magnetization of the Ising spin chain, with h being the inverse of the transverse magnetic field and L the length of the chain. In our case, \({{{\mathcal{A}}}}\) refers to the Fisher information Fω(t, β), with L and h being the system size β and the evolution time t respectively. From Eq. (20), it is clear that if we plot \({{{\mathcal{A}}}}{h}^{-a}\) against h/Lb for different values of L and h, all the curves collapse onto a single. Therefore, we can determine the unknown exponents a and b by numerical fitting of the data to Eq. (20) and find the best scaling collapse.

We can define appropriate measure of the quality of the data collapse to remove subjectiveness of the approach. If the scaling function f(x) is known, we can define the measure67

$${{{{\mathcal{M}}}}}_{{{{\rm{kn}}}}}(a,b)=\sqrt{\frac{1}{N}\mathop{\sum}\limits_{ij}{\left(\frac{{{{\mathcal{A}}}}({L}_{i},{h}_{j}){h}_{j}^{-a}-f({L}_{i}^{-b}{h}_{j})}{f({L}_{i}^{-b}{h}_{j})}\right)}^{2}}$$
(21)

which is minimized by the optimal a and b, with N the total number of data points. Notice that in Eq. (21) the division with respect to the scaling function f(x) is essential—otherwise \({{{{\mathcal{M}}}}}_{{{{\rm{kn}}}}}\) can be minimized for small values of the numerator which not necessarily indicates a good collapse.

In the general case the scaling function f(x) is not known. We can interpolate f(x) via any set of data points corresponding to a specific L. Denoting the different sets via the subscript p, we can measure the quality of the data collapse by comparing the data points of pairs of sets p1,2 in their overlapping regions and summing up all the contribution,

$${{{\mathcal{M}}}}(a,b)=\sqrt{\frac{1}{N}\mathop{\sum}\limits_{p}\mathop{\sum}\limits_{i\ne p}\mathop{\sum}\limits_{j,{{{\rm{ov}}}}}{\left(\frac{{{{\mathcal{A}}}}({L}_{i},{h}_{j}){h}_{j}^{-a}-{{{{\mathcal{E}}}}}_{p}({L}_{i}^{-b}{h}_{j})}{{{{{\mathcal{E}}}}}_{p}({L}_{i}^{-b}{h}_{j})}\right)}^{2}},$$
(22)

where \({{{{\mathcal{E}}}}}_{p}(x)\) is the interpolation function based on the basis set p and N is the total number of points in all overlapping regions. Here, the innermost index j runs over the overlapping region of the set p and another set i. It is clear that \({{{\mathcal{M}}}}\ge 0\) with the zero lower bound achieved only in case of perfect data collapse. As a result, it can be used via the variational principle for an automatic and objective extrapolation of the relevant critical exponents.

For our case, in order to examine the quality of the finite-size scaling analysis for different decoherence rates, as shown in Fig. 4(b) in the main text, we introduce the quality factor

$$Q=\frac{{{{{\mathcal{M}}}}}_{{{{\rm{id}}}}}^{F}}{{{{{\mathcal{M}}}}}^{{{{\rm{F}}}}}},$$
(23)

where \({{{{\mathcal{M}}}}}^{{{{\rm{F}}}}}={{{\mathcal{M}}}}(2,1)=\sqrt{\frac{1}{N}{\sum }_{p}{\sum }_{i\ne p}{\sum }_{j,{{{\rm{ov}}}}}{\left(\frac{F({t}_{j}{\beta }_{i}){t}_{j}^{2}-{{{{\mathcal{E}}}}}_{p}({\beta }_{i}{t}_{j})}{{{{{\mathcal{E}}}}}_{p}({\beta }_{i}{t}_{j})}\right)}^{2}}\) is the measure of the data collapse, and \({{{{\mathcal{M}}}}}_{{{{\rm{id}}}}}^{F}\) refers to the measure of the ideal noiseless case.