Abstract
In equilibrium systems amplitude and phase collective modes are decoupled, as they are mutually orthogonal excitations. The direct detection of these Higgs and Leggett collective modes by linearresponse measurements is not possible, because they do not couple directly to the electromagnetic field. In this work, using numerical exact simulations we show for the case of twogap superconductors, that optical pump–probe experiments excite both Higgs and Leggett modes out of equilibrium. We find that this nonadiabatic excitation process introduces a strong interaction between the collective modes, which is absent in equilibrium. Moreover, we propose a type of pump–probe experiment, which allows to probe and coherently control the Higgs and Leggett modes, and thus the order parameter directly. These findings go beyond twoband superconductors and apply to general collective modes in quantum materials.
Introduction
Collective excitation modes are a characteristic feature of symmetrybroken 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 symmetrybroken phase, such as superconductors, chargedensity waves or antiferromagnets. For example, superconductors exhibit an amplitude Higgs mode and a phase mode, which are the radial and angular excitations in the Mexicanhat potential of the free energy. In twoband superconductors there exists in addition a Leggett phase mode^{1}, which corresponds to collective fluctuations of the interband phase difference.
As we show in this work, outofequilibrium excitations of symmetrybroken 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 materials^{2,3,4,5,6,7,8,9,10,11,12,13}. In particular, this technique has been used to measure the oscillations of the amplitude Higgs mode of the onegap superconductor NbN. It has been shown, both theoretically^{14,15,16,17,18,19,20,21,22,23,24,25,26,27,28} and experimentally^{2,3,4}, that a short intense laser pulse of length τ much shorter than the dynamical time scale of the superconductor τ_{Δ}h/(2Δ) induces oscillations in the order parameter amplitude at the frequency ω_{H}=2Δ_{∞}/ħ, with Δ_{∞} the asymptotic gap value.
Although nonequilibrium collective modes in conventional singlegap superconductors are well understood, the investigation of collective excitations in unconventional nonequilibrium superconductors with multiple gaps, such as MgB_{2} or iron pnictides, is still in its infancy^{29,30,31}. These multicomponent superconductors have a particularly rich spectrum of collective excitations^{1,32,33}. In this study, we simulate the pump–probe process in a twogap superconductor using a seminumerical approach based on the densitymatrix formalism. This method is exact for meanfield Hamiltonians^{15,21}, 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 experiments^{2,3}. Twogap superconductors exhibit besides the amplitude Higgs^{34} and the phase modes^{35,36}, also a Leggett mode^{1}, which results from fluctuations of the relative phase of the two coupled gaps, that is, equal but opposite phase shifts of the twoorder 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 field^{37}. As a consequence, these excitations cannot be detected directly with standard linearresponsetype 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 systems^{38,39,40,41}.
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 nonadiabatic excitation process of the pump pulse induces an intricate coupling between the two chargeneutral modes, which pushes the frequency of the Leggett mode below the continuum of twoparticle 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
Pumpexcitation 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 Mexicanhat 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 pumppulseinduced changes in ℱ occur on a faster time scale than the intrinsic response time of the symmetrybroken state, the collective modes start to oscillate at their characteristic frequencies about the new freeenergy minimum (see Fig. 1a). In this work we study this nonadiabatic excitation mechanism for twoband superconductors perturbed by a short and intense pump pulse.
The Hamiltonian describing the twoband superconductor coupled to the pump laser field is given by H=H_{BCS}+H_{laser}, with the twoband BardeenCooperSchrieffer (BCS) meanfield 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 ɛ_{k l}≤ħω_{c}=50 meV, ω_{c} being the cutoff frequency. The gaps Δ_{1} and Δ_{2} in the two bands are determined at each temporal integration step from the BCS gap equations with the attractive intraband pairing interactions V_{1} and V_{2}, and the interband coupling V_{12}=V_{1}. Motivated by the numbers for MgB_{2} (ref. 42), we fix V_{1} and V_{2} such that the gaps in the initial state take on the values Δ_{1}(t_{i})=7 meV and Δ_{2}(t_{i})=3 meV, and study the dynamics of the twogap superconductor as a function of the relative interband coupling . H_{laser} 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 ħω_{0}=8 meV, pulse width τ=0.4 ps and lightfield amplitude A_{0}. 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 twoband superconductor with a short laser pulse of length τ τ_{Δ} excites a nonthermal distribution of Bogoliubov quasiparticles above the gaps Δ_{i}, which in turn leads to a rapid, nonadiabatic change in the freeenergy 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 Δ_{i} and of the phase difference Φ_{1}–Φ_{2} 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 ω_{H1}, ω_{H2} and ω_{L}. The two modes at ω_{H1} and ω_{H2} only exist in the dynamics of Δ_{1}(t) and Δ_{2}(t), respectively, and their peaks are located at the energy of the superconducting gaps , where denotes the asymptotic gap value^{14,15,16,17,18,19,20}. 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 ω_{H1} and ω_{H2} to the Higgs amplitude modes of the two gaps. The higher Higgs mode ω_{H1} is strongly damped, because it lies within the continuum of Bogoliubov quasiparticle excitations, which is lower bounded by . For the lower mode ω_{H2}, on the other hand, the decay channel to quasiparticles is small, as ω_{H2} is at the continuum threshold. This is similar to the nonequilibrium Higgs mode of the singlegap superconductor NbN, whose oscillations have recently been observed over a time period of about 10 ps by pump–probe measurements^{2,3}.
Interestingly, twoband superconductors exhibit a third collective mode besides the two Higgs modes at a frequency ω_{L} below the quasiparticle continuum. This mode is most clearly visible in the dynamics of the phase difference Φ_{1}–Φ_{2} (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 ω_{L} is completely absent. We thus identify ω_{L} 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 Δ_{1}(t) and Δ_{2}(t) (Figs 2b,c), which indicates that Higgs and Leggett modes are coupled in nonequilibrium 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 nonequilibrium Leggett mode ω_{L} shows a square root increase, which is in good agreement with the equilibrium Leggett frequency^{1,43}
where ρ_{1} and ρ_{2} denote the density of states on the two bands. Indeed, as displayed by the inset of Fig. 3, equation (2) represents an excellent parameterfree fit to the numerical data at low . For larger , on the other hand, the nonequilibrium Leggett mode deviates from the square root behaviour of equation (2). That is, as ω_{L} approaches the Bogoliubov quasiparticle continuum, it is repelled by the lower Higgs mode ω_{H2}, evidencing a strong coupling between them. As a result, the nonequilibrium 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, ω_{L} and its higher harmonics are observable not only in the phase difference Φ_{1}–Φ_{2} but also in the dynamics of the gap amplitudes Δ_{i}(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 Δ_{i}(t) and Φ_{1}–Φ_{2} as a function of integrated pump pulse intensity A_{0}^{2}τ. 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 ω_{H2} to resonance with ω_{L}, which strongly enhances the magnitude of the collectivemode oscillations (Fig. 5a,c,e). A similar enhancement is obtained when ω_{H2} 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 experiments^{2,3,4,5,6,7}, we take the probe pulse to be very short with width τ_{pr}=0.15 ps and centre frequency ħω_{pr}=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 nonequilibrium Leggett mode ω_{L} and the Higgsmode ω_{H2} 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, ω_{H2} and ω_{L}, 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 ω_{H2} and ω_{L} (and its higher harmonics) (Fig. 6b). We therefore predict that both the lower Higgs mode ω_{H2} and the Leggett mode ω_{L} 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 ω_{L}. The higher Higgs mode ω_{H1}, on the other hand, is not visible in the pump–probe signal, as it is strongly damped by the twoparticle continuum.
Discussion
Using a seminumerical method based on the density matrix approach, we have studied the nonequilibrium excitation of Higgs and Leggett modes in twoband superconductors. Although the amplitude Higgs and the Leggett phase mode are decoupled in equilibrium, we find that the outofequilibrium excitation process leads to a strong coupling between these two collective modes. As a result, the Leggett phase mode ω_{L} 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 ω_{H2} 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 pumppulse duration τ should be smaller than the intrinsic response time of the superconductor h/(2Δ_{i}), such that the collective modes are excited in a nonadiabatic manner; (ii) the pumppulse 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 ħωΔ_{i} are not populated; and (iii) the pumppulse 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 timeresolved pump–probe absorption spectra (Fig. 6). Similarly, we expect that collective mode oscillations are visible in other pump–probetype experiments, for example, in timeresolved photoemission spectroscopy or timeresolved Raman scattering.
Our findings apply beyond the scope of twoband superconductors to general collective modes in quantum materials. That is, we expect that outofequilibrium excitations lead to the coupling of collective modes in any symmetrybroken 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 hightemperature 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 modes^{44}. 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 H_{BCS} (see equation (1) in the main text) are given by^{45}
where N is the number of lattice points, V_{1} and V_{2} denote the intraband interactions and V_{12}=V_{1} is the interband coupling. The twoband superconductor is brought out of equilibrium via the coupling to a pump pulse, which is modelled by
where m_{l} is the effective electron mass of the lth band and A_{q}(t) represents the transverse vector potential of the pump laser. We consider a Gaussian pump pulse described by
with central frequency ω_{0}, pulse width τ, lightfield amplitude A_{0}=A_{0}ê_{y} and photon wave vector q_{0}=q_{0}ê_{x}.
Density matrix formalism
To simulate the nonequilibrium dynamics of the twoband superconductor (1), we use a seminumerical 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 α_{k,l} and β_{k,l}, with
where , and . We emphasize that the coefficients u_{k,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 timeindependent Bogoliubovde 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 meanfield 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 q_{pr}=q_{pr}ê_{x} are the vector potential and the wave vector of the probe pulse, respectively. With this, we obtain the pump–probe conductivity via^{23,46}
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 A_{pr}(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 kvalues where the attractive pairing interaction takes place. Furthermore, as the external electromagnetic field may add or subtract only momentum n q_{0}, it is sufficient to consider expectation values with indices (k, k + n q_{0}), where n∈ℤ. For small amplitudes the offdiagonal 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 10^{5} equations, which we are able to solve numerically using highefficiency 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 nonequilibrium superconductors. Nat. Commun. 7:11921 doi: 10.1038/ncomms11921 (2016).
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Acknowledgements
We gratefully acknowledge many useful discussions with A. Avella, S. Kaiser and R. Shimano. G.S.U. and H.K. acknowledge financial support by the DFG in TRR 160 and by the Stiftung Mercator. H.K. thanks the MaxPlanckInstitut FKF Stuttgart for its hospitality.
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The density matrix simulations were developed and run by H.K., N.B. and A.P.S. All authors contributed to the discussion and interpretation of the results, and to the writing of the paper.
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Krull, H., Bittner, N., Uhrig, G. et al. Coupling of Higgs and Leggett modes in nonequilibrium superconductors. Nat Commun 7, 11921 (2016). https://doi.org/10.1038/ncomms11921
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