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+{"text":"\\section{Introduction}\\label{intro}\nFollowing the phenomenal growth in the study of Dirac fermions in graphene and other (quasi-) two dimensional materials~\\cite{Novoselov2005,Martino2007,Park2008,Park2008_2,Yan2008,Zhu2010,Castro2009,Borisenko2014,Sun2016,Sun_2016}, there has been growing interest in recent years in investigating bosonic analogs of the same~\\cite{Chumak2015,Fransson2016,Sun2021,Xu2016,Pershoguba2018,Cheng2007,Wu2015,Lu2016,Wang2015}. These include both fundamental particles such as photons, as well as quasiparticles such as phonons and magnons. Magnons are low-energy quasiparticle excitations in quantum magnets and have long served as a versatile testbed for realizing bosonic analogs of fermionic phases.\n\nSince magnetic properties are easily controlled by external magnetic fields, magnonic bands offer a unique platform to explore the rich and still evolving fundamentals of the band theory. So far, several topological magnon phases have been proposed and material candidates have been identified~\\cite{Zhang2013,Owerre_2016,Cai_2019,Chen2018,Cai_2021,Nguyen_2021,McClarty2022}. Some key results include the characterization of the Chern magnon bands and linear touching points -- for example, the material \\ce{Cu[1,3-bdc]} with the kagome lattice exhibits a magnetic field-dependent thermal Hall conductivity, and the material \\ce{Cu3TeO6} shows a nodal topology~\\cite{Hirschberger2015, Yuan2020}.\n\nMagnonic systems are also ideal platforms for investigating interacting bosons~\\cite{Pershoguba2018, Mook2021}. \nIn contrast to the fermionic quasiparticles, the absence of a conservation law for the magnon particle number allows for nonconserved many-body interactions such as spontaneous decay of magnons~\\cite{Mook2021, Zhitomirsky2013}. However, so far, studies of magnonic band topology have largely neglected magnon-magnon interactions, treating the magnons within the single-particle approximation~\\cite{Owerre_2016}. This is typically justified by the argument that interaction effects are frozen out at low temperatures or are negligible as a small perturbative effect. However, the single-particle picture does not generally apply, and interactions become important when the magnon density is enhanced -- either due to increased temperature, or parametric amplification (as discussed later). \nIn Dirac fermions, Coulomb interaction leads to a logarithmic renormalization of the Fermi velocity~\\cite{Elias2011}, and similar analyses for interacting magnons have shown that both velocity and energy are renormalized due to interaction in the vicinity of the Dirac points~\\cite{Pershoguba2018}.\n\nIn this work, we have investigated the effects of interactions on {\\it topological} magnons in a Heisenberg ferromagnet with Dzyaloshinskii-Moriya interaction (DMI) on a honeycomb lattice. In the absence of DMI, the single-particle magnon spectrum features two bands that touch at the Dirac points with linear dispersion. DMI opens up a gap in the spectrum and imparts non-trivial topology to the bands. We have investigated the effects of interaction on the band topology using Green's function formalism up to the second-order in perturbation theory. Crucially, we are able to systematically probe the interaction-driven renormalization of the magnon bands in different parts of the magnetic Brillouin zone, and isolate the effects of interactions from thermal fluctuations, by employing the recently developed magnon amplification scheme to controllably tune magnon density at selective energy-momentum values~\\cite{Malz2019}.   Our results show that the magnon-magnon interaction leads to a significant momentum-dependent renormalization of the single-magnon dispersion -- interactions suppress the bandwidth, and lead to dissipative scattering of the magnons. More interestingly, interaction effects, in conjunction with DMI and magnon amplification can introduce a non-trivial Haldane term (complex next nearest neighbor hopping) -- and consequently, topological phase transition -- leading to magnon bands with tunable Chern numbers. In the same vein, the Berry curvature distribution of the magnon bands can be tailored by interactions. We argue from a microscopic perspective that effective gauge fields emerge in this bipartite system through their underlying interaction, and propose that a magnon-mediated anomalous transport - the thermal Hall effect - allows for the experimental investigation of this many-body effect~\\cite{Onose297}.\n   \nThe remainder of this paper is structured as follows. Section~\\ref{sec.II} introduces the basic theory of interacting magnons. The Green's function formalism is introduced in Sec.~\\ref{sec.II_A}. \nA low-temperature approximation is discussed in Sec.~\\ref{sec.II_B} as an intuitive introduction to the band renormalization, and finally, the renormalization expression is extended beyond the low-temperature approximation in Sec.~\\ref{sec.II_C}. Section~\\ref{sec.III} focuses on the main calculation results. The emergence of the gauge field and the modification of band geometry due to many-body interaction is discussed using the quantum geometric tensor in Sec.~\\ref{sec.III}, and the evolution of the spinor spectral function and the scattering rate with DMI using second-order self-energy is discussed in Sec.~\\ref{sec.IV}. Section~\\ref{sec.V} presents the results of magnon amplification via parametric instability and experimental feasibility of demonstrating the theoretical results in a real quantum magnet, with Sec.~\\ref{sec.V_A} showing how the magnon population can be amplified by the external light field and Sec.~\\ref{sec.V_B} examining the relationship between magnon populations and the thermal Hall conductivities. We end with a summary and conclusions in Sec.~\\ref{con}.\n\\begin{figure}[t!]\n\\includegraphics[width=8.5 cm]{fig1.pdf}\n\\caption{Honeycomb structure of ferromagnet and noninteracting spectrum of magnonic excitation. (a) Geometric structure of honeycomb ferromagnets with NN bonds $\\bm{\\delta}_n$ and NNN bonds $\\bm{\\sigma}_n$. (b) Calculated magnon spectrum by single-particle Hamiltonian, dashed green curves are the topological non-trivial bands with $D\/J=0.05$. One can see the DMI-induced gaps at the Dirac points. (c) Berry curvature distribution of acoustic branch with $D\/J=0.05$. (d) The density of states and partial density of states for gapped magnon bands. One can see the conserved sublattice symmetry from the equal distributions of A and B sites. }\n\\label{fig:1}\n\\end{figure}\n\n\\section{Dirac magnons in honeycomb ferromagnet}\\label{sec.II}\nThe Heisenberg model on the honeycomb lattice is described by the Hamiltonian:\n\\begin{equation}\n\\begin{split}\nH=-J\\sum_{\\langle ij\\rangle}\\bm{S}_i\\cdot\\bm{S}_j,\n\\end{split}\n\\end{equation}\nwhere $J>0$ denotes the strength of ferromagnetic nearest neighbour (NN) exchange coupling and $\\bm{S}_i$ denotes the local spin on the $i$-site of the lattice. The ground state is a ferromagnet, and low energy magnon excitations above the ground state are obtained through the Holstein-Primakoff (HP) transformation~\\cite{HP1940}, which reduces to the familiar linear spin-wave theory in its lowest order: \n\\begin{equation}\n\\begin{split}\nS^x_i+iS^y_i=\\sqrt{2S}a_i,\\quad S^x_i-iS^y_i=\\sqrt{2S}a^{\\dagger}_i,\\quad S^z_i=S-a^{\\dagger}_ia_i.\n\\end{split}\n\\end{equation}\n$a~(a^{\\dagger})$ is the magnon annihilation (creation) operator, and obey commutation relation $[a_i, a^{\\dag}_j]=\\delta_{ij}$. In the linear spin wave theory framework (neglecting any magnon-magnon interaction), the above Hamiltonian reduces to an effective tight-binding model of magnons:\n\\begin{equation}\n\\begin{split}\nH=-JS\\sum_{\\langle ij\\rangle}a^{\\dagger}_ia_j+3JS\\sum_{i}a^{\\dagger}_ia_i.\n\\end{split}\n\\label{eq:rsHamil}\n\\end{equation}\nHere, we have subtracted the ferromagnetic ground state energy $E_G=-3N_LJS^2$. $N_L$ is the number of unit cells\\cite{Fransson2016}. \n\nDMI in honeycomb ferromagnets is the dominant interaction driving the topological properties of the  magnons.  Here we introduce a physically relevant out-of-plane next nearest neighbor (NNN) DMI,  $\\bm{D}_{i'j'}\\cdot(\\bm{S}_{i'}\\times\\bm{S}_{j'})=\\sum_{\\alpha\\beta\\gamma}D\\epsilon_{\\alpha\\beta\\gamma}v^{\\alpha}_{ij}S^{\\beta}_iS^{\\gamma}_j$, where $\\bm{v}_{i'j'}=\\frac{\\bm{d}^1_{i'j'}\\times\\bm{d}^2_{i'j'}}{\\abs{\\bm{d}^1_{i'j'}\\times\\bm{d}^2_{i'j'}}}=(0,0,\\pm1)$, and $D$ is the strength of the DM vector. We redefine the HP boson operator with two flavors due to the pseudospin degree of freedom: $a_i\\to a_i, a_j \\to b_{i+\\bm{\\delta}_n}$. The $\\bm{\\delta}_{n}$ are the three NN bonds of magnon orbital with $\\bm{\\delta}_{1}=\\left(\\frac{1}{2},\\frac{1}{2\\sqrt{3}}\\right)$, $\\bm{\\delta}_{2}=\\left(\\frac{-1}{2},\\frac{1}{2\\sqrt{3}}\\right)$ and $\\bm{\\delta}_{3}=\\left(0,\\frac{-1}{\\sqrt{3}}\\right)$. We plot them as the green arrows shown in Fig.~\\ref{fig:1}(a). Fourier transformation of Eq.(\\ref{eq:rsHamil}, yields the single-magnon Hamiltonian in the momentum space: $H_0=\\sum_{\\bm{k}}\\psi_{\\bm{k}}^{\\dagger}M_{\\bm{k}} \\psi_{\\bm{k}}$ with $\\psi_{\\bm{k}}^{\\dagger}=\\left(a^{\\dagger}_{\\bm{k}},b^{\\dagger}_{\\bm{k}}\\right)$. The explicit expression of the $H_0$ is given by:\n\\begin{equation}\n\\begin{split}\nH_0=\\sum_{\\bm{k}}(a^{\\dagger}_{\\bm{k}},b^{\\dagger}_{\\bm{k}})\n\\begin{pmatrix} \n 3J-2DS\\beta_{\\bm{k}}&-JS\\gamma_{\\bm{k}}\\\\ \n -JS\\gamma^*_{\\bm{k}}&3JS+2DS\\beta_{\\bm{k}}\n\\end{pmatrix}\n\\begin{pmatrix} \n a_{\\bm{k}}\\\\ \n b_{\\bm{k}}\n\\end{pmatrix}\n\\end{split},\n\\end{equation}\nwhere $\\gamma_{\\bm{k}_i}=\\sum_n e^{i\\bm{k}_i\\cdot \\bm{\\delta}_n}$, $\\beta_{\\bm{k}}=\\sum_n Im\\left(e^{i\\bm{k}\\cdot \\bm{\\sigma}_n}\\right)$, and $\\bm{\\sigma}_n$ are the NNN bonds with $\\bm{\\sigma}_1=\\left(1,0\\right)$, $\\bm{\\sigma}_2=\\left(\\frac{-1}{2},\\frac{\\sqrt{3}}{2}\\right)$ and $\\bm{\\sigma}_3=\\left(\\frac{-1}{2},\\frac{-\\sqrt{3}}{2}\\right)$. The NNN bonds are represented by the dashed blue arrows, as shown in Fig.~\\ref{fig:1}(a). Using the canonical diagonalization process, we obtain the spectrum of the $H_0$:\n\\begin{equation}\n\\begin{split}\n\\omega_{\\bm{k}}=3JS\\pm\\sqrt{J^2S^2\\abs{\\gamma_{\\bm{k}}}^2+4D^2S^2\\beta^2_{\\bm{k}}}.\n\\end{split}\n\\end{equation}\nWhen $\\bm{k}\\to 0$, the asymptotic expression of lowest energy $\\omega_{\\bm{k}}\\approx\\frac{1}{2}JS\\bm{k}^2\\to0$ reveals the gapless nature of the magnon excitation. This gapless Goldstone mode is the consequence of the spontaneously broken symmetry in the ferromagnetic ground state. In the honeycomb lattice, magnonic excitations appear in two flavours, exhibiting a graphene-like band structure. The two bands touch linearly at the Dirac point energy when $D=0$. A nonzero $D$ provides an effective Haldane mass term that opens up a non-trivial band gap $\\Delta g=6\\sqrt{3}DS$ between the upper and lower branches at the Dirac points. The spectra are plotted in Fig.~\\ref{fig:1}(b). One can see the blue curves of the gapless bands with $D=0$ and dashed green curves of the gapped topological bands with $D\/J=0.05$. \n\nThe Berry curvature distributions of the lower band are shown in Fig.~\\ref{fig:1}(c), the finite values with the same sign around the gapped Dirac points contribute to the integer Chern number $C=1$ by k-space integration. Similar to the corresponding fermionic systems, the non-zero Berry curvature provides an effective gauge field for the bosonic magnons in the k-space and dominates the transport properties. One should note that a small, but non-zero, magnetic field is necessary for stabilizing the ferromagnetic phase and the magnon excitations because the Mermin-Wagner theorem rules out the long-distance ferromagnetic order in 2D systems at finite temperatures without the magnetic field~\\cite{Mermin1966}. However, the role of the magnetic field is just to stabilize the ground state; it does not contribute to the topological properties of the bands.\n\n\\subsection{Interaction renormalized Dirac magnon}\\label{sec.II_A}\nConsidering the first-order expansion of $\\frac{1}{\\sqrt{S}}$ in HP transformation, we include the magnon-magnon interaction in the real lattice space:\n\\begin{equation}\n\\begin{aligned}[b]\n&H'=\\frac{J}{4}\\sum_{i,n}a^{\\dagger}_i b^{\\dagger}_{i+\\bm{\\delta}_n}b_{i+\\bm{\\delta}_n}b_{i+\\bm{\\delta}_n}+b^{\\dagger}_{i+\\bm{\\delta}_n} b^{\\dagger}_{i+\\bm{\\delta}_n}a_ib_{i+\\bm{\\delta}_n}+a^{\\dagger}_i b^{\\dagger}_{i+\\bm{\\delta}_n}a_ia_i\\\\\n&+a^{\\dagger}_i a^{\\dagger}_ia_ib_{i+\\bm{\\delta}_n}-J\\sum_{i,n}a^{\\dagger}_ia_ib^{\\dagger}_{i+\\bm{\\delta}_n}b_{i+\\bm{\\delta}_n}.\n\\end{aligned}\n\\end{equation}\nThe interacting Hamiltonian in k-space can be obtained by using a Fourier transformation of the above real space interaction Hamiltonian:\n\\begin{equation}\\label{eq:5}\n\\begin{aligned}[b]\nH'=&\\frac{J}{4N_L}\\sum_{\\bm{k}_i}\\gamma_{\\bm{k}_1}a^{\\dagger}_{\\bm{k}_1}b^{\\dagger}_{\\bm{k}_2}b_{\\bm{k}_3}b_{\\bm{k}_4}+\\frac{J}{4N_L}\\sum_{\\bm{k}_i}\\gamma^{*}_{\\bm{k}_4}b^{\\dagger}_{\\bm{k}_1}b^{\\dagger}_{\\bm{k}_2}b_{\\bm{k}_3}a_{\\bm{k}_4}\\\\\n+&\\frac{J}{4N_L}\\sum_{\\bm{k}_i}\\gamma^{*}_{\\bm{k}_1}b^{\\dagger}_{\\bm{k}_1}a^{\\dagger}_{\\bm{k}_2}a_{\\bm{k}_3}a_{\\bm{k}_4}+\\frac{J}{4N_L}\\sum_{\\bm{k}_i}\\gamma_{\\bm{k}_4}a^{\\dagger}_{\\bm{k}_1}a^{\\dagger}_{\\bm{k}_2}a_{\\bm{k}_3}b_{\\bm{k}_4}\\\\\n-&\\frac{J}{N_L}\\sum_{\\bm{k}_i}\\gamma_{\\bm{k}_4-\\bm{k}_2}a^{\\dagger}_{\\bm{k}_1}b^{\\dagger}_{\\bm{k}_2}a_{\\bm{k}_3}b_{\\bm{k}_4}.\n\\end{aligned}\n\\end{equation}\nThe momentum is conserved in all of the above interaction terms by $\\frac{1}{N_L}\\sum_{i}e^{i(\\bm{k}_1+\\bm{k}_2-\\bm{k}_3-\\bm{k}_4)\\cdot\\bm{r}_i}=\\delta_{\\bm{k}_1+\\bm{k}_2,\\bm{k}_3+\\bm{k}_4}$. All the interaction terms can be expressed in a more compact form: $H'=\\sum_{\\{\\bm{k}_i\\}}V^{\\bm{k}_1,\\bm{k}_2}_{\\bm{k}_3,\\bm{k}_4}\\psi^{\\dagger}_{\\bm{k}_1}\\psi^{\\dagger}_{\\bm{k}_2}\\psi_{\\bm{k}_3}\\psi_{\\bm{k}_4}$, where the summation runs over all $(\\bm{k}_1,\\bm{k}_2,\\bm{k}_3,\\bm{k}_4)$ combinations constrained by the momentum conservation. \n\nWe employ a retarded Green's function formalism to study the renormalization of the magnon bands due to magnon-magnon interactions.  In the real space-time domain, a retarded Green's function is defined as $\\langle \\psi(\\bm{r},t);\\psi^{\\dagger}(\\bm{r}',t')\\rangle=-i\\theta(t-t')\\langle [\\psi(\\bm{r},t),\\psi^{\\dagger}(\\bm{r}',t') ]\\rangle$. Here $\\langle\\cdots \\rangle$ denotes the ensemble average, $\\theta(t-t')$ is the step function, and $\\psi^{\\dag}(\\bm{r},t)$ is the field operator of the spinor which can be written as $\\left(\\sum_{i}\\phi^{*}_{a,i}(\\bm{r})a^{\\dag}_{i}(t),\\sum_{i}\\phi^{*}_{b,i}(\\bm{r})b^{\\dag}_{i}(t)\\right)$. The HP boson operators of (A, B)-site in the bipartite honeycomb lattice are given by $a^{\\dag}_i(t)$ and $b^{\\dag}_i(t)$. The frequency-dependent retarded Green's function in $k-$space is given by $G_R(\\bm{k},\\bm{k}';\\omega)=\\langle \\psi_{\\bm{k}};\\psi^{\\dagger}_{\\bm{k}'}\\rangle_{\\omega}$. Using the Heisenberg picture, we write down the equation of motion of Green's function in the frequency domain: \n\\begin{equation}\\label{eq:eom}\n\\begin{split}\n\\omega G_R(\\bm{k},\\bm{k}';\\omega)=\\delta_{\\bm{k},\\bm{k}'}+\\langle [\\psi_{\\bm{k}},H_0];\\psi^{\\dagger}_{\\bm{k}'} \\rangle_{\\omega}+\\langle [\\psi_{\\bm{k}},H'];\\psi^{\\dagger}_{\\bm{k}'} \\rangle_{\\omega},\n\\end{split} \n\\end{equation}\nwhere  $[\\psi_{\\bm{k}},H']=\\sum_{\\{\\bm{k}_i\\}}V^{(\\bm{k},\\bm{k}_2)}_{\\bm{k}_3,\\bm{k}_4}\\psi^{\\dagger}_{\\bm{k}_2}\\psi_{\\bm{k}_3}\\psi_{\\bm{k}_4}$, which gives rise to many-body effects. In order to solve the above equation, we employ the mean-field approximation (MFA) and random phase approximation (RPA): $b^{\\dag}_{\\bm{k}_2}a_{\\bm{k}_3}b_{\\bm{k}_4}\\approx \\delta_{\\bm{k}_2,\\bm{k}_3}\\langle b^{\\dag}_{\\bm{k}_2}a_{\\bm{k}_3}\\rangle b_{\\bm{k}_4}+\\delta_{\\bm{k}_2,\\bm{k}_4}\\langle b^{\\dag}_{\\bm{k}_2}b_{\\bm{k}_4}\\rangle a_{\\bm{k}_3}$. The $\\delta-$function is for the RPA, and a short explanation of its validity is in order here. Generally, the expectation value $\\langle b^{\\dag}_{\\bm{k}_2}a_{\\bm{k}_3}\\rangle$ has a dominant time dependence $\\langle b^{\\dag}_{\\bm{k}_2}a_{\\bm{k}_3}\\rangle\\propto e^{i(\\omega_{\\bm{k}_2}-\\omega_{\\bm{k}_3})t}$. When considering summation over all $\\bm{k}_2$ and $\\bm{k}_3$, we neglect the terms with $\\bm{k}_2 \\neq \\bm{k}_3$, because these are averaged to zero due to the rapid oscillations in the time domain, and therefore we only retain the dominat contribution from $\\bm{k}_2 = \\bm{k}_3$ term. \n\nFrom the perturbation theory, we can derive the spinor Dyson equation for the interacting magnons up to the second order, which is written as:\n\\begin{equation}\n\\begin{aligned}[b]\nG_R(\\bm{k},\\bm{k}';\\omega)=&G^{(0)}_R(\\bm{k},\\bm{k}';\\omega)+G^{(0)}_R(\\bm{k},\\bm{k}';\\omega)\\Sigma^{(1)}_{\\bm{k}}G^{(0)}_R(\\bm{k},\\bm{k}';\\omega)\\\\\n&+G^{(0)}_R(\\bm{k},\\bm{k}';\\omega)\\Sigma^{(2)}_{\\bm{k}}(\\omega)G^{(0)}_R(\\bm{k},\\bm{k}';\\omega),\n\\end{aligned}\n\\end{equation}\nwhere $G_R(\\bm{k},\\bm{k}';\\omega)$ is the interacting spinor Green's function of the Dirac magnon, $G^{(0)}_R(\\bm{k},\\bm{k}';\\omega)=\\frac{\\delta_{\\bm{k},\\bm{k}'}}{\\omega-M(\\bm{k}')}$ is the free particle Green's function without magnon-magnon interactions. $\\Sigma^{(1)}_{\\bm{k}}$ and $\\Sigma^{(2)}_{\\bm{k}}$ are the first-order and second-order self-energies, respectively. We will show the explicit forms in the following sections. \n\nUsing MFA and RPA, we can simply expand $[\\psi_{\\bm{k}},H']$ in Eq.~(\\ref{eq:eom}), and obtain the Hartree-type self-energy, which is the first-order renormalization given by:\n\\begin{equation}\n\\begin{aligned}[b]\n\\Sigma^{(1)}_{H'}(\\bm{k})=\\sum_{\\bm{k}_2}V^{(\\bm{k},\\bm{k}_2)}_{(\\bm{k}_2,\\bm{k})}\\langle\\psi^{\\dagger}_{\\bm{k}_2}\\psi_{\\bm{k}_2}\\rangle^{(0)}, \n\\end{aligned}\n\\end{equation}\nwhere $V^{(\\bm{k},\\bm{k}_2)}_{(\\bm{k}_2,\\bm{k})}=2V^{\\bm{k},\\bm{k}_2}_{\\bm{k}_2,\\bm{k}}+2V^{\\bm{k},\\bm{k}_2}_{\\bm{k},\\bm{k}_2}$ is the interaction coefficient matrix, $\\langle\\psi^{\\dagger}_{\\bm{k}_2}\\psi_{\\bm{k}_2}\\rangle^{(0)}=\\begin{pmatrix} \n\\langle a^{\\dagger}_{\\bm{k}_2}a_{\\bm{k}_2}\\rangle^{(0)}&\\langle a^{\\dagger}_{\\bm{k}_2}b_{\\bm{k}_2}\\rangle^{(0)}\\\\ \n\\langle b^{\\dagger}_{\\bm{k}_2}a_{\\bm{k}_2}\\rangle^{(0)}&\\langle b^{\\dagger}_{\\bm{k}_2}b_{\\bm{k}_2}\\rangle^{(0)}\n\\end{pmatrix}$ is the zeroth order population matrix, which is basically determined by the thermal excitation or the magnon parametric amplifications~\\cite{Kamimaki2020,Malz2019}. \n\\begin{figure}[t!]\n\\includegraphics[width=8.5 cm]{fig2_2.pdf}\n\\caption{The first order renormalization on magnon band from thermal excitation. (a) The density of renormalization factor $a^T(\\bm{k})$, where $\\int_{BZ}d^{2}\\bm{k}a^T(\\bm{k})=\\alpha(T)$. (b) Renormalized magnon bands at $k_BT\/JS=0.8$ are shown by solid curves, which have smaller band widths compared to the pristine magnon bands in dashed curves.}\n\\label{fig:2}\n\\end{figure}\n\n\\subsection{Low temperature approximation}\\label{sec.II_B}\nIn the case of thermal excitation we usually use a low temperature approximation, which means that we only include finite density magnonic excitations around the lowest energy state at $\\Gamma$ point in the lower magnon band. The first-order self-energy $\\Sigma^{(1)}_{\\bm{k}}$ of the thermal excitation is given by~\\cite{Pershoguba2018}:\n\\begin{equation}\n\\begin{aligned}[b]\n\\Sigma^{(1)}_{H'}(\\bm{k})\\approx-\\frac{J}{2N_L}\\sum_{\\bm{k}_1}\\frac{\\omega_{\\bm{k}_1}}{3JS}f(\\omega_{\\bm{k}_1})\n\\begin{pmatrix} \n3&-\\gamma_{\\bm{k}}\\\\ \n-\\gamma^*_{\\bm{k}}&3\n\\end{pmatrix},\n\\end{aligned}\n\\end{equation}\nwhere $f(\\omega_{\\bm{k}_1})=\\frac{1}{e^{\\omega_{\\bm{k}_1}\\beta}-1}$ is the Bose-Einstein distribution of magnons at finite temperature $T$, and $\\beta=\\frac{1}{k_BT}$. We employ the quadratic approximation $\\omega_{\\bm{q}}\\approx c_2 q^2$ of the acoustic magnon band around the $\\Gamma$ point. The summation $\\frac{1}{N_L}\\sum_q \\omega_{q}f(\\omega_{q})$ in the above equation can be easily evaluated, and we obtain $\\frac{1}{N_L}\\sum_q c_2 q^2\\frac{1}{e^{\\omega_{q}\\beta}-1}=\\frac{A k^2_B}{4\\pi c_2}\\zeta(2)T^2$, where $A$ is the area of the primitive unit cell, and $\\zeta(2)$ is the Riemann zeta function. Finally, we have a renormalized magnon band modified by the first-order self-energy:\n\\begin{equation}\n\\begin{aligned}[b]\n\\omega_{\\bm{k}}=3JS(1-\\alpha(T))\\pm\\sqrt{(1-\\alpha(T))^2J^2S^2\\abs{\\gamma_{\\bm{k}}}^2+4D^2S^2\\beta^2_{\\bm{k}}},\n\\end{aligned}\n\\end{equation}\nwhere $\\alpha(T)=\\frac{A \\pi k^2_B}{24 J^2S^3}T^2\\propto T^2$ is defined as the renormalization factor, which is consistent with the results in \\cite{Pershoguba2018} (also see Appendix~\\ref{appen:B}). We plot the distribution of the renormalization factor from the thermal excitation at $k_BT\/JS=0.8$ in Fig.~\\ref{fig:2}(a), and the modified magnon band in Fig.~\\ref{fig:2}(b). The first-order renormalization at a low temperature ($T \\ll J$) does not change the non-trivial magnon gap, but does reduce the magnon bandwidth by a factor $1-\\alpha(T)$. \n\n\\subsection{Beyond the low temperature approximation}\\label{sec.II_C}\nThe above discussion is only valid at low temperatures. The thermally generated magnons are concentrated at the bottom of the band where the Berry curvature is minimum. As a result, the effects of non-trivial band topology are not manifested in any physical observable. Fortunately, recent studies have shown that magnons can be excited not only by thermal energy but also at any energy-momentum point of the spectrum by an external electromagnetic field~\\cite{Malz2019}.  Hence one can controllably generate magnons at specific points in the magnetic Brillouin zone with a large concentration of Berry curvature so that the simultaneous effects of non-trivial band topology and interactions on magnons can be studied systematically. This is the approach taken in the following.  To encompass all possible situations, we shall focus on the full expression of the first-order self-energy from the interacting Hamiltonian:\n\\begin{equation}\\label{eq:self}\n\\begin{aligned}[b]\n&\\Sigma^{(1)}_{H'}(\\bm{k})=\\frac{J}{2N_L}\\sum_{\\bm{k}_1}\\\\\n&\\begin{pmatrix} \n\\abs{\\gamma_{\\bm{k}_1}}\\Delta_2(\\bm{k}_1)-\\gamma_0\\Theta_b(\\bm{k}_1)&\\gamma_{\\bm{k}}\\Theta^+(\\bm{k}_1)-\\theta(\\bm{k},\\bm{k}_1)\\\\ \n\\gamma^*_{\\bm{k}}\\Theta^+(\\bm{k}_1)-\\theta^*(\\bm{k},\\bm{k}_1)&\\abs{\\gamma_{\\bm{k}_1}}\\Delta_2(\\bm{k}_1)-\\gamma_0\\Theta_a(\\bm{k}_1)\n\\end{pmatrix},\n\\end{aligned}\n\\end{equation}\nwhere we have the terms:\n\\begin{equation}\\label{eq:func}\n\\begin{aligned}[b]\n\\Delta_2(\\bm{k}_1)&=\\sqrt{1-\\frac{B^2}{A^2}}\\Theta^{-}(\\bm{k}_1),\\\\\n\\Theta_b(\\bm{k}_1)&=\\Theta^{+}(\\bm{k}_1)-\\frac{B}{A}\\Theta^{-}(\\bm{k}_1),\\\\\n\\Theta_a(\\bm{k}_1)&=\\Theta^{+}(\\bm{k}_1)+\\frac{B}{A}\\Theta^{-}(\\bm{k}_1),\\\\\n\\theta(\\bm{k},\\bm{k}_1)&=\\Delta_2(\\bm{k}_1)\\gamma_{\\bm{k}-\\bm{k}_1}e^{i\\phi_{\\bm{k}_1}},\n\\end{aligned}\n\\end{equation}\nand $A=\\sqrt{(\\frac{2D\\beta_{\\bm{k}}}{J})^2+\\abs{\\gamma_{\\bm{k}}}^2}$, $B=\\frac{2D\\beta_{\\bm{k}}}{J}$, $\\Theta^{+}(\\bm{k})=f(\\omega_{d_{\\bm{k}}})+f(\\omega_{u_{\\bm{k}}})$, $\\Theta^{-}(\\bm{k})=f(\\omega_{d_{\\bm{k}}})-f(\\omega_{u_{\\bm{k}}})$, $f(\\omega_{d_{\\bm{k}}})$ and $f(\\omega_{u_{\\bm{k}}})$ are the magnon populations at point $\\bm{k}$ of lower and upper bands, respectively. The quartic DM interactions are also considered, leading to a self-energy term given by $\\Sigma^{(1)}_{DM}(\\bm{k})=\\begin{pmatrix} \n\\sigma'_{11}&0\\\\ \n0&\\sigma'_{22}\n\\end{pmatrix}$.  The explicit form of each term is (more details in Appendix~\\ref{appen:B}):\n\\begin{equation}\n\\begin{aligned}[b]\n\\sigma'_{11}&=\\frac{D\\beta_{\\bm{k}}}{N_L}\\sum_{\\bm{k}_1}\\Theta^+(\\bm{k}_1)+\\frac{D}{N_L}\\sum_{\\bm{k}_1}\\frac{\\beta_{\\bm{k}_1}B}{A}\\Theta^-(\\bm{k}_1)\\\\\n\\sigma'_{22}&=-\\frac{D\\beta_{\\bm{k}}}{N_L}\\sum_{\\bm{k}_1}\\Theta^+(\\bm{k}_1)+\\frac{D}{N_L}\\sum_{\\bm{k}_1}\\frac{\\beta_{\\bm{k}_1}B}{A}\\Theta^-(\\bm{k}_1).\n\\end{aligned}\n\\end{equation}\nHence the renormalized Hamiltonian at the first-order level is given by:\n\\begin{equation}\\label{eq:re_H}\n\\begin{aligned}[b]\n&H_1=H_0+\\Sigma^{(1)}_{H'}(\\bm{k})+\\Sigma^{(1)}_{DM}(\\bm{k})\\\\\n&=\\begin{pmatrix} \n 3JS-2DS\\beta_{\\bm{k}}+P+\\delta_m&-(JS-Q)\\gamma_{\\bm{k}}-g(\\bm{k})\\\\ \n -(JS-Q)\\gamma^*_{\\bm{k}}-g^*(\\bm{k})&3JS+2DS\\beta_{\\bm{k}}+P-\\delta_m\n\\end{pmatrix},\n\\end{aligned}\n\\end{equation}\nwhere $P=\\frac{J}{2N_L}\\sum_{\\bm{k}_1}(\\Delta_1(\\bm{k}_1)-\\gamma_0\\Theta^+(\\bm{k}_1))$, $\\Delta_1(\\bm{k}_1)=\\abs{\\gamma_{\\bm{k}_1}}\\Delta_2(\\bm{k}_1)+\\frac{B^2}{A}\\Theta^{-}(\\bm{k}_1)$, and $Q=\\frac{J}{2N_L}\\sum_{\\bm{k}_1}\\Theta^+(\\bm{k}_1)$. These are all overall factors and $\\bm{k}$-independent. It is easy to see that the diagonal term $P$ mimics the scalar potential, and the interaction-induced off-diagonal term $g(\\bm{k})=\\frac{J}{2N_L}\\sum_{\\bm{k}_1}\\theta(\\bm{k},\\bm{k}_1)$ mimics the vector potential in the vicinity of the magnon Dirac point using minimal coupling with $\\gamma_{\\bm{k}}+g(\\bm{k})\\to \\bm{k}+g(\\bm{k}_D)$. \n\nThe most interesting term is the interaction-induced Haldane term $\\delta_m=\\frac{J}{2N_L}B(\\bm{k})\\sum_{\\bm{k}_1}\\Theta^{+}(\\bm{k}_1)$. Thus the interaction effect can be seen as the emergent gauge field in the weakly interacting magnon system. The summation of the value $\\bm{k} _1$ is limited to the first Brillouin zone. We diagonalize the new Hamiltonian to obtain the renormalized magnon spectrum as: \n\\begin{equation}\\label{eq:re_e}\n\\begin{aligned}[b]\n\\omega_{\\bm{k}}=3JS+P\\pm\\sqrt{\\abs{(JS-Q)\\gamma_{\\bm{k}}+g(\\bm{k})}^2+(2DS\\beta_{\\bm{k}}-\\delta_m)^2}\n\\end{aligned}\n\\end{equation}\nwhere the factor $Q$ affects the bandwidths of the magnon  consistent with the results in~\\cite{Pershoguba2018}, when using a low temperature approximation.\n\n\\begin{figure}[t!]\n\\includegraphics[width=8.5 cm]{fig3_2.pdf}\n\\caption{Population of excited magnon by electromagnetic field and distribution of $g(\\bm{k})$. (a) The anisotropic population of magnon by EM field amplification. Two excited magnon states with opposite $\\bm{k}$ vectors ($(0,b)$ and $(0,-b)$). (b) Distribution of the in-plane components of the vector $\\bm{g}$ shows the nonuniform of $g(\\bm{k})$ in BZ. (c) The isotropic population of magnon by EM field amplification. (d) Distribution of the in-plane components of the vector $\\bm{g}$ by isotropic populations in (c). }\n\\label{fig:3}\n\\end{figure}\n\n\\section{Band engineering from magnon-magnon interaction}\\label{sec.III}\n\\subsection{Shifting of Dirac points and quantum geometric tensor}\\label{sec.III_A} \nWe use Eqs.(\\ref{eq:re_H}) and (\\ref{eq:re_e}) to study the many-body effect for arbitrary magnon populations, in particular for magnons amplified by the electromagnetic field. The magnon amplification will be discussed in Sec.~\\ref{sec.V}. We argue that the treating the interaction within a mean-field theory gives rise to an effective potential, in which the diagonal part $P$ is a scalar potential, and the off-diagonal part $g(\\bm{k})$ is an effective vector potential in the vicinity of the Dirac point. The scalar potential $P$ for the energy shift will not change the physics around the Dirac point.\n\nIt is interesting that in the pseudospin space, $g(\\bm{k})$ acts as an effective in-plane Zeeman field on sublattice pseudospin $\\bm{\\sigma}$. In Eq.~(\\ref{eq:re_H}), we can rewrite the non-trivial part as $\\begin{pmatrix} \n \\delta_m&-g(\\bm{k})\\\\ \n -g^*(\\bm{k})&-\\delta_m\n\\end{pmatrix}=\\bm{g}\\cdot\\bm{\\sigma}$, It follows that we can reformulate the complex quantity $g(\\bm{k})$ and the interaction-induced Haldane mass into a vector form given by the expression $\\bm{g}=(-Re(g(\\bm{k})),Im(g(\\bm{k})),\\delta_m)$. For an anisotropic magnon density distribution $f(\\omega_{\\bm{k}})$ as shown in Fig.~\\ref{fig:3}(a), we have the distribution of vector $\\bm{g}$, showing a nonuniform vortex feature illustrated in Fig.~\\ref{fig:3}(b). $g(\\bm{k})$ breaks the $C_6$ rotational symmetry of the first BZ, and as a consequence the Dirac point will be shifted by the effective vector potential, as shown in Fig.~\\ref{fig:4}(a)~\\cite{Tarruell2012}. The black dots represent the original Dirac points, whereas the red dots are the new ones shifted by $\\bm{g}$. However, the gapless feature of Dirac point will still be preserved without DM interactions, since the original interaction terms in Eq~(\\ref{eq:5}) do not contain the symmetry breaking factors, e.g., broken time reversal and chiral symmetries. In addition, we do not include the anomalous paring term $\\langle\\psi^{\\dag}_{\\bm{k}}\\psi^{\\dag}_{-\\bm{k}}\\rangle$ ($\\langle\\psi_{\\bm{k}}\\psi_{-\\bm{k}}\\rangle$) in RPA, which breaks the $U(1)$ symmetry. Dirac point shifts are observed in many cases by strain engineering~\\cite{Feilhauer2015,Kim2021}, whereas we propose that many-body effects can induce the same effect without lattice deformation.\n\nThe shift of Dirac points results in the change of geometric properties of the magnon dispersion. We study how the effective gauge field $g(\\bm{k})$ modifies the geometric properties encoded in quantum geometric tensor (QGT). QGT consists of the Berry curvature and the quantum metric measuring the \"distance\" between the eigenstates in the Hilbert space. It is defined as follows:\n\\begin{equation}\\label{eq:qgt1}\n\\begin{split}\nQ_{ab}=G_{ab}-\\frac{i}{2}\\Omega_{ab},\n\\end{split}\n\\end{equation}\nwhere the real part $G_{ab}$ is the quantum metric, and $\\Omega_{ab}$ is the Berry curvature. We already know that Berry curvature is crucial for topological phases; \nthe quantum metric is associated with superfluidity in flat bands and orbital magnetic susceptibility. Finally, it should be noted that the QGT is not just an abstract mathematical construct, but has been measured directly~\\cite{Gianfrate2020}. Since the gauge-invariant QGT contains the structural information about the eigenstates of a parametrized Hamiltonian, we have the explicit expressions as below:\n\\begin{equation}\\label{eq:qgt2}\n\\begin{aligned}[b]\nG_{ab}&=Re\\left(\\sum_{m\\ne n}\\frac{\\mel{u_m}{\\partial_{k_a}H}{u_n}\\mel{u_n}{\\partial_{k_b}H}{u_m}}{(E_m-E_n)^2}\\right),\\\\\n\\Omega_{ab}&=i\\left(\\sum_{m\\ne n}\\frac{\\mel{u_m}{\\partial_{k_a}H}{u_n}\\mel{u_n}{\\partial_{k_b}H}{u_m}}{(E_m-E_n)^2}-(a\\to b)\\right).\n\\end{aligned}\n\\end{equation} \nIt's convenient to rewrite the Hamiltonian in Eq. ~(\\ref{eq:re_H}) with a pseudospin freedom $\\bm{\\sigma}$:\n\\begin{equation}\\label{eq:pss}\n\\begin{aligned}[b]\nH_1=(3JS+P)\\sigma_0+\\bm{h}(\\bm{k})\\cdot\\bm{\\sigma}+\\bm{g}(\\bm{k})\\cdot\\bm{\\sigma}\n\\end{aligned}\n\\end{equation}\nwhere effective field vector $\\bm{h}$ is $(-(JS-Q)\\abs{\\gamma_{\\bm{k}}}\\cos{\\phi_{\\bm{k}}}, (JS-Q)\\abs{\\gamma_{\\bm{k}}}\\sin{\\phi_{\\bm{k}}}, -2DS\\beta_{\\bm{k}})$, $\\bm{\\sigma}=(\\sigma_x, \\sigma_y, \\sigma_z)$ is the Pauli matrix, and $\\bm{g}(\\bm{k})=(-\\abs{g_{\\bm{k}}}\\cos{\\phi_{g_{\\bm{k}}}}, \\abs{g_{\\bm{k}}}\\sin{\\phi_{g_{\\bm{k}}}}, \\delta_m)$. \nWithin this representation, we put the band physics on a Bloch sphere, and all the geometric and topological properties are contained in the $\\bm{h}\\cdot\\bm{\\sigma}+\\bm{g}\\cdot\\bm{\\sigma}$ term. We plot the modified Berry curvature in Fig.~\\ref{fig:4}(b), the $xx$ and $yy$ components of quantum metric $G$ in Fig.~\\ref{fig:4}(c)-(d). We can see that the QGT is renormalized to a new distribution in BZ by interaction-induced Zeeman field $g_{\\bm{k}}$. The shifted Dirac points and the renormalized bands can be detected by well-established techniques including Brillouin light scattering, inelastic X-ray scattering, and inelastic neutron scattering~\\cite{Braicovich2010, Samuelsen1971, Cenker2021, Sobolev1994}.\n\\begin{figure}[t!]\n\\includegraphics[width=9.0 cm]{fig4.pdf}\n\\caption{Quantum geometric tensor in renormalised magnon bands. (a) The shift of Dirac points by anisotropic magnon amplification shown in Fig.~\\ref{fig:3}(a) without DM interaction. Black dots represent the original Dirac points, and the shifted Dirac points are shown in red dots. (b) The distribution of Berry curvature shows the broken $C_6$ symmetry by interaction effect. (c) Quantum metric $G_{xx}$ component. (d) Quantum metric $G_{yy}$ component.}\n\\label{fig:4}\n\\end{figure}\n\n\\subsection{Topological bands with tunable Chern numbers}\\label{sec.III_B}\nFrom Eq.~(\\ref{eq:pss}), we find that interaction gives rise to an effective mass $\\delta_m$, which can be understood as an out-of-plane Zeeman field on the pseudospin degrees of freedom. The effective mass term determining the bandgap is essential for the geometric effect and the non-trivial band topology. The competition between the DM term $2DS\\beta_{\\bm{k}}$ and $\\delta_m$ can lead to the different Chern numbers and thus distinct topology. One should note that $\\delta_m$ depends on DMI, as can be seen from the expression $\\delta_m=\\frac{JB}{2N_L}\\sum_{\\bm{k}_1}\\Theta^{+}(\\bm{k}_1)$, where the parameter $B\\propto DS$. When $D=0$, $\\delta_{m}$ vanishes.\n\nThe total mass term is given by $m_H=2DS\\beta_{\\bm{k}}-\\delta_m$. By tuning the magnon population, we can achieve topological phase transitions in the bottom band from $C=1$ to $C=-1$ or $C=-1$ to $C=1$. This is due to the fact that $\\delta_m$ is dominated by the total population at point $\\bm{k}_1$, which can be driven by pumping magnons in both branches or by thermal excitation. Our results show that the total mass term decreases linearly with increasing magnon population, as illustrated in Fig.~\\ref{fig:5}(a). The sign-change of $m_H$ reveals a topological phase transition with reversing of the Chern numbers. A recent study has proposed a similar topological phase transition by the thermal excitation~\\cite{Lu2021}. We argue that the band topology can be simply tuned using a magnon amplification approach, which will be discussed in Sec.~\\ref{sec.V}. The parametric amplification approach exhibits more advantages over the thermal excitation due to its flexibility~\\cite{Malz2019}. \n\nBy properly tuning the magnon population with the external EM field, we can increase the number of crossing points between the upper and lower magnon branches. This, in principle, can give rise to magnon bands with the higher Chern numbers. Mathematically, the interaction-induced $g(\\bm{k})$ can have any form. The general expression of $g(\\bm{k})$ consists of a linear combination of the $n$-th nearest geometric factors $\\gamma^{(n)}_{\\bm{k}}$, or $g(\\bm{k})=\\sum_{n}c_n\\gamma^{(n)}_{\\bm{k}}$, where $\\gamma^{(n)}_{\\bm{k}}=\\sum_j e^{i\\bm{k}\\cdot\\bm{r}^{(n)}_j}$, and $\\bm{r}^{(n)}_j$ is the $n$-th nearest bond. As a specific example, let us look at the model of $g(\\bm{k})=c\\sum_{n}e^{i\\bm{k}\\cdot\\bm{\\eta}_n}$, where $c$ is expressed as the strength of the vector potential, and $\\bm{\\eta}_n$ is the geometric vector. We find that when $\\eta_n$ is for the third-nearest bonds, there are three more Dirac points at each $\\bm{K}$ valley as illustrated in Fig.~\\ref{fig:5}(b). We introduce a nonzero DMI to gap all the Dirac points, and obtain the Berry curvature distribution of the lower band as shown in Fig.~\\ref{fig:5}(c). In this case the Chern number of the lower magnon band is given by $C=\\int_{BZ} \\Omega_{\\bm{k}}d\\bm{k}=-2$. \n\\begin{figure}[t!]\n\\includegraphics[width=9.0 cm]{fig5.pdf}\n\\caption{Renormalized magnon bands with high Chern number, we use $D\/J=0.05$. (a). Renormalized Haldane mass at Dirac point in magnon bands we use $D\/J=0.05$, $F(\\omega_{\\bm{k}})=\\frac{1}{4N_L}\\sum_{\\bm{k}_1}\\left(f(\\omega_{d_{\\bm{k}_1}})+f(\\omega_{u_{\\bm{k}_1}})\\right)$ is the quantity for the normalised magnon density. (b) The energy difference $\\Delta\\omega_{\\bm{k}}\/JS$ between upper and lower bands. The original Dirac point is labeled in black dot, and the three satellitic Dirac points are shown by red dots. (c) Berry curvature distributions.}\n\\label{fig:5}\n\\end{figure}\n\n\\subsection{Topological flat bands by parametric pumping}\\label{sec.III_C} \nUsing isotropic parametric pumping (see details in Sec.~\\ref{sec.V_A}), we can engineer a topologically flat magnon band with a very small bandwidth. We employ Eq.~(\\ref{eq:re_e}) with pumped magnons that are located in a circle with equal energy as shown in Fig.~\\ref{fig:3}(c). The energy of the pumped magnon is  $\\omega(\\bm{k})\\approx 1.6J$, for $D\/J=0.1$, and the population intensity $I_p$ is about 7.2.  \nThe resulting flat bands are shown in Fig.~\\ref{fig:6}(a) by the solid curves. The flatness of the topological bands can be simply characterized by the ratio $r_f=\\frac{w_p}{w_r}$, where $w_p$ is the bandwidth without magnon-magnon interactions, and $w_r$ is the bandwidth of the renormalized magnon spectrum. The computed value of $r_f \\approx 0.09$ implies that the renormalized bands are almost dispersionless. \n\nThe almost flat topological bands give rise to a more uniform Berry curvature and quantum metric tensors that mimic  Landau level physics of interacting particles~\\cite{Wang2021}. The quantum geometric properties are illustrated in Fig.~\\ref{fig:6}(b)-(d). The Berry curvature is redistributed and the hotspots are shifted from $K$ to $M$ points in the BZ. The elements $G_{xx}$ and $G_{yy}$ of the quantum metric tensor in Fig.~\\ref{fig:6}(c)-(d) are also strongly renormalized. \n\nThe renormalized magnon Bloch bands also conform to the ideal flat band condition that is provided by~\\cite{Claassen2015}:  \n\\begin{equation}\\label{}\n\\begin{aligned}[b]\n\\sqrt{\\det G(\\bm{k})}-\\frac{1}{2}\\abs{\\Omega(\\bm{k})}=0,\n\\end{aligned}\n\\end{equation}\nwhere $\\det G(\\bm{k})=G_{xx}G_{yy}-G_{xy}G_{yx}$ is the determinant of quantum metric tensor. The nearly dispersionless magnon bands with ideal flat conditions allow us to investigate strongly correlated behaviour, since the interaction strength between the magnons is comparable to the bandwidth. Thus, we can expect the superfluid-Bose insulator transition at the partially filled lowest band~\\cite{Fisher1989}. More interestingly, the band minimum is also modified, which shifts from $\\Gamma$ point to $K$ point, where there is a gapped Dirac cone with finite Berry curvatures and nonzero quantum metric. The induced flat band can potentially host a novel Bose-Einstein condensate that is stabilized by the nonzero quantum metric~\\cite{Julku2021}. Furthermore, the emergent Goldstone modes from this flat band condensation reveal a quantum geometric dependency~\\cite{Julku2021prl}. We argue that our interaction-induced flat magnon bands provide a realistic platform for studying this novel phenomenon, which deserves further theoretical study.\n\\begin{figure}[t!]\n\\includegraphics[width=9.0 cm]{fig5_6.pdf}\n\\caption{QGT in topologically flat bands. (a) The topologically flat band by magnon-magnon interactions. Solid curves are interaction induced flat bands, and the dashed curves are noninteracting magnon bands. (b) The Berry curvature is redistributed by the interaction effect. (c) Quantum metric $G_{xx}$ component. (d) Quantum metric $G_{yy}$ component.}\n\\label{fig:6}\n\\end{figure}\n\n\n\\section{Second order renormalization and magnon scattering}\\label{sec.IV}\nIn the previous section, we employed a mean-field approximation of the term $[\\psi_{\\bm{k}}, H']$, and directly obtained the first-order self-energy. In order to include the higher-order renormalization, we must consider the dynamics of the commutator  $[\\psi_{\\bm{k}},H']$. The second-order effect from the quartic DM interactions is neglected, due to its smallness in the order of $\\frac{D^2}{J^2}$. In this section, we focus on the second-order approximation by considering the equation of motion of retarded Green's function with the commutator $[\\psi_{\\bm{k}}, H']$:\n\\begin{equation}\\label{}\n\\begin{aligned}[b]\n\\omega \\langle [\\psi_{\\bm{k}},H'];\\psi^{\\dagger}_{\\bm{k}'} \\rangle_{\\omega}=\\langle [[\\psi_{\\bm{k}},H'],\\psi^{\\dagger}_{\\bm{k}'}] \\rangle+\\langle [[\\psi_{\\bm{k}},H'],H];\\psi^{\\dagger}_{\\bm{k}'} \\rangle_{\\omega}.\n\\end{aligned}\n\\end{equation}\n\nThe second-order self-energy from perturbation theory can be written as:\n\\begin{equation}\n\\begin{aligned}[b]\n\\Sigma^{(2)}_{\\bm{k}}=\\frac{1}{2}\\sum_{\\{\\bm{k}_i\\}}\\frac{V^{(\\bm{k},\\bm{k}_2)}_{(\\bm{k}_3,\\bm{k}_4)}V^{(\\bm{k}_4,\\bm{k}_3)}_{(\\bm{k}_2,\\bm{k})}(n_{\\bm{k}_2}(1+n_{\\bm{k}_3}+n_{\\bm{k}_4})-n_{\\bm{k}_3}n_{\\bm{k}_4})}{\\omega_{\\bm{k}}+i\\epsilon-\\mathcal{M}(\\bm{k}_2,\\bm{k}_3,\\bm{k}_4)},\n\\end{aligned}\n\\end{equation}\nwhere $V^{(\\bm{k},\\bm{k}_2)}_{(\\bm{k}_3,\\bm{k}_4)}=V^{\\bm{k},\\bm{k}_2}_{\\bm{k}_3,\\bm{k}_4}+V^{\\bm{k}_2,\\bm{k}}_{\\bm{k}_3,\\bm{k}_4}+V^{\\bm{k},\\bm{k}_2}_{\\bm{k}_4,\\bm{k}_3}+V^{\\bm{k}_2,\\bm{k}}_{\\bm{k}_4,\\bm{k}_3}$ is the interacting coefficient matrix. The nonzero matrix elements from the interacting Hamiltonian are $V^{1,2}_{1,2}=-\\frac{J}{N_L}\\gamma_{\\bm{k}_4-\\bm{k}_2}$, $V^{1,2}_{2,2}=\\frac{J}{4N_L}\\gamma_{\\bm{k}_1}$, $V^{2,1}_{1,1}=\\frac{J}{4N_L}\\gamma^*_{\\bm{k}_1}$, $V^{2,2}_{2,1}=\\frac{J}{4N_L}\\gamma^*_{\\bm{k}_4}$ and $V^{1,1}_{1,2}=\\frac{J}{4N_L}\\gamma_{\\bm{k}_4}$. We define $V^{a,b}_{c,d}\\triangleq V^{a\\bm{k}_1,b\\bm{k}_2}_{c\\bm{k}_3,d\\bm{k}_4}$, and $a$,$b$,$c$,$d$ $\\in \\{1,2\\}$ are the labels for the two components of spinor $\\psi_{\\bm{k}}(\\psi^{\\dag}_{\\bm{k}})$. The function $n_{\\bm{k}}$ denotes the zeroth order population matrix $\\langle\\psi^{\\dag}_{\\bm{k}}\\psi_{\\bm{k}}\\rangle^{(0)}$, which can be the equilibrium or non-equilibrium distribution function. $\\mathcal{M}(\\bm{k}_2,\\bm{k}_3,\\bm{k}_4)$ is a tensor dominated by the elements of $M(\\bm{k}_2)$, $M(\\bm{k}_3)$ and $M(\\bm{k}_4)$. The definition of $\\mathcal{M}(\\bm{k}_2,\\bm{k}_3,\\bm{k}_4)$ comes from the operation:\n\\begin{equation}\n\\begin{aligned}[b]\n\\mathcal{M}\\psi^{\\dag}_{\\bm{k}_2}\\psi_{\\bm{k}_3}\\psi_{\\bm{k}_4}&=\\psi^{\\dag}_{\\bm{k}_2}M(\\bm{k}_3)\\psi_{\\bm{k}_3}\\psi_{\\bm{k}_4}+\\psi^{\\dag}_{\\bm{k}_2}\\psi_{\\bm{k}_3}M(\\bm{k}_4)\\psi_{\\bm{k}_4}\\\\\n&-\\psi^{\\dag}_{\\bm{k}_2}M(\\bm{k}_2)\\psi_{\\bm{k}_3}\\psi_{\\bm{k}_4}\n\\end{aligned}\n\\end{equation}\nWe simplify this operation and obtain $Q'_{a,b,c}(\\bm{k}_2,\\bm{k}_3,\\bm{k}_4)=\\sum_{d,e,f}\\mathcal{M}^{a,b,c}_{d,e,f}(\\bm{k}_2,\\bm{k}_3,\\bm{k}_4)Q_{d,e,f}(\\bm{k}_2,\\bm{k}_3,\\bm{k}_4)$, where $Q'_{a,b,c}$ is the element in RHS of above equation, $Q_{a,b,c}$ is the element in $\\psi^{\\dag}_{\\bm{k}_2}\\psi_{\\bm{k}_3}\\psi_{\\bm{k}_4}$. Since $\\psi^{\\dag}_{\\bm{k}_2}\\psi_{\\bm{k}_3}\\psi_{\\bm{k}_4}$ has 8 elements, it gives rise to $2^6=64$ elements in $\\mathcal{M}$. Here we list the 8 diagonal elements of $\\mathcal{M}$:\n\\begin{equation}\n\\begin{aligned}[b]\n\\mathcal{M}^{111}_{111}&=-M^{11}(\\bm{k}_2)+M^{11}(\\bm{k}_3)+M^{11}(\\bm{k}_4),\\\\\n\\mathcal{M}^{222}_{222}&=-M^{22}(\\bm{k}_2)+M^{22}(\\bm{k}_3)+M^{22}(\\bm{k}_4),\\\\\n\\mathcal{M}^{112}_{112}&=-M^{11}(\\bm{k}_2)+M^{11}(\\bm{k}_3)+M^{22}(\\bm{k}_4),\\\\\n\\mathcal{M}^{121}_{121}&=-M^{11}(\\bm{k}_2)+M^{22}(\\bm{k}_3)+M^{11}(\\bm{k}_4),\\\\\n\\mathcal{M}^{122}_{122}&=-M^{11}(\\bm{k}_2)+M^{22}(\\bm{k}_3)+M^{22}(\\bm{k}_4),\\\\\n\\mathcal{M}^{211}_{211}&=-M^{22}(\\bm{k}_2)+M^{11}(\\bm{k}_3)+M^{11}(\\bm{k}_4),\\\\\n\\mathcal{M}^{212}_{212}&=-M^{22}(\\bm{k}_2)+M^{11}(\\bm{k}_3)+M^{22}(\\bm{k}_4),\\\\\n\\mathcal{M}^{221}_{221}&=-M^{22}(\\bm{k}_2)+M^{22}(\\bm{k}_3)+M^{11}(\\bm{k}_4).\\\\\n\\end{aligned}\n\\end{equation}\nThese diagonal elements are the most important terms for determining the magnon-magnon scattering channels. It is more convenient to work with the self-energy matrix in an eigen-mode basis, and transform the $\\begin{pmatrix}a_{\\bm{k}}\\\\b_{\\bm{k}}\\end{pmatrix}$ basis to the eigen-mode basis $\\begin{pmatrix}d_{\\bm{k}}\\\\u_{\\bm{k}}\\end{pmatrix}$ using a unitary transformation $U$, where $d_{\\bm{k}}$($u_{\\bm{k}}$) is the magnon operator for the lower (upper) band. As a consequence, $\\mathcal{M}$ has only diagonal components, and $M$ is a diagonal matrix of the eigenvalues. Thus we have the second-order self-energy matrix in the eigen-mode basis:\n\\begin{equation}\\label{eq:seself}\n\\begin{aligned}[b]\n\\tilde{\\Sigma}^{(2)}_{\\bm{k}}=\\frac{1}{2}\\sum_{\\{\\bm{k}_i\\}}\\frac{\\tilde{V}^{(\\bm{k},\\bm{k}_2)}_{(\\bm{k}_3,\\bm{k}_4)}\\tilde{V}^{(\\bm{k}_4,\\bm{k}_3)}_{(\\bm{k}_2,\\bm{k})}(\\tilde{n}_{\\bm{k}_2}(1+\\tilde{n}_{\\bm{k}_3}+\\tilde{n}_{\\bm{k}_4})-\\tilde{n}_{\\bm{k}_3}\\tilde{n}_{\\bm{k}_4})}{\\omega_{\\bm{k}}+i\\epsilon+\\omega_{\\bm{k}_2}-\\omega_{\\bm{k}_3}-\\omega_{\\bm{k}_4}}\n\\end{aligned}\n\\end{equation}\nwhere $\\tilde{V}^{(\\bm{k},\\bm{k}_2)}_{(\\bm{k}_3,\\bm{k}_4)}=U^{\\dag}_{\\bm{k}}U^{\\dag}_{\\bm{k}_2}V^{(\\bm{k},\\bm{k}_2)}_{(\\bm{k}_3,\\bm{k}_4)}U_{\\bm{k}_3}U_{\\bm{k}_4}$ (see details in Appendix~\\ref{appen:D}). The expression of unitary $U$ is given by:\n\\begin{equation}\\label{}\n\\begin{split}\n\\begin{pmatrix} \na_{\\bm{k}}\\\\\nb_{\\bm{k}}\n\\end{pmatrix}=\\frac{1}{\\sqrt{2}}\n\\begin{pmatrix} \n\\sqrt{1+\\frac{B}{A}}e^{i\\frac{\\phi_{\\bm{k}}}{2}}&\\sqrt{1-\\frac{B}{A}}e^{i\\frac{\\phi_{\\bm{k}}}{2}}\\\\ \n\\sqrt{1-\\frac{B}{A}}e^{-i\\frac{\\phi_{\\bm{k}}}{2}}&-\\sqrt{1+\\frac{B}{A}}e^{-i\\frac{\\phi_{\\bm{k}}}{2}}\n\\end{pmatrix}\n\\begin{pmatrix} \nd_{\\bm{k}}\\\\u_{\\bm{k}}\n\\end{pmatrix}\n\\end{split}\n\\end{equation}\nwhere $A$ and $B$ factors are given in Eq.~(\\ref{eq:func}), which are dominated by DM interactions. With the unitary transformations, we have all the interaction matrix elements on an eigenmode basis (see details in Appendix~\\ref{appen:D}). \n\nThe second-order self-energy term has a more complicated form, which not only modifies the bandwidth, but also leads to magnon damping. Replacing $\\omega$ by $\\omega_{\\bm{k}}+i\\epsilon$ in Eq.~(\\ref{eq:seself}), where the infinitesimal imaginary part $i\\epsilon$ is the argument of the retarded Green's function, we have $\\tilde{\\Sigma}^{(2)}_{\\bm{k}}=\\tilde{\\Sigma}^{(2)'}_{\\bm{k}}-i\\tilde{\\Sigma}^{(2)''}_{\\bm{k}}$, where the real part is the renormalization term and the imaginary part is the damping term or magnon-magnon scattering rate~\\cite{Mook2021}. The scattering rate contributes to the broadening of the magnon band, which gives rise to an upper bound of the magnon lifetime $\\tau =\\frac{\\hbar}{2\\tilde{\\Sigma}^{(2)''}_{\\bm{k}}}$. \n\n\\begin{figure}[t!]\n\\includegraphics[width=8.5 cm]{fig6_7.pdf}\n\\caption{Second-order renormalization of magnon dispersion, we use parameter $T\/J$=0.5, and $D\/J=0.1$. (a) Spectral function $A(\\bm{k},\\omega_{\\bm{k}})$. In the plot, we use 1\/10 of the maximal $A(\\bm{k},\\omega_{\\bm{k}})$ for the color mapping to make a better display of damping components. (b)-(c) The magnon scattering rates with different DMI. One pronounced feature is that DMI significantly enhances the scattering rate around Dirac point of the lower band, and gives rise to a faster decay dynamics. (d) Scattering rate in lower band by the magnon parametric amplifications. We here employ a very small temperature $T\/J=0.1$ to purely show the scattering effect from non-equilibrium magnon population.}\n\\label{fig:7}\n\\end{figure}\n\nWe define the spectral function $A(\\bm{k},\\omega)$, which describes quasiparticle properties in magnon spectrum (band information) and density of states. Essentially, $A(\\bm{k},\\omega)$ is the imaginary part of the retarded Green's function:\n\\begin{equation}\\label{}\n\\begin{aligned}[b]\nA(\\bm{k},\\omega)=-\\frac{1}{\\pi}\\Im(G_R(\\bm{k},\\omega))\n\\end{aligned}\n\\end{equation}\nThe renormalized magnon dispersion at temperature $T\/J=0.5$ with the second-order self-energy is shown in Fig.~\\ref{fig:7}(a). The blurred areas indicate the magnon damping effect from the imaginary component $\\tilde{\\Sigma}^{(2)''}_{\\bm{k}}$. We compare the scattering rate of the two cases with and without DMI. One can see that DMI strongly influences the scattering rate, and the effect is momentum dependent, as shown in Fig.~\\ref{fig:7}(b)-(d) -- while the rate is enhanced at the $K$-point, it is suppressed at the $M$-point. We now focus on the scattering rate in the lower band. The rate near the Dirac point with DMI is significantly enhanced for the lower band, as shown in Fig.~\\ref{fig:7}(c), indicating that the DMI facilitates scattering rate and results in faster magnon dynamics. The simple explanation for the DMI-enhanced scattering rate at the Dirac point is that the nonzero topological gap contributes to a finite density of states for the Dirac magnon, and thus provides more scattering channels for interacting magnons. \n\nWe also study the scattering effect from the magnon parametric amplification, which can dramatically modify the scattering rate by changing the energy position of the amplified magnons; the basic features are shown in Fig.~\\ref{fig:7} (d). With a fixed magnon population intensity $I_p$, we increase the energy position from $\\omega_{d_{\\bm{k}}}=1.2J$ to $1.8J$ for magnons out of equilibrium. The scattering rate along the k-points is increased except for the Dirac points.  There are pronounced peaks between $\\Gamma$ and $M$ points, which is due to the obvious scattering around the pumped magnons. We employ a very small temperature of $T\/J=0.1$ in the calculation to focus on the scattering effect from non-equilibrium magnon populations illustrating that the scattering rate can be flexibly tuned by the light-induced magnons. Magnon scattering rates are closely linked to the microscopic dynamics aspects, which provide powerful tools for measuring and controlling the magnetic order in the ultrafast time regime. This is a rapidly developing research area that is relevant for magnetic memory and spintronics~\\cite{Yang2020}.\n\n\n\\section{Parametric amplifications and thermal Hall effect}\\label{sec.V}\n\\subsection{Parametric magnon amplifications}\\label{sec.V_A} \nAs we mentioned in previous sections, we can populate magnon by an external electromagnetic field~\\cite{Chumak2009}, in addition to thermal excitations. The magnon population can be significantly enhanced by the so-called parametric instability while preserving the magnetic order in bulk state~\\cite{Malz2019}. This instability can be induced by coupling the magnetic material to the external field, whereby excited magnons are created in pairs. Here we employ a coherent driving electromagnetic (EM) field and start with the effective coupling Hamiltonian: \n\\begin{equation}\\label{}\n\\begin{aligned}[b]\nH_{int}=\\sum_{\\bm{k}}\\frac{g_{\\bm{k}}}{2}(d^{\\dagger}_{-\\bm{k}}d^{\\dagger}_{\\bm{k}} b+b^{\\dagger}d_{\\bm{k}}d_{-\\bm{k}}),\n\\end{aligned}\n\\end{equation} \nwhere the field operator $b\\approx\\beta\\exp(-i\\Omega_0t)$ is the external EM field or the pump photon, and $g_{\\bm{k}}\\approx g$ is the coupling coefficient between external field and magnon excitation that can be treated as momentum independent in a small bandwidth regime.  \n$H_{int}$ describes the process of creation of a pair of magnons by absorption of a photon (and, by hermiticity of the coupling Hamiltonian, the reverse process of annihilation of a pair of magnons accompanied by the emission of a photon). Magnons need to be created (or annihilated) in pairs with equal and opposite momenta to satisfy the conservation of momentum during the process (photons carry zero momentum). For the single-particle Hamiltonian, we have to include the scattering terms $H_{imp}$ caused by impurities and disorders. The magnon density is determined by the time-dependent Heisenberg equation of motion as follows~\\cite{Malz2019}:\n\\begin{equation}\\label{}\n\\begin{aligned}[b]\ni\\frac{d T_{\\bm{k}}(t)}{dt}=\\langle[\\hat{T}_{\\bm{k}},H]\\rangle=\\tilde{\\Omega}_{\\bm{k}} T_{\\bm{k}}(t),\\quad H=H_1+H_{int} + H_{imp},\n\\end{aligned}\n\\end{equation} \nwhere operator $\\hat{T}_{\\bm{k}}(t)=( d_{\\bm{k}}, d^{\\dagger}_{-\\bm{k}})$, and $T_{\\bm{k}}(t)=(\\langle d_{\\bm{k}} \\rangle, \\langle d^{\\dagger}_{-\\bm{k}} \\rangle)$ are the classical amplitudes of the magnon fields, and $\\tilde{\\Omega}_{\\bm{k}}$ is the dynamical matrix, which has the eigenvalues:\n\\begin{equation}\\label{}\n\\begin{aligned}[b]\n\\omega_{\\bm{k},\\pm}=\\frac{\\omega_{\\bm{k}}-\\omega_{-\\bm{k}}}{2}-\\frac{i\\gamma}{2}\\pm\\sqrt{\\frac{(\\omega_{\\bm{k}}+\\omega_{-\\bm{k}}-\\Omega_0)^2}{2}-\\epsilon^2},\n\\end{aligned}\n\\end{equation} \nwhere $\\gamma$ denotes a linear dissipative damping, which is dominated by $H_{imp}$, and $\\epsilon=g\\beta$ is the overall coupling strength. When the detuning term $\\omega_{\\bm{k}}+\\omega_{-\\bm{k}}-\\Omega_0\\approx0$, the imaginary part of $\\omega_{\\bm{k},+}$ becomes $\\epsilon-\\gamma\/2$, and the magnon density at $\\bm{k}$ is $\\langle d_{\\bm{k}} \\rangle \\propto e^{(\\epsilon-\\gamma\/2)t}$. When the coupling strength $\\epsilon$ exceeds the dissipation $\\gamma$, there is an exponential growth of the magnon density in mode $\\bm{k}$, and resonant amplification is achieved. This limits the growth in amplified magnon density can be modelled by a phenomenological nonlinear damping constant $\\eta$. Thus, the dynamical equation of magnon population $\\langle d_{\\bm{k}} \\rangle$ is given by:\n\\begin{equation}\n\\begin{aligned}[b]\n&i\\frac{dI_p}{dt}=\\\\\n&\\left(\\tilde{\\omega}_{\\bm{k}}-\\tilde{\\omega}_{-\\bm{k}}-i(\\gamma+\\eta I_p)+i\\sqrt{4\\epsilon^2-(\\tilde{\\omega}_{\\bm{k}}+\\tilde{\\omega}_{-\\bm{k}})^2}\\right)I_p,\n\\end{aligned}\n\\end{equation}\nwhere $I_p=\\abs{\\langle d_{\\bm{k}} \\rangle}^2$ is the magnon population intensity. When the resonance condition $\\omega_{\\bm{k}}+\\omega_{-\\bm{k}}=\\Omega_0$ is reached, we can easily solve the dynamical equation and obtain the typical solution as:\n\\begin{equation}\n\\begin{aligned}[b]\nI_p=\\frac{1}{\\frac{\\eta}{2\\epsilon-\\gamma}+c_0e^{-2(\\epsilon-\\gamma)t}}.\n\\end{aligned}\n\\end{equation}\nWhen $t\\to\\infty$, we have the steady-state magnon intensity $I_p=\\frac{2\\epsilon-\\gamma}{\\eta}$, which is determined by the magnon-photon coupling strength and the dissipation coefficients $\\gamma$ and $\\eta$. The amplification dynamics is shown in Fig.~\\ref{fig:8}(a). Thus, we can employ the amplification scheme to populate the magnon at specific momenta in the BZ, which will give rise to a tunable many-body effect on magnon quasiparticles and transport properties. \n\\begin{figure}[t!]\n\\includegraphics[width=8.5 cm]{fig8.pdf}\n\\caption{Thermal Hall effect with parametric amplifications  (a) Magnon intensity is a function of parametric amplifications. (b) The schematic plot for thermal Hall measurements. (c) The position of amplified magnon in Brillouin zone. (d) The thermal Hall conductivity with different magnetic fields. We set $T\/J=0.2$ in the calculations.}\n\\label{fig:8}\n\\end{figure}\n\n\\subsection{Thermal Hall effect for Dirac magnon}\\label{sec.V_B}\nOne direct application of the parametric amplification is the modification of the magnon thermal Hall effect and the use of this effect as an experimental tool for probing the topological character of the magnon bands and their renormalization due to interaction effects. The schematic plot of thermal Hall measurement is shown in Fig.~\\ref{fig:8}(b). With the thermal gradient along the x-direction, one can measure the thermal Hall conductance along the y-direction. The calculation of the thermal conductivity $\\kappa^{xy}$ in a slab geometry yields~\\cite{Romhanyi2015, Matsumoto2011, Sun2021}: \n\\begin{equation}\\label{}\n\\begin{aligned}[b]\n\\kappa^{xy}=-\\frac{k_{B}^2T}{(2\\pi)^2}\\sum_{n}\\int d^2\\bm{k} c_{2}\\left(f(\\omega_n(\\bm{k}))\\right)\\Omega^{xy}_n(\\bm{k}),\n\\end{aligned}\n\\end{equation}   \nwhere $k_B$ is Boltzmann constant,$f(\\omega_n(\\bm{k}))$ is the magnon density distribution that, in the present case, has contributions from both thermally excited magnons -- whose population is given by the Bose-Einstein (BE) distribution function, $f(\\omega_n(\\bm{k}))=\\frac{1}{e^{\\omega_n(\\bm{k})\\beta}-1}$ -- as well as those generated by EM radiation. The function $c_2(u)$ is given by~\\cite{Matsumoto2011}: \n\\begin{equation}\\label{eq:}\n\\begin{aligned}[b]\nc_2(u)=(1+u)\\left[\\ln\\left(1+\\frac{1}{u}\\right)\\right]^2-\\left[\\ln(u)\\right]^2-2\\text{Li}_2(-u),\n\\end{aligned}\n\\end{equation} \nwith $Li_2(u)=\\sum^{\\infty}_{k=1}\\frac{u^k}{k^2}$ as the polylogarithm function, and $\\Omega^{xy}_n(\\bm{k})$ is the Berry curvature of $n$-th band. \n\nIn thermal Hall transport with the magnon amplifications, the interaction between the thermally generated magnons and light-induced magnons is delicate. To fully solve this transport problem, we need to employ a self-consistent solution. However, this is beyond the scope of this paper. For simplicity, we consider the system at low temperature, so that the band renormalization effect from thermally excited magnons is negligible. \nFurthermore, we ignore the second-order scattering processes from magnon-magnon interactions in the transport measurement due to the low temperature.\n\n With these simplifications, we focus on the thermal transport in the mean-field renormalized band by amplified magnons. We apply a monochromatic external EM field with energy $\\Omega_0=3.2J$ that can only produce magnon populations at the energy $\\omega_{\\bm{k}}=\\frac{\\Omega_0}{2}$ in the lower band, as shown in Fig.~\\ref{fig:8} (c). The advantage of this scheme is that the amplified magnons with the energy $\\Omega_0=1.6J$ do not contribute to the thermal Hall signal due to the zero Berry curvature at their locations, thus we can only have the thermal Hall response from the thermally excited magnons. \nWe use the magnon intensity $I_p$ as the controllable variable, and the calculated thermal Hall conductivity is shown in Fig.~\\ref{fig:8}(d). One can see that each curve has a transition point $I^c_p$ where the thermal Hall signal is going to reverse the signs. This originates from the topological phase transitions that are discussed in Sec.~\\ref{sec.III_B}. The thermal Hall signals change the sign when the Chern number of the magnon band is reversed. An increase in the thermal Hall signal is expected to occur before the transition point due to the reduction of the bandwidth. When $I_p$ is close to the transition point, the topological gap is closed, thus the signal becomes zero. The thermal Hall signal sharply increases after the reopening of the topological gap.\n\nWe also show that the transition points are magnetic field-dependent. The increase of the magnetic field gives rise to a larger critical point $I^c_p$ as shown in Fig.~\\ref{fig:8}(d). Thus with a larger magnetic field, one may need a higher magnon intensity to achieve the topological phase transition. Therefore the thermal Hall measurement can be a useful experimental tool for exploring the interaction-induced topological phase transition.  \n\n\n\\section{Conclusions}\\label{con}\nTo summarize, we show much interesting physics arises from the magnon systems with non-trivial DM interaction, when the effect of magnon-magnon interaction is properly accounted for. Our perturbative calculation shows the possibility of magnon band engineering exploiting such interaction, especially when the magnon density can be flexibly induced in different parts of the BZ with parametric amplification by the electromagnetic field. Such band engineering allows us to experimentally tune the bandwidth and the Berry curvature distribution. With properly designed amplification schemes, we can even induce band inversions leading to topological phase transitions. The transport properties of magnons under either a magnetic field gradient or a thermal gradient are very sensitive to the geometric and topological properties of the band. This allows us to propose a number of scenarios to experimentally probe the interplay between geometry, topology, and many-body effects in magnon systems.\n\nThe interaction-induced reduction of bandwidth and shift of the Dirac points can be detected by well-established techniques including Brillouin light scattering, inelastic X-ray scattering, and inelastic neutron scattering. By combining the computed quantum geometric tensor and the interaction-induced Haldane mass term, the thermal Hall conductivity of the magnons provides a direct diagnosis of the geometric and topological properties of the magnonic Chern bands by the external pumping field. More importantly, we can achieve flat bands with non-trivial topology, that provides a realistic platform for the study of strongly correlated bosonic systems such as superfluid-insulator transitions and flat band Bose-Einstein condensation. \n\nBy applying a second-order perturbation theory, we also reveal that the magnon scattering rate is significantly enhanced near the Dirac point with the DMI, which has not been studied before. Such scattering from magnon-magnon interaction can play an important role in the stability of the Dirac magnons and their decay dynamics. This is especially the case for systems with the intrinsic large interaction and small bandwidth, and our study highlights that the magnonic Dirac materials could serve as useful platforms for many-body physics arising from lattice bosons with non-trivial band topologies.\n\n\\begin{acknowledgements}\nB.Y. would like to acknowledge the support from the Singapore National Research Foundation (NRF) under NRF fellowship award NRF-NRFF12-2020-0005, and a Nanyang Technological University start-up grant (NTU-SUG). P.S. acknowledges financial support from the Ministry of Education, Singapore through MOE2019-T2-2-119.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Introduction}\n\n\\indent \nCultural Heritage assets (monuments, artefacts and sites) suffer from on-going deterioration through natural disasters, climate change and human negligence or interventions. Monuments are defined as structures created by a person or event and they symbolize a historic period of the corresponding place due to its artistic \\cite{rallis2017extraction}, historical, political, technical or architectural importance \\cite{moropoulou2013non}. UNESCO considers as a first priority the preservation and valorisation of the tangible\/intangible Cultural Heritage and applies innovative techniques for the capturing, digitizing, documenting and preserving prestigious monuments \\cite{marrie2008unesco}, \\cite{adamopoulos2017multi}. \n\nEarly detection of decay and deterioration is essential to preserve monuments. Material degradation leads to the failure of the the buildings components. Non-destructive techniques utilized for detection of monument decay. Most of these techniques come from the scientific fields of computer vision, whose great goal is the extraction of information regarding regions of interest (ROIs) from images or sequences of images \\cite{forsyth2012computer}. \n\nThe state-of-the-art for 3D\/4D documentation and modelling of complex sites utilizes multiple sensors and  technologies (e.g., LIDAR, photogrammetry, in-situ surveying, hyperspectral sensors) in order to define the preservation status of the monument\\cite{maltezos2018building}. The efficient use of these tools can give significantly better material detection and object recognition, and thus to identify even the smallest differences in spectral signatures of various objects. This continuous development of new sensors, data capturing methodologies, computer vision algorithms,  multi-resolution 3D\/4D representations and the improvements of existing ones are contributing significantly to the growth of the interdisciplinary cultural heritage domain. \n\nHyperspectral images are more suitable than RGB ones since they provide a large amount of information (high-spatial and high spectral resolution), allowing identifying the screened materials based on their chemical composition rather than only their size, shape, and visible colour \\cite{makantasis2015deep}. Moreover, the recent advancement of sensors technologies has led to the development of hyperspectral imaging sensors with higher spectral and spatial resolution on-board various satellite, aerial, UAV and ground acquisition platforms.\n\n\nIn our study, we exploit image clustering techniques on hyperspectral images to detect and evaluate the corrosion of the stones on cultural heritage assets. In more details, an automated mechanism is proposed for the detection of the ROIs in an unsupervised way. In other words, the evaluation of specific ROIs identifiability, using unsupervised clustering techniques, is being attempted.\n\n\n\n\n\n\n\n\n\\section{Related work}\n\nA plethora of methods for assessing and detecting the deterioration of stone monuments are available to researchers \\cite{grilli2019classification}. Those methods are distinguished into (a) destructive and (b) non-destructive techniques. The main drawback of the destructive approach is that a valuable  piece of monument structure is taken \\cite{fitzner2002damage}. On the other hand, the non-destructive approaches utilize methods that extract features of the examined surface in order to detect cracks,  defects in the architectural surface and material degradation.  \n\nIn \\cite{moropoulou2013non}, the authors proposed a method to identifying the exterior and interior surface flaws. This approach provides elastic features of the architectural structure materials in order to detect  the crack and inclusion in the building taking into consideration the affected layer inside the material.\nIn \\cite{parida2018fuzzy}, the authors introduced a fuzzy clustering approach for extracting the local variance feature from an image. This method applied to define the transitional features implementing hybrid segmentation. \nIn \\cite{moropoulou2018multispectral}, the authors exploited infrared thermography to diagnose materials decay taking into account different historical periods.This approach is used as a tool in the diagnostic level, for the detection of invisible superficial cracks or\/and disparities, as well as the revelation of moisture presence within structures.\n\nIn the case of cultural heritage, techniques such as clustering can be applied from archaeological artifacts to the entire archaeological site. \nIn \\cite{Pozo} the authors used multispectral images from different geomatic sensors, trying to define different construction materials and the main pathologies of Cultural Heritage elements by combining active and passive sensors recording data in different range. During this study, the unsupervised clustering method K-means is proposed. The results shows that an ideal sensors calibration can provide more accurate clustering. In \\cite{APOLLONIO201889} the authors used 3D models and data mapping on 3D surfaces in the context of the restoration documentation of Neptune's Fountain in Bologna. In \\cite{oses2014image} the authors used machine learning classifiers, support vector machines and classification trees for the masonry classification. In \\cite{messaoudi2018ontological} the authors developed a correlation pipeline for the integration of semantic, spatial and morphological dimension of a built heritage.\n\n\n\n\\begin{figure}\n\\includegraphics[width=0.9\\linewidth]{imgs\/schematics\/clustering_proc.png} \n\\caption{The proposed methodology workflow. Firstly, the original images are being converted to flat image arrays. At a second step, these flat images are used as input to clustering methods. Then, the clustering labels are produced. Finally, the clustered images are being constructed by the clustering labels and the performance metrics are being calculated for each clustering method.}\n\\label{fig:methodology}\n\\end{figure}\n\n\\section{Methodology Overview}\n\nCluster based machine learning approaches have been used for the detection of the wall corrosion, regarding the aforementioned historic monument. As described in \\cite{HUANG2014293}, clustering is an unsupervised learning technique that is being applied to data in order to group them into clusters according to some common characteristics. Several well known clustering algorithms were applied to the images of this study like K-means, Meanshift, Spectral, Birch, DBSCAN and Optics \\cite{Slawomir}.  \n\nThe data set, used for the purposes of this study, contains hyperspectral images with 42 channels, which represent the several frequencies of the electromagnetic spectrum. The first step of the proposed approach is the conversion of the initial images to flat image arrays, as shown in Figure \\ref{fig:methodology}. At this stage, the image content is in the appropriate form defined by the clustering methods. The application of the clustering algorithms bring as result an array with labels that correspond to the produced clusters. Finally, these arrays of labels are used to build the so-called clustered images and to calculate the performance metrics of each clustering method.\n\n\\subsection{Our contribution}\nThe major outcome of this study is the development of an automated mechanism for the detection of several deterioration types on historical walls. To achieve this, a pipelined approach was followed for the decomposition of the initial images, the clustering fitting procedure, the construction of the clustered images and their comparison with the annotated ones. A pixel based processing was applied to the images, offering a more detailed analysis.  Another aspect of this study is the hyperspectral images selection as part of the dataset, since a correlation between the wall deterioration and the additional information from across the electromagnetic spectrum of the image was attempted. \n\n\n\n\\section{The experimental Setup}\nOur proposed dataset consists of 14 final hyperspectral images of the  Fort of Saint Nicholas, with 42 channels for each image. These measurements carried out using the HyperView \\cite{garea2016hyperview} multi sensor hyperspectral sensing platform by 3D-one. This HyperView system is a dual head system combining one Visual (VIS) snap-shot camera and one Near Infrared (NIR) snap-shot camera, which are connected on a EP-12 board. These cameras acquire only one band per pixel (instead of acquiring all spectral bands for every pixel) while they acquire all the bands in small windows, 4x4 for the VIS head and 5x5 for the NIR head. \n\nThen, the raw images turn into a low resolution hyperspectral image (1\/4th or 1\/5th of the initial resolution for VIS and the NIR camera respectively), an intermediate hyperspectral image  and the final pansharepened hyperspectral image. \n\n\n\n\\begin{figure}[h]\n\n\\begin{subfigure}{0.45\\textwidth}\n\\includegraphics[width=0.9\\linewidth, height=3cm]{imgs\/Images\/agiosnik002211_RGB_real.png} \n\\caption{The original image}\n\\label{fig:nonannot}\n\\end{subfigure}\n\\begin{subfigure}{0.45\\textwidth}\n\\includegraphics[width=0.9\\linewidth, height=3cm]{imgs\/Images\/agiosnik002211_RGB.png}\n\\caption{The annotated image}\n\\label{fig:annotated}\n\\end{subfigure}\n\\caption{Less than 30\\% of the wall is affected by the erosion. In the left image a small part of the wall looks brighter from the sunrays so is not considered as damaged.}\n\\label{fig:orivsanot}\n\\end{figure}\n\n\n\n\\subsection{The annotation process}\nFirst step of the annotation process was the selection of the  hyperspectral image channels that correspond to the natural colours. The appropriate channel combination is the following triplet: 15th channel for Red, 6th channel for Green, and 3rd channel for Blue. Next step was the highlighting of the ROIs, using white colour. The rest of the image was coloured black in order for the damaged areas to be more clearly distinctive. \nIn Figure \\ref{fig:orivsanot}, the annotation process that implemented to one of the hyperspectral images, is presented. In the left photo (Figure \\ref{fig:nonannot}) the image with the natural colours is shown, while in the right one (Figure \\ref{fig:annotated}) the corresponding annotated image is depicted.\n\n\n\\subsection{Clustering areas depiction}\nThe corrosion of the stones could be characterised by the different colour and the roughness of the relevant area surface on the corresponding images of the dataset. Thus, different clustering techniques could offer alternative views of the same data. \nIn Figure \\ref{fig:original}, the original image of the monument wall is depicted. Figure \\ref{fig:K-means} presents the outcome of the K-means clustering algorithm, applied to that specific image, while Figures \\ref{fig:birch}, \\ref{fig:spectral} depict the Birch and the Spectral partitioning accordingly.\n\\begin{figure}[h]\n\n\\begin{subfigure}{0.45\\textwidth}\n\\includegraphics[width=0.9\\linewidth, height=3cm]{imgs\/Images\/agiosnik003218_RGB.png} \n\\caption{The original image}\n\\label{fig:original}\n\\end{subfigure}\n\\begin{subfigure}{0.45\\textwidth}\n\\includegraphics[width=0.9\\linewidth, height=3cm]{imgs\/Images\/agiosnik003218_Kmeans.png}\n\\caption{The K-means clustering method}\n\\label{fig:K-means}\n\\end{subfigure}\n\n\\begin{subfigure}{0.45\\textwidth}\n\\includegraphics[width=0.9\\linewidth, height=3cm]{imgs\/Images\/agiosnik003218_Birch.png} \n\\caption{The Birch clustering method}\n\\label{fig:birch}\n\\end{subfigure}\n\\begin{subfigure}{0.45\\textwidth}\n\\includegraphics[width=0.9\\linewidth, height=3cm]{imgs\/Images\/agiosnik003218_Spectral.png}\n\\caption{The Spectral clustering method}\n\\label{fig:spectral}\n\\end{subfigure}\n\n\\caption{In Figure (a), the original hyperspectral image is presented, while in Figures (b), (c), (d) the images produced by methods K-means, Birch and Spectral are presented accordingly.}\n\\label{fig:clustering}\n\\end{figure}\n\n\n\\section{Evaluation of Clustering methods}\n\\subsection{Clustering algorithms characterization}\nEssential part of this study is the characterization of the several clustering techniques regarding their performance. Metrics such as Calinski\u2013Harabasz \\cite{Calinski}, Davies\u2013Bouldin \\cite{4766909} indexes and Silhouette value \\cite{Papakostas} were calculated for the initial evaluation of the clustering methods. \n\nFor the calculation of these metrics, some definitions and assumptions should be provided. Let K denotes the number of clusters $\\{C_k\\}, k=0, 1, 2, ..., K$. Let $X = \\{x_1, x_2, ..., x_N\\}$ be a vector containing N objects, where $x_{ij}$ denotes the jth element of $x_i$. The grouping of all objects $x_i, i=1,2,...,N$ in K clusters can be defined as follows:\n\\begin{equation} \\label{eq:w}\nw_{ki} = \\begin{cases}\n1, \\hspace{0.5cm} iff x_i \\in C_k \\\\\n0, \\hspace{0.5cm} otherwise.\n\\end{cases}\n\\end{equation} \nEq. \\ref{eq:w} ensures the uniqueness of the object to cluster association, which is a valid case for both hierarchical and partitioning cluster analysis\n\nThe Calinski\u2013Harabasz index (CHI) is described by the Eq. \\ref{eq:chi}:\n\\begin{equation}\\label{eq:chi}\nCHI(k) = \\frac{T_B\/(K-1)}{T_W\/(K-1)}\n\\end{equation}\n\n\\begin{align*}\nWhere \\hspace{0.2cm}T_B = \\sum_{k=1}^{K}|\\bar{C_K}|\\lVert C_K-\\bar{x}\\rVert, \\hspace{0.5cm}\nT_W = \\sum_{k=1}^{K}\\sum_{i=1}^N w_{ki}\\lVert x_i-\\bar{C_K}\\rVert ^2\n\\end{align*}\n\n\n\nAccording to \\cite{Papakostas}, the maximum CHI value is associated with the optimal partitioning of the given data. By using constant number of clusters for all clustering methods, the most fitting one gives the maximum CHI value. The Davies\u2013Bouldin index (DBI) is an internal evaluation scheme, where the quality of the clustering is being examined according to information extracted directly from the given dataset. The DBI is defined by the Eq. \\ref{eq:db}:\n\\begin{equation}\\label{eq:db}\nDB(k) = \\frac{1}{K} \\sum_{k=1}^K R_K\n\\end{equation}\n\n\\begin{align*}\nwhere \\hspace{0.2cm}R_K = max\\Bigg(\\frac{S_k+S_j}{d_{kj}}\\Bigg), j=1,2, ..., K, j\\neq K, \\\\\nand \\hspace{0.2cm} d_{kj} = \\lVert \\bar{x_k}-\\bar{x_j}\\rVert \\label{eq:d}, \\hspace{0.5cm} S_K = \\frac{1}{\\sum_{i=1}^{N}w_{ki}}\\sum_{i=1}^{N}w_{ki}\\lVert x_i-\\bar{x_k}\\rVert\n\\end{align*}\n\n\nThe minimum DBI value is related to the best partitioning solution. Thus, by using constant number of clusters for all clustering methods, the most fitting one gives the minimum DBI value. The silhouette value shows the similarity of an object regarding the cluster it belongs, compared to other clusters. The silhouette value is described by the Eq.\\ref{eq:sil}\n\\begin{equation}\\label{eq:sil}\ns(x_i) = \\frac{b(x_i)-a(x_i)}{max(b(x_i),a(x_i))}\n\\end{equation}\nwhere $a(x_i)$ represents the average dissimilarity of the object with all the other data in the same cluster and $b(x_i)$ represents the lowest average dissimilarity of the object to any other cluster. Since, Silhouette value ranges from -1 to 1, a value close to 1 ensures that the object is well matched to its own cluster. \n\n\n\\subsection{Ground truth verification}\nTo evaluate the outcome of the clustering methods, a set of annotated images was used that denote the ground truth regarding the corrosion area of the initial images. So, a comparison between the annotated and the clustered image was performed, using accuracy,  precision, recall and f1 scores \\cite{rallis2018spatio}. The detailed procedure is depicted in Figure \\ref{fig:groundTruth} and is distinguished into (a) conversion of annotated and clustered images to flat image arrays, (b) assignation of colour triplets (RGB) to specific identifiers that represent the clustering labels and the corresponding clustering colour, (c) accuracy, precision, recall and f1-scores calculation and (d) design of the corresponding graphs. \n\n\nIn Figure \\ref{fig:groundTruthComp}, a simple example case is shown which offers a more descriptive view of the evaluation process. The initial images are being converted to RGB arrays. Each distinct RGB triplet is being assigned to a unique identifier that represents a specific cluster label (1-6) or one of the two distinct areas of the annotated images (0, 10). The two single-dimensional arrays are being adapted to the current clustering label which is under examination. Each identifier with number 10 of the annotated image single-dimensional array is being replaced by the identifier of the current clustering label. Each position of the clustered image single-dimensional array is being set to zero except from these which contain the same identifier of the current clustering label. The final single-dimensional arrays are used for the calculation of accuracy, precision, recall and f1 scores. This procedure is being repeated for each of the clustering labels (1-6). The most matching clustering label was extracted.\n\\begin{figure}[h]\n\\includegraphics[width=0.9\\linewidth]{imgs\/schematics\/groundTruthComparison.png} \n\\caption{The clustered images evaluation. The annotated and the clustered images are being converted to flat RGB arrays. Each distinct RGB triplet gets associated to a unique identifier, representing the cluster label. The single dimensional arrays with the unique identifiers are combined for the calculation of accuracy, precision, recall and f1 scores}\n\\label{fig:groundTruth}\n\\end{figure}\n\n\\begin{figure}[h]\n\\includegraphics[width=0.9\\linewidth]{imgs\/schematics\/groundTruthComparisonImgs.png} \n\\caption{Results evaluation process. The annotated and the clustered images are being converted to flat RGB arrays. The annotated image RGB triplets get associated with identifiers 0 and 10, while the clustered image RGB triplets get associated with identifiers 1-6. These two single dimensional arrays are being adapted to the current clustering label identifier and the process is being repeated for each identifier.}\n\\label{fig:groundTruthComp}\n\\end{figure}\n\n\\section{Experimental results}\nAs it was mentioned above, the initial evaluation of the clustring techniques was performed using the cluster indexes (Calinski\u2013Harabasz, Davies-Bouldin, Silhouette), which are characterized as internal metrics. According to Davies-Bouldin metric, Meanshift clustering technique presents the best partitioning quality, since its value is the closest to 0. At the same time, Meanshift seems to achieve better similarity among the objects of a common cluster because the Silhouette value is closer to 1 than in any other case (Figure \\ref{fig:db}).\n\nA secondary, more practical approach was used for the evaluation of the clustering methods by comparing the clustered images with the annotated ones and calculating the performance scores accuracy, precision, recall and f1 (Figures \\ref{fig:af}, \\ref{fig:pr}). From this evaluation, it arises that DBSCAN was the most fitting technique, since it achieved the best scores. Consequently, despite the better internal performance metrics of Meanshift, DBSCAN proved to be the technique that identified more sufficiently the ROIs of the given images.  \n\n\\begin{figure}[th]\n\\includegraphics[width=0.8\\linewidth]{imgs\/plots\/db.png} \n\\caption{In this graph, the average cluster indexes (Calinski\u2013Harabasz, Davies-Bouldin, Silhouette) for each clustering technique is presented. As shown, Meanshift method achieved the best performance, since its DBI is closest to 0 and Silhouette value is the closest to 1.}\n\\label{fig:db}\n\\end{figure}\n\\begin{figure}[th]\n\\includegraphics[width=0.8\\linewidth]{imgs\/plots\/af.png} \n\\caption{In this graph, the average accuracy and f1 performance scores are presented. As shown, DBSCAN achieved the best results.}\n\\label{fig:af}\n\\end{figure}\n\\begin{figure}[th]\n\\includegraphics[width=0.8\\linewidth]{imgs\/plots\/pr.png} \n\\caption{In this graph, the average precision and recall performance scores are presented. As shown, DBSCAN achieved the best results.}\n\\label{fig:pr}\n\\end{figure}\n\n\\section{Conclusion}\nIn our approach, we investigated whether the spectral signatures suffice to distinct various ROIs  using trivial unsupervised machine learning techniques. Therefore, we investigate various clustering approaches to identify the feasibility of such methods.\nAccording to the results, unsupervised techniques can provide an early, still appropriate mechanism regarding the identification of certain regions of an image (monitoring, defect recognition).\n\n\\section*{Acknowledgement}\nThis paper is supported by the European Union Funded  project Hyperion \"Development of a Decision Support System for Improved Resilience \\& Sustainable Reconstruction of historic areas to cope with Climate Change \\& Extreme Events based on Novel Sensors and Modelling Tools\" under the Horizon 2020 program H2020-EU.3.5.6., grant agreement No 821054. \n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOver the last decades an extensive amount of video data is being recorded and stored. Since access to and exploitation of such data is difficult to monitor let alone prevent, appropriate measures need to be taken to ensure that such data is not misused and the privacy of people is adequately protected.\n\n\n\n\n\nA popular approach towards privacy protection in image and video data is the use of deidentification. Ribari\\'{c} et al. \\cite{Ribaric_Review2016} define deidentification as the process of concealing or removing personal identifiers from source content in order to prevent disclosure and use of data for unauthorized purposes. For video data, for example, this may translate to ``blurring'' or ``pixelation'' of the facial areas~\\cite{PixelizationAndBlurring}, both of which represent early deidentification examples. These naive methods are typically useful for preventing humans from recognizing subjects in videos, but are far less successful with automated recognition techniques, where repeating the (naive) deidentification process on the test data still enables automated recognition, i.e., parrot attack~\\cite{Newton_original}. Thus, for successful deidentification of images and videos, more advanced techniques are needed.\n\nAnother shortcoming of naive deidentification techniques is the fact that all information contained in the data is typically removed even if the information is not related to identity. This raises the question of data utility. If the deidentified data is to be useful for purposes that do not require identity information, but, for example, rely on gender or age information (e.g., customer-profiling applications in shopping malls), this information needs to be preserved even after deidentification. Recent deidentification approaches, therefore, focus on ways of removing identity information from images and videos, while still retaining other non-identity related information~\\cite{Gross_utility},~\\cite{Sim2015}.\n\nIn this paper we follow these recent trends and present a new deidentification approach exploiting generative neural networks (GNNs), which represent contemporary generative models capable of synthesizing photo-realistic artificial images of any object (see, e.g., \\cite{Goodfellow_GAN2014}, \\cite{Dosovitskiy_Chairs2015}, \\cite{VAE_GAN}) based on supplied high-level information. Similarly to existing deidentification techniques, we replace the original faces in the input data with surrogates generated from a small number of identities. However, instead of synthesizing the surrogate faces through pixel averaging as in prior work, we use a GNN to combine identities and generate artificial surrogates for deidentification. The flexibility of the GNN also allows us to parameterize the generation process with respect to various appearance-related characteristics and synthesize faces under different appearances (under varying pose, with different facial expressions, etc.). This property ensures that our deidentification approach is able to conceal the identity of individuals, but also to preserve the utility of the data.\n\nWe demonstrate the feasibility of the proposed deidentification pipeline through extensive experiments on the ChokePoint dataset~\\cite{Wong_Chokepoint2011}. Our experimental results show that GNNs are a viable solution for the problem of face deidentification and are able to generate realistic, visually convincing deidentification results. Furthermore, the deidentified faces offer a suitable level of privacy protection as evidenced by experiments with a number of contemporary recognition models as well as humans.   \nIn summary, we make the following contributions:\n\\begin{itemize}\n\\item We introduce a face deidentification pipeline that exploits GNNs to produce artificial surrogate faces for deidentification and offers a level of flexibility in the generation process that  is not available with existing deidentification approaches.\n\\item We present a qualitative evaluation of the proposed pipeline with challenging data captured in a real surveillance scenario and discuss the advantages and limitations of our deidentification approach.\n\\item We demonstrate the efficacy of the proposed pipeline in comprehensive quantitative experiments with several state-of-the-art recognition techniques from the literature and human annotators.\n\\end{itemize}\n\n\n\n\n\n\\section{Related work}\n\nIn this section we review the most important work related to our deidentification pipeline.\nFor a more comprehensive review please refer to the surveys by Ribari\\'{c} et al.~\\cite{Ribaric_Review2016},~\\cite{Ribaric_Re}.\n\n\n\nExisting approaches to deidentification often implement formal privacy protection models such as $k$-anonymity~\\cite{Sweeney_Kanonym}, $l$-diversity~\\cite{Machana_Ldiversity}, or $t$-closeness~\\cite{Li_Tcloseness}. Among these, the $k$-anonymity models have likely received the most attention in the area of face deidentification and resulted in the so-called $k$-same family of algorithms~\\cite{Newton_original},~\\cite{Gross_utility},~\\cite{Gross_MFM2008}. These algorithms operate on a closed set of static facial images and substitute each image in the set with the average of the closest $k$ identities computed from the same closed set of images. Because several images are replaced with the same average face, data anonymity of a certain level is guaranteed. A number of $k$-same variants was presented in the literature, including the original $k$-same algorithm~\\cite{Newton_original},  $k$-same-select~\\cite{Gross_utility}, and  $k$-same-model~\\cite{Gross_kSameM} to name a few. The majority of these techniques is implemented using Active Appearance Models (AAMs).\n\n\n\n\n\nAnother example of a deidentification technique using AAMs was recently presented by Joura\\-bloo et al. in \\cite{Jourabloo_AAM2015}. Here, the authors combine facial-attribute and face-verification classifiers in a joint objective function. By optimizing the objective function, optimal weights are estimated such that the deidentified  and the original image have as many common attributes as possible, but at the same time are classified as two different subjects.\n\nA $q$-far deidentification approach, which is also AAM based, but does not follow the $k$-same principle (since the surrogate faces are mutually different), was proposed by Samar\\v{z}ija and Ribari\\'{c} in~\\cite{Samarzija_2014} and combines face deidentification with pose estimation.  The authors cover different facial orientations by fitting multiple AAMs and achieve anonymity by replacing the original faces with surrogates that are sufficiently far (i.e., $q$-far) from the initial identities.\n\nSim and Zhang present a method for controllable face deidentification in \\cite{Sim2015}. They demonstrate a high  degree of control over different attributes (such as identity, gender, age, femininity or race) of the deidentified faces and similar to our approach are able to alter or retain specific aspects of the target appearance.\n\nAnother example related to the AAM-based techniques was recently proposed by Sun et al. \\cite{Sun2015}. Here, the authors propose the $k$-diff-furthest algorithm, which is related to the $k$-anonymity model and $k$-same family of techniques, but differs in its ability to  track individuals in the deidentified video, since the deidentified faces have distinguishable properties.\nThe experimental evaluation performed by the authors shows that the algorithm is capable of maintaining the diversity of the deidentified faces and keeps them as distinguishable as their original faces. However, the approach does not deal with the data utility aspect, e.g., expression preservation.\n\nDifferent from AAM-based deidentification approaches, Brki\\'{c} et al.~\\cite{Brkic_ArtBased2016} propose a deidentification method based on style-transfer. The authors describe a pipeline that enables altering the appearance of faces in videos in such a way that an artistic style replacement is performed on the input data, thus making automatic recognition more difficult. Another interesting deidentification approach was presented by Chriskos et al. in \\cite{Chriskos2015}, which, in contrary to most deidentification methods, hinders the recognition only for automatic recognition algorithms, but not human observers. The authors utilize projections on hyperspheres in order to defeat classifiers, while preserving enough visual information to enable human viewers to correctly identify individuals. \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Deidentification with generative deep neural networks}\n\nHere we present a detailed description of our deidentification pipeline, which exploits generative neural networks (GNNs) to conceal the identity of people in the image data.\n\n\\subsection{Overview}\n\nA block-diagram of our deidentification pipeline is presented in Fig.~\\ref{fig:pipe}. The procedure starts with a face detection step  that takes an image or video frame as input and then locates candidate facial regions in the input data for deidentification. For each detected region, a feature vector is computed using a state-of-the-art deep face recognition network, i.e., the VGG network from~\\cite{Parkhi_Recog2015}, and matched against a fixed gallery of $M$ subjects. Based on this matching procedure, the $k$-closest identities ($k\\ll M$) from the gallery data are selected and fed to our generative network to synthesize an artificial face with visual characteristics of the selected $k$ identities. Finally, the artificially generated (deidentified) face is blended into the input image (or frame) to conceal the original identity.      \n\n\t\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=\\columnwidth]{pipeline.pdf}\n\\caption{Block diagram of our deidentification pipeline. The procedure uses a generative neural network to generate synthetic faces that can be used for deidentification. Each generated face is a combination of $k$ identities from the gallery data that are closest (i.e., most similar in the feature space) to the input face.}\n\\label{fig:pipe}\n\\end{figure}\n\nOne appealing characteristic of our deidentification pipeline is the flexibility of the GNN, which is able to synthesize high-resolution, realistic-looking faces under various appearances. Here, the generation process is governed by a small number of appearance-related parameters that control the visual characteristics of the synthesized faces, such as pose, skin color, gender, identity, facial expression, and alike. Thus, with this setup, we are able to generate artificial faces with predefined identities, facial expressions, gender, and so forth, or alter any of these at the time. For instance, if our goal is to preserve facial expressions of faces, we could automatically recognize facial expressions from the input image and use the recognition result as input to the GNN. The network would then generate a synthetic image with the predefined expression. A similar procedure could be used to retain or alter any visual characteristic of the input faces and contribute towards the preservation of data utility, which is one of the main goals of contemporary deidentification technology. \n  \n  \nEven though our approach is similar in nature to the $k$-same family of algorithms~\\cite{Newton_original},~\\cite{Gross_utility},~\\cite{Ribaric_Re} that implement the $k$-anonymity protection model~\\cite{Sweeney_Kanonym}, there are important differences that invalidate some of the $k$-anonymity model assumptions. For example, our technique does not operate on a subject-specific set of images (with one image per subject only) nor is it limited to closed set scenarios. Thus, the anonymity guarantees associated with the $k$-same family of algorithms do not apply to our approach, so we use extensive experimental validation to demonstrate the feasibility of the developed deidentification pipeline. \n\n\n\\subsection{Face detection and target identity estimation}\n\nOur deidentification procedure starts with a standard face detection step using the off-the-shelf Viola-Jones face detector from OpenCV \\cite{ViolaJones_Detector2001}. The detector process the input image or video frame and returns bounding boxes of all detected faces. Each detected region is then processed separately in a sequential manner and a 4096-dimensional feature vector is extracted from each region using the pre-trained 16-layer VGG face network from~\\cite{Parkhi_Recog2015}. For this step, the output of the last fully-connected layer of the VGG face network is considered as a feature vector. Each computed feature vector is matched against a gallery of feature vectors using the cosine similarity.  \n\nThe matching procedure between the feature vector extracted from a region of the input image and the gallery of feature vectors results in an ordered list of similarity scores. Based on this list, we identify the $k$ most similar identities (where $k\\ll M$) in our gallery and feed these to the generative network for surrogate-face generation. The idea of generating a synthetic surrogate face based on $k$ closest identities is similar in essence to the established family of $k$-same family of algorithms, except for the fact that the final face is in our case entirely generated by a GNN.\n\nTo generate the gallery for our deidentification approach, we process the images from the Radboud Faces Database (RaFDB) \\cite{Langner_RAFDB2010} with the VGG network during an offline extraction step and store templates of all $M$ identities of the RaFDB dataset in the so-called feature database, FeatDB in Fig.~\\ref{fig:pipe}, of our pipeline. These feature vectors correspond to a finite set of facial identities that can be used for generating new (surrogate) faces for deidentification.  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Face generation \\label{sub:gen}}\n\nThe generative part used in our deidentification pipeline comprises a powerful GNN\nrecently introduced by Dosovitskiy et al.~ in \\cite{Dosovitskiy_Chairs2015} for generating 2D images from 3D objects under different viewpoint angles and various basic transformations. The same architecture was later extended to another application involving face generation\\footnote{https:\/\/github.com\/zo7\/deconvfaces} by Michael D. Flynn. In this work, we use the same approach (and architecture) to GNNs\nand train our own network for deidentification. \n\nAs already suggested in the previous section, we want the network to be able to generate surrogate faces of identities or mixtures of identities contained in the feature database of our deidentification pipeline. Thus, we use RaFDB for GNN training. Once fully trained, the network is able to generate new artificial faces in accordance with the supplied identities,\nbut also in line with other appearance-related parameters that are exposed during the training stage. The generation process can be described as follows:\n\\begin{equation}\n\\mathbf{x} = \\text{GNN}(\\mathbf{y},\\mathbf{z}),\n\\end{equation}\nwhere $\\mathbf{x}$ is the output of the generative network, GNN, $\\mathbf{y}$ stands for an identity-related parameter vector that encodes information about the $k$-closest identities returned by our matching procedure, and $\\mathbf{z}$ denotes a parameter vector that guides the generation process and affects specific characteristics of the visual appearance of the generated output. \n\nIn the above equation, the vector $\\mathbf{z}$ can in general relate to any appearance characteristic that is appropriately annotated in the training data. Since RaFDB is annotated with respect to different facial expressions, \nwe train our network to generate faces with different identities as well as facial expressions. However, the number of appearance-related parameters exposed by the network is not limited and is in general defined by the labels available with the training data.\n\nThe generative network consists of fully-connected and deconvolutional layers as described in detail in~\\cite{Dosovitskiy_Chairs2015}. Each deconvolution layer includes one upscaling layer followed by a convolution layer. Stabilization of the error during training is achieved by adding batch normalization layers and leaky-rectifier-unit-activation functions between the deconvolution layers. The loss of the training procedure is calculated as the pixel-wise mean square difference between the ground truth image and the artificially generated image. The training is performed with 456 images from the RaFDB dataset on a desktop PC with Intel(R) Core(TM) i7-5820K CPU (3.30GHz), 32GB of RAM, utilizing a TitanX GPU and takes around 24 hours.\n\nWith the current training dataset, our network is able to interpolate between different identities as well as different facial expressions. By combining identities, it can generate new averaged faces, almost without ghosting effect as shown in Fig.~~\\ref{fig:generated_outputs}. By using the appearance-related parameter vector, $\\mathbf{z}$, it can also preserve the utility of the input data, e.g., in the form of facial expressions, which can be retained or altered in regards to the original input. A few sample faces, generated with our GNN are shown in Fig.~\\ref{fig:generated_outputs}. Here, the first block of images shows artificial images generated based on a single identity (i.e., $k=1$), the second block shows images computed based on two identities (i.e., $k=2$) and the last block shows sample faces generated from four identities (i.e., $k=4$). Note that the generated faces get closer to an average appearance as the number of identities increases, yet they still appear realistic and feature no ghosting effects.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1\\textwidth]{figures\/generated-examples.pdf}\n\\caption{Sample outputs from the generative network -- identity mixing with $k$ identities using $k=1$, $k=2$ and $k=4$ is displayed from left to the right, respectively. With an increasing number of identities the generated faces converge towards an average appearance, yet they are still realistic and without ghosting effects. The first label below each images refers to the generated facial expression and the second to the identity \/ identities used during the generation process.}\n\\label{fig:generated_outputs}\n\\end{figure}\n\n\n\\subsection{Face replacement}\n\nThe last step of our deidentification pipeline is face replacement, during which the generated surrogate face is blended into the input image. The face replacement step starts with facial-landmark estimation using the approach from \\cite{Kazemi_One2014}. The landmarks are detected in both, the generated face and the detected original, input face. Using both sets of landmarks, we then estimate a perspective transformation that aligns the landmarks of the artificially generated face and the landmarks of the original face using RANSAC. The generated face image is then warped using this transformation in order to adjust the synthetic content over the landmarks of the original image. This correction is needed in all cases, where faces in the input images are not entirely frontal.\n\nFollowing the geometric corrections, we apply a second post-processing procedure that discards the background of the generated faces. During this step, simple skin-color segmentation is performed using the \\textit{upper} and \\textit{lower} boundaries in the HSV color-space that define the skin intensities, i.e., \\textit{lower}=$[0, 10, 20]$, \\textit{upper}=$[200, 255, 255]$. Pixels with values within the defined range are retained and the rest is discarded. Erosion and dilation are then used to remove possible isolated regions that do not belong to the facial area. With this step we make sure that most of the background around the generated facial area is removed and only the facial region without the gray-colored background, as seen in Fig. \\ref{fig:rep}, is swapped during deidentification.\n\nIn the last step, the warped and segmented synthetic face image is blended with the original image. Blending is performed with a Gaussian kernel mask: \n\\begin{equation}\ng(x,y) = e^{{{ - (\\left( {x - \\mu_x } \\right)^2 + \\left( {y - \\mu_y } \\right)^2 ) } \\mathord{\\left\/ {\\vphantom {{ - \\left( {x - \\mu } \\right)^2 } {2\\sigma ^2 }}} \\right. \\kern-\\nulldelimiterspace} {2\\sigma ^2 }}},\n\\end{equation}\nwhere $ \\mu_x = s\/2$, $\\mu_y = s\/2$, $\\sigma = s\/6$, $s=\\min(w, h)$, $w$ and $h$ stand for the dimensions of the generated image, and $x$ and $y$ denote image coordinates. This online generated kernel then serves as a weight mask when blending the original and generated image pixels. The kernel is warped using the same homography transformation in order to ensure the best possible face alignment and a suitable level of naturalness of the final output. The replacement procedure is illustrated in Fig.~\\ref{fig:rep}. Here, the first image shows the Gaussian weight mask, the second image shows the initial output of the generative network, the third image shows the Gaussian mask modified with the result of the geometric correction and segmentation step, the fourth image presents the adjusted (synthetic face) and the last two images depict an original and deidentified frame from our test dataset.\n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=1\\textwidth]{figures\/replacer.png}\n\\caption{Illustration of the replacement procedure (from left to right): the Gaussian mask, the artificially generated face image, the modified Gaussian mask, geometrically corrected synthetic face without background, sample frame, and deidentified frame.}\n\\label{fig:rep}\n\\end{figure}\n\n\n\n\\section{Experiments and results}\\label{sec:experimenti}\n\nIn this section we present experimental results aimed at demonstrating the merits of our deidentification pipeline. We first discuss the experimental dataset and performance metrics and then present qualitative as well as quantitative results.\n\n\\subsection{Dataset, experimental setup and performance measures}\\label{sec:dataset}\n\nTo evaluate the performance of our deidentification approach, we use the ChokePoint dataset~\\cite{Wong_Chokepoint2011}, which contains video footage captured in a typical surveillance scenario. People in the videos were recorded while walking through a portal above which an array of 3 cameras was placed. The ChokePoint videos exhibit variations across illumination conditions, pose, image sharpness, and alike and are well suited for studying the performance of deidentification technology.\n\nThe ChokePoint dataset contains 48 video sequences with a total of 64,204 frames. The videos feature 25 subjects walking through the first portal and 29 subjects walking through the second portal. For our experiments, we partitioned the video sequences into two distinct subsets. The first subset contained 24 sequences with people in mostly frontal poses, while the second subset contained the remaining 24 sequences with people in less frontal poses, i.e., profile frames. We refer to the former subset as \\textit{original} and to the latter as \\textit{profile}  from hereon. The video sequences from the original subset were subjected to our deidentification approach and stored for the experimental evaluation.\n\nTo measure the efficacy of the developed deidentification pipeline, we conduct four types of verification experiments with a 10-fold cross-validation protocol. During each fold, we  perform 300 legitimate (matching, client) and 300 illegitimate (non-matching, impostor) verification attempts. The different types of verification experiments are briefly outlined below:\n\\begin{itemize}\n\\item \\textbf{Original vs. original:} In this experiment we sample 300 image pairs for the legitimate verification attempts and 300 image pairs for the illegitimate verification attempts for each experimental fold from video sequences of the original subset. The goal of this experiment is to establish the baseline performance of the recognition techniques considered in our experiments. Since video frames are sampled from the same set of videos, this experiment may be biased towards higher performances, since the appearance variability between frames is limited.   \n\\item \\textbf{Original vs. profile:} In this experiment we construct the image pairs for the legitimate and illegitimate verification attempts of each fold from images taken from the original and profile subsets. Here, the first image in the pair is always sampled from the original subset and the second is always sampled from the profile subset. Because the two subsets contain distinct video sequences, this experiment better reflects the baseline performance of the recognition techniques considered in our experiments.     \n\\item \\textbf{Deidentified vs. original:} This experiment is equivalent to the \\textit{original vs. original} experiment with the difference that the first image of each image pair is replaced with its deidentified version. Thus, the experiment is meant to measure the efficacy and performance of the proposed deidentification procedure. All verification attempts of all 10 cross-validation folds in this experiment have a direct correspondence in the original vs. original experiment and, therefore, clearly demonstrate the effect of deidentification on the verification performance. \n\\item \\textbf{Deidentified vs. profile:} The last experiment follows the same approach as the \\textit{original vs. profile} experiment, but replaces the video frames from the original subset with its deidentified version. The goal of this experiment is  again to demonstrate the feasibility of our deidentification approach.   \n\\end{itemize}\n\nWe report performance with standard performance metrics and graphs. Specifically, we present Receiver Operating Characteristic (ROC) curves~\\cite{ROC_curves},~\\cite{Ziga_Hindawi}, which plot the value of the verification rate (VER) against the false acceptance rate (FAR) for different values of the decision threshold, and report a number of scalar performance metrics for all experiments, i.e., the equal error rate (EER), which is the operating point on a ROC curve, where FAR and 1-VER are equal, the verification rate at 1\\% FAR (VER-1) and the area under the ROC curve (AUC)~\\cite{Gajsek_ROC}. Because we use a 10-fold cross validation protocol, we report all metrics in the form of the mean and standard deviation computed over all experimental folds~\\cite{Neurocomputing}.   \n\n\\subsection{Qualitative evaluation}\n\nWe first demonstrate the efficacy of our deidentification approach with a few qualitative examples in Fig.~\\ref{fig:deid_examples_good}. Here, each row shows a few frames from a video sequence of the ChokePoint dataset and the corresponding deidentification result. In each image pair, the left image represents the original frame and the right image its deidentified counterpart. We can see that for the most part the deidenfied faces generated by the generative network appear natural and realistic.\nIn this case, substituted identities are generated from 2 most similar identities (i.e.  $k=2$). \n\n\\begin{figure}[tb]\n\\centering\n\\begin{minipage}{\\textwidth}\n  \\centering\n  \\includegraphics[width=.163\\linewidth]{orig11}\n  \\hspace{-0.2cm}\n  \\includegraphics[width=.163\\linewidth]{deid11}\n  \\includegraphics[width=.163\\linewidth]{orig12}\n  \\hspace{-0.2cm}\n  \\includegraphics[width=.163\\linewidth]{deid12}\n  \\includegraphics[width=.163\\linewidth]{orig13}\n  \\hspace{-0.2cm}\n  \\includegraphics[width=.163\\linewidth]{deid13}\n\\end{minipage}%\n\\vspace{0.05cm} \\\\\n\\begin{minipage}{\\textwidth}\n  \\centering\n  \\includegraphics[width=.163\\linewidth]{orig21g}\n  \\hspace{-0.2cm}\n  \\includegraphics[width=.163\\linewidth]{deid21g}\n  \\includegraphics[width=.163\\linewidth]{orig22g}\n  \\hspace{-0.2cm}\n  \\includegraphics[width=.163\\linewidth]{deid22g}\n  \\includegraphics[width=.163\\linewidth]{orig23g}\n  \\hspace{-0.2cm}\n  \\includegraphics[width=.163\\linewidth]{deid23g}\n\\end{minipage}%\n\\caption{Qualitative examples of deidentified frames. Each row shows a few example frames from a video sequence of the ChokePoint dataset (left image of each pair) and the corresponding deidentification results (right image of each pair). Note how the generative network is able to generate realistic renderings of faces for deidentification.}\n\\label{fig:deid_examples_good}\n\\end{figure}\n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=1\\textwidth]{figures\/emo_deid.pdf}\n\n \n \n\n \n \n \n \n\n \n \n\n\n\\caption{Deidentified frames rendered with different facial expressions. Images in the first column represent original frames from a video sequences of the ChokePoint dataset. The second, third, fourth and fifth columns show deidentified frames rendered with a ``happy'', ``angry'', ``surprised'' and ``neutral'' expression, respectively. As can be seen, our deidentification approach is highly flexible and is able to retain or alter specific aspects of the deidentified faces.}\n\\label{fig:deid_examples_happy}\n\\end{figure}\n\nOne key characteristic of our deidentification approach is the flexibility that the generative network offers when producing synthetic face images for deidentification. The generation process can be parameterized with respect to the desired target appearance of the synthetic face, which makes it possible to generate  faces with different characteristics (in terms of facial expression, skin color, gender, etc.) and is important when trying to retain non-identity-related information in the deidentified  data. In video conferencing applications, for example, one may want to protect the privacy of the conference participants by hiding their identity, but still preserve the information that facial expressions convey during the conversation. In customer-profiling applications, the focus is typically on the demographics of the customers (such as gender or age distributions) and not on the identity. With our deidentification approach we are able to conceal the identity of people in the image data and retain (or alter) certain aspects of the facial appearance. This characteristic is demonstrated in Fig.~\\ref{fig:deid_examples_happy}. Here, the first column shows a few sample frames from a video sequence of the ChokePont dataset and the second, third, fourth and fifth column show three different deidentification results that were rendered with a ``happy'', ``angry'', ``surprised'' and ``neutral'' facial expression, respectively. While we only show  results for different facial expressions, our deidentification pipeline is in general able to generate variations of synthetic faces in accordance with any appearance-related label of the training data. Thus, if the data used for training contains images annotated with respect to facial expressions, we are able to generate faces with different facial expressions, if the data contains labels for gender, we can synthesize faces belonging to males or females and so forth. The number of different appearance variations our approach can cover is only limited by the number of available labels. \n\n\nIn Fig.~\\ref{fig:deid_examples_bader} we show some examples of visually less pleasing (or problematic) deidentification results. The image artifacts visible here are a consequence of different scene conditions (see the fourth image in the second row of Fig.~\\ref{fig:deid_examples_bader} for an extreme example) and can be ascribed to our replacement procedure. These artifacts could be alleviated by a more elaborate face-replacement approach exploiting, for example, Poisson blending or color-profile matching. However, this would affect the speed of our pipeline, which currently runs at around 12 frames per second if processing the sequence with only one subject present at the time and around 5 frames per second if executing it on a sequence involving multiple subjects simultaneously present in a scene. These framerates were achieved on a desktop PC with Intel(R) Core(TM) i7-6700K CPU (4.00GHz) and 32GB of RAM. Another cause of image artifacts are extreme facial poses when people exit the scene (this is common to all sequences), which result in visible misalignment between the superimposed (deidentified) faces and the original facial areas.    \n\nAmong the main limitations of our deidentification approach is the persistence of identity in the deidentified data. As we are dealing with video footage, the deidentification procedure should ideally produce the same (consistent) result for all frames of the given video sequence. In other words, the facial area of a given subject should be replaced with an artificially generated face of the same target identity over the entire duration of each video. However, due to changes in facial appearance, our matching module occasionally returns inconsistent results and causes changes in the target identity of the deidentified frames. This effect is demonstrated in the first row of Fig.~\\ref{fig:deid_examples_bader}, where an identity change can be observed in the last image pair due to variations in the scene's illumination despite the fact that the same subject is being deidentified. Nevertheless, because the target identity for deidentification is determined with a state-of-the-art face recognition model (i.e., VGG~\\cite{Parkhi_Recog2015}), our procedure is able to assign a consistent target identity most of the time for all test videos considered in our experiments.\n\n\nIn Section \\ref{sec:con}, where we discuss possible directions for future work, we propose  some possible improvements of our deidentification pipeline, which address most of the existing issues of the current implementation.\n\n\n\\begin{figure}[tb]\n\\centering\n\\begin{minipage}{\\textwidth}\n  \\centering\n  \\includegraphics[width=.164\\linewidth]{orig31}\n  \\hspace{-0.2cm}\n  \\includegraphics[width=.164\\linewidth]{deid31}\n  \\includegraphics[width=.164\\linewidth]{orig32}\n  \\hspace{-0.2cm}\n  \\includegraphics[width=.164\\linewidth]{deid32}\n  \\includegraphics[width=.164\\linewidth]{orig33}\n  \\hspace{-0.2cm}\n  \\includegraphics[width=.164\\linewidth]{deid33}\n\\end{minipage}\n\\vspace{0.05cm} \\\\\n\\begin{minipage}{\\textwidth}\n  \\centering\n  \\includegraphics[width=.164\\linewidth]{orig41}\n  \\hspace{-0.2cm}\n  \\includegraphics[width=.164\\linewidth]{deid41}\n  \\includegraphics[width=.164\\linewidth]{orig42}\n  \\hspace{-0.2cm}\n  \\includegraphics[width=.164\\linewidth]{deid42}\n  \\includegraphics[width=.164\\linewidth]{orig43}\n  \\hspace{-0.2cm}\n  \\includegraphics[width=.164\\linewidth]{deid43}\n\\end{minipage}\n\\caption{Qualitative examples of problematic deidentification results. The upper row shows an identity switch in the last frame due to a change in the scenes illumination. The lower row shows difficulties due to the presence of multiple people, some of which occlude faces in the background. Misalignment between the original and surrogate faces is also visible in the second image pair of the lower row, which happens due to extreme viewing angles when people exit the scene.}\n\\label{fig:deid_examples_bader}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Automatic and manual reidentification}\n\nThe quality and efficacy of deidentification techniques is typically measured through reidentification experiments~\\cite{NIST_Simson}, where the goal is to evaluate the risk of successfully identifying a person from deidentified data. This risk is commonly assessed with automatic and manual recognition experiments. \n\nAs outlined in Section~\\ref{sec:dataset}, we perform a number of verification experiments in a 10-fold cross validation protocol towards the risk assessment and consider three state-of-the-art automatic recognition approaches from the literature. Specifically, we use \\textit{i)} the open-source implementation of the 16-layer VGG face network from~\\cite{Parkhi_Recog2015} \n-- VGG from hereon, \\textit{ii)} our own 24-layer implementation of the SqueezeNet network from~\\cite{SqueezeNet} trained on around 2.5 million images (i.e., the VGG network training data) \n-- SqueezeNet from hereon, \nand \\textit{iii)} the 4SF algorithm from OpenBR (version 1.1) \\cite{Klontz2013_OpenBR}\n-- OpenBR from hereon. For the two networks, we use the output from the last fully connected layer of each network as a feature vector and compute a similarity score for an image pair as the cosine angle between the two corresponding feature vectors. For OpenBR we use the default matching option. \n\nWe also conduct manual recognition experiments using a similar 10-fold cross-validation protocol as with automatic techniques, but limit the extend of comparisons to 5\\% of the automatic experiments. Thus, 30 comparisons are performed during each fold resulting in a total of 300 verification attempts (150 legitimate and 150 illegitimate experiments) in each experimental run. Each verification experiment was evaluated by one human evaluator, i.e., four evaluators covered four verification experiments. To produce similarity scores needed for generating performance metrics and ROC curves, we manually assign a similarity score from a five-point scale to each comparison in accordance with the methodology proposed in~\\cite{HumanAnot}. \n\nSimilar to other existing works on face deidentification, \nour approach tries to conceal the identity of people by replacing the detected facial areas with a synthetically generated surrogate faces. However, identity cues can also be extracted from contextual information that is not directly related to facial appearance. For example, the facial outline, hair-style, or even clothing can represent a give-away that recent recognition techniques based on deep models as well as humans may be able to pick up. To explore this issue, we conduct two sets of experiments: \n\\begin{itemize}\n\\item With context: here we feed the facial area to the recognition technique directly as it is detected by the face detector. Thus, the facial area also contains contextual information about the shape of the head, hair style and alike. A comparison of two images with context is illustrated in the last column of Table~\\ref{Tab:_numeric results} (first row). \n\\item Without context: here we trim the bounding box returned by the face detector on each side by 10\\%, the facial areas used for the recognition experiments are therefore cropped tighter and contain only little contextual information. A sample comparison of two images as used in this set of experiments is shown in the last column of Table~\\ref{Tab:_numeric results} (second row).   \n\\end{itemize}\n\n\\begin{table}[H]\n\\setlength{\\tabcolsep}{3pt}\n\\renewcommand{\\arraystretch}{1.2}\n\\caption{Quantitative results of the experiments. Average values and standard deviations over 10-fold are presented for all performance metrics.\n}\n\\footnotesize\n\\label{Tab:_numeric results}\n\\centering\n\\begin{tabular}{lrrrrrrrrrc}\n\\toprule\n\\multicolumn{2}{l}{{Test description}} & \\multicolumn{2}{c}{{Original-to-original}} & \\multicolumn{2}{c}{{Original-to-profile}} & \\multicolumn{2}{c}{{Deidentified-to-original}}& \\multicolumn{2}{c}{{Deidentified-to-profile}}& \\multirow{2}{*}{Context illustration}\\\\ \\cmidrule(l){1-10\n\\multicolumn{2}{l}{{Metric (in \\%)}}       & \\multicolumn{1}{c}{EER}              & \\multicolumn{1}{c}{VER-1}    & \\multicolumn{1}{c}{EER}              & \\multicolumn{1}{c}{VER-1}  & \\multicolumn{1}{c}{EER}              & \\multicolumn{1}{c}{VER-1}    & \\multicolumn{1}{c}{EER}              & \\multicolumn{1}{c}{VER-1} & \\\\\\midrule\n\\multirow{4}{*}{\\rotatebox[origin=c]{90}{Context}}    & VGG         & $8.7\\pm1.0$  & $70.7\\pm5.7$   &    $10.5\\pm 1.3$  & $56.0\\pm 10.3$& $34.4\\pm2.2$& $4.2\\pm3.0$   & $34.5\\pm1.2$   & $5.5\\pm3.2$  &\\multirow{3}{*}{\\vspace{8mm}\\centering\\includegraphics[width=2.6cm]{figures\/context.pdf}}\\\\\n                                 & SqueezeNet  & $40.8\\pm 2.8$  & $4.4\\pm2.1$   &    $40.9\\pm1.9$  & $3.3\\pm1.6$& $47.3\\pm2.3$& $0.8\\pm0.7$   & $47.4\\pm2.9$   & $1.2\\pm1.0$  &\\\\\n                                 & OpenBR      & $23.6\\pm2.2$  & $34.5\\pm6.8$   &    $28.3\\pm1.7$  & $24.2\\pm6.3$& $42.8\\pm1.8$& $2.8\\pm1.8$   & $45.3\\pm2.8$   & $2.1\\pm1.4$  &\\\\\n                                 & Human       & $2.0\\pm 2.8$  & $n\/a$   &    $1.0\\pm 2.2$  & $n\/a$& $42.0\\pm 7.6$& $n\/a$   & $41.8\\pm 7.2$   & $n\/a$  &\\\\  \\midrule\n\\multirow{4}{*}{\\rotatebox[origin=c]{90}{No Context}} & VGG         & $21.5\\pm2.9$  & $26.1\\pm6.3$   &    $21.8\\pm1.7$  & $13.1\\pm5.7$& $43.3 \\pm2.2$& $1.6\\pm1.3$   & $40.6 \\pm2.0$   & $3.4\\pm1.5$  &\\multirow{3}{*}{\\vspace{8mm}\\centering\\includegraphics[width=2.6cm]{figures\/no_context.pdf}}\\\\\n                                 & SqueezeNet  & $47.0\\pm2.1$  & $3.8\\pm1.3$   &    $47.1\\pm2.1$  & $1.8\\pm0.9$& $49.4\\pm1.8$& $0.9\\pm0.7$   & $48.4\\pm2.1$   & $1.4\\pm 1.3$  &\\\\\n                                 & OpenBR      & $27.8\\pm2.0$  & $19.1\\pm6.6$   &    $32.2\\pm3.1$  & $15.9\\pm5.7$& $43.9\\pm2.9$& $1.9\\pm1.5$   & $45.2\\pm2.9$   & $1.1\\pm0.9$  &\\\\\n                                 & Human       & $2.3\\pm 3.2$  & $n\/a$   &    $1.7\\pm 2.8$  & $n\/a$& $44.0\\pm 8.4$& $n\/a$   & $47.3\\pm 5.8$   & $n\/a$  &\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\nNumerical results of the experiments are presented in Table~\\ref{Tab:_numeric results}. Note that VER-1 values are not reported for the manual experiments (denoted as Human) because of an insufficient number of manually graded image comparisons. As expected, the results with contextual information are significantly better than those without contextual information for all experiments when non-deidentified images are used. When the verification attempts are conducted with deidentified images, contextual information still contributes to a higher performance in all experiments, but the differences between images with and without context are smaller. As also evidenced by the ROC curves of the experiments in Fig.~\\ref{fig:results_ROC}, the best performing automatic technique, the VGG network, is able to ensure a recognition performance well above random with an EER of 34.4\\% for the \\textit{deidentified-vs-original} experiment and an EER of 34.5\\% for the \\textit{deidentified-vs-profile} experiment when context is available. If no contextual information is present, the VGG performance drops to an EER of 43.3\\% and 40.6\\% for the same experiments, respectively. These observations suggest that contextual information is  important and may be exploited by contemporary recognition techniques to boost performance. Thus, care needs to be taken to appropriately conceal, modify or remove contextual information from the data as well.           \n\\begin{figure*}[!thb]\n\\begin{minipage}{0.5\\textwidth}\n \\centering\n \\includegraphics[width=1\\textwidth]{figures\/fig1.pdf}\n\\end{minipage}\n\\begin{minipage}{0.5\\textwidth}\n \\centering\n \\includegraphics[width=1\\textwidth]{figures\/fig3.pdf}\n\\end{minipage}\n\\begin{minipage}{0.5\\textwidth}\n \\centering\n \\includegraphics[width=1\\textwidth]{figures\/fig2.pdf}\n\\end{minipage}\n\\begin{minipage}{0.5\\textwidth}\n \\centering\n \\includegraphics[width=1\\textwidth]{figures\/fig4.pdf}\n\\end{minipage}\n\n\\caption{ROC curves of the verification experiments. The curves on the left show the results of the experiments with images with contextual information and the curves on the right show the results obtained without contextual information. The upper row shows experiments with unaltered and deidentified images from the \\textit{original} subset and the lower row shows experiments with unaltered and deidentified images from the \\textit{profile} subset. All results show that our approach is effective.}\n\\label{fig:results_ROC}\n\\end{figure*}\n\nAnother interesting observation that can be made from the ROC plots in Fig.~\\ref{fig:results_ROC} is the drop in performance for the manual experiments. On the unaltered images, human performance is close to perfect for all experiments. However, after deidentification human performance drops to (more or less) random if no contextual information is present and is only slightly better than chance if contextual information is available. \n\\begin{figure*}[htb]\n \\centering\n \\includegraphics[width=1\\textwidth]{figures\/BoxPlots1.pdf}\n\\caption{AUC values from the 10 experimental folds presented in the form of box plots for all assessed techniques as well as human experiments. The blue  plots show results with unaltered images, the red plots show   experiments with deidentified images. The left box plots present the results with contextual information and the right box plots present the results without context. Note that after deidentification (red plots) the result are very close to 0.5 which indicated random performance.}\n\\label{fig:results_Boxplots}\n\\end{figure*}\n\nThe observations made so far are also supported by the  box-and-whiskers plots of the AUC values computed from the 10 experimental folds in Fig.~\\ref{fig:results_Boxplots}. Here, a value of 0.5 indicates random performance. The blue box plots show results for unaltered images and the red box plots show the results for the deidentified images.\nIt needs to be noted that even in cases, when the performance after deidentification is not exactly random, it is still significantly lower than that obtained with unaltered images for all tested techniques. The highest median AUC value of any experiment after deidentification (AUC = 0.719) is achieved by the VGG network when contextual information is available. However, while this value is significantly above random it is of limited use to applications requiring reliable face recognition.  \n\nIn our last experiment, we compare our deidentification approach to existing deidentification techniques from the literature. Specifically, we report results for two naive methods, i.e., blurring and pixelation, which unlike techniques from the $k$-Same family can be applied to video data using the same experimental protocol as used in the previous experiments. The results of the comparison are generated with the best performing recognition approach from Table~\\ref{Tab:_numeric results}, i.e., the VGG network, and are presented in Table~\\ref{Tab:comparison}. As can be seen, the naive methods result in worse recognition performance than our approach and therefore appear to ensure better anonymity. However, these methods destroy most of the information content of the images and can to a certain extent also be bypassed as shown by the results of the parrot (or imitation) attack experiments.\n\nOn the right side of Table~\\ref{Tab:comparison} we show some qualitative deidentification examples on a closed set of images from the XM2VTS dataset \\cite{Messer2003_XM2VTS} (top row). Note that a closed set is required for the $k$-Same family of techniques to be applicable. In accordance with the $k$-anonymity scheme \\cite{Sweeney_Kanonym}, we replace clusters of (in this case $k=2$) images with the same surrogate face  generated by our GNN (the clusters are color-coded in the image). The results of our deidentification approach (last row) are visually convincing and feature no ghosting effects, such as the images generated by the original $k$-Same approach from \\cite{Newton_original} (fourth row). With our approach it is also possible to retain certain aspects of the original data, which is not necessarily true for the blurred and pixelated images, shown in the second and third row of the image, respectively.\n\n\n\n\\begin{table}[H]\n\\setlength{\\tabcolsep}{3pt}\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{\nDeidentification performance with the VGG network. The left part of the table shows a comparison of a few existing (naive) deidentification techniques and the proposed approach in experiments on the ChokePoint dataset. The right part of the table presents a qualitative comparison of our approach with competing techniques from the literature on a closed set of images.\n}\n\\footnotesize\n\\label{Tab:comparison}\n\\centering\n\\begin{tabular}{lrrrrr}\n\\toprule\n\\multirow{2}{*}{Deidentification technique \\hspace{2mm}} & \\multicolumn{2}{c}{{Context\\hspace{2mm}}} & \\multicolumn{2}{c}{{No Context}} & \\hspace{4mm}Qualitative comparison (closed set)\\hspace{4mm} \\\\ \\cmidrule(l){2-6}\n& EER (in \\%) \t& VER-1 (in \\%) &  EER (in \\%) \t& VER-1 (in \\%) & \\hspace{4mm}\\multirow{6}{*}{\\vspace{8mm}\\vspace{4mm}\\centering\\includegraphics[width=4.8cm]{figures\/DeId_examples.pdf}}\\\\\n\\cmidrule(l){1-5}\nPixelated\t\t\t\t   \t& $45.1\\pm 1.8$\t\t\t& $1.7\\pm 0.9$\t\t\t&\t$47.3\\pm 2.7$\t\t\t& $1.3\\pm 0.8$\t\t\t&\\\\\nBlurred\t\t\t\t\t\t& $37.0\\pm 1.4$\t\t\t& $1.7\\pm 1.2$\t\t\t&  $43.0\\pm 2.1$\t\t\t& $1.5\\pm 1.5$\t\t\t&\\\\\nPixelated (parrot attack)\t& $38.0\\pm 1.4$\t\t\t& $3.0\\pm 1.6$\t\t\t&  $39.4\\pm 2.6$\t\t\t& $2.7\\pm 1.3$\t\t\t&\\\\\nBlurred (parrot attack)\t\t& $32.4\\pm 1.7$\t\t\t& $13.9\\pm 4.2$\t\t\t&  $35.8\\pm 2.1$\t\t\t& $8.6\\pm 3.4$\t\t\t&\\\\\nOurs\t\t\t\t\t\t& $34.3\\pm2.2$ \t& $4.2\\pm3.0$\t&  $43.2\\pm2.2$\t& $1.6\\pm1.3$\t&\\\\\t\n\\cmidrule(l){1-5}\nNo deidentification\t\t\t& $8.7\\pm1.0$\t& $70.7\\pm5.7$ \t& $21.5\\pm2.5$ \t& $26.1\\pm6.3$ \t&\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\section{Conclusion \\label{sec:con}}\nIn this paper we have presented a novel approach to face deidentification using generative neural networks. The proposed approach was evaluated on the ChokePoint dataset with highly encouraging results. Our evaluation suggests that generative networks are a viable tool for face deidentification and that a high degree of anonymity can be ensured by swapping the original faces by artificially generated surrogate faces. Furthermore, our experiments show that due to the flexibility of the generative network it is possible to control the appearance of the generated surrogate faces and thus retain (or alter) only specific aspects of the input images -- contributing significantly to the utility of the deidentified faces.  \n\nWhile our deidentification results are visibly convincing, additional improvements are possible. As part of our future work, we plan on \nincluding additional generator parameters to further capitalize on the utility of the deidentified faces. Other possible improvements include a better blending procedure that would improve the overall naturalness of the deidentified faces and remove artifacts. We will also consider incorporating a tracking scheme, which would improve the applicability of our approach on video data.\n\n\\section*{Acknowledgement}\n\nThis research was supported in parts by the ARRS (Slovenian Research Agency) Research Programme P2-0250 (B) Metrology and Biometric Systems, the ARRS Research Programme P2-0214 (A) Computer Vision,  by TUBITAK project no. 113E067, by a Marie Curie FP7 Integration Grant within the 7th EU Framework Programme, by the Croatian HRRZ project DePPSS 6733 De-identification for Privacy Protection in Surveillance Systems and COST Action IC 1206 on De-identification for privacy protection in multimedia content.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nIn the last two decades, the design and implementation of devices for quantum information processing has been a major goal of condensed matter physics. An essential requirement of the quantum information paradigm is the possibility for two \\textit{qubits} to interact coherently in a controlled fashion, in order to achieve controlled gate operations. This must in principle be possible for each arbitrarily chosen pair of qubits in the system. Most of the technologies, however, employ qubits which are at all times spatially separated and do not interact directly \\cite{DiCarlo2009,Riebe2004,Weperen2011}. The interaction can then be achieved by means of a \\textit{quantum bus}, namely a spatially extended degree of freedom interacting with all localized qubits. In a more general picture, these spatially extended degrees of freedom might even form a quantum network connecting distant quantum information systems.\\cite{Kimble2008} A quantum bus can be of several kinds -- two common examples being phonons in chains of trapped ions\\cite{Cirac_1995} and microwave photons in superconducting circuits.\\cite{Sillanpaeae_2007, Majer_2007}. Photons are the most natural choice in a solid-state system, given their low decoherence rate, high velocity, and the recent advances in the on-chip photonic technology, especially in the photonic crystal (PHC) domain \\cite{Noda2007, Notomi2010}. There, extremely high-$\\mathcal{Q}$ cavities with modal volumes of the order of $(\\lambda\/n)^3$ have been fabricated both in silicon\\cite{Kuramochi_2006, Taguchi_2011} and in GaAs\\cite{Combrie_2008}, as well as waveguides allowing for low-loss, long-range photon transfer with a controllable group velocity.\\cite{Baba_2008} The advance in PHC technology opened the way to several experimental breakthroughs, including low power all-optical switching,\\cite{Husko_2009, Nozaki_2010} and the dynamic control of the strong coupling between two distant cavities\\cite{Sato_2012} -- highlighting the extreme level of control of light which is currently achievable.\n\nSemiconductor quantum dots (QDs) have long been considered as viable qubit candidates,\\cite{Loss_1998} as they naturally fulfill the criteria of scalability and integrability required in a quantum information technology. Facing the remarkable advance made in the system of spin qubits in lateral QDs \\cite{Weperen2011} -- where electron spins are controlled electronically with ohmic contacts -- optical excitations in self-organized QDs have only recently caught up in the race towards controlled quantum operations. On one hand, in fact, full single-qubit optical control has been successfully demonstrated.\\cite{Patton_2005, Press2008, Berezovsky_2008, Greilich_2009, Greilich_2011, Poem_2011, Muller_2012, Godden_2012, Kodriano_2012} On the other, integrating QDs in photonic structures has made significant progress, and both single-dot Purcell enhancement in cavities\\cite{Gerard_1998, Solomon_2001, Happ_2002, Ramon_2006, Munsch_2009} and waveguides\\cite{Viasnoff-Schwoob_2005, Lund-Hansen_2008, Thyrrestrup_2010, Schwagmann_2011, Hoang_2012, Laucht_2012, Laucht_2012a}, and strong coupling to a cavity mode\\cite{Yoshie_2004, Reithmaier_2004, Peter_2005} have been demonstrated. Single-dot coupling to light modes is in itself important for practical applications, as suggested by the possibility of non-classical light generation,\\cite{Faraon_2008, Kasprzak_2010,Dousse2010, Reinhard_2012} or single-photon optical switching.\\cite{Volz_2012} Beyond that, short-distance coupling in quantum dot ``molecules'' has been demonstrated,\\cite{Bayer_2001, Borri_2003, Gerardot_2005, Krenner_2005, Stinaff_2006,Robledo2008} where however the coupling is enforced by the direct overlap of the QD wave-functions and\/or the electrostatic F\\\"{o}rster dipole-dipole interaction,\\cite{Govorov_2003, Govorov_2005} rather than by any long-distance mechanism. Altogether, these advances suggest that the field has reached the milestone, following which the process of \\textit{long-distance}, photon-mediated interaction between two or more quantum dots should also be addressed. It has been shown that the light-matter interaction between a QD and the electromagnetic modes of a non-structured photonic environment is very weak,\\cite{Parascandolo_2005, Scheibner_2007, Tarel_2008, Kasprzak_2011} thus photonic structures are needed in order to tailor the density of optical modes and thus enhance radiative coupling between spatially separated quantum dots. Indeed, short-range radiative coupling has already been achieved in several experiments involving small optical cavities, where strong coupling of two quantum dots to the same cavity mode was detected,\\cite{Reitzenstein_2006, Laucht_2010, Kim_2011} and, most recently, its coherent nature was demonstrated.\\cite{Albert_2012} A photonic structure has also brought the experimental demonstration of long-distance transfer of photons emitted by an embedded QD.\\cite{Englund2007}\n\nOn the theoretical side, specific aspects of structures with one or more quantum dots in a photonic environment have been studied. These include the strong coupling regime and emission spectrum of one\\cite{Andreani_1999, Khitrova_2006, Milde2008, Tarel_2010,Hughes2011} or more\\cite{Kessler_2008, Laussy_2011, Auffeves_2011} dots in a microcavity, as well as the possibility of performing cavity-mediated qubit operations through coherent excitation exchange in such a system.\\cite{Agarwal_1998, Imamoglu_1999, Piermarocchi_2002, Quinteiro_2006, Xu_2011} In addition, the spontaneous emission enhancement of one dot coupled to a single waveguide mode has been estimated\\cite{Hughes_2004, Rao_2007, Lecamp_2007}, and non-trivial dynamics of single-dot cavity-QED in presence of coupling to a second, distant cavity, have been predicted\\cite{Hughes_2007}. There are, however, only a few studies of the dot-dot interaction at a mesoscopic (i.e. more than one wavelength) inter-dot distance -- which is a main focus of this work. Most notably, the possibility to generate entangled states between distant QDs in a coupled-cavity system was recently demonstrated,\\cite{Yao_2009} as well as the non-trivial decay dynamics\\footnote{Here we prefer not to use the term ``superradiant'', in order to avoid ambiguity with the concept of Dicke superradiance, which has a radically different physical nature.} of two distant dots in a photonic crystallite.\\cite{Kristensen_2011} However, a general formalism accounting for an arbitrary number of quantum dots coupled to arbitrarily many photonic modes is still lacking, and in particular, the distance dependence of the radiative interaction, the influence of fabrication disorder, and the competition between excitation transfer at-a-distance and radiation  losses still remain open questions. To address those, a microscopic description of light-matter coupling with a realistic description of the photonic modes is needed.\n\nIn this paper, we lay down the semi-classical linear response theory for a system of $N$ distinct, spatially localized excitonic transitions in QDs, coupled to $M$ photonic modes of an arbitrary photonic structure. In particular, we frame the underlying Maxwell equations into an eigenvalue problem, describing the polariton modes of the system in analogy with the polariton formalisms for a bulk semiconductor\\cite{Hopfield_1958}, for quantum wells\\cite{Tassone_1992}, and for QDs in an unstructured photonic environment.\\cite{Parascandolo_2005,Tarel_2008} For the computation of the photonic modes, the Bloch-mode expansion method is employed,\\cite{Savona_2011} although any other method which provides reliable field profiles (e.g., finite-element\nmethod (FEM), finite-difference time-domain (FDTD)) can also be used. Even though the modeling of radiative effects in presence of fabrication disorder lies beyond the scope of the present work, the Bloch-mode expansion is particularly well-suited for treating large, disordered photonic structures\\cite{Savona_2011, Minkov_2012} and was thus an obvious choice in view of a future extension to disordered PHCs. We apply the formalism to $Ln$ cavities and $W\\mathit{1}$ waveguides based on a PHC slab. We show how known single-dot radiative properties -- such as the vacuum Rabi splitting in a microcavity and the Purcell enhancement and $\\beta$-factor in a waveguide -- are well reproduced. The main focus of the work however is the quantitative characterization of radiative coupling between \\textit{two} dots in those same structures. To this purpose, we characterize the spectra of the polariton eigenmodes, the time-evolution of a starting excitation in one of the dots, and the distance-dependence of the radiative excitation transfer in a spatially extended structure. Our simulations provide a comprehensive picture of the effective dot-dot radiative coupling, and show that, with realistic PHC and QD parameters, a sizable interaction can be expected at \u0430 mesoscopic distance. \n\nThe work is organized as follows. In Section II we derive the main theoretical formalism, while in Section III we thoroughly discuss the values of the parameters entering the model, for realistic InAs\/GaAs-based semiconductor nanostructures. Section IV contains the main results obtained from the application of the model to the study of one and two QDs embedded in $Ln$ cavities and $W\\mathit{1}$ waveguides. In Section V we present our conclusions and an outlook of future work.\n \n\\section{Theoretical formalism}\n\nStarting from Maxwell's equations with the assumptions of a non-magnetic medium and no free charges, the electric field in the frequency domain obeys the equation (written in Gaussian units) \n\n\\begin{equation}\n\\nabla \\times \\nabla \\times \\mathbf{E}(\\mathbf{r}, \\omega) - \\frac{\\omega^2}{c^2}\\left(\\varepsilon(\\mathbf{r}) \\mathbf{E}(\\mathbf{r}, \\omega) + 4 \\pi \\mathbf{P}(\\mathbf{r}, \\omega) \\right) = 0 \\, .\n \\label{starting}\n\\end{equation}\nIn particular, here the spatial dependence of the dielectric constant, $\\varepsilon(\\mathbf{r})$, completely characterizes the underlying photonic structure, while the optical response of the quantum dots is included in the polarization vector through a non-local susceptibility tensor\\cite{Kubo_1957}, such that \n\n\\begin{equation}\n\\mathbf{P}(\\mathbf{r},\\omega) = \\int \\mathrm{d}\\mathbf{r}' \\hat{\\chi}(\\mathbf{r}, \\mathbf{r}', \\omega) \\mathbf{E}(\\mathbf{r}', \\omega) \\, .\n\\end{equation} \n\nIn what follows, we will consider the specific case of excitons originating from the heavy-hole band of a semiconductor with cubic symmetry (e.g. InAs), for which only the $\\mathbf{x}-$ and $\\mathbf{y}-$components of the polarization couple to the electromagnetic field according to the following susceptibility tensor\\cite{Tassone_1990,Tassone_1992,Andreani_1994}\n\n\\begin{equation}\n\\hat{\\chi}(\\mathbf{r}, \\mathbf{r}', \\omega) = \\frac{\\mu_{cv}^2}{\\hbar} \\sum_{\\alpha = 1}^N \\frac{\\Psi_{\\alpha}^*(\\mathbf{r}) \\Psi_{\\alpha}(\\mathbf{r}')}{\\omega^{\\alpha} - \\omega} \n\\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 0 \n \\end{pmatrix} \\, . \n \\label{susc}\n\\end{equation}\nThe formalism can be easily generalized to different forms of the susceptibility tensor. Here, $\\alpha$ runs over all QDs, $\\mu_{cv}^2$ is the squared dipole matrix element of the inter-band optical transition, $\\Psi_{\\alpha}(\\mathbf{r}) = \\Psi_{\\alpha}(\\mathbf{r}_e = \\mathbf{r}, \\mathbf{r}_h = \\mathbf{r})$, and $\\Psi_{\\alpha}(\\mathbf{r}_e, \\mathbf{r}_h)$ is the excitonic wave-function, normalized as\n\n\\begin{equation}\n\\int \\mathrm{d} \\mathbf{r}_e \\int \\mathrm{d} \\mathbf{r}_h \\vert \\Psi_{\\alpha}(\\mathbf{r}_e, \\mathbf{r}_h)\\vert ^2 = 1 \\, .\n\\label{psinorm}\n\\end{equation}\nWe denote the frequencies of the bare excitons by a \\textit{superscript} $\\alpha$, in order to distinguish them from the frequencies of the photonic resonances, which we will later on index with \\textit{subscripts}, e.g. as $\\omega_m$. Notice also that here all frequencies are assumed to be complex quantities, e.g.  $\\omega^{\\alpha} = \\Re(\\omega^{\\alpha}) - i\\frac{\\gamma^{\\alpha}}{2}$, where $\\gamma^{\\alpha}$ represents the overall decay rate of the exciton state, including any possible non-radiative mechanism and the rate of radiative decay into photon modes that are \\textit{not} included among the $M$ modes treated exactly.\n\nIn order to turn the Maxwell equation into a self-adjoint form, we introduce the quantities \\cite{Sakoda_2001} $\\mathbf{Q}(\\mathbf{r}, \\omega) = \\sqrt{\\varepsilon(\\mathbf{r})} \\mathbf{E}(\\mathbf{r}, \\omega)$. Eq. (\\ref{starting}) then becomes\n\n\\begin{align}\n\\Upsilon & \\mathbf{Q}(\\mathbf{r}, \\omega) - \\frac{\\omega^2}{c^2} \\mathbf{Q}(\\mathbf{r}, \\omega) =  \\label{eqforq} \\\\  \n&\\frac{4 \\pi}{\\sqrt{\\varepsilon(\\mathbf{r})}} \\frac{\\omega^2}{c^2}  \\int \\mathrm{d}\\mathbf{r}' \\hat{\\chi}(\\mathbf{r}, \\mathbf{r}', \\omega) \\frac{\\mathbf{Q}(\\mathbf{r}', \\omega)}{\\sqrt{\\varepsilon(\\mathbf{r}')}}. \\nonumber\n\\end{align}\nwhich is an inhomogeneous differential equation defined for the \\textit{self-adjoint} differential operator\n\n\\begin{equation}\n\\Upsilon = \\frac{1}{\\sqrt{\\varepsilon(\\mathbf{r})}} \\nabla \\times \\nabla \\times \\frac{1}{\\sqrt{\\varepsilon(\\mathbf{r})}} \\; \\; .\n\\end{equation}\nThe susceptibility tensor as given in Eq. \\ref{susc} decouples the $\\mathbf{z}$-polarized fields. We then define the two-dimensional field $\\mathbf{Q} = (Q_x, Q_y)$. We can solve the problem using a Green's function approach\\cite{Martin_1998}, in which the formal solution to Eq. (\\ref{eqforq}) is\n\n\\begin{align}\n\\mathbf{Q}(\\mathbf{r}, \\omega) &= \\mathbf{Q}_0(\\mathbf{r}, \\omega) + \\label{dyson}\\\\ \n \\frac{4\\pi}{\\sqrt{\\varepsilon(\\mathbf{r})}}\\frac{\\omega^2}{c^2}&\\int \\mathrm{d}\\mathbf{r}' \\int \\mathrm{d}\\mathbf{r}'' \\hat{G}(\\mathbf{r}, \\mathbf{r}', \\omega) \\frac{\\hat{\\chi}(\\mathbf{r}', \\mathbf{r}'', \\omega)}{\\sqrt{\\varepsilon(\\mathbf{r}'')}} \\mathbf{Q}(\\mathbf{r}'', \\omega). \\nonumber\n\\end{align}\n\nThe Green's tensor can be expanded onto the basis of field eigenmodes using the resolvent representation, following Fredholm's theory\\cite{Economou_2006}\n\n\\begin{equation}\n\\hat{G}(\\mathbf{r}, \\mathbf{r}', \\omega) = \\sum_{m} \\frac{\\mathbf{Q}_m(\\mathbf{r})\\otimes \\mathbf{Q}_m^*(\\mathbf{r}')}{\\frac{\\omega_m^2}{c^2} - \\frac{\\omega^2}{c^2}} \\, , \n\\label{green}\n\\end{equation}\nwhere the $\\mathbf{Q}_m$-s are the \\textit{orthonormal} eigenfunctions of $\\Upsilon$ corresponding to eigenvalues $\\omega_m^2\/c^2$, and $\\otimes$ is an outer product defined as\n\n\\begin{equation}\n\\mathbf{A}\\otimes\\mathbf{B} = \n\\begin{pmatrix}\nA_x B_x & A_x B_y \\\\\nA_y B_x & A_y B_y\n\\end{pmatrix} \\, .\n\\end{equation}\nThe sum in Eq. (\\ref{green}) runs in principle over the infinite set of eigenmodes. In most situations of interest, however, this sum is dominated by the resonant modes of the photonic crystal that are closest to the frequency range characterizing the excitonic transitions. In addition, in all structures of interest (e.g. a PHC\\cite{Yoshie_2004, Badolato_2005}, pillar cavity \\cite{Reithmaier_2004} or a microdisc \\cite{Peter_2005}), the dots are typically embedded within the dielectric medium, i.e. their wave-functions are non-negligible only in a region where $\\varepsilon(\\mathbf{r}) = \\varepsilon_{\\infty}$, the permittivity of the semiconductor. Thus, as the $\\mathbf{r}$-dependence of all quantities will eventually enter through overlap integrals with the QD wave-functions, in eq. (\\ref{dyson}) we can safely substitute  $\\sqrt{\\varepsilon(\\mathbf{r})} = \\sqrt{\\varepsilon(\\mathbf{r}'')} = \\sqrt{\\varepsilon_{\\infty}}$. Finally, in typical situations, all QD transition frequencies lie within a small range originating from the inhomogeneous distribution of QD sizes. A very good approximation consists then in replacing the $\\omega$ on the r.h.s. of (\\ref{dyson}), as well as the $(\\omega_m+\\omega)\/2$ obtained by factoring the denominator in (\\ref{green}), with an average exciton transition frequency $\\omega_0$. In order to compute the complex frequency poles, corresponding to the resonances of the coupled system, we consider the homogeneous problem associated with Eq. (\\ref{dyson}). Then, by defining \n\n\\begin{equation}\n\\mathbf{Q}^{\\alpha}(\\omega) = \\int \\mathrm{d}\\mathbf{r} \\Psi_{\\alpha}(\\mathbf{r}) \\mathbf{Q}(\\mathbf{r}, \\omega) \\, ,\n\\label{overlap}\n\\end{equation}\nwe obtain\n\n\\begin{equation*}\n\\mathbf{Q}(\\mathbf{r}, \\omega) = \\frac{2\\pi \\omega_0}{\\varepsilon_{\\infty}}\\frac{\\mu_{cv}^2}{\\hbar} \\sum_{\\alpha=1}^N \\sum_{m = 1}^{M}\\frac{\\mathbf{Q}_m(\\mathbf{r})\\otimes \\mathbf{Q}_m^{\\alpha*}}{(\\omega_n- \\omega)(\\omega^{\\alpha} - \\omega)} \\mathbf{Q}^{\\alpha}(\\omega).\n\\end{equation*}\nBy integrating Eq. \\ref{dyson} with $\\int \\mathrm{d}\\mathbf{r} \\Psi_{\\beta}(\\mathbf{r})$ and defining additionally $\\tilde{\\mathbf{Q}}^{\\alpha}(\\omega) = \\mathbf{Q}^{\\alpha}(\\omega)\/(\\omega^{\\alpha} - \\omega)$, we finally obtain a set of equations (labeled by $\\beta$) for the complex frequency poles\n\n\\begin{equation}\n(\\omega^{\\beta} - \\omega)\\tilde{\\mathbf{Q}}^{\\beta}(\\omega) = \\frac{2\\pi \\omega_0}{\\varepsilon_{\\infty}}\\frac{\\mu_{cv}^2}{\\hbar} \\sum_{\\alpha=1}^N \\sum_{m = 1}^{M}\\frac{\\mathbf{Q}_m^{\\beta}\\otimes \\mathbf{Q}_m^{\\alpha*}}{(\\omega_n- \\omega)} \\tilde{\\mathbf{Q}}^{\\alpha}(\\omega) \\, .\n\\label{alphbet}\n\\end{equation}\n\nWe now define the quantities \n\n\\begin{equation}\n\\mathbf{g}_m^{\\alpha} = ({g}_{m, x}^{\\alpha}, {g}_{m, y}^{\\alpha}) =  \\left(\\frac{2\\pi \\omega_0}{\\varepsilon_{\\infty}}\\frac{\\mu_{cv}^2}{\\hbar}\\right)^{1\/2}\\mathbf{Q}_m^{\\alpha} \\, ,\n\\label{coupling}\n\\end{equation}\nwhich should be interpreted as the coupling strengths between the $m$-th mode of the PHC and the $\\alpha$-th QD. To this end, we notice that the $2N$ equations in (\\ref{alphbet}) can be solved only for those values of $\\omega$ for which the $N\\times N$ matrix \n\n\\begin{widetext}\n\\begin{equation}\n\\Lambda_1 = \n\\begin{pmatrix}\n\\omega^{1}_x - \\omega - \\sum_{m=1}^M \\frac{g_{m,x}^{1} g_{m,x}^{1*}}{\\omega_m - \\omega} & - \\sum_{m=1}^M \\frac{g_{m,x}^{1} g_{m,y}^{1*}}{\\omega_m - \\omega} & \\cdots & - \\sum_{m=1}^M \\frac{g_{m,x}^{1} g_{m,x}^{N*}}{\\omega_m - \\omega} & - \\sum_{m=1}^M \\frac{g_{m,x}^{1} g_{m,y}^{N*}}{\\omega_m - \\omega} \\\\\n - \\sum_{m=1}^M \\frac{g_{m,y}^{1} g_{m,x}^{1*}}{\\omega_m - \\omega} & \\omega^{1}_y - \\omega - \\sum_{m=1}^M \\frac{g_{m,y}^{1} g_{m,y}^{1*}}{\\omega_m - \\omega} & \\cdots & - \\sum_{m=1}^M \\frac{g_{m,y}^{1} g_{m,x}^{N*}}{\\omega_m - \\omega} & - \\sum_{m=1}^M \\frac{g_{m,y}^{1} g_{m,y}^{N*}}{\\omega_m - \\omega} \\\\\n\\vdots & \\cdots & \\ddots & \\cdots & \\vdots \\\\\n- \\sum_{m=1}^M \\frac{g_{m,y}^{N} g_{m,x}^{1*}}{\\omega_m - \\omega} & - \\sum_{m=1}^M \\frac{g_{m,y}^{N} g_{m,y}^{1*}}{\\omega_m - \\omega} & \\cdots & - \\sum_{m=1}^M \\frac{g_{m,y}^{N} g_{m,x}^{N*}}{\\omega_m - \\omega}& \\omega^{N}_y - \\omega  - \\sum_{m=1}^M \\frac{g_{m,y}^{N} g_{m,y}^{N*}}{\\omega_m - \\omega}\n\\end{pmatrix}\n\\label{matrixone}\n\\end{equation}\n\\end{widetext}\nis singular. This is a nonlinear equation, but we notice that it can be transformed into a more familiar form, since it is mathematically equivalent to finding the eigenvalues of the matrix \n\n\\begin{equation}\n\\Lambda_2 = \n\\begin{pmatrix}\n\\omega_{x}^1 & 0 & \\cdots & 0 & g_{1,x}^1 & \\cdots & g_{M, x}^1 \\\\\n0 & \\omega_{y}^1 & \\cdots & 0 & g_{1,y}^1 & \\cdots & g_{M, y}^1 \\\\\n\\vdots & \\cdots & \\ddots & \\vdots & \\vdots & \\cdots & \\vdots \\\\\n0 & 0 & \\cdots & \\omega_{y}^N & g_{1,y}^N & \\cdots & g_{M,y}^N \\\\\ng_{1, x}^{1*} & g_{1, y}^{1*} & \\cdots & g_{1, y}^{N*} & \\omega_1 & \\cdots & 0 \\\\\n\\vdots & \\cdots & \\ddots & \\vdots & \\vdots & \\cdots & \\vdots \\\\\ng_{M, x}^{1*} & g_{M, y}^{1*} & \\cdots & g_{M, y}^{N*} &0& \\cdots & \\omega_M  \n\\end{pmatrix}\\, .\n\\label{fullmat}\n\\end{equation}\nMore precisely, solving $\\det(\\Lambda_1) = 0$ is equivalent to solving $\\det(\\Lambda_2 - \\omega I_{(2N\\times M)\\times(2N\\times M)}) = 0$, whenever $\\omega \\neq \\omega_m \\, \\forall \\, m = 1 \\dots M$. The proof can be easily obtained by, on one hand, multiplying the equation for $\\Lambda_1$ by $\\prod_{m=1}^M (\\omega_m - \\omega)$, and on the other, using in the eigenvalue problem for $\\Lambda_2$ the following identity for the determinant of a block-matrix:\n\n\\begin{equation}\n\\det\\begin{pmatrix}A& B\\\\ C& D\\end{pmatrix} = \\det(D) \\det(A - B D^{-1} C)\\, .\n\\end{equation}\n\nThe poles $\\omega = \\omega_m$ will generally exist as solutions only when a photonic mode $\\mathbf{Q}_m$ is fully decoupled from the system, i.e. when $\\mathbf{g}_m^{\\alpha} = 0 \\, \\forall \\, \\alpha$, in which case this mode can safely be excluded from the very beginning. The $2N + M$ complex eigenvalues of $\\Lambda_2$ then define the frequencies (real part) and the loss rates ($-2 \\times $ imaginary part) of the {\\em polariton modes} of the system, while the eigenvectors \n\n\\begin{equation}\n\\bm{\\lambda} = (\\lambda_{x}^1, \\lambda_{y}^1, \\dots \\lambda_{x}^N, \\lambda_{y}^N, \\lambda_1 \\dots, \\lambda_M)\n\\label{eigvec}\n\\end{equation}\ndefine the corresponding Hopfield coefficients\\cite{Hopfield_1958}, which, for each eigenstate, give the probability amplitude of finding an excitation in the corresponding bare-exciton or bare-photon mode. Notice in addition that the matrix of Eq. (\\ref{fullmat}) corresponds to a Tavis-Cummings Hamiltonian\\cite{Tavis_1968} in the weak excitation regime, when only transitions from the ground state to the manifold of states with a single excitation are considered. Thus, notice that our approach has a straightforward extension to treating non-linear quantum dot dynamics, as the coupling constants in the off-diagonal terms of (\\ref{fullmat}) can be used to write the Tavis-Cummings Hamiltonian in its most general from, i.e. including transitions among all excitation-number manifolds. This describes the system whenever the quantum dots behave as two-level systems, which is indeed the case for small dots under resonant excitation. \n\nThe present formalism applies to a large variety of photonic structures and to an arbitrary spatial distribution of QDs. In this sense, it generalizes the results that were obtained for specific configurations\\cite{Andreani_1999, Hughes_2004, Rao_2007, Lecamp_2007, Hughes_2007, Yao_2009, Kristensen_2011}. As an illustrating application, in section \\ref{applications} we present results obtained for the case of two quantum dots embedded in several of the most widely studied photonic crystal structures: the $L3$ and $Ln$ cavities\\cite{Akahane_2003}, and the $W1$ waveguide.\n\n\\section{Model parameters}\n\n\\label{parameters}\n\nIn order to quantify the susceptibility (\\ref{susc}), we need an appropriate model of the exciton wave-function evaluated at equal electron and hole positions, $\\Psi_{\\alpha}(\\mathbf{r}) = \\Psi_{\\alpha}(\\mathbf{r}_e = \\mathbf{r}, \\mathbf{r}_h = \\mathbf{r})$. This function is \\textit{not} properly normalized as a function of $\\mathbf{r}$ (the correct normalization is over $\\mathbb{R}^3 \\times \\mathbb{R}^3$ as given in Eq. (\\ref{psinorm})). In fact, similarly to the quantum well case \\cite{Tassone_1990, Andreani_1994}, the oscillator strength of the exciton transition in the QD depends on the dimensionless quantity\n\n\\begin{equation}\nC^2 = \\left| \\int \\mathrm{d} \\mathbf{r} \\Psi_{\\alpha}(\\mathbf{r}) \\right| ^2 \\, .\n\\end{equation}\n\nThe particular \\textit{shape} of the wave-function enters through the overlap integrals with the electric field, as given in Eq. (\\ref{overlap}). As long as the size of the QDs is much smaller than the characteristic wavelength, the electric field varies very weakly in the region where $\\Psi_{\\alpha}$ is non-negligible, and thus the point dipole assumption, $\\Psi_{\\alpha} (\\mathbf{r}) = C \\delta(\\mathbf{r} - \\mathbf{r}_{\\alpha})$, is a very good approximation. In what follows we will mostly use parameters typical of self-organized InGaAs QDs \\cite{Bimberg_1999}, whose size lies in the $10-20 \\mathrm{nm}$ range, with a typical exciton recombination energy of $1.3 \\mathrm{eV}$ ($\\lambda \\approx 950 \\mathrm{nm}$). For these values, we checked that assuming a Gaussian shape for $\\Psi_{\\alpha}(\\mathbf{r})$ introduces little change with respect to the Dirac-delta assumption. Notice, however, that the strong dependence\\cite{Gil_2002, Langbein_2004} of the QD oscillator strength with its size is still present, carried by the normalization constant $C$. One way to estimate this constant is through a microscopic model of $\\Psi_{\\alpha}(\\mathbf{r})$\\cite{Wang_1999, Stier_1999}. Here, instead, we take a more pragmatic approach, and compute $C$ based on the measured radiative decay rate of QDs. Following Ref. [\\onlinecite{Parascandolo_2005}], this is given by twice the imaginary part of the quantity \n\n\\begin{equation}\nG_{\\alpha} = i\\frac{2\\pi^2\\mu_{cv}^2}{\\hbar \\varepsilon_{\\infty}} \\int_0^{\\infty} \\mathrm{d}k \\vert \\Psi_{\\alpha\\mathbf{k}} \\vert^2 \\frac{k(2k_0^2 - k^2)}{k_z} \\, ,\n\\end{equation}\nwhere $\\Psi_{\\alpha\\mathbf{k}}$ is the Fourier transform of $\\Psi_{\\alpha} (\\mathbf{r})$. With the assumption $\\Psi_{\\alpha} (\\mathbf{r}) = C \\delta(\\mathbf{r} - \\mathbf{r}_{\\alpha})$, the decay rate is thus\n\n\\begin{equation}\n\\Gamma^{\\alpha} = \\frac{4}{3} \\frac{ k_0^3}{\\hbar \\varepsilon_{\\infty}} d^2 \\, , \n\\label{loss}\n\\end{equation}\nwhere $k_0 = (\\omega_0\/c)\\sqrt{\\varepsilon_{\\infty}}$, and we defined the dipole moment $d$ of the dot (also labeled  $\\mathcal{D}$\\cite{Thranhardt_2002} or $\\mu$\\cite{Khitrova_2006})  as \n\\begin{equation}\nd^2 = \\mu_{cv}^2 C^2 \\, .\n\\label{dipole}\n\\end{equation}\nEq. (\\ref{loss}) coincides with the expression that is commonly adopted\\cite{Andreani_1999, Thranhardt_2002, Khitrova_2006}. For typical QDs\\cite{Hennessy_2007, Faraon_2008, Reinhard_2012}, with radiative lifetime of $1 \\mathrm{ns}$ and exciton transition energy $\\hbar \\omega^{\\alpha} \\approx 1.3 \\mathrm{eV}$, we obtain a squared dipole moment $d^2 \\approx 0.51 \\mathrm{eV \\times nm^3}$.\n\nThe last requirement of the problem is the knowledge of the modes of the PHC structure, i.e. the set of orthonormal functions $\\{\\mathbf{Q}_m(\\mathbf{r})\\}$ and their corresponding eigenfrequencies $\\omega_m$. Here, PHC modes are computed using the Bloch-mode expansion method \\cite{Savona_2011}, which consists in expanding the modes on the basis of the Bloch modes of a regular waveguide. These latter were in turn computed using an expansion over the guided modes of a uniform dielectric slab.\\cite{Andreani_2006} This approach turns out to be particularly well suited for elongated PHC cavities as considered in the present work. The computation was carried out over a finite supercell $S$ in the plane of the crystal, and infinite space along the orthogonal, $z-$direction. The orthogonality relation is then given by\n\n\\begin{equation}\n\\int_S \\mathrm{d}^2\\rho \\int_{-\\infty}^{\\infty} \\mathrm{d}z \\; \\mathbf{Q}_m(\\bm{\\rho}, z) \\mathbf{Q}_n^*(\\bm{\\rho}, z) = \\delta_{mn} \\, .\n\\end{equation}\n\nAll the photonic crystals we consider are based on a triangular lattice of circular holes etched in a dielectric slab suspended in air. The specific parameters we chose are relevant to GaAs structures,\\cite{Hennessy_2007, Faraon_2008,Reinhard_2012} namely: lattice constant $a = 260 \\mathrm{nm}$, hole radius $65 \\mathrm{nm}$, and slab thickness $120 \\mathrm{nm}$, with a real part of the refractive index $\\sqrt{\\varepsilon_{\\infty}}=3.41$. In this work we consider only ideal PHC structures, in the absence of any disorder that would arise from the fabrication process. Disorder in PHCs has two important effects. First, it determines extrinsic radiation loss rates of otherwise fully guided modes in waveguides, and strongly suppresses the quality factors of high-quality PHC cavities. This effect is here taken into account through the inclusion of a constant phenomenological loss rate for the modes under study, related to their quality factor $\\mathcal{Q}$ by $\\gamma = \\omega\/\\mathcal{Q}$. For the $L3$ cavity of section \\ref{L3}, we set $\\mathcal{Q} = 10000$ or $30000$. For the longer $Ln$ cavities (section \\ref{Ln}) and the $\\mathit{W1}$ waveguide (section \\ref{W1}), we set $\\mathcal{Q} = 50000$ for all modes. The second way disorder affects the results is by modifying the spatial profiles of the electric field modes, especially in the case of waveguides. This effect lies beyond the scope of the  present work -- although a brief discussion is given in section \\ref{discussion} -- and will be the object of a future work.\n\n\\section{Applications}\n\n\\label{applications}\n\nIn this section, we apply the formalism to the prototypical cases of one or two QDs embedded in elongated $Ln$ cavities or in a $\\mathit{W1}$ waveguide.\n\n\\subsection{Application to an $\\mathit{L3}$ cavity}\n\\label{L3}\n\nThe system of one quantum dot coupled to an $\\mathit{L3}$ cavity has been widely studied\\cite{Yoshie_2004, Hennessy_2007, Reinhard_2012} and is thus a good starting point for testing the present formalism. \n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=8cm, trim = 1in 0in 0in 0in, clip = true]{L3_profiles.eps}}\n\\caption{(Color online) Imaginary part of the electric field of the fundamental mode of an $\\mathit{L3}$ cavity, (a): $\\Im(Q_x(\\mathbf{r}))$ and (b): $\\Im(Q_y(\\mathbf{r}))$. In the one-QD simulation, the dot was placed in the central maximum of the y-field (dot position marked by a white cross). For the two-QD simulations, the dots were placed in the corresponding secondary maxima (positions marked by white stars).}\n\\label{L3_profiles}\n\\end{figure}\n\nThe cavity is a modified $L3$ cavity\\cite{Hennessy_2007, Reinhard_2012}, where the two holes on each side of the cavity are shifted outwards by $0.15a$, and their radii are decreased by $80\\%$. This design improves the quality factor by more than one decade compared to that of a standard $L3$ cavity, while changing the field profile only marginally. We include in the computation only the fundamental cavity mode, shown in Fig. \\ref{L3_profiles}. We further assume the QD to lie on the in-plane symmetry axis of the cavity, where $Q_x = 0$. The diagonalization of the matrix (\\ref{fullmat}) is then equivalent to the well-known expression\n\n\\begin{equation}\n\\det \\begin{pmatrix}\n\\omega_y - \\omega & g_c \\\\\ng^*_c & \\omega_c  - \\omega \n\\end{pmatrix} = 0 \\, .\n\\label{detL3}\n\\end{equation}\n\nThe coupling constant $g_c$, through Eq. (\\ref{coupling}), is\n\n\\begin{equation}\ng_c = \\left(\\frac{2\\pi \\omega_0}{\\varepsilon_{\\infty} \\hbar}\\right)^{1\/2} d \\,Q_y(\\mathbf{r}_{\\alpha}) \\, ,\n\\end{equation}\nwhich matches previous theoretical results \\cite{Andreani_1999} when the dot is sitting in the center $\\mathbf{r}_0$ of the cavity and the mode volume is defined as $\\frac{1}{V} = \\vert {Q}_y(\\mathbf{r}_0) \\vert^2$. As expected from Eq. (\\ref{detL3}), for $ \\vert g_c \\vert^2 > \\vert \\gamma_c - \\gamma_y\\vert ^2 \/16$, vacuum-field Rabi splitting appears between two polariton modes. The energy splitting at zero dot-cavity detuning is given by $2\\hbar\\Omega$, where the Rabi frequency $\\Omega$ is\n\n\\begin{equation}\n\\Omega = \\sqrt{\\vert g_c \\vert ^2 - \\frac{(\\gamma_c - \\gamma_y)^2}{16}} \\, .\n\\label{Rabi}\n\\end{equation}\nUsing the PHC and QD parameters we already introduced, the coupling constant was computed to be $\\hbar \\vert g_c \\vert = 147 \\mathrm{\\mu eV}$, which compares perfectly with the most recently reported result for that system\\cite{Reinhard_2012}.\n\nAfter showing the way the standard one-dot cavity-QED results are reproduced with our formalism, we now proceed to the situation of two dots coupled to the same cavity mode (see Fig. \\ref{L3_profiles}), and so radiatively coupled to each other. \n\nWe assume a symmetric spatial configuration of the two dots with respect to the cavity center (see Fig. \\ref{L3_profiles}), resulting in equal coupling constants $\\hbar \\vert g_c \\vert = 125 \\mathrm{\\mu eV}$ for the two dots. Since usually $\\gamma_c \\gg \\gamma_y$, i.e. the losses through the cavity mode are significantly larger than the QD losses through other channels (both non-radiative and radiative through modes other than the cavity mode), we set here and in all following sections $\\gamma_y^{1,2} = 0$. Given the phenomenological way these rates enter the formalism, calculations can easily be generalized to include finite QD loss rates. Let us first consider the case of zero dot-dot detuning $\\delta=\\omega_y^1-\\omega_y^2$. The relevant exciton states are in this case the symmetric and antisymmetric linear combinations of the two QD states, whose coupling to the cavity mode depends on the symmetry of the electric field profile. As discussed extensively in Ref. [\\onlinecite{Portalupi_2011}], the $\\mathit{L3}$ cavity symmetry is described by the $D_{2h}$ point group, and its fundamental mode belongs to the $B_{2u}$ irreducible representation, which is even with respect to the $\\hat{\\sigma}_{yz}$ symmetry operation (mirror reflection with respect to the $\\mathit{yz}$ plane) -- as can also be seen from Fig. \\ref{L3_profiles}. Hence, the antisymmetric QD state remains dark, while the symmetric one behaves as a single exciton with a coupling constant $\\sqrt{2} g_c $. \n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=8cm, trim = 0.7in 0in 0in 0.7in, clip = true]{L3_2QD_delta0.eps}}\n\\caption{(Color online) (a): Eigenfrequencies (solid lines) and radiative rates (dashed lines) for two QDs with no dot-dot detuning, strongly coupled to an $\\mathit{L3}$ cavity mode with $\\mathcal{Q} = 10000$. With a dashed-dotted line, the bare cavity resonance is also indicated. The Hopfield coefficients for each solution, correspondingly color-coded, are presented in panels (b): equal (in absolute value) QD coefficients and (c): cavity coefficient.}\n\\label{L3_2D_det0}\n\\end{figure}\n\nIn Fig. \\ref{L3_2D_det0} (a) we plot the eigenfrequencies of the system as a function of the detuning between the exciton resonance frequency $\\omega_y$ (same for both dots) and the cavity resonance frequency $\\omega_c$, as computed for a cavity quality factor of $\\mathcal{Q} = 10000$. We observe vacuum Rabi splitting between an upper and a lower polariton in exactly the same way we would for a single dot coupled to the cavity, but in addition we see a dark mode which is a trivial solution, $\\omega = \\omega_y$. The splitting between the lower and the upper polaritons at zero dot-cavity detuning is $2 \\hbar \\Omega_c = 347 \\mathrm{\\mu eV}$, which for $\\mathcal{Q} = 10000$ corresponds exactly to an effective coupling constant of  $\\sqrt{2} \\times 125\\mathrm{\\mu eV}$. The system is further characterized in panels (b) and (c), where we plot the Hopfield coefficients for each of the three eigenmodes (correspondingly color-coded). This clear collective behavior has been observed experimentally in a QD-cavity system\\cite{Reitzenstein_2006, Laucht_2010, Kim_2011, Albert_2012}, while the more general dependence of the effective coupling constant with the number of coupled two-level systems $N$ -- given by $\\sqrt{N} \\vert g_c \\vert$ -- has also been observed in a circuit-QED system\\cite{Fink_2009}. It is very important to remark that this dependence has nothing to do with the $\\sqrt{N} \\vert g_c \\vert$ energy splitting  of different rungs in a Jaynes-Cummings model, where $N$ would be the number of photons in the system: on the contrary, as discussed before, here we restrict to the linear response only, which holds in the limit of vacuum electromagnetic field. The effect in our case is simply due to the collective behavior of the $N$ resonant quantum dots.\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=8cm, trim = 0.6in 0in 0in 0.5in, clip = true]{L3_2QD_delta300.eps}}\n\\caption{(Color online) (a): Eigenfrequencies (solid lines) and radiative rates (dashed lines) for two QDs with a dot-dot detuning of $300 \\mathrm{\\mu eV} $, strongly coupled to an $\\mathit{L3}$ cavity mode with $\\mathcal{Q} = 10000$. With dashed-dotted lines, the bare excitons and the bare cavity resonances are also shown. The Hopfield coefficients for each solution, correspondingly color-coded, are presented in panels (b): first exciton coefficient, (c): second exciton coefficient, and (d): cavity coefficient.}\n\\label{L3_2D_det300}\n\\end{figure}\n\nThe major experimental challenges to the radiative coupling of two spatially separated quantum dots is achieving both spatial control (to ensure strong overlap between each of the dots and the cavity mode) and spectral control (to ensure as small dot-dot and dot-cavity detuning as possible). Typically, QDs are characterized by an inhomogeneous distribution of exciton energies with a width of several meV. Then, two QDs are very likely to be detuned. In Fig. \\ref{L3_2D_det300}, we study the same system, but assuming a detuning $\\delta = 300 \\mathrm{\\mu eV}$. Close to resonance, all of the eigenmodes acquire a finite component from the cavity mode. Additionally, they have both a significant $\\vert \\lambda_y^1 \\vert$ coefficient (panel (b)), and a significant $\\vert \\lambda_y^2 \\vert$ coefficient (panel (c)), implying that there is a sizable radiative coupling present. The radiative coupling is expected to vanish as the cavity-dot detunings become much larger than the coupling constant, and an expression for an effective coupling strength in this limit was derived in Ref. [\\onlinecite{Imamoglu_1999}] and [\\onlinecite{Gywat_2006}]. Concerning the spatial control, it is important to note that our approach allows for a statistical analysis of the effect of an imperfect positioning of the dots, although such an analysis lies beyond the scope of the present work. \n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=8cm, trim = 0.7in 0in 0in 0.5in, clip = true]{L3_timedep.eps}}\n\\caption{(color online) Time evolution of the probability of an excitation in one dot to be transferred to the second dot or to the cavity. (a): $\\delta = 0$, $\\mathcal{Q} = 10000$; (b): $\\delta = 0$, $\\mathcal{Q} = 30000$; (c) $\\hbar \\delta = 300\\mathrm{\\mu eV}$, $\\mathcal{Q} = 10000$; (d) $\\hbar \\delta = 300\\mathrm{\\mu eV}$, $\\mathcal{Q} = 30000$.}\n\\label{time_L3}\n\\end{figure}\n\nWe now address the question of how the excitation transfer process depends on time. This aspect is of particular importance to assess the usefulness of the radiative excitation transfer as a coupling mechanism between different qubits in a semiconductor-based quantum gate architecture. In the present case, when polaritonic features are spectrally resolved, one correspondingly expects the excitation to oscillate between the different basis states, including the photon state. To illustrate this aspect, we compute the time-dependent amplitudes of the various basis states, assuming that one QD is excited at $t=0$. From these amplitudes, we extract time-dependent probabilities of finding the excitation in each of the basis modes, expressed in vector form as\n\n\\begin{equation}\n\\mathbf{P}(t) = \\left|e^{-i\\Lambda_2 t} \\bm{\\lambda}_{in} \\right|^2 \\, ,\n\\label{prob}\n\\end{equation}\nwhere $\\Lambda_2$ is the matrix of Eq. (\\ref{fullmat}). These probabilities are properly normalized if one accounts also for the probability $P_{out}(t)$ of the excitation to have radiated out of the system, i.e. $\\sum P_i(t) = 1 - P_{out}(t)$. In Fig. \\ref{time_L3} we plot these time-resolved probabilities for a starting excitation in one of the QDs, i.e. $\\bm{\\lambda}_{in} = (1, 0, 0)$. We study four different cases: either zero dot-dot and dot-cavity detuning, or $\\hbar \\delta = 300 \\mathrm{\\mu eV}$ (with the cavity frequency tuned at the average of the two exciton frequencies), and cavity $\\mathcal{Q}$-factor equal to either $10000$ or $30000$. In panels (a) and (b), where $\\delta = 0$, the probabilities never decay to zero due to the presence of a dark state and the fact that no non-radiative decay mechanism was included. In panels (c) and (d) a dark state no longer exists, and a clear decay of the excitation with a characteristic lifetime depending on the $\\mathcal{Q}$-factor is visible. All plots show that the excitation oscillates between the three possible states, on a time scale defined through the radiative coupling strength. In particular, the probability of finding the system in an excited state of the second QD remains sizable over several oscillation periods, showing that a significant dot-dot interaction can be achieved with experimentally feasible parameters. \nThese results generally agree with specific setups of radiatively coupled QDs in photonic crystals, that have been recently studied in the literature.\\cite{Yao_2009, Kristensen_2011}\n\n\\subsection{Application to $Ln$ cavities}\n\\label{Ln}\n\nRecently, using $Ln$ cavities with $n > 3$ to achieve light-matter coupling has spurred interest\\cite{Choi_2007, Felici_2010, Surrente_2011}, as these cavities generally have a larger quality factor than the $\\mathit{L3}$ ones -- though at the expense of a larger mode volume and thus a smaller dot-cavity coupling strength. \n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=8cm, trim = 1.8in 0in 0in 0in, clip = true]{L11_profiles.eps}}\n\\caption{(Color online) $\\Im(Q_y(\\mathbf{r}))$ for the four lowest-energy modes of the $\\mathit{L11}$ cavity; (a): Fundamental mode $M_1$, at $\\hbar \\omega_1 = 1.3065 \\mathrm{eV}$, (b): $M_2$, $\\hbar \\omega_2 = 1.3125 \\mathrm{eV}$, (c): $M_3$, $\\hbar \\omega_3 = 1.3269 \\mathrm{eV}$, (d): $M_4$, $\\hbar \\omega_4 = 1.3565 \\mathrm{eV}$. The positions of the quantum dots are marked with white stars.}\n\\label{L11_profiles}\n\\end{figure}\n\nHere, we investigate cavities of varying length $\\mathit{n}$ with a common setup, illustrated in Fig. \\ref{L11_profiles} for $\\mathit{n} = 11$. In the figure, we show the first four modes, $M_{1-4}$, of the $\\mathit{L11}$ cavity, with resonant energies $1.3065 \\mathrm{eV}, 1.3125    \\mathrm{eV}, 1.3269 \\mathrm{eV}, $ and $1.3565\\mathrm{eV}$, respectively. In all the results to follow, for all $\\mathit{n}$, the two dots were placed in the center of an elementary cell on each side of the center of the defect (i.e. at a distance $a$ from the center of the cavity and so $2a$ from each other), where the coupling constants for each of them in the $\\mathit{n} = 11$ case are $\\vert \\hbar g_1 \\vert = 94 \\mathrm{\\mu eV}$, $\\vert \\hbar g_2 \\vert = 55 \\mathrm{\\mu eV}$, $\\vert \\hbar g_3 \\vert = 65 \\mathrm{\\mu eV}$, and $\\vert \\hbar g_4 \\vert = 89 \\mathrm{\\mu eV}$. Since the smallest energy difference between the cavity resonances in this case is between $\\omega_1$ and $\\omega_2$, and is $\\approx 6 \\mathrm{meV}$, i.e. much larger than all the coupling strengths, it is reasonable to expect that the dots will never couple significantly to more than one mode. Thus, the phenomenology of the system will be, qualitatively, the same as the one described in section \\ref{L3}, which was also verified by our computations.\n\nThe situation should change significantly when increasing the length $n$ of the photonic defect. Then, we expect the energy spacing between the resonant frequencies of the $Ln$ cavity to decrease and eventually become comparable to the typical coupling strength. In this situation, the radiative transfer process is no longer mediated by an isolated cavity mode, and a smooth transition to a multi-mode coupling regime is expected. In order to determine at which cavity length this crossover occurs, one should also consider the fact that the coupling of a dot to each individual mode decreases with the increase of the mode volume. As a result, the crossover length is increased with respect to what would be given by a simple assumption of constant coupling strength per mode.\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=8cm, trim = 1in 0in 0in 0.8in, clip = true]{Ln_comparison.eps}}\n\\caption{(Color online) (a): Black lines -- coupling constants between one QD and the ten lowest modes of an $Ln$ cavity, vs. $n$; green line -- energy separation between the lowest two cavity modes. (b)-(d): Hopfield coefficients of one polariton eigenstate as a function of the bare exciton frequency $\\omega^1_y$ with no dot-dot  detuning, for $n = 71$, $n = 141$, $n = 211$ (the values marked by dashed vertical lines in (a)). The red line shows the dot coefficients, while the blue lines belong to the many cavity modes. (e)-(g): Same as (b)-(d) but for another polariton state.}\n\\label{Ln_comparison}\n\\end{figure}\n\nIn Fig. \\ref{Ln_comparison}, we plot the minimum mode-separation $\\omega_2 - \\omega_1$ vs. the length $n$ of the cavity, and in addition show the coupling strengths $\\vert g_m^{1} \\vert$ for $m = 1\\dots 10$. For all $n$, the dots were placed as in Fig. \\ref{L11_profiles} -- at a distance $a$ on each side of the center of the cavity. The fact that half of the coupling constants decay much faster as a function of $n$, is again explained by the particular symmetry of the field profiles. It turns out that for every $n$, the modes alternate between symmetric and antisymmetric w.r.t. $\\hat{\\sigma}_{yz}$, as can be seen in Fig. \\ref{L11_profiles} for the $\\mathit{L11}$ case. In the limit of large $n$, the antisymmetric modes have a small amplitude at the QD positions close to the node, resulting in a small radiative coupling strength. \n\nThe crossover from a single-mode to a many-mode regime occurs around $n = 150$, as clearly visible in Fig. \\ref{Ln_comparison}. In panels (b)-(g), we show the corresponding Hopfield coefficients for three different values of $n$, given by $n=71$, $n=141$, and $n=211$, also indicated by dashed lines in panel (a), and for two different polariton modes.\nConsequently, for $n = 71$, the Hopfield coefficients of two different polariton eigenstates, shown in panels (b) and (e) respectively, are still largely dominated by one cavity and one dot component. On the other hand, for $n = 211$ -- panels (d) and (g) -- the value of several photonic fractions $\\lambda_m$ is non-negligible. \n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=8cm, trim = 0.8in 0in 0in 0.8in, clip = true]{L141_2QD_delta0.eps}}\n\\caption{(Color online) Eigenfrequencies for two QDs with no dot-dot detuning in an $\\mathit{L141}$ cavity ($\\mathcal{Q} = 50000$ for each mode) vs. the resonant frequency of the excitons. The insets show close-ups over two selected regions.}\n\\label{L141_2QD_det0}\n\\end{figure}\n\nIn Fig. \\ref{L141_2QD_det0} we plot the polariton energies as a function of QD-exciton energy in the case of the $\\mathit{L141}$ cavity, for $\\delta = 0$. As mentioned already, the photon modes alternate between symmetric and anti-symmetric w.r.t. the $\\hat{\\sigma}_{yz}$ operator, hence coupling to either the symmetric or anti-symmetric linear combination of the QD states is present. In the figure, the polaritons due to a combination of symmetric states are denoted by blue lines, while the anti-symmetric combinations are represented by red lines. In the symmetric case, the exciton-photon coupling strength is always large, and anti-crossing occurs at every mode. In the anti-symmetric case, the results show a transition from weak coupling (close to the lowest $\\omega_m$) to strong coupling (anti-crossing is visible in the higher-$\\omega$ inset), due to the fact that the coupling strengths there become larger than $\\omega_m\/4\\mathcal{Q}$. It is clear both from Figs. \\ref{Ln_comparison} and \\ref{L141_2QD_det0} that for $n\\rightarrow\\infty$, the dots couple to a  structured continuum of photon modes, reproducing the physics of a W1 waveguide. This situation is studied in the next section. \n\n\\subsection{Application to a W1 waveguide} \n\\label{W1}\n\nThe results obtained for a $Ln$ cavity indicate that radiative coupling is still sizable in very long structures and might be effective even at very long distance between the two QDs. Here, we investigate this possibility in more detail, by considering QDs embedded in a W1 photonic crystal waveguide. \n\nCoupling of a single dot to a W1 waveguide (or a similar structure) with the purpose of spontaneous emission enhancement (and the potential application as a single-photon source) has already been widely discussed theoretically\\cite{Rao_2007, MangaRao_2007, Lecamp_2007}, and achieved experimentally\\cite{Viasnoff-Schwoob_2005, Lund-Hansen_2008, Thyrrestrup_2010, Schwagmann_2011, Hoang_2012, Laucht_2012, Laucht_2012a}. The fact that it is already possible to couple efficiently a dot to the guided modes of the waveguide is promising in view of achieving radiative coupling between \\textit{two} dots that could -- due to the spatial extension of the structures and the modes they support -- extend to inter-dot distances for which targeting each dot individually by a laser pulse is possible. \n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=8cm, trim = 1in 0in 0in 1in, clip = true]{W1_profiles.eps}}\n\\caption{(Color online) (a): Band structure of the $W1$ waveguide; the dashed line shows the light cone. The QD resonant energies are close to the band-edge energy of the main guided band (blue). The field profiles of four guided modes in that spectral region are shown, over a small stretch of the waveguide: (b), (c): the two degenerate modes at $\\hbar \\omega_k = 1.30308\\mathrm{eV}$ (anti-symmetric combination in (b), symmetric in (c)); (d): the symmetric mode at $\\hbar \\omega_k = 1.30224\\mathrm{eV}$ (e): the symmetric mode at $\\hbar \\omega_k = 1.30218\\mathrm{eV}$ (at the band edge). In all computations, one QD was placed in the center of the waveguide (white cross), while the second one was placed in the center of one of the successive elementary cells (white stars).}\n\\label{W1_profiles}\n\\end{figure}\n\nWe begin our study by looking at the modes of the W1 waveguide. The band structure is presented in Fig. \\ref{W1_profiles} (a), where two guided bands in the band-gap of the regular crystal are visible. Strongest dot-PHC coupling is typically achieved for the smallest group velocity (largest local density of states of the photonic modes), and so the spectral range we concentrate on is around the band-edge of the main guided band (blue line), where the group velocity of the ideal photonic structure vanishes. The second guided band is spatially odd with respect to a $\\hat{\\sigma}_{xz}$ reflection,\\cite{Andreani_2006} and would not couple to the exciton state of a QD located at the center of the waveguide. In the simulations below, we compute the $W1$ modes for $2048$ $k$-points in the interval $(-\\pi\/a, \\pi\/a]$, which is equivalent to simulating a waveguide of length 2048 elementary cells with periodic boundary conditions (PBC). In panels (b)-(e) of Fig. \\ref{W1_profiles}, we show the electric field profiles of four modes lying close in energy to the band edge of the main guided band. As is the case with all structures we considered so far, this band has vanishing $Q_x$ component on the symmetry axis of the waveguide, allowing us again to include the $y$-polarized fields only. Furthermore, modes at $\\pm k$ are degenerate -- one propagating and one counter-propagating -- with real-space profiles proportional to $\\exp(ikx)$ and $\\exp(-ikx)$, respectively. As basis states, we take the symmetric and the anti-symmetric combination of the degenerate guided modes, representing the fields by their ``standing wave'' profiles: one with a maximum and one with a zero amplitude in the center of the guide (compare panels (b)-(c)). Without loss of generality (due to the PBC), we place one dot at that position, so that it couples to one of the modes only. The second dot is then placed in the center of a successive elementary cell, and will, in general, couple to every mode in the basis thus constructed, so even for zero dot-dot detuning, no fully dark state is present.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=8cm, trim = 0.6in 0in 0in 0.6in, clip = true]{W1_2QD_delta0.eps}}\n\\caption{(Color online) (a): Eigenfrequencies for two QDs in the waveguide, with no dot-dot detuning. The Hopfield coefficients for the red line of (a) are shown in (b), where the green line shows the QD coefficients $\\vert \\lambda_y^1 \\vert = \\vert \\lambda_y^2 \\vert$, and the black lines show all the waveguide coefficients $\\vert \\lambda_m \\vert$. The same in (c), but for the blue line of (a). The loss rate of each mode is $\\gamma_{\\mathrm{w}} = \\omega_y^1\/50000 \\approx 26 \\mathrm{\\mu eV}$.}\n\\label{W1_2QD_det0}\n\\end{figure}\n\nIn Fig. \\ref{W1_2QD_det0} (a), we show the polariton structure in the spectral range close to the band edge of two dots with dot-dot detuning $\\delta = 0$, with the second dot placed at the closest possible distance, $a$, from the first one. The quality factor of each of the photonic modes was again set to $\\mathcal{Q} = 50000$ for all modes. While a strong dependence of the $W1$ loss rates on the group velocity close to the band edge has been shown in transmission measurements\\cite{OFaolain_2007}, this dependence is heavily influenced by back-scattering due to disorder. In our case, we model stationary modes rather than transport, and the only relevant radiative loss is the one out of the plane of the PHC slab. Then, the assumption of approximately constant $\\mathcal{Q}$-s is realistic, as seen from microscopic modeling of extrinsic disorder-induced losses \\cite{Savona_2011, Gerace_2004}. Polariton modes originating from antisymmetric photon modes are essentially uncoupled and are not displayed (although, they were still included in the computation). The coupling constants of each of the dots to each of the symmetric modes varies very little, and is $\\hbar \\vert g_m^{1,2} \\vert \\approx 7 \\mathrm{\\mu eV}$. The $\\omega = \\omega_y^1$ solution (straight diagonal in panel (a)) is due to the anti-symmetric QD combination, which is almost dark. The strongest anti-crossing behavior is exhibited by the polariton lying below the band edge (blue line), whose Hopfield coefficients are given in panel (c). The remaining polariton modes display similar behavior, so the Hopfield coefficients of just one of them (the red line of (a)) are given in panel (b). For completeness, the same plots but for $\\hbar \\delta = 100 \\mathrm{\\mu eV}$ are given in Fig. \\ref{W1_2QD_det100}. In this case, no dark modes are present and Hopfield coefficient corresponding to the two QDs are generally different from each other, as seen in panels (b) and (c). In both Figs.  \\ref{W1_2QD_det0} and \\ref{W1_2QD_det100} we observe that anti-crossings are still present -- though characterized by a very small energy splitting -- where the exciton becomes resonant with the various guided modes. This situation can be understood as the precursor to the structured continuous spectrum of modes that would arise in the limit of infinite waveguide length, analogous, for example, to the polariton modes arising from the interaction between an exciton in a two-dimensional quantum well and the three-dimensional continuum of electromagnetic modes.\\cite{Tassone_1990, Tassone_1992}\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=8cm, trim = 0.6in 0in 0in 0.8in, clip = true]{W1_2QD_delta100.eps}}\n\\caption{(Color online) (a): Eigenfrequencies for two QDs in the waveguide, with dot-dot detuning $\\hbar \\delta = 100 \\mathrm{\\mu eV}$ and $\\gamma_{\\mathrm{w}}$ as in Fig. \\ref{W1_2QD_det0}. The Hopfield coefficients for the red line of (a) are shown in (b), where the two green lines show the QD coefficients $\\vert \\lambda_y^{1, 2} \\vert$, and the black lines, all the waveguide coefficients $\\vert \\lambda_m \\vert$. Same in (c), but for the blue line of (a).}\n\\label{W1_2QD_det100}\n\\end{figure}\n\nThe present formalism provides a detailed quantitative account of the effect of the guided electromagnetic field on the radiation properties of few QDs. In particular, we derive below the Purcell enhancement of the radiative rate characterizing a single QD, and the distance-dependence of the radiative excitation transfer process between two distant QDs. We compute these properties both numerically, and analytically. To this purpose, let us consider the elements of the matrix $\\Lambda_1$ introduced by Eq. (\\ref{matrixone}):\n\n\\begin{equation}\n\\Lambda_1^{\\alpha \\beta} = (\\omega^{\\alpha} - \\omega)\\delta_{\\alpha \\beta} - G^{\\alpha \\beta}(\\omega) \\, ,\n\\end{equation}\nwhere the coupling matrix elements $G^{\\alpha \\beta}$ are proportional to the Green's function of eq. \\ref{green}:\n\n\\begin{equation}\nG^{\\alpha \\beta}(\\omega) = \\sum_{m=1}^M \\frac{g_{m}^{\\alpha} g_{m}^{\\beta *}}{\\omega_m - \\omega} = d^2 \\frac{2 \\pi}{\\epsilon_{\\infty} \\hbar} \\frac{\\omega^2}{c^2} G(\\mathbf{r}_{\\alpha}, \\mathbf{r}_{\\beta}, \\omega).\n\\label{lamab}\n\\end{equation}\nFor a structure with no sharp resonances -- like the waveguide -- we can take advantage of the exciton-pole approximation and substitute $\\omega = \\omega_0$ in the denominator, in which case $G^{\\alpha \\alpha}(\\omega_0)$ is the self-interaction energy of each dot, while $|G^{12}(\\omega_0)|$ is an effective coupling constant for the case of two dots with zero dot-dot detuning, i.e. $\\omega^1_y = \\omega^2_y = \\omega_0$. In order to derive an analytical expression for the coupling, let us replace the sum with an integral over $k = k_x$, and use the fact that, in accordance with Bloch's theorem, when $\\mathbf{r}_{\\alpha}$ and $\\mathbf{r}_{\\beta}$ are in the center of an elementary cell, $g_k(\\mathbf{r}_{\\beta}) = \\exp{(-\\mathrm{i}kx)} g_k(\\mathbf{r}_{\\alpha})$, to write\n\n\\begin{equation}\nG^{\\alpha \\beta}(\\omega_0) = \\frac{a}{2\\pi} \\int_{-\\frac{\\pi}{a}}^{\\frac{\\pi}{a}}\\mathrm{d}k \\frac{\\left| g_k \\right|^2 e^{\\mathrm{i}kx}}{\\omega(k) - \\omega_0} \\, .\n\\end{equation}\n\nA few simplifications are due. First, we write $\\omega_k = \\Re(\\omega(k))$ and $\\gamma_{\\mathrm{w}} = -2\\Im(\\omega(k))$, and assume the latter is constant, equal to $\\omega_0\/\\mathcal{Q}$. Furthermore, we assume $\\left| g_k \\right|^2 = \\left| g \\right|^2$, i.e. the coupling strength has weak dependence on $k$. This feature is due to the small spatial extension of the exciton wave function, resulting in a very broad distribution in Fourier space with approximately constant overlap with all guided modes, and is also confirmed by our numerical results. Finally, by taking $k_0$ as the positive Bloch momentum for which the guided mode is resonant with the exciton frequency $\\omega_0$, and defining the corresponding group velocity\n\n\\begin{equation}\nv_g = -\\left. \\frac{\\mathrm{d} \\omega_{k}}{\\mathrm{d} k} \\right|_{k_0} \\, ,\n\\end{equation} \nwe get\n\n\\begin{equation}\nG^{\\alpha \\beta}(\\omega_0) \\approx \\frac{a}{\\pi} \\frac{\\left| g \\right|^2}{v_g} \\int_{0}^{\\frac{\\pi}{a}}\\mathrm{d}k \\frac{ \\cos{(kx)}}{k - k_0 - \\mathrm{i}\\frac{\\gamma_{\\mathrm{w}}}{2v_g}} \\, . \n\\label{lamapprox}\n\\end{equation}\n\nThis expression holds in the limit where the resulting spectral linewidth is small enough so that the group velocity is still well defined. It can now be applied for example to obtain the radiative lifetime of a single dot embedded in the waveguide as $\\Gamma^{\\alpha} = 2\\Im(G^{\\alpha \\alpha})$, and so\n\n\\begin{equation}\n\\Gamma^{\\alpha} = \\frac{2a}{\\pi}\\frac{\\left| g \\right|^2}{v_g} \\left. \\tan^{-1}{\\left(\\frac{2(k-k_0)v_g}{\\gamma_{\\mathrm{w}}}\\right)}\\right|_{k = 0}^{k = \\pi\/a} \\, .\n\\label{w1loss}\n\\end{equation}\nThe Purcell factor for the enhancement of the single-dot spontaneous emission rate is then given by the ratio between Eq. (\\ref{w1loss}) and Eq. (\\ref{loss}). This result takes into account the detailed structure of the photonic environment resulting from the waveguide. In this respect, it generalizes the result obtained by assuming that only one Bloch mode at wave vector $k=k_0$ determines the radiation loss process.\\cite{Rao_2007, Lecamp_2007} This simplified result is recovered by taking the limit $\\gamma_{\\mathrm{w}} \\rightarrow 0$ in the integral (\\ref{lamab}), namely by assuming that the guided Bloch mode has vanishing extrinsic radiation loss rate. The emission rate $\\Gamma^{l}$ of the dot into leaky modes can also be estimated numerically by restricting the summation in Eq. (\\ref{lamab}) to the modes which lie above the light-cone only. Then, the $\\beta$-factor in the absence of non-radiative decay mechanisms can also be computed as\n\n\\begin{equation}\n\\beta = \\frac{\\Gamma^{\\alpha}}{\\Gamma^{\\alpha} + \\Gamma^{l}} \\, ,\n\\end{equation}\nand a further generalization to the case in which non-radiative processes are also present follows straightforwardly.\n\nAs a development from the previous works which consider just one dot in the waveguide, we now proceed to quantify the radiative excitation transfer process between \\textit{two} QDs and its dependence on inter-dot distance. The closed-form expression for the cross-coupling term $G^{12}$, obtained by carrying out the integral (\\ref{lamapprox}) reads\n\n\\begin{widetext}\n\\begin{equation}\nG^{12} = \\frac{a}{\\pi}\\frac{\\left| g \\right|^2}{v_g} \\left. \\left[\\cosh\\left(\\frac{x}{r_{12}} - \\mathrm{i}k_0 x\\right) \\mathrm{Ci}\\left(-\\mathrm{i}\\frac{x}{r_{12}} + (k - k_0)x\\right) + \\mathrm{i} \\sinh\\left(\\frac{x}{r_{12}} - \\mathrm{i}k_0 x\\right) \\mathrm{Si}\\left(\\mathrm{i}\\frac{x}{r_{12}} - (k - k_0)x\\right)\\right]  \\right|_{k = 0}^{k = \\frac{\\pi}{a}} \\, ,\n\\label{lam12}\n\\end{equation} \n\\end{widetext}\nwhere $\\mathrm{Ci}(z)$ and $\\mathrm{Si}(z)$ are respectively the cosine integral and the sine integral functions, and we defined $r_{12} = 2v_g\/\\gamma_{\\mathrm{w}}$. The quantity $r_{12}$ is simply the decay length associated to the propagation of light along the resonant guided mode. We expect this decay to characterize also the distance dependence of the radiation transfer process. Indeed, under the ideal assumption of vanishing radiation loss rate for the guided mode, in a one-dimensional geometry one would expect the radiative transfer process to be \\textit{independent of the distance}. For comparison, as has already been shown, the coupling strength decays as $R_{\\alpha \\beta}^{-1}$ in 3D bulk semiconductor\\cite{Parascandolo_2005}, and as $R_{\\alpha \\beta}^{-1\/2}$ in a 2D planar cavity system\\cite{Tarel_2008}. In Fig. \\ref{W1_distance}, we display the absolute value of $G^{12}$ computed numerically through Eq. (\\ref{lamab}), for four different values of the exciton frequency $\\omega_0$ of the two QDs, in a waveguide of length $2048a$. This quantity is compared to the result obtained from the analytical model of Eq. (\\ref{lam12}) and to the simpler assumption of an exponential dependence $|G^{12}| = |G^{11}| e^{-x\/r_{12}}$. In panel (d), where $\\omega_0$ is taken to lie below the edge of the guided band, the group velocity cannot be properly defined, and thus the analytical model does not apply.\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=8cm, trim = 1.5in 0in 0in 1in, clip = true]{W1_distance.eps}}\n\\caption{(Color online) Absolute value of the off-diagonal term of the matrix in Eq. (\\ref{matrixone}), in the exciton-pole approximation, computed numerically for a finite-size waveguide (solid line), analytically through Eq. (\\ref{lam12}) (dashed-dotted line), and through an exponential decay model with characteristic distance $r_{12} = 2v_g\/\\gamma_{\\mathrm{w}}$ (dashed line, $\\gamma_{\\mathrm{w}}$ as in Fig. \\ref{W1_2QD_det0}). In (a): $\\hbar \\omega_0 = 1.30353\\mathrm{eV}$, $n_g = 74$, (b): $\\hbar \\omega_0 = 1.30240\\mathrm{eV}$, $n_g = 195$, (c): $\\hbar \\omega_0 = 1.30220\\mathrm{eV}$, $n_g = 525$, (d): $\\hbar \\omega_0 = 1.30208 \\mathrm{eV}$, i.e. 100 $\\mathrm{\\mu eV}$ below the band edge.}\n\\label{W1_distance}\n\\end{figure}\nApart from this case, it is clear that the distance dependence of the inter-dot coupling is perfectly captured by the simple exponential decay model. The oscillations of the numerical curve in panel (a) are due to the finite length of the waveguide and reproduce the spatial behavior of the Bloch mode at $k=k_0$ that dominates the transfer process. These oscillations cannot obviously be reproduced by the analytical model that implicitly assumes an infinitely extended waveguide. As anticipated, the numerical results show that the distance dependence of the transfer rate is expressed by the decay associated to the light propagation, and quantified by the decay length $r_{12}$. It is interesting to note that even for very small group velocities, e.g. $v_g < c\/500$, the interaction distance is still of the order of $100a = 26 \\mu \\mathrm{m}$, i.e. of mesoscopic scale, thus confirming the potential of the $\\mathit{W1}$ for very long-distance dot-dot coupling. \n\nMore generally, Eq. (\\ref{lam12}) suggests that there is a compromise, enforced by the group velocity, between strength and distance dependence of the transfer process. The overall strength of the transfer rate depends inversely on the group velocity. This expresses the magnitude of the local density of states at the QD exciton frequency or, in a more suggestive picture, the fact that slow light interacts with a QD over a longer time lapse. However, a smaller group velocity also implies a shorter characteristic decay length $r_{12}$, as we are assuming a constant radiation loss rate. In a realistic system, including disorder, we further expect the group velocity picture to break down at frequencies close to the band edge, where disorder-induced localization of light dominates and the spatial decay associated to the localization length becomes shorter than $r_{12}$. This calls for an analysis including disorder effects, that we will consider in a future work.\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=8cm, trim = 1.1in 0in 0in 0.8in, clip = true]{W1_timedep.eps}}\n\\caption{(Color online) Time evolution of the probability of an excitation in one dot to be transfered to the second dot or to the many PHC modes (blue lines). The dot-dot detuning is $\\hbar \\delta = \\mathrm{100 \\mu eV}$. Horizontally across the panels, the inter-dot distance changes from $260 \\mathrm{nm}$ to $2.6 \\mathrm{\\mu m}$ to $5.2 \\mathrm{\\mu m}$. Vertically across the panels, the exciton frequency of the first dot changes from (a)-(c): $\\hbar \\omega_y^1 = 1.30224\\mathrm{eV}$ (close to the band edge), through (d)-(f): $\\hbar \\omega_y^1 = 1.30218\\mathrm{eV}$ (at the band edge), to (g)-(i): $\\hbar \\omega_y^1 = 1.30208\\mathrm{eV}$ ($100 \\mathrm{\\mu eV}$ below the band edge energy, which is then resonant with $\\hbar \\omega_y^2$).}\n\\label{W1_timedep}\n\\end{figure}\n\nWe conclude this section by studying the time-dependent probability amplitudes of the excitation lying in each mode. These quantities are plotted in Fig. \\ref{W1_timedep}, assuming that one QD is excited at $t=0$, for three different inter-dot distances and three different values of $\\omega_y^1$. In all cases, $\\hbar \\delta = 100 \\mathrm{\\mu eV}$ was imposed. As discussed above, the transfer mechanism is driven by several light modes. The plots show that the radiative transfer process still occurs and, in particular, the marked oscillations are characterized on average by a period that can be associated to an effective transfer rate $\\hbar \\Omega = 50-60 \\mathrm{\\mu eV}$. As in short $Ln$ cavities, this rate is quite sizable and should be observable in state-of-the-art GaAs-based photonic structures.\\cite{Viasnoff-Schwoob_2005, Surrente_2011, Lund-Hansen_2008, Thyrrestrup_2010, Schwagmann_2011, Hoang_2012, Laucht_2012, Laucht_2012a}\n\n\\section{Discussion and Outlook}\n\\label{discussion}\n\nWe have developed a general formalism of linear radiation-matter coupling in systems of many QDs embedded in a photonic crystal structure. The formalism is an extension of the exciton-polariton formalism well known for bulk semiconductors and quantum wells. It provides a quantitative account of a variety of radiative effects, starting from the basic microscopic parameters of the QD-PHC system. It is important to establish a relation between the present approach and  previous works that use the photonic Green's function\\cite{Hughes_2004, Rao_2007, Hughes_2007, Yao_2009, Kristensen_2011}. The equations obtained there have the advantage of highlighting the importance of each single mode in determining the effects under study, but, on the other hand, incorporate either single-mode approximations or perturbative expansions. Our approach is in a sense complementary, with the main advantage coming from the fact that the problem is framed into a simple matrix diagonalization form, and that we make use of the Bloch-mode expansion to obtain the exact electric field profile for each mode, which allows us to compute the couplings independently of any approximations. As examples of application, the main results presented in this work concerned radiative effects in the systems of one or two QDs embedded in $Ln$ cavities and the $\\mathit{W1}$ waveguide. In the case of one QD, we recover the known results for the Purcell enhancement of the radiative rate and the vacuum Rabi splitting in the strong coupling regime. In the two-QD case, we quantify the strength of the radiative excitation transfer between \\textit{spatially separated} QDs, which lies in the $100~\\mu\\mbox{eV}$ range at short distance. The comparison of the single-mode coupling strength and the energy spacing between modes in $Ln$ cavities of increasing length clearly shows that a crossover occurs -- around $n=150$ for GaAs-based systems -- between single-mode and multi-mode radiative coupling. In the multi-mode case, the radiative coupling strength through each photonic mode is smaller but the overall effective excitation transfer rate still ranges at about $50~\\mu\\mbox{eV}$, thus suggesting that the $W\\mathit{1}$ is an ideal structure for the realization of long-range radiative dot-dot coupling.\n\nThese results suggest that the QD-PHC system could be a candidate system to operate as a quantum bus and achieve controlled entangling interaction between distant qubits. This perspective is corroborated by the two following remarks. First, semiconductor QDs have recently seen a tremendous progress\\cite{Patton_2005, Press2008, Berezovsky_2008, Greilich_2009, Greilich_2011, Poem_2011, Muller_2012, Godden_2012, Kodriano_2012} towards the physical implementation of qubits that rely on the electron or hole spin as the computational degree of freedom, and on the interband optical transition as the main handle for single-qubit operations. Second, the optical quantum bus technology has already been successfully applied to achieve controlled two-qubit operations in the system of superconducting qubits.\\cite{DiCarlo2009} The controlled operation in that case has been achieved by moving in and out of the anti-crossing region in the polariton spectrum arising from radiation-matter coupling. In view of a similar development in the semiconductor QD case, at least three steps are still needed. First, the ability to fabricate site-controlled QDs, in order to position them with respect to the PHC structure. This is nowadays possible thanks to various kinds of growth on a patterned substrate.\\cite{Kiravittaya2006,Martin-Sanchez2009,Mehta2007,Mereni2009,Schneider2009} Second, a clear experimental proof of the radiative excitation transfer mechanism at long distance, that might only come from ad-hoc technique such as, for example, the single-QD two-dimensional four-wave-mixing spectroscopy.\\cite{Kasprzak_2011} Third, a reliable scheme for dynamically controlling the exciton-photon detuning at sufficiently high speed. For this latter task, extremely promising results are already available on the optical control of the resonant frequency of high-Q cavities, particularly using carrier-induced optical nonlinearity.\\cite{Notomi2010,Sato_2012}\n\nAn additional challenge is represented by the task of understanding and optimizing disorder effects in the light propagation throughout PHC structures. Apart from small variations of the field profiles\\cite{Portalupi_2011}, that should only marginally influence the magnitude of the coupling between one QD and a photon mode, the main effect of disorder is the localization of light in long PHC structures.\\cite{Topolancik_2007, Sapienza_2010, Savona_2011, Huisman2012} For the line defects studied in this work in particular, localization is known to compete with the ability of the waveguide to support slow-light propagation,\\cite{LeThomas2009, Mazoyer2009} and thus with the enhancement of radiative effects expected in these structures. More specifically, when approaching the band edge in a $W1$ waveguide, the localization length becomes shorter than the decay length related to extrinsic radiation losses. In this limit, there is no more light propagation and the concept of a group velocity of light no longer holds. As an illustrative example of this dramatic effect on light propagation, the statistical fluctuations of the transmission through a waveguide in the light-localization regime increase and the transmission coefficient of a finite-length waveguide takes values uniformly distributed between zero and one, for nominally identical samples and arbitrarily small variations of the frequency.\\cite{Mazoyer2010} It is therefore important to accurately characterize the radiative coupling mechanism between distant QDs in a disordered PHC structure. The formalism presented here only relies on the knowledge of the spectrum and field maps of the PHC modes. Together with the possibility to simulate photon modes in very long PHC structures -- offered by the recently developed Bloch-mode expansion method\\cite{Savona_2011} -- it therefore represents the election method to carry out a systematic study of disorder effects on the radiative properties of QDs in PHC structures.\n\n\\begin{acknowledgments}\nThis work was supported by the Swiss National Science Foundation through Project No. $200020\\_132407$.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\subsubsection*{Preliminaries}\nLet us assume a routing change, e.g., an announcement of a new prefix by an AS, in the network at time $t_{0}=0$. Our goal is to calculate the \\textit{BGP convergence time}, i.e., the time needed till all ASes\/routers in the network have the final (i.e., shortest, conforming to policies) BGP routes for this prefix.\n\nTo this end, using Assumption~\\ref{assumption:exponential-lambda}, we can model the dissemination of the BGP updates in the network with the Markov Chain (MC) of Fig.~\\ref{fig:mc-nodes}, where each state corresponds to the number of ASes\/routers that have the final BGP routes. At time $t_{0}=0$ the system is at state $0$, while the state $N$ denotes the BGP convergence. When an AS in the SDN cluster receives the BGP update, all the nodes in the SDN cluster are informed (through the controller); thus, we have a transition, e.g., from state $i$ to state $k+i$. The transition rates in the MC, as we discuss in detail later, depend on the network topology.\n\nThe Markov Chain of Fig.~\\ref{fig:mc-nodes} is transient, and the BGP convergence time is the time needed to move from state $0$ to state $N$.\n\nFor notation brevity, in the remainder we use the MC of Fig.~\\ref{fig:mc-steps}, which is equivalent to the MC of Fig.~\\ref{fig:mc-nodes}. Here, the states represent the \\textit{number of transitions} in the MC of Fig.~\\ref{fig:mc-nodes}. For example, the state\/step $1$ corresponds to the state $1$ or $k$ of the MC of Fig.~\\ref{fig:mc-nodes}, while the state\/step $i$ corresponds to the state $i$ or $k+i-1$ in the MC of Fig.~\\ref{fig:mc-nodes}. The states $0$ are equivalent in both MCs, while the state\/step $C$ denotes the BGP convergence, and, thus, corresponds to the state $N$ in the MC of Fig.~\\ref{fig:mc-nodes}. \n\nIf we denote with $x$ the step at which -for the first time- an AS in the SDN cluster receives the BGP update, then the transitions rates $\\lambda_{i}^{'}$ in the MC of Fig.~\\ref{fig:mc-steps} are given by\n\\begin{equation}\n\\lambda_{i}^{'} = \\left\\{\n\\begin{tabular}{ll}\n$\\lambda_{i,i+1}+\\lambda_{i,i+k}$\t\t&$, i\\leq x$\\\\\n$\\lambda_{k+i-1, k+i}$\t\t\t\t\t&$, i>x$\n\\end{tabular}\n\\right.\n\\end{equation}\n\n\n\\begin{figure}\n\\subfigure[Markov Chain (number of nodes)]{\\includegraphics[width=\\linewidth]{.\/figures\/MC_nb_of_nodes1.eps}\\label{fig:mc-nodes}}\n\\subfigure[Markov Chain (number of transitions, or \\textit{steps})]{\\includegraphics[width=\\linewidth]{.\/figures\/MC_steps.eps}\\label{fig:mc-steps}}\n\\caption{Markov Chains where the states correspond to (a) the number of nodes that have updated BGP routes, and (b) the number of transitions, or \\textit{steps}, of the BGP update dissemination process.}\n\\label{fig:markov-chains}\n\\end{figure}\n\n\n\nWe now proceed to calculate the rates $\\lambda_{i}^{'}$. \nThe ASes that have received the BGP updates, will then send BGP updates to some of their neighboring ASes, according to their routing policies. We refer to a neighbor to which the update will be forwarded as a \\textit{bgp-eligible} neighbor. \n\n\\begin{definition}\\label{def:bgp-degree}\nWe define as the \\texttt{bgp-degree} at step $i$, $D(i)$, the number of the ASes that are bgp-eligible neighbors with \\underline{any} of the ASes that have received the BGP updates at step $i$. \n\\end{definition}\n\nAlthough an AS might receive the same BGP update from more than one neighbors, the final BGP route will correspond to only one of the received updates (i.e., shortest path). Hence, to calculate the transition rate $\\lambda_{i}^{'}$, we take into account only one BGP connection (corresponding to the shortest path) per bgp-eligible neighbor. Since the BGP update times are exponentially distributed with rate $\\lambda$ (see, Assumption~\\ref{assumption:exponential-lambda}), it follows that $\\lambda_{i}^{'}$ will be given by\\footnote{The transition time is the minimum of $D(i)$ i.i.d. exponentially distributed times.}\n\\begin{equation}\\label{eq:lambda-prime}\n\\lambda_{i}^{'} = \\lambda\\cdot D(i)\n\\end{equation}\n\nKnowing the rates $\\lambda_{i}^{'}$, we can calculate the transition delays in each step. Adding the delays in each step, we can derive Theorem~\\ref{thm:expected-delay-generic}, which gives the BGP convergence time, i.e., the time to move from state $0$ to state $C$. \n\n\n\n\\begin{theorem}\\label{thm:expected-delay-generic}\nThe expectation of the BGP convergence time $T$ in a hybrid SDN\/BGP inter-domain topology is given by\n\\begin{equation}\\label{eq:theorem-expected-delay-generic}\nE[T] = \\frac{1}{\\lambda}\\cdot \\sum_{x=0}^{N-k}\\sum_{i=1}^{N-k}\\frac{1}{D(i|x)}\\cdot P_{sdn}(x)\n\\end{equation}\nwhere $D(i|x)$ is the \\texttt{bgp-degree } of the network at step $i$ given that the SDN cluster receives the update at step $x$, and $P_{sdn}(x)$ is the probability that the SDN cluster receives the update at step $x$.\n\\end{theorem}\n\\begin{proof}\nTo calculate $E[T]$, we first apply the conditional expectation\n\\begin{equation}\\label{eq:ET-sum-ETx}\nE[T] = \\sum_{x=0}^{N-k}E[T|x]\\cdot P_{sdn}(x)\n\\end{equation}\nwhere $E[T|x]$ denotes the expected convergence time, given that the SDN cluster receives the update at step $x$. Under this condition, the bgp-degrees at each step are $D(i|x)$, $i\\in[1,N-k]$. Hence, taking also into account \\eq{eq:lambda-prime}, it follows that the transition delay from a step $i$ to a step $i+1$, $T_{i,i+1}$, is exponentially distributed with rate $\\lambda_{i}^{'} = \\lambda\\cdot D(i|x)$, and its expectation is given by\n\\begin{equation}\nE[T_{i,i+1}|x] = \\frac{1}{\\lambda\\cdot D(i|x)}\n\\end{equation}\n\\underline{Remark:} The state\/step $0$ does not correspond to a real state of the system; it is only used for presentation purposes. Thus, we set $T_{0,1}=0$.\n\nAs mentioned earlier, the BGP convergence delay is the time needed to move from step $0$ to step $C$, and thus it is given by the sum of the transition delays of all the intermediate steps, i.e., \n\\begin{align}\\label{eq:ETx-sum-Dix}\nE[T|x] \t&= E\\left[\\sum_{i=1}^{N-k}T_{i,i+1}|x\\right] \\nonumber\\\\\n\t\t&= \\sum_{i=1}^{N-k}E[T_{i,i+1}|x] \\nonumber\\\\\n\t\t&= \\sum_{i=1}^{N-k}\\frac{1}{\\lambda\\cdot D(i|x)}\n\\end{align}\nNow, the expression of \\eq{eq:theorem-expected-delay-generic} follows by substituting \\eq{eq:ETx-sum-Dix} to \\eq{eq:ET-sum-ETx}.\n\\end{proof}\n\n\nIn the following sections we calculate the quantities ${D(i|x)}$ and $P_{sdn}(x)$ for important network topologies.\n\n\n\\subsection{Full-Mesh Network Topology}\nWe first consider a basic topology: a full-mesh network, where every AS-pair is connected.\n\n\\begin{theorem}\\label{thm:P-sdn}\nThe probability that the SDN cluster receives the update at step $x$ is given by\n\\begin{equation}\\label{eq:P-sdn}\nP_{sdn}(x) = \\frac{k}{N-x}\\cdot \\prod_{j=0}^{x-1}\\left(1-\\frac{k}{N-j}\\right)\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe SDN cluster comprises $k$ (out of the total $N$) ASes. Since we consider that the prefix announcement is made by a (random) AS in the network, the probability that the announcing AS is in the SDN cluster (and thus $x=0$) is \n\\begin{equation}\nP_{sdn}(0) = \\frac{k}{N}\n\\end{equation}\nIf the announcing AS is not in the SDN cluster, then $x>0$, and thus\n\\begin{equation}\\label{eq:Psdn-x-larger-0}\nP_{sdn}(x>0) = 1-P_{sdn}(0) = 1-\\frac{k}{N}\n\\end{equation}\nThe probability $P_{sdn}(1)$ is given by\n\\begin{equation}\nP_{sdn}(1) = P_{sdn}(1|x>0)\\cdot P_{sdn}(x>0)\n\\end{equation}\nwhere $P_{sdn}(1|x>0)$ denotes the probability that the SDN cluster receives the BGP update at step $1$, given that it has not received it before. If $x>0$, then at step $1$ the remaining ASes  without the update are $N-1$, of which $k$ belong to the SDN cluster. Since the BGP update processes are distributed with the same rate $\\lambda$, the probability that the next AS to get the update belongs to the SDN cluster is $\\frac{k}{N-1}$. Therefore, and taking into account \\eq{eq:Psdn-x-larger-0}, it holds that \n\\begin{equation}\nP_{sdn}(1) = P_{sdn}(1|x>0)\\cdot P_{sdn}(x>0) = \\frac{k}{N-1}\\cdot \\left(1-\\frac{k}{N}\\right)\n\\end{equation}\nand, respectively,\n\\begin{align}\nP_{sdn}(x>1) &= \\left(1-P_{sdn}(1|x>0)\\right)\\cdot P_{sdn}(x>0) \\nonumber\\\\\n\t\t\t&= \\left(1-\\frac{k}{N-1}\\right)\\cdot \\left(1-\\frac{k}{N}\\right)\n\\end{align}\n\nProceeding similarly for the next steps $i=2,...,N-k$, it can be shown that\n\\begin{equation}\nP_{sdn}(i|x>i-1) = \\frac{k}{N-i}\n\\end{equation}\nand\n\\begin{align}\nP_{sdn}(x>i-1) \t&= \\left(1-P_{sdn}(i-1|x>i-2)\\right)\\cdot ... \\cdot P_{sdn}(x>0) \\nonumber\\\\\n\t\t\t\t& = \\left(1-\\frac{k}{N-(i-1)}\\right)\\cdot...\\cdot  \\left(1-\\frac{k}{N}\\right) \\nonumber\\\\\n\t\t\t\t& = \\prod_{j=0}^{i-1}\\left(1-\\frac{k}{N-j}\\right)\n\\end{align}\nand, therefore,\n\\begin{align}\nP_{sdn}(i) &= P_{sdn}(i|x>i-1)\\cdot P_{sdn}(x>i-1)\\nonumber\\\\\n\t\t\t\t&= \\frac{k}{N-i}\\cdot \\prod_{j=0}^{i-1}\\left(1-\\frac{k}{N-j}\\right)\n\\end{align}\nwhich is the expression of \\eq{eq:P-sdn}.\n\\end{proof}\n\n\nTheorem~\\ref{thm:Dix-full-mesh} gives the bgp-degrees $D(i|x)$ in a mesh network as a function of $n(i|x)$, which is defined as the number of nodes with updated BGP information at step $i$, given that the SDN cluster received the update at step $x$\n\\begin{equation}\nn(i|x) = \\left\\{\n\\begin{tabular}{ll}\n$i$\t& $, i\\leq x$ \\\\\n$i+k-1$\t& $, i>x$\n\\end{tabular}\n\\right.\n\\end{equation}\n\n\n\\begin{theorem}\\label{thm:Dix-full-mesh}\nThe \\texttt{bgp-degree } $D(i|x)$, $i\\in[1,N-k], x\\in[0,N-k]$, in a full-mesh network topology is given by\n\\begin{equation}\nD(i|x) = N-n(i|x)\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nIn a mesh network, since every AS-pair is directly connected, only the BGP messages sent by the announcing AS (i.e., shortest path) need to be considered. In step $i$, the announcing AS has $N-n(i|x)$ neighbors that have not received the BGP updates, and thus, it follows that $D(i|x) = N-n(i|x)$.\n\\end{proof}\n\n\n\n\\subsection{Random Graph Network Topologies}\n\nIn networks that are not full-meshes, ASes can be connected in many different ways and policies. Since it is not possible to study every single topology, we use two classes of random graphs to capture the effects of routing centralization in non full-mesh networks. In this first approach, we consider unconstrained routing policies\n\n\n\n\\subsubsection{Poisson (Erdos-Renyi) Graph}\nWe first consider the case of a Poisson random graph, where a link between each AS-pair exists with probability $p$. Varying the value of $p$ we can capture different levels of sparseness.\n\nUsing similar arguments as in the full-mesh case, it is easy to show that the probabilities $P_{sdn}(x)$ are given by Theorem~\\ref{thm:P-sdn}. The \\textit{expected} bgp-degrees, which can be used (as an approximation) instead of $D(i|x)$ in Theorem~\\ref{thm:expected-delay-generic}, are given by the following Theorem. \n\\begin{theorem}\\label{thm:Dix-poisson}\nThe expectation of the \\texttt{bgp-degree } $D(i|x)$, $i\\in[1,N-k], x\\in[0,N-k]$, in a Poisson graph network topology is\n\\begin{equation}\nE[D(i|x)] = \\left(N-n(i|x)\\right) \\cdot \\left(1-(1-p)^{n(i|x)}\\right)\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nIn a non full-mesh network, some ASes are not directly connected to the announcing AS. Thus, in the calculation of the $D(i|x)$ we need to consider the bgp-eligible neighbors of \\textit{all} the ASes that have received the update.\n\nLet assume that we are at step $i$, and $n(i)$ nodes have received the BGP updates; we denote the set of these nodes as $S_{i}$. A node $m\\notin S_{i}$ is a bgp-eligible neighbor with a node $j\\in S_{i}$ with probability\n\\begin{equation}\nP(m,j) = p\n\\end{equation}\nsince every pair of nodes is connected with probability $p$ (by the definition of a Poisson graph). The probability that $m$ is \\textit{not} a bgp-eligible neighbor with \\textit{any} of the nodes $j\\in S_{i}$, is given by\n\\begin{align}\n1-P(m,S_{i}) = \\prod_{j\\in S_{i}}(1-P(m,j)) = \\prod_{j\\in S_{i}}(1-p) = (1-p)^{n(i)}\n\\end{align}\nsince $|S_{i}| = n(i)$. It follows easily that the complementary event, i.e., $m$ is a bgp-eligible neighbor with \\textit{any} of the nodes $j\\in S_{i}$, happens with probability\n\\begin{align}\nP(m,S_{i}) = 1- (1-p)^{n(i)}\n\\end{align}\n\nThere are $N-n(i)$ ASes without the update, with each of them being a bgp-eligible neighbor with any of the nodes $j\\in S_{i}$ with (equal) probability $P(m,S_{i})$. Hence, the total number of bgp-eligible neighbors (or, as defined in Def.~\\ref{def:bgp-degree}, the \\textit{bgp-degree} $D(i)$) is a binomially distributed random variable, whose expectation is given by \n\\begin{equation}\nE[D(i)] = (N-n(i))\\cdot (1-(1-p)^{n(i)})\n\\end{equation}\n\\end{proof}\n\n\n\\begin{corollary}\nUsing the expectation of $D(i|x)$ in Theorem~\\ref{thm:expected-delay-generic}, underestimates the BGP convergence time, i.e.,\n\\begin{equation}\nE[T] \\geq \\frac{1}{\\lambda}\\cdot \\sum_{x=0}^{N-k}\\sum_{i=1}^{N-k}\\frac{1}{E[D(i|x)]}\\cdot P_{sdn}(x)\n\\end{equation}\n\\end{corollary}\n\\begin{proof}\nThe bgp-degree $D(i|x)$ in non full-mesh networks, is a random variable that can take different values, depending on the BGP updates dissemination process (i.e., the exact set of nodes that have received the BGP updates at step $i$, and their links to the rest of the nodes). Thus, we can write for the transition delay \n\\begin{equation}\\label{eq:transition-delay-generic}\nE[T_{i,i+1}|x] = \\sum_{y} \\frac{1}{y} \\cdot P\\{D(i|x) = y\\} = E\\left[\\frac{1}{D(i|x)}\\right]\n\\end{equation}\nCalculating the exact value of the expectation $E\\left[\\frac{1}{D(i|x)}\\right]$ is difficult, thus, we use a well known approximation in \\eq{eq:transition-delay-generic}:\n\\begin{equation}\nE[T_{i,i+1}|x] = E\\left[\\frac{1}{D(i|x)}\\right] \\approx \\frac{1}{E[D(i|x)]}\n\\end{equation}\nwhere the calculation of $E[D(i|x)]$ is much easier (see, e.g., proof of Theorem~\\ref{thm:Dix-poisson}). Then the BGP convergence delay is approximately given by\n\\begin{equation}\nE[T] \\approx \\frac{1}{\\lambda}\\cdot \\sum_{x=0}^{N-k}\\sum_{i=1}^{N-k}\\frac{1}{E[D(i|x)]}\\cdot P_{sdn}(x)\n\\end{equation}\nHowever, applying Jensen's bound for the expectation of a convex function (here, $f(x) = \\frac{1}{x}$) of a random variable (here, $D(i|x)$) on \\eq{eq:transition-delay-generic}, gives\n\\begin{equation}\nE[T_{i,i+1}|x] = E\\left[\\frac{1}{D(i|x)}\\right] \\geq \\frac{1}{E[D(i|x)]}\n\\end{equation}\nwhich proves the Corollary.\n\\end{proof}\n\n\\subsubsection{Arbitrary Degree Sequence Random Graph}\nThe structure of networks where the degrees (i.e., the number of connections) of the ASes are largely heterogeneous, e.g., power-law graphs, can be better described with a Configuration-Model Random Graph (CM-RG) rather than a Poisson graph. In the CM-RG model, a random graph is created by connecting randomly the nodes (i.e., ASes), whose degrees are given~\\cite{Newman:Networks-book}. Hence, we can use the CM-RG to model a network with \\textit{any arbitrary degree sequence} with mean value $\\mu_{d}$ and variance $\\sigma_{d}^{2}$ (and, $CV_{d} = \\frac{\\sigma_{d}}{\\mu_{d}}$).\n\nIf the participation of an AS in the SDN cluster is independent of its degree, then the probabilities $P_{sdn}(x)$ are given by Theorem~\\ref{thm:P-sdn}, and the bgp-degrees $D(i|x)$ are given by the following Result\\footnote{We use the notation ``Result'', instead of ``Theorem'', because the provided expression is an approximation.}.\n\\begin{result}\nThe expectation of the \\texttt{bgp-degree } $D(i|x)$, $i\\in[1,N-k], x\\in[0,N-k]$, in a CM-RG network topology is given by\n\\begin{equation}\\label{eq:Dix-CM}\nE[D(i|x)] = D(1|x)\\cdot \\prod_{j=1}^{i-1}A(j|x) + \\sum_{j=1}^{i-1}\\left(\\mu_{d}(j|x)-1\\right)\\cdot \\prod_{m=j+1}^{i-1}A(m|x)\n\\end{equation}\nwhere\n\\begin{align}\\label{eq:CM-D1x}\nD(1|x)&= \\left\\{\n\\begin{tabular}{ll}\n$\\mu_{d}$\t& $, x>0$ \\\\\n$(N-k)\\cdot \\mu_{d} \\cdot \\ln\\left(\\frac{N}{N-k}\\right)$\t& $, x=0$\n\\end{tabular}\n\\right.\\\\\n\\mu_{d}(j|x) \t&= \\mu_{d}\\cdot \\prod_{m=1}^{j-1}\\left(1-\\frac{CV_{d}^{2}}{N-n(m|x)-1}\\right) \\label{eq:average-out-degree}\\\\\nA(j|x) \t\t\t&= 1-\\frac{\\mu_{d}(j|x)}{N-n(j|x)-1}\n\\end{align}\n\\end{result}\n\\begin{proof}\nThe main difference with the Poisson case is that in the CM-RG case, it is more probable that the ASes with the higher degrees will receive the BGP updates faster. For instance, let us assume that the announcing AS, e.g., AS-1, does not belong to the SDN cluster. If we denote with $d_{1},d_{2}$, and $d_{3}$ the degrees of AS-1, AS-2 and AS-3 (where AS-2 and AS-3, have not received yet the BGP update), a property of a CM-RG says that AS-1 is directly connected with AS-2 and AS-3 with probabilities\n\\begin{equation}\nP(1,2) = c\\cdot d_{1}\\cdot d_{2}~~~~and~~~P(1,3) = c\\cdot d_{1}\\cdot d_{3}\n\\end{equation}\nrespectively, where $c$ a normalizing constant. In other words, the AS with with the higher degree has a higher probability to be directly connected to AS-1. Consequently, ASes with higher degrees have a higher probability to get the BGP update faster.\n\nNow, let us first derive \\eq{eq:CM-D1x}. If $x>0$, the announcing AS does not belong to the SDN cluster ($n(1|x>0)=1$), and thus the bgp-degree will be equal to the degree of the announcing AS. Since the average degree of a node is $\\mu_{d}$, it follows easily that expectation of the bgp-degree in this case, is given by\n\\begin{equation}\nE[D(1|x>0)] = \\mu_{d}\n\\end{equation}\n\nIf $x=0$, the announcing AS belongs to the SDN cluster, and, thus, all the $k$ nodes in the SDN cluster have the BGP updates. In this case, the bgp-degree is the number of all bgp-eligible neighbors of these $k$ nodes. Let us denote with $S_{1}$ the set of nodes in the SDN cluster. Since the fact that a node belongs to the SDN cluster and its degree are independent, the probability that an edge coming out from a node $m\\notin S_{1}$ is connected to a node $j\\in S_{1}$, is equal to $\\frac{k}{N}$. Hence, a node $m\\notin S_{1}$, with degree $d_{m}$, is not connected to any of the $k$ nodes in the SDN cluster with probability\n\\begin{equation}\n1-P(m,S_{1}) = \\left(1-\\frac{k}{N}\\right)^{d_{m}}\n\\end{equation}\nand, respectively\n\\begin{equation}\nP(m,S_{1}) = 1-\\left(1-\\frac{k}{N}\\right)^{d_{m}}\n\\end{equation}\nThe above equation holds $\\forall m\\notin S_{1}$; the degrees $d_{m}$ can have different values for each $m$. Since, there are $N-k$ nodes that do not belong to the SDN cluster (and, do not have the BGP updated route), the expected bgp-degree is given by\n\\begin{equation}\nE[D(1|0)] = (N-k)\\cdot E\\left[1-\\left(1-\\frac{k}{N}\\right)^{d}\\right]\n\\end{equation}\nwhere the expectation is taken over $d$, i.e., over all the degrees $d_{m}, m\\notin S_{1}$. To calculate this expectation is difficult, thus we approximate it using a Taylor series approximation, i.e, \n\\begin{align}\nE\\left[1-\\left(1-\\frac{k}{N}\\right)^{d}\\right] &= 1-E\\left[\\left(1-\\frac{k}{N}\\right)^{d}\\right]  \\nonumber\\\\\n\t\t&\\approx 1-\\left(1+E[d]\\cdot \\ln\\left(1-\\frac{k}{N}\\right)\\right) \\nonumber\\\\\n\t\t&= - E[d]\\cdot \\ln\\left(1-\\frac{k}{N}\\right)\\nonumber\\\\\n\t\t&= E[d]\\cdot \\ln\\left(\\frac{N}{N-k}\\right) \\nonumber \\\\\n\t\t&= \\mu_{d}\\cdot \\ln\\left(\\frac{N}{N-k}\\right)\n\\end{align}\nwhich completes the derivation of \\eq{eq:CM-D1x}.\n\nTo compute the bgp-degrees $D(i|x)$ of the steps $i=2,...,N-k$, we follow a methodology similar to~\\cite{pavlos-conf-model}. Let $D(i-1)$ be the bgp-degree at step $i-1$ and $\\mu_{d}(i-1)$ the average degree of the nodes that have not received the BGP update by step $i-1$  (i.e., the nodes $m, m\\notin S_{i-1}$). The average degree $\\mu_{d}(i-1)$ is not equal to $\\mu_{d}$, in general; this is due to the fact that nodes with higher degrees receive faster the BGP updates and thus the remaining nodes are nodes with lower degrees (for a more detailed argumentation see~\\cite{pavlos-conf-model}). Following similar arguments as in~\\cite{pavlos-conf-model}, it can be shown that the average degrees $\\mu_{d}(i)$ are approximately given by \\eq{eq:average-out-degree}.\n\nWe calculate the bgp-degree of the step $i$, based on the quantities $D(i-1)$ and $\\mu_{d}(i-1)$, as follows\n\\begin{equation}\\label{eq:Di_Di-1}\nD(i) = D(i-1) - 1 + \\mu_{d}(i-1)\\cdot \\left(1- \\frac{D(i-1)}{N-n(i-1)}\\right)\n\\end{equation}\nThe term $D(i-1) - 1$ is the bgp-degree of the previous step minus $1$, which denotes the $i^{th}$ node that received the BGP update (i.e., at the transition between step $i-1$ and $i$). To this quantity, we need to add the number of nodes that are bgp-eligible neighbors of the $i^{th}$ node, but are not bgp-eligible neighbors of any of the nodes $\\in S_{i-1}$. The total nodes without the updated BGP routes at step $i-1$ are $N-n(i-1)$; $D(i-1)$ of these nodes are bgp-eligible neighbors of at least one node $\\in S_{i-1}$. Hence, the probability that a bgp-eligible neighbor of the $i^{th}$ node is not a bgp-eligible neighbor of any of the nodes $\\in S_{i-1}$, is given by $\\left(1- \\frac{D(i-1)}{N-n(i-1)}\\right)$. Since, the $i^{th}$ node has (on average) $\\mu_{d}(i-1)$ bgp-eligible neighbors, it follows that the number we need to add to the quantity $(D(i-1) - 1)$ is \n\\begin{equation}\n\\mu_{d}(i-1) \\cdot \\left(1- \\frac{D(i-1)}{N-n(i-1)}\\right)\n\\end{equation} \n\nFinally, calculating recursively \\eq{eq:Di_Di-1}, after some algebraic manipulations, we derive \\eq{eq:Dix-CM}. \n\n\\end{proof}\n\n\n\\section{Introduction}\\label{sec:intro}\n\\input{Abstract}\n\n\\section{Model}\\label{sec:model}\n\\input{Model}\n\\section{Analysis: BGP Convergence Time}\\label{sec:analysis}\n\\input{Analysis}\n\\section{Validation and Discussion}\\label{sec:discussion}\n\\input{Discussion}\n\\section{Case Study: BGP Convergence at the Internet Core Network}\\label{sec:case-study}\n\\input{CaseStudy}\n\n\\bibliographystyle{ieeetr}\n\n\\subsection{Network Model}\n\\textbf{Network Model.} We consider a network (e.g., the Internet) composed of $N$ \\textit{domains} or \\textit{autonomous systems} (ASes). We represent each AS as a single node, i.e., a single BGP router (similarly to~\\cite{Kotronis-Routing-Centralization-ComNets-2015}). Such an abstraction allows to hide the details of the internal structure of ASes, and focus on inter-domain routing.\n\n\nWe assume that $k\\in[1,N]$ ASes cooperate in order to centralize their inter-domain routing: there exists a \\textit{multi-domain SDN controller}, which is connected to the BGP routers of these $k$ ASes\\footnote{This system abstraction can capture the main functionality of most of the previously proposed approaches.}. In the remainder, we refer to the set of the $k$ ASes, as the \\textit{SDN cluster}.\n\n\n\n\\textbf{BGP Updates.} As in the Internet, ASes use BGP to exchange information and establish routing paths.\nWhen a BGP edge router of an AS receives a BGP update, it (i) calculates the updates (if any) for its BGP routing table, (ii) sends updates to the other BGP edge routers within the same AS (e.g., with iBGP), and (iii) sends updates to the BGP routers of the neighboring ASes. The time needed for this process may vary a lot among different connections since it depends on a number of factors, like the employed technology (hardware\/software), routers' configuration (e.g., MRAI timers), intra-domain network, etc. \nTo this end, in order to be able to analytically study the BGP updates dissemination (given the uncertainty and complexity), we model the time between the reception and forwarding of a BGP update in a probabilistic way.\n\n\\begin{assumption}\\label{assumption:exponential-lambda}\nThe time between the reception of a BGP update in an AS\/router and its forwarding to a neighbor AS\/router, is exponentially distributed with rate $\\lambda$.\n\\end{assumption}\nDespite the simplicity of the above assumption, our results can capture the behavior of real\/emulated networks (see Section~\\ref{sec:discussion}).\n\n\n\n\n\\textbf{Inter-domain SDN routing.} Each AS belonging to the SDN cluster informs the SDN controller upon the reception of a BGP update. The SDN controller, which is aware of the topology of the SDN cluster (neighbors, policies, paths, etc.), calculates the changes in the routing paths and installs the updated routes in each router\/AS belonging to the SDN cluster. ASes react to updates from the SDN controller, as in regular BGP updates, and, thus, forward them to their (non SDN) neighbors.\n\nLet $T_{sdn}$ be the time needed for an AS to inform the SDN controller and the controller to install the updated routes in every AS in the SDN cluster. This time can be expected to be in the order of few seconds~\\cite{Kotronis-Routing-Centralization-ComNets-2015}, and much lower than the BGP updating process (cf., the default value for MRAI timers in Cisco routers is $30sec.$), thus, for simplicity, we assume here that $T_{sdn}=0$.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}