Non-Hermitian Quantum Sensing: Fundamental Limits and Non-Reciprocal Approaches
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Hoi-Kwan Lau ∗ and Aashish A. Clerk Institute for Molecular Engineering, University of Chicago,5640 South Ellis Avenue, Chicago, Illinois 60637, U.S.A. (Dated: May 31, 2018)Unconventional properties of non-Hermitian systems, such as the existence of exceptional points,have recently been suggested as a resource for sensing. The impact of noise and utility in quantumregimes however remains unclear. In this work, we analyze the parametric-sensing properties of linearcoupled-mode systems that are described by effective non-Hermitian Hamiltonians. Our analysisfully accounts for noise effects in both classical and quantum regimes, and also fully treats a realisticand optimal measurement protocol based on coherent driving and homodyne detection. Focusingon two-mode devices, we derive fundamental bounds on the signal power and signal-to-noise ratiofor any such sensor. We use these to demonstrate that enhanced signal power requires gain, but notnecessarily any proximity to an exceptional point. Further, when noise is included, we show thatnon-reciprocity is a powerful resource for sensing: it allows one to exceed the fundamental boundsconstraining any conventional, reciprocal sensor. We analyze simple two-mode non-reciprocal sensorsthat allow this parametrically-enhanced sensing, but which do not involve exceptional point physics.
Among the most powerful and ubiquitous measure-ment techniques is dispersive measurement, where a pa-rameter of interest shifts the frequency of a resonant elec-tromagnetic mode. Dispersive measurement is used ina myriad of tasks, including in settings where quantumnoise and quantum limits are relevant. Examples rangefrom the sensing of biomolecules and nanoparticles [1–3], to the measurement of superconducting qubits [4, 5],quantum optomechanical measurements of mechanicalmotion [6], and gravitational wave detection [7–9].Given its widespread utility, methods for improvingdispersive measurements are of immense practical andfundamental interest. In this regard, there has beenconsiderable recent interest in exploiting non-Hermitiandynamics in linear coupled-mode systems to enhancedispersive-style measurements [10–16]. Such systems aredescribed by an effective non-Hermitian Hamiltonian ma-trix, and can exhibit exceptional points (EPs), where asa function of parameters two eigenvalues of the Hamilto-nian coalesce and the matrix becomes defective. Nearsuch EPs, the system eigenvalues have an extremelystrong dependence on small changes in parameters. Inthe simplest two-mode realization [17, 18], a parameter ǫ which enters the Hamiltonian linearly is able to shifteigenmode frequencies by an amount √ ǫ . For small ǫ , thissuggests an extremely strong response, and the possibil-ity of enhanced sensing. The first experiments probingthis extreme sensitivity of mode frequencies to paramet-ric changes have recently been reported [19, 20].To date, almost all work on EP-based sensing focuseson frequency shifts whose magnitude is at least compa-rable to mode linewidths. It is however also interest-ing to ask whether non-Hermitian sensing methods areeffective in the common weak dispersive regime, where ∗ [email protected] frequency shifts are smaller than linewidths; this is thegoal of our work. Analyzing this regime involves address-ing several general questions about non-Hermitian sens-ing. First, most studies focus exclusively on character-izing parametric shifts of mode frequencies; the processof how such shifts are measured is not fully analyzed.This is problematic, as a realistic sensing protocol maybe sensitive to the parametric dependence of both theeigenvalues and eigenvectors of the system Hamiltonian;this latter dependence could conceivably counteract theparameter dependence of the eigenvalues [21]. Second,the impact of fluctuations has not been discussed. In thecoupled mode settings of interest, non-Hermitian effec-tive dynamics always corresponds to dissipative dynam-ics which will generically be accompanied by noise. Thisnoise can limit the ability to resolve parameter changes.This is especially crucial in quantum settings, where onecan never ignore the effects of vacuum noise, especially ifthe dissipative dynamics involves amplification processes.In this paper, we address both these sets of issues. Weanalyze a generic linear non-Hermitian sensing setup bymapping it to a fully probability-conserving open quan-tum system. This allows us to fully account for fluctua-tion effects in both classical and quantum regimes of op-eration. This mapping is not unique, implying that thereare many possible ways to realize a given non-HermitianHamiltonian, each having different levels of fluctuations.Among these, we show there is an optimal low-noise re-alization that is ideal for sensing.Further, we go beyond simply characterizing the para-metric dependence of eigenvalues, and explicitly modela full measurement protocol. We consider a standardapproach which is in fact optimal: to detect changes insystem eigenvalues, the system is driven coherently viaan input-output waveguide, and the reflected signal ismeasured using homodyne interferometry (see Fig. 1(a)).Focusing on the well-studied case of a two-mode sensingsetup, our general approach allows us to derive funda- classical drivehomodynereadoutwaveguide mode-modemode-lossmode-gainmode-readout (a) (b)(c) FIG. 1. (a) General dispersive measurement setup consisting of resonant modes (circles) that interact via a parameter-dependentnon-Hermitian Hamiltonian. Standard analyses only consider the non-Hermitian dynamics of mode amplitude (region insidegrey rectangle). In this work, we instead treat the system as an open quantum system, where non-Hermitian dynamics isgenerated by coupling to gain/loss baths (red/blue rectangles) and a readout waveguide. The coupling rates to the variousbaths (dotted lines) is characterized by matrices Y and Z as defined in Eq. (5); H describes the Hermitian direct couplingsbetween modes (solid lines). A classical drive is injected into the readout waveguide, which couples only to mode 1. Its reflectedfield is measured by homodyne detection. A parametric change in the Hamiltonian (e.g. coupling between modes 1 and 2 here)changes the state of modes as well as that of the reflected field. (b) Integrated homodyne current ˆ m at a certain measurementtime τ , as a function of the detuning ∆ of the drive frequency from the cavity 1 resonance frequency. The shaded area denotesuncertainty due to measurement noise, and the two curves are for two values of the parameter to be sensed. A parametricchange can be optimally detected by measuring at a single detuning, e.g. ∆ (dashed vertical line). (c) Time variation ofintegrated homodyne current for a fixed detuning ∆ . The signal induced by the perturbation to be sensed ( √S , pink arrow)scales linearly as τ , while the uncertainty ( √N , green arrow) has a weaker scaling, √ τ . Therefore, any small perturbation canbe resolved for sufficiently long τ . mental bounds on both the signal power generated by thisprotocol, as well as on the signal-to-noise ratio (SNR). Asany setup could be improved indefinitely by simply boost-ing drive power, we focus on the physically-motivatedcase where the total circulating power (i.e. intracavityphoton number) used for the measurement is kept fixed.For a reciprocal system, we show explicitly that the onlyway to parametrically boost the signal without also boost-ing the circulating power is to build an amplifier, i.e. sig-nals should be reflected from the system with gain. Weshow that this can be accomplished without having totune a system near an EP. Further, we show that the SNRis fundamentally bounded by intracavity photon numberalone, reflecting the unavoidable noise associated withgain processes.Finally, we show that there is an alternate approachfor exploiting non-Hermitian dynamics for sensing, onethat does not require proximity to an EP. By construct-ing an optimal non-reciprocal, non-Hermitian system,we demonstrate that one can have measurement pro- tocols that arbitrarily surpass the fundamental boundsconstraining any reciprocal system. Non-reciprocity isalso useful outside the regime of weak-dispersive mea-surements: we show that it can also be used to dramat-ically enhance sensing schemes based on mode-splitting,without any need for an EP. We analyze an implementa-tion of these ideas that should be accessible in a varietyof different experimental platforms. I. PARAMETER SENSING WITH ANON-HERMITIAN COUPLED-MODENETWORKA. General setup
We consider a generalized version of the non-Hermitiansensing system studied in previous works [10, 13–15, 19,20, 22]: M resonant modes interact as described by thelinear and Markovian coupled-mode equations:˙ α i ( t ) = − i X j ˜ H ij [ ǫ ] α j ( t ) . (1) α j ( t ) denotes the amplitude of mode j , and the M × M matrix ˜ H is an effective non-Hermitian Hamiltoniandescribing both coherent and dissipative linear dynamics.The Hamiltonian depends on a parameter ǫ , and the goalis to sense an infinitesimal change in ǫ . We assume thatthis parameter only changes non-dissipative terms in ˜ H ,and thus write ˜ H ij [ ǫ ] = ˜ H ij [0] + ǫV ij (2)where the Hermitian matrix V describes the coupling ofthe parameter to the dynamics. We take ǫ to have unitsof frequency, and hence V is dimensionless.Unlike many works, we explicitly analyze the protocolused to measure the parametric dependence of ˜ H . Ageneral strategy is to couple mode 1 to an input-outputwaveguide or transmission line, and then use this port todrive this system with a coherent tone at a frequency ω dr .The reflected signal is then measured, and used to infer ǫ . Coupling to the waveguide introduces extra dampingof mode 1, and hence ˜ H ij → ˜ H ij − i ( κ/ δ i δ j , where κ is the coupling rate to the waveguide. Working in arotating frame at the drive frequency, the coupled modeequations now become:˙ α i = i ∆ α i − i X j ˜ H ij [ ǫ ] α j − iδ i √ κβ , (3)where β is the amplitude of the coherent drive. WLOGwe take β to be real and positive, and choose a frequencyreference such that Re ˜ H [0] = 0. This implies that ∆represents the detuning of the drive frequency from themode-1 resonance frequency.In addition to fully treating the measurement, we alsowant to consistently describe noise effects associated withthe dissipative dynamics encoded in ˜ H . Dissipativedynamics correspond to the anti-Hermitian part of ˜ H ,which can always be written in terms of the difference oftwo positive-definite matrices. We thus write ˜ H − ˜ H † i ≡ Y Y † − ZZ † − (1 / κ , (4)where ( κ ) ij = κ δ i δ j . The matrix Y Y † representsgain processes, i.e. processes that tend to cause expo-nential growth in time; correspondingly, ZZ † representsloss processes (beyond the loss associated with the input-output waveguide). For definiteness, we take Y to be a M × N Y matrix, and Z to be a M × N Z matrix. We alsodefine H = ( ˜ H + ˜ H † ) / ˜ H ,which describes non-dissipative dynamics).We can now view the coupled-mode equation in Eq. (3)as the noise-averaged version of a fully probability-conserving linear Markovian open quantum system. This description is useful even in the classical regime if onewants to account for the effect of thermal noise. Thenon-Hermitian dynamics in ˜ H are generated by couplingto N Y + N Z distinct dissipative environments, with spe-cific mode-bath coupling constants given by the matrices Y , Z . Letting ˆ a i denote the canonical bosonic annihila-tion operator of the i th mode, the full system is describedby the Heisenberg-Langevin equations˙ˆ a i = i ∆ˆ a i − i X j ˜ H ij [ ǫ ]ˆ a j − iδ i √ κβ − iδ i √ κ ˆ B in − i √ N Y X j =1 Y ij ˆ C in † j + N Z X j =1 Z ij ˆ D in j . (5)The first line here has the same structure as in Eq. (3),and describes the linear dynamics of our system and itscoherent driving. The terms on the second line insteaddescribe zero-mean noise driving our system. ˆ B in is noiseentering from the input-output waveguide, whereas ˆ C in j ( ˆ D in j ) are noises entering from the dissipative baths usedto realize the gain (loss) parts of the dissipative dynam-ics encoded in ˜ H . Consistent with the linear, Marko-vian nature of our system, these noise operators repre-sent (operator-valued) Gaussian white noise. Quantummechanically, they cannot be zero: at best, they describevacuum fluctuations. In this case, we have: h ˆ Q in ( t ) ˆ Q in † ( t ′ ) i = (¯ n th Q + 1) δ ( t − t ′ ) (6) h ˆ Q in † ( t ) ˆ Q in ( t ′ ) i = ¯ n th Q δ ( t − t ′ ) (7) h ˆ Q in ( t ) ˆ Q in ( t ′ ) i = 0 (8)where Q ∈ { B, C j , Z j } , and there are no correlationsbetween different noise operators. The averages aboverepresent averages over different realizations of the noiseprocess, or equivalently, over the state of the bath degreesof freedom. ¯ n th Q represents the thermal occupancy of bath Q ; we focus on the case where there is only vacuum noise,and these occupancies vanish (though our formalism canalso easily treat the classical case ¯ n th Q ≫ α i ≡ h ˆ a i i recoversEq. (3). The additional noise effects encoded in Eq. (5)will however be important in determining our ability tomake a measurement. We stress that these MarkovianHeisenberg-Langevin equations are standard in the studyof open quantum systems; a derivation is provided in Ap-pendix A, and pedagogical treatments are given in [5, 23].A crucial observation here is that the system-bath cou-pling matrices Y , Z in Eq. (4) are not uniquely deter-mined by ˜ H . This ambiguity corresponds to a simplephysical fact: there are many different ways to couple todissipative baths to realize a given non-Hermitian dynam-ics . As is perhaps obvious, noise will play a crucial rolein determining the measurement sensitivity of ǫ ; hence,the sensitivity will depend on the particular choice ofbaths and bath couplings used to realize ˜ H . This leadsto two important conclusions: (i) ˜ H on its own does notcompletely specify the performance of our detector, and(ii) for a given non-Hermitian Hamiltonian, an optimalmeasurement will require using an optimized choice ofdissipative baths and bath couplings. B. Homodyne measurement and measurement rate
We now discuss how the information on ǫ in the re-flected field leaving mode 1 can be extracted [24]. We willcharacterize the measurement sensitivity using standardmetrics that are well established in describing a weak,continuous linear measurement; see, e.g., [5] for a peda-gogical discussion. This will allow us to directly comparethe non-Hermitian sensing protocols to more establishedmethods.The amplitude of the reflected field in the waveguideis described by an operator ˆ B out . Using standard input-output theory [23], we haveˆ B out ( t ) = (cid:16) β + ˆ B in ( t ) (cid:17) − i √ κ ˆ a ( t ) . (9)The first term describes the incident field on mode 1that is promptly reflected, whereas the second term de-scribes the field emitted from mode 1. Note that the re-flected field in our geometry is completely equivalent tothe transmitted field in standard setups where an opticalfiber is coupled to a whispering-gallery mode resonator[1–3].For small ǫ , the average value of the output field willhave a linear dependence on ǫ . We will be interestedthroughout this paper on long measurement times, andhence will focus on the steady state (time-independent)value of this average. We thus write h ˆ B out i ǫ ≃ h ˆ B out i + λ ǫ (10)where λ is a (possibly complex)linear response coefficient.We throughout use h .. i z to denote an average calculatedusing Eq. (5) with ǫ = z .Letting φ = − arg λ , it is clear that all the informa-tion on ǫ in the output field is contained in the real partof e iφ ˆ B out . An optimal measurement strategy is thus tomeasure this quantity directly. This corresponds to onequadrature of the output field, and the necessary mea-surement is known as homodyne detection. The time-dependent measurement signal (i.e. the homodyne cur-rent) is described by the operator ˆ I ( t ):ˆ I ( t ) ≡ r κ (cid:16) e iφ ˆ B out ( t ) + e − iφ ˆ B out † ( t ) (cid:17) (11)Note the factor of √ κ is included in the homodyne cur-rent for convenience, as it makes ˆ I have the units of arate. The homodyne current will be subject to shot noisefluctuations which will obscure our ability to extract ǫ .This noise is described by a spectral density [5]:¯ S II [ ω ] = 12 Z ∞−∞ dte iωt h{ δ ˆ I ( t ) , δ ˆ I (0) }i , (12)where δ ˆ I ≡ ˆ I − h ˆ I i . As we are considering the effects ofan infinitesimal perturbation ǫ , we can characterize ourmeasurement sensitivity using the noise spectral densitycalculated to zeroth order in ǫ .To estimate ǫ , the homodyne current is integrated from t = 0 to t = τ to average away the effects of noise.The time-integrated measurement is thus described bythe operator: ˆ m ( τ ) ≡ Z τ dt ˆ I ( t ) . (13)Considering the long- τ limit, the “power” associated withthe signal induced by the perturbation is: S = [ h ˆ m ( τ ) i ǫ − h ˆ m ( τ ) i ] = 2 κǫ | λ | τ (14)We have assumed a measurement time τ that is longenough that we can ignore any transient effects in thebehaviour of h ˆ I ( t ) i . Note also that with our definitions, S is dimensionless.Similarly, the noise power associated with the inte-grated homodyne current in the long time limit is: N ≡ h δ ˆ m ( τ ) δ ˆ m ( τ ) i = τ ¯ S II [0] , (15)where δ ˆ m ≡ ˆ m − h ˆ m i .Combing these results, we see that the power signal tonoise ratio associated with the homodyne measurementgrows linearly with time:SNR( τ ) ≡ SN = 2 κǫ τ | λ | ¯ S II [0] ≡ ǫ κ τ Γ meas . (16)We have defined the long-time linear growth of the SNRin terms of a measurement rate Γ meas . This is a standardmetric for quantifying the resolving power of weak con-tinuous measurements; ( κ/ǫ ) Γ − represents the min-imum time required to distinguish ǫ = ǫ from ǫ = 0.The measurement rate defined here is also directly re-lated to the another standard metric for sensitivity, theimprecision noise spectral density, see [5].More fundamentally, one could ask whether homodynemeasurement is truly the optimal way to use the out-put field to estimate ǫ . While heuristically this seemsclear from Eq. (10), one can ask the question more for-mally. The maximum amount of information available inthe output field considering all possible measurements isquantified by the quantum Fisher information [25]. Thisquantity can be calculted exactly for our linear, Gaussiansystem [26]. In Appendix C, we show that this metric co-incides with the SNR given above in the limit where thedriving field β is sufficiently large. As such, the homo-dyne measurement strategy here is indeed the optimalstrategy.We stress that our measurement scheme involves driv-ing the system at a single frequency only. This is incontrast to most works on EP sensing [10, 13, 19, 20],which involve probing the system over a wide range offrequencies to measure a full output field spectrum. Forthe small ǫ regime we consider, there is no advantage forsuch multi-tone driving, as the information generated ateach frequency is independent. It is thus optimal to probethe system with a single coherent tone whose frequencyis chosen to optimize Γ meas , see Fig. 1. We provide a rig-orous proof of this statement (in terms of the quantumFisher information) in Appendix C 2. C. General expressions and constraints for a linearsystem
While the definition of the SNR and measurement ratein Eq. (16) is generally applicable, things simplify enor-mously for our system given the linearity of the dynamics.For a stable system, the Langevin equations in Eq. (5)can be solved in the Fourier domain in terms of the di-mensionless system susceptibility matrix ˜ χ defined as ˜ χ [ ω ; ∆; ǫ ] ≡ iκ h ( ω + ∆) I − ˜ H [ ǫ ] i − , (17)where I is the M × M identity matrix. Using the input-output relation in Eq. (9) and taking average values, weimmediately find that the steady-state average homodynecurrent is given by h ˆ I i = √ κ Re (cid:2) e iφ β (1 − ˜ χ [0; ∆; ǫ ]) (cid:3) (18)Note that the homodyne current depends on ǫ through ˜ χ , which in turn depends on both the eigenvalues and eigenvectors of ˜ H . The zero-frequency susceptibility ma-trix can in general be written in terms of the eigenvaluesΩ j of ˜ H as ˜ χ [0; ∆; ǫ ] = − iκ adj( − ∆ I + ˜ H [ ǫ ]) Q j ( − ∆ + Ω j [ ǫ ]) , (19)where adj( · ) is the adjugate matrix. The basis of manysensing techniques is that the eigenvalues Ω j generallyhave a dependence on ǫ , which directly influences thesusceptibility and hence output field. However, to get acomplete description of the measurement, one must alsoworry about the numerator in this expression: the adju-gate matrix (e.g. right and left eigenvectors of ˜ H ) willalso in general depend on ǫ , which can serve to suppressthe overall sensitivity to ǫ . In what follows, we thus focuson the entire susceptibility matrix, and not just on theeigenfrequencies of ˜ H .Returning to Eq. (18) and considering small ǫ , onereadily finds a direct expression for the linear response coefficient λ in Eq. (10). Defining χ (∆) ≡ ˜ χ [0; ∆; 0] asthe zero-frequency, unperturbed susceptibility matrix, wehave λ = − β d ˜ χ [0; ∆; ǫ ] dǫ (cid:12)(cid:12)(cid:12) ǫ =0 = i βκ ( χV χ ) . (20)We will implicitly assume χ is evaluated at ∆ unless spec-ified.Using this expression, it is straightforward to calcu-late the signal power associated with the time-integratedhomodyne current (c.f. Eq. (14)): S ( ǫτ ) = 2 β κ | ( χV χ ) | = 2¯ n tot | ( χV χ ) | ( χ † χ ) . (21)In the second equality, we have expressed S in terms ofthe total average photon number in all modes induced bythe coherent drive:¯ n tot ≡ X i h ˆ a † i ih ˆ a i i = β κ X i | χ i | = β κ ( χ † χ ) . (22)Our motivation here is that S can always be increasedindefinitely by simply increasing the drive power. For ameaningful metric, one thus needs to ask how much sig-nal is generated given a fixed number of photons usedfor the measurement. In many situations, the photonsto worry about are the intracavity photons described by¯ n tot : if this photon number becomes too large, a vari-ety of problems typically ensue (e.g. unwanted heatingeffects, breakdown of linearity, etc.) [27].Turning to the fluctuations in the homodyne current, astraightforward calculation using Eqs.(5) and (15) yields:¯ S II [0] = κ (cid:16) κ ( χY Y † χ † ) (cid:17) . (23)The first term here represents the unavoidable shot noisein the homodyne current. The second term describes ad-ditional noise emanating from the dissipative baths thatgenerate the gain processes in ˜ H . This extra noise cor-responds to the amplification of zero-point fluctuations,and is connected to the fact that quantum mechanically,phase-insensitive linear amplification cannot be noiseless[28]. We stress that for a fixed ˜ H [0], the choice of Y is not unique; thus, the noise properties of our setup is not directly determined by ˜ H [0], but will depend cru-cially on how the dissipative dynamics is realized usingexternal baths.For a fixed ˜ H [0] (and hence fixed χ ), we can findthe optimal choice of baths and bath couplings thatminimizes the noise in the homodyne current (see Ap-pendix B). We find:¯ S II [0] min = κ (cid:16) (cid:2) | − χ | − (cid:3) ( | − χ | − (cid:17) , (24)where Θ[ z ] is the Heaviside step function. Again, thisresult reflects the well known quantum limits on addednoise of linear amplifiers [28]. Here, if our system hasreflection gain (i.e. | − χ | > ˜ H [0] and corresponding susceptibility matrix χ ,it is is always possible to construct a realization of thedissipative dynamics (in terms of bath couplings Y , Z )that attains this minimum possible noise level (see Ap-pendix F).Combining these results gives us a general bound onthe measurement rate of any linear system:Γ meas ≤ Γ opt (25) ≡ κ ¯ n tot ( χ † χ ) | ( χV χ ) | (cid:2) | − χ | − (cid:3) ( | − χ | − . We are now in a position to quantitatively ask whethersystems exploiting non-Hermitian physics (such as EP-based sensing schemes) truly offer advantages over moreconventional sensing schemes, including simple sensingschemes based on a linear amplifier.
II. TWO-MODE NON-HERMITIAN SENSORS
The results of Sec. I are extremely general, applying toany non-Hermitian sensing setup described by Eq. (5). InAppendix E, we consider the simple case where the pa-rameter of interest simply shifts the resonant frequencyof mode 1. Here however, we apply our results to thespecific kind of system that has been extensively studiedin the literature on EP sensing [10, 13–15, 19, 20]: a two-mode system described by a non-Hermitian Hamiltonian ˜ H [ ǫ ], where the parameter to be determined is a Hermi-tian coupling between the modes. This corresponds to acoupling matrix V = (cid:18) / / (cid:19) (26)in Eq. (2).The signal power in the homodyne current follows di-rectly from Eq. (21) and is given by S = 116 | χ | | χ + χ | | χ | + | χ | S ǫ . (27)where S ǫ ≡ ǫ τ ¯ n tot . (28)is the signal power associated with a standard, single-mode dispersive measurement (see Appendix E). A. Reciprocal sensors
Consider first a reciprocal system [29] [30], where themagnitude of the coupling between the two modes does - - - - FIG. 2. (Left) Signal power S and (right) measurementrate Γ meas against drive detuning ∆ for three 2-mode non-Hermitian sensors. Blue dot-dashed: Reciprocal EP systemwithout any gain, described by Eq. (31) with γ = 0 , γ =0 . κ, J = 0 . κ . Blue solid: Reciprocal EP system with gain,described by Eq. (31) with γ = 0 , γ = − . κ, J = 0 . κ .Despite a higher signal power, introducing gain does not en-hance the measurement rate due to the corresponding in-creased level of measurement noise. Neither of these systemsbeat the fundamental reciprocal-system bound in Eq. (30)(green dotted). Red: Non-reciprocal system in Eq. (35)( γ = κ, γ = 0 . κ, J = 1 . κ, ν = 0). It yields a measure-ment rate which appreciably exceeds the reciprocal-systembound for a wide range of ∆. not have any directionality, i.e. | ˜ H | = | ˜ H | . This im-mediately implies that | χ | = | χ | , and allows us tobound the maximum value of S : S recip ≤ S ǫ | χ | . (29)Thus, for a reciprocal system, the only way to paramet-rically increase the signal power (at fixed measurementtime τ and intracavity photon number ¯ n tot ) is to make | χ | large. This implies that the system is an ampli-fier: signals incident in the coupling waveguide will bereflected with gain.Including now the effects of measurement noise, theabove bound on signal power for a reciprocal two-modesystem, when combined with Eq. (24), immediatelyyields a bound on the measurement rate:Γ meas , recip ≤ κ ¯ n tot . (30)We see that Γ meas for a reciprocal sensor is fundamen-tally bounded by the intra-cavity photon number and thecoupling rate κ to the waveguide; unlike signal power, itcannot be made arbitrarily large by increasing | χ | . Asdiscussed in Appendix D, achieving this bound requires χ = 2, implying the absence of reflection gain. If one in-stead increases | χ | ≫ κ ¯ n tot . - - EPno EP
Detuning S i gna l po w e r FIG. 3. Signal power versus drive detuning for two reciprocaltwo-mode sensors: an EP system (blue, described by Eq. (31)with κ + γ = κ , γ = − . κ , J = 0 . κ ), and a simpletwo-mode amplifier system that never has an EP (orange,Eq. (31) with κ + γ = γ = 0 . κ , J = 0 . κ ). The twosystems have similar peak signal powers. Dotted lines denotebound on signal power for both systems as given by Eq. (29). These results apply directly to the kind of non-Hermitian two-mode sensors that have been studied ex-tensively in the literature [14–16, 19]. These systemsgenerically involve a sensing parameter that couples asper Eq. (26), and a reciprocal two-mode effective Hamil-tonian of the form ˜ H recip [0] = (cid:18) − i κ + γ JJ − i γ (cid:19) . (31)Here J is the Hermitian coupling between the modes,whereas γ , γ describe possible gain/loss processes (de-pending on the sign) acting locally on each mode. Asalways, κ represents the coupling rate between the input-output waveguide and mode 1; note that this couplinghas mostly been neglected in previous work.The eigenvalues of ˜ H [0] in this case are:Ω ± [0] = − i κ + γ + γ ± r J −
14 ( κ + γ − γ ) . (32)It thus exhibits a stable EP when J = ( κ + γ − γ ) / κ + γ + γ >
0. For this tuning of J , the modeeigenvalues behave as Ω ± [ ǫ ] = ±√ Jǫ − i ( κ + γ + γ ) / ǫ .Despite the large sensitivity of mode frequencies to ǫ at the EP, the signal power and measurement rate forthis setup remain bounded by Eqs. (29) and (30). Thisis shown explicitly in Fig. 2, where the signal power andmeasurement rate for this system is plotted as a functionof the drive frequency. These quantities never exceed thefundamental bounds. Note that in many applications, it is only the signalpower that is relevant, as the measurement noise will belimited by non-intrinsic effects (e.g. following amplifiersand detector inefficiency). It is thus interesting to notethat the signal-power performance of the two-mode EPsystem in Eq. (31) can be matched with a simple two-mode amplifier setup, where the first mode is subjectlocally to gain, and the total damping rate of mode 2 ismade to match that of mode 1. While this system neverpossesses an EP, its performance matches the EP system,see Fig. 3. Thus, in terms of signal power at fixed photonnumber, there is no fundamental utility here to using aEP system. B. Non-reciprocal sensors
The above discussion shows that for a reciprocal sys-tem, tuning to an EP does not provide special advantagesfor measurement. We now consider another means ofexploiting non-Hermitian physics: a sensor whose effec-tive Hamiltonian breaks reciprocity, i.e. | ˜ H | 6 = | ˜ H | .Breaking reciprocity allows one to parametrically ex-ceed the bounds in Eqs. (29) and (30) that constrainany reciprocal two-mode sensing system. Synthetic non-reciprocity in driven photonic systems is an active areaof current research (see Ref. [31] and references therein),with experimental demosntrations in photonic platformsas well as superconducting quantum circuits and optome-chanical systems. While most work in this area has fo-cused on achieving non-reciprocal scattering to build de-vices such as isolators and circulators, we show here thatnon-reciprocity can also be a powerful resource for en-hanced sensing.To see how non-reciprocity changes our sensing prob-lem, consider again Eq. (27) for the signal power, in theextreme directional limit where χ = 0, but χ = 0.This describes a situation where mode 2 influences mode1 but not vice-versa. The signal power for this fully di-rectional setup becomes independent of χ : S dir = 116 S ǫ | χ | . (33)The signal power could now in principle be increased in-definitely by increasing χ while keeping the intracavityphoton number and reflection gain fixed.The benefits of non-reciprocity are more apparentwhen we consider noise and the full expression for themeasurement rate. In a non-reciprocal system we can in-crease the signal power indefinitely (by making χ large) without having to have a large χ and hence reflectiongain. This implies that the output noise can stay at theshot noise level. For a non-reciprocal system, we thushave: Γ meas , dir ≤ κ ¯ n tot | χ | (34)For | χ | >
4, this exceeds the fundamental bound onthe measurement rate of a reciprocal system given inEq. (30). We thus see that non-reciprocity is a resourcefor enhanced sensing; moreover, it does not require a sys-tem that is tuned to an EP.It is helpful to consider a concrete example of a fullynon-reciprocal setup. Consider a non-Hermitian Hamil-tonian ˜ H dir [0] = (cid:18) − i κ + γ J ν − i γ (cid:19) , (35)where γ and γ describe local damping or anti-dampingof the two modes, ν is the frequency detuning of thetwo modes, and J describes a (complex) non-reciprocalmode-mode coupling. Such directional couplings can berealized in many different ways, e.g. by using parametricdriving and engineered dissipation as discussed in [32].The susceptiblity matrix is readily found. One has χ = 0. For a drive that is resonant with mode 1(i.e. ∆ = 0), the remaining elements are χ = 2 κκ + γ , χ = − χ Jν − iγ / . (36)As desired, one can make χ arbitrarily large by increas-ing J without requiring that χ also become large. Asa result, one can reach the upper bound on the mea-surement rate given in Eq. (34) for γ ≥
0. The perfor-mance of this non-reciprocal sensor is shown in Fig. 2,where its performance is compared against reciprocalnon-Hermitian systems. One clearly sees the violation ofthe reciprocal-system bound on the measurement rate.Note that at an EP, the Jordan normal form of a 2 × H dir [0] areΩ − [0] = − i κ + γ , Ω + [0] = ν − i γ . (37)The system has an EP only when the parameters areprecisely tuned to ν = 0 and κ + γ = γ . In contrast,the large enhancement of the measurement rate we ob-tain only requires | J | ≫ p ν + γ /
4. This condition isclearly unrelated to the presence of an EP.While the simple non-reciprocal sensing setup inEq. (35) is capable of reaching the fundamental bound inEq. (34), whether or not this occurs depends on exactlyhow the dissipative dynamics encoded in ˜ H dir is realizedthrough couplings to external baths. Here, it is possibleto achieve the needed non-Hermitian Hamiltonian usingonly passive dissipation (i.e. no coupling to gain baths, Y = 0 in Eq. (4)). The simplest realization would involvecoupling both modes to an effective chiral waveguide, asdepicted in Fig. 4; the (positive) coupling rate betweenmode j and the waveguide is denoted γ j . Focusing on thecase of two modes with identical frequencies (i.e. ν = 0), FIG. 4. Implementation of a simple non-reciprocal two modesensor, where both modes are coupled to a single effectivechiral waveguide; no coupling to gain baths is required. Thissystem is capable of arbitrarily exceeding the fundamentalbound on the measurement rate of any reciprocal two-modesensor. The required chiral waveguide could be realized us-ing circulators, dynamic modulation [31] or by using drivenparametric interactions and external dissipation [32]. and using standard cascaded quantum systems theory[23] to describe this setup, we realize the non-HermitianHamiltonian in Eq. (35) with J = − i √ γ γ . Further,as there are no couplings to gain baths, the homodynecurrent noise is always given by its minimal shot noisevalue. This setup then realizes the optimal value for themeasurement rate for a non-reciprocal setup as given inEq. (34). Setting γ = κ , we have:Γ meas = 4 κ ¯ n tot (cid:18) γ γ (cid:19) (38)Comparing against Eq. (30), we see that this systembeats the reciprocal-system measurement rate boundwhenever γ < (1 / γ .We note that there are a variety ways of implementinga coupling to an effective chiral waveguide. These rangefrom conventional approaches based on the use of circu-lators, to realizations of chiral waveguides using topolog-ical photonic systems [33], to methods that mimic chi-ral propagation by using dynamic modulation and engi-neered dissipation [31, 32, 34, 35]. We stress that suchengineered non-reciprocal interactions have been experi-mentally realized in photonic setups [36–38], classical mi-crowave circuits [39, 40], optomechanical systems [41, 42]and superconducting circuits [43, 44]. While the motiva-tion for these experiments was largely to build circulatorsand isolators, our work shows that such systems couldalso be exploited for enhanced sensing. C. Non-reciprocal sensors and the mode-splittingtechnique
Up to this point, our work has focused exclusively onsensing parametric changes in ǫ that are small enough to - - J = 0 J = 20 J = 50 Detuning O u t pu t f i e l d i n t en s i t y FIG. 5. Drive-detuning dependence of the output field in-tensity P [∆] (c.f. Eq. (39)), for a non-reciprocal system de-scribed by Eq. (35). We have taken γ = 0 . κ , γ = κ , ν = 4 κ . (Upper panel) Spectrum for ǫ = 0, when the systemis fully non-reciprocal. Even though the system has two non-degenerate eigenvalues Ω ± [0], only one resonance is seen, asnon-reciprocity makes the Ω + [0] dark to the incident drive.Note that this spectrum is independent of J . (Lower panel)Black dashed: Spectrum where the parameter to be sensed ǫ = 0 . κ , but without non-reciprocity ( J = 0). The spec-trum only has a single dip, and almost identical to the ǫ = 0spectrum. Red and green: Spectra with the same perturba-tion ǫ = 0 . κ but with non-reciprocal couplings J = 20 κ and J = 50 κ respectively. Two resonances are clearly observed,and their separation increases with the strength of the non-reciprocal coupling J . allow the use of a perturbative, linear response approach;this typically requires ǫ to be smaller than relevant modelinewidths. Non-reciprocity enhanced sensing is howeveralso highly effective for larger, non-perturbative changesin ǫ . We consider the same general setting as recentworks on EP sensing [10, 13, 19, 20] that aim to detecta relatively large change in ǫ by directly measuring thefrequency-splitting of two normal modes [45–47]. Thisinvolves first measuring the output field intensity as afunction of drive frequency, and then fitting this curve toextract a mode splitting.For a sufficiently strong classical drive the contributionof amplified vacuum fluctuations can be ignored, and theintensity of the waveguide output field ˆ B out (c.f. Eq. (9)) is P [∆] ≡ (cid:28)(cid:16) ˆ B out (cid:17) † ˆ B out (cid:29) ≈ h ˆ B out i ∗ h ˆ B out i = β | − χ | . (39)where ∆ is as always the detuning of the drive frequencyfrom the cavity 1 resonance frequency, and β the (real)amplitude of incident driving field.As discussed in Sec. I C, the magnitude of χ is largewhen ∆ is close to an eigenfrequency of ˜ H , hence P [∆]will generically exhibit a resonance feature (peak or dip)near these values. If a non-zero ǫ lifts the degeneracyof eigenvalues, it will thus manifest itself by the appear-ance of new resonances in the intensity spectrum. Wenote again that the transmitted field in standard setupswhere a nearby readout object (e.g. prism or fiber) iscoupled to an optical resonator [20, 45, 47, 48] is com-pletely equivalent to the reflected field in our geometry.As we now show, non-reciprocity has two distinct ben-efits to frequency-splitting detection. First, a pertur-bation to a non-reciprocal system can induce new res-onances in P [∆] even if there is no degeneracy in theunperturbed system. This allows the frequency-splittingtechnique to be implemented in a wider range of sys-tems. Second, non-reciprocity can dramatically increasethe parametric, ǫ -dependent splitting of resonances: onecan obtain the same √ ǫ type splitting as a system tunedto an EP, without actually needing to be at an EP. Again,this greatly reduces the fine tuning needed to achievesuch strong parametric mode splittings (and also demon-strates that EP is not a necessary ingredient for suchbehaviour).Consider the first point above: with non-reciprocity,the mode-splitting technique can be used even if the un-perturbed system has degenerate eigenvalues. The reasonis simple: because of non-reciprocity, a given eigenmodeof the system may fail to be excited by the incident mea-surement drive when ǫ = 0, irrespective of ∆. If howevera non-zero ǫ breaks the system’s non-reciprocity, these“dark” modes may become visible in the output intensityspectrum. Further, breaking non-reciprocity can lead toparametrically large mode splittings, much larger thanwould be possible without non-reciprocity.To illustrate both the above points, we again considerthe simple two-mode non-reciprocal sensor described byEq. (35). As usual, the parameter to be sensed corre-sponds to a Hermitian coupling between the two modes(c.f. Eq. (26)). For ǫ = 0, we have a purely non-reciprocalcoupling between the modes: mode 2 can influence mode1, but not vice-versa. One finds that the eigenvaluesof ˜ H dir [0], as given in Eq. (37), are in general non-degenerate in both real and imaginary parts. Note alsothat the eigenvalues are completely independent of thecoupling J , reflecting the lack of any coherent oscillationsbetween mode 1 and 2.When perturbation ǫ is non-zero, reciprocity is lost, asmode 1 can now influence mode 2. The eigenvalues of0 ˜ H dir [ ǫ ] areΩ ± [ ǫ ] = ν − i κ + γ + γ ± r J ǫ ǫ ν i κ + γ − γ . (40)For ǫ > J , the frequencysplitting demonstrates a square root dependence on ǫ , i.e.Ω + [ ǫ ] − Ω − [ ǫ ] ≈ √ Jǫ . We see that the non-reciprocalcoupling J amplifies the effect of ǫ , even though it has noimpact on the unperturbed eigenvalues. The √ ǫ splittingdependence resembles that of EP sensing schemes; herehowever, the unperturbed modes are not required to betuned to a degeneracy, and the unperturbed system is not at an EP. The enhanced splitting obtained here can bemuch larger than mode linewidths, and directly manifestsitself in the output intensity spectrum P [∆]. An exampleis shown in Fig. 5. III. CONCLUSION
We have provided a comprehensive analysis of weakdispersive-style measurements made using coupled modesystems described by effective non-Hermitian Hamiltoni-ans. Our work goes beyond previous analyses of non-Hermitian sensing techniques, in that we fully treat fluc-tuation effects, and fully treat the entire measurementprocess. We derive fundamental bounds on any recipro-cal two-mode non-Hermitian sensor, and show that theyalso constrain systems that are tuned to an exceptionalpoint (EP). Generically, we find that amplification is thenecessary ingredient for generating large signal powers,and this can be achieved without any proximity to anEP. However, amplification process must incorporate ex-tra noise that will fundamentally limit the quantum mea-surement rate of any reciprocal sensor. Our results high-light the fact that the efficacy of a non-Hermitian sensingscheme is not completely described by the parametric de-pendence of mode eigenvalues. Considering fluctuationeffects, the particular dissipative implementation of thedynamics is crucial as this will set noise levels.We also discussed a new method for enhancing disper-sive measurement using effective non-Hermitian physics,namely the use of non-reciprocity to enhance sensing. Weshow that non-reciprocity allows one to arbitrarily exceedthe fundamental bound on the measurement rate of a re-ciprocal sensor, and discussed a simple implementationthat does not require any amplification processes. Wealso show that non-reciprocity can enhance the sensitiv-ity of mode-splitting type sensor.Finally, we note that the general theory developedin this work could be easily applied to more generalkinds of sensing problems. For example, the same for-malism could be used to understand the performance ofnon-Hermitian sensors when thermal noise dominates (aswould be the case for systems deep in the classical limit). The formalism could also be extended to study the sens-ing of time-dependent perturbations.We thank Liang Jiang, Douglas Stone, and MengzhenZhang for useful conversations. This work was supportedby a grant from the Simons Foundation (Award Number505450, AC).
Note added–
During the completion of this work, webecame aware of work by Zhang et. al. on a relatedtopic.
Appendix A: Derivation of non-HermitianHamiltonian
Our measurement setup consists of a readout waveg-uide that interacts with only one cavity mode. This cou-pled cavity mode can interact with other modes as wellas arbitrary gain and loss baths. The interaction be-tween cavities is limited to be photon number conserving,i.e. only hopping. The total Hamiltonian of the systemis ˆ H = M X i,j =1 H ij ˆ a † i ˆ a j + Z dk (cid:16) ω b,k ˆ b † k ˆ b k (cid:17) + N Y X j =1 Z dk (cid:16) ω c,j,k ˆ c † j,k ˆ c j,k (cid:17) + N Z X j =1 Z dk (cid:16) ω d,j,k ˆ d † j,k ˆ d j,k (cid:17) + Z dk √ π g ( k ) (cid:16) ˆ a ˆ b † k + ˆ a † ˆ b k (cid:17) + M X i =1 N Y X l = j Z dk √ π (cid:16) Y ∗ ij ( k )ˆ a i ˆ c j,k + Y ij ( k )ˆ a † i ˆ c † j,k (cid:17) + M X i =1 N Z X j =1 Z dk √ π (cid:16) Z ∗ ij ( k )ˆ a i ˆ d † j,k + Z ij ( k )ˆ a † i ˆ d j,k (cid:17) . (A1)ˆ b k , ˆ c j,k , and ˆ d j,k are the annihilation operator of themode with wave number k in the readout waveguide,gain bath, and loss bath respectively. Mode operatorsof different bath commute, and that of the same bathfollows [ ˆ O k , ˆ O † k ′ ] = δ ( k − k ′ ), where O ∈ { b k , c j,k , d j,k } .We have chosen a unit that ~ = 1, and k has a unit offrequency. For simplicity, we assume all baths have lin-ear dispersion relation, i.e. ω b,k = ω c,j,k = ω d,j,k = k ,and homogeneous mode-bath coupling, i.e. g ( k ) = √ κ , Y ij ( k ) = Y ij , and Z ij ( k ) = Z ij .Following standard procedures [5, 23], the Langevinequation in Eq. (5) can be derived. The input field op-1erators are defined asˆ B in ( t ) = Z dk √ π ˆ b k ( t ) e − ik ( t − t ) (A2)ˆ C in j ( t ) = Z dk √ π ˆ c j,k ( t ) e − ik ( t − t ) (A3)ˆ D in j ( t ) = Z dk √ π ˆ d j,k ( t ) e − ik ( t − t ) , (A4)where t → −∞ . Appendix B: Minimum noise
For a given ˜ H , the measurement noise depends on thechoice of gain and loss baths. We can optimize the choiceto obtain a minimum measurement noise. We first rec-ognize in Eq. (23) that ( χY Y † χ † ) ≥
0, because
Y Y † is positive semi-definite. By using Eq. (4), we can obtainanother relation( χY Y † χ † ) = 12 i (cid:16) ( χ ˜ Hχ † ) − ( χ ˜ H † χ † ) (cid:17) + κ | χ | + ( χZZ † χ † ) (B1) ≥ − κ χ ∗ + χ ) + κ | χ | , (B2)where we employ the fact that ZZ † is positive semi-definite in the last relation. After rearrangement, we getthe minimum noise in Eq. (24). Appendix C: Quantum Fisher information1. Single drive frequency
We want to characterize the maximum amount of in-formation available on ǫ in the reflected output mode inour waveguide. As we are interested in the limit of longintegration times τ , the relevant temporal mode of theoutput field is described by an annihilation operatorˆ B ( τ ) ≡ √ τ Z τ ˆ B out ( t ) dt . (C1)This is a standard bosonic annihilation operator satisfy-ing [ ˆ B ( τ ) , ˆ B † ( τ )] = 1.Changing the parameter ǫ will change both our non-Hermitian Hamiltonian ˜ H as well as the state of the tem-poral mode ˆ B . To sense this change, one would measuresome property of ˆ B , described by an observable ˆ M . Thepossible outcomes z of the measurement would be de-scribed by a probability distribution P ǫ [ z ] which dependsparametrically on ǫ . Our goal is to maximize the the sta-tistical distance between P ǫ [ z ] and P [ z ]. For small ǫ ,standard definitions and arguments yield that this dis-tance ds is given by ǫ F , where F is the Fisher infor-mation [49]. Optimizing F over all possible choices of measurement observables ˆ M yields the quantum Fisherinformation, F QFI [25, 50].In our case, because of the linear nature of our systemand the Gaussian nature of the relevant noise, ˆ B is alwaysin a Gaussian state, and F QF I can be computed exactly[26]. For infinitesimal ǫ , one finds: F QFI = (cid:18) d~u ǫ dǫ W − ǫ d~u Tǫ dǫ (cid:19) (cid:12)(cid:12)(cid:12) ǫ → + Ξ , (C2)where ~u ǫ ≡ ( h ˆ q i ǫ , h ˆ q i ǫ ) and ( W ǫ ) jl ≡ h{ δ ˆ q j , δ ˆ q l }i ǫ arerespectively the first and second moments of the Gaus-sian state; ˆ q ≡ ( ˆ B + ˆ B † ) / √ q ≡ i ( − ˆ B + ˆ B † ) / √ B mode quadratures; δ ˆ q j ≡ ˆ q j − h ˆ q j i ǫ [51]. Ξ isa scalar that depends on only the ǫ -dependence of thesecond moment. The first (second) term in Eq. (C2)can be viewed as the information associated with the ǫ -induced change in the first (second) moment of the tem-poral mode B .After solving the Langevin equation in Eq. (5), the firstand second moment for our linear sensor can be evaluatedin the long- τ limit of interest: ~u ǫ = √ τ β (1 − Re ˜ χ [0; ∆; ǫ ] , − Im ˜ χ [0; ∆; ǫ ])(C3) W = ¯ S II [0] κ (cid:18) (cid:19) (C4)Due to the linearity of Eqs. (5) and (9), the classicaldrive affects only the first but not the second momentof the output field. This can be seen from the fact thatthe first moment in Eq. (C3) scales as β , while the sec-ond moment in Eq. (C4) is independent of β . For suf-ficiently strong drive, the QFI will be dominated by thefirst, drive-dependent term in Eq. (C2), and the contri-bution from Ξ can be neglected.One can now confirm that the SNR for an optimalhomodyne measurement (as given in in Eq. (16)) coin-cides with the quantum Fisher information, i.e. SNR = ǫ F QFI . This implies that homodyne detection is the op-timal measurement for dispersive sensing because it ex-tracts the maximum information about ǫ from B mode.
2. Multiple drive frequencies
In the main text and the subection above, we con-sidered the case where the system is driven at a singlefrequency. One might naturally ask if the measurementrate can be increased by driving and measuring the sys-tem using multiple drive tones, each at a different fre-quency. For sufficiently many frequencies, this methodis equivalent to probing the full spectral response of thesystem.As mentioned in the main text, this multi-tone ap-proach is no better that simply probing the system witha single tone with an optimally chosen driving frequency.We make this statement rigorous here. We here showthat if the total intra-cavity photon number is restricted,2then probing the entire spectral response (via multi-tonedriving) does not provide more information (as quanti-fied by the quantum Fisher information) than an optimalsingle-tone measurement.We first consider a generalized coherent driving fieldon mode 1 that consists of N B distinct frequencies: β ( t ) = N B X j =1 β j e − i ∆ j t . (C5)In the long time limit, the output field state becomes adynamic steady state that consists of components in eachtone ∆ j . Each component can be viewed as the state ofa temporal mode:ˆ B j ≡ τ Z τ ˆ B out ( t ) e i ∆ j t dt . (C6)It is easy to check that temporal modes are independentbosonic modes at τ → ∞ , i.e. [ ˆ B j , ˆ B l ] = 0 and [ ˆ B j , ˆ B † l ] = δ jl .Because the system is linear, the multi-mode outputstate is Gaussian. We again assume each drive is suffi-ciently strong that QFI is dominated by the first termin Eq. (C2). The multi-mode first moment is ~u ǫ =( h ˆ q , i ǫ , h ˆ q , i ǫ , h ˆ q , i ǫ , h ˆ q , i ǫ , . . . ), where the quadra-ture operators of each mode are ˆ q ,j ≡ ( ˆ B j + ˆ B † j ) / √ q ,j ≡ i ( − ˆ B j + ˆ B † j ) / √ h ˆ B j i ǫ = √ τ β j (cid:0) − ˜ χ [0; ∆ j ; ǫ ] (cid:1) . (C7)In the long time limit, we find that the second momentis block diagonal, i.e. W = L N B j =1 W ( j )0 , and eachblock corresponds to the second moment of each tem-poral mode: W ( j )0 = ˜ S II [∆ j ] κ (cid:18) (cid:19) (C8)where˜ S II [∆ j ] = κ (cid:18) κ (cid:16) χ (∆ j ) Y Y † χ † (∆ j ) (cid:17) (cid:19) , (C9)and χ (∆ j ) ≡ ˜ χ [0; ∆ j ; 0]. Note that ˜ S II [∆] ≡ S II [0] inEq. (23) because all dynamics in the main text is evalu-ated at the rotating frame of the single drive frequency.To fairly compare this multi-tone approach with otherschemes, we again constrain the problem to have a fixedtotal photon number. Here the time-averaged total pho-ton number is¯ n tot = N B X j =1 ¯ n j ≡ N B X j =1 β j κ (cid:0) χ † (∆ j ) χ (∆ j ) (cid:1) . (C10)The multi-tone Fisher information can be evaluated as F mt = N B X j =1 τκ ¯ n j ˜Γ(∆ j ) (C11) where ˜Γ(∆ j ) is the measurement rate per coherent pho-ton for a single-tone measurement at detuning ∆ j :˜Γ(∆ j ) ≡ κ ˜ S II [∆ j ] | ( χ (∆ j ) V χ (∆ j )) | ( χ † (∆ j ) χ (∆ j )) . (C12)Following the argument in Appendix. B, we know thateach S II [∆ j ] is lower bounded by˜ S II [∆ j ] ≤ ˜ S II [∆ j ] min ≡ κ (cid:16) (cid:0) | − χ (∆ j ) | − (cid:1) × (cid:0) | − χ (∆ j ) | − (cid:1)(cid:17) , (C13)and so the maximum per-photon measurement rate is˜Γ(∆ j ) ≤ ˜Γ opt (∆ j ) ≡ κ ˜ S II [∆ j ] min | ( χ (∆ j ) V χ (∆ j )) | ( χ † (∆ j ) χ (∆ j )) . (C14)We note that there might not be a single set of baththat could optimize all ˜ S II [∆ j ] to saturate the bound inEq. (C13). In general the bath can only be optimizedwith respect to a specific detuning ∆ j .We can then see that the multi-tone QFI is upper-bounded by the maximum single-tone QFI at the optimaldetuning: F mt ≤ N B X j =1 τκ ¯ n j ˜Γ opt (∆ j ) ≤ N B X j =1 τκ ¯ n j max ∆ j (cid:16) ˜Γ opt (cid:17) = τκ ¯ n tot max ∆ j (cid:16) ˜Γ opt (cid:17) = max ∆ j ( F QFI ) . (C15)Recall again that the QFI is the maximum informationobtainable from any detection scheme. Our result thusshows that probing the entire spectral response of thesystem (via multi-tone driving) does not provide moreinformation about the parameter ǫ than simply drivingwith a single (optimally chosen) tone and performing ahomodyne measurement. Appendix D: Bound on the measurement rate of areciprocal two-mode system
We focus here on a two-mode system where the param-eter to be sensed corresponds to the Hermitian couplingbetween the modes (as given in Eq. (26) in the main text).For any such two-mode system, the maximum measure-ment rate obtainable by using an optimized bath isΓ meas ≤ Γ opt = κ ¯ n tot | χ + χ | | χ | + | χ | f ( χ ) (D1)where f ( χ ) is a positive-valued function of a complex χ : f ( χ ) ≡ | χ | | − χ | − | − χ | − . (D2)3 FIG. 6. Shaded blue region indicates the allowable values of f ( χ ) (c.f. Eq. (D2)) as a function | χ | . f ( χ ) sets themaximum possible measurement rate both for sensing a para-metric change in mode coupling (c.f. Eq. (D1)) and for sens-ing a parametric change in the resonance frequency of mode1 (c.f. Eq. (E3)). The dashed horizonal line shows the max-imum possible value of f ( χ ), which is achieved only when χ = 2. Note that for larger | χ | ≥ f ( χ ) < Due to reciprocity, i.e. | χ | = | χ | , the sum ofanti-diagonal susceptibility entries is bounded by | χ + χ | ≤ | χ | . This condition bounds the optimal mea-surement rate asΓ opt ≤ κ ¯ n tot f ( χ ) ≤ κ ¯ n tot . (D3)The first inequality is exploited when | χ | ≫ | χ | .The second inequality is imposed by the maximum valueof f ( χ ). As illustrated in Fig. 6, the maximum ismax { f ( χ ) } = 4, which is attainable when χ = 2.These inequalities complete the reciprocal-system boundin Eq. (30). Appendix E: Bounds on sensing a change in thefrequency of mode 1
Consider a general M mode setup where the parameterof interest is a simple shift in the resonance frequency ofmode 1, i.e. V ij = δ i δ j . In this case, one finds fromEq. (21) that the signal power S is given by S = 14 | χ | ( χ † χ ) S ǫ ≤ | χ | S ǫ , (E1)where S ǫ is defined in Eq. (28). The last relation be-comes an equality in the special case where χ j = 0 for j = 1. In this case, the coherent drive only induces a co-herent photon population in mode 1 (and not in modes2 through M ).If we further specialize to a system with just one mode( M = 1), the Hamiltonian and susceptibility are simple scalars,˜ H one [0] = − i κ + γ , χ = iκ ∆ + i ( κ + γ ) / . (E2)If we further assume a resonant drive (∆ = 0) and noextra gain or loss ( γ = 0), we have the usual setup foran ideal dispersive measurement (see e.g. [5]). One has χ = 2, implying that the signal power is just S = S ǫ .If one now allows for gain (i.e. γ < S ǫ without increasingthe intracavity photon number ¯ n tot : one simply increases | χ | above 2. We stress that this enhancement does notrequire EP, but simply the introduction of gain, and alsoapplies to the case of a multimode system M > meas associated with detecting a mode-1 frequency shift;as discussed, Γ meas considers both the signal power andthe impact of intrinsic noise in the homodyne current.For a general M mode sensor, we find that Γ meas is upper-bounded byΓ meas ≤ κ ¯ n tot f ( χ ) ≤ κ ¯ n tot , (E3)where f ( χ ) is defined in Eq. (D2). As shown in Fig. 6,the measurement rate is maximum when χ = 2. Thisoptimal value is achieved by the simple one mode sensorin Eq. (E2) in the case where the dissipation of mode1 is solely due to the waveguide coupling, i.e. γ = 0.It is interesting to note that this bound is identical tothe bound on a reciprocal two-mode sensor, c.f. Eq. (30)In contrast, a non-reciprocal two-mode sensor could haveΓ meas arbitrarily larger than this bound, c.f. Eq. (34).Because a simple one-mode system achieves the opti-mal value of Γ meas for sensing a parametric change in thefrequency of mode 1, using a multi-mode system is un-necessary for this problem. This is true even if one usesa multi-mode system tuned to an EP where eigenvaluesexhibit a √ ǫ scaling. As a concrete example, considerthe reciprocal two-mode system given in Eq. (31). When J = ( κ + γ − γ ) /
4, the system exhibits EP. The unscaledJordan normal form can be obtained in an appropriatebasis:12 (cid:18) − i − i (cid:19) ˜ H recip [0] (cid:18) ii (cid:19) = (cid:18) Ω[0] 2 J (cid:19) , (E4)where the unperturbed degenerate eigenvalue is Ω[0] ≡− i ( κ + γ + γ ) /
4. In this basis, the perturbation matrixhas non-vanishing anti-diagonal entry, i.e.12 (cid:18) − i − i (cid:19) V (cid:18) ii (cid:19) = (cid:18) / i/ − i/ / (cid:19) , (E5)and so the eigenvalue of ˜ H [ ǫ ] has √ ǫ dependence at small ǫ , i.e. Ω[ ǫ ] ≈ Ω[0] ± √− iǫJ . (E6)As usual, one might be tempted to conclude fromthis strong dependence of eigenvalues on ǫ that this sys-tem should out perform the simple one-mode system in4Eq. (E2). This is however not true: the two-mode EP sys-tem still has a measurement rate fundamentally boundedby Eq. (E3). It is interesting to note that this bound isachieved by the two-mode EP system only when mode 2has non-zero loss, i.e. γ >
0, and when the gain of mode1 is tuned to γ = − ( √ κ − √ γ ) . Appendix F: Systematic construction of minimumnoise bath
In Eq. (24), we show the lower bound of measurementnoise for a given ˜ H . Here we outline a systematic con-struction of baths that attains the minimum measure-ment noise. We first recall that the coupling to gain andloss bath is specified by the positive semi-definite M × M matrices Y Y † and ZZ † . For a given ˜ H , those matricesare not unique because Eq. (4) is unchanged if we changethe baths as Y Y † → Y Y † + K and ZZ † → ZZ † + K ,for any positive semi-definite K . Our aim is to find a Y Y † such that the gain noise term is( χY Y † χ † ) = max ((cid:16) χ (cid:0) ˜ H − ˜ H † + i κ i (cid:1) χ † (cid:17) , ) . (F1)For convenience, we define the Hermitian matrix h ≡ χ (cid:0) ˜ H − ˜ H † + i κ i (cid:1) χ † . (F2)Then Eq. (F1) becomes a conditional equation of h only. In the following, we separately consider the casesof h < h >
1. Bath construction when h < For negative h , our aim is to construct Y Y † suchthat ( χY Y † χ † ) = 0 . (F3)We first construct a positive semi-definite Hermitian ma-trix X − :( X − ) i = ( X − ) ∗ i = − h i = − h ∗ i , (F4)( X − ) jj = − M | h j | h , (F5)( X − ) ij = 0 otherwise. (F6)It is easy to see that h + X − has vanishing entries in firstrow and first column, i.e. ( h + X − ) i = ( h + X − ) i =0. Because h + X − is Hermitian, it can always be diag-onalized by a unitary matrix U : (cid:0) U ( h + X − ) U † (cid:1) ij = Λ i δ ij , (F7) where the eigenvalues Λ i are real. Due to the vanishingfirst row and first column in h + X − , we can require U i = U i = δ i and Λ = 0.Next we decompose h + X − = X +2 − X − as the dif-ference of two positive semi-definite Hermitian matrices X +2 and X − , which are constructed as( U X ± U † ) ij ≡ | Λ i | ± Λ i δ ij . (F8)Combining the matrices and Eq. (F2), we have χ (cid:0) ˜ H − ˜ H † + i κ i (cid:1) χ † = X +2 + ( − X − − X − ) , (F9)where the first (second) term in R.H.S. is positive (neg-ative) semi-definite and thus corresponds to gain (loss)bath. By using Eqs. (4) and (17), we can construct thegain and loss baths as Y Y † = 1 κ (cid:16) ∆ I − ˜ H [0] (cid:17) X +2 (cid:16) ∆ I − ˜ H † [0] (cid:17) , (F10) ZZ † = 1 κ (cid:16) ∆ I − ˜ H [0] (cid:17)(cid:16) X − + X − (cid:17)(cid:16) ∆ I − ˜ H † [0] (cid:17) . (F11)It is easy to verify that Eq. (F3) is satisfied.
2. Bath construction when h > For positive h , our aim is to construct Y Y † suchthat ( χY Y † χ † ) = h . (F12)We first construct a positive semi-definite Hermitian ma-trix X +1 : ( X +1 ) i = ( X +1 ) ∗ i = h i = h ∗ i , (F13)( X +1 ) jj = M | h j | h , (F14)( X +1 ) ij = 0 otherwise. (F15)It is easy to see that h − X +1 has vanishing entries in firstrow and first column, i.e. ( h − X +1 ) i = ( h − X +1 ) i =0. Because h − X +1 is Hermitian, it can always be diag-onalized by a unitary matrix U : (cid:0) U ( h − X +1 ) U † (cid:1) ij = Λ i δ ij , (F16)where the eigenvalues Λ i are real. Due to the vanishingfirst row and first column in h − X +1 , we can require U i = U i = δ i and Λ = 0.Next we decompose h − X +1 = X +2 − X − as the dif-ference of two positive semi-definite Hermitian matrices X +2 and X − , which are constructed as( U X ± U † ) ij ≡ | Λ i | ± Λ i δ ij . (F17)ombining the matrices and Eq. 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