## A nonlinear peek into stripes

In many theoretical models of high-temperature superconductors, remnants of superconductivity persist to temperatures higher than the transition temperature, *T*_{C}. Rajasekaran *et al.* used nonlinear terahertz spectroscopy to probe this region of the phase diagram of a cuprate superconductor that is well known for a stripe phase that appears for certain doping levels (see the Perspective by Ergeçen and Gedik). For a sample deep in the stripe phase, a large nonlinear signal persisted from the superconducting region up to temperatures much higher than *T*_{C}. The findings suggest the formation of a peculiar spatially modulated superconducting state called the pair-density wave.

## Abstract

Unconventional superconductivity in the cuprates coexists with other types of electronic order. However, some of these orders are invisible to most experimental probes because of their symmetry. For example, the possible existence of superfluid stripes is not easily validated with linear optics, because the stripe alignment causes interlayer superconducting tunneling to vanish on average. Here we show that this frustration is removed in the nonlinear optical response. A giant terahertz third harmonic, characteristic of nonlinear Josephson tunneling, is observed in La_{1.885}Ba_{0.115}CuO_{4} above the transition temperature *T*_{c} = 13 kelvin and up to the charge-ordering temperature *T*_{co} = 55 kelvin. We model these results by hypothesizing the presence of a pair density wave condensate, in which nonlinear mixing of optically silent tunneling modes drives large dipole-carrying supercurrents.

Single-layer cuprates of the type La_{2–}_{x}_{–}* _{y}*(Ba,Sr)

*(Nd,Eu)*

_{x}*CuO*

_{y}_{4}exhibit an anomalous suppression of the superconducting transition temperature

*T*

_{c}for doping levels near 12.5% (

*1*). Several studies have shown that this suppression coincides with the formation of “stripes,” one-dimensional chains of charge rivers separated by regions of oppositely phased antiferromagnetism (

*2*,

*3*). A schematic phase diagram of La

_{2–}

*Ba*

_{x}*CuO*

_{x}_{4}, adapted from (

*1*), is shown in Fig. 1, along with a sketch of the stripe geometry.

Recently, high-mobility in-plane transport was found in some of these striped phases at temperatures above the bulk superconducting *T*_{c} (*4*). The existence of a striped superfluid state with a spatially oscillating superconducting order parameter, a so-called pair density wave (PDW) state, was hypothesized (*5*) to explain the anomalously low in-plane resistivity.

Superfluid stripes are difficult to detect. Scanning tunneling microscopy (STM) experiments have reported spatial modulations in the superconducting condensate strength of Bi_{2}Sr_{2}CaCu_{2}O_{8+}* _{x}* (

*6*). However, STM is not sensitive to the phase of the order parameter, and these low-temperature measurements did not clarify the broader question of whether finite momentum condensation may be taking place above

*T*

_{c}. Also, according to the pair density wave model, superfluid transport perpendicular to the planes is frustrated owing to the stripe alignment (Fig. 1, inset). Hence, these stripes are invisible in linear

*c*-axis optical measurements (

*7*).

In this report, we show that superfluid stripes display characteristic signatures in the nonlinear terahertz frequency response. In La_{1.885}Ba_{0.115}CuO_{4}, we detect superfluid stripes above *T*_{c} = 13 K and up to *T*_{co }= 55 K.

Figure 2 displays the linear and nonlinear terahertz reflectivity spectra for two La_{(2–}_{x}_{)}Ba* _{x}*CuO

_{4}crystals (

*x*= 9.5 and 15.5%), measured with single-cycle pulses (

*8*) polarized along the

*c*axis [see section S1 of (

*9*)] and covering the spectral range between 150 GHz and 2.5 THz [see section S2 of (

*9*)]. At these doping levels, the material exhibits homogeneous superconductivity and only weak stripe order, with transition temperatures of

*T*

_{co}~

*T*

_{c}= 34 K (

*x*= 9.5%) and

*T*

_{co}~ 40 K >

*T*

_{c}= 32 K (

*x*= 15.5%).

The low-temperature linear reflectivities (*T* = 5 K < *T*_{c}), show Josephson plasma edges at ω_{JP0} = 500 GHz and ω_{JP0} = 1.4 THz for *x *= 9.5 and 15.5%, respectively (see dashed curves in Fig. 2, A and C). The reflectivity edges shifted to lower frequencies with increasing temperature, indicating a decrease in superfluid density [Fig. 2, B and D, and section S2 of (*9*)].

Nonlinear reflectivities, measured at field strengths between 20 and 80 kV/cm (see Fig. 2, A and C) displayed two characteristic effects of nonlinear interlayer Josephson coupling: (i) a field-dependent red shift of the plasma edge (*10*–*12*) and (ii) a reflectivity peak at the third harmonic of the pump field ω ≈ 3ω_{pump} (*13*). The third-harmonic field amplitude *E*(3ω_{pump}) scaled with the cube of the incident field strength *E*(ω_{pump}) (Fig. 2E), decreased in strength with temperature, and disappeared for *T *= *T*_{c}, tracking the superfluid density ωσ_{2}(ω→0) (see Fig. 2, C, D, and F).

These observations are well understood by a semiquantitative analysis of the interlayer phase dynamics of a homogeneous layered superconductor (*10*, *12*, *14*). For a *c*-axis electric field *E*(*t*) = *E*_{0}sin(ω_{pump}*t*) the interlayer phase difference θ(*t*) advances in time, *t*, according to the second Josephson equation , where *d* is the interlayer spacing (~1 nm), 2*e* is the Cooper pair charge, and *ћ* is Planck’s constant *h* divided by 2π (*15*). Because the *c*-axis superfluid density ρ* _{c}* scales with the order parameter phase difference ρ

*cosθ and because ρ*

_{c}*ω*

_{c}_{J}

^{2}, the plasma frequency renormalizes as , where θ

_{0}= 2

*edE*

_{0}/

*ћ*ω

_{pump}, an average red shift of the equilibrium plasma resonance ω

_{JP0}is hence estimated as . Secondly, a tunneling supercurrent,

*I*, is excited at the first and third harmonic of the driving field, , giving rise to third harmonic radiation. Finally, because the third-harmonic signal is proportional to

*I*

_{c}

*,*it is expected to follow the same temperature dependence as the superfluid density.

More comprehensive numerical simulations, based on space (*x*, not to be confused with the symbol for doping concentration)– and time (*t*)–dependent one-dimensional sine-Gordon equation for the Josephson phase θ(*x*,*t*) [see section S3 of (*9*)] (*12*, *14*, *16*, *17*), were used to obtain the electromagnetic field at the surface of the superconductor and to calculate the reflectivity for arbitrary field strengths. These simulations, reported in Fig. 3, reproduce the experimental data closely.

We next turn to the key results of this report, which provide evidence for superfluidity in the normal-state striped phase in La_{1.885}Ba_{0.115}CuO_{4} (*x* = 11.5%). In this compound, striped charge order coexists with superconductivity below *T*_{c} = 13 K and extends into the normal state up to *T*_{co} = 55 K (*1*).

Figure 4A displays the results for *T* < *T*_{c} = 13 K. Note that in this material, the equilibrium Josephson plasma resonance was at lower frequency than in the other two compounds with higher *T*_{c} (at ~150 GHz) and could not be observed [see section S2 of (*9*)]. However, a third harmonic at ω = 3ω_{pump} ≈ 1.4 THz was clearly observed. Notably, the third-harmonic signal remained finite also for *T* > *T*_{c} = 13 K and up to *T* ~ *T*_{co} = 55 K (Fig. 4, C and D). A giant third harmonic, amounting to several percent of the driving field, can be understood only in the presence of superconducting tunneling above *T*_{c} and up to *T*_{co}.

In the following paragraphs, we show that a pair density wave, which does not show features of superfluidity in the linear optical properties, retains the large nonlinear optical signal of the homogeneous condensate. As shown in Fig. 4E, the order parameter phase of the PDW, encoded by the vector angle (black arrow), changes between neighboring stripes and rotates by 90° from one plane to the next (*18*), resulting in a checkerboard lattice of π/2 and –π/2 Josephson junctions. Hence, the equilibrium PDW supports a lattice of staggered tunneling supercurrents, which average out to zero (thick red arrows in Fig. 4E).

The interlayer phase fluctuations probed in the optical response are described by two normal modes, termed here ϕ_{0} and ϕ_{π}. The ϕ_{0} mode is optically active. For an optical field *E*(*t*) = *E*_{0}sin(ω_{pump}*t*), one has identical phase excursions at each site (shading under black arrows in Fig. 4F). However, current fluctuations δ*I*_{0}(*t*) of equal magnitude, but opposite sign, at neighboring π/2 and –π/2 junctions, makes this mode silent (red arrows in Fig. 4F). The second ϕ_{π} mode consists of phase excursions δθ_{π}(*t*) occurring in opposite directions at neighboring sites and is optically inactive (see Fig. 4F).

In the nonlinear regime, the optical response of the PDW is no longer zero. Because ϕ_{0} is odd and ϕ_{π} is even, a nonlinear expansion of the Josephson energy (*U*) can be written as *U*(ϕ_{0},ϕ_{π}) *J*_{0}ϕ_{π}^{2} – *J*″ϕ_{π} + *C*ϕ_{0}^{2}ϕ_{π}, where *J*_{0} and *J*″ are the in-plane and out-of-plane Josephson energies and *C* = *J*″/2, with *J*_{0 }* J*″, where *C* is the coupling constant between the two normal modes [see section S5 of (*9*)].

A large third-harmonic signal is readily predicted from the ϕ_{0}^{2}ϕ_{π} coupling term. For an optical field of the type* E*(*t*) = *E*_{0}sin(ω_{pump}*t*), which acts only on the mode ϕ_{0}, the equations of motion of the phase are and . These coupled equations imply not only phase oscillations in ϕ_{0} mode at ω_{pump}, i.e., δθ_{0}(ω_{pump}), but also indirect excitation of the optically inactive ϕ_{π} mode. Because in the equation of motion for ϕ_{π} the driving force is proportional to ϕ_{0}^{2}, the phase of this mode δθ_{π} is driven at 2ω_{pump}. Because the total nonlinear current *I*_{tot} = Σ*I*_{junctions} contains terms of the type ~ 2*I _{c}*ϕ

_{0}ϕ

_{π}, the two phase coordinates ϕ

_{0}(ω

_{pump}) and ϕ

_{π}(2ω

_{pump}) are mixed and produce current oscillations at the difference and sum frequencies ω

_{pump}and 3ω

_{pump}. A numerical solution of the two equations of motion with realistic parameters for the Josephson coupling energies

*J*

_{0}and

*J*ʺ displays oscillatory currents at the fundamental and third harmonic [Fig. 4G, also see section S5 of (

*9*) for details on the simulations].

Finally, from the model above, the third-harmonic current is predicted to scale linearly with the out-of-plane critical current and, hence, the superfluid density of the stripes. From the plot in Fig. 4D, the local superfluid density at spin- and charge-ordering temperature *T*_{so} is found to be 60% of that measured below *T*_{c} and subsequently decreases continuously as the temperature is increased further, before vanishing at *T*_{co}.

Although the model discussed above provides a plausible description of the experimental observations, other hypotheses for the origin of the third-harmonic signal for *T* > *T*_{c} should be considered. The measured and simulated third harmonic is far larger than, and hence easily distinguished from, the effect of noncondensed quasiparticles. The nonlinear susceptibility detected in the present experiments χ^{(3)} ~ 10^{–15}*m*^{2}/*V*^{2} (obtained from the electric field strengths at the fundamental and third harmonic) is several orders of magnitude bigger than typical cubic nonlinearities (χ^{(3)} ~ 10^{–18} – 10^{–20}*m*^{2}/*V*^{2}) (*19*–*21*). Furthermore, first-principle calculations show that, for this compound, the value of χ^{(3)} from quasiparticle transport within anharmonic bands is at least three orders of magnitude smaller than what is measured here [see section S6 of (*9*)].

A second alternative may involve the sliding of a charge density wave along the *c* axis, as discussed in (*22*) for blue bronze. However, the efficiency of the sliding of a charge density wave, reported in (*22*) for kilohertz frequency excitation, is expected to reduce strongly at higher excitation frequencies and can be ruled out for the terahertz irradiation [see section S7 of (*9*)]. The results for the 15.5% sample, where the third-harmonic signal disappears at *T*_{c} = 32 K (<*T*_{co} = 40 K), further indicate that the third harmonic results from superconducting tunneling rather than charge ordering.

The observation of a colossal third-harmonic signal in the stripe-ordered state of La_{1.885}Sr_{0.115}CuO_{4} provides compelling experimental evidence for finite momentum condensation in the normal state of cuprates and underscores the power of nonlinear terahertz optics as a sensitive probe of frustrated excitations in quantum solids. A natural direction for this line of research involves the study of other forms of charge order that compete or coexist with superconductivity, such as those found in YBa_{2}Cu_{3}O_{6+}* _{x}* (

*23*,

*24*). One may also find application of these techniques in other regimes of the cuprate pseudogap, with finite superfluid density, with vanishing range phase correlations (

*25*,

*26*), or where other forms of density waves (

*27*,

*28*) have been discussed.

## Supplementary Materials

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

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- ↵Materials and methods are available as supplementary materials.
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- ↵(1) In addition to the two features, a reflectivity spike at ω
_{JP0}is observed for*x*= 9.5% doping. This feature corresponds to a parametric amplification of the Josephson plasma waves [see (*14*)]. (2) The central frequency of the terahertz pulses for experiments in*x*= 9.5 and 11.5% was ω_{pump}= 0.45 THz. Note that to obtain a clear third-harmonic signal without interference from the reflectivity edge at the plasma frequency in*x*= 15.5% sample, terahertz pulses with ω_{pump}= 0.7 THz were used. - ↵
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- ↵For the self-consistent computations, we sample the Brillouin zone with a
*k*point mesh of 15 by 15 by 7, set the angular momentum truncation of the potential and wave function inside the muffin tins to*l*_{max(APW)}= 12 and set the product of the average muffin tin radius and the maximum reciprocal lattice vector to 7.5 (APW, augmented plane method). Moreover, we set the number of empty states to 15 for each atom within the unit cell. The self-consistent calculations are reiterated until the root mean square change of the potential and the total energy becomes smaller than 10^{–7}and 10^{–8}Hartree, respectively. Finally, we determine the band structure from non–self-consistent calculations using the previous converged potentials and a fine*k*point mesh grid of 600*k*points along the*c** direction.

**Acknowledgments:**We thank S. A. Kivelson and J. M. Tranquada for helpful feedback on the manuscript. We thank D. Nicoletti for providing us with the linear reflectivity data of the

*x*= 11.5% sample. The research leading to these results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) and ERC grant agreement no. 319286 [Frontiers in Quantum Materials’ Control (Q-MAC)]. Work performed at Brookhaven was supported by the U.S. Department of Energy, Division of Materials Science under contract no. DE-AC02-98CH10886. The data from the terahertz measurements and the simulations are kept at the Cavalleri Laboratory at Max Planck Institute of Structure and Dynamics, Hamburg, and are available from the authors upon request.