Higgs Oscillations Research Articles

Higgs Oscillations in a Unitary Fermi Superfluid.

Symmetry-breaking phase transitions are central to our understanding of states of matter. When a continuous symmetry is spontaneously broken, new excitations appear that are tied to fluctuations of the order parameter. In superconductors and fermionic superfluids, the phase and amplitude can fluctuate independently, giving rise to two distinct collective branches. However, amplitude fluctuations are difficult to both generate and measure, as they do not couple directly to the density of fermions and have only been observed indirectly to date. Here, we excite amplitude oscillations in an atomic Fermi gas with resonant interactions by an interaction quench. Exploiting the sensitivity of Bragg spectroscopy to the amplitude of the order parameter, we measure the time-resolved response of the atom cloud, directly revealing amplitude oscillations at twice the frequency of the gap. The magnitude of the oscillatory response shows a strong temperature dependence, and the oscillations appear to decay faster than predicted by time-dependent Bardeen-Cooper-Schrieffer theory applied to our experimental setup.

Collective excitations in three-band superconductors

We investigate equilibrium states, magnetic response and the normal oscillations of internal degrees of freedom (Higgs modes and Goldstone modes) of three-band superconductors accounting the terms of both internal proximity effect and the ``drag'' effect (intergradient interaction) in the Lagrangian. Both the Goldstone mode and the Higgs mode are split into three branches each: common mode oscillations and two modes of anti-phase oscillations, which are analogous to the Leggett mode in two-band superconductors. It is demonstrated that the second and third branches are nonphysical, and they can be removed by special choice of coefficients at the ``drag'' terms in Lagrangian. As a result, three-band superconductors are characterized by only single coherence length. Spectrum of the common mode Higgs oscillations has been obtained. The magnetic penetration depth is determined with densities of superconducting electrons in each band, although the drag terms renormalize the carrier masses.

Comment on "Calculation of an Enhanced A_{1g} Symmetry Mode Induced by Higgs Oscillations in the Raman Spectrum of High-Temperature Cuprate Superconductors".

In a recent manuscript Phys. Rev. Lett. 127, 197001 (2021) Puviani et al. claim that the Higgs mode gives an enhancement of the Raman response in the symmetric $A_{1g}$ Raman channel of cuprates. Here we report a series of technical mistakes in their derivation that completely invalidate this conclusion. In particular, the Raman response function contains two renormalized vertices, with a clearly overcounting of diagrams and uncontrolled results for the Higgs-mode contribution.

Probing two Higgs oscillations in a one-dimensional Fermi superfluid with Raman-type spin—orbit coupling

We theoretically investigate the Higgs oscillation in a one-dimensional Raman-type spin-orbit-coupled Fermi superfluid with the time-dependent Bogoliubov-de Gennes equations. By linearly ramping or abruptly changing the effective Zeeman field in both the Bardeen-Cooper-Schrieffer state and the topological superfluid state, we find the amplitude of the order parameter exhibits an oscillating behaviour over time with two different frequencies (i.e., two Higgs oscillations) in contrast to the single one in a conventional Fermi superfluid. The observed period of oscillations has a great agreement with the one calculated using the previous prediction [Volkov and Kogan, J. Exp. Theor. Phys. 38, 1018 (1974)], where the oscillating periods are now determined by the minimums of two quasi-particle spectrum in this system. We further verify the existence of two Higgs oscillations using a periodic ramp strategy with theoretically calculated driving frequency. Our predictions would be useful for further theoretical and experimental studies of these Higgs oscillations in spin-orbit-coupled systems.

Incoherent excitation of coherent Higgs oscillations in superconductors

We investigate theoretically the excitation of Higgs oscillations of the order parameter in superconductors by incoherent short pulses of light with frequency much larger than the superconducting gap. The excitation amplitude of the Higgs mode is controlled by a single parameter which is determined by the total number of quasiparticles excited by the pulse. This fact can be traced back to the universality of the shape of the light-induced quasiparticle cascade at energy below the Debye frequency and above the gap.

Collective Excitations in Two-Band Superconductors

We investigate the eigen oscillations of internal degrees of freedom (Higgs mode and Goldstone mode) of two-band superconductors using the extended time-dependent Ginzburg-Landau theory, formulated in a work Grigorishin (2021) \cite{grig2}, for the case of two coupled order parameters by both the internal proximity effect and the drag effect. It is demonstrated, that the Goldstone mode splits into two branches: common mode oscillations with the acoustic spectrum, which is absorbed by the gauge field, and anti-phase oscillations with the energy gap (mass) in the spectrum determined by the interband coupling, which can be associated with the Leggett mode. The Higgs mode splits into two branches also: a massive one, whose energy gap vanishes at the critical temperature $T_{c}$, another massive one, whose energy gap does not vanish at $T_{c}$. It is demonstrated, that the second branch of the Higgs mode is nonphysical, and it, together with the Leggett mode, can be removed by the special choice of coefficient at the "drag" term in the Lagrangian. In the same time, such a choice leaves only one coherence length, thereby prohibiting so-called type-1.5 superconductors. We analyze experimental data about the Josephson effect between two-band superconductors. In particular, it is demonstrated, that the resonant enhancement of the DC current through a Josephson junction at a resonant bias voltage $V_{\mathrm{res}}$, when the Josephson frequency matches the frequency of some internal oscillation mode in two-band superconductors (banks), can be explained with the coupling between AC Josephson current and Higgs oscillations. Thus, explanation of this effect does not need the Leggett mode.

Higgs-mode signature in ultrafast electron dynamics in superconducting graphene

We theoretically investigate the effect of superconductivity on the ultrafast electron dynamics in graphene interacting with an ultrashort linearly polarized optical pulse. The optical pulse, with frequency greater than 1 THz, quadratically couples to the Higgs (amplitude) mode and therefore can excite the Higgs-mode oscillations in superconductors, leading to quenching superconducting excitation energy, even in $s$-wave pairing symmetry. Since the duration of the used pulse is less than the electron scattering time in graphene (10--100 fs), the electron dynamics driven by the electric field of pulse remains coherent and is described within the tight-binding model of graphene. We show that the electron-electron and hole-hole transitions from valence to conduction bands are substantially irreversible, with a large residual population of the conduction band, which implies quantum electron dynamics being highly nonadiabatic. Such a feature of the system allows us to clearly project the effect of Higgs oscillations on the electric dipole moment created by the electric field of the pulse between the conduction and valence bands. In particular, we study the impact of the superconducting pair potential on the conduction band population, where the strong pulse causes electronlike and holelike quasiparticles transitions between the conduction and valence bands. The conduction band electron redistribution in honeycomb Dirac points results in almost asymmetric hot spots in two different $K$ and ${K}^{\ensuremath{'}}$ valleys after the pulse ends, which may be interpreted as the valley polarization effect due to the presence of superconductivity. The redistribution of hot spots around Dirac points is in good agreement with the Higgs-mode oscillations (corresponding to the electric dipole moment pattern), and the number of spots weakly depends on the superconducting gap, whereas it has a strong dependence on the ultrashort-pulse intensity.

Anomalous Higgs oscillations mediated by Berry curvature and quantum metric

Higgs spectroscopy, the study of Higgs bosons of a superconductor, is an emerging field in studying superconductivity. Here we show that the Berry curvature and the quantum metric of bands play a central role in the Higgs mode generation. They allow the detection of Higgs bosons even when the conventional contribution from the band curvature vanishes. Furthermore, we show that the Higgs mode can couple to the external electromagnetic field linearly when mediated by the Berry connection. As a result, we predict the existence of a second harmonic generation, in addition to the well-known third harmonic generation. We elucidate our theory further by considering a flat band superconductor realized by the Harper-Hubbard model and show that the geometric Higgs mode is lower bounded by the band Chern number. We demonstrate in twisted bilayer graphene the existence of the geometrically induced Higgs modes when superconductivity is realized in the nearly flat band at the magic angle.

Calculation of an Enhanced A_{1g} Symmetry Mode Induced by Higgs Oscillations in the Raman Spectrum of High-Temperature Cuprate Superconductors.

In superconductors the Anderson-Higgs mechanism allows for the existence of a collective amplitude (Higgs) mode which can couple to eV light mainly in a nonlinear Raman-like process. The experimental nonequilibrium results on isotropic superconductors have been explained going beyond the BCS theory including the Higgs mode. Furthermore, in anisotropic d-wave superconductors strong interaction effects with other modes are expected. Here we calculate the Raman contribution of the Higgs mode from a new perspective, including many-body Higgs oscillations effects and their consequences in conventional, spontaneous Raman spectroscopy. Our results suggest a significant contribution to the intensity of the A_{1g} symmetry Raman spectrum in d-wave superconductors. In order to test our theory, we predict the presence of measurable characteristic oscillations in THz quench-optical probe time-dependent reflectivity experiments.

Time-resolved optical conductivity and Higgs oscillations in two-band dirty superconductors

Recent studies have emphasized the importance of impurity scattering for the optical Higgs response of superconductors. In the dirty limit, an additional paramagnetic coupling of light to the superconducting condensate arises which drastically enhances excitation. So far, most work concentrated on the periodic driving with light, where the third-harmonic generation response of the Higgs mode was shown to be enhanced. In this work, we additionally calculate the time-resolved optical conductivity of single- and two-band superconductors in a two-pulse quench-probe setup, where we find good agreement with existing experimental results. We use the Mattis-Bardeen approach to incorporate impurity scattering and calculate explicitly the time-evolution of the system. Calculations are performed both in a diagrammatic picture derived from an effective action formalism and within a time-dependent density matrix formalism.

Momentum-resolved analysis of condensate dynamic and Higgs oscillations in quenched superconductors with time-resolved ARPES

Higgs oscillations in nonequilibrium superconductors provide an unique tool to obtain information about the underlying order parameter. Several properties like the absolute value, existence of multiple gaps and the symmetry of the order parameter can be encoded in the Higgs oscillation spectrum. Studying Higgs oscillations with time-resolved angle-resolved photoemission spectroscopy (ARPES) has the advantage over optical measurements that a momentum-resolved analysis of the condensate dynamic is possible. In this paper, we investigate the time-resolved spectral function measured in ARPES for different quench protocols. We find that analyzing amplitude oscillations of the ARPES intensity in the whole Brillouin zone allows to understand how the condensate dynamic contributes to the emerging of collective Higgs oscillations. Furthermore, by evaluating the phase of these oscillations the symmetry deformation dynamic of the condensate can be revealed, which gives insight about the ground state symmetry of the system. With such an analysis, time-resolved ARPES experiments might be used in future as a powerful tool in the field of Higgs spectroscopy.

Theory of driven Higgs oscillations and third-harmonic generation in unconventional superconductors

Higgs spectroscopy is a new field in which Higgs modes in nonequilibrium superconductors are analyzed to gain information about the ground state. One experimental setup in which the Higgs mode in s-wave superconductors was observed is periodic driving with THz light, which shows resonances in the third-harmonic generation (THG) signal if twice the driving frequency matches the energy of the Higgs mode. We derive expressions of the driven gap oscillations for arbitrary gap symmetry and calculate the THG response. We demonstrate that the possible Higgs modes for superconductors with non-trivial gap symmetry can lead to additional resonances if twice the driving frequency matches the energy of these Higgs modes and we disentangle the influence of charge density fluctuations (CDF) to the THG signal within our clean-limit analysis. With this we show that THG experiments on unconventional superconductors allow for a detection of their Higgs modes. This paves the way for future studies on realistic systems including additional features to understand the collective excitation spectra of unconventional superconductors.

Classification and characterization of nonequilibrium Higgs modes in unconventional superconductors

Recent findings of new Higgs modes in unconventional superconductors require a classification and characterization of the modes allowed by nontrivial gap symmetry. Here we develop a theory for a tailored nonequilibrium quantum quench to excite all possible oscillation symmetries of a superconducting condensate. We show that both a finite momentum transfer and quench symmetry allow for an identification of the resulting Higgs oscillations. These serve as a fingerprint for the ground state gap symmetry. We provide a classification scheme of these oscillations and the quench symmetry based on group theory for the underlying lattice point group. For characterization, analytic calculations as well as full scale numeric simulations of the transient optical response resulting from an excitation by a realistic laser pulse are performed. Our classification of Higgs oscillations allows us to distinguish between different symmetries of the superconducting condensate.

Higgs oscillations in time-resolved optical conductivity

Driving superconductors out of equilibrium is a promising avenue to study their equilibrium properties as well as to control the superconducting state. Non-equilibrium superconductors are often studied using time resolved optical conductivity measurements. Thus, the characterization of a superconducting state in a pump driven non-equilibrium state requires careful attention in the time domain. We calculate time-resolved optical conductivity of a pumped superconducting state using a non-equilibrium Keldysh approach. Through functional derivation, the optical conductivity is obtained with full vertex corrections and used to characterize the transient superconducting state. The transient optical conductivity shows the suppression of the superconducting order parameter in the time domain. The subsequent recovery of the order parameter exhibits oscillatory behavior that corresponds to the Higgs amplitude mode, and may be seen in several parts of the spectrum.

Excitation of Higgs Mode in Superfluid Fermi Gas in BCS–BEC Crossover

In quantum many-body systems with spontaneous breaking of continuous symmetries, Higgs modes emerge as collective amplitude oscillations of order parameters. Recently, Higgs mode has been observed in the ultracold Fermi gas. In the present paper, we use the time-dependent Bogoliubov-de Gennes equations to investigate Higgs amplitude oscillations of the superfluid order parameter in a Fermi gas induced by a rapid change of the ${\it s}$-wave scattering length. In particular, we investigate the Higgs mode with different values of the initial scattering length. We find that the energy of the Higgs mode coincides with the threshold energy of the pair-breaking excitation, and exponent of the power-low decay of the Higgs mode $\gamma$ continuously changes between $\gamma=-1/2$ and $\gamma=-3/2$ through the Bardeen-Cooper-Schrieffer-Bose-Einstein condensation (BCS-BEC) crossover. Moreover, we propose the optimal ramp speed of the scattering length for observing the clearest Higgs oscillations.

Generation of the Higgs condensate and its decay after inflation

We investigate the dynamics of the Standard Model higgswith a minimal coupling to gravity during and after inflation. Inthe regime where the Standard Model vacuum is stable, we find thatthe higgs becomes a light spectator field after about 30 efolds ofinflation, irrespectively of its initial value. Once the higgs hasbecome light, its root-mean-square value h* relaxes toequilibrium in about 85 efolds for the inflationary scale ofH* = 104 GeV and in 20 efolds for H* = 1010 GeV. Theequilibrium value is given by h* ∼ 0.36λ*−1/4H*,where λ* = 0.09…0.0005 is the higgs self coupling at thescales H* = 104…1010 GeV. We show that the main decaychannel of the higgs condensate after inflation is the resonantproduction of Standard Model gauge bosons. For a set of parameterswe find that a significant part of the condensate has decayed inbetween 340 and 630 Hubble times after the onset of higgsoscillations, depending on H* in a non-trivial way. The higgsperturbations correspond to isocurvature modes during inflation butthey could generate significant adiabatic perturbations at a laterstage for example through a modulation of the reheating stage.However, this requires that the inflaton(s) decay no later than afew hundred Hubble times after the onset of higgs oscillations.

Baryon number nonconservation and phase transitions at preheating

Certain inflation models undergo pre-heating, in which inflaton oscillations can drive parametric resonance instabilities. We discuss several phenomena stemming from such instabilities, especially in weak-scale models; generically, these involve energizing a resonant system so that it can evade tunneling by crossing barriers classically. One possibility is a spontaneous change of phase from a lower-energy vacuum state to one of higher energy, as exemplified by an asymmetric double-well potential with different masses in each well. If the lower well is in resonance with oscillations of the potential, a system can be driven resonantly to the upper well and stay there (except for tunneling) if the upper well is not resonant. Another example occurs in hybrid inflation models where the Higgs field is resonant; the Higgs oscillations can be transferred to electroweak (EW) gauge potentials, leading to rapid transitions over sphaleron barriers and consequent B+L violation. Given an appropriate CP-violating seed, we find that preheating can drive a time-varying condensate of Chern-Simons number over large spatial scales; this condensate evolves by oscillation as well as decay into modes with shorter spatial gradients, eventually ending up as a condensate of sphalerons. We study these examples numerically and to some extent analytically. The emphasis in the present paper is on the generic mechanisms, and not on specific preheating models; these will be discussed in a later paper.

Induced gravity inflation in the SU(5) GUT.

We investigate the cosmological consequences of a theory of induced gravity in which the scalar field is identified with the Higgs field of the first symmetry breaking of a minimal SU(5) GUT. The mass of the X boson determines a great value for the coupling constant of of gravity-particle physics. Because of this fact, a ``slow'' rollover dynamics for the Higgs field is not possible in a ``new'' inflation scenario and, moreover, a contraction era for the scale factor in the early Universe exists, after which inflation follows automatically; ``chaotic'' inflation is performed without problems. Inflation is successfully achieved due to the relationship among the masses of particle physics at that scale: the Higgs-boson, X-boson, and Planck masses. As a result the particle physics parameter \ensuremath{\lambda} is not fine-tuned as usual in order to predict acceptable values of reheating temperature and density and gravitational wave perturbations. Moreover, if the coherent Higgs oscillations did not decay they could explain the missing mass problem of cosmology.