# Gauge-invariant theory of optical response to THz pulses in -wave and (+)-wave superconducting semiconductor quantum wells

###### Abstract

We investigate the optical response to the THz pulses in the -wave and (+)-wave superconducting semiconductor quantum wells by using the gauge-invariant optical Bloch equations, in which the gauge structure in the superconductivity is explicitly retained. By using the gauge transformation, not only can the microscopic description for the quasiparticle dynamics be realized, but also the dynamics of the condensate is included, with the superfluid velocity and the effective chemical potential naturally incorporated. We reveal that the superfluid velocity itself can contribute to the pump of quasiparticles (pump effect), with its rate of change acting as the drive field to drive the quasiparticles (drive effect). Specifically, the drive effect can contribute to the formation of the blocking region for the quasiparticle, which directly suppresses the anomalous correlation of the Cooper pairs. We find that both the pump and drive effects contribute to the oscillations of the Higgs mode with twice the frequency of the optical field. However, it is shown that the contribution from the drive effect to the excitation of Higgs mode is dominant as long as the driven superconducting momentum is less than the Fermi momentum. This is in contrast to the conclusion from the Liouville or Bloch equations in the literature, in which the drive effect on the anomalous correlation is overlooked with only the pump effect considered. Furthermore, in the gauge-invariant optical Bloch equations, the charge neutrality condition is consistently considered based on the two-component model for the charge, in which the charge imbalance of quasiparticles can cause the fluctuation of the effective chemical potential. It is predicted that during the optical process, the quasiparticle charge imbalance can be induced by both the pump and drive effects, leading to the fluctuation of the chemical potential. This fluctuation of the chemical potential is further demonstrated to directly lead to a relaxation channel for the charge imbalance even with the elastic scattering due to impurities. This is contrast to the previous understanding that in the isotropic -wave superconductivity, the impurity scattering cannot cause any charge-imbalance relaxation. Furthermore, it is revealed that when the momentum scattering is weak (strong), the charge-imbalance relaxation is enhanced (suppressed) by the momentum scattering. Finally, we predict that in the (+)-wave superconducting (100) quantum wells, with the vector potential parallel to the quantum wells, the optical field can cause the total spin polarization of Cooper pairs, oscillating with the frequency of the optical field. The direction of the total Cooper-pair spin polarization is shown to be parallel to the vector potential.

###### pacs:

74.40.Gh, 74.25.Gz, 74.25.N-, 73.21.Fg## I Introduction

In recent decades, the nonequilibrium property of superconductors has attracted much attention for providing new understandings in superconductivityTinkham_book (); Kopnin (); Chandrasekhar (); Triplet_S_F (); Buzdin_S_F (); Gray (); Larkin (); Pethick_review () and/or exploring novel phases or regimes.Leggett_book (); Liao (); Ming_Gong (); Twisting_Anderson (); Soliton (); Critical () Among them, the optical response plays an important role in both linearMattis (); Nam (); Schrieffer (); two_fluid (); Lee () and nonlinear regimes.MgB2_1 (); MgB2_2 (); YBCO (); BSCCO (); voltage (); Earliest (); Matsunaga_1 (); Matsunaga_2 (); Matsunaga_3 (); Matsunaga_4 () The former has been well established from the understanding of the optical conductivity in the linear response of the superconducting state, which sheds light on the determination of the pairing symmetry of the superconducting order parameter.Mattis (); Nam (); Schrieffer (); two_fluid () The latter is inspired by the recently-developed THz technique, whose frequency lies around the superconducting gap.MgB2_1 (); MgB2_2 (); YBCO (); BSCCO (); voltage (); Earliest (); Matsunaga_1 (); Matsunaga_2 (); Matsunaga_3 (); Matsunaga_4 () With an intense THz optical field, the superconductor can be even excited to the states far away from the equilibrium, opening a window to reveal the dynamical properties of both the Bogoliubov quasiparticles and the condensate.MgB2_1 (); MgB2_2 (); YBCO (); BSCCO (); voltage (); Earliest (); Matsunaga_1 (); Matsunaga_2 (); Matsunaga_3 (); Matsunaga_4 ()

In the linear regime, in the dirty limit at zero temperature, Mattis and BardeenMattis () revealed that the optical absorption is realized by breaking the Cooper pairs into the quasi-electron and quasi-hole when the photon energy is larger than twice the magnitude of the superconducting gap.Nam (); Schrieffer () Nevertheless, in the early-stage work,Mattis () a physical optical conductivity is established only for a specific gauge with transverse vector potential and zero scalar potential.Mattis (); Nam (); Schrieffer () A gauge-invariant description with charge conservation for the optical conductivity tensor is later established by Nambu based on the generalized Ward’s identity,Nambu_gauge (); Goldstone () in which the collective excitation is revealed to cancel the unphysical longitudinal current.Schrieffer (); Nam (); Lee () Furthermore, Ambegaokar and KadanoffAmbegaokar () showed that in the long wave limit, the collective mode can be actually described as a state in which the superconducting phase of the order parameter varies periodically in time and space.Ambegaokar (); Altland (); conjugacy (); Lee (); Schrieffer (); Enz (); Griffin () Actually, without considering the response of the order parameter to the optical field, the absence of the charge conservation naturally arises because the particle number is not a conserved quantity in the mean-field description of the superconductor with a global symmetry spontaneously broken.Altland (); conjugacy (); Lee (); Schrieffer (); Nambu_gauge ()

When the photon energy is far below the superconducting gap, a simple physical picture for the optical response can be captured based on the two-fluid model, in which the optical conductivity at finite frequency readstwo_fluid (); MgB2_1 (); MgB2_2 (); YBCO (); BSCCO ()

(1) |

Here, and denote the normal-fluid and super-fluid densities in the equilibrium state, respectively; is the effective mass of the electron; and represents the momentum relaxation time. Based on Eq. (1), the optical absorption can be well understood from the electric current driven by the optical field.Mattis (); Nam (); Schrieffer (); two_fluid () In the clean limit, the optical conductivity is purely imaginary with the phase difference between the induced current and the optical field being exactly , and hence no optical absorption is expected. Nevertheless, in the dirty sample, the real part of the optical conductivity arises due to the existence of the normal-fluid, which contributes to the electric current in phase to the optical field, and hence, the optical absorption. Thus, in the pump-probe measurement, after strongly excited by the pump field, the non-equilibrium normal-fluid and super-fluid densities can be estimated from the optical response to the probe field with photon energy far below the superconducting gap.MgB2_1 (); MgB2_2 (); YBCO (); BSCCO () However, to the best of our knowledge, a microscopic theoretical-description for the evolution of the normal- and super-fluids from the equilibrium state to the non-equilibrium ones is still lacking.

In the nonlinear regime, in which the superconducting state can be markedly influenced by the optical field, the experimentalEarliest (); Matsunaga_1 (); Matsunaga_2 (); Matsunaga_3 (); Matsunaga_4 () and theoreticalSoliton (); Critical (); Axt1 (); Axt2 (); Anderson1 (); Higgs_e_p (); multi_component (); Leggett_mode (); Xie (); Tsuji_e_p (); multiband (); path_integral () studies are still in progress. Very recently, it was reported in several experiments in the film of the conventional superconducting metal that the oscillations of the Higgs mode, i.e., the fluctuation of the order-parameter magnitude, can be excited by the intense THz field.Earliest (); Matsunaga_1 (); Matsunaga_2 (); Matsunaga_3 (); Matsunaga_4 () It is revealed that the oscillation frequency of the Higgs mode is twice the frequency of the THz field, no matter the photon energy is larger or smaller than twice the magnitude of the superconducting gap.Matsunaga_3 (); Matsunaga_4 () Moreover, a large THz third-harmonic generation was reported when the photon energy is tuned to be resonant with the superconducting gap.Matsunaga_3 (); Matsunaga_4 () Finally, it was discovered that there exists plateau for the Higgs mode after the THz pulse in most situations, whose value increases with the increase of the field intensity.Matsunaga_1 (); Matsunaga_2 () These observations indicate that there exists strong optical absorption with the quasiparticles considerably excited by the strong optical field.Matsunaga_1 (); Matsunaga_2 (); Matsunaga_3 (); Matsunaga_4 ()

These experimental findings have been theoretically clarified based on the Liouville equationAxt1 (); Axt2 (); Higgs_e_p () or the Bloch equationAnderson1 (); multi_component (); Leggett_mode (); Xie (); Tsuji_e_p (); multiband (); Matsunaga_3 (); Matsunaga_4 () derived in the Anderson pseudospin representationAnderson_origin () in the clean limit. Specifically, the optical absorption in the clean limit is naturally understood by the nonlinear term proportional to , with standing for the vector potential of the optical field. It is shown that this non-linear term contributes to the precessions between the quasi-electron and quasi-hole states,Axt1 (); Axt2 () which directly contribute to the excitation of the quasiparticles (pump effect).Soliton (); Critical (); Axt1 (); Axt2 (); Anderson1 (); Higgs_e_p (); multi_component (); Leggett_mode (); Xie (); Tsuji_e_p (); multiband (); path_integral () Thus, the optical absorption is realized in the clean limit due to this pump effect, from which the Cooper pairs are broken into the quasi-electrons and quasi-holes.MgB2_1 (); MgB2_2 (); YBCO (); BSCCO (); voltage (); Earliest (); Matsunaga_1 (); Matsunaga_2 (); Matsunaga_3 (); Matsunaga_4 () Furthermore, because the frequency of is , the pump effect contributes to the oscillation of the Higgs mode with twice the frequency of the optical field.Axt1 (); Axt2 (); Anderson1 (); Higgs_e_p (); multi_component (); Leggett_mode (); Xie (); Tsuji_e_p (); multiband () Moreover, it is revealed that the Higgs mode can be resonant with the optical field when the photon energy equals to the superconducting gap, which is further shown to contribute to the large third harmonic generation.Matsunaga_3 (); Matsunaga_4 (); path_integral ()

However, there still exist several difficulties inherited in the LiouvilleAxt1 (); Axt2 (); Higgs_e_p () or BlochAnderson1 (); multi_component (); Leggett_mode (); Xie (); Tsuji_e_p (); multiband () equations used in the literature. Firstly, the anomalous correlation is calculated between the two electrons with momenta and , no matter the optical field is slowly or rapidly varied. This means that it is preconceived that no center-of-mass momentum of the Cooper pairs can be excited.Axt1 (); Axt2 (); Higgs_e_p (); Anderson1 (); multi_component (); Leggett_mode (); Xie (); Tsuji_e_p (); multiband () Nevertheless, in the nonlinear regime, with a strong electric field applied, a large supercurrent is expected to be induced, which should arise from the center-of-mass momentum of the Cooper pairs. It has been well understood that in the static situation, a large contributes to the Doppler shift in the energy spectra of the elementary excitation, which can lead to the formation of the blocking region with the anomalous correlation of the Cooper pairs significantly suppressed.Tao_2 (); FF (); LO (); FFLO_Takada (); supercurrent (); Yang (); Doppler_1 (); Doppler_2 () Nevertheless, the induction of the center-of-mass momentum for the Cooper pairs and its further influence on the superconducting state are absent in the description of the Liouville equation or the Bloch equation in the Anderson pseudospin representation.Axt1 (); Axt2 (); Anderson1 (); Higgs_e_p (); multi_component (); Leggett_mode (); Xie (); Tsuji_e_p (); multiband () In fact, in the Liouville equation, the generalized coordinate, i.e., the momentum , is treated to be time-independent or fixed, whereas the velocity field located at the generalized coordinate varies with time. This is similar to the Euler description in the fluid mechanics, in contrast to the Lagrangian description with time-dependent generalized coordinate.Landau () Thus, the anomalous correlation is always described between and in the Liouville or Bloch equations used in the literature.Axt1 (); Axt2 (); Anderson1 (); Higgs_e_p (); multi_component (); Leggett_mode (); Xie (); Tsuji_e_p (); multiband ()

Secondly, the scattering effect, which is inevitable in the dirty superconducting metal,Anderson1 (); Higgs_e_p () cannot be simply included in the Liouville equation in the presence of the optical field.Jianhua_Liouville () Moreover, a simple inclusion of the elastic scattering with the Boltzmann descriptionKopnin (); Tinkham_book () in the Liouville equation does not influence the calculated results, because the pump effect is isotropic in the momentum space.Axt1 (); Axt2 (); Anderson1 (); Higgs_e_p (); multi_component (); Leggett_mode (); Xie (); Tsuji_e_p (); multiband (); path_integral () However, this is un-physical because the normal-fluid can still be scattered. Finally, the gauge invarianceNambu_gauge (); Altland (); Schrieffer () in the Liouville or the Bloch equations used in the literature is not clearly addressed.Axt1 (); Axt2 (); Anderson1 (); Higgs_e_p (); multi_component (); Leggett_mode (); Xie (); Tsuji_e_p (); multiband () On one hand, two quantities in the vector potential, scalar potential and superconducting phase are simultaneously taken to be zero.Nambu_gauge (); Altland (); Schrieffer () Specifically, with the vector potential chosen, the resulted physical current is shown to be proportional to , which is not a gauge-invariant physical quantity unless a transverse gauge for is further restricted.Schrieffer (); Mattis () On the other hand, from different choices of gauge, different forms of the equation can be expected. Specifically, with only the scalar potential, the -term vanishes and the electric field contributes to the drive field; whereas with only the superconducting phase , its rate of change can also contribute to a drive field.Nambu_gauge (); Altland (); Schrieffer (); Stephen ()

In fact, as pointed out by Nambu,Nambu_gauge () the absence of the gauge invariance in the theoretical description is equivalent to the breaking of the charge conservation.Altland (); conjugacy (); Lee (); Schrieffer () By restoring the gauge invariance, in the linear regime, Nambu revealed a collective excitation stimulated in the optical process,Nambu_gauge () which was further shown by Ambegaokar and KadanoffAmbegaokar () to be described by a state with the period variations in time and space for the superconducting phase in the long-wave limit.Ambegaokar (); Altland (); conjugacy (); Lee (); Schrieffer (); Enz (); Griffin () The temporal and spacial variations of the superconducting phase can further contribute to the effective chemical potential and superconducting velocity.Ambegaokar (); Altland (); conjugacy (); Lee (); Schrieffer (); Enz (); Griffin () Then, it is inspired by this schemeAltland (); conjugacy (); Lee (); Schrieffer (); Nambu_gauge () that with the gauge invariance retained in the kinetic equation, the collective excitation can also arise naturally.path_integral () Specifically, by noting that in the mean-field description based on the Bogoliubov-de Gennes (BdG) Hamiltonian, only the dynamics of the quasiparticle is considered. It has been suggested that the “condensate” can respond to the dynamics of the quasiparticles from the consideration of the gauge structure in superconductor,Nambu_gauge () with the charge conservation restored by the fluctuation of the chemical potential.Tinkham1 (); Tinkham2 (); Tinkham_condensate (); Takahashi_SHE1 (); Takahashi_SHE2 (); Hirashima_SHE (); Takahashi_SHE3 (); Takahashi_SHE_exp (); Hershfield (); Spivak (); kinetic_book ()

One way to understand the interplay between the particle charge and chemical potential is based on the two-component model for the charge.Tinkham1 (); Tinkham2 (); Takahashi_SHE2 (); Tinkham_book (); Pethick_review (); Larkin (); Gray (); Tinkham_condensate () In the two-component model, the electron charge is treated to be carried by the quasiparticle and condensate, respectively. This can be easily seen in the electrical injection process. In that process, the injection of one electron with charge into the conventional superconductor can add a quasiparticle with charge and one Cooper pair with charge , respectively. Here, and comes from the Bogoliubov transformation with , indicating the charge conservation in the electrical injection process.Tinkham1 (); Tinkham2 (); Takahashi_SHE2 () Thus, the fluctuation of the quasiparticle charge is associated with the fluctuation of the condensate density.Tinkham1 (); Tinkham2 (); Takahashi_SHE2 (); Tinkham_book (); Pethick_review (); Larkin (); Gray (); Tinkham_condensate () This is consistent with the conjugacy relationship between the particle number and superconducting phase.conjugacy (); Nambu_gauge ()

Furthermore, in the dynamical process, the charges for the quasiparticle and condensate can both be deviated from their equilibrium values. This is referred to as the charge imbalance,Tinkham1 (); Tinkham2 (); Takahashi_SHE2 (); Tinkham_book (); Pethick_review (); Larkin (); Gray (); Clarke_first (); Clarke_thermal () which has been measured for both the quasiparticleTinkham1 (); Tinkham2 (); Pethick_review (); Clarke_first (); Clarke_thermal () and condensate.voltage () For the quasiparticle, due to the momentum-dependence of the charge, its non-equilibrium distribution can lead to the charge imbalance, whose creation and relaxation are intensively studied in the electrical experiment.Tinkham_book (); Tinkham1 (); Tinkham2 (); Takahashi_SHE2 (); Clarke_first (); Clarke_thermal () It is so far widely believed that for the isotropic -wave superconductor, the elastic scattering due to the impurity cannot cause the relaxation of the charge imbalance.Tinkham_book (); Tinkham1 (); Tinkham2 (); Pethick_review () This is because there exists the coherence factor in the scattering potential, where and are the initial and final momenta during the scattering, due to which the elastic scattering cannot exchange the electron-like and hole-like quasiparticles.Tinkham_book (); Tinkham1 (); Tinkham2 (); Pethick_review () However, in that relaxation process, the condensate is assumed to be in its equilibrium state, meaning that the charge conservation or neutrality is not explicitly considered in the literature. Moreover, the correlation between the quasi-electron and quasi-hole is often neglected.Tinkham2 (); Takahashi_SHE2 (); Tinkham_book (); Pethick_review (); Larkin (); Gray () Thus, it is essential to check the influence of the condensate on the charge-imbalance relaxation in the framework of charge neutrality. Furthermore, although the charge imbalance including its creation and relaxation is intensively studied in the electrical experiment,Tinkham1 (); Tinkham2 (); Takahashi_SHE2 (); Tinkham_book (); Pethick_review (); Larkin (); Gray (); Clarke_first (); Clarke_thermal () it has yet been well investigated in the optical process.voltage ()

So far we have addressed the experimental and theoretical investigations on the optical response in the conventional superconductivity, in which the Cooper pairs do not carry any net spin. As the optical method is often used to create and manipulate the electron spins in semiconductorsJinluo (); Jianhua_Liouville (); Ivchenko (); wureview () or topological insulatortopological_0 () by inducing an effective spin-orbit coupling (SOC), it is intriguing to consider the possibility of the creation and manipulation for the Cooper-pair spin polarization. This is possible in the triplet superconductivity as the triplet Cooper pairs can carry net spin polarization.Sigrist (); Supercurrents (); Superconducting_spintronics (); Eschrig_supercurrent (); Linder1 (); Linder2 () It is noticed that the inclusion of the optical field can break the time-reversal symmetry. Previous works have shown that the breaking of the time-reversal symmetry by the Zeeman fieldLinder_SFS () or the supercurrentTkachov () can induce the Cooper-pair spin polarization. In the former situation, for the conventional -wave superconductor in proximity to a ferromagnet, the triplet Cooper pairing can be induced in the superconductor-ferromagnet interface.Buzdin_S_F (); Triplet_S_F (); Linder1 (); Tokatly_PRLB (); exp_realization1 (); exp_realization2 (); Berezinskii () Then Jacobsen et al. showed that in the superconductor-ferromagnet-superconductor Josephson junction, when there exists the SOC in the superconductor-ferromagnet interface, the net spin polarization of triplet Cooper pairs can be created, and a superconducting spin flow with spin-flip immunity can be realized.Linder_SFS () In the latter situation, Tkachov pointed out that in noncentrosymmetric superconductors, a nonunitary triplet pairingLeggett (); Leggett_book (); Sigrist () can be induced by the supercurrent, which contributes to the spin polarization of triplet Cooper pairs and can be detected by the magnetoelectric Andreev effect.Tkachov () It is further noted that the optical field can also induce a supercurrent in noncentrosymmetric superconductivity, which is expected to dynamically generate the Cooper-pair spin polarization.

Recently, the proximity-induced superconductivity has been realized in InAsInAs_1 (); InAs_2 (); Doppler_2 () and GaAsGaAs_1 (); GaAs_2 (); GaAs_3 () heterostructures. Thus, based on the well-developed techniques in semiconductor optics,Haug (); Axt_optics (); Rossi_optics () the superconducting semiconductor quantum wells (QWs) can provide an ideal platform to study the optical response of superconductivity. Compared to the film of the superconducting metal, the QWs can be synthesized to be extremely clean. Furthermore, the material parameters in the QWs, e.g., the electron density, the strength of the SOC and the interaction strengths including the Coulomb, electron-phonon and electron-impurity interactions, can be easily tuned. Moreover, in the QWs, the simple Fermi surface and exactly-known interaction forms can significantly reduce the difficulties in the comparison between the theory and experiment. Finally, the predictions revealed in the superconducting QWs can still shed light on the optical response in the superconducting metal even with complex Fermi surfaces.

In the present work, we investigate the optical response to the THz pulses in both the -wave and (+)-wave superconducting semiconductor QWs. The gauge-invariant optical Bloch equations are set up via the gauge-invariant nonequilibrium Green function approach,Tao_2 (); Haug (); Levanda (); GKB (); BoYe () in which the gauge-invariant Green function with the Wilson lineWilson_line (); Haug (); Levanda () is constructed by using the gauge structure revealed by Nambu.Nambu_gauge () In the optical Bloch equations, the structure can be easily captured by a special gauge, in which the superconducting phase is chosen to be zero among the vector potential, scalar potential and superconducting phase. This gauge is referred to as the -gauge here, with being the superfluid momentum driven by the optical field. It is noted that this superfluid momentum directly contributes to the center-of-mass momentum of Cooper pairs. Furthermore, in the -gauge, not only can the microscopic description for the quasiparticle dynamics be realized, but also the dynamics of the condensate is included, with the superconducting velocity and the effective chemical potential naturally incorporated. Then in the derived gauge-invariant optical Bloch equations, this superconducting velocity is shown to directly contribute to the pump of the quasiparticles (pump effect), whose rate of change induces a drive field to drive the quasiparticle (drive effect). We find that both the pump and drive effects contribute to the oscillation of the Higgs mode with twice the frequency of the optical field. However, it is shown that the contribution from the drive effect to the excitation of Higgs mode is dominant as long as the superconducting momentum is smaller than the Fermi momentum , thanks to the efficient suppression of the pump effect by the Pauli blocking. This is in sharp contrast to the results from the LiouvilleAxt1 (); Axt2 (); Higgs_e_p () or BlochAnderson1 (); multi_component (); Leggett_mode (); Xie (); Tsuji_e_p (); multiband () equations in the literature, where only the pump effect is considered and the effects of the center-of-mass momentum on the superconducting state are overlooked. The influence of the electron-impurity scattering is also addressed, which is shown to further suppress the Cooper pairing on the basis of the drive effect.

The physical picture for the suppression of the anomalous correlation of Cooper pairs by the optical field can be understood as follows. Thanks to the drive of the optical field, the electron states are drifted, obtaining exactly the center-of-mass momentum in the impurity-free situation. The drift states of electrons are schematically presented in Fig. 1 with the Fermi surface labeled by the red chain curve. In Fig. 1, without loss of generality, the superconducting momentum is taken to be along the -direction, i.e., , with . It can be seen that with the drift of the electron states, a blue region labeled by “B” arises, in which the electrons deviate from their equilibrium states. Actually, these electrons are directly excited to be the quasiparticles, whose population can be close to one.FF (); LO (); FFLO_Takada (); supercurrent (); Tao_2 (); Yang () By using the terminology in the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state,FF (); LO (); FFLO_Takada () this blue region populated by the quasiparticles is referred to as the blocking region.

Furthermore, it is noted that the applied optical field breaks the time-reversal symmetry. Thus, the paired electrons do not necessarily come from two time-reversal partners with momenta and . On the contrary, due to the induction of the center-of-mass momentum for the Cooper pairs by the applied optical field, the two electrons with momenta and are paired together. Nevertheless, once the electrons are excited in the blocking region, they no longer participate in the Cooper pairing.FF (); LO (); FFLO_Takada (); supercurrent (); Tao_2 (); Yang () One typical example is shown in Fig. 1, in which the electron labeled by “N” cannot pair with its old partner labeled by “M” in the blocking region, which has been excited to be the quasiparticle. Consequently, the anomalous correlation in the blocking region is significantly suppressed, directly leading to the suppression of the magnitude of the order parameter.FFLO_Takada (); Tao_2 (); Yang () This is responsible for the oscillation of the Higgs mode. Nevertheless, at high frequency, this oscillation is suppressed due to the suppression of the drift effect and hence the range of the blocking region. This picture is consistent with the the static case when the center-of-mass momentum of the Cooper pairs emerges due to either the spontaneous symmetry-breakingFF (); LO (); FFLO_Takada () or the supercurrent.supercurrent (); Tao_2 (); Yang ()

In the derived optical Bloch equations, the charge neutrality condition is consistently considered based on the two-component model for the charge, in which the induction of the charge imbalance of quasiparticles can cause the fluctuation of the condensate chemical potential.Tinkham1 (); Tinkham2 (); Takahashi_SHE2 (); Tinkham_book (); Pethick_review (); Larkin (); Gray () We predict that during the optical process, the charge imbalance can be created by both the pump and drive effects, with the former arising from the AC Stark effect and the latter coming from the breaking of Cooper pairs by the electrical field. The induction of the charge imbalance directly leads to the fluctuation of the chemical potential. This fluctuation is further found to directly provide a relaxation channel for the charge imbalance even with the elastic scattering due to impurities. This is in contrast to the previous understanding that in the isotropic -wave superconductivity, the impurity scattering cannot cause any charge-imbalance relaxation.Tinkham1 (); Tinkham2 (); Pethick_review () Specifically, we reveal that when the momentum scattering is weak (strong), the charge-imbalance relaxation is enhanced (suppressed) by the momentum scattering.

We demonstrate that the fluctuation of the condensate chemical potential can first induce the quasiparticle correlation between the quasi-electron and quasi-hole, which then provides the charge-imbalance relaxation channel for the quasiparticle populations in the presence of the elastic momentum scattering. In the previous works, it was revealed that in the presence of the impurities, the charge-imbalance relaxation is induced by the direct scattering of quasiparticles between the electron- and hole-like branches,Tinkham1 (); Tinkham2 (); Pethick_review () during which the quasiparticle number is conserved. Nevertheless, this is demonstrated to be forbidden in the isotropic -wave superconductors.Tinkham1 (); Tinkham2 (); Pethick_review () Differing from this charge-imbalance relaxation channel,Tinkham1 (); Tinkham2 (); Pethick_review () in this work, the charge-imbalance relaxation is actually caused by the direct annihilation of the quasiparticles in the quasi-electron and quasi-hole bands, in which the quasiparticle-number conservation is broken. These two charge-imbalance relaxation channels are schematically shown in Fig. 2, labeled by “1⃝” and “2⃝”, respectively. Specifically, process 1⃝ represents the direct scattering of quasiparticles between the electron- and hole-like branches. Whereas in process 2⃝, the quasi-electron and quasi-hole, labeled by “M” and “N”, become correlated due to the fluctuation of the effective chemical potential, which then annihilate into one Cooper pair due to the momentum scattering.

Actually, it is overlooked in the previous studiesTinkham1 (); Tinkham2 (); Pethick_review () that the non-equilibrium effective chemical potential itself can induce the precession between the quasi-electron and quasi-hole states and hence the quasiparticle correlation.Tinkham1 (); Tinkham2 (); Pethick_review () The quasiparticle correlation is crucial to induce the quasiparticle-number fluctuation. As addressed in our previous work,Tao_2 () the induction of the quasiparticle correlation is related to the process of the condensation with two quasiparticles binding into one Cooper pair in the condensate, or vice versa.Bardeen (); Tinkham2 (); Josephson_B () These processes can directly cause the annihilation of the extra quasiparticles in the quasi-electron or quasi-hole bands, inducing the charge-imbalance relaxation for the quasiparticles. Meanwhile, with the condensation or breaking of the Cooper pairs in the condensate, the fluctuation of the effective chemical potential is also induced. If only the induction of the quasiparticle correlation was not influenced by the momentum scattering, the charge-imbalance relaxation rate would be proportional to the electron-impurity scattering strength. Nevertheless, it is further revealed that the induction of the quasiparticle correlation can be suppressed by the impurity scattering. Thus, the competition between the relaxation channels due to the quasiparticle correlation and population leads to the non-monotonic dependence on the momentum scattering for the charge-imbalance relaxation.

Finally, we predict that in the (+)-wave superconducting InSb (100) QWs, with the vector potential being along the -direction, the optical field can cause the spin polarization of Cooper pairs, which is also along the -direction and oscillates with the frequency of the optical field. Specifically, in our previous work, it has been revealed that in InSb (100) QWs in proximity to an -wave superconductor, due to the Rashba-like SOC, there exists -wave triplet Cooper correlation in -type,Tao_1 () represented by with denoting the Pauli matrices. In the equilibrium state, the -vector of the triplet Cooper correlation is parallel to the effective magnetic field due to the SOC in the momentum space. Here, the -vector is defined from

(2) |

with representing the anomalous correlations of triplet Cooper pairs.Leggett_book ()

Actually, the anomalous correlations , calculated by the optical Bloch equations in this work, are just the Fourier components of the wavefunction of triplet Cooper pairs in the spatial space.Leggett_book () By further considering the spin space, the wavefunction of the triplet Cooper pairs is expressed asLeggett_book ()

(3) |

Here, , and denote the wavefuctions of the triplet Cooper pairs with total spin , and ,Leggett_book () respectively, with being the relative coordinate for the two electrons (labeled by “1” and “2”) in the Cooper pairs. Thus, with the -direction of the spin operator chosen to be perpendicular to the QWs, in the equilibrium state of superconducting InSb (100) QWs, and .Tao_1 () From the Cooper-pair wavefunction, the total spin polarization of Cooper pairs is determined by

(4) |

with being the total spin operator by the sum of the spin operators and of two electrons.

When the optical field with the vector potential along the -direction is applied to the superconducting system, the superconducting velocity is induced, which is shown to contribute to an effective SOC along the -direction. This effective SOC can cause the precession of the -vectors, with a component perpendicular to induced. Thus, with the Cooper-pair spin vector defined as ,Leggett (); Sigrist (); Tkachov (); non_unitary () whose momentum integral contributes to the total Cooper-pair spin polarization [refer to Eq. (4)], the -component of can be induced. Specifically, the -component of the Cooper-pair spin polarization is . Accordingly, one finds that the excitation of the -component of the Cooper-pair spin polarization is the reflection of the optical-induction of the triplet Cooper-pair wavefunction with . Actually, the Fourier component of is exactly .Leggett (); Sigrist (); Tkachov (); non_unitary () Furthermore, we reveal that the Cooper-pair spin polarization is proportional to the superconducting velocity, which oscillates with the frequency of the optical field.

This paper is organized as follows. We first focus on the -wave superconducting semiconductor QWs in Sec. II, whose framework is then generalized to the (+)-wave one in (100) QWs in Sec. III. Specifically, for the -wave [(+)-wave] superconducting QWs, we present the Hamiltonian in Sec. II.1 (Sec. III.1); then in Sec. II.2 (Sec. III.2), the optical Bloch equations are derived via the gauge-invariant non-equilibrium Green function approach; the numerical results are presented in Sec. II.3 (Sec. III.3). We conclude and discuss in Sec. IV.

## Ii -wave superconducting QW

In this section, we investigate the optical response to the THz pulses in the -wave superconducting QWs, which can be realized in the GaAs QWs in proximity to an -wave superconductor with negligible SOC. We first present the Hamiltonian, in which the gauge structure is emphasized (Sec. II.1). Then the optical Bloch equations via the nonequilibrium Green function method with the generalized Kadanoff-Baym (GKB) ansatz are set up, in which the gauge invariance is retained explicitly by using the gauge-invariant Green function (Sec. II.2).Tao_2 (); Haug (); Levanda (); GKB (); BoYe () Finally, we numerically calculate the optical response by solving the optical Bloch equations including the THz-field–induced oscillations of the Higgs mode and THz-field–induced charge imbalance, in which a novel charge-imbalance relaxation channel due to the elastic momentum scattering is revealed (Sec. II.3).

### ii.1 Hamiltonian and Gauge Structure

In the -wave superconducting QWs with negligible SOC, the Hamiltonian is composed by the free BdG Hamiltonian and the interaction Hamiltonian including the electron-electron Coulomb, electron-phonon and electron-impurity interactions , and . Specifically, is written as ( throughout this paper)

(5) |

in which with being the time-space point, denoting the vector potential and representing the chemical potential of the system; is the particle field operator in the Nambu space; denotes the scalar potential; and stand for the -wave order parameter and the superconducting phase. The electron-electron, electron-phonon and electron-impurity interactions are written as

(6) | |||

(7) | |||

(8) |

respectively. Here, represent the Pauli matrices in the Nambu space; and denote the screened Coulomb potentials whose expressions have been derived in Ref. Tao_1, ; is the phonon field operator; and stand for the electron-phonon interactions due to the deformation potential in the LA branch and piezoelectric coupling including LA and TA branches, with denoting the corresponding phonon branch.e_p_formula1 (); e_p_formula2 () Their Fourier components are explicitly given in Refs. e_p_formula1, ; e_p_formula2, .

The gauge structure in the -wave superconductivity was first revealed by Nambu.Nambu_gauge (); Stephen (); Altland () By performing the gauge transformation, i.e.,

(9) |

the gauge invariance of the BdG Hamiltonian [Eq. (5)] requires the vector potential, scalar potential and superconducting phase transforming asNambu_gauge (); Stephen (); Altland ()

(10) | |||

(11) | |||

(12) |

From Eqs. (10-12), one can construct the gauge-invariant physical quantitiesNambu_gauge (); Stephen (); Altland ()

(13) | |||

(14) |

which represent the superconducting momentum and effective chemical potential. It is noted that the above two gauge-invariant quantities are related by the acceleration relationNambu_gauge (); Stephen (); Altland ()

(15) |

which is valid under any circumstances. Thus, with an optical field applied to the superconducting system, Eq. (15) shows that in the homogeneous limit, a time-dependent superconducting momentum can be induced, which is always a transverse physical quantity in the presence of the optical field.Mattis (); Nambu_gauge ()

### ii.2 Optical Bloch Equations

In this section, we derive the optical Bloch equations in the -wave superconducting QWs via the nonequilibrium Green function method with the GKB ansatz.Haug (); wureview (); GKB (); Tao_2 () From Sec. II.1, one notices that there exists a nontrivial gauge structure in the BdG Hamiltonian. To account for this gauge structure, the gauge-invariant Green function is used to obtain the gauge-invariant kinetic equations.Haug (); BoYe (); Levanda ()

#### ii.2.1 Gauge-invariant Green function

The optical Bloch equations can be constructed from the “lesser” Green function , in which represents the time-space point and denotes the ensemble average.Haug (); wureview (); Tao_2 () With the gauge transformation in Eq. (9), the “lesser” Green function transforms as . As in the kinetic equations in the quasiparticle approximation,Haug () only the center-of-mass coordinates are retained, the gauge structure cannot be easily realized in the kinetic equations constructed from .Levanda (); Haug () Nevertheless, the gauge invariance can be retained by introducing the Wilson line to construct the gauge-invariant Green function,Levanda (); Haug (); Wilson_line () which is constructed as

(16) |

In Eq. (16), , are the center-of-mass coordinates, and “” indicates that the line integral is path-dependent. Then by the gauge transformation in Eq. (9), the gauge-invariant Green function is transformed as , in which the transformed phase only depend on the center-of-mass coordinates.

Finally, by choosing the path to be the straight line connecting and ,Levanda (); Haug () the gauge-invariant Green function reads

(17) |

in which are the relative coordinates.

#### ii.2.2 Derivation on the optical Bloch equations

In this part, we derive the optical Bloch equations in the -wave superconducting QWs, with special attention paid to the gauge structure. Accordingly, we do not specify any gauge in the beginning of the derivation, and finally choose a special gauge for the convenience of physical analysis and numerical calculation. Thus, in the derived equations, there exist , and , which are not physical quantities.

We begin from the two Dyson equations,Haug (); wureview (); Tao_2 ()

(18) | |||

(19) |

in which “” and “” label the retarded and advanced Green functions, and are the self-energies contributed by the electron-electron and electron-impurity interactions.Haug (); wureview (); Tao_2 () In Eqs. (18) and (19),

(20) |

and

(21) |

We first present the derivation of the free terms in the kinetic equations including the coherent, pump, drive and diffusion terms, in which the gauge-invariant scheme is used. Specifically, from the left-hand side of Eqs. (18) and (19), one obtains the equations for the gauge-invariant Green function . Then by using the gradient expansion, the kinetic equations are derived from the Fourier component of the gauge-invariant Green function . Finally, after the integration over the frequency, one obtains the optical Bloch equations for the density matrix in the Nambu space

(22) |

whose diagonal terms represent the distributions of electron and hole, and off-diagonal terms denote the anomalous correlations. Finally, the optical kinetic equations are written as

(23) |

with . Here, and represent the commutator and anti-commutator, respectively. It is noted that in the equation, the gradient expansion has been performed to the second order in , i.e., the sixth term on the left-hand side in Eq. (23), to retain the gauge-invariance structure in the optical kinetic equations.

In Eq. (23), on the left-hand side, the second and third terms represent the coherent terms contributed by the kinetic energy and the order parameter, respectively; the fourth term describes the pump term, as addressed in the Liouville equation in the literature;Soliton (); Critical (); Axt1 (); Axt2 (); Anderson1 (); Higgs_e_p (); multi_component (); Leggett_mode (); Xie (); Tsuji_e_p (); multiband (); path_integral () the fifth term is the drive term, which can directly induce the center-of-mass momentum of the Cooper pairs [Eq. (15)];Altland (); Nambu_gauge (); Schrieffer () the diffusion terms are contributed by the sixth to the ninth terms. On the right-hand side of the equation, and represent the Hartree-Fock (HF) term contributed by the Coulomb interaction and scattering term due to the electron-impurity and electron-phonon interactions, which are derived from the right-hand side of Eqs. (18) and (19). The gauge-invariant versions of the scattering terms are complex.Haug (); BoYe (); Levanda () Nevertheless, these terms can be approximated by the ones without gauge-invariant treatments as long as the applied field is not very strong with the driven center-of-mass momentum of the system being much smaller than the Fermi momentum .Haug (); BoYe (); Stephen () In this situation, the energy spectra is not significantly disturbed. The gauge structure of Eq. (23) is then checked by the gauge transformation . The same gauge structures as Eqs. (10), (11) and (12) are obtained for the vector potential, scalar potential and superconducting phase.

For the convenience of the physical analysis and numerical calculation, a specific gauge is chosen. It is noted that generally one cannot choose two quantities in the vector potential, scalar potential and superconducting phase to be zero. Nevertheless, in the Liouville and Bloch equations used in the literature, both the scalar potential and superconducting phase are taken to be zero.Soliton (); Critical (); Axt1 (); Axt2 (); Anderson1 (); Higgs_e_p (); multi_component (); Leggett_mode (); Xie (); Tsuji_e_p (); multiband (); path_integral () Here, we choose a special gauge referred to as the -gauge, in which the superconducting phase is zero.Kerson_Huang (); Altland () This can be realized by the gauge transformation in Eq. (23). Then by using the definition of the superconducting momentum [Eq. (13)] and effective chemical potential [Eq. 14], the optical Bloch equations become

(24) |

where is the total chemical potential in the system including the contribution from the rate of change of the superconducting phase.

It is noted that in Eq. (24), the electric force is replaced by according to the acceleration relation [Eq. (15)]. Accordingly, in Eq. (24), only the gauge invariant physical quantities and appear. In fact, in the gauge-invariant framework, from any specific gauge at the beginning of the derivation, one can obtain Eq. (24) with the existence of both the pump and drive terms.Haug () Moreover, in Eq. (24), with and describing the kinetics of the condensate, Eq. (24) not only describes the dynamics of the quasiparticle, bot also includes the influence of the condensate. This is consistent with the two-component description for the charge, in which there exists interplay between the quasiparticle and condensate.Tinkham1 (); Tinkham2 (); Takahashi_SHE2 (); Tinkham_book (); Pethick_review (); Larkin (); Gray ()

When considering the optical excitation by the THz pulses in the superconductor, Eq. (24) can be significantly simplified. Often the spacial dependence in the optical field can be neglected, and hence Eq. (24) can be solved in the homogeneous limit. Specifically, with , and being independent on , the optical Bloch equations [Eq. (24)] are reduced to

(25) |

It is addressed that Eq. (25) is different from the LiouvilleAxt1 (); Axt2 (); Higgs_e_p () or BlochAnderson1 (); multi_component (); Leggett_mode (); Xie (); Tsuji_e_p (); multiband (); Matsunaga_3 (); Matsunaga_4 () equations used in the literature in several aspects. Firstly, the momenta of the two electrons participating in the anomalous correlation are no longer and during the evolution. This is because in the optical kinetic equation here, similar to the Boltzmann equation,Haug (); Kadanoff_Baym (); Hershfield (); Spivak (); kinetic_book (); Kopnin () the Lagrangian description is used, in which the generalized coordinate evolves with time.Landau () Thus, with the anomalous correlation represented by in which is the annihilation operator of the electron, the center-of-mass momentum of the Cooper pairs . Then, with , the acceleration relation in the homogeneous limit [Eq. (15)] can be directly recovered. One sees that it is natural to include the contribution of the center-of-mass momentum in the anomalous correlation in our description. Secondly, in the homogeneous limit, with and being transverse in the presence of the optical field [Eq. (15)], the obtained electrical current is perpendicular to the propagation direction of the optical field. Moreover, the obtained physical quantities are naturally gauge-invariant due to the gauge invariance in and . Furthermore, the effective chemical potential naturally arises from the gauge-invariant treatment in the derivation, which corresponds to the collective excitation, evolving with time in the homogeneous limit.Nambu_gauge (); Ambegaokar (); Enz (); Griffin () Finally, the scattering term can be simply included in our description which is similar to its setup in the Boltzmann equation,Haug (); Kadanoff_Baym (); Hershfield (); Spivak (); kinetic_book (); Kopnin () with the details addressed as follows.

In Eq. (25), and are derived in the GKB ansatz.Tao_1 (); Tao_2 () For the HF term, it is written as

(26) |

In Eq. (26), it is assumed that the renormalization energy due to the Coulomb interaction has been included in the free BdG Hamiltonian [Eq. (5)], and hence the density matrix in the equilibrium state appears in the HF self-energy. Accordingly, the fluctuation of the order parameter is represented by

(27) |

which can be treated as the Higgs mode when the phase fluctuation can be neglected.Soliton (); Critical (); Axt1 (); Axt2 (); Anderson1 (); Higgs_e_p (); multi_component (); Leggett_mode (); Xie (); Tsuji_e_p (); multiband (); path_integral ()

For the scattering terms, both the electron-impurity and electron-phonon interactions are considered, which are written as

(28) | |||

(29) |

In Eq. (28), is the impurity density; in which with ; represent the projection operators. Here,

(30) |

is the unitary transformation matrix from the particle space to the quasiparticle one with and . In Eq. (29), is the -branch–phonon energy with momentum ; represents the phonon distribution function; .

#### ii.2.3 Charge neutrality condition

Equation (23) provides the microscopic description for the quasiparticle dynamics. Moreover, in the -gauge, both the superfluid momentum and the effective chemical potential which are associated with the dynamics of the condensate, appear in Eq. (23), although and still needs to be determined. Thus, the two-component picture naturally arises in our description, in which there exists the interplay between the quasiparticle and condensate.Tinkham1 (); Tinkham2 (); Takahashi_SHE2 (); Tinkham_book (); Pethick_review (); Larkin (); Gray () Actually, this can be directly seen from the modified Bogoliubov transformation in which the creation and annihilation of the Cooper-pair operators and are added,Bardeen (); Tinkham2 (); Josephson_B ()

(31) |

Here, () is the creation operator for the quasi-electron (quasi-hole). From Eq. (31), one has and . By noting that annihilates one Cooper pair with charge , one obtains that () corresponds to create a quasi-electron (quasi-hole) with charge (). Furthermore, one observes that the creation of one quasi-electron and one quasi-hole is associated with the creation and annihilation of the Cooper pair with probability and , respectively. Thus, the net creation of the Cooper pair is , which is positive (negative) when (). Accordingly, when , both quasiparticles and Cooper pairs are created; whereas when , the quasiparticles are created by breaking Cooper pairs.

The above physical picture suggests that in the dynamical process, to maintain the charge neutrality or charge conservation, the Cooper pair condensate has to respond to the dynamics of the quasiparticles.Takahashi_SHE1 (); Takahashi_SHE2 (); Hirashima_SHE (); Takahashi_SHE3 (); Takahashi_SHE_exp (); Hershfield (); Spivak (); kinetic_book () That is to say, in the dynamical process, once the charge imbalance for the quasiparticle is created, the chemical potential of the condensate reacts to screen the extra charge due to the charge imbalance. Hence it is suggested that in Eq. (25), the effective chemical potential is determined from the charge neutrality condition, which actually has been used in the dynamical problem in superconductivity.Takahashi_SHE1 (); Takahashi_SHE2 (); Hirashima_SHE (); Takahashi_SHE3 (); Takahashi_SHE_exp (); Hershfield (); Spivak (); kinetic_book () Specifically, in the quasiparticle space, the particle number with momentum is expressed as

(32) |

with treated as the distribution function of the condensate.Takahashi_SHE1 (); Takahashi_SHE2 (); Hirashima_SHE (); Takahashi_SHE3 (); Takahashi_SHE_exp (); Hershfield (); Spivak (); kinetic_book () When the system is near zero temperature and the equilibrium state, to keep charge neutrality, the chemical potential for the condensate is suggested to be varied .Tinkham_condensate (); Hershfield (); Spivak (); kinetic_book () Then the time evolution of the effective chemical potential can be obtained by solving the self-consistent equation with the quasiparticle density matrix obtained from Eq. (23),Tinkham_condensate (); Hershfield (); Spivak (); kinetic_book ()

(33) |

Here, is the total electron density. From Eq. (33), it can be seen that not only the non-equilibrium quasi-electron and quasi-hole distributions but also the correlation between quasi-electron and quasi-hole states contribute to the charge imbalance.

The superfluid momentum can be obtained from Eq. (15) in the homogeneous limit with the electrical field in the optical pulse known. With the propagation direction of the optical field assumed to be perpendicular to the QWs, i.e., the -direction, the direction of the electrical field is taken to be along the -direction without loss of generality. Thus,

(34) | |||

(35) |

Here, is the strength of the effective electrical field in the superconductorAmbegaokar () and represents the duration time of the optical pulse. In the numerical calculation, .

Finally, we address that Eqs. (25), (33-35) provide the consistent equations to solve the optical response to the THz pulses. Here, the condensate is assumed to react to the quasiparticles simultaneously due to the charge neutrality.Bardeen (); Tinkham2 (); Josephson_B () In our previous work in the study of the quasiparticle spin dynamics with small spin imbalances, it is assumed that the condensation rate is slower than the spin relaxation one and hence the framework with the quasiparticle-number conservation is used.Tao_2 () Therefore, different assumptions for the condensate dynamics can lead to different schemes. Nevertheless, for the problem near the equilibrium, the induced change imbalance is expected to be small and these two schemes can even give similar physical results.

### ii.3 Numerical Results

In this subsection, we present the numerical results by solving the optical Bloch equations [Eqs. (25), (33-35)] in a specific material GaAs QW in proximity to an -wave superconductor. All parameters used in our computation are listed in Table 1.material_book ()

(cm) | |||
---|---|---|---|

In Table 1, for the material parameters, stands for the relative dielectric constant; denotes the well width; and is the mass density of the crystal. For the parameters associated with the electron-phonon interaction, denotes the deformation potential; represents the piezoelectric constant; and are the velocities of LA and TA phonons, respectively.e_p_formula1 (); e_p_formula2 () Finally, is the environment temperature.

With these parameters, we directly estimate the contribution of the electron-AC-phonon interaction in the scattering term at K, compared to the one of the electron-impurity interaction with the typical impurity density . In Eq. (29), at low temperature, . Thus, the electron-AC-phonon interaction is approximately determined by its strength . We explicitly calculate the electron–AC-phonon interaction strength due to the deformation potential in the LA branch and piezoelectric coupling including LA and TA branches, which are found to be about three orders of magnitude smaller than . Thus, the electron-AC-phonon interaction is negligible in our computation.

#### ii.3.1 Excitations of Higgs mode

Recently, it was reported in several experiments in the conventional superconducting metals that the Higgs mode can be excited by the intense THz field, which oscillates with twice the frequency of the THz field.Earliest (); Matsunaga_1 (); Matsunaga_2 (); Matsunaga_3 (); Matsunaga_4 () These experiments also show that there exists plateau for the Higgs mode after the THz pulse in most situations, whose value increases with the increase of the field intensity.Matsunaga_1 (); Matsunaga_2 () Previously, the oscillation of the Higgs mode has been explained by the pump effect from the Anderson pseudo-spin picture, in which the drive effect on the superconducting state is absent.Soliton (); Critical (); Axt1 (); Axt2 (); Anderson1 (); Higgs_e_p (); multi_component (); Leggett_mode (); Xie (); Tsuji_e_p (); multiband () Here, we aim to distinguish the contribution of the pump and drive effects to the evolution of the Higgs mode in GaAs QW in proximity to an -wave superconductor.

##### Different pump regimes

Before we present the numerical results, we first analyze the behavior of the pump effect from a simplified model, from which different regimes are divided according to the pump strength. In the pump term in Eq. (25), with slowly varying with time. The analytical calculation is simplified for high optical frequency , with which the rotation-wave approximationHaug () can be applied with . In this situation, in the free situation without the drive and HF terms, the optical Bloch equations in the quasiparticle space read [refer to Eq. (79)]

(36) |

With the initial state being the equilibrium distribution, the population for the quasi-electron is

(37) |

Here, represents the equilibrium distribution for the quasi-electron with being the Boltzmann constant; , from which it can be seen that directly contributes to the AC stark effect in the energy spectrum.Jinluo (); Jianhua_Liouville ()

According to the behavior of , which is further expressed as

(38) |

one can separate different pump regimes. When , the minimum value of lies at , indicating that the quasi-electron distribution evolves around . This regime with is referred to as the weak-pump regime. Whereas when , the minimum values of are realized when , which is smaller than zero. This indicates that during the pump process, the quasi-electron population mainly arises at and hence the hole-like quasi-electrons are mainly pumped. This regime with is referred to as the strong-pump regime. Actually, in the experiments, with meV for the metal NbN and , meV when the peak electric field is kV/cm, indicating that the experiments lie in the strong-pump regime.Matsunaga_1 (); Matsunaga_2 (); Matsunaga_3 (); Matsunaga_4 ()

##### Weak-pump regime

We first focus on the weak-pump regime. In Figs. 3(a), (b) and (c), the temporal evolutions of the Higgs mode are plotted in the clean (blue solid curves) and dirty (red chain and green dashed curves) samples with different pump frequencies of the optical field , and , respectively ( meV THz). The electric field strength kV/cm. Thus, for , meV is much smaller than , indicating that the system lies in the weak-pump regime. With this electric field strength, the temporal evolutions of the superconducting momentum , which are driven by the optical field [Eq. (34)], are presented in Fig. 3(d) with (the red chain curve) and (the blue solid curve), respectively. It can be seen in Fig. 3(d) that when , the induced supercurrents by the THz pulse is small in magnitude with

. By comparing the oscillation frequencies of the Higgs mode [Figs. 3(a), (b) and (c)] with the ones of the supercurrent [Fig. 3(d)], one finds that the Higgs mode oscillates with twice the frequency of the THz field when both the pump and drive effects exist. Then, the contributions of the pump and drive effects to the Higgs mode are compared in Figs. 3(a) and (b) in the impurity-free situation. It can be seen that without the pump effect, the Higgs modes, plotted by the yellow dotted curves, coincide with the one with both the pump and drive effects, represented by the blue solid curves. This shows that the pump effect is marginal for the excitation of Higgs mode in the weak-pump regime. Moreover, it is found that there always exist plateaus for the Higgs mode after the THz pulse, which are suppressed with the increase of the optical-field frequency, as shown in Figs. 3(a), (b) and (c). Finally, the role of the electron-impurity scattering is addressed. It is shown in Fig. 3(a) [or (b), (c)] by the blue solid, red chain, and green dashed curves that with the increase of the impurity density, the oscillation amplitude of the Higgs mode is suppressed and the plateau value of the Higgs mode increases. These rich features can be understood as follows.