Abstract
We report the effect of quantum and thermal fluctuations on stability of mutual phase locking in one-dimensional long Josephson junction devices, involving layered superconductors. Accounting for both the induction coupling and the charging effect, we determined the zero-temperature $(T=0)$ phase diagram, using renormalization-group analysis, and found that the in-phase mode is stable, but some out-of-phase modes are unstable against quantum fluctuations. At finite T, all stable phase-locking modes (at $T=0)$ are unstable, but stability is still maintained within a finite length, which decreases inversely with T. In ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+y},$ this length for the in-phase mode is roughly 1000 \ensuremath{\mu}m at 1 K.
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