Induction of p-wave and d-wave order parameters in s-wave superconductors with light pulses

TL;DR

This work develops a symmetry-based, time-dependent Ginzburg-Landau framework to dynamically induce lower-symmetry superconducting order parameters in centrosymmetric $s$-wave materials using microwaves. By incorporating Lifshitz-type gradient invariants that become allowed in the presence of spin-orbit coupling, the model enables coupling between $s$-wave and $p$- and $d$-wave channels via the vector potential, yielding localized triplet and singlet admixtures. Analytic and numerical results show that linearly polarized light induces oscillatory triplet components with zero average, while circular polarization yields a nonzero time-averaged triplet component and rectified singlet components, with spatially localized effects for beam geometries. The findings point to a platform for Floquet-engineered topological superconductivity and the broader concept of quantum printing, wherein light's gauge structure imprints on and controls the superconducting state, albeit within the TDGL regime and requiring microscopic verification.

Abstract

We construct a generalized time-dependent Ginzburg-Landau model to demonstrate the possibility of inducing p- and d-wave components in an originally pure s-wave centrosymmetric superconductor via microwave radiation. In this framework, specializing to $O_h$ point-group symmetry, we introduce gradient terms that couple the s-wave superconducting order parameter with other symmetry-allowed components. The singlet-to-singlet gradient terms are quadratic in spatial derivatives, while, in the presence of spin-orbit coupling, linear-in-derivatives terms coupling singlet and triplet order parameters are also permitted. Through the minimal substitution procedure, these terms enable coupling between different superconducting order parameters via the vector potential, thereby leading to the generation of p-wave, d-wave, and other symmetry-allowed components. Such a manipulation of the superconducting state locally via a microwave beam could be considered as one more facet of the concept of quantum printing.

Figures (12)

• Figure 1: Schematic diagram of gradient couplings between different SC OPs. Red and blue colors stand for the singlet-to-singlet and singlet-to-triplet couplings, respectively. Thicker lines correspond to couplings to the $s$-wave OP, which is the only non-zero OP in the absence of the vector potential.
• Figure 2: (a) $\abs{\eta_{A_{1g}}}$ ($s$-wave); (b) $\Im(\eta_{T_{1u}}^1)$ ($p$-wave) ; (c) $\Re(\eta_{E_{g}}^1)$ ($d$-wave). Red and blue curves correspond to the solutions of \ref{['eq: TDGL A1g', 'eq: TDGL T1u', 'eq: TDGL T1u2', 'eq: TDGL T1u3', 'eq: TDGL Eg', 'eq: TDGL Eg2', 'eq: TDGL T1g', 'eq: TDGL T1g2', 'eq: TDGL T1g3', 'eq: TDGL T2g', 'eq: TDGL T2g2', 'eq: TDGL T2g3']} in the case of a Gaussian and uniform linearly polarized beams, respectively. Black curve corresponds to the approximate solution of \ref{['eq: simplified eqs.', 'eq: simplified eqs.2', 'eq: simplified eqs.3', 'eq: simplified eqs.4', 'eq: simplified eqs.5', 'eq: simplified eqs.6']} in the case of a uniform beam. Dashed lines correspond to the approximate average values given by \ref{['eq: average values']} around which the approximate solutions oscillate. The TDGL parameters are given in \ref{['tab: Coefficients']}, the dimensionless frequency and amplitude of the microwaves are $\omega=1$ and $A_{0}=0.5$, respectively, the waist of the Gaussian beam is $w_0=4 \xi$, and the side length of the thin square film is $l=20\xi$. (d) Time-dependence of the values of $\eta_{A_{1g}}$ ($s$-wave), $\eta_{T_{1u}}^1$ ($p_y$-wave in the 2D limit), and $\eta_{T_{1u}}^2$ ($p_x$-wave in the 2D limit) order parameters at the center of the circularly polarized Gaussian beam with waist $w_0=4 \xi$. The TDGL parameters are given in \ref{['tab: Coefficients']}, the dimensionless frequency and amplitude of the beam are $\omega=1$ and $A_{0}=0.5$, respectively, the waist of the beam is $w_0=4 \xi$, and the side length of the thin square film is $l=20\xi$.
• Figure 3: Time-dependence (according to \ref{['eq: TDGL A1g', 'eq: TDGL T1u', 'eq: TDGL T1u2', 'eq: TDGL T1u3', 'eq: TDGL Eg', 'eq: TDGL Eg2', 'eq: TDGL T1g', 'eq: TDGL T1g2', 'eq: TDGL T1g3', 'eq: TDGL T2g', 'eq: TDGL T2g2', 'eq: TDGL T2g3']}) of $\eta_{A_{1g}}$ ($s$-wave) (a) and $\eta_{T_{1u}}^1$ ($p_y$-wave in the 2D limit) (b) OPs for different values of $r_{T_{1u}}$ ranging from $-30$ to $0.9$ under irradiation by linearly polarized uniform microwave. The TDGL parameters are given in \ref{['tab: Coefficients']}, except for $c_{A_{1g}T_{1u}}^2=0$ and values of $r_{T_{1u}}=r$; the dimensionless frequency and amplitude of the microwaves are $\omega=1$ and $A_{0}=0.5$, respectively. We demonstrate that the induced triplet component can be as large as half of the initial value of the $s$-wave component.
• Figure 4: Time-dependence (according to \ref{['eq: TDGL A1g', 'eq: TDGL T1u', 'eq: TDGL T1u2', 'eq: TDGL T1u3', 'eq: TDGL Eg', 'eq: TDGL Eg2', 'eq: TDGL T1g', 'eq: TDGL T1g2', 'eq: TDGL T1g3', 'eq: TDGL T2g', 'eq: TDGL T2g2', 'eq: TDGL T2g3']}) of $\eta_{A_{1g}}$ ($s$-wave) (a) and $\eta_{T_{1u}}^1$ ($p_y$-wave in the 2D limit) (b) OPs for different values of $r_{T_{1u}}$ ranging from $-30$ to $0$ under irradiation by linearly polarized uniform microwave. The TDGL parameters are given in \ref{['tab: Coefficients']}, except for $c_{A_{1g}T_{1u}}^2=0$, $c_{A_{1g}T_{1u}}^1=0.1$, and values of $r_{T_{1u}}=r$; the dimensionless frequency and amplitude of the microwaves are $\omega=1$ and $A_{0}=0.5$, respectively. In this case, the induced triplet component can be as large as about a half of the initial value of the $s$-wave component, and the $s$-wave can be enhanced transiently above its initial (equilibrium) value.
• Figure 5: Density plots for the absolute values of $s$-wave $\eta_{A_{1g}}$, $d$-wave $\eta_{Eg}^{1,2}$, and $p$-wave $\eta_{T_{1u}}^{1,2,3}$ OPs under incident linearly polarized microwave beam at approximately quarter of the microwave cycle, $t/T\approx 0.25$, where $T$ is the period of microwave. Coordinates on the thin film are given in units of $\xi$.