non-hermitian topological sensors · 2020. 9. 25. · ponentially small olr (see eqs. (3-11)), thus...

Non-Hermitian Topological Sensors

Jan Carl Budich1∗ and Emil J. Bergholtz2†1Institute of Theoretical Physics, Technische Universität Dresden and

Würzburg-Dresden Cluster of Excellence ct.qmat, 01062 Dresden, Germany2Department of Physics, Stockholm University, AlbaNova University Center, 106 91 Stockholm, Sweden

(Dated: November 3, 2020)

We introduce and study a novel class of sensors whose sensitivity grows exponentially with the sizeof the device. Remarkably, this drastic enhancement does not rely on any fine-tuning, but is foundto be a stable phenomenon immune to local perturbations. Specifically, the physical mechanismbehind this striking phenomenon is intimately connected to the anomalous sensitivity to boundaryconditions observed in non-Hermitian topological systems. We outline concrete platforms for thepractical implementation of these non-Hermitian topological sensors (NTOS) ranging from classicalmeta-materials to synthetic quantum-materials.

Introduction.— High-precision sensors represent a keytechnology that is of ubiquitous importance in both sci-ence and everyday life. In the quest for future sensors,a main challenge is to identify viable scenarios, wherebasic physical observables become highly susceptible tochanges in the quantity to be detected, yet in a control-lable fashion. Prominent examples along these lines in-clude microcavity sensors [1–5], metal oxide semiconduc-tor field-effect transistor (MOSFET) based sensors [6],mechanical transducers [7], graphene sensors [8], super-conducting quantum interference devices (SQUIDS) [9]and magnetometers [10].

Dissipative systems effectively described by non-Hermitian (NH) Hamiltonians can exhibit a high suscep-tibility to small perturbations of their complex energyspectrum that has no counterpart in closed Hermitiansettings [11–18]. As a first step towards using the uniquealgebraic properties of NH matrices [11–13] for sensing[19], an enhancement of precision in sensors operatingat NH spectral degeneracies known as exceptional points(EPs) has been claimed [20, 21]. Furthermore, in NHlattice systems with many degrees of freedom, a strikingspectral sensitivity has recently been found in the con-text of NH topological systems, where phase transitionsdriven by small changes in the boundary conditions havebeen predicted and observed [14–17, 22–27].

Here, drawing intuition from NH topological phases[27–33], we identify and study a widely applicable mech-anism for drastically enhancing the sensitivity of a noveltype of devices coined non-Hermitian topological sensors(NTOS). These systems are designed such that the phys-ical quantity to be detected (measurand) effectively cou-ples to the boundary conditions of an extended system of2N − 1 lattice sites [34], e.g. by modifying the couplingΓ between the ends of the NTOS with a tunneling bar-rier (see Fig. 1). Quite remarkably, in this scenario theenergy E0 of a characteristic bound state is then foundto exhibit the exponential sensitivity

S = ∂E0∂Γ

= κ exp(αN), (1)

FIG. 1: Left: Illustration of the non-Hermitian topologicalsensor (NTOS) setup consisting of one-dimensional chain with2N − 1 sites in open ring geometry, described by an effectiveNH Hamiltonian. The measurand interferes with the couplingΓ between the ends of the open ring (sites 1 and 2N − 1).The simplest measurement signal is the energy shift ∆E ofthe lowest eigen-mode. Right: Exponential scaling of |∆E|with system size N of the NTOS based on the NH topologicalinsulator model Eq. (5) for different values of Γ. Dots areexact numerical energies which are in striking agreement withthe analytical perturbative expression Eq. (11) (solid lines).Parameters are t1 = t2 = 2γ = 1. In a wide parameterregime, the quotient of |∆E/Γ| gives a good estimate for thesensitivity S of the NTOS.

where α, κ are model-specific constants. Below, wedemonstrate both analytically and numerically that theexponential amplification of the sensitivity correspond-ing to Re[α] > 0 is a generic and robust phenomenonthat does not rely on any fine-tuning or symmetry and isas such immune to local perturbations including randomdisorder in the NTOS. Furthermore, we show how Eq. (1)manifests in the response signal of an NTOS device, thuselevating the exponential sensitivity to an enhanced pre-cision in noise-limited experiments.

To illustrate the broad range of physical platformsin which NTOS may be realized from existing build-ing blocks, we outline several implementations in theclassical as well as in the quantum realm, including adetailed analysis of as system of coupled optical ring-resonators, as well as a discussion of topological electric

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circuits [23, 24, 35], a quantum condensed matter settingof coupled quantum dots [36–38], and a topological coldatom systems with dissipation [39–41].

Exponential amplification mechanism.— We now de-rive and explain with theory the exponential scaling insystem size parameter N (cf. Eq. (1)), i.e. in the num-ber of physical degrees of freedom of the NTOS. To thisend, we discuss a generic class of NH topological insulatormodels that exhibit the following basic phenomenology,noting that a specific case study will be presented fur-ther below. We consider systems that for an odd numberof sites 2N − 1 exhibit a localized boundary state withenergy E0(Γ), satisfying E0(0) = 0, i.e. an exact zero-energy edge mode in the limiting case of open bound-ary conditions [14, 42]. The corresponding unnormalizedright- and left-eigenvectors are then of the general ex-ponential form expressed in terms of the position (site)j = 1, . . . , 2N − 1 in the lattice with non-vanishing com-ponents

〈j|ΨR,0〉 = (rR)j+12 , 〈ΨL,0|j〉 = (rL)

j+12 , j odd, (2)

where µ = R,L denote right- and left-eigenstates local-ized around j = 1 if |rµ| < 1 and around j = 2N − 1 if|rµ| > 1, and where the amplitude of the boundary modevanishes for even values of j [42]. It is worth emphasiz-ing that, in stark contrast to the Hermitian realm where|ΨL,0〉 = |ΨR,0〉, right- and left-eigenvectors genericallydiffer in NH systems. In fact, the extreme scenario of anoverlap

OLR = 〈ΨL,0|ΨR,0〉/(‖ΨL,0‖‖ΨR,0‖) (3)that is exponentially small in system size (N) representsa crucial ingredient for the exponential amplification atthe heart of the proposed NTOS. An intuitive picturefor this phenomenon is that in the considered NH sce-nario, a single zero-energy state may completely changeits spatial location when switching from a right- to a left-eigenstate description.

To see this more quantitatively and derive Eq. (1),we now consider a coupling between the first (j = 1)and last (j = 2N − 1) sites with a matrix element Γ,i.e. ∆H = Γ(|1〉〈2N− 1|+ |2N− 1〉〈1|). A straight for-ward calculation to leading order perturbation theory inΓ yields [42]

∆E≈ 〈ΨL,0|∆H |ΨR,0〉〈ΨL,0|ΨR,0〉=

Γ(rLrR−1)(rN−1L +rN−1R )

(rLrR)N − 1N�1−→ Γκ eαN , (4)

where the combination of the unperturbed right- and left-eigenstates entering the biorthogonal expectation valuesoccurring in Eq. (11) is crucial, thus representing a nat-ural generalization of perturbation theory to the non-Hermitian regime. The final expression in Eq. (11) de-scribes the leading asymptotic behavior of ∆E in the

limit of large system size that is readily derived bycomparing the competing powers of rL and rR [42].While |κ(rL, rR)| > 0, the real part of the crucialparameter α(rL, rR) changes its sign corresponding todifferent physical regimes: As long as both the left-and right-eigenstates are localized at the same end, i.e.sign(log(|rR|)) = sign(log(|rL|)) (cf. Eq. (2)), Re[α] isnegative and ∆E decays exponentially with system size,while OLR remains of order 1. This is the behavior in-tuitively expected and familiar from Hermitian systems.However, as soon as the left- and right-eigenstates arelocalized at opposite boundaries, i.e. sign(log(|rR|)) 6=sign(log(|rL|)), Re[α] becomes positive, driven by an ex-ponentially small OLR (see Eqs. (3-11)), thus hallmark-ing a response that grows exponentially with system size(cf. Eq. (1)).

Case study of microscopic model.— To illustrate thisphenomenology with a concrete microscopic model, andto quantify the validity regime of the perturbative result(11), we consider a NH Su-Schrieffer-Heeger (SSH) chain[43, 44] with alternating nearest neighbor hopping ampli-tudes t1±γ and t2, respectively. However, we stress thatour main findings are of relevance beyond this specificsetting [45]. In reciprocal space, the model with periodicboundaries is described by the Bloch Hamiltonian

HS(k) = (t1 + t2 cosk, t2 sink + iγ, 0) · σ, (5)where length is measured in units of the lattice spacing,and σ is the vector of Pauli matrices acting on a sublat-tice degree of freedom with sublattices consisting of thesites with odd (even) index j. For an odd total numberof sites 2N−1 and open boundary conditions, there is anexact E = 0 state of the form (2) with rR = −(t1−γ)/t2and rL = −(t1 + γ)/t2 [14, 42]. To the open chain, weagain add the coupling ∆H between the end sites.

In Fig. 1, we plot the absolute value of the energyshift ∆E = E0(Γ)−E0(0) due to the presence of a finitecoupling Γ of the mode with smallest absolute value ofenergy for various values of N and Γ. We have chosen theparameters t1 = t2 = 2γ = 1, noting that all qualitativeresults are robust against changing these parameters (seediscussion below). The exact numerical energies (dots)are found to be essentially identical to the approximateanalytical result (11) for a wide range of N and Γ, yield-ing an exponentially amplified signal over many orders ofmagnitude. This quantitatively corroborates to strikingaccuracy our above general analysis.

The perturbative energy shift in Eq. (11) is real-valuedfor the NH SSH model, and the numerically obtained ex-act energies share this property to a remarkable precisionuntil the energy shift reaches the scale set by the bulkgap. This is reflected in Fig. 1, where deviations form theperturbative results (solid lines) occur at roughly similar∆E independent of the coupling strength Γ.

Generally, we find an exponential enhancement of thesensitivity S whenever ||t1| − |t2|| < |γ| < ||t1|+ |t2||. In-

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FIG. 2: Scaling of the boundary-condition induced energyshift |∆E| with system size 2N − 1 in the presence of com-plex disorder, for various values of Γ and system sizes up to119 sites. Data for a single disorder realization (solid dots)are compared to a disorder average (circles) over M = 104

disorder realizations at every N [47]. The disorder strengthis w = 0.3 in both plots. Solid lines indicate the perturbativeresult (see Eq. (11)) in the absence of disorder.

terestingly, this desirable regime precisely corresponds tothe parameter range with a non-trivial spectral windingnumber [28]

ν =1

2πi

∮ π−π

dk∂

∂klog (det[HS(k)]) . (6)

The genuinely non-Hermitian integer topological invari-ant ν characterizes the bulk band-structure of the NTOSby measuring how often the complex phase of the deter-minant of HS(k) winds around the origin of the complexplane. Specifically, in Eqs. (1),(11), we obtain Re[α] > 0,i.e. the desired exponential amplification of the sensitiv-ity S, if |ν| = 1, and Re[α] < 0 implying exponentialdamping of S if ν = 0. We note that winding numbersaround energies shifted from the origin of the complexplane may readily be defined [28]. For the particularlyexperimentally relevant case of a constant imaginary shiftto all energies corresponding to a passive setting with lossonly [46], this redefinition does not affect the spatial lo-calization and existence of boundary modes.

While the qualitative change in the response corre-sponds to the bulk topological invariant ν — and thusoccurs at parameters where the Bloch bands touch — itdoes not entail gap closings in the open boundary sys-tem. In fact, from the perspective of energy gaps there isa well-documented breakdown of the conventional bulk-boundary correspondence: in the open boundary system,gap closings occur at |t21−γ2| = |t2|2 [14, 17, 48], instead

of the above condition for the change of ν. The originof this disparity can be traced back to the zero-modesin Eq. (2): the response phase transition (sign change ofRe[α]) occurs when the right or left eigenstates changetheir localization, i.e. when |rR| = 1 or |rL| = 1, whereasthe spectral phase transitions are tied to changes in thebi-orthogonal localization occurring at |rLrR| = 1 [14].

Stability against local perturbations.— We now demon-strate the robustness of our main findings against localperturbations. To this end, we add (complex) Gaussianon-site disorder to our above SSH model Hamiltonian ofthe NTOS. The disorder strength is quantified by w > 0.In a wide parameter range, in particular as long as w issignificantly smaller than the absolute value of bulk en-ergy gap, we find that such local random perturbationsdo not have any qualitative effect on the exponential am-plification of the sensitivity S with system size. Notably,even though the considered disorder breaks the chiralsymmetry of the above NH SSH model resulting in a shiftof the lowest (in absolute value) energy eigenvalue awayfrom zero, the difference in energy ∆E = E0(Γ)− E0(0)induced by a change in boundary conditions is still foundto exhibit a similar exponential sensitivity as in the ab-sence of disorder. In Fig. 2, we corroborate and exem-plify this result: Even at the level of individual disorderrealizations, the exponential growth of the sensitivity S isvery clear, unsurprisingly with visible fluctuations. How-ever, we emphasize that these fluctuations are irrelevantfor the operation principle of a given NTOS device forwhich the system size N as well the disorder realizationare fixed. In this sense, our data shows that the expectedvalue for the sensitivity of an individual device followsthe same rules as without disorder, which becomes clearwhen averaging over sufficiently many disorder realiza-tions for every system size N .

Platforms for implementation.— Making available keyingredient for the proposed NTOS, the general physics ofanomalous boundary modes in NH topological systemshas recently been observed in a range of experimentalsettings, including mechanical metamaterials [22], elec-trical circuits [23, 24], and photonic quantum walks [25].For mechanical metamaterials, a detailed dynamical re-sponse theory has been developed and shown to possessharp transitions in the response to external excitations[18]. Furthermore, coupling the boundaries of NH sys-tems has been shown to lead to anomalous dynamics [49].Motivated by these exciting developments, we now out-line several scenarios for the implementation of NTOS.

Our main focus is on an array of coupled optical ring-resonators [50], as inspired by experiments on NH EPsensors in such systems [20] as well as optical implemen-tations of topological materials [51, 52]. Combining ex-isting technology [20, 53–55], the NH SSH model (5) forour NTOS may be realized as follows [42]: The sublatticedegree of freedom is represented by the clockwise/anti-clockwise mode of each ring-resonator. The asymmet-

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ric coupling between these modes, parameterized by γ inEq. (5), has been experimentally achieved in Refs. [20, 54]by introducing two scatterers into the mode-volume of aring-resonator. While realizing an EP sensor requiresfine-tuning of γ to compensate the symmetric couplingstrength t1, e.g. by adjusting the angle that the scatterersform with the center of the resonator [20], we stress thatthe proposed NTOS works in a broad parameter range offinite asymmetry γ [42], thus highlighting the topologicalrobustness of our approach. The final ingredient, namelythe chirality-selective coupling between neighboring ring-resonators (t2 in Eq. (5)), may be achieved by chiral cou-plers (see Ref. [55] for a review). Again, far from perfectchirality of the coupling is acceptable to enter the righttopological phase of the NTOS, characterized by a non-vanishing winding number ν (cf. Eq. (6)) [42]. In thisscenario, the NTOS may be seen as an extension of anEP sensor, alleviating the requirement of fine-tuning andenhancing the attainable precision exponentially in thenumber coupled ring-resonators (cf. Eq. (1)). Regardingpower-consumption and response times, the microscopicsimilarity of these two devices, in both cases limited bya comparable overall energy scale of t1, γ (∼ 10 MHz inRef. [20]), allows for their direct comparison. In particu-lar this implies their operation at similar time-scales andsimilar dissipated energy-per-resonator, respectively.

Beyond this optical realization, in various classicalmeta-materials, NTOS devices may be constructed fromthe available toolbox. The probably simplest platformare topological electrical circuits [23, 24], where basicallyany topological insulator model in arbitrary geometrymay be realized, since arbitrary connectivity of the cir-cuit can readily be achieved. The weak link Γ can bedesigned at any point of a circuit in ring geometry, anda measurand which either couples to an inductance or acapacitor of the device may be detected. By downsizingthis architecture to the nano-scale, it may complementthe existing toolbox of microelectronic devices includingMOSFET sensors.

The proposed NTOS may also be realized in syntheticquantum materials, such as ultracold atoms in opticallattices [39–41] or coupled arrays of quantum dots [36–38]. In this quantum many-body context, NH Hamilto-nians may emerge as an effective description of an un-derlying quantum master equation [56, 57]. In particu-lar, the spectral sensitivity of NH matrices has recentlybeen shown [58] to carry over to a full quantum master-equation description corresponding to NH models similarto the above NH SSH model (see Eq. (5)). These insightspave the way towards implementing NTOS in chains ofcoupled atoms or quantum dots, where the weak link Γcoupling to a measurand is simply provided by a hightunnel barrier between two neighboring sites. For theexample of quantum dot arrays, such a tunnel barriermay consist of an electrostatic potential, which renderscharge measurements a natural application of this quan-

tum version of the proposed NTOS.

Discussion.— Harnessing the susceptibility of NHtopological phases to changes in the boundary conditions,we have identified a robust and widely applicable mech-anism for high-precision sensors. In particular, the pro-posed NTOS devices, designed in an open ring geometry(cf. Fig. 1), are capable of measuring any observable af-fecting the coupling between their ends with a sensitivitythat increases exponentially with the number of degreesof freedom of the sensor. We have demonstrated thiscentral result by a simple perturbative analysis that isaccurately confirmed by the numerically exact solutionof a microscopic model, where our findings are largelyinsensitive to random perturbations in the form of disor-der. Furthermore, several platforms for realizing NTOSsystems with state of the art experimental methods havebeen outlined.

On a related note, in a number of recent studies on NHsystems, the possibility and implementation of sensorsoperating at exceptional points, i.e. parameter pointswhere a NH matrix becomes non-diagonalizable, has beenextensively discussed [19–21]. There, the energy split-ting around an EP of order m at energy Ee scales as∆E ∼ �1/m, with � denoting the deviation from the EP,thus also enhancing the sensitivity with respect to � expo-nentially strongly in m. However, at least two crucial dif-ferences of the EP scenario compared to our present pro-posal are worth mentioning: First, to prepare an EP oforder m, an extensive number of (2m−2) real parameters[59, 60] need to be fine-tuned. By contrast, the NTOSsystems studied in our present work are largely insensi-tive to changes in system parameters, except the couplingΓ between the boundaries, which is designed to be sen-sitive to the measurand. Second, the spectral sensitivityof EP sensors has been criticized for not being a suit-able quantity to assess the precision of an EP sensor ina noise-limited measurement [61, 62]. In particular, thesimultaneous �-dependence of the coalescing eigenstatesand ∆E may conspire to yield a linear field response in�, despite the sub-linear ∆E [61]. By contrast, the sen-sitivity of the proposed NTOS is not stymied by such aninterplay between eigenstates and eigenvalues. This isquite intuitive since our scheme relies on the frequency-shift of an energetically isolated mode rather than thecoalescence of two modes at an EP. This intuition is sup-ported by a microscopic analysis of the NH SSH model[42] along the lines of [61], clearly showing that the fieldresponse of the NTOS system exhibits a similar exponen-tial enhancement with system size as the frequency-shiftitself. In this sense, the spectral sensitivity of the NTOSis a suitable quantity to assess the signal to noise ratiothat can be experimentally achieved in principle. Thispoint in principle being made, we note that due to theapplicability of the NTOS mechanism to a wide rangeof physical systems, a detailed analysis of the relevantsources of noise for each individual implementation is

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beyond the scope of this first study, and thus remainsa subject of more specialized future work.

The spectral sensitivity at the heart of the NTOS isalso intriguing from a viewpoint of boundary modes intopological materials. Normally, in Hermitian systems,hybridization of the edge modes localized at the endsof a topological insulator leads to a mini-gap thatdecays exponentially with system size. In the consid-ered NH SSH model instead, coupling the ends of thesystem leads to a shift in the energy of the boundarymode that increases exponentially in system size inthe parameter regime where the complex bulk energyspectrum of the NTOS winds around the origin of thecomplex plane. In this sense, non-Hermitian topolog-ical sensors are based on a reversed mini-gap mechanism.

Acknowledgments.— We would like to thank FloreKunst, Alexander Palatnik, Hannes Pichler, and CarstenTimm for discussions. J.C.B. acknowledges financialsupport from the German Research Foundation (DFG)through the Collaborative Research Centre SFB 1143 andthe Cluster of Excellence ct.qmat. E.J.B. is supported bythe Swedish Research Council (VR) and the WallenbergAcademy Fellows program of the Knut and Alice Wallen-berg Foundation.

∗ Electronic address: [email protected]† Electronic address: [email protected]

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7

Supplementary Material for “Non-Hermitian Topological Sensors”

In this supplementary material, we provide details on our derivations and arguments presented in the main text.First, the analysis of boundary states in NH lattice models and its application to the NH SSH model discussed in themain text is explicated. Second, we analyze a model with on-site dissipation only (i.e. without asymmetric hoppingterms) which is found to nevertheless possess an exponential boundary sensitivity, as mentioned in the main text.Third, we detail the implementation of the proposed NTOS devices with coupled optical ring-resonators. Finally, wecalculate the response of an NTOS setting, thus confirming our main finding that the exponential amplification of thespectral sensitivity carries over to the full dynamical response.

Boundary states in NH lattice models

We begin by considering generic nearest neighbor hopping models, in particular allowing for asymmetric hoppingamplitudes. This includes the non-Hermitian SSH chain studied in the main text (see Fig. 3(a) for an illustration) asa special case, which is detailed at the end of this section. For periodic boundary conditions, in reciprocal space, theconsidered models are described by the Bloch Hamiltonian

H(k) = d(k) · σ with dx(k)± idy(k) ≡ f± + g±e∓ik, where f±, g± ∈ C , (7)corresponding to a generic nearest neighbour hopping model with chiral symmetry σzH(k)σz = −H(k) as long asdz(k) = 0. Let us now consider the sensor setup with 2N − 1 sites, for which the Hamiltonian matrix, expanded inthe local site basis, is

H =

0 f+ 0 0 0 · · · Γf− 0 g+ 0 0 · · · 00 g− 0 f+ 0 · · · 00 0 f− 0

. . .. . . 0

0 0 0. . .

. . . f+ 0...

......

. . . f− 0 g+Γ 0 0 · · · 0 g− 0

, (8)

where Γ denotes the hopping between the end sites induced by the measurand. At Γ = 0 there is an exact zero energy(E = 0) eigenstate of the form

|ΨR,0〉 := NR

10rR0r2R0r3R...0

rN−1R

; 〈ΨL,0| := NL

(1 0 rL 0 r

2L 0 r

3L · · · 0 rN−1L

), (9)

with rR = − f−g+ and rL = −f+g−

. The overlap between the left and right states is

〈ΨL,0|ΨR,0〉 = NLNR(rLrR)

N − 1rLrR − 1

. (10)

Using this result it is straightforward to calculate the shift in energy induced by the measurand via a coupling of theend sites according to ∆H = Γ(|1〉〈2N− 1|+ |2N− 1〉〈1|). To first order perturbation theory in Γ, we obtain

∆E ≈ 〈ΨL,0|∆H |ΨR,0〉〈ΨL,0|ΨR,0〉= Γ

(rLrR − 1)(rN−1L + rN−1R )(rLrR)N − 1

N�1−→ Γκ eαN . (11)

8

This result, as shown in the main text, provides a very accurate approximation of the energy shift as long as ∆E issmaller than the energy gap separating the perturbed zero mode from the rest of the spectrum. Specifically, for thelarge N limit we find that κ 6= 0 throughout the considered parameter space, and we identify

α = log(max(rR, rL)) when log(|rRrL|)) < 0, and α = − log(min(rR, rL)) when log(|rRrL|)) > 0 , (12)

where max(rR, rL) (min(rR, rL)) selects the maximum (minimum) value as sorted by the absolute value. Crucially,one finds

Re[α] > 0 when sign(log(|rR|)) 6= sign(log(|rL|)), and Re[α] < 0 when sign(log(|rR|)) = sign(log(|rL|)) (13)

implying exponential amplification precisely in the parameter region characterized by a skin effect, i.e. where theperturbed zero mode has right and left eigenvectors located at different ends of the chain.

We end this section by applying our general analysis to the non-Hermitian SSH model with the asymmetric hoppingterms (see Fig. 3(a)), used in our case study in the main text. In reciprocal space, the model with periodic boundariesis described by the Bloch Hamiltonian

HS(k) = (t1 + t2 cosk, t2 sink + iγ, 0) · σ where t1, t2, γ ∈ R . (14)

In the sensor setup with 2N − 1 sites the Hamiltonian matrix is

H =

0 t1 − γ 0 0 0 · · · Γt1 + γ 0 t2 0 0 · · · 0

0 t2 0 t1 − γ 0 · · · 00 0 t1 + γ 0

. . .. . . 0

0 0 0. . .

. . . t1 − γ 0...

......

. . . t1 + γ 0 t2Γ 0 0 · · · 0 t2 0

, (15)

and at Γ = 0 there is an exact E = 0 boundary state of the form of Eq. (9) with rR = − t1−γt2 and rL = −t1+γt2

. Forthis case, the region of exponential signal amplification becomes ||t1| − |t2|| < |γ| < ||t1|+ |t2|| as quoted in the maintext, and which follows from Eq. (13).

Sensor with on-site non-Hermitian terms only

An alternative lattice model platform for the NTOS exclusively featuring diagonal NH terms is shown in Fig. 3(b).With periodic boundary conditions the Bloch Hamiltonian is given by

H̃(k) = (t1 + t2 cos k, 0, t2 sin k + iγ) · σ . (16)

While H̃ has its non-Hermiticity reflected only in a staggered on-site gain and loss pattern, it is unitarily equivalentto the NH SSH model with asymmetric hopping studied in the main text by virtue of the transformation

H̃ = U†HSU with U =1√2

(1 ii 1

)(17)

and these two models thus have identical spectra for periodic boundary conditions. The unitary equivalence alsoholds for open boundaries for an even number of sites [S1], but is not identically satisfied for an odd number of sites.There, the Hamiltonian matrix describing the sensor setup in real space reads as

H̃ =

−iγ t1 it2/2 t2/2 0 · · · Γt1 iγ t2/2 −it2/2 0 · · · 0

−it2/2 t2/2 −iγ t1 it2/2 · · · 0t2/2 it2/2 t1 iγ

. . .. . . 0

0 0 −it2/2. . .

. . . t1 it2/2...

......

. . . t1 iγ t2/2Γ 0 0 · · · −it2/2 t2/2 −iγ

, (18)

9

FIG. 3: NTOS models (a) shows the asymmetric hopping model discussed in the main text and (see Eq. (15)) (b) shows amodel whose non-Hermiticity is reflected in an alternating gain and loss (see Eq. (18)). For an even number of sites the twomodels are unitary equivalent, while the sensors based on lattices with an odd number of sites behave identically for largeenough systems as discussed in text and demonstrated in Fig. 4. (c) Illustration of the implementation of the couplings t1, t2, γof the NH SSH model shown in (a) with optical ring resonators. Chiral couplers [S5] (here shown as a spherical shape piercedby a double-arrow indicating the coupled modes) mediate the direction-selective coupling t2 between neighboring resonators,while the asymmetry γ in the intra-resonator coupling of opposite chiralities is experimentally achieved by two scatterers (perring-resonator) entering the mode volume of the resonators [S3].

0 50 100 150

10-9

10-6

0.001

1

0 50 100 150

10-9

10-6

0.001

1

Spectral measurement2

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Measurand

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100AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0ls8eNW8OKxomkLbSib7aZdutmE3Y1QQn+CFw+KePUXefPfuEmDqPXBwOO9GWbm+TFnStv2p1VaWV1b3yhvVra2d3b3qvsHHRUlklCXRDySPR8rypmgrmaa014sKQ59Trv+9Drzuw9UKhaJez2LqRfisWABI1gb6c6x7WG1ZtftHGiZOAWpQYH2sPoxGEUkCanQhGOl+o4day/FUjPC6bwySBSNMZniMe0bKnBIlZfmp87RiVFGKIikKaFRrv6cSHGo1Cz0TWeI9UT99TLxP6+f6ODSS5mIE00FWSwKEo50hLK/0YhJSjSfGYKJZOZWRCZYYqJNOpU8hKsM598vL5POWd1p1Bu3zVqrWcRRhiM4hlNw4AJacANtcIHAGB7hGV4sbj1Zr9bborVkFTOH8AvW+xdrGo1S

150AAAB6nicbVDLSsNAFL2pr1pfVZduBovgqiS29bEruHFZ0T6gDWUynbRDJ5MwMxFK6Ce4caGIW7/InX/jJA2i1gMXDufcy733eBFnStv2p1VYWV1b3yhulra2d3b3yvsHHRXGktA2CXkoex5WlDNB25ppTnuRpDjwOO160+vU7z5QqVgo7vUsom6Ax4L5jGBtpDunYQ/LFbtqZ0DLxMlJBXK0huWPwSgkcUCFJhwr1XfsSLsJlpoRTuelQaxohMkUj2nfUIEDqtwkO3WOTowyQn4oTQmNMvXnRIIDpWaBZzoDrCfqr5eK/3n9WPuXbsJEFGsqyGKRH3OkQ5T+jUZMUqL5zBBMJDO3IjLBEhNt0illIVylOP9+eZl0zqpOrVq7rVea9TyOIhzBMZyCAxfQhBtoQRsIjOERnuHF4taT9Wq9LVoLVj5zCL9gvX8BcrONVw==

|�E|AAAB8XicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY/BB3iMYB6YLGF2MkmGzM4uM71CSPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9djIrq2vrG9nN3Nb2zu5efv+gZqJEM15lkYx0I6CGS6F4FQVK3og1p2EgeT0YXE/9+hPXRkTqAYcx90PaU6IrGEUrPY5bN1wiJbfjdr7gFt0ZyDLxUlKAFJV2/qvViVgScoVMUmOanhujP6IaBZN8kmslhseUDWiPNy1VNOTGH80unpATq3RIN9K2FJKZ+ntiRENjhmFgO0OKfbPoTcX/vGaC3Ut/JFScIFdsvqibSIIRmb5POkJzhnJoCWVa2FsJ61NNGdqQcjYEb/HlZVIrFb2zYun+vFC+SuPIwhEcwyl4cAFluIMKVIGBgmd4hTfHOC/Ou/Mxb8046cwh/IHz+QMScpCF


AAAB6XicbVDLSsNAFL2pr1pfVZduBovgqiS29bEruHFZxT6gDWUynbRDJ5MwMxFK6B+4caGIW//InX/jJA2i1gMXDufcy733eBFnStv2p1VYWV1b3yhulra2d3b3yvsHHRXGktA2CXkoex5WlDNB25ppTnuRpDjwOO160+vU7z5QqVgo7vUsom6Ax4L5jGBtpLuGPSxX7KqdAS0TJycVyNEalj8Go5DEARWacKxU37Ej7SZYakY4nZcGsaIRJlM8pn1DBQ6ocpPs0jk6McoI+aE0JTTK1J8TCQ6UmgWe6Qywnqi/Xir+5/Vj7V+6CRNRrKkgi0V+zJEOUfo2GjFJieYzQzCRzNyKyARLTLQJp5SFcJXi/PvlZdI5qzq1au22XmnW8ziKcATHcAoOXEATbqAFbSDgwyM8w4s1tZ6sV+tt0Vqw8plD+AXr/QsDJ40c


150AAAB6nicbVDLSsNAFL2pr1pfVZduBovgqiS29bEruHFZ0T6gDWUynbRDJ5MwMxFK6Ce4caGIW7/InX/jJA2i1gMXDufcy733eBFnStv2p1VYWV1b3yhulra2d3b3yvsHHRXGktA2CXkoex5WlDNB25ppTnuRpDjwOO160+vU7z5QqVgo7vUsom6Ax4L5jGBtpDunYQ/LFbtqZ0DLxMlJBXK0huWPwSgkcUCFJhwr1XfsSLsJlpoRTuelQaxohMkUj2nfUIEDqtwkO3WOTowyQn4oTQmNMvXnRIIDpWaBZzoDrCfqr5eK/3n9WPuXbsJEFGsqyGKRH3OkQ5T+jUZMUqL5zBBMJDO3IjLBEhNt0illIVylOP9+eZl0zqpOrVq7rVea9TyOIhzBMZyCAxfQhBtoQRsIjOERnuHF4taT9Wq9LVoLVj5zCL9gvX8BcrONVw==

1AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlptcvV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucPe+uMuQ==

10�3AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgxbCbCHoMevEYwTwgWcPsZJIMmZ1dZnqFsOQjvHhQxKvf482/cZLsQRMLGoqqbrq7glgKg6777eTW1jc2t/LbhZ3dvf2D4uFR00SJZrzBIhnpdkANl0LxBgqUvB1rTsNA8lYwvp35rSeujYjUA05i7od0qMRAMIpWannuY3pRnfaKJbfszkFWiZeREmSo94pf3X7EkpArZJIa0/HcGP2UahRM8mmhmxgeUzamQ96xVNGQGz+dnzslZ1bpk0GkbSkkc/X3REpDYyZhYDtDiiOz7M3E/7xOgoNrPxUqTpArtlg0SCTBiMx+J32hOUM5sYQyLeythI2opgxtQgUbgrf88ippVspetVy5vyzVbrI48nACp3AOHlxBDe6gDg1gMIZneIU3J3ZenHfnY9Gac7KZY/gD5/MHP8GO2w==

10�6AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgxbAbRT0GvXiMYB6QrGF2MkmGzM4uM71CWPIRXjwo4tXv8ebfOEn2oIkFDUVVN91dQSyFQdf9dnIrq2vrG/nNwtb2zu5ecf+gYaJEM15nkYx0K6CGS6F4HQVK3oo1p2EgeTMY3U795hPXRkTqAccx90M6UKIvGEUrNT33MT27nHSLJbfszkCWiZeREmSodYtfnV7EkpArZJIa0/bcGP2UahRM8kmhkxgeUzaiA962VNGQGz+dnTshJ1bpkX6kbSkkM/X3REpDY8ZhYDtDikOz6E3F/7x2gv1rPxUqTpArNl/UTyTBiEx/Jz2hOUM5toQyLeythA2ppgxtQgUbgrf48jJpVMreeblyf1Gq3mRx5OEIjuEUPLiCKtxBDerAYATP8ApvTuy8OO/Ox7w152Qzh/AHzucPRFCO3g==

10�9AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgxbAbBfUW9OIxgnlAsobZySQZMju7zPQKYclHePGgiFe/x5t/4yTZgyYWNBRV3XR3BbEUBl3328mtrK6tb+Q3C1vbO7t7xf2DhokSzXidRTLSrYAaLoXidRQoeSvWnIaB5M1gdDv1m09cGxGpBxzH3A/pQIm+YBSt1PTcx/TsetItltyyOwNZJl5GSpCh1i1+dXoRS0KukElqTNtzY/RTqlEwySeFTmJ4TNmIDnjbUkVDbvx0du6EnFilR/qRtqWQzNTfEykNjRmHge0MKQ7NojcV//PaCfav/FSoOEGu2HxRP5EEIzL9nfSE5gzl2BLKtLC3EjakmjK0CRVsCN7iy8ukUSl75+XK/UWpepPFkYcjOIZT8OASqnAHNagDgxE8wyu8ObHz4rw7H/PWnJPNHMIfOJ8/SN+O4Q==

|�E|AAAB8XicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY/BB3iMYB6YLGF2MkmGzM4uM71CSPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9djIrq2vrG9nN3Nb2zu5efv+gZqJEM15lkYx0I6CGS6F4FQVK3og1p2EgeT0YXE/9+hPXRkTqAYcx90PaU6IrGEUrPY5bN1wiJbfjdr7gFt0ZyDLxUlKAFJV2/qvViVgScoVMUmOanhujP6IaBZN8kmslhseUDWiPNy1VNOTGH80unpATq3RIN9K2FJKZ+ntiRENjhmFgO0OKfbPoTcX/vGaC3Ut/JFScIFdsvqibSIIRmb5POkJzhnJoCWVa2FsJ61NNGdqQcjYEb/HlZVIrFb2zYun+vFC+SuPIwhEcwyl4cAFluIMKVIGBgmd4hTfHOC/Ou/Mxb8046cwh/IHz+QMScpCF

10�3AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgxbCbCHoMevEYwTwgWcPsZJIMmZ1dZnqFsOQjvHhQxKvf482/cZLsQRMLGoqqbrq7glgKg6777eTW1jc2t/LbhZ3dvf2D4uFR00SJZrzBIhnpdkANl0LxBgqUvB1rTsNA8lYwvp35rSeujYjUA05i7od0qMRAMIpWannuY3pRnfaKJbfszkFWiZeREmSo94pf3X7EkpArZJIa0/HcGP2UahRM8mmhmxgeUzamQ96xVNGQGz+dnzslZ1bpk0GkbSkkc/X3REpDYyZhYDtDiiOz7M3E/7xOgoNrPxUqTpArtlg0SCTBiMx+J32hOUM5sYQyLeythI2opgxtQgUbgrf88ippVspetVy5vyzVbrI48nACp3AOHlxBDe6gDg1gMIZneIU3J3ZenHfnY9Gac7KZY/gD5/MHP8GO2w==

10�6AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgxbAbRT0GvXiMYB6QrGF2MkmGzM4uM71CWPIRXjwo4tXv8ebfOEn2oIkFDUVVN91dQSyFQdf9dnIrq2vrG/nNwtb2zu5ecf+gYaJEM15nkYx0K6CGS6F4HQVK3oo1p2EgeTMY3U795hPXRkTqAccx90M6UKIvGEUrNT33MT27nHSLJbfszkCWiZeREmSodYtfnV7EkpArZJIa0/bcGP2UahRM8kmhkxgeUzaiA962VNGQGz+dnTshJ1bpkX6kbSkkM/X3REpDY8ZhYDtDikOz6E3F/7x2gv1rPxUqTpArNl/UTyTBiEx/Jz2hOUM5toQyLeythA2ppgxtQgUbgrf48jJpVMreeblyf1Gq3mRx5OEIjuEUPLiCKtxBDerAYATP8ApvTuy8OO/Ox7w152Qzh/AHzucPRFCO3g==


1AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlptcvV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucPe+uMuQ==

� = 10�1AAAB9XicbVBNSwMxEM36WetX1aOXYBG8WHaroBeh6EGPFewHtNsym2bb0CS7JFmlLP0fXjwo4tX/4s1/Y9ruQVsfDDzem2FmXhBzpo3rfjtLyyura+u5jfzm1vbObmFvv66jRBFaIxGPVDMATTmTtGaY4bQZKwoi4LQRDG8mfuORKs0i+WBGMfUF9CULGQFjpU77FoSAK8/tpKfeuFsouiV3CrxIvIwUUYZqt/DV7kUkEVQawkHrlufGxk9BGUY4HefbiaYxkCH0actSCYJqP51ePcbHVunhMFK2pMFT9fdECkLrkQhspwAz0PPeRPzPayUmvPRTJuPEUElmi8KEYxPhSQS4xxQlho8sAaKYvRWTASggxgaVtyF48y8vknq55J2Vyvfnxcp1FkcOHaIjdII8dIEq6A5VUQ0RpNAzekVvzpPz4rw7H7PWJSebOUB/4Hz+AByhkZs=

� = 10�3AAAB9XicbVBNSwMxEJ2tX7V+VT16CRbBi2W3FfQiFD3osYL9gHZbsmnahibZJckqZen/8OJBEa/+F2/+G9N2D9r6YODx3gwz84KIM21c99vJrKyurW9kN3Nb2zu7e/n9g7oOY0VojYQ8VM0Aa8qZpDXDDKfNSFEsAk4bwehm6jceqdIslA9mHFFf4IFkfUawsVKnfYuFwFee20nOypNuvuAW3RnQMvFSUoAU1W7+q90LSSyoNIRjrVueGxk/wcowwukk1441jTAZ4QFtWSqxoNpPZldP0IlVeqgfKlvSoJn6eyLBQuuxCGynwGaoF72p+J/Xik3/0k+YjGJDJZkv6sccmRBNI0A9pigxfGwJJorZWxEZYoWJsUHlbAje4svLpF4qeuVi6f68ULlO48jCERzDKXhwARW4gyrUgICCZ3iFN+fJeXHenY95a8ZJZw7hD5zPHx+rkZ0=

� = 10�5AAAB9XicbVBNSwMxEM3Wr1q/qh69BIvgxbJbFb0IRQ96rGA/oN2W2TRtQ5PskmSVsvR/ePGgiFf/izf/jWm7B219MPB4b4aZeUHEmTau++1klpZXVtey67mNza3tnfzuXk2HsSK0SkIeqkYAmnImadUww2kjUhREwGk9GN5M/PojVZqF8sGMIuoL6EvWYwSMldqtWxACrjy3nZycjzv5glt0p8CLxEtJAaWodPJfrW5IYkGlIRy0bnpuZPwElGGE03GuFWsaARlCnzYtlSCo9pPp1WN8ZJUu7oXKljR4qv6eSEBoPRKB7RRgBnrem4j/ec3Y9C79hMkoNlSS2aJezLEJ8SQC3GWKEsNHlgBRzN6KyQAUEGODytkQvPmXF0mtVPROi6X7s0L5Oo0jiw7QITpGHrpAZXSHKqiKCFLoGb2iN+fJeXHenY9Za8ZJZ/bRHzifPyK1kZ8=

� = 10�7AAAB9XicbVBNSwMxEJ2tX7V+VT16CRbBi2W3CvUiFD3osYL9gHZbsmnahibZJckqZen/8OJBEa/+F2/+G9N2D9r6YODx3gwz84KIM21c99vJrKyurW9kN3Nb2zu7e/n9g7oOY0VojYQ8VM0Aa8qZpDXDDKfNSFEsAk4bwehm6jceqdIslA9mHFFf4IFkfUawsVKnfYuFwFee20nOypNuvuAW3RnQMvFSUoAU1W7+q90LSSyoNIRjrVueGxk/wcowwukk1441jTAZ4QFtWSqxoNpPZldP0IlVeqgfKlvSoJn6eyLBQuuxCGynwGaoF72p+J/Xik3/0k+YjGJDJZkv6sccmRBNI0A9pigxfGwJJorZWxEZYoWJsUHlbAje4svLpF4qeufF0v1FoXKdxpGFIziGU/CgDBW4gyrUgICCZ3iFN+fJeXHenY95a8ZJZw7hD5zPHyW/kaE=

� = 10�9AAAB9XicbVBNSwMxEM3Wr1q/qh69BIvgxbJbBfUgFD3osYL9gHZbZtO0DU2yS5JVytL/4cWDIl79L978N6btHrT1wcDjvRlm5gURZ9q47reTWVpeWV3Lruc2Nre2d/K7ezUdxorQKgl5qBoBaMqZpFXDDKeNSFEQAaf1YHgz8euPVGkWygcziqgvoC9ZjxEwVmq3bkEIuPLcdnJyOe7kC27RnQIvEi8lBZSi0sl/tbohiQWVhnDQuum5kfETUIYRTse5VqxpBGQIfdq0VIKg2k+mV4/xkVW6uBcqW9Lgqfp7IgGh9UgEtlOAGeh5byL+5zVj07vwEyaj2FBJZot6MccmxJMIcJcpSgwfWQJEMXsrJgNQQIwNKmdD8OZfXiS1UtE7LZbuzwrl6zSOLDpAh+gYeegcldEdqqAqIkihZ/SK3pwn58V5dz5mrRknndlHf+B8/gAoyZGj

� = 10�11AAAB+HicbVBNSwMxEM36WetHVz16WSyCF8umCnoRih70WMF+QLuW2TRtQ5PskmSFuvSXePGgiFd/ijf/jWm7B219MPB4b4aZeWHMmTa+/+0sLa+srq3nNvKbW9s7BXd3r66jRBFaIxGPVDMETTmTtGaY4bQZKwoi5LQRDq8nfuORKs0ieW9GMQ0E9CXrMQLGSh230L4BIeAS+w/pCcbjjlv0S/4U3iLBGSmiDNWO+9XuRiQRVBrCQesW9mMTpKAMI5yO8+1E0xjIEPq0ZakEQXWQTg8fe0dW6Xq9SNmSxpuqvydSEFqPRGg7BZiBnvcm4n9eKzG9iyBlMk4MlWS2qJdwz0TeJAWvyxQlho8sAaKYvdUjA1BAjM0qb0PA8y8vknq5hE9L5buzYuUqiyOHDtAhOkYYnaMKukVVVEMEJegZvaI358l5cd6dj1nrkpPN7KM/cD5/AAfskgc=

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FIG. 4: Exponential dependence of the boundary mode for the model with onsite gain and loss as defined in Eq. 18 andillustrated in Fig. 3(b). The parameters in this example are set to t1 = t2 = 1 and γ = 0.7. The solid lines represent theperturbative analytical result for the asymmetric hopping model—which, as demonstrated here, becomes essentially exact alsofor the model with onsite dissipation when the system size is not too small.

and does not have a simple exact form of the boundary modes, which for finite systems slightly (exponentially littlein system size) differ from zero energy (mini-gap). However, the results from the sensitivity analysis of the NH SSHmodel still carries over to this system, even to high quantitative accuracy, once the considered systems are not toosmall. This finding is illustrated in Fig. 4, where the analytical results (solid lines) derived for the NH SSH modelwith asymmetric hopping become essentially identical to the numerical results (dots) for the alternative model (H̃)with staggered gain and loss.

10

FIG. 5: Phase diagram for the optical ring-resonator implementation of the NH SSH model (see Eq. (19)) as a function of thehopping asymmetry γ and the imperfection of the chiral coupler δ. The parameter point discussed in the main text is markedwith a red dot. The required phase for operating the NTOS, characterized by the finite winding spectral number ν = 1 isshown in light color. The Hermitian hoppings are chosen as t1 = t2 = 1.

Implementation of NH SSH model with optical ring resonators

Elaborating on the discussion in the main text, we now describe how the NH SSH model with asymmetric couplingsrepresenting the main microscopic model discussed in the main text (cf. Fig. 4(a)) can be realized with state of the artexperimental techniques. The key ingredients of the proposed implementation are illustrated in Fig. 4(c) for a segmentof two resonators (unit cells of the NH SSH model). The sublattice degree of freedom of the NH SSH is representedby the two chiralities, clockwise (CL) and counter-clockwise (CO) of the ring-resonator. In the absence of scatterersentering the mode-volume of the resonators, the CL and CO modes may for practical purposes be considered asperfectly decoupled and isolated [S2]. However, by introducing two scatterers into the mode-volume, an asymmetriccoupling (γ 6= 0) has been experimentally achieved [S3], where the strength of the asymmetry depends on the differencein the angle between the two scatterers as seen from the center of the ring-resonator. This technology is well undercontrol and even fine-tuning to an exceptional point (t1 = γ for a given isolated resonator) has been achieved to highaccuracy [S3, S4]. The core non-Hermitian ingredient γ of the proposed NTOS may thus be realized by completeanalogy to (multiple copies of) the previously realized EP sensor [S4], however without any need for fine-tuning (seediscussion of phase diagram below). The energy scale of this asymmetric coupling (∼ 10 MHz [S3, S4]) is the limitingfactor for the overall time- and energy-scales of both the EP sensor and an NTOS realized on this platform. The(Hermitian) coupling t2 between neighboring resonators is achieved by chiral couplers, a common element from thetoolbox of chiral quantum optics [S5]. Also here, no fine tuning is required, i.e. the nearest neighbor coupling byno means needs to be perfectly chirality-selective to enter the topologically non-trivial NTOS phase, as quantifiedbelow in the discussion of the phase diagram (cf. Fig. 5). These benign aspects of the implementation highlight thetopological robustness of our present approach to NH sensing.

Regarding the practical measurement scheme that may be realized with a ring-resonator based NTOS, we note thata straight-forward generalization of the experimental setting of the aforemementioned EP sensors [S4], amounting toobserving a scatterer that perturbs a weak coupling Γ� t2 realized between two of the ring-resonators (interpreted asthe ends of a broken ring geometry), is only one of many conceivable applications of our NTOS. Generally, the weaklink Γ between the end sites may be represented by any optically impenetrable region imposing a tunneling barrier,i.e. a region with a finite frequency gap hosting only evanescent modes. The measurand then only needs to weaklyaffect this tunnel barrier, e.g. by slightly changing the refractive index of the impenetrable medium.

11

Robustness regarding imperfections in the implementation

To assess and quantify the topological robustness of the NTOS with respect to an imperfect experimental realizationof the NH SSH model with the aforementioned ring-resonators (cf. Figs. 4(a),(c)), we compute the topological phasediagram (as characterized by the spectral winding number ν). Specifically, we vary the strength of the asymmetryfrom the value γ = t1/2 = 0.5 chosen for our simulations in the main text, and consider an imperfection δ thatquantifies the breaking of the chiral selectivity of the nearest neighbor coupling t2. Regarding the latter, the couplingδ still respect the chiral symmetry of the NH SSH model that for open boundary conditions ensures zero energy E0 = 0of the end state, but couples the CL in a given resonator to the CO mode in its right neighbor, thus amounting to athird-nearest neighbor coupling in Fig. 4(a). Overall, in reciprocal space the extended NH Bloch Hamiltonian of theperturbed model reads as

H(k) = (t1 + (t2 + δ) cosk, (t2 − δ) sink + iγ, 0) · σ. (19)

The phase diagram characterized by the value of the spectral winding number ν (see Eq. (6) in the main text) isshown in Fig. 5. Clearly, the topologically non-trivial phase (ν = 1) required for the operation of the NTOS extendsover a wide parameter regime, basically only requiring that the (at least in the experimentally most relevant regimeγ ≤ t1) desirable hopping asymmetry γ exceeds the undesirable imperfection of the chiral coupler δ.

In principle, there may also be connections between modes of the same chirality, e.g. a coupling δ2 connecting theCL modes in neighboring resonators among each other as well as δ′2 connecting the CO modes, both amounting tonext-nearest neighbor couplings in the language of Fig. 4(a). Such couplings explicitly break the chiral symmetry of

the NH SSH model and lead to additional termsδ2+δ

′2

2 σ0 +δ2−δ′2

2 σz in the Bloch Hamiltonian (19). We stress that thetopological invariant ν does not rely on chiral symmetry and the NTOS would thus clearly exhibit robustness againstsuch perturbations in a finite parameter range as well (see also the related discussion on complex disorder aroundFig. 2 in the main text). However, in the proposed implementation the symmetry breaking couplings δ2, δ

′2 can even

be avoided by simple directional couplers, as has experimentally been demonstrated to very high precision [S2], thuslimiting the practical experimental relevance of such perturbations for the proposed implementation.

Sensor response

Adapting the analysis from Ref. S6, we now calculate the response of an NTOS to an external probe, thus confirmingthat the exponential spectral sensitivity studied in the main text is directly visible in the dynamical response of thesystem. Consider the state |S(t)〉 of the system with an initial excitation |x(t)〉 which implies

|S(t)〉 =∫ t−∞

e−iH(t−t′) |x(t′)〉 dt′ , (20)

where the system is assumed to be unperturbed as t→ −∞. Decomposing this in the eigenmodes of the system yields

|S(t)〉 =∑n

∫ t−∞

e−iEn(t−t′) |ΨR,n〉〈ΨL,n|〈ΨR,n|ΨL,n〉

|x(t′)〉 dt′ , (21)

Since the relevant signal of the NTOS results from the perturbed zero mode of the system, we can project to thelow-energy sector by considering only

|S(Γ, t)〉 ≈∫ t−∞

e−iE0(t−t′) |ΨR,0〉〈ΨL,0|〈ΨR,0|ΨL,0〉

|x(t′)〉 dt′ , (22)

where the we have explicitly indicated the crucial dependence on the boundary coupling Γ. This approximationbecomes exact in the limit of a large gap to all other states. For an instant excitation |x(t)〉 = δ(t) |x0〉, we have

|S(Γ, t)〉 = θ(t)e−iE0t |ΨR,0〉〈ΨL,0|〈ΨR,0|ΨL,0〉|x0〉 . (23)

which is a state (a vector) describing the excited system. To see the strength of the response of the system we needto relate this to a scalar which can for instance be done by defining

12

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non-hermitian topological sensors · 2020. 9. 25. · ponentially small olr (see eqs. (3-11)), thus...

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