Control of Yu-Shiba-Rusinov States through a Bosonic Mode

We investigate the impact of a bosonic degree of freedom on Yu-Shiba-Rusinov (YSR) states emerging from a magnetic impurity in a conventional superconductor. Starting from the Anderson impurity model, we predict that an additional p-wave conduction band channel opens up if a bosonic mode is coupled to the tunnelling between impurity and host, which implies an additional pair of odd-parity YSR states. The bosonic mode can be a vibrational mode or the electromagnetic field in a cavity. The exchange couplings in the two channels depend sensitively on the state of the bosonic mode (ground state, few quanta or classically driven Floquet state), which opens possibilities for phononics or photonics control of such systems, with a rich variety of ground and excited states.

Here,Ĥ d describes the impurity orbital (with fermion operatorsĉ † dσ ,n dσ =ĉ † dσĉ dσ and spin index σ) at the energy level d well below the Fermi energy µ = 0, and with Coulomb repulsion U . For simplicity, we consider the particle-hole symmetric case, with d = −U/2.
(An asymmetric model would lead to an additional potential scattering term in the low energy Hamiltonian, which plays a subordinate role here, since it implies an asymmetry in the bound state wavefunction but does not create intra-gap states by itself.) The superconductor is included in Eq. (1) as a BCS mean-field HamiltonianĤ host , written in terms of Bogoliubov quasiparticle operatorsα † kσ,γ = u kâ † kσ,γ − σv kâ−k−σ,γ with excitation energy E k = ( k − µ) 2 + |∆| 2 , where k is the single-particle energy and u k and v k are the usual coherence factors. The termĤ hyb describes the tunnelling between the host and the impurity. We consider the coupling to an inversion-odd mode with displacementQ = (b † +b)/ √ 2, such as the electric field or vector potential of the cavity, or an infrared active phonon;Ĥ ω 0 refers to the free Hamiltonian of the bosonic mode ( = 1). The tunnelling matrix element V k (gQ) between the impurity orbital and the host states k depends onQ, where g is an overall dimensionless coupling strength. Under inversion symmetry [V k (r) = V −k (−r)], it can be decomposed into the even and odd channel, leading to the hybridisation term in Eq. (4). For g = 0, the impurity therefore hybridises only with the even conduction band channel, while the bosonic mode activates the coupling to the odd parity channel.
Since the interaction is antiferromagnetic J < 0 (due to d < 0), an antiparallel alignment of the electron spin is favoured. For sufficiently strong exchange scattering, the bound state energies cross zero at the critical coupling constant J C = − [0.5πρ ( F ) S] −1 which marks a quantum phase transition (QPT). Beyond this point (J/J C > 1, E < 0), the ground state (BCS state with free impurity spin S d = Sẑ) and the excited state (YSR state) interchange roles. The impurity then localises a spin-down quasiparticle in the ground state, which partially screens the impurity spin S d = (S − 1/2)ẑ, leaving one unpaired electron in the superconductor [5,50]. A ferromagnetic interaction J > 0, which will be encountered later, favours a parallel spin alignment, so that for J > |J C | the localised quasiparticle increases the impurity spin, S d = (S + 1/2)ẑ.
When the bosonic mode activates the coupling to the odd parity channel (−), we obtain two pairs of YSR states, one in each channel. For a high frequency ω 0 J, we can project the low energy HamiltonianĤ eff onto a fixed boson number n ≥ 0,Ĥ eff =Ĥ n,n eff |n n|. Virtual tunnelling can now go together with the virtual absorption or emission of bosons.
Since V k,+(−) (gQ) contains only even (odd) powers ofQ, the coupling to the even (odd) parity combination of the bath involves an even (odd) number of virtual bosons. The Hamiltonian can be split into the Hamiltonian for the even (+) and odd (−) channel,Ĥ n,n eff =Ĥ n,n + +Ĥ n,n Here,Ŝ kk γ = 1 2 σσ â † kσ,γ σ σσ â k σ ,γ is the spin of conduction electrons with Pauli vector σ = (σ x , σ y , σ z ), and the exchange interaction with the coupling constant J kk n,γ is given in terms of the matrix elements V n,n+l k,γ = n| V k,γ (gQ) |n + l ; l ≥ −n is the number of absorbed Results -In order to obtain quantitative results, we consider a Peierls-type coupling, for which the matrix element in Eq. (6) takes the form with the matrix element n| e iγg(b+b † ) |n + l = i |l| γ l j n,n+l . The latter decays rapidly with |l|, such that the coupling between different photon bands decays quickly to zero [36]. The contribution to the exchange scattering is antiferromagnetic (ferromagnetic) if the intermediate state lies higher (lower) in energy than the ground state, corresponding to a negative  Figure 1 shows the two-photon exchange constants J 2,γ as a function of |ω 0 / d | and g; red (blue) corresponds to antiferromagnetic (ferromagnetic) couplings.
For g = 0, the impurity couples only to the even channel, J 2,− = 0, while J 2,+ reduces to the bare exchange constant J < 0. For g > 0, the coupling to the odd channel opens up.
In the even (odd) channel, a ferromagnetic contribution is obtained right above the two- In the context of YSR states, the strength of the interactions controls their energies, whereas the sign of the exchange coupling constant determines the spin-polarisation of the bound quasiparticles. Figure 2 shows the energies E 2,γ in the even and odd channel (Eq. (7) with the couplings J 2,γ ), for a bare exchange constant below the critical coupling J/J C < 1 (taking J C < 0) (see (a),(b)) and above the critical coupling (J/J C > 1, (c),(d)). With the coupling to the mode (g > 0), in both cases we can tune the energy of the YSR states in both channels to any value within the gap ∆, and also into the QPT (E 2,γ = 0) where ground and excited state interchange roles. For J/J C > 1, the reduction of exchange scattering in the even channel further away from the resonance pushes its energy towards the QPT at around g = 0.3 and finally to +∆ (Fig. 2 (c)).
These observations can be easily transferred to distinguish between different ground states, which we label by the impurity spin S d = S dẑ and the number of bound quasiparticles within each channel q = 0, 1 + , 1 − , 2 (see Fig. 3 on AFM/STM tips [51], which provide a tunable cavity setting. Such picocavities have already been used to strongly couple atomically confined light to vibrational modes of single molecules (see, e.g., [52,53]), and to exciton-based single quantum emitters [54], with a volume compression V mode /λ 3 below 10 −6 . Moving towards lower photon frequencies allows larger mode volumes but results in a reduced controllability within the present setting, since the largest effect on the exchange couplings is obtained for near resonant couplings.
An alternative route to control the couplings is to consider a classically driven system, where the strength of the light-matter interaction can be controlled by the amplitude of the drive. In this classical limit, the state of the light field is unaffected by the solid and it can be written asĤ In the high-frequency limit, the exchange of bosons with the solid occurs only as intermediate states and we can project the Hamiltonian onto the sector of a fixed boson number n ≥ 0. The Hamiltonian readŝ H n,n eff |n n| := n|Ĥ eff |n |n n| =P 1,nĤ0P1,n + 1 2P 1,n Ŝ ,Ĥ hyb P 1,n .
Neglecting constant energy terms and considering inversion-symmetry around the impurity, we obtain two decoupled channels, withŜ kk γ = σσ 1 2â † kσ,γ σ σσ â k σ ,γ and the even and odd parity combinationsâ kσ,γ given in the main text. We consider an impurity level d well below the Fermi surface and a small superconducting gap (compared to d ). Therefore, we used E k ± d ≈ ± d in the derivation.
In addition, we have used that V n,n+l k,γ V n,n+l k ,−γ * = 0 holds, since V k,+(−) gQ contains only even (odd) powers ofQ V n,n+l k,+(−) = 0 for l = odd (even) . Taking in addition the relation V n,n+l For the classically driven system, gQ is replaced by a time-dependent field A cos (ω 0 t), with dimensionless amplitude A. This gives rise to a time-periodic HamiltonianĤ (t) = H (t + T ) with period T = 2π/ω 0 . The stroboscopic dynamics is described by the timeindependent Floquet Hamiltonian. Like in the undriven case, this Floquet Hamiltonian can be projected onto a given low energy subspace (the singly occupied impurity) using a time-periodic Schrieffer-Wolff transformation [17,25,[56][57][58]. Alternatively, the Floquet Hamiltonian can be obtained within the Floquet block matrix approach, in which the periodically driven system is mapped to a time-independent Hamiltonian in an extended Hilbert space with an additional discrete Floquet band index. In the high-frequency limit, we can then obtain the Floquet Hamiltonian by projecting to the lowest Floquet band [25,58].
This Floquet Hamiltonian for the Peierls coupling is reproduced from the quantum Floquet formalism by taking the limit n → ∞, g → 0 with g √ n being a finite number, which was already observed in [36,38]. In this limit, j n,n+l converges to the |l|th Bessel function of the first kind J |l| (2g √ n), We obtain the effective Hamiltonian given in Eqs. (18), (21) and (22)  The obtained results are very similar to the exchange constants for two photons in the cavity (see Fig. 1 in the main text). However, a resonance at |ω 0 / d | = 1/3 is shown in the odd channel due to the infinite number of available photons. In the quantum formalism, this resonance appears for at least three photons in the cavity. These statements are transferred to the expressions for the energies.