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arXiv:1906.02751 · Physical review. B./Physical review. B

Higgs modes in proximized superconducting systems

Vasilii Vadimov, Ivan M. Khaymovich, Alexander S. Mel'nikov

Published Sep 16, 2019• 10 citations
arXivHTMLPDF

Abstract

The proximity effect in hybrid superconducting--normal-metal structures is shown to affect strongly the coherent oscillations of the superconducting order parameter $\mathrm{\ensuremath{\Delta}}$ known as the Higgs modes. The standard Higgs mode at frequency $2\mathrm{\ensuremath{\Delta}}$ is damped exponentially by the quasiparticle leakage from the primary superconductor. Two new Higgs modes with the frequencies depending on both the primary and induced gaps in the hybrid structure are shown to appear due to the coherent electron transfer between the superconductor and the normal metal. Altogether, these three modes determine the long-time asymptotic behavior of the superconducting order parameter disturbed either by the electromagnetic pulse or the quench of the system parameters and thus are of crucial importance for the dynamical properties and restrictions on the operating frequencies for superconducting devices based on the proximity effect used, e.g., in quantum computing, in particular, with topological low-energy excitations.

DOI
10.1103/physrevb.100.104515

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The proximity effect in hybrid superconducting--normal-metal structures is shown to affect strongly the coherent oscillations of the superconducting order parameter \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ known as the Higgs modes. The standard Higgs mode at frequency 2\ensuremathΔ2\mathrm{\ensuremath{\Delta}}2\ensuremathΔ is damped exponentially by the quasiparticle leakage from the primary superconductor. Two new Higgs modes with the frequencies depending on both the primary and induced gaps in the hybrid structure are shown to appear due to the coherent electron transfer between the superconductor and the normal metal. Altogether, these three modes determine the long-time asymptotic behavior of the superconducting order parameter disturbed either by the electromagnetic pulse or the quench of the system parameters and thus are of crucial importance for the dynamical properties and restrictions on the operating frequencies for superconducting devices based on the proximity effect used, e.g., in quantum computing, in particular, with topological low-energy excitations.
We consider a three-terminal system consisting of a quantum dot strongly coupled to two superconducting reservoirs in the infinite-gap limit and weakly coupled to a normal metal. Using a real-time diagrammatic approach, we calculate the dynamics of the proximity-induced pair amplitude on the quantum dot. We find that after a quench the pair amplitude shows pronounced oscillations with a frequency determined by the coupling to the superconductors. In addition, it decays exponentially on a timescale set by the coupling to the normal metal. Strong oscillations of the pair amplitude occur also when the system is periodically driven both in the adiabatic and fast-driving limit. We relate the dynamics of the pair amplitude to the Josephson and Andreev current through the dot to demonstrate that it is an experimentally accessible quantity.
This thesis reports on theoretical investigations of the ultrafast nonequilibrium dynamics in superconductors. In particular, two nonequilibrium scenarios are modeled by combining several analytical and numerical methods: transient superconducting phase transition and collective excitations of a superconducting condensate. In the first part of the thesis the possibility to induce singlet superconducting correlations far from equilibrium is explored by applying time-dependent Lanczos algorithm to the one-dimensional extended Hubbard model. Here, the system is driven into a nonequilibrium state by (i) interaction quench and (ii) quench by pulse. In both cases the buildup of the superconducting correlations is studied in view of several correlation functions. Moreover, a transient Meissner effect, which reflects fingerprints of the induced superconductivity, is observed in calculations of the important for experiments time-dependent optical conductivity. In addition, the differences between nonequilibrium dynamics of the extended Hubbard model initiated by interaction quench and by pulse quench are discussed in detail. The second part is devoted to the excitation of collective modes in a superconductor out of equilibrium. First of all, focus is given to the properties of collective modes close to equilibrium conditions, where a linearized response can be expected. Within the framework of the (Nambu-) matrix kinetic theory in the clean limit a comprehensive gauge-invariant description of the collective modes associated with phase fluctuations of the order parameter is analytically formulated for the generalized case of two- band superconducting systems. Particular emphasis is on the application to the recently discovered noncentrosymmetric superconductors. A detailed analysis of the collective modes includes specification of the gauge mode (common to all superconductors) and the Leggett mode (collective fluctuations of the interband phase difference), and investigation of their subtle interplay in view of validity of the charge conservation law and the participation in the Anderson-Higgs mechanism. Further, the nonequilibrium excitation of the superconducting gaps for the case of two-band superconductors with singlet pairing correlations is investigated in a view of optical pump-probe experiments. For this purpose a semi-numerical method based on the density- matrix theory is formulated. In this case additional amplitude (Higgs) modes were identified, which allow to directly detect the properties of the superconducting condensates as a function of time. Moreover, a new interplay between the phase (Leggett) and amplitude (Higgs) modes is studied in detail. In particular it was found that the nonadiabatic excitation process leads to strong coupling between both modes. Finally, based on calculations of the time-dependent optical conductivity the prediction is made that the coupled Higgs and Leggett modes can be clearly visible in the pump-probe absorption spectra.
In superconductors gauge U(1) symmetry is spontaneously broken. According to Goldstone theorem this breaking of a continuous symmetry establishes the existence of the Bogoliubov phase mode while the gauge-invariant response also includes the amplitude fluctuations of the order parameter. The latter, which are also termed ’Higgs’ modes in analogy with the standard model, appear at the energy of the spectral gap 2Δ when the superconducting ground state is evaluated within the weak coupling BCS theory and are therefore damped. Previously we have shown that in the time-dependent Gutzwiller approximation (TDGA) Higgs modes appear inside the gap with a finite binding energy relative to the quasiparticle continuum. Here we show that the binding energy of the Higgs mode becomes exponentially small in the weak coupling limit converging to the BCS solution. On the other hand, well-defined undamped amplitude modes exist in strongly coupled superconductors when the interaction energy becomes of the order of the bandwidth.
We study the effect of strong electron-phonon interactions on the damping of the Higgs amplitude mode in superconductors by means of nonequilibrium dynamical mean-field simulations of the Holstein model. In contrast to the BCS dynamics, we find that the damping of the Higgs mode strongly depends on the temperature, becoming faster as the system approaches the transition temperature. The damping at low temperatures is well described by a power law, while near the transition temperature the damping shows exponential-like behavior. We explain this crossover in terms of a temperature-dependent quasiparticle lifetime caused by the strong electron-phonon coupling, which smears the superconducting gap edge and makes the relaxation of the Higgs mode into quasiparticles more efficient at elevated temperatures. We also reveal that the phonon dynamics can soften the Higgs mode, which results in a slower damping.
The performance of many superconducting devices is degraded in presence of non-equilibrium quasiparticles in the superconducting part. One promising approach towards their evacuation is the use of normal-metal quasiparticle traps, where normal metal is brought into good metallic contact with the superconductor. A voltage biased normal-metal--insulator--superconductor junction equipped with such a trap is used to investigate on the trapping performance and the part played by the superconducting proximity effect therein. This involves an appropriate one-dimensional model of the junction and the numerical solution of Usadel equations describing the non-equilibrium state of the superconductor. The functionality of the trap is determined by the density of states (DOS) at the tunnel barrier. Herein, the proximity effect leads to two antagonistic characteristics affecting the trapping performance: the beneficial reduction of the DOS at an energy ∣E∣=ΔBCS|E| = Δ_{\text{BCS}}∣E∣=ΔBCS​ versus the contraction of the spectral energy gap causing quasiparticle poisoning. For both effects the trap position is decisive, which needs to be taken into account for optimizing the trapping performance. In addition, the conversion between dissipative normal and supercurrent inside the superconducting part with its impact on the quasiparticle density is studied.
We extend a recently developed projective circuit quantisation approach to incorporate superconducting Higgs modes associated to gap dynamics. This approach starts from a microscopic fermionic Hamiltonian for mesoscopic superconductors, and projects the system onto its low-energy "BCS" Hilbert space. We derive analytical results for the superconducting Higgs mass, "spring" constant, and oscillation frequency of the gap dynamics, which we validate numerically. We compute anharmonic corrections to the Higgs frequency for higher excitations of small superconducting islands, and compare our results to previous long-wavelength calculations.