Dynamical Cooper pairing in non-equilibrium electron-phonon systems

Michael Knap,1 Mehrtash Babadi,2 Gil Refael,2 Ivar Martin,3 and Eugene Demler4 1Department of Physics, Walter Schottky Institute, and Institute for Advanced Study, Technical University of Munich, 85748 Garching, Germany 2Institute for Quantum Information and Matter, Caltech, Pasadena, CA 91125, USA 3Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA 4Department of Physics, Harvard University, Cambridge MA 02138, USA (Dated: December 16, 2016) We analyze Cooper pairing instabilities in strongly driven electron-phonon systems. The light-induced non- equilibrium state of phonons results in a simultaneous increase of the superconducting coupling constant and the electron scattering. We demonstrate that the competition between these effects leads to an enhanced supercon- ducting transition temperature in a broad range of parameters. Our results may explain the observed transient enhancement of superconductivity in several classes of materials upon irradiation with high intensity pulses of terahertz light, and may pave new ways for engineering high-temperature light-induced superconducting states.

The application of a strong electromagnetic drive has phonons into squeezed quantum states, which can have sig- emerged as a powerful new way to manipulate material prop- nificantly enhanced coupling to electrons, and, finally, the pe- erties [1,2]. Long-range charge density wave order has been riodic modulation of system parameters leading to Floquet melted by light [3–6], insulators have been destroyed [7– states. Of these effects, we find that phonon squeezing uni- 9], and the breaking of superconducting pairs has been ob- versally leads to an enhancement of Tc, by potentially a large served [10–14]. Even more remarkably, long range order can factor [29]. The periodic modulation of the system parame- be not only destroyed but also created by exciting samples ters (Floquet) can also enhance Cooper pairing, via a super- with light. For instance, the dynamic emergence of transient conducting proximity effect in time rather than space. By spin-density wave [15], charge-density wave [16], as well as contrast, the static renormalization of the system parameters superconducting order [17–21] has been demonstrated. These can, depending on material-specific details, either enhance or observations lead to the very fundamental theoretical ques- suppress Tc. tions of the origin of the light induced order, including: What Driving phonons into a highly excited state, unfortunately, is the mechanism for the emergence of transient collective also leads to an increased electron-phonon scattering rate, behavior? What determines the lifetime of transient ordered which weakens Cooper pairing. We analyze the competition states? How robust and universal are the observed phenom- between the enhanced Cooper pair formation and Cooper pair ena? Answers to these questions may hold the key to a breaking and show that the enhancement of pairing can dom- novel route for achieving ordered many-body states by peri- inate in a broad parameter range resulting in signatures of su- odic driving as opposed to cooling; a subject that has attracted perconductivity that appear at higher temperatures compared considerable theoretical attention recently [22–28]. to equilibrium. We note that the predicted enhancement of We propose a general mechanism for making a normal Tc should be understood as a transient phenomenon, since in- conducting metal unstable toward Cooper-pair formation by elastic scattering of electrons with excited phonons will even- irradiation with light. The key ingredient is the nonlinear tually heat the system and destroy the superconducting order. coupling between optically active infrared phonons and the Even though our study is motivated by specific experiments, Raman phonons that mediate electron-electron attraction, re- the minimal models that we introduce and study are intended sponsible for superconductivity (Fig.1). Depending on the to elucidate the qualitative origins of these effects, rather than form of the nonlinearity (Tab.I), several effects can arise. to provide detailed material-specific predictions. These include the parameter renormalization of the time- Furthermore, the proposed mechanism can be readily ap- averaged Hamiltonian, the dynamic excitation of the Raman plied to many other types of long-range order such as charge- density and spin-density waves, by performing the instability analysis in the appropriate channels.

TABLE I. Types of the phonon nonlinearities and their static and arXiv:1511.07874v3 [cond-mat.supr-con] 15 Dec 2016 non-equilibrium effects on the system. The renormalization of the I. ROLE OF THE PHONON NONLINEARITY parameters leads to an effectively static Hamiltonian, while phonon squeezing and the periodic Floquet modulation of system parameters are purely dynamical. We now describe the consequences of different types of static renorm. dynamical periodic optically-accessible phonon nonlinearities. The pump pulse of parameters squeezing Floquet directly couples to infrared-active phonon modes, which have IR 2 R a finite dipole moment. Since the photon momentum is negli- I: (Q0 ) Q0 X × X IR 2 R R gible compared to the reciprocal lattice vector, the drive cre- II: (Q0 ) QkQ−k XXX ates a coherent phonon state at zero momentum, QIR (t) = IR R R q=0 III: Q0 QkQ−k × XX E cos Ωt, where Ω is the drive frequency and E is proportional 2

5 enhancement of Tc >100% pump pulse 1 80% parametric coherently driven resonance infrared phonons 2 60%

Raman modes excited 40% by nonlinear coupling 3 to infrared phonons 20% pump frequency enhancement of SC temporal coupling 4 proximity effect 0% < -10% t pump time driving amplitude A

FIG. 1. Dynamical enhancement of the superconducting transition temperature. A schematic representation of the physical processes leading to light-induced superconductivity: (1) The pump pulse couples coherently to (2) an infrared-active phonon mode which in turn (3) via nonlinear interactions drives Raman phonons that are responsible for superconducting pairing. The non-equilibrium occupation of the Raman phonons (4) universally enhances the superconducting coupling strength γ, which is a product of the density of states at the Fermi level NF and the induced attractive interaction between the electrons U, and hence (5) increases the transition temperature Tc of the superconducting state. We calculate the relative enhancement of Tc compared to equilibrium (Tc − Tc,eq)/Tc,eq by taking into account the competition between dynamical Cooper pair-formation and Cooper pair-breaking processes, as a function of the pump frequency Ω/ω¯ and the driving amplitude A. The data is evaluated for linearly dispersing phonons with mean frequency ω¯, relative spread ∆ω/ω¯ = 0.2, and negative quartic couplings of type II between the Raman and infrared-active modes; Tab.I. Moreover, the electron-phonon interaction strength is chosen to give an equilibrium effective attractive interaction U/W = 1/8 that is weak compared to the bare electronic bandwidth W . The static renormalization of Raman modes leads to the uniform increase of Tc with increasing driving amplitude A, the squeezed phonon state manifests in the strong enhancement near parametric resonance Ω ∼ ω¯, and the temporal proximity effect dominates near Ω/ω¯ ∼ 0. to the drive amplitude. In the presence of phonon nonlinear- II. A MINIMAL ELECTRON-PHONON MODEL ities, the driven infrared-active phonon mode couples to Ra- R man modes Qq of the crystal [18, 22, 30, 31], which in turn We illustrate the aforementioned effects by using a model can couple to the conduction electrons. with Frohlich-type¨ electron-phonon interactions that couple with strength gk to the displacement of the Raman phonons R R Qk (Pk is the conjugate momentum) to the local electron den- sity niσ: There are three leading types of phonon nonlinearities X X Hˆ = −J c† c + (P RP R + ω2QRQR ) which can have static and dynamic effects (Tab.I). A static el-ph 0 iσ jσ k −k k k −k renormalization of the Hamiltonian parameters arises from hiji,σ k IR r phonon nonlinearities that involve even powers of Q , since X 2ωk 0 + g eikri QR n . (1) these terms have finite time averages. For phonon nonlin- V k k iσ earities of type I (Tab.I), this leads to a static displacement ikσ of the zero-momentum Raman phonon while for nonlineari- We consider dispersive optical phonons ωk with mean fre- ties of type II the frequency of the Raman phonons is modi- quency ω¯ and spread ∆ω. Here, J0 is the bare electron hop- fied, cf. App.A. Both affect the superconducting instability, ping matrix element and V the volume of the system. Further, causing either enhancement or suppression; the analysis can we assume a quartic nonlinearitiy of type II. The phonon drive be done with a standard equilibrium Migdal-Eliashberg for- term, introduced for t > 0, reads malism [32]. Here we will focus, however, primarily on the dynamical effects. There are two types of dynamical effects ˆ X 2 R R Hdrv(t) = − ωkAk(1 + cos 2Ωt)Qk Q−k, (2) that can be distinguished: First, a simple modulation of sys- k tem parameters, which makes system instantaneously more 2 2 or less superconducting, and, second, dynamical squeezing of where Ak = −ΛkE /2ωk and E is the amplitude of the driven phonons, which is an explicitly quantum effect. Notably, both infrared-active phonon. dynamical effects lead to an increased superconducting insta- The effect of the drive is twofold. First, there is a static bility temperature. contribution, which renormalizes Raman phonon frequencies. 3

For negative nonlinearities, Λk < 0, phonons are softened c,eq (a) dyn. with pair breaking 2 dyn. \wo pair breaking by ωk(1 − Ak). The mode softening suppresses the electron tunneling, increases the density of states and thus the over- equilibrium all pairing strength. Second, the time dependent part of the drive (2) realizes a parametrically driven oscillator which dy- namically generates phonon squeezing correlations and can strongly suppress the electron tunneling matrix element [33]. T Im[E]/T quasienergy: While the mode softening effect on c depends on the sign of (b) the nonlinearity, the dynamical generation of squeezing corre- c,eq lations universally leads to enhancment of Tc. Effective Hamiltonian.—We will now derive an effective electronic Hamiltonian by integrating out phonons. This is analogous to the standard derivation of the Bardeen-Cooper- Schrieffer (BCS) Hamiltonian starting from the Frohlich¨

model, however, with the phonons being driven strongly out Im[E]/T quasienergy: of equilibrium. As a consequence, the Hamiltonian will be T/T explicitly time-dependent. In addition, it will contain resid- c,eq ual electron-phonon scattering that contribute to the electron FIG. 2. Dynamical Cooper instability. The dynamical Cooper scattering rate. instability evaluated within the Floquet BCS theory including non- We first perform the Lang-Firsov transformation (App.B), equilibrium pair-breaking processes, red solid line, is compared to which eliminates the electron-phonon interaction term in the one without pair breaking, black dotted line, and the BCS solu- Eq. (1) but introduces an effective electron-electron interac- tion for the equilibrium problem, blue dashed line. Data is evaluated tion and dresses the electron tunneling with phonons. This for mean phonon frequency ω/W¯ = 1/8 and effective attractive in- dressing, which depends on the phonon squeezing, suppresses teractions U/W = 1/8 that are weak compared to the bare electronic the electron tunneling and modulates it in time. We (i) take bandwidth W , a phonon frequency spread of ∆ω = 0.2¯ω, driving into account the softening of the Raman modes by a static strength A = 0.1, and driving frequency (a) Ω/ω¯ = 0.001 and (a) renormalization of the phonon coordinates and (ii) treat the Ω/ω¯ = 0.8. The non-equilibrium Cooper pair-formation rate domi- nates over the pair breaking, hence, leading to an enhanced transition dynamic excitation of finite-momentum phonons by trans- temperature. forming the system into a rotating frame. Since we are con- sidering optical phonons, we assume their thermal occupation to be negligible before the drive was switched on. Tracing out effective time-dependent Hamiltonian the phonons, leads to the dressed electron tunneling matrix el- ˆ P † ˜ −ζ X † X ement Hkin → J(t) ijσ ciσcjσ. (iii) We compute the rate H(t) = Jeqe ciσcjσ − U(1 + A cos 2Ωt) ni↑ni↓ of non-equilibrium Cooper pair breaking processes resulting ij i from dynamical excitations of the Fermi sea induced by the + Hˆ , (4) drive and take into account the competition between the en- el-ph scatt. hanced pair formation and pair breaking. This treatment is ˆ where Hel-ph scatt. represent electron-phonon scattering terms justified by showing a posteriori that pair formation domi- that vanish upon tracing out the phonons in the rotating frame. nates pair breaking. The drive, Eq. (2), has thus several effects: (i) a suppression of Solving the full problem numerically we find that J(t) os- the electron tunneling by a factor exp[−ζ], (ii) a dynamic Flo- cillates with 2Ω around its mean value quet contribution from modulating the interaction energy by A cos 2Ωt, and (iii) an enhancement of the electron scattering −ζ hJ(t)i = Jeqe , (3) due to the non-equilibrium phonon occupation. Dynamical Cooper instability.—We study the dynamical which is suppressed by mode softening and phonon squeez- Cooper instability toward pair formation in Hamiltonian (4) ing, as parametrized by ζ. Near parametric resonance the by combining the BCS approach [34] with a Floquet analy- squeezing correlations increase in time leading to a decrease sis [35]. We also take into account the finite electron life- in J(t) and an increase in the amplitude of its oscillations. time τ due to the non-equilibrium phonon occupation, by in- Since we are studying transient phenomena, we compute the troducing an imaginary self-energy correction i/τ [calculated average over the first ten driving cycles to extract the effec- in (App.B) using Floquet Fermi’s Golden rule]. In contrast tive electron tunneling. We take the electron-phonon cou- to elastic scattering on a time-reversal symmetry preserving 2 pling gk such that gk/ωk = const., which leads to a local potential, where Anderson’s theorem [36, 37] shows that the Hubbard-type electron-electron interaction U. This assump- thermodynamics of a superconductor remains unchanged, the tion is reasonable for phonons with wavevectors above the inelastic scattering processes arising here indeed alter the su- Thomas-Fermi screening length [32]. We emphasize however perconducting properties. that this assumption is not crucial for our analysis. For tech- To carry out the BCS Floquet analysis of the pairing insta- nical convenience, we transform the oscillatory part from the bility, we use the equations of motion technique [38]. We kinetic to the interaction term by rescaling time, yielding the introduce infinitesimal pairing amplitudes ak = hck↑c−k↓i 4 and determine whether the system is stable or unstable to the oscillations in the interaction. Then we solve the Floquet the growth of ak [39, 40] by finding the eigenfequencies of BCS problem. In Fig.2 we show the dynamical Cooper insta- the corresponding collective mode. We decompose ak(t) into bility including pair-breaking processes for driving Ω/ω¯ = −iEt P∞ 2inΩt Floquet modes ak(t) ∼ e n=−∞ akne , where E {0.001, 0.8} and Ak = A = 0.1 constant for all k, red is the energy which has to be determined self-consistently solid line, and compare them to the instability without pair from the Floquet BCS gap equation (App.C) breaking, black dotted line, and the equilibrium solution, blue dashed line. A 0 = (U −1 + F )∆ + F (∆ + ∆ ) (5a) n n 2 n n−1 n+1 The reduction of the superconducting transition tempera- 1 X 1 − 2nk ture due to the finite electron lifetime is of the Abrikosov- F = . (5b) n Gorkov form Tc = Tc,0 − 1/τ, where Tc is the dynamical V E + 2nΩ − 2(k + i/τ − µ − Uρ) k transition temperature including pair-breaking and Tc,0 the The instability of the system manifests in the appearance of one without pair-breaking. We find an enhancement of the an eigenmode with negative imaginary part of E which we dynamical Tc compared to equilibrium, Fig.2, and hence that obtain by searching for the zeros of the determinant of (5) in the pair-formation rate dominates the pair-breaking rate. In U P ∗ the high-frequency limit Ω/ω¯ = 0.8, (b), the enhancement the complex plane of E [39, 40]. Here, ∆n = V k akn are the Floquet harmonics of the gap, ρ is the electron density results mainly from the efficient suppression of the electron of a single spin-component, µ the chemical potential, nk the tunneling near parametric resonance Ω ∼ ω¯ and in the slow Fermi-Dirac distribution of the electrons determined by the drive limit Ω → 0, (a), from a combination of mode softening temperature of the undriven system, and k the electron dis- and the temporal proximity effect. The relative enhancement persion. The assumed thermal state of the electrons is justified of Tc, taking into account pair-breaking processes, is shown in the weak-coupling regime and since we are only interested in Fig.1 for an extended range of drive amplitudes A and fre- in transient effects that follow the drive [41]. We determine quencies Ω/ω¯. Softening of the Raman modes leads to the the critical Tc by locating the highest temperature at which uniform increase of Tc with driving amplitude for all values we find an unstable solution of the Floquet BCS gap equations of the pump frequency, while the effect of phonon squeez- (5). Equations with similar structure are obtained for spatially ing is most prominant near parametric resonance Ω ∼ ω¯ and inhomogeneous superconductors [42]. can dominate mode softening by about an order of magnitude. The high and low frequency limits of the Floquet BCS gap As a result, the enhancement of Tc displays an intricate non- equations (5) can be understood from perturbative arguments. monotonic dependence on the driving frequency. In the high frequency limit, Ω → ∞, we use a Magnus expan- sion to derive the stroboscopic Floquet Hamiltonian [35]. To zeroth order in 1/Ω, the Floquet Hamiltonian is given by the time averaged Hamiltonian. Thus for the harmonic drive in Eq. (4), the contribution ∼ A drops out and we obtain an equi- III. EXPERIMENTAL IMPLICATIONS librium BCS gap equation with interaction U and the reduc- tion of the electron tunneling J exp[−ζ] due to the squeezed eq Even though we are not studying a specific material from phonon state. Enhancement of Tc due to the suppression of the effective electron bandwidth has also been suggested in a first principles, let us estimate the typical order of magnitude model without phonons [25]. of the discussed effects for a recent experiment in which tran- In the low frequency limit Ω → 0, the core of the gap equa- sient non-equilibrium superconductivity has been explored in tions (5) barely depends on the Floquet index n leading to K3C60 fullerides [21]. Typical energy scales in K3C60 are [U −1 +F (1+A)]∆ = 0. The maximally enhanced transition the following [43]: bare bandwidth W ∼ 0.6eV, intramolec- ular phonon frequencies ω ∼ 0.03 − 0.2eV, and interaction temperature Tc is thus determined by an equilibrium BCS gap 2 equation with U(1 + A) which is the largest instantaneous at- U = g /ω ∼ 0.1eV. Thus, the parameters we chose are tractive interaction. Hence, in the slow drive limit, the pairing representative for K3C60 . In the experiment [21] the driving induced by the strongest instantaneous interaction dominates frequency Ω ranges from a tenth to a third of the bandwidth. the Cooper pair formation which can be interpreted as a su- Translating to our scenario, the main enhancement of Tc in perconducting proximity effect in time rather than in space. this experiment should result from a combination of the static We verify these perturbative predictions by numerically mode softening and the dynamic phonon squeezing. solving the Floquet BCS gap equations (5) on a two- Our calculations suggest that the increase of Tc is related dimensional square lattice away from half-filling. The conver- to a dynamical enhancement of the effective electron mass, gence of the results with system size and number of Floquet Eq. (3). Such a dynamical renormalization of the electronic bands is checked. We choose linearly dispersive phonons with dispersion can, for instance, be determined experimentally mean frequency ω/W¯ = 1/8, interaction energy U/W = 1/8 by time-resolved ARPES measurements, see e.g. [3, 11– weak compared to the bare electron bandwidth W , and set 13]. Contributions from mode softening and phonon squeez- the width of the phonon branch to ∆ω = 0.2¯ω; a typical ing can, in principle, be experimentally distinguished by the value for optical phonons. We first solve the driven phonon driving-strength dependence of the mass renormalization, that problem from which we determine the average suppression of we predict to be linear in the former and quadratic in the latter the electron tunneling e−ζ and the effective amplitude A of case. 5

IV. SUMMARY AND OUTLOOK ACKNOWLEDGMENTS

We studied the competition between the increased Cooper pair formation and Cooper pair breaking in strongly-driven electron-phonon systems and found that the latter need not to inhibit the dynamic enhancement of Tc. Even though we We thank E. Berg, A. Cavalleri, I. Cirac, U. Eckern, D. focused on superconducting instabilities, the proposed mech- Fausti, A. Georges, S. Gopalakrishnan, B.I. Halperin, A. anism for achieving a larger coupling constant is generic and Imamoglu, S. Kaiser, C. Kollath, M. Norman, T. Shi for use- can be directly applied to other forms of long-range order such ful discussions. We acknowledge support from the Technical as spin or charge density waves. University of Munich - Institute for Advanced Study, funded The present analysis addresses the transient dynamics of by the German Excellence Initiative and the European Union the system. At long times, the inelastic scattering of electrons FP7 under grant agreement 291763, DFG grant No. KN from excited phonons will increase their temperature and will 1254/1-1, Harvard-MIT CUA, NSF grant No. DMR-1308435 tend to destroy superconducting order, as seen in the exper- and DMR-1410435, AFOSR Quantum Simulation MURI, the iments. At intermediate times, heating of the electrons can ARO-MURI on Atomtronics, Humboldt foundation, Dr. Max give rise to nonequilibrium distribution functions which po- Rossler,¨ the Walter Haefner Foundation, the ETH Foundation, tentially enhances superconducting coherence as experimen- the Simons foundation, as well as the Institute for Quantum tally seen by microwave irradiation of superconducting sam- Information and Matter, an NSF Physics Frontiers Center with ples [44–49]. The analysis of complete thermalization due to support of the Gordon and Betty Moore Foundation. This feedback effects in a fully self-consistent and conserving cal- work was supported by the U.S. Department of Energy, Of- culation is an exciting future direction. fice of Science, Materials Sciences and Engineering Division.

Appendix A: Type I nonlinearity: Uniform lattice displacement

IR 2 R R A nonlinear phonon coupling of form I, Λ(Q0 ) Q0 (cf. Tab. 1 in the main text), acts as a classical force on Q0 proportional IR 2 IR 2 2 2 E2 to (Q0 ) [18, 22, 30, 31]. The drive (Q0 ) = E cos Ωt = 2 (1 + cos 2Ωt) can be separated in a static and an oscillatory contribution. We take into account a simple electron-phonon model in which the phonons modulate electron hopping processes R P † † −(J0 + g Q0 ) hiji,σ ciσcjσ (g is the electron-phonon coupling and ciσ the electron creation operator) as for example in a Su-Schrieffer-Heeger (SSH) model [51] X 1 X H = −(J + g QR) c† c + (P RP R + ω2QRQR ) + Λ(QIR)2QR. (A1) SSH 0 0 iσ jσ 2 k −k k k −k 0 0 hiji,σ k IR The infrared phonon is coherently driven Q0 = E cos Ωt. Neglecting the feedback of the electrons on the phonons, we find for the phonon equation of motion ΛE2 Q¨R + ω2QR = (1 + cos 2Ωt). (A2) 0 0 0 2 R ˜ ˜ ΛE2 ΛE2 This equation can be solved analytically with Q0 (t) = Q + δQ cos 2Ωt, where Q = 2 and δQ = 2 2 . The phonon 2ω0 2(ω0 −4Ω ) displacement thus oscillates with twice the pump frequency 2Ω around a mean value Q˜. Plugging this into the SSH Hamiltonian, we find X 1 X H = −[J + g(Q˜ + δQ cos 2Ωt)] c† c + (P RP R + ω2QRQR ). (A3) SSH 0 iσ jσ 2 k −k k k −k hiji,σ k6=0 The finite displacement of the lattice along the coordinates of the Raman mode renormalizes the electron tunneling by a term ∝ gΛE2. The effective electronic bandwidth is reduced when gΛ < 0; a condition that is material specific. Reducing the bandwidth results in a higher density of states and hence an enhanced superconducting transition temperature Tc. For gΛ > 0 contrary is the case. These general considerations may account for the physical origin of the enhancement of the transition temperature which has been seen in ab initio calculations for specific materials [18]. On top of the enhancement due to the time-averaged displacement of the Raman phonon mode, the strong oscillations at 2Ω can give rise to an additional dynamic enhancement of Tc by the temporal proximity effect as we discuss in this work.

Appendix B: Type II and III nonlinearities: Finite-momentum phonon excitations

An alternative form of the nonlinear phonon interaction contains pairs of Raman phonons at finite but opposite momenta. Here, we analyze both the quartic type II and the cubic type III nonlinearities (cf. Tab. 1 in the main text for their classification). 6

(a) >100% (b) >100%

80% 80% parametric parametric resonance resonance 60% 60%

40% 40%

20% 20% pump frequency pump frequency temporal proximity effect 0% 0% < -10% < -10% - - - - - driving amplitude A driving amplitude A

FIG. 3. Phonon squeezing as a universal mechanism for the enhancement of Tc. The relative change of the superconducting transition temperature is shown for (a) quartic type II phonon nonlinearities with positive Λk > 0 and (b) cubic type III phonon nonlinearities, taking into account the competition between Cooper pair formation and pair breaking. (a) Quartic phonon nonlinearities that couple to finite momentum Raman modes have two distinguished effects on the bandwidth renormalization: (i) the static displacement of the Raman modes and (ii) phonon squeezing. For positive nonlinearities, the static displacement hardens the frequency of the Raman phonons. This leads to the uniform suppression of Tc with increasing magnitude of the driving amplitude A. However, near parametric resonance Ω ∼ ω¯, the phonon squeezing dominates over the static displacement, leading to the universal enhancement of Tc. (b) Cubic nonlinearities exhibit only the squeezing mechanism, and hence irrespectively of the sign of the coupling lead to an enhanced transition temperature. Furthermore, the shape of the enhancement diagram is similar to the one for quartic interaction with the main difference that in the case of quartic nonlinearities the resonance condition is Ω ∼ ω¯ while in the cubic case it is Ω ∼ 2¯ω. The data is evaluated for ω/W¯ = 1/8, U/W = 1/8, ∆ω = 0.2¯ω.

Type II: Quartic nonlinearities.—The lowest order nonlinearity in centro-symmetric crystals containing pairs of Raman IR 2 R R phonons is of the quartic form Λk(Q0 ) Qk Q−k (type II). In this case, the coherently driven infrared phonon excites Raman phonons in pairs and with opposite finite momenta, leading to quantum correlations between k and −k modes. These states are referred to as squeezed states in quantum optics [52]. The driven phonon Hamiltonian has the form

ˆ X R R 2 R R X 2 R R ˆ ˆ Hph = (Pk P−k + ωkQk Q−k) − ωkAk(1 + cos 2Ωt)Qk Q−k = Hph,0 + Hdrv., (B1) k k √ 2 2 with Ak = −ΛkE /2ωk, which describes a parametric oscillator with resonance condition Ω = ωk 1 − Ak. The drive can be again separated into a static and an oscillatory contribution with the implications of the static displacement being crucially dependent on the sign of the nonlinearity Λk. For Λk < 0 the frequencies of the Raman modes ωk are softened 2 and hence the effective electron-electron interaction, which is typically of the form gk/ωk, is increased, while for Λk > 0 the opposite is the case. By contrast, we find that irrespective of the sign of the couplings, the oscillatory term squeezes the Raman modes which results in a polaronic suppression of the electron tunneling matrix element compared to equilibrium and hence in an enhanced density of states, Fig.3 (a). Such an enhancement in the density of states in turn increase the superconducting coupling constant, enabling dynamic Cooper pair formation at higher temperatures where equilibrium Cooper pairing would be impossible. Type III: Cubic nonlinearities.—In non-centrosymmetric crystals, the lowest order phonon nonlinearity is of the cubic form IR R R III, Q0 Qk Q−k, as the zero-momentum infrared active phonon mode can linearly couple to the displacement of the Raman modes. Coupling to pairs of Raman modes at opposite momentum gives rise to phonon squeezing that leads to an enhanced Tc. Hence, phonon squeezing is largely insensitive to the microscopic details of the phonon nonlinearities and can occur in a variety of nonlinear couplings. The effective phonon Hamiltonian arising from such a nonlinearity is

ˆ X R R 2 R R Hph = Pk P−k + ωk(1 + Ak cos Ωt)Qk Q−k, (B2) k with Ak = ΛkE, which also describes a parametric oscillator but with resonance condition Ω = 2ωk. In the case of cubic IR interactions, direct mode softening cannot occur because the the time average of Q0 vanishes. Nonetheless, the dynamic generation of squeezing near the parametric resonance is universally present also for cubic nonlinearities and qualitatively similar enhancement diagrams can be also obtained in that case, cf. Fig.3 (b). 7

For studying phonon nonlinearities which generate squeezed states, we consider the Holstein model with Frohlich¨ electron- phonon interactions where the phonon displacement couples to the charge density

1 X † X † X † ikri Hel-ph = −J0 ciσcjσ + ωkbkbk + √ gk(bk + b−k)e niσ. (B3) ijσ k V ik

We expect our results to be insensitive to the fine details of the model and expect for example similar results for the SSH model. We pursue the following strategy:

1. Using a Lang-Firsov transformation we remove the electron-phonon interaction.

2. We determine the unitary transformation W that transforms the system into a rotating frame in which the phonon driving is absent. This leads to a driven Floquet BCS problem.

3. We estimate the dynamic Cooper pair breaking rate due to the non-equilibrium scattering of phonons and due to oscil- lations in the microscopic interaction parameter using Floquet Fermi’s Golden rule. The Cooper pair-breaking rate is self-consistently taken into account by adding an imaginary self-energy correction to the Floquet BCS equations.

4. Solving the driven Floquet BCS problem, we determine the enhancement of Tc relatively to the undriven one.

1. Lang-Firsov transformation

˜ ˜† Considering the Holstein model (B3), we introduce operators bk and b−k which diagonalize the static phonon Hamiltonian P R R 2 R R † k Pk P−k + ωk(1 − Ak)Qk Q−k. These operators are related to the bk, b−k operators which diagonalize Hph,0 e.g. by † 1 † 4 ˜ ˜ (b−k −bk) = (1−Aq) (b−k −bk). The electron-phonon interaction in Eq. (B3) can be removed by a Lang-Firsov transformation 1 g S −S X q iqrj ˜ ˜† Hel-ph → e Hel-phe with S = −√ p e (bq − b−q)njσ. (B4) V qjσ ωq 1 − Aq

Applying it to the Hamiltonian and switching on the drive we obtain

X † X X R R 2 R R H = − Jijciσcjσ + Uijniσnjσ0 + Pk P−k + ωk(1 − Ak − Ak cos 2Ωt)Qk Q−k. (B5) ijσ ijσσ0 k

g ikr † − √1 P k (eikri −e j )(b −b ) V k ω (1−A )3/4 k −k with dressed tunneling matrix element Jij = J0e k k and attractive electron-electron inter- 2 1 P −ik(rj −ri) gk action Uij = − e √ . V k ωk 1−Ak

2. Rotating phonon frame

We construct a unitary transformation to remove the phonon driving following Ref. [53]. We introduce the undriven and driven phonon Hamiltonian, respectively, as

2 † † H0 = PqP−q + ωq QqQ−q = ωq(bqbq + b−qb−q) (B6a) 1 1 H = P P + ω2Ω2Q Q = (Ω2 + 1)ω (b†b + b b† ) + (Ω2 − 1)ω (b†b† + b b ). (B6b) 1 q −q q t q −q 2 t q q q −q −q 2 t q q −q q −q Applying the unitary transformation, we find

d dW d dW i W |ψ i = i |ψ i + W i |ψ i = i |ψ i + WH |ψ i = H W |ψ i (B7) dt 0 dt 0 dt 0 dt 0 0 0 1 0 yielding

dW i W † + WH W † = H . (B8) dt 0 1 8

† Multiplying this equation from left by W and from right by W and identifying H0 = H0,ph and H1 = Hph,0 + Hdrv, we obtain dW W †(Hˆ + Hˆ )W − iW † = Hˆ . (B9) ph,0 drv dt ph,0 Our goal is to construct a mapping of the quantum problem onto a classical Mathieu equation which determines the transfor- mation W [53] uniquely. To this end, we introduce the Ansatz

−2iωq t ∗ 2iωq t W (t) = eξe K+−ξ e K− e−2iK0φ, (B10) where 1 K = (b†b + b b† ),K = b†b† ,K = b b , (B11) 0 2 q q −q −q + q −q − q −q

− + 0 ± ± which obey SU(2) algebra [K ,K ] = 2K0, [K ,K ] = ±K . The time dependent factors in Eq. (B10) are chosen such iωq t ˆ that bqe = bq(t) are invariants of the undriven problem H0, which are defined by requiring that they commute with the ˆ ˆ ˆ corresponding action, i.e., [bq(t), i∂t − H0] = 0. For a given invariant bq(t), bq(t) |ψi remains an eigenstate of H0 provided |ψi ˆ ˆ is an eigenstate, since 0 = [bq(t), i∂t − H0] |ψi = (i∂t − H0)bq(t) |ψi. We furthermore introduce the invariants of the driven problem as

[ˆaq(t), i∂t − H1] = 0, (B12)

† which will be generated by the unitary transformation aˆq(t) = W bq(t)W . By differentiation and subsequent integration we obtain for the invariants

ξe−iωq t aˆ (t) = W b (t)W † = eiφ(cosh |ξ|b eiωq t − sinh |ξ|b† ) ≡ χ(t)b + λ∗(t)b† (B13a) q q q |ξ| −q q −q ξ∗eiωq t aˆ† (t) = W b† (t)W † = e−iφ(cosh |ξ|b† e−iωq t − sinh |ξ|b ) ≡ χ∗(t)b† + λ(t)b . (B13b) −q −q −q |ξ| q −q q

We do not explicitly write the q dependence in χ and λ, as these functions are symmetric in q, i.e., χq = χ−q. Next, we compute the invariants i∂taˆq(t) = [H, aˆq(t)],(B12), using relations (B13) dχ ω ω = i q (Ω2 + 1)χ − i q (Ω2 − 1)λ∗ (B14a) dt 2 t 2 t dλ∗ ω ω = −i q (Ω2 + 1)λ∗ + i q (Ω2 − 1)χ. (B14b) dt 2 t 2 t We transform α = χ − λ∗, β = χ + λ∗ which yields the Mathieu equation

d2α dα + ω2Ω2α = 0 = iω β. (B15) dt2 q t dt q From the initial conditions that require χ(0) = 1, λ(0) = 0, we find

α(0) = 1, α˙ (0) = iωq. (B16)

Using the definition of χ, λ we find the relations between ξ(t), φ(t) from the unitary transformation Eq. (B10) and the parameters of the Mathieu equation

e−iωt iα˙ cosh |ξ|eiφ = (α − ) (B17a) 2 ωq ξ eiωt iα˙ sinh |ξ| eiφ = (α + ). (B17b) |ξ| 2 ωq

2 Plugging in the driving from Eq. (B5), we find Ωt = (1 − Aq − Aq cos 2Ωt). Even though the Mathieu equation can not be solved analytically, for this form of the driving√ its solution is well understood, as it realizes the parametric oscillator, which displays a parametric resonance when Ω = ω 1 − A. On resonance the phonon squeezing |ξ| increases linearly in time while off resonance it oscillates around a mean value, Fig.4. 9

on resonance off resonance phonon squeezing

√ FIG. 4. Mathieu equation. Numerical√ solution of the Mathieu equation at parametric resonance (A = 0.1, Ω = ω 1 − A), blue solid line, and off resonance (A = 0.1, Ω = 0.9ω 1 − A), red dashed line. On resonance the phonon squeezing |ξ| increases linearly in time, while off resonance it oscillates around a finite mean value.

Now we turn again to the electron-phonon problem. Using the unitary transformation W , we remove the driving in Eq. (B5). Following equation Eq. (B9) we have to transform the phonon operators in the dressed kinetic energy according to W † · W

ξe−2iωq t W †b W = cosh |ξ|eiφb + sinh |ξ|e−iφb† (B18a) q q |ξ| −q ξ∗e2iωq t W †b† W = cosh |ξ|e−iφb† + sinh |ξ|eiφb . (B18b) −q −q |ξ| q Assuming, that the drive is adiabatically switched on, the system remains in phonon vacuum in the rotating frame from which we find

g2 1 P q 2 − (2−2 cos q(ri−rj )) |αq | 2V q ω2(1−A )3/2 J(t) = −J0e q k , (B19)

ξ∗e2iωq t iφ where αq = (cosh |ξ| − |ξ| sinh |ξ|)e . Upon time averaging this equation yields equation (3) where we parametrize the relative suppression of the electron tunneling compared to the one at equilibrium by the exponential factor exp[−ζ].

3. Rescaling of time

It is convenient to transform the time evolution from the kinetic energy to the interaction energy. Once, we evaluated the phonon dynamics, Eq. (B19), we obtain the effective electron Hamiltonian of the form

H(t) = J(t)HˆT − UHˆU , (B20) where we used HˆT and HˆU as short hand notation for the kinetic energy and the interaction energy, respectively, and J(t) is the −ζ time dependent hopping matrix element. We introduce J(t) = Jeqe j(t) where j(t) is an oscillating function with mean value one. In order to move the time dependence from the kinetic energy to the interaction energy, we consider Z t Z t Z t0 H(t)dt = H˜ (t) j(t)dt = H˜ (f(t0))dt0 (B21) 0 0 | {z } 0 dt0 where U H˜ (τ) = J e−ζ Hˆ − Hˆ (B22) eq T j(τ) U and Z t0 Z t t0 = dt0 = j(t)dt = f −1(t). (B23) 0 0 10

(a) 3 (b) 3

2 2

1 1

0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 time time

FIG. 5. Relative strength of the oscillations in the microscopic parameters. We show the time dependence of U/J(t) relatively to the undriven system for ω¯ = U = W/8, ∆ω = 0.2¯ω, A = 0.1, (a) off resonance Ω/ω¯ = 0.1 and (b) near the parametric resonance Ω/ω¯ = 0.9 (blue solid line). Initially oscillations at multiple frequencies appear, however, all oscillations except for the driving frequency are washed out in time because for a dispersive phonon the only common frequency of all modes is the driving frequency. Oscillations with Ω are indicated by the red dashed line which we choose to approximate the dynamics of the microscopic parameters. The time average of U/J(t), black dashed line, is enhanced near resonance and with increasing driving strength.

The inverse of this equation cannot be calculated analytically, however, for an oscillating function with amplitude small compared to the mean of j(t) follows that t ∼ t0 which holds because we chose j(t) to oscillate around one. Thus we can directly transform the time dependent part from J(t) to the interaction and arrive at

U H˜ (t) = J e−ζ Hˆ − Hˆ . (B24) eq T j(t) U

As we show in the next section, the oscillations in the interaction can be well described by a single harmonic.

4. Time dependence of the microscopic parameters

For a dispersive phonon with spread ∆ω the time evolution of U/J(t) typically oscillates at driving frequency Ω which is a common oscillation frequency for the phonon modes at all wave vectors. We approximate the oscillations of U/J(t) with a single harmonic of frequency Ω and strength A, see Fig.5. In case of strongly off-resonant driving, the interaction oscillates around its equilibrium value, while near the parametric resonance on average it is enhanced, as a result of the polaronic suppression −ζ −ζ of the bandwidth Jeqe . The mean suppressed bandwidth Jeqe and strength A of the oscillations in U/J(t) are extracted numerically, cf. Fig.6.

5. Floquet Fermi’s Golden Rule

Fluctuations around the initial phonon state and the temporal modulation of the interactions reduce the quasiparticle lifetime which in turn decrease the superconducting transition temperature. Below, we compute the quasiparticle scattering rate from the phonon fluctuations 1/τph and the modulated interactions 1/τint by Floquet Fermi’s Golden Rule. Due to the time dependence of the coupling constants, energy required to create particle-hole excitations can be borrowed from the drive. This enhances the scattering rate compared to equilibrium. The total dynamic Cooper pair breaking rate 1/τ = 1/τph + 1/τint is shown in Fig.7 for ω¯ = U = W/8. For low driving frequencies, the pair-breaking rate is small, since only higher order Floquet harmonics can provide the required energy and the corresponding matrix elements are small. When the driving frequency is near parametric resonance with phonon pair excitations, the effective interactions and hence the decay rate increases since the drive can efficiently provide the required energy to create particle-hole excitations. In broad regime of parameters we find that the enhanced pair formation rate dominates over the enhanced pair breaking rate, c.f. Fig. 1 in the main text. 11

(a) (b) effective amplitude effective enh. of effective mass enh. of effective

driving amplitude A driving amplitude A

pump frequency pump frequency

FIG. 6. Relative enhancement of the effective mass in the driven system compared to the undriven one and the amplitude of coherent oscillations. The data is shown for linearly dispersing phonons ω¯ = U = W/8, and ∆ω = 0.2¯ω. Near Ω/ω¯ ∼ 1 we find a strong enhancement of (a) the effective mass exp[ζ] and (b) the effective amplitude A due to the efficient phonon squeezing. Furthermore, a weaker enhancement is observed at the higher order resonances where Ω/ω¯ ∼ 1/n with n being an integer. In addition to the enhancement of the effective mass near the parametric resonance, it generically increases with increasing driving amplitude A due to the softening of the phonon modes. pair breaking rate pair breaking

driving amplitude A

pump frequency

FIG. 7. Cooper pair breaking rate. The Cooper pair breaking rate due to phonon fluctuations and the modulation of the effective electron- electron interactions evaluated from Floquet Fermi’s golden rule for ω¯ = U = W/8.

The full form of the effective Hamiltonian (4) is

X P ∗ ∗ † P † X ¯ −ζ − k αk(t)Γkbk k αk(t)Γkbk H = −Jeqe e e ciσcjσ − U(1 + A cos 2Ωt) ni↑ni↓, (B25) ijσ i

√1 ikri ikrj gk where Γk = − (e − e ) 3/4 . Taking the phonon vacuum expectation value of Eq. (B25), we obtain the V ωk(1−Ak) first two terms of equation (4). The electron-phonon scattering term, represented by the last term in equation (4) is given by P ∗ ∗ † P † ˆ −ζ P − k αk(t)Γkbk k αk(t)Γkbk Hel-ph scatt. = −Jeqe ijσ(e e − 1)ciσcjσ which as discussed vanishes upon taking the phonon vacuum expectation value. Using a Floquet Fermi’s Golden Rule analysis we estimate the Cooper pair-breaking rate which orig- inates from both (i) phonon fluctuations in the kinetic energy and (ii) modulation of the effective electron-electron interactions.

a. Phonon Fluctuations. Fluctuations around the phonon vacuum require energy. This energy can be borrowed from the ˆ time dependence of the drive. Expanding the exponentials of the electron phonon scattering term Hel-ph scatt. to first order, we 12 obtain

1 X 2gq X Hˆ ∼ −J e−ζ √ [ cos k − cos(k − q )](α (t)b − α∗ (t)b† )c† c el-ph scatt. eq ω (1 − A )3/4 d d d q q −q −q kσ k−qσ V kqσ q q d 1 X = [F α (t)b − F ∗ α∗(t)b† ] c† c (B26) V qk q q qk q −q kσ k−qσ kqσ with X gq F = −2J e−ζ [ cos k − cos(k − q )] . (B27) kq eq d d d ω (1 − A )3/4 d q q Higher order terms in the series expansion of the exponentials correspond to multi-phonon processes which are energetically suppressed. The finite lifetime of quasiparticles at the Fermi energy can be calculated from the imaginary part of the retarded phonon- fluctuation self energy 1 = Im Σph,+. (B28) kF τph To leading order the corresponding greater and lesser components are

ph, i X Σ ≷(t , t ) = = |F |2α (t )α∗(t )G≷ (t , t )D≷(t , t ). (B29) k 1 2 V kq q 1 q 2 k−q 1 2 q 1 2 q

≷ ≷ Here, Gk (t1, t2) is the bare electron and Dq (t1, t2) the bare phonon propagators, which are functions of the time difference as the unperturbed Hamiltonian is static. Thus, in Fourier space they are given by > Gk (ω) ≈ −2πiδ(ω − k + µ)[1 − nk], < Gk (ω) ≈ +2πiδ(ω − k + µ)nk, > Dq (ω) ≈ −iπδ(ω − ωq), < Dq (ω) ≈ −iπδ(ω + ωq), where we considered the phonons to be in the vacuum state and thus neglected their distribution functions in D> and D<. The self-energy contains the drive αq(t) and therefore is a fully non-equilibrium object which is a function of two times, Eq. (B29). 1 We rewrite the self-energy using the average time T = 2 (t1 + t2) and the time difference t = t1 − t2

ph, i X Σ ≷(t, T ) = |F |2α (T + t/2)α∗(T − t/2)G≷ (t)D≷(t). (B30) k V kq q q k−q q q In order to estimate a quasi-particle lifetime, we integrate over the “slow” timescale T which yields for the effective coupling π 2 Ω R Ω ∗ 2 |α¯q(t)| = π 0 dT αq(T + t/2)αq (T − t/2). The effective coupling |α¯q(t)| , distinguished from the bare coupling by the bar, 2 is only a function of the time difference t. From a Fourier transform we obtain its Floquet components |α¯qn| , where n ranges + from −∞ to ∞. We calculate the retarded self-energy Σph,k using Z dω0 Σph,>(ω0) − Σph,<(ω0) Σph,+(ω) = i k k , (B31) k 2π ω − ω0 + i0+ where ω is the conjugate variable to the time difference t, and find for the lifetime at the Fermi surface 1 π X = |F |2|α¯ |2{(1 − n )δ(2nΩ − E − ω) + n δ(2nΩ − E + ω)}. (B32) τ 2V qkF qn kF −q kF −q kF −q kF −q ph qn

In order to obtain a semi-analytical estimate for the pair breaking rate, we neglect the weak wavevector dependence of gq/ωq(1− 3/4 Aq) and replace them by their mean. We replace wavevector summations by integrals over energies with a constant density of states, yielding 1 π X eζ g¯2 = (2nΩ − ω¯)2Θ(2nΩ − ω¯){α Θ(|E + 2nΩ − ω¯| − W ) τ 2 8J ω¯2(1 − A)3/2 n F ph n>0 eq

+ α−nΘ(|EF − 2nΩ +ω ¯| − W )}, (B33) −ζ where W = 4Jeqe is half of the electronic bandwidth. 13

b. Modulated interactions. The temporal modulation of the effective electron-electron interaction leads to another source for decreasing the quasiparticle lifetime. Similarly as in the case of phonon fluctuations, we estimate the interaction decay rate by computing the imaginary part of the leading order self-energy contribution

int, 1 X Σ ≷(t , t ) = = U 2(1 + A cos 2Ωt )(1 + A cos 2Ωt )× k 1 2 V 2 1 2 ql

≷ ≷ ≶ Gk+l−q(t1, t2)Gq (t1, t2)Gl (t2, t1). (B34)

Performing a Floquet Fermi’s Golden Rule analysis, we first integrate over the “slow” timescale T to obtain an effective coupling as a function of the time difference t and then compute the Floquet components of the coupling. Plugging this into the expression for the retarded self-energy and taking the imaginary part, we obtain the decay rate

1 U 2A2 X = 2 [(1 − nkF +l−q) (1 − nq) nl + nkF +l−q nq (1 − nl)] τint 4V ql

[δ(2Ω − (EkF +q−l + Eq − El) + δ(2Ω + (EkF +q−l + Eq − El)]. (B35)

We evaluate both quasiparticle decay rates numerically, cf. Fig.7. The total pair-breaking rate 1/τ = 1/τph + 1/τint, which we consider as an imaginary self-energy correction in the Floquet BCS equations, is in a wide parameter range much smaller than the Cooper pair formation rate and thus only slightly shifts the transition temperature to lower values.

Appendix C: Floquet BCS approach

† † We first evaluate the equations of motion for ck↑c−k↓ from the rescaled Hamiltonian equation (4) taking into account the Cooper pair breaking rate 1/τ, computed in Sec.B5, as an imaginary self-energy correction

d U(1 + A cos 2Ωt) X c† c† = 2i( + i/τ − µ)c† c† − i c† c† (c c† − c† c ) (C1) dt k↑ −k↓ k k↑ −k↓ V m↑ q↓ m+q−k↓ −k↓ k↑ m+q+k↑ mq and factorize the quartic term using a mean-field decoupling

d hc† c† i = 2i( + i/τ − µ)hc† c† i dt k↑ −k↓ k k↑ −k↓ U(1 + A cos 2Ωt) X − i hc† c† i(n + n ) + hc† c† i(1 − n − n ). (C2) V k↑ −k↓ q↑ q↓ q↑ −q↓ k↑ k↓ q

1 P Using spin symmetry and defining 2ρ = V q(nq↑ + nq↓) we obtain

d 1 X hc† c† i = 2i( + i/τ − µ − U(1 + A cos 2Ωt)ρ)hc† c† i − iU(1 + A cos 2Ωt)(1 − 2n ) hc† c† i. (C3) dt k↑ −k↓ k k↑ −k↓ k V q↑ −q↓ q

† † Next, we remove the term 2iUρA cos 2Ωthckc−ki by an appropriate unitary transformation of the form UρA hc† c† i = a∗ exp[−i sin 2Ωt] (C4) k↑ −k↓ k Ω which gives:

d 1 X a∗ = 2i( + i/τ − µ − Uρ)a∗ − iU(1 + cos 2Ωt)(1 − 2n ) a∗. (C5) dt k k k k V q q

Using the Floquet Ansatz

∗ iEt X ∗ i2nΩt ak(t) = e akne (C6) n 14 we obtain " # U X A X [E + 2nΩ − 2( + i/τ − µ − Uρ)]a∗ = −(1 − 2n ) a∗ + (a∗ + a∗ ) . (C7) k kn k V qn 2 qn+1 qn−1 q q

Dividing by E + 2nΩ − 2(k + i/τ − µ − Uρ), and summing over k, we find the Floquet BCS gap equation: " # 1 X ∗ 1 X 1 − 2nk U X ∗ A X ∗ ∗ akn = − aqn + (aqn+1 + aqn−1) . (C8) V V E + 2nΩ − 2(k + i/τ − µ − Uρ) V 2 k k q q | {z } =Fn U P ∗ Defining the gap ∆n = V k akn, we obtain the simple system of equations A (U −1 + F )∆ + F (∆ + ∆ ) = 0. (C9) n n 2 n n−1 n+1 The fact that we used a single harmonic to describe the time evolution of U(t) reflects in the gap equation having only a single side band. More complicated functions would lead to further side bands which would give quantitative differences but our conclusions will not be altered on the qualitative level. The BCS Floquet equations have a nontrivial solution, when the determinant is zero, which we determine by scanning E in the complex plane. The Cooper pair formation rate is characterized by the imaginary part of E.

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