Introduction

Antiferromagnetic spin fluctuations are a promising candidate for mediating high-temperature superconductivity (HTSC) in cuprates1; however, the precise relationship between magnetism and superconductivity (SC) across the cuprate phase diagram remains unclear despite extensive studies. This uncertainty is due, in part, to the fact that modeling the competition between local moment formation and itinerant quasiparticles with doping is a tremendously difficult problem.

The quasi-one-dimensional (q-1D) spin ladders (Fig. 1) serve as an ideal platform to tackle this issue. Doped cuprate ladders share several similarities with two-dimensional (2D) cuprates, including high-energy spin-fluctuations2, d-wave pairing3,4,5, non-Fermi-liquid transport behavior6, and large magnetic exchange couplings7. A SC phase has also been observed at elevated pressure (1.5–8 GPa) in the hybrid chain-ladder materials \({{{\mathrm{Sr}}}}_{14 - {{{\mathrm{x}}}}}{{{\mathrm{Ca}}}}_{{{\mathrm{x}}}}{{{\mathrm{Cu}}}}_{24}{{{\mathrm{O}}}}_{41}\) (SCCO) for 11 < x < 148,9,10. Importantly, q-1D spin ladders are also more amenable to modeling3,4,5. Characterizing the magnetic excitations in these materials will, therefore, help us better understand HTSC in cuprates.

Fig. 1: Experimental overview and schematic of ladder triplon excitations.
figure 1

Schematics of (a) RIXS experimental geometry with the orientation of the two-leg Cu2O3 ladders in SCCO. The ladder-leg direction is along c-axis, with lattice constant cleg ∼3.9 Å. b S = 1 triplon excitations (orange dashed-line ellipse) in two-leg quantum spin ladders. The local antiferromagnetic interactions along the ladder leg and rung are mediated by spins (blue arrows) with equal probability pointing along any directions (opposite direction denoted as gray hollow arrows). The rung-singlets (purple ellipses) interact through ladder-leg exchange coupling. c Main resonances of Cu L3-edge XAS. 3d9 white line is marked by the black dashed-line. d Cu L3-edge RIXS spectra taken at 3d9 peak at \(q_{{{{\mathrm{leg}}}}} = - 0.31\) (rlu) in σ polarization. Here, a negative (positive) momentum transfer indicates the RIXS data taken in grazing incidence (emission) geometry.

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Magnetic excitations in even-leg ladders can be viewed as coupled spinons, i.e. triplons, due to the exchange interactions across their rungs (Fig. 1b)3,4,5. In the undoped ladders, these triplon excitations are gapped3,4,5,11,12, and their spectra near the zone-boundary closely resemble the prototypical magnon-like excitations observed in 2D SC cuprates13,14,15. Conversely, experimental studies of triplons in doped ladders are sparse. Inelastic neutron scattering (INS) experiments discovered that the one-triplon gap at \({{{\mathbf{q}}}} \equiv \left( {q_{{{{\mathrm{leg}}}}},q_{{{{\mathrm{rung}}}}}} \right) = \left( {{{{\mathrm{\pi }}}},{{{\mathrm{\pi }}}}} \right)\) survives up to x = 12.2 in SCCO, and is nearly unchanged compared to x = 016,17. Nuclear magnetic resonance (NMR) studies on SCCO, on the other hand, find that the triplon gap decreases with increasing Ca content, which develops into a gap collapse at elevated pressure18,19,20. Although dispersing one- and two-triplon excitations have been reported in the nearly undoped ladder \({{{\mathrm{Sr}}}}_{10}{{{\mathrm{La}}}}_4{{{\mathrm{Cu}}}}_{24}{{{\mathrm{O}}}}_{41}\) using INS12, comprehensive studies of their corresponding doping-dependence are lacking. While these studies have provided some insight into the doping-driven changes in the magnetic excitations of ladder systems, a systematic assessment of the triplon response across a large portion of Brillouin zone as a function of doping has yet to be carried out.

Owing to the strong spin-orbit coupling (SOC) in Cu 2p core-levels, Cu L3-edge resonant inelastic X-ray scattering (RIXS) studies on cuprate materials can probe the momentum-dependent magnetic excitations in the dominant spin-flip channel (ΔS = 1)21,22. Due to an order of magnitude difference in the magnetic exchange of the ladder (100 meV) and chain (10 meV) subsystems, a previous Cu L3-edge RIXS study on \({{{\mathrm{Sr}}}}_{14}{{{\mathrm{Cu}}}}_{24}{{{\mathrm{O}}}}_{41}\) (Sr14) showed that the high-energy magnetic excitations at around 100–350 meV are dominated by the ladder triplons when probed at the Cu 3d9 resonance23.

In this letter, we present a Cu L3-edge RIXS study of the ladders in SCCO with varied Ca content (x = 0 and x = 12.2). Since superconductivity occurs in SCCO samples with 11 < x < 14 under hydrostatic pressure8,9,10, our measurements contrast the magnetic excitations of non-superconducting and superconducting samples. We observe a clear evolution in the low-energy RIXS spectra from collective two-triplon excitations to weakly-dispersive excitations as Sr is replaced by Ca. By comparing with numerical calculations of different disordered Hubbard ladder models, we find that the evolution of the spin excitations is consistent with hole-localization induced by off-ladder impurities. This conclusion is supported by a polarization analysis that uncovers spectral depletion of the charge-like excitations probed in the spin-conserving ΔS = 0 scattering channel, reflective of suppressed carrier mobility. Our results naturally explain the observation of low-temperature insulating phases for high Ca content at ambient pressure, despite the expected increase of ladder holes6,8,9. Since SCCO serves as a bridge between q-1D and 2D cuprates, our study also helps elucidate the interplay between spin fluctuations, chemical pressure (i.e. Ca doping), electronic correlations, and SC in cuprate materials.

Results

Overview of RIXS spectra

Figure 1d shows the Cu L3-edge RIXS spectra at \(q_{{{{\mathrm{leg}}}}} = - 0.31\) (rlu), where magnetic excitations appear around 200–400 meV. These magnetic excitations have reduced spectral weight and increased broadening when Sr is replaced by Ca. Both RIXS spectra are taken at the 3d9 main resonance close to 930 eV indicated in Fig. 1c. (The high-energy inelastic structures, including the local crystal-field splitting of the Cu 3d orbitals [dd excitations] from −1.5 to −3 eV loss, and the broad charge-transfer excitations around −4 eV loss and above, are shown in Supplementary Figure 1)

Figures 2a and d display the momentum-dependent RIXS spectra along \({{{\mathbf{q}}}} = \left( {q_{{{{\mathrm{leg}}}}},q_{{{{\mathrm{rung}}}}} = 0} \right)\). For Sr14 (Fig. 2a), we observe magnetic excitations corresponding to collective two-triplon excitations with a band minimum at the zone center, in accordance with the previous Cu L3-edge work23. The excitations are broadened in \({{{\mathrm{Sr}}}}_{1.8}{{{\mathrm{Ca}}}}_{12.2}{{{\mathrm{Cu}}}}_{24}{{{\mathrm{O}}}}_{41}\) (Sr1.8Ca12.2) (Fig. 2d) and exhibit a flattened dispersion centered around 280 meV loss that extends across a large portion of the Brillouin zone.

Fig. 2: Experimental and theoretical RIXS spectra with Ca-doping effects.
figure 2

a–c (d–f) Cu L3-edge RIXS results for Sr14 (Sr1.8Ca12.2). The elastic line in each of the experimental spectra is subtracted for clarity. a, d Momentum-dependent RIXS experimental spectra for Sr14 and Sr1.8Ca12.2 in σ polarization with bicubic interpolation. b Calculated ΔS = 1 RIXS spectra on an undoped ladder cluster for Sr14. c, e Calculated ΔS = 1 RIXS spectra on doped ladder clusters with hole doping p close to the experimentally determined values from O K-edge XAS results, which are about 6% and 11% for Sr14 and Sr1.8Ca12.2, respectively. The following set of parameters are applied for Sr14 (Sr1.8Ca12.2) on a 32 × 2 site cluster with spectral convergence: \(t_{{{{\mathrm{leg}}}}} = 340\) meV, \(U = 8t_{{{{\mathrm{leg}}}}}\), and \(r = 0.85\) (\(t_{{{{\mathrm{leg}}}}} = 300\) meV, \(U = 9t_{{{{\mathrm{leg}}}}}\) and \(r = 1.1457\)). f Calculated ΔS = 1 RIXS spectra for Sr1.8Ca12 on a 16 × 2 site doped ladder cluster with an impurity potential \(V_{{{{\mathrm{imp}}}}} = 0.5\,U\,\). We make additional assumptions that the on-site impurities (green circles) are screened poorly, and their potential extends to neighboring sites with strength (diameter size) scaled as \(\propto 1/r\). The same tleg, U and r parameters as in (e) are adapted in (f), with a hole doping of 12.5%.

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Quantifying the carrier concentrations

Before proceeding to an analysis of the data, we first establish the effective doping level of the ladder subsystem as this will play a crucial role in the subsequent modeling. Despite extensive studies, the precise experimental quantification of hole doping and carrier mobility with respect to the Ca content remains challenging in the cuprate ladders. Early experiments on SCCO observed an increase in the low-energy Drude-like weight and enhanced metallicity in samples with elevated Ca content, which led to a speculated hole transfer between the chains and ladders upon isovalent Sr-Ca substitution24,25,26. This transfer was rationalized in a ladder-dominant picture of the transport in SCCO due to the 180° Cu-O-Cu bond angle in the ladders24,25,26. Ca also has a smaller atomic radius, which induces local structural changes in the Cu-O-Cu bond distances and can generate interactions between the chain and ladder subsystems27. Recent studies of Sr14, however, revealed that errors up to a factor of 2–5 can occur when quantifying the ladder hole content if charge and magnetic correlations in the chains are not taken into consideration28.

Given these uncertainties, we estimated the nominal ladder hole densities in Sr14 and Sr1.8Ca12.2 using polarization-dependent O K-edge XAS measurements. This procedure is facilitated by the atomic sensitivity of XAS to the distinct Cu-O bonding environments in the edge-sharing CuO2 chain and two-leg Cu2O3 ladder subsystems29. By aligning the linearly polarized light vector parallel to the rung or leg direction, the XAS data can be used to extract the respective O 2p hole character of the chain and ladder hole subsystems by linear dichroism in Sr14 (H1 and H2 of 527.5 and 528.2 eV, respectively, in Fig. 3a, b, see caption)28,29,30,31. To extend this analysis to the Ca-doped compounds, it was assumed that the isovalent Sr-Ca substitution would not alter the total number of doped holes, but rather exchanges the respective weight of chain (H1) and ladder (H2) hole-doping levels29 due to the chain-ladder charge transfer inferred in optical and transport experiments24,25,26. Therefore, one can extract the chain and ladder hole concentration by measuring their relative hole peak fraction H1 or H2 to the total weight of holes H1+H2 using polarization-resolved O K-edge XAS (see Supplementary Note 2). Using the model in ref. 29, we obtain excess ladder hole concentrations of 0.06 and 0.11 (holes per Cu atom) for Sr14 and Sr1.8Ca12.2, respectively, in line with previous literature (see Supplementary Note 2).

Fig. 3: Evaluated ladder hole concentration with Ca-doping effects.
figure 3

O K-edge XAS spectra with fitting components for (a–b) Sr14 and (c–d) Sr1.8Ca12.2. A linear background (BG) is applied in the spectral fitting. The spectral weight at 527.5, 528.2 and 529.25 eV, respectively, represent the chain hole content H1, the ladder hole content H2, and the upper Hubbard band28,29,31. The polarization of incident X-rays is parallel to the leg c-direction in (a, c), whereas (b, d) are measured in the rung a-direction configuration.

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Localized magnetic excitations due to off-ladder plane impurities

To better understand our experimental results, we modeled the low-energy RIXS response to lowest order in the ultrashort core-hole lifetime expansion32,33,34 using density matrix renormalization group (DMRG) calculations for doped Hubbard ladders35,36. This approximation has proven accurate for Cu L-edge measurements on 3d9 layered 2D cuprates and strongly correlated ladders33,37. For Sr14, we adopted parameters resulting in magnetic exchange parameters close to the isotropic limit with \(r = J_{{{{\mathrm{rung}}}}}/J_{{{{\mathrm{leg}}}}} = 0.85\)37. For Sr1.8Ca12.2, we estimated \(r \approx 1.15\) for the highest Ca concentrations using perturbation theory and the available structural data (see Supplementary Fig. 13 and full description in Supplementary Note 3).

Figure 2 compares the experimental Cu L3-edge RIXS data with DMRG calculations of the dynamical spin structure factor. Here, we consider both undoped and “nominally” doped ladders, as determined by our O K-edge XAS results. As in previous studies, the ΔS = 1 scattering channel dominates the spectra21. The RIXS spectra for Sr14 are reasonably well described by both the undoped (Fig. 2b) and doped (Fig. 2c) models; however, we observe a slight increase in the broadening of the two-triplon excitations towards the zone boundary in the 6 % hole-doped ladders, which may be caused by the mixing with charge or Stoner-like continuum excitations like in 2D cuprates38,39. Overall, the experimental Sr14 Cu L3-edge RIXS spectra are better described by the DMRG calculation of the 6% nominally doped ladder.

The spin excitations for an 11% hole-doped ladder in Fig. 2e are more diffused compared to the undoped ladder calculations for Sr14 in Fig. 2b, consistent with experiments. However, the increased hole doping also produces a pronounced downturn in the calculated dispersion close to the zone center, which does not occur in our experimental data for Sr1.8Ca12.2. Despite residual low-energy weight, the overall magnetic excitations for Sr1.8Ca12.2 (Fig. 2d) are clearly gapped and relatively dispersionless.

Our perturbation theory analysis in the Supplementary Note 3 indicate that \(r \approx 1.15\) for the Ca-doped samples. This estimate explicitly rules out large values of the rung coupling which would produce a flattened magnetic excitation spectrum. Another mechanism must therefore be responsible for the large change in magnetic excitations. Early transport studies found that the resistivity for SCCO with x > 11 decreased linearly down to 100–150 K, followed by an abrupt upturn with further cooling8,9. The onset of this insulating behavior was attributed to carrier localization at low temperature9. We, therefore, consider the effects of localization on the low-energy excitations in Sr1.8Ca12.2. To this end, we performed DMRG calculations on ladders with additional impurity potentials distributed randomly throughout the ladder. The physical picture is that Ca doping introduces impurity potentials outside the ladder planes, which are poorly screened by the excess ladder charges. There are two possible scenarios here, which cannot be distinguished in our experiments. One is based on current SCCO literature, where it is widely accepted that Sr-Ca substitution transfers electrons from ladders to the edge-sharing chains6,24 where they tend to localize40 and exert an attractive potential on the ladder holes. (This scenario can be viewed as a sort of excitonic formation between the particle-hole pair introduced by the charge transfer process.) Alternatively, Ca could be acquiring excess charges from the Sr sites, however, this is unlikely for two isovalent elements. Lastly, chemical doping in q-1D materials is often associated with structural distortions that lead to varied magnetic exchange couplings41,42. We tested the possibility of such a scenario and found it inconsistent with our estimated ladder-rung couplings in Sr1.8Ca12.2. (see Supplementary Fig. 14)

To examine the feasibility of the localization proposal, we modeled the unscreened off-ladder impurities by introducing an extended impurity potential with 12.5% coverage of ladder sites (see Supplementary Note 3). The on-site value of the potential was set to \(V_{{{{\mathrm{imp}}}}} = 0.5\,U\,\) while the values on the neighboring sites up to next-nearest-neighbors was scaled as \(\propto 1/r\), as sketched in the inset of Fig. 2f. The resulting simulated RIXS signal in the ΔS = 1 channel after averaging over eight disorder configurations is shown in Fig. 2f. The downward dispersing spectral weight near zone center is suppressed, while the overall dispersion flattens close to the zone boundary. These modeled spectra are in closer agreement with the experiments and demonstrate that local impurities and charge localization are a plausible explanation for the flattened magnetic excitations. However, we also note that there remains a non-negligible difference between the theory and experiment at small momentum transfers. This may be due to the absence of RIXS dipole matrix elements, or the overall simplicity of our model. A third possibility could be finite size effects in our calculations. Due to the large computational time required by our DMRG simulations, we have found a compromise between the largest cluster size compatible with the nominal doping extracted from the XAS analysis (~11%), the highest coverage of impurities (up to 12.5%), disorder averaging (up to 8 configurations), and the largest attainable precision (see Supplementary Note 3). Nevertheless, our results suggest that other factors beyond simple charge doping such as localization need to be invoked for explaining the evolution of the magnetic RIXS response with increasing Ca content.

Polarization-resolved RIXS analysis

To gather additional support for our interpretation of localized holes in the ladders of Sr1.8Ca12.2, we performed a polarimetric analysis of the RIXS measurements to assess the character of charge and spin scattering in the low-energy excitations33,43,44,45. Here, we apply the method established in ref. 43 to disentangle ΔS = 1 and ΔS = 0 channels of the RIXS response, which was previously demonstrated for the two-spinon excitations of \({{{\mathrm{CaCu}}}}_2{{{\mathrm{O}}}}_3\). (see Supplementary Note 4) This approach is based on the idea that the RIXS intensity can be expressed as the product of a dynamical structure factor \(G\left( {\Delta S,\varepsilon ,q,\omega } \right)\) and a form factor \(F\left( {\Delta S,\varepsilon ,q} \right)\) such that \(I\left( {\varepsilon ,q} \right) = G\left( {\Delta S,\varepsilon ,q,\omega } \right)F\left( {\Delta S,\varepsilon ,q} \right)\)22,46. Here, \(\omega\), \(\varepsilon\) (\(\sigma\) or π), and q are the energy loss, incident X-ray polarization, and momentum-transfer, respectively. By calculating the form factors \(F\left( {\Delta S,\varepsilon ,q} \right)\) for respective ΔS = 1 and ΔS = 0 scattering channels, we extract the spin-resolved structure factors \(G\left( {\Delta S,\varepsilon ,q,\omega } \right)\) from the measured \(I\left( {\varepsilon ,q} \right)\) in polarization-resolved RIXS experiments. In this work, we follow the single-ion picture in ref. 43 where the local spin-flip and charge scattering probabilities \(P\left( {\Delta S,\varepsilon ,q} \right)\) are used to approximate the form factors \(F\left( {\Delta S,\varepsilon ,q} \right)\).

Figure 4 compares the ΔS = 1 and ΔS = 0 signal for \(q_{{{{\mathrm{leg}}}}} = 0.38\) and 0.14 rlu. We find that the ΔS = 1 channel indeed provides the major contribution to the low-energy RIXS response for both Sr14 and Sr1.8Ca12.2, confirming a magnetic origin for these excitations. In Sr14 (Fig. 4a, b), the ΔS = 0 scattering is dominated by a sharp mode at slightly reduced energy compared to the ΔS = 1 channel. Progress has recently been made in understanding the RIXS response in ΔS = 0 channel of the ladder system, which indeed predicts a sharp bound two-triplon state in this channel37,47. Additionally, a broad component centered around 400–600 meV appears in this channel, which is more pronounced near the zone boundary, coinciding with the momentum region where the excitation profiles show a stronger high-energy tail (Fig. 2a). The energy scale of these excitations is comparable to the theoretically predicted charge and ΔS = 0 multi-triplon excitations11,37. Additionally, these modes resemble the 500 meV peak in the ΔS = 0 RIXS channel of 2D cuprates48,49,50,51, while its nature and connection to ladder excitations remains an open question.

Fig. 4: Disentangled spin and charge dynamics by polarization-resolved RIXS.
figure 4

Cu L3-edge RIXS spectra with ΔS = 1 (blue-shadowed) and ΔS = 0 (red line) channels disentangled for (a–b) Sr14 and (c–d) Sr1.8Ca12.2 taken at \(q_{{{{\mathrm{leg}}}}} = 0.38\) rlu (a, c) and \(q_{{{{\mathrm{leg}}}}} = 0.14\) rlu (b, d). Experimental data shown here are taken with π polarization in grazing emission geometry. We remind the reader that the RIXS intensity \(I\left( {\varepsilon ,q} \right)\) can be expressed as combination of ΔS = 1 and ΔS = 0 dynamical structure factors \(G\left( {\Delta S,\varepsilon ,q,\omega } \right)\) multiplied by form factors \(F\left( {\Delta S,\varepsilon ,q} \right)\) of the scattering processes22,46,60.

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The ΔS = 0 channel in Sr1.8Ca12.2, on the other hand, is essentially a structure-less background below −1 eV loss, as shown in Fig. 4c, d. Since these excitations arise from low-energy charge and ΔS = 0 multi-triplon excitations, their suppression with Ca doping is consistent with localization. Previous theoretical studies on 2D 3d9 cuprates and ladders have demonstrated that the Cu L3-edge RIXS signal is dominated by single spin-flip excitations, with additional intensity modulations induced by the dipole matrix elements33. At the same time, the leading contribution in the ΔS = 0 channel of the doped system comes from charge (particle-hole) excitations33,37. With mobile carriers, the ΔS = 0 response is dominated by the charge continuum with excitations extending to zero-energy33,37. Conversely, with localized charges, the charge excitations are gapped with weight transferred to higher-energy33,37. This behavior is also captured by the calculated modified dynamical charge structure factor that reflects the RIXS intensity in this scattering channel. (see Supplementary Fig. 15) We, therefore, conclude that the observed non-dispersive and damped magnetic excitations in Sr1.8Ca12.2 can be understood using a hole-immobilized ladder picture, despite an increased carrier density.

Superconducting correlations

We now explore the implications of our results for superconductivity in the Ca-rich samples. Here, we envision that the localized carriers with Sr-Ca substitution will be released as hydrostatic pressure is applied, eventually forming a superconducting condensate. We again model the effects of Ca doping by introducing the same random impurity configurations used to simulate the spectra discussed in the previous sections. We then assume that the primary effect of the hydrostatic pressure is to reduce the lattice bond distances, thus effectively increasing the hopping integrals while holding the impurity potentials fixed. (For simplicity, we assume that all hopping integrals increase uniformly in the ladders with pressure.) This procedure thus captures the competition between the localizing potentials introduced by Ca-doping and the increase of bandwidth introduced by pressure. We then compute the pair correlations (see methods) for the clean system and for the averaged impurity configurations with increased hopping, as shown in Fig. 5. Without impurities, we obtain a power-law dependence in the pair correlations, indicating strong pairing tendencies. The correlations are exponentially suppressed when we introduce the impurity potentials. As hydrostatic pressure is applied and the hopping increases, the pair correlations begin to recover the robust behavior of the clean system.

Fig. 5: Calculated pair correlations with Ca-doping and static pressure effects.
figure 5

Rung-singlet pair correlations plotted as a function of distance along the ladder-leg direction in logarithmic scale. For simplicity, the increasing hopping integral for simulating static pressure is uniform over the ladder cluster. The Hubbard repulsion and impurity potential Vimp are kept fixed in these calculations.

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Discussion

Our results have important consequences for understanding the starting condition from which the SC state in SCCO is established with high Ca content and elevated pressures. Despite the idealized assumptions of our model, our results imply that off-plane impurity potentials can significantly shape the low-energy magnetic and superconducting properties of the cuprate ladders. The ingredients of a complete theory must now reconcile the roles of Sr-Ca substitution, magnetic and charge fluctuations, and the observed localization in Ca-doped SCCO.

We believe that this issue should also be re-visited in 2D cuprates, where impurities in the charge reservoir layers are poorly screened. Noticeably, INS studies have revealed similar localized multi-magnon excitations in the ΔS = 1 channel at ∼130 meV in 2D superconducting cuprates. With elevated hole concentration, the excitation energy was interpreted as scaled with the pseudogap energy upon doping52. We connect this observation to our measured localized magnetic excitations because of a pseudogap-like feature ∼100 meV observed in x = 11 SCCO samples using optical sepctroscopy53. Unfortunately, this mode could be obscured by our instrumental resolution of ∼95 meV. To further elucidate the overall effects of Sr-Ca replacement in SCCO, we suggest future experiments using INS at high-pressure or higher-resolution RIXS for measuring momentum-resolved magnetic excitations. This will help to clarify the underlying electronic correlations connected to pairing in SCCO as well as the microscopic interactions of other competing electronic phases, e.g. pseudogap formation53.

In summary, we have performed Cu L3-edge RIXS measurements on the spin ladder in SCCO. We observed a crossover in the magnetic excitation spectrum from the collective 2 T excitations in Sr14 to a high-energy incoherent gapped magnetic excitations in Sr1.8Ca12.2. By comparing these with spectra evaluated using DMRG calculations on a model Hamiltonian, we conclude that the observed reorganization of the magnetic excitations reflects a tendency towards carrier-immobility in the Ca-doped system. This conclusion was supported by polarization-dependent RIXS measurements, where we extracted a clear suppression of the ΔS = 0 channel from well-defined charge and multi-triplon excitations (x = 0) to a featureless background (x = 12.2). We further calculated the pair correlations, and our results indicated that (Sr,Ca) layer impurities could impact the low-energy properties of the ladder subsystem. Our work calls for future studies re-examining the role of off-plane impurities in 2D cuprates in the doped regime where superconductivity, pseudogap, as well as charge and spin orders compete. Finally, our results on localized excitations with Ca-doping could be useful to understand pair (pre-) formation in the pseudogap regime. On the other hand, our work gives valuable information on the dynamics of q-1D cuprate ladders, as well as their relationship with various competing electron-pairing hypotheses.

Methods

RIXS experiment

Single crystals of \({{{\mathrm{Sr}}}}_{14}{{{\mathrm{Cu}}}}_{24}{{{\mathrm{O}}}}_{41}\) (Sr14) and \({{{\mathrm{Sr}}}}_{1.8}{{{\mathrm{Ca}}}}_{12.2}{{{\mathrm{Cu}}}}_{24}{{{\mathrm{O}}}}_{41}\) (Sr1.8Ca12.2) were grown with the traveling-solvent floating zone method54,55. The samples were cut and polished to form the a-c plane, with the b-axis pointing out-of-plane. Subsequently, top-post cleavage was performed in-situ before all measurements in a vacuum pressure of better than 5 × 10−10 mbar. Cu L3-edge RIXS and X-ray absorption spectroscopy (XAS) measurements were performed at the Advanced Resonant Spectroscopies (ADRESS) beamline at the Swiss Light Source (SLS), Paul Scherrer Institut56,57,58. The total energy resolution of the RIXS experiment was ∼95 meV at the Cu L3-edge (930 eV). The RIXS spectrometer was fixed at a scattering angle 2θ = 130° while the experimental geometry was set such that the crystallographic b- and c-axis of the sample lay in the scattering plane (Fig. 1a). This geometry allows no momentum-transfer along the ladder rungs and selectively enhances the multi-triplon RIXS response with even-parity along \({{{\mathbf{q}}}} = \left( {q_{{{{\mathrm{leg}}}}},q_{{{{\mathrm{rung}}}}} = 0} \right)\)23,59. RIXS measurements were acquired with 15 min per spectrum, and normalized to the total spectral weight of the crystal-field excitations. XAS spectra were recorded in total fluorescence yield (TFY) mode. σ polarization was employed for the incident X-rays, unless specified otherwise. All measurements were taken at base temperature 20 K.

DMRG calculation

We use a single band Hubbard ladder with nearest neighbor hopping to model the system (see Supplementary Note 3). The lowest order contributions at the Cu L3-edge are given by the dynamical spin structure factor S(q, ω), and a modified dynamical charge structure factor \(\tilde N\left( {q, \omega} \right)\)33. These structure factors are calculated as:

$$S\left( {q,\omega} \right) = \mathop {\sum }\limits_{f,\sigma} \left| {\left\langle f \right|S_{q,\sigma}^z\left| g \right\rangle} \right|^2\delta \left( {E_f - E_g + \omega} \right)$$
(1)

and

$$\tilde N\left( {q,\omega} \right) = \mathop {\sum}\limits_{f,\sigma} \left| {\left\langle f \right|\tilde n_{q,\sigma }\left| g \right\rangle} \right|^2\delta \left( {E_f - E_g + \omega} \right),$$
(2)

where

$$\tilde n_{q,\sigma } = n_{q,\sigma } - n_{q,\sigma }n_{q,\sigma ^\prime }$$
(3)

Here, q and ω are the net (1D) momentum and energy transfer to the system, \(\sigma\) is a spin index, g and f are the initial and final states of the scattering process, respectively, Eg and Ef are their respective energies, and \(S_{q,\sigma }^z\) and \(\tilde n_{q,\sigma }\) are the Fourier transforms of the local spin and charge operators, respectively. For Sr14, we use a 32 × 2 ladder cluster and keep up to m = 1000 states and a maximum truncation error of 10−7. The artificial broadening parameter is set to \(\eta = 47.5\) meV to match the experimental resolution. For Sr1.8Ca12.2, we use a 16 × 2 ladder with the same m = 1000 states and artificial broadening parameter \(\eta\) as was used for Sr14. The maximum truncation error is reduced to 10−8. The impurity potentials \(V_{{{{\mathrm{imp}}}}} = 0.5\,U\,\) are introduced at four randomly chosen sites in the ladders and extended to nearest neighbors with appropriate rescaling of strength (see Supplementary Note 3 and Fig. 9). To understand the implications of hole localization in the pairing tendencies exhibited by the carriers in the ladder, we further calculate the averaged rung singlet pair correlation function P(d) as a function of distance d

$$P\left( d \right) = \frac{1}{{L - d}}\mathop {\sum}\limits_{j = 1}^{L - d} {\left\langle {{{\Delta}}_j^{\dagger} {{\Delta }}^{\phantom\dagger}_{j + d}} \right\rangle},$$
(4)

where \(\Delta _i^{\dagger}\) is defined2 as

$$\frac{1}{{\sqrt 2 }}\left( {c_{i,0, \uparrow }^{\dagger} c_{i,1, \downarrow }^{\dagger} - c_{i,0, \downarrow }^{\dagger} c_{i,1, \uparrow }^{\dagger}} \right).$$
(5)