Higgs mode stabilization by photoinduced long-range interactions in a superconductor

We show that low-lying excitations of a 2D Bardeen–Cooper–Schrieffer superconductor are significantly altered when coupled to an externally driven cavity, which induces controllable long-range attractive interactions between the electrons. We find that they combine nonlinearly with intrinsic local interactions to increase the Bogoliubov quasiparticle excitation energies, thus enlarging the superconducting gap. The long-range nature of the driven-cavity-induced attraction qualitatively changes the collective excitations of the superconductor. Specifically, they lead to the appearance of additional collective excitations of the excitonic modes. Furthermore, the Higgs mode is pushed into the gap and now lies below the Bogoliubov quasiparticle continuum such that it cannot decay into quasiparticles. This way, the Higgs mode's lifetime is greatly enhanced.

Figure 1

(a) Setup: Electrons (spheres) in a 2D BCS superconductor (the green envelopes represent Cooper pairing) are evanescently coupled to a complementary split-ring cavity (the gray structure on top) [48]. The blue shading symbolizes the cavity field. The coupled system is driven by a laser field (schematically shown by the red shading) with wave-vector ${\mathbf{q}}_L$ in the vertical direction and frequency ${\omega }_L$, which is red-detuned from the cavity frequency, ${\omega }_c$. The inset shows the top view. (b) The coupling to the cavity and the driving fields result in long-range attractive interactions (blue arc) being induced on top of preexisting short-range attractions (red link).

Figure 2

(a) The zero-temperature gap functions with two different long-range-interaction strengths are plotted along the radial direction in k-space with green lines. The $x$ axis is rescaled by the Fermi velocity and normalized by the intrinsic gap size to show the electron dispersion ${\xi }_{\mathbf{k}}/{\mathrm{\Delta }}^{\text{in}}$ at those wave vectors. We also plot the Bogoliubov quasiparticle excitation energies for the system with dashed blue lines. We use solid lines for $\tilde{U}/{\mathrm{\Delta }}^{\text{in}}\approx 0.056$, and semitransparent lines for $\tilde{U}/{\mathrm{\Delta }}^{\text{in}}\approx 0.085$. The gap function without the long-range interactions is plotted with a gray dash-dotted line. (b) We plot the values of the various excitation energies of the system against $\tilde{U}$; $2E_{k_F}$ is the lowest Bogoliubov quasiparticle excitation energy, ${\varepsilon }_{k_F}^{\text{exc}}$ is the lowest excitonic excitation energy, $2{\mathrm{\Delta }}_{{\epsilon }_{\text{D}}/v_F}$ is twice the smallest value the gap function takes, and ${\varepsilon }^{\text{H}}$ is the zero-momentum Higgs excitation energy. The dashed blue, red, and green lines give the linear dependence of the corresponding energies on $\tilde{U}$ obtained from first-order perturbation theory. The results are calculated for $T=0\mathrm{K}$. In (a) and (b) we have used ${\epsilon }_{\text{D}}/{\mathrm{\Delta }}^{\text{in}}=113$. (c) An energy diagram showing the electron-pair binding energy (from considering the Cooper problem) as a function of the long-range interaction strength. Without the long-range interactions ($\tilde{U}/{\mathrm{\Delta }}^{\text{in}}=0$), the only one two-electron bound state is in the $s$-wave channel. Long-range interactions cause states with higher angular momenta (e.g., $d$ wave) to be bound [75]. Cooper pairs in the condensate can be excited to these higher energy bound states to excite excitonic (aka Bardasis-Schrieffer) modes.

Figure 3

Normalized oscillations of the Higgs mode as a function of time following small perturbations of the superconducting gap with (in red) and without (in blue) the long-range interactions. Without the long-range interactions, the Higgs oscillation decays as $t^{-1/2}$ [46] as it is on the edge of the quasiparticle continuum. In contrast, in the presence of the long-range interactions, the Higgs oscillation does not decay. The intrinsic parameters for the superconductor used here are the same as in Fig. 2. The initial perturbations of the order parameters are about 2% of their equilibrium values. Due to the finite size of the initial perturbation, the gap oscillates around a value below the ground-state value; hence the oscillations are offset by a small amount [5, 78, 79]. The offset vanishes when the perturbation strength tends to zero, when the pseudospin equation of motion can be linearized.