Mode separation of the Josephson plasma in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\mathrm{\delta }\mathrm{}}$

The Josephson plasma resonance in single-crystalline ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\mathrm{\delta }\mathrm{}}$ has been investigated at a microwave frequency of 35 GHz in a ${\mathrm{TE}}_{102}$ cavity resonator as functions of external magnetic field $({\mathit{H}}_{\mathrm{ext}}\Vert c$ axis), temperature, and sample dimension $L$ of the $\mathrm{ab}$ plane. A sharp resonance is observed in a perpendicular oscillating electric field $\left({\mathit{E}}_{\mathrm{rf}}\Vert c\right)$, whereas multipeaks are found in a parallel oscillating magnetic field $\left({\mathit{H}}_{\mathrm{rf}}\Vert ab\right)$. The former is independent of the sample dimension, while the latter shifts to higher fields as the sample size $L$ is reduced, and it disappears when $L$ becomes smaller than the critical length $L^{*}$. At 35 GHz, the value of $L^{*}$ is obtained to be 1.6 mm. The size-dependent resonance observed in ${\mathit{H}}_{\mathrm{rf}}\Vert \mathrm{ab}$ can be explained by the strong dispersion relation of the transverse Josephson plasma, while the size independence of the resonance in ${\mathit{E}}_{\mathrm{rf}}\Vert c$ can be described by the longitudinal Josephson plasma as predicted by the recent theory of Josephson plasma of Tachiki et al. Since the longitudinal plasma mode is the Nambu-Goldstone mode in a superconductor, the experimental distinction between the longitudinal and the transverse mode leads to the conclusion that the existence of the Nambu-Goldstone mode as predicted by Anderson was experimentally confirmed by direct observation of the Josephson plasma resonance with longitudinal excitations. The finite gap found in the Josephson plasma resonance also provides a direct proof of the Anderson-Higgs mechanism within the context of the spontaneously broken phase symmetry of the gauge-field theory in a superconductor.