Non-Markovian bath spectroscopy of discrete two-level systems in a fluxonium qubit

Recent studies have shown that qubit environment, especially TLS bath, can be very long-lived. Thus, measuring qubit relaxation dynamics without knowledge of environment can be problematic, especially when doing T₁ spectroscopy studies in a dense TLS bath. Here we introduce two-time-scale relaxometry, a technique to probe the qubit and environment relaxation simultaneously and efficiently. In addition to 'delay time' in regular T₁ measurement, we further consider the concept of 'environment response time', which records the time scale of qubit decay rate changing while TLS is polarized. The polarization effect is achieved by repeated qubit initialization. We apply this technique to a fluxonium in a frequency range of 200-400 MHz, and reveals the discrete spectrum of long-lived TLS, on top of a background qubit T₁ of around 2ms. Our spectrum shows that TLS in this frequency range has a density of 0.4-0.6/GHz/μm², and their projected charge dipole of ~1eÅ, which are similar to previous TLS studies in several GHz range. This technique should overtake traditional qubit T₁ measurement when qubit is potentially affected by complicated TLS environment. Studying and reducing TLS environment in the junction chain is crucial to reach expected long coherence time in fluxonium and use it to build large-scale quantum processor.