Quantum-regime parametric amplification of terahertz Josephson plasma waves: squeezing of the superconducting phase, single-photon THz devices above the millikelvin regime, and control of fluctuating superconductivity

Parametric amplification of terahertz Josephson plasma waves — driven by the intrinsic cubic nonlinearity of the Josephson tunneling between the superconducting planes of layered cuprates — was demonstrated in the classical regime in 2016. The authors of that work identified three forward directions that remain open: generating genuine quantum squeezing of the superconducting order-parameter phase, building phase-sensitive THz devices that manipulate single THz photons at temperatures far above the millikelvin regime of microwave Josephson parametric amplifiers, and extending parametric control to fluctuating superconductivity, possibly even above $T_c$. Recent experiments have shown phonon-mediated parametric amplification of coherent pairs of opposite-momentum Josephson plasma polaritons, identified as a squeezed state of the Josephson plasma, but this operates through classical coherent observables and a different (phonon-pumping) mechanism. The open problem is to determine whether the direct Josephson-nonlinearity parametric process can be pushed into the quantum regime and into regimes of fluctuating superconductivity — what squeezing, gain, noise, and temperature performance are achievable, and what dissipation, multimode-coupling, and materials constraints bound them.

Background

In a layered superconductor such as the cuprate $\mathrm{La}_{2-x}\mathrm{Sr}_x\mathrm{CuO}_4$, superconducting planes are coupled along the out-of-plane (c) axis by Josephson tunneling: Cooper pairs tunnel between adjacent planes, and each pair of planes acts as an intrinsic Josephson junction with capacitance and a phase-dependent inductance. Because the Josephson inductance depends on the interlayer phase difference $\phi$ between the order parameters of adjacent planes (roughly as $1/\cos\phi$), it is nonlinear. A collective excitation results: the Josephson plasma wave (JPW), a transverse electromagnetic wave at terahertz frequencies in which the interlayer phase oscillates, propagating along the planes through the capacitively coupled junction stack. Its dynamics are governed, in the standard one-dimensional description, by the sine-Gordon equation — a prototypical nonlinear wave equation for the phase field $\phi(z,t)$. Parametric amplification is a process in which a strong periodic modulation of a system parameter (here, the effective Josephson plasma frequency, modulated by an intense pump field at nearly twice the plasma frequency) converts pump photons into signal photons, amplifying a weak seed wave in a phase-sensitive way: the seed quadrature aligned with the pump is amplified while the orthogonal quadrature is de-amplified. In quantum optics, this degenerate parametric amplification is the standard mechanism for producing squeezed states — states in which the fluctuations of one quadrature of a field or, here, of the superconducting phase are reduced below their vacuum (zero-temperature) level. Squeezing of the order-parameter phase means suppressing fluctuations of the interlayer phase difference below this quantum floor. At microwave frequencies and millikelvin temperatures, superconducting parametric amplifiers exploiting the same Josephson nonlinearity in artificial junction circuits are a mature technology [3], enabling quantum-limited amplification used for qubit readout. Whether the intrinsic Josephson nonlinearity of a high-$T_c$ layered superconductor can do the analogous job at THz frequencies and at temperatures of order 1 K or above — where thermal energies still lie well below the THz plasmon energy but where conventional quantum circuits cannot operate — is the question at stake. A related target is fluctuating superconductivity: in underdoped cuprates, superconducting correlations with short-lived phase coherence persist above $T_c$, and light-induced superconductivity experiments have shown that mid-infrared driving can transiently strengthen interlayer coherence; parametric manipulation of the phase is a proposed control knob for such fluctuating states.

Problem Statement

The demonstrated parametric amplification of terahertz Josephson plasma waves by the intrinsic Josephson nonlinearity of layered superconductors is classical. Determine whether and how this process can be extended into the quantum regime and into regimes of fluctuating superconductivity. Specifically: (a) Quantum phase squeezing — under what pump strengths, temperatures, dissipation rates, and multimode-coupling conditions (including coupling to phonons and to the in-plane plasma mode) can the Josephson-nonlinearity parametric process produce genuine quantum squeezing of the superconducting order-parameter phase, i.e., interlayer phase fluctuations below the vacuum level, going beyond the pair-amplification phenomenology already observed via coherent classical spectroscopy? (b) THz quantum devices — can the process be engineered into phase-sensitive THz parametric amplifiers or single-THz-photon manipulation devices operating at temperatures far above the millikelvin regime (the source work targets operation above 1 K, versus the millikelvin, microwave Josephson parametric amplifiers that currently define the state of the art), and what gain, bandwidth, added-noise, materials, and cavity or metamaterial architectures would that require? (c) Fluctuating superconductivity — can parametric control be extended to superconducting fluctuations above $T_c$, where phase coherence is short-lived, and with what field strengths, temporal windows, and materials parameters? These three directions share one objective: extending the already-demonstrated classical parametric control of the interlayer phase to the quantum-noise level and to temperatures above both the millikelvin quantum-circuit regime and $T_c$. A complete answer would address all three directions, each in either experimental-demonstration or rigorous-theoretical-delimitation form: for the quantum-regime directions (a) and (b), quantitative characterization of quadrature noise, phase-sensitive gain, bandwidth, and operating temperature relative to vacuum and thermal references, together with the materials and cavity or metamaterial architecture required, or a rigorous delimitation of the achievable quantum-regime performance, including the bounds imposed by dissipation, finite temperature, and multimode coupling; for the fluctuating-superconductivity direction (c), quantitative characterization of the parametric control achievable above $T_c$ (field strengths, temporal windows, and materials parameters), or a rigorous delimitation of that achievable control.

References

1. S. Rajasekaran, E. Casandruc, Y. Laplace, D. Nicoletti, G. D. Gu, S. R. Clark, D. Jaksch, A. Cavalleri. Parametric amplification of a superconducting plasma wave. Nature Physics 12, 1012-1016 (2016). DOI: 10.1038/nphys3819 (https://doi.org/10.1038/nphys3819); preprint arXiv:1511.08378, preprint title "Parametric Amplification of a Terahertz Quantum Plasma Wave" (https://arxiv.org/abs/1511.08378).
2. S. J. Denny, S. R. Clark, Y. Laplace, A. Cavalleri, D. Jaksch. Proposed Parametric Cooling of Bilayer Cuprate Superconductors by Terahertz Excitation. Physical Review Letters 114, 137001 (2015). DOI: 10.1103/PhysRevLett.114.137001 (https://doi.org/10.1103/PhysRevLett.114.137001).
3. C. Macklin, K. O'Brien, D. Hover, M. E. Schwartz, V. Bolkhovsky, X. Zhang, W. D. Oliver, I. Siddiqi. A near-quantum-limited Josephson traveling-wave parametric amplifier. Science 350, 307-310 (2015). DOI: 10.1126/science.aaa8525 (https://doi.org/10.1126/science.aaa8525).
4. M. H. Michael, A. von Hoegen, M. Fechner, M. Först, A. Cavalleri, E. Demler. Parametric resonance of Josephson plasma waves: A theory for optically amplified interlayer superconductivity in YBa2Cu3O6+x. Physical Review B 102, 174505 (2020). DOI: 10.1103/PhysRevB.102.174505 (https://doi.org/10.1103/PhysRevB.102.174505).
5. M. Buzzi, G. Jotzu, A. Cavalleri, J. I. Cirac, E. A. Demler, B. I. Halperin, M. D. Lukin, T. Shi, Y. Wang, D. Podolsky. Higgs-Mediated Optical Amplification in a Nonequilibrium Superconductor. Physical Review X 11, 011055 (2021). DOI: 10.1103/PhysRevX.11.011055 (https://doi.org/10.1103/PhysRevX.11.011055).
6. F. Gabriele, M. Udina, L. Benfatto. Non-linear Terahertz driving of plasma waves in layered cuprates. Nature Communications 12, 752 (2021). DOI: 10.1038/s41467-021-21041-6 (https://doi.org/10.1038/s41467-021-21041-6).
7. S. R. Ul Haque, M. H. Michael, J. Zhu, Y. Zhang, L. Windgätter, S. Latini, J. P. Wakefield, G.-F. Zhang, J. Zhang, A. Rubio, J. G. Checkelsky, E. Demler, R. D. Averitt. Terahertz parametric amplification as a reporter of exciton condensate dynamics. Nature Materials 23, 796-802 (2024). DOI: 10.1038/s41563-023-01755-2 (https://doi.org/10.1038/s41563-023-01755-2).
8. S. Kalhor, S. Savel'ev, K. Delfanazari. Engineering ultrastrong coupling between Josephson plasmon polaritons and subwavelength microcavity arrays in silicon/van der Waals layered superconductor heterostructure for terahertz hybrid circuit cavity quantum electrodynamics. Physical Review B 106, 245140 (2022). DOI: 10.1103/PhysRevB.106.245140 (https://doi.org/10.1103/PhysRevB.106.245140).
9. N. Taherian, M. Först, A. Liu, M. Fechner, D. Pavicevic, A. von Hoegen, E. Rowe, Y. Liu, S. Nakata, B. Keimer, E. Demler, M. H. Michael, A. Cavalleri. Probing amplified Josephson plasmons in YBa2Cu3O6+x by multidimensional spectroscopy. npj Quantum Materials 10, 54 (2025). DOI: 10.1038/s41535-025-00776-1 (https://doi.org/10.1038/s41535-025-00776-1); preprint arXiv:2401.01115, preprint title "Squeezed Josephson plasmons in driven YBa2Cu3O6+x" (https://arxiv.org/abs/2401.01115).
10. A. Liu, D. Pavicevic, M. H. Michael, A. G. Salvador, P. E. Dolgirev, M. Fechner, A. S. Disa, P. M. Lozano, Q. Li, G. D. Gu, E. Demler, A. Cavalleri. Probing inhomogeneous cuprate superconductivity by terahertz Josephson echo spectroscopy. Nature Physics 20, 1751-1756 (2024). DOI: 10.1038/s41567-024-02643-5 (https://doi.org/10.1038/s41567-024-02643-5).
11. C.-J. Yang, J. Li, M. Fiebig, S. Pal. Terahertz control of many-body dynamics in quantum materials. Nature Reviews Materials 8, 518-532 (2023). DOI: 10.1038/s41578-023-00566-w (https://doi.org/10.1038/s41578-023-00566-w).