Quantifying dynamics and interactions of individual spurious low-energy fluctuators in superconducting circuits

Understanding the nature and dynamics of material defects in superconducting circuits is of paramount importance for improving qubit coherence and parameter stability and much needed for implementing large-scale quantum computing. Here we present measurements on individual highly coherent environmental two-level systems (TLS). We trace the spectral diffusion of specific TLS and demonstrate that it originates from the TLS coupling to a small number of low energy incoherent fluctuators. From the analysis of these fluctuations, we access the relevant parameters of low energy fluctuators: dipole moments, switching energies, and, more importantly, interaction energies. Our approach opens up the possibility of deducing the macroscopic observables in amorphous glassy media from direct measurements of local fluctuator dynamics at the microscopic level—a route towards substantiating commonly accepted, but so far phenomenological, models for the decohering environment.

Figure 1

Planar superconducting resonator as a sensor of fluctuating environmental defects. (a) Sketch of the sample setup: Tuning resonator frequency (via the applied current $I_{\mathrm{b}}$) and external electrostatic gate $V_{\mathrm{g}}$ allow us to select individual coherent TLS for study. The resonator is read out by measuring the transmission $S_{21}$ between ports 1 and 2. (b) The TLS (blue) couples to a few neighboring incoherent TLFs (red) within the TLS interaction volume (blue circles). The random switching of TLFs causes the telegraphlike fluctuation of TLS excitation energy. The larger bath of background TLFs (gray) affects the TLS linewidth. The TLS couple to the resonator via the microwave electric field $E_{\mathrm{RF}}$, and the DC electric field $E_{\mathrm{DC}}$ arising from the applied gate voltage $V_{\mathrm{g}}$ allows us to select specific TLS to be in resonance with the resonator.

Figure 2

Spectral and electric field mapping of TLSs. The measured variation of the quality factor $\delta Q_i=(Q_i-\langle Q_i\rangle )/\langle Q_i\rangle$ of a 4.86 GHz resonator as a function of resonator frequency and applied gate voltage ($V_{\mathrm{g}}$). Numerous individual TLS trace out the hyperbolas associated with additional loss (lower Q). Data is taken at 10 mK and average photon number $\sim 10.$

Figure 3

(a) The time-averaged resonator frequency around its resonance with a strongly coupled TLS tuned by the gate. (b) Temporal fluctuations of the resonance frequency when resonator and TLS are tuned in (red) and off (black) resonance. Corresponding gate voltages are indicated by the dashed lines in (a). The measurements performed at $T=30\mathrm{mK}$. (c) Histograms of the two time traces in (b). (d) Sketch of the corresponding microscopic TLF-TLS configuration discussed in the text. $g_{\mathrm{FF}}$ and $g_{\mathrm{SF}}$ show the couplings between the different entities. (e) The fitted TLF energy scales for a range of temperatures.

Figure 4

(a) Schematic (not to scale) of the sample and its enclosure outlining the location of the gate electrode and ground potentials with respect to the sample. (b) Simulated electric field strength (for 1 V applied to the gate) along the two surfaces indicated by the red and blue contours in (a). (c) Normalized histogram showing the probability of finding a TLS, randomly placed at the MS or MA interfaces, at a point with a certain electric field strength.

Figure 5

(a) Assuming the TLS is located at either the MA or SA interface we can from the fitted hyperbola and the $E$-field distribution obtain a probability distribution for the dipole moment of the TLS. (b) The probability distribution of the TLF-TLS separation $r_0$ assuming the rotation angle $\varphi =0$. A larger $\varphi$ would shift the distribution towards smaller $r_0$. (c) Measured distribution of TLS voltage coupling coefficients $\gamma$ obtained by fitting a large number of hyperbolas observed across a number of cooldowns. Only hyperbolas that are well defined (e.g., by the TLS minimum energy within or near the measured frequency window) are included in the statistics. The solid lines show the expected distribution obtained from the surface $E$-field distribution and assuming a uniform angular distribution of TLS orientations.

Figure 6

Distribution of resonator frequency for the coupled resonator-TLS-TLF system together with Gaussian peak fits at different temperatures.