Dissipative quantum mechanics of Andreev bound states

arXiv:2609.10691 · cond-mat.supr-con · Submitted 2026-09-09 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai, and with me are Mira and Lev, guest researcher.

Mira: Today's paper: "Dissipative quantum mechanics of Andreev bound states".

Kai: Dissipative quantum mechanics of Andreev bound states proposes a microscopic scheme to describe the ac Josephson effect in superconducting junctions by focusing on the dissipative quantum dynamics of subgap Andreev…

Mira: First, who's behind it and why it matters.

Title and authors: Kai: We've seen how this paper tackles the core issue of describing ac Josephson effect in junctions using these dissipative quantum dynamics of Andreev bound states, and it’s really about going beyond simple descriptions.

Mira: It summarizes the standard model for a single-channel symmetric non-magnetic superconducting junction with transmission D biased by a voltage V(t) by focusing on the two Andreev bound states inside the junction.

Lev: So, what's the actual physical picture they are building? What are these bound states doing in this setup?

Kai: The physical picture centers on two Andreev bound states with phase-dependent energies, where the energy of these subgap bound states is given by epsilon A(chi) = q

one - D squared (chi/two): <ref:2609.10691#pg0>.

Mira: When a small external bias voltage V is applied, this voltage sweeps the Josephson phase chi(t), which in turn drives a quasiparticle along the lower Andreev level.

Lev: That sounds like the standard setup for studying transport across a junction, but what makes it dissipative in this context?

Kai: The dissipation arises because when the Josephson phase approaches two pi n, the dynamics become non-unitary, and these bound states start "talking" to the continuous spectrum <ref:2609.10691#pg0>.

Mira: This interaction allows quasiparticles to easily escape into that continuum, which is what generates the dissipative contribution we’re looking at.

Lev: So, it’s not just a simple ohmic loss; it's a dynamic process tied directly to how the phase evolves near those specific points.

Kai: Precisely; this dissipation is periodic in chi and remains significant as long as delta chi(t), the distance from two pi n, is less than that specific threshold we mentioned earlier <ref:2609.10691#pg0>.

Mira: The paper summarizes that this total current, I

chi(t): , isn't just the non-dissipative Josephson current, I J

chi(t): , but a sum of both terms: I

chi(t): = I J

chi(t): + I diss

chi(t): .

Lev: So, the key summary here is that the low bias regime requires us to treat these dynamics with this combined functional approach because the simple Hamiltonian-like approach breaks down.

Kai: That’s right; they use these exact operator identities to handle the complexities of (W+a)-one which is necessary for a full description of non-equilibrium evolution <ref:2609.10691#pg1>.

Mira: The paper summarizes that this method successfully recovers the total current, I

chi(t): , by first solving an integral equation for the wave functions and then substituting those into Eq. (forty-six) for the inverse operator.

Lev: That’s a pretty complete summary of what they’ve done mathematically to bridge the gap between standard models and this more detailed quantum picture.

The paper's summary: Kai: Moving on to how they improve things, the paper suggests that their primary improvement is developing a microscopic scheme that allows them to describe the ac Josephson effect through dissipative quantum dynamics.

Mira: The authors suggest this approach is particularly useful for highly transparent junctions at subgap bias voltages where the standard models become less accurate.

Lev: So, what specific improvements are they proposing beyond just applying these existing formalisms to a new system? Are they fixing some inherent limitations?

Kai: They improve the description by identifying and incorporating the role of Landau-Zener tunneling between Andreev levels, which causes coherent oscillations on the current-phase relation.

Mira: Furthermore, they introduce a sub-Ohmic phase-dependent dissipative contribution to the current that is controlled by quasiparticle dynamics near the superconducting gap edge in low bias.

Lev: That sounds like a big step because it means we aren't just looking at static dissipation; we are modeling dynamic, phase-dependent loss mechanisms.

Kai: Exactly; they show that this dissipative contribution is periodic and remains appreciable even when the phase deviates significantly from two pi n, provided that deviation stays within a certain range <ref:2609.10691#pg0>.

Mira: They also improve the evaluation by showing how to handle regions where the dynamics become non-unitary because Andreev bound states begin merging with the continuum as they approach two pi n <ref:2609.10691#pg0>.

Lev: If we're thinking about hardware, this implies that for error correction schemes, we need models that can capture these coherent oscillations stemming from tunneling effects.

Kai: So, the improvement is having a tool to predict and analyze those oscillations as a function of junction parameters like transmission R and bias V.

Mira: This allows for much better parameter optimization because an AI system could map inputs like transparency or temperature to the resulting CPRs.

Lev: It sounds like they’re giving us a more detailed picture of how the system behaves when pushed out of its simple adiabatic limits.

The paper's improvements: Kai: So, to wrap up this paper "Dissipative quantum mechanics of Andreev bound states," it really boils down to recovering the total current I

chi(t): by combining the non-dissipative and dissipative parts.

Mira: The main implication is that this approach provides a rigorous microscopic quantum description of the non-equilibrium evolution of Andreev states, which is essential because those dynamics become nonunitary and dissipative when a bias voltage is applied.

Lev: From an error correction viewpoint, it means we have a more detailed way to understand the dynamics that can inform how we design robust superconducting components.

Kai: The paper shows that while Landau-Zener tunneling leads to coherent oscillations, there’s still this additional sub-Ohmic phase-dependent dissipative contribution deep in the subgap regime.

Mira: This finding is significant because it means we have a quantifiable way to measure and control energy loss based on the phase deviation from integer multiples of two pi <ref:2609.10691#pg0>.

Lev: We can use this to better predict noise characteristics in devices where quasiparticle escape is a key factor in error manifestation.

Kai: So, the study of "Dissipative quantum mechanics of Andreev bound states" gives us a full picture combining coherent tunneling and phase-dependent energy loss.

Conclusion: Kai: So, to recap, this paper "Dissipative quantum mechanics of Andreev bound states" successfully recovers the total current I

chi(t): by combining the non-dissipative Josephson current and that subOhmic dissipative correction tied to quasiparticle dynamics near two pi n.

Mira: Exactly, Kai; it’s really about showing how you can rigorously model those non-unitary transitions where the Andreev bound states start interacting with the continuum, which is crucial for understanding transport at high transparency.

Lev: For error correction research, this means we have a more complete picture of the noise landscape we might encounter when designing superconducting circuits that operate in these highly transparent regimes.

Kai: It gives us a concrete tool to predict those coherent oscillations caused by Landau-Zener tunneling, which is something we can actually try to engineer into our experimental setups using controlled bias voltages.

Mira: I agree; the way they handle the energy gap developing between levels when R is non-zero shows that this framework is robust enough to handle real material imperfections, not just idealized clean systems.

Lev: If we were running this on hardware, the main challenge would be accurately measuring those phase-dependent dissipative terms in real time, but having a solid theory like this gives us the target to aim for with our measurements.

Kai: It’s exciting because it moves us past just fitting data; now we have a microscopic mechanism explaining *why* we see these complex current-phase relations in experiments.

Mira: This paper really solidifies the theoretical foundation for designing components that are specifically aware of their dissipation profile, which is something we need as device complexity increases.

Lev: I think the next step is applying this framework to model more complex junction geometries, perhaps those with multiple Andreev reflections where these effects overlap.

Kai: Definitely; moving from a single channel to multi-channel physics using this formalism would be the logical next step for experimental validation.

Mira: It’s fascinating how they connect the quasiparticle escape probability directly to the phase deviation near two pi n, which is a very specific and useful physical constraint.

Lev: That constraint gives us a clear boundary condition to test; we can now design experiments specifically tuned to probe that threshold where dissipation kicks in.

Kai: Well, that brings us right up until we have covered the core of "Dissipative quantum mechanics of Andreev bound states." We’ve seen how this research provides the tools to model non-equilibrium transport with great detail.

Mira: It truly is a strong piece of condensed matter theory because it successfully links the microscopic dynamics—the bound states—to macroscopic observables like current-phase relations.

Lev: I think having this level of theoretical rigor will make our error correction simulations much more realistic when we start translating these concepts into actual circuit designs.

Kai: Indeed, the ability to predict those phase-dependent dissipative terms is a huge asset for anyone trying to build next-generation quantum hardware.

Mira: We’re looking forward to seeing how this framework interacts with other quantum phenomena we’ve been discussing, like the entropic characterization of Haag duality.

Lev: That entropic criterion might actually provide a complementary way to check our assumptions about the underlying structure of these superconducting systems.

Kai: Stay tuned; next time, we'll be taking a look at how this new formalism helps us characterize those spin systems in an entropic sense.

I.E.Tamm Department of Theoretical Physics, P.N.Lebedev Physical Institute · National Research University Higher School of Economics

cond-mat.supr-con

Submitted: 2026-09-09

Updated: 2026-09-09

Comments: 16 pages, 7 figures

Journal ref: Phys. Rev. B 114, 214502 (2026)

DOI: 10.1103/64s2-jm2r

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 77/100

The gist: Dissipative quantum mechanics of Andreev bound states proposes a microscopic scheme to describe the ac Josephson effect in superconducting junctions by focusing on the dissipative quantum dynamics of

Key concepts

Andreev Bound States (ABS): These are special quantum states that exist inside a superconducting junction's energy gap. They are crucial because they govern the behavior of charge carriers near zero bias and dictate how current flows when a voltage is applied across the junction.
Josephson Current Components: The total current flowing through the junction is split into two parts: a non-dissipative part ($I_J$) derived from adiabatic approximations, and a dissipative part ($I_{ ext{diss}}$). The latter arises when the phase of the junction changes rapidly, indicating energy loss due to quasiparticle dynamics.
Landau-Zener Tunneling: This is a quantum tunneling mechanism where an electron transitions between two different energy levels as a parameter (like the Josephson phase) is swept through a region. In this context, it describes how quasiparticles can jump between the two Andreev bound states in non-transparent junctions.
Sub-Ohmic Dissipative Contribution: This refers to an additional current term that depends on the junction's phase and is not proportional to the voltage in a simple linear way. It becomes significant deep within the superconducting gap, showing a complex, phase-dependent energy loss mechanism driven by quasiparticle escape.

Terminology

Andreev Bound States — 25× in this paper · explained in 1 episode
Effective Hamiltonian — 6× in this paper · explained in 2 episodes
Adiabatic Evolution — 2× in this paper · explained in 2 episodes
Core — 2× in this paper · explained in 2 episodes
Trace — 1× in this paper · explained in 2 episodes

Summary

Dissipative quantum mechanics of Andreev bound states proposes a microscopic scheme to describe the ac Josephson effect in superconducting junctions by focusing on the dissipative quantum dynamics of subgap Andreev bound states, which is particularly useful for highly transparent junctions at subgap bias voltages where Landau-Zener tunneling and quasiparticle escape play an important role.

The gist

This approach evaluates the non-equilibrium current-phase relation of superconducting junctions at arbitrary transmissions and identifies a sub-Ohmic phase-dependent dissipative contribution to the current controlled by quasiparticle dynamics near the superconducting gap edge in the low bias regime.

Model and Physical Picture

The standard model for a single-channel symmetric non-magnetic superconducting junction with transmission D is described using quasiclassical Eilenberger-Keldysh formalism supplemented by Zaitsev boundary conditions. The physical picture centers on two Andreev bound states inside the junction with phase-dependent energies, where the energy of these subgap bound states is given by εA(χ) = ∆ q [1 − D sin2 (χ/2)]. When a small external bias voltage V is applied, the Josephson phase χ(t) is swept, driving a quasiparticle along the lower Andreev level.

Josephson Current Components

The total phase-dependent current across the junction at subgap bias voltages is given by the sum of two functionals of χ(t): I[χ(t)] = IJ [χ(t)] + Idiss[χ(t)]. The non-dissipative Josephson current, IJ [χ(t)], is derived from an adiabatic approximation, while the dissipative current, Idiss[χ(t)], arises when the Josephson phase approaches 2πn. This dissipative contribution is periodic in χ and strongly phase-dependent, remaining non-zero provided the Josephson phase deviates from 2πn by not more than ∼ (eV /∆)1/3.

Dynamics and Tunneling Mechanisms

The physics governing these effects involves several key quantum mechanical processes:

In the regime far from 2πn, quasiparticle dynamics can be described within a Hamiltonian-like approach where the probability of excitation from Andreev bound states into the continuum is exponentially small, neglecting dissipation.
As the Josephson phase approaches 2πn, quasiparticle dynamics become essentially non-unitary because Andreev bound states start talking to the continuous spectrum and eventually merge with it, allowing quasiparticles to easily escape into the continuum, causing a dissipative contribution.
For not fully transparent junctions (where R ≠ 0), the energy gap between two Andreev levels develops, and quasiparticles may jump between these subgap bound states due to Landau-Zener tunneling, causing pronounced coherent oscillations on the CPR in a certain parameter range.

Evaluation of Dissipative Contribution

The dissipative current Idiss[χ(t)] is evaluated by considering the escape of quasiparticles into the continuum. The total current I[χ(t)] is expressed as a combination of two functionals: I[χ(t)] = IJ [χ(t)] + Idiss[χ(t)]. The dissipative contribution to the current (60) contains the Landau-Zener tunneling probability d2 and is proportional to δχ(t), where δχ(t) denotes the distance between χ(t) and the nearest point 2πn. This contribution remains appreciable for phase values where δχ(t) ≪ 4eV/D∆1/3.

Results and Significance

The analytical results demonstrate that the total current I[χ(t)] is given by Eq. (B43), which combines the non-dissipative Josephson contribution (52) and the dissipative correction (60). The paper shows that this total current in Fig. 3 and Fig. 4 is in excellent agreement with numerical results, even beyond the strict applicability range of eV ≪ ∆. The dissipative contribution contains the probability of Landau-Zener tunneling d(t)2, indicating that a quasiparticle climbing in energy from −∆ to ∆ inevitably performs Landau-Zener tunneling between the two Andreev levels. This provides a rigorous microscopic quantum description of non-equilibrium evolution of Andreev states, which is necessary because the dynamics become nonunitary and dissipative when a bias voltage is applied.

Conclusion

The approach successfully recovers the total current I[χ(t)] by first recovering the proper wave functions through solving the integral equation (47) and then substituting them into Eq. (46) for the inverse operator, which is valid for all phase values, including those close to 2πn. The final result shows that while Landau-Zener tunneling leads to coherent oscillations, a rigorous treatment yields an additional sub-Ohmic phase-dependent dissipative contribution that remains appreciable deep in the subgap regime.

Improvements for AI systems

As a fastidious and diligent researcher, I have analyzed this paper, Dissipative quantum mechanics of Andreev bound states, which proposes a microscopic scheme to describe the ac Josephson effect in superconducting junctions using dissipative quantum dynamics of subgap Andreev bound states.

The improvements suggested below are targeted at enhancing AI systems by integrating the physical principles derived from this research into their architecture and capabilities.

Here are the specific improvements and what an improved AI system could achieve:

)

Improved AI System Capability: Real-time, Predictive Modeling of Quantum Transport in Superconducting Devices
System Improvement: Integration of a Dissipative Quantum Dynamics Engine based on the framework described in Section II, III, and IV (Effective Hamiltonian approach).
Specific Functionality: The improved AI system can perform high-fidelity simulations and predictive modeling for non-equilibrium transport phenomena in superconducting junctions under arbitrary bias voltages.
Specific Capabilities Derived from the Paper:

)

Predicting Non-Equilibrium Current-Phase Relations (CPR): The AI can accurately predict the time-dependent current across a junction, including both the coherent Josephson current and a subOhmic, phase-dependent dissipative contribution (Eq. 8). This is critical for designing superconducting circuits where non-sinusoidal CPRs are expected due to high transparency or bias.
Modeling Landau-Zener Tunneling Effects: The AI can explicitly model the role of Landau-Zener tunneling between Andreev levels, which causes coherent oscillations in CPR (Section V, Eq. 52). This allows the system to predict and analyze these oscillations as a function of junction parameters (transmission R) and bias voltage (V), which is vital for understanding quantum interference effects in superconducting devices.
Characterizing Dissipative Contributions: The AI can quantify the sub-Ohmic dissipative current, which depends on the phase deviation from integer multiples of 2π (Eq. 60). This enables the design of dissipation-aware superconducting components that minimize unwanted energy loss or maximize specific quantum noise characteristics.
Extrapolation Beyond Validity Limits: The system can be trained to use the generalized inverse operator approach (Section V, Eq. 46) to evaluate current even when the phase approaches critical points (2πn), effectively capturing physics beyond the strict adiabatic approximation and predicting dissipative effects in regions where standard models fail.
Parameter Optimization for Device Design: By mapping input parameters (e.g., junction transparency D, bias V, temperature T) to output CPRs, the AI can optimize device parameters to achieve specific current-phase characteristics or minimize dissipation under operational constraints.

Abstract

We propose a microscopic scheme that allows to describe ac Josephson effect in superconducting junctions in terms of dissipative quantum dynamics of subgap Andreev bound states. This approach is particularly useful for highly transparent junctions at subgap bias voltages in which case a non-trivial combination of Landau-Zener tunneling between Andreev levels and their instability due to quasiparticle escape into continuum may play an important role. In the low bias regime, we evaluate the non-equilibrium current-phase relation of superconducting junctions at arbitrary transmissions and identify a sub-Ohmic phase-dependent dissipative contribution to the current controlled by quasiparticle dynamics near the superconducting gap edge.