A. Bardin, F. Lorenzi, and L. Salasnich have corrected the equation of motion governing Bosonic Josephson junctions by incorporating both thermal and quantum fluctuations, moving beyond simpler models. For all the dynamical quantities, the quantum fluctuations play an opposite role with respect to the thermal fluctuations. Josephson frequency is decreased by thermal fluctuations and both the critical strengths of macroscopic quantum self trapping and spontaneous symmetry breaking are increased. The researchers assessed experimentally accessible regimes and found the expected conditions are dominated by quantum fluctuations, suggesting these subtle effects are observable in real-world systems.

Thermal and Quantum Corrections to Josephson Junction Dynamics

This work quantifies how these fluctuations compete to influence the dynamics of the system, offering a more complete picture than previously available. Recent experimental realizations of Bosonic Josephson junctions are predicted to operate in a regime dominated by quantum fluctuations, according to the researchers’ assessment of relevant parameters.

The team’s calculations indicate that, while thermal fluctuations can become relevant at sufficiently high temperatures or low densities, the typical conditions of these experiments favor the dominance of quantum effects. The study extended a two-mode description of the junction to include both quantum and thermal fluctuations in three spatial dimensions, addressing a complexity often overlooked in prior analyses.

The researchers acknowledge that their model relies on the assumption of a stationary non-condensed fraction, neglecting its transport across the junction, and that extending the formalism to include the dynamics of this thermal cloud represents a natural progression for future work. This future research could illuminate dissipative effects and relaxation processes within the junction, further refining the understanding of its behavior. The interplay between quantum and thermal fluctuations provides a consistent explanation for how different sources of fluctuations affect the nonlinear dynamics of the junction.

“Specifically, thermal fluctuations lead to a reduction of the Josephson frequency and to an increase of both critical strengths, while quantum fluctuations act in the opposite direction,” the authors write in their published work. This opposing behavior is a key finding, offering a subtle understanding of the complex interplay between these fundamental forces.

The team’s approach offers a semianalytical method for assessing the relative importance of quantum and thermal fluctuations in different regimes, providing a framework for interpreting experimental observations and refining theoretical models. This detailed analysis of the interplay between these fluctuations represents a step toward a more complete understanding of Bosonic Josephson junction dynamics and their potential applications.

Two-Mode Formalism for Bosonic Josephson Junctions

This approach moves beyond earlier models that relied on mean-field descriptions, which often overlook the subtle effects of these fluctuations. The resulting corrections allow for predictions of how these fluctuations impact key characteristics of the junction, such as the Josephson frequency and the thresholds for self-trapping and symmetry breaking. An assessment of experimentally accessible regimes, based on parameters from recent Bosonic Josephson junction realizations, indicates that quantum fluctuations currently dominate.

The researchers demonstrated that the expected operating regime of these junctions falls squarely within the domain of quantum fluctuation dominance, though thermal fluctuations could become more relevant at higher temperatures or lower densities. The work’s accuracy is further enhanced by its ability to predict the range of validity of mean-field approximations, providing a benchmark for assessing the limitations of simpler models. The team’s approach offers a valuable tool for researchers seeking to explore the interplay between quantum and thermal effects in Bosonic Josephson junctions and related systems.

Impact of Fluctuations on Josephson Frequency

Calculations of Josephson frequency corrections reveal a strong dependence on both temperature and gas parameter, indicating that the influence of thermal and quantum fluctuations is not uniform across all conditions. This interplay allows them to predict the effects of both types of fluctuations and assess their relative importance in different regimes in a semianalytical fashion.

This competition between fluctuations is not merely theoretical; the researchers assessed experimentally accessible regimes using data from recent experiments with Bosonic Josephson junctions, including those by Singh et al. (2025) using 87Rb, Pezzè et al. (2024) using 6Li and Bernhart et al. (2024) using 23Na.

Analysis of these experiments, alongside data from Albiez et al. (2005), indicates that the expected operating regime for these junctions is dominated by quantum fluctuations, with the study reporting that mean-field theory remains accurate within a 1%-error margin when gas parameter values are below 3.77 x $10^{- 5}$ and temperatures are below 0.103. This detailed understanding of fluctuation effects is made possible by a two-mode formalism that approximates fluctuations in each well as those of a homogeneous Bose gas, retaining a tractable model while incorporating beyond-mean-field corrections in a semi-analytical form.

The approach allows for the derivation of modified expressions for the Josephson frequency, the critical strength for macroscopic quantum self-trapping, and the threshold for spontaneous symmetry breaking, extending previous zero-temperature Gaussian fluctuation results to finite temperatures. “At low temperature $T^{*}$ and high gas parameter γ, quantum fluctuations are stronger than thermal fluctuations; thus, the beyond-mean-field Josephson frequency is greater than the mean-field one,” the paper states, illustrating the clear dominance of quantum effects in specific regimes.

Spontaneous Symmetry Breaking in Extended Junctions

The resulting corrections to the equation of motion allow for predictions of how both types of fluctuations impact the junction’s behavior across different regimes. The study highlights a phenomenon where a fixed phase difference between the junction sites, specifically at π, can lead to spontaneous symmetry breaking when the on-site interaction exceeds a certain value.

This manifests as a shift from a single stationary point at zero population imbalance to a pair at ± $z_{s}$ , effectively breaking the symmetry between the two sites. The researchers found that increasing temperature elevates the critical interaction strength required to induce the symmetry break. The dependence of the correction on the initial imbalance is weak, as shown in the study’s analysis. The framework employed relies on a two-site approximation, discarding the spatial structure of the superfluid to create an effective zero-dimensional model.

While acknowledging limitations associated with this simplification, particularly for stronger interactions, the researchers maintain that it provides an effective description of many dynamical features of the junctions. This method builds upon previous work with zero-temperature Gaussian fluctuations, extending the analysis to include finite temperature effects.

👉 More information
🗞 Bosonic Josephson junction dynamics: Interplay between quantum and thermal fluctuations
✍️ A. Bardin, F. Lorenzi and L. Salasnich
🧠 DOI: http://link.aps.org/doi/10.1103/1y19-1z4q

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