Hybrid light-matter networks of Majorana zero modes

Topological excitations, such as Majorana zero modes, are a promising route for encoding quantum information. Topologically protected gates of Majorana qubits, based on their braiding, will require some form of network. Here, we propose to build such a network by entangling Majorana matter with light in a microwave cavity QED setup. Our scheme exploits a light-induced interaction which is universal to all the Majorana nanoscale circuit platforms. This effect stems from a parametric drive of the light-matter coupling in a one-dimensional chain of physical Majorana modes. Our setup enables all the basic operations needed in a Majorana quantum computing platform such as fusing, braiding, the crucial T-gate, the read-out and, importantly, the stabilization or correction of the physical Majorana modes.

Majorana quasiparticles in condensed matter systems have been the subject of intense experimental work for almost a decade [1][2][3][4][5] , for their potential in defining topologically protected qubits and gates 6 . However, experimental realizations have not succeeded so far in measuring the expected non-abelian statistics of these exotic excitations. Several protocols have been proposed for performing such advanced experiments through electronic transport measurements [6][7][8][9][10][11][12][13][14] . They all require a microscopic control and fine tuning of the experimental platforms, a 2D or at least a network geometry and an invasive transport based read-out.
Cavity photons have appeared as a major toggle for manipulating, coupling and reading out the quantum state of superconducting circuits 15,16 . However, the direct application of Circuit QED techniques to Majorana fermions is hindered by their self-adjoint property which forbids a direct energy exchange between an isolated Majorana doublet and a cavity 17 . It was proposed to probe the presence and parity of a given Majorana doublet by observing charge transitions to supplementary states [17][18][19] or by using a charge sensitive Josephson circuit 20 . In principle, one can also detect the dynamical phase resulting from the braiding of Majorana fermions by probing the cavity field 21 . However, the above proposals are elusive regarding the manipulation and coupling of Majorana states through the photonic degree of freedom. This is why circuit QED could not be envisionned as a full platform for performing all the requested operations for fusing and braiding the MBSs, so far.
In this paper, we propose a hybrid Majorana-cavity platform which fills these gaps. We show that by modulating the Majorana-photon coupling at the cavity frequency, one can fuse and braid two MBSs or perform T-gates in a four to six MBSs linear chain. This resource is obtained because the modulation produces an effective 2D network out of a 1D chain. Finally, we show how we can preserve the topological protection of the Majorana modes using an active stabilization based on the joint action of the cavity photons and the modulation of the coupling.
We consider a linear chain of MBSs hosted in a nanoconductor, as represented in figure 1a. The nanoconductor may be implemented in various physical platforms. It is capacitively coupled to a microwave cavity, which we describe as a single photonic modeâ with frequency ω c /2π. Each MBS is associated with a self-adjoint creation operatorγ i as depicted in figure 1a. A small overlap between two neighboring MBSs gives rise to energy splittings jj+1 , which are exponentially suppressed with the distance between the two MBSs. The low-energy effective Hamiltonian of the system can be written as 22 :H chain = j g jj+1 (â +â † )iγ jγj+1 + j jj+1 iγ jγj+1 + ω câ †â (1) One can associate to each Majorana pair (j, k) a topological charge with a parity operatorP jk = iγ jγk . Unless otherwise specified, we assume that our chain has already well developped Majorana modes with energies jj+1 much smaller than ω c .
One of the main results of our work is how we can shape the above hamiltonian to manipulate, read-out and stabilize Majorana modes under a parametric drive. The electron-photon couplings can be locally modulated at microwave frequencies ω RF ω c through a modulation of local gate electrodes (which modulate the MBS overlap) such that: g jj+1 (t) =ḡ jj+1 +g jj+1 cos(ω RF t + φ jj+1 ). In the following, we consider different types of parametric drives to implement the different Majorana operations. In all cases, we can transform the above hamiltonian into a quasi-static one by going into the rotating frame of the cavity field and/or performing a suitable dispersive unitary transformation. Figure 1: Hybrid light-matter network of Majorana zero modes. a We consider a chain of Majorana quasiparticleŝ γj embedded inside a microwave cavity. The cavity is represented the two mirrors (shaded black) concentrating a photonic field (light yellow) around the circuit. The Majorana modes are represented as turquoise balls. The microwave drive of each section (j, j + 1) is represented as orange vertical wavy lines. The resulting effective interaction is represented as turquoise lines, turning the chain into an elementary network suitable for fusing, braiding and stabilizing Majorana modes. b Nearest neighbour interaction and its signature in the trajectory of a coherent state in the quadratures I-Q plane of the cavity field. c Second nearest neighbour interaction and its signature in the trajectory of a coherent state in the quadratures I-Q plane of the cavity field.
This gives: where f nm is a linear combination ofâ,â † ,â †â which depends on the operation considered, and δ = ω c − ω RF is the detuning between the drive and the cavity. The change from 1D to 2D is one of the main resources which we exploit in this paper. As shown in figure 1b, 1c 1d and 1e, the read-out of the parityP jj+1 , orP jj+2 of pairs (j, j + 1) or (j, j + 2) can be implemented by choosing appropriate gate voltage pulses (See Methods). The specific trajectories of the coherent cavity field carrying the information on the MBSs parity in the two elementary cases is shown in figure 1c and e. This information can be retrieved by measuring the field leaking out of the cavity with microwave techniques, as shown in the input-output theory section in the methods. Unless otherwise specified, we now assume that the chain considered has a given total parity. In addition, we will omit for clarity the j index of each Majorana until the discussion of the Majorana stabilization, replacing j + 1..j + 6 by 1..6. The fusion of two MBSs j and k is the projective measurement of their parityP jk 23 . Measuring the coherent field spots in the quadratures I-Q plane of the cavity field is projective for separated spots like those sketched in figure 1c or 1e. Hence, our proposed setup enables to fuse pairs of MBSs. Strikingly, such a scheme also gives direct access to the fusion rules which are directly linked to the non-abelian algebra of the MBSs 23 . The full sequence for establishing the fusion rules is represented in figure 22 for an odd total parity. One can measure the parityP 12 and thenP 23 (panel a) or the parityP 12 and thenP 34 (panel c). The results are expected to be qualitatively different whetherP 23 orP 34 is measured. In the first case, random spots with equal weight should appear in the I-Q plane along the axis defined by the first parity measurement whereas perfectly anti-correlated spots should appear in the second case. Specifically, the second parity measurement of the sequence of figure 2a shows directly that the fusion creates an equal weight coherent superposition of states. These constitute a direct signature of the fusion rules of MBS1 and MBS2.
The braiding of two MBSs is the coherent exchange of them. Performing such an exchange in 1D is a challenge. It has been suggested to make use of anyon teleportation by strong parity measurements 23-25 rather than moving in real space or in phase space the MBSs. These ideas have not been implemented so far. One important roadblock is that one needs to read-out the parity corresponding to distant Majorana's such as 2 and 4. The conventional wisdom is that this still requires a network geometry since it seems difficult to "jump over" the intermediate Majorana (here Majorana 3) in a 1D setup 12 . However, this can be done thanks to the microwave cavity by using two phase shifted modulation pulses (optimally by π/2) withg 23 = 0 andg 34 = 0, turning effectively our 1D system into a synthetic, light-induced, 2D system (see methods eq. (8)). The braiding of MBS 1 and 4 can be performed by using the anyon teleportation protocol enabled by our hybrid light-Majorana platform. One has to first measureP 23 and postselect the +1 eigenvalue as an initialization step, thenP 21 , thenP 24 and finallyP 23 again (also postselecting the +1 eigenvalue, see methods section) to perform the braiding 12,24 . The non-abelian nature of the braiding can be directly seen by changing the order of the parity measurementsP 21 andP 24 and obtaining different measurement outcomes for the total wave function of the chain. In the four MBSs chain, the change in the total wave function is the geometrical phase ±π/4, which cannot be sensed directly by the cavity photons. Importantly, this geometrical phase has a measurable signature if we enlarge the chain to six MBSs, as described in figure 3, to make the clockwise and anticlockwise braiding paths interfere. For that purpose, we enrich the anyon teleportation protocol by the initialization of the state through the measurement of P 46 and the postselection of the |0 46 parity state, starting from a two fermion-state, e.g.|1 12 1 34 0 56 ). In the latter case, this gives the initial state |Ψ init = 1 √ 2 (|1 12 1 34 0 56 + i|1 12 0 34 1 56 ), which we write in the natural basis formed by the eigenstates ofP 12 ,P 34 andP 56 . We restrict the discussion to the even total parity (the discussion for odd total parity is very similar). The initial state |Ψ init creates a superposition of two different parities in the subspace associated with the four MBSs 1 to 4. Since they live in different parity subspaces, they pick up opposite π/4 phases during the braiding operation. The choice of the initial state and the pulse sequence makes them interfere like in a polarizer/analyzer setup with birefringent media. The corresponding pulse sequences for B 14 and B 41 are displayed in figure 33a. After the initialization sequence, one should measureP 23 ,P 21 , thenP 24 and thenP 23 for B 41 , postselecting the +1 eigenvalue.
For B 14 , one should measureP 23 ,P 24 , thenP 21 and thenP 23 , postselecting the +1 eigenvalue. After the braiding (see methods), we obtain the state |Ψ braided 14 = e −iπ/4 2 (|0 12 0 34 0 56 + |1 12 1 34 0 56 − |1 12 0 34 1 56 − |0 12 1 34 1 56 ) for the clockwise braiding and |Ψ braided 41 = e iπ/4 2 (|0 12 0 34 0 56 + |1 12 1 34 0 56 + |1 12 0 34 1 56 + |0 12 1 34 1 56 ) for the anti-clockwise braiding. The non-abelian character of the operation becomes therefore directly visible in the different outcomes of the coherent field spots in the I-Q plane for the parityP 45 measurement which is carried out at the last step in our protocol (see figure 3a). The clockwise braiding corresponds to the blue spot (P 45 = −1) whereas the anti-clockwise Figure 3: Braiding protocol in a chain of 6 MBSs a Pulse sequence enabling the "clockwise" or "counterclockwise" braiding depending on the order of pulse III or IV. The first pulse is an initialization and the last pulse is the readout. b Result of the clockwise and counterclockwise braiding as observed in the I-Q plane trajectory of the coherent state spot (blue or red spots) after pulse VI. The qualitative difference of the cavity field in the two possible braids would be a direct observation of the non-abelian braiding.
The above methods for fusion or braiding can be extended to more complex gates. In particular, the T-gate (also called π/8 gate) can be implemented in a 6 MBSs chain similar to that of figure 3. It relies on a parity measurement involving simultaneously both the I and Q quadratures of the cavity field (i.e. along an arbitrary angle φ in the I-Q plane), each of them being coupled to iγ 2γ3 and iγ 3γ4 (more details can be found in the methods section). Such aP φ measurement should be inserted in the place of the measurement ofP 24 in the sequence proposed for braiding. While such a gate is not topologically protected, it could be made exponentially accurate using mitigation techniques 26 .
The previous discussion relies on the fact that we electrically manipulate, couple and read-out coupled MBSs, which seems incompatible with topological protection because of electrical noise or disorder in the jj+1 's. We now show another crucial consequence of the form (2) which implies that even for a chain of MBSs with finite overlap between the MBSs, one can induce with the cavity light a robust topological phase with stabilized, or self-corrected MBSs i.e with exponential protection. The principle of this exponential protection is to induce thanks to the cavity field and the gate modulation a synthetic, light-induced, Kitaev hamiltonian as sketched in figure 4 4a. Like for error correction protocols 27 , this scheme requires some degree of redundancy and therefore longer chains than the ones considered so far. Let us first assume that we work with a chain with N MBS sections {0..N }. We assume that a gate modulation is applied every other section, starting from section (1,2). In such a condition, the hamiltonian (2) becomes: where α is the static classical part of the cavity field in the rotating frame and jj+1 is the residual overlap between physical MBSs. Assuming that the phases φ jj+1 and the modulationsg jj+1 are tuned to φ andg and that the phase of the coherent field α is θ,H stab can be divided into a Kitaev hamiltonian H K and a doping hamiltoninan H D and has a topological phase transition with exponentially localized MBSs at sites 0 and N (see figure 4a), for J =g|α| cos (φ − θ) −2 jj+1 . These end MBSs are now stabilized because their overlap can be made exponentially small using macroscopic 'knobs'. The parametersg, |α| and φ − θ are these 'knobs' and set the topological gap of our synthetic Kitaev hamiltonian as shown in figure 4c. This principle can be used on bigger chains to produce 4-to 6logical Majorana chains as needed by the previously introduced protocol.
In writing the above hamiltonian, we have neglected two terms: one time dependent classical field term δH (1) = g|α| cos (2ω RF t + φ + θ) jodd iγ jγj+1 and one term arising from quantum fluctuations of the cavity field δH (2) = g cos (ω RF t + φ − θ) jodd iγ jγj+1 (b +b † ). The quantum fluctuations of the cavity field are defined by the operator b. Since both perturbations are periodic in time, it is convenient to use the Floquet formalism (see Methods and Supplementary). Noting that all the parities p jj+1 for the (j, j + 1) sections with j odd are good quantum numbers in the Kitaev chain, the matrix elements arising in the perturbation theory depend now on p = p jj+1 which is an integer directly linked to the occupation of the chain and m which is an integer arising from the Floquet ladder (see Methods and Supplementary). The first term is a fast oscillating term at roughly twice the cavity frequency. It (see methods). They can be safely negelected because they are of for smallg|α|/ω RF which is a very realistic condition. It is also essential to evaluate the effect of quantum noise on the topological protection of our scheme. Defining the polaronic shift E 0 =g 2 ω c /2(ω 2 RF −ω 2 c ), we can write the quasi-energy of the driven chain as : The result of perturbation theory on the Floquet space is twofold. First, any local perturbation η flipping one of the p jj+1 can only induce an exponentially small coupling between the end stabilized Majorana's at sites 0 and N of order η (N +1)/2 , thus preserving the topological protection. Second, the drive tends to shift the cavity field entangled with the state of the chain of quasienergy E K at different spots in the I-Q plane for different states of the chain with total quantum number p or p because δH (2) is a drive term proportional to p. The quasi-orthogonality of two coherent states with different amplitudes quenches exponentially the transition to excited states. The corresponding matrix element reads approximately: (16) in the supplementary). This exponential polaronic protection which further protects the topological phase is presented in figure 44b.
In summary, we have presented circuit QED protocols based on the parametric modulation of light-matter coupling for performing advanced quantum gates for Majorana zero modes. Such an approach can also be used for a parametric stablization of the Majorana zero modes, enhancing the topological protection of a given physical platform. This should allow one to perform advanced operations with exponentially protected Majorana zero modes.

SUPPLEMENTARY MATERIAL Nearest neighbours light bonds and strong parity measurement
We now assume that the electron-photon couplings can be locally modulated at microwave frequencies through a modulation of the Majorana quasiparticles overlap: jj+1 (t) =¯ jj+1 +˜ jj+1 cos(ω RF t + φ jj+1 ) which leads to g jj+1 (t) =ḡ jj+1 +g jj+1 cos(ω RF t + φ jj+1 ), with j an integer. This can be done thanks to the use of RF gates, each being capacitively coupled to one (j, j + 1) section of the circuit. We specialize the discussion to section (1,2). A cavity field grows in the cavity when, for example, g 12 is modulated at the cavity frequency (ω RF = ω c ). It reveals directly the parityP 12 . Omitting all the other sections for the sake of simplicity, the low-energy effective Hamiltonian of the system can be reduced to: We can rewrite the Hamiltonian in a rotating frame at ω c . We obtain: where the static term proportional toḡ 12 is neglected as a fast oscillating term, under the Rotating Wave Approximation (RWA). Equation (6) shows an effective coupling between the Majorana's 1 and 2 which can be used to measure their parityP 12 = iγ 1γ2 via the cavity field as shown in figure 2 1. This measurement follows the same principle as the longitudinal coupling read-out for qubits 28 . The parity eigenstate is read-out from the position of the coherent state spots in the I-Q plane associated to theP 12 = ±1 eigenvalues. The contrast for the coherent state spots in the I-Q plane isg 12 / √ κ, where κ is the damping rate of the cavity (see Input-Output section of the methods), which can be made much larger than the width of the gaussian spots of the coherent states even deep in the topological regime whereg 12 → 0 for small enough κ. Similarly, one can also measureP 23 = iγ 2γ3 . This is simply done by lettingg 23 non zero keeping the other modulating terms negligible. This gives a concrete protocol to fuseγ 1 andγ 2 of figure 12, and to detect the fusion rules through the use of cavity photons.

Second nearest neighbours light bonds
We now show specifically on a 4 Majorana chain (1, 2, 3, 4) how we can obtain a second nearest neighbours photon mediated interaction between MBSs 2 and 4. Starting again from hamiltonian (1), we now assume that the RF signal acting on the gates (2, 3) and (3,4) is detuned from the cavity and performs the combined unitary transformation: The first unitary transformation is the RWA in the frame of the gate drives and the second is the dispersive transformation which implies that ω RF − ω c >g 23 ,g 34 . The outcome of these two transformations is: Performing the T(or π/8)-gate Other useful forms of the hamiltonian (2) can be derived. A particularly important one enables the implementation of a T-gate which corresponds to a π/8 geometrical phase during the unitary evolution of the system. We specialize to the 4 MBS chain again for the sake of simplicity and assume that¯ 34 = 0 andg 23 = 0. For the unitary transformation U = e i(ωcâ †â +¯ 34 iγ3γ4t) (interacting picture), the effective hamiltonian becomes: For ω RF = ω c + 2¯ 34 , retaining only the resonant terms, we get: Such a form shows that the two different directions corresponding to iγ 2 γ 3 or "σ x " and to iγ 2 γ 4 or"σ y " become coupled with the two quadratures of the cavity field (respectively I and Q). This allows us to perform a T gate simply by measuring the cavity field along the bisector between I and Q. If such a measurement is inserted instead of the measurement ofP 24 in the braiding sequence, the unitary evolution of the wave function will pick up a π/8 geometrical phase instead of the π/4 of the braiding.

Floquet formalism for the stabilized Majorana modes
In deriving the effective HamiltonianH stab , the cavity field has been replaced by its resonant component in the rotating frame. As we have shown, this procedure generates a static Kitaev Hamiltonian H K . The goal of this section is to demonstrate that, crucially, the remarkable topological protection of Majorana edge mode degeneracy which is garanteed by H K also extends to the periodically driven situation considered in the present work without relying on the rotating frame approximation.
We first write the cavity field as a sum:â = |α|e −i(ω RF t+θ) +b (11) Assuming that the coupling between the Majorana chain and the cavity is modulated only on the (j, j + 1) bonds with j odd, we get the time periodic coupling Hamiltonian: Here we have set J =J cos ϕ, withJ =g|α| and ϕ = φ − θ. This has the form: Besides the static Kitaev Hamiltonian already derived earlier using the rotating wave approximation, we get two time-periodic perturbations δH (1) (t) and δH (2) (t). The former induces a time-periodic modulation of the Kitaev coupling, J being replaced by J ef f (t) = J +J cos (2ω RF t + φ + θ). The later couples the Majorana modes to quantum fluctuations of the cavity field. A key feature of this model is that both δH (1) (t) and δH (2) (t) commute with H K , and even more importantly, with its local conserved operatorsp jj+1 = iγ jγj+1 for odd j. Since the existence of conserved local operators lies at the heart of topological protection, the persistence of this property in the full H c (t) is of course essential for our purpose here. The first key ingredient to achieve topological protection is a large energy gap, compared to the strength of the static perturbation jj+1 . Here lies a potential fragility of the present proposal, because inelastic interactions due to the periodic driving may strongly reduce the value of the effective gap below its static value 2J. This concern is particularly clear for the δH (1) (t) perturbation because J ef f (t) vanishes twice in each period π/ω RF (or just once if ϕ is an integer multiple of π).
To address this issue, we have to extend the analysis of topological protection to situations where the reference Hamiltonian is time-periodic. We should first understand the Floquet spectrum of H c (t) and then investigate the effect of the static perturbation H D = j jj+1 iγ jγj+1 . To make the discussion clearer, we shall discuss separately the Floquet spectra when either δH (1) (t) or δH (2) (t) is added to H K .
Let us denote by |τ, {p jj+1 } a state of the Majorana chain such that: Here, each eigenvalue p jj+1 and τ can be ±1. The Floquet eigenstates of H K + δH (1) (t) have the form: so their Floquet quasi-energy is pJ, which is defined modulo ω RF .
To study the effect of H D , we view it as a perturbation of the operator Topological protection means that the effective coupling between Majorana end modes generated by the static perturbation H D is exponentially small in N . The existence of the local conserved operatorsp jj+1 (for odd j) implies that such an effective coupling, proportional to iγ 0γN , occurs only at order (N + 1)/2 in perturbation theory. Indeed, the lowest order product of Majorana operators which contains bothγ 0 andγ N , and which commutes with allp jj+1 operators (for odd j) is l iγ 2lγ2l+1 , where l runs from 0 to (N − 1)/2. Each term of the product corresponds to a local perturbation i 2l,2l+1γ2lγ2l+1 . Let us assume that it connects state |τ, {p jj+1 }; m to state |τ , {p jj+1 }; m . In this case, one has p jj+1 = ±p jj+1 , the minus sign occurring only if j = 2l − 1 or j = 2l + 1. To each intermediate state is associated an energy denominator (p GS − p)J + m ω RF , where p GS = −(N − 1)/2 since p j,j+1 = −1 for any odd j in any of the two-fold degenerate ground-states of H K . Compared to the static case, we see that the large gap proportional to J is replaced by the smaller value min {m} (J − m ω RF ). Therefore, a necessary condition for topological protection to survive in the presence of a periodic modulation of J ef f is that inelastic transitions to states with a non-zero value of m should be strongly suppressed. It is thus crucial to examine in more detail the matrix elements of the perturbation.
Using the time dependence of unperturbed Floquet eigenstates given by Eq. (16), we get: and this matrix element vanishes if m − m is odd. In Eq. (17), J m−m 2 is the usual Bessel function of the first kind.
Since the above matrix element is proportional to (J/ω RF ) |m−m | 2 at smallJ/ω RF , we see that inelastic transitions to states with a non-zero value of m are suppressed whenJ << ω RF , i.e. when the driving frequency is large compared to the time averaged gap of the effective Kitaev chain.
Although this argument is quite compelling, a potential danger lies in the fact that the ordering between the (N + 1)/2 local perturbations i 2l,2l+1γ2lγ2l+1 is arbitrary, so we have ((N 2 − 1)/8)! terms at order (N + 1)/2. In the static case, this factorial growth is compensated by the large value of typical energy denominators. In the periodically modulated case, no exact solution in the presence of the static perturbation H D is available, and to establish rigorously that the effective coupling between boundary Majorana modes decays exponentially with N would require a more involved analysis, which is beyond the scope of the present work.
Let us now turn to the Floquet eigenstates of H K + δH (2) (t). Since this Hamiltonian commutes with the conserved operators of H K , we can put the Majorana chain in one of the states |τ, {p jj+1 } for all times t. The quantum oscillator mode of the cavity if then subjected to the Hamiltonian: The Floquet spectrum of H cav is discussed in the Supplementary Material section. Let us first consider the nonresonant case, when the detuning δ = ω c − ω RF is larger than the cavity damping rate Γ. Combining the Majorana chain and the cavity, the eigenstates of the operator L 2 = H K + δH (2) (t) − i d dt , acting in the Hilbert space H per can be written as |τ, {p jj+1 }; n; m , where n is a non-negative integer associated to the cavity oscillator and, as before, m labels Fourier modes in the auxiliary space of periodic functions of time. The corresponding eigenvalues are pJ + p 2 E 0 + n ω c − m ω RF .
The effect of the static perturbation H D is similar to the previous case. The local perturbation term i 2l,2l+1γ2lγ2l+1 acts only on the Majorana chain, where it connects state |τ, {p jj+1 }; m to state |τ , {p jj+1 }; m . The new feature with δH (2) (t), compared to δH (1) (t), is that the transition between these two states of the chain also modifies the amplitude of the periodic driving seen by the cavity mode. The analysis of these matrix elements is presented in the Supplementary Material section in the case where the detuning δ is small. One of the main features is the approximate selection rule n − m = n − m. This implies that the energy denominators (p GS − p)J + (p 2 GS − p 2 )E 0 − n ω c + m ω RF in the leading contributions to the effective coupling between boundary Majorana modes are close to the values (p GS − p)J governing the static case. This gives strong support to our claim that topological protection is achieved in this model of a driven Majorana chain. Another bonus provided by the driven model comes from the fact that the matrix elements of H D are proportional to the overlap between coherent states: The gaussian factor in Eq. (19) may be significantly smaller than 1, which would enhance the protection of the ground-state degeneracy with respect to residual static perturbations such as H D . This is analogous to the reduction of a polaron hopping amplitude, due to its strong coupling to lattice vibration modes. In this analogy, the polaron becomes the Majorana chain and the vibration modes are replaced by the cavity oscillator.
In the case of a finite cavity damping κ, it is necessary to take into account the coupling of the cavity oscillator to a continuum of environmental modes. In the limit of a small damping ω c κ, it is shown in the Supplementary Material section that the driving term in H cav (t) couples mostly to the dressed modes near the cavity frequency ω c . Therefore, most of the previous analysis of topological protection in the limit of small detuning survives in the case of a small but finite damping.

Input-Output theory
We show here how one can capture the cavity based measurement processes for the parity of the Majorana chain using an input-output theory. This method gives results which agree with equation (2) of the main text but it also allows to capture the dissipative dynamics related to the projective measurement of the system. The equations of motion for the photonic field in the cavityâ with loss rate κ and for the input and output fields,â in andâ out , read, for a pair of MBSs (1, 2): withP 12 (t) constant since [H,P 12 ] = 0.
In the semi-classical regime (α =<â >) and in absence of a cavity drive, the output field is given by: The modulation of g 12 populates the cavity field, as represented in figure 1b. By measuring the occupation of the cavity along the appropriate quadrature (more specifically, with a measurement phase of φ meas = φ 12 + π/2 [π]), one can therefore perform a measurement of the parity. The SNR in this measurement depends ong 12 √ κ and on the measurement time τ . It is given by 28 : We now show how the parityP 24 can be measured as a dispersive shift of the cavity resonant frequency as explained in the main text, using: .
The coupled equations of motion are: In the rotating frame (at ω c ), and keeping resonant terms in the RWA, we get a first reduced equation on the cavity field: We additionally suppose |ω RF − ω c | << ω c so that we can neglect the time evolution ofγ 2γ4 , as well as the one of a in the above integral. This gives: One sees here again that the optimum is φ 34 − φ 23 = π/2. Experimental requirements and distinction with accidental Andreev bound states Let us now estimate the feasibility of this scheme, and more specifically whether it could indeed allow for single-shot readout of the parity. With coplanar waveguide (CPW) resonators, cavity loss can be extremely low. However, in the scenario considered we also need to measure the cavity output in a time much smaller compared to the parity lifetime. Since the measurement time is of the order of a few κ, we assume a loss of κ = 2π × 1M Hz to be conservative (we thus require more than tens of microseconds for the parity lifetime). We then need to estimate how strongly the coupling strength can be modulated. In electrical circuits, coupling strength between a charge and a cavity of the order of g = 2π × 100M Hz are now achievable 29 , and would still be compatible with the condition jj+1 , g jj+1 ω c . Assuming that this coupling strength can by modulated by a factor 10%, either by modulating the position of the Majorana pair or by modifying the shape of the electromagnetic mode, our scheme enables single-shot cavity readout of the parity. It is important to stress that our setup can also distinguish between Majorana modes and accidental Andreev bound states. Whereas braiding is a priori the most unambiguous way of distinguishing between accidental Andreev bound states and Majorana modes, establishing the fusion rules should be enough for a large class of situations. An Andreev bound state is expected to give rise to a transverse coupling which yields a trajectory of the type of figure 1e. Measuring several sections of the chain which display only trajectories of the type of figure 1c in the I-Q plane (i.e. longitudinal coupling) in the fusion rule setup (with 4 nodes) should constraint very much the models with accidental Andreev bound states (if any exists) yielding the same signature.
State sequence in the braiding protocol We recall first the measurement based braiding protocol and specifically apply it to our scheme. Let us consider again a linear chain of four Majorana quasiparticles,γ j=1..4 . The set of measurements needed for performing a braiding operationB 14 between two Majoranaγ 1 andγ 4 stems from the identity 12,24 : The operatorΠ jk projects the electronic state onto the subspace with parityP jk = 1.
For the state sequence presented in this work, we start by measuringP 46 giving an intial state |Ψ init = We therefore arrive at the result of the main text : |Ψ braided 14 is an eigenvector ofP 45 for the clockwise braiding with eigenvalue −1, yielding the blue spot in the I-Q plane and |Ψ braided 41 is an eigenvector ofP 45 for the anti-clockwise braiding with eigenvalue +1, yielding the red spot in the I-Q plane. The reasoning for the even total parity is exactly the same.

Perturbation theory for Floquet Hamiltonians
Let us consider a time-periodic Hamiltonian H(t) = H 0 (t) + H 1 (t). Its time period is T = 2π/ω RF , where ω RF is the driving frequency. We start with a Floquet eigenstate |Ψ(t) = e −iE0t/ |χ 0 (t) for the unperturbed Hamiltonian H 0 (t). The Floquet energy E 0 is defined modulo ω 0 , and the state |χ(t) is periodic in t with period T . Perturbation theory in is more conveniently implemented within the infinite dimensional Hilbert space H per of time-periodic wave-functions|χ(t) . H per can be described as the tensor product of the physical Hilbert space H phys and the space of T -periodic functions of t. We first notice that |χ 0 (t) is an eigenvector of the operator L 0 = H 0 (t) − i d dt acting in H per , with the eigenvalue E 0 . Determining Floquet eigenstates for the full Hamiltonian H(t) is equivalent to finding eigenvectors of L = L 0 + H 1 (t) in H per . Since L is a hermitian operator for the hermitian scalar product in H per defined by χ 1 |χ 2 = 1 T T 0 χ 1 (t)|χ 2 (t) , we can use the standard procedure of time-independent perturbation theory for L. Doing this, we have to keep in mind that each Floquet eigenstate |χ(t) of L with eigenvalue E generates an infinite ladder of Floquet eigenstates |χ; m of L with eigenvalues E − m ω RF . The notation emphasizes that |χ; m should be regarded as a vector in H per , and the corresponding time-periodic wave-function is |χ m (t) = e −imω RF t |χ(t) .
Floquet spectrum for a periodically driven oscillator We consider a single oscillator mode with a time-periodic and linear driving, described by the Hamiltonian (18). To simplify notations, we shall assume that φ = 0. The dynamics of this driven oscillator is easily solved by considering the dressed annihilation operatorb(t), solution of the evolution equation i d dtb (t) = [H(t),b(t)], and subjected to the quasi-periodicity conditionb(t + T ) = e iωcTb (t). Note that this time-dependent operator should not be confused with the Heisenberg picture of theb operator. A simple calculation shows thatb(t) = e iωct (b + z(t)), with: We introduce the time-periodic stationary state |S(t) , which is a solution of the Schrödinger equation, such that it is annihilated byb(t) at all time t. |S(t) is thus a coherent state, and more precisely: with E 0 =g 2 ωc 2 (ω 2 RF −ω 2 c ) . The complete Floquet spectrum of this driven oscillator is given by the states (b † (t)) n |S(t) , n non-negative integer, whose Floquet quasi-energies are E n = p 2 E 0 + n ω c .
Since the ratio z (t)/z(t) = p /p is a real number, the overlap z (t)|z(t) is also real. Let us concentrate on the time evolution of phase factors entering the integral (35). For arbitrary values of the detuning δ, it is difficult to make precise statements. So let us assume now that δ is still finite, but small. Therefore, the first term in the right-hand side of Eq. (33) dominates, and the argument of z(t) is close to −ω RF t. If the driving amplitudeg/ is sufficiently small, so that the quadratic correction β(t) (whose time-averaged value is zero) can also be neglected, we see that the argument in the R.H.S. of (35) is well approximated by (m − m − n + n)ω RF t. So we see that we have an approximate selection rule for the matrix elements of the static perturbation between eigenstates of the L 2 operator: Floquet spectrum in the presence of cavity damping At resonance (ω c = ω RF ), the stationary state |S(t) is ill-defined, because z(t) defined in Eq. (33) becomes infinite. To recover a stationary state, we need to take into account cavity damping. A simple way to model such a situation is to consider the following Hamiltonian: where the creation and annihilation operatorsĉ † (ω),ĉ(ω) refer to the continuum of environmental modes, and x(ω) is an arbitrary coupling function. This Hamiltonian can be diagonalized using dressed operatorsγ † (ω),γ(ω), which satisfy [H,γ † (ω)] = ωγ † (ω). The cavity oscillator operatorb can be written as a linear combination of the dressed operatorsγ(ω):b where Σ R (ω) is the retarded self-energy defined by: When the cavity lifetime τ = κ −1 is long (ω c τ 1), the spectral decomposition (39) is peaked around the cavity frequency ω c . In the vicinity of ω c , we can Taylor expand the self energy, which gives: Denoting by u(ω) the coefficient ofγ(ω) in the spectral decomposition (39), we see that u(ω) Zx(ω c )/(ω−ω c +iκ/2).
In the presence of the periodic driving described by the Hamiltonian (18), each dressed mode of the continuum reaches its own stationary state |S ω , constructed by replacingb † byγ † (ω) and multiplyingg/ by the factor u(ω) in Eq. (34). Since u(ω) is peaked around the cavity frequency ω c when ω c τ 1, we see that the driving term couples mostly to the dressed modes near the cavity frequency ω c .