Topological quantum fluctuations and travelling wave amplifiers

It is now well-established that photonic systems can exhibit topological energy bands; similar to their electronic counterparts, this leads to the formation of chiral edge modes which can be used to transmit light in a manner that is protected against back-scattering. While it is understood how classical signals can propagate under these conditions, it is an outstanding important question how the quantum vacuum fluctuations of the electromagnetic field get modified in the presence of a topological band structure. We address this challenge by exploring a setting where a non-zero topological invariant guarantees the presence of a parametrically-unstable chiral edge mode in a system with boundaries, even though there are no bulk-mode instabilities. We show that one can exploit this to realize a topologically protected, quantum-limited travelling-wave parametric amplifier. The device is naturally protected both against internal losses and back-scattering; the latter feature is in stark contrast to standard travelling wave amplifiers. This adds a new example to the list of potential quantum devices that profit from topological transport.

The quantization of the electromagnetic field introduces a fundamentally new phenomenon into physics: vacuum fluctuations that permeate all of space. These fluctuations were initially seen as a basic unalterable feature of space-time, before it was realized that they could be engineered to great effect. Simply modifying geometric boundary conditions changes the size of the fluctuations as a function of position and frequency, leading to phenomena such as the Purcell enhancement of spontaneous emission. The introduction of nonlinear optical materials gives rise to an even greater level of control, leading to the possibility of squeezed vacuum states [1], with important applications to sensing beyond the limits usually set by quantum mechanics [2][3][4].
In recent years, new approaches for altering the dynamics of wave fields have gained prominence, based on engineering periodic materials to elicit topological properties. Topologically protected unidirectional wave propagation was originally discovered in the study of 2D electrons in strong magnetic fields, and underlies the robust quantization of the Hall conductance [5]. The engineering of topological photonic materials has been the focus of intense theoretical investigation [6], and various experimental platforms have already been developed [7][8][9][10]. Phononic topological states have also attracted recent attention [11][12][13][14][15] and the first experimental steps at the macroscopic scale have been taken [15][16][17].
Despite this considerable work in topological photonics and phononics, using topology to address the engineering of vacuum fluctuations has not been addressed. Most photonic and phononic topological systems are based on a single particle Hamiltonian which conserves particle number. These topological states mimic well known electronic topological phases such as the Quantum Hall phase [7,13,[18][19][20][21][22][23][24][25], or the spin Hall phase [10,16,[26][27][28] and have a trivial vacuum. In order to modify the properties of the vacuum one has to introduce particle nonconserving terms to the Hamiltonian which can coher-ently add and remove pairs of particles from the system; these terms have a formal similarity to pairing terms in the mean-field description of a fermionic superconductor. If the amplitude of these terms is sufficiently weak, the system remains stable; even in this regime, the bosonic nature of the particles makes the topological properties of such Hamiltonians very different from their fermionic (topological superconductor) counterparts [29][30][31][32]. An even starker difference occurs when the parametric terms lead to dynamical instabilities [33,34]. These instabilities have no fermionic analogue, and are akin to the parametric instability in an oscillator whose spring constant is modulated in time.
Here, we consider a situation where parametric driving is introduced to a system where photons hop on a lattice in the presence of a synthetic gauge field (see Fig. 1a). We show how to realize an exotic situation where all bulk modes are stable, but where the topologically-protected chiral edge modes that exist in a system with a boundary are unstable. This leads to an unusual spatially-depedent modification of vacuum fluctuations: when the system is stabilized by dissipation, quantum fluctuations in the bulk are only weakly perturbed, whereas those along the system edge are strongly distorted. The result is not just an unusual driven-dissipative quantum state, but also a unique kind of photonic device: as we show in detail, the system serves both as a topologically-protected, nonreciprocal, quantum-limited amplifier, as well as a source of chiral squeezed light. It thus represents a potentially powerful new kind of application of topological materials.

I. IDENTIFYING UNSTABLE MODES
Before delving into the details of our proposal, it is useful to discuss the underlying theoretical ideas in a general setting. Our main goal is to exploit topological features of a dynamically unstable Hamiltonian, adding dissipation to realize a non-thermal steady state. In the absence of topological considerations, this is a situation that is ubiquitous in quantum optics. The simplest bosonic Hamiltonian exhibiting instability is the single-mode squeezing Hamiltonian:Ĥ whereâ is a bosonic annihilation operator. Heuristically, H S describes photons in a single cavity mode (effective energy ∆) which are subject to coherent two-particle driving (with amplitude ν). Without dissipation,Ĥ S becomes unstable and cannot be diagonalized when the driving amplitude exceeds the energy cost for creating a pair of photons, i.e. when ν > |∆|. In this regime, the dynamics corresponds to an ever-growing, exponential accumulation of entangled pairs of bosonic particles: there is no stationary state.
If we now add dissipation, stability can be restored by offsetting the effective two-particle driving described bŷ H S against the decay rate κ of the mode; one requires κ > 2 √ ν 2 − ∆ 2 . The result is a non-thermal stationary state having a steady flux of excitations flowing from the driven mode to the dissipative bath (which could be a waveguide serving as an input-output port). This is precisely the situation realized in a standard parametric amplifier: the linear-response properties of this drivendissipative steady-state allow for quantum-limited amplification of an additional signal drive. The required two-photon driving in Eq.(1) is generically realized by using a nonlinearity and parametric down-conversion of a driven pump mode.
With these preliminaries, we now consider a very general quadratic Hamiltonian describing bosons on a lattice subject to parametric driving: (2) Here the ladder operatorâ k,s annihilates a boson with quasimomentum k in the state s, where s, s = 1, . . . , N label polarization and/or sublattice degrees of freedom. The first set of terms describes the hopping of photons on the lattice, and explicitly conserves both particle number and quasimomentum. It could be diagonalized to yield a standard band structure: for each quasimomentum k, we would have N band eigenstates. The second set of parametric driving terms break particle number conservation, and in general also break the conservation of quasimomentum: the two-photon driving injects pairs with a net quasimomentum k p , implying that quasimomentum is only conserved modulo k p . For a realization based on a driven χ (2) medium, the two-photon driving terms correspond to the down-conversion of pump photons with quasimomentum k p into a pair of photons with quasimomenta k and k p − k, respectively. Having a nonzero quasimomentum for injected pairs will be a crucial resource when we attempt to control parametric instabilities.
Analogous to the simple Hamiltonian in Eq.(1), the lattice Hamiltonian in Eq. (2) exhibits instabilities when the amplitude for creating a pair of photons exceeds the energy of the pair. Formally, one can introduce a generalized normal mode decomposition of this generic Hamiltonian which explicitly separates out stable modes and unstable modes. One obtains [see Appendix A]: For each quasimomentum k in the first Brillouin zone, we will now have both a set of stable modes (indexed by n ∈ S k ), and a set of unstable modes (n ∈ U k ). The first n-sum in Eq.(3) describes the stable modes: they are described by canonical bosonic anihiliation operatorsβ n,k , and enter the Hamiltonian in the standard manner, as a real energy times a number operatorn n,k =β † n,kβ n,k . The unstable modes can also be described by canonical bosonic anihiliation operatorsβ n,k . They however enter the Hamiltonian via unstable two-mode squeezing (parametric amplifier) Hamiltonians: where E n,k = −E n,kp−k and λ n,k = λ n,kp−k are both real. This effective Hamiltonian for the unstable modes has a simple interpretation: pairs of quasiparticles with opposite energies ±E n,k are created with an amplitude λ n,k . We stress that for any non-zero λ n,k ,Ĥ n,k is unstable (as the total energy for creating the relevant pair of excitations is always zero). The quasiparticle operatorŝ β † n,k in Eqs. (3,4) are a complete set of Bogoliubov ladder operators.

II. PARAMETRICALLY DRIVEN HOFSTADTER MODEL
Having established the necessary theoretical framework, we will now show how to engineer a Hamiltonian whose only unstable Bogoliubov modesβ † n,k are chiral states co-propagating along the physical boundary of a topological system. We consider photons hopping on a 2D square lattice in the presence of a synthetic magnetic field flux, which are also subject to parametric twophoton driving on each site (see Fig. 1(c)). Writing the Hamiltonian in the position basis, we havê whereâ j is the photon annihilation operator on site j = (j x , j y ), and ω j are the corresponding onsite energies. As usual, the synthetic gauge field is encoded in the pattern of phases φ ij of the nearest-neighbor hopping rates, J ij = J exp(iφ ij ). We take the synthetic flux per plaquette to be Φ = π/2. Working in the Landau gauge, we then have φ ij = 0 for vertical hopping and φ ij = −πj y /2 for rightwards hopping. The parametric driving amplitude on a given site j is written νe iθ j ; we take the phase to vary as θ j = k p j x , implying the injection of pairs with a quasimomentum k p = k p e x . For a realization based on a driven nonlinear medium, ν ∝ I p where I p is the power of the pump laser, and k p e x would be the quasimomentum of the pump laser photons. Note that a gauge transformationâ j =â j exp[if j ] would modify both pattern of phases φ ij and θ j .
When the laser is switched off, ν = 0, and there is no disorder, ω j = ω 0 , we have the well known Hofstadter model [35]. The band structure of a semi-infinite strip (extending to the lower-half 2D-plane) is shown in Fig.  2a. The continuous bulk band structure consists of four bands (one for each site in the magnetic unit cell). The top and bottom bands are flat Landau levels separated from the two central bands by topological band gaps. Because of the boundary, one finds inside each topological band gap an edge state (dark line). The net number of these edge states (the number weighted by the sign of their slope) is a topologically protected quantity which does not depend on the shape of the edge and can be calculated from the bulk Hamiltonian [36].
We now turn on the parametric driving such that the resulting Hamiltonian can exhibit instability. Our goal is twofold: we want the system to be unstable only if we have a boundary, and in this case, the dominant unstable Bogoliubov modes should be chiral excitations localized at the system's boundary. We do this by choosing the parametric drive parameters so that the only pairs of photons that can be created in an energy and quasimomentum conserving fashion correspond to edge state excitations of the original (ν = 0) model. For concreteness, we will focus on exciting the edge mode in the lower topological band gap (dispersion ε E (k)). In the lab frame, we will thus tune the pump photon frequency ω p and quasimomentum k p so that a single pump photon can be converted into two edge excitations with frequency ω p /2 and quasimomentum k p /2. In the rotating frame we use to write our Hamiltonians, this requirement reduces to ε E (k p /2) = 0. If this resonance condition is met, an arbitrarily weak parametric drive ν will cause instability of the edge mode. The required tuning is shown in Fig. 2a.
Because of the approximately linear dispersion relation of the edge mode, the above tuning guarantees that the parametric driving can resonantly create a pair of edge mode photons having momenta k p /2 ± δk, see the hollow circles in Figure 2(b). Thus, even for a weak parametric drive amplitude, the edge state will exhibit instability over a range of quasimomenta near k p /2 (corresponding to a finite bandwidth around ω p /2 in the lab frame), see Figure 2(c).
Conversely, the energies of two bulk excitations always This guarantees the bulk stability (even in the presence of disorder) for a driving amplitude ν below the minimal value of the bulk detuning.

III. TOPOLOGICAL NON-RECIPROCAL AMPLIFIER
Having shown how to realize an unstable topological edge mode, we now want to understand how one can use it. More precisely, we show that a finite size array of nanocavities coupled to simple waveguides can be used as a new kind of topologically-protected, phase-sensitive, quantum-limited amplifier. The role of the waveguides is two-fold: they serve as amplifier input-output ports and they stabilize the dynamics.
We consider a realization of our system using a 30 × 12 array of nanocavities, and additionally include three coupling waveguides. Each waveguide is coupled to a site at the edge of the sample, as shown in Fig. 3a-b. This coupling is described using standard input/output theory, and is entirely characterized by the three rates κ in , κ out , and κ sink , see Appendix C. In addition, we take each cavity to have an internal-loss decay rate κ.
When the small decay rate κ is neglected, and without parametric driving, the array can be operated as an ideal circulator where a signal from any waveguide is entirely transmitted into the next waveguide, see Figure 3c. Indeed, it is always possible to match the impedances at each port to cancel the back-reflection by tuning the corresponding coupling rate (κ in , κ out , or κ sink ). Once inside the array a wave in a topological band gap has no alternative but to chirally propagate along the edge. In addition, the impedance matching ensures that a wave impinging on a waveguide from the edge channel will be entirely transmitted.
We harness the robust non-reciprocity of this topological circulator to design an amplifier. We use the waveguide on the upper right (left) as input (output) port of the amplifier. We choose parametric driving parameters similar as in Fig. 2. In the finite geometry, the quasimomentum matching will be approximately realized only on the upper edge. Thus, the amplification occurs mainly in the region between the input and the output port.
The linear response of the amplifier is investigated numerically in Fig. 3. A signal injected into the array from the input port propagates chirally until it leaves the array through the output port, see Fig. 3a. Depending on its phase, it can be amplified or de-amplified along the way. Treating the amplifier as a phase-sensitive amplifier, we find that the power gain for the amplified signal quadrature is flat over a large bandwidth, corresponding to the frequency range over which the edge state dispersion is purely linear (see panel d). At the same time, any signals incident upon the output port will be almost entirely dumped into the sink port, and not reach the input port, see 3b. The residual reverse gain and input reflection are much smaller than unity, see panel e, ensuring the protection of a potentially fragile signal source (e.g. a qubit). Crucially, this strongly non-reciprocal amplifying behavior is of topological origin and is thus robust against disorder. We demonstrate this resilience by including moderate levels of disorder in our simulations (see Fig. 3).
Our numerical results are in qualitative agreement with analytical results for a model in which a 1D chiral edge state is coupled to three waveguides, see Appendix D. In this context, we find simple expressions for the impedence matching condition and the maximum power gain respectively. Here, u(j y ) is the transverse edge state wavefunction, v is the edge state velocity and L is the number of sites separating the input and the output ports. Thus, we see that the gain is the exponential of the rate 2ν of creation of down-converted pairs times the time of flight L/v from the input to the output port. While we have focused on operation as a phase-sensitive amplifier, for frequencies different from ω p /2, one could also use the device as a quantum-limited phase preserving (i.e. non-degenerate) amplifier.

Quantum-limited amplification
The noise floor of the amplifier is the frequencyresolved noise of the amplified output quadrature, see Appendix C. It is plotted in dark (light) red for a clean (disordered) sample in Fig. 4a. The quantum limit on a phase-sensitive amplifier is to have zero added noise, implying that the noise floor is simply set by the amplification of the vacuum fluctuations entering the input port. The added noise (expressed as an equivalent number of input quanta) is plotted in Fig. 4b; despite disorder and noise associated with internal loss, the amplifier is nearly quantum limited over the entire amplification bandwidth.

IV. TOPOLOGICAL STATIONARY STATE
Next, we investigate the quantum fluctuations in the stationary state that arise from the steady flow of photons from the parametric pump to the amplifier ports in the form of down-converted radiation. Physically, such a flow arises because vacuum fluctuations entering the input port (within the amplification bandwidth) are amplified inside the array before exiting through the output port. The resulting stationary state of each cavity is Gaussian, and can be represented by a Wigner function; these are visualized as a set of noise ellipses in Fig.  4c. Each noise ellipse completely characterizes the steady state of the corresponding site once all remaining sites and the waveguides have been traced out. The areas of the ellipses are constrained from below by the Heisenberg uncertainty principle and assume the minimal possible value for pure states. The bulk sites are all in the trivial quantum groundstate, which is characterized by circular ellipses with the minimal area and zero photons. On the other hand, the eccentricity, the area and the average onsite photon number (color code) grow while moving from the input to the output port along the upper edge (the major axis corresponds to the amplified quadrature). We emphasize that in a thermal equilibrium setting, the area of the ellipses would be equal on all sites and directly reflect the environment temperature. Here, the excess noise of the sites on the upper edge has a quantum origin. This phenomenom has been termed quantum heating [37][38][39].
Due to quantum heating, the stationary state of each cavity along the edge corresponds to a thermal squeezedn state, implying that the squeezed quadrature has a larger variance than required by the uncertainty principle. Nonetheless, the frequency-resolved output noise is strongly squeezed below the vacuum level (for frequencies within the amplification bandwidth). Remarkably, the quality of the output squeezing is not deteriorated in the presence of disorder. For the parameters considered here, more than 15dB of output squeezing are predicted both in the case of a clean and a disordered sample, see the blue curves in Fig. 4a.

V. IMPLEMENTATION
Photonic gauge fields have been already realized in several experimental platforms [7,9,10]. The only additional ingredient of our proposal is the parametric pumping with a finite quasimomentum. In the setup of Ref.
[10], one could fabricate the microrings from a nonlinear optical χ (2) material and drive them with a single laser impinging at a finite angle. The implementation of parametric pumping [32] and synthetic gauge fields [13,20,22,25,32] is in principle possible in any cavity array platform where a nonlinear resource is available. These include photonic crystals microcavities [40] fabricated from nonlinear optical χ (2) materials [41][42][43] or piezoelectric materials, optomechanical arrays based on optomechanical crystals [44,45], or lattices of superconducting resonators with embedded Josephson nonlinearities [46]. We also note that very recently, a proposal for realizing topological insulator physics in a classical optical network with nonlinearities was put forward [47]; such a setup could also be adapted to implement our scheme, as it contains all the necessary ingredients.

VI. CONCLUSIONS AND OUTLOOK
In this work, we have introduced a means to tie the squeezing and amplification of vacuum fluctuations to topological properties of a band structure. Our work represents a new design principle for a non-reciprocal quantum-limited amplifier which has topological protection. Non-reciprocal amplifiers have the potential to revolutionize experiments with superconducting qubits, as they could eliminate the need for ferrite-based circulators and the accompanying insertion losses which limit current experiments. A variety of (non-topological) designs based on multiple parametric interactions have been proposed recently [48,49] and even realized experimentally [50,51], including a travelling wave parametric amplifier (TWPA) built using an array of over 2000 Josephson junctions [52]. In such a conventional TWPA, the reverse transmission is at best unity, and even small amounts of disorder can lead to large amounts of unwanted reflection gain. In contrast, the topological underpinnings of our design ensure reverse transmission and input reflection coefficients that are well below unity even in the presence of disorder.
More generally, our topological amplifier differs markedly from other proposed topological devices such as isolators or non-amplifying circulators [6], in that it has some protection against internal losses: in the large gain limit, only the loss (and corresponding noise) in the immediate vicinity of the input port hinder quantum limited operations, as it is only this noise which is amplified to any significant degree (see Appendix E for a quantitative discussion of this point).
In conclusion, our work shows how utilizing the topological properties of an unstable bosonic Hamiltonian provides a new route for both engineering electromagnetic vacuum fluctuations, and correspondingly, constructing a new kind of topologically-protected, nonreciprocal quantum amplifier. It opens the door to future studies, both pursuing other kinds of novel applications, as well as more fundamental issues (e.g. the effects of additional photon-photon interactions in such systems).
We consider the generic Hamiltonian Eq.
(2). We group all ladder operators with quasimomentum k in a vector of ladder operators, |â k = (â k,1 , . . . ,â k,N ,â † kp−k,1 , . . . ,â † kp−k,N ) T . The Heisenberg equation of motion for |â k reads where In the following, we explicitly construct a complete set of Bogoliubov operatorsβ n,k , leading to the generalized normal mode decomposition Eq. (3), from the solutions of the eigenvalue problem σ z h k |k n,l = Λ k,n,l |k n,l . (A2) We note that we need to find only N annihilation operators while the eigenvalue problem has dimension 2N . However, the equations for quasimomentum k p − k are not independent from the equations for quasimomentum k. The ones can be obtained from the others by taking the adjoint. This doubling of the degrees of freedom accompanied by an embedded particle-hole symmetry occurs because we are effectively applying a single-particle formalism to a problem where the number of excitations is not conserved. From Eq. (A2) it is easy to prove that eigenvalues which are not related by complex conjugation Λ n,l,k = Λ * n ,l ,k have σ z -orthogonal eigenvectors, k n,l |σ z |k n ,l = 0. Moreover, the eigenvectors with real eigenvalues have a non-zero symplectic norm, k n,l |σ z |k n,l = 0 (which can also be negative). We assign the label l = + to the positive norm eigenvectors.
We construct a set of orthonormal Bogoliubov creation operators from these positive norm solutions with the definition,β n,k ≡ k n,+ |σ z |â k . (A3) We note that the scalar product between a standard vector and a vector of operators is an operator. Moreover, we have to normalize the positive vectors |k n,l according to k n,+ |σ z |k n,+ = 1 such that [β n,k ,β † n ,k ] = δ nn . By taking the time derivative of Eq. (A3) and plugging Eq. (A1) and the adjoint of Eq. (A2) we immediately finḋ β n,k = −i k n,+ |σ z σ z h k |â k = −iΛ n,+,kβn Thus,β n,k is the annihilation operator of a harmonic oscillator with energy E n,k = Λ n,+,k . In the same way, one could construct a set of creation operatorsβ † n,kp−k from the negative norm eigenvectors |k n,− . However, it is possible to focus only on the positive norm solutions because of the particle-hole symmetry: the information encoded in the negative norm solutions |k n,− is also encoded in the positive norm solutions |(k p − k) n,+ .
They appear whenever the Hamiltonian is unstable. In this case, the matrix σ z h k has pairs of complex conjugated eigenvalues Λ n,+,k = Λ * n,−,k . For concreteness, we indicate with the label + the eigenvalues with positive imaginary part. The pair of eigenvectors |k n,± are not ortoghonal to each other, k n,+ |σ z |k n,− = 0. In this case, we define a pair of commuting ladder operators as, The bosonic commutation relations are recovered by requiring the normalization k n,− |σ z |k n,+ = i. By taking the time derivative of Eq. (A4) and using Eq. (A1), the transpose of Eq. (A2), and Eq. (A5) we finḋ β n,k = −iRe[Λ n,+,k ]β n,k + Im[Λ n,+,k ]β † n,kp−k .
The corresponding Hamiltonian is the two-mode squeezing Hamiltonian (Eq.(4)) with energies E n,k = −E n,kp−k = Re[Λ n,+,k ] and amplification amplitude λ n,k = Im[Λ n,+,k ]. When the matrix σ z h is diagonalizable, the set of Bogoliubov annihilation operators defined in Eqs. (A3) and (A4) is complete. The pathological case where the matrix σ z h is not diagonalizable occurs only exactly at the threshold of an instability.
Appendix B: Details of the calculation of the band structure

Stability of the bulk Hamiltonian
In the main text, we have explained that a bosonic Hamiltonian with anomalous pairing terms is unstable when it allows the creation of a pair of Bogoliubov excitations without any net energy change. In other words, the sum of two quasiparticle energies should be zero. We have also claimed that for the parameters of Figure 2 the bulk Hamiltonian is stable because no combination of bulk states whose energies add up to zero exists. This is not immediately obvious from the plot of the standard band structure. In order to visually illustrated the absence of such combination of bulk states one has rather to plot the corresponding Bogolibov de Gennes band structure, see Fig. 5. For each value of k x , the latter is formed by both the quasiparticle energies E n,kx (blue bands) as well as the energies −E n,kp−kx (grey bands). In analogy to the descriptions of quasiparticles in fermionic superconductors, one refers to the former (latter) as the particle (the hole) bands of the system. The absence of crossings between particle and hole bands for the chosen pump laser frequency implies that there is no combination of a pair of bulk quasiparticles whose energies add up to zero while at the same time the corresponding quasimomenta add up to k p . In the limit of vanishing parametric driving (ν = 0), the minimal distance between particle and hole bands is the minimal detuning of a parametric transition involving two bulk states.
In the presence of weak disorder, there is no selection rule for the quasimomenta of the pair of quasiparticles created in a parametric transition. Nevertheless, all possible parametric transitions are still detuned because the band gap separating particle and hole bands is not merely locally defined (for a fixed k x ) but rather extends to the whole Brillouin zone. Thus, the stability of the bulk Hamiltonian is a robust feature.
As explained in Appendix A, the Bogoliubov de Gennes band structure can be calculated by diagonalizing the matrix σ z h(k x ) where h(k x ) is the first-quantized Bogoliubov de Gennes Hamiltonian equivalent to the second quantized Hamiltonian Eq. (5) of the main text in the presence of periodic boundary conditions both in the x-and y-directions.
Band structure of a strip Figure 2 of the main text represents the band structure and the amplification amplitudes of a semi-infinite strip. Those have been derived from the the Bogoliubov de Gennes band structure and the amplification rates of a strip with two physical edges as explained below.
We have simulated a finite strip of width M = 40 magnetic unit cells. For each value of the quasimomentum k x , the set of energies E n,kx and −E n,kp−kx forming the BdG band structure and the corresponding amplification rates λ n (k x ) are calculated by diagonalizing the relevant 4M × 4M matrix σ z h(k x ). While strictly speaking the energy spectrum is discrete, the spacing between subsequent bulk states is not visible on the figure scale. The resulting band structure [panels (b-c)] and the corresponding amplification rates [panel (d)] are shown in Fig. 5. By inspecting the corresponding wavefunctions one can easily distiguish between particle and hole bands (plotted in blue and grey, respectively), edge and bulk states (plotted by the thick dark and thin light lines, respectively) and upper-and lower-edge states (solid and dashed lines, respectively).
Since the hole bands are not independent from the particle bands and refer to the same set of normal modes, we have not displayed the hole energies in the plots in the main text. Moreover, we have shown only the edge states localized on the upper edge. Removing the edge states localized on the lower edge effectively corresponds to plotting the band structure of a semi-infinite strip extending to the lower half plane. We note that the particle (hole) edge states cross the hole (particle) band. Thus, a pair formed by an edge and a bulk excitation can in principle also be resonantly excited. However, the corresponding matrix element is very small and the resulting amplification rate is not visible on the scale of the figure.

Analytical derivation of Edge state dispersion and amplification rate
Here, we outline a direct derivation of the edge state dispersion for a semi-infinite strip.
Since, the strip extends to the lower-half plane the j y -th sum runs over the negative integers. By definition of the normal modesα n,kx and eigenenergies E n [k x ] we haveĤ = n,kx E n [k x ]α † n,kxα n,kx , or equivalently [α n,kx ,Ĥ] = E n [k x ]α n,kx . By plugging the ansatzα † n,kx = jy u n,kx [j y ]â † jy,kx , we arrive to the firstquantized time-independent Schrödinger equation which we set in the matrix form, Here, we have defined n [k x ] = (E n [k x ] − ω 0 )/J. In principle, the above equation holds only for j y ≤ −2 but not for j y = −1 because there is no row corresponding to j y = 0. One can circumvent this problem by formally adding the row j y = 0 together with the boundary condition u n,kx [0] = 0. Thus, one immediately finds Next, we focus on the edge state solutions. In the following, we drop the subscript n because there is only one edge state in each band gap, see below. For the edge states we have to enforce the boundary condition Equivalently, one can require where M ( [k x ]) is the transfer matrix by a full magnetic unit cell (four sites), is fullfilled if and only if the vector (1, 0) is an eigenvector of the 4-site transfer matrix M ( [k x ]) whose eigenvalue has modolous smaller than unity. In other words, the edge state dispersion is determined by the conditions, Equation (6) is a third order polynomial in the dimensionless energy . Thus, it has three roots for each value of k x . Each root corresponds to a solution inside one of the three band gaps (lower, middle, or upper). By plugging the analytical expression of each root of Equation (6) into Equation (7) we find the range of k x where the corresponding edge state is defined: the interval −π < k x < 0 for the upper edge state, 0 < k x < π for the lower edge state, and the two intervals −π/2 < k x < 0 and π/2 < k x < π for the edge state in the local band gap between the two central bands. In order to enforce the bosonic commutation relations [α kx ,α † kx ] = 1, we have to appropriately normalize the single-particle wavefunctions jy |u kx [j y ]| 2 = 1. This condition fixes the modolous of the wavefunction on the initial site u kx [−1]. The complex phase of u kx [−1] is arbitrary. For concreteness, we choose u kx [−1] to be real.
Next, we show how the edge state is modified by the parametric pump. The pump drive written in terms of the annihilation operatorsâ jy,kx reads, jy,kp−kx −â jy,kxâjy,kp−kx .
(B8) Before pursuing an exact numerical solution we adopt a semi-analytical treatment. We focus on the lower band gap edge state which is resonantly driven for the parameters of Fig. 2 of the main text. As in the main text, we consider the case where the frequency and quasimomentum of the pump laser are chosen to resonantly excite pair of edge photons with quasimomentum k p /2, In this case, it is convenient to introduce the quasimomentum δk = k x −k p /2 counted off from k p /2. By rewriting the full Hamiltonian (including the pump) in terms of the edge states ladder operatorsα δk and keeping only the terms that acts on the lower edge state (we neglect an off-resonant parametric coupling to the bulk modes and the other edge states), we find Here, we have grouped all excitation conserving terms into two contributions whose amplitudesẼ n (δk) and ∆ n (δk) are an odd an even function of δk, repectively, Note that ∆(δk) is the quasimomentum dependent detuning of the parametric transition creating pairs of photons with quasimomentum k p /2 ± δk. The corresponding parametric coupling is given by It is easy to show that it can be expanded in terms of even powers of δk and that the leading order is ν The edge state is unstable over the finite quasimomentum interval where V (δk) > |∆(δ)|. Following the general procedure presented in Appendix A, we find the eigenenergies and amplification amplitude of the Bogoliubov edge state normal modes in the unstable and stable quasimomentum ranges, respectively. We can recover the edge state dispersion and lineshape of the amplification amplitude obtained numerically and shown in Fig 2 of the main text by keeping the leading order contributions in Eqs. (B11,B12,B14), Appendix C: Input-output formalism We include the effects of dissipation using the standard input-output formalism [1]. Each site is described by the Langevin equatioṅ Here, κ j is the decay rate on site j. On the sites coupled to waveguides, the decay rates κ in , κ out , and κ sink are induced by the coupling to the waveguides and are chosen to achieve impedance matching. In this case, the input fieldsâ (in) j describe the field impinging on site j from the corresponding waveguide including the field vacuum fluctuations. The fieldâ (out) j leaking out of the waveguide is given by the input-output relationŝ On the remaining sites (i.e. those not coupled to iwaveguides), we assume a small uniform decay rate κ (κ j = κ) corresponding to internal loss;â (in) j describes the corresponding incident vacuum fluctuations associated with this loss port.
The linear response and the noise properties of the array are simple functions of the retarded Green's functions We calculate the above Green's functions numerically for a finite array.

Response Ellipses
In Fig. 3a (Fig. 3b) we show the linear response of the pair of quadratureŝ to a classical field of frequency ω p /2 injected at the input (output) port as a function of the field phase ϑ, j indicates the site attached to the relevant injection waveguide (j = j in for Fig. 3a, j = j out for Fig. 3b).
The response on each site is represented by the ellipse are the major and minor semiaxes of the ellipse, and is the angle parametrizing the orientation of the major semiaxis. In the figure, the reference angle γ jj = 0 is rotated by π/2 compared to the page vertical. The maximum (minimum) response r + jj (r − jj ) at site j to an input field at the port j is obtained for the driving phase ϑ = η jj (ϑ = η jj + π/2 ), η jj = π/2 − arg G E (0, j, j )G I (0, j, j ) 2. (C7) Gain, Reverse Gain, and Input Reflection As we analyze our amplifier in the phase-sensitive mode of operation, it is useful to consider its scattering properties in a quadrature representation, using an optimal quadrature basis fields in the input and output waveguides. The relevant scattering between these waveguides is described bŷ where we have omitted writing terms describing contributions from vacuum noise incident from the internal loss ports, as well as terms describing the reflection of noise incident from the output waveguide (due to slightly imperfect impedance matching). We define the quadratures of the output waveguide aŝ while the quadratures of the input-waveguide are defined asÛ We will define both the quadrature basis in each coupling waveguide so that |s X ,U [ω = 0]| is maximal. This implies that the largest amplification of a narrow-band signal centered at zero frequency (ω p /2 in the lab frame) occurs when the incident signal is in theÛ in quadrature of the input waveguide, with the amplified output being contained in the output fieldX out quadrature of the output waveguide. It also follows naturally that for vacuum noise inputs,Ŷ out will be the optimally squeezed quadrature. An explicit calculation of the scattering matrix in terms of the Green's functions introduced above shows that the angles defining these preferred input and output quadratures are given by see Eqs. (C6,C7). The fact that in general θ in = θ out is a simple reflection of the fact that our system can both perform phase-sensitive amplification of incident signals, as well as simply rotate incident signals in phase space. The frequency dependent power gain is then by definition given by the transmission probability: The reverse gain is obtained similarily by exchanging the two indices of the Green's functions in the above formula and in Eq. (C8). Finally, the input reflection probability describes the reflection of signals incident in the (ampli-fied) U quadrature of the input port

4.
Added noise and squeezing The symmetrized frequency-resolved noise in the amplified and de-amplified output-waveguide output field quadratures shown in Figure 4(a) are defined as respectively. The number of added noise quanta shown in Figure 4(b) is defined by,

Noise Ellipses
In Figure 4(c), the stationary state of each site j is represented as a noise ellipse. The noise ellipse on a particular site j is obtained by diagonalizing the corresponding covariance matrix and identifying the square root of its eigenvaluesr + j and r − j as the ellipse semi-axes and the angleγ j as the ellipse rotation angle (the reference angleγ j = 0 is rotated by π/2 compared to the page vertical). In this supplementary note we want to derive an effective quantum field theory for the edge state coupled to a waveguide. We want to model a finite system which will then support a single edge state with a ring geometry (i.e. periodic boundary conditions). We adopt the simplest possible approach valid when the input signal has a bandwidth well within the amplifier bandwidth. In this case, we can approximate the edge state velocity and amplification rates to be constant,Ẽ ≈ vδk, ∆ ≈ 0, and V (δk) ≈ ν, respectively. Moreover, we can neglect the quasimomentum dependence of the edge state transverse wavefunction. We thus arrive at the edge state ladder operator where N is the number of sites along the edge, and the indexes j and j ⊥ (j ⊥ ≥ 1) label the position in the directions parallel and longitudinal to the edges (for the edge state along the upper edge described above j = j x and j ⊥ = −j y ). Moreover, u(j ⊥ ) is the transverse edge state wavefunction, u(j ⊥ ) ≡ u kp/2 (−j ⊥ ) see Eqs. (B3,B7). The periodic boundary conditionsĉ(j ) =ĉ(N + j ) are enforced by the quasimomentum quantization δk n = 2πn/N .
We consider a waveguide attached to a single site along the edge in the logitudinal position j ≡ j ex . The coupling to the waveguide as described by standard inputoutput theory is entirely characterized by the decay rate κ. Since, the edge state is the only state within the amplifier bandwidth the coupling of the waveguide to a single site is equivalent to a direct coupling to the edge state at the same longitudinal position j ex with the renormalized decay rate κ = κ|u[j ⊥ = 1]| 2 . The resulting Langevin equation readṡ with the input-output relationŝ Next we take the continoum limit by defining the chiral edge fieldĉ where L is the edge length and the periodic boundary conditionsĉ(z) =ĉ(z + L) follow from the quasimomentum quantization δk n = 2πn/L. From Eq. (B10) with E = vδk, ∆ = 0, and V (δk) = ν, and Eq. (D2) we find the field equations (D5) We note that due to the point-interaction with the waveguide the fieldĉ(z) is not continous at the position z = z ex where the waveguide is attached,ĉ(z ex +0 + ) =ĉ(z ex +0 − ) (0 + and 0 − are infinitesimal positive and negative numbers, respectively). Keeping this in mind, the input output relation in the continous limit readŝ Before investigating the interaction with the waveguide, we first discuss the propagation inside the ring. For concreteness we consider v > 0 such thatĉ(z ex + 0 + ) (ĉ(z ex + 0 − )) is the field immediately after (before) the interaction with the waveguide. A signal travels an almost complete round trip from z = z ex + 0 + to z = L + z ex + 0 + = z ex + 0 − in the time t = L/v. During this time no interaction with the waveguide takes place. From Eq. (D5), one readily findŝ Here, we have introduced the amplified and de-amplified quadratures,X = (ĉ † +ĉ)/ √ 2 andŶ = i(ĉ † −ĉ)/ √ 2. We can conclude that a signal with the right phase during a complete round trip inside the ring experience a power gain Next, we discuss the interaction with the waveguide. An input signal from the waveguide will be partly reflected and partly transmitted into the ring at z = 0 + . Then, it will propagate inside the ring until it has completed a round trip. A signal with the right phase will be amplified along the way. Before starting a new round trip, part of the amplified signal returns into the waveguide. If the signal remaining in the ring at the beginning of the second round trip is smaller compared to the signal in the ring at the beginning of the first round trip, the signal will decay after few round trips. In this regime, the waveguide stabilizes the edge state. By integrating the Heisenberg equations (D5) close to z = 0 we find where r and t are the reflection and transmission probability amplitudes, To prevent instability, we require that the transmission t be large enough that the field in the edge-mode ring does not grow with each round trip. The simplest case is where we tune the decay rate κ so that all of the incident wave in the edge mode ends up in the coupling waveguide, i.e. r = 0, t = 1. This requires g = 2, or in terms of the decay rate κ, the edge state velocity v, and the edge state transversal wavefunction at the edge u(j ⊥ = 1), we find κ = 4v |u(j ⊥ = 1)| 2 . (D10) If this impedance matching condition is met, signals incident from the waveguide in theX quadrature will be reflected back with a power gain G = G Max independent of frequency. The impedance matching ensures that multiple traversals of the ring are impossible, which both precludes instability, but also prevents the formation of standing wave resonances and a strongly frequencydependent gain. We note that the effective model described above can be straightforwardly extended to describe a chiral edge state coupled to several waveguides. In particular, we consider a set up with two impedance matched waveguides at position z = z in and z = z out (z out − z in = L), respectively. A signal from the first waveguide entering the edge state at position z = z in is entirely transmitted into the second waveguide at position z = z out . For the appropriate phase of the input signal, we havê whereÛ (in) is the input quadrature. We denoteX out , the corresponding amplified quadrature at z = z out , X out (t) = − √ vX(z out + 0 − , t).
From Eq. (D5) one finds where the gain G is given by Eq. (D7).

Comparison between finite size simulations and effective model
In the regime where our effective theory applies (for signals well within the amplifier bandwidth), it can reproduce well the simulations of finite arrays. There are some quantitative differences due to well understood finite size effects. For instance, the edge state velocity is not constant in the finite size array but rather decreases close to the edges where the edge state propagation changes direction. For this reason the gain does not depend only on the number of sites separating the input and the output port but also on the precise position of the ports. We have placed the input and output port close to the edges to enhance the gain. Indeed, the gain in the finite size simulations is slightly larger than predicted by Eq. (D7). Due to the position dependent edge state velocity also the value of the decay rate required to obtain impedance matching depends on the position where the waveguide is attached (and is lower at the edges).
We note that the analytic continuum model can be extended to capture the frequency dependence of the gain and noise of our travelling wave amplifier; one needs however to incorporate into the model the leading-order quasimomentum dependence of the edge state velocity (which creates an effective pump detuning for the relevant parametric process). This is will be presented in a future work.