Terahertz phase slips in striped ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x\mathrm{Cu}{\mathrm{O}}_4$

Interlayer transport in high-$T_C$ cuprates is mediated by superconducting tunneling across the ${\mathrm{CuO}}_2$ planes. For this reason, the terahertz frequency optical response is dominated by one or more Josephson plasma resonances and becomes highly nonlinear at fields for which the tunneling supercurrents approach their critical value $I_C$. These large terahertz nonlinearities are in fact a hallmark of superconducting transport. Surprisingly, however, they have been documented in ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x\mathrm{Cu}{\mathrm{O}}_4$ (LBCO) also above $T_C$ for doping values near $x=\frac{1}{8}$ and interpreted as an indication of superfluidity in the stripe phase. Here, electric-field-induced second harmonic is used to study the dynamics of time-dependent interlayer voltages when LBCO is driven with large-amplitude terahertz pulses, in search of other characteristic signatures of Josephson tunneling in the normal state. We show that this method is sensitive to the voltage anomalies associated with 2π Josephson phase slips, which near $x=\frac{1}{8}$ are observed both below and above $T_C$. These results document a regime of nonlinear transport that shares features of fluctuating stripes and superconducting phase dynamics.

Figure 1

(a) Schematic representation of a layered superconductor in the presence of a single-cycle terahertz driving field polarized along the out-of-plane direction. Depending on the instantaneous electric field carried by the pulse (color coded from blue through red to green), deformations of different magnitude in the space- and time-dependent order-parameter phase $\phi$ are achieved. According to the second Josephson equation, these phase gradients lead to the development of a voltage $V$ across the layers. (b) Such a scenario can be visualized in terms of a Josephson junction under the effect of a current bias, characterized by a tilted washboard potential. Each value of the electric field carried by the pulse corresponds to a certain tilting amplitude. Depending on the peak electric field, different regimes can be explored, starting from the linear one (blue), characterized by a response at the Josephson plasma frequency ${\omega }_{\mathrm{JPR}}$ to a nonlinear perturbative regime (red), until approaching the critical current (green), a condition in which one expects to observe phase slips. (c) Calculated space- and time-dependent interlayer phase $\phi \left(x,t\right)$ for a single extended Josephson junction illuminated by a terahertz pulse as that in (a), with a spectrum in resonance with ${\omega }_{\mathrm{JPR}}$ and different terahertz peak field values $E_{\mathrm{peak}}=1$, 30, and 160 kV/cm. Here, $x$ denotes the in-plane direction along the pulse propagation. In the rightmost panel, the occurrence of a phase slip is apparent. (d) Corresponding space- and time-dependent interlayer voltages $V\left(x,t\right)\propto \partial \phi \left(x,t\right)/\partial t$. (e) Fourier transforms of the voltages in (d), calculated at the junction boundary ($x=0$) for different $E_{\mathrm{peak}}$ values. The peaks at $\sim {\omega }_{\mathrm{drive}}$ and $\sim 3{\omega }_{\mathrm{drive}}$ are shaded in red and green, respectively. All spectra at $\omega >1.2\mathrm{THz}$ have been multiplied by 30. (f) Terahertz electric-field-dependent voltage, estimated at the peak frequency in (d) by scanning $E_{\mathrm{peak}}$ continuously. A marked jump indicative of a phase slip is apparent.

Figure 2

(a) Fourier transform of the second harmonic intensity measured in superconducting ${\mathrm{La}}_{1.905}{\mathrm{Ba}}_{0.095}{\mathrm{CuO}}_4$ at $T=5\mathrm{K}<T_C$ for different terahertz peak driving fields (see legend), after subtraction of the same quantity measured at $T>T_C$. The spectra have been vertically offset, maintaining their relative amplitude. Uncertainty bars are standard errors estimated from different measurement sets. The peaks at $\sim {\omega }_{\mathrm{drive}}$ and $\sim 3{\omega }_{\mathrm{drive}}$ are shaded in red and green, respectively. Inset: Schematic of the experimental geometry, in which a single-cycle terahertz pulse polarized along the $c$ axis is shone onto the sample surface together with a near-infrared (NIR) probe pulse (also polarized along $c$). The radiation generated at the second harmonic of the probe is then detected. The clocks in the layers represent amplitude (diameter) and phase (hand angle) of the superconducting order parameter. (b) Light red: Peak electric field dependence of the $\sim {\omega }_{\mathrm{drive}}$ spectral component. Error bars are extracted here from the multi-Gaussian fits in (a). Dark red: Same quantity, measured in ${\mathrm{La}}_{1.885}{\mathrm{Ba}}_{0.115}{\mathrm{CuO}}_4$ at $T=5\mathrm{K}<T_C$, for which full sets of spectra are reported in Fig. S3(a) in the Supplemental Material [15]. Dashed lines are linear guides to the eye, while the red shading highlights the phase-slip regime. (c) Same quantities as in (b), extracted for the $\sim 3{\omega }_{\mathrm{drive}}$ component. Here, the dashed lines are $\propto E^3$ fits. The vertical scales in (b) and (c) are mutually calibrated. Inset: Temperature-doping phase diagram of ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x\mathrm{Cu}{\mathrm{O}}_4$, where the exact location of the investigated compounds is indicated by blue circles. Here, $T_{\mathrm{CO}},T_{\mathrm{SO}},\mathrm{and}{T}_C$ are the charge-order, spin-order, and superconducting transition temperature, respectively.

Figure 3

(a) Temperature dependence of the $\sim {\omega }_{\mathrm{drive}}$ spectral component in the second harmonic intensity measured for both ${\mathrm{La}}_{1.905}{\mathrm{Ba}}_{0.095}{\mathrm{CuO}}_4$ (light red) and ${\mathrm{La}}_{1.885}{\mathrm{Ba}}_{0.115}{\mathrm{CuO}}_4$ (dark red), for a constant terahertz peak driving field of $\sim 165$ kV/cm. Error bars are extracted from multi-Gaussian fits to the experimental spectra reported in Fig. S4 in the Supplemental Material [15], while dashed lines are mean field fits. The horizontal temperature axis has been normalized by the superconducting $T_{\mathrm{C}}$ of each compound. While for ${\mathrm{La}}_{1.905}{\mathrm{Ba}}_{0.095}{\mathrm{CuO}}_4$ the response vanishes at $T_C=T_{\mathrm{CO}}=$ 32 K, in ${\mathrm{La}}_{1.885}{\mathrm{Ba}}_{0.115}{\mathrm{CuO}}_4$, it persists well above $T_C=$ 13 K, all the way up to $\sim T_{\mathrm{CO}}=$ 53 K. (b) Same quantities as in (a), extracted for the $\sim 3{\omega }_{\mathrm{drive}}$ component. (c) Peak electric field dependence of the $\sim {\omega }_{\mathrm{drive}}$ component measured in ${\mathrm{La}}_{1.885}{\mathrm{Ba}}_{0.115}{\mathrm{CuO}}_4$ at $T_C<T=30\mathrm{K}<T_{\mathrm{SO}},T_{\mathrm{CO}},$ for which full sets of spectra are reported in Fig. S3(b) in the Supplemental Material [15]. Dashed lines are linear guides to the eye, while the red shading highlights the phase-slip regime. Inset: Schematic of the experimental geometry. The charge order pattern in the layers is displayed as colored stripes. (d) Same quantity as in (c), extracted for the $\sim 3{\omega }_{\mathrm{drive}}$ component. Here, the dashed lines are $\propto E^3$ fits. The vertical scales in (a)–(d) are mutually calibrated. Inset: Temperature-doping phase diagram of ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x\mathrm{Cu}{\mathrm{O}}_4$, where the exact location of the investigated compound is indicated by a blue circle.