Measurement-Induced Qubit State Mixing in Circuit QED from Up-Converted Dephasing Noise

We observe measurement-induced qubit state mixing in a transmon qubit dispersively coupled to a planar readout cavity. Our results indicate that dephasing noise at the qubit-readout detuning frequency is up-converted by readout photons to cause spurious qubit state transitions, thus limiting the nondemolition character of the readout. Furthermore, we use the qubit transition rate as a tool to extract an equivalent flux noise spectral density at $f\sim 1\mathrm{GHz}$ and find agreement with values extrapolated from a $1/f^{\alpha }$ fit to the measured flux noise spectral density below 1 Hz.

Figure 1

Experimental setup and readout trace. The transmon qubit is coupled to an asymmetric microwave readout cavity. Noise or coherent tones can be injected into the qubit loop via a weakly coupled fast flux line with 2.2 GHz bandwidth (gray dashed lines indicate flux coupling). The readout signal is amplified by a superconducting paramp, enabling continuous high-fidelity monitoring of the qubit state. The inset shows a sample output trace (solid red line) and the corresponding extracted qubit state (dotted black line).

Figure 2

Spurious excitation with coherent fast flux tone. The steady-state qubit population during measurement with a coherent microwave tone applied to the fast flux line is shown as a function of ${\omega }_{\mathrm{ff}}$ and $\overline{n}$. The panels correspond to rms added fluxes of $300\mu {\Phi }_0$ (a) and $550\mu {\Phi }_0$ (b). The black x’s denote the values of ${\Delta }_{\mathrm{ro}}(\overline{n})$ extracted independently from qubit spectroscopy. Regions with no data (at the lowest and highest frequencies) are shown in uniform dark blue. Panel (c) shows the qubit population versus ${\omega }_{\mathrm{ff}}$ for $\overline{n}=6$ and an rms added flux of $420\mu {\Phi }_0$ (black dotted line and diamonds), along with the corresponding qubit spectroscopy trace (red solid line). The horizontal axes are aligned such that the detuning between readout and spectroscopy frequencies is the same as the fast flux frequency. Panel (d) shows the center (solid line) and widths (dotted lines) of the qubit excited state population response for the data in (b). The red (bold) traces are experimental data, while the blue (narrow) traces are from numerical simulations.

Figure 3

Qubit population with added noise. Part (a) shows the steady-state qubit population as a function of added flux noise at $|{\Delta }_{\mathrm{ro}}(\overline{n})|$ and the number of measurement photons $\overline{n}$ in the readout cavity. Part (b) shows the qubit population immediately after the readout has fully energized. A thermal population of 1.4% is visible, and is equivalent for all values of added flux noise and $\overline{n}$.

Figure 4

Spectral density of flux noise versus frequency. We plot equivalent flux noise extracted from Ramsey fringes (thick black line), Rabi oscillation decay (red circles), and our measurements of spurious excitation (blue squares). The thin red line is a fit to the Ramsey fringe data using a $1/f^{\alpha }$ power law, with the gray shaded area representing the 95% confidence interval on the fitted value of $\alpha =0.57$. The inset view shows a detail of the Ramsey fringe data and fit line.