Exceptional points in oligomer chains

Symmetry underpins our understanding of physical law. Open systems, those in contact with their environment, can provide a platform to explore parity-time symmetry. While classical parity-time symmetric systems have received a lot of attention, especially because of the associated advances in the generation and control of light, there is much more to be discovered about their quantum counterparts. Here we provide a quantum theory which describes the non-Hermitian physics of chains of coupled modes, which has applications across optics and photonics. We elucidate the origin of the exceptional points which govern the parity-time symmetry, survey their signatures in quantum transport, study their influence for correlations, and account for long-range interactions. We also find how the locations of the exceptional points evolve as a function of the chain length and chain parity, capturing how an arbitrary oligomer chain transitions from its unbroken to broken symmetric phase. Our general results provide perspectives for the experimental detection of parity-time symmetric phases in one-dimensional arrays of quantum objects, with consequences for light transport and its degree of coherence.


Introduction
While the eigenvalues of a Hermitian Hamiltonian are always real, the Hermicity condition is more stringent than is strictly necessary [1]. It was shown by Bender and co-workers that Hamiltonians that obey parity-time (PT ) symmetry can both admit real eigenvalues and describe physical systems [2,3]. The condition of combined space and time reflection symmetry has immediate utility for some open systems, where there is balanced loss into and gain from the surrounding environment. The application of the concept of PT symmetry into both classical and quantum physics has already led to some remarkable advances and unconventional phenomena, which cannot be captured with standard Hermitian Hamiltonians [4][5][6][7][8].
An important concept within PT symmetry is that of exceptional points. Let us consider the simplest case of a pair of coupled oscillators, each of resonance frequency ω 0 and interacting via the coupling constant g. The two resulting eigenfrequencies ω 2 and ω 1 are given by ω 2,1 = ω 0 ± g. After including gain at a rate κ into the first oscillator and an equivalent loss κ out of the second oscillator, the renormalized eigenfrequencies ω ′ 2 and ω ′ 1 of this PT -symmetric setup become ω ′ 2,1 = ω 0 ± g 2 − κ 2 4 [see Supplementary Note 1]. The exceptional point (for this N = 2 oscillator system) is which defines the crossover between the unbroken PT phase with wholly real ω ′ 2,1 , and the broken phase with complex ω ′ 2,1 . Therefore, by modulating the ratio g κ one can induce a plethora of (sometimes unexpected) phenomena intrinsically linked to PT symmetry, for example in light transport where amplification and attenuation readily arise [4][5][6][7][8].
Inspired by the pioneering experiments of Hodaei and coworkers with chains of ring-shaped optical resonators [46], we develop a simple theory of short oligomer chains in an open quantum systems approach. In particular, we study dimer (N = 2), trimer (N = 3) and quadrimer (N = 4) chains in detail [see also Supplementary Notes 1 and 2]. We derive the locations of the exceptional points, and explore the influence of the PT symmetry phase on both the population dynamics (revealing regions of amplification) and for correlations (showing areas of perfect coherence and incoherence). Our open quantum systems approach follows in the wake of a number of recent theoretical works [47][48][49][50][51][52][53][54] which employ the concept of PT symmetry with quantum master equations. We note that a related and pioneering experiment with superconducting qubits has latterly been reported [55], highlighting the timeliness of quantum PT -symmetry. We also uncover how the exceptional point of Eq. (1) is generalized for an oligomer chain of an arbitrary size N , where there is gain into the first oscillator and an equivalent loss out of the last oscillator, with neutral oscillators in between. Using a transfer matrices approach, we derive an interesting scaling with N of (g κ) N , and we find a feature due to the parity of the oligomer which provides tantalizing opportunities for experimental detection. Finally, we investigate the emergent and rich PT symmetry phase diagrams when long-range coupling (beyond nearest-neighbor) is taken into account, which crucially determines whether the exceptional point is of higher order (compared to the dimer case) or not.
The PT -symmetric trimer. Panel (a): a sketch of the PT -symmetric trimer chain (colored spheres), where each oscillator is of resonance frequency ω0, and the coupling constant is g. The first oscillator (green sphere) is subject to gain κ (yellow arrow), and the final oscillator (cyan sphere) to loss κ (purple arrow). Panel (b): the real parts of the eigenfrequencies ω ′ n of the trimer, measured from ω0, as a function of g, in units of κ [Eq. (11)]. Panel (c): the corresponding imaginary parts of the eigenfrequencies ω ′ n . Vertical dashed lines: exceptional points marking broken-unbroken PT symmetry phases [Eq. (13)].

Results and Discussion
Trimer chain: model Here we look at simplest nontrivial chain of oscillators, the trimer chain (N = 3 sites). The trimer [sketched in Fig. 1 (a)] already displays some interesting phenomena which is common across all odd-sited oligomers, and yet it retains some beauty due to its simplicity. The Hamiltonian operator for the trimer chain reads (we set ̵ h = 1 throughout) where the bosonic creation (annihilation) operators on oscillator n is denoted by b † n (b n ). Each oscillator is associated with the resonance frequency ω 0 , and the nearest-neighbor coupling between sites is given by g. The three eigenfrequencies ω n of the trimer read revealing a solitary eigenfrequency ω 2 , which is unshifted from the bare resonance ω 0 , while the two other eigenfrequencies ω 3 and ω 1 display a splitting of √ 2g from the central resonance. Incoherent processes in the trimer chain are taken into account via the quantum master equation [56][57][58] in terms of the Lindblad superoperators The unitary evolution is supplied by the commutator term on the right-hand-side of Eq. (4), where the Hamiltonian operator H is given by Eq. (2). Losses into the heat bath are tracked using the first Lindbladian term, where γ n ≥ 0 is the damping decay rate of the nth oscillator. Incoherent gain processes, where P n ≥ 0 is the pumping rate into oscillator n, are modeled by the final term in Eq. (4). In order to probe the mean-field dynamics, we exploit the property ⟨O⟩ = Tr (Oρ), for any operator O, with the master equation of Eq. (4). This procedure leads to the following Schrödinger-like equation for the first moments of the trimer chain with the three-dimensional Bloch vector and where the dynamical matrix H of first moments reads In Eq. (8), the renormalized damping decay rate Γ n of each oscillator, due to the incoherent pumping P n , is Let us consider the configuration where the first oscillator is subject to gain via P 1 = κ (with P 2 = P 3 = 0), and the final oscillator can be described by the loss γ 3 = κ (with γ 1 = γ 2 = 0). Then the mean-field theory of Eq. (8) implies the PTsymmetric Hamiltonian which is invariant under the necessary twin transformations of time and space, essentially that means i → −i and (1, 3) → (3, 1). This PT -symmetric arrangement of the trimer chain is sketched in Fig. 1 (a), which highlights that the first oscillator is subject to gain κ (yellow arrow), and the final oscillator to an equivalent loss κ (purple arrow). The three eigenfrequencies ω ′ n from Eq. (10) are where we have introduce the frequency Equation (11) which marks the border between the regime when the PT Hamiltonian is in its unbroken phase with real eigenvalues, g ≥ κ (2 √ 2), and the broken phase with complex eigenvalues, g < κ (2 √ 2). We plot the eigenfrequencies of ω ′ n in Fig. 1 using Eq. (11), where real parts are given in panel (b) and the imaginary parts in panel (c). The exceptional point of Eq. (13) is denoted by the dashed gray line, making explicit the broken and unbroken phases of the system. There are several features of Fig. 1 which are shared amongst all odd-sited oligomers, namely: the purely real resonance frequency ω 0 (orange line) is always an eigenfrequency, two eigenfrequencies always become complex in the broken PT -symmetric phase, and these two aforementioned eigenfrequencies are always the two eigenfrequencies closest to ω 0 (neglecting the trivial ω 0 solution). Under the popular classification where an n-th order exceptional point refers to when n eigenvalues coalesce at the exceptional point [46], Fig. 1 (b, c) exposes a higher order exceptional point of the 3rd order (compared to 2nd order for a dimer, see Supplementary Note 1). These remarks are further justified in Supplementary Note 2, where analogous behavior with the quadrimer chain (N = 4) is analyzed in detail, and some features associated with all even-sited oligomers are discussed in Supplementary Note 3.

Trimer chain: dynamics
The equation of motion for the second moments of the trimer gives access to the mean populations ⟨b † n b n ⟩ along the trimer chain. Similar to the calculation leading to Eq. (6), we obtain for the 9-vector of correlators u and drive term P, where  where 0 n is the zero matrix (of n-rows and a single column), and with the sub-vectors of u The matrix M of second moments in Eq. (14) reads where the on-diagonal sub-matrices comprising M are where Γ n is defined in Eq. (9), while the two off-diagonal sub-matrices of M are defined by In Eq. (17), the symbols * , † and T represent taking the conjugate, conjugate transpose, and transpose respectively.
Let us consider the PT -symmetric arrangement of the trimer, as sketched in Fig. 1 (a). In this special configuration, the nontrivial eigenvalues of the matrix M in Eq. (17) are ±i 8g 2 − κ 2 and ±i 8g 2 − κ 2 2, recovering the criticality first expounded at the level of the non-Hermitian Hamiltonian in Eq. (13). Using the frequency Ω as defined in Eq. (12), we find the following analytic expressions for the populations We plot the populations ⟨b † n b n ⟩ of the first, second and third oscillators (n = 1, 2, 3) as the thick green, medium orange and thin cyan lines in Fig. 2, using the solutions of Eq. (20). In Fig. 2 (a), where g = κ, a high frequency population cycle is observed, which is maintained over time due to the balanced loss and gain in the system. In panels (b) and (c), where the coupling strength is reduced to g = 3κ 4 and g = κ 2 respectively, the frequency of the population cycle is successively reduced, while the maxima of the mean populations are increased due to the closening proximity to the exceptional point [cf. Eq. (13)]. The broken PT phase is 2), which displays the characteristically diverging population dynamics associated with breakdown beyond the exceptional point.

Trimer chain: correlations
The temporal coherence can be quantified using the first-order correlation function [56] g (1) where τ is the time delay, and where the normalization is taken over a long time scale t → ∞. This quantity has the property that perfect coherence is associated with g n (τ ) = 1 and complete incoherence corresponds to g (1) n (τ ) = 0, while intermediate cases specify the degree of partial coherence. The manipulations resulting in Eq. (6), and an application of the quantum regression theorem, lead to an equation for the first desired two-time correlator ⟨b † 1 (t)b 1 (t + τ )⟩, via with the 3 × 3 regression matrix Similar equations may be derived for ⟨b † 2 (t)b 2 (t + τ )⟩ and ⟨b † 3 (t)b 3 (t + τ )⟩. The solution of Eq. (22), along with the definition of Eq. (21), leads to the sought after first-order correlation functions. In the PT -symmetric setup of trimer, as drawn in Fig. 1 (a), one finds the neat expressions g (1) which characteristically include the harmonic component e −iω0τ , representing a monochromatic field cen-   tered on ω 0 , and a pre-factor accounting for the specific PT -symmetric setup of the trimer. In the limit of Ω → 0, that is approaching the exceptional point g → κ (2 √ 2), Eq. (24) tends towards the quadratically divergent results g   2 (τ ) of the first and second oscillators as the thick green and thin orange lines in Fig. 3, using the solutions of Eq. (24). In Fig. 3 (a), well above the exceptional point at g = κ, the PT symmetry ensures an undamped periodic response, with rapid oscillations and a dynamic behaviour satisfying 0 < g 2), there is initially regular, high frequency oscillations due to short time behaviour being essentially dominated by the zeroth order term g (1) n (τ ) ≃ e −iω0τ . Once the quadratic correction in κt becomes non-negligible, the divergence characteristic of the broken PT symmetric phase finally emerges.
Trimer chain: long-range coupling Let us now consider the effects of going beyond the nearest- neighbor coupling approximation employed in Eq. (2). To do so, we introduce the second-nearest neighbor coupling constant h, which connects the first and third oscillators, via the generalized , whereĤ is defined in Eq. (2). This extension leads to a generalization of the eigenfrequencies of Eq. (3) toω n , wherẽ The associated PT -symmetric setup of trimer chain is sketched in Fig. 4 (a) [cf. Fig. 1 (a)]. The resulting eigenfrequenciesω ′ n are [cf. Eq. (11)] where we have introduced the quantity The inclusion of second-nearest neighbor coupling h leads to a significantly richer phase diagram than with nearestneighbor coupling only, as is demonstrated in Fig. 4 (b). Notably, when h = 0 Eq. (13) is recovered, so that above this threshold strength of 1 (2 √ 2) the system is in its unbroken phase. With increasing h, the exceptional point (g κ) 3 increases in value, up until h = κ 2. Above this critical point, the unbroken phase can be explored either with weak enough g, or strong enough g, with a region of broken phase in between. This causes a green stripe in the phase diagram of Fig. 4 (b), which notably contains the equal coupling (h = g) ring-like limit. The aforementioned broken-unbroken transitions from above and from below can be explicitly seen in Fig. 5, where the real and imaginary parts ofω ′ n are shown, as a function of g κ, in the upper and lower rows respectively. In the first column of Fig. 5, one notices how a nonzero secondnearest neighbor coupling (h = κ 4) has led to a larger exceptional point of (g κ) 3 ≃ 0.660, compared to the nearestneighbor coupling case when (g κ) 3 ≃ 0.353. The middle column, at the critical point of h = κ 2, shows the onset of a new region of unbroken PT phase for vanishingly small g. This novel region is even more apparent in the final column of Fig. 5, where h = 3κ 4 and the exceptional point is well below the nearest-neighbor value, being (g κ) 3 ≃ 0.295. Across all of these cases, it is most apparent that the higher (3rd order) exceptional point of the trimer with nearest-neighbor coupling only [cf. Fig. 1 (b, c)] has been downgraded to a standard 2nd order exceptional point in Fig. 5. This is due to the longrange interactions perturbing the eigensolution otherwise residing exactly at ω 0 . Similarly rich features due to long-range interactions are also seen in the quadrimer chain (N = 4), as is demonstrated in Supplementary Note 2.

Oligomer chains
We have seen some fundamental properties of short PT - where the index n ∈ [1, N ] labels each mode [such that the specific results for N = 3 reproduce Eq. (3)]. If we generalize the PT -symmetric arrangement of Fig. 1 (a), that is if we allow for gain into the first oscillator and an equivalent loss out of the N th oscillator in the chain, so that the setup is We display graphically the formula of Eq. (29) in Fig. 6. Most notably, for oligomers of even size N (red circles), the exceptional point is constant at (g κ) N = 1 2, and is of 2nd order. However, oligomers of odd size N (green circles) are associated with smaller exceptional points than the celebrated dimer result, and are of higher (3rd) order. These exceptional points are bounded by the limiting cases of the trimer result of (g κ) 3 = 1 (2 √ 2) ≃ 0.353... and the infinitely long chain result of (g κ) ∞ = 1 2, as shown in Fig. 6 for chains up to N = 20 oscillators. In particular, the even-odd behavior shown in Fig. 6 is ripe for future experimental detection, as is the trend for increasing large values of the exceptional point with increasingly long odd-numbered chains, following the trend encapsulated by Eq. (29), and its inverse-linear asymptotics (g κ) N ≃ (1 − N −1 ) 2 for large N . While we do not account for disorder, or for dimerization of the chain (which may be interesting from a topological point of view [62]), such extensions can be readily taken care of within this framework.
The addition of next-nearest neighbor hoppings to oligomers of an arbitrary length allows us to generalize our investigation of long-range interactions in a short trimer chain [cf. Fig. 4 (b)]. Similar to the N = 3 case, we can map the phase diagram marking the regions of broken (colored) and unbroken (white) PT symmetric phase, as is shown in Fig. 7 (a, b, c, d) for oligomers of length N = {4, 5, 6, 7}. The two relevant parameters are the first and second-neighbor coupling strengths g and h, such that the vertical axis (h = 0) is marked with analytic results from Eq. (29). Away from this point, the influence of nonzero next-nearest neighbor hopping is rather profound: leading to seas (and even enclaves) of unbroken PT symmetry in a variety of geometries. Recent advances with so-called programmable interactions in atomic arrays suggest that the experimental exploration of such phase diagrams is increasingly accessible [63], aside from the demonstrated tunable interaction ranges in trapped atomic ions [64,65].

Conclusions
We have considered some fundamental properties of oligomers of an arbitrary size which satisfy PT symmetry due to having gain into the first oscillator and an equivalent loss out of the final oscillator. We have unveiled analytically the behavior of the exceptional points as a function of the chain length, which governs the stability of the population dynamics in the system and the presence of amplification. In particular, we have reported an even-odd effect for oligomers of increasing size, derived the bounds on all possible exceptional points, and mapped the relevant phase diagrams when long-range interactions are taken into account.
Focusing on short oligomers, we have provided simple quantum theories locating their exceptional points, and in doing so we found unconventional population dynamics and interesting first-order coherences near to the unbroken-broken PT -symmetric phases. We have also discussed effects beyond nearest-neighbor coupling, which leads to rich PT symmetry phase diagrams. In particular, we have shown that reaching the unbroken PT symmetric phase is no longer purely dependent on going above a threshold value of coupling-to-dissipation strength g κ, rather one may also go below a different threshold value, such that the broken phase can live in a sweet-spot in-between.
Our versatile theory is relevant across a number of optical and photonic platforms, including coupled ring resonators [66], coupled cavities [67], coupled waveguides [68,69] and meta-atoms [70]. Our theoretical results provide a route-map for the scaling up of PT -symmetric systems, and paves the way for the observation of cooperative effects in arbitrarily large systems. There are clear perspectives for the experimental detection of our predictions, including finite size effects, even-odd behaviors, unconventional light transport and correlations, and long-range interactions leading to sweet spot regions of PT symmetry phase breakdown.  29)]. The N = 3 case was already presented in Fig. 4 (b).

Methods
In this theoretical work, the methods used are quantum master equations (as described in the main text [cf. Eq. (4)] and Supplementary Note 1), and an extended transfer matrices method (as detailed in Supplementary Note 3).

Data availability
Data sharing is not applicable to this article as no datasets were generated or analysed during the current theoretical study.