Introduction

The experimental ability to probe and manipulate complex electronic correlations at the femtosecond time scale has opened unprecedented opportunities for the study of non-equilibrium quantum phenomena in strongly correlated or topological materials1,2,3,4,5. The development of next-generation spintronics and quantum computing applications requires the control of magnetic environments not only on similar time scales, but also on nanoscopic length scales. The latter has been achieved by using scanning tunneling spectroscopy (STS) techniques that have enabled the writing of magnetic skyrmions6, the tuning of local magnetic exchange couplings7,8,9, and, in combination with electron spin resonance (ESR) techniques, the rotation of individual magnetic moments in impurity clusters10. These techniques were successfully applied9 to tune the energy of Yu–Shiba–Rusinov (YSR) states11,12,13—induced by magnetic impurities placed on the surface of an s-wave superconductor—and thus to drive the system through a quantum phase transition14,15,16,17. While the required control of magnetic environments on electronic time scales has yet to be achieved, recent progress10 has raised the question not only of how quantum phenomena—such as quantum phase transitions—can be manipulated on the (electronic) femtosecond or picosecond time scale, but also of how such time-dependent phenomena can be described theoretically and visualized experimentally.

In this article, we address this question by investigating the manipulation of YSR states11,12,13 on electronic time scales. By using two different theoretical methods based on the non-equilibrium Keldysh formalism18,19,20 we study the non-equilibrium emergence and manipulation of YSR states in response to external perturbations of the magnetic environment. We show that the time evolution of YSR states can be visualized through the time-dependent differential conductance, dI(t)/dV, which, in analogy to the equilibrium case, is proportional to a non-equilibrium local density of states (LDOS), Nneq. These findings allow us to study the time-dependent phase transition of the system from a singlet to a doublet ground state by subjecting it to a time-dependent change in either the strength of a magnetic impurity’s exchange coupling, or in the relative orientation of moments in a magnetic dimer, or by externally driving a periodic precession of an impurity’s magnetic moment. Moreover, we show that the extent to which the system is driven out-of-equilibrium is controlled by the time scale over which perturbations occur, and is directly reflected in the time and frequency dependence of Nneq. Finally, our formalism reveals the transient behavior between static and periodically driven magnetic structures, allowing us to visualize the emergence of Floquet YSR states21,22,23. Our work thus provides a theoretical framework to study the emergence of non-equilibrium phenomena on electronic time and nanoscopic length scales in complex materials.

Results

Theoretical method

To study the non-equilibrium emergence and manipulation of YSR states, we consider magnetic impurities that are placed on the surface of an s-wave superconductor, with the exchange coupling or orientation of magnetic moments being time-dependent. Such a system is described by the Hamiltonian \({{{{{{{\mathcal{H}}}}}}}}={{{{{{{{\mathcal{H}}}}}}}}}_{0}+{{{{{{{\mathcal{U}}}}}}}}(t)\), where

$${{{{{{{{\mathcal{H}}}}}}}}}_{0} = -{t}_{{{{{{{{\rm{e}}}}}}}}}\mathop{\sum}\limits_{\langle {{{{{{{\bf{r}}}}}}}},{{{{{{{{\bf{r}}}}}}}}}^{\prime}\rangle ,\sigma }{c}_{{{{{{{{\bf{r}}}}}}}},\sigma }^{{{{\dagger}}} }{c}_{{{{{{{{{\bf{r}}}}}}}}}^{\prime},\sigma }-\mu \mathop{\sum}\limits_{{{{{{{{\bf{r}}}}}}}}}{c}_{{{{{{{{\bf{r}}}}}}}},\sigma }^{{{{\dagger}}} }{c}_{{{{{{{{\bf{r}}}}}}}},\sigma }\\ \quad +{\Delta }_{0}\mathop{\sum}\limits_{{{{{{{{\bf{r}}}}}}}}}\left({c}_{{{{{{{{\bf{r}}}}}}}},\uparrow }^{{{{\dagger}}} }{c}_{{{{{{{{\bf{r}}}}}}}},\downarrow }^{{{{\dagger}}} }+h.c.\right),\\ {{{{{{{\mathcal{U}}}}}}}}(t) = \,J(t)\mathop{\sum}\limits_{{{{{{{{\bf{R}}}}}}}},\alpha ,\beta }{c}_{{{{{{{{\bf{R}}}}}}}},\alpha }^{{{{\dagger}}} }{\left[{{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{\bf{R}}}}}}}}}(t)\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}\right]}_{\alpha \beta }{c}_{{{{{{{{\bf{R}}}}}}}},\beta },$$
(1)

where −te is the nearest-neighbor hopping parameter on a square lattice, μ is the chemical potential, Δ0 is the superconducting s-wave order parameter, and \({c}_{{{{{{{{\bf{r,\sigma }}}}}}}}}^{{{{\dagger}}} }\) creates an electron with spin σ at site r. J is the magnetic exchange coupling and SR is the impurity spin at site R with the last sum running over all impurity positions. Since the hard superconducting s-wave gap suppresses Kondo screening17,24, we take the magnetic impurity spin as classical in nature17, i.e., \({{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{\bf{r}}}}}}}}}=S(\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta )\), where θ and ϕ are the polar and azimuthal angles, respectively. Finally, we note that the perturbation \({{{{{{{\mathcal{U}}}}}}}}(t)\) is purely magnetic in nature, and thus does not break particle–hole symmetry.

To theoretically study the electronic response of the system to a time-dependent J or Sr, we recall that the primary experimental probe in the study of YSR states is the differential conductance, dI/dV, measured in STS experiments17. We therefore compute the time-dependent current flowing between an STS tip and the superconductor using a formulation of the non-equilibrium Keldysh formalism in the interaction representation18,19,25, yielding

$$I(V,t)=\frac{e}{\hslash }(-{t}_{{{{{{{{\rm{tun}}}}}}}}})\mathop{\sum}\limits_{\sigma }\left[{G}_{ts}^{{ < }}(\sigma ,t,t)-{G}_{st}^{{ < }}(\sigma ,t,t)\right],$$
(2)

where ttun is the tunneling amplitude between the tip and the superconductor, V is the applied bias, and \({G}_{ts}^{\,{ < }}\) is the spin-dependent, equal-time lesser Green’s function between the tip (t) and the site in the superconductor (s) that the electrons tunnel into25. Due to the picometer spatial resolution achieved in STS experiments26, we here assume that the electrons tunnel into a single site of the superconductor only. To compute \({G}_{ts}^{\,{ < }}\), we employ the Keldysh Dyson equations in real time, given by

$${\bar{G}}_{ts}^{\,{ < }}(t,t) = \,{\bar{g}}_{ts}^{\,{ < }}(t,t)+\int \,d{t}_{1}{\bar{g}}_{ts}^{{{{{{{{\rm{r}}}}}}}}}(t,{t}_{1})\bar{{{{{{{{\mathcal{U}}}}}}}}}({t}_{1}){\bar{G}}_{ss}^{\,{ < }}({t}_{1},t)\\ \quad +\int \,d{t}_{1}{\bar{g}}_{ts}^{\,{ < }}(t,{t}_{1})\bar{{{{{{{{\mathcal{U}}}}}}}}}({t}_{1}){\bar{G}}_{ss}^{{{{{{{{\rm{a}}}}}}}}}({t}_{1},t)\\ {\bar{G}}_{ts}^{{{{{{{{\rm{a}}}}}}}}}({t}^{\prime},t )= \,{\bar{g}}_{ts}^{{{{{{{{\rm{a}}}}}}}}}({t}^{\prime},t)+\int \,d{t}_{1}{\bar{g}}_{ts}^{{{{{{{{\rm{a}}}}}}}}}({t}^{\prime},{t}_{1})\bar{{{{{{{{\mathcal{U}}}}}}}}}({t}_{1}){\bar{G}}_{ss}^{{{{{{{{\rm{a}}}}}}}}}({t}_{1},t),$$
(3)

where \({\bar{g}}^{\,{ < }\,,r,a}\) are the equilibrium lesser, retarded and advanced Greens function matrices of the unperturbed system in Nambu space, and \(\bar{{{{{{{{\mathcal{U}}}}}}}}}(t)\) is the time-dependent perturbation matrix arising from Eq. (1). By discretizing time, Eq. (3) transforms into a set of coupled matrix equations, allowing us to obtain a closed form for \({G}_{ts}^{\,{ < }}(t,t)\) (for details, see Supplementary Note 1). Below all times are given in units of τe = ℏ/te which implies that for typical values of te of a few hundred meV, τe is of the order of a few femtoseconds.

As the calculation of dI/dV using Eq. (2) is computationally intensive, we also employ an alternative, computationally more efficient approach using the Heisenberg representation of the Keldysh formalism19,20. This approach utilizes the fact that in equilibrium, the differential conductance is proportional to the local density of states \(N({{{{{{{\bf{r}}}}}}}},\omega )=-{{{{{{{\rm{Im}}}}}}}}\,{g}^{{{{{{{{\rm{r}}}}}}}}}({{{{{{{\bf{r,\; r}}}}}}}},\omega )/\pi\). To define an analogous quantity out-of-equilibrium, we note that in the presence of a time-dependent perturbation, \({G}^{{{{{{{{\rm{r}}}}}}}}}(t,{t}^{\prime})\) depends explicitly on t and \({t}^{\prime}\). Setting \({t}^{\prime}=t-\Delta t\) and performing a Fourier transformation with regards to Δt, we obtain a time and frequency dependent Greens function Gr(t, ω), allowing us to define a "non-equilibrium” density of states, Nneq, via

$${N}_{{{{{{{{\rm{neq}}}}}}}}}({{{{{{{\bf{r}}}}}}}},\sigma ,t,\omega )=-\frac{1}{\pi }{{{{{{{\rm{Im}}}}}}}}{G}^{{{{{{{{\rm{r}}}}}}}}}({{{{{{{\bf{r}}}}}}}},{{{{{{{\bf{r}}}}}}}},\sigma ,t,\omega ),$$
(4)

where Gr(t, ω) is obtained from

$$\left[{{{{{{{\rm{i}}}}}}}}\frac{d}{dt}+\omega +i\Gamma -\hat{H}(t)\right]{\hat{G}}^{{{{{{{{\rm{r}}}}}}}}}\left(t,\omega \right)=\hat{1},$$
(5)

for an arbitrary time dependence of the Hamiltonian. Here, \(\hat{H}(t)\), \({\hat{G}}^{{{{{{{{\rm{r}}}}}}}}}(t,\omega )\) and \(\hat{1}\) are matrices in real and Nambu space (for details, see Supplementary Note 2), and τΓ = ℏ/Γ is the quasi-particle lifetime17. We demonstrate below that Nneq is proportional to the time-dependent dI(V, t)/dV, and thus represents a physical observable that describes the non-equilibrium time evolution of the superconductor’s electronic structure.

Non-equilibrium emergence of YSR states

We begin by considering the question of how a YSR state emerges when a single magnetic impurity is placed on the surface of an s-wave superconductor at time t = 0. To this end, we consider a perturbation of the form

$${{{{{{{\mathcal{U}}}}}}}}(t\ge 0)=JS\mathop{\sum}\limits_{\alpha ,\beta }{c}_{{{{{{{{\bf{r}}}}}}}},\alpha }^{{{{\dagger}}} }{\sigma }_{\alpha \beta }^{z}{c}_{{{{{{{{\bf{r}}}}}}}},\beta },$$
(6)

where S is the magnitude of the impurity spin, and \({{{{{{{\mathcal{U}}}}}}}}(t\, < \,0)=0\). In Fig. 1a, we present the resulting time and bias dependence of dI(V, t)/dV as obtained from Eq. (2) (due to the significant computational resources required, the calculation of dI/dV is limited to t ≤ 100τe). At t = 0, the LDOS is that of an unperturbed s-wave superconductor without a magnetic impurity, exhibiting a hard gap and coherence peaks at ± Δ0. As the magnetic impurity is placed on the surface, spectroscopic weight from the gap edges begins to be transferred into the gap, creating at first a broad peak centered around the energy of the emerging YSR state (see black arrows in Fig. 1a). This shift of spectroscopic weight occurs in a wave-like form, as reflected in the stripe-like patterns of dI(V, t)/dV in Fig. 1a27,28. With increasing time, the width of the peaks decreases, while their height increases, which is a direct consequence of the uncertainty principle: tΔE ≥ ℏ implies that as the time over which the impurity is located on the surface increases, the YSR state’s energy uncertainty, i.e., the peaks’ width, decreases.

Fig. 1: Non-Equilibrium Emergence of Yu-Shiba-Rusinov States.
figure 1

Time evolution of a the differential conductance dI/dV, and b, c the non-equilibrium local density of states Nneq(r, t, ω) at the site of the magnetic impurity (we set ℏ = 1 such that ω has the units of energy). The colorbar in a represents the normalized dI/dV, the colorbar next to c represents the normalized Nneq in b, c. d Time evolution of the deviation of the Yu–Shiba–Rusinov (YSR) peaks' height from equilibrium, ΔI, as obtained from Nneq in c. e Comparison of the equilibrium local density of states (LDOS) in the presence of a magnetic impurity, and Nneq(r, t, ω) at t = 400τe. Here, τe = ℏ/te with the nearest-neighbor hopping parameter − te. The calculation of dI/dV and Nneq used identical parameters sets with (μ, JS, Δ0) = ( − 2.0, 1.8, 0.4)te, and Γ = 0.01te. Here, μ is the chemical potential, J is the magnetic coupling, S is the spin of the adatoms and Δ0 is the superconducting order parameter. For the calculation of dI/dV, we used ttun = 0.01te for the tunneling between the system and the tip, corresponding to the weak-tunneling limit. The same parameters are also used in the following figures. The results for Nneq were obtained for a 50 × 50 site real space system.

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To elucidate the relation between dI/dV and Nneq, we note that for the perturbation of Eq. (6), the solution of Eq. (5) is given by

$${\hat{G}}^{{{{{{{{\rm{r}}}}}}}}}(t,\omega ) = \left[1-{e}^{{{{{{{{\rm{i}}}}}}}}\left(\omega \hat{1}-{\hat{H}}^{+}\right)t}{e}^{-\Gamma t}\right]{\hat{g}}_{+}^{{{{{{{{\rm{r}}}}}}}}}(\omega )\\ \quad +{e}^{{{{{{{{\rm{i}}}}}}}}\left(\omega \hat{1}-{\hat{H}}^{+}\right)t}{e}^{-\Gamma t}{\hat{g}}_{0}^{{{{{{{{\rm{r}}}}}}}}}(\omega )$$
(7)

where \({\hat{g}}_{0,+}^{{{{{{{{\rm{r}}}}}}}}}\) are the equilibrium Green’s functions for a system without and with an impurity, respectively, and \({\hat{H}}^{+}\) is the full Hamiltonian for t ≥ 0. We thus have \({\hat{G}}^{{{{{{{{\rm{r}}}}}}}}}(t=0,\omega )={\hat{g}}_{0}^{{{{{{{{\rm{r}}}}}}}}}(\omega )\) and \({\hat{G}}^{{{{{{{{\rm{r}}}}}}}}}(t\to \infty ,\omega )={\hat{g}}_{+}^{{{{{{{{\rm{r}}}}}}}}}(\omega )\). A comparison of Nneq obtained from Eqs. (4) and (7) (see Fig. 1b) with dI/dV (see Fig. 1a) shows remarkable quantitative agreement (up to an overall scaling factor) in the (ω, t)-plane (a more detailed comparison is provided in Supplementary Movie 1). This suggests that dI(V, t)/dV ~ Nneq(t, ω = eV), implying that Nneq represents a physical observable that describes the non-equilibrium time evolution of the superconductor’s electronic structure. Since Nneq is computed in the absence of a tip, we expect dI/dV ~ Nneq to hold only in the weak-tunneling regime, similar to the equilibrium case17. As the calculation of Nneq is computationally less demanding than that of dI/dV, we can now study the time-evolution of the system up to much larger times using Nneq, as shown in Fig. 1c, where we present Nneq(t, ω) up to t = 400τe. Eq. (7) suggests that when ω = ± EYSR, with EYSR being the YSR state energy, the oscillatory term, \(\sim {e}^{i\left(\omega \hat{1}-\hat{{H}^{+}}\right)t}\) is identical to one, and Nneq relaxes exponentially to the static case for t → ∞, with the relaxation time given by τΓ28, which is identical to the lifetime of the YSR state. This is confirmed by a log-plot of the deviation of the YSR peaks’ height from the equilibrium value, ΔI, as obtained from Nneq in Fig. 1c, as a function of time presented in Fig. 1d, where the dashed line corresponds to ~ e−Γt/ℏ. Finally, a comparison of the equilibrium LDOS in the presence of a static magnetic impurity with Nneq(t, ω) at t = 400τe (see Fig. 1e) shows very good agreement, demonstrating that this formalism allows us to study the system during the entire equilibration process. We note that the agreement between dI/dV and Nneq is independent of the specific value of Γ chosen (see Supplementary Movie 2), as Γ only controls the relaxation time of the process. Moreover, for more realistic parameters sets, the relaxational dynamics discussed above can be observed on time scales up to 100 ps (see Supplementary Fig. 1).

Dynamical tuning of quantum phase transitions

An intriguing phenomenon associated with the presence of a magnetic impurity is the possibility to drive the superconductor through a quantum phase transition between a singlet and doublet ground state by either tuning J through a critical value, \({J}_{{{{{{{{\rm{c}}}}}}}}}={(\pi {N}_{0}S)}^{-1}\),14,15,16,17, where N0 is the normal state density of states at the Fermi energy, or by changing the distance or relative spin alignment of two or more impurities29,30. This phase transition is accompanied by a zero-energy crossing of the YSR peaks in the LDOS. To investigate whether such a phase transition can also be induced using non-equilibrium techniques31,32, we begin by considering the case of a time-dependent exchange coupling J, which is motivated by the experimental ability to significantly vary J using STS techniques7,8,9. Specifically, we increase the exchange coupling from an initial value Ji < Jc at t = 0 to a final value Jf > Jc at a time t = Δt, using the time evolution \(J(t)={J}_{{{{{{{{\rm{i}}}}}}}}}+({J}_{{{{{{{{\rm{f}}}}}}}}}-{J}_{{{{{{{{\rm{i}}}}}}}}}){\sin }^{2}\left(\frac{\pi t}{2\Delta t}\right)\) for 0 ≤ t ≤ Δt as shown in Fig. 2a. A plot of the equilibrium LDOS for J = Ji,f in Fig. 2b demonstrates that the zero energy crossing of the YSR peaks at the phase transition exchanges the spin projection of their particle- and hole-like branches14,15,16,17, which can be mapped using spin-polarized STS33. The time evolution of the phase transition can thus be best visualized by considering the spin-resolved Nneq(σ, t, ω), shown in Fig. 2c, d for σ =  ↑ and σ =  ↓, respectively. While the change in J over a finite time Δt leads to significant oscillations in Nneq, a zero-energy crossing of the YSR peaks, indicating a time-dependent phase transition14,15,16,17, can still be identified. Moreover, Nneq reveals several noteworthy features. First, the spectral weight of the YSR peaks is decreased during the zero-energy crossing, concomitant with the phase transition. Second, a substantial redistribution of the YSR state’s spectral weight begins to occur only after t ≈ 50τe = Δt/2, indicating a delayed response of the system to the imposed change in J. This is particularly evident from comparison of the time-dependent YSR peak height at the final energy position in Fig. 2e with J(t) shown in Fig. 2a. Third, substantial oscillations in Nneq at the energy of the original YSR peaks persist up to time scales significantly longer than Δt, and the spectral weight of these oscillations is shifted to the final energy position of the YSR states in a wave-like pattern, as evidenced by the stripe-like pattern in Nneq (see Fig. 2c, d). Finally, a comparison of dI/dV and Nneq shown in Supplementary Movie 3 again reveals good quantitative agreement between them, further supporting our conclusion that dI/dV ~ Nneq.

Fig. 2: Dynamical tuning of quantum phase transitions.
figure 2

a Time dependence of the magnetic coupling J(t) with an initial value of Ji = 1.8te < Jc, a final value of Jf = 2.8te > Jc, where a quantum phase transition occurs at the critical magnetic coupling of Jc = 2.45te, and the transition from Ji to Jf occurs over the time Δt = 100τe. b Spin-resolved equilibrium local density of states (LDOS) in the presence of a single magnetic impurity with Ji < Jc and Jf > Jc. Time dependence of the spin-resolved non-equilibrium local density of states Nneq(σ, t, ω) for c spin σ = ↑, and d σ = ↓ arising from J(t) in a. The colorbar represents the normalized Nneq. e Height of the Yu–Shiba–Rusinov (YSR) peaks as a function of time, representing the relaxation of the system. The results for Nneq were obtained for a 24 × 24 site real space system.

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Twisting a spin dimer

The superconductor can also be tuned through a quantum phase transition by changing the relative alignment of the spins of two impurities in close proximity from antiparallel to parallel29, even if each of the two impurities possesses an exchange coupling smaller than the critical value, Jc. Such a change in the spins’ relative alignment in an impurity dimer has been achieved using ESR techniques10. For antiparallel alignment, the YSR states associated with each impurity cannot hybridize as the spin quantum number of their particle- and hole-like branches are opposite. The LDOS thus exhibits two pairs of degenerate YSR states with opposite spin quantum numbers29, as shown in Fig. 3a, b. When the alignment is rotated to parallel, the YSR states hybridize, leading to an energy splitting between them and the emergence of four peaks in the LDOS (see Fig. 3a, b). Plotting the equilibrium LDOS as a function of the angle θ between the spins (Fig. 3c, d), corresponding to the adiabatic limit of the rotation, we find that the energy splitting increases with decreasing θ, leading to a zero energy crossing of two of the YSR peaks, and a concomitant phase transition from a singlet to a doublet ground state. The question naturally arises as to the time dependence of such a phase transition when the relative alignment is changed over a finite time Δt, as was done in ref. 10 through the application of an ESR π pulse. To address this question, we consider for concreteness the rotation of one of the two impurity spins through the xz plane described by \({{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{\bf{r}}}}}}}}}=S[\sin \theta (t),0,\cos \theta (t)]\), with the time-dependent polar angle given by

$$\theta (t)=\pi \,{\cos }^{2}\left(\frac{\pi }{2\Delta t}t\right),\quad {{{{{{{\rm{for}}}}}}}}\,0\le t\le \Delta t.$$
(8)

In Fig. 3e, f we present the resulting spin-resolved Nneq(σ, t, ω) for Δt = 200τe, with the spins being antiparallel for t = 0 and parallel for t = Δt. As before, we can identify a time-dependent zero-energy crossing of the YSR peaks (see yellow arrows in Fig. 3e, f) indicating a phase transition from a singlet to a doublet ground state. We note that the angular dependence of the equilibrium LDOS shown in Fig. 3c, d, corresponds to the time dependence of Nneq between 0 ≤ t ≤ Δt (see white dashed line in Fig. 3e, f) in the adiabatic limit of Δt → ∞. A comparison of the equilibrium LDOS with Nneq in Fig. 3e, f thus reveals that a finite Δt leads to significant time-dependent oscillations in Nneq, in particular near the energy positions of the YSR peaks at t = 0, \(\pm {E}_{{{{{{{{\rm{YSR}}}}}}}}}^{(0)}\). To investigate how the strength of these oscillations depends on Δt, we plot in Fig. 3g, h\({N}_{{{{{{{{\rm{neq}}}}}}}}}({t}_{\Delta },\pm {E}_{{{{{{{{\rm{YSR}}}}}}}}}^{(0)})\) as a function of time after the rotation is completed, i.e., tΔ = t − Δt, for two different values of Δt. We find that the amplitude of the oscillations in Nneq(tΔ, ω) increases with decreasing Δt as the system is more strongly driven out of equilibrium. This result is consistent with the expectation that in the quasi-static, adiabatic limit, Δt → ∞, these oscillations vanish, and Nneq becomes identical to the equilibrium LDOS in Fig. 3c, d, when plotted as a function of angle.

Fig. 3: Twisting a spin dimer.
figure 3

Equilibrium a spin-↑ and b spin-↓ local density of states (LDOS) for parallel (↑↑) and antiparallel (↑↓) alignment of the spins of two magnetic impurities, located on nearest-neighbor sites. The LDOS is computed at the site of the spin-↑ magnetic impurity. Quasi-static evolution of the c spin-↑, d spin-↓ LDOS as a function of the angle θ between the two magnetic moments. Spin-resolved non-equilibrium local density of states Nneq(σ, t, ω) for e σ = ↑, and f σ = ↓ arising from the relative rotation of the impurity moments. The colorbar represents the normalized Nneq. Nneq(σ, tΔ, ω) for g σ = ↑, and h σ = ↓ as a function of the time after the perturbation tΔ (see text). The results for Nneq were obtained for a 24 × 24 site real space system.

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Emergence of Floquet YSR states

The formalism presented above can also be employed to investigate periodically driven systems, which are typically studied using the Floquet formalism21,22,23, and the transient behavior of quantum systems between the static and periodically driven limits. The latter is of particular interest since heating effects in interacting systems34 could potentially destroy novel quantum states, such as Floquet topological insulators22 or Floquet topological superconductors35 even before they are fully formed. To study such transient behavior, we consider the case of a single magnetic impurity, and study the time evolution from a single static impurity spin at t = 0 to an impurity spin that rotates periodically in the xz-plane with a driving frequency of ω0 = 2π/T. We note that the periodic rotation of the impurity’s spin leads to a splitting of the (static) YSR peaks in the density of states36, with the splitting given by ω0. This is shown in Fig. 4a where we present the evolution of the Floquet LDOS NFl with increasing driving frequency ω0 (for details, see Supplementary Note 3). Since, J = 2.8te > Jc, and since for ω → ∞ the YSR states merge with the continuum, as the scattered electrons see a vanishing effective J, a zero-energy crossing of the YSR peaks in NFl occurs at some critical value of the driving frequency, ωc, indicating a phase transition (see yellow arrow in Fig. 4a) from a doublet to a singlet ground state. To investigate the transient behavior from a static to a periodically rotating impurity spin, we employ a time-dependent polar angle given by \(\theta (t)={\omega }_{0}t\tanh \left(5{\omega }_{0}t/\pi \right)\) for 0 ≤ t ≤ T/2 and θ(t) = ω0t for t ≥ T/2. The resulting Nneq shown in Fig. 4b for ω0 = 0.2te/ℏ clearly reveals the emergence of the YSR peak splitting and of four YSR peaks with increasing time. Since ω0 > ωc and J > Jc, we find that with increasing time, there is a substantial transfer of spectral weight across zero energy, indicating a time-dependent phase transition (see black arrow in Fig. 4b). We note that for t ≫ T, the system is described by a Hamiltonian that is periodic in time, such that Nneq is expected to coincide with the Floquet LDOS NFl. We find that already for t = 400τe, Nneq is in very good quantitative agreement with NFl (see Supplementary Fig. 3 and Supplementary Note 4) that supports the validity not only of the non-equilibrium formalism presented above to describe periodically driven systems, but also the definition of Nneq as an observable physical quantity. Moreover, we note that while the total, spin-summed Nneq(t, ω) does not exhibit any signature of the rotation period (consistent with a time-independent total NFl), the spin-resolved Nneq(σ, t, ω) shown in Fig. 4c, d does, an effect refereed to as micromotion37,38. As expected from a rotation of the spin in the xz-plane, we find that Nneq for σ = ↑ and σ = ↓ are out of phase, i.e., shifted by half a period T. A comparison of dI/dV and Nneq shown in Supplementary Movie 4 again reveals good quantitative agreement between these quantities, further supporting our conclusion that dI/dV ~ Nneq. Due to the numerical complexity in computing dI/dV, we were restricted to time scales t ≤ 20τe for this comparison.

Fig. 4: Emergence of Floquet Yu-Shiba-Rusinov States.
figure 4

a Floquet local density of states (LDOS) NFl as a function of energy for a continuously rotating spin with driving frequency ω0, with l = 2 as the cutoff in the Floquet–Sambe matrix. Note that the energy splitting of the Yu–Shiba–Rusinov (YSR) states (see red arrow) is given by ω0. The colorbar represents the normalized Floquet LDOS. Time evolution of b the non-equilibrium local density of states Nneq(t, ω) summed over both spin orientations, and c the spin-resolved Nneq(↑, t, ω) and d Nneq(↓, t, ω) for ω0 = 0.2te/ℏ, corresponding to T = 10πτe. The colorbar next to c represents the normalized Nneq in b–d. Here, the magnetic coupling J = 2.8te > Jc is larger than the critical value Jc, where a phase transition occurs. The results for Nneq were obtained for a 24 × 24 site real space system.

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Discussion

The ability to understand the non-equilibrium response of complex quantum systems to external perturbations on electronic (femtosecond) timescales and nanoscopic length scales is crucial not only for advancing our theoretical understanding of these systems, but also for the development of next generation spintronics and quantum computing applications. Here, we have shown that time-dependent manipulations of magnetic impurities—leading to changes in their exchange coupling or relative spin alignment—not only allow us to study the emergence of quantum phenomena, such as the formation of YSR states, but can also give rise to new intriguing phenomena, such as a time-dependent phase transition. Studying and understanding the response of complex quantum system to external perturbations will be crucial for realizing topological quantum gates, and the required braiding of Majorana fermions. Indeed, the formalism we have presented here provides a suitable approach to study the non-equilibrium response of topological superconductors, and in particular Majorana modes39,40,41.

One could envision a few different approaches to realizing experiments capable of probing the relaxational dynamics of the quantum phenomena described above. To this end, experiments would have to be performed on the scale of the lifetime τΓ of YSR states, which for more realistic parameters (see Supplementary Fig. 1) might extend up to 100ps. For example, ESR/STS techniques could in general be used to rotate magnetic moments (see ref. 10, which is relevant for the case discussed in Fig. 2 of our manuscript) or change the magnetic exchange interaction (see refs. 7,8,9, relevant for the case of Fig. 3) while simultaneously measuring the differential conductance dI/dV. The time scale for spin-flip processes in ESR/STS experiments is currently on the order of 20 ns10, but could plausibly be further reduced by factors of 10–100 by increasing local magnetic fields, for example, created by additional magnetic impurities (Lutz, C. P., private communication). Moreover, the quantum state of individual molecules located in a scanning tunneling microscope (STM) cavity can be manipulated using THz irradiation, with the STM measuring the resulting changes using inelastic tunneling spectroscopy42. Similarly, it might be possible to use light-spin interactions to induce spin-oscillations on the ps time scale43 in magnetic atoms located in an STM cavity. Which of these possible experimental approaches will ultimately be successful can only be determined by future work. Finally, we note that in interacting systems (in contrast to the non-interacting system considered here) non-equilibrium perturbations can lead to detrimental heating effects, which are likely more relevant for periodic drives, such as the one considered in Fig. 4, than for the cases discussed in Figs. 1–3. However, it was shown that even for periodic drives, there exists a pre-thermalization regime44,45 in which non-equilibrium quantum states can be formed before they are destroyed by heating effects at larger time scales34. How heating effects and the pre-thermalization regime can be quantum engineered46 in the context of YSR states is an interesting question for future work.

Methods

In order to obtain the differential conductance, we employ Eq. (2) to calculate the current between the tip and the system for a given set of bias voltages and numerically take the derivative with respect to the bias voltage. The details regarding the calculation of the Green’s functions using the Keldysh formalism can be found in Supplementary Note 1. To calculate the non-equilibrium local density of states, we employ Eqs. (4) and (5). The local density of states in the static case is computed by representing the Hamiltonian of Eq. (1) as a matrix in real and Nambu space and using Supplementary Eq. (S31) to calculate the retarded Green’s function matrix \({\hat{g}}^{r}\), from which the LDOS is obtained as \(N({{{{{{{\bf{r}}}}}}}},\sigma ,\omega )=-{{{{{{{\rm{Im}}}}}}}}\left[{\hat{g}}^{r}({{{{{{{\bf{r}}}}}}}},\sigma ;{{{{{{{\bf{r}}}}}}}},\sigma ,\omega )\right]/\pi\). The Floquet LDOS NFl is calculated using the Floquet formalism, which is described in Supplementary Note 3.