Introduction

Collective excitation modes are a characteristic feature of symmetry-broken phases of matter. These collective excitations are due to amplitude and phase fluctuations of the order parameters, which are decoupled in equilibrium systems, as they represent mutually orthogonal excitations. The properties of collective modes are of fundamental interest, as they are a distinguishing feature of any symmetry-broken phase, such as superconductors, charge-density waves or antiferromagnets. For example, superconductors exhibit an amplitude Higgs mode and a phase mode, which are the radial and angular excitations in the Mexican-hat potential of the free energy. In two-band superconductors there exists in addition a Leggett phase mode1, which corresponds to collective fluctuations of the interband phase difference.

As we show in this work, out-of-equilibrium excitations of symmetry-broken phases lead to a direct coupling between phase and amplitude modes, an effect which is absent in equilibrium systems. Furthermore, we demonstrate that ultrafast pump–probe measurements allow to directly probe and coherently control collective excitation modes. Pump–probe measurements have recently become a key tool to probe the temporal dynamics and relaxation of quantum materials2,3,4,5,6,7,8,9,10,11,12,13. In particular, this technique has been used to measure the oscillations of the amplitude Higgs mode of the one-gap superconductor NbN. It has been shown, both theoretically14,15,16,17,18,19,20,21,22,23,24,25,26,27,28 and experimentally2,3,4, that a short intense laser pulse of length τ much shorter than the dynamical time scale of the superconductor τΔh/(2|Δ|) induces oscillations in the order parameter amplitude at the frequency ωH=2Δ∞/ħ, with Δ∞ the asymptotic gap value.

Although non-equilibrium collective modes in conventional single-gap superconductors are well understood, the investigation of collective excitations in unconventional non-equilibrium superconductors with multiple gaps, such as MgB2 or iron pnictides, is still in its infancy29,30,31. These multicomponent superconductors have a particularly rich spectrum of collective excitations1,32,33. In this study, we simulate the pump–probe process in a two-gap superconductor using a semi-numerical approach based on the density-matrix formalism. This method is exact for mean-field Hamiltonians15,21, captures the coupling between the superconductor and the electromagnetic field of the pump laser at a microscopic level and allows for the calculation of the pump–probe conductivity, as measured in recent experiments2,3. Two-gap superconductors exhibit besides the amplitude Higgs34 and the phase modes35,36, also a Leggett mode1, which results from fluctuations of the relative phase of the two coupled gaps, that is, equal but opposite phase shifts of the two-order parameters, see Fig. 1b. In equilibrium superconductors, the Higgs and Leggett modes are decoupled, as they correspond to mutually orthogonal fluctuations. In contrast to the phase mode, both Higgs and Leggett modes are charge neutral and therefore do not couple to the electromagnetic field37. As a consequence, these excitations cannot be detected directly with standard linear-response-type measurements. Observation of these modes has only been possible in special circumstance, for example, when they couple to another order parameter, such as in charge density wave systems38,39,40,41.

Figure 1: Illustration of Leggett and Higgs modes.
figure 1

(a) Illustration of the excitation process for a one band superconductor. The pump laser pulse modifies the free energy ℱ on different time scales depending on the pulse duration τ. For τ h/(2|Δ|) the superconductor can follow the change in ℱ adiabatically, resulting in a monotonic lowering of the order parameter |Δ| (orange trace in inset (II)). For short pulses with τ≲h/(2|Δ|), on the other hand, the superconductor is excited in a non-adiabatic manner, which results in oscillations of |Δ| about the new minimum of ℱ (red trace in inset (I)). The blue and cyan lines in the two insets represent the Gaussian profiles of the pump pulses. (b) Effective free-energy landscape ℱ for a two-gap superconductor, with green and red representing the Mexican-hat potentials of the smaller and larger gaps, respectively. The amplitude Higgs modes and the phase modes are indicated by red and blue/black arrows, respectively. The Leggett mode corresponds to out-of-phase fluctuations of the phase difference between the two gaps.

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Here we show that in a pump–probe experiment both Leggett and Higgs modes can be excited out of equilibrium, and directly observed as oscillations in the absorption spectra at their respective frequencies. We find that the non-adiabatic excitation process of the pump pulse induces an intricate coupling between the two charge-neutral modes, which pushes the frequency of the Leggett mode below the continuum of two-particle excitations. Moreover, the frequencies of the Leggett and Higgs modes and the coupling between them can be controlled by the fluence of the pump pulse. Hence, by adjusting the laser intensity the two modes can be brought into resonance, which greatly enhances their oscillatory signal in the pump–probe absorption spectra.

Results

Pump-excitation process

In pump–probe measurements, the pump laser pulse excites a high density of quasiparticles above the gap of the order parameter, thereby modifying the Mexican-hat potential of the free energy ℱ (Fig. 1). As a result, the amplitude of the order parameter decreases, reducing the minimum of the free energy. If the pump-pulse-induced changes in ℱ occur on a faster time scale than the intrinsic response time of the symmetry-broken state, the collective modes start to oscillate at their characteristic frequencies about the new free-energy minimum (see Fig. 1a). In this work we study this non-adiabatic excitation mechanism for two-band superconductors perturbed by a short and intense pump pulse.

The Hamiltonian describing the two-band superconductor coupled to the pump laser field is given by H=HBCS+Hlaser, with the two-band Bardeen-Cooper-Schrieffer (BCS) mean-field Hamiltonian

where denotes the normal state Hamiltonian and creates electrons with momentum k, band index l and spin σ. The first sum in equation (1) is taken over the set W of momentum vectors with |ɛk l|≤ħωc=50 meV, ωc being the cutoff frequency. The gaps Δ1 and Δ2 in the two bands are determined at each temporal integration step from the BCS gap equations with the attractive intraband pairing interactions V1 and V2, and the interband coupling V12=V1. Motivated by the numbers for MgB2 (ref. 42), we fix V1 and V2 such that the gaps in the initial state take on the values Δ1(ti)=7 meV and Δ2(ti)=3 meV, and study the dynamics of the two-gap superconductor as a function of the relative interband coupling . Hlaser represents the interaction of the pump laser with the superconductor and contains terms linear and quadratic in the vector potential of the laser field, which is of Gaussian shape with central frequency ħω0=8 meV, pulse width τ=0.4 ps and light-field amplitude |A0|. We determine the dynamics of Hamiltonian (1) by means of the density matrix approach and solve the resulting equations of motion using Runge–Kutta integration (see Methods).

Pump response

Pumping the two-band superconductor with a short laser pulse of length τ τΔ excites a non-thermal distribution of Bogoliubov quasiparticles above the gaps Δi, which in turn leads to a rapid, non-adiabatic change in the free-energy landscape ℱ (Fig. 1). As a result, the collective modes of the superconductor start to oscillate about the new minima of ℱ. This is clearly visible in Fig. 2, which shows the temporal evolution of the gap amplitudes |Δi| and of the phase difference Φ1–Φ2 between the two gaps. From the Fourier spectra in Figs 2d–f we can see that three different modes (and their higher harmonics) are excited at the frequencies ωH1, ωH2 and ωL. The two modes at ωH1 and ωH2 only exist in the dynamics of Δ1(t) and Δ2(t), respectively, and their peaks are located at the energy of the superconducting gaps , where denotes the asymptotic gap value14,15,16,17,18,19,20. This holds for all parameter regimes, even as the laser fluence is increased far beyond the linear absorption region (see Fig. 5). We therefore assign the peaks at ωH1 and ωH2 to the Higgs amplitude modes of the two gaps. The higher Higgs mode ωH1 is strongly damped, because it lies within the continuum of Bogoliubov quasiparticle excitations, which is lower bounded by . For the lower mode ωH2, on the other hand, the decay channel to quasiparticles is small, as ωH2 is at the continuum threshold. This is similar to the non-equilibrium Higgs mode of the single-gap superconductor NbN, whose oscillations have recently been observed over a time period of about 10 ps by pump–probe measurements2,3.

Figure 2: Leggett phase mode and amplitude Higgs mode oscillations.
figure 2

Numerical simulation of the gap dynamics of a two-gap superconductor after a non-adiabatic excitation by a short intense laser pulse of width τ=0.4 ps, pump energy ħω0=8 meV and light-field amplitude |A0|=10 × 10−8 Js(Cm)−1.(a) Phase difference Φ1–Φ2 between the two gaps as a function of time t for various interband coupling strengths . (d) Fourier spectrum of the oscillations in a. The frequency of the non-equilibrium Leggett mode oscillation is indicated by ωL. (b,c) Gap amplitudes |Δ1| and |Δ2| as a function of time t for different interband couplings . (e,f) Fourier spectra of the amplitude mode oscillations in b,c, which display the following frequencies: ωH1 and ωH2 are the frequencies of the non-equilibrium Higgs modes of gap Δ1 and Δ2, respectively; ωL is the frequency of the non-equilibrium Leggett mode; and higher harmonics of the non-equilibrium Leggett mode are denoted by 2ωL and 4ωL.

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Figure 3: Leggett phase mode oscillations versus relative interband coupling.
figure 3

Fourier spectrum of the phase mode Φ1–Φ2 as a function of relative interband coupling v for a two-band superconductor perturbed by the same laser pulse as in Fig. 2. The amplitude of the phase fluctuations is indicated by the colour scale with dark red and light yellow representing the highest and lowest amplitudes, respectively. The blue open circles mark the frequency of the non-equilibrium Leggett mode ωL. The blue solid line represents the frequency of the equilibrium Leggett mode described by equation (2). The dashed grey line indicates the frequency of the Higgs mode ωH2, which coincides with the boundary to the continuum of Bogoliubov quasiparticle excitations, given by twice the asymptotic gap value of the second band . The inset shows a zoom-in of the blue frame in the main panel.

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Figure 4: Amplitude mode oscillations versus relative interband coupling.
figure 4

Fourier spectrum of the amplitude mode oscillations as a function of relative interband coupling for (a) the superconducting gap Δ1 in the first band and (b) the superconducting gap Δ2 in the second band. The parameters of the laser pump pulse are the same as in Fig. 2. The amplitude of the oscillations is indicated by the colour scale with dark red and light yellow representing the highest and lowest amplitudes, respectively. The open circles represent the frequencies of the non-equilibrium Leggett mode ωL and its higher harmonics denoted by 2ωL and 4ωL. The frequencies of the non-equilibrium Higgs mode of the first and second band, ωH1 and ωH2, are indicated by the grey open squares and triangles, respectively. The blue solid line is the frequency of the equilibrium Leggett mode given by equation (2).

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Figure 5: Fluence dependence of gap dynamics.
figure 5

(a,b) Fourier spectrum of the phase mode Φ1–Φ2 as a function of laser fluence (integrated pulse intensity) |A0|2τ for two different interband couplings with pulse energy ħω0=8 meV and pulse width τ=0.4 ps. The amplitude of the phase fluctuations is represented by the colour scale with dark red and light yellow indicating high and low amplitudes, respectively. The open circles mark the frequencies of the non-equilibrium Leggett mode ωL and its higher harmonic 2ωL. The grey dotted lines display the Higgs mode ωH2, which coincides with the boundary to the continuum of Bogoliubov quasiparticle excitations. (c–f) Gap amplitudes |Δ1| and |Δ2| as a function of time t for two different interband couplings v and integrated pulse intensities |A0|2τ.

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Interestingly, two-band superconductors exhibit a third collective mode besides the two Higgs modes at a frequency ωL below the quasiparticle continuum. This mode is most clearly visible in the dynamics of the phase difference Φ1–Φ2 (Fig. 2a) and displays a striking dependence on interband coupling strength . With decreasing its frequency rapidly decreases, whereas its intensity grows. In the limit of vanishing , however, the third mode ωL is completely absent. We thus identify ωL as the Leggett phase mode, that is, as equal but opposite oscillatory phase shifts of the two coupled gaps. Remarkably, the Leggett phase mode is also observable in the time dependence of the gap amplitudes Δ1(t) and Δ2(t) (Figs 2b,c), which indicates that Higgs and Leggett modes are coupled in non-equilibrium superconductors.

To obtain a more detailed picture, we plot in Figs 3 and 4 the energies of the amplitude and phase mode oscillations against the relative interband coupling . This reveals that for small the non-equilibrium Leggett mode ωL shows a square root increase, which is in good agreement with the equilibrium Leggett frequency1,43

where ρ1 and ρ2 denote the density of states on the two bands. Indeed, as displayed by the inset of Fig. 3, equation (2) represents an excellent parameter-free fit to the numerical data at low . For larger , on the other hand, the non-equilibrium Leggett mode deviates from the square root behaviour of equation (2). That is, as ωL approaches the Bogoliubov quasiparticle continuum, it is repelled by the lower Higgs mode ωH2, evidencing a strong coupling between them. As a result, the non-equilibrium Leggett mode is pushed below the continuum and remains nearly undamped for a wide range of , which is considerably broader than in equilibrium. Moreover, owing to the dynamical coupling among the collective modes, ωL and its higher harmonics are observable not only in the phase difference Φ1–Φ2 but also in the dynamics of the gap amplitudes Δi(t) (blue and green circles in Fig. 4).

A key advantage of measuring collective modes by pump–probe experiments is that the frequencies of the Higgs modes can be adjusted by the pump fluence. This is demonstrated in Fig. 5, which plots the dynamics of Δi(t) and Φ1–Φ2 as a function of integrated pump pulse intensity |A0|2τ. With increasing pump fluence, more Cooper pairs are broken up and superconductivity is more and more suppressed, as reflected in the reduction of the gap amplitudes. At the same time, the frequency of the Higgs oscillations decreases, as it is controlled by the superconducting gaps after pumping. Hence, it is possible to tune the lower Higgs mode ωH2 to resonance with ωL, which strongly enhances the magnitude of the collective-mode oscillations (Fig. 5a,c,e). A similar enhancement is obtained when ωH2 is brought into resonance with twice the frequency of the Leggett mode (Fig. 5b,d,f).

Pump–probe signal

Finally, let us discuss how the Higgs and Leggett modes can be observed in pump–probe spectroscopy. In view of the recent THz pump–THz probe experiments of refs 2, 3, 4, we focus on the dynamics of the optical pump–probe conductivity σ(δt, ω)=j(δt, ω)/[iωA(δt,ω)], where δt is the delay time between pump and probe pulses, j(δt, ω) denotes the current density and A(δt, ω) represents the vector potential of the probe pulse. As the probe pulse has a weak intensity, we neglect terms of second order and higher in the probe field A(δt, ω). Similar to recent experiments2,3,4,5,6,7, we take the probe pulse to be very short with width τpr=0.15 ps and centre frequency ħωpr=5.5 meV (see Methods). With this choice, the probe pulse has a broad spectral bandwidth such that the dynamics of the superconductor is probed over a very wide frequency range.

In Fig. 6a we plot the real part of the pump–probe response Re[σ(δt, ω)] as a function of frequency for fixed δt. The non-equilibrium Leggett mode ωL and the Higgs-mode ωH2 reveal themselves in the pump–probe signal as sharp peaks. Figure 6b shows Re[σ(δt, ω)] versus delay time δt and frequency ω. Clear oscillations are seen as a function of delay time δt. These are most prominent at the frequencies of the lower Higgs and the Leggett modes, ωH2 and ωL, where σ(δt, ω) displays sharp edges as a function of ω (Fig. 6a). Fourier transforming with respect to δt shows that the dominant oscillations are ωH2 and ωL (and its higher harmonics) (Fig. 6b). We therefore predict that both the lower Higgs mode ωH2 and the Leggett mode ωL can be observed in THz pump–THz probe experiments as oscillations of the pump–probe conductivity, in particular at the gap edge and the Leggett mode frequency ωL. The higher Higgs mode ωH1, on the other hand, is not visible in the pump–probe signal, as it is strongly damped by the two-particle continuum.

Figure 6: Pump–probe spectrum.
figure 6

(a) Real part of the pump–probe response Re[σ(δt,ω)] as a function of frequency ω for a two-band superconductor excited by the same pump pulse as in Fig. 2. Both the energy of the non-equilibrium Leggett mode ωL and the Higgs mode ωH2 are visible in the pump–probe signal as sharp peaks. (b) Temporal evolution of Re[σ(δt,ω)]. The intensity of the pump–probe signal is represented by the colour scale with dark red and dark violet, indicating high and low intensities, respectively. (c) Fourier spectrum of the pump–probe signal. The pump–probe spectrum oscillates as a function of pump–probe delay time δt with the frequencies of the non-equilibrium Higgs mode ωH2 (dashed gray), the frequency of the non-equilibirum Leggett mode ωL (dashed blue) and the frequencies of higher harmonics of the Leggett mode 2ωL (dashed green) and 4ωL (dashed red).

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Discussion

Using a semi-numerical method based on the density matrix approach, we have studied the non-equilibrium excitation of Higgs and Leggett modes in two-band superconductors. Although the amplitude Higgs and the Leggett phase mode are decoupled in equilibrium, we find that the out-of-equilibrium excitation process leads to a strong coupling between these two collective modes. As a result, the Leggett phase mode ωL is pushed below the Bogoliubov quasiparticle continuum and remains undamped for a wide range of interband couplings (Figs 3 and 4). Likewise, the lower Higgs mode ωH2 is only weakly damped, as its frequency is at the threshold to the quasiparticle continuum. To maximize the oscillatory signal of these collective modes in the pump–probe spectra, it is necessary to choose the experimental parameters as follows: (i) the pump-pulse duration τ should be smaller than the intrinsic response time of the superconductor h/(2|Δi|), such that the collective modes are excited in a non-adiabatic manner; (ii) the pump-pulse energy needs to be of the order of the superconducting gap (that is, in the terahertz regime), so that Bogoliubov quasiparticles are excited across the gap, but modes at higher energies ħω|Δi| are not populated; and (iii) the pump-pulse intensity must not exceed a few nJ cm−2, to ensure that the superconducting condensate is only partially broken up, but not completely destroyed. We have predicted that under these conditions both Higgs and Leggett modes can be observed as clear oscillations in the time-resolved pump–probe absorption spectra (Fig. 6). Similarly, we expect that collective mode oscillations are visible in other pump–probe-type experiments, for example, in time-resolved photoemission spectroscopy or time-resolved Raman scattering.

Our findings apply beyond the scope of two-band superconductors to general collective modes in quantum materials. That is, we expect that out-of-equilibrium excitations lead to the coupling of collective modes in any symmetry-broken phase. It would be particularly intriguing to study this in more detail for the case of unconventional exotic superconductors, where several competing orders are present, such as heavy fermion superconductors or high-temperature cuprate and pnictide superconductors. In these systems the pump pulse could be used to induce a transition from one competing order to another. Furthermore, the unconventional pairing symmetries of these superconductors, such as the -wave pairing of the cuprates, give rise to a multitude of new Higgs modes44. Our work indicates that pump–probe experiments will allow to coherently excite and control these novel Higgs modes, which await to be further explored both theoretically and experimentally.

Methods

Model definition

The gap equations for the BCS Hamiltonian HBCS (see equation (1) in the main text) are given by45

where N is the number of lattice points, V1 and V2 denote the intraband interactions and V12=V1 is the interband coupling. The two-band superconductor is brought out of equilibrium via the coupling to a pump pulse, which is modelled by

where ml is the effective electron mass of the lth band and Aq(t) represents the transverse vector potential of the pump laser. We consider a Gaussian pump pulse described by

with central frequency ω0, pulse width τ, light-field amplitude A0=|A0|êy and photon wave vector q0=q0êx.

Density matrix formalism

To simulate the non-equilibrium dynamics of the two-band superconductor (1), we use a semi-numerical method based on the density matrix formalism. This approach involves the analytical derivation of equations of motions for the Bogoliubov quasiparticle densities , , and , which are then integrated up numerically using a Runge Kutta algorithm. The Bogoliubov quasiparticle densities are defined in terms of the fermionic operators αk,l and βk,l, with

where , and . We emphasize that the coefficients uk,l and k,l do not depend on time, that is, the temporal evolution of the quasiparticle densities is computed with respect to a fixed time-independent Bogoliubov-de Gennes basis in which the initial state is diagonal. The equations of motion for the quasiparticle densities are readily obtained from Heisenberg’s equation of motion. Since equation (1) represents a mean-field Hamiltonian, this yields a closed system of differential equations and hence no truncation is needed (for details, see refs 17, 29, 23, 25).

Pump–probe response

All physical observables, such as the current density , can be expressed in terms of the quasiparticle densities. For the current density we find that

where and qpr=|qpr|êx are the vector potential and the wave vector of the probe pulse, respectively. With this, we obtain the pump–probe conductivity via23,46

where j(δt,ω) and A(δt,ω) denote the Fourier transformed y components of the current density and the vector potential , respectively. To compute the effects of the probe pulse, we neglect terms of second order and higher in the probe field Apr(t), as the probe pulse has a very weak intensity.

Numerical discretization and integration

To keep the number of equations of motion manageable, we have to restrict the number of considered points in momentum space. The first restriction is that we only take expectation values with indices k and k+q∈W into account. This means that we concentrate on the k-values where the attractive pairing interaction takes place. Furthermore, as the external electromagnetic field may add or subtract only momentum n q0, it is sufficient to consider expectation values with indices (k, k + n q0), where n∈ℤ. For small amplitudes the off-diagonal elements of the quasiparticle densities decrease rapidly as n increases, as . Thus, we set all entries with n>4 to 0. With this momentum–space discretization, we obtain of the order of 105 equations, which we are able to solve numerically using high-efficiency parallelization. Further, technical details can be found in refs 23, 25.

Data availability

All relevant numerical data are available from the authors upon request.

Additional information

How to cite this article: Krull, H. et al. Coupling of Higgs and Leggett modes in non-equilibrium superconductors. Nat. Commun. 7:11921 doi: 10.1038/ncomms11921 (2016).