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 nonreciprocity and directional amplification in drivendissipative cavity arrays. Specifically, we unveil a onetoone correspondence between a nonzero topological invariant defined on the spectrum of the dynamic matrix and regimes of directional amplification, in which the endtoend 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 phasepreserving and phasesensitive multimode directional amplifiers.
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 nonreciprocity, allows to select the direction of propagation while blocking signals in the reverse^{1,2}. Nonreciprocity is of wideranging practical value; for instance, it simplifies the construction of photonic networks^{3,4,5}, enhances the information capacity in communication technology^{6,7}, and can be a resource for (quantum) sensing^{8}. Combining nonreciprocity and amplification, directional amplifiers allow for the detection of weak signals while protecting them against noise from the readout electronics. For these reasons, these devices have become important components for promising quantum information platforms such as superconducting circuits^{9}.
In response to this demand, many proposals and realizations of nonreciprocal and amplifying devices have appeared in the recent literature. Isolators and circulators based on magnetooptical effects have become the conventional choice, but they are bulky and require undesired magnetic fields to explicitly break timereversal symmetry. Josephson junctions^{10,11,12} have been investigated as an alternative. Other approaches include refractive index modulation^{13,14}, interfering parametric processes^{15}, and optomechanics^{16,17,18}. An elegant solution is provided by reservoir engineering^{19,20,21,22,23,24,25,26,27}, where nonreciprocity is achieved by interfering coherent and dissipative processes^{20,22}. Based on this approach, several fewmode isolators and directional amplifiers have been proposed^{19,20,22,26} and demonstrated^{21,23,24,25,27}.
On the other hand, chiral edge states of topological photonic systems^{28} give rise to the directional transport of photons and phonons^{29,30}, which has been used to design traveling wave amplifiers^{31} and topological lasers^{32,33,34,35,36}. 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 nonHermitian system has previously appeared in the study of the SuSchriefferHeeger (SSH) laser^{39,40}.
In this paper, we unify the plethora of ad hoc proposals for directional amplifiers by uncovering an organizing principle underlying directional amplification in drivendissipative cavity arrays: the nontrivial 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 drivendissipative cavity chains as the one depicted in Fig. 1a, featuring both coherent and dissipative couplings between modes. Nontrivial topology coincides with directional amplification and arises from the competition of local and nonlocal 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 drivendissipative 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 onetoone correspondence between nontrivial topology and directional amplification leading to one of our main results: the analytic expression for the scattering matrix in nontrivial 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 nontrivial 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 finetuned point to the whole topologically nontrivial regime. The generality of our results becomes clear in the last section of Results, in which we examine with our topological framework scaledup versions of different models for phase preserving and phase sensitive amplifiers that have appeared in the literature^{20,22,41}. 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 nontrivial topology, formally defined only in the thermodynamic limit, even in very small systems, which makes our work relevant for stateofthe 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 circuits^{10,42}, optomechanical systems^{43}, and topolectric circuits^{44,45}. Finally, our work also has direct relevance for the study of the topology of nonHermitian Hamiltonians^{46,47}, for which similar topological invariants have been proposed^{48,49}, leading to the recent classification in terms of 38 symmetry classes^{50}. 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 drivendissipative chain
Let us start by introducing the system that will guide us through the general discussion and illustrate our results. We consider a drivendissipative 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)
which describes photons hopping with uniform amplitude J along the chain. The dissipation consists of both local and nonlocal contributions and is described by the master equation
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^{−iθ}a_{j+1} couples dissipatively neighboring cavities with rate Γ^{20,47}, 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 setup^{23,26,43}, in which the mechanical mode is adiabatically eliminated giving rise to the nonlocal 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 inputoutput 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}〉
In these Eqs. (3), we have chosen J real, which is always possible due to gauge freedom^{20}. The input 〈a_{j,in}(t)〉 enters as a coherent drive in the frame rotating with the cavity frequency. Note that the nonlocal dissipator contributes both to the coupling terms and to the local decay rate. The phase θ is crucial for the nonreciprocity 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 iJ = −e^{iθ}Γ/2, i.e., \(\theta =\frac{3\pi }{2}\), in Eq. (3), each cavity j in Fig. 1a only couples to its righthand 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 right^{20,22} and corresponds to standard cascaded quantum systems theory^{53,54}. 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 matrixvector 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
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 drivendissipative 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 Nth cavity to a strongly amplified output at the first cavity, while transmission in the opposite direction is suppressed. Formally, nonreciprocity 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 drivendissipative chain, the gain relates the input at the first (last) to the output at the last (first) cavity as follows
Conversely, the reverse gain pertains to the transmission in the opposite propagation direction
An efficient directional amplifier obeys \({\mathcal{G}}\gg 1\) and \(\bar{{\mathcal{G}}}\ll 1\).
For convenience, we introduce
with M(0) = H and dub it dynamic matrix at frequency ω. We also define its inverse as the susceptibility matrix
which is related to the scattering matrix through
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 model^{55}, 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 nonHermitian Hamiltonians.
In general, the dynamic matrix of a translational invariant 1D system, such as our drivendissipative 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 \(\leftk\right\rangle =\frac{1}{\sqrt{N}}{\sum }_{j}{e}^{{\rm{i}}kj}\leftj\right\rangle \) with k = 2πr/N, r = 0, 1, …, N − 1
with the generating function h(k) ≡ ∑_{ℓ}μ_{ℓ}e^{ikℓ}. 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 \(\{\leftj\right\rangle \}\) and plane wave basis \(\{\leftk\right\rangle \}\), 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 2π, it describes a closed curve in the complex plane, cf. Fig. 2. This enables us to define a winding number from the argument principle^{48}
where we have introduced z = e^{ik} 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 2π. 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 nearestneighbor interactions between cavity modes. Mathematically, this translates into generating functions of the form
permitting only ν = 0, ±1. In the Hermitian case, a Hamiltonian without any additional symmetries would be topologically trivial. However, in the case of nonHermitian operators, one complex band is enough to obtain nontrivial values of a topological invariant^{48}. Note that Eq. (11) connects the winding number to the number of zeros h(z) encloses within the unit circle. For nearestneighbor interactions, the zeros are given by
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 nontrivial topological regimes are always linked to the competition of local, i.e. μ_{0}, and nonlocal terms, μ_{1}μ_{−1}.
Topological regimes of the drivendissipative 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 onsite decay rate (γ + 2Γ − κ)/2 and we introduce a rescaled hopping constant Λ ≡ 4J/(γ + 2Γ − κ) and a cooperativity \({\mathcal{C}}\equiv 2\Gamma /(\gamma +2\Gamma \kappa )\) defined analogous to^{20}. \({\mathcal{C}}\) is the ratio between the nonlocal dissipative contributions Γ in Eq. (3) and the overall onsite decay rate (γ + 2Γ − κ)/2. We refer to these two terms as nonlocal and local dissipation, respectively.
With these definitions, the generating function (12) obtained from Eqs. (3) becomes
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 nonzero constant. Figure 2a and b illustrates h(k) in the complex plane in topologically nontrivial 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 welldefined in the presence of dissipation. The imaginary part of h(k) in Eq. (14) takes both positive and negative values, so any nonvanishing 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 nonlocal dissipative interaction needs to exceed this local contribution to include the origin within h(k), cf. Fig. 2. A nontrivial winding number therefore always requires
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 nontrivial 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 antidamping 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 \(\theta =\frac{\pi }{2},\frac{3\pi }{2}\). Endtoend amplification sets in for \({\mathcal{C}}\;> \;1\) as we enter the topologically nontrivial 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 nontrivial winding numbers coincide with directional amplification. Note that within topologically nontrivial 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 nontrivial 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 nontrivial regime, which is advantageous for a directional amplifier.
Our drivendissipative chain not only cancels the signal in the reverse direction, it also ensures that any field entering the output cavity is not backreflected and mixedin 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 property^{20,22,26}. 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 available^{56}, 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 finetuned EP to the entire nontrivial 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 nontrivial topological regimes is independent of the coherent coupling Λ ≠ 0. This changes, for the nonresonant response ω ≠ 0. Rescaling also ω accordingly, \(\tilde{\omega }\equiv 2\omega /(\gamma +2\Gamma \kappa )\), we obtain
Local and nonlocal contributions in both real and imaginary parts compete to yield a nonzero winding number. The condition for nontrivial topology reads
This amounts to shifting the two lobes ν = ±1 against each other whereby the overlapping region becomes trivial, see Fig. 3d.
Onetoone correspondence of nontrivial topology and directional amplification
We now rigorously prove the existence of a onetoone correspondence between nontrivial values of the winding number and directional amplification for generic 1D systems with nearestneighbor 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 nontrivial topological regimes, the corrections that arise from moving to OBC, lead to directional amplification by several orders of magnitudes. While we focus on nearestneighbor couplings and generating functions of the form (12), our technique can also be employed beyond nearestneighbor 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
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
Inserting this expression into Eq. (18) allows us to evaluate the sum over k. Since z = e^{ik} and k = 2πr/N takes discrete values, we can write
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
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 principle^{57} that
with
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
subtracting the corners of the matrix corresponding to PBC. To calculate the influence of this change in boundary conditions, we import the following mathematical result^{58}: The matrix inverse of the sum of an invertible matrix M and a rankone 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 \({E}_{1}={\mu }_{1}\left1\right\rangle \left\langle N\right\) and \({E}_{2}={\mu }_{1}\leftN\right\rangle \left\langle 1\right\), we obtain an analytic expression for \({\chi }_{{\rm{obc}}}={M}_{{\rm{obc}}}^{1}\). Within topologically nontrivial regimes, it simplifies to
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
and for ν = −1 corresponding to ∣z_{±}∣ > 1
with
As we show in Fig. 4c, d, the above expansions for χ_{obc} converge exponentially fast to the exact result within the whole topologically nontrivial 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
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
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 nontrivial 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
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 drivendissipative chain (3), however, the results of Eqs. (23) to (29) apply more generally to systems with nearestneighbor couplings. We can map any system with a generating function of the form (12) to the parameters of the drivendissipative 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 drivendissipative 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 Clerk^{20,22} and sketched in Fig. 5b. We consider the generalization of their twomode 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 nonlocal 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
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
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 to^{59}. 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 x_{out} = S_{x}x_{in} and p_{out} = S_{x}p_{in} 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 in^{20,22}. It couples the field quadratures via the coherent interaction λp_{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
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
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
together with onsite dissipator \(\gamma {\mathcal{D}}[{a}_{j}]\) gives rise to the following equations of motion for the system’s quadratures
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 nontrivial regimes. Defining \({\mathcal{C}}\equiv 2\Delta /\gamma \) and Λ ≡ 2J/γ, the generating functions are
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 nontrivial 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 drivendissipative 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 nontrivial values of the winding number have a directly observable consequence expressed in the scattering matrix (4), and we have established a onetoone correspondence between nontrivial topology and directional amplification. One of our main results is the ‘phase diagram’ for the scattering matrix, Fig. 3b, that associates topologically nontrivial 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 nearestneighbor couplings and have revealed an exponential scaling of the gain with the number of sites within topologically nontrivial 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 nontrivial regime. Therefore, directional amplification can be seen as a proxy of nontrivial topology, formally defined only in the limit N → ∞, even in very small systems, which makes our work relevant for stateofthe 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 circuits^{10,42}, optomechanics^{43}, photonic crystals^{28} and nanocavity arrays^{60}, as well as topolectric circuits^{44,45} and mechanical metamaterials^{61,62,63}. On a fundamental level, our analysis sheds light on the role of topology in open quantum systems^{64} and is of direct relevance for the study of nonHermitian topology^{46,48,49,50}, 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 onetoone correspondence between a nontrivial winding number and directional amplification. χ_{obc} is crucially determined by \({I}_{n}\equiv \mathop{\sum }\nolimits_{m = 1}^{0}\frac{1}{2\pi {\rm{i}}}{\oint }_{ \tilde{z} = 1}{\rm{d}}\tilde{z}\ \frac{{\tilde{{z}}^{nmN1}}}{h(\tilde{z})}\), 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
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 eigenvectors^{56}, 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 nonzero. 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 Nfold 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 drivendissipative cavity chain
Here, we discuss the stability of the drivendissipative 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 by^{56}
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 \({\mathcal{G}}(\omega )\) using^{65}. 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
and the reverse gain
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 satisfy^{66}
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
in which the ω_{j} can be determined from Eq. (50). All Lorentzians have the same width set by the effective onsite 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 gainbandwidth 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 Eq. (29).
Data availability
No datasets were generated or analyzed during the current study.
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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).
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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.
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Wanjura, C.C., Brunelli, M. & Nunnenkamp, A. Topological framework for directional amplification in drivendissipative cavity arrays. Nat Commun 11, 3149 (2020). https://doi.org/10.1038/s41467020168639
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