Logarithmically slow heat propagation in a clean Josephson-junction chain

We consider a clean Josephson-junction chain coupled by one of its extremities to a thermal bath through a resistance. Considering the Langevin dynamics in the classical regime, in the case of Josephson energy much smaller than charging energy, we find that heat propagates logarithmically slowly through the system, rather than diffusively, as highlighted by the logarithmic increase in time of a thermalization length we define and by the logarithmically slow increase in time of the energy. This behavior – typical of quantum Anderson or many-body localized systems – is observed here also in a clean classical glassy Hamiltonian system. We argue that this phenomenon might imply strong robustness to the effect of ergodic inclusions for the nonergodic behavior in the charge-quantized regime.

💡 Research Summary

In this work the authors investigate heat transport in a clean one‑dimensional Josephson‑junction (JJ) chain whose dynamics are governed by classical Langevin equations. The chain consists of superconducting islands with capacitance C (charging energy E_C) coupled by Josephson junctions of energy E_J. Only the leftmost island is attached to a resistor R that acts as a thermal bath at temperature T; the associated Johnson‑Nyquist noise ξ(t) is Gaussian, zero‑mean and satisfies ⟨ξ(t)ξ(t′)⟩=2k_B T/(RC)δ(t−t′). The equations of motion are

  ˙q_j = −∂{θ_j}H − (1/RC) q_1 δ{j1} + δ_{j1} ξ(t),
  ˙θ_j = ∂_{q_j}H,

with the Hamiltonian H = (E_C/2)∑_j q_j² − E_J∑j cos(θ_j−θ{j+1}). The study focuses on the regime E_J ≪ E_C, i.e. deep in the charge‑quantized limit where the Josephson coupling is a weak perturbation.

Numerically the authors integrate the stochastic equations using a Verlet scheme with time step Δt = 10⁻⁴, averaging over 990 independent noise realizations. Initial conditions are q_j = 0 and random phases θ_j ∈