Quantum probe of an on-chip broadband interferometer for quantum microwave photonics

With the advent of superconducting circuits as a key player in the field of quantum science and technology, the quantum properties of the microwave signals emitted from these circuits have become a popular object of study. The related experiments can be divided into two major groups. In one of them, the focus is put on the implementation of continuous-variable quantum protocols [1–6]. The other group aims at realizing scattering experiments of microwave photons of a quantum system using either microwave photons in a discrete-variable description or quasi-classical coherent states as probe signals [7–13]. This latter set of experiments, to which also this work contributes, is closely related to the concepts of generation, routing, manipulation, and detection. These areas have been actively explored in photonics at optical frequencies and are therefore often referred to as microwave quantum photonics [14, 15].

In this work, we are interested in the scattering of an incident microwave field off an artificial atom placed in a broadband, but nevertheless carefully engineered, open quantum circuit. In theory, such a system can be described in the framework of the spin-boson model (SBM) [16, 17]. In this model description the relaxation and dephasing rates of the qubit are determined by the spectral function of the electromagnetic environment. Therefore, measuring the qubit relaxation and dephasing rates over a wide range of the qubit transition frequency allows one to obtain valuable information on the environment. Getting such information on the environmental spectral function is crucial both for designing quantum technology and studying fundamental properties of quantum coherence. In particular, one can experimentally determine the microwave environment provided by the implemented circuit and verify whether or not the actually measured environment coincides with the designed one. Most importantly, identifying the origin of differences between the realized and designed environment provides a tool for deliberate environment engineering. This is highly relevant when it comes to the implementation of complex quantum circuits, as required for quantum information processing (QIP) with propagating photons in the spirit of all-optical QIP [18–21]. In addition, one may get valuable information on external sources of decoherence and how to remove them. In this way, quantum coherence, one of the key figures of merit in quantum technology, can potentially be improved.

In our specific experiment, we implement a Mach–Zehnder interferometer based on two on-chip microwave beam splitters and place a transmon qubit in one interferometer arm. The qubit acts as a highly sensitive spectrometer, whose relaxation rates are expected to be dominated by the coupling to the broadband electromagnetic environment provided by the interferometer (see figure 1). The analysis of the experimental results confirms this expectation. Our work makes a major step forward towards the controlled implementation and characterization of complex broadband on-chip circuits required for advanced microwave quantum photonics.

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This article is structured as follows: first, in section 2, we give an overview of the experimental setup and discuss preliminary measurements used to characterize both the interferometer and the transmon qubit via transmission microwave measurements with the help of a vector network analyzer. In section 3, we present the spectral behavior of the composite qubit-interferometer system and provide a detailed data analysis. In particular, we describe three fundamental regimes of operation and the mathematical model used for analyzing our data. Fitting the experimental data provides two key quantities characterizing the qubit environment. First, the linear increase of the qubit relaxation rate Γ₁ with increasing qubit transition frequency in the range between $4\,\mathrm{GHz}$ and $7.5\,\mathrm{GHz}$ clearly shows that the environment provided by the interferometer arm can be well described by an Ohmic spectral density J(ω) ∝ ω associated with memory-less damping Markovian dynamics. Above $7.5\,\mathrm{GHz}$, we find deviations from this simple behavior, most likely due to weak parasitic resonant modes on the chip. Second, we analyze the pure dephasing rate Γ_φ, which indicates the presence of 1/f-noise. Finally, in section 4, we conclude and give an outlook on future experiments and promising applications based on our platform.

In this section, we present details on circuit fabrication, measurement setup, and precharacterization.

Fabrication: Beam splitters and interferometer are fabricated from a $100\,\mathrm{nm}$ thick superconducting Nb film sputter-deposited on top of a $525\,\mu {\rm{m}}$ thick Si substrate covered with $50\,\mathrm{nm}$ of thermal oxide. Both the microwave resonator used for precharacterization and all coplanar waveguide (CPW) structures on the interferometer chip are patterned by optical lithography and reactive ion etching. The transmon qubits are fabricated with aluminum technology and shadow evaporation [22].

Millikelvin setup: For all measurements involving a qubit, the sample chip is placed inside a copper box and mounted at the base temperature stage ($30\,\mathrm{mK}$) of a dilution refrigerator. To minimize the influence of noise leaking to the sample via cabling, attenuators thermally anchored at every temperature stage of the refrigerator are used in the input lines. The output signals are amplified with a chain of cryogenic HEMT and room temperature RF amplifiers. Two circulators, one in each output line, protect the sample from the amplifier noise. More details on the measurement setup can be found in appendix A.

Precharacterization: Before we investigate the coupled qubit-interferometer system, we characterize the two system components independently. We start with the characterization of the Mach–Zehnder type microwave interferometer [23] composed of two equivalent quadrature hybrid beam splitters as shown in figures 2(a) and (c) [24–26].

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We cool down both the beam splitter and the interferometer to $4.2\,{\rm{K}}$ in a liquid helium bath cryostat [25, 26] and use a vector network analyzer to measure the transmission magnitude $| {S}_{{ij}}|$. The indices i and j identify the output and the input port, respectively.

The measured transmission spectra shown in figures 2(e)–(f) agree well with those obtained from analytical theory [24] and from finite element simulations⁸ . The interferometer shows a typical isolation of more than $20\,\mathrm{dB}$ near its designed working frequency ${\omega }_{\mathrm{IF}}/2\pi =5.746\ \mathrm{GHz}$. At $\omega /2\pi =5.9\ \mathrm{GHz}$ we find the maximum isolation of approximately $30\,\mathrm{dB}$.

In order to obtain independent reference data for the qubit parameters and the coupling strength between the qubit and the interferometer, we fabricate a transmon qubit [27] with the same design parameters and place it at the voltage-antinode of a λ/4 resonator with a resonance frequency ${\omega }_{\mathrm{res}}/2\pi =6.54\,\mathrm{GHz}$. This resonator has a CPW design with the same geometry as the transmission line arms connecting the two beam splitters forming the interferometer in the coupled qubit–interferometer system. To access the qubit transition frequency far away from the resonator frequency we perform two-tone measurements [28] using a vector network analyzer and an additional microwave source. The results are shown in figures 3(b) and (c). As expected from the chosen qubit design, the transition frequency of the transmon qubit is tuneable by an applied magnetic flux over the wide frequency range of ${\omega }_{01}=4\mbox{--}9\,\mathrm{GHz}$. Fitting the data, we derive charging energy ${E}_{C}/h=592.4\,\mathrm{MHz}$, a Josephson coupling energy ${E}_{J}/h=20\,\mathrm{GHz}$, and a qubit–resonator coupling $g/(2\pi )=70\,\mathrm{MHz}$.

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Transmon in interferometer: The sample under study in section 3 of this work consists of a transmon qubit placed in one arm of a Mach–Zehnder type interferometer. Both the geometry of the transmission lines forming the interferometer arms as well as the design and fabrication parameters of the transmon qubit are similar to those of the resonator and transmon qubit used for precharacterization. Details on the sample layout can be found in figure 4.

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In this section, we discuss the properties of the qubit in-interferometer system. First, we qualitatively describe the transmission spectrum (section 3.1) and then develop a quantitative model based on transfer-matrices (section 3.2). We then fit the measurement data to this model in order to extract the qubit transition frequency ω₀₁, the relaxation rate Γ₁, and the dephasing rate Γ_φ. We use these results to confirm the validity of our model. In section 3.3, we use the qubit as a sensitive spectrometer of its environment and take a closer look at its decoherence properties. By analyzing the derived Γ₁ values for a wide range of qubit transition frequencies we can conclude that the qubit predominantly interacts with an Ohmic bath provided by the broadband transmission line. Additionally, we find evidence for a broad parasitic mode at a frequency above $8\,\mathrm{GHz}$. Finally, exploiting ${{\rm{\Gamma }}}_{\varphi }$ measurements, we discuss dephasing noise at frequencies much smaller than ω₀₁ in the framework of the Ornstein–Uhlenbeck model [30, 31].

3.1. Qualitative analysis

We probe the qubit in a resonance fluorescence experiment. To this end, we irradiate it with an attenuated coherent microwave signal with an average photon number of much less than one, via one interferometer port and record the magnitude and phase of the through- (S₃₂ and S₁₄) and the cross- (S₁₂ and S₃₄) transmission signals as a function of frequency (see figure 2 for definitions). By varying the magnetic flux threading the dc-SQUID loop of the transmon qubit, we can vary the qubit transition frequency over a wide range and record transmission spectra for different qubit frequencies. Typical spectra are shown in figures 5(b) and (c), where we plot the transmission amplitude and phase as a function of the probe frequency in a narrow frequency windows close to three different qubit transition frequencies ω₀₁. In these spectroscopic measurements, the qubit absorbs and reemits microwave photons propagating along the transmission line. The presence of the interferometer modifies the interference between the incident and the reemitted signal. Depending on the shape of the resonance fluorescence signal, we can identify three different regimes as shown in figure 5:

• peak-dip (${\omega }_{01}/2\pi \lesssim 5\,\mathrm{GHz}$)
• dip ($5\,\mathrm{GHz}\lesssim {\omega }_{01}/2\pi \lesssim 6.5\,\mathrm{GHz}$)
• peak-dip (${\omega }_{01}/2\pi \gtrsim 6.5\,\mathrm{GHz}$)

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These regimes can be understood in an intuitive picture: near its center frequency ω_IF, the interferometer is $50\,{\rm{\Omega }}$-matched, and therefore, the qubit acts as if it was placed in a bare transmission line [8]. In this case the presence of the qubit results in the typical symmetric dip in the transmission amplitude due to resonance fluorescence. However, away from the center frequency the interferometer is no longer impedance matched, causing a finite scattering amplitude at the interferometer ports. The resulting constructive and destructive interference effects result in Fano-like shapes of the transmission spectra [32].

3.2. Transfer matrix model

The qualitative discussion of the transmission spectra in the previous section shows that a quantitative analysis must take into account all scattering and interference effects on the sample chip. A powerful technique to appropriately model this situation is the transfer matrix approach, where each circuit part is modeled by an individual transfer matrix [33]. In our experiments, we measure the transmitted and reflected amplitude of the complete circuit composed of the beam splitter, the transmission lines forming the interferometer arms and the transmon qubit. Therefore, in order to relate the incoming and outgoing modes on one side of the sample with those on the other side in our model approach, we have to construct the total transfer matrix ${ \mathcal M }$ representing the whole circuit. This can be done in a straightforward way by modeling each circuit component by its individual scattering matrix (for details, see appendix C). In this way the total matrix ${ \mathcal M }$ can be expressed as

$$\begin{eqnarray}\begin{array}{rcl}{ \mathcal M } & \equiv & { \mathcal M }(\omega ,{\omega }_{\mathrm{IF}},{\omega }_{01},{{\rm{\Gamma }}}_{1},{{\rm{\Gamma }}}_{\varphi })\\ & = & {M}_{\mathrm{BS}}\cdot {M}_{\mathrm{TL}}\cdot {M}_{{\rm{Q}}}\cdot {M}_{\mathrm{TL}}\cdot {M}_{\mathrm{BS}}.\end{array}\end{eqnarray} \tag{ 1 }$$

Here, M_BS, M_TL, and M_Q are the transfer matrices representing the two beam splitters, the transmission lines left and right of the qubit, and the transmon qubit, respectively. Detailed expressions of the different transfer matrices are derived in appendix C.

The transfer matrix of the transmon qubit is that of a local scatterer. The coefficients of the scattering matrix are determined by the relaxation and dephasing rate as well as the detuning δω = ω − ω₀₁ between the probe signal and the qubit transition frequency. The relaxation and dephasing rates of the qubit can be modeled within the framework of the SBM, describing a single two-level system (TLS) interacting with a bosonic bath. We can apply this model to the transmon qubit interacting with the interferometer, since in the low-probe-power limit the former can be well modeled by an effective TLS, whereas the latter can be described by a broad spectrum of harmonic oscillators [24]. Since these oscillators are expected to provide memory-less Ohmic damping for the qubit, we expect Γ₁ = α ω₀₁ [16]. The reason for this linear scaling is the fact that Ohmic dissipation results in an effective bath spectral density which is linear in frequency, J(ω) = βω (with high-frequency cutoff ω_c), with β corresponding to a friction constant. Note that the dimensionless parameter α reflects the strength of dissipation which in a physical system depends on the amplitude of the noise and its coupling strength. Further details are given in the appendix D.

Analyzing the measured transmission spectra by the transfer matrix model, we can derive important qubit parameters in a quantitative way. From a numerical fit of (1) to the measured cross-transmission data S₃₄ and S₁₂ (see figures 5(a) and (b)), we extract the qubit transition frequency as well as the relaxation and dephasing rates. Doing so, we discard S₁₄ and S₃₂, as the almost vanishing signal (see figures 5(a) and (c)) leads to calibration issues and unreliable fits.

Within a 95% confidence interval, the statistical error of the extracted qubit transition frequency is below 2%. For Γ₁ and Γ_φ, we typically observe statistical errors below 33%. Thus, we obtain a reliable set of data for the decoherence properties of the transmon qubit over a wide range of qubit transition frequencies ($4\mbox{--}8.5\,\mathrm{GHz}$). The excellent fit quality of the individual spectra for most qubit transition frequencies and the reproduction of the regimes described in section 3.1 (see figure 5), provides strong evidence for the validity of the applied transfer matrix model. Additionally, as expected by design, the interferometer predominantly dictates the transmission spectrum of the system, except in a small region of approximately $100\,\mathrm{MHz}$ near the qubit transition frequency.

3.3. Environment

In order to gain information about the local electromagnetic environment of the qubit, we use the transmon qubit as a broadband spectrometer in this section. Following a Golden rule argument, the relaxation rate Γ₁ is proportional to the noise power spectral density at the qubit transition frequency S(ω₀₁). Hence, the measurement of Γ₁ as a function of qubit transition frequency allows us to obtain information on S(ω).

Therefore, we vary the transition frequency of the transmon qubit between 4 and $8.5\,\mathrm{GHz}$ by changing the coil current and, hence, the magnetic flux threading the dc-SQUID loop of the tunable junction.

From the transmission spectra recorded for every particular qubit frequency we can derive the relaxation and dephasing rates over a wide frequency regime. From this data we can in turn derive valuable information on the interaction of the qubit with its electromagnetic environment.

More specifically, we perform fits of our transmission model to each recorded spectrum as described in section 3.2 and extract ω₀₁, Γ₁, and Γ_φ. For an Ohmic environment the qubit relaxation rate Γ₁(ω₀₁) is expected to follow ${{\rm{\Gamma }}}_{1}({\omega }_{01})\,=\alpha \cdot {\omega }_{01}$, $\alpha ={\tilde{d}}^{2}/({\rm{\hslash }}c{\epsilon }_{0})$, with the reduced Planck constant ${\rm{\hslash }}$, the speed of light c, the electrical vacuum permittivity ${\epsilon }_{0}$, and the qubit dipole moment divided by the cross-sectional area of the CPW $\tilde{d}$ (see appendix D). Figure 6 shows that our Γ₁ data follows a linear trend for frequencies up to about 7 GHz. This clearly supports our starting assumption that the transmission line coupled to the qubit provides an Ohmic bath. Interestingly, deviations from the Ohmic environment are rather small in the range between $4$–$7\,\mathrm{GHz}$ although the coupling strength is low in comparison to other experiments [17]. For frequencies above $8\,\mathrm{GHz}$, we observe a pronounced rise in Γ₁ which provides a hint to the presence of an additional on-chip mode coupling to the qubit. In a first-order approximation, we model this mode by an additional Lorentzian (center frequency ${\omega }_{{\rm{L}}0}/2\pi \,=8.3\,\mathrm{GHz}$, full width half maximum ${\rm{\Gamma }}/2\pi =1.5\,\mathrm{GHz}$) on top of the linear Ohmic background. By fitting the data, we find $\alpha =(1.7\pm 0.3)\times {10}^{-4}$, corresponding to $\tilde{d}=(6.9\pm 2.7)\,\times {10}^{-21}\,{\rm{A}}\,{\rm{s}}$. In order to find a more quantitative evidence for the transmission line to be the dominant bath for qubit relaxation, we also determine ${\alpha }_{\mathrm{res}}=\pi {({g}_{\mathrm{res}}/{\omega }_{\mathrm{res}})}^{2}=(3.6\pm 0.04)\,\times {10}^{-4}$ (see appendix D for details) in the qubit–resonator system used for the precharacterization in section 2. The good agreement between α and α_res clearly confirms the validity of the SBM-based data analysis.

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Finally, we characterize the noise causing pure dephasing of the transmon qubit inside the interferometer circuit. For the subsequent analysis, we only consider Γ_φ values with less than 33% statistical error (see section 3.2). It is well established that flux noise through the dc-SQUID loop is a dominant source for the fluctuation of the transmon qubit transition frequency ω₀₁, leading to dephasing [34–37]. As a consequence, we expect a strong dependence of Γ_φ(ω₀₁) on the first derivative of ω₀₁ with respect to flux Φ [35]. Indeed, our data is well fitted with the ansatz ${{\rm{\Gamma }}}_{\varphi }\propto {(\partial {\omega }_{01}/\partial {\rm{\Phi }})}^{\eta }$, as shown in figure 6. Interestingly, the exponent η ≃ 1.64 ± 0.21 suggests that the observed flux noise may be appreciably correlated rather than simple white noise, for which an exponent of 2 is expected [35]. To further characterize the properties of the observed flux noise, we fit the Γ_φ(ω₀₁) data with a model of Gaussian colored noise (Ornstein–Uhlenbeck process) [30, 31, 38], which allows us to smoothly interpolate between the limits of fast (white) and slow noise. To first order, the transmon transition frequency fluctuates as ω₀₁(t) = ω₀₁ + δω₀₁(t), where the deviations δω₀₁(t) are related to random flux fluctuations via the first derivative as

$$\begin{eqnarray}&&\delta {\omega }_{01}(t)=\displaystyle \frac{\partial {\omega }_{01}}{\partial {\rm{\Phi }}}\delta {\rm{\Phi }}(t).\end{eqnarray} \tag{ 2 }$$

In addition, the colored Gaussian noise model relies on a specific autocorrelation function for the random flux fluctuations,

$$\begin{eqnarray}&&\langle \delta {\rm{\Phi }}(0)\delta {\rm{\Phi }}(\tau )\rangle ={\sigma }^{2}{e}^{-\kappa | \tau | }.\end{eqnarray} \tag{ 3 }$$

Here, σ describes the flux noise amplitude and κ is a rate describing the temporal range of the correlations or 'speed of noise'. The noise spectrum corresponding to this model is $S(\omega )\,=\int {e}^{i\omega \tau }\langle \delta {\omega }_{01}(0)\delta {\omega }_{01}(\tau )\rangle d\tau$. For $\kappa \to \infty$, we expect fast noise, because ${\mathrm{lim}}_{\kappa \to \infty }S(\omega )={(\partial {\omega }_{01}/\partial {\rm{\Phi }})}^{2}(2{\sigma }^{2}/\kappa )$ becomes constant. The model smoothly connects this white noise limit to the opposite case, $\kappa \to 0$. Here, one obtains colored quasi-static Gaussian noise, because ${\mathrm{lim}}_{\kappa \to 0}S(\omega )={(\partial {\omega }_{01}/\partial {\rm{\Phi }})}^{2}(2\pi {\sigma }^{2})\delta (\omega )$ diverges for $\omega \to 0$. This limit would correspond to a Gaussian decay envelope in a Ramsey or spin echo type time domain experiment [34]. From a numerical fit of the Ornstein–Uhlenbeck model [31] to the dephasing data Γ_φ(ω₀₁), we extract σ = (79 ± 9)μΦ₀ with Φ₀ being the flux quantum. We further find that κ/2π vanishes within a statistical uncertainty of $52\,\mathrm{kHz}$. Hence, this noise speed is negligible with respect to the noise strength $| \partial {\omega }_{01}/\partial {\rm{\Phi }}| (\sigma /2\pi )$, which is on the order of a few megahertz. We conclude that the noise in our device is well described by colored Gaussian noise in the quasi-static limit. This is also consistent with our previous assessment based on $\eta \ne 2$ and with a noise spectrum diverging at $\omega /2\pi \simeq 0\,\mathrm{Hz}$. A possible source for such noise can be e.g. TLS ensembles produced by surface defects in dielectric materials [39, 40]. We can directly relate the quantity sigma to the strength of the 1/f-noise typically produced by such ensembles [34, 35, 37, 36]. The standard treatment [34] provides us with an upper bound of approximately 100 μΦ₀, which is well compatible with the values on the order of a few μΦ₀ found in many other works [34, 35, 41, 42].

We have investigated a complex, engineered on-chip open quantum system consisting of a microwave interferometer with a transmon qubit in one of its arms. The interferometer works over a broad frequency range of $4\mbox{--}8\,\mathrm{GHz}$ in the microwave domain and the qubit transmission frequency is also tunable over this range. The measured transmission properties are well described by a transfer matrix model, where the scattering matrix of the transmon qubit is derived in the framework of the SBM. Using this model, we extract the relaxation rate Γ₁ and the pure dephasing rate Γ_ϕ of the qubit and discuss their dependence on the qubit transition frequency.

The linear behavior of Γ₁ with respect to the qubit transition rate ω₀₁ confirms our expectation that the transmission line acts as an Ohmic bath in the frequency range mentioned above. Above $8.5\,\mathrm{GHz}$, our sensitive spectrometer detects a weak enhancement of the spontaneous emission, which we attribute to a spurious on-chip resonance. We have further analyzed the qubit dephasing rate using a model based on the Ornstein–Uhlenbeck process [31]. We find that our circuit QED open quantum system is dominated by slow, colored Gaussian noise. Future experiments with higher frequency-resolution or, equivalently, high-resolution time domain experiments, would even allow for the investigation of the microscopic nature of the noise sources.

Beyond the current results, our work is also relevant for the analysis of more complex open microwave quantum systems and higher order multi-photon processes [43], as they would be required for quantum computing with microwaves in the spirit of all-optical quantum computing [20]. Such experiments would provide a novel approach for QIP, combining established techniques from the optical domain with the advantages of the strong nonlinear elements provided by Josephson junctions in the microwave domain. In this context, the present work already implements key elements, such as beam splitters and nonlinearities.

The authors thank all collaborators, especial Edwin Menzel for his efforts in sample fabrication. The authors acknowledge support from the German Research Foundation through FE 1564/1-1, the IMPRS 'Quantum Science and Technology', from the MINECO/FEDER Project FIS2015-70856-P, and CAM PRICYT Research Network QUITEMAD + S2013/ICE-2801. Toms Ramos further acknowledges the Juan de la Cierva fellowship FJCI-2016-29190. We also acknowledge support from the doctorate program ExQM and the EU project PROMISCE.

While the precharacterization of the beam splitter and interferometer samples are done in a liquid helium bath cryostat, the transmission measurements of the composite system are done in a dry dilution refrigerator at a base temperature of $T\simeq 30\,\mathrm{mK}$. Figure A1 shows a photograph of the sample box and the coil and figure A2 shows a schematic of the setup.

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The interferometer and transmon qubit properties analyzed in the main text are also accessible in the composite system. This means that fitting the transmission spectrum at each qubit operating point with our transfer matrix model gives us access to all relevant parameters such as the interferometer frequency ω_IF, the qubit transition frequency ω₀₁, the relaxation rate Γ₁, and the pure dephasing rate Γ_Φ.

The transmission spectrum of the interferometer alone is shown in figure B1. It is obtained by measuring the transmission magnitude with the qubit transition frequency tuned outside the accessed frequency range.

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By analyzing the transmission spectra as discussed in section 3.2 and shown in figure 5, we obtain the flux-dependence of the qubit transition frequency. We fit the theoretically expected behavior [34] to the data and obtain ${E}_{C}/h=571\pm 450\,\mathrm{MHz}$, and ${E}_{J}/h=19.5\,\pm 13\,\mathrm{GHz}$ (see figure B2). These values are in good agreement with those measured for the transmon qubit in the transmission line resonator in pre-characterization experiments.

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The transfer matrix relates the incoming and outgoing modes a_in and a_out on one side of the scatterer to the outgoing and incoming modes b_in and b_out on the other side. In contrast, a scattering matrix usually connects the incoming modes a_in and b_in on both sides of the scatterer to the outgoing modes a_out and b_out. In the following, we stick to the transfer matrix formalism to model the transmission response of the qubit-interferometer system (see figure C1). For our sample, the total transfer matrix ${ \mathcal M }$ connects the complex input signals $({a}_{1,\mathrm{in}},{a}_{2,\mathrm{in}},{a}_{3,\mathrm{in}},{a}_{4,\mathrm{in}})$ to the complex output signals $({a}_{4,\mathrm{out}},{a}_{3,\mathrm{out}},{a}_{2,\mathrm{out}},{a}_{1,\mathrm{out}})$:

$$\begin{eqnarray}&&\left(\begin{array}{c}{a}_{1,\mathrm{out}}\\ {a}_{1,\mathrm{in}}\\ {a}_{3,\mathrm{out}}\\ {a}_{3,\mathrm{in}}\end{array}\right)={ \mathcal M }\left(\begin{array}{c}{a}_{4,\mathrm{in}}\\ {a}_{4,\mathrm{out}}\\ {a}_{2,\mathrm{in}}\\ {a}_{2,\mathrm{out}}\end{array}\right).\end{eqnarray} \tag{ C.1 }$$

This property is different from the scattering matrix, which connects the incoming and the outgoing modes at each port. As shown in figure C1, ${ \mathcal M }$ can be decomposed into a product of the matrices M_BS, M_TL, and M_Q for the beam splitter, transmission line and transmon qubit, respectively. One then obtains (1). We set a_{1, in} = a_3,in = 0 assuming that no signal enters the interferometer via port 1 and port 3 on the right side. Technically this assumption is assured by isolators (see figure A2). For the inputs a_2,in, and a_4,in, we assume coherent microwave signals. By solving the system of linear equations, we obtain a model function for the different measurement paths illustrated in figure 2(b). In the following subsections, we derive expressions for the individual matrices M_TL, M_BS, and M_Q.

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C.1. Transmission line

For the transmission line, we take into account the real and imaginary exponent of each signal, thus absorption and phase. This leads to

$$\begin{eqnarray*}{M}_{\mathrm{TL}}=\left(\begin{array}{cccc}{e}^{{{ic}}_{1}t} & 0 & 0 & 0\\ 0 & {e}^{{{ic}}_{2}t} & 0 & 0\\ 0 & 0 & {e}^{{{ic}}_{3}t} & 0\\ 0 & 0 & 0 & {e}^{{{ic}}_{4}t}\end{array}\right),\end{eqnarray*}$$

where c_i = (i r_i + ϕ_i). Here, r_i describes the absorption which is treated as a fitting parameter (figure B1). In contrast, ϕ_i is the phase which is determined by the length of the transmission line.

C.2. Beam splitter

The transfer matrix of the beam splitter provides a 50:50 splitting of an input signal incident at an input port into the two output ports. It also adds a 180° phase shift to one of the output signals, resulting in

$$\begin{eqnarray*}{M}_{\mathrm{BS}}=\left(\begin{array}{cccc}-\displaystyle \frac{i}{\sqrt{2}} & 0 & -\displaystyle \frac{1}{\sqrt{2}} & 0\\ 0 & \displaystyle \frac{i}{\sqrt{2}} & 0 & -\displaystyle \frac{1}{\sqrt{2}}\\ -\displaystyle \frac{1}{\sqrt{2}} & 0 & -\displaystyle \frac{i}{\sqrt{2}} & 0\\ 0 & -\displaystyle \frac{1}{\sqrt{2}} & 0 & \displaystyle \frac{i}{\sqrt{2}}\end{array}\right).\end{eqnarray*}$$

C.3. Transmon qubit

The transfer matrix has to reflect the fact that the qubit is placed only in one arm arm of the interferometer. Thus the scattering properties of the qubit have to be taken into account only for one arm of the interferometer, whereas the coefficients for the other arm are just unity:

$$\begin{eqnarray*}{M}_{{\rm{Q}}}=\left(\begin{array}{cccc}-\displaystyle \frac{{r}^{2}}{t} & \displaystyle \frac{r}{t} & 0 & 0\\ -\displaystyle \frac{r}{t} & \displaystyle \frac{1}{t} & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{array}\right).\end{eqnarray*}$$

Following [8], the respective transmission and reflection amplitudes characterizing the scattering by the qubit are given by

$$\begin{eqnarray*}\begin{array}{rcl}r & = & {r}_{0}\displaystyle \frac{1-i(\delta \omega /{{\rm{\Gamma }}}_{2})}{1+{(\delta \omega /{{\rm{\Gamma }}}_{2})}^{2}+{{\rm{\Omega }}}^{2}/{{\rm{\Gamma }}}_{1}{{\rm{\Gamma }}}_{2}},\\ t & = & 1-r,\end{array}\end{eqnarray*}$$

where Ω is the qubit Rabi frequency, Γ₂ ≡ Γ₁/2 + Γ_φ is the qubit decoherence rate and r₀ is the maximum reflection amplitude. δω ≡ ω − ω₀₁ is the detuning between the drive and the qubit transition frequency.

Here, we briefly derive the coupling strength g of a transmon qubit with transition frequency ω₀₁ to the Ohmic bath formed by a broadband transmission line in the framework of the SBM. The spin-boson Hamiltonian reads:

$$\begin{eqnarray*}\begin{array}{rcl}{H}_{\mathrm{SBM}} & = & \mathop{\underbrace{\displaystyle \sum _{k}{\rm{\hslash }}{\omega }_{k}{\hat{a}}_{k}^{\dagger }{\hat{a}}_{k}}}\limits_{{H}_{\mathrm{bath}}}+\mathop{\underbrace{\displaystyle \frac{{\rm{\hslash }}{\omega }_{01}}{2}\hat{{\sigma }^{z}}}}\limits_{{H}_{\mathrm{qb}}}\\ & & +\mathop{\underbrace{\displaystyle \sum _{k}({g}_{k}{\hat{\sigma }}^{+}\hat{{a}_{k}}+{g}_{k}^{* }{\hat{\sigma }}^{-}{\hat{{a}_{k}}}^{\dagger })}}\limits_{{H}_{\mathrm{int}}},\end{array}\end{eqnarray*}$$

where H_bath, H_qb and H_int denote the bath, qubit and interaction part of the Hamiltonian, respectively.

The coupling g_k of a qubit to a single photon with frequency ω_k is given by a dipole interaction [44], and thus

$$\begin{eqnarray}&&{g}_{k}=\displaystyle \frac{d\cdot {E}_{0}}{{\rm{\hslash }}},\end{eqnarray} \tag{ D.1 }$$

where d is the electric dipole moment of the artificial atom (qubit) and E₀ is the root-mean-square electric field due to zero point fluctuations. The electric field contains half of the energy of the transmission line, while the other half is stored in the magnetic field. Therefore,

$$\begin{eqnarray}&&\displaystyle \frac{{\epsilon }_{0}}{2}\int {E}^{2}(x)A(x)\,{dx}=\displaystyle \frac{{\epsilon }_{0}}{2}{E}_{0}^{2}{A}_{0}L=\displaystyle \frac{1}{2}\displaystyle \frac{{\rm{\hslash }}{\omega }_{k}}{2},\end{eqnarray} \tag{ D.2 }$$

where ${\epsilon }_{0}$ is the vacuum permittivity, L is the length of the waveguide, and A₀ is the cross-sectional area perpendicular to the propagation direction. In other words, we determine an effective electric field which is constant inside the mode volume A₀ · L and zero elsewhere.

Solving for E₀ in (D.2), the coupling g_k reads

$$\begin{eqnarray}&&{g}_{k}={g}_{0}\sqrt{\displaystyle \frac{{\omega }_{k}}{2L}},\quad \mathrm{with}\quad {g}_{0}=\sqrt{\displaystyle \frac{{\tilde{d}}^{2}}{{\rm{\hslash }}{\epsilon }_{0}}}.\end{eqnarray} \tag{ D.3 }$$

Here, $\tilde{d}=d/\sqrt{{A}_{0}}$ is a qubit dipole moment divided by the cross-sectional area of the CPW A₀.

The relaxation rate of the qubit is given in the Markov and rotating wave approximations [45] by

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{1}=J({\omega }_{01}),\end{eqnarray} \tag{ D.4 }$$

where the spectral function of the SBM reads,

$$\begin{eqnarray}&&J(\omega )=2\pi \displaystyle \sum _{k}| {g}_{k}{| }^{2}\delta (\omega -{\omega }_{k}).\end{eqnarray} \tag{ D.5 }$$

Substituting (D.5) and (D.3) into (D.4), taking the continuum limit, and assuming a linear dispersion relation ${\omega }_{k}=c| k|$, we finally obtain

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{1}=\alpha {\omega }_{01},\quad \mathrm{with}\quad \alpha =\displaystyle \frac{{g}_{0}^{2}}{c}.\end{eqnarray} \tag{ D.6 }$$

We see that in this case of an Ohmic bath, the relaxation rate Γ₁ is proportional to the resonance frequency of the qubit ω₀₁. The authors remark that the α defined here is not the Kondo parameter, also commonly called α [17]. If we denote the Kondo parameter as α_K, then α_K = α/(2π).