Dissipation due to bulk localized low-energy modes in strongly disordered superconductors

This paper presents a novel microscopic theory explaining that low-temperature microwave dissipation in strongly disordered superconductors is dominated by bulk localized collective modes arising from spatial inhomogeneity, thereby resolving the limitations of standard Mattis-Bardeen theory and offering strategies to mitigate losses in superconducting quantum devices.

The Gist

The Big Picture: The "Leaky" Superconductor

Imagine you are building a super-fast, ultra-precise clock (a quantum computer). To make it work, you need a material that acts like a perfect, frictionless slide for electricity. In the world of quantum physics, this material is a superconductor.

Usually, if you cool a metal down enough, it becomes a perfect slide. But scientists have been using "strongly disordered" superconductors (materials that are messy and full of impurities) because they have a special property: they act like a very stiff spring, which is great for making tiny, compact quantum devices.

The Problem: These messy materials have a hidden flaw. Even when they are super cold, they "leak" energy. It's like trying to slide down a frictionless slide, but the slide is actually covered in tiny, invisible patches of sticky mud. This energy loss (dissipation) ruins the clock's accuracy.

For a long time, scientists used an old rulebook (called the Mattis–Bardeen theory) to predict how much energy would leak. But this rulebook failed for these messy materials. It couldn't explain why the energy loss was so high, even when the temperature was near absolute zero.

The New Discovery: The "Sticky Patches"

The authors of this paper developed a new theory to solve this mystery. Here is what they found, using an analogy:

1. The Material is a Patchwork Quilt
Imagine the superconductor isn't a smooth, uniform sheet of ice. Instead, it's a giant quilt made of thousands of tiny patches.

• Most patches are thick, strong ice (strong superconducting regions).
• A few rare patches are very thin, weak ice (weak spots).

2. The "Sticky Patches" (Low-Energy Modes)
In the old theory, scientists thought energy loss came from breaking pairs of electrons (Cooper pairs) apart. But in these messy materials, the "weak spots" in the quilt are so thin that they don't need to break the pairs to let energy through.

Instead, these weak spots act like tiny, localized trampolines.

• When you send a microwave signal (a wave of energy) through the material, it mostly passes over the strong ice patches without trouble.
• However, when it hits a "weak spot," it gets stuck on the trampoline. The trampoline bounces up and down, absorbing the energy and turning it into heat.

3. The "Two-Level" Behavior
The paper explains that these weak spots behave like simple light switches (or two-level systems). They can be in one of two states: "off" or "on."

• At very low temperatures, these switches are mostly "off."
• As you slightly warm the material, the switches start flipping "on" and "off" randomly, absorbing energy. This explains why the energy loss increases as the temperature rises, even slightly.

Why Frequency Matters (The "Tuning" Analogy)

The paper also discovered something surprising about the frequency (the pitch) of the energy waves.

• Low Pitch (Low Frequency): The "trampolines" are hard to find. The energy wave glides over them easily. The device works well.
• High Pitch (High Frequency): As you increase the pitch, the energy wave starts hitting more and more of these weak trampolines. It's like shaking a box of marbles; if you shake it gently, they stay still. If you shake it violently (high frequency), they all start rattling and absorbing your energy.

The authors found that the energy loss grows very quickly as the frequency goes up. This is because the "weak spots" in the material are distributed in a specific way: there are very few strong spots, but a "tail" of many, many weak spots that only show up when you look closely (high frequency).

The Solution: Tuning the Clock

The paper offers a practical tip for engineers building these quantum devices: Turn down the volume (frequency).

Because the energy loss is so sensitive to frequency, simply lowering the operating frequency of the device can reduce the energy loss by a huge amount (potentially ten times better). This doesn't require changing the material; it just requires tuning the device to a lower pitch where the "sticky patches" are less likely to catch the energy.

Summary

• The Mystery: Messy superconductors leak energy in ways old physics couldn't explain.
• The Cause: The material is a patchwork of strong and weak areas. The weak areas act like tiny, energy-absorbing trampolines (collective modes).
• The Mechanism: These trampolines act like simple switches that flip on and off, soaking up microwave energy.
• The Fix: By running the device at a lower frequency, you avoid hitting these trampolines, making the quantum device much more stable and efficient.

This theory helps scientists understand exactly why these materials lose energy and gives them a clear strategy to build better quantum computers using the materials they already have.

Technical Summary

Problem Statement
Strongly disordered superconductors (SDSCs), such as amorphous InO$_x$, TiN, and NbN, are critical materials for superconducting quantum devices (qubits, resonators, detectors). However, their performance is limited by intrinsic microwave dissipation at low temperatures, which restricts coherence times. The standard Mattis–Bardeen theory, which successfully describes dissipation in conventional superconductors via quasiparticle excitations, fails in SDSCs. This failure occurs because SDSCs exhibit a hard single-particle pseudogap ($\Delta_P$) that persists well above the superconducting transition temperature ($T_c$), and the superconducting order parameter ($\Delta$) is highly spatially inhomogeneous below $T_c$. Consequently, the low-energy electromagnetic response cannot be explained by simple disorder-added quasiparticle models. The central unresolved question is identifying the microscopic objects responsible for absorbing microwave photons in the regime where $\hbar\omega, T \ll \Delta \leq \Delta_P$.

Methodology
The authors develop a novel microscopic theory based on a pseudospin Hamiltonian describing localized, preformed Cooper pairs. The model treats the system as an interaction graph of Anderson-localized single-particle states, where the local superconducting order parameter varies significantly due to strong disorder.

• Theoretical Framework: The authors employ the "Belief Propagation" (BP) technique, justified by the locally tree-like structure of the interaction graph in the limit of strong disorder. This allows for the calculation of local physical quantities by solving a self-consistency equation for order parameter fields ($h_{i \to j}$) on directed edges of the graph.
• Network Model (NM): To compute macroscopic conductivity, the authors utilize a numerical Network Model. This involves generating large instances of the locally tree-like graph, solving the self-consistency equations for order parameters, computing local current correlators ($R_{ij}$), and solving Kirchhoff equations for superconducting phases to determine the total Joule heat dissipation.
• Analytical Approximation: The authors derive an approximate analytical expression for the real part of the conductivity, $\text{Re}\sigma(\omega)$, linking it to the probability distribution of the order parameter, $P(h)$. This derivation assumes that dissipation is dominated by rare "weak spots" (regions with low local order parameter) where localized collective modes exist.

Key Contributions and Results

1. Identification of Dissipation Mechanism: The paper identifies that low-frequency dissipation in SDSCs is dominated by a new type of bulk localized collective modes. These modes arise from the spatial inhomogeneity of the superconducting state, specifically within rare regions where the order parameter is significantly suppressed ("weak spots"). These modes correspond to low-energy rearrangements of Cooper pairs with electric dipole moments determined by the size of the weak spots, rather than by pair breaking.
2. Microscopic Theory of $Q(\omega, T)$: The authors derive a closed-form expression for the resonator quality factor $Q(\omega, T)$ in the regime $\hbar\omega, T \ll \Delta \leq \Delta_P$:
   $$\frac{1}{Q} = C \frac{\omega}{2\Delta_0} \tanh\left(\frac{\omega}{2T}\right) \int_0^\omega dh P(h) \arccos\left(\frac{2h}{\omega}\right)$$
   where $P(h)$ is the distribution of the local order parameter.
3. Frequency and Temperature Dependence:
   • Frequency: The theory predicts a strong decrease in $Q$ with increasing frequency $\omega$. This is a direct consequence of the steep profile of the low-value tail of the order parameter distribution $P(h)$.
   • Temperature: For $T \ll T_c$, $Q$ exhibits a two-level-system (TLS)-like growth with temperature, governed by the $\tanh(\omega/2T)$ factor, reflecting the thermal activation of these local modes. However, this growth is eventually slowed by the temperature-induced broadening of the low-value tail of $P(h)$.
4. Validation: The theoretical results, particularly the steep frequency dependence and the magnitude of $Q$, show qualitative agreement with recent experimental data on InO$_x$ resonators. The numerical solution of the Network Model validates the analytical approximations, though discrepancies at very low frequencies are attributed to the breakdown of the Belief Propagation approximation due to long-range correlations in the limit of vanishing order parameter.

Significance and Claims
The paper claims to provide the first microscopic understanding of intrinsic microwave losses in SDSC-based quantum devices that accounts for the hard pseudogap and order parameter inhomogeneity.

• Experimental Probe: The derived relationship between $Q(\omega)$ and $P(h)$ suggests that microwave spectroscopy can be used as a direct probe to reconstruct the distribution of the order parameter in rare weak regions, a quantity previously difficult to measure directly.
• Mitigation Strategies: The results suggest a practical strategy for improving coherence times: lowering the operating frequency of the device. Because $P(h)$ is steep, halving the frequency can increase the quality factor $Q$ by an order of magnitude for the same film, provided the disorder is sufficient for the theory to apply.
• Scope: The authors note that while their theory explains the main trends in InO$_x$, it does not fully resolve the spatial structure of the localized modes or the temperature dependence of the superfluid stiffness at higher frequencies. They also acknowledge that while similar dissipation mechanisms may exist in granular aluminum, the physical origin of localization differs (grain structure vs. Anderson localization).

In summary, the work establishes that the intrinsic loss in strongly disordered superconductors is governed by the statistical tail of the order parameter distribution, offering a unified framework to interpret existing experimental data and guiding the design of lower-loss quantum circuits.