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ABSTRACT Directional amplification, in which signals are selectively amplified depending on their propagation direction, has attracted much attention as key resource for applications,

including quantum information processing. Recently, several, physically very different, directional amplifiers have been proposed and realized in the lab. In this work, we present a unifying

framework based on topology to understand non-reciprocity and directional amplification in driven-dissipative cavity arrays. Specifically, we unveil a one-to-one correspondence between a

non-zero topological invariant defined on the spectrum of the dynamic matrix and regimes of directional amplification, in which the end-to-end gain grows exponentially with the number of

cavities. We compute analytically the scattering matrix, the gain and reverse gain, showing their explicit dependence on the value of the topological invariant. Parameter regimes achieving

directional amplification can be elegantly obtained from a topological ‘phase diagram’, which provides a guiding principle for the design of both phase-preserving and phase-sensitive

multimode directional amplifiers. SIMILAR CONTENT BEING VIEWED BY OTHERS ROBUST TEMPORAL ADIABATIC PASSAGE WITH PERFECT FREQUENCY CONVERSION BETWEEN DETUNED ACOUSTIC CAVITIES Article Open

access 17 February 2024 TUNABLE OPTICAL NONRECIPROCITY IN DOUBLE-CAVITY OPTOMECHANICAL SYSTEM WITH NONRECIPROCAL COUPLING Article Open access 27 January 2025 NON-ORTHOGONAL CAVITY MODES NEAR

EXCEPTIONAL POINTS IN THE FAR FIELD Article Open access 05 January 2024 INTRODUCTION Controlling amplification and directionality of electromagnetic signals is one key resource for

information processing. Amplification allows to compensate for attenuation losses and to read out signals while adding a minimal amount of noise. Directionality, also known as

non-reciprocity, allows to select the direction of propagation while blocking signals in the reverse1,2. Non-reciprocity is of wide-ranging practical value; for instance, it simplifies the

construction of photonic networks3,4,5, enhances the information capacity in communication technology6,7, and can be a resource for (quantum) sensing8. Combining non-reciprocity and

amplification, directional amplifiers allow for the detection of weak signals while protecting them against noise from the read-out electronics. For these reasons, these devices have become

important components for promising quantum information platforms such as superconducting circuits9. In response to this demand, many proposals and realizations of non-reciprocal and

amplifying devices have appeared in the recent literature. Isolators and circulators based on magneto-optical effects have become the conventional choice, but they are bulky and require

undesired magnetic fields to explicitly break time-reversal symmetry. Josephson junctions10,11,12 have been investigated as an alternative. Other approaches include refractive index

modulation13,14, interfering parametric processes15, and optomechanics16,17,18. An elegant solution is provided by reservoir engineering19,20,21,22,23,24,25,26,27, where non-reciprocity is

achieved by interfering coherent and dissipative processes20,22. Based on this approach, several few-mode isolators and directional amplifiers have been proposed19,20,22,26 and

demonstrated21,23,24,25,27. On the other hand, chiral edge states of topological photonic systems28 give rise to the directional transport of photons and phonons29,30, which has been used to

design traveling wave amplifiers31 and topological lasers32,33,34,35,36. Transport phenomena in dissipative systems characterized by a topological winding number have been studied in refs.

37,38. A generalized winding number applied to a non-Hermitian system has previously appeared in the study of the Su-Schrieffer-Heeger (SSH) laser39,40. In this paper, we unify the plethora

of ad hoc proposals for directional amplifiers by uncovering an organizing principle underlying directional amplification in driven-dissipative cavity arrays: the non-trivial topology of the

matrix governing the time evolution of the cavity modes. Based on this notion of topology, we develop a framework to understand directional amplification in multimode arrays and provide a

recipe to design novel devices. The systems we consider are driven-dissipative cavity chains as the one depicted in Fig. 1a, featuring both coherent and dissipative couplings between modes.

Non-trivial topology coincides with directional amplification and arises from the competition of local and non-local dissipative terms while the Hamiltonian describing the evolution of the

closed system features a topologically trivial band structure. We build our analysis on the scattering matrix illustrated in Fig. 1b. The scattering matrix characterizes the isolating

properties as well as the amplification of a weak probe across the chain. Next, we introduce a topological invariant, the winding number, see Fig. 2, which is defined on the spectrum of the

dynamic matrix governing the evolution of the cavity amplitudes and enters directly in the scattering matrix. We then employ the winding number to discuss the topological regimes of the

driven-dissipative chain leading to the topological ‘phase diagram’ for the scattering matrix, Fig. 3, which at the same time defines the directionally amplifying parameter regimes. We go on

to rigorously prove the one-to-one correspondence between non-trivial topology and directional amplification leading to one of our main results: the analytic expression for the scattering

matrix in non-trivial topological regimes, Eq. (23). This result already holds for systems consisting of as few as two modes in the vicinity of the exceptional point (EP), where it is exact,

and converges to the exact result exponentially fast within the whole topologically non-trivial regime. From Eq. (23) we find the exponential scaling of the amplifier gain with the chain

length, Eq. (28), while signals in the reverse direction are exponentially suppressed, Eq. (29). Therefore, increasing the chain length enlarges the parameter range for which directional

amplification occurs, from a fine-tuned point to the whole topologically non-trivial regime. The generality of our results becomes clear in the last section of Results, in which we examine

with our topological framework scaled-up versions of different models for phase preserving and phase sensitive amplifiers that have appeared in the literature20,22,41. We demonstrate how we

can predict the different amplifying regimes of these devices, compute gain and reverse gain, and obtain the scattering matrix from our topological framework by inspecting the winding

number. Directional amplification can be seen as a proxy of non-trivial topology, formally defined only in the thermodynamic limit, even in very small systems, which makes our work relevant

for state-of-the art devices such as ref. 27. Our analysis serves as a general recipe for designing multimode amplifiers that can be integrated in scalable platforms, such as superconducting

circuits10,42, optomechanical systems43, and topolectric circuits44,45. Finally, our work also has direct relevance for the study of the topology of non-Hermitian Hamiltonians46,47, for

which similar topological invariants have been proposed48,49, leading to the recent classification in terms of 38 symmetry classes50. In this context, our work provides a direct way to

detect topological features, e.g., extract the value of the topological invariant, which has previously been challenging. RESULTS DIRECTIONAL AMPLIFICATION IN A DRIVEN-DISSIPATIVE CHAIN Let

us start by introducing the system that will guide us through the general discussion and illustrate our results. We consider a driven-dissipative chain of _N_ identical cavity modes _a__j_

as depicted in Fig. 1a. Its coherent evolution in a frame rotating with respect to the cavity frequency is governed by the Hamiltonian (_ℏ_ = 1) $${\mathcal{H}}=\sum _{j}(J{a}_{j}^{\dagger

}{a}_{j+1}+{J}^{* }{a}_{j}{a}_{j+1}^{\dagger }),$$ (1) which describes photons hopping with uniform amplitude _J_ along the chain. The dissipation consists of both local and non-local

contributions and is described by the master equation $$\dot{\rho }=-{\rm{i}}[{\mathcal{H}},\rho ]+\sum _{j}\left(\Gamma {\mathcal{D}}[{z}_{j}]\rho +\gamma {\mathcal{D}}[{a}_{j}]\rho +\kappa

{\mathcal{D}}[{a}_{j}^{\dagger }]\rho \right)$$ (2) for the system density matrix _ρ_. The first dissipator \({\mathcal{D}}[{z}_{j}]\rho ={z}_{j}\rho {z}_{j}^{\dagger

}-\frac{1}{2}\{{z}_{j}^{\dagger }{z}_{j},\rho \}\) with _z__j_ ≡ _a__j_ + _e_−i_θ__a__j_+1 couples dissipatively neighboring cavities with rate Γ20,47, the second describes photon decay into

the wave guide with rate _γ_, while the last is an incoherent pump at rate _κ_. This last term can be implemented with the help of a parametrically coupled auxiliary mode which is

subsequently adiabatically eliminated from the equations of motion. The phase _θ_ can for instance be obtained in a driven optomechanical setup23,26,43, in which the mechanical mode is

adiabatically eliminated giving rise to the non-local dissipator. The controllable phase of the pumps is imprinted onto the amplitude of the coherent state inside the cavities and therefore

transferred to the optomechanical coupling constant. This gives rise to the phase _θ_. Our main interest will be in the fields entering 〈_a__j_,in(_t_)〉 and exiting 〈_a__j_,out(_t_)〉 the

cavities through the wave guides, which are connected via the input-output boundary conditions \(\langle {a}_{j,{\rm{out}}}\rangle =\langle {a}_{j,{\rm{in}}}\rangle +\sqrt{\gamma }\langle

{a}_{j}\rangle \)51,52. Following the standard procedures, we obtain the following equations of motion for the cavity amplitudes 〈_a__j_〉 $$\langle {\dot{a}}_{j}\rangle = \, \frac{\kappa

-\gamma -2\Gamma }{2}\langle {a}_{j}\rangle -\sqrt{\gamma }\langle {a}_{j,{\rm{in}}}\rangle \\ -\left({\rm{i}}J+\frac{{e}^{-{\rm{i}}\theta }\Gamma }{2}\right)\langle {a}_{j+1}\rangle

-\left({\rm{i}}J+\frac{{e}^{{\rm{i}}\theta }\Gamma }{2}\right)\langle {a}_{j-1}\rangle \\ \equiv \, \sum _{j}{H}_{j,\ell }\langle {a}_{\ell }\rangle -\sqrt{\gamma }\langle

{a}_{j,{\rm{in}}}(t)\rangle.$$ (3) In these Eqs. (3), we have chosen _J_ real, which is always possible due to gauge freedom20. The input 〈_a__j_,in(_t_)〉 enters as a coherent drive in the

frame rotating with the cavity frequency. Note that the non-local dissipator contributes both to the coupling terms and to the local decay rate. The phase _θ_ is crucial for the

non-reciprocity of the chain: since coherent and dissipative couplings between neighboring modes form a closed path, these processes can interfere constructively or destructively depending

on the phase _θ_. For example, setting i_J_ = −_e_i_θ_Γ/2, i.e., \(\theta =\frac{3\pi }{2}\), in Eq. (3), each cavity _j_ in Fig. 1a only couples to its right-hand side neighbor (_j_ + 1),

but not to the cavity (_j_ − 1) on its left. This leads to the complete cancellation of the transmission from left to right20,22 and corresponds to standard cascaded quantum systems

theory53,54. These are also the EPs of the system as we show in Methods. As we can see from the last line of Eqs. (3), the evolution equations can be conveniently expressed as matrix-vector

product with _H_ the dynamic matrix. _H_ plays an important role in characterizing the transmitting and amplifying properties of the system. This is because it determines the scattering

matrix _S_(_ω_), which linearly links the input 〈_a__j_,in(_ω_)〉 to the output fields 〈_a__j_,out(_ω_)〉 in frequency space $${{\bf{a}}}_{{\rm{out}}}=[{\mathbb{1}}+\gamma {({\rm{i}}\omega

{\mathbb{1}}+H)}^{-1}]{{\bf{a}}}_{{\rm{in}}}\equiv S(\omega ){{\bf{a}}}_{{\rm{in}}},$$ (4) where we set \({{\bf{a}}}_{{\rm{in/out}}}\equiv {(\langle {a}_{1,{\rm{in}}/{\rm{out}}}\rangle

,\ldots ,\langle {a}_{N,{\rm{in}}/{\rm{out}}}\rangle )}^{{\rm{T}}}\). Figure 1b illustrates the role of the scattering matrix for the driven-dissipative chain. As we can see, the chain acts

as a directional amplifier in the case shown: the dominant top right corner of _S_(_ω_) relates a weak input signal at the _N_th cavity to a strongly amplified output at the first cavity,

while transmission in the opposite direction is suppressed. Formally, non-reciprocity between modes _j_ and _ℓ_ corresponds to the condition ∣_S__j_,_ℓ_∣ ≠ ∣_S__ℓ_,_j_∣ and practically

useful amplification to ∣_S__j_,_ℓ_∣ ≫ 1. Indeed, one of the key quantities used to characterize amplifiers is the gain \({\mathcal{G}}\)52, which we define as the scattering matrix element

with the largest absolute value. For the driven-dissipative chain, the gain relates the input at the first (last) to the output at the last (first) cavity as follows $${\mathcal{G}}(\omega

)\equiv \left\{\begin{array}{ll}| {S}_{N,1}(\omega ){| }^{2}:&\theta \in (0,\pi )\\ | {S}_{1,N}(\omega ){| }^{2}:&\theta \in (\pi ,2\pi ).\end{array}\right.$$ (5) Conversely, the

reverse gain pertains to the transmission in the opposite propagation direction $$\bar{{\mathcal{G}}}(\omega )\equiv \left\{\begin{array}{ll}| {S}_{1,N}(\omega ){| }^{2}:&\theta \in

(0,\pi )\\ | {S}_{N,1}(\omega ){| }^{2}:&\theta \in (\pi ,2\pi ).\end{array}\right.$$ (6) An efficient directional amplifier obeys \({\mathcal{G}}\gg 1\) and \(\bar{{\mathcal{G}}}\ll

1\). For convenience, we introduce $$M(\omega )\equiv {\rm{i}}\omega {\mathbb{1}}+H$$ (7) with _M_(0) = _H_ and dub it dynamic matrix at frequency _ω_. We also define its inverse as the

susceptibility matrix $$\chi (\omega )\equiv {({\rm{i}}\omega {\mathbb{1}}+H)}^{-1},$$ (8) which is related to the scattering matrix through $$S(\omega )={\mathbb{1}}+\gamma \chi (\omega

).$$ (9) It is clear that _M_(_ω_) determines the properties of _S_(_ω_) and we use it to define a topological invariant. THE WINDING NUMBER In this section, we introduce a topological

invariant akin to the winding number of the canonical SSH model55, but defined on the complex spectrum of the dynamic matrix (in reciprocal space). The same topological invariant was

recently studied by Gong et al.48 for non-Hermitian Hamiltonians. In general, the dynamic matrix of a translational invariant 1D system, such as our driven-dissipative chain, has the form

_M__j_,_j_+_ℓ_ ≡ _μ__ℓ_ for all _j_. Our strategy is to employ periodic boundary conditions (PBC) to probe the bulk properties and to define a meaningful topological invariant—the winding

number. We will see that the system is extremely sensitive to changes of the boundary conditions. Indeed, moving to open boundary conditions (OBC) leads to the directional amplification we

want to characterize. Under PBC, _M_pbc is diagonal in the plane wave basis \(\left|k\right\rangle =\frac{1}{\sqrt{N}}{\sum }_{j}{e}^{{\rm{i}}kj}\left|j\right\rangle \) with _k_ = 

2_π__r_/_N_, _r_ =  0, 1, …, _N_ − 1 $${M}_{{\rm{pbc}}} = \sum _{\ell }{\mu }_{\ell }\sum _{j}\left|j\right\rangle \left\langle (j+\ell )\,\mathrm{mod}\,\,N\right|\\ = \sum _{k}\sum _{\ell

}{\mu }_{\ell }{e}^{{\rm{i}}k\ell }\left|k\right\rangle \left\langle k\right|\equiv \sum _{k}h(k)\left|k\right\rangle \left\langle k\right|,$$ (10) with the generating function _h_(_k_) ≡ 

∑_ℓ__μ__ℓ__e_i_k__ℓ_. Equivalently, _h_(_k_) generates the entries \({\mu }_{\ell }=\frac{1}{2\pi }\mathop{\int}\nolimits_{0}^{2\pi }{\rm{d}}k\ h(k){e}^{-{\rm{i}}k\ell }\) of _M_. We have

adopted a Dirac notation for referring to the (cavity) site basis \(\{\left|j\right\rangle \}\) and plane wave basis \(\{\left|k\right\rangle \}\), respectively. _h_(_k_) can be regarded as

an energy band in the 1D Brillouin zone; only that now, _h_(_k_) takes complex values since _M_ ≠ _M_†. As _h_(_k_) is periodic in _k_ with period 2_π_, it describes a closed curve in the

complex plane, cf. Fig. 2. This enables us to define a winding number from the argument principle48 $$\nu \equiv \frac{1}{2\pi {\rm{i}}}\int_{0}^{2\pi }{\rm{d}}k\ \frac{h^{\prime}

(k)}{h(k)}=\frac{1}{2\pi {\rm{i}}}{\oint }_{| z| = 1}{\rm{d}}z\ \frac{\frac{\partial }{\partial z}h(z)}{h(z)},$$ (11) where we have introduced _z_ = _e_i_k_ in the last step. The winding

number is an integer counting the number of times _h_ wraps around the origin as _k_ changes from 0 to 2_π_. While Gong et al.48 define the winding number w.r.t. an arbitrary base point, we

choose the origin as special point for the physically relevant scattering matrix: as we will see later from Eq. (18), it is the pole of the scattering matrix under PBC. In the following, we

focus on nearest-neighbor interactions between cavity modes. Mathematically, this translates into generating functions of the form $$h(k)={\mu }_{0}+{\mu }_{1}{e}^{{\rm{i}}k}+{\mu

}_{-1}{e}^{-{\rm{i}}k}={\mu }_{0}+{\mu }_{1}z+{\mu }_{-1}\frac{1}{z}$$ (12) permitting only _ν_ = 0, ±1. In the Hermitian case, a Hamiltonian without any additional symmetries would be

topologically trivial. However, in the case of non-Hermitian operators, one complex band is enough to obtain non-trivial values of a topological invariant48. Note that Eq. (11) connects the

winding number to the number of zeros _h_(_z_) encloses within the unit circle. For nearest-neighbor interactions, the zeros are given by $${z}_{\pm }\equiv \frac{-{\mu }_{0}\pm \sqrt{{\mu

}_{0}^{2}-4{\mu }_{1}{\mu }_{-1}}}{2{\mu }_{1}}.$$ (13) The values of _ν_ defining different topological regimes correspond to having two zeros within the unit circle (_ν_ = +1), one zero

(_ν_ = 0), or none (_ν_ = −1), see Fig. 3(c). Due to the form of Eq. (13) it is clear that non-trivial topological regimes are always linked to the competition of local, i.e. _μ_0, and

non-local terms, _μ_1_μ_−1. TOPOLOGICAL REGIMES OF THE DRIVEN-DISSIPATIVE CHAIN We first consider the resonant response _ω_ = 0, i.e., when the probe frequency equals the cavity frequency.

It is convenient to rescale all parameters by the on-site decay rate (_γ_ + 2Γ − _κ_)/2 and we introduce a rescaled hopping constant Λ ≡ 4_J_/(_γ_ + 2Γ − _κ_) and a cooperativity

\({\mathcal{C}}\equiv 2\Gamma /(\gamma +2\Gamma -\kappa )\) defined analogous to20. \({\mathcal{C}}\) is the ratio between the non-local dissipative contributions Γ in Eq. (3) and the

overall on-site decay rate (_γ_ + 2Γ − _κ_)/2. We refer to these two terms as non-local and local dissipation, respectively. With these definitions, the generating function (12) obtained

from Eqs. (3) becomes $$h(k)\propto -1-{\mathcal{C}}\cos (k+\theta )-{\rm{i}}\Lambda \cos k.$$ (14) We have dropped the proportionality factor (_γ_ + 2Γ − _κ_)/2 since the winding number is

unchanged by the multiplication of the generating function with a non-zero constant. Figure 2a and b illustrates _h_(_k_) in the complex plane in topologically non-trivial and trivial

regimes, respectively. Equation (14) shows that the imaginary part of _h_(_k_) pertains to the coherent evolution, while the real part encodes the dissipation. Therefore, the winding number

(11) is only well-defined in the presence of dissipation. The imaginary part of _h_(_k_) in Eq. (14) takes both positive and negative values, so any non-vanishing Λ can lead to _ν_ ≠ 0.

However, the real part in Eq. (14) contains a constant shift (−1), which is due to local dissipation. This implies that the oscillating contribution \({\mathcal{C}}\cos (k+\theta )\) from

the non-local dissipative interaction needs to exceed this local contribution to include the origin within _h_(_k_), cf. Fig. 2. A non-trivial winding number therefore always requires

$${{\mathcal{C}}}^{2}{\sin }^{2}\theta\;> \;1$$ (15) for _ν_ ≠ 0. This yields the ‘phase diagram’ Fig. 3b with the two orange lobes _ν_ = ±1. We note that _ν_ ≠ 0 is inaccessible for

reciprocal dynamics (_θ_ = 0, _π_). In this case, _h_(_k_) degenerates into a line in the complex plane and _ν_ = 0, unless it crosses the origin, in which case the winding number becomes

undefined. Entering the non-trivial topological regime is only possible with the help of the incoherent pump \({\mathcal{D}}[{a}_{j}^{\dagger }]\) of rate _κ_ in Eq. (2) featuring as local

anti-damping in Eqs. (3). Condition (15) implies that we require at least \(1\;<\;{\mathcal{C}}=1/(1+\frac{\gamma -\kappa }{2\Gamma })\), which is equivalent to _κ_ > _γ_. Hence, the

modes _a__j_ have to be coupled to a bath which is out of equilibrium to obtain _ν_ ≠ 0. The system response is captured by the scattering matrix _S_(0), for which we show some

representative examples under OBC within different regimes as insets in Fig. 3b. Indeed, we can associate a scattering matrix with each point in the ‘phase diagram’ and obtain qualitatively

the same behavior within one topological regime. Figure 3a shows gain and reverse gain under OBC for \(\theta =\frac{\pi }{2},\frac{3\pi }{2}\). End-to-end amplification sets in for

\({\mathcal{C}}\;> \;1\) as we enter the topologically non-trivial regime, while transmission in the reverse direction is strongly suppressed. The sign of _ν_ sets the propagation

direction: _ν_ = +1 (_ν_ = −1) leads to amplification from right (left) to left (right). In regimes with _ν_ = 0, the gain dominates over the reverse gain, but no amplification takes place.

This is a clear indication that non-trivial winding numbers coincide with directional amplification. Note that within topologically non-trivial regimes the gain grows exponentially with _N_,

\({{\mathcal{G}}}_{\nu = \pm 1}\propto | {z}_{\mp }{| }^{-2\nu N}\) (for _N_ ≫ 1)—a result we will derive in the next section. At the transition from the trivial to the non-trivial regime,

the corresponding _z_± is located on the unit circle, see Fig. 3c. Therefore, the gain is asymptotically independent of _N_ and \({\mathcal{O}}(1)\), see Fig. 3a. Within regimes _ν_ ≠ 0, the

gain increases with \({\mathcal{C}}\) while the reverse gain decreases until we reach the EP \({\mathcal{C}}=\Lambda \), and \(\theta =\frac{\pi }{2}\) or \(\frac{3\pi }{2}\), at which

\(\bar{{\mathcal{G}}}=0\). Note that Λ sets the position of the EP on the lines \(\theta =\frac{\pi }{2}\) and \(\theta =\frac{3\pi }{2}\). For Λ > 1 it is located within the

topologically non-trivial regime, which is advantageous for a directional amplifier. Our driven-dissipative chain not only cancels the signal in the reverse direction, it also ensures that

any field entering the output cavity is not back-reflected and mixed-in with the output signal since we can choose _γ_ in Eq. (9) such that _S_1,1 = _S__N_,_N_ = 0 (impedance matching)

whenever \(\theta =\frac{\pi }{2}\) or \(\frac{3\pi }{2}\) in the stable regime, see insets in Fig. 3b. At the EP, the condition for impedance matching can be found analytically as _γ_ = 2Γ 

− _κ_. This is a significant advantage over other proposals for directional amplifiers which do not necessarily have this property20,22,26. Among other things, it means that the amplifier is

phase preserving even if signals are scattered back from other devices behind the amplifier. The gain continues to increase with larger \({\mathcal{C}}\) beyond the EP until we reach the

parametric instability at which one eigenvalue of _M_obc is zero. We have an analytic expression for the eigenvalues under OBC available56, which we provide in Methods and use to plot the

unstable regime in Fig. 3a, b; all other regimes are stable. Crucially, a longer chain also leads to the suppression of the reverse gain. Indeed, the reverse gain scales inversely with

respect to \({\mathcal{G}}\), i.e., \({\overline{{\mathcal{G}}}}_{\nu = \pm 1}\propto | {z}_{\pm }{| }^{2\nu N}\), and \(\overline{{\mathcal{G}}}\) vanishes at the EP, see Eq. (29) and Fig. 

3a. This improves the isolation considerably, and in the thermodynamic limit, _N_ → _∞_, extends the parameter regime over which we obtain completely directional amplification from the

fine-tuned EP to the entire non-trivial topological regime. Directional amplification is induced by the transition from PBC to OBC, which can intuitively be understood as follows: For PBC

and _ν_ ≠ 0, excitations travel around the ring in a given direction gaining energy, see Fig. 2c. In this case, the dynamics are unstable, since the eigenvalues _h_(_k_) need to have both

positive and negative real part to encircle the origin, see Fig. 2a. Removing one link (OBC) can lead to stable dynamics and to the accumulation of excitations at one end of the chain, which

translates into amplified steady state cavity amplitudes ∣〈_a__ℓ_〉∣2, see Fig. 2d. For reciprocal dynamics, OBC only lead to local changes and no directional amplification, see Fig. 2e. On

resonance, the existence of non-trivial topological regimes is independent of the coherent coupling Λ ≠ 0. This changes, for the non-resonant response _ω_ ≠ 0. Rescaling also _ω_

accordingly, \(\tilde{\omega }\equiv 2\omega /(\gamma +2\Gamma -\kappa )\), we obtain $$h(k)\propto -1+{\rm{i}}\tilde{\omega }-{\mathcal{C}}\cos (k+\theta )-{\rm{i}}\Lambda \cos k.$$ (16)

Local and non-local contributions in both real and imaginary parts compete to yield a non-zero winding number. The condition for non-trivial topology reads

$${\left(\frac{1}{{\mathcal{C}}\sin \theta }-\frac{\tilde{\omega }}{\Lambda \tan \theta }\right)}^{2}+\frac{{\tilde{\omega }}^{2}}{{\Lambda }^{2}}\;<\;1.$$ (17) This amounts to shifting

the two lobes _ν_ = ±1 against each other whereby the overlapping region becomes trivial, see Fig. 3d. ONE-TO-ONE CORRESPONDENCE OF NON-TRIVIAL TOPOLOGY AND DIRECTIONAL AMPLIFICATION We now

rigorously prove the existence of a one-to-one correspondence between non-trivial values of the winding number and directional amplification for generic 1D systems with nearest-neighbor

interactions that give rise to a dynamic matrix of Toeplitz form with uniform coupling constants. To establish the correspondence, we study the susceptibility _χ_(_ω_) = _M_−1(_ω_), first

under PBC and then under OBC. Within non-trivial topological regimes, the corrections that arise from moving to OBC, lead to directional amplification by several orders of magnitudes. While

we focus on nearest-neighbor couplings and generating functions of the form (12), our technique can also be employed beyond nearest-neighbor interactions. Under PBC, calculating _χ_pbc is

straightforward. For clarity, we omit the argument _ω_ in what follows. Since we are ultimately interested in the scattering matrix, we express \({\chi }_{{\rm{pbc}}}={M}_{{\rm{pbc}}}^{-1}\)

in the site basis $${\chi }_{{\rm{pbc}}}=\sum _{k}\frac{1}{h(k)}\left|k\right\rangle \left\langle k\right|=\sum _{j,\ell }\frac{1}{N}\sum _{k}\frac{{e}^{{\rm{i}}k(j-\ell

)}}{h(k)}\left|j\right\rangle \left\langle \ell \right|.$$ (18) We see now, why the origin is a special point in the complex plane: it constitutes the pole of the scattering matrix.

Rewriting the sum over _k_, we make the connection to the zeros of the generating function and hence _ν_. For this purpose, we expand _z__j_−_ℓ_/_h_(_z_) into a Laurent series around _z_ = 0

$$\frac{{z}^{j-\ell }}{h(z)}=\frac{1}{2\pi {\rm{i}}}\mathop{\sum }\limits_{n=-\infty }^{\infty }{z}^{n}{\oint }_{| \tilde{z}| = 1}{\rm{d}}\tilde{z}\ \frac{{\tilde{{z}}^{(j-\ell

)-n-1}}}{h(\tilde{z})}.$$ Inserting this expression into Eq. (18) allows us to evaluate the sum over _k_. Since _z_  = _e_i_k_ and _k_ = 2_π__r_/_N_ takes discrete values, we can write

$${\chi }_{{\rm{pbc}}}=\sum _{j,\ell }\mathop{\sum }\limits_{n=-\infty }^{\infty }\mathop{\sum }\limits_{r=1}^{N}\frac{{e}^{{\rm{i}}\frac{2\pi nr}{N}}}{N}\frac{1}{2\pi {\rm{i}}}{\oint }_{|

\tilde{z}| = 1}{\rm{d}}\tilde{z}\ \frac{{\tilde{{z}}^{(j-\ell )-n-1}}}{h(\tilde{z})}\left|j\right\rangle \left\langle \ell \right|.$$ Using \(\frac{1}{N}\mathop{\sum }\nolimits_{r =

1}^{N}{e}^{{\rm{i}}\frac{2\pi nr}{N}}={\delta }_{n,mN}\) for \(m\in {\mathbb{Z}}\) gives rise to the overall expression $${\chi }_{{\rm{pbc}}}=\sum _{j,\ell }\mathop{\sum

}\limits_{m=-1}^{\infty }\frac{1}{2\pi {\rm{i}}}{\oint }_{| \tilde{z}| = 1}{\rm{d}}\tilde{z}\ \frac{{\tilde{{z}}^{(j-\ell )-mN-1}}}{h(\tilde{z})}\left|j\right\rangle \left\langle \ell

\right|.$$ (19) Here, we have used the fact that since _h_(_z_) can at most have _N_ zeros, the sum only starts from _m_ = −1. It follows from Cauchy’s principle57 that $${\chi

}_{{\rm{pbc}}}=\sum _{j,\ell }[{I}_{j-\ell }+{\varepsilon }_{j-\ell }(N)]\left|j\right\rangle \left\langle \ell \right|$$ (20) with $${I}_{n}\equiv \mathop{\sum

}\limits_{m=-1}^{0}\frac{1}{2\pi {\rm{i}}}{\oint }_{| \tilde{z}| = 1}{\rm{d}}\tilde{z}\ \frac{{\tilde{{z}}^{n-mN-1}}}{h(\tilde{z})}$$ (21) and \({\varepsilon

}_{n}(N)={\mathcal{O}}({c}^{-N})\) an exponentially small correction with some ∣_c_∣ > 1. We have obtained exact expressions for _I__n_ and _ε__n_ with the residue theorem for generating

functions of the form (14), and we give the results in Methods. _I__n_ is a function of the zeros of _h_(_z_), cf. Eq. (13), and thus of the winding number (11), since the number of zeros

within the unit circle determines the contributions to the integral (21), cf. Fig. 3(c). This directly connects _χ_pbc to the winding number. _I__n_ is at most \({\mathcal{O}}(1)\) and is

illustrated in Fig. 4, so no significant amplification takes place under PBC. Moving on to OBC, we express $${M}_{{\rm{obc}}}={M}_{{\rm{pbc}}}-({\mu }_{1}\left|1\right\rangle \left\langle

N\right|+{\mu }_{-1}\left|N\right\rangle \left\langle 1\right|)$$ (22) subtracting the corners of the matrix corresponding to PBC. To calculate the influence of this change in boundary

conditions, we import the following mathematical result58: The matrix inverse of the sum of an invertible matrix _M_ and a rank-one matrix _E__j_ can be calculated from

\({(M+{E}_{j})}^{-1}={M}^{-1}-\frac{1}{1+{g}_{j}}({M}^{-1}{E}_{j}{M}^{-1})\) with \({g}_{j}={\rm{tr}}\ ({M}^{-1}{E}_{j})\). Applying the formula recursively in two stages, with

\({E}_{1}={\mu }_{1}\left|1\right\rangle \left\langle N\right|\) and \({E}_{2}={\mu }_{-1}\left|N\right\rangle \left\langle 1\right|\), we obtain an analytic expression for \({\chi

}_{{\rm{obc}}}={M}_{{\rm{obc}}}^{-1}\). Within topologically non-trivial regimes, it simplifies to $$S(\omega )-{\mathbb{1}}\propto {\chi }_{{\rm{obc}}}= \, \underbrace{\mathop{\sum

}\limits_{j,\ell = 1}^{N}{I}_{j-\ell }\left|j\right\rangle \left\langle \ell \right|}_{{{\rm{PBC}}\;{\rm{background}}}}\\ \, +\underbrace{\mathop{\sum }\limits_{j,\ell =

1}^{N}\left[\frac{{\mu }_{1}{I}_{j-N}{I}_{1-\ell }}{1+{g}_{1}}+\frac{{\mu }_{-1}{I}_{j-1}{I}_{N-\ell }}{1+{g}_{2}}\right]\left|j\right\rangle \left\langle \ell

\right|}_{{{\rm{directional}}\;{\rm{amplification}}}}\\ \, +\underbrace{\mathop{\sum }\limits_{j,\ell }{\mathcal{O}}\left({z}_{\pm }^{\nu N+[\nu (j-\ell

)+N]{\mathrm{mod}}\,N}\right)\left|j\right\rangle \left\langle \ell \right|}_{{{\rm{exponentially}}\;{\rm{small}}\;{\rm{correction}}}}$$ (23) with _g_1 = −_μ_1(_I_1−_N_ + _ε_1−_N_(_N_)) and

_g_2 = −_μ_−1(_I__N_−1 + _ε__N_−1(_N_)). Equation (23) is one of our central results. The susceptibility _χ_obc has three contributions: a PBC background equal to _χ_pbc, cf. Eq. (20), a

term giving rise to directional amplification, and an exponentially small correction. For _N_ ≫ 1 only the second term dominates due to the division by (1 + _g__j_): for _ν_ = +1 the term (1

 + _g_1) is exponentially small, while \((1+{g}_{2})={\mathcal{O}}(1)\), and vice versa for _ν_ = −1. This traces back to the values of _μ_1_I_1−_N_ and _μ_−1_I__N_−1 in the definitions of

_g__j_, which sensitively depend on _ν_. One of the _g__j_ is exactly  − 1 if _ν_ ≠ 0 only leaving _ε_1−_N_ or _ε__N_−1 in the denominator. This exponentially small denominator gives rise to

amplification. We obtain the following expressions for _ν_ = +1 corresponding to ∣_z_±∣ < 1 $$\begin{array}{ccc}{\chi }_{{\rm{obc}}}&=&\mathop{\sum }\limits_{j,\ell

=1}^{N}{I}_{j-\ell }\left|j\right\rangle \left\langle \ell \right|-\frac{1}{{\varepsilon }_{1-N}}\mathop{\sum }\limits_{j,\ell =1}^{N}{I}_{j-N}{I}_{1-\ell }\left|j\right\rangle \left\langle

\ell \right|\\ &&+\mathop{\sum }\limits _{j,\ell }{\mathcal{O}}\left({z}_{-}^{j-\ell +N-1}\right)\left|j\right\rangle \left\langle \ell \right|,\end{array}$$ (24) and for _ν_ = −1

corresponding to ∣_z_±∣ > 1 $$\begin{array}{ccc}{\chi }_{{\rm{obc}}}&=&\mathop{\sum }\limits_{j,\ell =1}^{N}{I}_{j-\ell }\left|j\right\rangle \left\langle \ell

\right|-\frac{1}{{\varepsilon }_{N-1}}\mathop{\sum }\limits_{j,\ell =1}^{N}{I}_{j-1}{I}_{N-\ell }\left|j\right\rangle \left\langle \ell \right|\\ &&+\mathop{\sum }\limits _{j,\ell

}{\mathcal{O}}\left({z}_{+}^{j-\ell -N+1}\right)\left|j\right\rangle \left\langle \ell \right|\end{array}$$ (25) with $$\frac{1}{{\varepsilon }_{\nu (1-N)}}=\nu {\mu

}_{1}({z}_{+}-{z}_{-}){\left[\frac{{z}_{+}^{\nu (N+1)}}{1-{z}_{+}^{\nu N}}-\frac{{z}_{-}^{\nu (N+1)}}{1-{z}_{-}^{\nu N}}\right]}^{-1}.$$ (26) As we show in Fig. 4c, d, the above expansions

for _χ_obc converge exponentially fast to the exact result within the whole topologically non-trivial regime, and already yield high accuracy for systems as small as _N_ = 2 in the vicinity

of the EP, where they become exact. For instance, at _N_ = 2 for \(\theta =\frac{\pi }{2}\), \({\mathcal{C}}=2.06\), and Λ = 2 the relative error of \(| {({\chi }_{{\rm{obc}}})}_{N,1}| \) is

only 3.3%. The region of small relative error, Fig. 4d, rapidly extends as _N_ increases, converging faster within the dynamically stable regime and more slowly close to the boundary.

Expanding _ε__ν_(1−_N_) of Eq. (26) for large _N_ and ∣_z_+∣ sufficiently different from ∣_z_−∣, we obtain $$\frac{1}{{\varepsilon }_{\nu (1-N)}}\cong {\mu }_{1}({z}_{+}-{z}_{-})\ {z}_{\pm

}^{-\nu (N+1)}$$ (27) in which we choose _z_+ in the expansion for _ν_ = +1 and _z_− for _ν_ = −1. The susceptibility _χ_obc determines the behavior of _S_(_ω_) according to Eq. (9). We

identify 1/_ε__ν_(1−_N_) as the contribution giving rise to amplification, as it is directly related to the gain (5), which asymptotically grows exponentially with the system size

$${{\mathcal{G}}}_{\nu = \pm 1}\cong \ {\gamma }^{2}\ \frac{| {\mu }_{0}^{2}-4{\mu }_{1}{\mu }_{-1}| }{| {\mu }_{\pm 1}{| }^{4}}\ | {z}_{\pm }{| }^{-2\nu (N+1)},$$ (28) and at the EP,

\({{\mathcal{G}}}_{\nu = \pm 1}\cong \frac{{\gamma }^{2}}{| {\mu }_{\pm 1}{| }^{2}}{\left|\frac{{\mu }_{\pm 1}}{{\mu }_{0}}\right|}^{2N}\). In the thermodynamic limit, _N_ → _∞_,

\({\mathcal{G}}\) diverges within non-trivial regimes, but stays finite in trivial regimes. We can also give the asymptotic expression for the reverse gain. The leading order contribution

stems from the PBC background, _I__ν_(_N_−1), and therefore \(\overline{{\mathcal{G}}}\) decreases exponentially with _N_ $${\overline{{\mathcal{G}}}}_{\nu = \pm 1}\cong {\gamma }^{2}\

\frac{1}{| {\mu }_{0}^{2}-4{\mu }_{1}{\mu }_{-1}{| }^{2}}| {z}_{\mp }{| }^{2\nu (N+1)},$$ (29) and at the EP, \(\overline{{\mathcal{G}}}=0\) exactly. These expressions also converge

exponentially fast and are most practical starting from _N_ ≈ 5. In general, the individual elements of _χ_obc and therefore the scattering matrix (9) are formed by the terms

_I__j_−_N__I_1−_ℓ_, and _I__j_−1_I__N_−_ℓ_, according to Eqs. (24) and (25), respectively, which give rise to directionality. Since _I__n_ decreases approximately exponentially with _n_ and

is defined modulo _N_, the products of the different _I__n_ only leave one matrix element that contributes significantly, see Fig. 4. This is the one determining the gain (28). In trivial

topological regimes we obtain more cumbersome combinations of _I__n_ and 1/(1 + _g__j_), but \((1+{g}_{j})={\mathcal{O}}(1)\), so no amplification takes place. However, as we can see from

the scattering matrices displayed in Fig. 3b, directionality is still possible. APPLICATIONS—DESIGN OF MULTIMODE DIRECTIONAL AMPLIFIERS So far, we have focused on the driven-dissipative

chain  (3), however, the results of Eqs. (23) to (29) apply more generally to systems with nearest-neighbor couplings. We can map any system with a generating function of the form (12) to

the parameters of the driven-dissipative chain, i.e., \({\mathcal{C}}\), Λ, \(\tilde{\omega }\) and _θ_, and apply all of our previous results. However, the physical interactions giving rise

to amplification and indeed the amplified observables may be very different from those of the driven-dissipative chain. We illustrate this by applying our topological framework to several

models for phase preserving and phase sensitive amplifiers. Remarkably, the expressions for the scattering matrix (23), the ‘phase diagram’ Fig. 3b, the gain (28) and the reverse gain (29)

in Fig. 3a apply _mutatis mutandis_. Hence, we obtain the same exponential growth and attenuation with _N_ for gain and reverse gain, respectively, without any explicit calculations. First,

we focus on the phase preserving amplifier proposed by Metelmann and Clerk20,22 and sketched in Fig. 5b. We consider the generalization of their two-mode proposal to a chain of _N_ cavities.

Two neighboring modes _a__j_ and _a__j_+1 are coupled both via the coherent parametric interaction \(\lambda {a}_{j}^{\dagger }{a}_{j+1}^{\dagger }+{\lambda }^{* }{a}_{j}{a}_{j+1}\) and

through the non-local dissipator \({\mathcal{D}}[{a}_{j}+{e}^{-{\rm{i}}\theta }{a}_{j+1}^{\dagger }]\). Gauge freedom allows us to absorb the phase into _λ_; however, we focus on the case of

imaginary _λ_, i.e., _λ_ = i∣_λ_∣, which ensures that the amplifier does not couple different quadratures and therefore is phase insensitive. The equations of motion for the field

quadratures \({x}_{j}\equiv ({a}_{j}+{a}_{j}^{\dagger })/\sqrt{2}\) and \({p}_{j}\equiv -{\rm{i}}({a}_{j}-{a}_{j}^{\dagger })/\sqrt{2}\) are then given by $$\begin{array}{ccc}\langle

{\dot{x}}_{\ell }\rangle =-\frac{\gamma }{2}\langle {x}_{\ell }\rangle &+&\left(| \lambda | -\frac{\Gamma }{2}\right)\langle {x}_{\ell +1}\rangle \\ &+&\left(| \lambda |

+\frac{\Gamma }{2}\right)\langle {x}_{\ell -1}\rangle -\sqrt{\gamma }\langle {x}_{\ell ,{\rm{in}}}\rangle \end{array}$$ (30) $$\begin{array}{ccc}\langle {\dot{p}}_{\ell }\rangle

=-\frac{\gamma }{2}\langle {p}_{\ell }\rangle &-&\left(| \lambda | -\frac{\Gamma }{2}\right)\langle {p}_{\ell +1}\rangle \\ &-&\left(| \lambda | +\frac{\Gamma

}{2}\right)\langle {p}_{\ell -1}\rangle -\sqrt{\gamma }\langle {p}_{\ell ,{\rm{in}}}\rangle .\end{array}$$ (31) From the equations above we can directly read off the generating function for

the two quadratures. Introducing \({\mathcal{C}}\equiv 4| \lambda | /\gamma \) and Λ ≡ 2Γ/_γ_, we find $${h}_{x}(k)\propto -1+{\mathcal{C}}\cos k+{\rm{i}}\Lambda \sin k$$ (32)

$${h}_{p}(k)\propto -1-{\mathcal{C}}\cos k-{\rm{i}}\Lambda \sin k,$$ (33) Notice that _x_ and _p_ quadratures have the same generating function up to the sign of the oscillating terms, which

reflects the phase conjugating property of the amplifier: _x_ and _p_ quadratures are amplified with the same gain, but the _p_ quadrature exits with a _π_ phase shift, i.e. a negative

sign, at the output. Nevertheless, the amplifier is still considered to be phase insensitive according to59. The minus sign has no impact on the topological regimes, since \(\cos k\) in Eq.

(33) takes both positive and negative values as _h__p_ winds around the origin, and we obtain the same regimes for _x_ and _p_ quadratures according to Eq. (15): _ν_ = 0 for

\({\mathcal{C}}\;<\;1\), _ν_ = +1 for \({\mathcal{C}}\,> \, 1\) and Λ > 0. We have set \(\theta =\frac{\pi }{2}\) for _ν_ = −1 and \(\theta =\frac{3\pi }{2}\) for _ν_ = +1 in Eq.

(15), since _θ_ is defined as the phase difference between real and imaginary part. Therefore, gain and reverse gain for the quadratures of the phase insensitive amplifier, Eqs. (30) and

(31), are given by Fig. 3a with \({\mathcal{C}}=4| \lambda | /\gamma \). Furthermore, the scattering matrices _S__x_(_ω_) and _S__p_(_ω_) linking Xout = _S__x_Xin and Pout = _S__x_Pin with

\({{\bf{x}}}_{{\rm{in}}/{\rm{out}}}\equiv {(\langle {x}_{1,{\rm{in}}/{\rm{out}}}\rangle ,\ldots ,\langle {x}_{N,{\rm{in}}/{\rm{out}}}\rangle )}^{{\rm{T}}}\) and

\({{\bf{p}}}_{{\rm{in}}}/{\rm{out}}\equiv {(\langle {p}_{1,{\rm{in}}/{\rm{out}}}\rangle ,\ldots ,\langle {p}_{N,{\rm{in}}/{\rm{out}}}\rangle )}^{{\rm{T}}}\) are given by Eq. (23). Since the

generating functions are the same up to the sign conjugation, ∣_S__x_(0)∣2 = ∣_S__p_(0)∣2; off resonance, analogous considerations lead to ∣_S__x_(_ω_)∣2 = ∣_S__p_(_ω_)∣2. Beyond that, the

scattering matrices ∣_S__x_(0)∣2, ∣_S__p_(0)∣2 take the same form as the insets in Fig. 3b with \(\theta =\frac{\pi }{2}\) and \(\theta =\frac{3\pi }{2}\). Furthermore, the asymptotic

scaling of the gain is given by \({{\mathcal{G}}}_{\nu = \pm 1}\propto | {z}_{\mp }{| }^{-2\nu N}\) and of the reverse gain by \({\overline{{\mathcal{G}}}}_{\nu = \pm 1}\propto | {z}_{\pm

}{| }^{2\nu N}\) according to Eqs. (28) and (29), respectively. This demonstrates the power of the framework: we can determine the properties of a physically very different amplifier

consisting now generally of _N_ modes without numerically calculating the scattering matrix. Next, we examine the phase sensitive amplifier proposed in20,22. It couples the field quadratures

via the coherent interaction _λ__p__j__x__j_+1 and the dissipator \(\Gamma {\mathcal{D}}[{x}_{j+1}+{\rm{i}}{p}_{j}]\). We again consider the generalization to a chain of _N_ modes and

obtain the equations of motion $$\langle {\dot{x}}_{\ell }\rangle =-\frac{\gamma }{2}\langle {x}_{\ell }\rangle -(\Gamma -\lambda )\langle {x}_{\ell +1}\rangle -\sqrt{\gamma }\langle

{x}_{\ell ,{\rm{in}}}\rangle $$ (34) $$\langle {\dot{p}}_{\ell }\rangle =-\frac{\gamma }{2}\langle {p}_{\ell }\rangle -(\Gamma +\lambda )\langle {p}_{\ell -1}\rangle -\sqrt{\gamma }\langle

{p}_{\ell ,{\rm{in}}}\rangle .$$ (35) The equations for _x_ and _p_ quadratures decouple and therefore, we consider them separately. Defining \({{\mathcal{C}}}_{\pm }\equiv 2(\Gamma \pm

\lambda )/\gamma \) with the positive sign for _p_ and the negative sign for _x_, the generating functions take the form $${h}_{x}(k)\propto -1-{{\mathcal{C}}}_{-}\cos

k-{\rm{i}}{{\mathcal{C}}}_{-}\sin k$$ (36) $${h}_{p}(k)\propto -1-{{\mathcal{C}}}_{+}\cos k+{\rm{i}}{{\mathcal{C}}}_{+}\sin k.$$ (37) We obtain the following topological regimes from

condition (15) with \(\theta =\frac{3\pi }{2}\): _ν__x_ = +1 for \(| {{\mathcal{C}}}_{-}|\;> \;1\), _ν__x_ = 0 for \(| {{\mathcal{C}}}_{-}|\;<\;1\); and with \(\theta =\frac{\pi

}{2}\): _ν__p_ = −1 for \(| {{\mathcal{C}}}_{+}|\;> \;1\), _ν__p_ = 0 for \(| {{\mathcal{C}}}_{+}|\;<\;1\), where _ν__x_ and _ν__p_ refer to the winding numbers for _x_ and _p_

quadratures, respectively. As we illustrate in Fig. 6a, depending on the regime, both quadratures, only one of them, or none, are amplified. The amplification direction for _x_ and _p_

quadratures is the reverse. We again calculate the scattering matrices _S__x_ and _S__p_ for _x_ and _p_ from Eq. (23) and show some as insets in Fig. 6a. Analogously, the gain and the

reverse gain are obtained from Eqs. (28) and (29), respectively. The gain follows the same behavior as Fig. 3a. Finally, we consider the ‘bosonic Kitaev chain’ proposed in ref. 41 and

illustrated in Fig. 5d, for which _x_ and _p_ quadratures are amplified in opposite directions. This also follows straightforwardly from our topological framework. The Hamiltonian

$${\mathcal{H}}=\frac{1}{2}\sum _{j}[(\Delta -J){x}_{j+1}{p}_{j}+(\Delta +J){p}_{j+1}{x}_{j}]$$ (38) together with on-site dissipator \(\gamma {\mathcal{D}}[{a}_{j}]\) gives rise to the

following equations of motion for the system’s quadratures $$\langle {\dot{x}}_{\ell }\rangle = -\frac{\gamma }{2}\langle {x}_{\ell }\rangle +\frac{J+\Delta }{2}\langle {x}_{\ell -1}\rangle

-\frac{J-\Delta }{2}\langle {x}_{\ell +1}\rangle -\sqrt{\gamma }\langle {x}_{\ell ,{\rm{in}}}\rangle \\ \equiv \sum _{j}{H}_{\ell ,j}\langle {x}_{j}\rangle -\sqrt{\gamma }\langle {x}_{\ell

,{\rm{in}}}\rangle$$ (39) $$\langle {\dot{p}}_{\ell }\rangle =-\frac{\gamma }{2}\langle {p}_{\ell }\rangle +\frac{J-\Delta }{2}\langle {p}_{\ell -1}\rangle -\frac{J+\Delta }{2}\langle

{p}_{\ell +1}\rangle -\sqrt{\gamma }\langle {p}_{\ell ,{\rm{in}}}\rangle \\ \equiv \sum _{j}{(-{H}^{{\rm{T}}})}_{\ell ,j}\langle {p}_{j}\rangle -\sqrt{\gamma }\langle {p}_{\ell

,{\rm{in}}}\rangle.$$ (40) We have added coherent driving to obtain the input terms in Eqs. (39) and (40) and cast them into the same form as Eqs. (3). As we can see from the last lines of

Eqs. (39) and (40), the dynamic matrix governing the evolution of the _p_ quadratures is the negative transpose of that of the _x_ quadratures. On the level of the generating functions, this

translates into a change in the sign of the winding number within topologically non-trivial regimes. Defining \({\mathcal{C}}\equiv 2\Delta /\gamma \) and Λ ≡ 2_J_/_γ_, the generating

functions are $${h}_{x}(k)\propto -1+{\mathcal{C}}\cos k-{\rm{i}}\Lambda \sin k$$ (41) $${h}_{p}(k)\propto -1-{\mathcal{C}}\cos k-{\rm{i}}\Lambda \sin k.$$ (42) Assuming Λ > 0, we obtain

from condition (15): _ν__x_ = 0 and _ν__p_ = 0 for \(| {\mathcal{C}}|\;<\;1\), _ν__x_ = −1 and _ν__p_ = +1 for \({\mathcal{C}}\;> \;1\), _ν__x_ = +1 and _ν__p_ = −1 for

\({\mathcal{C}}\;<-\!1\), cf. Fig. 6b. The non-trivial cases correspond to setting \(\theta =\frac{\pi }{2}\) for _x_ and \(\theta =\frac{3\pi }{2}\) for _p_ quadratures for

\({\mathcal{C}}\;> \;1\), or vice versa for \({\mathcal{C}}\;<\,-\!1\), in the ‘phase diagram’ Fig. 3b and the gain Fig. 3a. As for the previous examples, we obtain the scattering

matrices from Eq. (23) and illustrate them in Fig. 6b. Since the winding numbers for _x_ and _p_ quadratures have opposite sign they are amplified in reverse directions. Gain and reverse

gain follow from Eqs. (28) and (29), respectively. DISCUSSION In this work we have developed a framework based on the topology of the dynamic matrix to predict and describe directional

amplification in driven-dissipative systems. In contrast to topological states of matter for closed systems, we have introduced the winding number (11) as topological invariant based on the

spectrum of the dynamic matrix—the generating function (12). We have shown that non-trivial values of the winding number have a directly observable consequence expressed in the scattering

matrix (4), and we have established a one-to-one correspondence between non-trivial topology and directional amplification. One of our main results is the ‘phase diagram’ for the scattering

matrix, Fig. 3b, that associates topologically non-trivial parameter regimes with directional amplification. We have obtained an analytic expression for the scattering matrix (9) in Eq.

(23), the gain (28) and the reverse gain (29) in the case of nearest-neighbor couplings and have revealed an exponential scaling of the gain with the number of sites within topologically

non-trivial phases, while the reverse gain is exponentially suppressed. In the limit of an infinite chain, completely directional amplification is obtained within the whole topological

regime. Our result for the scattering matrix (23) already yields high accuracy for systems as small as _N_ = 2 in the vicinity of the EP, where it is exact, and it converges exponentially

fast within the whole topologically non-trivial regime. Therefore, directional amplification can be seen as a proxy of non-trivial topology, formally defined only in the limit _N_ → _∞_,

even in very small systems, which makes our work relevant for state-of-the art devices such as ref. 27. Furthermore, we have demonstrated the generality of our results and shown how four

systems each with different coherent and dissipative interactions can be analyzed with our topological framework. One of our key assumption is translational invariance. However, we still

expect our results to serve as good approximation when the terms breaking translational invariance are sufficiently small. Another way to go beyond our assumptions is, for instance, to add

parametric interactions to Eq. (3). This yields two rather than one complex band, and we have to modify our main result (23). Interactions beyond nearest neighbors yield yet another form of

the dynamic matrix which leads to higher winding numbers. This necessaitates additional terms in our decomposition of the scattering matrix, Eq. (23). These extensions will be addressed in

future work. Suitable platforms for implementation include superconducting circuits10,42, optomechanics43, photonic crystals28 and nanocavity arrays60, as well as topolectric circuits44,45

and mechanical meta-materials61,62,63. On a fundamental level, our analysis sheds light on the role of topology in open quantum systems64 and is of direct relevance for the study of

non-Hermitian topology46,48,49,50, where our framework predicts immediate physical and observable consequences for a topological invariant. METHODS EXACT EXPRESSIONS We give here the exact

expressions for _I__n_ and _ϵ__n_ arising in the derivation of our main results Eqs. (23) to (29)—the one-to-one correspondence between a non-trivial winding number and directional

amplification. _χ_obc is crucially determined by \({I}_{n}\equiv \mathop{\sum }\nolimits_{m = -1}^{0}\frac{1}{2\pi {\rm{i}}}{\oint }_{| \tilde{z}| = 1}{\rm{d}}\tilde{z}\

\frac{{\tilde{{z}}^{n-mN-1}}}{h(\tilde{z})}\), see Eq. (23). We can calculate it exactly for generating functions (12) using the residue theorem. For that purpose, we use the general Leibniz

rule and find the residues with \(r(n)\equiv (\nu n+N)\,\mathrm{mod}\,\,N\) * _ν_ ≠ 0, i.e., either ∣_z_±∣ > 1 or ∣_z_±∣ < 1, and _z_+ ≠ _z_− $${I}_{n}=\frac{\nu }{{\mu

}_{1}}\frac{{z}_{+}^{\nu | r(n)| }-{z}_{-}^{\nu | r(n)| }}{{z}_{+}-{z}_{-}}$$ (43) * _ν_ ≠ 0 and _z_+ = _z_− $${I}_{n}=\left\{\begin{array}{ll}\frac{1}{{\mu }_{1}}| r(n)| {z}_{+}^{\nu |

r(n)| -1}:&n\,\ne\, 0\\ 0&\hskip -13pt:\hskip 13ptn=0\end{array}\right.$$ (44) * _ν_ = 0: ∣_z_+∣ < 1 and ∣_z_−∣ > 1 or ∣_z_+∣ > 1 and ∣_z_−∣ < 1

$${I}_{n}=\left\{\begin{array}{ll}\pm \frac{1}{{\mu }_{1}}\frac{{z}_{\pm }^{| n| }}{{z}_{+}-{z}_{-}}&:n\,\ge\, 0\\ \pm \frac{1}{{\mu }_{1}}\frac{{z}_{\mp }^{-| n|

}}{{z}_{+}-{z}_{-}}&:n\,<\, 0.\end{array}\right.$$ (45) One important feature of this expression within topological regimes is _I_0 = 0. This allows us to simplify _χ_obc to yield Eq.

(23). We also employ the residue theorem to calculate the correction _ε__n_(_N_) exactly rewriting the sum as geometric series and inserting the calculated residues $${\varepsilon }_{n}(N)

\equiv \,\mathop{\sum }\limits_{m=1}^{\infty }\frac{1}{2\pi {\rm{i}}}{\oint }_{| \tilde{z}| = 1}{\rm{d}}\tilde{z}\ \frac{{\tilde{{z}}^{n-mN-1}}}{h(\tilde{z})}\\ =\, \frac{\nu }{{\mu

}_{1}({z}_{+}-{z}_{-})}\left(\frac{{z}_{+}^{\nu (N+r(n))}}{1-{z}_{+}^{\nu N}}-\frac{{z}_{-}^{\nu (N+r(n))}}{1-{z}_{-}^{\nu N}}\right)\\ \quad \,\frac{1}{{\varepsilon }_{\nu (1-N)}} \cong

{\mu }_{1}({z}_{+}-{z}_{-}){z}_{\pm }^{-\nu (N+1)},$$ (46) in which  ±  is chosen according to the winding number: _z_+ for _ν_ = +1 and _z_− for _ν_ = −1. DETERMINING THE EP The value of

the EP can be extracted analytically for all _N_. At the EP, eigenvalues and eigenvectors coalesce. The dynamic matrix, Eq. (3), is a Toeplitz matrix, for which there exists an analytic

expression for both eigenvalues and eigenvectors56, see Eq. (47). From this expression it is clear, that the eigenvalues can only coalesce when either \({\rm{i}}J=-\frac{{e}^{{\rm{i}}\theta

}\Gamma }{2}\) or \({\rm{i}}J=-\frac{{e}^{-{\rm{i}}\theta }\Gamma }{2}\), in which case the dynamic matrix becomes an upper (lower) triangular matrix with only the diagonal and

super-(sub-)diagonal non-zero. Since all the entries on the respective diagonal and super-(sub-)diagonal are the same, the matrix has rank 1 and these are indeed EPs. We obtain the _N_-fold

degenerate right eigenvectors from Gaussian elimination to be either (1, 0, …, 0, 0)T in the former case or (0, 0, …, 0, 1)T in the latter case. STABILITY AND BANDWIDTH OF THE

DRIVEN-DISSIPATIVE CAVITY CHAIN Here, we discuss the stability of the driven-dissipative chain as well as the gain \({\mathcal{G}}(\omega )\) as a function of _ω_. Stability requires the

real part of all eigenvalues _λ__m_ of the dynamic matrix _M_obc to be negative. The analytic expression for _λ__m_ is given by56 $$\begin{array}{ccc}{\lambda }_{m}&=&\frac{| 2\Gamma

+\gamma -\kappa | }{2}\left[-1+{\rm{i}}\tilde{\omega }\right.\\ &&+\left.\sqrt{{{\mathcal{C}}}^{2}-{\Lambda }^{2}+2{\rm{i}}{\mathcal{C}}\Lambda \cos \theta }\cos\, \left(\frac{m\pi

}{N+1}\right)\right]\end{array}$$ (47) for _κ_ < 2Γ + _γ_ and _m_ = 1, …, _N_. Larger values of Λ extend the stable regime to larger \({\mathcal{C}}\). In order to obtain a regime which

is both stable and amplifying, we require Λ > 1. The eigenvalues also determine the bandwidth of the gain \({\mathcal{G}}(\omega )\). We can write the exact expression for

\({\mathcal{G}}(\omega )\) using65. Together with Eq. (9) and denoting \({\mu }_{0}(\omega )=-1+{\rm{i}}\tilde{\omega }\) and \({\mu }_{\pm 1}=-{\rm{i}}\Lambda -{\mathcal{C}}{e}^{\mp

{\rm{i}}\theta }\), we write the gain $${{\mathcal{G}}}_{\nu = \pm 1}(\omega )={\left|\frac{{\mu }_{\mp 1}^{N-1}}{{({\mu }_{1}{\mu }_{-1})}^{N/2}}\frac{1}{{U}_{N}\left(\frac{{\mu

}_{0}(\omega )}{2\sqrt{{\mu }_{1}{\mu }_{-1}}}\right)}\right|}^{2}$$ (48) and the reverse gain $${\overline{{\mathcal{G}}}}_{\nu = \pm 1}(\omega )={\left|\frac{{\mu }_{\pm 1}^{N-1}}{{({\mu

}_{1}{\mu }_{-1})}^{N/2}}\frac{1}{{U}_{N}\left(\frac{{\mu }_{0}(\omega )}{2\sqrt{{\mu }_{1}{\mu }_{-1}}}\right)}\right|}^{2},$$ (49) in which _U__N_ denotes the Chebyshev polynomial of the

second kind. This expression diverges at the zeros of the Chebyshev polynomials, see peaks in Fig. 3b, which satisfy66 $$\frac{{\mu }_{0}(\omega )}{2\sqrt{{\mu }_{1}{\mu }_{-1}}}=\cos

\left(\frac{m+1}{N}\pi \right),$$ (50) with _m_ = 1, 2, …, _N_. This is equivalent to _λ__m_ = 0 for at least one eigenvalue, cf. (47). In principle, this equation has _N_ solutions,

however, since \({\rm{Re}}\ {\mu }_{0}(\omega )=-1\), the above condition cannot be fulfilled for all parameters Λ and _θ_, and we only obtain ⌊(_N_/2)⌋ zeros, see Fig. 7. A factorization in

terms of these zeros lets us write \({\mathcal{G}}(\omega )\) as product of Lorentzians $${\mathcal{G}}(\omega )\propto \mathop{\prod }\limits_{j=1}^{\lfloor (N/2)\rfloor

}\frac{{\left(\frac{\gamma +2\Gamma -\kappa }{2}\right)}^{2}}{{(\omega -{\omega }_{j})}^{2}+{\left(\frac{\gamma +2\Gamma -\kappa }{2}\right)}^{2}},$$ (51) in which the _ω__j_ can be

determined from Eq. (50). All Lorentzians have the same width set by the effective on-site dissipation (_γ_ + 2Γ − _κ_). However, if the Lorentzians are centered around distinct _ω__j_,

which is the case if \(\theta\,\ne\,\frac{\pi }{2}\) and \(\theta\,\ne\,\frac{3\pi }{2}\), the peak is broadened, see Fig. 7. Therefore, the amplifier has no conventional gain-bandwidth

product, which will be the subject of future research. The reverse gain has the same line shape, but is suppressed by many orders of magnitude—it is attenuated exponentially with _N_, see

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Download references ACKNOWLEDGEMENTS We would like to thank Katarzyna Macieszczak and Daniel Malz for insightful discussions. C.C.W. acknowledges the funding received from the Winton

Programme for the Physics of Sustainability and the EPSRC (Project Reference EP/R513180/1). A.N. holds a University Research Fellowship from the Royal Society and acknowledges additional

support from the Winton Programme for the Physics of Sustainability. We are grateful for the funding received from the European Union’s Horizon 2020 research and innovation programme under

Grant No. 732894 (FET Proactive HOT). AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE, UK Clara C. Wanjura, Matteo Brunelli 

& Andreas Nunnenkamp Authors * Clara C. Wanjura View author publications You can also search for this author inPubMed Google Scholar * Matteo Brunelli View author publications You can

also search for this author inPubMed Google Scholar * Andreas Nunnenkamp View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS A.N. initiated and

directed the project. C.C.W. derived the analytical results with input from M.B. All authors contributed to the writing of the manuscript. CORRESPONDING AUTHOR Correspondence to Clara C.

Wanjura. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing interests. ADDITIONAL INFORMATION PEER REVIEW INFORMATION _Nature Communications_ thanks Henning Schomerus

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