Non-Hermitian Topological Sensors

We introduce and study a novel class of sensors whose sensitivity grows exponentially with the size of the device. Remarkably, this drastic enhancement does not rely on any fine-tuning, but is found to be a stable phenomenon immune to local perturbations. Specifically, the physical mechanism behind this striking phenomenon is intimately connected to the anomalous sensitivity to boundary conditions observed in non-Hermitian topological systems. We describe concrete platforms for the practical implementation of these non-Hermitian topological sensors (NTOS) in classical meta-materials and synthetic quantum-materials.

Introduction.-High-precision sensors represent a key technology that is of ubiquitous importance in both science and everyday life. In the quest for future sensors, a main challenge is to identify viable scenarios, where basic physical observables become highly susceptible to changes in the quantity to be detected, yet in a controllable fashion. Prominent examples along these lines include microcavity sensors [1][2][3][4][5], metal oxide semiconductor field-effect transistor (MOSFET) based sensors [6], mechanical transducers [7], graphene sensors [8], superconducting quantum interference devices (SQUIDS) [9] and magnetometers [10].
Dissipative systems effectively described by non-Hermitian (NH) Hamiltonians can exhibit a high susceptibility to small perturbations of their complex energy spectrum that has no counterpart in closed Hermitian settings [11][12][13][14][15][16][17][18]. As a first step towards using the unique algebraic properties of NH matrices [11][12][13] for sensing [19], an enhancement of precision in sensors operating at NH spectral degeneracies known as exceptional points (EPs) has been demonstrated [20,21]. Furthermore, in NH lattice systems with many degrees of freedom, a striking spectral sensitivity has recently been found in the context of NH topological systems, where phase transitions driven by small changes in the boundary conditions have been predicted and observed [14][15][16][17][22][23][24][25][26].
Here, drawing intuition from NH topological phases [26][27][28][29][30][31], we identify and study a widely applicable mechanism for drastically enhancing the sensitivity of a novel type of devices coined non-Hermitian topological sensors (NTOS). These systems are designed such that the physical quantity to be detected (measurand) effectively couples to the boundary conditions of an extended system of 2N − 1 lattice sites, e.g. by modifying the coupling Γ between the ends of the NTOS with a tunneling barrier (see Fig. 1). Quite remarkably, in this scenario the energy E 0 of a characteristic bound state is then found to exhibit the exponential sensitivity where α, κ > 0 are model-specific constants. Below, we demonstrate both analytically and numerically that this exponential amplification of the sensitivity is a generic and robust phenomenon that does not rely on any finetuning or symmetry and is as such immune to local perturbations including random disorder in the NTOS.
To illustrate the broad range of physical platforms in which NTOS may be realized from existing building blocks, we outline several implementations in the classical as well as in the quantum realm. The former include a setup of coupled optical resonators, where the two ends of the NTOS are separated by an almost impenetrable region that is perturbed by the presence the measurand, as well as a system of topological electric circuits [23,24,32] where a capacitive or inductive coupling to the measur-and effectively changes Γ. Examples of the latter (quantum) kind include a quantum condensed matter setting of coupled quantum dots [33][34][35] in which the measurand represents a charge degree of freedom that electrostatically couples to a tunnel barrier between the outermost quantum dots (sites j = 1, 2N − 1), as well as topological cold atom systems with dissipation [36][37][38].
Exponential amplification mechanism.-We now derive and explain with theory the exponential scaling in system size parameter N (cf. Eq. 1), i.e. in the number of physical degrees of freedom of the NTOS. To this end, we discuss a generic class of NH topological insulator models that exhibit the following basic phenomenology, noting that a specific case study will be presented further below. Building up on the general analysis in Ref. 14, we consider systems that for an odd number of sites 2N − 1 exhibit a localized boundary state with energy E 0 (Γ), satisfying E 0 (0) = 0, i.e. an exact zeroenergy edge mode in the limiting case of open boundary conditions. The corresponding unnormalized right-and left-eigenvectors are then of the general exponential form expressed in terms of the position (site) j = 1, . . . , 2N −1 in the lattice with non-vanishing components where µ = R, L denote right-and left-eigenstates localized around j = 1 if |r µ | < 1 and around j = 2N − 1 if |r µ | > 1, and where the amplitude of the boundary mode is zero on even sites j = 2n [14]. It is worth emphasizing that, in stark contrast to the Hermitian realm where |Ψ L,0 = |Ψ R,0 , right-and left-eigenvectors generically differ in NH systems. In fact, the extreme scenario of an exponentially small overlap represents a crucial ingredient for the exponential amplification at the heart of the proposed NTOS. An intuitive picture for this phenomenon is that in the considered NH scenario, a single zero-energy state may completely change its spatial location when switching from a rightto 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 forward calculation to leading order perturbation theory in Γ yields where the combination of the unperturbed right-and left-eigenstates entering the biorthogonal expectation values occurring in Eq. (4) is crucial, thus representing a natural generalization of perturbation theory to the non-Hermitian regime. As long as both the left-and right-eigenstates are localized at the same end, i.e. sign(log(|r R |)) = sign(log(|r L |)), α is negative and the response, ∆E, decays exponentially with system size while O LR is of order 1. This is the behavior intuitively expected and familiar from Hermitian systems.
Non-Hermitian SSH model: a case study.-To illustrate this phenomenology with a concrete microscopic model, and to quantify the validity regime of the perturbative result (4), we consider a non-Hermitian SSH chain [39,40] with alternating nearest neighbor hopping amplitudes t 1 ± γ and t 2 , respectively. In reciprocal space, the model with periodic boundaries is described by the Bloch Hamiltonian where length is measured in units of the lattice spacing, and σ is the vector of Pauli matrices acting on a sublattice degree of freedom with sublattices consisting of the sites with odd (even) index j. For an odd total number of sites 2N −1 and open boundary conditions, there is an exact E = 0 state of the form (2) with r R = −(t 1 − γ)/t 2 and r L = −(t 1 + γ)/t 2 [14]. To the open chain, we again add the coupling ∆H = Γ(|1 2N − 1| + |2N − 1 1|) between the end sites. In Fig. 1, we plot the absolute value of the energy shift ∆E = E 0 (Γ) − E 0 (0) due to the presence of a finite coupling Γ of the mode with smallest absolute value of energy for various values of N and Γ. We have chosen the parameters t 1 = t 2 = 2γ = 1, noting that all qualitative results are robust against changing these parameters (see discussion below). The exact numerical energies (dots) are found to be essentially identical to the approximate analytical result (4) for a wide range of N and Γ, yielding an exponentially amplified signal over many orders of magnitude. This quantitatively corroborates to impressive accuracy our above general analysis.
It is worth noting that the perturbative energy shift in Eq. (4) is entirely real for the considered NH SSH model. The numerically obtained exact energies share this property to a remarkable precision until there is a sudden transition once the energy shift reaches the energy scale set by the gap in the system. This is reflected in Fig. 1, where deviations form the perturbative results (solid lines) occur at roughly the same value of ∆E independent of the coupling strength Γ.
Generally, we find an exponential enhancement of the sensitivity S whenever ||t 1 |−|t 2 || < |γ|. Interestingly, this desirable regime precisely corresponds to the parameter range with a non-trivial spectral winding number [27] The genuinely non-Hermitian integer topological invariant ν characterizes the bulk band-structure of the NTOS by measuring how often the complex phase of the determinant of H S (k) winds around the origin of the complex energy plane. Specifically, in Eqs. (1),(4), we obtain α > 0, i.e. the desired exponential amplification of the sensitivity S, if |ν| = 1, and α < 0 implying exponential damping of S if ν = 0. While the qualitative change in the response corresponds to a bulk topological invariant-and thus occurs at points in parameter space where the Bloch bands close-it does not correspond to gap closings in the open boundary system. In fact, from the perspective of energy gaps there is a well-documented breakdown of the conventional bulk-boundary correspondence: in the open boundary system, gap closings occur at |t 2 1 − γ 2 | = |t 2 | 2 [14,17,42], instead of the above condition for the change of ν. The origin of this disparity can be traced back to the properties of the zero-modes in Eq. (2): the response phase transition (sign change of α) occurs when the right or left eigenstates change their localization, i.e. when |r R | = 1 or |r L | = 1, whereas the spectral phase transitions are fundamentally tied to changes in the biorthogonal localization occurring at |r L r R | = 1 [14].
Stability against local perturbations.-We now demonstrate the robustness of our main findings against local perturbations. To this end, we add (complex) Gaussian on-site disorder to our above SSH model Hamiltonian of the NTOS. The disorder strength is quantified by the parameter w > 0. In a wide parameter range, in particular as long as w is significantly smaller than the absolute value of bulk energy gap of the NTOS, we find that such local random perturbations do not have any qualitative effect on the exponential amplification of the sensitivity S with system size. Notably, even though the considered disorder breaks the chiral symmetry of the above NH SSH model resulting in a shift of the lowest (in absolute value) energy eigenvalue away from zero, the difference in energy ∆E = E 0 (Γ) − E 0 (0) induced by a change in boundary conditions is still found to exhibit a similar exponential sensitivity as in the absence of disorder. In Fig. 2, we corroborate and exemplify this result: Even at the level of individual disorder realizations, the exponential growth of the sensitivity S with system size is very clear, unsurprisingly with visible fluctuations. However, we emphasize that these fluctuations are irrelevant for the operation principle of a given NTOS device for which the system size N as well the disorder realization are fixed. In this sense, our data simply shows that the expected value for the sensitivity of an individual device follows the same rules as in the case without disorder, as becomes clear when averaging over sufficiently many disorder realizations for every system size N .
Platforms for implementation.-Making available the key ingredient for the proposed NTOS, the general physics of anomalous boundary modes in NH topological systems has recently been observed in a range of experimental settings, including mechanical metamaterials [22], electrical circuits [23,24], and photonic quantum walks [25]. Directly applied to the mechanical metamaterial setting, a detailed dynamical response theory has been developed and shown to posses sharp transitions in the response to external excitations [18]. Furthermore, coupling the boundaries of NH systems has been shown to lead to anomalous dynamics [43]. Motivated by these exciting developments, we now outline several scenarios of immediate experimental relevance for the implementation of NTOS. In, particular we discuss two distinct classes of physical settings: classical meta-materials and synthetic quantum matter.
In various classical meta-materials, the realization of NTOS is viable with existing technology. The probably simplest platform are topological electrical circuits [23,24], where the flexibility for realizing basically any topological insulator model in arbitrary geometry is highest, since arbitrary connectivity between the unit cells of the circuit can readily be achieved. There, the weak link corresponding to Γ can be designed at any point of a circuit in ring geometry, and a measurand which either couples to an inductance or a capacitor of the device may be detected with high precision. By downsizing this electrical circuit NTOS to the nano-scale, it may complement the existing toolbox of microelectronic devices including MOSFET sensors. Another classical implementation with coupled optical resonators is inspired by the combination of the NH EP sensors realized in Refs. [20,21] with optical implementations of topological materials [44]. A practical way of constructing an optical NTOS in this setting is to implement the weak coupling Γ between the end sites (resonators) [45] as an optically impenetrable region, i.e. a region with a finite frequency gap with only evanescent modes imposing a tunneling barrier. The measurand then only needs to weakly affect this tunnel barrier, e.g. by slightly changing the refractive index of the impenetrable medium.
The proposed NTOS may also be realized in the context of synthetic quantum materials, such as ultracold atoms in optical lattices [36][37][38] or coupled arrays of quantum dots [33][34][35]. In this quantum many-body context, NH Hamiltonians may emerge as an effective description of an underlying quantum master equation [46,47]. In particular, the spectral sensitivity of NH matrices that is at the heart of our present analysis has recently been shown [48] to carry over to a full quantum master-equation description corresponding to NH models similar to the above NH SSH model (see Eq. (5)). These insights pave the way towards implementing NTOS in chains of coupled atoms or quantum dots, where prominent sources of dissipation include dephasing and loss of particles, and the weak link Γ coupling to a measurand is simply provided by a high tunnel barrier between two neighboring sites. For the example of quantum dot arrays, such a tunnel barrier simply consists of an electrostatic potential, which renders charge measurements, e.g. on another quantum dot representing the measurand, a natural application of this quantum version of the proposed NTOS.
Concluding discussion.-Harnessing the susceptibility of non-Hermitian topological insulators to changes in the boundary conditions, we have identified a robust and widely applicable mechanism for devising high-precision sensors. In particular, the proposed NTOS devices, designed in an open ring geometry (cf. Fig. 1), are capable of measuring any observable affecting the coupling between their ends with a sensitivity that increases exponentially with the number of degrees of freedom of the sensor. We have demonstrated this central result by a simple perturbative analysis that is accurately confirmed by the numerically exact solution of a microscopic model, where all findings have been shown to be largely insensitive to random perturbations in the form of disorder. Furthermore, several readily conceivable platforms for realizing NTOS systems with state of the art experimental methods have been outlined.
We have focused our analysis on revealing the general principles of the proposed NTOS devices, assuming that a spectral measurement of the lowest energy mode can be done efficiently. Depending on the specific physical platform, dynamical simulations determining the influence of noise and assessing the response times of a given NTOS implementation are an interesting subject of future work.
On a loosely related note, in a number of recent studies on NH systems, the possibility and implementation of sensors operating at exceptional points, i.e. parameter points where a NH matrix becomes non-diagonalizable, has been extensively discussed [19][20][21]. There, the energy spectrum around an EP of order m at energy E e scales as (E − E e ) 1/m , thus also enhancing the sensitivity exponentially strongly in m. However, the crucial difference of the EP scenario compared to our present proposal is that in order to prepare an EP of order m, an extensive number of parameters, more precisely (m 2 + m − 2)/2 parameters [49], need to be fine-tuned. By contrast, the NTOS systems studied in our present work are largely insensitive to changes in any system parameters, except the coupling Γ between the boundaries, which is designed to be sensitive to the measurand.
The spectral sensitivity at the heart of the NTOS is also quite intriguing from a viewpoint of boundary modes in topological materials. Normally, in Hermitian systems, hybridization of the edge modes localized at the ends of a topological insulator leads to a mini-gap that decays exponentially with system size. In the considered NH SSH model instead, coupling the ends of the system leads to a shift in the energy of the boundary mode that increases exponentially in system size in the parameter regime where the complex bulk energy spectrum of the NTOS winds around the origin of the complex plane. In this sense, non-Hermitian topological sensors are based on a reversed mini-gap mechanism.