Photoinduced Electron Pairing in a Driven Cavity

We demonstrate how virtual scattering of laser photons inside a cavity via two-photon processes can induce controllable long-range electron interactions in two-dimensional materials. We show that laser light that is red (blue) detuned from the cavity yields attractive (repulsive) interactions whose strength is proportional to the laser intensity. Furthermore, we find that the interactions are not screened effectively except at very low frequencies. For realistic cavity parameters, laser-induced heating of the electrons by inelastic photon scattering is suppressed and coherent electron interactions dominate. When the interactions are attractive, they cause an instability in the Cooper channel at a temperature proportional to the square root of the driving intensity. Our results provide a novel route for engineering electron interactions in a wide range of two-dimensional materials including AB-stacked bilayer graphene and the conducting interface between ${\text{LaAlO}}_3$ and ${\text{SrTiO}}_3$.

Figure 1

(a) Setup: a 2D material (indicated by the square lattice) on a substrate (yellow) is coupled to a complementary split-ring cavity (gray) [39], whose electric field is represented by the blue arrows with thickness proportional to the field strength. The coupled system is driven by a laser field (schematically shown by the red shading) with wave vector ${\mathbf{q}}_L\parallel z$ axis and frequency ${\omega }_L$ detuned from the cavity frequency ${\omega }_c$. (b) The mechanism of inducing interactions between two electrons (green spheres). Laser (cavity) photons are represented by red (blue) wavy arrows and the green balls are electrons. The red (blue) shading symbolizes the laser (cavity) field.

Figure 2

(a) Relevant energy levels for the two-photon diamagnetic interactions involving the driving (red) and cavity (blue) fields in the lab frame (top) and in a rotating frame (bottom). The electron dispersion is plotted in black. ${\epsilon }_F$ is the Fermi level. (b) The real part of the screened interaction potential $V_{\mathrm{ret}}^{\mathrm{RPA}}/\mathcal{S}$ is shown in red. The real part of $V_{\mathrm{ret}}^{\mathrm{bare}}/\mathcal{S}$ (blue dashed line) is shown for comparison. The plasmon resonance ${\omega }_{\mathrm{pl}}(q_0)$ is indicated by the grey vertical dashed line. The small irregularity below ${\omega }_{\mathrm{p}\mathrm{l}}$ comes from the discontinuity of the 2D electron polarizability given in Ref. [59]. The cavity parameters used are ${\omega }_c=2\pi \times 0.3\mathrm{THz}$, ${\delta }_c=0.1{\omega }_c$, quality factor $Q=500$ and $\mathrm{\Lambda }=2.5/8\times {10}^{-5}$. We set $\mathcal{S}\approx 2.1\times {10}^{-5}{\lambda }^2$, where the proportionality constant is obtained from Ref. [39]. The driving intensity is $I_d=3{\mathrm{MW}\mathrm{cm}}^{-2}$. We use an electron density $n_e={10}^{12}{\mathrm{cm}}^{-2}$ and $m^{*}=2m$ consistent with various 2D materials such at (gated) LAO/STO and monolayer TMDs [60, 61, 62]. The plasmon damping rate is set to 10% of plasmon frequency [63]. We consider a ${\text{SrTiO}}_3$ substrate with a large ${\epsilon }_r\sim {10}^4$ at low temperatures [64].

Figure 3

Maximum $T_c\mathrm{s}$ (with respect to driving and cavity parameters) for the Cooper channel instability (top panel) and the optimal detunings, ${\delta }_c^{\mathrm{op}}$, (bottom panel) are plotted for a range of driving intensities. We calculate them for four sets of electron parameters. All of them follow the dashed green line (same in both panels) representing Eq. (5) at large enough driving intensities. The color map shows the cavity photon number $N_{\mathrm{ph}}$ for different $I_d$ with the optimal driving and cavity parameters at $T_c^{\max }$ using electron and substrate parameters appropriate for gated LAO/STO [same as those used in Fig. 2]. We used $n_e=3\times {10}^{11}$, $5\times {10}^{10}{\mathrm{cm}}^{-2}$, ${\epsilon }_r=10$, 13 for dual-gated BLG and GaAs based quantum well (QW), respectively [36, 68, 69]; $m^{*}=0.07m$ for GaAs QW [36]. AB-stacked BLG dispersion is calculated as in Ref. [70] with a gate-induced band gap of 50 meV. We used ${\omega }_c=10{\delta }_c^{\mathrm{op}}$.