Transiently enhanced interlayer tunneling in optically driven high-$T_c$ superconductors

Recent pump-probe experiments reported an enhancement of superconducting transport along the $c$ axis of underdoped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+\delta }$ (YBCO), induced by a midinfrared optical pump pulse tuned to a specific lattice vibration. To understand this transient nonequilibrium state, we develop a pump-probe formalism for a stack of Josephson junctions, and we consider the tunneling strengths in the presence of modulation with an ultrashort optical pulse. We demonstrate that a transient enhancement of the Josephson coupling can be obtained for pulsed excitation and that this can be even larger than in a continuously driven steady state. Especially interesting is the conclusion that the effect is largest when the material is parametrically driven at a frequency immediately above the plasma frequency, in agreement with what is found experimentally. For bilayer Josephson junctions, an enhancement similar to that experimentally is predicted below the critical temperature $T_c$. This model reproduces the essential features of the enhancement measured below $T_c$. To reproduce the experimental results above $T_c$, we will explore extensions of this model, such as in-plane and amplitude fluctuations, elsewhere.

Figure 1

(a) Schematic depiction of YBCO. Superconducting ${\mathrm{CuO}}_2$ layers (gray) form a stack of bilayer Josephson junctions. THz pulses (wavy lines) excite apical oxygen atoms (circles) that induce oscillations of $j_1$ and $j_2$. (b) Typical time-dependent voltage response $V\left(t\right)$ (solid lines) for different probe pulses $I(t-t_p)$ (dashed lines) at $t_p=-50,0$ and 50. The driving amplitude $A\left(t\right)$ is also depicted.

Figure 2

Transient imaginary conductivity $Im\sigma (\omega ,{\tau }_e)$ and $J_{\text{eff}}\left({\tau }_e\right)$ for continuous driving at $T=0$. (a) Blue-detuned case ${\omega }_e=1.2{\omega }_{\text{Jp}}$. (b) Red-detuned case ${\omega }_e=0.8{\omega }_{\text{Jp}}$.

Figure 3

(a) $J_{\text{eff}}\left({\tau }_e\right)$ for several pump widths ${\mathrm{\Delta }}_e$ for $\gamma =0.1$ at $T=0$. (b) $J_{\text{eff}}\left({\tau }_e\right)$ for different damping factors $\gamma$ for ${\mathrm{\Delta }}_e=10$ at $T=0$.

Figure 4

(a) Temperature dependence of $J_{\text{eff}}\left({\tau }_e\right)$ for continuous driving. We use $\gamma =0.1, {\mathrm{\Delta }}_e=15$, and ${\omega }_e=1.2$. (b) Temperature dependence of $J_{\text{eff}}$ for $A_0=0$ (undriven), $A_0=0.8$ (steady-state), and $A_0=0.8$ (transient value at ${\tau }_e=19$).

Figure 5

$J_{\text{eff}}/J_0$ in the driven steady state with additional second harmonic driving $|A_1|cos(2{\omega }_{e}t+{\varphi }_1)$ as a function of $|A_1|$ and ${\omega }_e$ at $\gamma =0.1$, and $A_0=0.2$ (top) or 0.6 (bottom). The phase difference ${\varphi }_1$ is taken to be 0.

Figure 6

The effective Josephson energy $J_{\text{eff}}$ for various phase differences ${\varphi }_1$ at $A_0=0.4, |A_1|=0.1$, and $\gamma =0.1$. The solid lines are obtained by taking 21 modes, and the dashed line by 3 modes, Eq. (15).

Figure 7

Schematic description of the relationship between the optical pump pulse that excites the apical oxygen atoms and the driving amplitude given in Eq. (19) (only the envelop parts are depicted).

Figure 8

Changes in transient imaginary conductivity $Im\mathrm{\Delta }\sigma (\omega ,{\tau }_e)$. The simulated conductivity is rescaled to fit the value at $\omega =0.5{\omega }_{\text{Jp1}}$ in equilibrium to the experimental one. (a) Experimental data from Ref. [13] at $T=10\text{K}\ll T_c$. (b) A simulated result at $\gamma =0.1, a_1=0.3$, and $a_2=0.6$ at $T=0$.