Criticality Enhanced Quantum Sensing via Continuous Measurement

Theodoros Ilias,1, ∗ Dayou Yang,1, ∗ Susana F. Huelga,1 and Martin B. Plenio1

1Institut fur Theoretische Physik and IQST, Universitat Ulm,Albert-Einstein-Allee 11, D-89069 Ulm, Germany

(Dated: August 17, 2021)

Present protocols of criticality enhanced sensing with open quantum sensors assume direct mea-surement of the sensor and omit the radiation quanta emitted to the environment, thereby omittingpotentially valuable information. Here we propose a protocol for criticality enhanced sensing viacontinuous observation of the emitted radiation quanta. Under general assumptions, we establisha scaling theory for the global quantum Fisher information of the joint system and environmentstate at a dissipative critical point. We demonstrate that it obeys universal scaling laws featuringtransient and long-time behavior governed by the underlying critical exponents. Importantly, suchscaling laws exceed the standard quantum limit and can in principle satuarate the Heisenberg limit.To harness such advantageous scaling, we propose a practical sensing scheme based on continuousdetection of the emitted quanta. In such a scheme a single interrogation corresponds to a (stochas-tic) quantum trajectory of the open system evolving under the non-unitary dynamics dependent onthe parameter to be sensed and the back-action of the continuous measurement. Remarkably, wedemonstrate that the associated precision scaling significantly exceeds that based on direct mea-surement of the critical steady state, thereby establishing the metrological value of detection of theemitted quanta at dissipative criticality. We illustrate our protocol via counting the photons emittedby the open Rabi model, a paradigmatic model for the study of dissipative phase transition withfinite components. Our protocol is applicable to diverse open quantum sensors permitting continu-ous readout, and may find applications at the frontier of quantum sensing such as human-machineinterface, magnetic diagnosis of heart disease and zero-field nuclear magnetic resonance.

I. INTRODUCTION

A most ambitious vista of quantum sensing is to en-hance the sensor precision from the standard quantumlimit (SQL) [1–3] of independent, uncorrelated measure-ments towards the Heisenberg limit (HL) [4, 5], the ul-timate precision allowed by quantum mechanics. Givena fixed number N of the sensor components and a to-tal time t of the sensing interrogation, the SQL rep-resents a (classical) scaling of the estimation error as

∼ 1/√Nt, whereas the HL features a quantum-enhanced

scaling ∼ 1/Nt [6]. Such enhancement, on the one hand,can in principle be achieved by preparing individual sen-sor components in entangled quantum states, e.g., theGreenberger–Horne–Zeilinger (GHZ) state [7] and spin-squeezed states [6, 8]. Nevertheless, such states are tech-nically challenging to scale up and are fragile when ex-posed to noise [9], and is thus far restricted to a smallnumber of components, see recent advances [10–16]. Onethe other hand, alternative strategies without direct useof entangled states open up intriguing and promisingroutes towards quantum-enhanced sensing with afford-able technological demand, see Ref. [17] for an overview.

A recent highlight among these is criticality enhancedsensing [18–24]. The key ingredient there is the univer-sally divergent susceptibility of the ground state to smallvariation of parameters of Hamiltonians at quantum criti-cal points (CPs). Such divergence translates directly into

∗ T. I. and D. Y. contributed equally to this work.

a divergent Quantum Fisher information (QFI). Proto-cols of criticality enhanced sensing typically adopt one ofthe following two approaches. The first approach [19, 24]is based on the time evolution of the ground state follow-ing a quench of the Hamiltonian parameters across theCP, i.e., exploiting the dynamic susceptibility. The pre-cision is quantified by the QFI of the time-evolved state,which obeys a sub-Heisenberg scaling ∼ t2Nα with α ≤ 2determined by the critical exponents of the underlyingCP. The second approach [18, 23] exploits the static sus-ceptibility of the ground state, by adiabatically switchingthe Hamiltonian parameters across the CP. The precisionis quantified by the QFI of the ground state, which mayobey an apparent super-Heisenberg scaling with respectto N . This, however, comes at the price of a divergentinterrogation time to maintain adiabaticity close to theCP [20]. Taking into account such time, the ground-stateQFI obeys the same scaling ∼ t2Nα as in the quenchapproach [20]. Therefore, the two approaches are equiv-alent and both provide a promising avenue towards theHL based on engineering short-range interacting many-body systems close to quantum criticality. Experimen-tally, such a capability has been demonstrated in variousquantum-optical setups well isolated from the environ-ment [25–28].

Open quantum systems represent another interestingcandidate for criticality enhanced sensing. These sys-tems support dissipative criticality [32–37], defined viagap closing in the lowest (real) spectrum of the Liouvillesuper-operator for the reduced density matrix of the sys-tem. At a dissipative CP, the thermodynamic proper-ties of the steady density matrix manifest divergent scal-

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1

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101 102 103 104 105

102

104

106

108

1010

1012

Iω (t)

Fω (t)∼ t3

∼ t

∼ t2

∼ t

0 1 2−1

0

1

(a)

D(t,0)θ = ω Processor

ωest

Var(ωest) ≥ Fω(t)−1 ≥ Iω(t)−1

Continuous MeasurementDissipative CP

(b)

D(t,0

)κt Long-Time RegimeTransient Regime

κt

⟨ c⟩st/

ηeiϕ

⟨ n⟩c(t)

g/gc

Precision Bounds

κ

0 10 20 3001

Var(ωest)

κt* ≃ η

∼ tη2

∼ t3

∼ t2

∼ tη

HL ∼ t2η

FIG. 1. (a) The open Rabi model as a model system demonstrating criticality enhanced sensing via continuous measurement.We consider the sensing of the cavity mode frequency ω as an illustration. The cavity leaks photons at a rate κ, countedcontinuously by a photon detector. The model parameters are tuned to a dissipative CP gc, where its steady state undergoesa continuous phase transition (see text). Here η is the effective system size, φ = arctan(2κ/ω). A sensing interrogationcorresponds to a quantum trajectory consisting of the continuously detected signal D(t, 0) and the associated conditionalevolution of the open system, as exemplified by the cavity mode occupation 〈n〉c(t). Processing D(t, 0) (e.g., via Bayesianinference [29–31]) provides an estimator ωest with an imprecision (variance) Var(ωest). (b) Scaling of the precision boundsat dissipative criticality, including the global QFI of the joint system-environment Iω(t), and the FI Fω(t) of the detectedsignal. Both quantities manifest algebraic scaling with respect to t and η, and for illustration we show the numerical resultsfor η = 500. With respect to the interrogation time t, the precision bounds obeys (super-)Heisenberg scaling in the transientregime, t . t∗ ' η/κ, whereas linear scaling in the long-time regime, t t∗. The scaling should be contrasted with the HL∼ t2η (thick dashed blue) and the SQL ∼ t

√η (not shown). The scaling exponents of the global QFI can be related to the

critical exponents z = 1, ∆n = −1/2 and the spatial dimension d = 0 via the general formulas (13) and (14).

ing behavior, bearing similarities to the behavior of theground state at a quantum CP.

State-of-the-art protocols [21–23] for critical open sen-sors quantify the sensor precision limit via the QFI ofthe reduced density matrix of the system. This, however,only represents the optimal precision achieved by directmeasurement of the system alone. In stark contrast toclosed systems, open dissipative systems continuously ex-change radiation quanta with their environments, whichcarry information about the system and therefore, aboutthe parameter to be sensed. Such radiation quanta maybe detected continuously in time via mature experimen-tal techniques in quantum optics, e.g., photon countingand homodyne measurement [38–41], accomplishing in-direct monitoring of the system via measurement of theenvironment. Such a unique opportunity makes opensensors a natural platform for implementing continuous-measurement–based sensing schemes [29–31, 42–44] and,from a theoretical point of view, requires the use of theglobal QFI of the joint system-environment state |Ψ(t)〉

Iθ(t) = 4[〈∂θΨ(t)|∂θΨ(t)〉 − |〈Ψ(t)|∂θΨ(t)〉|2] (1)

as the ultimate precision bound, which can in princi-ple be achieved by the most general measurement of thejoint system-environment (implementing such measure-ment in practice, however, may be challenging). Thishas been emphasized in sensing with non-interactingopen system [29–31, 42] and systems manifesting first-order dynamic phase transitions [45], as well as in sce-narios of noisy quantum sensing with observed environ-ments [46, 47].

In contrast to the existing studies [29–31, 42, 45–47],

here we are interested in open quantum many-body sen-sors at (continuous) dissipative CPs. Two questionsemerge naturally in this context. (i) Does the globalQFI (1) manifest criticality-enhanced universal scaling ata dissipative CP? (ii) If so, can we access such enhancedprecision scaling via realistic measurement schemes of theemitted radiation quanta? In this manuscript, we pro-vide positive answers to both questions. We show thatthe global QFI (1) for a generic Markovian open sen-sor obeys universal scaling laws at dissipative criticality,which are governed by the universal critical exponentsof the underlying CP. Such scaling exceeds the SQL, andcan in principle saturate the HL. Moreover, as a practicalscheme to access the criticality-enhanced precision, weanalyze a continuous-measurement—based sensing pro-tocol applicable to the the sensing of arbitrary parame-ters at generic dissipative CPs.

Our key findings are illustrated in Fig. 1 at the exam-ple of the open Rabi model (see Refs. [48] and below),a paradigmatic model for the study of dissipative phasetransition with finite components. Panel 1 (a) presents aschematic of our sensing protocol. We assume the opensystem is tuned to a dissipative CP. The quantity wewish to sense, θ, is encoded as a parameter in the systemHamiltonian. A single sensing interrogation consists ofinitializing the system at t0 = 0 in the same (arbitrary)state and subsequent detection and evolution spanning[0, t). We emphasize such evolution is subjected to mea-surement back-action randomly driving the system awayfrom the steady state, in stark contrast with the steady-state based scenario [21, 23]. As such, our scheme has thenatural advantage that it does not require steady-state

3

preparation which typically suffers critical slowing down.Detection of the radiation quanta continuously in timeprovides us with measurement signals D(t, 0) dependenton the system dynamics, that is, on θ. This allows us toconstruct an estimator θest. The associated precision canquantified by the Fisher information (FI) of the detectedsignal

Fθ(t) =∑D(t,0)

P [D(t, 0)] ∂θlnP [D(t, 0)]2 , (2)

where P [D(t, 0)] is the probability of the continuouslydetected signal D(t, 0). According to the Cramer-Raoinequality [49], the FI sets a lower bound to the varianceof any (unbiased) estimator of θ, i.e., Var(θest) ≥ 1/Fθ(t).

The universal scaling of the precision bounds is exem-plified in Panel. 1(b) by the sensing of the cavity modefrequency, θ = ω. We find that the time dependence ofboth the global QFI and the FI can be divided into tworegimes, the transient regime and the long-time regime,separated by a characteristic time scale t∗ ' η/κ, with ηthe effective size of the system and κ the cavity damp-ing rate. In the transient regime, t . t∗, the global QFIobeys a super-Heisenberg scaling with respect to timeIω(t) ∼ t3 whereas the FI obeys the Heisenberg scalingFω(t) ∼ t2. In the long-time regime, t t∗, both quan-tities grow linearly in time, and depend algebraically onthe system size η, Iω(t) ∼ η2t and Fω(t) ∼ ηt. The scal-ing exponents of the global QFI can be expressed analyt-ically in terms of the critical exponents of the underlyingCP.

The rest of the manuscript is organized as follows.In Sec. II, we discuss the universal scaling behaviorof the global QFI (1) at a generic dissipative criticalpoint. Afterwards, we analyze in Sec. III a continuous-measurement based sensing protocol for accessing thecriticality-enhanced precision, illustrated at the exampleof the open Rabi model. We conclude in Sec. IV with asummary of our results and an outlook.

II. UNIVERSAL SCALING OF THE GLOBALQUANTUM FISHER INFORMATION

Let us start our discussion by analyzing the precisionlimit of a generic open quantum sensor at dissipative crit-icality. Consider the sensor as an open many-particle sys-tem with spatial extension L defined on a d-dimensionallattice, consisting of N = Ld interacting particles as indi-vidual sensor components. We assume the open systemis coupled to a Markovian environment at zero tempera-ture, and the reduced density matrix of the unobservedsystem evolves according to a Lindblad master equation(LME) [50–53] (we set ~ = 1 hereafter)

ρ = Lρ ≡ −i[H(θ), ρ] +∑`

(J`ρJ

†` −

1

2J†` J`, ρ

). (3)

We consider the sensing of a single quantity θ which,without the loss of generality, is assumed to be encoded as

a parameter of the many-body Hamiltonian H(θ). J`represents a set of (single- or many-body) jump opera-tors resulting from system-environment coupling, with `the set index. To be specific, we focus on the experimen-tally common situation where θ is encoded in single-bodyterms,

H(θ) =

N∑i=1

hi(θ) +

N∑i<j

hij + . . . , (4)

i.e., we assume all n-body (n ≥ 2) terms describing inter-particle interactions are θ-independent.

The sensor precision is upper-bounded by the globalQFI (1), which may be achieved by the most generalmeasurement performed on the joint system-environmentstate |Ψ(t)〉. Interestingly enough, assuming the validityof the quantum regression theorem, such a global QFIcan be expressed solely in terms of the system auto-correlators [42],

Iθ(t) = 2

∫ t

0

dτ

∫ t

0

dτ ′〈δO(τ ′), δO(τ)〉. (5)

Here, O := ∂θH(θ) is the Hermitian operator which en-codes θ, the angle bracket 〈· · · 〉 := tr[· · · ρ(0)] denotesan expectation with respect to the initial (pure) system

density matrix ρ(0) = |ψ(0)〉〈ψ(0)|, and δO(t) := O(t)−〈O(t)〉. Under our assumption Eq. (4), O =

∑Ni=1 oi is

an extensive single-body operator, with oi := ∂θhi(θ) thelocal operator of the i-th particle. The global QFI (5)can therefore be expressed as

Iθ(t, L) = 8Ld∫ t

0

dτ

∫ t−τ

0

ds S(τ, s, L), (6)

S(τ, s, L) =1

Ld

N∑i,j=1

R[〈δoi(τ + s)δoj(τ)〉], (7)

in which R denotes the real part, and we have indicatedexplicitly the dependence of the global QFI on the sys-tem size L. Equations (6) and (7) provide us with auseful connection between the global QFI and the auto-correlators of the open quantum sensor. In particular, aswe show below, they lead to explicit quantitative predic-tions at continuous dissipative CPs, where the universalscaling laws of the auto-correlators translate directly intothe scaling of the global QFI.

A. Criticality and universal scaling

Let us assume that the LME (3) supports a continuousdissipative CP in its steady state ρst, i.e., the solution ofthe stationary LME Lρst = 0. We assume h is a parame-ter of the LME (3) that drives the system across the phasetransition (h can be independent of θ), and the CP is lo-cated at hc = 0. Close to the CP, the static and dynamicproperties associated with ρst obey universal scaling laws

4

governed by a small number of critical exponents. In thefollowing we show that the global QFI (6) manifests suchuniversal scaling behavior.

To begin with, we introduce

Sst(s, L) =1

Ld

N∑i,j=1

R[〈δoi(s)δoj(0)〉st], (8)

with 〈· · · 〉st := tr(ρst · · · ) an expectation value with re-spect to the steady state. Note that Sst(0, L) is the staticstructure factor of the steady state at zero momentum.Assuming translational invariance, and taking the ther-modynamic limit L→∞, Sst(0, L) obeys the well-knownscaling behavior (see, e.g., Ref. [54]) Sst(0, L) ∼ ξd−2∆o ,provided d− 2∆o > 0, where ∆o is the scaling dimensionof the operator oi, defined via 〈δoi(0)δoi+r(0)〉st ∼ r−2∆o

at the CP hc = 0. ξ is the correlation length of the sys-tem, which diverges according to ξ ∼ h−ν close to the CP,with ν the correlation-length critical exponent. In the op-posite case of d− 2∆o ≤ 0, Sst(0, L) ∼ const. dependenton the short-distance (ultra-violet) cut-off, i.e., it is notuniversal. For correlations of non-equal time s 6= 0 andof finite system-size L, the above scaling behavior can begeneralized to a dynamic scaling form [55, 56]

Sst(s, L) = Λd−2∆oφst(Λ1/νh,Λzs−1,ΛL−1), (9)

in which Λ is a cut-off length scale determined by the rel-ative strength of the two perturbations to the CP, h andL−1, φst is a universal function and z is the dynamic criti-cal exponent. If h is sufficiently small, h L−1/ν , the in-verse system size L−1 is the most relevant perturbation tothe CP, i.e., the finite-size effect most severely drives thesystem away from criticality. Correspondingly, Λ ' L.In the opposite limit h L−1/ν , h is the dominant per-turbation to the CP. Correspondingly, Λ ' ξ ∼ h−ν .

We now analyze the scaling behavior of S(τ, s, L), cf.Eq. (7), via analogy with Eq. (9). Note that the quan-tum regression theorem prescribes 〈δoi(τ + s)δoj(τ)〉 =tr[δoie

LsδojeLτρ(0)], and therefore, the auto-correlator

S(τ, s, L)→ Sst(s, L) in the limit τ →∞. Consequently,the parameter τ−1 can be regarded as another pertur-bation to the CP that controls the scaling behavior ofS(τ, s, L). Assuming τ−1 is a relevant perturbation (inthe renormalization group sense), and d − 2∆o > 0, wewrite down a dynamic scaling ansatz

S(τ, s, L) = Λd−2∆oφ(Λ1/νh,Λzτ−1,Λzs−1,ΛL−1),(10)

where the cut-off length scale Λ is determined by the rel-ative strength of the relevant perturbations h, τ−1 andL−1. Equation (10) provides us with a simple yet pow-erful scaling form for auto-correlators close to dissipativecriticality, which leads directly to the scaling laws of theglobal QFI via Eq. (6), as shown in the next section.

We emphasize that at this stage Eq. (10) should beviewed as a tentative ansatz, the validity of which shouldbe checked, e.g., via numerical finite-size scaling once the

model is specified. In appendix A, we provide such nu-merical validation for the model system (the open Rabimodel, to be introduced in Sec. III A) considered in thepresent work.

B. Scaling laws of the global quantum Fisherinformation

We now study the scaling laws of the global QFI (6),based on the scaling ansatz (10). First, we note thatEq. (10) allows us to identify different regimes wherethe cutoff length scale Λ is set by different perturbationsto the CP among h, L−1 and τ−1, and correspondinglyS(τ, s, L) obeys different scaling forms. For criticalityenhanced sensing, we consider h = 0, i.e., the systemparameter is tuned at the CP. Consequently, S(τ, s, L)picks up the following scaling forms

S(τ, s, L) = τ (d−2∆o)/zφτ (τs−1, τL−z), τ . Lz, (11)

= Ld−2∆oφL(Lzτ−1, Lzs−1), τ Lz. (12)

in which φτ and φL are universal scaling functions inher-ited from φ in Eq. (10).

Plugging these scaling forms into Eq. (6), we see imme-diately that the global QFI manifests two distinct scalingbehavior dependent on the total interrogation time t com-pared to the system size L. (i) t . Lz, which is referredto as the transient regime hereafter, where the integrandS(τ, s, L) obeys the scaling form (11). (ii) t ≥ Lz, whichis referred to as the long-time regime hereafter, whereS(τ, s, L) picks the form (12) in most of the integrationinterval in Eq. (6). Completing the integration in Eq. (6)in the two regimes we find

Iθ(t, L) ∼ t(d−2∆o)/z+2Ld, t . Lz, (13)

∼ tL2d−2∆o+z, t Lz. (14)

In arriving at Eq. (14), we have made the approxi-

mation∫ t−τ

0ds '

∫∞0ds, justified by the fact that

φL(Lzτ−1, Lzs−1)→ 0 sufficiently fast at large s/Lz as aresult of the finite correlation time in a finite-size system.

Equations (13) and (14) serve as the central formulas ofthis section, which provide us with the scaling laws of theglobal QFI (1) of a generic (Markovian) open quantumsensor at dissipative criticality. Such scaling depends onthe spatial dimension d, the dynamic critical exponent zand the scaling dimension of the local operator oi thatencodes the unknown parameter.

C. Super-Heisenberg scaling and consistency withthe Heisenberg limit

Let us discuss a few peculiar features of the scalinglaws in Eqs. (13) and (14), cf. Fig. 2. First, we note thatthey surpass the SQL, and are improved further by cou-pling θ to operators with a small scaling dimension ∆o.

5

t* ∼ Lz

Global QFI

log(t)

∼ t2N2

∼ t2N2−2Δ o /dTransient Regime Long-Time Regime

∼ tβN

∼ tNγ

∼ tN

log(Pr

ecisio

nBou

nds)

SQL

HL

FIG. 2. Illustration of the universal scaling laws of the globalQFI at a generic dissipative CP, Eqs. (13) and (14), withrespect to the interrogation time t and the particle numberN ≡ Ld, assuming d ≥ 1. The global QFI (thick red solid),its tight bound Eq. (15) (light green solid), are contrastedwith the HL (dark blue dashed) and the SQL (black doted).The characteristic time scale t∗ ∼ Lz separates the transientregime and the long-time regime, where the global QFI obeysdifferent scaling laws. The scaling exponents are determinedby the spatial dimension d, the dynamic critical exponent zand the scaling dimension ∆o of the local operator oi thatencodes the unknown parameter, via β = 2+(d−2∆o)/z andγ = 2 + (z − 2∆o)/d.

Such a condition is met typically by relevant operators(in the renormalization group sense) of low-dimensionalCPs [20].

Second, we compare the scaling laws in Eqs. (13) and(14) to the HL widely discussed in the context of inter-ferometric sensing [4, 5]. We focus on the case that oiis a single-body operator, and we temporarily assumed ≥ 1. Therefore, the HL adopts the familiar expressionHL ∼ t2N2 ≡ t2L2d. We comment on the special cased = 0 at the end of this section.

One interesting aspect is the apparent super-Heisenberg scaling with respect to the interrogation timet or the particle number N in different regimes. In thetransient regime t . Lz, the global QFI [cf. Eq. (13)]manifests super-Heisenberg scaling with respect to t [notethat we assume d− 2∆o > 0 for the validity of Eqs. (13)and (14)] whereas linear scaling with respect to N . In thelong-time regime t Lz, the global QFI [cf. Eq. (14)]manifests linear scaling in t and super-Heisenberg scalingin N , provided by the condition z − 2∆o > 0, which canbe fulfilled by encoding θ in operators with small scalingdimensions.

Despite such apparent super-Heisenberg scaling, theglobal QFI is indeed upper-bounded by the HL. To seethis, let us introduce the precision scaling

I∗θ (t, L) ∼ t2L2d−2∆o . (15)

We note that for continuous CPs defined in spatial di-mensions d ≥ 1, the operator scaling dimensions arepositive, ∆o ≥ 0. Therefore, the scaling (15) is sub-Heisenberg. Dividing both sides of Eqs. (13) and (14) by

I∗θ (t, L),

Iθ(t, L)/I∗θ (t, L) ∼ (tL−z)(d−2∆o)/z, t . Lz, (16)

∼ Lzt−1, t Lz. (17)

Therefore, Iθ(t, L) . I∗θ (t, L) in both the transient andthe long-time regime. Only at the sweet spot t∗ ' Lz,Iθ(t∗, L) saturate I∗θ (t, L). This confirms that the globalQFI is Heisenberg limited in d ≥ 1.

In d = 0, the critical open system does not possess arigorous spatial extension L. Consequently, the HL can-not be expressed as t2L2d, and should be analyzed interms of the actual resources involved in the sensing pro-tocol. This is illustrated in the next section via the openRabi model, for which we show the associated global QFIactually saturates the HL [cf., Fig. 1(b) and Sec. III B].

III. CRITICALITY ENHANCED PRECISIONVIA CONTINUOUS MEASUREMENT

The universal scaling of the global QFI opens up anavenue towards enhanced precision scaling harnessingdissipative criticality. Saturating the global QFI, how-ever, requires a most general measurement of the jointsystem-environment, which may be practically challeng-ing. Do practical measurement schemes provide accessto the criticality-enhanced scaling (which, in general, islower than the scaling of the global QFI)? In this sectionwe provide a positive answer to this question, by ana-lyzing a readily implementable sensing protocol based oncontinuous measurement of the radiation quanta emittedby the open sensor. While our protocol is generally ap-plicable to critical open systems in diverse setups thatpermit optical readout, to be specific in the following weillustrate it via the open Rabi model [48], a light-matterinteracting model featuring a continuous dissipative CPin zero spatial dimension. Besides its conceptual sim-plicity, its finite-component nature facilitates numericalsimulation, making it possible to extract the scaling ofthe (classical) Fisher information as the precision boundof our sensing scheme. As such, it allows for direct com-parison between the precision scaling of our protocol andthat of a recent study [23] based on direct, instantaneousmeasurement of the critical steady state of the open Rabimodel, therefore demonstrating the advantages of ourscheme.

A. Model system: the open Rabi model

Let us consider a cavity mode coupled to a qubit [cf.Fig. 1(a)] as described by the quantum Rabi Hamiltonian

H = ωc†c+Ω

2σz − λ(c+ c†)σx. (18)

Here, c(c†) denotes the annihilation (creation) operatorof the cavity mode, σz,x are the Pauli matrices of the

6

qubit, ω is the cavity mode frequency, Ω is the qubittransition frequency and λ is the coupling strength. Thecavity leaks photons to the external electromagnetic en-vironment at a rate κ. Without monitoring the environ-ment, the dynamics of the open cavity-qubit system canbe described by a standard LME

ρ = −i[H, ρ] + κ

(cρc† − 1

2c†c, ρ

). (19)

Equation (19) conserves a Z2 parity symmetry c →−c, σx → −σx. Remarkably, in the soft-mode limitω/Ω→ 0, the steady state of Eq. (19) can spontaneouslybreak such a symmetry, resulting in a continuous dissi-pative phase transition [48].

Being zero-dimensional, the model (19) does not pos-sess a rigorously defined system-size. Nevertheless, fol-lowing [57] and [48], we can introduce the frequency ra-tio η = Ω/ω as the ‘effective size’ of the model, withη → ∞ corresponding to the thermodynamic limit [48].We further introduce a dimensionless coupling constantg = 2λ/

√Ωω. In the limit η → ∞, the system un-

dergoes a continuous phase transition at a critical pointgc =

√1 + (2κ/ω)2 [48], cf. Fig. 1(a). At g < gc, the

system is in the normal phase, characterized by the orderparameter 〈c〉st = 0, whereas at g > gc, the system entersthe superradiant phase, characterized by 〈c〉st 6= 0. Fi-nite η plays the role of a finite-size cutoff, which rendersthe phase transition to a smooth crossover.

The critical properties of the CP has been extractednumerically via finite-size(i.e., finite-η) scaling of variousthermodynamic quantities [48], which provides z = 1 asthe dynamic and ν = 2 as the correlation length criticalexponent. As such, the open Rabi model lies in the sameuniversality class as the open Dicke model [58, 59]. Atthe CP g = gc, the mean occupation of the cavity modediverges according to 〈n〉st ∼ η1/2, in which n := c†c. Therefore, the scaling dimension of n is ∆n = −1/2,which satisfies the condition d− 2∆n > 0 (cf. Sec. II B).As such, the sensing of the cavity mode frequency ωserves as a ideal demonstration of criticality enhancedsensing, which we analyze in detail below.

B. Scaling laws of the global quantum Fisherinformation

In Fig. 3 we show the numerical finite-size scaling ofthe global QFI for the sensing of ω, for different systemsizes η in both the transient regime κt . ηz [Fig. 3(a)]and in the long-time regime κt ηz [Fig. 3(b)], assumingthe model is tuned at the CP g = gc. The perfect datacollapse indicates the following scaling behavior of theglobal QFI

Iω(t, η) = (κt)3fI (κt/η) , κt . η, (20)

Iω(t, η) = const.× κtη2, κt η, (21)

in which fI(κt/η) is a universal function reflecting thefinite-size correction. This validates the predictions of

10−1 100

κt/η

10−1

100

η : 100 200 300 400 500 600 700 800 900 1000

I ω(t

,η)

(κt)

3.0

102 103 104

κt/η

10−8

10−6

10−4

∝ ( κtη )−2

(a) (b)

FIG. 3. Finite-size scaling of the global QFI of the criticalopen Rabi model plus the environment, in both (a) the tran-sient regime κt . η and (b) the long-time regime κt η. Theopen system is at a dissipative critical point g = gc, and η isits effective size (see text).

the general formulas (13) and (14) when the relevant ex-ponents d = 0, z = 1 and ∆n = −1/2 are plugged in.

The scaling laws (20) and (21) are Heisenberg-limited.As a zero-dimensional model, the HL of the open Rabimodel can be analyzed in terms of the actual resourcesinvolved, that is, the interrogation time t and the meanoccupation of the cavity mode 〈n〉st. Hereby HL ∼t2〈n〉2st ∼ t2η. From Eqs. (20) and (21), we haveIω(t, η) ∼ (κt/η) × HL in the transient regime κt . η,whereas Iω(t, η) ∼ (η/κt) × HL in the long-time regimeκt η. Therefore, the global QFI manifests super-Heisenberg scaling with respect to the interrogation timet(the cavity photon number 〈n〉st) in the transient(long-time) regime, nevertheless is always upper bounded bythe HL. We further note that Iω(t∗, η) ∼ HL at t∗ = η/κ,i.e., the global QFI saturates the HL in the asymptoticlimit t, η → ∞ with κt ' η kept fixed. These featuresare illustrated in Fig. 1(b).

C. Photon counting and the scaling laws of theFisher Information

Let us now analyze a continuous-measurement–basedsensing protocol for accessing the criticality-enhancedprecision scaling with open quantum sensors. As a con-crete illustration, we consider the sensing of the cavitymode frequency ω of the open Rabi model via photoncounting. Generalization to other types of continuousmeasurement, e.g., homodyning [31], is straightforward.

We assume the photons leaked from the cavity are di-rected to and counted by a photon detector [cf. Fig. 1(a)].For simplicity, we assume unit detection efficiency (finitedetection efficiency reduces the FI by an overall factornevertheless does not change its scaling, see [60]). Theevolution of the joint cavity-qubit system is thereforesubjected to the measurement back-action conditionedon a specific series of photon detection events. Specifi-cally, in an infinitesimal time interval dτ , the detection

7

1 2κt/η

0.075

0.100

0.125

η : 200 300 400 500 600F ω

(t,η

)(κ

t)2.

0

50 100κt/η

0.005

0.010

0.015

0.020

∝ ( κtη )−1

(b)(a)

FIG. 4. Finite-size scaling of the FI Fω(t, η) [cf., Eq. (24)], forthe photon counting signals of the critical open Rabi model(g = gc), in both (a) the transient regime κt . η and (b)the long-time regime κt η. Each data point represents anaverage of 105 independent trajectories in (a), whereas 104

independent trajectories in (b). The sampling numbers arechosen such that the error associated with finite sampling issufficiently small, cf. Fig. 5.

of a photon leads to the collapse of the (unnormalized)conditional state of the cavity-qubit system according to|ψc〉 →

√κdtc|ψc〉, while if no photon is detected, the

system evolves according to the non-unitary dynamics|ψc〉 → M0|ψc〉 with M0 = 1 − dτ(iH + κc†c/2) [51–53, 61]. For an infinitesimal time interval dτ , the proba-bility of the detection of a photon is p1 = κ〈c†c〉cdτ with

〈· · · 〉c := 〈ψc| · · · |ψc〉/〈ψc|ψc〉, whereas the probability of

the detection of no photon is p0 = 〈M†0M0〉c = 1 − p1.Repeating such stochastic evolution for each time step[τ, τ + dτ), in which τ ∈ [0, t), defines a quantum trajec-tory consisting of the photon detection signals up to timet and the associated conditional quantum state |ψc(t)〉.

Mathematically, this can be formulated rigorously interms of a (Ito) stochastic Schrodinger equation [51–53]

d|ψc〉 = −(iH +

κ

2c†c)dt|ψc〉+ dN(t)(

√κdtc− 1)|ψc〉.

(22)Here dN(t) is a stochastic Poisson increment that takestwo values: dN(t) = 0 with probability p0, and dN(t) = 1with probability p1. The last term of Eq. (22) accountsfor the back-action of photon counting by updating thesystem state conditioned on the detection of a photonor not. Corresponding to Eq. (22), the photon countingsignal up to time t for a specific trajectory is D(t, 0) :=dN(ndt), ..., dN(dt), dN(0) where we identify t ≡ ndt.The probability of this trajectory is [51–53]

P [D(t, 0)] = pdN(ndt) · · · pdN(0) = 〈ψc(t)|ψc(t)〉. (23)

An ensemble average over all conditional states leads tothe definition of a density operator of the cavity-qubitsystem, ρ(t) =

∑D(t,0) |ψc(t)〉〈ψc(t)|, which evolves ac-

cording to the LME (19).A single interrogation of our sensing protocol, there-

fore, corresponds to a quantum trajectory consisting of

the continuously detected signal D(t, 0) and the associ-ated conditional evolution of the open system. Process-ing D(t, 0) via standard means (e.g., via Bayesian in-ference [29–31]) provides an estimator ωest of the cavitymode frequency. The associated precision is quantifiedby the Fisher information (FI) of the detected signal

Fω(t) =∑D(t,0)

P [D(t, 0)] ∂ωlnP [D(t, 0)]2 . (24)

The FI (24) can be extracted numerically, via approxi-mating the ensemble average

∑D(t,0) by a statistical av-

erage over sufficient (but finite) numbers of sampled tra-jectories. For each trajectory, we extract P [D(t, 0)] viaEq. (23) following the numerical propagation of Eq. (22),and ∂ωP [D(t, 0)] via calculating P [D(t, 0)] at a slightlydifferent ω and subsequent numerical differentiation.

In Fig. 4 we show the results of our numerical finite-size scaling of the FI for different system sizes η in boththe transient regime κt . ηz [Fig. 4(a)] and in the long-time regime κt ηz [Fig. 4(b)], assuming the modelis tuned at the CP g = gc. The perfect data collapseindicates, similar to the global QFI, that the FI obeysscaling behavior at criticality,

Fω(t, η) = (κt)2fF (κt/η), κt . η, (25)

Fω(t, η) = const.× κtη, κt η. (26)

where fF (κt/η) is a universal function reflecting thefinite-size correction.

As a validation of the accuracy of our numerics, weshow in Fig. 5 the convergence of the approximated FIwith respect to Ntraj, the number of sampled trajecto-ries. As can be seen, the approximated FI graduallyconverges to the predicted scaling form (cf. Fig. 4) asNtraj increases, while convincing data collapse typicallyrequires Ntraj ≥ 104. As the simulation time of eachtrajectory scales polynomially in the system size at crit-icality, the extraction of such FI scaling for generic dis-sipative criticality in high spatial dimensions may repre-sent a computational challenge. Here, thanks to its zero-dimensional, finite-component nature, our model systemallows for extracting such scaling behavior directly byquantum trajectory simulation.

D. Discussion

In contrast to the global QFI, we are not able to relateanalytically the scaling laws of the FI (25) and (26) to thecritical exponents of the underlying CP. Nevertheless, wemanage to develop a physical understanding of these scal-ing laws by analyzing them with respect to the resourcesinvolved in our sensing protocol. First, the long-time be-havior Eq. (26) can be expressed as Fω(t, η) ∼ κ2tt∗with t∗ ' η/κ as defined previously in Sec. III B. Thisshould be compared to the long-time behavior of theglobal QFI (21), which can be expressed similarly as

8

0.5 1.0 1.50.10

0.15

0.20

0.25

η : 200 300 400 500 600

F ω(t

,η)

(κt)

2.0

0.5 1.0 1.5

25 50 75κt/η

0.01

0.02

0.03

F ω(t

,η)

(κt)

2.0

25 50 75κt/η

(a) (b)

(c) (d)

FIG. 5. Convergence of the FI (24) with respect to the numberof sampled trajectories Ntraj. In (a) and (c) Ntraj = 103, whilein (b) and (d) Ntraj = 104. The error bars are estimatedfrom the variance of 10 independent samplings, and are notshown if they are smaller than the data point size. The FIconverges more slowly in the transient regime [(a) and (b)]than in the long-time regime [(c) and (d)], as representedby larger error bars in the former. By sampling a sufficientnumber of trajectories we reduce the sampling error in the FI,leading to the caling collapse as shown in Fig. 4.

Iω(t, η) ∼ κ2tt∗〈n〉2st (note that 〈n〉st ∼ η1/2 at the CP, asintroduced in Sec. III A). Such an expression illustratesthat the global QFI (21) behaves the same as that of astandard noiseless interferometric scheme involving 〈n〉stentangled photons and spanning a time window t∗ in eachinterrogation. The time scale t∗ therefore characterizesthe correlation time of the emitted photons—two pho-ton detection events that are separated by more than t∗are essentially uncorrelated and therefore are analogousto two independent interrogations. Such an interpreta-tion extends naturally to the FI—in comparison to theglobal QFI, it lacks the contribution ∼ 〈n〉2st from thecavity photons, as it is obtained by measurement of theenvironment alone.

In contrast to these long-time scalings, the transientscaling Eq. (25) is due to dynamic critical behavior thatlacks an analog in conventional interferometric schemes.Nevertheless, it can be recovered from the long-time scal-ing Eq. (26) by the simple replacement η → (κt)1/z, asthey both originate from the dynamic criticality of theunderlying CP, with η−1 [(κt)−1/z] being the dominantperturbation in the long-time (transient) regime respec-tively. Similarly for the global QFI, Eq. (20) can be re-

lated to (21) by the same replacement.Finally, we compare the precision bounds of our

continuous-measurement–based sensing protocol withthat based on direct measurement of the critical steadystate of the open Rabi model [23]. The latter schemediscards the emitted radiation quanta and, therefore, theachievable sensing precision of it is upper-bounded bythe QFI of the reduced density metrix of the cavity-qubit system. Such a QFI was shown [23] to obey theSQL, i.e., ∼ κt〈n〉st ∼ κtη1/2, where the linear depen-dence on t results from repeated preparation of the steadystate in every interrogation. By continuously countingthe emitted radiation quanta, the precision scaling is en-hanced as quantified by the FI in the long-time regimeFω(t, η) ∼ κtη. Moreover, in our scheme the global QFIin the long-time regime obeys a further enhanced scalingIω(t, η) ∼ κtη2, which can in principle be achieved by ajoint measurement of the open system and the environ-ment.

IV. CONCLUSION AND OUTLOOK

In contrast to the previous state-of-the-art, we haveestablished a protocol for criticality enhanced sensing viacontinuous observation of the radiation quanta emittedby critical open sensors. The resulting precision achievessignificantly enhanced scaling, thereby establishing themetrological usefulness of the emitted radiation quanta.To achieve this, we have followed a two-fold approach.

First, we establish a scaling theory for the global QFIat a continuous dissipative CP. Under general assump-tions, we demonstrate that the global QFI obeys tran-sient and long-time scaling laws governed by the univer-sal critical exponents of the underlying CP. Such scalingcan be super-Heisenberg with respect to the interroga-tion time t or the particle number N , but not both. Weshow that the global QFI is close to and can in prin-ciple saturate the HL at dissipative criticality, thereforeproviding rich opportunities towards criticality enhancedquantum sensing. To achieve such a precision limit, how-ever, requires the most general measurement of the jointsystem-environment, which may be challenging in prac-tice.

In view of this, we present a feasible sensing schemefor approaching such criticality-enhanced precision scal-ing, based on continuous measurement of the radiationquanta emitted by the open sensor. We illustrate ourprotocol via counting photons emitted by a critical opensensor—the open Rabi model. The relatively simplestructure of this model allows us to extract the FI ofthe detected signal, as a key parameter quantifying theachievable precision of our protocol. Similar to the globalQFI, the FI manifests (transient and long-time) scalingbehavior at the CP which, importantly, exceeds the SQLand therefore is criticality-enhanced. Moreover, as theQFI of the reduced density matrix of the open Rabimodel obeys the SQL [23], our protocol outperforms any

9

protocol based on direct measurement of the open sen-sor alone. Such a continuous measurement based sens-ing scheme can be applied to various open quantum sen-sors permitting continuous readout, thereby establishinga general and practical strategy for criticality enhancedsensing with open quantum sensors.

The quantum sensing framework established in thepresent work is timely and feasible in view of the signifi-cant experimental progresses in recent years towards theintegration of synthetic many-body systems as quantumsensors. Promising candidates for directly implement-ing our model system include the recent realization ofthe critical Rabi Hamiltonian in trapped-ion setups [62],where continuous readout is a well-established tool. Sev-eral experimental platforms have demonstrated the inti-mately related open Dicke model [63, 64], which lies inthe same universality class as the open Rabi model andtherefore shares the same criticality enhanced precision.Other relevant systems where our general theory may ap-ply include optically addressable spins in solid, e.g., colorcenters in diamond [65], two-dimensional surfaces on dia-mond [66], driven-dissipative atomic gas [67, 68], polari-ton condensates [69] and many-body cavity QED [70] andcircuit QED [71] setups. With its broad applicability, oursensing protocol may lead to significant improvements tothe design of ultimate sensing devices on the basis of in-teracting many body systems that may, for example, findapplications in human-machine interfaces [72] or heartdiagnostics [73].

The present work raises a few interesting theoreticalquestions as well. For example, while the general scalingtheory of the global QFI (cf. Sec. II) established here arevalidated numerically for the open Rabi(Dicke) univer-sality class, we believe they and the underlying scalingassumptions hold for a broad class of dissipative CPs,including CPs in higher spatial dimensions. Extendingsuch analysis to models belonging to other universalityclasses and evaluating the associated metrological preci-sion limit therefore represents an attractive theoreticalproblem. In the broad context of open quantum sen-sors, remaining open questions include how to achievethe criticality-enhanced scaling of the global QFI via re-alistic measurement schemes, and the possible generaliza-tion of present results to non-Markovian scenarios [74].Finally, as the QFI is a witness of multipartite entan-glement [75, 76], the universal scaling of the global QFIat dissipative criticality may lead to interesting implica-tions, e.g., to the study of the dynamics of entanglementand other correlations in critical open quantum systems.

ACKNOWLEDGMENTS

This work was supported by the ERC Synergy grantHyperQ (Grant No. 856432), the EU projects HY-PERDIAMOND (Grant No. 667192) and AsteriQs(Grant No. 820394), the QuantERA project NanoSpin(13N14811) and the BMBF project DiaPol (13GW

10−2 10−1 100 101

κτ/η

10−4

10−3

10−2

S(τ,

s,η

)κτ

κs = 0.1η

κs = 1η

κs = 2.4ηη : 100 200 300 400 500 600 700 800 900 1000

∝ ( κτη )−1

10−1 100√η |h|

10−7

10−6

10−5

10−4

S st(

s,η

)×

|h|2

∝ (√

η |h|)2

10−2 10−1 100 101

κτ/η

10−4

10−3

10−2

S(τ,

s,η

)κτ

κs = 0.1η

κs = 1η

κs = 2.4ηη : 100 200 300 400 500 600 700 800 900 1000

∝ ( κτη )−1

(a)

(b)

FIG. 6. Numerical verification of the dynamic scaling ansatz(10) for the open Rabi model, which has z = 1, ν = 2 and∆n = −1/2. The perfect scaling collapse demonstrates thevalidity of Eq. (10) along specific axis in the parameter space(see text).

0281C). We acknowledge support by the state of Baden-Wurttemberg through bwHPC and the German ResearchFoundation (DFG) through grant no INST 40/575-1FUGG (JUSTUS 2 cluster). Part of the numerical simu-lations were performed using the QuTiP library [77].

Appendix A: Numerical validation of the dynamicscaling ansatz

Here we provide a validation of the dynamic scalingansatz (10) for the open Rabi model. Equation (10)prescribes that the correlator S(τ, s, L) obeys a univer-sal scaling form determined by three relevant perturba-tions h, τ−1 and L−1, which can be validated via finite-size scaling collapse in the (h, τ−1, L−1, s−1) parameterspace. For illustration, in the following we fix either τ−1

or L−1, and show numerical finite-size scaling of the cor-relator with respect to the other two perturbations. Tobe specific, we choose O = n which has scaling dimension∆n = −1/2 (cf. Sec. III A).

10

First, we fix τ−1 = 0, which corresponds to taking theinfinitely long time limit, at which S(τ, s, η) = Sst(s, η).In Fig. 6 (a), we show the results of our numerical finite-size scaling of Sst(s, η) for different η and h = g − gc atvarious κs (which serves as an irrelevant perturbation).The perfect scaling collapse indicates, at sufficiently large|h|, |h| η−1/ν , h is the dominant perturbation andcorrespondingly Sst(s, η) ∼ |h|2ν∆n = h−2. In contrast,at small h the inverse system size η−1 is the dominantperturbation, and correspondingly Sst(s, η) ∼ η−2∆n =η. These demonstrate the validity of Eq. (10) along theaxis τ−1 = 0.

Next, we fix h = 0, and perform finite-size scal-ing of S(τ, s, η) for different η and τ at various κs, asshown in Fig. 6 (b). The perfect scaling collapse indi-cates, at sufficiently small τ−1, the inverse system sizeη−1 is the dominant perturbation and correspondinglyS(τ, s, η) ∼ η−2∆n = η, whereas at large τ−1 the scalingform becomes Sst(s, η) ∼ τ−2∆n/z = τ . These demon-strate the validity of Eq. (10) along the axis h = 0.

Similar procedure can be carryied out for all the pa-rameter space (h, τ−1, L−1, s−1) surrounding the CP,which validates Eq. (10).

Appendix B: Numerical calculation of the globalquantum Fisher information

We follow an efficient method proposed in Ref. [42] tonumerically calculate the global QFI of the joint system-environment, which we summarize here to keep our workself-contained. The starting point is an alternative ex-pression of the QFI equivalent to Eq. (1) of the maintext,

Iθ(t) = 4∂θ1∂θ2(log | 〈Ψθ1(t)|Ψθ2(t)〉|)|θ1=θ2=θ, (B1)

where |Ψθ1(t)〉 is the pure global state of the system plusthe environment at time t. Although a complete knowl-edge of the global quantum state is impractical as thenumber of photons in the environment increases withtime, in [42] the authors describe an efficient way to cal-culate the QFI without accessing the full quantum state.The main idea is that as long as the Born-Markov ap-proximation is valid we can descretize time such thatin every time interval [ti, ti + δt) the system interactswith independent environmental degrees of freedom. Asa result, at time t = Nδt, the state of the joint system-bath |Ψ(t)〉 can be expressed as an entangled state in thetensor-product Hilbert space of the open system and Nenvironmental subspaces

|Ψ(t)〉 = UtN−1UtN−2

· · · Ut0 |ψ(0)〉 |0N−1, · · · 00〉 , (B2)

where |ψ(0)〉 is the initial state of the system, Uti arethe unitary operators acting on the open system and theenvironment Hilbert space associated to time ti. Here wehave also associated an L-dimensional Hilbert space foreach environment subspace |mi〉,m = 0, 1, . . . , L − 1,and assumed that each of them are initialized in the vac-uum |0i〉 before the interaction. In turn, we can intro-

duce ‘measurement effect operators’, Mmiwhich define

the evolution of the open quantum system with the asso-ciated transfer of the environment from state |0i〉 to |mi〉and write

|Ψ(t)〉 =

M−1∑m0···mN−1=0

MmL−1· · · Mm0

|ψ(0)〉

⊗ |mN−1, · · ·m0〉 (B3)

A particular choice of m0 · · ·mN−1 which correspondsto a single term in Eq. (B3) defines the quantum stochas-

tic trajectory |ψc(t)〉 ∝ MmN−1· · · Mm0 |ψ(0)〉. The un-

known parameter, θ, that we want to estimate is encodedin the unitary operators Uti and hence the measurementeffect operators Mmi . After tracing out the environmentdegrees of freedom we obtain the reduced density ma-trix that describes the open quantum system and evolesaccording to the equation:

dρ(t)

dt=

(L−1∑m=0

Mmρ(t)M†m − ρ(t)

)/δt (B4)

For infinitecimal time step δt this gives rise tothe standard Lindblad master equation as describedin the main text. Going back to Eq.(B1) noticethat the inner product 〈Ψθ1 |Ψθ2〉 can be written asTrsys,env|Ψθ1〉 〈Ψθ2 | = Trsysρθ1,θ2 where the actionof the operators MmN−1

(θ1) · · ·Mm0(θ1) from the left

and M†m0(θ2) · · ·M†mN−1

(θ2) from the right, has been ab-sorbed in the definition of ρθ1,θ2 . This is similar toEq. (B4) and thus ρθ1,θ2 is the solution of a general-ized master equation (ME) dρ/dt = Lθ1,θ2ρ. Specificallyto our case that the unknown parameter is encoded inthe Hamiltonian of the open system, the generalized MEcan be derived to be

Lθ1,θ2ρ = −iH(θ1)ρ+iρH(θ2)+∑`

(J`ρJ

†` −

1

2J†` J`, ρ

).

We can solve the generalized ME numerically for (θ1, θ2)in the neighborhood of (θ, θ) and determine the globalQFI by numerical difference via Eq. (B1).

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