Anomalous Dissipation in Current Biased Josephson Systems

Technical Summary: Anomalous Dissipation in Current Biased Josephson Systems

Problem Statement
The paper investigates the dynamical behavior of a current-biased Josephson junction (JJ) embedded in a specific circuit environment characterized by "anomalous dissipation." While standard models (such as the Resistively and Capacitively Shunted Junction, or RCSJ) typically account for Ohmic dissipation coupling to the junction phase, this work focuses on a configuration where a resistor ($R_g$) is placed in series with a capacitor ($C$) and connected in parallel to the junction. This specific topology causes the electromagnetic environment to couple effectively to the junction's charge rather than solely its phase. Since phase and charge are canonically conjugate variables in a Josephson junction, this coupling is analogous to momentum dissipation (or anomalous dissipation). The authors aim to understand how this environment affects the nonlinear dynamics of the junction, specifically the decay rate out of a metastable zero-voltage state and the subsequent re-trapping behavior, particularly in the regime of strong bias currents ($I_B/I_C > 0.9$).

Methodology
The study employs a combination of quantum field theory techniques and semiclassical analysis:

1. Hamiltonian Formulation and Langevin Equations: The system is modeled using a Hamiltonian where the dissipative resistors ($R_S$ and $R_g$) are represented by semi-infinite transmission lines (Rubin models). By integrating out the bath degrees of freedom, the authors derive quantum Langevin equations for both conventional phase coupling (via $R_S$) and charge coupling (via the $R_g-C$ branch). They prove that the $R_g-C$ branch leads to a Langevin equation where the damping kernel couples to the acceleration (second time derivative of the phase) rather than the velocity, confirming its nature as anomalous dissipation.
2. Semiclassical Dynamics: In the quasiclassical regime, the authors analyze the deterministic trajectories of the phase particle in a tilted washboard potential. They examine the interplay between the standard Ohmic damping ($R_S$) and the state-dependent damping/amplification introduced by $R_g$.
3. Metastable Decay Rates: To calculate the escape rates from the metastable well, the authors utilize the imaginary part of the free energy (ImF) approach within the Euclidean path integral formalism. They derive effective actions including kernels for both phase and charge dissipation. The analysis covers both the thermal activation regime (high temperature) and the quantum tunneling regime (low temperature), determining the crossover temperature between these two mechanisms.
4. Numerical Simulations: The theoretical framework is applied to realistic circuit parameters (e.g., $I_C = 21 \, \mu\text{A}$, $C_{tot} = 6 \, \text{pF}$) to simulate voltage dynamics, re-trapping probabilities, and energy landscapes.

Key Contributions and Results

• State-Dependent Damping and Amplification: The analysis reveals that the resistor $R_g$ introduces a damping term that depends on the phase difference $\phi(t)$ via a $\cos(\phi(t))$ factor. Consequently, the environment can act as a damper or an amplifier depending on the particle's position in the washboard potential. Specifically, when $\cos(\phi) < 0$, the charge dissipation accelerates the phase particle.
• Enhanced Quantum Escape: A primary finding is that anomalous dissipation significantly enhances the quantum tunneling rate out of the metastable well at low temperatures. Unlike conventional Ohmic dissipation, which typically suppresses tunneling, the coupling to the charge variable via $R_g$ increases the escape rate. This enhancement persists over a wide temperature range, pushing the crossover temperature (where thermal activation dominates over tunneling) to higher values.
• The Phase Diffusive Regime: The authors identify a new dynamical regime where a particle, after tunneling out of a metastable well, is accelerated by the anomalous environment but subsequently re-trapped in the next potential minimum. This occurs deterministically for specific combinations of $R_g$ and $R_S$, even at high bias currents ($I_B \approx I_C$) where re-trapping is usually impossible without fluctuations. This leads to a "phase diffusive" behavior characterized by sequences of tunneling, acceleration, and re-trapping.
• Voltage Pulse Amplification: The re-trapping process is preceded by a substantial voltage pulse. The anomalous environment amplifies the voltage oscillations generated by the phase particle's motion in the running state before it is captured by the next well.
• Position-Dependent Activation Energy: The study demonstrates that the effective activation energy for escape becomes position-dependent due to the sign-changing nature of the charge dissipation. Fluctuations can significantly alter the final state (running vs. trapped) depending on the precise escape point and the value of $R_g$.

Significance and Claims
The paper claims to provide a comprehensive theoretical framework for engineering dissipative dynamics in nonlinear quantum systems using anomalous environments. By demonstrating that specific circuit topologies can enhance quantum tunneling rates and induce deterministic re-trapping, the work suggests new possibilities for reservoir engineering in circuit QED.

The authors explicitly state that this class of circuits could be leveraged for:

• Detecting microwave photons.
• Facilitating dissipative quantum annealing processes.

The study emphasizes that these effects are most prominent in the strong bias regime ($I_B/I_C > 0.9$) and at low temperatures where quantum tunneling dominates. The work bridges the gap between abstract concepts of momentum dissipation and practical circuit implementations, offering a detailed picture of how anomalous environments alter the escape and re-trapping dynamics of Josephson junctions.