Quasiparticle two-level systems from superconducting gap fluctuations: quantitative strength, dirty-limit trap theory, microscopic origin, and population dynamics

Superconducting quantum devices host two-level-system (TLS) defects that limit qubit coherence. A 2020 experiment (de Graaf et al., Science Advances 6, eabc5055) identified a subset of highly coherent TLS - with a low (~300 mK) reconfiguration temperature, a strongly non-uniform density of states, and a splitting that tracks the superconducting gap in magnetic field - and proposed that they are quasiparticles (QPs) trapped in shallow subgap states formed by spatial fluctuations of the superconducting order parameter ('qTLS'). The proposal rests on a clean-limit quasiclassical theory whose key input, the gap-fluctuation strength $g$, is unknown for the thin films actually used in qubits, and it leaves open the trap physics in the dirty limit, the microscopic origin of the fluctuations, and the trapping dynamics that govern qTLS populations and device recovery. The problem is to quantitatively establish the qTLS mechanism in device-relevant superconductors: determine $g$ in qubit films (notably aluminum) together with a dirty-limit theory of gap-fluctuation traps, identify the fluctuation origin through a discriminating spectroscopic signature, and quantify QP capture, escape, and recombination rates well enough to predict qTLS densities, frequency windows, and recovery after ionizing-radiation events. Adjacent progress exists on radiation-event QP dynamics modeling, subgap-state energy scales in Al and Nb, single-defect imaging, and material surveys, but the central quantitative theory-experiment loop remains open.

Background

Superconducting circuits are a leading platform for quantum computing: a transmon qubit, the workhorse device, stores quantum information in the quantized energy levels of a Josephson junction - two superconductors separated by an insulating barrier through which Cooper pairs tunnel coherently. Two materials-related mechanisms dominate its decoherence. First, two-level systems (TLS): atom-like tunneling defects in the amorphous dielectrics of the circuit (junction barriers, surface oxides), which resonate with the qubit and absorb energy. In the standard tunneling model, a TLS is a particle or group of atoms tunneling between two nearly degenerate configurations in a double-well potential; in glasses its density of states is roughly energy-independent. Second, nonequilibrium quasiparticles (QPs): single-fermion excitations produced when Cooper pairs are broken, for example by stray radiation or high-energy phonons; QPs tunnel across the junction, causing relaxation and dephasing.

de Graaf et al. (Science Advances 6, eabc5055 (2020)) mapped TLS-induced frequency fluctuations in gate-tunable superconducting resonators and isolated a subset of TLS with properties unlike the glassy population: detected linewidths down to about 70 kHz (coherence times of microseconds), an irreversible reshuffling of the TLS landscape upon warming to only ~300 mK, a density of states strongly suppressed at low frequency, and a splitting that is suppressed in parallel magnetic field on the same scale as the superconducting order parameter $\Delta$ - suggesting these TLS are tied to superconductivity itself. They proposed that these 'qTLS' are QPs bound in shallow subgap states: spatial fluctuations $\delta\Delta(\mathbf{r})$ of the gap $\Delta(\mathbf{r}) = \Delta_0 + \delta\Delta(\mathbf{r})$ occasionally produce wells deep enough to host discrete levels below the gap edge, and a QP occupying such a well behaves as a highly coherent, electrically dipolar two-level system. Modeling $\delta\Delta$ as Gaussian noise with correlator $\langle \delta\Delta(\mathbf{r})\,\delta\Delta(\mathbf{r}') \rangle = g\,\delta(\mathbf{r}-\mathbf{r}')$, where $g$ sets the fluctuation strength and $\xi_0$ is the BCS coherence length (of order $\hbar v_F/\Delta_0$, with $v_F$ the Fermi velocity), the theory - an optimal-fluctuation calculation (finding the rarest, energetically cheapest gap fluctuation capable of hosting a bound level), in the spirit of Lifshitz-tail problems (the exponentially small density of states below a band edge produced by rare disorder fluctuations) - predicts a strongly non-uniform qTLS density of states, peaking near 6-10 GHz for their NbN parameters and cutting off below a material-dependent frequency set by the maximum trap depth. The mechanism implies that QPs localize in these traps essentially indefinitely at millikelvin temperatures, with consequences for qubit recovery after ionizing radiation, when a particle impact generates phonons that break Cooper pairs across the chip.

The framework has clear antecedents. Subgap states induced by gap inhomogeneity in dirty superconductors were computed by Silva and Ioffe (Physical Review B 71, 104502 (2005)) in the context of Josephson-qubit dephasing, and engineered quasiparticle traps - normal-metal or lower-gap islands that capture QPs - have been studied as a mitigation tool (for example, Riwar and Catelani, Physical Review B 100, 144514 (2019)). A related proposal by Bespalov et al. (Physical Review Letters 117, 117002 (2016)) argued that QP localization in subgap states could slow recombination and explain the long-standing 'excess quasiparticle' problem, namely that measured QP densities in cold devices far exceed thermal-equilibrium expectations.

The 2020 work nevertheless left the qTLS mechanism unquantified for real devices, for five linked reasons. (i) The strength of gap fluctuations $g$ - the parameter controlling the qTLS density of states, its frequency window, and its upper frequency bound - was not known for thin films used in qubits, notably aluminum. (ii) The derivation assumes the clean limit, in which the elastic mean free path $\ell$ exceeds $\xi_0$ so that a QP moves ballistically inside a trap; but the disordered polycrystalline films used for qubits are typically in the dirty limit ($\ell \ll \xi_0$), where strong impurity scattering rules out the elongated optimal traps of the clean theory and the trap spectrum must be recomputed. (iii) The microscopic origin of $\delta\Delta$ - surface magnetic moments, film-thickness variations, disorder-induced density-of-states fluctuations - was not identified, and on-chip electron spin resonance (ESR, a microwave spectroscopy that detects spins through their resonant absorption) had not found any signature uniquely associated with qTLS. (iv) The rates of QP capture by, escape from, and recombination at gap-fluctuation traps in realistic device geometries - needed to predict qTLS populations and device recovery times after radiation events - were unmeasured. (v) Consequently, no materials or device strategy could yet be designed to suppress the mechanism, and its prevalence across superconducting materials, and the scaling of the qTLS frequency window and density with material parameters ($\Delta_0$, $\xi_0$, the Fermi wave vector $k_F$), were unknown.

Problem Statement

Quantitatively establish the quasiparticle-TLS (qTLS) mechanism in superconducting quantum devices: close, at the level of measured microscopic parameters, the loop between order-parameter gap fluctuations in device-relevant superconducting films and the density of states, frequency window, and population dynamics of the two-level systems they produce. With the common objective of a predictive, materials-specific account of qTLS-induced decoherence, an answer must resolve all of the linked questions left open by de Graaf et al. (2020), each of which is a necessary input to that account; resolving only a subset does not by itself constitute a resolution:

1. Determine the strength of order-parameter gap fluctuations $g$ (equivalently, the subgap density of states they generate on the coherence-length scale) in the thin superconducting films used for qubits - aluminum as the mandatory target, with the other common electrode materials (Nb, Ta, TiN, NbN) an optional extension - well enough to predict the qTLS density of states and its upper frequency bound, and check the prediction against measured TLS populations in devices, including the predicted strong suppression of the qTLS density of states at low frequency.

2. Extend the trap theory from the clean limit to the dirty limit ($\ell \ll \xi_0$) that describes real films: recompute the optimal fluctuation shape and the resulting trap spectrum and qTLS density of states, and state how the frequency window and areal density scale with material parameters ($\Delta_0$, $\xi_0$, $k_F$, disorder strength).

3. Identify the microscopic origin(s) of the gap fluctuations - disentangling the relative contributions of surface magnetic moments, film-thickness variations, and disorder-induced density-of-states fluctuations - and isolate a spectroscopic signature (for example in on-chip electron spin resonance or an equivalent probe) that is uniquely associated with qTLS, so that the mechanism can be recognized and separated from conventional glassy TLS and from paramagnetic spins.

4. Quantify the population dynamics: the rates of quasiparticle capture by and escape from gap-fluctuation traps, and the recombination of trapped quasiparticles, in realistic device geometries and following ionizing-radiation events, sufficiently to predict qTLS populations and device recovery times.

A complete answer establishes how qTLS densities and frequency windows depend on material and fabrication parameters - thereby enabling targeted mitigation (materials processing or device architectures that reduce gap fluctuations or keep trapped quasiparticles away from sensitive circuit elements) - and settles, from the aluminum fluctuation strength together with the scaling laws of Question 2, whether qTLS are a ubiquitous, designable decoherence channel across superconducting platforms or specific to particular materials.

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