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. Open quantum systems governed by non-Hermitian dynamics provide a platform to explore PT symmetric phases. Here, a general theory is derived to describe the location of exceptional points in a 1D oligomer chain of arbitrary length.

W hile 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 0 2 and ω 0 1 of this PT -symmetric setup become ω 0 [see Supplementary Note 1]. The exceptional point (for this N ¼ 2 oscillator system) is which defines the crossover between the unbroken PT phase with wholly real ω 0 2;1 , and the broken phase with complex ω 0 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 ¼ 2), trimer (N ¼ 3) and quadrimer (N ¼ 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 ðg=κÞ 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.

Results and discussion
Trimer chain: model. Here, we look at the simplest nontrivial linear chain of harmonic oscillators: the trimer chain (that is, a N ¼ 3 site oligomer). The trimer [which is sketched in Fig. 1a] 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 operatorĤ for this system is (we set ℏ = 1 throughout this manuscript) where b y n and b n , which satisfy bosonic commutation relations, are the creation and annihilation operators, respectively for site n. All of the oscillators are associated with the identical resonance frequency ω 0 , while the nearest-neighbor coupling strength between sites is given by g. Diagonalization of Eq. (2) leads to the Fig. 1 The PT -symmetric trimer and its eigenfrequencies. a A cartoon of the three-site chain (colored balls), where each oscillator has the resonance frequency ω 0 . The left oscillator (green sphere) is subject to gain κ (yellow arrow), while the right oscillator (cyan sphere) suffers an equivalent loss κ (purple arrow), such that the arrangement fulfills PT symmetry. The coupling strength is g. b The real parts of the eigenfrequencies ω 0 n , as a function of g [Eq. (11)]. c The imaginary parts. Dashed lines: exceptional points at the transition between the broken and unbroken PT -symmetric phases [Eq. (13)]. three eigenfrequencies ω n , which read This analysis reveals a solitary eigenfrequency ω 2 , which is unshifted from the bare resonance ω 0 , while the two other eigenfrequencies ω 3 and ω 1 exhibit a splitting of ffiffi ffi 2 p g from the central resonance. Incoherent processes in the chain are taken into account via a quantum master equation in Lindblad form [56][57][58] where the two Lindblad superoperators are Lb n ¼ 2b n ρb y n À b y n b n ρ À ρb y n b n ; ð5aÞ The unitary evolution is supplied by the commutator term on the right-hand-side of Eq. (4), where the Hamiltonian operatorĤ is given by Eq. (2). The external environment surrounding the chain allows for energy exchange. Losses are tracked by the first Lindbladian term in Eq. (4), where γ n ≥ 0 is the damping decay rate of the nth oscillator into its heat bath. Incoherent gain processes, where P n ≥ 0 is the pumping rate into oscillator n, are similarly modeled by the final term in Eq. (4).
In order to probe the mean-field dynamics of the chain, we exploit the property hOi ¼ Tr Oρ À Á for some operator O. The cyclic properties of the trace operator, along with the quantum master equation introduced Eq. (4), leads to the following Schrödinger-like equation for the first moments 〈b n 〉 of the chain where the three-dimensional Bloch vector ψ reads and with the 3 × 3 dynamical matrix H, given by In Eq. (8) we have introduced the renormalized damping decay rate Γ n for each oscillator n, which is necessary due to the incoherent pumping P n and the bosonic statistics. Explicitly, this quantity reads Let us now consider the specific chain configuration where the left oscillator is subject to gain via P 1 = κ (and γ 1 = 0), while the right oscillator is described by the equivalent loss γ 3 = κ (and P 3 = 0). The central oscillator is neutral (since P 2 = γ 2 = 0). Then the mean-field theory of Eq. (8) implies a PT -symmetric HamiltonianĤ 0 may be written down aŝ Equation (10)  unchanged. This PT -symmetric arrangement of the trimer chain is portrayed in Fig. 1a. Upon diagonalizing Eq. (10), the three eigenfrequencies ω 0 n are [cf. Eq. (3) for the closed system] ω 0 where we have introduced the frequency Ω, where Equation (11) reveals the renormalization of the upper and lower eigenfrequencies ω 0 3 and ω 0 1 , as compared to ω 3 and ω 1 in the closed system modeled in Eq. (3). This is due to the incoherent processes captured by κ. In particular, there is now an exceptional point located at g κ N ¼3 which marks the border between the regime when the PT HamiltonianĤ 0 is in its unbroken phase with real eigenvalues, g ≥ κ=ð2 ffiffi ffi 2 p Þ, and the broken phase with complex eigenvalues, . We plot the PT -symmetric regime eigenfrequencies ω 0 n in Fig. 1 using Eq. (11). The real parts are given in panel b, while the imaginary parts are displayed in panel c. The exceptional point of Eq. (13) is marked by the dashed gray line, and makes explicit the broken and unbroken PT -symmetric phases. 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 a valid 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 aforementioned, guaranteed ω 0 eigensolution). Under the popular classification where an nth order exceptional point refers to when n eigenvalues coalesce at the exceptional point 46 , Fig. 1b, 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 ¼ 4) is analyzed in detail, and some features associated with all evensited oligomers are discussed in Supplementary Note 3.
Trimer chain: dynamics. The equation of motion for the second moments hb y n b m i of the trimer gives access to the mean populations along the chain, hb y n b n i. Similar to the calculation leading to Eq. (6), we obtain the first-order matrix differential equation for the 9-vector of correlators u and the inhomogenous pumping term P, where where 0 n is the zero matrix (of n-rows and a single column). The sub-vectors of u read The matrix M of second moments in Eq. (14) is where the on-diagonal sub-matrices comprising M are where Γ n is defined in Eq. (9), while the two off-diagonal submatrices 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. 1a. In this special configuration, the nontrivial eigenvalues of the matrix M in Eq. (17) are ± i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8g 2 À κ 2 p and ± i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8g 2 À κ 2 p =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 hb y n b n i 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. 2a, 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 exemplified in panel d, where g ¼ 0:35κ < κ=ð2 ffiffi ffi 2 p Þ, 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 ð1Þ n ðτÞ ¼ lim 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 jg ð1Þ n ðτÞj ¼ 1 and complete incoherence corresponds to jg ð1Þ n ðτÞj ¼ 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 hb y 1 ðtÞb 1 ðt þ τÞi, via with the 3 × 3 regression matrix Similar equations may be derived for hb y 2 ðtÞb 2 ðt þ τÞi and which characteristically include the harmonic component e Àiω 0 τ , representing a monochromatic field centered on ω 0 , and a prefactor accounting for the specific PT -symmetric setup of the trimer. In the limit of Ω → 0, that is approaching the exceptional point g ! κ=ð2 ffiffi ffi 2 p Þ, Eq. (24) tends towards the quadratically divergent results g ð1Þ 1;3 ðτÞ ! f1 À κ 2 τ 2 =16ge Àiω 0 τ and g ð1Þ 2 ðτÞ ! f1 À κ 2 τ 2 =24ge Àiω 0 τ . For coupling strengths below the exceptional point the trigonometric functions are replaced with hyperbolic functions, indicating exponentially divergent behavior. We plot the real parts of the coherences g ð1Þ 1 ðτÞ and g ð1Þ 2 ðτÞ 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. 3a, well above the exceptional point at g = κ, the PT symmetry ensures an undamped periodic response, with rapid oscillations and a dynamic behavior satisfying 0 < jg ð1Þ n ðτÞj < 1. Exactly at g = κ/2, where all three coherences are accidentally equal as shown in panel b, a well-defined wave envelope develops. In panel c, at the exceptional point g ¼ κ=ð2 ffiffi ffi 2 p Þ, there is initially regular, high frequency oscillations due to short time behavior being essentially dominated by the zeroth order term g ð1Þ n ðτÞ ' e Àiω 0 τ . Once the quadratic correction in κt becomes non-negligible, the divergence characteristic of the broken PT symmetric phase finally emerges.
ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00757-3 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 secondnearest neighbor coupling constant h, which connects the first and third oscillators, via the generalized Hamiltonian (2). This extension leads to a generalization of the eigenfrequencies of Eq.

Eq. (11)]ω
where we have introduced the quantity The inclusion of second-nearest neighbor coupling h leads to a significantly richer phase diagram than with nearest-neighbor 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=ð2 ffiffi ffi 2 p Þ 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. 4b, 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 0 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 second-nearest neighbor coupling (h = κ/4) has led to a larger exceptional point of (g/κ) 3 ≃ 0.660, compared to the nearest-neighbor coupling case when (g/κ) 3 ≃ 0.353. The middle column, at the critical point of h = κ/2, shows the onset of a region of unbroken PT phase for vanishingly small g. This 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. 1b, c] has been downgraded to a standard 2nd order exceptional point in Fig. 5. This is due to the long-range 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 ¼ 4), as is demonstrated in Supplementary Note 2.
Oligomer chains. We have seen some fundamental properties of short PT -symmetric oligomer chains (specifically for N ¼ 3, and for N ¼ 2 and N ¼ 4 in the Supplementary Notes 1 and 2). Let us now consider a general oligomer of arbitrary size N , with nearest-neighbor coupling only. The eigenfrequencies read where the index n 2 ½1; N labels each mode [such that the specific results for N ¼ 3 reproduce Eq.
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 ðg=κÞ N ¼ 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 ðg=κÞ 3 ¼ 1=ð2 ffiffi ffi 2 p Þ ' 0:353::: and the infinitely long chain result of (g/κ) ∞ = 1/2, as shown in Fig. 6 for chains up to N ¼ 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 inverselinear asymptotics ðg=κÞ N ' ð1 À N À1 Þ=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. Fig. 4 The effect of long-range interactions. a A sketch of the PT -symmetric trimer (colored spheres) beyond nearest-neighbor coupling, where the first-neighbor coupling constant is g, and the second-neighbor coupling constant is h. The first oscillator (green sphere) is subject to gain κ (yellow arrow), and the final oscillator (cyan sphere) to loss κ (purple arrow). b The PT symmetry phase diagram of the trimer, given by the evolution of the exceptional point (g/κ) 3  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. 4b]. Similar to the N ¼ 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. 7a-d for oligomers of length N ¼ f4; 5; 6; 7g. 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 firstorder 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.

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).  26)]. d-f The imaginary parts, corresponding to the real parts in a-c. Dashed lines: the exceptional points denote the border between broken and unbroken PT -symmetric phases. The results for the first, second and third eigenfrequenciesω 0 n are denoted by the thin green, medium orange and thick cyan lines respectively [see the legend in a, which applies to the whole figure]. In the first, second, and third columns, the second-nearest neighbor coupling constant h = κ/4, κ/2, and 3κ/4, respectively.

Data availability
The data that support the findings of this study are available from the corresponding author C.A. Downing upon reasonable request.