Cavity quantum electrodynamics with mesoscopic topological superconductors

Cavity quantum electrodynamics with mesoscopic topological superconductors

Abstract

We study one-dimensional -wave superconductors capacitively coupled to a microwave stripline cavity. By probing the light exiting from the cavity, one can reveal the electronic susceptibility of the -wave superconductor. We analyze two superconducting systems: the prototypical Kitaev chain, and a topological semiconducting wire. For both systems, we show that the photonic measurements, via the electronic susceptibility, allows us to determine the topological phase transition point, the emergence of the Majorana fermions, and the parity of their ground state. We show that all these effects, which are absent in effective theories that take into account the coupling of light to Majorana fermions only, are due to the interplay between the Majorana fermions and the bulk states of the superconductors.

pacs:
74.20.Mn, 42.50.Pq, 03.67.Lx

I Introduction

Condensed matter systems are an endless resource of emergent physical phenomena and associated quasiparticles. Majorana fermions, which are particles that are their own antiparticles and which have been first proposed as particles in the context of high energy physics, emerge beautifully as zero energy excitations in condensed matter setups Kitaev (2001); Alicea (2012). Specifically, they are predicted to occur as zero energy excitations in solid-state systems, such as genuine -wave superconductors Read and Green (2000); Ivanov (2001); Sato and Fujimoto (2009), or engineered from topological insulators Fu and Kane (2008), semiconductor wires in a magnetic field Oreg et al. (2010); Lutchyn et al. (2010); Mourik et al. (2012), or in chains of magnetic atoms Nadj-Perge et al. (2013); Pientka et al. (2013a); Klinovaja et al. (2013); Braunecker and Simon (2013); Vazifeh and Franz (2013); Kim et al. (2014); Nadj-Perge et al. (2014), all in the proximity of -wave superconductors. These exotic objects are robust against local perturbations and, moreover, they obey non-Abelian statistics Ivanov (2001); Nayak et al. (2008); Alicea et al. (2011) under braiding operations, thus recommending them as qubits for the implementation of topological quantum computation.

Electronic transport is the foremost experimental tool for investigating the Majorana fermions physics but alternative, non-invasive, methods that preserve the quantum states would be highly desired to address these objects. Cavity quantum electrodynamics (cavity QED) has been established as an extremely versatile tool to address equilibrium and out-of-equilibrium electronic and spin systems non-invasively Wallraff et al. (2004); Blais et al. (2004); Majer et al. (2007); DiCarlo et al. (2009); Trif et al. (2008); Delbecq et al. (2011); Frey et al. (2012); Petersson et al. (2012); Putz et al. (2014). Majorana fermions, too, have been recently under theoretical scrutiny in the context of cavity QED physics Hassler et al. (2011); Trif and Tserkovnyak (2012); Schmidt et al. (2013); Pekker et al. (2013); Cottet et al. (2013); Müller et al. (2013); Hyart et al. (2013); Ohm and Hassler (2014); Yavilberg et al. (2015); Ohm and Hassler (2015). However, most of these studies dealt with effective low energy models that involved Majorana fermions only, leaving the bulk physics, which is at the heart of the Majorana physics, largely unexplored.

Figure 1: A sketch of the system: a one dimensional system (red rectangle) is placed at the maximum of the electrical field (green straight arrows) inside a superconducting microwave cavity (blue). The electromagnetic field inside the cavity is probed by sending input fields of amplitude and phase and , respectively, and measuring the field at the end with and . The difference between the two gives a direct access to the electronic correlation function in the wire (see text). The presence of Majorana end modes in the finite wire (black curves) is also signaled in the cavity response.

The basic idea behind cavity QED with electronic system is that it allows one to extract various properties of the latter, such as its spectrum and its electronic distribution function, from photonic transport measurements, as opposed to electronic transport. Such photonic transport is quantified by the complex transmission coefficient that relates the output and input photonic fields as depicted in Fig. 1. In the weakly coupled limit, one finds Cottet et al. (2011); Schiró and Le Hur (2014), Appendix B:

(1)

where and are the frequency and the escape rate of the cavity, respectively, while is an electronic correlation function that depends on the actual coupling between the two systems, and which contains information about the spectrum of the electronic system. The phase and amplitude response of the cavity close to resonance are related to the susceptibility as follows: and , where , , and () is the real (imaginary) part of the susceptibility.

In this paper, we evaluate the function first for the simple case of a one-dimensional (1D) -wave superconductor described by the Kitaev chain and then for more realistic model of a 1D topological semiconducting wire in proximity of a superconductor. We assume in both cases that these 1D systems are coupled to a microwave cavity, as showed schematically in Fig. 1. We address various physical situations for this coupling and show that such a method allows us to ascertain the topological phase transition point, the occurrence of Majorana fermions, and the parity of the ground state, all in a global and non-invasive fashion. The paper is structured as follows. In Sec. II, we describe our model Hamiltonians for the two systems under consideration and discuss the coupling between the microwave photons and the electrons in the 1D topological systems. In Sec. III, we show how the optical transmission through the cavity is able to probe the topological phase transition. In IV, we demonstrate that the cavity allows to detect the occurrence of Majorana fermions and the parity of the Majorana fermionic state in a non-invasive fashion. Finally, in Sec. V we provide a brief summary of our results. Technical details of the calculations are given in the appendices.

Ii Model Hamiltonian

In the following we will consider various models of -wave superconductors coupled to a microwave (superconducting) cavity, such as the Kitaev -wave superconductor model, and the spin-orbit coupled nanowire subjected to a magnetic field and in the proximity of an -wave superconductor.

The general Hamiltonian for the one-dimensional systems we consider here is of the form:

(2)

being the sum of the electronic Hamiltonian, its capacitive coupling to the cavity, and the free photon field, respectively. While the electronic term is model specific, and it will be discussed below, the last two terms read:

(3)

and

(4)

In Eq. (3), is the photon creation (annihilation) operator, respectively. is the electron-photon coupling constant that couples to the charge density . This merely acts as to shift the chemical potential. In Eq. (4), is the frequency of the photonic mode (setting throughout). Such a model could be realized experimentally by coupling a spin-orbit nanowire in the presence of a Zeeman field to a nearby -wave superconductor Lutchyn et al. (2010); Oreg et al. (2010). In the present setup, which is based on a microwave superconducting stripline cavity, the -wave superconductor that induces superconducting correlations in the wire could be a part of the underlaying cavity. For example, the nanowire could be tunnel-coupled to the central superconducting material showed in Fig. 1. We have a considered a global capacitive coupling between the electronic nanosystem and the cavity electric field. Such a coupling can be justified by a full microscopic approach (see Appendix A for details and also Ref. Cottet et al. (2015) that provides a microscopic description of the electric coupling between electrons in a nanocircuit and cavity photons).

By solving the equation of motion for the photonic field iteratively up to second order in with respect to the cavity frequency  Schiró and Le Hur (2014), we find for the correlation function in Eq. (1) in the time domain

(5)

being the total charge susceptibility of the electronic system (which can be here a 1D wave superconductor or a topological 1D wire). In Eq. (5), we introduced , with being the total number of electrons operator and the evolution operator for the electronic system. We assume zero temperature limit () so that the average is taken over the superconducting ground state. Note that and that in the absence of superconductivity (), i.e. there are no effects from such a coupling for a wire in the normal state. We detail below the models for both topological 1D systems we consider in this paper.

ii.1 Kitaev chain

The simplest model of a -wave superconductor that hosts Majorana fermions is the Kitaev chain Kitaev (2001). Therefore, we first consider for the electronic part in Eq. (2), the Kitaev Hamiltonian that reads:

(6)

where is the hopping parameter, is the -wave superconducting pairing potential, is the chemical potential, and is the total number of sites. Also, is the creation (annihilation) electronic operator at the site . Note that the electronic operators are spinless, and the electronic density is given by . In the present setup, which is based on a microwave superconducting stripline cavity, the -wave superconductor that induces superconducting correlations in the wire could be a part of the underlaying cavity. For example, the nanowire could be tunnel-coupled to the central (super-)conductor showed in Fig. 1. The fact that microwave photons effectively couple only to electrons of the Kitaev chain is accounted for in Appendix A.

The Kitaev Hamiltonian in Eq. (6) can be easily diagonalized. The susceptibility in this case can be found by simply substituting the expression for the density in Eq. (5) with the one corresponding to the Kitaev model. We will discuss its physical content in Sec. III and Sec. IV.

ii.2 Spin-orbit coupled nanowire

A realistic system that can emulate, in some limits, the Kitaev chain consists of a nanowire with a spin-orbit interaction, subjected to an external magnetic field, and coupled by proximity effect to an -wave superconductor Oreg et al. (2010); Lutchyn et al. (2010); Mourik et al. (2012). The entire system is then assumed to be (capacitively) coupled to the microwave cavity. The tight-binding Hamiltonian for the nanowire with spin-orbit (SO) interaction in the presence of the magnetic field reads Rainis et al. (2013)

(7)

where, as before, and are the hopping amplitude and the chemical potential, respectively, is the spin-flip hopping amplitude (or the spin-orbit coupling), is the -wave pairing potential induced by proximity, is the Zeeman splitting energy (, with and being the -factor and external magnetic fields, respectively). Also, () are the annihilation (creation) operators for electrons at site and spin , and , with are the Pauli matrices that act in the spin space. This model accounts thus for spinfull electrons. Note that we assumed the spin-orbit field and the magnetic field to be orthogonal. The coupling to the cavity is again capacitive, and the density reads in this case . In order to find the susceptibility, we need to substitute this expression for the electronic density in Eq. (5), and we will discuss the various cases in the following sections.

Iii Topological phase transition

Next we will show that the topological phase transition can be inferred from the cavity response from the transmission via the susceptibility . This function can be calculated straightforwardly in the case of a closed ring, i.e. for periodic boundary conditions (PBCs), so that for the Kitaev chain ( for the SO nanowire).

iii.1 Kitaev chain

For PBCs, we can utilize the Fourier description for the electronic operators: , with (assuming the lattice spacing thereon), with . For more details see Appendix C. By doing so, we can readily write down the electronic Hamiltonian , with

(8)

where are Pauli matrices that act in the Nambu (particle-hole) space, i.e. on the vectors . The coupling to the cavity, on the other hand, simply reads

(9)

so that the susceptibility in the time domain can be written as:

(10)

with . Utilizing this description, after some lengthy but straightforward calculations, we obtain for the susceptibility (in the space):

(11)

where is the Bogoliubov spectrum of the 1D -wave superconductor [from diagonalizing the BdG Hamiltonian in Eq. (8)] Kitaev (2001) and is a small positive number that accounts for causality. For , the imaginary part acquires a simple analytical form, and it is given by

(12)

for and is zero otherwise. The topological phase transition takes place at , with the system being in the topological (trivial) phase for (). In Fig. 2 we plot (main plot) and (inset) as a function of the chemical potential for various values of the cavity frequency . We see that this function shows a large peak at the transition point (), which becomes narrower and more pronounced for smaller (compared to the gap ). Physically, this is due to the fact that the electronic levels close to the zero energy have larger curvatures, i.e. they are more susceptible close to the phase transition point. The real part also serves for detecting the phase transition, although not as directly as the imaginary part, as shown in Fig. 2, where the phase transitions are inferred from the kinks in this function. We have checked that the same peak structure holds for the cases when , too, the only modification being a shift in the scale for , which should be of the order of .

Figure 2: The imaginary part of the density-density correlation function [] as a function of for the Kitaev model. The topological phase transition takes place at , where this function reaches its maximum, indicating the transition point. Inset: The real part of the density-density correlation function [], which also shows features (kinks) around the topological phase transition point. The full, dashed, dot-dashed curves correspond to the , , and , respectively. We assumed , and all energies are expressed in terms of .

iii.2 Spin-orbit (SO) coupled nanowire

The case of a realistic SO coupled nanowire is more complicated that the Kitaev model showed above, and so is the evaluation of susceptibility. This is so because the SO coupled wire has four bands (because of the spin), instead of two, and a more complicated quasiparticle spectrum. Nevertheless, writing the electronic operators in the Fourier space as , we can write again the electronic Hamiltonian as , with

(13)

and the coupling to the cavity the same as in Eq. (9). However, the expression for becomes rather cumbersome for the general case and to get some analytical insights we need to resort to approximations. For that, the Hamiltonian can be put in a different form by the use of a unitary transformation (see Appendix D):

(14)

while the stays unchanged. Progress can be made if we assume the limit of large magnetic field, , in which case we can neglect the last term in the above Hamiltonian. By doing so, we recover two copies of the Kitaev chain, for . The susceptibility becomes:

(15)

where are given by the Kitaev spectrum with:

(16)
(17)

All the results from the previous section apply to this case but with the -dependent parameters showed above. The system is in the topological nontrivial (trivial) regime for (). In Fig. 3 we plot the imaginary part of the susceptibility as a function of the Zeeman splitting for two different values of the cavity frequency . We see a similar behavior as in the case of the Kitaev chain: a peak emerges in at the topological phase transition point, which becomes narrower as omega becomes smaller. However, an extra peak emerges at a larger , and it is due to the resonance condition with the gaps around the (external gaps in the SO coupled nanowire spectrum).

Figure 3: The imaginary part of the density-density correlation function [] as a function of the Zeeman splitting for a SO coupled nanowire in case of PBCs for parameters m, eV, meV, eV, meV, , and meV ( meV in the Inset). The topological transition takes place for meV, for which the susceptibility reaches its maximum. The emergence of a second peak is due to the resonance condition around the external gaps ().

Iv Majorana fermions detection

In this section, we consider a finite 1D topological system coupled to the cavity (therefore with open boundary conditions, or OBCs), so that there are two Majorana fermions emerging in the topological region, each localized at one of the two ends of the chain. Taken together, they give rise to a zero-energy fermionic state in the infinite wire limit, which can be either empty or occupied, thus labeling the parity of a 1D -wave superconductor Alicea et al. (2011). The Majorana wavefunctions decay exponentially in the wire on the scale of the superconducting correlation length , and for a finite wire it can lead to a finite energy splitting of the initially zero energy fermionic state Kitaev (2001). In the following, we will show that both the presence of the Majorana fermions and the parity of the Majorana fermionic state can be inferred from the susceptibility .

In the finite chain case we cannot obtain exact results for anymore, therefore we proceed to calculate this quantity numerically (see Appendix E). We will treat the two models, the Kitaev chain and the SO coupled wire on equal footing, showing that they give similar results.

For starters, the electronic Hamiltonian can be casted in the following form:

(18)

with

(19)

where counts internal degrees of freedom, such as spin, band index, etc. For the Kitaev chain (and thus we can disregard it), while for the SO coupled nanowire . Here, is a matrix Kitaev (2001), and we can write , with

(20)

is a unitary matrix () whose columns are the eigenvectors of  Kitaev (2001). Also, , with are the eigenenergies of the electronic Hamiltonian, including the Majoranas (if present). Thus, the electronic Hamiltonian can be re-written as

and , where

(21)

with () are the creation (annihilation) operators for the Bogoliubov quasiparticles in the finite wire, with labeling the energy levels. Finally, we can write

(22)

and also define the spinorial wavefunction for the state of energy at position as , where are the electron (hole) components of the wavefunction at position in the wire.

The electron-cavity coupling Hamiltonian can be then written in the new basis as follows:

(23)

where are coefficients that depend on the transformation from the electronic basis to the Bogoliubov basis and read Trif and Tserkovnyak (2012); Cottet et al. (2015):

(24)

Here the pseudo-spin acts in the Nambu (or particle-hole) subspace. In general, all for and , thus there are couplings between all the levels (and bands) via the cavity field, and that includes transitions between the Majorana and the bulk (or gaped) modes. This in turn affects the correlation function in Eq. (5), which can be written as , being the sum of the terms that contain only bulk states (bulk-bulk, or BB), cross terms between Majorana and the bulk (bulk-Majorana or BM), and Majorana contributions only (Majorana-Majorana or MM), respectively. However,  Cottet et al. (2013) due to the fact that the cavity cannot mix different parities, and in consequence the only contribution from the Majorana modes comes through the cross terms . We have found that for the contribution is given by the one obtained from the PBCs in the first part of the paper, i.e., , while , up to exponentially small terms in . We note in passing that in a real wire, the smallness of the compared to is measured by , with being the Fermi wavelength and the length of the wire.

Figure 4: Dependence of on the chemical potential . The blue (dot-dashed), red (full), and the black (dashed) lines correspond to the susceptibility for PBCs, OBCs for parity , and OBCs for parity , respectively. Lower Inset: the relative strength of the susceptibility as a function of in logarithmic scale, for (red-full), (green-dashed), and (blue-dot-dashed). The size of is exponentially reduced as a function of . We used , , , , and all energies are expressed in terms of .

In the following, we analyze the cross-terms contribution . For , with , we obtain:

(25)

where and are the occupations of the bulk and Majorana states, respectively. This is one of our main results. Inspecting the above expression, we see that it is strongly dependent on the Majorana state parity . Assuming that for and for in the ground state, we obtain that () for (). To get more physical insight into the resulting susceptibility, we write the coefficients in the following way:

(26)

Let us analyze the implication of the above result. When , we also have , and thus , since electron and hole contributions are the same in the Majorana state. However, for a finite energy splitting , and thus we have that , which in turn results in . All these suggest that the susceptibility , via should allow us to infer both the parity of the ground state and the zeros in the Majorana energy , assuming their spatial overlap is large enough.

Figure 5: Dependence of on the Zeeman splitting . Above the topological transition (indicated in the figure), the red (full) and black (dashed) curves correspond to for parity and , respectively. shows oscillations as a function of , that are different in amplitude for the two parities (but having the same period), crossing at points where (see Inset). Below the topological transition the susceptibility reproduces well the one for PBCs. The parameters we took are Ben-Shach et al. (2015): m, eV, eV, eV, eV, eV, and .

In the main plot in Fig. 4, we show the real part for the Kitaev chain as a function of the chemical potential for the two parities as well as the bulk value for PBCs. First of all, the values for in case of periodic and OBCs are different because of , as this contribution has a different dependence on and from the bulk states. Second of all, the open BCs wire susceptibility shows oscillations as a function of on top of the average value, of the form , with corresponding to (), i.e. they are opposite in sign for the two parities. Here is the Fermi wavevector of the electronic system, and for the range of parameters considered is  Kitaev (2001). This means that the cavity field can access the parity of the Majorana fermions non-invasively and without locally accessing the wire. Moreover, the oscillations disappear below the phase transition point , the susceptibility acquires the same value as for the PBCs wire which signals that the Majorana fermions exist only above the topological phase transition. In order to get a closer look at the oscillations of , in the lower inset in Fig. 4 we show the real part of the relative difference between the two parities, , for different values of . We see that the oscillations have the same periodicity as the Majorana energy splitting . Notice that the oscillations of the Majorana splitting with the chemical potential has been studied in detail Prada et al. (2012); Das Sarma et al. (2012) together with the fact that the magnitude of the oscillations becomes exponentially suppressed in  Pientka et al. (2013b); Zyuzin et al. (2013).

In the main figure in Fig. 5, we plot the real part of the susceptibility for a 1D topological wire as a function of the Zeeman splitting for the two parities . The susceptibility for that figure was computed using realistic parameters that might be appropriate for an InSb wire such as in the experiments in Ref. Mourik et al., 2012. We find similar features as for the Kitaev toy model, namely oscillations as a function of the Zeeman splitting above the topological transition. These oscillations around the ground state have opposite sign and different amplitudes for each parity. Like for the Kitaev model, they have the same periodicity as the Majorana energy (see the inset of Fig. 5) and cross at points where . Notice that if the parity is not conserved in the system (for example, due to the quasi-particle poisoning), will follow the ground state and exhibit therefore sharp cusps as a function at the crossing points where (see also Ref. Väyrynen et al., 2015 for similar features in a topological Josephson junction). As for the Kitaev chain, we thus find that the cavity phase shift is thus able to detect the Majorana fermions and the parity of the ground state of a realistic topological wire.

Figure 6: Dependence of on the cavity frequency for . The full-red (dashed-black) line corresponds to the susceptibility in the topological regime for , while the dot-dashed-blue (dotted-green) corresponds to the non-topological regime with open (periodic) BCs with , so that the effective gap is the same in both regions. Inset: a zoom in the region where the Majorana peak emerges. For all the plots we used , and all energies are expressed in terms of .

The imaginary part of gives us also information on the presence of Majorana fermions. In Fig. 6, we show the dependence of on for the Kitaev chain, both in the topological and non-topological regimes, for . We see that the Majorana fermions, through , give rise to an extra peak in the susceptibility at half the effective superconducting gap in the topological regime, while such a peak is absent for the same effective gap , but in the non-topological case. For completeness, we also show the result for PBCs, in which case there are no Majorana fermions.

Finally, let us give some estimates for , and in particular for and the resulting phase shift in the exiting photonic signal. We assume typical experimental values for the cavity frequency, eV, and with a quality factor , which results in photon escape rate eV. For an estimate of the capacitive coupling we refer, for example, to the case of carbon nanotubes, which have been under experimental scrutiny in the context of cavity QED Cottet et al. (2015); Delbecq et al. (2011). There, it was found that eV, and we believe similar values should be relevant for semiconductor nanowires too. The phase shift of the radiation exiting the cavity satisfies so that we obtain which is a sizeable value.

V Conclusions and outlook

We studied two paradigmatic examples of 1D topological superconducting systems capacitively coupled to a microwave superconducting stripline cavity: the Kitaev chain and a 1D nanowire with strong SO interaction in the presence of a magnetic field and in proximity of a superconductor. We analyzed the electronic charge susceptibility of these systems that is revealed in the photonic transport through the microwave cavity via its transmission . We showed that this electronic susceptibility can actually be used to detect the topological phase transition, the occurrence of Majorana fermions and the parity of the Majorana fermionic state in a non-invasive fashion. Such effects are due to the interplay between the bulk and Majorana states, either via virtual or real transitions taking place between the two, and which are mediated by the photonic field. As an outlook, it would be interesting to use the same cavity QED setup to access the physics associated with the fractional Josephson effect.

Acknowledgements.
Acknowledgments— We acknowledge discussions with the LPS mesoscopic group in Orsay, and in particular insightful suggestions by Helene Bouchiat, and we thank Jukka Vayrynen for valuable correspondence. This work is supported by a public grant from the “Laboratoire d’Excellence Physics Atom Light Matter” (LabEx PALM, reference: ANR-10-LABX-0039) and the French Agence Nationale de la Recherche through the ANR contract Dymesys.

Appendix A Derivation of the effective Kitaev Hamiltonian in the presence of the cavity field

In this section, we provide theoretical arguments for the wire Hamiltonian utilized in Eq. (6), and the effective electron-cavity Hamiltonian used in the Main Text (MT). In a continuum description, the natural way to account for the interaction between the electrons and the electromagnetic field is via the minimal coupling, i.e. in the electronic Hamiltonian, with being the electromagnetic field vector potential and being the momentum of the electrons in the material. In a tight-binding picture instead, one accounts for the coupling between light and matter by performing the Peierls substitution to the hopping parameters between neighboring sites and , namely

(27)

with being the electromagnetic field vector potential at position , and the integration is performed between the sites and . We will focus on the derivation of the effective Kitaev model in the tight-binding picture, as the microscopic, continuum model was described in great detail very recently in Cottet et al. (2015). We thus refer the reader to that paper for a detailed calculation of the cavity effects, as well as the derivation of the capacitive coupling starting from the minimal coupling.

Here we give some details on the derivation of Eq. (6) in the MT starting from a non-superconducting nanowire coupled to a bulk -wave superconductor with such a coupling being assisted by the cavity field. For simplicity, we assume the bulk to be not , but -wave paired, thus the presence of spin-orbit coupling in the wire is not a necessary ingredient. However, the present calculations can be straightforwardly generalized to more realistic system, such as nanowires with SOI. The total Hamiltonian of the system reads:

(28)

where

(29)

with , and (no intrinsic superconductivity in the wire), and the -wave pairing in the bulk superconductor. Here, () and are the electronic annihilation (creation) operator at position and the hopping parameter in system , respectively. The tunneling Hamiltonian in the presence of the cavity reads:

(30)

where , with , , , , and (), being the cavity vector potential, the coupling strength, the cavity frequency, and the cavity photon annihilation (creation) operators, respectively. Note that we assumed that the cavity field points perpendicularly to the wire, and it has no component along it. If instead such components would exists, we should have modified the wire Hamiltonian too in order to account for the cavity induced phase factors. In the following, we will assume that , namely it is constant along the entire wire. Finally, the Hamiltonian of the cavity reads:

(31)

with being the (fundamental) frequency of the cavity. Before deriving an effective wire Hamiltonian, it is instructive to switch to the Fourier space, for both the bulk and wire Hamiltonians. We get:

(32)
(33)
(34)

where , with the chemical potential in the system.

Next we perform the so called Lang-Firsov transformation on the system Hamiltonian, which means with chosen as follows:

(35)

After some lengthy, but straightforward calculation we obtain the system Hamiltonian as follows:

(36)

which implies we excluded the photonic field from the tunneling term at the expense of adding photon-dependent chemical potential shift in the wire (third term) as well as an interaction term (fourth term). Note that for , the transformation does not affect the spectrum, as it can be simply undone. However, as will see in the following, in the presence of the tunneling term the photonic field in the form of the capacitive coupling can lead to real effects.

In the following, we aim at finding an effective Hamiltonian describing the wire only by integrating the bulk superconductor degrees of freedom up to second order in the tunneling . We choose to do so by employing the Schrieffer-Wolff transformation formalism, which means, as before, that we unitary rotate the system Hamiltonian as

(37)

and choose

(38)

or , with being a superoperator whose action is defined as , . This is equivalent to the following identity:

(39)

This term excludes the tunneling Hamiltonian in leading order (assuming there is no diagonal contribution caused by such a term). Then, we neglect the contributions of the higher order terms on the wire spectrum by averaging over the bulk ground state in order to derive a purely (renormalized) wire Hamiltonian:

(40)

In order to find from Eq. (39) explicitly, let us perform Bogoliubov transformation for the bulk -wave superconductor defined as

(41)
(42)

where and

(43)

with the phase of the superconducting condensate (that we choose from now on) and the spectrum. We can then express the bulk Hamiltonian in terms of the and operators:

(44)

Utilizing the fact that:

(45)

and

(46)

we can readily find the transformation matrix as follows (assuming also that , since we are interested in the energies well inside the band gap of the bulk superconductor):

(47)

Utilizing this expression for , we can calculate the expectation values for the different commutators in Eq. (40). We obtain:

(48)

which can be interpreted as follows: the first term renormalizes the single particle spectrum in the wire, while the second term is responsible for the induced superconductivity in the wire. The full wire Hamiltonian thus becomes:

(49)

with