Charged Two-Level Fluctuators (TLFs)

• Charged two-level fluctuators (TLFs) are localized quantum systems that switch between two charge configurations, producing discrete changes in the electromagnetic environment.
• They generate characteristic low-frequency noise, including 1/f and telegraph noise, which plays a crucial role in decoherence and error patterns in superconducting and nanoscale devices.
• TLFs are characterized through microscopic models, spectral analysis, and spatial mapping, informing strategies to mitigate noise and improve quantum device performance.

Charged two-level fluctuators (TLFs) are localized quantum systems in condensed matter environments that can switch between two charge configurations, producing discrete changes in local potential or electromagnetic environment. These entities play a central role as noise sources in advanced solid-state and superconducting devices, affecting electronic, spin, and bosonic degrees of freedom via a variety of direct and indirect coupling pathways. Their stochastic or coherent dynamics, spatial distributions, and interactions with baths and each other yield characteristic low-frequency $1/f$ noise, telegraph noise, decoherence phenomena, and spatially correlated error patterns relevant to quantum computation and precision metrology.

1. Microscopic Models and Physical Realizations

TLFs are most generally described as localized quantum two-level systems with charge, typically associated with charged defects, impurity dipoles, or electron trap pairs in dielectric, oxide, or interface regions. The prototypical Hamiltonian for a single TLF in the charge basis is

$$\hat{H}_{TLF} = \frac{1}{2} \epsilon\, \sigma_z + \frac{1}{2} \Delta\, \sigma_x$$

where $\epsilon$ is the energy bias between charge configurations, and $\Delta$ is the tunneling amplitude. In the energy (pseudospin) eigenbasis, the energy splitting is $\Omega = \sqrt{\epsilon^2 + \Delta^2}$. TLFs may arise as bistable charge traps, atomic-scale dipoles, or as small clusters—commonly observed near gate dielectrics (Si/SiO₂, AlOₓ), in the substrate or tunnel barriers, or at interfaces in nanodevices (Zaretskey et al., 2013, Ye et al., 25 Jan 2024). Their switching corresponds to stochastic occupation of two potential wells, with rates set by their local environment and coupling to baths.

Ensembles of TLFs can be realized as distributions of such microscopic dipoles or traps, with parameters $(\epsilon, \Delta)$ spanning a wide range, and spatially distributed throughout interfaces or dielectrics. In quantum dot and spin qubit systems, spectroscopically and time-domain-resolved measurements have isolated individual TLFs and characterized their switching times, sensitivities to temperature/gating, and their microscopic locations (Ye et al., 25 Jan 2024, Rojas-Arias et al., 9 May 2025). In superconducting devices, TLFs may be particularly relevant inside thin tunnel barriers, leading to direct modulation of critical currents or gate charges (Zaretskey et al., 2013).

2. Noise Generation and Spectral Properties

The canonical signature of TLFs is broadband low-frequency noise, with ensembles of fluctuators giving rise to characteristic $1/f$ or $1/f^2$ power spectral densities (PSDs) in device observables. Each individual TLF produces a Lorentzian noise spectrum,

$$S_{TLF}(\omega) = \frac{4\gamma v^2}{4\gamma^2 + \omega^2}$$

with $v$ the coupling constant and $\gamma$ the switching rate. Random telegraph switching of TLFs leads to temporally discrete charge jumps and non-Gaussian temporal statistics when the ensemble is sparse (Mehmandoost et al., 29 Apr 2024, Rojas-Arias et al., 9 May 2025).

If the distribution of TLF switching rates $P(\gamma)$ exhibits a log-uniform form $P(\gamma) \sim 1/\gamma$ over several decades, their ensemble-averaged spectrum recovers a $1/f$ law: $$S_{tot}(\omega) \sim \int_{\gamma_{min}}^{\gamma_{max}} d\gamma\, \frac{1}{\gamma} \frac{4\gamma v^2}{4\gamma^2 + \omega^2} \sim \omega^{-1}$$ in the frequency interval $\gamma_{min} \ll \omega \ll \gamma_{max}$. Deviations from power-law PSDs arise when individual TLFs dominate, resulting in Lorentzian “bumps” or strong time-localized fluctuations (Mehmandoost et al., 29 Apr 2024, Rojas-Arias et al., 9 May 2025). In high-quality qubits with sparse fluctuator baths, $S(f)$ can remain robustly $1/f$ even as the actual decoherence varies strongly between devices, being controlled by a small number of exceptional slow TLFs.

Temperature-dependent studies further reveal that TLF noise spectra can exhibit saturation, crossovers (e.g., $1/f$ to $1/f^2$ as $T$ decreases), and strong dependence on TLF activation energies distributed across 0.35–0.65 eV (Noel et al., 2018).

3. Dynamics, Decoherence, and Device Impact

TLFs produce both dephasing and dissipative effects on quantum and classical systems. They may couple to qubit states (charge, spin, phase) via Ising-type longitudinal interactions ($\sim \sigma_z \tau_z$), leading to random shifts in energy splitting, or via transverse channels, enabling relaxation. TLFs can be modeled as sources of non-Gaussian, non-Markovian noise, leading to coherence decay approximated as compressed exponentials $e^{-(t/T_2)^{\alpha}}$, with the stretching parameter $\alpha$ and $T_2$ inheriting their temperature dependences from the microscopic coupling and bath structure (Beaudoin et al., 2014).

TLF-mediated decoherence mechanisms include:

• Free induction (pure dephasing), with $T_2^*$ determined by the strongest and slowest TLFs (Mehmandoost et al., 29 Apr 2024).
• Echo decay, which can be limited by TLFs with switching rates close to the timescale of the measurement (Mehmandoost et al., 29 Apr 2024, Beaudoin et al., 2014).
• TLF-assisted spectral diffusion, in which high-frequency TLS/qubit lifetimes fluctuate slowly due to low-frequency TLFs, leading to $T_1$ instability (You et al., 2022, Zhu et al., 16 Sep 2024).

In superconducting qubits, TLFs residing in tunnel barriers can couple simultaneously to charge and Josephson energy, enabling direct visualization in spectroscopic measurements as “quadrupled” transitions (Zaretskey et al., 2013). In Si/SiGe quantum dot systems, TLF-induced charge noise manifests as random shifts in Zeeman energies through position-dependent magnetic gradient coupling, with spatial “triangulation” possible via cross-correlation analysis (Rojas-Arias et al., 9 May 2025).

In resonant and magnonic systems (YIG), TLFs result in additional damping contributions, saturated at high drive but dominant at cryogenic temperatures and low excitation (Kosen et al., 2019). In single-electron transistors, TLF ensembles can be probed via the long-timescale drift of the offset charge following a gate voltage step, with drift scaling logarithmically in time—diagnostic for tunneling-mediated rather than thermally activated transitions (Pourkabirian et al., 2014).

4. Spatial, Temporal, and Statistical Characterization

Individual TLFs can be isolated and characterized with high time-resolution methods (e.g., waiting time distribution analysis, factorial hidden Markov models), allowing direct extraction of switching rates, occupation probabilities, and spatial sensitivities to control parameters (Jenei et al., 2019, Ye et al., 25 Jan 2024, Rojas-Arias et al., 9 May 2025). Sensitivity to temperature, gate voltages, and even pulse-driven heating—where TLF occupation can serve as a local thermometer—has been demonstrated (Ye et al., 13 Sep 2025).

Spatial mapping of correlated noise via cross-correlation functions (e.g., magnitude-squared coherence $C_{xy}(f)$) reveals the influence range of individual TLFs and clarifies mitigation strategies via geometric device design (Donnelly et al., 6 May 2024). Interacting models (e.g., spin glass simulations for quantum dots) and analytical triangulation procedures have shown that noise correlations decay exponentially with separation, tying noise cross-correlations to specific physical gate or interface regions (Mickelsen et al., 2023, Rojas-Arias et al., 9 May 2025).

Sparsity of the TLF bath (where $d = n/w \ll 1$ with $n$ TLFs in log-width $w$) fundamentally alters the statistical distribution of decoherence in multi-qubit arrays: the noise spectrum remains similar, but sample-to-sample $T_2^*$ and $T_2$ exhibit large variability controlled by the rare slowest fluctuators (Mehmandoost et al., 29 Apr 2024).

5. Dissipation, Device Nonidealities, and Engineering Implications

Beyond charge fluctuations, the interaction of TLFs with conduction electrons can directly influence the electronic properties of disordered superconductors through kinetic inductance fluctuations, resulting in resonator jitter and reduced $Q$ (Sueur et al., 2018). The presence of a dc supercurrent further enhances the TLF coupling to external fields, leading to parametric enhancement of ac conductivity and equilibrium current $1/f$ noise when the TLS relaxation-time distribution is broad (Liu et al., 18 Jan 2025).

In the context of adiabatic passage techniques (STIRAP/CTAP), TLFs can resonantly degrade transfer efficiency when the system’s instantaneous energy gap matches the TLF resonance, and the effect can only be mitigated by carefully choosing pulse amplitudes or broadening the TLF spectral width (Vogt et al., 2012). Device design must thus ensure that swept system energies avoid or are less sensitive to active TLF resonances.

Mitigation strategies for TLF-induced decoherence include (i) reduction of TLF densities through improved materials and fabrication (e.g., better oxides, encapsulation, substrate choices), (ii) device geometries that reduce electromagnetic participation or enhance thermalization, (iii) spectral engineering of the environment (applying longitudinal or transverse engineered noise to suppress or depolarize TLSs or TLFs) (You et al., 2022), (iv) minimizing active gate area near electron accumulations to reduce pulse heating effects (Ye et al., 13 Sep 2025), and (v) active identification and targeted deactivation/passivation of the rare, most-harmful fluctuators in sparsely defective regimes (Mehmandoost et al., 29 Apr 2024).

6. Quantum Information and Emerging Frontiers

Emerging research directly links TLF dynamics to temporal and spatial fluctuation of qubit properties (e.g., $T_1$, $T_2$), correlated error rates in multiqubit devices, and failure modes of quantum error correction under spatially correlated, non-Gaussian noise (Donnelly et al., 6 May 2024, Rojas-Arias et al., 9 May 2025). Accurate noise models, triangulation of active TLF locations, and strategies for suppressing correlated noise will remain crucial for the scalable implementation of quantum computing architectures.

Future work is poised to explore the ultimate limits of decoherence in low-defect-density devices, the interplay of TLFs with quasiparticle and bosonic baths, further statistical modeling of rare-event noise, and materials characterization that identifies, localizes, and selectively controls individual TLFs.

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Table 1. Representative TLF Hamiltonian Structures and Couplings

Type	Hamiltonian	Coupling Mechanisms
Charge TLF	$$\frac{1}{2}\epsilon \sigma_z + \frac{1}{2}\Delta \sigma_x$$	Charge, dipole, $\Delta J$, $V_g$
Qubit Coupling	$\lambda_q \sigma_z \tau_z$ (Ising-type)	Longitudinal, transverse
Bath Interaction	$\sum_j \lambda_{j} \sigma_z b_j + h.c.$	Boson, electron, phonon
TLF-TLF Coupling	$\sum_{i,j} \mu_{i,j} \sigma_z^i \sigma_z^j$	Strain-mediated, dipole-dipole

Table 2. Experimental Probes and Characterizations

Method	Information Extracted	Implications
Power/cross spectra (PSD)	$1/f$ scaling, TLF localization	Noise source distribution
Time-domain (Allan, WTD)	TLF rates, occupation bias	Telegraphed noise, bath sparsity
Spectroscopic line splitting	Coupling to charge, $I_c$	Junction barrier, defect modulation
Bath engineering/control	TLF suppression/stabilization	Enhanced coherence, TLF mitigation

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In summary, charged two-level fluctuators represent a universal and essential ingredient in the decoherence landscape of solid-state quantum, mesoscopic, and classical devices, generating noise with complex spectral, spatial, and statistical structure. Theoretical, experimental, and engineering advances continue to refine their characterization, coupling dynamics, and mitigation, with central implications for device performance, scaling, and error correction paradigms in quantum technologies.