Distinguishing Coherent and Thermal Photon Noise in a Circuit Quantum Electrodynamical System

In the cavity-QED architecture, photon number fluctuations from residual cavity photons cause qubit dephasing due to the ac Stark effect. These unwanted photons originate from a variety of sources, such as thermal radiation, leftover measurement photons, and cross talk. Using a capacitively shunted flux qubit coupled to a transmission line cavity, we demonstrate a method that identifies and distinguishes coherent and thermal photons based on noise-spectral reconstruction from time-domain spin-locking relaxometry. Using these measurements, we attribute the limiting dephasing source in our system to thermal photons rather than coherent photons. By improving the cryogenic attenuation on lines leading to the cavity, we successfully suppress residual thermal photons and achieve $T_1$-limited spin-echo decay time. The spin-locking noise-spectroscopy technique allows broad frequency access and readily applies to other qubit modalities for identifying general asymmetric nonclassical noise spectra.

Figure 1

Simplified measurement schematic and noise generation. (a) Device and measurement setup. A CSFQ is capacitively coupled to a $\lambda /2$ transmission line cavity. Besides qubit control (red) and readout (black) pulses, a weak, uniform, cavity drive (green) is applied to create a coherent state. To mimic thermal photon noise, we amplitude modulate this drive with a broadband white noise (blue). (b) Microwave implementation of the experiment. The qubit control sequence executes the spin-locking protocol, during which the photon noise generation signal is kept on. A strong readout pulse follows for dispersively measuring the qubit state. (c) Schematic diagram of coherent or thermal photon noise generation. The black curve indicates the cavity response. A single-frequency drive creates coherent photons in the cavity. A broadband noise (bandwidth $\gg \kappa$) mimics the thermal photon noise [32]. (d) Schematic of coherent or thermal photon noise spectrum. Coherent photons produce a noise spectrum symmetric with respect to $\omega =-\mathrm{\Delta }$ and generally lead to unequal emission and absorption spectral densities (red bars) and, hence, unequal relaxation and excitation rates for the dressed qubit (inset). Thermal photons produce a symmetric spectrum, and the width is twice that of coherent photons.

Figure 2

$T_{1\rho }$ decay characterization. (a) $T_{1\rho }$ decay with coherent photons. The selected traces are measured at cavity drive detuning $\mathrm{\Delta }/2\pi =+1$, 0, $-1\mathrm{MHz}$, respectively, and with the same cavity and qubit drive power ($\mathrm{\Omega }/2\pi =1\mathrm{MHz}$). The solid lines are exponential fits. (b) Symmetrized noise PSD with coherent photons vs. $\mathrm{\Omega }$ and $\mathrm{\Delta }$, derived from the fitted decay constants by using Eq. (6). The markers denote the cases shown in (a). (c) Steady-state polarization with coherent photons vs $\mathrm{\Omega }$ and $\mathrm{\Delta }$, extracted directly from the fit. The color scale, blue (red), indicates that the dressed qubit predominantly emits (absorbs) energy.

Figure 3

Coherent and thermal photon noise spectra. (a) Two-sided (nonclassical) PSD $\mathcal{S}(\omega ,\mathrm{\Delta })$ with coherent photons at various $\mathrm{\Delta }$, reconstructed from data shown in Figs. 2 and 2. (b) Example slices of reconstructed $\mathcal{S}(\omega )$. Plotted are three typical cases with coherent drive at $\mathrm{\Delta }/2\pi =+1$, 0, $-1\mathrm{MHz}$ and one thermal case which has no $\mathrm{\Delta }$ dependence. The solid lines are Lorentzian fits. (c) Extracted center frequencies in both coherent and thermal cases. (d) Extracted HWHMs. The mean value of HWHMs with coherent photons is 0.83 MHz. The HWHM with thermal photons is 1.61 MHz. The green lines in (c) and (d) are included as a guide to the eye, since the thermal photons are generated with broadband noise and have no $\mathrm{\Delta }$ dependence.

Figure 4

Decoherence mitigation. (a) Photon noise spectra with the second device. Plotted are spectra with injected coherent photons (black), injected thermal photons (green), and no injected photons (blue). Solid lines are a Lorentzian fit. Arrows indicate the peak width spanned by the corresponding $-3\mathrm{dB}$ points (dashed lines) for the thermal and ambient cases ($\mathrm{HWHM}\approx 2.0\mathrm{MHz}$) and for the coherent case ($\mathrm{HWHM}\approx 1.1\mathrm{MHz}$). Because the injected noise disrupts the spin-locking condition when the locking drive is weak, the measured PSDs at the low-frequency end ($<0.4\mathrm{MHz}$) are consistently lower and, hence, excluded during fitting. (b) Histogram of measured $T_{2\text{Echo}}$ statistics before and after adding extra attenuation on the input side of the cavity. In both cases, the fluctuation mainly comes from temporal $T_1$ instability due to quasiparticles [37]. The gray area indicates the spread of the $2T_1$ limit measured with $-43\mathrm{dB}$ attenuation.