Theory of Laser-Controlled Competing Superconducting and Charge Orders

We investigate the nonequilibrium dynamics of competing coexisting superconducting (SC) and charge-density wave (CDW) orders in an attractive Hubbard model. A time-periodic laser field $\overrightarrow{A}(t)$ lifts the SC-CDW degeneracy, since the CDW couples linearly to the field ($\overrightarrow{A}$), whereas SC couples in second order (${\overrightarrow{A}}^2$) due to gauge invariance. This leads to a striking resonance: When the photon energy is red detuned compared to the equilibrium single-particle energy gap, CDW is enhanced and SC is suppressed, while this behavior is reversed for blue detuning. Both orders oscillate with an emergent slow frequency, which is controlled by the small amplitude of a third induced order, namely $\eta$ pairing, given by the commutator of the two primary orders. The induced $\eta$ pairing is shown to control the enhancement and suppression of the dominant orders. Finally, we demonstrate that light-induced superconductivity is possible starting from a predominantly CDW initial state.

Figure 1

Laser-controlled order. (a) ${\mathrm{\Delta }}_{\mathrm{SC}}(t)$ and ${\mathrm{\Delta }}_{\mathrm{CDW}}(t)$, and total half gap $\sqrt{{\mathrm{\Delta }}_{\mathrm{SC}}^2(t)+{\mathrm{\Delta }}_{\mathrm{CDW}}^2(t)}$, for a driving field with $\omega =19\mathrm{meV}$ (red detuned from $2{\mathrm{\Delta }}_0$). (b) The corresponding ${\mathrm{\Delta }}_{\eta }(t)$ (“ETA”). (c) Depiction of the dynamics in the ${\mathrm{\Delta }}_{\mathrm{SC}}\text{−}{\mathrm{\Delta }}_{\mathrm{CDW}}$ plane with enhanced CDW. (d)–(f) The same for a driving field with $\omega =21\mathrm{meV}$ (blue detuned from $2{\mathrm{\Delta }}_0$) with enhanced SC. In all cases, dashed colored lines show the respective time averages.

Figure 2

Nonequilibrium orders controlled by gap resonance. Oscillation frequency ${\omega }_{\mathrm{slow}}$ (black circles), obtained from the time-dependent $\mathrm{\Delta }$’s, as well as amplitude ${\mathrm{\Delta }}_{\eta ,\max }$ of the time-dependent ${\mathrm{\Delta }}_{\eta }$, plotted as a function of the driving frequency $\omega$. The red vertical line indicates the $\omega =2{\mathrm{\Delta }}_0$ resonance.

Figure 3

Light-induced superconductivity. (a) ${\mathrm{\Delta }}_{\mathrm{SC}}(t)$ and ${\mathrm{\Delta }}_{\mathrm{CDW}}(t)$ for a driving field with $\omega =21\mathrm{meV}$ and $A_{\max }=5\times {10}^{-5}$, starting from an initial state with mostly CDW order. (b) The corresponding ${\mathrm{\Delta }}_{\eta }(t)$ (“ETA”). (c), (d) The same for $A_{\max }=10\times {10}^{-5}$. (e), (f) The same for $A_{\max }=20\times {10}^{-5}$. Dashed black line indicates the total half gap.