Critical parametric quantum sensing

Critical quantum systems are a promising resource for quantum metrology applications, due to the diverging susceptibility developed in proximity of phase transitions. Here, we assess the metrological power of parametric Kerr resonators undergoing driven-dissipative phase transitions. We fully characterize the quantum Fisher information for frequency estimation, and the Helstrom bound for frequency discrimination. By going beyond the asymptotic regime, we show that the Heisenberg precision can be achieved with experimentally reachable parameters. We design protocols that exploit the critical behavior of nonlinear resonators to enhance the precision of quantum magnetometers and the fidelity of superconducting qubit readout.


INTRODUCTION
Criticality is a compelling resource, commonly used in classical sensing devices such as transition-edge detectors and bolometers 1 .However, these devices do not follow optimal sensing strategies from the quantum mechanical point of view.A promising approach to quantum sensing exploits quantum fluctuations in the proximity of the criticality to improve the measurement precision.Despite a critical slowing down at the phase transition, theoretical analyses of many-body systems [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] show that critical quantum sensors can achieve the optimal scaling of precision 18 , both in the number of probes and in the measurement time 8,17 .Furthermore, it has been shown 19 that finite-component phase transitions [20][21][22][23][24] -where the thermodynamic limit is replaced by a scaling of the system parameters [25][26][27][28][29] -can also be applied in sensing protocols.Surprisingly, quantum criticalities are versatile sensing resources that do not require the complexity of manybody system, as demonstrated by efficient dynamical protocols 30 , the inclusion of quantum-control methods 31 or ancillary probes 32 , the design of multiparameter estimation protocols 33 and of a critical quantum-thermometer 34 , and by first experimental implementations 35 .
Finite-component critical sensors have hitherto been designed for light-matter interacting models where the atomic levels introduce a nonlinearity 36 .Despite their high experimental relevance in quantum optics and information [37][38][39][40][41][42][43][44][45][46][47] , driven resonators with nonlinear photon-photon interactions have so far been overlooked for applications in critical quantum metrology.These systems display a broad and exotic variety of critical phenomena, and their nontrivial dynamics and steady states depend on both the system and bath parameters 26,48,49 .
Here, we introduce the critical parametric quantum sensor, a measurement apparatus based on the second-order drivendissipative phase transition of a parametric nonlinear (Kerr) resonator.We apply tools of quantum parameter estimation, quantum hypothesis testing, and non-linear quantum optics to characterize the potential of this instrument for finite-component critical sensing.Our treatment uses the analytical solutions of the driven-dissipative Kerr resonator model [50][51][52] , together with exact numerical calculations to: (i) Evaluate the quantum Fisher information (QFI) for frequency estimation, analyzing its scaling in the thermodynamic limit of small-but finite-Kerr nonlinearity.We provide the parameter set maximizing the QFI, and show that homodyne detection virtually saturates the optimal precision bound.Importantly, the whole analysis considers the role of dissipation in these driven transitions.This allows us to design a highly-sensitive magnetometer, that can be built with state-of-theart circuit QED technology.(ii) Compute the optimal and homodyne-based error probabilities in distinguishing the normal and the symmetry-broken phases.We apply this result to the dispersive qubit readout task in circuit-QED.Our approach goes beyond the semi-classical approximation 53,54 , and allows one to recognize the set of parameters minimizing the average error probability.We find that the optimal working point lies in proximity of the critical point, in a region where semi-classical or Gaussian approximation can not be applied.

Kerr resonator model
Our starting point is the Kerr-resonator model, whose Hamiltonian is This Z 2 -symmetric model can be realized in various photonic platforms.In particular, we consider the case of a circuit-QED implementation, where a resonator at frequency ω r is coupled with a superconducting quantum interference device (SQUID) element 53,55 .If the resonator is pumped at a frequency ω p ≃ 2ω r , then Eq. ( 1) describes effectively the system, by interpreting ω = ω r − ω p /2 as the pump-resonator detuning, ϵ as the effective pump-power, and χ as the SQUID-induced nonlinearity.We consider the system embedded in a Markovian thermal bath at zero temperature, described by the Lindblad dissipation superoperator L D ½Á ¼ _Γ½2â Á ây À fâ y â; Ág, where Γ ≥ 0 is loss rate induced by the system-bath coupling.Such a dissipator leaves the model Z 2 invariant (see the Supplemental Material).With no loss of generality, we take ϵ positive.For χ = 0, the model is Gaussian and its phenomenology can be easily explained.In the absence of noise, for Γ = 0, the model has a ground state only for ϵ < |ω|.This is a squeezed vacuum state with squeezing approaching infinity in the ϵ/|ω| → 1 limit.When the bath is turned on, for Γ > 0, the diverging point is shifted.In this case, the steady-state is a squeezed thermal-state and exists only for ϵ< ϵ c , with purity approaching zero when ϵ/ϵ c → 1.The effect of the nonlinearity χ > 0 is to regularize the model for all parameter values, thus erasing the divergences.In the scaling limit χ → 0 a second-order dissipative phase transition (DPT) emerges, associated with the spontaneous breaking of the Z 2 -symmetry of the model 26 (see the Supplemental Material).The steady-state is still Gaussian for ϵ < ϵ c .Beyond the critical point, for ϵ > ϵ c , the steadystate is double-degenerate, and it is given by a statistical mixture of two equiprobable displaced squeezed thermal-states 27 , see Fig. 1.Since χ can be made small in a circuit QED implementation, we can exploit the presence of this DPT for both quantum parameter estimation and discrimination.On the one hand, we can use the large susceptibility of the steady state in the proximity of the critical point, in order to get a good estimation of ω.In turn, as the resonator frequency has a steep dependence on the external magnetic field threading the SQUID loop, the DPT can be applied in the design of a critical magnetometer.On the other hand, the presence of the DPT allows one to faithfully discriminate between two discrete values of ω, each corresponding to a different phase, in a single-shot measurement.

Quantum parameter estimation
Given an observable Ô, we can define the signal-to-noise ratio (SNR) for estimating the parameter ω as where , and the expectation values are computed in the steady-state manifold.This standard definition of SNR is useful for parameter estimation protocols because it is directly related to the mean-square error of the estimator 56 .The corresponding precision over M measurements is Δω 2 ' ½MS ω À1 .In this paper, we consider the SNR for three important measurements: homodyne, heterodyne, and the quantum-mechanical optimal given by the QFI.Homodyne detection consists in projecting on the rotated quadrature operator xφ ¼ cosðφÞx þ sinðφÞp.Due to the Z 2 -symmetry of the system, we consider the observable x2 φ , and define the homodyne SNR as S Hom Heterodyne detection corresponds to a noisy measurement of the conjugate quadratures, with outcomes X and P. We consider the SNR for the outcome X 2 + P 2 , which can be written as , see "Methods".Finally, if we maximize the SNR in Eq. ( 2) among all the observables, we obtain the QFI: I ω ¼ max Ô S ω ½ Ô.This can be expressed as 57 where is the fidelity between the steady-states ρ ω and ρ ω 0 .
The normal phase (χ → 0).To begin with, we consider the case χ → 0, which provides us with a good approximation of the steady-state when we are far enough from the DPT.The model in Eq. ( 1) with χ = 0 has a steady-state solution only for ϵ < ϵ c , corresponding to the normal phase.Using the analytical formula for Gaussian states 58 , we compute the QFI with respect to the parameter ω, in the steady-states manifold: where is the number of photons [see Fig. 2a].We have two possible diverging scaling for ϵ/ϵ c → 1.For ω ≠ 0 we retrieve the Heisenberg scaling I ω = O(N 2 ), while for ω = 0 one has I ω = O(N).Notice that here we focused on the scaling with respect to the number of photons, which is the most relevant figure for the relevant regime of parameters.However, even if the Gaussian model presents a critical slowing down, the Heisenberg scaling can in principle be achieved also with respect to time 17,19 .We notice also that the divergence rate I ω /N 2 is maximal at ω = Γ.In the following, we focus at this point, where the QFI is maximal for low-enough χ.
The symmetry-broken phase (χ → 0).The model is invariant under the transformation â !Àâ, resulting in a Z 2 -symmetry.In the χ → 0 limit, and for ϵ > ϵ c , such a symmetry is broken resulting in a second-order DPT.The symmetry-broken solutions are wellapproximated by Gaussian states that can be obtained by displacing the field â !â þ α, with α 2 C 29 .For nonzero χ, the steady state is well-approximated by a statistical mixture of two Gaussian states 50 .Indeed, a Gaussian approximation leads to Here, ρ ± are the steadystates for À ω and jϵ 0 j ¼ ϵ c .Namely, α is the solution of ωα + ϵα * + 2χ|α| 2 α − iΓα = 0, see "Methods".By setting Fig. 1 Wigner function of the system steady-state.This has been obtained with numerical simulations of the full quantum model (colormap) and half-height contours (dashed blue circles) of the corresponding analytical solutions obtained under semi-classical approximation (see "Methods").The four sub plots are obtained taking ω = 1Γ, χ = 0.04Γ and for increasing values of the pump strength ϵ.The figure shows the transition from the normal (a) to the symmetry-broken [(c) and (d)] phases, taking place around the semiclassical prediction The system is highly susceptible in the proximity of the criticality, and so it can be exploited in high-sensitivity magnetometry.Moreover, the system shows two highly distinguishable phases, corresponding to a vacuum-like (a) and displaced state (d), a feature that can be exploited in high-fidelity qubit readout.α = |α|e iϕ , we find the two solutions, holding for ϵ > ϵ c : Notice that the Hamiltonians H ± are the same at the zeroth order in χ.Therefore, ρ + ≃ ρ − and the steady-state solutions consist in a mixture of two identical squeezed-thermal states displaced in opposite directions 50 .The QFI shows a divergence at ϵ → ϵ c , as seen in the normal phase.This confirms that in the proximity of the transition the QFI diverges for χ → 0. Instead, for sufficiently large ϵ, the QFI value is solely determined by the response of α to the ω's changes.Using Eq. ( 5), one can easily see that The full model (finite χ).We are now ready to show our results beyond the Gaussian approximation.1][52] , while the steady-state density matrix are obtained solving the equation Ài½ ĤKerr ; ρ ss þ L D ½ρ ss ¼ 0 via sparse LU decomposition 59 .We then compute the QFI using Eq. ( 3).The effect of the Kerr term is to regularize the model, eliminating the divergences that appear in the Gaussian approximation.As expected, the QFI increases with ϵ up to a maximum point, then it starts to decrease.The maximum is reached for ϵ = ϵ c in the χ → 0 limit.Let us consider the quantity S ω ¼ max ϵ S Hom ω , and focus on the ω = Γ point.With a numerical fit, we find that S ω ' cðχΓÞ À1 for χ/ Γ ≲ 10 −2 , where c ≃ 0.55 (see Fig. 2a).Since N ¼ Θð ffiffiffiffiffiffiffi χ À1 p Þ holds, the Heisenberg scaling is reached already for χ/Γ ≲ 10 −2 .In Fig. 2b, we show that homodyne detection virtually saturates the QFI already for χ/Γ = 0.04.In fact, one can easily see that homodyne detection is optimal in the χ → 0 limit, see Methods.

Magnetometry
We now consider an application of our results for the quantum estimation of magnetic flux.Let us consider a SQUID coupled with a λ/4 resonator.A magnetometer can be designed by coupling the magnetic field to the SQUID loop.The effective Hamiltonian is given in Eq. ( 1).Here, the resonator frequency ω r depends on the external magnetic flux as ω r ðΦÞ ' ω λ=4 = ½1 þ γ 0 =j cosðΦÞj, where ω λ/4 is the bare resonant frequency in the absence of the SQUID, Φ = πΦ ext /Φ 0 is the applied magnetic flux Φ ext in unit of the flux quantum Φ 0 , and γ 0 is the ratio of SQUID inductance at zero external magnetic flux and the geometric inductance of the resonator.For π/4 ≲ Φ < π/2, where the pump-induced non-linearity is small, the non-linearity depends on the magnetic field as χðΦÞ ' χ 0 ω λ=4 γ 3 0 =jcos 3 ðΦÞj, where χ 0 = πZ 0 e 2 /(2ℏ) is a constant dependent on the resonator characteristic impedance Z 0 (see the Supplemental Material).Assuming a typical value Z 0 ≃ 50Ω 55 , we have χ 0 ≃ 0.02.It is convenient to work at the point Φ ≃ π/4, where χ is minimized.We can also assume χ to be independent on Φ, by working in the limit χ 0 γ 2 0 ( 1, which ensures the condition ∂χ From the input-output theory, we have that the resonator output mode is âout ¼ ffiffiffiffiffi 2Γ p â À âin , where âin is the input mode assumed to be in the vacuum 60 .By applying the right temporal filter at the output mode, one can retrieve the same statistics of the intracavity mode 61,62 .With this premise, the SNR for the output mode is the same as the one derived for the intracavity mode.A change of Φ by δΦ induces the shift ω !ω þ ∂ωr ∂Φ δΦ.Therefore, the uncertainty over M independent measurements is Fig. 2 Metrological performance of the critical parametric quantum sensor.(a) QFI for the estimation of ω as a function of ϵ, computed for ω/Γ = 1 and various values of χ/Γ.In the Gaussian case (χ → 0), the QFI diverges at ϵ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi . For finite values of χ, the QFI has a maximum.In the inset, we show that

Dispersive qubit readout
We now discuss an application of the Kerr resonator for superconducting-qubit readout.By dipersively coupling a qubit to the resonator Hamiltonian in Eq. ( 1), the Hamiltonian becomes Here, δω = g 2 /Δ is a frequency-shift that depends on the qubitresonator coupling g and the qubit-to-resonator detuning Δ 64 .When the qubit is in its excited state e j i, a frequency-shift is induced onto the resonator.The Hamiltonian Ĥdisp can be derived by applying perturbation theory to the full qubitresonator Hamiltonian, for g/Δ ≪ 1.The dispersive approximation holds as long as g 2 N/(4Δ 2 ) ≡ η ≪ 1, where N is the number of photons in the resonator.Notice that a small η also minimizes the disturbance induced to the qubit by the readout scheme (see the Supplemental Material).In the following, we show how the presence of a DPT leads to two highly distinguishable quantum states, that can be used to perform high-fidelity qubit readout.A similar setup, with an unoptimized set of parameters, has been experimentally investigated in ref. 54 .Here, the authors map the qubit discrimination problem to distinguish between the vacuum and a classical state of ~200 photons, where the dispersive approximation is clearly not valid anymore.As a result, the qubit significantly suffers from additional dissipation processes mediated by the readout resonator.Therefore, one must reduce the number of photons, bringing the system closer to the critical point, where the semiclassical approximation does not hold, and quantum fluctuations shall unavoidably be taken into account.In the following, we conduct this performance analysis in a systematic way using the full quantum model, identifying the set of parameters that maximizes the readout fidelity, while still respecting the dispersive approximation.
Generally speaking, the method consists in discriminating between two density matrices, i.e., ρ g and ρ e , corresponding to the steady-states when the qubit is in the state e j i or g j i respectively.We limit ourselves to the case of discrimination via a single measurement of the mode â (single-shot readout).The average error probability is bounded by P err P opt err , also known as the Helstrom bound 57 , where is the trace norm (the qubit readout fidelity can be defined as F = 1 − P err ).The optimal error probability P opt err is in principle achievable by measuring in the eigenbasis of ρ e − ρ g .In Fig. 3a, we show a map of the P opt err values with respect to the frequency-shift δω and the pump strength ϵ, for χ = 0.08Γ.For a given value of η, the graph shows the presence of a sweet spot where the error probability is minimized.The value P opt err is always attainable, and gives us a bound on what error probabilities can be in principle reached.However, the measurement can be complicated to implement, so we consider also a practical strategy based on homodyne detection.Let us define the probability density functions P g,e (x) = ∫W g,e (x, p)dp, where W g,e (x, p) are the Wigner functions of the resonator steadystate in the case of qubit in the g j i or e j i states.We declare that the state of the qubit is g j i if our measurement outcome belongs to {x|P g (x) > P e (x)} and e j i otherwise.The error probability for this discrimination strategy is Z min x fP g ðxÞ; P e ðxÞgdx: This procedure can be further optimized by considering a rotated homodyne measurement.Notice that we have used the Wigner function as a tool to find the best threshold value distinguishing between the two qubit states, given a single homodyne measurement.Indeed, our strategy does not rely on the reconstruction of the resonator Wigner function.In Fig. 3b, we show that the homodyne strategy, although not saturating the optimal strategy, achieves error probability values of the order of 10 −3 .Notice, however, that the position of the minimum of the error probability with the homodyne strategy coincides with that of P opt err .In the Supplemental Material, we show that for η = 10 −2 there is no backaction on the qubit states (see the Supplemental Material).Experimentally achievable values attaining the optimal value for η = 10 −2 are: Γ ≃ 2π × 1 MHz, χ/Γ ≃ 0.08, g/Γ ≃ 10 2 , ω r /Γ ≃ 8 × 10 3 , ω q /Γ ≃ 6 × 10 3 .In this case, the resonator has at most N ≃ 30 photons at the steady-state.

METHODS
The Gaussian approximation for χ → 0 Here, we find the Gaussian approximation for χ → 0, for both the regimes ϵ < ϵ c (normal phase) and ϵ > ϵ c (symmetry-breaking phase).
The normal phase (ϵ < ϵ c ).For ϵ < ϵ c , we set χ = 0 and look for the steady-state solutions.It is convenient to rewrite the master Fig. 3 Qubit detection performance with a critical parametric quantum sensor.(a) Error probability map with respect to δω/ Γ = g 2 /(ΓΔ) and ϵ/Γ, for ω = 0 and χ/Γ = 0.08.The dashed lines represent different values of the dispersive parameter η = Nδω 2 / (4g 2 ), where N ¼ maxfN g j i ; N e j i g and we have fixed g/Γ = 10 2 to be in the strong-but not ultrastrong-coupling regime.For η = 10 −2 , we can reach error probability values as low as 10 −4 with the optimal measurement.(b) Error probability for homodyne detection at the optimal points and optimal angle φ for different values of η.
The inset shows the separation in time of hx 2 φ i for the normal and symmetry-broken phases.The steady-state value is reached at Γt ≃ 10. equation as Fokker-Planck equation, in the Wigner function formalism: . Since this equation is quadratic in x and p, it can be solved by a Gaussian ansatz where r = (x, p) and the Wigner function is normalized to one.The covariance matrix σ is defined as σ ij ¼ hfr i ;r j gi À 2hr i ihr j i.
From ( 9) we get the linear system of equations where σ L ¼ I 2 and B ¼ 0 ðω À ϵÞ Àðω þ ϵÞ 0 ! .We find the steady state by solving ∂ t σ ss = 0.The solution is which corresponds to a physical state only for . This sets the critical value in the χ → 0 limit.In this limit, the number of photons is which diverges for ϵ → ϵ c .
The symmetry-broken phase (ϵ > ϵ c ).Let us derive an effective quadratic Hamiltonian for ϵ > ϵ c .We follow the approach developed in ref. 29 .The idea is that for small χ, the model is well approximated by a double-well potential, and that the lowenergy physics can be described with a quadratic expansion around each minimum.In order to center the reference frame on one of the two minima, let us apply a displacement operation such that U y âU ¼ â þ α.We obtain an effective Hamiltonian where The dissipator L D , instead, is left unchanged.The quadratic part of the displaced Hamiltonian ( 14) is well-defined, i.e., it has normal modes with positive frequency and is bounded from below.Accordingly, far from the critical point the steady state will have bounded quantum fluctuations, and the norm of the creation/annihilation operators on the steady state will be bounded.In the limit of small χ, and of large α, higher-order terms are negligible and the model is well-approximated by a Gaussian approximation which includes only terms quadratic in ây and â [see the solutions below in Eq. ( 18)].Of course this approximation will break in a small region for ϵ !ϵ þ c , and the size of the critical region is proportional to χ (the smaller the nonlinearity, the more reliable the Gaussian approximation even as the critical point is approached).
Setting α = |α|e iϕ we find two solutions for ϵ > ϵ c : We find the effective Hamiltonians in the symmetry-broken phase by plugging the solution into Eq.( 14).We get with Notice that the Hamiltonians H ± are the same for χ → 0. Therefore, in this limit the two solutions are degenerate.Increasing the pump power ϵ corresponds to an effective growth of the pump-resonator detuning, since ω 0 $ ϵ for large ϵ.Instead, the effective squeezing parameter ϵ 0 remains constant in modulus, while its argument changes until reaching the value Therefore, the effect of increasing the pump is to displace the state to the new equilibrium points, and to rotate and reduce the squeezing of each of the resulting states.

Quantum parameter estimation
Here, we derive the quantum parameter estimation results for the full model.

Signal-to-noise ratio (SNR).
The SNR induced by the observable Ô in the task of estimating the parameter ω is defined as where ω and the index ω indicates the expectation value computed on the steady-state ρ ω .The SNR computed in ω = ω 0 should be interpreted as the precision achievable for estimating the parameter ω when its value is close to ω 0 , through the relation Δω 2 jω'ω0 ' ½M S ω0 À1 , where M ≫ 1 is the number of measurements.Generally speaking, if an experimentalist is able measure the expectation value of a class of observables fOð r !Þg, with r !¼ ðr 1 ; ; r K Þ, they would like to maximize the SNR with respect to r ! in order to obtain a better precision rate (call r !max the maximizing set of parameters).This in principle requires the preknowledge of ω 0 .If this knowledge is not provided, then they can implement a two-step adaptive protocol, where first they measure the expectation value of A 2 fOð r !Þg such that the function f(ω) = 〈A〉 ω is invertible in the range of values where ω belongs, obtaining a first order estimation of ω, i.e., ω 0 .Then they find r !max and measure Oð r !max Þ.We are particularly interested in the following SNRs.

•
Homodyne detection: This is defined by the POVM X Hom , where x φ is an eigenstate of the rotated quadrature We consider this SNR S ω ½x 2 φ to evaluate the perfomance of homodyne detection for finite χ.This will be clear in the following, when we will evaluate the classical Fisher information.We also consider the best phase choice, i.e., • Heterodyne detection: This is defined by the POVM , where γ j i is a coherent state.The heterodyne measurement can be modeled as the signal â entering in a beamsplitter with a thermal mode ĥ as the other input, obtaining the two modes b ffiffi ffi 2 p are finally measured.This is equivalent to measure the complex envelope operator Indeed, from the measurement outcomes one can estimate the moments of Ŝ, and then invert Eq. ( 27) to obtain an estimation of the moments of â.In the quantum-limited case, when ĥ is a vacuum mode, the moments of Ŝ can be easily computed as h Ŝym Ŝn i ¼ hâ n âym i, since ½ Ŝ; Ŝy ¼ 0 65,66 .The SNR for the obser- where pðx φ jωÞ ¼ x φ ρ ω x φ is the probability density function of the outcome.
• Quantum Fisher information (QFI): Generally speaking, the QFI provides the precision for the optimal unbiased estimator allowed by quantum mechanics.This is indeed defined as I ω ¼ max Ô S ω ½ Ô, that can be generally expressed as 57 where Á 2 is the fidelity between the two density matrices ρ ω and ρ ω−dω , and ρ ω (ρ ω−dω ) is the steady-state of the model with pump-resonator detuning ω (ω − dω).This expression can be easily evaluated numerically by considering a dω smaller and smaller, and by doing a convergence check.In the paper, this procedure has been used to compute the QFI in the steady-state manifold for the χ > 0 case.Indeed, Eq. ( 30) has been evaluated numerically by letting the system evolve to the steady-state for two values of ω close to each other.
Quantum parameter estimation for χ → 0 (Normal phase) Here, we derive the formulas for the QFI and the FI for the Gaussian model.We show that homodyne detection saturates the QFI for χ → 0.
QFI for the Gaussian model.The QFI for the Gaussian model can be analytically calculated using the covariance matrix formalism 58 , using the solutions of Eq. ( 12): where μ ¼ ðdet σÞ À1=2 is the purity of the quantum state, and we have used that the displacements are zero for all parameter values.The solution has been computed using a symbolic computation software, obtaining This expression can be cast as in Eq. ( 4) using the relation We recall that the quantum Fisher information provides an upper bound to the achievable SNR as defined in Eq. ( 2), which can be saturated when the optimal measurement is implemented.

•
For ω ≠ 0, the FI scales differently with respect to ϵ c − ϵ.We exemplify the calculation for ω = Γ, that is the point where the QFI shows the maximal divergence rate.Here, we have that (35) In the ϵ/ϵ c → 1 limit, we have that 2 for any φ.This means that F ω¼Γ ðX Hom φ Þ=I ω¼Γ ! 1, and homodyne detection saturates the QFI for any φ, Quantum parameter estimation for χ > 0 Let us evaluate the QFI scaling in two regimes: (i) ϵ/ϵ c ≫ 1, and (ii) ϵ close to the criticality.
• ϵ/ϵ c ≫ 1.In this regime, we have seen that the Wigner function becomes a mixture of two equiprobable coherent states, symmetrically displaced with respect to the center.These states are uniquely determined by |α(ω)| 2 , as the phase ϕ in Eq. ( 19) does not depend on ω.Therefore, for symmetry reasons, the optimal observable is the photon-number operator.This gives rise to the optimal SNR scaling • ϵclose to the criticality.In this case, we have analyzed numerically the scaling of S ω ¼ max ϵ S Hom ω , computed in ω = Γ, where the QFI is maximal for low-enough χ.We find that S ω ðω ¼ ΓÞ $ cðχΓÞ À1 for χ/Γ ≲ 0.01, where c ≃ 0.55.In addition, since in the same regime we have that N ¼ Θð ffiffiffiffiffiffiffi χ À1 p Þ, that the Heisenberg scaling is reached.Let us focus on I ω ¼ max ϵ I ω .On the one hand, we always have that I ω ðω ¼ ΓÞ !S ω ðω ¼ ΓÞ.On the other hand, in practice homodyne detection already saturates the QFI for χ/Γ = 0.04, meaning that one should expect I ω ðω ¼ ΓÞ ' S ω ðω ¼ ΓÞ already in this regime, since homodyne performs optimally for χ → 0.
Optimal parameter choice.For each sample, the state of the resonator collapse either on ρ e or ρ g .Discriminating between these two states gives us the measurement result.Fixing a value of χ/Γ, one can draw a (δω, ϵ)-dependent map of the optimal error probability for discriminating ρ e and ρ g , i.e., P opt err ¼ 1 À kρ e À ρ g k=2 Â Ã =2.Since ϵ is monotone with respect to N, for each value of (δω, ϵ, η), one can find a value of g and Δ satisfying the conditions δω = g 2 /Δ and η = g 2 N/(4Δ 2 ).However, since g/Γ cannot be too large, otherwise we go to the ultrastrong regime where the counter-rotating terms appear, we have fixed g/Γ = 10 2 , and choose ω r and ω q such that g= minfω q ; ω r g ( 1.We have then drawn the lines for η equals to 10 −2 and 0.5 × 10 −2 , see Fig. 3a. Published in partnership with The University of New South Wales npj Quantum Information (2023) 23 ∂Φ( ∂ω ∂Φ to hold.The protocol consists in: (i) Apply a constant magnetic flux bias Φ ≃ π/4 to the SQUID.(ii) Apply a pump at frequency ω p ≃ 2[ω r (π/ 4) − Γ].This allows to work at ω ≃ Γ, where the QFI is maximal.(iii)Perform homodyne detection of the output signal.