Redistribution of phase fluctuations in a periodically driven cuprate superconductor

We study the thermally fluctuating state of a bilayer cuprate superconductor under the periodic action of a staggered field oscillating at optical frequencies. This analysis distills essential elements of the recently discovered phenomenon of light-enhanced coherence in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$, which was achieved by periodically driving infrared active apical oxygen distortions. The effect of a staggered periodic perturbation is studied using a Langevin and Fokker-Planck description of driven, coupled Josephson junctions, which represent two neighboring pairs of layers and their two plasmons. In a toy model including only two junctions, we demonstrate that the external driving leads to a suppression of phase fluctuations of the low-energy plasmon, an effect which is amplified via the resonance of the high-energy plasmon. When extending the modeling to the full layers, we find that this reduction becomes far more pronounced, with a striking suppression of the low-energy fluctuations, as visible in the power spectrum. We also find that this effect acts on the in-plane fluctuations, which are reduced on long length scales. All these findings provide a physical framework to describe light control in cuprates.

Figure 1

Simplified representation of the crystal structure of YBCO. The copper oxide layers are shown as red and blue, and some of the atoms in the insulating layers are shown. The distortion of this structure, due to the motion of the apical oxygen atoms of the infrared-active $B_{1u}$ mode, discussed in Ref. [18], is driven periodically in time. This results in an external potential in the CuO layers that is periodic in time and staggered in the $c$ direction, represented by the red and blue coloring, changing periodically in time.

Figure 2

Time evolution of the variances of the weak (red dashed line) and the strong (blue dashed line) junction relative to their equilibrium value, $\Delta V_w\equiv V_w-V_{w,th}$ and $\Delta V_s\equiv V_s-V_{s,th}$, respectively, as percent of their equilibrium values $V_{w,th}$ and $V_{s,th}$, respectively. $V_w$ is also represented on the scale of an effective temperature $T_{\mathrm{eff}}$ on the right-hand side, based on Eq. (6). The solid lines are the time evolution smoothed via Gaussian averaging with a time scale of $1.8$ ps.

Figure 3

Time-averaged variances of the weak junction (top, red lines), ${\overline{V}}_w$, and the strong junction (bottom, blue lines), ${\overline{V}}_s$, plotted against the driving frequency. Equilibrium states are shown as dashed lines, and driven steady states as continuous lines. We find a reduction of ${\overline{V}}_w$ near the resonance of the strong junction, where the strong junction acts as an amplifier of the external driving, and a similar feature near ${\omega }_w$, due to direct driving of the weak junction.

Figure 4

(a) Power spectrum of the weak junction, $S_w\left(\omega \right)$, in arbitrary units on a logarithmic scale. The blue line is the thermal spectrum, and the red line is the spectrum of the driven steady state. (b) Difference of the power spectrum of the driven state and the equilibrium spectrum. We find that the low-frequency fluctuations are reduced, whereas the fluctuations near the driving frequency are enhanced. Therefore, driving leads to an up-conversion of spectral weight.

Figure 5

(a) Time evolution of $V_w\left(t\right)$ and $V_s\left(t\right)$ of the bulk system for a driving frequency of ${\omega }_m=2\pi \times 10.4$ THz, a driving amplitude of $A_0=4.3$ meV, and a temperature of $T=100$ K. The system is a $256\times 256\times 4$ lattice. (b) ${\overline{V}}_w$ of the bulk system, shown as a red, continuous line, plotted against the driving frequency, for a driving amplitude of $A_0=4.3$ meV, for a $256\times 256\times 4$ system. For comparison, we show the nondriven value as a dashed line. As a second comparison, we show ${\overline{V}}_w$ of the toy model as a black continuous line, with a temperature chosen such that the thermal magnitude of the bulk system is reproduced. We use a driving amplitude of $A_0=4.3$ meV, which is near the optimal driving amplitude for the toy model. We note that there is a large reduction of ${\overline{V}}_w$ near the resonance of the large plasmon frequency, and that the bulk system has a much stronger reduction than the toy model. Furthermore, the reduction due to direct driving of the weak junctions around ${\omega }_w$ is washed out due to the strong additional damping in the bulk. (c) Power spectrum of the total current. The system is a $128\times 128\times 4$ lattice at $T=100$ K, with a driving amplitude $A_0=4.3$ meV, and for a driving frequency of ${\omega }_m=7.9$ THz, which is near the minimum of ${\overline{V}}_w$. We again see a reduction of the low-frequency fluctuations when the system is driven. (d) Difference of the power spectrum of the driven state and the equilibrium spectrum.

Figure 6

Current fluctuations in a plane of weak junctions, for a driving frequency of ${\omega }_m=8.3$ THz, and for a driving amplitude of $A_0=4.3$ meV. On the left, we see a thermal state; on the right, we show a cycle during the steady state of the driven system.

Figure 7

Current fluctuations in the steady state at half a cycle, for three values of $A_0$. The first panel is the same as in Fig. 6 at half a cycle. $A_0=12.9$ meV is near the optimal driving. At $A_0=38.8$ meV, the magnitude of ${\overline{V}}_w$ has reached approximately its equilibrium value again, as can be seen from Fig. 12.

Figure 8

We show the equilibrium in-plane correlation function $G\left(\mathbf{r}\right)$ and the time-averaged correlation function $\overline{G}\left(\mathbf{r}\right)$ of the driven state, for two driving amplitudes. Weak driving corresponds to $A_0=4.3$ meV and optimal driving to $A_0=12.9$ meV.

Figure 9

(a) Time-averaged variances ${\overline{V}}_w$ and ${\overline{V}}_s$ of the steady state of the toy model for increasing driving strength $A_0$. (b) The time-averaged variance ${\overline{V}}_w$ for the bulk system, as a function of the driving frequency, for several values of $A_0$.

Figure 10

(a) Time evolution of $V_w$ for the effective, single-oscillator model, Eq. (10). The temperature is $T=1$ K, the damping $\gamma =2.1$ THz, the driving frequency ${\omega }_m=10.4$ THz, $U$ is 5000 K, and $F_0/\left(\gamma {\omega }_m\right)=0.3$. The red line shows the numerical solution, and the blue line the analytical solution in Eq. (B19). (b) Time-averaged ${\overline{V}}_w$ for the single-oscillator model, as a function of the driving amplitude. The red lines are numerical results, and the blue line is the analytical result in Eq. (14).

Figure 11

Effective temperature as a function of $x$. The dashed lines indicate the asymptotic values for $x\to 0$ and $x\to \infty$.

Figure 12

We show ${\overline{V}}_w$ as a function of the driving amplitude $A_0$, for the bulk system of $256\times 256\times 4$ sites, at $T=100$ K, at a driving frequency of ${\omega }_m=8.3$ THz. Additionally, we show the prediction of the effective model, calculated via the high-temperature expansion of the FP equation described in Sec. B 3. The result for the bulk simulation is shown as a continuous red line, with the nondriven case as a dashed red line. The result for the effective single-junction model is shown in blue. $J_{w,\mathrm{eff}}$ has been chosen so that the equilibrium value of $V_w$ is reproduced. $F_0$ has been chosen to be proportional to $V_0$, with a proportionality coefficient such that the minima match up. We indeed see that the large reduction of the phase fluctuations is approximately captured by the effective single-oscillator model.

Figure 13

Time evolution of the current correlation function, defined in Eq. (18).