Abstract
We experimentally investigate a superconducting qubit coupled to the end of an open transmission line, in a regime where the qubit decay rates to the transmission line and to its own environment are comparable. We perform measurements of coherent and incoherent scattering, on and offresonant fluorescence, and timeresolved dynamics to determine the decay and decoherence rates of the qubit. In particular, these measurements let us discriminate between nonradiative decay and pure dephasing. We combine and contrast results across all methods and find consistent values for the extracted rates. The results show that the pure dephasing rate is one order of magnitude smaller than the nonradiative decay rate for our qubit. Our results indicate a pathway to benchmark decoherence rates of superconducting qubits in a resonatorfree setting.
Introduction
Superconducting circuits are promising building blocks for implementing quantum computers^{1,2,3}. In those devices, the key elements are superconducting artificial atoms made by Josephson junctions which induce a strong and engineerable nonlinearity. Such artificial atoms are also used in the field of superconducting waveguide quantum electrodynamics (waveguide QED)^{4,5}, where they interact with a continuum of light modes in a 1D waveguide. In the past decade, many quantum effects from atomic physics and quantum optics have been demonstrated in waveguide QED, e.g., the Mollow triplet^{6}, giant crossKerr effect^{7}, and cooperative effects^{5,8,9}. Other recent experiments have shown phenomena which are currently beyond the reach of atomic physics, such as ultrastrong^{10,11} and superstrong coupling^{12} between light and matter. Waveguide QED is also an enabling quantum technology. One of the key applications is to generate^{13,14,15,16,17,18,19} and detect^{4,20,21,22,23,24,25,26} single photons. It has been proposed to use waveguide QED to create bound states^{27,28,29} and implement quantum computers^{30,31,32}.
The performance of quantum computers and waveguideQED devices is often limited by the coherence of the Josephson circuits. For example, the efficiency of producing and detecting single photons, the lifetime of bound states, and the fidelity of logical gates can all be improved by enhancing the coherence. In a waveguideQED setup, decoherence can be due to decay into the waveguide, pure dephasing, and nonradiative decay into other modes. However, the rates for pure dephasing and nonradiative decay are typically not explored separately. An understanding of which one is dominant will give an insight into the decoherence mechanisms, and thus how the device performance can be improved.
In this work, we probe a superconducting transmon qubit coupled directly to the end of an open transmission line. In previous realizations^{6,7,8,33,34,35}, the coupling rates were much larger than intrinsic decoherence mechanisms of the qubit, so the effects of nonradiative decay and pure dephasing were small and could not be well characterized. Here, we investigate a qubit whose radiative decay rate into the transmission line is larger than, yet comparable to, other decoherence mechanisms. This allows us to explore the pure dephasing rate Γ_{ϕ}, the radiative decay rate Γ_{r} from the capacitive coupling to the waveguide, and the nonradiative decay rate Γ_{n}. The total relaxation and decoherence rates are given by Γ_{1} = Γ_{r} + Γ_{n} and Γ_{2} = Γ_{1}/2 + Γ_{ϕ}, respectively. We demonstrate different methods to extract the different rates and find consistent results. Similar to the results in circuit QED^{36,37,38,39,40}, our methods can also enable the evaluation of the decoherence of qubits over a broad range of frequencies. They provide a pathway to investigate Josephson junctions or superconducting quantum interference devices (SQUIDs) without any resonator. In addition, we also consider it important to study Γ_{n} and Γ_{ϕ} separately. For instance, this could help to improve the Purcell enhancement factor, \(\frac{{{{\Gamma }}}_{{\rm{r}}}}{{{{\Gamma }}}_{{\rm{n}}}+2{{{\Gamma }}}_{\phi }}\), in devices such as the one presented in ref. ^{9}. Moreover, the spontaneousemission factor β, which is customarily quoted in other waveguideQED platforms^{41,42,43,44} is also related to Γ_{n}, namely, \(\beta =\frac{{{{\Gamma }}}_{{\rm{r}}}}{{{{\Gamma }}}_{{\rm{r}}}+{{{\Gamma }}}_{{\rm{n}}}}\) in our case.
The paper is structured as follows: first, we characterize the coherent scattering of the device and obtain the radiative decay rate and the decoherence rate of the qubit as a reference for later measurements. Then, we exploit the fluorescence of the qubit under coherent excitation to find the nonradiative decay rate and the pure dephasing rate. The resonance fluorescence spectrum at strong driving develops into the Mollow triplet^{45}, which has been widely used to probe quantum properties in systems based on superconducting qubits such as coherence^{6,8} and vacuum squeezing^{46}. The resonance spectrum is symmetric around the central peak. However, if pure dephasing exists, the offresonant spectrum becomes asymmetric, something which has been studied experimentally in quantum dots^{47,48}. We take advantage of this fact to extract the pure dephasing rate. We also measure the nonradiative decay rate under a continuous coherent drive, where coherently and incoherently scattered photons provide information about the different decay channels. Afterward, we apply a pulse to the qubit to both obtain the decay rates and find the stability of the qubit frequency and coherence as a function of time. In contrast to other methods, we use the phase information of emitted photons from the qubit to investigate the qubitfrequency stability in superconducting waveguide QED. Finally, we summarize the measured results and compare the advantages and disadvantages of the different methods.
In this paper, we have investigated waveguide QED with a qubit weakly coupled to a transmission line which is an unexplored regime. In this regime, it becomes possible to quantify the nonradiative decay rate of the qubit. We use several different methods to extract the different decay rates, provide a thorough comparison of the different methods, and find consistent results. Among these methods, those based on the offresonant Mollow triplet and the power loss analysis are original. Moreover, we show that timedomain analysis of decay rates can also be done in waveguide QED.
Results
Device characterization
The device used in our experiment (see Fig. 1a) is a magneticfluxtunable Xmontype transmon qubit^{49}, capacitively coupled to the open end of a onedimensional transmission line with characteristic impedance Z_{0} ≃ 50 Ω. The circuit is equivalent to an atom in front of a mirror in 1D space. The device is fabricated from aluminum on a silicon substrate using the same fabrication recipe as in ref. ^{38}. We denote \(\left0\right\rangle \), \(\left1\right\rangle \) and \(\left2\right\rangle \) as the ground state, the first and the second excited states of the qubit, respectively. The \(\left0\right\rangle \left1\right\rangle \) transition energy is \(\hslash {\omega }_{01}\approx \sqrt{8{E}_{{\rm{J}}}({{\Phi }}){E}_{{\rm{C}}}}{E}_{{\rm{C}}}\), where E_{C} = e^{2}/(2C_{∑}) is the charging energy, e is the elementary charge, C_{∑} is the total capacitance of the qubit, and E_{J}(Φ) is the Josephson energy. The Josephson energy can be tuned from its maximum value \({E}_{{\rm{J}},\max }\) by an external magnetic flux Φ using a coil: \({E}_{{\rm{J}}}({{\Phi }})={E}_{{\rm{J}},\max } \cos (\pi {{\Phi }}/{{{\Phi }}}_{0}) \), where Φ_{0} = h/(2e) is the magnetic flux quantum.
Figure 1a illustrates the simplified experimental setup for measuring the reflection coefficient of a probe signal coming from a vector network analyzer (VNA), after interacting with the qubit. The probe signal at frequency ω_{pr} and a pump at frequency ω_{p} are combined and attenuated before being fed into the transmission line. Then, the VNA receives the reflected signal to determine the complex reflection coefficient.
Figure 1b shows the magnitude of the reflection coefficient, ∣r∣, for a weak probe (with an intensity Ω_{pr} < Γ_{2}) as a function of the external flux Φ. Afterward, all the other measurements are taken at the fluxsweet spot. We use twotone spectroscopy to determine the anharmonicity of the qubit, α = (ω_{12} − ω_{01})/\(2\pi\), where ω_{12} is the frequency of the \(\left1\right\rangle  \left2\right\rangle\) transition. Specifically, we apply a strong pump (with an intensity Ω_{p} ≫ Γ_{2}) at ω_{01} to saturate the \(\left0\right\rangle \left1\right\rangle \) transition and measure the reflection coefficient as a function of probe frequency. The result is shown in Fig. 1c: a dip appears in the reflection at ω_{pr} = ω_{12} due to the photon scattering from the \(\left1\right\rangle \left2\right\rangle \) transition. At a higher pump power, the Autler − Townes splitting is observed^{50}. From Fig. 1b, c, we find E_{C}/h ≈ α = 252 MHz, and then, \({E}_{{\rm{J}},\max }/ h=16.56\ {\rm{GHz}}\) by fitting the data in Fig. 1b to the equation between the qubit frequency and the external flux mentioned previously. In order to obtain the radiative decay and decoherence rates, we perform singletone spectroscopy with a weak probe (Ω_{pr} ≪ Γ_{2}). Figure 1d, e are the corresponding magnitude and phase response of r, where we obtain Γ_{r}/2π = 227 kHz and Γ_{2}/2π = 141 kHz by using the circle fit technique from ref. ^{51}. The corresponding fitting equation is
where the phase ϕ ≈ 0.150 ± 0.004 quantifies the impedance mismatches in the line (for a perfectly matched line, ϕ = 0).
Atomic fluorescence
Even though measuring the reflection coefficient can give the decoherence and radiative decay rates, it cannot distinguish between pure dephasing and nonradiative decay. In order to distinguish them, we study the atomic fluorescence for different pump intensities and frequencies. For this measurement, the VNA is turned off, the pump is used to drive the system, another 50 dB of attenuation is added between the directional coupler and the pump, and the output signal is sampled by a digitizer (see Fig. 1a). When the qubit is pumped, its state evolves at a Rabi frequency Ω. With a Rabi frequency much larger than the natural linewidth of the qubit (Ω ≫ Γ_{2}), the energy levels of the qubit become dressed, leading to three distinct spectral components known as the Mollow triplet^{45}. In particular, the spectrum contains the elastic “Rayleigh” line in the middle in which the scattered wave has the same frequency as the incident wave, with two inelastic sidebands positioned symmetrically on both sides of the central peak.
To measure the power spectrum for the atomic fluorescence, we use a digitizer to sample the amplified signal as V(t) in the time domain. The sample rate and the number of samples are from 2 MHz to 6 MHz and from 500 to 1000, respectively, for the different measured spectra in Figs. 2 and 3. After that, we calculate the power spectrum \({S}_{{\rm{i}}}^{{\rm{meas.}}}(\omega )\) according to the Welch method^{52}, after normalization to the system gain G. In order to increase the corresponding signaltonoise ratio (SNR), a number of averages are required to obtain \(\langle {S}_{{\rm{i}}}^{{\rm{meas.}}}(\omega )\rangle \). For different results shown below, the number of averages varies from 2 × 10^{6} to 10^{8}.
Onresonant Mollow triplet
When the qubit is driven by a resonant pump (Δ = ω_{p} − ω_{01} = 0), as shown in Fig. 2a, the splitting of the spectrum between the sidebands and the central peak increases as a function of the pump power P_{p}. The splitting equals the Rabi frequency and obeys \({{\Omega }}=2\sqrt{A{{{\Gamma }}}_{{\rm{r}}}{P}_{{\rm{p}}}/(\hslash {\omega }_{01})}\). By fitting the extracted Rabi splitting ∣Ω∣ in Fig. 2b, and using Γ_{r} from the previous measurement in the section “Device characterization”, we extract a total attenuation A = −145 dB of which about −125 dB attenuation is from attenuators and directional couplers, −7 dB from an Eccosorb filter and the rest is due to cable loss. This allows us to renormalize all applied powers to either the power at the qubit, or the corresponding Rabi frequency. The total gain in the output line of the measurement setup can be calibrated by tuning the qubit away and measuring the power at the output port at room temperature. This results in a total gain G = 115 dB, of which approximately 44 dB gain comes from a high electron mobility transistor (HEMT) amplifier, and the rest is from the room temperature amplifiers and the preamplifiers of the digitizer.
The Rabi rate can be made much larger than all the decay rates of the qubit (Ω ≫ Γ_{1}, Γ_{2}). Consequently, the overlap in the frequency domain between the sideband emission and the central peak becomes negligible. In Fig. 2c, we use an input power to the qubit P_{q} ≈ −116 dBm, equivalent to Ω/2π ≈ 9 MHz. The incoherent part of the corresponding power spectral density (PSD) is given by
(see more details in Supplementary section A), where the half width at half maximum of the central peak and the sidebands are Γ_{2} and Γ_{s} = (Γ_{1} + Γ_{2})/2, respectively. The solid curves in Fig. 2c are fits to 2πS_{i}(ω) which is the PSD expressed in linear frequency. We obtain Γ_{2}/2π = 141 ± 2 kHz for the central peak, Γ_{s,red}/2π = 210 ± 3 kHz and Γ_{s,blue}/2π = 206 ± 4 kHz for sidebands. By taking the average of Γ_{s,red} and Γ_{s,blue}, we obtain Γ_{1}/2π = 275 ± 7 kHz. We note that the extracted Γ_{2}/2π value is fully consistent with the result from the reflectioncoefficient measurement in the section “Device characterization”. From that measurement, we also know Γ_{r}/2π = 227 ± 1 kHz. Thus, we can now extract both the nonradiative decay rate, Γ_{n}/2π = 48 ± 7 kHz, and the pure dephasing rate, Γ_{ϕ}/2π = 3 ± 4 kHz.
By integrating the PSD of each peak in the Mollow triplet we can compare their relative weights. After normalization with \(\hbar\)ω_{01}Γ_{r}, the results are about 0.254, 0.116 and 0.124 for the middle peak, the red and blue sidebands, respectively. According to Eq. (2), we would expect these numbers to be 0.250, 0.125 and 0.125, respectively, for a fully saturated qubit.
Offresonant Mollow triplet
We also study the offresonant Mollow triplet at a variety of pump powers and frequency detunings between the pump and the qubit. In Fig. 3a, the pump power at the qubit is swept from −150 dBm to −130 dBm at detuning Δ/2π = 790 kHz. We find that the PSD is weaker than the onresonant case, implying that the qubit is less excited. In Fig. 3b, as we sweep the frequency detuning between the pump and the qubit, the spectrum is nearly symmetric over 1 MHz bandwidth, therefore, the extracted pure dephasing rates by the offresonant fluorescence are insensitive to the frequency detuning Δ. We can either choose a large Δ which will lead to a small excitation of the qubit, or a small Δ which results in an unresolved spectrum between the central peak and sidebands.
Compared to the onresonant case, the offresonant Mollow triplet carries additional information in its sideband asymmetry and the approximation used in Eq. (2) is no longer valid. Therefore, the full expression for the PSD must be used, shown in Eq. (16) in Supplementary section A which is an extension of ref. ^{53}. The fit of the data in Fig. 3c to Eq. (16) yields Γ_{ϕ}/2π = 3 ± 3 kHz (pink solid line). The symmetry of the sidebands around the central peak is due to the relatively small pure dephasing rate. In Fig. 3d, the sample was measured in an earlier cooldown. There, we observed a larger asymmetry for both positive (green dots) and negative detunings (brown dots). In the case of positive detuning, the red sideband is closer to the qubit original frequency than the blue sideband, whereas the blue sideband is closer when the sign of the detuning is changed. We fit the two sets of data simultaneously to obtain \(\Gamma_{\phi}/2\pi = \left(7 \pm 2\right)\) kHz, which is slightly larger than the second cooldown. This is likely due to that we used only two isolators in the first cooldown, and four isolators in the second cooldown, leading to less thermal photons from the transmission line in the second case.
The mechanism by which the pure dephasing gives rise to an asymmetry in the Mollow triplet can be explained as follows (for details, see Supplementary section B). Relaxation from the qubit will cause transitions between dressed states \(n,\pm \rangle\) (see Fig. 4a) that contain different numbers n of drive photons. As shown in Fig. 4b, these transitions will either be between or within the + and − subspaces. In equilibrium, if the pure dephasing rate is zero, the probabilities P_{±} for the system to be in these subspaces are given by the detailedbalance condition
i.e., the number of emitted photons causing transitions from + to − (the blue sideband) must equal the number of emitted photons causing transitions from − to + (the red sideband). However, the interaction causing pure dephasing has a nonzero matrix element for transitions between \(\leftn,+\right\rangle \) and \(\leftn,\right\rangle \), which leads to a modified detailedbalance condition:
As Γ_{ϕ} increases, this will push the occupation probabilities towards P_{+} = P_{−}. For offresonant driving Γ_{+−} ≠ Γ_{−+} and thus the number of emitted photons in the two sidebands becomes different: Γ_{+−}P_{+} ≠ Γ_{−+}P_{−}. The larger number of photons will be emitted at the frequency corresponding to the larger of the two transition rates Γ_{+−} and Γ_{−+}; from transitionmatrix elements, this can be seen to be the frequency that is closer to the qubit frequency.
From Fig. 3c, we have Γ_{1}/2π = 275 ± 6 kHz and Γ_{2}/2π = 140 ± 3 kHz. Again, from the measured radiative decay rate, by subtracting Γ_{r} from Γ_{1}, we obtain Γ_{n}/2π = 48 ± 6kHz. Based on the results in this section, the on/offresonant Mollow spectra allow us to extract the pure dephasing rate and nonradiative rate of a qubit. Specifically, for our qubit in this environment, we find that the nonradiative decay rate is one order of magnitude larger than the pure dephasing rate. Compared to the onresonant Mollow triplet, the offresonant Mollow triplet allows us to characterize the qubit decay rates at a lower pump power.
Photon scattering by the qubit
To verify the extracted decay rates above, we can also measure the power scattered by the qubit and the dissipated power due to the nonradiative decay channel directly. We normalize all the powers by the singlephoton energy \(\hbar\)ω_{01}. The pump is on resonance with the qubit. The output power then consists of a coherent part and an incoherent part, P_{out} = P_{coh} + P_{incoh}, where \({P}_{{\rm{coh}}}=\frac{{{{\Omega }}}^{2}}{4{{{\Gamma }}}_{{\rm{r}}}}{(1\frac{{{{\Gamma }}}_{1}{{{\Gamma }}}_{{\rm{r}}}}{{{{\Omega }}}^{2}+{{{\Gamma }}}_{2}{{{\Gamma }}}_{{\rm{1}}}})}^{2}\) and \({P}_{{\rm{incoh}}}=\frac{{{{\Gamma }}}_{{\rm{r}}}}{2}\frac{{{{\Omega }}}^{2}({{{\Gamma }}}_{1}{{{\Gamma }}}_{\phi }+{{{\Omega }}}^{2})}{{({{{\Gamma }}}_{1}{{{\Gamma }}}_{2}+{{{\Omega }}}^{2})}^{2}}\) (see Supplementary section D).
For our qubit, the pure dephasing rate was verified to be around 3 kHz, i.e., much less than other rates and therefore negligible, so, the expression for the incoherent power can be further simplified to \({P}_{{\rm{incoh}}}\simeq 2{{{\Gamma }}}_{{\rm{r}}}{\rho }_{11}^{2}\), where ρ_{11} is the population of the firstexcited state of the qubit. In this case, the expression for the dissipated power due to the nonradiative decay is then \({P}_{{\rm{loss}}}={{{\Gamma }}}_{{\rm{n}}}{\rho }_{11}={{{\Gamma }}}_{{\rm{n}}}\frac{{{{\Omega }}}^{2}}{2({{{\Gamma }}}_{1}{{{\Gamma }}}_{2}+{{{\Omega }}}^{2})}\).
Experimentally, we use about 4.2 × 10^{9} averages to measure all the powers. We denote the measured voltage V and the pump power P_{in}. The subscripts "off” and "on” used in the following contexts mean that the qubit is off/on resonance with the pump, respectively. When the qubit is tuned away, it is offresonant with the pump; we will have \({P}_{{\rm{in}}}={\langle V\rangle }_{{\rm{off}}}^{2}\) because of the coherence of the pump. Besides the pump power, the system noise will also make a contribution to the total measured power, \({P}_{{\rm{in}}}^{{\rm{meas.}}}\). Therefore, we have \({P}_{{\rm{in}}}^{{\rm{meas.}}}={\langle {V}^{2}\rangle }_{{\rm{off}}}={P}_{{\rm{in}}}+P_{\rm{noise}}\), where P_{noise} is the noise power added by the amplification chain over the measurement bandwith. When instead the qubit is on resonance with the pump, the total measured output power \({P}_{{\rm{out}}}^{{\rm{meas.}}}={\langle {V}^{2}\rangle }_{{\rm{on}}}={P}_{{\rm{out}}}+P_{\rm{noise}}\), where \({P}_{{\rm{coh}}}={\langle V\rangle }_{{\rm{on}}}^{2}\) (Fig. 5, black circles). Therefore, P_{loss} (blue crosses) is obtained by taking \({P}_{{\rm{loss}}}={P}_{{\rm{in}}}^{{\rm{meas.}}}{P}_{{\rm{out}}}^{{\rm{meas.}}}={P}_{{\rm{in}}}{P}_{{\rm{out}}}\) with P_{incoh} = P_{in} − P_{coh} − P_{loss} (green circles). Figure 5 shows all the types of measured power as a function of the Rabi frequency. There, we find three interesting regions:

(i)
At high input power, when \({{\Omega }}\, > \,(1+\frac{1}{\sqrt{2}}){{{\Gamma }}}_{{\rm{r}}}\approx 2\pi * 391\ {\rm{kHz}}\), to the right of the dashed line in Fig. 5, the qubit starts to be saturated. The outgoing field is then mainly coherent from the pump itself. By increasing the input power further, the qubit is completely saturated, leading to \({P}_{{\rm{in}}}\approx {P}_{{\rm{coh}}}\approx \frac{{{{\Omega }}}^{2}}{4{{{\Gamma }}}_{{\rm{r}}}}\), \({P}_{{\rm{incoh}}}\approx \frac{{{{\Gamma }}}_{{\rm{r}}}}{2}\) and \({P}_{{\rm{loss}}}\approx \frac{{{{\Gamma }}}_{{\rm{n}}}}{2}\). In this case, almost all the incoming photons are reflected by the mirror.

(ii)
In the lowpower region \({{\Omega }}\, < \,\sqrt{\frac{{{{\Gamma }}}_{1}{{{\Gamma }}}_{2}{{{\Gamma }}}_{{\rm{n}}}}{{{{\Gamma }}}_{{\rm{r}}}{{{\Gamma }}}_{{\rm{n}}}}}\approx 2\pi * 103\,{\rm{kHz}}\) derived from P_{incoh} = P_{loss}, to the left of the dashdotted line, the scattering process is dominated by the interaction between the qubit and the incoming photons. The incoherent scattering is proportional to \({\rho }_{11}^{2}\) whereas the power loss depends linearly on the excitation probability. Therefore, the incoherent power can be less than the power loss when \({\rho }_{11}<\frac{{{{\Gamma }}}_{{\rm{n}}}}{2{{{\Gamma }}}_{{\rm{r}}}}\approx 0.11\). Besides the incoherent photons, there is a small coherent scattering by the qubit. Compared to the loss, the coherent power is smaller if the nonradiative decay is large enough, namely if \({{{\Gamma }}}_{{\rm{n}}}\, > \,\frac{{{{\Gamma }}}_{1}{({{{\Gamma }}}_{{\rm{r}}}{{{\Gamma }}}_{2})}^{2}}{2{{{\Gamma }}}_{{\rm{r}}}{{{\Gamma }}}_{2}}\approx 2\pi * 34\ {\rm{kHz}}\).

(iii)
In the intermediatepower region where \(\sqrt{\frac{{{{\Gamma }}}_{1}{{{\Gamma }}}_{2}{{{\Gamma }}}_{{\rm{n}}}}{{{{\Gamma }}}_{{\rm{r}}}{{{\Gamma }}}_{{\rm{n}}}}}\, < \,{{\Omega }}\, < \,(1+\frac{1}{\sqrt{2}}){{{\Gamma }}}_{{\rm{r}}}\), both the mirror and the qubit make substantial contributions to the scattering process. The photons reflected by the mirror interfere destructively with those scattered by the qubit, resulting in a suppression of the coherent part of the output field. In particular, the dip around Ω/2π ≈ 160 kHz in the coherent power appears due to the fully destructive interference. In addition, the qubit excitation is not small anymore and the incoherent power is larger than the loss because Γ_{r} > Γ_{n}. We note that this region can be nonexistent when either the nonradiative decay or the pure dephasing is sufficiently large.
We also fit the data in Fig. 5 to obtain all the decay rates. The result for the incoherent power indicates Γ_{r}/2π = 229 ± 2 kHz with Γ_{1}Γ_{2}/4π^{2} = 39590 ± 211 kHz^{2} and Γ_{1}Γ_{ϕ}/4π^{2} = 281 ± 281 kHz^{2}. From fits to the power loss, we find Γ_{n}/2π = 49 ± 1 kHz and Γ_{1}Γ_{2}/4π^{2} = 41260 ± 4750 kHz^{2}. Therefore, with Γ_{r} and Γ_{n}, we obtain Γ_{1}/2π = (Γ_{n} + Γ_{r})/2π = 278 ± 2 kHz. Then, Γ_{ϕ}/2π ≃ 1 kHz. The coherent power yields Γ_{r}/2π = 229 ± 2 kHz, Γ_{n}/2π = 48 ± 8 kHz and Γ_{ϕ}/2π = 1 ± 1 kHz. Then, with Γ_{1} and Γ_{ϕ}, we have Γ_{2}/2π = 140 ± 1 kHz.
From the discussion on region (i), at the highest Rabi frequency Ω/2π = 1119 kHz in Fig. 5, Γ_{n}/Γ_{r} ≈ P_{loss}/P_{incoh}. Then, we obtain Γ_{n}/Γ_{r} = [0.1971, 0.2385, 0.2227, 0.2297], by dividing the total measured data into four pieces. Combined with Γ_{r} from the reflection coefficient in the section “Device characterization”, we find Γ_{n}/2π ≈ 45, 54, 51, 52 kHz, respectively. The mean value is about 50 kHz with 3 kHz as the standard deviation. According to Eqs. (40) and (41) in Supplementary section D, as Γ_{ϕ} is small for our qubit, we have \({{{\Gamma }}}_{{\rm{n}}}=2{P}_{{\rm{loss}}}(1+\frac{{{{\Gamma }}}_{1}{{{\Gamma }}}_{2}}{{{{\Omega }}}^{2}})\) and \({{{\Gamma }}}_{{\rm{r}}}\simeq 2{P}_{{\rm{incoh}}}{(1+\frac{{{{\Gamma }}}_{1}{{{\Gamma }}}_{2}}{{{{\Omega }}}^{2}})}^{2}\). Due to \(\frac{{{{\Gamma }}}_{1}{{{\Gamma }}}_{2}}{{{{\Omega }}}^{2}}\approx 3 \% \), we have Γ_{n} ≈ 2.06P_{loss} and Γ_{r} ≈ 2.12P_{incoh}. Therefore, the estimated value for Γ_{n} has a systematic error of about 3%. Since \({{{\Gamma }}}_{\phi }={{{\Gamma }}}_{2}\frac{{{{\Gamma }}}_{{\rm{r}}}+{{{\Gamma }}}_{{\rm{n}}}}{2}\) and Γ_{1} = Γ_{r} + Γ_{n}, the pure dephasing and total relaxation rates are 2π*(2 ± 2) kHz and 2π*(277 ± 2) kHz, respectively. Additionally, because P_{loss} = 2π*0.0243*10^{6} and P_{incoh} = 2π*0.1056*10^{6}, we have Γ_{n}/2π ≈ 49 kHz and Γ_{r}/2π ≈ 224 kHz, respectively.
The results shown here agree well with the values from other sections in the paper, which implies that it is possible to take the Γ_{r} value from the reflection coefficient measurement as a reference for the Mollow triplet in order to separate the nonradiative decay rate and the pure dephasing rate.
Timeresolved dynamics
All measurements described previously span time ranges from several hours to tens of hours. It is noteworthy that qubit decay rates extracted by different methods agree relatively well. However, the long duration means that any fluctuations of the decay rates are averaged out. Recently, several groups have characterized such fluctuations in circuit QED, using Rabi pulses, Ramsey interference measurements, and dispersive qubit readout^{36,37,38,39,54}. To probe the decoherence of the qubit with a temporal resolution of 7 minutes, we prepare the qubit in a superposition of the ground and the firstexcited state and monitor its spontaneous emission into the waveguide by recording both quadratures of the output field with a digitizer as a complex trace in the time domain, i.e., V(t) = I(t) + iQ(t). We measure for 4.27 × 10^{5} s (approximately 119 hours) with 975 repetitions, and each trace has 2.30 × 10^{6} averages. The averaged trace, 〈V(t)〉, is proportional to the qubit emission. In order to generate a pulse at the qubit frequency, a signal generator at the frequency 250 MHz higher than the qubit frequency is mixed with a pulse with a 250 MHz carrier frequency. This pulse is from an arbitrary waveform generator (AWG) where the bandwidth of the AWG is 1.25 giga samples per second.
In detail, after the 50 ns long π/2pulse, the qubit superposition state evolves in time τ as \(\frac{1}{\sqrt{2}}(\left0\right\rangle +{{\rm{e}}}^{{{{\Gamma }}}_{2}\tau {\rm{i}}\delta {\omega }_{01}\tau }\left1\right\rangle )\) with δω_{01} = ω_{01} − ω_{pulse}. The emitted field carries information about the qubit operator \(\langle {\sigma }_{}\rangle ={{\rm{e}}}^{{{{\Gamma }}}_{2}\tau }{{\rm{e}}}^{{\rm{i}}\delta {\omega }_{01}\tau }\), where the amplitude response and the phase response show the decoherence and the qubit frequency shift with τ, respectively.
Figure 6a, b shows the magnitude and phase response of a single trace where the decay of the magnitude is fitted to an exponential curve and the phase of the photons emitted from the qubit grows linearly with time due to the free evolution of the qubit, where the slope determines δω_{01}/2π = 125 kHz. Figures 6c, d are histograms of Γ_{2} and δω_{01} for all the repetitions. Both histograms can be fitted to a Gaussian with parameters shown in the figures. In comparison with the decoherence rates extracted from other measurements, we find that the standard deviation here is larger than the previously measured error bar. This shows that the dynamics of the qubit on a short time differs slightly from that over a long measurement time. By taking the average of all the traces in Fig. 6d, we fit to an exponential decay and get an averaged Γ_{2}/2π = 145 ± 1 kHz.
To study Γ_{1}, we instead send a πpulse to fully flip the qubit, and then measure the emission from the qubit. The corresponding output power, \(P(\tau )=(\hslash {\omega }_{01}{{{\Gamma }}}_{{\rm{r}}}/2)(1+\langle {\sigma }_{z}\rangle ){e}^{{{{\Gamma }}}_{1}\tau }\)^{55} allows us to determine Γ_{1}. The trace is measured with 1.92 × 10^{9} averages, shown in Fig. 6e. A fit to an exponential decay with Γ_{1}/2π = 273 ± 11 kHz agrees well with the data. Combining these numbers with Γ_{r} from section “Device characterization”, we can also calculate Γ_{n} and Γ_{ϕ} from these measurements. The resulting values can are summarized in Table 1.
Discussion
We have shown several methods to determine different decay rates of a qubit placed in front of a mirror. In principle, these methods can also be used when the qubit couples to a transmission line without a mirror, except for the scattering method, where the corresponding measurement taken on both the input and output ports is required.
The measured rates are consistent between methods within the error bars of two standard deviations corresponding to 95% confidence. The results are summarized in Table 1. The reflection measurement is the baseline to provide the value of Γ_{r} to extract the nonradiative decay rate of the qubit for measurements, except for the scattering measurement. These different methods have advantages and disadvantages that we summarize below:

(i)
The fastest way to obtain the nonradiative decay rate is to send a strong pump on resonance with the qubit so that the central peak and the sidebands of the Mollow triplet do not overlap. The drawback is that the pump power needed here is much stronger than for the other methods and that may change the rates slightly.

(ii)
In the second method, we measure the offresonant Mollow triplet by detuning the pump frequency slightly from the qubit frequency. The sidebands will be asymmetric around the central peak if the pure dephasing rate is nonnegligible. In this case, only weak probe power is required. However, the corresponding measurement time is increased by almost a factor of three.

(iii)
The most accurate way to measure the nonradiative decay rate is to measure the difference between the input and output power, labeled as Scattering in Table 1. Using this method, we can obtain not only the power loss but also the coherent and incoherent power scattered by the qubit. However, the measurement time is much longer. In addition, the attenuation between the sample and the input line, as well as the gain between the detector and the sample have to be calibrated at the beginning, in order to get the absolute power values from the qubit. To simplify the measurement, as we described in “Photon scattering by the qubit”, we can see that when the pump saturates the qubit, we get \({P}_{{\rm{incoh}}}\approx \frac{{{{\Gamma }}}_{{\rm{r}}}}{2}\) and \({P}_{{\rm{loss}}}\approx \frac{{{{\Gamma }}}_{{\rm{n}}}}{2}\). Then, the ratio of the nonradiative decay rate to the radiative decay rate can be obtained from the ratio of the lost power to the incoherent power. Knowing the value of Γ_{r} from the reflection measurement, we can obtain the nonradiative decay rate. Therefore, in principle, we do not need to sweep the pump power as was done in Fig. 5. This simple way is labeled as SinglePoint in Table 1.

(iv)
Finally, pulses can be applied to excite the qubit. Afterward, the exponential decay of the emission and the emitted power trace from the qubit can be recorded to extract the total relaxation rate and the decoherence rate with a much larger measurement bandwidth. The distortion on the scattered photons due to the nonflat frequency response will affect the extracted values of the decay rates. This may be the reason why decay rates from this method are slightly different from those measured by other methods. However, the advantage of this method is that it allows us to study the shorttime dynamics of the qubit.
The measurement time for these methods is from 2 to 63 h. The coherent measurement is related to the first moment (amplitude) whereas other methods are related to the second moment (power). In order to estimate the system noise N, we measured the background PSD by turning the drive off (not shown) and comparing the result with the measurement of Fig. 2c. We found N ≈ 49 photons. However, we expect that using a quantumlimited Josephson travelingwave amplifier^{56} would reduce the system noise to about two photons. The corresponding SNR is improved by a factor of 5. Therefore, this would result in a reduction of the measurement time by factors of 25 and 625, respectively, for the coherent measurement and the other methods^{57}.
From the measured result, our qubit is T_{1}limited, i.e., the radiative decay dominates the interaction. However, the nonradiative decay rate is one order of magnitude larger than the pure dephasing rate. The corresponding spontaneousemission factor is β ≈ 85%, which is typically close to 100% when we engineer the radiative decay much larger than the nonradiative decay. Therefore, reducing the nonradiative decay rate will be the next step to improve the intrinsic coherence of our qubit. In addition, it is worthwhile to investigate why the nonradiative decay rate of our qubit is one order of magnitude larger than the qubit coupled to a resonator which was fabricated on the same wafer^{38} in the future.
Our methods allow us to analyze all the decay channels in detail. This will be useful to study and engineer decay channels of the qubit, which is the critical element in superconducting circuits. For example, engineering the decay channels can improve the quantum efficiency of generating single photons, which is set by Γ_{r}/2Γ_{2}. Also, the fidelity of detecting a single photon can be increased by extending the qubit coherence time. More importantly, compared to circuit QED where a resonator dispersively couples to a qubit, our study provides a straightforward way to investigate superconducting qubits, which are crucial elements in superconducting quantum computers.
Data availability
The data that supports the findings of this study are available from the corresponding authors upon reasonable request.
Code availability
The code that supports the findings of this study is available from the corresponding authors upon reasonable request.
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Acknowledgements
The authors acknowledge the use of Nanofabrication Laboratory (NFL) at Chalmers. We wish to express our gratitude to David Niepce and Marco Scigliuzzo for insightful discussions. This work was supported by the Knut and Alice Wallenberg Foundation via the Wallenberg Center for Quantum Technology (WACQT) and by the Swedish ResearchCouncil and the EUproject OpenSuperQ.
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Open Access funding provided by Chalmers University of Technology.
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Y.L. and P.D. planned the project. Y.L. performed the measurements with the input from A.B., J.B., S.G., and B.S. A.B. and Y.L. designed the sample, A.B. fabricated the device. A.F.R. worked on the recipe development and characterization. Y.L., E.W., A.F.K., S.G., and G.J. developed the theoretical expressions. All authors contributed to discussions and the interpretation of results. The manuscript was written by Y.L. with help from all authors. P.D. supervised this work.
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Lu, Y., Bengtsson, A., Burnett, J.J. et al. Characterizing decoherence rates of a superconducting qubit by direct microwave scattering. npj Quantum Inf 7, 35 (2021). https://doi.org/10.1038/s41534021003675
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