Non-Markovian Effects of Two-Level Systems in a Niobium Coaxial Resonator with a Single-Photon Lifetime of 10 milliseconds

Understanding and mitigating loss channels due to two-level systems (TLS) is one of the main cornerstones in the quest of realizing long photon lifetimes in superconducting quantum circuits. Typically, the TLS to which a circuit couples are modeled as a large bath without any coherence. Here we demonstrate that the coherence of TLS has to be considered to accurately describe the ring-down dynamics of a coaxial quarter-wave resonator with an internal quality factor of $0.5\times {10}^9$ at the single-photon level. The transient analysis reveals effective non-Markovian dynamics of the combined TLS and cavity system, which we can accurately fit by introducing a comprehensive TLS model. The fit returns an average coherence time of around $T_2^{\ast }\approx 0.3\mu \mathrm{s}$ for a total of $N\approx {10}^9$ TLS with power-law distributed coupling strengths. Despite the shortly coherent TLS excitations, we observe long-term effects on the cavity decay due to coherent elastic scattering between the resonator field and the TLS. Moreover, this model provides an accurate prediction of the internal quality factor’s temperature dependence.

Figure 1

High-$Q$ coaxial resonator: The key element of the particular resonator design is the exponential attenuation of the resonator mode’s fields along the circular waveguide section of length $L$. This is revealed by finite-element simulations where the electric field magnitude is plotted scaled to a peak value of $E_{max}$ on the tip of the quarter-wave stub. Experimental setup: The cavity is excited either by a cw signal (blue signal path) or a rectangular excitation pulse (red signal path). The input signal enters a cryogenic environment with temperatures down to $T=20\mathrm{mK}$ and is routed to the cavity through a coupling pin. The cavity reflected signal is then digitized and analyzed in either the frequency or time domain. Data acquisition: The distinctive reflected power response in the time domain of an overcoupled resonator to a resonant rectangular drive pulse is depicted on the screen of the spectrum analyzer. The magnified plot displays additionally the rectangular drive pulse, the cavity ring-up fit function, the pure exponential decay, and the reflected power levels directly after switching on the cavity drive and after reaching the steady state (yellow marks). The frequency-domain equivalent is shown on the monitor of the vector network analyzer with phase and magnitude of the reflection scattering parameter near the resonance. Both domains allow the extraction of all characteristic parameters.

Figure 2

(a) Internal quality factor obtained by steady-state (black), ring-up (mint), and ring-down (purple to orange) measurements as a function of average cavity population. Brighter colors for ring-down traces indicate higher saturation pulse powers, see (d). For readability, we only plot three of the ring-up traces for various pulse powers with the same color. Lower cavity population leads to a lower quality factor. The ring-down measurements clearly differ from the ring-up and VNA measurements. Solid lines show the expected ring-down behavior predicted by the TLS model. (b) Time-dependent cavity linewidth $\kappa$ for the exponential decay of the cavity field after the drive pulse is switched off. Colors are chosen to match (a),(d). Dots are a rolling average of 50 data points with error bars indicating the standard deviation of the mean ($1\sigma$). The total cavity linewidth is rising as power decays. The prediction of a simultaneous fit of the ten time traces is plotted by solid lines. (c) Sampled TLS distribution for parameters returned by the fit. Only six TLS classes with coupling strengths up to $110\mathrm{Hz}$ have a TLS number larger than 1. (d) Total number of TLS returned by the fit as a function of pulse power applied to the cavity port. Colors are chosen to indicate the powers in (a),(b). The blue line indicates the scaling expected from the linewidth change of the cavity while assuming a constant spectral density of TLS. The error bars indicate 1 standard deviation.

Figure 3

(a) Normalized frequency shift as a function of the cryostat mixing chamber temperature. Increasing temperature increases the kinetic inductance, which translates to a decreased resonance frequency of the cavity. (b) Internal quality factor as a function of temperature. TLS saturation leads to an increase in quality factor for increasing temperature. A plateau arises between 1 and 2 K due to a temperature-independent dephasing rate of the TLS. For high temperatures, we observe an exponential decrease due to quasiparticle excitations. Horizontal error bars indicate the estimated uncertainty for temperature stabilization and vertical error bars the uncertainty returned by the circle-fit routine in both figures. Vertical error bars are smaller than point size.

Figure 4

Population and coherences for TLS classes with different coupling strength during the exponential decay for highest used pulse power. Too weakly coupled TLS ($g\le {10}^{-1}$) do not get populated by the pulse at all. The time-domain fit from the main part revealed that there are no TLS with coupling strengths larger than ${10}^2$. During the slow power decay in the cavity, each TLS class has a characteristic time to stop being saturated and build up coherences during the decay.

Figure 5

Circle fit for single-photon power level (data, blue; fit, red) and for ${10}^{10}$ photons (data, green; fit, orange) VNA measurements. The top two plots show magnitude and phase of the reflected signal as a function of frequency. The bottom plot shows the imaginary and real plane of the complex scattering parameters, together with the fitted circle. An increase in radius indicates increasing internal quality factor, if the coupling stays the same. The internal quality factor rises here from $5.3\times {10}^8$ to $9.4\times {10}^8$ for a 100-dB increase in power.

Figure 6

Schematic cavity response to a resonant rectangular pulse $P_f$ in the under-, critically, and overcoupled regime. $P_r$ is the reflected power signal and $V_r$ the reflected wave amplitude. Adapted from [21, Figure 8.5].

Figure 7

Transient cavity behavior in the overcoupled regime after switching on a resonant (left) or slightly detuned (right) cavity drive. $P_r$ describes the cavity’s reflected power, and $\delta$ the detuning between the cavity drive and the resonance frequency. Except for $\delta$, all other parameters are identical in both cases.

Figure 8

Cavity reflected power trace in the time domain (blue) responding to a resonant rectangular drive pulse (green). Moreover, the cavity ring-up fit function (red dashed line) and a pure exponential decay (cyan dashed line) are shown. The decay constant, combined with the reflected power levels directly after switching on the cavity drive and after reaching a steady state (yellow marks), as well as the ring-up fit lead to all relevant resonator parameters. Additionally, the ring-up fit detects a cavity drive detuning as the minimum of the transient response gets lifted from the noise floor.

Figure 9

Power dependence of coupling and internal quality factors obtained from steady-state and ring-up measurements at different cavity drive strengths in units of photons $⟨n_{\mathrm{ph}}⟩$. We fit a full circle to VNA scattering parameter data in the complex plane to extract all relevant resonator parameters in the frequency domain. In the time domain, the distinctive reflected power ring-up trace is fitted with an equivalent, analytically derived model. As we fit one curve to the whole ring up, the quality factor results can be interpreted as averaged values, linked to the maximum of the inner cavity circulating photon number after reaching a steady state.

Figure 10

$Q_{\mathrm{int}}$ power dependence of the aged niobium coaxial quarter-wave resonator as a function of power in the cavity in units of photons $⟨n_{\mathrm{ph}}⟩$. Black dots are VNA measurements, and colored dots show ring-down measurements for low (red) to high (orange) pulse powers.