Probing the memory of a superconducting qubit environment

Abstract

Achieving fault tolerance with superconducting quantum processors requires qubits to operate within the regime of threshold theorems based on the Born-Markov approximation. This approximation, which models dissipation as constant energy decay into a memoryless environment, breaks down when qubits couple to long-lived two-level systems (TLSs) that become polarized during operation and retain memory of past qubit states. Here, we show that non-Poissonian quantum jump traces carry the information required to distinguish long-lived TLSs from the standard Markovian bath. By fitting the Solomon equations to measured quantum jumps dynamics arising naturally due to thermal fluctuations, we can disentangle the coupling of the qubit to the two environments. Sweeping the qubit frequency reveals distinct peaks, each associated with a TLS that outlives the qubit, providing a handle to understand their microscopic origin.

Figures (10)

• Figure 1: Non-Markovian qubit dynamics induced by coupling to a long-lived TLS.(a) Schematics of the coupling of the qubit to both a TLS and a thermal bath. The coupling rates of the qubit to the TLS and the thermal bath are $\mathrm{\Gamma_{qt}}$ and $\mathrm{\Gamma_{q}}$, respectively. When the coupling rate of the TLS to the thermal bath $\mathrm{\Gamma_{t}}$ is smaller or comparable to $\mathrm{\Gamma_{qt}}$, the TLS becomes polarizable and we refer to it as a long-lived TLS. (b) Quantum jumps correlations from Solomon equation based simulation (dashed line) and from measured data (orange). As expected for a single photon emitter within the Born-Markov approximation (black dashed line), the $\mathrm{g_2}$ correlation function (see Eq. \ref{['equ:g2']}) shows anti-bunching of quantum jumps at short time scales and tends exponentially to uncorrelated jumps. In contrast, in the presence of a long-lived TLS, the correlation function shows quantum jump bunching persisting longer than the $T\mathrm{_1}$ of the qubit (orange). (c) IQ quadrature of 5000 single-shot readouts of the fluxonium qubit, plotted in the complex plane. The radius of the circles indicate 2 standard deviations for the distribution of the points. (d) Representative and cropped sample of the I quadrature over a 3 ms time trace, with the corresponding extracted state indicated by the continuous line. Orange shaded areas highlight three examples of 0.5 ms long time traces, post selected after an energy relaxation event and contributing to the trace averaging. (e,f) Qubit dynamics shown as averaged trajectories conditioned on ground-state selection (green) and jump-down events (orange). The data are well described by fits to the single-TLS Solomon model (eq. \ref{['equ:Solomon']}). (f) Zoom-in and re-scaling on the averaged dynamics after the ground state selection scenario, showing deviations from the expected exponential decay.
• Figure 2: Unraveling of the hidden TLS dynamics. Qubit (a) and hidden TLS (b) populations for the qubit at $\mathrm{\Delta}t$ = 0 in the ground state (green) or immediately after a jump down event (orange). The filled markers show measurements. Crosses show populations calculated from quantum jumps simulations, using the $\mathrm{\Gamma_{qt}}$, $\mathrm{\Gamma_{q}}$ and $\mathrm{\Gamma_{t}}$ extracted from the Solomon equations fit of the measured $\mathrm{P_q}$ for $\mathrm{\Delta t}$$>$ 0. The dashed line shows populations from quantum jumps simulations assuming a Born-Markov environment.
• Figure 3: Electric field susceptibility of the long-lived TLSs. (a) Single electric field correlation spectroscopy vs qubit frequency at 0 V/m. The Markovian environment of the qubit is nearly featureless, as shown by the qubit relaxation rate $\Gamma_q$ (b), while the qubit-TLS decay rate $\Gamma_{qt}$ shows distinct peaks susceptible to electric field bias (c). Insets show the qubit dynamics at the same frequency $f_q = 2.045GHz$ preformed under two different electric fields, corresponding to a mostly Markovian environment ($\circ$) and a long-lived TLS in resonance ($\diamond$), respectively. To aid the discussion in the main text the long-lived TLSs are labeled A-H.
• Figure 4: Granular aluminum fluxonium qubit. (a) False-colored optical microscope picture of the qubit sample. The stack is composed of two aluminum layers (blue) and one granular aluminum layer (red). (b) Equivalent circuit of the fluxonium qubit, capacitively coupled to an aluminum lumped-element LC resonator for readout. The resonator is measured in reflection via a circulator, and the signal is amplified by a low-noise Josephson Parametric Amplifier (JPA) winkel_nondegenerate_2020 (c) Qubit spectrum versus external flux $\Phi_\mathrm{ext}$. Black dots denote the measured $0-1$ transition frequencies, solid line depicts the fit to the fluxonium Hamiltonian.
• Figure 5: Fitting with more than one TLS (a) and (b) show a fit using the Solomon equations from the main text. (c) and (d) show a fit using 2 TLSs coupled to the qubit.