PHYSICAL REVIEW B 94, 214504 (2016)

Dynamical Cooper pairing in nonequilibrium 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, California 91125, USA3Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA

4Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA(Received 20 July 2016; revised manuscript received 10 October 2016; published 8 December 2016)

We analyze Cooper pairing instabilities in strongly driven electron-phonon systems. The light-inducednonequilibrium state of phonons results in a simultaneous increase of the superconducting coupling constantand the electron scattering. We demonstrate that the competition between these effects leads to an enhancedsuperconducting transition temperature in a broad range of parameters. Our results may explain the observedtransient enhancement of superconductivity in several classes of materials upon irradiation with high intensitypulses of terahertz light, and may pave new ways for engineering high-temperature light-induced superconductingstates.

DOI: 10.1103/PhysRevB.94.214504

I. INTRODUCTION

The application of a strong electromagnetic drive hasemerged as a powerful new way to manipulate materialproperties [1,2]. Long-range charge-density wave order hasbeen melted by light [3–6], insulators have been destroyed[7–9], and the breaking of superconducting pairs has beenobserved [10–14]. Even more remarkably, long-range ordercan be not only destroyed but also created by exciting sampleswith light. For instance, the dynamic emergence of transientspin-density wave [15], charge-density wave [16], as well assuperconducting order [17–21] has been demonstrated. Theseobservations lead to the very fundamental theoretical questionsof the origin of the light induced order, including: What is themechanism for the emergence of transient collective behavior?What determines the lifetime of transient ordered states? Howrobust and universal are the observed phenomena? Answers tothese questions may hold the key to a novel route for achievingordered many-body states by periodic driving as opposed tocooling, a subject that has attracted considerable theoreticalattention recently [22–28].

We propose a general mechanism for making a normalconducting metal unstable toward Cooper-pair formation byirradiation with light. The key ingredient is the nonlinearcoupling between optically active infrared phonons and theRaman phonons that mediate electron-electron attraction,responsible for superconductivity (Fig. 1). Depending onthe form of the nonlinearity (Table I), several effects canarise. These include the parameter renormalization of thetime-averaged Hamiltonian, the dynamic excitation of theRaman phonons into squeezed quantum states, which canhave significantly enhanced coupling to electrons, and, finally,the periodic modulation of system parameters leading toFloquet states. Of these effects, we find that phonon squeezinguniversally leads to an enhancement of Tc, by potentiallya large factor [29]. The periodic modulation of the systemparameters (Floquet) can also enhance Cooper pairing, via asuperconducting proximity effect in time rather than space. Bycontrast, the static renormalization of the system parameterscan, depending on material-specific details, either enhance orsuppress Tc.

Driving phonons into a highly excited state, unfortunately,also leads to an increased electron-phonon scattering rate,which weakens Cooper pairing. We analyze the competitionbetween the enhanced Cooper pair formation and Cooperpair breaking and show that the enhancement of pairing candominate in a broad parameter range resulting in signatures ofsuperconductivity that appear at higher temperatures comparedto equilibrium. We note that the predicted enhancement of Tc

should be understood as a transient phenomenon, since inelas-tic scattering of electrons with excited phonons will eventuallyheat the system and destroy the superconducting order. Eventhough our study is motivated by specific experiments, theminimal models that we introduce and study are intended toelucidate the qualitative origins of these effects, rather than toprovide detailed material-specific predictions.

Furthermore, the proposed mechanism can be readilyapplied to many other types of long-range order such ascharge-density and spin-density waves, by performing theinstability analysis in the appropriate channels.

II. ROLE OF THE PHONON NONLINEARITY

We now describe the consequences of different types ofoptically accessible phonon nonlinearities. The pump pulsedirectly couples to infrared-active phonon modes, whichhave a finite dipole moment. Since the photon momentumis negligible compared to the reciprocal-lattice vector, thedrive creates a coherent phonon state at zero momentum,QIR

q=0(t) = E cos �t , where � is the drive frequency and E isproportional to the drive amplitude. In the presence of phononnonlinearities, the driven infrared-active phonon mode couplesto Raman modes QR

q of the crystal [18,22,30,31], which in turncan couple to the conduction electrons.

There are three leading types of phonon nonlinearitieswhich can have static and dynamic effects (Table I). A staticrenormalization of the Hamiltonian parameters arises fromphonon nonlinearities that involve even powers of QIR

0 , sincethese terms have finite time averages. For phonon nonlinear-ities of type I (Table I), this leads to a static displacement ofthe zero-momentum Raman phonon while for nonlinearities

2469-9950/2016/94(21)/214504(13) 214504-1 ©2016 American Physical Society

KNAP, BABADI, REFAEL, MARTIN, AND DEMLER PHYSICAL REVIEW B 94, 214504 (2016)

enhancement of Tc

pump pulse 1

coherently driveninfrared phonons 2

Raman modes excitedby nonlinear couplingto infrared phonons

3

enhancement of SCcoupling 4

5

0%

40%

>100%

20%

60%

80%

< -10%

parametricresonance

driving amplitude A

pum

p fr

eque

ncy

temporalproximity effect

timetpump

FIG. 1. Dynamical enhancement of the superconducting transition temperature. A schematic representation of the physical processesleading to light-induced superconductivity: (1) The pump pulse couples coherently to (2) an infrared-active phonon mode which in turn (3) vianonlinear interactions drives Raman phonons that are responsible for superconducting pairing. The nonequilibrium occupation of the Ramanphonons (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 superconductingstate. We calculate the relative enhancement of Tc compared to equilibrium (Tc − Tc,eq)/Tc,eq by taking into account the competition betweendynamical Cooper pair-formation and Cooper pair breaking processes, as a function of the pump frequency �/ω and the driving amplitude A.The data are evaluated for linearly dispersing phonons with mean frequency ω, relative spread �ω/ω = 0.2, and negative quartic couplingsof type II between the Raman and infrared-active modes (Table I). Moreover, the electron-phonon interaction strength is chosen to give anequilibrium effective attractive interaction U/W = 1/8 that is weak compared to the bare electronic bandwidth W . The static renormalizationof Raman modes leads to the uniform increase of Tc with increasing driving amplitude A, the squeezed phonon state manifests in the strongenhancement near parametric resonance � ∼ ω, and the temporal proximity effect dominates near �/ω ∼ 0.

of type II the frequency of the Raman phonons is modified,see Appendix A. Both affect the superconducting instability,causing either enhancement or suppression; the analysiscan be done with a standard equilibrium Migdal-Eliashbergformalism [32]. Here we will focus, however, primarily on thedynamical effects. There are two types of dynamical effectsthat can be distinguished: First, a simple modulation of systemparameters, which makes the system instantaneously moreor less superconducting, and, second, dynamical squeezingof phonons, which is an explicitly quantum effect. Notably,both dynamical effects lead to an increased superconductinginstability temperature.

TABLE I. Types of the phonon nonlinearities and their static andnonequilibrium effects on the system. The renormalization of theparameters leads to an effectively static Hamiltonian, while phononsqueezing and the periodic Floquet modulation of system parametersare purely dynamical.

Static renormalization Dynamical Periodicof parameters squeezing Floquet

I:(QIR

0

)2QR

0 � × �II:

(QIR

0

)2QR

k QR−k � � �

III: QIR0 QR

k QR−k × � �

III. A MINIMAL ELECTRON-PHONON MODEL

We illustrate the aforementioned effects by using a modelwith Frohlich-type electron-phonon interactions that couplewith strength gk to the displacement of the Raman phononsQR

k (P Rk is the conjugate momentum) to the local electron

density niσ :

Hel-ph = −J0

∑〈ij〉,σ

c†iσ cjσ +

∑k

(P R

k P R−k + ω2

kQRk QR

−k

)

+∑ikσ

√2ωk

Vgke

ikri QRk niσ . (1)

We consider dispersive optical phonons ωk with mean fre-quency ω and spread �ω. Here, J0 is the bare electron hoppingmatrix element and V is the volume of the system. Further,we assume a quartic nonlinearity of type II. The phonon driveterm, introduced for t > 0, reads

Hdrv(t) = −∑

k

ω2kAk(1 + cos 2�t)QR

k QR−k, (2)

where Ak = −�kE2/2ω2k and E is the amplitude of the driven

infrared-active phonon.The effect of the drive is twofold. First, there is a static

contribution, which renormalizes Raman phonon frequencies.For negative nonlinearities, �k < 0, phonons are softenedby ω2

k(1 − Ak). The mode softening suppresses the electron

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DYNAMICAL COOPER PAIRING IN NONEQUILIBRIUM . . . PHYSICAL REVIEW B 94, 214504 (2016)

tunneling and increases the density of states and thus theoverall pairing strength. Second, the time-dependent part of thedrive [Eq. (2)] realizes a parametrically driven oscillator whichdynamically generates phonon squeezing correlations and canstrongly suppress the electron tunneling matrix element [33].While the mode softening effect on Tc depends on the signof the nonlinearity, the dynamical generation of squeezingcorrelations universally leads to enhancement of Tc.

A. Effective Hamiltonian

We will now derive an effective electronic Hamiltonian byintegrating out phonons. This is analogous to the standardderivation of the Bardeen-Cooper-Schrieffer (BCS) Hamil-tonian starting from the Frohlich model, however, with thephonons being driven strongly out of equilibrium. As a con-sequence, the Hamiltonian will be explicitly time dependent.In addition, it will contain residual electron-phonon scatteringthat contributes to the electron scattering rate.

We first perform the Lang-Firsov transformation(Appendix B), which eliminates the electron-phonon inter-action term in Eq. (1) but introduces an effective electron-electron interaction and dresses the electron tunneling withphonons. This dressing, which depends on the phonon squeez-ing, suppresses the electron tunneling and modulates it in time.We (i) take into account the softening of the Raman modesby a static renormalization of the phonon coordinates and(ii) treat the dynamic excitation of finite-momentum phononsby transforming the system into a rotating frame. Since weare considering optical phonons, we assume their thermaloccupation to be negligible before the drive was switched on.Tracing out the phonons leads to the dressed electron tunnelingmatrix element Hkin → J (t)

∑ijσ c

†iσ cjσ . (iii) We compute

the rate of nonequilibrium Cooper pair breaking processesresulting from dynamical excitations of the Fermi sea inducedby the drive and take into account the competition between theenhanced pair formation and pair breaking. This treatment isjustified by showing a posteriori that pair formation dominatespair breaking.

Solving the full problem numerically we find that J (t)oscillates with 2� around its mean value

〈J (t)〉 = Jeqe−ζ , (3)

which is suppressed by mode softening and phonon squeezing,as parametrized by ζ . Near parametric resonance the squeezingcorrelations increase in time leading to a decrease in J (t) andan increase in the amplitude of its oscillations. Since we arestudying transient phenomena, we compute the average overthe first ten driving cycles to extract the effective electrontunneling. We take the electron-phonon coupling gk such thatg2

k/ωk = const, which leads to a local Hubbard-type electron-electron interaction U . This assumption is reasonable forphonons with wave vectors above the Thomas-Fermi screeninglength [32]. We emphasize, however, that this assumption isnot crucial for our analysis. For technical convenience, wetransform the oscillatory part from the kinetic to the interactionterm by rescaling time, yielding the effective time-dependent

Hamiltonian

H (t) = Jeqe−ζ

∑ij

c†iσ cjσ − U (1 + A cos 2�t)

∑i

ni↑ni↓

+ Hel-ph scatt., (4)

where Hel-ph scatt. represent electron-phonon scattering termsthat vanish upon tracing out the phonons in the rotatingframe. The drive, Eq. (2), has thus several effects: (i) asuppression of the electron tunneling by a factor exp[−ζ ], (ii) adynamic Floquet contribution from modulating the interactionenergy by A cos 2�t , and (iii) an enhancement of the electronscattering due to the nonequilibrium phonon occupation.

B. Dynamical Cooper instability

We study the dynamical Cooper instability toward pairformation in Hamiltonian (4) by combining the BCS approach[34] with a Floquet analysis [35]. We also take into accountthe finite electron lifetime τ due to the nonequilibrium phononoccupation, by introducing an imaginary self-energy correc-tion i/τ (calculated in Appendix B using Floquet Fermi’sGolden rule). In contrast to elastic scattering on a time-reversalsymmetry preserving potential, where Anderson’s theorem[36,37] shows that the thermodynamics of a superconductorremains unchanged, the inelastic scattering processes arisinghere indeed alter the superconducting properties.

To carry out the BCS Floquet analysis of the pairinginstability, we use the equations of motion technique [38].We introduce infinitesimal pairing amplitudes ak = 〈ck↑c−k↓〉and determine whether the system is stable or unstable tothe growth of ak [39,40] by finding the eigenfrequencies ofthe corresponding collective mode. We decompose ak(t) intoFloquet modes ak(t) ∼ e−iEt

∑∞n=−∞ akne

2in�t , where E is theenergy which has to be determined self-consistently from theFloquet BCS gap equation (Appendix C):

0 = (U−1 + Fn)�n + A2

Fn(�n−1 + �n+1), (5a)

Fn = 1

V

∑k

1 − 2nk

E + 2n� − 2(εk + i/τ − μ − Uρ). (5b)

The instability of the system manifests in the appearanceof an eigenmode with negative imaginary part of E which weobtain by searching for the zeros of the determinant of Eq. (5)in the complex plane of E [39,40]. Here, �n = U

V

∑k a∗

kn arethe Floquet harmonics of the gap, ρ is the electron densityof a single spin component, μ is the chemical potential, nk

is the Fermi-Dirac distribution of the electrons determinedby the temperature of the undriven system, and εk is theelectron dispersion. The assumed thermal state of the electronsis justified in the weak-coupling regime and since we are onlyinterested in transient effects that follow the drive [41]. Wedetermine the critical Tc by locating the highest temperatureat which we find an unstable solution of the Floquet BCS gapequation (5). Equations with similar structure are obtained forspatially inhomogeneous superconductors [42].

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KNAP, BABADI, REFAEL, MARTIN, AND DEMLER PHYSICAL REVIEW B 94, 214504 (2016)

The high- and low-frequency limits of the Floquet BCS gapequation (5) can be understood from perturbative arguments. Inthe high-frequency limit, � → ∞, we use a Magnus expansionto derive the stroboscopic Floquet Hamiltonian [35]. To zerothorder in 1/�, 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 equilibriumBCS gap equation with interaction U and the reduction of theelectron tunneling Jeq exp[−ζ ] due to the squeezed phononstate. Enhancement of Tc due to the suppression of the effectiveelectron bandwidth has also been suggested in a model withoutphonons [25].

In the low-frequency limit � → 0, the core of the gapequations (5) barely depends on the Floquet index n leading to[U−1 + F (1 + A)]� = 0. The maximally enhanced transitiontemperature Tc is thus determined by an equilibrium BCS gapequation with U (1 + A) which is the largest instantaneousattractive interaction. Hence, in the slow drive limit, the pairinginduced by the strongest instantaneous interaction dominatesthe Cooper pair formation which can be interpreted as asuperconducting proximity effect in time rather than in space.

We verify these perturbative predictions by numericallysolving the Floquet BCS gap equation (5) on a two-dimensional square lattice away from half filling. The conver-gence of the results with system size and number of Floquetbands is checked. We choose linearly dispersive phonons withmean frequency ω/W = 1/8, interaction energy U/W = 1/8weak compared to the bare electron bandwidth W , and setthe width of the phonon branch to �ω = 0.2ω, a typicalvalue for optical phonons. We first solve the driven phononproblem from which we determine the average suppressionof the electron tunneling e−ζ and the effective amplitudeA of the oscillations in the interaction. Then we solve theFloquet BCS problem. In Fig. 2 we show the dynamicalCooper instability including pair breaking processes fordriving �/ω = {0.001, 0.8} and Ak = A = 0.1 constant forall k, red solid line, and compare them to the instability withoutpair breaking, black dotted line, and the equilibrium solution,blue dashed line.

The reduction of the superconducting transition tempera-ture due to the finite electron lifetime is of the Abrikosov-Gorkov form Tc = Tc,0 − 1/τ , where Tc is the dynamicaltransition temperature including pair breaking and Tc,0 is theone without pair breaking. We find an enhancement of thedynamical Tc compared to equilibrium, Fig. 2, and hence thatthe pair-formation rate dominates the pair breaking rate. In thehigh-frequency limit �/ω = 0.8, (b), the enhancement resultsmainly from the efficient suppression of the electron tunnelingnear parametric resonance � ∼ ω and in the slow drive limit� → 0, (a), from a combination of mode softening and thetemporal proximity effect. The relative enhancement of Tc,taking into account pair breaking processes, is shown in Fig. 1for an extended range of drive amplitudes A and frequencies�/ω. Softening of the Raman modes leads to the uniformincrease of Tc with driving amplitude for all values of thepump frequency, while the effect of phonon squeezing is mostprominent near parametric resonance � ∼ ω and can dominatemode softening by about an order of magnitude. As a result,the enhancement of Tc displays an intricate nonmonotonicdependence on the driving frequency.

(a)

quas

iene

rgy:

Im

[E]/

Tc,

eq

T/Tc,eq

(b)

equilibrium

dyn. with pair breakingdyn. w\o pair breaking

quas

iene

rgy:

Im

[E]/

Tc,

eq

FIG. 2. Dynamical Cooper instability. The dynamical Cooperinstability evaluated within the Floquet BCS theory includingnonequilibrium pair breaking processes, red solid line, is comparedto the one without pair breaking, black dotted line, and the BCSsolution for the equilibrium problem, blue dashed line. Data areevaluated for mean phonon frequency ω/W = 1/8 and effectiveattractive interactions U/W = 1/8 that are weak compared to the bareelectronic bandwidth W , a phonon frequency spread of �ω = 0.2ω,driving strength A = 0.1, and driving frequency (a) �/ω = 0.001and (b) �/ω = 0.8. The nonequilibrium Cooper pair-formation ratedominates over the pair breaking, hence, leading to an enhancedtransition temperature.

IV. EXPERIMENTAL IMPLICATIONS

Even though we are not studying a specific materialfrom first principles, let us estimate the typical order ofmagnitude of the discussed effects for a recent experimentin which transient nonequilibrium superconductivity has beenexplored in K3C60 fullerides [21]. Typical energy scales inK3C60 are the following [43]: bare bandwidth W ∼ 0.6 eV,intramolecular phonon frequencies ω ∼ 0.03 − 0.2 eV, andinteraction U = g2/ω ∼ 0.1 eV. Thus, the parameters wechose are representative for K3C60 . In the experiment [21]the driving frequency � ranges from a tenth to a third of thebandwidth. Translating to our scenario, the main enhancementof Tc in this experiment should result from a combination ofthe static mode softening and the dynamic phonon squeezing.

Our calculations suggest that the increase of Tc is relatedto a dynamical enhancement of the effective electron mass,Eq. (3). Such a dynamical renormalization of the electronicdispersion can, for instance, be determined experimentallyby time-resolved ARPES measurements; see, e.g., [3,11–13].Contributions from mode softening and phonon squeezingcan, in principle, be experimentally distinguished by thedriving-strength dependence of the mass renormalization, thatwe predict to be linear in the former and quadratic in the lattercase.

V. SUMMARY AND OUTLOOK

We studied the competition between the increased Cooperpair formation and Cooper pair breaking in strongly driven

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DYNAMICAL COOPER PAIRING IN NONEQUILIBRIUM . . . PHYSICAL REVIEW B 94, 214504 (2016)

electron-phonon systems and found that the latter neednot inhibit the dynamic enhancement of Tc. Even thoughwe focused on superconducting instabilities, the proposedmechanism for achieving a larger coupling constant is genericand can be directly applied to other forms of long-range ordersuch as spin- or charge-density waves.

The present analysis addresses the transient dynamics of thesystem. At long times, the inelastic scattering of electrons fromexcited phonons will increase their temperature and will tendto destroy superconducting order, as seen in the experiments.At intermediate times, heating of the electrons can give riseto nonequilibrium distribution functions which potentiallyenhances superconducting coherence as experimentally seenby microwave irradiation of superconducting samples [44–49].The analysis of complete thermalization due to feedbackeffects in a fully self-consistent and conserving calculationis an exciting future direction.

ACKNOWLEDGMENTS

We thank E. Berg, A. Cavalleri, I. Cirac, U. Eckern,D. Fausti, A. Georges, S. Gopalakrishnan, B. I. Halperin,A. Imamoglu, S. Kaiser, C. Kollath, M. Norman, and T. Shi foruseful discussions. We acknowledge support from the Tech-nical University of Munich—Institute for Advanced Study,funded by the German Excellence Initiative and the EuropeanUnion FP7 under Grant No. 291763, Deutsche Forschungsge-meinschaft Grant No. KN 1254/1-1, Harvard-MIT CUA, NSFGrants No. DMR-1308435 and No. DMR-1410435, AFOSRQuantum Simulation MURI, the ARO-MURI on Atomtronics,Humboldt foundation, Dr. Max Rossler, the Walter HaefnerFoundation, the ETH Foundation, the Simons foundation, aswell as the Institute for Quantum Information and Matter, anNSF Physics Frontiers Center with support of the Gordon andBetty Moore Foundation. This work was supported by the USDepartment of Energy, Office of Science, Materials Sciencesand Engineering Division.

APPENDIX A: TYPE I NONLINEARITY—UNIFORMLATTICE DISPLACEMENT

A nonlinear phonon coupling of form I, �(QIR0 )2QR

0 (seeTable I in the main text), acts as a classical force on QR

0proportional to (QIR

0 )2 [18,22,30,31]. The drive (QIR0 )2 =

E2 cos2 �t = E2

2 (1 + cos 2�t) can be separated in a staticand an oscillatory contribution. We take into account asimple electron-phonon model in which the phonons modulateelectron hopping processes −(J0 + g QR

0 )∑

〈ij〉,σ c†iσ cjσ (g is

the electron-phonon coupling and c†iσ is the electron creation

operator) as, for example, in a Su-Schrieffer-Heeger (SSH)model [51]:

HSSH =−(J0 + g QR

0

) ∑〈ij〉,σ

c†iσ cjσ

+ 1

2

∑k

(P R

k P R−k + ω2

kQRk QR

−k

) + �(QIR

0

)2QR

0 .

(A1)

The infrared phonon is coherently driven QIR0 = E cos �t .

Neglecting the feedback of the electrons on the phonons, wefind for the phonon equation of motion

QR0 + ω2

0QR0 = �E2

2(1 + cos 2�t). (A2)

This equation can be solved analytically with QR0 (t) = Q +

δQ cos 2�t , where Q = �E2

2ω20

and δQ = �E2

2(ω20−4�2)

. The phonon

displacement thus oscillates with twice the pump frequency2� around a mean value Q. Plugging this into the SSHHamiltonian, we find

HSSH =−[J0 + g(Q + δQ cos 2�t)]∑〈ij〉,σ

c†iσ cjσ

+ 1

2

∑k =0

(P R

k P R−k + ω2

kQRk QR

−k

). (A3)

The finite displacement of the lattice along the coordinatesof the Raman mode renormalizes the electron tunnelingby a term ∝g�E2. The effective electronic bandwidth isreduced when g� < 0, a condition that is material specific.Reducing the bandwidth results in a higher density of statesand hence an enhanced superconducting transition temperatureTc. For g� > 0 the contrary is the case. These generalconsiderations may account for the physical origin of the en-hancement of the transition temperature which has been seen inab initio calculations for specific materials [18]. On top ofthe enhancement due to the time-averaged displacement ofthe Raman phonon mode, the strong oscillations at 2� cangive rise to an additional dynamic enhancement of Tc by thetemporal proximity effect as we discuss in this work.

APPENDIX B: TYPE II AND IIINONLINEARITIES—FINITE-MOMENTUM

PHONON EXCITATIONS

An alternative form of the nonlinear phonon interactioncontains pairs of Raman phonons at finite but oppositemomenta. Here, we analyze both the quartic type II and thecubic type III nonlinearities (see Table I in the main text fortheir classification).

(i) Type II: Quartic nonlinearities. The lowest-order nonlin-earity in centrosymmetric crystals containing pairs of Ramanphonons is of the quartic form �k(QIR

0 )2QRk QR

−k (type II). Inthis case, the coherently driven infrared phonon excites Ramanphonons in pairs and with opposite finite momenta, leading toquantum correlations between k and −k modes. These statesare referred to as squeezed states in quantum optics [52]. Thedriven phonon Hamiltonian has the form

Hph =∑

k

(P R

k P R−k + ω2

kQRk QR

−k

)−

∑k

ω2kAk(1 + cos 2�t)QR

k QR−k

= Hph,0 + Hdrv., (B1)

with Ak = −�kE2/2ω2k , which describes a parametric oscil-

lator with resonance condition � = ωk

√1 − Ak .

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KNAP, BABADI, REFAEL, MARTIN, AND DEMLER PHYSICAL REVIEW B 94, 214504 (2016)

0%

40%

>100%

20%

60%

80%

< -10%

parametricresonance

driving amplitude A

pum

p fr

eque

ncy

temporalproximity effect

(b)

0%

40%

>100%

20%

60%

80%

< -10%

driving amplitude A

pum

p fr

eque

ncy

(a)

- - - - -

parametricresonance

FIG. 3. Phonon squeezing as a universal mechanism for the enhancement of Tc. The relative change of the superconducting transitiontemperature is shown for (a) quartic type-II phonon nonlinearities with positive �k > 0 and (b) cubic type-III phonon nonlinearities, taking intoaccount the competition between Cooper pair formation and pair breaking. (a) Quartic phonon nonlinearities that couple to finite momentumRaman 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 uniformsuppression of Tc with increasing magnitude of the driving amplitude A. However, near parametric resonance � ∼ ω, the phonon squeezingdominates over the static displacement, leading to the universal enhancement of Tc. (b) Cubic nonlinearities exhibit only the squeezingmechanism, and hence irrespectively of the sign of the coupling lead to an enhanced transition temperature. Furthermore, the shape of theenhancement diagram is similar to the one for quartic interaction with the main difference that in the case of quartic nonlinearities the resonancecondition is � ∼ ω while in the cubic case it is � ∼ 2ω. The data are evaluated for ω/W = 1/8, U/W = 1/8, �ω = 0.2ω.

The drive can be again separated into a static and anoscillatory contribution with the implications of the staticdisplacement being crucially dependent on the sign of thenonlinearity �k . For �k < 0 the frequencies of the Ramanmodes ωk are softened and hence the effective electron-electron interaction, which is typically of the form g2

k/ωk ,is increased, while for �k > 0 the opposite is the case. Bycontrast, we find that irrespective of the sign of the couplings,the oscillatory term squeezes the Raman modes which resultsin a polaronic suppression of the electron tunneling matrixelement compared to equilibrium and hence in an enhanceddensity of states, Fig. 3(a). Such an enhancement in thedensity of states in turn increases the superconducting couplingconstant, enabling dynamic Cooper pair formation at highertemperatures where equilibrium Cooper pairing would beimpossible.

(ii) Type III: Cubic nonlinearities. In noncentrosymmetriccrystals, the lowest-order phonon nonlinearity is of the cubicform III, QIR

0 QRk QR

−k , as the zero-momentum infrared activephonon mode can linearly couple to the displacement of theRaman modes. Coupling to pairs of Raman modes at oppositemomentum gives rise to phonon squeezing that leads to anenhanced Tc. Hence, phonon squeezing is largely insensitiveto the microscopic details of the phonon nonlinearities and canoccur in a variety of nonlinear couplings. The effective phononHamiltonian arising from such a nonlinearity is

Hph =∑

k

P Rk P R

−k + ω2k(1 + Ak cos �t)QR

k QR−k, (B2)

with Ak = �kE , which also describes a parametric oscillatorbut with resonance condition � = 2ωk . In the case of cubicinteractions, direct mode softening cannot occur because the

the time average of QIR0 vanishes. Nonetheless, the dynamic

generation of squeezing near the parametric resonance is uni-versally present also for cubic nonlinearities and qualitativelysimilar enhancement diagrams can be also obtained in thatcase, see Fig. 3(b).

For studying phonon nonlinearities which generatesqueezed states, we consider the Holstein model with Frohlichelectron-phonon interactions where the phonon displacementcouples to the charge density:

Hel-ph =−J0

∑ijσ

c†iσ cjσ +

∑k

ωkb†kbk

+ 1√V

∑ik

gk(bk + b†−k)eikri niσ . (B3)

We expect our results to be insensitive to the fine details ofthe model and expect, for example, similar results for the SSHmodel.

We pursue the following strategy:(a) Using a Lang-Firsov transforma we remove the electron-

phonon interaction.(b) We determine the unitary transforma W that transforms

the system into a rotating frame in which the phonon drivingis absent. This leads to a driven Floquet BCS problem.

(c) We estimate the dynamic Cooper pair breaking ratedue to the nonequilibrium scattering of phonons and due tooscillations in the microscopic interaction parameter usingFloquet Fermi’s Golden rule. The Cooper pair breaking rateis self-consistently taken into account by adding an imaginaryself-energy correction to the Floquet BCS equations.

(d) Solving the driven Floquet BCS problem, we determinethe enhancement of Tc relatively to the undriven one.

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1. Lang-Firsov transformation

Considering the Holstein model (B3), we introduce oper-ators bk and b

†−k which diagonalize the static phonon Hamil-

tonian∑

k P Rk P R

−k + ω2k(1 − Ak)QR

k QR−k . These operators are

related to the bk , b†−k operators which diagonalize Hph,0, e.g.,

by (b†−k − bk) = (1 − Aq)14 (b†−k − bk). The electron-phonon

interaction in Eq. (B3) can be removed by a Lang-Firsovtransformation:

Hel-ph → eSHel-phe−S

with S = − 1√V

∑qjσ

gq

ωq

√1 − Aq

eiqrj (bq − b†−q)njσ . (B4)

Applying it to the Hamiltonian and switching on the drive weobtain

H =−∑ijσ

Jij c†iσ cjσ +

∑ijσσ ′

Uijniσ njσ ′

+∑

k

P Rk P R

−k + ω2k(1 − Ak − Ak cos 2�t)QR

k QR−k

(B5)

with dressed tunneling matrix element Jij =J0e

− 1√V

∑k

gk

ωk (1−Ak )3/4 (eikri −eikrj )(bk−b

†−k)

and attractive electron-electron interaction Uij = − 1

V

∑k e−ik(rj −ri ) g2

k

ωk

√1−Ak

.

2. Rotating phonon frame

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

H0 = PqP−q + ω2qQqQ−q = ωq(b†qbq + b−qb

†−q), (B6a)

H1 = PqP−q + ω2q�

2t QqQ−q

= 12

(�2

t + 1)ωq(b†qbq + b−qb

†−q)

+ 12

(�2

t − 1)ωq(b†qb

†−q + bqb−q). (B6b)

Applying the unitary transformation, we find

id

dtW |ψ0〉 = i

dW

dt|ψ0〉 + Wi

d

dt|ψ0〉

= idW

dt|ψ0〉 + WH0 |ψ0〉 = H1W |ψ0〉 (B7)

yielding

idW

dtW † + WH0W

† = H1. (B8)

Multiplying this equation from left by W and from right by W †

and identifying H0 = H0,ph and H1 = Hph,0 + Hdrv, we obtain

W †(Hph,0 + Hdrv)W − iW † dW

dt= Hph,0. (B9)

Our goal is to construct a mapping of the quantum problemonto a classical Mathieu equation which determines thetransformation W [53] uniquely. To this end, we introduce

the ansatz

W (t) = eξe−2iωq tK+−ξ∗e2iωq tK−e−2iK0φ, (B10)

where

K0 = 12 (b†qbq + b−qb

†−q), K+ = b†qb

†−q, K− = bqb−q,

(B11)

which obey SU(2) algebra [K−,K+] = 2K0, [K0,K±] =±K±. The time-dependent factors in Eq. (B10) are chosensuch that bqe

iωq t = bq(t) are invariants of the undrivenproblem H0, which are defined by requiring that they commutewith the corresponding action, i.e., [bq(t),i∂t − H0] = 0. Fora given invariant bq(t), bq(t) |ψ〉 remains an eigenstate ofH0 provided |ψ〉 is an eigenstate, since 0 = [bq(t),i∂t −H0] |ψ〉 = (i∂t − H0)bq(t) |ψ〉. We furthermore introduce theinvariants of the driven problem as

[aq(t),i∂t − H1] = 0, (B12)

which will be generated by the unitary transformation aq(t) =Wbq(t)W †. By differentiation and subsequent integration weobtain for the invariants

aq(t) = Wbq(t)W †

= eiφ

(cosh |ξ |bqe

iωq t − ξe−iωq t

|ξ | sinh |ξ |b†−q

)≡ χ (t)bq + λ∗(t)b†−q, (B13a)

a†−q(t) = Wb

†−q(t)W †

= e−iφ

(cosh |ξ |b†−qe

−iωq t − ξ ∗eiωq t

|ξ | sinh |ξ |bq

)≡ χ∗(t)b†−q + λ(t)bq. (B13b)

We do not explicitly write the q dependence in χ and λ, asthese functions are symmetric in q, i.e., χq = χ−q .

Next, we compute the invariants i∂t aq(t) = [H,aq(t)],Eq. (B12), using relations (B13):

dχ

dt= i

ωq

2

(�2

t + 1)χ − i

ωq

2

(�2

t − 1)λ∗, (B14a)

dλ∗

dt=−i

ωq

2

(�2

t + 1)λ∗ + i

ωq

2

(�2

t − 1)χ. (B14b)

We transform α = χ − λ∗, β = χ + λ∗ which yields theMathieu equation

d2α

dt2+ ω2

q�2t α = 0,

dα

dt= iωqβ. (B15)

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 betweenξ (t), φ(t) from the unitary transformation Eq. (B10) and the

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KNAP, BABADI, REFAEL, MARTIN, AND DEMLER PHYSICAL REVIEW B 94, 214504 (2016)

off resonance

on resonance

phon

on s

quee

zing

FIG. 4. Mathieu equation. Numerical solution of the Mathieuequation at parametric resonance (A = 0.1, � = ω

√1 − A), blue

solid line, and off resonance (A = 0.1, � = 0.9ω√

1 − A), reddashed line. On resonance the phonon squeezing |ξ | increases linearlyin time, while off resonance it oscillates around a finite mean value.

parameters of the Mathieu equation

cosh |ξ |eiφ = e−iωt

2

(α − iα

ωq

), (B17a)

sinh |ξ | ξ

|ξ |eiφ = eiωt

2

(α + iα

ωq

). (B17b)

Plugging in the driving from Eq. (B5), we find �2t = (1 −

Aq − Aq cos 2�t). Even though the Mathieu equation cannotbe solved analytically, for this form of the driving its solutionis well understood, as it realizes the parametric oscillator,which displays a parametric resonance when � = ω

√1 − A.

On resonance the phonon squeezing |ξ | increases linearly intime while off resonance it oscillates around a mean value,Fig. 4.

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

W †bqW = cosh |ξ |eiφbq + ξe−2iωq t

|ξ | sinh |ξ |e−iφb†−q,

(B18a)

W †b†−qW = cosh |ξ |e−iφb†−q + ξ ∗e2iωq t

|ξ | sinh |ξ |eiφbq.

(B18b)

Assuming that the drive is adiabatically switched on, thesystem remains in phonon vacuum in the rotating frame fromwhich we find

J (t) = −J0e− 1

2V

∑q [2−2 cos q(ri−rj )]

g2q

ω2q (1−Ak )3/2 |αq |2

, (B19)

where αq = (cosh |ξ | − ξ∗e2iωq t

|ξ | sinh |ξ |)eiφ . Upon time aver-aging this equation yields Eq. (3) where we parametrize the

relative suppression of the electron tunneling compared to theone at equilibrium by the exponential factor exp[−ζ ].

3. Rescaling of time

It is convenient to transform the time evolution from thekinetic energy to the interaction energy. Once we evaluate thephonon dynamics, Eq. (B19), we obtain the effective electronHamiltonian of the form

H (t) = J (t)HT − UHU, (B20)

where we used HT and HU as short hand notation for thekinetic energy and the interaction energy, respectively, and J (t)is the time-dependent hopping matrix element. We introduceJ (t) = Jeqe

−ζ j (t) where j (t) is an oscillating function withthe mean value one. In order to move the time dependencefrom the kinetic energy to the interaction energy, we consider∫ t

0H (t)dt =

∫ t

0H (t) j (t)dt︸ ︷︷ ︸

dt ′

=∫ t ′

0H [f (t ′)]dt ′ (B21)

where

H (τ ) = Jeqe−ζ HT − U

j (τ )HU (B22)

and

t ′ =∫ t ′

0dt ′ =

∫ t

0j (t)dt = f −1(t). (B23)

The inverse of this equation cannot be calculated analytically;however, for an oscillating function with amplitude smallcompared to the mean of j (t) it follows that t ∼ t ′, whichholds because we chose j (t) to oscillate around one. Thus wecan directly transform the time-dependent part from J (t) tothe interaction and arrive at

H (t) = Jeqe−ζ HT − U

j (t)HU . (B24)

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

4. Time dependence of the microscopic parameters

For a dispersive phonon with spread �ω the time evolutionof U/J (t) typically oscillates at driving frequency � whichis a common oscillation frequency for the phonon modes atall wave vectors. We approximate the oscillations of U/J (t)with a single harmonic of frequency � and strength A,see Fig. 5. In case of strongly off-resonant driving, theinteraction oscillates around its equilibrium value, while nearthe parametric resonance on average it is enhanced, as a resultof the polaronic suppression of the bandwidth Jeqe

−ζ . Themean suppressed bandwidth Jeqe

−ζ and strength A of theoscillations in U/J (t) are extracted numerically, see Fig. 6.

5. Floquet Fermi’s Golden Rule

Fluctuations around the initial phonon state and the tem-poral modulation of the interactions reduce the quasiparticlelifetime which in turn decrease the superconducting transitiontemperature.

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DYNAMICAL COOPER PAIRING IN NONEQUILIBRIUM . . . PHYSICAL REVIEW B 94, 214504 (2016)

timetime

(a) (b)

0 10 20 30 40 50 600

1

2

3

0 10 20 30 40 50 600

1

2

3

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

Below, we compute the quasiparticle scattering rate fromthe phonon fluctuations 1/τph and the modulated interactions1/τint by Floquet Fermi’s Golden Rule. Due to the timedependence of the coupling constants, energy required tocreate particle-hole excitations can be borrowed from the drive.This enhances the scattering rate compared to equilibrium. Thetotal dynamic Cooper pair breaking rate 1/τ = 1/τph + 1/τint

is shown in Fig. 7 for ω = U = W/8. For low drivingfrequencies, the pair breaking rate is small, since only higher-order Floquet harmonics can provide the required energy andthe corresponding matrix elements are small. When the drivingfrequency is near parametric resonance with phonon pairexcitations, the effective interactions and hence the decay rateincreases since the drive can efficiently provide the required

energy to create particle-hole excitations. In a broad regimeof parameters we find that the enhanced pair formation ratedominates over the enhanced pair breaking rate, see Fig. 1 inthe main text.

The full form of the effective Hamiltonian (4) is

H =−Jeqe−ζ

∑ijσ

e− ∑k α∗

k (t)�∗k b

†k e

∑k αk (t)�kbk c

†iσ cjσ

−U (1 + A cos 2�t)∑

i

ni↑ni↓, (B25)

where �k = − 1√V

(eikri − eikrj ) gk

ωk(1−Ak )3/4 . Taking the phononvacuum expectation value of Eq. (B25), we obtain the first

effe

ctiv

e am

plit

ude

enh.

of e

ffec

tive

mas

s

(a) (b)

driving amplitude A

pump frequency

driving amplitude A

pump frequency

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

214504-9

KNAP, BABADI, REFAEL, MARTIN, AND DEMLER PHYSICAL REVIEW B 94, 214504 (2016)pa

ir b

reak

ing

rate

pump frequency

driving amplitude A

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

two terms of Eq. (4). The electron-phonon scattering term,represented by the last term in Eq. (4), is given by Hel-ph scatt. =−Jeqe

−ζ∑

ijσ (e−∑k α∗

k (t)�∗k b

†k e

∑k αk (t)�kbk − 1)c†iσ cjσ which as

discussed vanishes upon taking the phonon vacuum expec-tation value. Using a Floquet Fermi’s Golden Rule analysiswe estimate the Cooper pair breaking rate which originatesfrom both (i) phonon fluctuations in the kinetic energy and (ii)modulation of the effective electron-electron interactions.

(i) Phonon fluctuations. Fluctuations around the phononvacuum require energy. This energy can be borrowed from thetime dependence of the drive. Expanding the exponentials ofthe electron phonon scattering term Hel-ph scatt. to first order,we obtain

Hel-ph scatt. ∼−Jeqe−ζ 1√

V

∑kqσ

2gq

ωq(1 − Aq)3/4

×[∑

d

cos kd − cos(kd − qd )

]

× [αq(t)bq − α∗−q(t)b†−q]c†kσ ck−qσ

= 1

V

∑kqσ

[Fqkαq(t)bq − F∗qkα

∗q (t)b†−q] c

†kσ ck−qσ

(B26)

with

Fkq =−2Jeqe−ζ

[∑d

cos kd − cos(kd − qd )

]gq

ωq(1 − Aq)3/4.

(B27)

Higher-order terms in the series expansion of the exponentialscorrespond to multiphonon processes which are energeticallysuppressed.

The finite lifetime of quasiparticles at the Fermi energy canbe calculated from the imaginary part of the retarded phonon-

fluctuation self-energy:

1

τph= Im �

ph,+kF

. (B28)

To leading order the corresponding greater and lesser compo-nents are

�ph,≷k (t1,t2) = i

V

∑q

|Fkq |2αq(t1)α∗q (t2)G≷

k−q(t1,t2)D≷q (t1,t2).

(B29)

Here, G≷k (t1,t2) is the bare electron and D

≷q (t1,t2) are the bare

phonon propagators, which are functions of the time differenceas the unperturbed Hamiltonian is static. Thus, in Fourier spacethey are given by

G>k (ω) ≈ −2πiδ(ω − εk + μ)[1 − nk],

G<k (ω) ≈ +2πiδ(ω − εk + μ)nk,

D>q (ω) ≈ −iπδ(ω − ωq),

D<q (ω) ≈ −iπδ(ω + ωq),

where we considered the phonons to be in the vacuum stateand thus neglected their distribution functions in D> and D<.The self-energy contains the drive αq(t) and therefore is afully nonequilibrium object which is a function of two times,Eq. (B29). We rewrite the self-energy using the average timeT = 1

2 (t1 + t2) and the time difference t = t1 − t2:

�ph,≷k (t,T ) = i

V

∑q

|Fkq |2αq(T + t/2)

×α∗q (T − t/2)G≷

k−q(t)D≷q (t). (B30)

In order to estimate a quasiparticle lifetime, we integrate overthe “slow” time scale T which yields for the effective coupling

|αq(t)|2 = �π

∫ π�

0 dT αq(T + t/2)α∗q (T − t/2). The effective

coupling |αq(t)|2, distinguished from the bare coupling by thebar, is only a function of the time difference t . From a Fouriertransform we obtain its Floquet components |αqn|2, where n

ranges from −∞ to ∞. We calculate the retarded self-energy�+

ph,k using

�ph,+k (ω) = i

∫dω′

2π

�ph,>

k (ω′) − �ph,<

k (ω′)ω − ω′ + i0+ , (B31)

where ω is the conjugate variable to the time difference t , andfind for the lifetime at the Fermi surface

1

τph= π

2V

∑qn

|FqkF|2|αqn|2{(1 − nkF −q)δ(2n� − EkF −q−ω)

+ nkF −qδ(2n� − EkF −q + ω)}. (B32)

In order to obtain a semianalytical estimate for the pairbreaking rate, we neglect the weak wave-vector dependenceof gq/ωq(1 − Aq)3/4 and replace them by their mean. Wereplace wave-vector summations by integrals over energies

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DYNAMICAL COOPER PAIRING IN NONEQUILIBRIUM . . . PHYSICAL REVIEW B 94, 214504 (2016)

with a constant density of states, yielding

1

τph= π

2

∑n>0

eζ

8Jeq

g2

ω2(1 − A)3/2(2n� − ω)2�(2n� − ω)

×{αn�(|EF + 2n� − ω| − W )

+α−n�(|EF − 2n� + ω| − W )}, (B33)

where W = 4Jeqe−ζ is half of the electronic bandwidth.

(ii) Modulated interactions. The temporal modulation of theeffective electron-electron interaction leads to another sourcefor decreasing the quasiparticle lifetime. Similarly as in thecase of phonon fluctuations, we estimate the interaction decayrate by computing the imaginary part of the leading-orderself-energy contribution:

�int,≷k (t1,t2) = 1

V 2

∑ql

U 2(1 + A cos 2�t1)(1 + A cos 2�t2)

×G≷k+l−q(t1,t2)G≷

q (t1,t2)G≶l (t2,t1). (B34)

Performing a Floquet Fermi’s Golden Rule analysis, we firstintegrate over the “slow” time scale T to obtain an effectivecoupling as a function of the time difference t and then computethe Floquet components of the coupling. Plugging this intothe expression for the retarded self-energy and taking theimaginary part, we obtain the decay rate

1

τint= U 2A2

4V 2

∑ql

[(1 − nkF +l−q

)(1 − nq) nl

+ nkF +l−q nq (1 − nl)]

× {δ[2� − (

EkF +q−l + Eq − El

)]+ δ

[2� + (

EkF +q−l + Eq − El

)]}. (B35)

We evaluate both quasiparticle decay rates numerically, seeFig. 7. The total pair breaking rate 1/τ = 1/τph + 1/τint, whichwe consider as an imaginary self-energy correction in theFloquet BCS equations, is in a wide parameter range muchsmaller than the Cooper pair formation rate and thus onlyslightly shifts the transition temperature to lower values.

APPENDIX C: FLOQUET BCS APPROACH

We first evaluate the equations of motion for c†k↑c

†−k↓ from

the rescaled Hamiltonian Eq. (4) taking into account theCooper pair breaking rate 1/τ , computed in Sec. B5, as animaginary self-energy correction

d

dtc†k↑c

†−k↓ = 2i(εk + i/τ − μ)c†k↑c

†−k↓ − i

U (1 + A cos 2�t)

V

×∑mq

c†m↑c

†q↓(cm+q−k↓c

†−k↓ − c

†k↑cm+q+k↑)

(C1)

and factorize the quartic term using a mean-field decoupling

d

dt〈c†k↑c

†−k↓〉 = 2i(εk + i/τ − μ)〈c†k↑c

†−k↓〉

− iU (1 + A cos 2�t)

V

×∑

q

〈c†k↑c†−k↓〉(nq↑ + nq↓)

+〈c†q↑c†−q↓〉(1 − nk↑ − nk↓). (C2)

Using spin symmetry and defining 2ρ = 1V

∑q(nq↑ + nq↓) we

obtain

d

dt〈c†k↑c

†−k↓〉 = 2i(εk + i/τ − μ − U (1 + A cos 2�t)ρ)

×〈c†k↑c†−k↓〉 − iU (1 + A cos 2�t)(1 − 2nk)

× 1

V

∑q

〈c†q↑c†−q↓〉. (C3)

Next, we remove the term 2iUρA cos 2�t〈c†kc†−k〉 by anappropriate unitary transformation of the form

〈c†k↑c†−k↓〉 = a∗

k exp

[− i

UρA�

sin 2�t

](C4)

which gives

d

dta∗

k = 2i(εk + i/τ − μ − Uρ)a∗k

− iU (1 + cos 2�t)(1 − 2nk)1

V

∑q

a∗q . (C5)

Using the Floquet ansatz

a∗k (t) = eiEt

∑n

a∗kne

i2n�t (C6)

we obtain

[E + 2n� − 2(εk + i/τ − μ − Uρ)]a∗kn

=−(1 − 2nk)U

V

[∑q

a∗qn + A

2

∑q

(a∗qn+1 + a∗

qn−1)

].

(C7)

Dividing by E + 2n� − 2(εk + i/τ − μ − Uρ), and sum-ming over k, we find the Floquet BCS gap equation:

1

V

∑k

a∗kn = − 1

V

∑k

1 − 2nk

E + 2n� − 2(εk + i/τ − μ − Uρ)︸ ︷︷ ︸=Fn

× U

V

[∑q

a∗qn + A

2

∑q

(a∗qn+1 + a∗

qn−1)

].

(C8)

Defining the gap �n = UV

∑k a∗

kn, we obtain the simple systemof equations

(U−1 + Fn)�n + A2

Fn(�n−1 + �n+1) = 0. (C9)

The fact that we used a single harmonic to describe the timeevolution of U (t) reflects in the gap equation having only asingle side band. More complicated functions would lead tofurther side bands which would give quantitative differencesbut our conclusions will not be altered on the qualitative level.

214504-11

KNAP, BABADI, REFAEL, MARTIN, AND DEMLER PHYSICAL REVIEW B 94, 214504 (2016)

The BCS Floquet equations have a nontrivial solution, whenthe determinant is zero, which we determine by scanning

E in the complex plane. The Cooper pair formation rate ischaracterized by the imaginary part of E.

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