Terahertz phase slips in striped La$_{2-x}$Ba$_x$CuO$_4$

Interlayer transport in high-$T_C$ cuprates is mediated by superconducting tunneling across the 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 La$_{2-x}$Ba$_x$CuO$_4$ also above $T_C$ for doping values near $x=1/8$, and interpreted as an indication of superfluidity in the stripe phase. Here, Electric Field Induced Second Harmonic (EFISH) is used to study the dynamics of time-dependent interlayer voltages when La$_{2-x}$Ba$_x$CuO$_4$ is driven with large-amplitude terahertz pulses, in search of other characteristic signatures of Josephson tunnelling in the normal state. We show that this method is sensitive to the voltage anomalies associated with 2$\pi$ Josephson phase slips, which near $x=1/8$ are observed both below and above $T_C$. These results document a new regime of nonlinear transport that shares features of fluctuating stripes and superconducting phase dynamics.

The c-axis terahertz-frequency electrodynamics of high-" cuprates [1] can be described by stacks of extended Josephson junctions [2,3] with distributed tunnelling inductance # ( , , ) between capacitively coupled planes ( and are here the inplane spatial coordinates and is the time). For low fields, # = ℏ %&' ! (where is the electron charge and " the junction's critical current) is independent of space and time and a Josephson plasma resonance (JPR) is found at #() = 2 3 # ⁄ ( is the equivalent capacitance of the planes). In most cuprates, #() ranges between gigahertz [4,5] and terahertz [6] frequencies. As characteristic for a plasmonic response, for > #() the superconductor is transparent, corresponding to a positive dielectric function, * ( ) > 0, whereas for < #() it is a perfect reflector with = 1 and * ( ) < 0.
At high electric fields, the Josephson electrodynamics become nonlinear [7,8,9]. As radiation at photon energies below the average superconducting gap couples weakly to the order parameter amplitude, | ( , , )|, the electrodynamics are primarily determined by changes of the order parameter phase. The phase difference between adjacent layers, Δ (henceforth referred to simply as ), is therefore the relevant parameter, and the voltage that develops across the planes is expressed, according to the second Josephson equation, as = ℏ %& +, +-(see Fig. 1(a)).
For an intuitive understanding of these physics, we consider here the case of a Josephson junction under a DC current bias, [10]. In this condition, the phase dynamics of the junction can be modelled as the motion of a fictitious particle in the "washboard" potential ( ) = − Fig. 1(b) [11]. These dynamics are well understood, both in the classical and in the quantum regime [12,13].
For ≪ " the potential has local minima where the phase particle is trapped and oscillates at the JPR frequency (blue). An increase of has the effect of tilting the potential and decreasing the barrier between two neighbouring minima, progressively entering a nonlinear regime (red). For ~" the phase "escapes" from the well and a net voltage develops at the junction's edges (green). By decreasing the bias current, the potential tilt is reduced and the particle will be retrapped in the new potential minimum [11].
A more comprehensive description of the effect of an intense transient THz field on the extended Josephson coupled cuprate planes (i.e., with dimensions that exceed the London penetration depth) can be obtained by simulating the electrodynamics with the sine-Gordon equation [14,15], which in one dimension reads Here, is the in-plane coordinate along the propagation direction of the pulse, the relative dielectric permittivity, the speed of light in vacuum, and a damping coefficient which accounts for the tunneling of normal quasiparticles.
Representative results of these simulations are shown in Fig. 1(c), where we report the time-and space-dependent Josephson phase in presence of an out-of-plane polarized driving field, with the same pulse shape shown in Fig. 1 [16,17,18,19].
As the junction's current reaches the critical value, " , 2 phase slips are expected to develop across the junction, as already qualitatively captured by the analogy with the washboard potential depicted in Fig. 1(b). In the right panel of Fig. 1(c), the occurrence of a phase slip in the material is apparent.
It should be noted here that for the damping values, , and the shape of the THz pulse used for our experiments and included in these simulations, we do not expect soliton modes like those reported in Ref. [17]. Therein, these could be excited because the phase was driven with different THz pulse shapes, that is narrowband multi-cycle pulses from a free-electron laser.
As shown in Fig. 1 , as a function of the driving electric field ( Fig. 1(f)). Here, one observes an abrupt, discontinuous increase of the signal, which we shall take as a univocal fingerprint of the occurrence of a Josephson phase slip in the material.
We set out to measure the field scaling of the c-axis Josephson voltage, which develops at the surface of high-" superconductors of the La2-xBaxCuO4 (LBCO) family, when driven by strong field single-cycle THz pulses. La2-xBaxCuO4 is an extensively studied "214" cuprate, exhibiting an anomalous suppression of the transition temperature, " , for doping levels near 1/8 (see schematic phase diagram in the inset of Fig. 2(c)) [20], related to the formation of "stripes", a peculiar charge-and spin-order pattern within the CuO2 planes, consisting of one-dimensional chains of doped holes separated by antiferromagnetically ordered regions [21]. Recent studies [22] support the existence of a striped superfluid state at > " with a spatially modulated superconducting order parameter, a so-called pair-density-wave (PDW) state [23,24,25]. However, superfluid stripes are difficult to detect with conventional techniques (e.g., scanning tunnelling microscopy), which are not sensitive to the order parameter phase, nor are they visible in linear -axis optical measurements, due to their cross alignment in neighbouring CuO2 layers (see e.g. inset of Fig. 3(c)) [24].
A recent experiment on LBCO showed that this frustration is removed in the nonlinear optical response [19]. A giant THz third harmonic, characteristic of nonlinear Josephson tunnelling, was observed in La1.885Ba0.115CuO4 above the superconducting transition temperature, and up to the charge-ordering temperature, "8 . Such response was modelled by assuming the presence of a PDW condensate, in which nonlinear mixing of optically silent tunnelling modes drives large dipole-carrying supercurrents.
Here, we make use of a new experimental probe technique to confirm the generation of odd harmonics of the phase and to measure voltage signatures of Josephson phase dynamics.
The single-cycle THz pump pulses were generated in LiNbO3 by the tilted-pulse front technique [26] with a spectrum peaked at ~0.5 THz, and were focused at normal incidence on the sample surface, with polarization along the out-of-plane crystallographic axis and maximum peak fields up to ~165 kV/cm [15].
The Electric Field Induced Second Harmonic (EFISH) [27] was sampled by 100-fs-long near-infrared (NIR) probe pulses with 800-nm wavelength, which were also polarized along the c axis, and scanned in time through the profile of the THz frequency pump. In equilibrium and without pump, no second harmonic was found.
The pump-induced 400-nm second harmonic intensity, 9:; , generated at the surface was detected with a photomultiplier. By scanning the time delay between THz pump and NIR probe, the full second harmonic temporal profile could be determined.
As the NIR pulse duration is much shorter than the period of the THz field, and the frequency dependence of (?) (ω >') , ω >') , ω <:= ) within the THz spectral bandwidth This type of measurement is different from that reported in Ref. [28]. Therein, coherent oscillations in the second harmonic signal, persisting over much longer time windows than the duration of the pump pulse, were attributed to an effective second-order susceptibility, which tracked the time-dependent oscillations of symmetry-odd, infrared-active modes. In our case, instead, we are dealing with an effect that is present only while the THz driving field excites the material, and therefore can be fully described in terms of an EFISH process.
In Fig. 2(a) we report the results of the experiment carried out in the superconducting phase at the lowest temperature ( = 5 K) in LBCO 9.5%. Therein, we show the Fourier transform of the measured EFISH time trace for various peak amplitudes of the THz driving field, having subtracted the same quantity measured at high temperature. The exact same type of measurements was also performed in the superconducting state of LBCO 11.5% (see Fig. S3(a) [15]).
In both compounds, the response was almost identical, characterized by a strong peak at the fundamental driving frequency, 4567& ≃ 0.5 THz (which in LBCO 9.5% overlaps with the equilibrium #() [29]), whose intensity scaled with the THz field amplitude.
Concurrently, another contribution developed for A&BC ≳ 100 kV/cm, with oscillations appearing at the third harmonic frequency, 3 4567& ≃ 1.5 THz and, only for the highest field data ( A&BC = 165 kV/cm), signatures of a 5 4567& ≃ 2.5 THz peak were also found.
The latter, however, is comparable with the noise floor of our measurement and we refrain from analyzing it further, focusing on the 4567& and 3 4567& terms.
The frequencies of these contributions did not depend strongly on A&BC . We found instead a systematically higher signal amplitude in LBCO 9.5% compared to LBCO 11.5%, likely related to the detuning of 4567& compared to the resonance at #() , which in the latter material is #() ≃ 0.2 THz. Note that this frequency was contained in the drive spectrum but not at its peak (see Fig. S1 [15]). dependence, the fundamental peak shows a linear behavior only for low fields. A clear discontinuity is observed around 150 kV/cm in LBCO 9.5% and at ~130 kV/cm in LBCO 11.5%. This trend is reminiscent of that found in the voltage simulations of Fig. 1(f), and is indicative of Josephson phase slips in the superconducting phase of both materials when sufficiently high transient THz fields are applied. It is also not surprising that this effect is observed for slightly higher A&BC values in LBCO 9.5%, a superconductor with higher critical temperature and larger phase rigidity.
The temperature dependence, which we report for both samples in Fig. 3(a) and Fig.   3(b) at a constant A&BC ≃ 165 kV/cm (see Fig. S4 [15] for the original spectra), provides additional information on these phenomena. While in the weakly-striped superconductor LBCO 9.5% the doubly-peaked EFISH response progressively reduces with increasing temperature approaching " ≃ 98 ≃ "8 , and then completely disappears in the stripe-free normal state, a much more striking effect is observed in LBCO 11.5%. Here, both the 4567& and 3 4567& peaks clearly survive at > " , all the way up to the charge-ordering temperature, "8 .
The presence of phase slips in the stripe-ordered normal state of LBCO 11.5% is further underscored by the data in Fig. 3(c) and Fig. 3(d). These figures have the same structure as Fig. 2(b) and Fig. 2(c) (see Fig. S3(b) [15] for full spectra), with the difference that the data here were taken at = 30 K, a temperature almost three times higher than " in this compound. The response is virtually identical to that measured in the superconducting state, with a clear discontinuity in the field dependence of the fundamental peak (Fig. 3(c)) around ~130 kV/cm, a behavior which is very reminiscent of that predicted by the simulation of Fig. 1(f).
The observation of phase slips in the stripe-ordered state of La1.885Ba0.115CuO4 complements the colossal third-harmonic signal previously reported in Ref. [19], providing new experimental evidence for interlayer coherence, and possibly for finite momentum condensation in the normal state of this cuprate. Whilst we are able to associate the voltage anomalies with phase slips in a finite momentum condensate, a comprehensive theory for this phenomenon has not yet been formulated.
A natural evolution for this field of study should address other forms of charge order that compete or coexist with superconductivity, such as those found in YBa2Cu3O6+x [30,31]. This same technique also has great potential to be applied in other regions of the cuprate phase diagram, where for example finite superfluid density and vanishing range phase correlations [32,33] are present, or other forms of density waves [34,35] have been discussed.

S1. Simulation of the Josephson phase dynamics with the sine-Gordon equation
To simulate the dynamics of the Josephson junction we solved the one-dimensional sine-Gordon equation using a finite difference approach. Hereby, we assumed a single Josephson junction with semi-infinite layers, having the extended dimension along the direction. The evolution of the Josephson phase, ( , ), is described by (S1) Here, "#$ is the plasma frequency, the speed of light, the relative dielectric permittivity, and a damping factor accounting for the quasiparticle tunneling current.
We used tabulated values for "#$ and , and set as a fitting parameter.
We incorporated the THz driving field, ( ), by setting the spatial and temporal phase evolution at the interface to the following condition: with ( = Φ ' "#$ /(2 ), where Φ ' is the flux quantum and the distance between adjacent superconducting layers. Note that this is the same approach as that used in our previous works (see e.g., Ref. [i]). For the simulations shown in Fig. 1 of the main text, we used the following values: "#$ = 0.53 THz, = 25, = 0.37 THz, and = 10 Å.
The temporal and spatial grid have been tested for stability and convergence and finally set to Δ = 1 µm and Δ = 4 fs.

S2. Experimental methods
Large single crystals of La2-xBaxCuO4 with = 0.095 and = 0.115 (~4 mm diameter), grown by transient solvent method, were studied here. These crystals belonged to the same batch of samples as reported in earlier works i,ii,iii , and were cut and polished along the ac surface.
Laser pulses with 800-nm wavelength, ~100-fs duration and ~3.5 mJ energy were split into 2 parts (99% -1%) with a beam splitter. The most intense beam was used to generate terahertz (THz) pulses by optical rectification in LiNbO3 with the tilted pulse front technique iv . These pump pulses were collimated and focused at normal incidence onto the sample, and they were s-polarized (i.e., perpendicular to the plane of In order to eliminate the effect of any slow drift in the signal, which could have originated for example from the accumulation of ice on the sample surface, we have undertaken a measurement procedure which involved the acquisition of scans at a given temperature interspersed with reference scans at a temperature higher than all relevant temperatures in the given compound. These reference scans were acquired at ,-. = 50 K > / , )0 , /0 for LBCO 9.5% and at ,-. = 70 K > / , )0 , /0 for LBCO 11.5%, after verifying in both materials that the EFISH signal was independent of temperature for all > ,-. . Each measurement at these temperature pairs was then repeated several times, and the data were statistically analyzed to estimate the error bars shown, for example, in Fig. 2(a) of the main text.
An example of data processing on a single set of measurements is reported in Fig. S2.
Therein, we show how time traces acquired at a certain < ,-. and at = ,-. are first subtracted in time domain and then Fourier transformed. Virtually identical results were obtained by performing the subtraction in frequency domain.

S3. Additional data sets
In this section we report additional data sets to complement those shown in the main text. In particular, these are the original spectra from which we extracted, via multi-Gaussian fits, the dependence of 1 and 21 on the peak driving THz field for LBCO 11.5% at = 5 K (Fig. 2(b) and Fig. 2(c)) and = 30 K (Fig. 3(c) and Fig. 3(d)), as well as the temperature dependences of the same quantities for both compounds ( Fig.   3(a) and Fig. 3(b)).
In Fig. S3 we report THz field dependent spectra acquired in LBCO 11.5% at two temperatures ( < / and / < < /0 ), displaying a behavior virtually identical to that measured in the superconducting phase of LBCO 9.5% ( Fig. 2(a) of the main text). Fig. S4 shows instead the variation of the second harmonic spectra as a function of temperature in both compounds, all acquired at the same peak THz field value of ~165 kV/cm. As discussed for Fig. 3(a) and Fig. 3(b) of the main text, while in LBCO 9.5% the response disappears at / = 32 K, in LBCO 11.5% it extends all the way to /0 .    905Ba0.095CuO4 at different temperatures, for a peak THz field of ~165 kV/cm, after subtraction of the same quantity measured at = 50 K > ! . The spectra have been vertically offsetted maintaining their relative amplitude. Uncertainty bars are standard errors estimated from different measurement sets. Inset: Temperature-doping phase diagram of La2-xBaxCuO4, where the measured temperatures are indicated by circles ( !& , %& , and ! are the chargeorder, spin-order, and superconducting transition temperature, respectively). The graphics on the right represent the progressive reduction of the amplitude of the superconducting order parameter (clock diameter) with increasing temperature and its disappearance above ! . (b) Same quantity as in (a) measured in La1.885Ba0.115CuO4 at different temperatures across ! , %& , and !& , for a ~165 kV/cm driving field. Normalization is done here by = 70 > ! , %& , !& . Inset: Temperature-doping phase diagram as in (a). The graphics on the right represent the disappearance of the macroscopic superconducting order parameter (clock) when crossing ! , as well as the gradual fainting of the stripe order, which coexists with superconductivity at < ! , and survives all the way up to = !& .