Introduction-Asymmetric/nonreciprocal devices such as isolators and circulators are routinely used in communications, radar and LiDAR technologies, and integrated photonic circuits at microwave and optical frequencies [1]. Their operational principle relies on the violation of reciprocity, which is typically achieved (in linear structures) by using an odd-vector bias (e. g. an external magnetic field) [2,3] or by violating the timeinvariance via a spatiotemporal modulation of the propagating medium [4][5][6][7][8][9][10][11][12][13][14][15]. Utilizing nonlinearities as a means to achieve asymmetric transport (AT) is a promising alternative approach . In this framework, an upper bound for the maximum transmittance asymmetry has been achieved based on time-reversal symmetry considerations [34]. These studies analyze AT in simple nonlinear systems [34][35][36][37] without paying attention to the geometrical complexity of the underlying structures, nor to the presence of losses that violates time-reversal symmetry. The challenge now is to develop a general theory of nonlinearity-induced AT in typical complex wave settings that lack geometric or time-reversal symmetries.

Here, we develop a general theory of AT from complex wave scattering systems without any geometric symmetries, where nonlinearities coexist with losses. We confirm our predictions by utilizing an experimental platform of wave transport in complex systems, i.e., complex networks of coaxial cables (graphs) , see Fig. 1a. Specifically, we have derived a general expression for the upper bound of the transmittance asymmetry occurring in nonlinear wave media in terms of losses and other structure-specific characteristics of the underlying linear system. The necessary conditions to exceed the transmission asymmetry bound given by the corresponding lossless analogues [34][35][36][37] are identified. Furthermore, our analysis highlights an intimate relation between the AT properties of a nonlinear wave system and the structural asymmetry factor (SAF) that is determined by the geometric complexity of the underlying linear structure. We find that SAF dictates the asymmetric intensity range (AIR) defined as the ratio of input powers injected from opposite directions which leads to the same transmittance (see Fig. 1b). We demonstrate experimentally that the presence of losses does not degrade the transmission asymmetry at all -instead, it enhances the AIR.

Experimental Implementation -A nonlinear microwave graph consists of coaxial cables (bonds) coupled together via n = 1, • • • , N Tee-junctions (vertices). For the tetrahedron graph shown in Fig. 1a, N = 4. We have chosen the length of the bonds not to be commensurable -thus avoiding spectral degeneracies. Additionally, we made sure that our choice of bond lengths allows us to measure a sufficient number of resonances in the frequency range where the nonlinearity is activated. The local nonlinearity is always incorporated at the N -th vertex. It is implemented by substituting the Tee-junction with a cylindrical resonator which is inductively coupled to a metallic ring that is short-circuited to a diode, see inset of Fig. 1a. The nonlinear resonator is designed to operate at 6.1-6.5 GHz. It is coupled with the rest of the graph via "kink" antennas (see Supplemental Material) [64]. The system is coupled to transmission lines (TL) attached to n = 1, 2 vertices of the graph. Each transmission line supports a single propagating mode and it is connected to a port of the Vector Network Analyzer (VNA).

Scattering Theory for Nonlinear Graphs-The theoretical analysis utilizes a standard open quantum graph description [44] with the modification that the N -th vertex is now nonlinear. We assume that the bond lengths l B are taken from a box distribution centered around some mean value l. The position x nm ≡ x on bond B = (n, m) is x = 0(l B ) on vertex n(m). The scattering field on each bond satisfies the Helmholtz equation agating wave with angular frequency ω, n r is the relative index of refraction of the coaxial cable, c is the speed of light, λ n is the dielectric coefficient at node n, δ nN is the Kronecker delta function, and the superscript α = 1, 2 indicates the lead from which the incident wave has been injected. The losses in the coaxial cables are modeled by a complex-valued refraction index n r , while losses at the vertices are modeled by complex λ n . The scattering field ψ (α) B (x) is expressed in terms of its value at the vertices ψ

with χ K , χ s and z 1 being complex variables.

The wavefunction at a vertex n satisfies the continuity and the current conservation relations which can be written as (see Supplemental Material)

where we have introduced the scattering vector field Φ (α) = (φ

N ) T . The two-dimensional vector I (α) has components I (α) µ = A µ δ α,µ and describes the amplitude of the incident field of the channel α that has been used to inject the wave. Finally W α,n = δ α,n is a 2 × N matrix describing the connection between the αth lead and the vertices n = 1, 2. The

incorporates information about the metric and the connectivity of the graph, where A is the adjacency matrix having elements zero (whenever two vertices are not connected) and one (whenever two vertices are connected) [44]. Finally, (M N L ) n,m = kf (|φ

Using Eq. ( 2) we find that the field intensity at the nonlinear vertex x α ≡ |φ

where the coefficients b = b({M nm }, {W α,n }) and c α = c α ({M nm }, {W α,n }) depend on the metric and connectivity of the linear graph solely, which are encoded in the matrix elements of the matrix M (see Eq. ( 3)), and on the coupling matrix W (for the precise definition refer to the Supplemental Material Eqs. (SC.18) and (SC.19), respectively). In addition, c α incorporates the information about the TL α = 1, 2 which is used to inject the incident wave. Further manipulations allow us to turn Eq. ( 4) to a cubic algebraic equation for x α which can be solved exactly using Cardano's formula (see Supplemental Material). Substituting the value of x α back in Eq. ( 2) allows us to evaluate the rest of the components of the scattering vector field Φ (α) . Specifically, the field amplitude φ (α)

where the constant q 1,2 = q 2,1 = q = q({M nm }, {W α,n }) encodes information about the structure (metric and connectivity) of the graph and the vertices where the TLs are attached (see Supplemental Material).

Evaluation of Nonlinear Transmittance -The continuity condition at the vertex n enforces that the transmitted wave has the same amplitude given by Eq. ( 5).

Consequently, the transmittance is

where

(see Supplemental Material for a generalization to complex-valued nonlinearities). Figure 1b shows the measured transmittance T 1 (T 2 ) from TL 1 (2) to TL 2 (1) for a fixed frequency as a function of the input power. We find a strong nonlinear dependence of the transmissions on the input power.

Transmittance Asymmetry -Equation ( 6) indicate that two waves, with the same amplitude A 1 = A 2 and wavenumber k, that are injected from different ports α = 1, 2 can lead to

The latter occurs when the roots x α of Eq. ( 4) that describe the field intensities at the nonlinear vertex, differ for α = 1, 2 due to the dependence of c α on the incident TL α (see rhs of Eq. ( 4)). This non-reciprocal response does not require any form of external bias: the excitation field itself acts as a bias and triggers the system into a "high-transmission" or "low-transmission" state depending on the incident TL. In Fig. 1c we show the measured transmission difference ∆T = T 2 -T 1 as a function of the input power and frequency ν. These measurements compare nicely with the results from the graph modeling Eqs. (2,3), see Fig. 1d. More refined modeling that takes into consideration the resonant nature of the nonlinear vertex provides an even better description of the asymmetric transport, see Fig. 1e. We will refer to it as resonant-graph modeling (see Supplemental Material).

Further analysis of Eq. ( 4) allows us to identify the amplitude range for which asymmetric transport occurs. Specifically, from the right-hand-side of this equation we conclude that the scattering field intensity x α at the nonlinear vertex is the same for a left (α = 1) and a right (α = 2) incident waves as long as they satisfy the relation

The latter equality shows that the field intensity x α at the nonlinear vertex (and therefore the nonlinear electric potential) for injected waves from port 2 is equal to the one from port 1, if the input power from port 1 is SAF ≡ c2 c1 2 times larger than that from port 2. This condition leads to the same transmission coefficients for waves injected from different ports. The ratio of these input powers that lead to the same transmission defines the AIR ≡ max A1 A2 2 ; A2 A1 2 (see Fig. 1b). Within the AIR, the graph largely breaks Lorentz reciprocity, since the transmission levels in opposite directions are different for the same input power and frequency. It follows that AIR = SAF.

Bounds for Transmission Asymmetry-The maximum transmittance can be used as an upper bound for the transmission asymmetry since T ≥ 0 in all cases and, therefore, ∆T max = T max -T min ≤ T max .

From Eq. ( 6) we derive an upper bound for the transmittance by maximizing T α with respect to X α . For realvalued nonlinearities we have

where Λ = c1c2 qk ( b k ) (for a more general case of complex nonlinearities see Supplemental Material). Equation ( 7), together with Eq. (SD.10) of the Supplemental Material, are the main results of this paper. They provide guidance on the dependence of AT on the parameter Λ which encodes the structural characteristics of the graph.

The special case of lossless graph is retrieved from the above expression for (Λ) = 1 (see Supplemental Material). In this case, Eq. ( 7) simplifies to

This expression is nicely confirmed from our numerical data for a lossless graph with Kerr (open blue circles) and saturable nonlinearities (not shown) in Fig. 2a. A further investigation reveals that there is an interlinked relation between the maximum transmittance achieved for a specific incident power and the SAF (or equivalently of the AIR [34,36,37]). This is reflected in the three examples shown in the inset of Fig. 2a, where we report the transmittances T 1 , T 2 associated with the same incident wave being injected from channels α = 1 and α = 2, respectively, versus the incident power. We find that an increase in the AIR (=SAF) is associated with a decrease in the maximum transmittance and vice-versa as expected by Eq. ( 8).

Necessary conditions for the violation of an upper bound for asymmetric transport-Equation ( 8) has been previously derived as the upper bound of nonlinear AT. Its derivation assumed nonlinear Fano resonators with time-reversal symmetry (i.e. no losses) and has utilized the coupled-mode theory (CMT) framework [34,36,65]. Here, however, we have derived Eq. ( 8) for a generic nonlinear wave system, where SAF explicitly refers to specific bulk asymmetries pertaining to the topology and metrics of the graph. Given the technological importance of AT, it is natural to investigate and establish (necessary) conditions which enforce the violation of Eq. ( 8) and allow for an enhanced AIR (for a fixed T max ) or enhanced transmission asymmetry bound (for a fixed AIR) given by Eq. ( 7).

As discussed above, Eq. ( 8) does not hold when losses are introduced in the system. However, the lossy elements need to be strategically placed either on the bonds of the graph or at vertices not connected to the two TLs or the nonlinear vertex, i.e. n loss = 1, 2, N (see Supplemental Material). In the opposite case of losses located at the nonlinear vertex, a simple renormalization of the nonlinearity (so that it incorporates the lossy term) results in an upper bound given by Eq. ( 8). Similarly, when the losses are implemented on a vertex connected to the TLs, a new bound is found which is a stricter version of Eq. ( 8) (see Supplemental Material). We find that the interference between (at least) two nearby resonance modes can result in a of Eq. ( 8) as in the case of AT due to the presence of a magnetic field [66] (see Supplemental Material). Finally, from Eq. ( 7) we speculate that if (Λ) < 1, the lossy graph configurations might violate the lossless bound given by Eq. (8). Detailed numerical analysis has confirmed that the above inequality is a necessary but not sufficient condition for violating the lossless limit (see Supplemental Material).

A numerical example where the violation of Eq. ( 8) occurs for a tetrahedron graph with losses at the vertex n 0 = 3, is shown in Fig. 2b. Such targeted arrangement of loss is effectively equivalent to a new graph configuration where a third (fictitious) channel is attached to the node n 0 , thus changing the topology of the graph and indirectly affecting the coupling between this vertex and the other vertices. While Eq. ( 8) is violated for intermediate values of loss, it is still respected in the two limiting cases of zero and very large losses at the n 0 -vertex. The second limit is understood as an impedance-mismatch phenomenon: due to the large imaginary "electric potential", the n 0 -vertex is decoupled from the rest of the graph, which now acts as a lossless system with N -1 vertices and thus again satisfies the bound of Eq. ( 8). In Fig. 2b we demonstrate the trajectory of the maximum transmittance versus AIR as the losses at the vertex n 0 = 3 of a tetrahedron graph increase. The numerical data (light blue cycles) for T max nicely match the theoretical results (dashed black line) of Eq. ( 7), indicating that the deterioration of T max for increasing losses occurs at a slower rate than the enhancement of AIR. At some loss-strength, the AIR reaches its maximum value. Further increase of loss results in a decrease (increase) of AIR (T max ) towards its "impedance-mismatch" limit.

In Fig. 2c we report our measurements (purple triangles) for the graph of Fig. 1, with uniformly distributed losses at the bonds of the graph. A violation of Eq. ( 8) is evident and it is qualitatively supported by our simulations (green cycles) using a resonant-graph modeling for the same bond configurations. The difference between the simulations and the experiment is primarily due to dispersion in the real and imaginary part of the refraction index and other loss-sources, such as the T-junctions, which have not been taken into account. The insets in Fig. 2c report the experimental transmittances T 1 , T 2 for two cases with the same SAF-the upper one exceeds the bound, while the lower case corresponds to a configuration that respects the bound (see black arrows).

All the above conclusions have been further confirmed by a nonlinear random matrix theory (NLRMT) which can model typical nonlinear chaotic cavities (see Supplemental Material). An example of the violation of Eq. ( 8) for overlapping resonances is shown in Fig. 3.

Conclusions -Using a generic nonlinear complex wave system, we have established experimentally and theoreti- cally, an exact expression for the upper bound of transmittance asymmetry in the presence of losses. We have identified necessary conditions for enhanced asymmetry intensity range (for a fixed T max ) or enhanced transmission asymmetry bound (for a fixed AIR). It will be interesting to extend these studies toward a universal statistical description of transmission asymmetries using NLRMT.

Another interesting direction would be the derivation of the bounds Eqs. (7,8) in the case of microwave graphs using semiclassical tools. This will allow us to unveil the influence of non-universal features (e.g. wave-scars), which are present in any typical dynamical system, to the nonlinearity-induced AT.