Probing and dressing magnetic impurities in a superconductor

TL;DR

This paper introduces a microwave-based method to probe and control magnetic impurity interactions in superconductors by utilizing Yu-Shiba-Rusinov states, enabling insights into magnetic order and potential applications in spin glasses.

Contribution

It presents a novel technique to manipulate and measure magnetic impurity interactions in superconductors using microwave radiation and YSR states, offering new experimental possibilities.

Findings

Microwave radiation can induce ferro- or antiferromagnetic interactions depending on detuning.

Measurement of microwave response reveals magnetic order of impurities.

Estimated microwave absorption and interaction strength are feasible for experimental realization.

Abstract

We propose a method to probe and control the interactions between an ensemble of magnetic impurities in a superconductor via microwave radiation. Our method relies upon the presence of sub-gap Yu-Shiba-Rusinov (YSR) states associated with the impurities. Depending on the sign of the detuning, radiation generates either a ferro- or antiferromagnetic contribution to the exchange interaction. This contribution can bias the statistics of the random exchange constants stemming from the RKKY interaction. Moreover, by measuring the microwave response at the YSR resonance, one gains information about the magnetic order of the impurities. To this end, we estimate the absorption coefficient as well as the achievable strength of the microwave-induced YSR-interactions using off-resonant radiation. The ability to utilize microwave fields to both probe and control impurity spins in a superconducting host may open new paths to studying metallic spin glasses.

Equations

$${\int }_0^{2\Delta /\hslash }d\omega \sigma (\omega )=\bar{F}(\{\mathbf{S}(\mathbf{r})\})\Sigma ,\Sigma \sim {\sigma }_n\frac{n^2}{{\nu }^2\Delta }\frac{1/n{\xi }^2}{l+1/n{\xi }^2},$$

$$\frac{⟨J_{\mathrm{YSR}}^{\mathrm{ind}}⟩}{⟨J_{\mathrm{RKKY}}⟩}\sim \frac{\Delta }{2E_{\mathrm{YSR}}-\omega }{\left(\frac{d}{\xi }\right)}^3{\left(\frac{e\mathcal{E}\xi }{\omega }\right)}^2,$$

$$\mathcal{H}={ϵ}_p{\tau }_z+\Delta {\tau }_x-\sum _i{J}_i{\text{S}}_i\cdot \mathbf{\sigma }\delta (\text{r}-{\text{R}}_i),$$

$${\Phi }_{i,{\text{S}}_i}^{+}(\text{r})=\frac{1}{{\mathcal{N}}_i}\frac{e^{-\text{absolutevalue}\sin 2{\delta }_i{r}_i/\xi }}{k_F{r}_i}U({\hat{\text{S}}}_i)\left(\begin{array}{c}\sin (k_F{r}_i+{\delta }_i)|\uparrow ⟩\\ \sin (k_F{r}_i-{\delta }_i)|\uparrow ⟩\end{array}\right),$$

$$\Psi (\text{r})=\sum _i\left[{\Phi }_{i,{\text{S}}_i}^{+}(\text{r}){\gamma }_i+{\Phi }_{i,{\text{S}}_i}^{-}(\text{r}){\gamma }_i^{†}\right],$$

$${\text{J}}_{1,2}(\text{r})=\frac{e\hslash }{2mi}\left\{{\Phi }_{1,{\text{S}}_1}^{+}(\text{r})\nabla {\Phi }_{2,{\text{S}}_2}^{-}(\text{r})-{\Phi }_{2,{\text{S}}_2}^{+}(\text{r})\nabla {\Phi }_{1,{\text{S}}_1}^{-}(\text{r})\right\}.$$

$$\mathcal{M}_{\text{YSR}}^{1,2}=\frac{1}{2}(\textbf{A}\cdot\hat{\textbf{R}})\,\hat{\textbf{R}}\cdot\!\int d^{3}\textbf{r}\big{[}\textbf{J}_{1,2}(\textbf{r})+\textbf{J}_{1,2}(-\textbf{r})\big{]}\,,$$

$$|{\mathcal{M}}_{\text{YSR}}^{1,2}|\approx$$

$$\sigma (\omega )=\frac{2}{\mathcal{V}{\mathcal{E}}^2}\sum _{i>j}2\pi \omega |{\mathcal{M}}_{\text{YSR}}^{i,j}{|}^2\delta (\omega -E_{\text{YSR}}^i-E_{\text{YSR}}^j),$$

$$\displaystyle\bigg{\langle}\frac{\sigma(\omega)}{\sigma_{n}}\bigg{\rangle}=$$

$$\times \frac{\xi }{l}\frac{1}{2W\Delta }g(\hslash \omega -2{\bar{E}}_{\text{YSR}},W),$$

$$H_{\text{eff}}=\delta E_{\text{GS}}=\frac{|{\mathcal{M}}_{\text{YSR}}{|}^2}{2E_{\text{YSR}}-\omega }=J_{\text{YSR}}^{\text{ind}}\text{leavevmode}{\text{S}}_1\cdot {\text{S}}_2,$$

Full text

The nature of interactions between magnetic impurities embedded in a metallic host gives rise to an intriguing state of matter: a spin glass Fischer and Hertz (1991). The RKKY exchange interaction between the impurities is carried by itinerant electrons and alternates in sign, depending on the inter-impurity separation  Ruderman and Kittel (1954); Kasuya (1956); Yosida (1957). The random position of the impurities with respect to each other results in a random-sign exchange interaction, frustrating the magnetic order in a system of localized spins. The efforts to understand the resulting low-temperature spin glass phase and the corresponding phase transition have led to the introduction of several important concepts in condensed matter physics, including the Edwards-Anderson Edwards and Anderson (1975) and functional Parisi (1980) order parameters. Moreover, these efforts have also motivated a ever-expanding toolset of quantum control techniques aimed at directly controlling the interactions between magnetic impurities.

Remarkably, even the simplest spin glass model introduced by Sherrington and Kirkpatrick Sherrington and Kirkpatrick (1975) in direct analogy to the Curie-Weiss model of a ferromagnet turns out to be extremely rich and, unlike the Curie-Weiss model, not amenable to a straightforward mean-field theory treatment Thouless et al. (1977); de Almeida and Thouless (1978). The frustration of the magnetic moments manifests itself in both the thermodynamic and electron transport properties of a normal metal with a magnetic element dissolved in it. Starting with magnetic susceptibility measurements on AuFe alloys Cannella and Mydosh (1972), there are a substantial number of such studies performed on bulk samples Cannella et al. (1971); Cannella and Mydosh (1973). With the development of mesoscopic systems, electron transport through mesoscale-sized alloys also received their fair share of attention; for example, the remanence of the resistance (i.e. its dependence on the cooling protocol) of a mesoscopic AgMn device was investigated in Capron et al. (2011), while quantum interference effects in the conductance of CuMn and AgMn were studied respectively in de Vegvar et al. (1991) and Capron et al. (2013).

In this Letter, we explore a novel method for investigating mesoscopic spin glasses using techniques recently perfected in the development of superconducting qubit technologies Devoret and Schoelkopf (2013). In particular, we consider the possibility of utilizing microwave radiation to directly probe and possibly control the many-body state of an ensemble of magnetic moments embedded in a thin superconducting bridge (Fig. 1).

Much as in a normal metal, magnetic impurities in a superconductor also exhibit random-sign RKKY interactions. This RKKY interaction is hardly modified by superconductivity, so long as the typical distance, $d$, between the impurities is shorter than the superconducting coherence length, $\xi$. For impurities separated by larger distances, the interaction is instead antiferromagnetic and dominated by a virtual process involving YSR states Yao et al. (2014); however, at such distances, the interactions are typically weak since they decay exponentially with $d/\xi$. Herein lies intuition behind our approach: To utilize microwave driving to enhance the virtual hybridization between the superconducting condensate and the YSR states.

With respect to such microwave excitation, there are two main differences between normal-metal and superconducting hosts. The first is that there exists a gap, $\Delta$, in the spectrum of excitations in a superconductor. In the absence of a magnetic field and impurities, a conventional s-wave superconductor, such as aluminum, possesses time-reversal symmetry. As a result, the gap is “hard”: At low temperatures there is a frequency threshold, $\omega_{\rm th}=2\Delta/\hbar$, for the absorption of electromagnetic radiation. The second difference is that a magnetic impurity in a superconductor creates a localized Yu-Shiba-Rusinov (YSR) state with energy $E_{\rm YSR}$ within the gap Luh (1965); Shiba (1968); Rusinov (1969); Ruby et al. (2016); Heinrich et al. (2018); Ruby et al. (2015, 2018); Island et al. (2017); Farinacci et al. (2018). A single YSR state may host no more than one quasiparticle, and therefore cannot facilitate absorption from the condensate. However, a pair of YSR states separated by distances $\lesssim\xi_{\text{YSR}}$ creates a discrete-energy state for an electron pair where $\xi_{\rm YSR}=\xi\sqrt{\Delta/(\Delta-E_{\rm YSR})}$ is the characteristic length-scale of a YSR state. To this end, at low temperatures, the sub-gap absorption results from a process in which a microwave photon transfers a Cooper pair from the condensate onto the pair of YSR states, leading to an absorption line centered at $\omega=2E_{\rm YSR}/\hbar$.

Crucially, the magnitude of this absorption by a YSR pair depends on the mutual orientation of the magnetic moments. For moments oriented in parallel, the associated pair of YSR states cannot accept a singlet Cooper pair (for simplicity, we assume that there is no spin-orbit coupling). Thus, the absorption is maximized for antiparallel moments and varies as $F[\mathbf{S}(\mathbf{R}_{1}),\mathbf{S}(\mathbf{R}_{2})]=1-\hat{\textbf{S}}_{1}\cdot\hat{\textbf{S}}_{2}$, where $\hat{\textbf{S}}_{1,2}=\textbf{S}_{1,2}/S$ are the unit vectors indicating the orientation of the magnetic moments (treated classically). Therefore, the sub-gap absorption coefficient provides information regarding ferromagnetic order at the scale, $|\mathbf{R}_{1}-\mathbf{R}_{2}|\lesssim\xi_{\text{YSR}}$. The absorption line width and its detailed shape depend on the inevitable spread of the contact exchange interaction  Yazdani et al. (1997); Ji et al. (2008); Cornils et al. (2017, 2017); Kezilebieke et al. (2018a) and the overlap between the YSR states Fominov et al. (2011); Kezilebieke et al. (2018b); Ruby et al. (2018); Choi et al. (2018).

For an ensemble of moments with density $n\gtrsim\xi^{-3}$, the many-body ferromagnetic order can be deduced from $\bar{F}(\{\mathbf{S}(\mathbf{r})\})=\langle\sum_{i\neq j}K(|\mathbf{R}_{i}-\mathbf{R}_{j}|)F[\mathbf{S}(\mathbf{R}_{i}),\mathbf{S}(\mathbf{R}_{j})]\rangle/N$ where $\langle\dots\rangle$ denotes averaging over YSR state energies, $K(r)\sim\Theta(\xi_{\text{YSR}}-r)$ [$\Theta$ is the Heaviside step function], $N\sim n^{2}\xi^{3}\mathcal{V}(w/\xi)^{3-D}$ is a normalization factor and $\mathcal{V}$ is the volume of the sample which we think of as either a wire ($D=1$) or as a film ($D=2$) with transverse dimension $w\ll\xi$. This quantity is related to the dissipative part of the conductivity integrated over the absorption line:

[TABLE]

where $n$ is the impurity concentration, $\sigma_{n}$ and $\nu$ are, respectively, the normal-state conductivity and the density of states at the Fermi-level, and $l$ is the electron elastic mean free path; note that the last factor in Eq. (1) extrapolates between the regimes of long and short mean free paths.

As aforementioned, in the absence of microwaves, there are two components of the inter-impurity interaction carried by virtual excitations of the itinerant electrons. The first (and dominant at $d\lesssim\xi$) one comes from the continuum of Bogoliubov quasiparticles and is responsible for the conventional indirect exchange coupling, i.e., RKKY interaction in normal metals and its counterpart in superconductors. The second one owes to the discrete YSR states and is specific to superconductors. Borrowing the idea of “off-resonant dressing” from quantum optics Cohen-Tannoudji et al. (1992), we note that off-resonant microwave radiation creates an additional channel for virtual transitions of Cooper pairs onto a pair of YSR states. For sufficiently strong drives, the corresponding amplitude may successfully compete with the one existing in the absence of radiation Yao et al. (2014), while the conventional RKKY component is only weakly affected, so long as the radiation remains far detuned from the gap edge. The sign of the effective interaction induced by the off-resonant drive depends on the sign of the detuning, $\omega-2E_{\rm YSR}$, and is ferromagnetic for $\omega-2E_{\rm YSR}>0$.

One can estimate the strength of the microwave-induced interaction, $J_{\rm YSR}^{\rm ind}$, relative to the RRKY component $J_{\rm RKKY}$ as

[TABLE]

where $\mathcal{E}$ is the electric field of the microwave. In order for superconductivity to remain intact, the last factor here must be small; in order to avoid resonant absorption, the denominator of the first factor must exceed the YSR absorption line width. Together, these two conditions set a limit for the strength of the “dressed” interaction. Intriguingly, in a dilute system, at $d\sim\xi$, the “dressed” interaction may compete with the conventional RKKY component opening up the possibility of studying the de Almeida – Thouless line de Almeida and Thouless (1978) in a spin glass.

Model—Our starting point is the BCS Hamiltonian of an s-wave superconductor with magnetic impurities. The Bougoliubov-de Gennes (BdG) Hamiltonian takes the form

[TABLE]

where $\epsilon_{p}=-\frac{1}{2m}\nabla^{2}-\mu$ is the kinetic energy (we set $\hbar=1$), $\mu$ the chemical potential, and $\Delta$ the superconducting order parameter. The Hamiltonian $H=\int d\textbf{r}\Psi^{\dagger}\mathcal{H}\Psi/2$ is written in conventional Nambu spinor notation, where $\Psi=[\psi_{\uparrow},\psi_{\downarrow},\psi_{\downarrow}^{\dagger},-\psi_{\uparrow}^{\dagger}]^{T}$ and $\tau(\sigma)$ are Pauli matrices acting on the particle-hole (spin) space. The last term in the Hamiltonian represents the contact interaction between electrons and the impurity spins, where $J_{i}$ characterizes the coupling strength and $\textbf{S}_{i}$ is the spin of the i${}^{\textrm{th}}$ impurity located at position $\textbf{R}_{i}$.

The energy of the sub-gap YSR bound state localized around impurity $i$ is $E^{i}_{\text{YSR}}=\Delta{(1-\alpha_{i}^{2})}/{(1+\alpha_{i}^{2})}$ Luh (1965); Shiba (1968); Rusinov (1969), where $\alpha_{i}=\nu\pi J_{i}S/2$ is a dimensionless exchange coupling. We treat the impurity spins classically assuming they have equal magnitudes $S$ but in general different orientations. The 4-component eigenspinors of the BdG Hamiltonian $\mathcal{H}$ corresponding to the sub-gap YSR states are given by

[TABLE]

and $\Phi^{-}_{i,\textbf{S}_{i}}=CT\Phi^{+}_{i,\textbf{S}_{i}}$ where a particle ${(+)}$ and hole ${(-)}$ states $\Phi_{i,\textbf{S}_{i}}^{\pm}$ have energies $\pm E^{i}_{\text{YSR}}$ and are related by the antiunitary symmetry transformation $CT=\tau_{y}\sigma_{y}K$ where $K$ denotes complex conjugation; $\mathcal{N}_{i}^{2}=2\pi\nu\Delta\alpha_{i}/(1+\alpha_{i}^{2})$ is a normalization factor, $r_{i}={|\textbf{r}-\textbf{R}_{i}|}$ the relative distance from the impurity, $\delta_{i}=\tan^{-1}(\alpha_{i})$ the phase shift, $\ket{\uparrow}$ is the $+1$ eigenstate of $\sigma_{z}$ and $U(\hat{\textbf{S}}_{i})$ is a unitary rotation operator aligning the quantization axis of the Nambu spinor $\Psi$ with the direction of the impurity spin. As we are interested in the physics resulting from low energy microwave excitation, we will project the electron field operator onto these sub-gap YSR states,

[TABLE]

where $\gamma_{i}$ is the annihilation operator of a YSR state located at the i${}^{\textrm{th}}$ impurity. The projected Hamiltonian then becomes $H=\sum_{i}E_{i}\gamma^{\dagger}_{i}\gamma_{i}+\sum_{i,j}(\mathcal{M}_{ij}\gamma_{i}^{\dagger}\gamma_{j}^{\dagger}+\mathcal{M}^{\prime}_{ij}\gamma_{i}^{\dagger}\gamma_{j}+h.c.)$. To be specific, we focus on the low-density regime, $n\xi_{\text{YSR}}^{3}\ll 1$, which exhibits the same qualitative behavior as the high-density regime. In the low-density regime, we can neglect the hopping and non-radiative pair-creation terms from $H$ which simplifies the calculation significantly.

Radiation matrix element—We now turn to calculating the matrix element, $\mathcal{M}_{\text{YSR}}$, corresponding to radiation-assisted YSR-pair-creation. When the system is coupled to a weak microwave field, the vector potential $\tilde{\textbf{A}}$ enters as $\frac{\hbar}{i}\nabla\rightarrow\frac{\hbar}{i}\nabla+\frac{e}{c}\tilde{\textbf{A}}$ and the superconducting order parameter is generally both complex and spatially dependent, i.e. $\Delta(\textbf{r})=|\Delta|e^{i\theta(\textbf{r})}$. We choose to work in the London gauge where the order parameter is real and the integral of the new vector potential, $\textbf{A}=\tilde{\textbf{A}}-\frac{\hbar c}{e}\nabla\theta(\textbf{r})$, yields a gauge-invariant phase difference that gives rise to the supercurrent in a superconductor.

The electromagnetic perturbation to the BdG Hamiltonian is given by: $H_{\text{EM}}=\int d^{3}\textbf{r}\,(e/2)\Psi^{\dagger}(\textbf{A}\cdot\textbf{v}+\textbf{v}\cdot\textbf{A})\Psi$, where v is the velocity operator. Using Eq. (5), the projected perturbation Hamiltonian takes form $H_{\text{EM}}\approx\mathcal{M}_{\text{YSR}}^{1,2}\gamma_{1}^{\dagger}\gamma_{2}^{\dagger}+h.c.$ where $\mathcal{M}_{\text{YSR}}^{1,2}=\int d^{3}\textbf{r}\,\textbf{J}_{1,2}\cdot\textbf{A}$. We assume a negligible thermal population of YSR states and thereby ignore hopping terms $\gamma_{1}^{\dagger}\gamma_{2}$. The current density $\textbf{J}_{1,2}$ is given by

[TABLE]

Since the relevant microwave frequencies ($\omega\leq 2\Delta$) corresponds to wavelengths significantly longer than both the superconducting coherence length and the characteristic YSR length-scale, A can be treated as position-independent. Moreover, since the integration domain contains all of space, the integral of the anti-symmetric portion of the integrand vanishes. The integrand can therefore be symmetrized,

[TABLE]

where we have also used the fact that the current density, Eq. (6), is rotationally symmetric around the axis connecting the two impurities, $\textbf{R}=\textbf{R}_{2}-\textbf{R}_{1}$ and $\hat{\textbf{R}}=\textbf{R}/|\textbf{R}|$, and therefore only the component of the vector potential that is parallel to this axis contributes to absorption.

Owing to the rotational symmetry, the integral is effectively two-dimensional and can be done in elliptical coordinates Note (1), $r_{+}=(r_{1}+r_{2})/2$, $r_{-}=r_{1}-r_{2}$, where the integral measure becomes $\int d^{3}\textbf{r}\rightarrow\pi\int_{R/2}^{\infty}dr_{+}\int_{-R}^{R}dr_{-}\{r_{+}^{2}-\frac{r_{-}^{2}}{4}\}$. In these new coordinates, $r_{+}^{2}-r_{-}^{2}/4=r_{1}r_{2}$, exactly canceling out the power-law decay of the YSR wave function!

For two identical impurities, the current density respects an additional reflection symmetry about the plane perpendicular to and bisecting R. This symmetry would imply that the integral in Eq. (7) vanishes. Thus, a non-zero radiation matrix element requires the breaking of this reflection symmetry. In practice Cornils et al. (2017), this is always the case as one invariably observes fluctuations in the exchange coupling strengths, suggesting that the reflection symmetry is naturally broken by disorder effects. Therefore, we assume hereafter that the impurities are not identical, $\alpha_{1}\neq\alpha_{2}$.

After a straightforward but tedious calculation, the general expression of the matrix element can be obtained. Assuming that $|\,|\sin 2\delta_{1}|-|\sin 2\delta_{2}|\,|\ll\xi/R$, the matrix element can be expanded to first order in the coupling difference, $|\alpha_{1}-\alpha_{2}|$, and $1/(k_{F}R)$ as 111See Supplementary Material for a derivation of the pair transition matrix element, the integrated conductivity in the clean dilute limit as well as calculation of the ratio $J_{\text{YSR}}^{\text{ind}}/J_{\text{RKKY}}$.

[TABLE]

where $\vec{\mathcal{E}}=-\partial\textbf{A}/\partial t$ is the electric field of the applied microwaves and $\alpha=(\alpha_{1}+\alpha_{2})/2$, the average coupling strength of the two impurities. One important and intriguing observation: Due to the longer intrinsic YSR length scale $\xi_{\text{YSR}}=\xi/|\sin 2\delta|$ as well as the absence of a power-law decay in the matrix element, the corresponding microwave-induced interaction has a significantly longer range than both the RKKY interaction and the bare YSR interaction Yao et al. (2014) (in the absence of microwaves).

Experimental Implementation—In this section, we propose an experimental implementation based upon superconducting circuits, which enables one to utilize microwave fields to both probe the spin state of the impurities as well as control their effective interactions. Since our proposed microwave-dressed interactions require both a superconducting background and a supercurrent, a natural setup is an S-s-S junction created from two large superconducting leads linked by a constriction (Fig. 1); such a setup has previously been used in experiments probing Andreev bound states Levenson-Falk et al. (2014). An oscillating bias potential, $V\cos{\omega t}$, can then be applied to the leads, to create a time-dependent supercurrent $j\propto\frac{\hbar e}{m}\nabla\theta$ governed by the Josephson relation $\partial_{t}\theta=\frac{2eV}{\hbar}\sin{\omega t}$, where $\theta$ is the gauge-invariant phase difference between the leads and $\omega$ is the microwave frequency.

To probe the many-body state of the impurities, we propose to apply a resonant microwave drive so that the ordering can be inferred from the integrated dissipative part of the sub-gap conductivity, $\int_{0}^{2\Delta}\sigma(\omega)d\omega$. The dissipative conductivity $\sigma$ is related to the energy absorption rate $\omega\Gamma=\sigma\mathcal{E}^{2}/2$, where $\Gamma$ is the transition rate obtained by plugging Eq. (8) into Fermi’s golden rule. At low temperatures, the initial state consists of unoccupied YSR states. The integrated sub-gap conductivity depends on the average distance between impurities and the elastic mean free path $l$. It can be analyzed in various limits that are detailed below and summarized in Fig. 2.

In the low-density limit ${n\xi_{\text{YSR}}^{3}\lesssim 1}$, the YSR states are well-localized so that the hybridization-induced energy splitting of the YSR states is negligible compared to $\Delta$. The sub-gap conductivity can be written as a sum of pair-creating transitions,

[TABLE]

where $\mathcal{M}_{\text{YSR}}^{i,j}$ is the radiation-assisted matrix element to create a pair of YSR quasiparticles on the i${}^{\textrm{th}}$ and j${}^{\textrm{th}}$ impurity. Assuming a uniform distribution of uncorrelated YSR levels in a narrow band, $W\ll 2E_{\text{YSR}}$, and ensemble-averaging the conductivity, we find  Note (1)

[TABLE]

where $\bar{\alpha}$ is the mean value of $\alpha_{i}$ and $\bar{E}_{\text{YSR}}$ the average YSR energy. The normalized distribution function $g(\omega,W)=2(1-|\omega|/W)^{3}\,\Theta(W-|\omega|)/W$ (specific to a uniform YSR band) characterizes the energy-dependence of the absorption and has a peak of width $W$. We normalize the conductivity by its normal-state value $\sigma_{n}=2e^{2}\nu\frac{1}{2}v_{F}l$, where $l$ is the electron mean free path from non-magnetic impurities; Eq. (10) is valid in the clean limit $l\gg\xi$. The dirty limit $l\ll\xi$ is obtained by replacing $\xi\to l$ in the second line of Eq. (10) Fominov et al. (2011); Kiendl et al. (2017).

In the high-density regime, $n\xi_{\text{YSR}}^{3}\gg 1$, the YSR states will strongly hybridize. The conductivity in this case is derived in Fominov et al. (2011). At a qualitative level, the high-density limit can be obtained from Eq. (10) by replacing $\xi$ (in the second line) by an effective mean free path arising from magnetic impurities, $\xi\to 1/(n\xi^{2})$. When the above conductivity is integrated over the sub-gap states, one naturally recovers Eq. (1).

Finally, we now estimate the achievable strength of the dressed interactions induced by an off-resonant microwave field. The relevant energy levels and matrix element are depicted in Fig. 1b. At leading order, one finds that the radiation-assisted YSR pair creation results in an effective spin-spin interaction originating from a spin-dependent AC stark shift to the ground state energy,

[TABLE]

where we have neglected a spin-independent overall shift. To compare the relative strength of this dressed interaction with the RKKY component, we use Eq. (8) and the expression for the RKKY interaction from Ref. Yao et al. (2014), resulting Note (1) in the estimate presented in Eq. (2).

In summary, we have shown that the electronic sub-gap states hosted by magnetic impurities in a superconductor provide a new way to access magnetic order. This opens the possibility to probe and control metallic spin glass physics in a superconducting narrow-bridge junction by using microwave driving (Fig. 1a). Keeping the transverse size $w$ of the bridge thinner than the London length ensures that the supercurrent is approximately uniform and couples to the magnetic moments in the full volume. Unlike conventional Andreev bound states, the YSR levels are insensitive to a static phase difference across the junction which provides a simple way to distinguish the respective contributions to the dissipative conductivity. Looking forward, our work also opens the door to an intriguing quantum information platform where magnetic impurities in a superconducting host play the role of quantum memories, while microwave driving can lead to on-demand long-range gates Levy (2002). Here, the absence of a power-law decay in the radiation matrix element could enable all-to-all connectivity between qubits as well as multi-body interactions, both of which are important for reducing the gate depth of certain quantum algorithms Bernstein and Vazirani (1997); van Dam et al. (2006).

Acknowledgments—We gratefully acknowledge the insights of and discussions with K. Murch, I. Siddiqi, and T. Ojanen. The authors would particularly like to thank G. Zarand for previous collaborations on related work and for pointing out the impact of reflection symmetry. This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under award AWD00003522, the David and Lucille Packard Foundation and the Sloan Foundation. JS acknowledges support from the NSF-DMR1555135 (CAREER). ED acknowledges support from the Harvard-MIT CUA, the AFOSR-MURI Photonic Quantum Matter (award FA95501610323), and the DARPA DRINQS program (award D18AC00014). LG acknowledges support from the NSF DMR Grant No. 1603243.