Josephson Plasmon Dynamics

• Josephson plasmon is the collective charge-and-phase oscillation generated by Josephson tunneling that converts a neutral phase mode into a charged plasma mode in layered superconductors and junction arrays.
• Its dispersion is derived from a phase-only formulation coupled to electromagnetic potentials, leading to observable plasma poles in density-sensitive probes such as RIXS and EELS.
• Josephson plasmons manifest in varied settings—from bilayer splitting and polarization-hidden modes to surface-bound and cavity-hybridized excitations—offering rich insights into superconducting electrodynamics.

A Josephson plasmon is the collective charge-and-phase oscillation that arises when superconducting layers or electrodes are coupled by Josephson tunneling and the superconducting phase mode is promoted into a charged plasma mode by electrodynamic coupling. In layered cuprates it is the characteristic $c$-axis collective excitation of Cooper-pair supercurrents between CuO$_2$ planes, whereas in Josephson junction arrays and extended junctions it denotes quantized plasma oscillations of a spatially distributed superconducting phase field. Bilayer systems support multiple branches and polarization patterns, including longitudinal, transverse, and staggered variants, and related excitations also occur at surfaces, in confined geometries, and in cavity-QED architectures (Sellati et al., 2024, Sellati et al., 2023, Grankin et al., 2023).

1. Phase-mode origin and electrodynamic formulation

In the phase-only description of a layered superconductor, the Josephson plasmon is obtained by starting from the Gaussian action for the superconducting phase and then coupling that phase to the electromagnetic scalar and vector potentials. For a single-layer system, the phase action can be written as

$$S_G[\theta]=\frac{d}{8}\sum_q\big[\kappa_0\Omega_m^2+D_{ab}k_{ab}^2+D_ck_c^2\big]|\theta(q)|^2,$$

with $\kappa_0$ the compressibility, $D_{ab}$ and $D_c$ the in-plane and out-of-plane superfluid stiffnesses, and $k_c=2/d\,\sin(q_cd/2)$. After the minimal substitution

$$\Omega_m\theta\to\Omega_m\theta-2e\phi,\qquad i\mathbf k\,\theta\to i\mathbf k\,\theta+\frac{2e}{c}\mathbf A,$$

and integration over the Coulomb field, the neutral phase pole becomes a plasma pole (Sellati et al., 2024).

The resulting longitudinal Josephson plasmon has dispersion

$$\omega_\text{RPA}^2(\mathbf q)=\omega_L^2(\mathbf q)(1+\alpha|\mathbf k|^2),$$

with

$$\omega_L^2(\mathbf q)=\omega_{ab}^2\frac{k_a^2}{|\mathbf k|^2}+\omega_c^2\frac{k_c^2}{|\mathbf k|^2},$$

where $_2$0, $_2$1, and $_2$2. In experiment, this mode appears as a peak in the charge response,

$_2$3

so the Josephson plasmon is not merely a formal pole of the phase propagator but a directly observable collective excitation in density-sensitive probes (Sellati et al., 2024).

The same logic appears in a more isotropic formulation: a neutral superconductor supports an Anderson–Bogoliubov sound mode, but long-range Coulomb interaction lifts it to a plasma oscillation. In layered systems, the essential difference is that the $_2$4-axis stiffness is generated by Josephson tunneling rather than by a conventional three-dimensional superfluid response, so the low-energy plasma scale is set by interlayer tunneling and anisotropic screening (Sellati et al., 2023).

2. Layered and bilayer superconductors

For a single-layer stack of superconducting planes, the simplest Josephson term generates an out-of-plane stiffness $_2$5 and a Josephson plasma frequency $_2$6. In bilayer cuprates, however, the unit cell contains two inequivalent Josephson links. One may parameterize them by $_2$7 and $_2$8, or equivalently by two couplings $_2$9 and $$S_G[\theta]=\frac{d}{8}\sum_q\big[\kappa_0\Omega_m^2+D_{ab}k_{ab}^2+D_ck_c^2\big]|\theta(q)|^2,$$0, which yield two $$S_G[\theta]=\frac{d}{8}\sum_q\big[\kappa_0\Omega_m^2+D_{ab}k_{ab}^2+D_ck_c^2\big]|\theta(q)|^2,$$1-axis plasma frequencies. Because there are two superconducting layers per unit cell, the phase field becomes a two-component object and the plasma problem becomes a matrix secular equation rather than a single-mode dispersion (Sellati et al., 2023, Sellati et al., 2024).

The doubled $$S_G[\theta]=\frac{d}{8}\sum_q\big[\kappa_0\Omega_m^2+D_{ab}k_{ab}^2+D_ck_c^2\big]|\theta(q)|^2,$$2-axis unit cell breaks translational symmetry and backfolds the original single-layer plasmon into the reduced Brillouin zone. In the phase-only treatment, the bilayer modes follow from

$$S_G[\theta]=\frac{d}{8}\sum_q\big[\kappa_0\Omega_m^2+D_{ab}k_{ab}^2+D_ck_c^2\big]|\theta(q)|^2,$$3

producing two branches $$S_G[\theta]=\frac{d}{8}\sum_q\big[\kappa_0\Omega_m^2+D_{ab}k_{ab}^2+D_ck_c^2\big]|\theta(q)|^2,$$4 and $$S_G[\theta]=\frac{d}{8}\sum_q\big[\kappa_0\Omega_m^2+D_{ab}k_{ab}^2+D_ck_c^2\big]|\theta(q)|^2,$$5. In the generalized electromagnetic formulation, the low-energy sector contains a high-energy in-plane plasma mode $$S_G[\theta]=\frac{d}{8}\sum_q\big[\kappa_0\Omega_m^2+D_{ab}k_{ab}^2+D_ck_c^2\big]|\theta(q)|^2,$$6 together with an upper and a lower Josephson plasma mode, $$S_G[\theta]=\frac{d}{8}\sum_q\big[\kappa_0\Omega_m^2+D_{ab}k_{ab}^2+D_ck_c^2\big]|\theta(q)|^2,$$7 and $$S_G[\theta]=\frac{d}{8}\sum_q\big[\kappa_0\Omega_m^2+D_{ab}k_{ab}^2+D_ck_c^2\big]|\theta(q)|^2,$$8, which reduce at $$S_G[\theta]=\frac{d}{8}\sum_q\big[\kappa_0\Omega_m^2+D_{ab}k_{ab}^2+D_ck_c^2\big]|\theta(q)|^2,$$9 to the two inequivalent $\kappa_0$0-axis plasma frequencies (Sellati et al., 2024, Sellati et al., 2023).

Bilayer electrodynamics also contains a transverse Josephson plasmon. Within the Josephson superlattice model, its frequency is

$\kappa_0$1

and this mode appears as a peak in the real part of $\kappa_0$2, in contrast to longitudinal modes, which appear as resonances in the loss function or as reflectivity edges (Hu et al., 2016). A microscopic derivation shows that the finite-frequency conductivity peak originates from out-of-phase $\kappa_0$3-axis oscillations inside the bilayer and that its spectral weight is proportional to $\kappa_0$4; it therefore vanishes when the two Josephson links become identical, which is why the mode is absent in the single-layer limit (Sellati et al., 2023).

This bilayer structure is the basis for several distinct experimental phenomena. In equilibrium optics it explains the coexistence of longitudinal plasma edges and a finite-frequency transverse peak. Under ultrafast phonon pumping, it also enables spectral-weight redistribution between inequivalent interlayer channels rather than a simple rigid shift of a single plasma edge (Hu et al., 2016).

3. Polarization structure and the ghost Josephson plasmon

A central refinement in the bilayer problem is that not every Josephson plasmon couples to density probes with comparable strength. The bilayer density response is a matrix,

$\kappa_0$5

and the measured total response contains two poles,

$\kappa_0$6

Although both branches are legitimate collective modes, the lower branch can become almost invisible in charge response near $\kappa_0$7 because its spectral weight vanishes there (Sellati et al., 2024).

In the compact $\kappa_0$8 limit, the lower-mode weight contains factors such as $\kappa_0$9, and these vanish at $D_{ab}$0. At the same momentum, the upper branch sits at $D_{ab}$1, making the remaining contribution cancel as well, so

$D_{ab}$2

This is the basis of the “ghost” Josephson plasmon: the lower branch is present as a true pole of the response, yet it is hidden in small-$D_{ab}$3 density probes such as RIXS or EELS (Sellati et al., 2024).

The physical explanation is polarization rather than weakness. Introducing the gauge-invariant current-like fields

$D_{ab}$4

one finds that the upper mode behaves as an ordinary longitudinal plasmon with in-phase current oscillations and longitudinal projection $D_{ab}$5. By contrast, the lower mode satisfies

$D_{ab}$6

so the out-of-plane currents in the intrabilayer and interbilayer links flow in opposite directions. The resulting staggered current pattern is virtually transverse at small physical $D_{ab}$7, even though it remains longitudinal with respect to a “reminiscent” large $D_{ab}$8 inherited from the backfolded single-layer plasmon at the original Brillouin-zone edge (Sellati et al., 2024).

This distinction resolves a common misconception. The lower bilayer plasmon is not absent and not merely weak; it is polarization-hidden. Its spectral weight grows only as $D_{ab}$9 approaches the zone boundary, where the mode acquires enough longitudinal character to couple efficiently to density fluctuations. In that framework, the Ca-YBCO RIXS signal at $D_c$0 is interpreted as most naturally corresponding to the lower Josephson plasmon, while the upper branch is likely overdamped by the quasiparticle continuum (Sellati et al., 2024).

4. Surface, confined, and cavity realizations

Josephson plasmons also occur as surface-bound and geometrically confined modes. In a layered cuprate, the weak $D_c$1-axis Josephson tunneling and strong in-plane response produce surface Josephson plasma waves (SJPWs), which are electromagnetic modes localized at an interface between a dielectric and a uniaxial superconductor. For an interface with isotropic permittivity $D_c$2 and superconducting dielectric tensor components $D_c$3 and $D_c$4, the surface dispersion is

$D_c$5

These modes are surface bound, acoustic-like at small in-plane momentum, and asymptotically approach a frequency close to the screened Josephson plasma frequency $D_c$6 at large $D_c$7 (Lu et al., 2020).

An experimental realization was reported in an ultrathin 13 nm single-crystal film of $D_c$8, using cryogenic AFM-scattering-type near-field optical microscopy at $D_c$9. In that work, $k_c=2/d\,\sin(q_cd/2)$0 at $k_c=2/d\,\sin(q_cd/2)$1, the probe frequency was $k_c=2/d\,\sin(q_cd/2)$2, and the superconducting near-field signal could be visualized with spatial resolution better than $k_c=2/d\,\sin(q_cd/2)$3. The measured enhancement below $k_c=2/d\,\sin(q_cd/2)$4 was attributed to resonant near-field coupling of the AFM tip to SJPWs rather than to Carlson–Goldman modes, Higgs modes, Bardasis–Schrieffer modes, or pair-breaking quasiparticles (Lu et al., 2020).

Spatial confinement leads to localized Josephson surface plasmons. For a small spherical anisotropic cuprate particle in the quasistatic limit, the multipolar surface modes satisfy

$k_c=2/d\,\sin(q_cd/2)$5

In the dipolar sector, the $k_c=2/d\,\sin(q_cd/2)$6 mode is dominated by $k_c=2/d\,\sin(q_cd/2)$7-axis Josephson dynamics, while the $k_c=2/d\,\sin(q_cd/2)$8 branch is quasi-two-dimensional and in-plane-plasmon-like. This yields a confined surface spectrum richer than that of an isotropic Drude metal because both $k_c=2/d\,\sin(q_cd/2)$9 and $$\Omega_m\theta\to\Omega_m\theta-2e\phi,\qquad i\mathbf k\,\theta\to i\mathbf k\,\theta+\frac{2e}{c}\mathbf A,$$0 enter nontrivially (Alpeggiani et al., 2013).

Cavity electrodynamics provides a further extension. A proposed oxide heterostructure composed of insulating layers, a thin cuprate film, a conducting bottom layer, and patterned top metallic patches supports sub-millimeter cavities with electric field polarized along the $$\Omega_m\theta\to\Omega_m\theta-2e\phi,\qquad i\mathbf k\,\theta\to i\mathbf k\,\theta+\frac{2e}{c}\mathbf A,$$1-axis. When the cavity resonance is tuned through the Josephson plasma resonance, the spectrum shows an avoided crossing and a Rabi splitting $$\Omega_m\theta\to\Omega_m\theta-2e\phi,\qquad i\mathbf k\,\theta\to i\mathbf k\,\theta+\frac{2e}{c}\mathbf A,$$2, producing Josephson plasmon polaritons. The effective Hamiltonian is

$$\Omega_m\theta\to\Omega_m\theta-2e\phi,\qquad i\mathbf k\,\theta\to i\mathbf k\,\theta+\frac{2e}{c}\mathbf A,$$3

and the proposed regime is one of ultrastrong coupling, with the further possibility of cooling superconducting phase fluctuations by cavity back-action (Laplace et al., 2015).

5. Arrays, rings, and multimode circuit platforms

Outside layered cuprates, Josephson plasmons are the elementary phase-only excitations of Josephson networks. In a quasi-one-dimensional multiladder array that is infinite along $$\Omega_m\theta\to\Omega_m\theta-2e\phi,\qquad i\mathbf k\,\theta\to i\mathbf k\,\theta+\frac{2e}{c}\mathbf A,$$4 and finite in the transverse direction, linearization of the RCSJ equations yields a spectrum of exactly $$\Omega_m\theta\to\Omega_m\theta-2e\phi,\qquad i\mathbf k\,\theta\to i\mathbf k\,\theta+\frac{2e}{c}\mathbf A,$$5 branches. In the unbiased case $$\Omega_m\theta\to\Omega_m\theta-2e\phi,\qquad i\mathbf k\,\theta\to i\mathbf k\,\theta+\frac{2e}{c}\mathbf A,$$6, there is an $$\Omega_m\theta\to\Omega_m\theta-2e\phi,\qquad i\mathbf k\,\theta\to i\mathbf k\,\theta+\frac{2e}{c}\mathbf A,$$7-fold degenerate perfectly flat band at the normalized Josephson plasma frequency $$\Omega_m\theta\to\Omega_m\theta-2e\phi,\qquad i\mathbf k\,\theta\to i\mathbf k\,\theta+\frac{2e}{c}\mathbf A,$$8; under uniform dc bias, one branch remains exactly flat while the other $$\Omega_m\theta\to\Omega_m\theta-2e\phi,\qquad i\mathbf k\,\theta\to i\mathbf k\,\theta+\frac{2e}{c}\mathbf A,$$9 previously degenerate flat branches become weakly dispersive and remain near the plasma frequency for small bias (Bukatova et al., 2021).

In a closed ring of $$\omega_\text{RPA}^2(\mathbf q)=\omega_L^2(\mathbf q)(1+\alpha|\mathbf k|^2),$$0 identical Josephson junctions, the harmonic excitations above each persistent-current state are plasmon modes with dispersion

$$\omega_\text{RPA}^2(\mathbf q)=\omega_L^2(\mathbf q)(1+\alpha|\mathbf k|^2),$$1

Quantum phase slips hybridize both the persistent-current states and the one-plasmon sector. Because phase slips through different junctions acquire charge-dependent phases, the resulting splittings exhibit an Aharonov–Casher effect for plasmons. In the weakly dispersive case with random induced charges, phase slips can localize nearly all one-plasmon excitations in individual junctions, in sharp contrast to the delocalized plasmons of the phase-slip-free ring (Süsstrunk et al., 2013).

Extended Josephson junctions represent a distinct multimode regime. When the junction length $$\omega_\text{RPA}^2(\mathbf q)=\omega_L^2(\mathbf q)(1+\alpha|\mathbf k|^2),$$2 becomes comparable to the Josephson penetration length $$\omega_\text{RPA}^2(\mathbf q)=\omega_L^2(\mathbf q)(1+\alpha|\mathbf k|^2),$$3, the phase can no longer be treated as a single lumped variable. Expanding the phase field in spatial eigenmodes,

$$\omega_\text{RPA}^2(\mathbf q)=\omega_L^2(\mathbf q)(1+\alpha|\mathbf k|^2),$$4

and quantizing the quadratic theory yields a tower of bosonic plasmon modes with

$$\omega_\text{RPA}^2(\mathbf q)=\omega_L^2(\mathbf q)(1+\alpha|\mathbf k|^2),$$5

The Josephson cosine generates self-Kerr, cross-Kerr, and parametric exchange terms between these modes, while the spatial profiles $$\omega_\text{RPA}^2(\mathbf q)=\omega_L^2(\mathbf q)(1+\alpha|\mathbf k|^2),$$6 produce parity-based selection rules for coupling to resonator photons. In this setting, a single physical junction functions as a multimode nonlinear resonator rather than as a single transmon-like oscillator (Grankin et al., 2023).

A broader scaling interpretation has also been proposed for small one- and two-dimensional arrays near the superconductor–insulator transition. In that framework, the experimentally observed transport scaling is described by interacting superconducting plasmons, with dephasing length

$$\omega_\text{RPA}^2(\mathbf q)=\omega_L^2(\mathbf q)(1+\alpha|\mathbf k|^2),$$7

and microscopic starting scale $$\omega_\text{RPA}^2(\mathbf q)=\omega_L^2(\mathbf q)(1+\alpha|\mathbf k|^2),$$8, where $$\omega_\text{RPA}^2(\mathbf q)=\omega_L^2(\mathbf q)(1+\alpha|\mathbf k|^2),$$9 is the Coulomb screening length. This interpretation treats Josephson plasmons as the critical fluctuations controlling the quantum phase transition, rather than regarding phase slips, vortices, or charge solitons as the only useful effective degrees of freedom in the scaling regime (Feldman et al., 2024).

6. Driven dynamics, synchronization, and interpretive debates

Strong driving can promote Josephson plasmons from equilibrium probes to actively amplified and mode-converting degrees of freedom. In bilayer $$\omega_L^2(\mathbf q)=\omega_{ab}^2\frac{k_a^2}{|\mathbf k|^2}+\omega_c^2\frac{k_c^2}{|\mathbf k|^2},$$0, a theory of apical-oxygen-phonon pumping describes a three-wave interaction in which modulation of the in-plane superfluid stiffness allows one driven phonon to decay into a pair of Josephson plasmon polaritons with opposite momenta. The parametric resonance condition is

$$\omega_L^2(\mathbf q)=\omega_{ab}^2\frac{k_a^2}{|\mathbf k|^2}+\omega_c^2\frac{k_c^2}{|\mathbf k|^2},$$1

and the instability selects a most strongly growing lower/upper plasmon pair near $$\omega_L^2(\mathbf q)=\omega_{ab}^2\frac{k_a^2}{|\mathbf k|^2}+\omega_c^2\frac{k_c^2}{|\mathbf k|^2},$$2 and $$\omega_L^2(\mathbf q)=\omega_{ab}^2\frac{k_a^2}{|\mathbf k|^2}+\omega_c^2\frac{k_c^2}{|\mathbf k|^2},$$3. In that picture, parametrically driven plasmons account for sharpened and shifted Josephson plasma edges below $$\omega_L^2(\mathbf q)=\omega_{ab}^2\frac{k_a^2}{|\mathbf k|^2}+\omega_c^2\frac{k_c^2}{|\mathbf k|^2},$$4, a new peak near $$\omega_L^2(\mathbf q)=\omega_{ab}^2\frac{k_a^2}{|\mathbf k|^2}+\omega_c^2\frac{k_c^2}{|\mathbf k|^2},$$5, and the re-emergence of overdamped plasma edges above $$\omega_L^2(\mathbf q)=\omega_{ab}^2\frac{k_a^2}{|\mathbf k|^2}+\omega_c^2\frac{k_c^2}{|\mathbf k|^2},$$6 (Michael et al., 2020).

The interpretation of photoinduced Josephson-plasmon-like features above $$\omega_L^2(\mathbf q)=\omega_{ab}^2\frac{k_a^2}{|\mathbf k|^2}+\omega_c^2\frac{k_c^2}{|\mathbf k|^2},$$7 remains contested. One proposal attributes them to pump-created long-range coherent superconductivity. An alternative argues that the lower Josephson plasmon in bilayer YBCO depends mainly on capacitive coupling between bilayers and does not require a nonzero inter-bilayer Josephson current. In that view, setting $$\omega_L^2(\mathbf q)=\omega_{ab}^2\frac{k_a^2}{|\mathbf k|^2}+\omega_c^2\frac{k_c^2}{|\mathbf k|^2},$$8 does not eliminate the mode or its parametric amplification, so the observed reflectivity edge and second-harmonic generation can be explained by local intra-bilayer pairing and short-ranged phase correlations already present in the pseudogap regime. This suggests that the experiments indicate local pairing amplitude up to $$\omega_L^2(\mathbf q)=\omega_{ab}^2\frac{k_a^2}{|\mathbf k|^2}+\omega_c^2\frac{k_c^2}{|\mathbf k|^2},$$9, not necessarily pump-induced global phase coherence (Michael et al., 6 May 2025).

Josephson plasmons also mediate synchronization phenomena. In Bi2212 intrinsic-junction mesa stacks, the coupling between macroscopic Josephson oscillations in two simultaneously biased mesas is interpreted as being carried by a small-amplitude Josephson plasmon inside the superconducting substrate. Polarization analysis and frequency-dependent phase delay indicate that the interaction propagates inside the crystal rather than through free space, and the phase shift follows the substrate propagation estimate

$_2$00

In that interpretation, the Josephson plasmon is the internal channel that entrains separate macroscopic oscillators into a common terahertz-emitting state (Kobayashi et al., 2021).

Nonequilibrium plasmon kinetics has been examined directly in a long, strongly inductive Josephson junction chain. There, multimode spectroscopy reveals a progression from pairwise four-wave-mixing couplings at weak drive to cascaded higher-order processes at strong drive. Under broadband incoherent driving, interactions among hundreds of modes produce near-continuum internal dynamics, nonthermal occupations, and nonlocal redistribution of energy in response to weak perturbations. The resulting state is described as a strongly interacting nonequilibrium plasmon fluid rather than as a set of weakly coupled standing-wave resonances (Bubis et al., 13 Apr 2025).

Taken together, these developments show that the Josephson plasmon is not a single narrowly defined resonance but a family of collective superconducting excitations whose precise form depends on geometry, anisotropy, coupling topology, and probe channel. In layered superconductors it organizes the $_2$01-axis electrodynamics; in bilayers it splits into inequivalent branches and can become polarization-hidden; at surfaces and in cavities it hybridizes with confined electromagnetic fields; in arrays and circuits it forms flat bands, charge-sensitive ring modes, and multimode nonlinear resonators; and far from equilibrium it mediates synchronization, parametric amplification, and plasmonic many-body kinetics (Sellati et al., 2024, Sellati et al., 2023).