Quantum action of the Josephson dynamics

TL;DR

The paper addresses quantum corrections to the mean-field Josephson dynamics of the relative phase between two coupled condensates. It develops a covariant background-field approach to derive a one-loop, only-phase quantum effective action, yielding a local equation of motion with a photon-like $\phi$-dependent mass and a quantum-corrected potential. The resulting dynamics predict a modest but observable shift in the Josephson frequency ($\approx$1–3%) and show substantially improved agreement with exact two-site Bose-Hubbard dynamics across a wide range of $U/J$ and $N$, while accurately capturing fast oscillations and highlighting limits where Gaussian approximations fail. This framework provides a systematic method to incorporate leading quantum fluctuations into semiclassical descriptions of Josephson junctions and lays the groundwork for extensions to more collective variables and finite temperature.

Abstract

We study the beyond-mean-field Josephson dynamics of the relative phase between two coupled macroscopic quantum systems. Using a covariant background field method, we derive the one-loop only-phase quantum effective action and the corresponding equation of motion for the quantum average of the phase. These analytical results are benchmarked against the exact quantum dynamics of the two-site Bose-Hubbard model, demonstrating a relevant improvement over the standard mean-field predictions across a wide range of interaction strengths.

Figures (4)

• Figure 1: Effective mass (left panel) and effective potential as functions of $\Phi$ for $U=J=1.0$ and $N=50$ (green lines), 100 (orange lines), and 200 (blue lines) [Eqs. \ref{['meff']}-\ref{['veff']}]. The dashed lines represent the corresponding classical results [Eq. \ref{['mV']}].
• Figure 2: Comparison between the exact dynamics [Eq. \ref{['phi_ex']}] (solid black line), the mean-field dynamics [Eq. \ref{['eqss']}] (dashed-dotted blue line), and the quantum-corrected dynamics [Eqs. \ref{['eomeff']}-\ref{['veff']}] (dashed red line) of the relative phase, for $N=80$, $U=J=1.0$, $\phi_0=0.1$, and $z_0=0$. (Units: $\hbar=1$).
• Figure 3: Range of validity of the quantum-corrected dynamics in terms of $U/J$ and $N$. In Region I (yellow) the quantum-corrected dynamics improves the mean field by bringing it closer to the exact dynamics. In Region II (red) the quantum-corrected dynamics performs worse than the mean field due to the breakdown of the initial Gaussian integration over $z$. In Region III (blue) the exact dynamics is dominated by strong anharmonicity, and neither the mean field nor the quantum-corrected dynamics provide an accurate description. The transition line between Regions I and II fits $\Lambda = UN/2J=10$ (black dashed-dotted line).
• Figure 4: Comparison between the exact dynamics (solid black line), the mean-field dynamics (dashed-dotted blue line), and the quantum-corrected dynamics (dashed red line) of the relative phase, for $N=50$, $J=1.0$, $\phi_0=0.1$, and $z_0=0$, for several values of $U$ in the regions of parameter space where the quantum-corrected dynamics loses accuracy. (a) $U=5.0$, Region III. (b) $U=2.5$, boundary between Region I and Region III. (c) $U=0.4$, boundary between Region I and Region II. (d) $U=0.05$, Region II. (Units: $\hbar=1$).