Quantum sensors for microscopic tunneling systems

The study addresses the long-standing problem of understanding microscopic tunneling two-level systems (TLS) responsible for universal low-temperature anomalies in amorphous solids. Traditional measurements probe large ensembles and cannot resolve individual TLS properties. With the emergence of superconducting qubits—sensitive, controllable quantum systems—there is an opportunity to detect, control, and characterize individual TLS which limit coherence in quantum processors. The paper proposes and demonstrates a qubit-based quantum sensor that integrates an arbitrary thin-film material as the dielectric of a dedicated sample capacitor shunting a transmon qubit. The research question is how to detect, locate, and characterize individual TLS (including their dipole moments, strain/electric-field tunability, and mutual interactions) within specific materials, in order to inform materials development for low-loss superconducting quantum circuits.

The Standard Tunneling Model (STM) explains low-temperature anomalies in amorphous materials via TLS characterized by asymmetry energy ε and tunneling energy Δ0, but it neglects TLS–TLS interactions and fails in the 1–10 K range. Refinements include interaction effects, different classes of TLS, and specific potential energy distributions. Prior experiments on bulk materials averaged over inhomogeneous ensembles, limiting microscopic insight. Superconducting qubits have revealed individual TLS effects such as avoided level crossings, decoherence via coupling to phonons or quasiparticles, and noise from interacting TLS. Methods to tune TLS include mechanical strain and DC-electric fields, enabling localization and characterization, with prior work demonstrating qubit-mediated coherent control, temperature dependence, and interactions between TLS.

Device concept: A transmon qubit (capacitively shunted DC SQUID) is augmented with a small 'sample capacitor' whose dielectric is the material under test. The qubit couples electrically to TLS within this dielectric; individual TLS can be detected via their resonant interaction with the qubit.

Capacitor designs: (1) Overlap (plate) sample capacitor with 50 nm AlOx dielectric (used in this work). (2) Alternative 'nanogap' coplanar electrodes separated by tens of nm, optionally covered by sample material (or left bare to probe native oxides and adsorbates). Coupling to TLS occurs via the fringing field in the nanogap.

Coupling model: TLS in a double-well have transition energy E = sqrt(Δ0^2 + ε^2), with dipole moment projection p = p0 (Δ0/E). Qubit–TLS coupling g_iq = p · F_p, where F is the qubit’s electric field at the TLS. Detectability criterion: single TLS are resolvable if g ≳ 1/T1. For an overlap capacitor, selecting dielectric thickness d satisfies d ≈ p T1 V_rms/ħ, with V_rms ≈ ħ ω10/(2 C_tot). Using ω10/2π ≈ 6.2 GHz, C_tot ≈ 100 fF, V_rms ≈ 4.5 μV, T1 ≈ 1 μs, and minimum p ≈ 0.1 eÅ leads to d ≈ 70 nm; chosen d = 50 nm. The implemented sample capacitor area is (0.25 × 0.3) μm^2, giving Cs ≈ 0.15 fF ≪ C_tot to keep added dielectric loss small.

TLS localization via tunability: TLS location is inferred by tracking their frequency response to (i) local DC bias on the sample capacitor top electrode (V_s) producing a well-defined field V_s/d inside the sample dielectric, (ii) a global DC electrode above the chip (global E-field), and (iii) mechanical strain via a piezo actuator. TLS in the sample dielectric respond to V_s and strain; surface/interface TLS respond to global E-field and strain; TLS inside tunnel barriers respond to strain but not to applied E-fields.

Spectroscopy protocol: Swap spectroscopy is used to map qubit T1 versus qubit frequency while sweeping control parameters (V_s, global bias, piezo voltage V_p). Dark traces indicate enhanced relaxation due to resonant TLS coupling. Each TLS trace (a hyperbola vs bias) is fitted to asymmetry ε(V) = ε0 + γ_g V_g + γ_s V_s + γ_p V_p, yielding coupling constants γ and, if within qubit tuning range, tunneling energy Δ0. For sample TLS, the coupling dipole p_γ follows from 2 p_γ V_s/d = γ_g V_g (with the well-defined field in the sample capacitor), enabling direct extraction of the projected dipole without ambiguity from the Δ0/E matrix element.

Fabrication: Qubit chips (KIT) with three Xmon qubits: two shunted by sample capacitors and one reference. 100 nm Al film patterned for electrodes and resonators; Josephson junctions via e-beam lithography and shadow evaporation. For small overlap sample capacitors, the bottom electrode was made simultaneously with the junctions; 50 nm AlOx dielectric deposited by e-beam evaporation of Al in O2 (3×10^-4 mbar, 5 sccm) at 0.2 nm s^-1, then in situ capped by 100 nm Al top electrode. A filter capacitor (Cg ≈ 250 fF) in the top electrode acts as a DC break for biasing. Larger sample capacitors (0.3 × 2.1 μm^2) were also fabricated for dielectric loss measurements.

Experimental setup: Measurements at 30 mK in a light-tight, magnetically shielded dilution refrigerator. Microwave lines are attenuated and filtered; qubit state read out via dispersive shift of a notch-type resonator with homodyne detection. Global DC gate implemented as a copper/Kapton foil electrode above the chip, filtered at 1 K and 30 mK. Sample capacitor top electrode is accessible via a bias line for applying V_s. Strain tuning via a piezo actuator beneath the chip.

Performance baseline: Small-sample-capacitor qubits exhibited T1 = 3.3–4.2 μs, comparable to a reference qubit (T1 = 4.3 μs), indicating negligible added loss. Larger sample capacitors reduced T1, enabling extraction of AlOx loss tangent.

• Demonstrated a qubit-based sensor enabling spectroscopy and control of individual TLS in an arbitrary thin-film dielectric integrated as a sample capacitor.
• Using swap spectroscopy while sweeping mechanical strain, global E-field, and local sample-capacitor bias, 138 TLS were characterized on two identical qubits in one cooldown; 13 TLS were identified within the sample AlOx dielectric.
• Sample TLS spectral density: 4.1 GHz^-1 within the sampled bandwidth; inferred volume density ρ0 ≈ 1800 μm^-3 GHz^-1 for the 50 nm AlOx layer (field-free dielectric volume estimated as 0.15 × 0.3 × 0.05 μm^3).
• Average projected electric dipole moment for sample TLS: p_γ = (0.4 ± 0.2) eÅ.
• Loss tangent of deposited AlOx (εr ≈ 10) from TLS statistics: tan δ0 ≈ π ρ0 p_γ^2/(3 ε0 εr) ≈ 1×10^-3, consistent with direct device-loss measurements.
• Direct loss measurement with larger sample capacitors yielded tan δ ≈ (1.7 ± 0.2) × 10^-3 for the AlOx dielectric.
• TLS in stray Josephson junction tunnel barriers (area ≈ 17.17 μm^2, thickness 1.5–2 nm) showed volume densities ρ0 ≈ 270–360 μm^-3 GHz^-1, about six times lower than in the thicker deposited AlOx, consistent with higher qubit field amplitude in the sample capacitor (≈90 V m^-1 vs ≈15 V m^-1 in tunnel barriers) and potential growth/defect-environment differences.
• Observed coherent TLS–TLS interactions inside the sample dielectric as avoided level crossings in electric-field spectroscopy; by jointly tuning strain and local E-field, the avoided crossing was shifted through the TLS symmetry point, allowing separation of longitudinal and transverse coupling components with fitted strengths g_x ≈ 19 μs^-1 and g_z ≈ 25 μs^-1.
• Classification by tunability indicates that decoherence was predominantly limited by TLS in the tunnel barriers of large-area stray junctions (fabrication artefacts), which could be mitigated by shorting these junctions in an added lithography step.

The results show that a transmon qubit with an integrated sample capacitor can selectively probe and control individual TLS in a target dielectric, overcoming ensemble-averaging limitations. Electric-field and strain tunability provide a location fingerprint, enabling discrimination among TLS in junction barriers, surface/interface oxides, and the sample dielectric. The ability to extract projected dipole moments directly from a calibrated local field in the sample capacitor gives quantitative microscopic parameters for TLS in specific materials. The observed coherent interactions between TLS, and the capacity to independently tune and characterize longitudinal versus transverse coupling, further illuminate the microscopic dynamics and interactions within amorphous dielectrics. Practically, the study identifies stray junction TLS as a dominant decoherence channel in the tested devices, underscoring fabrication strategies (e.g., shorting stray junctions) to improve qubit coherence. The measured TLS densities and loss tangents for deposited AlOx align with and complement prior ensemble measurements, providing a pathway to benchmark and optimize dielectric deposition processes for quantum circuits.

The study introduces and validates a quantum sensor based on a superconducting transmon qubit for the detection, localization, and quantitative characterization of individual TLS in thin-film materials. By integrating the material as a small sample capacitor and applying controlled electric fields and strain, the method measures TLS dipole moments, identifies their positions within the device, and reveals coherent TLS–TLS interactions with separable longitudinal and transverse components. These capabilities enable targeted materials spectroscopy for developing low-loss dielectrics critical to advancing superconducting quantum processors and other quantum/nanofabricated devices. Future work can extend the approach to diverse materials (including bulk via nanogap capacitors), temperature dependence, correlation with fabrication parameters, and real-time monitoring of TLS dynamics to further unravel amorphous-solid physics and inform process optimization.

• Detection sensitivity is limited by the qubit’s coherence (T1) and the coupling strength g; TLS with very small dipole moments may remain undetected.
• The statistics reported are for specific AlOx films and device geometries; generalization to other materials and deposition methods requires further studies.
• Larger sample capacitors introduce additional loss and reduce qubit T1, constraining the range of capacitor sizes usable without degrading sensitivity.
• Some TLS remained unclassified due to limited visibility across all control-parameter sweeps.
• Measurements are performed at millikelvin temperatures and in specific electromagnetic environments; environmental differences may affect observed TLS behavior and densities.