Cavity-enhanced superconducting response in an underdoped cuprate

Abstract

Superconductors carry electrical current without resistance when paired electrons condense into a coherent macroscopic quantum state. In underdoped cuprates, evidence suggests that pairing-related correlations and superconducting fluctuations can survive above the temperature at which global coherence is lost, pointing to phase fluctuations as a key limitation on superconductivity in this regime. Motivated by recent demonstrations of cavity-modified collective states in quantum materials, we investigate whether superconducting coherence can be stabilized by engineering the electromagnetic environment of the superconductor. We study an underdoped YBa₂Cu₃O₇₋δ thin film in a tunable terahertz cavity formed with a semi-transparent gold mirror. From temperature-dependent terahertz transmission measurements, we find that the cavity enhances the superconducting response below the critical temperature, with an increase of the inferred superfluid weight. The effect becomes more pronounced at smaller cavity lengths and is accompanied by an upward shift of the superconducting onset temperature. Calculations based on a cavity-coupled model for phase-fluctuating superconductors capture these trends and support an interpretation in terms of cavity-enhanced phase stiffness. These results showcase the potential of cavity engineering for designing emergent functionalities in correlated systems.

Main text

The control of quantum materials through cavity electrodynamics has seen a rapid development in recent years. A growing body of theoretical work has proposed that modifying the electromagnetic environment can reshape collective states of matter, while a series of experiments has begun to show that cavity confinement can alter the properties of correlated electronic systems. These advances have established cavity quantum materials as a promising route to manipulating many-body phases beyond conventional tuning parameters such as doping, pressure or external fields.

Superconductivity has only recently entered this experimental landscape. Initial studies of cavity-embedded superconductors have shown that the superconducting condensate can be directly affected by confinement of the electromagnetic field. Strong reduction of the superfluid density has been reported in both an organic superconductor and in NbN coupled to optical cavities. These results provide clear evidence that cavity electrodynamics can influence superconducting ground states, but they also suggest that the effect is not generically favourable to superconductivity. This naturally raises the question of whether cavity electrodynamics can, under appropriate conditions, enhance rather than suppress superconductivity.

Underdoped cuprates provide an exceptionally compelling platform in this respect. In these materials, electron pairing and long-range phase coherence are widely thought to develop on different temperature scales rather than emerging simultaneously. In particular, the pseudogap phase appearing above the critical temperature (Tc) in underdoped cuprates is widely discussed as a state in which strong pairing correlations persist up to a higher temperature T*, even though macroscopic superconducting coherence is weakened or lost (Figure 1a). From this perspective, a possible route to enhancing superconductivity would be to suppress the mechanisms that disrupt phase coherence rather than strengthening the pairing itself. This possibility is especially intriguing in light of theoretical proposals that cavity photons can mediate effective long-range interactions and stabilize collective order in quantum matter.

Here, we investigate whether a cavity-modified electromagnetic environment can influence superconducting coherence in a phase-fluctuation-dominated cuprate. We study a 100-nm-thick underdoped YBa₂Cu₃O₇₋δ (YBCO) film embedded in a tunable cavity formed by the film itself and a semi-transparent gold mirror (Figure 1b). The tunable distance between the superconducting film and the mirror sets the fundamental mode of the cavity and provides a tunable photonic environment for YBCO.

Figure 1. Cavity control of phase coherence in underdoped YBa₂Cu₃O₇₋δ. a, Phase fluctuations of the order parameter are widely discussed as an important factor limiting superconductivity in high-Tc cuprates. In underdoped (UD) samples, evidence for incoherent local pairing has been reported throughout the pseudogap phase. We investigate whether the presence of a mirror in proximity to the superconductor could effectively act as a cavity, promote phase coherence and enhance the superconducting response. b, The sketch illustrates the cavity geometry. Terahertz (THz) time-domain spectroscopy provides a direct way to track the optical path of photons inside the cavity: the delay between the main (m=0) and first (m=1) transmitted peaks yields the cavity round-trip time and thus a robust calibration of the cavity length. The colormap shows the room-temperature THz transmission as a function of cavity length.

Time-domain terahertz (THz) spectroscopy is a particularly well-suited technique to this platform for two reasons. First, it provides a direct, in situ and non-invasive measure of the cavity length: the delay between successive transmitted pulses yields the photon round-trip time inside the cavity, and therefore the optical path between the film and the mirror. Second, THz spectroscopy is highly sensitive to the onset of superconductivity. The low-frequency complex conductivity directly reveals the inductive response of the condensate through its imaginary component. For this reason, THz conductivity measurements have long been used to track the superconducting transition in thin film cuprates.

The room-temperature THz electric field transmitted by the cavity assembly is shown in Figure 1b for different cavity lengths. As the mirror position is tuned with nanometric precision, the first cavity reflection (m=1) shifts systematically in time. This delay provides a robust calibration of the cavity round-trip time and hence of the cavity length (see Methods for a detailed description of the experimental setup).

With the cavity geometry calibrated at room temperature, we next investigate its response across the superconducting transition. Thermally driven changes do not measurably modify the cavity length over the temperature range explored here, so the mirror-film separation can be treated as constant during cooling (Supplementary Information Sec. S8). For a fixed mirror position (L = 435 µm), we record the transmitted THz electric field as a function of temperature (Figure 2).

The superconducting transition is immediately visible in the temperature-dependent map, occurring at approximately 85 K, in agreement with the sample specifications (Methods). Across the transition, the transmitted THz field undergoes two pronounced changes: i) its amplitude is strongly reduced on entering the superconducting state, ii) the transmitted pulse acquires a clear phase shift. These changes are illustrated more clearly by representative traces measured above (orange) and below (purple) the critical temperature (inset in Figure 2b). Below Tc, the waveform develops a more derivative-like shape, reflecting the emergence of a strong inductive response. In electrodynamics terms, this behaviour arises from the rapid increase of the imaginary conductivity (σ2) driven by the condensate, which enhances reflectivity and shifts the peak of the transmitted field in time.

Additional structures are visible in the map. In particular, the feature at about 2.9 ps corresponds to the first cavity reflection (m=1), whose temperature dependence follows that of the main transmitted pulse. Other weaker features at shorter time delays originate from the LaAlO₃ (LAO) substrate on which the YBCO film is grown and are not discussed further here (Supplementary Information, Sec. S2).

Figure 2. Enhanced superconducting response in a terahertz cavity. a, Temperature-dependent transmitted THz electric field for a representative cavity length (L = 435 µm). THz traces recorded above Tc (orange) and below Tc (purple) are plotted in the inset of panel (b). On cooling below Tc, the THz waveform shows both a reduction in transmission and a shift in arrival time of the peak, consistent with the emergence of an inductive superconducting response. b, Frequency-averaged imaginary conductivity, σ̄2, extracted from the THz response and plotted as a function of temperature. The frequency averaging range is 0.4 THz to 1 THz. As a measure of the condensate-related superconducting response, σ̄2 is systematically larger inside the cavity (solid coloured curve) than outside it (dashed black curve) below Tc, indicating a cavity-induced enhancement. Inset, normalized temperature derivatives of the curves in the main panel, shown for the film outside the cavity (dashed line) and inside the cavity (solid line).

To quantify the cavity-induced change in the superconducting response, we convert the temperature-dependent THz transmission into the complex conductivity of the YBCO film using the thin-film Tinkham formalism. The analysis is performed with respect to a reference measurement of the empty cavity having the same length, consisting of the semi-transparent mirror and the LAO substrate without the superconducting film. Importantly, the analysis explicitly accounts for the temperature dependence of all cavity elements as well as for propagation through the full structure (Supplementary Information, Sec. S5). From the extracted conductivity, we evaluate the frequency-averaged imaginary part σ̄2, the integral of σ2(ω) over the averaging window divided by its width, which is proportional to the superfluid weight and therefore provides a figure of merit of the superconducting condensate response.

Figure 2b shows this quantity as a function of temperature for the film inside (solid coloured curve) and outside the cavity (dashed black curve). In both cases, σ̄2 rises sharply below Tc, as expected for the onset of superconductivity, but its magnitude is larger in the cavity, indicating an enhanced superfluid response. We stress that this conclusion does not rely on a single analysis protocol: independent approaches based on direct time-domain fitting and on the phase shift of the transmitted THz pulse yield consistent results (Supplementary Information, Sec. S3). Taken together, these observations provide evidence that, for this cavity geometry, the superconducting condensate response is enhanced inside the cavity.

A closer inspection of Figure 2b shows that σ̄2 starts to increase at slightly higher temperature inside the cavity, with a small but reproducible finite response visible above the critical temperature of the film without the cavity. The normalized temperature-gradients of the curves, plotted in the inset of Figure 2b for a temperature range around the transition, highlight this trend, suggesting that the cavity promotes the emergence of the superconducting condensate response at slightly higher temperature.

We performed several control measurements to rule out trivial origins of this effect, including thermal drifts, differences in thermal contact, direction of the temperature scan (heating or cooling) and THz-induced heating (Supplementary Information, Sec. S4). In all the tests, the upward shift of the transition temperature is consistently reproduced. Crucially, the shift disappears when the metallic gold layer is removed from the mirror, and only the mirror dielectric substrate is brought in proximity to the YBCO film (Supplementary Information, Sec. S4.3).

Figure 3. Cavity-length dependence of the enhanced superconducting response. Frequency-dependent σ2 (and σ1 in the insets) below (a, T = 66 K) and above Tc (b, T = 90 K), measured in free space (black curve) and inside the cavity (yellow, L = 20 µm). The most pronounced cavity-induced change occurs in σ2 below the superconducting transition, where the inductive response is enhanced. The shaded areas indicate the propagated standard uncertainties. c, Frequency-averaged σ2 (and σ1 in the inset) as a function of temperature for different cavity lengths, compared to the one for YBCO in free space (black curve). The enhancement of the superconducting response is larger at shorter cavity lengths.

To resolve how the cavity modifies the electrodynamics across the superconducting transition, we next vary the mirror-film separation and focus on the optical conductivity close to Tc. For the shortest cavity length studied (L = 20 µm), Figures 3a and 3b show the frequency-dependent optical conductivity extracted below and above the transition, respectively. The shaded areas indicate the propagated standard uncertainty in the extraction of the optical conductivity. Below Tc, the imaginary conductivity σ2(ω) displays the characteristic increase towards low frequency, going as the inverse of frequency, expected for the inductive response of the superconducting condensate. Importantly, this response is larger inside the cavity (yellow) than outside the cavity (black), as emphasized by the hatched area between the two curves. By contrast, the real conductivity σ1(ω), shown in the insets, exhibits only minor cavity-dependent variations. Above Tc, σ2(ω) becomes negligible both inside and outside the cavity, while σ1(ω) shows no systematic cavity-induced change. The most pronounced cavity-induced effect therefore appears in σ2 below the superconducting transition, indicating that the cavity selectively enhances the inductive, condensate-related response without significantly affecting the dissipative channel.

This behaviour becomes even clearer when the conductivity is averaged over frequency for different cavity lengths as a function of temperature (Figure 3c). The inset shows that the averaged σ1 remains essentially independent of cavity length and follows the same temperature dependence inside and outside the cavity. By contrast, the averaged σ2 in the main panel displays a clear and systematic cavity-length dependence: below the transition, it is consistently enhanced in the cavity, and the enhancement grows as the mirror is brought closer to the film.

Figure 4. Signatures of enhanced phase stiffness in the cavity. a, Normalized temperature derivatives of the data in Figure 3c, highlighting the evolution of the superconducting transition inside the cavity (coloured curves) relative to the film outside the cavity (black curve). The transition shows a small upward shift of the onset temperature, indicated by the horizontal arrow. b, Results of calculations for a cavity-coupled XY and BKT model. The model captures the two main experimental trends: an increase in phase stiffness J below the superconducting transition and a cavity-dependent shift of the TBKT. c, Experimental values of ωσ2 in the limit of zero frequency as a function of cavity length (fixed T = 66 K). The increase for shorter cavities is in qualitative agreement with the calculated cavity-enhanced phase stiffness (purple shaded area, T/T* = 0.55). Error bars represent the propagated standard uncertainties.

Figure 4a summarizes the temperature evolution of the condensate response for all cavity lengths. By plotting the normalized temperature-derivatives of the averaged σ2 referenced to the bare film, two features become apparent. First, the superconducting response remains systematically larger in the cavity over the full transition region. Second, the maximum shifts to higher temperature by up to about 1 K, as highlighted by the horizontal arrow. The black and yellow vertical lines indicate the transition temperatures in free space and for L = 20 µm, respectively, estimated from the second-derivatives of the curves (Supplementary Information, Sec. S9). These trends suggest that cavity confinement affects not only the magnitude of the condensate response in the superconducting phase, but also its onset.

As a minimal framework for interpreting these observations, we use a phenomenological description of a phase-fluctuating superconductor in which the superconducting order parameter can be treated, at a coarse-grained level, as having a nearly fixed amplitude but a fluctuating phase. In underdoped cuprates, this description is consistent with the phase diagram in Figure 1, where pairing-related correlations emerge below T*, while long-range superconducting coherence develops only at the lower temperature Tc. For a low-dimensional layered system such as YBCO in the weak interlayer-coupling limit, this picture implies a superconducting transition of the Berezinskii-Kosterlitz-Thouless (BKT) type, driven by the unbinding of vortex-antivortex pairs.

It should be stressed that this approach neglects the full three-dimensional character of YBCO and it is not intended here as a microscopic description of the material. Rather, it provides a natural minimal framework for addressing how the presence of the cavity may affect phase coherence in superconductors controlled by phase fluctuations.

Motivated by this picture, we describe the superconducting film within a long-wavelength approximation to the XY model, which is equivalent to the coarse-grained hydrodynamic London framework (see Supplementary Information, Sec. S6 for details). This model has a nonlinearity (mode-coupling term) that allows thermal and quantal phase fluctuations to renormalize the static long wavelength phase stiffness that is relevant to the BKT transition.

We consider the film inside a Fabry-Pérot cavity and couple the space and time derivatives of the superconductor’s phase field to the electromagnetic field in the usual way. A crucial point of the analysis is that within the plane of the film, the vector potential associated with fluctuations of the cavity modes has a longitudinal component that couples strongly to the superconducting phase at nonzero frequency and is strongly affected by the cavity. The latter acts to increase the superfluid stiffness, reducing the amplitude of the thermal and quantal fluctuations of the phase field and thus reducing the mode-coupling contribution to the BKT stiffness. This reduction appears first as an enhancement of the stiffness J, which in turn suppresses long-wavelength phase fluctuations below the transition. At the same time, the larger stiffness raises the energetic cost of vortex excitations, thereby strengthening vortex-antivortex binding and shifting the BKT transition to higher temperatures. The model therefore provides a qualitative framework that is consistent with the experimental observations.

The physical understanding of the phase stiffening within this model is the following. The controllable photon gap created by the cavity mirrors reduces the hybridization of the vector potential (that is, the photons contributing to non-static electric field) with the superconducting low-energy phase correlations, protecting the latter from electric-field fluctuations and thus increasing the stiffness compared with free space (Supplementary Information, Sec. S6.1).

Figure 4b shows the cavity-renormalized stiffness J, relative to its free-space value, for a few representative cavity lengths chosen to match the experimental range. In the calculations, cavity confinement enhances the phase stiffness below the transition temperature of the film outside the cavity. When vortex fluctuations are included, this increase is accompanied by an upward shift of the estimated BKT transition temperature TBKT, consistent with stronger phase rigidity and delayed vortex unbinding.

Figure 4c compares the measured cavity-length dependence of the low-frequency limit of ωσ2(ω) below Tc with the corresponding dependence of the calculated cavity-dependent phase stiffness. In the experiment, the zero-frequency limit of ωσ2(ω), which is proportional to the superfluid density in the London regime, increases as the cavity is shortened. Over the same range of cavity lengths, the calculations follow the same trend. We note that at the longest cavity length (L = 4 mm) the measured response remains above the free-space value, whereas the minimal model would suggest the cavity-induced enhancement should already be suppressed. The residual response likely reflects ingredients that are not fully captured by the simplified phase-only description, including the effective treatment of the experimental cavity geometry and additional electrodynamic contributions.

In the present configuration, although the mirror positioning can in principle reach smaller nominal separations, the minimum reproducible mirror-film distance is ultimately limited by alignment. At sufficiently small spacing, contact between the mirror and the film can occur, leading to an apparent shift of the transition temperature caused by local heating (Supplementary Information, Sec. S7). Accessing shorter cavity lengths will therefore require, for instance, the deposition of cavity-like heterostructures embedding the superconducting layer. Cavity assemblies in which the layer is positioned closer to the field antinode, or in general more symmetrically within the cavity, may also enhance the effective light-matter coupling and increase the magnitude of the phase-stiffness renormalization.

The same strategy adopted here is likely to be relevant beyond YBCO. More anisotropic cuprates such as Bi₂Sr₂CaCu₂O₈₊δ are especially promising, because their stronger two-dimensional and weaker interlayer coupling make superconducting phase fluctuations more pronounced. In such materials, cavity-induced modifications of phase coherence may therefore be even more visible, providing a compelling test of the phase-fluctuation scenario discussed here. This perspective is also consistent with the recent study of cavity-altered superconductivity, where pronounced superfluid suppression was observed both under resonant interfacial coupling in an organic superconductor and non-resonant conditions in a conventional superconductor. Taken together with our results, this contrast suggests that cavity schemes acting primarily through dressing of pairing-related modes do not necessarily reinforce the condensate response, whereas acting on the phase sector may offer a more favorable route to promoting and sustaining superconducting phase coherence in fluctuation-dominated cuprates. By controlling the emergence of long-range coherence, this approach paves the way to light-matter hybrids exhibiting superconductivity at higher temperature.

Methods

Cavity assembly

The cavity assembly was designed and realized in-house, inspired by a previously reported setup. It consists of a semi-reflecting gold mirror facing a YBCO thin film grown on a dielectric substrate, forming a planar Fabry-Pérot-like geometry. The mirror-sample separation is controlled by three piezoelectric actuators (Physik Instrumente), enabling nanometric tuning of the cavity length. The entire assembly is integrated into a closed-cycle liquid-helium cryostat (ARS) with vibration damping.

The cavity mirror is fabricated by evaporating a Ti/Au bilayer (5/10 nm) onto a 2-mm-thick z-cut quartz substrate. It is mounted on a copper disk using conductive silver paint and mechanically connected to a copper plate actuated by the piezo positioners. Thermalization is achieved via a copper braid connected to the cryostat cold finger. To ensure reliable operation of the piezo actuators at low temperature, they are thermally decoupled from the cold stage using a PEEK spacer.

The YBCO sample is a 100-nm-thick film grown on a LaAlO₃ substrate (10 × 5 × 0.5 mm³) with a 10-nm CeO₂ buffer layer (Ceraco). The film has a critical temperature Tc = 85 K and a specified critical current density Jc of about 3 MA/cm². The sample is mounted with conductive silver paint onto a copper disk featuring a 6 × 3 mm² aperture for transmission measurements. The disk is housed in a copper holder allowing both azimuthal and polar alignment, and is in direct thermal contact with the cryostat cold finger to ensure efficient heat dissipation.

The temperatures of the mirror and the sample are monitored independently using silicon diode sensors, while a third diode mounted on the cold finger is used for PID temperature control. The base temperature at the cold finger is 9 K. Due to finite thermal resistances, temperature offsets of approximately 1.5 K at the sample and up to about 20 K at the mirror are observed (Supplementary Information, Sec. S2). All the temperatures reported in the main text are the ones measured at the sample’s position.

The cavity alignment is performed at room temperature using a visible laser beam collinear with the THz path. The back-reflected beams from the mirror and the sample are overlapped in the far field to ensure parallelism. For measurements without the cavity, the mirror is removed without disturbing the thermal contacts or the sample mounting. A new reference measurement is acquired whenever the thermal configuration is modified.

Terahertz time-domain spectroscopy

The superconducting response of the YBCO film is probed by temperature-dependent terahertz time-domain spectroscopy in transmission geometry. Broadband THz pulses are generated using a large-area GaAs photoconductive antenna excited by ultrashort optical pulses (800 nm, 20 fs, 12 µJ) from a commercial laser system (Carbide plus Orpheus-N 2H, Light Conversion) operating at 40 kHz.

The THz radiation is focused onto the sample using a gold-coated parabolic mirror, yielding a beam waist of approximately 1 mm at the sample position. The transmitted THz field is detected via electro-optic sampling in a 0.5-mm-thick ZnTe crystal using a synchronized probe pulse (800 nm, 20 fs, less than 50 nJ). The orthogonal polarization components of the probe beam are measured with a balanced photodetector (Thorlabs), and the differential signal is recorded using lock-in detection referenced to the photoconductive antenna bias.

The signal-to-noise ratio of the measurement exceeds 2 × 10⁴. The THz generation and detection paths are enclosed in a nitrogen-purged environment to suppress absorption by atmospheric water vapor.

Acknowledgements

We thank Lara Benfatto and Martin Eckstein for helpful discussions. We are also grateful to Jürgen Linzmayer and Klaus Wölfel for their technical support and for the mechanical construction of the cavity assembly. This work was mainly supported by the Gordon and Betty Moore Foundation through the Grant CENTQC (No. GBMF12213). DF acknowledges support from the European Union’s Horizon Europe research and innovation programme under the Marie Skłodowska-Curie grant No HORIZON-MSCA-2023-DN-01 101169225 - SPARKLE. FP acknowledges funding from the Munich Quantum Valley within the Hightech Agenda Bayern Plus supported by the State Ministry of Science and the Arts. VP was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under the Walter Benjamin Programme (No. 566401345). DF, MAS, JF and DMK acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 531215165 (Research Unit "OPTIMAL"). MAS was funded by the European Union (ERC, CAVMAT, No. 101124492).

(Reference list and the nine sections of Supplementary Information omitted for length; the complete text is at the source. On this site, the companion measurement that puts two-dimensional NbSe2 in a terahertz resonator is at /library/stm-812175a230, and the Strasbourg experiment on superconductivity under strong coupling to the vacuum electromagnetic field is at /library/stm-21d2922b6e.)