Spectroscopy and Coherent Control of Two-Level System Defect Ensembles Using a Broadband 3D Waveguide

TL;DR

The paper develops and deploys Broadband Cryogenic Transient Dielectric Spectroscopy (BCTDS) in a broadband $3-6~\mathrm{GHz}$ 3D waveguide to probe coherent dynamics of TLS defect ensembles across materials and temperatures. By driving TLS baths and performing time-resolved, phase-coherent homodyne readout, it reveals distinctive phase-$V$ spectral signatures, Floquet-dressed eigenmodes, and memory effects, enabling direct measurement of the defect spectral density (e.g., $84~\mathrm{GHz}^{-1}$ over $4.1-4.6~\mathrm{GHz}$ in silicon). The work combines experiment, Floquet theory, and perturbative analytics to show how amplitude, phase, and inter-pulse spacing control bath dynamics, with implications for mitigating decoherence and engineering many-body effects in disordered quantum systems. Practically, BCTDS provides a versatile, device-agnostic diagnostic platform for defect-rich materials, offering pathways to optimize superconducting circuits and related quantum technologies with broadband, non-destructive characterization and control of TLS baths.

Abstract

Defects in solid-state materials play a central role in determining coherence, stability, and performance in quantum technologies. Although narrowband techniques can probe specific resonances with high precision, a broadband spectroscopic approach captures the full spectrum of defect properties and dynamics. Two-level system (TLS) defects in amorphous dielectrics are a particularly important example because they are major sources of decoherence and energy loss in superconducting quantum devices. However, accessing and characterizing their collective dynamics remains far more challenging than probing individual TLS defects. Building on our previously developed Broadband Cryogenic Transient Dielectric Spectroscopy (BCTDS) technique, we study the coherent control and time-resolved dynamics of TLS defect ensembles over a wide frequency range of 3-5 GHz without requiring full device fabrication, revealing quantum interference effects, memory-dependent dynamics, and dressed-state evolution within the TLS defect bath. The spectral response reveals distinct V-shaped structures corresponding to the bare eigenmode frequencies. Using these features, we extract a TLS defect spectral density of 84 GHz^-1 for a silicon sample, across a 4.1-4.6 GHz span. Furthermore, we systematically investigate amplitude- and phase-controlled interference fringes for multiple temperatures and inter-pulse delays, providing direct evidence of coherent dynamics and control. A driven minimal spin model with dipole-dipole interactions that qualitatively capture the observed behavior is presented. Our results establish BCTDS as a versatile platform for broadband defect spectroscopy, offering new capabilities for diagnosing and mitigating sources of decoherence, engineering many-body dynamics, and exploring non-equilibrium phenomena in disordered quantum systems.

Figures (16)

• Figure 1: BCTDS measurement setup and homodyne detection of polychromatic emission. a, Schematic of the broadband waveguide setup, with samples (cyan) mounted in the middle of the waveguide. b, Conceptual illustration of TLS defects in a sample, represented as a spin bath containing various defects at their respective resonance frequencies (color). Under a pulsed monochromatic drive, these defects become dressed and subsequently emit polychromatic radiation with frequency components described by Floquet-like sidebands and quasi-energies. c, Simple example of homodyne detection for a single emitter at angular frequency $\omega_1$. Mixing the signal with a local oscillator (LO) at $\omega_1$ removes the fast carrier oscillation, producing the in-phase ($I$) and quadrature ($Q$) signals from which we extract the amplitude and phase of the homodyne signal. In the case of monochromatic emission, the amplitude and phase remain constant when the local oscillator frequency matches the emitted signal. d, Homodyne detection of a superposed signal from three independent emitters at angular frequencies $\omega_1$, $\omega_2$, and $\omega_3$. Homodyne detection at $\omega_1$ results in an oscillating amplitude, and the linear phase evolution results in a characteristic sawtooth pattern due to the phase wrapping. The amplitude and phase carry information about all of the emitted frequencies that we can reconstruct through a Fast Fourier Transform (FFT).
• Figure 2: BCTDS result for a silicon sample with silicon oxide under different drive durations. a, Logarithmic amplitude of the homodyne signal arising from the transient dielectric response. b, FFT of the phase of the transient dielectric response, where V-shaped structures centered at the bare eigenfrequencies of TLS defect ensembles are clearly visible. Panels a,b are shown for three different pulse durations: 20 ns (i), 50 ns (ii), and 200 ns (iii). In aiii, we highlight a representative braiding pattern (white box) and single emission pattern (magenta box) resulting from emissions of different detunings. The drive amplitudes across the span are calibrated using a method described in BCTDS Appendix E, to ensure constant amplitude driving across all frequencies. c, Zero-time transient response at different pulse durations. Horizontal slices (lime dashed lines) from the logarithmic amplitude plots (a) are taken at $t = t_{\text{off}} =0$ for a continuous sweep of pulse durations from 5 to 200 ns, showing sharpening interference patterns as the pulse duration increases. d, Base location of phase Vs at different pulse durations. Horizontal slices at 0 MHz (cyan dashed lines) from the phase FFT plots (b) are taken for the 5-200 ns duration sweep. Bright lines marking the eigenfrequencies of the driven system start to emerge, and their positions remain constant as the pulse duration increases. e, Zero-frequency slices of phase FFT magnitude, averaged over different pulse durations. Prominent peaks correspond to constant phase V base locations at various pulse durations, while incoherent fluctuations average out. Counting all peaks with magnitude $>$120 a.u. (marked by gray triangles) yields a quantitative estimate of the TLS defect spectral density of $84$ GHz$^{-1}$, across the 4.1-4.6 GHz span.
• Figure 3: Coherent control of TLS defect ensembles in different samples using pulse amplitude and spacing. a, Sapphire sample with a thin 2 nm AlOx deposit via Atomic Layer Deposition (ALD). b, Silicon samples with a native oxide layer (same sample as Fig. \ref{['fig:phase_V']}). Sub-panel i shows the single pulse BCTDS spectroscopy result for the two samples over a 3-5 GHz range. Sub-panels ii-iv show the response under two pulses with different spacings: 10 ns (ii), 30 ns (iii), 50 ns (iv). We show the responses at a particular frequency (3.657 GHz for the AlOx sample (a), and 4.254 GHz for the silicon oxide sample (b)), selected due to their long ring-downs. We sweep the amplitude of the first pulse from 0 to 30000 arbitrary units, keeping the amplitude of the second pulse fixed at 10000. All drive amplitudes are calibrated (See Ref. BCTDS Appendix E), and the phase of both pulses is fixed. The result is coherent control of the non-Markovian collapse and revival responses from the ensemble of TLS defects. This behavior is observed consistently across both types of samples.
• Figure 4: Coherent control of TLS defect ensembles in different samples using pulse phase and spacing, setup is the same as Fig. \ref{['fig:spacing_amp']}. a, Sapphire sample with a thin 2 nm AlOx deposit via ALD. b, Silicon samples with a native oxide layer. Sub-panels i-iii show the response at a frequency cut (3.657 GHz for the AlOx sample (a), and 4.254 GHz for the silicon oxide sample (b)) under two pulses with different spacing: 10 ns (i), 30 ns (ii), and 50 ns (iii). The magnitudes of the two pulses are kept unchanged for both samples, with the first pulse being 3 times stronger than the second, and all drive amplitudes are calibrated (see Ref. BCTDS Appendix E). We sweep the phase of the first pulse from 0 to 360 degrees, keeping the phase of the second pulse fixed, resulting in coherent control of the non-Markovian responses from the ensemble of TLS defects, similar to Fig. \ref{['fig:spacing_amp']}. This behavior is observed consistently across both types of samples.
• Figure 5: Systematic study of the memory effect as a function of inter-pulse delay, while varying the phase of the first pulse relative to the second. We perform phase coherent control on sapphire samples with spin-coated Shipley 1813 photoresist at 3.4 GHz (similar to Fig. \ref{['fig:spacing_phase']}), sweeping the inter-pulse delay from 10 to 600 ns. For short delays, residual ring-down from the first pulse is still resolvable, indicating incomplete relaxation and a remaining memory that interferes with the second pulse. As the delay increases significantly beyond the extracted decay lifetime of the single pulse ring-down ($\sim$100 ns), the interference patterns diminish and coherent control is lost.