Non-equilibrium Dynamics of Two-level Systems directly after Cryogenic Alternating Bias

TL;DR

The paper addresses how cryogenic alternating bias alters two-level systems (TLSs) in amorphous Al$_2$O$_3$ barriers coupled to a superconducting LC oscillator. It employs a strong TLS-oscillator coupling platform and in-situ Cryogenic Alternating Bias Stimulation (CABS) to spectroscopically image TLSs, extract dipole moments and densities, and monitor loss tangent behavior. The main finding is that CABS suppresses the steady-state TLS spectral signatures and induces transient, minute-scale TLS frequency fluctuations, while the overall TLS density remains unchanged and the loss tangent is preserved; thermal cycling above 10 K reverses the spectral disruption, suggesting a reversible, non-equilibrium energy buildup in the oxide. The results offer a path toward time-dependent TLS modeling and inform strategies to mitigate TLS-related loss in quantum devices, by linking non-equilibrium energy release to TLS dynamics and their recoverability through thermal processing.

Abstract

Two-level systems (TLSs) are tunneling states commonly found in amorphous materials that electrically couple to qubits, resonators, and vibrational modes in materials, leading to energy loss in those systems. Recent studies suggest that applying a large alternating electric field changes the oxide structure, potentially improving the performance of qubits and resonators. In this study, we probe the effect of alternating bias at cryogenic temperatures on TLS dynamics within amorphous oxide parallel-plate capacitors operating in the strongly coupled regime. We bias the TLSs in the capacitors using an electric field. This allows us to spectroscopically image TLSs and extract their densities and dipole moments. When an in-situ alternating bias is applied, the steady-state spectra from the standard TLS model disappear. Post-alternating bias TLS spectroscopy reveals transient behavior, in which the TLS frequency fluctuates on the order of minutes. Thermal cycling above 10 K reverses these effects, restoring the TLS spectrum to its original state, indicating a reversible mechanism. Importantly, the intrinsic loss tangent of the LC oscillator remains unchanged before and after the application of the alternating bias. We propose that the disappearance of the steady-state spectrum are caused by non-equilibrium energy build up from strain in the oxide film introduced by the pulsed voltage bias sequence. Understanding this non-equilibrium energy could inform future models of time-dependent TLS dynamics.

Figures (9)

• Figure 1: (a) False-colored optical microscope image of the LC oscillator consisting of a Si substrate (orange), TiN (blue), Al$_2$O$_3$ (purple), and Al (yellow) layers. A voltage bias is applied along the thin, vertical Al channel. The TiN inductor is connected to four capacitive elements. The black box depicts two capacitors connected in a bias bridge. When biased, the capacitors have a constant dc field, E$_g$=$V_g$/2$d_o$, where $d_o$ = 49 nm is the dielectric thickness. (b) A circuit diagram for the lumped-element resonator. The color coding depicts the same galvanic connections as in (a). (c) An SEM image of the two capacitors highlighted in (a), acquired on a different oscillator of the same design. (d) A bright field-STEM image of the trilayer TiN-$\rm{Al}_{2}\rm{O}_{3}$-Al capacitor cross-section, capped with a Pt protective layer.
• Figure 2: (a) Transmission (S$_{21}$) of a oscillator interacting with a TLS at $V_g$ = 9.4 mV. The width of the TLS-induced avoided crossing is highlighted with black dashed lines, which is the outcome of the oscillator's coupling to the TLS. The TLS-oscillator coupling strength, $g/2\pi$, is calculated to be 1.3 MHz which matches the width of the avoided-crossing. (b) Transmission (S$_{21}$) as a function of applied voltage ($V_g$) and oscillator frequency. The TLS-oscillator interactions appear as hyperbolic shapes following Eq. \ref{['eq1:TLS_energy']}. An example is shown with the white curve corresponding to an extracted dipole moment $p_z$ = 0.24 e$\rm{\AA}$. The individual Lorentzian spectrum shown in (a) is a line cut at the red dashed line. Data processing for this figure is described in Supplemental Material III.
• Figure 3: Extracted dipole moments for TLS spectra collected after different treatments. The histograms represent the following: (a) control (blue, initial measurement), (b) control-aged, (orange, after aging 5 months), (c) 10 K temperature cycle after CABS (red), (d) 300 K temperature cycle after CABS (purple), and (e) after a 353 K alternating bias assisted annealing treatment (brown). We don't present any data on electric dipole moments directly after CABS because there were no TLS-induced hyperbolas found in this scan. (f) The extracted TLS densities for the different processing steps. $^*$(Calculated value of TLS density from the main paper after CABS treatment.)
• Figure 4: (a) The transmission (S$_{21}$) versus applied voltage ($V_g$) and oscillator frequency spectrum after the CABS treatment reveals sharp features during the voltage scan. Post-CABS treatment reveals no TLS-induced hyperbolas. Instead, the TLS spectrum appears to exhibit frequency shifts, with transient avoided crossings lasting less than 5 minutes. Data processing for this figure is described in Supplemental Material III. (b-e) show individual tranmission spectra at the points indicated in (a). Avoided crossing with a width of approximately 1 MHz can be observed.
• Figure 5: Measured transmission spectrum (S$_{21}$) after thermal cycling to 10 K. Hyperbolas can be seen in this spectrum. Data processing for this figure is described in Supplemental Material III.