⚛️ quantum physics

Dissipative Effects in Transmission Line Analogues of Hawking Radiation

This paper analyzes two superconducting circuit analogues of Schwarzschild black holes to determine their viability for observing Hawking radiation, concluding that while a tunable dc-SQUID transmission line can achieve distinguishable Hawking temperatures around 113 mK under realistic dissipative conditions, the solitonic SNAIL-based model requires further optimization.

Original authors: Elián Urtubey, Fernando C. Lombardo, Paula I. Villar

Published 2026-06-30

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Elián Urtubey, Fernando C. Lombardo, Paula I. Villar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to study a black hole. In reality, they are so far away and so cold that their "radiation" (a faint whisper of particles predicted by physics) is impossible to detect with our current tools. It's like trying to hear a whisper in a hurricane.

To solve this, scientists build analogues: tiny, controlled laboratory models that behave like black holes, but on a scale we can touch and measure. This paper explores two such models built using superconducting circuits (essentially, very high-tech electrical wires that carry electricity without resistance).

Here is a simple breakdown of what the researchers did and found:

1. The Setup: Creating a "Black Hole" in a Wire

In a normal wire, signals (like sound or light) travel at a constant speed. To make a black hole analogue, the researchers created a wire where the "speed limit" for signals changes as they move along it.

The Analogy: Imagine a river flowing downstream. If the water flows faster than a fish can swim upstream, the fish can't escape. That point where the water speed matches the fish's swimming speed is the "event horizon" (the point of no return).
The Experiment: They used two different types of superconducting wires to create this "river":
- Model A (The Tunable Wire): A wire made of tiny loops (SQUIDs) where they can tweak the speed of signals by applying a magnetic field. They created a "pulse" moving through the wire that acts like the flowing water.
- Model B (The Soliton Wire): A wire using special components (SNAILs) that naturally create a self-reinforcing wave (a soliton). This wave creates the "flow" needed to trap signals.

2. The Goal: Hearing the "Hawking Whisper"

According to theory, when a signal gets trapped at this horizon, it should emit a faint, thermal glow of particles (Hawking radiation). The researchers wanted to see if they could detect this glow in their circuits.

The Problem: Real-world labs are messy. Everything is connected to a "bath" of heat and noise (like static on a radio). This noise can easily drown out the tiny Hawking signal, making it look like just random background fuzz.
The Question: Can the Hawking signal survive long enough in these noisy wires to be measured before the environment wipes it out?

3. The Test: The "Distinguishability" Check

The researchers didn't just count particles; they used a clever mathematical tool called the Hilbert-Schmidt distance.

The Metaphor: Imagine you have two jars of marbles. One jar has a specific pattern of colors (the Hawking signal), and the other is a random mix (the thermal noise). The researchers asked: "How long can we look at the patterned jar before it starts looking so much like the random jar that we can't tell them apart?"
They calculated how long the signal stays "loud" enough to be distinguished from the background noise.

4. The Results: Who Won the Race?

The study compared the two models to see which one could produce a signal strong enough to beat the noise.

The Tunable Wire (Model A): This was the clear winner.
- It can generate a "temperature" for the Hawking radiation of about 113 millikelvin (a tiny fraction of a degree above absolute zero).
- The researchers found that as long as the signal is warmer than about 73 millikelvin, it stays distinguishable from the noise for a few microseconds.
- Verdict: This is well within the reach of current technology. We can likely build this and see the effect soon.
The Soliton Wire (Model B): This one struggled.
- Even with optimization, the best signal it could produce was around 1.9 millikelvin (or up to 42 mK with extreme tuning).
- This is far too cold to be easily distinguished from the background noise in a typical lab setting.
- Verdict: While theoretically interesting, this model needs much more work and better equipment before it can actually show us Hawking radiation.

Summary

The paper concludes that while building a black hole in a lab is tricky because of environmental noise, it is possible using the "Tunable Wire" design. The signal is strong enough to survive the noise and be detected with current superconducting technology. The "Soliton Wire" is a cool idea, but it's currently too weak to be heard over the static.

In short: We have a blueprint for a black hole in a wire that we can actually build and test today, provided we use the right design.

Technical Summary: Dissipative Effects in Transmission Line Analogues of Hawking Radiation

Problem Statement
Hawking radiation, a fundamental prediction of quantum field theory in curved spacetime, remains unobserved in astrophysical contexts due to the extremely low temperatures associated with black hole masses. While analogue gravity systems (e.g., Bose-Einstein condensates, fluid dynamics) have successfully reproduced horizon physics, the direct observation of the quantum nature of Hawking radiation is often obscured by environmental noise and dissipation. Circuit Quantum Electrodynamics (cQED) offers a highly controllable platform for such analogues, yet previous theoretical proposals for transmission-line black holes often neglected the impact of environmental coupling and thermal noise. This work addresses the critical question of whether analogue Hawking radiation in superconducting circuits can survive decoherence and dissipation long enough to be experimentally distinguishable from the thermal background.

Methodology
The authors analyze two specific superconducting circuit architectures proposed as Schwarzschild black hole analogues:

Tunable dc-SQUID Transmission Line: An array of dc-SQUIDs where an external magnetic flux modulates the propagation velocity of electromagnetic flux, creating an effective horizon via a subsonic-supersonic transition.
SNAIL-Based Solitonic Transmission Line: A transmission line utilizing Superconducting Nonlinear Asymmetric Inductive eLements (SNAILs) to support solitonic solutions of the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations, which generate effective horizons.

The study employs an open quantum systems approach to model environmental effects. The system is described using the Lindblad master equation, incorporating relaxation and dephasing rates representative of state-of-the-art superconducting qubits. Key methodological steps include:

Metric Derivation: Establishing the effective Painlevé-Gullstrand-Lemaître (PGL) metric for both architectures, ensuring the inclusion of transverse dimensions to resolve conformal ambiguities in the 1+1 dimensional description.
Dissipative Modeling: Assuming that dissipative effects are negligible during the excitation's transit from the near-horizon region to the asymptotic flat region (a timescale of $\sim$ 100 ps, much shorter than decoherence timescales of $\mu$ s). Dissipation is modeled as acting on the field modes once they reach the asymptotic region where particle interpretation is well-defined.
Distinguishability Metrics: To assess observability, the authors propose a dual-criteria approach:
1. Particle number measurements.
2. The Hilbert-Schmidt distance between the evolved state of the transmission line and the thermal bath state. A threshold distance ( $D=0.05$ or $0.01$) is established to define when the Hawking signal remains distinguishable from the thermal background over a measurement window of 2 $\mu$ s.

Key Contributions

Dissipative Framework: The paper extends the theoretical description of transmission-line black hole analogues by explicitly incorporating environmental coupling and thermal noise, moving beyond idealized unitary evolution.
Quantitative Detectability Thresholds: By combining particle number analysis with the Hilbert-Schmidt distance, the authors establish practical limits for the minimum Hawking temperature required for experimental detection under realistic noise conditions.
Comparative Viability Analysis: The work provides a rigorous comparison between the tunable dc-SQUID architecture and the solitonic SNAIL architecture, evaluating their respective capacities to generate observable signals.

Results

Tunable Transmission Line: This architecture supports Hawking temperatures up to approximately 113 mK (theoretical limit $\sim$ 120 mK) for realistic circuit parameters. The analysis indicates that Hawking temperatures above 73 mK (under standard conditions) or 48 mK (under optimistic conditions with lower bath temperatures and longer coherence times) remain distinguishable from the thermal bath for the required measurement duration.
Solitonic Transmission Line: The KdV soliton configuration yields the highest temperatures among solitonic models, reaching approximately 1.9 mK with optimized parameters (or 1.2 mK for representative parameters). The mKdV configurations yield significantly lower temperatures ( $\sim$ 0.01 mK). Even with optimization, the solitonic model's achievable temperatures (e.g., 42 mK with specific parameter tuning) remain below the robust detection thresholds established for the tunable line and require significantly more demanding experimental conditions.
Frequency Dependence: The study finds that higher-frequency components of the Hawking spectrum become indistinguishable from the thermal bath more rapidly than lower-frequency components. Consequently, the detectability of the full spectrum is limited by the highest-frequency modes.

Significance and Claims
The paper claims that while both architectures theoretically generate analogue Hawking radiation, the tunable dc-SQUID transmission line is the only architecture currently viable for experimental observation using available superconducting circuit technology. The solitonic model, while theoretically sound, requires further optimization and experimental conditions that exceed current capabilities to produce a signal distinguishable from thermal noise.

The authors emphasize that their analysis establishes a "practical detectability threshold," demonstrating that Hawking temperatures above $\sim$ 73 mK are robust against realistic dissipation and thermal noise. This suggests that the tunable transmission line architecture is a feasible platform for studying the quantum nature of Hawking radiation in a laboratory setting, provided the system parameters are tuned to maximize the temperature and minimize environmental coupling. The work does not claim to have observed Hawking radiation but rather provides the theoretical criteria and feasibility assessment necessary for such an observation.

1. The Setup: Creating a "Black Hole" in a Wire

2. The Goal: Hearing the "Hawking Whisper"

3. The Test: The "Distinguishability" Check

4. The Results: Who Won the Race?

Summary

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