Main

Unconventional superconductivity in cuprates emerges when either electrons or holes are doped into the insulating parent compound. Because superconductivity is optimized far away from optimal stoichiometric composition, disorder of the host lattice is unavoidable and may result in substantial electronic inhomogeneities1,2. Measurements of the electronic properties of cuprates using scanning tunnelling microscopy3 reveal disorder of the superconducting gap on nanometre length scales4,5, whose variations are further correlated with the distribution of dopant atoms6. Yet the potential presence of co-existing and competing orders complicates the assignment of observables to superconductivity. Modern techniques based on tunnelling from a superconducting tip7 can now isolate the superconducting response8,9,10,11 but are still limited to sampling surfaces amenable to tip-based techniques and temperatures typically below that of liquid helium. Additional probes of the role of these inhomogeneities on the superconducting order parameter are therefore necessary to obtain a complete picture of the role of disorder in these materials.

The tunnelling resonance arising from Josephson coupling between adjacent superconducting CuO2 planes, the so-called Josephson plasma resonance12,13, provides a direct measure of the superconducting order parameter and of the c-axis transport14,15. The Josephson plasma resonance has been extensively studied by linear spectroscopy, in which the c-axis reflectivity exhibits a pronounced plasma edge in the superconducting state (Fig. 1a). Van der Marel and Tsvetkov first pointed out16 that a distribution of Josephson plasma frequencies, reflecting disorder of the superconducting order parameter, manifests as a distortion of the Josephson plasmon loss function17. Corresponding distortion of the reflectivity edge profile was then exploited by Dordevic et al.18 to quantify the underlying disorder in La2−xSrxCuO4.

Fig. 1: Linear and nonlinear spectroscopies of plasma resonances.
Fig. 1: Linear and nonlinear spectroscopies of plasma resonances.
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a, A schematic of a linear reflectivity measurement, in which an incident field Ein induces a linear response χφ(1). The co-propagating reflected field Erefl and the linear response χφ(1) interfere to produce a characteristic edge in the reflectivity shown (inset). b, The generation of a nonlinear electric field ENL by two excitation pulses EA and EB of comparable amplitude. The amplitude and phase of ENL depend on the inter-pulse time delay τ. c, A schematic of nonlinear multidimensional spectroscopy, in which Fourier transform of ENL along the two time variables {τ, t} resolves a nonlinear optical response χφ(3) in a 2D spectrum. In the non-collinear excitation geometry shown, various components of χφ(3) are emitted in unique phase-matched directions shown.

However, this approach suffers from various shortcomings. For example, in ref. 18, the lineshape fits based on a two-fluid model are generally not completely constrained and require assumptions on both the normal fluid conductivity and the functional form of the plasma frequency distribution. Reliable fits are also only possible at low temperatures, in which a sharp plasma edge is observed.

To circumvent these limitations, techniques capable of resolving higher-order plasmon correlation functions are required. These higher-order plasmon correlations may be measured by nonlinear spectroscopy19,20, an example of which is shown schematically in Fig. 1b where two excitation fields EA and EB cooperatively generate a nonlinear electric field ENL. The depicted nonlinear process is governed by a third-order optical response χφ(3), with multiple frequency and wavevector components. These may be resolved in a two-dimensional (2D) spectrum21,22 (Fig. 1c) by Fourier transforming ENL along the inter-pulse time delay τ and the emission time t. So-called 2D spectroscopy has been implemented at terahertz frequencies in collinear excitation geometries to study a variety of material systems23,24,25,26,27,28,29,30,31,32,33,34,35, including superconductors more recently36,37,38.

Previous applications of 2D terahertz spectroscopy have all implemented a collinear geometry, in which the excitation fields and nonlinear signal emission all possess identical wavevectors. While convenient, such a collinear geometry cannot isolate specific nonlinearities and further precludes study of opaque materials. We therefore implement a non-collinear excitation geometry with which different components of χφ(3), corresponding to different peaks in the 2D spectrum, are emitted in unique phase-matched directions (Fig. 1c).

The non-collinear phase-matching geometry is shown in Fig. 2a, in which two quantities are conserved. In addition to the momentum magnitude determined by the Josephson plasma frequency, only in-plane momentum is conserved due to the interface39. For fixed excitation and detection geometry, the nonlinearity arriving at the detector may be chosen by rotating the sample and thereby the in-plane momentum of each excitation beam (Fig. 2b).

Fig. 2: Phase-matching resolves individual plasmon nonlinearities.
Fig. 2: Phase-matching resolves individual plasmon nonlinearities.
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a, A schematic of wavevector phase-matching, in which momentum is only conserved along the in-plane (∥) direction and in its magnitude |kfp|. b, Sample rotation changes the in-plane momenta of each excitation field and determines the nonlinearity that arrives at the detector. c, Experimental measurements of the constituent components of χφ(3) in La1.83Sr0.17CuO4 are shown with the corresponding sample rotation angles. The (ft, fτ) = (2, ±2) THz peaks are measured with EB arriving first, followed by EA. The (2, 0) and (2, 4) THz peaks are measured with EA arriving first, followed by EB. The (2, −2) THz peak corresponds to an emitted Josephson echo. The full phase-matching conditions for both time orderings are described in Supplementary Section 2.

Source data

We demonstrate this principle in Fig. 2c on near-optimally doped La2−xSrxCuO4 (x = 0.17, LSCO), which exhibits a Josephson plasma resonance with resonance frequency fp ≈ 2 THz at temperatures far below the phase transition (T ≪ Tc). All four components of χφ(3) that radiate at the Josephson plasma frequency40,41 are resolved at their respective phase-matching conditions (described in Supplementary Section 2). In contrast to linear spectra, which exhibit broad features due to the presence of both normal and superconducting optical responses, the optical nonlinearities of superconducting carriers are orders of magnitude stronger than those of the normal carriers, resulting in sharp peaks that reflect solely the superconducting response. Here, we focus on the nonlinearity appearing at (ft, fτ) = (2, −2) THz, which corresponds to a terahertz frequency ‘Josephson echo’.

The advantage of Josephson echoes in measuring disorder is illustrated by Fig. 3, in which two cartoons of homogeneous and disordered interlayer tunnelling are shown in Fig. 3a. If intrinsic spectral broadening of the resonance (for example, due to quasiparticle screening) is comparable to a distribution of Josephson plasma frequencies (due to variations in the interlayer tunnelling response), comparable optical responses are observed by linear optical spectroscopy (simulated comparisons between homogeneous and disordered spectra for reflectivity, loss function and optical conductivity are presented in Supplementary Section 6) and it is difficult to distinguish between these two physically distinct situations as shown in Fig. 3b.

Fig. 3: Disorder in schematic one- and two-dimensional spectra.
Fig. 3: Disorder in schematic one- and two-dimensional spectra.
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a, A cartoon of homogeneous and disordered Josephson tunnelling in a layered superconductor. b, Homogeneous and disordered Josephson plasma resonance in one-dimensional spectra, which exhibit ambiguous lineshapes (black curves) in the presence of comparable levels of intrinsic and disorder broadening. c, Homogeneous and disordered Josephson plasma resonance in 2D Josephson echo spectra, in which disorder is evident through asymmetry of the Josephson echo peak.

This ambiguity is eliminated in a 2D spectrum, specifically in the spectral lineshape of the Josephson echo peak42. In the case of a homogeneous Josephson plasma resonance, the peak is symmetric with identical peak widths along the two frequency axes. In the presence of disorder, however, the Josephson echo peak develops a marked asymmetry from projecting disorder line-broadening into an orthogonal direction from the intrinsic linewidth of the resonance. We emphasize that the Josephson echo peak is the only nonlinearity that separates intrinsic and extrinsic broadening mechanisms, which may be understood by referring to its frequency coordinates that are opposite in sign. Microscopically, this reflects an effective time-reversal operation induced by the second excitation pulse that cancels extrinsic dephasing due to disorder (that is, from a distribution of Josephson plasma frequencies).

Measured 2D spectra of the Josephson echo in LSCO are shown in Fig. 4 for increasing temperatures approaching Tc ≈ 36 K. At the lowest measured temperature of 6 K, the Josephson echo peak is asymmetric with a characteristic ‘almond’ shape that reveals a finite degree of disorder42,43. At higher temperatures, the peak becomes increasingly symmetric, suggesting a crossover into the regime of dominant intrinsic line-broadening. Next, we quantify this statement by extracting the individual intrinsic and disorder contributions to the Josephson plasma resonance peak width.

Fig. 4: Two-dimensional Josephson echo spectra of La1.83Sr0.17CuO4 for increasing temperatures.
Fig. 4: Two-dimensional Josephson echo spectra of La1.83Sr0.17CuO4 for increasing temperatures.
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Two-dimensional spectra in the (ft, fτ) = (fp, −fp) quadrant with equal-value contours plotted on top of the raw data. At the lowest temperature of 6 K, ‘almond’ asymmetry of the peak is clearly observed that indicates the presence of disorder. At higher temperatures, the peak lineshape becomes increasingly symmetric due to a lesser relative importance of disorder.

Source data

In 2D spectra, the disorder linewidth γdisorder is projected into a characteristic direction along the ‘diagonal’ (| ft| = | fτ|) line, in contrast to the intrinsic linewidth γintrinsic that results in symmetric broadening. One may intuitively understand this behaviour by interpreting the vertical and horizontal frequency axes as the frequencies of initial terahertz absorption and final terahertz emission respectively, causing each excitation in the continuum of Josephson couplings to appear at a unique position along the diagonal. In the limit of dominant disorder (γdisorder ≫ γintrinsic), the lineshapes along the ‘diagonal’ and perpendicular ‘anti-diagonal’ directions are decoupled42,44 and depend solely on γdisorder and γintrinsic, respectively42. In the present case of comparable intrinsic and disorder broadening, however, the two lineshapes must be simultaneously fit43 to extract γdisorder and γintrinsic.

Slices of the Josephson echo peak at 6 K are plotted in Fig. 5a, along the directions shown in the insets. The lineshape along the ‘diagonal’ direction (orange) is broader than the lineshape along the ‘anti-diagonal’ direction (blue), and the difference between the two peak widths indicates disorder line-broadening. Simultaneous fits of the two lineshapes (to phenomenological functional forms described and justified in Supplementary Section 3) were performed, from which we extract values of γdisorder = 0.08 THz and γintrinsic = 0.38 THz. This value for γdisorder is comparable to that reported by Dordevic et al.18 from linear reflectivity, which was extracted under an assumed value for γintrinsic.

Fig. 5: Fitting the Josephson echo peak lineshape to extract intrinsic and disorder linewidths.
Fig. 5: Fitting the Josephson echo peak lineshape to extract intrinsic and disorder linewidths.
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a, Slices of the Josephson echo peak at 6 K along the ‘diagonal’ (orange) and ‘anti-diagonal’ (blue) directions shown (inset). The lineshapes of both slices are broadened by intrinsic damping and disorder, and fits of the two lineshapes are thus performed simultaneously to extract γdisorder and γintrinsic. b, The temperature dependence of γdisorder and γintrinsic, in which disorder of the Josephson plasma resonance remains roughly constant while its intrinsic damping increases rapidly with temperature. The lines are guides to the eye, and the 2D spectra shown above are simulated with the fitted parameters at 6 K, 15 K and 25 K from left to right, respectively.

Source data

In the linear optical response, the reflectivity edge associated with superconductivity rapidly fades into a featureless background with increasing temperature14. However, 2D terahertz spectroscopy distils the superfluid response from this normal fluid background and uniquely enables us to follow disorder in the superconducting transport even at temperatures approaching the phase transition. Extracted values of γdisorder and γintrinsic are shown in Fig. 5b as a function of increasing temperature. First, the intrinsic linewidth γintrinsic increases rapidly with temperature owing to thermally activated quasiparticle excitations or perhaps even topological defects45. More surprising is the behaviour of the disorder linewidth γdisorder, which remains roughly constant up to a temperature of 25 K ≈ 0.7Tc. Above this temperature, the Josephson plasma resonance becomes lifetime limited and the disorder becomes unmeasurable.

One may expect the disorder to increase with temperature as the coherence length (reflecting the spatial extent of the Cooper wavefunction46) shrinks47. The independence of γdisorder with respect to temperature thus raises the question of whether the disorder linewidth already reflects the full extent of the underlying electronic disorder. We compare in Fig. 6 the measured distribution of plasma frequencies (of standard deviation γdisorder) with the previously measured superconducting gap distribution in LSCO5. The large difference between the two distribution widths indicates, however, that in-plane disorder of the superconducting order parameter is not directly reflected in disorder of the out-of-plane transport.

Fig. 6: Comparison of disorder in the superconducting gap and in the superconducting transport.
Fig. 6: Comparison of disorder in the superconducting gap and in the superconducting transport.
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A probability distribution for the Josephson plasma frequencies fp (solid line) is compared with that of the superconducting gap Δ (shaded bars) measured using scanning tunnelling microscopy5, measured at 6 K and 4.2 K, respectively. The standard deviation of the plasma frequency distribution γdisorder = 78 GHz is far smaller than the standard deviation of the superconducting gap γΔ = 2.8 THz.

We may intuitively understand the difference between the in-plane disorder of the superconducting order parameter to resultant disorder of the out-of-plane tunnelling resonance by considering its broadening effect on the linear optical response, from which the nonlinear optical response partially descends. Namely, assuming that the superfluid density directly inherits identical disorder from the underlying gap fluctuations (due to the short coherence length48 in LSCO49), one expects a disorder broadening of the linear optical response suppressed by the plasma frequency anisotropy \({({\omega }_{\mathrm{p}}/{\omega }_{ab})}^{2}\ll 1\).

Our preliminary theoretical analysis of γdisorder, which will be presented elsewhere, indicates that the many-body nature of Josephson plasmons introduces additional physics beyond the well-known interpretation of molecular photon echoes or spin echoes. Most notably, the momentum dependence that arises in extended systems prevents complete rephasing of collective excitations, resulting in an additional ‘homogeneous-like’ background. We emphasize however, that the primary rephased component still exhibits the elongated peak lineshape characteristic of disorder echoes and observed in our measurements.

Recent advances in tunnelling methods8,9,10,11 will also provide new data to corroborate our conclusions and to systematically connect these two methods. We note, however, that local probes are effective only at the lowest base temperatures, whereas the optical method demonstrated here is already applied up to 0.7Tc in this work. Nonlinear multidimensional responses are even observed immediately above Tc, although fitting of their corresponding Josephson echo signal is currently prevented by small signal amplitudes and signal-to-noise ratio. Suitable technical improvements will enable nonlinear probing of the partially coherent normal state and may provide precious new information on the nature of the pseudogap phase. We also note how new and frequency-agile terahertz sources, which are becoming available due to improvements in laser instrumentation50, will enable systematic measurements throughout the phase diagram of a single compound and across many families of cuprates, where frequencies of the plasma resonance vary between 100 GHz and 15 THz (ref. 12).

In summary, we have made use of 2D terahertz spectroscopy to quantify the role of disorder in the interlayer tunnelling and, thereby, in the superconducting condensate. By measuring the terahertz Josephson echo, we observe an interlayer tunnelling response that is largely immune to the underlying electronic disorder, which remains true even as temperature approaches the phase transition. This demonstration of terahertz echoes27 from a collective excitation provides us with a method to study inhomogeneities in a vast range of quantum materials, ranging from incipient ferroelectrics51 to spin liquids52. Furthermore, the inherent ultrafast nature of this method makes it applicable to multidimensional probes of light-induced non-equilibrium phenomena53,54 to understand the role of disorder in transient states and their formation mechanisms.

Methods

The sample used in our experiment was a single crystal of La1.83Sr0.17CuO4, grown using the travelling-solvent floating-zone method. The crystal was cut and mechanically polished using diamond lapping film along an ac-oriented surface ∼4 mm in size.

To perform 2D terahertz spectroscopy, two intense terahertz pulses were generated by optical rectification of 100 fs, 1,300 nm pulses in two OH1 (2-{3-(4-hydroxystyryl)-5,5-dimethylcyclohex-2-enylidene}malononitrile) organic crystals. The two terahertz pulses are then focused in a non-collinear geometry onto the sample with a parabolic mirror of focal length 76.2 mm (ref. 55), resulting in peak electric fields of ~25 kV cm−1 and ~10 kV cm−1 at the sample surface (Supplementary Section 1) that together are comparable to the threshold electric field for reaching a nonlinear regime Ethresh = (hfp)/2ed ≈ 40 kV cm−1 (ref. 40), where h is Planck's constant, 2e is the Cooper pair charge and d is the interlayer spacing. The emitted nonlinear electric field was then detected by conventional electro-optic sampling using 100 fs, 800 nm pulses in a ZnTe crystal. A differential chopping scheme, in which EA and EB were modulated at 500 Hz and 333 Hz, respectively, was used to isolate the nonlinear electric field from the excitation fields.