Superconducting flux qubit capacitively coupled to an LC resonator

Dispersive readout is widely used for superconducting qubits, because it potentially enables high-fidelity, fast and non-destructive readout [1]. In this readout scheme, the superconducting qubits are coupled to an electromagnetic resonator (LC resonator) or transmission line, and the type of coupling, i.e. what kind of degree of freedom to use, and its strength should be properly chosen based on various aspects, such as readout fidelity and backaction on the qubits. Note that even for the same type of qubit, we can still choose either capacitive (voltage) or inductive (current) coupling. For example, in charge (or transmon) and phase qubits, both capacitive and inductive couplings have been studied [2–6]. Transmon and phase qubits are (dc-) charge insensitive devices. However, as pointed out by Koch et al [7], this does not mean they are insensitive to the ac voltage field, and the strong coupling regime has indeed been achieved in these devices using capacitive coupling [2, 8].

For the superconducting flux qubits, on the other hand, inductive coupling has typically been used [9, 10]. This may be related to the fact that the states of the flux qubit are often associated with the clockwise and the counterclockwise circulating currents, besides that the flux qubits are also (dc-) charge insensitive devices. Again, however, this does not mean that capacitive coupling is impossible for the flux qubits⁶. We demonstrated in [13] that we can achieve strong coupling between a conventional flux qubit and an LC resonator coupled via a capacitance. In the paper, we also reported an enhancement of the dispersive shift of the cavity resonance induced by the third level of the flux qubit. This effect was first predicted for the transmon qubit [7] to occur when a specific level configuration, ω₀₁ < ω_r < ω₁₂, is realized, where ω_ij is the qubit transition frequency between the ith and the jth levels and ω_r is the cavity resonant frequency. They call this a straddling regime, in which cooperative interplay between 0–1 and 1–2 transitions gives rise to the enhancement of the dispersive shift. In [13], we show that the magnitude of the observed dispersive shift, which was more than four times larger than that estimated from a simple two-level approximation, is quite consistent with the theory which takes into account the higher levels of the flux qubit. We note that the effect of the higher levels of the artificial atom on the dispersive shift has recently been investigated in the fluxonium qubit too  [14, 15].

The present paper is an extension of our previous paper [13] and motivated mainly by the following three questions related to the enhancement in the dispersive shift. (i) Why had such a large enhancement never been observed in much more extensively studied devices with inductive coupling? (ii) Does the enhanced dispersive shift affect the coherence of the qubit? (iii) Can the enhanced dispersive shift be applied to anything besides the readout of the qubit? To answer these questions, the rest of the present paper is organized as follows: in section 2, we describe the details of the sample fabrication and the measurements. In the present paper, we study three different devices. In all the devices, a flux qubit is coupled to an LC resonator, but with different magnitude of the coupling capacitance. In section 3, we measure the dispersive shift in the three devices, and compare it with the calculation using the device parameters independently determined by the spectroscopy measurements. Section 4 is related to the first question. We calculate the energy band of the two circuits in the straddling regime, where the flux qubit is coupled to a resonator either via a capacitance or an inductance. Based on that, we discuss the difference in the enhancement of the dispersive shift. Section 5 is to answer the second question, in which we study the coherence property of our devices. The results show that the energy relaxation time is not limited by the coupling at present. Section 6 is related to the third question, where we clearly observe the discrete ac-Stark effect, which is a hallmark of the strong-dispersive regime. We show that the data are well reproduced by the theory. Finally, in section 7, we summarize our results.

In this work, we study the system where a superconducting flux qubit is capacitively coupled to a superconducting coplanar waveguide (CPW) resonator (figure 1). The CPW resonator is made of a 50 nm thick Nb film sputtered on a 300 μm thick undoped Si wafer covered by a 300 nm thick thermal oxide. It is patterned by electron-beam (EB) lithography using the ZEP520A-7 resist and CF₄ reactive ion etching. The flux qubit is a conventional three-Josephson-junction (3JJ) flux qubit, in which one junction is made smaller than the other two by a factor of α. It is fabricated by EB lithography and double-angle evaporation of Al using PMMA/Ge/MMA trilayer resist. The thicknesses of the bottom and the top Al layers are 20 and 30 nm, respectively. In order to realize a superconducting contact between Nb and Al, the surface of Nb is cleaned by Ar ion milling before the evaporation of Al.

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In this work, we study the three devices with different coupling capacitances C_c between the qubit and the resonator. We call them devices A, B and C, which have designed C_c of 2, 3 and 4 fF, respectively. All of the devices have a λ/2-type CPW resonator of the same design. They have the fundamental resonant frequency at around 10.25 GHz, and the loaded quality factor Q of ∼650, which is limited by the input capacitance C_in designed to be 15 fF. The qubits have almost the same design in all of the devices, but with small adjustment in α to compensate for the effect of C_c on ω₀₁. The loop has a size of ∼2.0 × 2.4 μm², and is coupled to the control line by a mutual inductance of ∼0.1 pH.

The devices were mounted in a sample holder made of gold-plated copper, which is thermally anchored to the mixing chamber of a dilution refrigerator and cooled to ∼10 mK. The sample holder is covered by three-layer (one superconducting and two μ-metal) magnetic shields, but we do not use radar-absorbent materials [16] in this work. For the spectroscopy measurements, we used a vector network analyzer. For the measurements of the dispersive shift and qubit coherence, we used a pulsed microwave and for the heterodyne detection of the time-domain data we used a commercial analogue-to-digital converter [13].

Before investigating the dispersive shift, we need to experimentally determine the device parameters. As shown in [13], this was done by probing the energy bands of the system by the spectroscopy measurements and fitting the result with the energy bands calculated from the model Hamiltonian (equations (1)–(4) in the next section). Figure 2 shows an example of the spectroscopy measurement for device B. In figure 2(a), we plot the phase of the reflection coefficient Γ as a function of the probe frequency ω_p and the flux bias for the qubit f ≡ Φ/Φ₀, where Φ is the flux through the qubit loop and Φ₀ is the flux quantum. We observe clear vacuum Rabi splitting indicating that the strong coupling regime is achieved. In figure 2(b), on the other hand, we measure Γ at a fixed probe frequency ω_p = ω_r = 2π × 10.274 GHz, where ω_r is the resonant frequency of the resonator when the qubit is in state |0〉, while applying an additional continuous microwave field at ω_d to the qubit control port. In the figure, |Γ| is plotted as a function of ω_d and f. Figures 2(c) and (d) are the same as figures 2(a) and (b), but the fitting curves are plotted together. As seen in the figures, the data are well fitted by the Hamiltonian (equation (1)), and from this fitting, we extracted the device parameters, which are listed in table 1 for all the devices. In the table, we also listed the parameters of the device used in [13]. The obtained C_c are quite consistent with the designs.

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Next, we investigate the dispersive shift. Using a pulsed readout, we measure Γ as a function of ω_p. By applying a π-pulse to the qubit before the readout, we can measure Γ corresponding to the qubit state |1〉. The pulse sequence is shown in the inset of figure 3. We use 100 ns long time trace data of the reflected readout pulse, which starts after a delay t_d from the termination of the π-pulse, to extract the amplitude and the phase. The delay t_d is adjusted to give a maximum contrast between the qubit states |0〉 and |1〉. Figure 3 shows the amplitude of the normalized reflection coefficient Γ' as a function of ω_p. Γ' is defined as Γ' ≡ (Γ − Γ_on)/|Γ_off − Γ_on|, where Γ_on and Γ_off are the reflection coefficients obtained by the on and the off resonance of the resonator, respectively [13]. For each device, we measure |Γ'| with and without the π-pulse, where the qubit is biased at f = 0.5. The shifts of the dip frequency correspond to the dispersive shifts. The observed dispersive shifts 2χ and ω_r are summarized in table 2, together with ω₀₁ determined by the spectroscopy measurements. In the table, we also listed the theoretical parameters obtained from the energy band calculation using the fitting parameters listed in table 1. The measured dispersive shifts are close to the theoretical predictions which are obtained by calculating the difference between the cavity resonant frequencies when the qubit is in state |0〉 and state |1〉. Their magnitudes are much larger than the values estimated under the two-level approximation for the qubit, namely, |g²₀₁/Δ₀₁|, where Δ₀₁ = ω₀₁ − ω_r. Here, g₀₁ represents the matrix element $\langle 1|\mathcal {H}_{{\rm cq}}^{{\rm c}}|0 \rangle$, where |0〉 and |1〉 are the ground and the first-excited states of the uncoupled qubit Hamiltonian (equation (2)), respectively, and $\mathcal {H}_{{\rm cq}}^{{\rm c}}$ is the coupling Hamiltonian $\mathcal {H}_{{\rm c}}^{{\rm c}}$ (equation (4)) restricted onto the qubit subspace. This enhancement is due to the effect of the straddling regime [7]. As seen in table 2, the condition for the straddling regime ω₀₁ < ω_r < ω₁₂ is satisfied in all the devices.

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As shown in the previous section and in [13], we observed a large enhancement of the dispersive shift of the cavity resonance due to the effect of the straddling regime [7]. It was observed in the system shown in figure 4(a), where a flux qubit is coupled to an LC resonator. However, a natural question is why this had not been reported before in much more extensively studied flux-qubit circuits with inductive coupling (figure 4(b)). Since the flux qubits typically have large anharmonicity, satisfying the condition for the straddling regime, namely, ω₀₁ < ω_r < ω₁₂, is quite easy. In fact, we even unintentionally satisfied this condition in our previous study using inductive coupling, but we did not observe such a large enhancement [17]. The purpose here is to answer this question.

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The Hamiltonian of the system where a flux qubit is capacitively coupled to an LC resonator as shown in figure 4(a) is given by

$$\begin{equation} \mathcal{H}_{\mathrm{C}}=\mathcal{H}_{{\rm q}}^{\mathrm{c}}+\mathcal{H}_{{\rm r}}^{\mathrm{c}}+\mathcal{H}_{{\rm c}}^{\mathrm{c}}, \end{equation} \tag{ 1 }$$

which consists of the qubit part, the resonator part and the coupling part. Each term of the Hamiltonian is given as below [13]:

$$\begin{equation} \fl \mathcal{H}_{{\rm q}}^{\mathrm{c}}=4E_{{\rm c}} \left[\frac{V}{X} (n_1^2 + n_2^2)+2\frac{W}{X}n_1n_2 \right]-E_{{\rm J}}\left[ \cos{\delta_1}-\cos{\delta_2}- \alpha \cos(\delta_1-\delta_2+2\pi f) \right], \end{equation} \tag{ 2 }$$

$$\begin{equation} \mathcal{H}_{{\rm r}}^{\mathrm{c}}=E_{{\rm r}}\sqrt{\frac{Y}{X}}\left(a^\dagger a+\frac{1}{2}\right), \end{equation} \tag{ 3 }$$

$$\begin{equation} \mathcal{H}_{{\rm c}}^{\mathrm{c}}=-2\mathrm{i}\frac{\sqrt{\beta_{\mathrm{c}}\gamma}}{Y^{1/4}X^{3/4}} \sqrt{E_{{\rm r}}E_{{\rm c}}}(n_1 - n_2)(a^\dagger - a), \end{equation} \tag{ 4 }$$

where E_J = I₀Φ₀/2π, f = Φ_ex/Φ₀, Φ₀ = h/2e, E_c = e²/2C_J, $E_{{\rm r}}=\hbar /\sqrt {L_{{\rm r}}C_{{\rm r}}}$, V =(1 + α)(1 + γ) + β_c, W = α(1 + γ) + β_c, X = (1 + 2α)(1 + γ) + 2β_c, Y =1 + 2α + 2β_c, β_c = C_c/C_J and γ = C_c/C_r. Here, I₀ and C_J are the critical current and the capacitance of the larger Josephson junction of the qubit, respectively. δ_i and n_i (i = 1, 2) are the phase differences across the larger junction, and its conjugate variable representing the charge number, respectively. a (a^†) is the annihilation (creation) operator of the photons in the resonator. Finally, L_r and C_r are the equivalent inductance and the capacitance of the resonator, respectively.

Next, we consider the Hamiltonian of the system where a flux qubit is inductively coupled to an LC resonator as shown in figure 4(b). The derivation is given in appendices A and B. The Hamiltonian is given by

$$\begin{equation} \mathcal{H}_L=\mathcal{H}_{{\rm q}}^{l}+\mathcal{H}_{{\rm r}}^{l}+\mathcal{H}_{{\rm c}}^{{l}}, \end{equation} \tag{ 5 }$$

and each term is given as follows:

$$\begin{eqnarray} &&\fl \mathcal{H}_{{\rm q}}^{l} = 2E_{{\rm c}}\Bigl( n_{{\rm a}}^2 + \frac{1}{1+2\alpha}n_{{\rm s}}^2 + \frac{2\alpha}{1+2\alpha}n_{{\rm t}}^2 \Bigr) - E_{{\rm J}}[\cos(\delta_{{\rm a}} - \delta_{{\rm s}} - \delta_{{\rm t}}) + \cos(\delta_{{\rm a}} + \delta_{{\rm s}} + \delta_{{\rm t}}) \nonumber \\ &&+\, \alpha \cos(2\pi f - \delta_{{\rm t}}/\alpha + 2\delta_{{\rm s}})]+ \frac{E_{{\rm J}}(1+2\alpha)}{2\alpha \beta_L} \delta_{{\rm t}}^2, \end{eqnarray} \tag{ 6 }$$

where β_L = αL_q/[(2α + 1)L_J], L_J = Φ₀/(2πI₀) and L_q is the loop inductance of the flux qubit. Here, we used the following variable transformations for δ (phase across the loop inductance of the qubit) and δ_i (i = 1,2,3 phases across the junction):

$$\begin{equation} \delta_{{\rm t}} = \frac{-\alpha}{1+2\alpha}\delta, \end{equation} \tag{ 7 }$$

$$\begin{equation} \delta_{{\rm s}} = \frac{1}{2(1+2\alpha)}[2\alpha (\delta_3 - 2\pi f) - \delta_1 - \delta_2], \end{equation} \tag{ 8 }$$

$$\begin{equation} \delta_{{\rm a}} = (\delta_1 - \delta_2)/2, \end{equation} \tag{ 9 }$$

and n_t, n_s and n_a are the corresponding conjugate variables. $\mathcal {H}_{{\rm r}}^l$ is given by

$$\begin{equation} \mathcal{H}_{{\rm r}}^{l} = E_{{\rm r}}\Bigl(a^{\dagger} a + \frac{1}{2} \Bigr). \end{equation} \tag{ 10 }$$

$\mathcal {H}_{{\rm c}}^{l}$ is given by

$$\begin{equation} \mathcal{H}_{{\rm c}}^{l} = -\frac{\pi}{\beta}\frac{E_{{\rm J}}}{E_{{M}}}\sqrt{E_{{\rm r}}E_{L{{\rm r}}}} (\hat{a}^{\dagger} + \hat{a}) \delta_{{\rm t}}, \end{equation} \tag{ 11 }$$

where E_M = Φ²₀/(2M) and E_Lr = Φ²₀/(2L_r).

Based on these, we compare the energy band structure of the two systems. Figures 5(a)–(c) show the excitation energy from the ground state, the coupling strength |g_ij| and the dispersive shift χ_ij, respectively, for the capacitive coupling case. Here, $g_{ij}=\langle i|\mathcal {H}_{{\rm cq}}^{\mathrm {c}}|j\rangle$, where |i〉 represents the ith eigenstate of the uncoupled qubit Hamiltonian (equation (2)), and χ_ij = |g_ij|²/(ω_ij − ω_r), where ω_ij = ω_j − ω_i. The parameters are taken from device C, namely, E_J = 148, E_c = 3.29 GHz, α = 0.613, C_c = 4.19 fF and ω_r/2π = 10.298 GHz. Figures 5(d)–(f) show the corresponding calculations for the inductive coupling case. The qubit parameters, E_J, E_c and α, are kept the same as those for the capacitive coupling case. The mutual inductance M = 8.90 pH and the resonator frequency ω_r = 13.80 GHz are chosen in such a way that |g₀₁| and ω₀₁ − ω_r, and hence, χ₀₁ become almost equal for both of the coupling cases. By comparing figures 5(c) and (f), we see that the total dispersive shift χ_total for the inductive coupling case is much smaller than that for the capacitive coupling case. Here, χ _total is obtained by calculating $\chi _{01}-\chi _{10}+\sum _{j=2}^M(\chi _{j1}-\chi _{1j}-\chi _{j0}+\chi _{0j})/2$ with M = 4, and confirmed to be very close to the result obtained by the energy band calculation [13]. χ_total at f = 0.5 is 8.6 MHz for the inductive coupling, while that for the capacitive coupling is 35.9 MHz. Note that the condition for the straddling regime, ω₀₁ < ω_r < ω₁₂, is satisfied in both cases. Indeed, the magnitude of χ_total is larger than that of χ₀₁, meaning that it is enhanced by the effect of the straddling regime. However, the degree of the enhancement is very different for the two cases.

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There are two main reasons that make this difference. Firstly, by comparing figures 5(a) and (c), we see that ω₁₂ is much smaller for the capacitive coupling case. Since the qubit parameters are the same, this reduction is due to C_c. This leads to smaller |Δ₁₂| = |ω_r − ω₁₂|, and hence larger χ₁₂. Secondly, by comparing figures 5(b) and (e), we see that |g₁₂|/|g₀₁| is larger for the capacitive coupling case. The |g₁₂|/|g₀₁| at f = 0.50 is 2.4 for the capacitive coupling, while that for the inductive coupling is 0.77. This also leads to larger χ₁₂ in the capacitive coupling case. Thus, we conclude that the capacitive coupling plays an essential role in the observed large enhancement of the dispersive shift.

We can also choose the parameters in such a way that the qubit transition energies (ω₀₁ or both ω₀₁ and ω₁₂) are similar in the capacitive and the inductive coupling cases. The results are shown in appendix C. The conclusion that |g₁₂|/|g₀₁| and χ_total are larger for the capacitive coupling case is the same.

We investigate the coherence properties of the flux qubit capacitively coupled to the resonator. Figure 6 shows an example of the measurements of (a) the energy relaxation and (b) the echo dephasing in device C, where the qubit is biased at f = 0.5004. The heterodyne voltage V₀, which is offset in such a way that it becomes zero when the qubit is in the ground state, is plotted as a function of the delay time. The pulse sequence for each measurement is shown in the inset of the figure. The data for the energy relaxation measurement are well fitted by an exponential decay of $\exp \,(-t/T_1)$ and give the qubit energy relaxation time T₁. The oscillation observed in the echo measurement is due to the fact that the rotation axis of the second π/2 pulse is rotated at 100 MHz. It is not so clear whether the decay envelope is exponential or Gaussian. In the figure, we fitted the data by an oscillating Gaussian, namely, $\exp \,(-t/2T_1)\exp \,[-(t/\tau _{{\rm echo}})^2]\sin \,(\omega t + \phi )$ and extracted the echo decay time τ _echo.

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We measured the flux bias dependence of T₁ and τ_echo in devices B and C, and plot them in figures 7(a) and (c), respectively. T₁ is independent of f in the measured range, and the magnitude is almost the same for both devices in spite of their difference in coupling capacitance. In the figure, we also plot the theoretical predictions based on the Purcell effect, κ(g₀₁/Δ₀₁)² [7]. The observed T₁ are more than one order of magnitude below the Purcell limit, and probably limited by something which commonly exists in both devices, such as the two-level systems at the surface of the substrate [18, 19].

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The observed τ_echo also looks very similar in the two devices. It becomes maximum at f = 0.5 and rapidly decreases as we move away from this bias point. To account for this flux dependence of τ_echo, we assumed the dephasing due to the 1/ν flux noise (ν is the frequency), which was carefully studied for the flux-qubit circuit with the dc-SQUID (superconducting quantum interference device) readout [20]. We fitted 1/τ_echo by the derivative of ω₀₁ with respect to f, namely,

$$\begin{equation} 1/\tau_{{\rm photon}} = \frac{\sqrt{A\ln{2}}}{\hbar}\left|\frac{\partial \omega_{01}}{\partial f}\right|, \end{equation} \tag{ 12 }$$

as shown in figures 7(b) and (d). In both devices, the fitting looks good, and the fitting parameter A is determined to be 6.6 × 10⁻⁶ and 6.4 × 10⁻⁶ for devices B and C, respectively. These values are of the same order as those reported previously [20–22], supporting the scenario of dephasing due to the 1/ν flux noise.

As shown in the figure, τ_echo saturates at ∼1 μs at f = 0.5. Note that τ_echo is the pure dephasing time and should not be limited by 2T₁. One possible source of the limit is the dephasing induced by the fluctuations of the photon number in the resonator [1, 23–25]. In the present devices, the cavity decay rate (κ/2π = f_r/Q) is about 15 MHz, and the qubit decay rate (γ/2π = T⁻¹₁/2π) is about 230 kHz. Thus, devices B and C are in the strong-dispersive regime [26], which means that 2χ > κ,γ. As shown in [25], the dephasing rate due to the photon number fluctuations in the strong-dispersive regime is given by

$$\begin{equation} \tau_{{\rm photon}}^{-1}=\kappa[(\langle n \rangle+1)N + \langle n \rangle(N+1)], \end{equation} \tag{ 13 }$$

where 〈n〉 is the mean photon number in the resonator and N corresponds to the N-photon resonator state. Since the present data were measured using the fundamental qubit transition frequency (N = 0),⁷ τ⁻¹ _photon is simply given by κ〈n〉, which does not depend either on χ or f. To account for the τ_echo of 1 μs with given κ, 〈n〉 of 0.01 (or the effective temperature T_eff of the resonator of 0.1 K with the assumption that $\langle n \rangle = [\exp \,(\hbar \omega _{{\rm r}}/k_{{\rm B}}T_{{\rm eff}})-1]^{-1}$) is needed, although at present we do not precisely know the actual value of 〈n〉.

As shown in the previous section, devices B and C are in the strong-dispersive regime [26]. The hallmark of this regime is the appearance of separate spectral lines of the qubit transition frequency, each of which corresponds to a particular photon number state of the resonator. This effect can be applied to characterize the photon statistics in the resonator [26]. Here, we investigate this effect using device C.

We measured the reflection coefficient Γ of the probe field (frequency ω_p/2π and power P_p) applied via the readout port, by varying the frequency ω_d/2π and the power P_d of the qubit drive field applied via the control port. Figure 8 shows |Γ| as a function of P_d and ω_d, where ω_p is fixed at ω_r/2π = 10.274 GHz. P_p is −140 dBm for figure 8(a) and −130 dBm for figure 8(b), which correspond to the mean photon number in the resonators of 0.06 and 0.6, respectively. In the strong-dispersive regime, the qubit transition frequency depends on the resonator photon number N as ω₀₁ − 2Nχ (N = 0,1,2,...), and we expect to observe separate spectral lines at these frequencies. In figure 8(a), we observe a spectral line at ω _d/2π = 5.350 GHz at low P_d. This frequency is equal to ω₀₁/2π (see table 2) and corresponds to the zero-photon state. As we increase P_d, another spectral line at 5.279 GHz appears. This frequency is equal to (ω₀₁ − 2χ)/2π and corresponds to the one-photon state. The result of the same measurement with 10 dB higher P_p is shown in figure 8(b). The larger P_p produces the population of higher photon-number states. Indeed, we observe more spectral lines up to the two-photon state.

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Note that the observed spectral lines are dips, not peaks. Without the qubit drive, a shallow (< 1 dB) dip is observed at ω_r (data not shown) due to the finite internal quality factor (∼ 3 × 10⁴) of the resonator. Thus, a small increase in |Γ| is naively expected when the qubit excitation induces the dispersive shift of the cavity resonance, although the expected magnitude of the signal (< 1 dB) is anyway much smaller than that observed here. The origin of these deep dips is inelastic scattering [27], which is intuitively understood as follows.

Figure 9 shows the level structure of the qubit–resonator system in the dispersive regime in a frame rotating at the qubit drive frequency ω_d. Here, we consider only the lowest two levels for the qubit, |0〉_q and |1〉_q: the higher levels do not play an essential role here, except that they enhance the magnitude of the dispersive level shift [13]. Figure 9(a) represents the energy diagram when the qubit drive is turned off. The label for each level |m,n〉 represents the state where the qubit is in the mth state and the resonator is in the nth state. Note that |m,n〉 is not a simple product of Fock states of the uncoupled qubit and the resonator due to the coupling g_ij. However, in the dispersive regime, g_ij mixes the two states |i〉_q|n + 1〉_r and |j〉_q|n〉_r only slightly and produces a dispersive level shift. As a result, the level spacings of the left ladder (|0〉_q) and the right ladder (|1〉_q) in figure 9(a) become different. Since there is no qubit drive, the microwave transition is of cavity origin and occurs vertically within each ladder. The probe field, which is adjusted at the transition frequency of the left ladder, is then scattered elastically.

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In contrast, when the qubit drive is turned on, state mixing between the two ladders can happen. For example, when ω_d = ω₀₁, |0,0〉 and |1,0〉 become degenerate (figure 9(b), yellow box), and these two states mix strongly. This gives rise to a new radiative decay path from the first excited state of the left ladder (originally |0,1〉) to the ground state of the right ladder (originally |1,0〉). Accordingly, the probe field can induce the Raman transition (red arrow in figure 9(b)). This inelastic scattering results in the reduction of the reflection coefficient. We can expect a similar reduction in |Γ| when ω_d is adjusted to be ω₀₁ − 2Nχ as shown in figures 9(c) and (d) for N = 1 and 2, respectively. As we increase the probe power and accordingly the mean photon number in the resonator, we are able to see the effect for larger N, which is what we observed in figure 8(b).

To understand the experimental results more quantitatively, we analyzed the microwave response of this system using a full-quantum theoretical model in which the qubit, the resonator and the propagating microwave modes are treated quantum mechanically, and calculated the reflection coefficient |Γ| (figures 8(c) and (d)) assuming the same parameter values in the experiment. The details of the calculation are summarized in appendix D. We observe fairly good agreement between the experiment and the theory.

In conclusion, we studied systematically the system where a superconducting flux qubit is capacitively coupled to an LC resonator. In all three devices with different coupling capacitances, the dispersive shifts are enhanced by the effect of the straddling regime, which is quantitatively reproduced by the theory. We showed by numerical calculation that the capacitive coupling has two effects on the dispersive shift, namely, to reduce the qubit anharmonicity and to produce the large matrix element g₁₂, both of which lead to the large enhancement in χ. We investigated the coherence properties in these devices. Even in the device with the largest C_c (device C) the observed T₁ is one order below the Purcell limit. We also showed that the flux dependence of the echo dephasing is consistent with the low-frequency flux noise. We also demonstrated the discrete ac-Stark effect using the device in the strong-dispersive regime. The results can be explained by the inelastic scattering of the microwave incident on the resonator, and well reproduced by the numerical calculations.

The authors are grateful to F Yoshihara for a useful discussion. Sputtered Nb films were fabricated in the clean room for analogue–digital superconductivity (CRAVITY) in the National Institute of Advanced Industrial Science and Technology (AIST). This work was partly supported by the Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST), Project for Developing Innovation Systems of MEXT, MEXT KAKENHI (grant numbers 21102002 and 25400417), SCOPE (111507004) and National Institute of Information and Communications Technology (NICT).

Here, we derive the Hamiltonian and the central equation for a single flux qubit, where we explicitly take into account the loop inductance. The derivation is based on [28], but with a difference that the periodicity of the potential terms in the Hamiltonian is taken into account. Probably because of this difference, the calculated energy bands do not have the doublet structure reported in [28].

First, we consider a single 3JJ flux qubit with a loop inductance as shown in figure A.1(a). From the flux quantization,

$$\begin{equation} \delta_1+\delta_2+\delta_3-\delta=2\pi f. \end{equation} \tag{ A.1 }$$

We introduce a dimensionless parameter representing the ratio between the loop inductance and the Josephson inductance

$$\begin{equation} \beta_{{\rm L}} \equiv \frac{2\pi L_{{\rm q}}}{\Phi_0}\Bigl( \frac{1}{I_0}+\frac{1}{I_0}+\frac{1}{\alpha I_0}\Bigr)^{-1} \end{equation} \tag{ A.2 }$$

$$\begin{equation} \hphantom{\beta_{{\rm L}}} = \frac{\alpha}{2\alpha+1} \frac{L_{{\rm q}}}{L_{{\rm J}}}, \end{equation} \tag{ A.3 }$$

where

$$\begin{equation} L_{{\rm J}} \equiv \frac{\Phi_0}{2\pi}\frac{1}{I_0}. \end{equation} \tag{ A.4 }$$

Following [28], we perform the following variable transformations:

$$\begin{equation} \delta_{{\rm t}} = \frac{-\alpha}{1+2\alpha}\delta, \end{equation} \tag{ A.5 }$$

$$\begin{equation} \delta_{{\rm s}}= \frac{1}{2(1+2\alpha)}[2\alpha (\delta_3 - 2\pi f) - \delta_1 - \delta_2], \end{equation} \tag{ A.6 }$$

$$\begin{equation} \delta_{{\rm a}} = (\delta_1 - \delta_2)/2, \end{equation} \tag{ A.7 }$$

or equivalently

$$\begin{equation} \delta_1 = \delta_{{\rm a}} - \delta_{{\rm t}} - \delta_{{\rm s}}, \end{equation} \tag{ A.8 }$$

$$\begin{equation} \delta_2 = - (\delta_{{\rm a}} + \delta_{{\rm t}} + \delta_{{\rm s}}), \end{equation} \tag{ A.9 }$$

$$\begin{equation} \delta_3 = 2\pi f - \delta_{{\rm t}}/\alpha + 2\delta_{{\rm s}}. \end{equation} \tag{ A.10 }$$

The potential energy of the JJs is expressed by

$$\begin{equation} \fl U_{{\rm J}} = -E_{{\rm J}}[\cos(\delta_{{\rm a}} - \delta_{{\rm s}} - \delta_{{\rm t}}) + \cos(\delta_{{\rm a}} + \delta_{{\rm s}} + \delta_{{\rm t}}) + \alpha \cos(2\pi f - \delta_{{\rm t}}/\alpha + 2\delta_{{\rm s}})] \end{equation} \tag{ A.11 }$$

$$\begin{eqnarray} &&\fl \hphantom{U_{{\rm J}}} = -\frac{E_{{\rm J}}}{2}\Bigl[ \mathrm{e}^{\mathrm{i}(\delta_{{\rm a}}+\delta_{{\rm s}})}\mathrm{e}^{\mathrm{i}\delta_{{\rm t}}} + \mathrm{e}^{-\mathrm{i}(\delta_{{\rm a}}-\delta_{{\rm s}})}\mathrm{e}^{\mathrm{i}\delta_{{\rm t}}} + \mathrm{e}^{\mathrm{i}(\delta_{{\rm a}}-\delta_{{\rm s}})}\mathrm{e}^{-\mathrm{i}\delta_{{\rm t}}} + \mathrm{e}^{-\mathrm{i}(\delta_{{\rm a}}+\delta_{{\rm s}})}\mathrm{e}^{-\mathrm{i}\delta_{{\rm t}}} \Bigr] \nonumber\\ &&\fl \hphantom{U_{{\rm J}} =} -\alpha \frac{E_{{\rm J}}}{2}\Bigl[ \mathrm{e}^{\mathrm{i}2\delta_{{\rm s}}}\mathrm{e}^{\mathrm{i}(2\pi f - \delta_{{\rm t}}/\alpha)} + \mathrm{e}^{-\mathrm{i}2\delta_{{\rm s}}}\mathrm{e}^{-\mathrm{i}(2\pi f - \delta_{{\rm t}}/\alpha)} \Bigr], \end{eqnarray} \tag{ A.12 }$$

where E_J = Φ₀I₀/(2π). The energy of the loop inductance is

$$\begin{equation} U_{\rm l} = \frac{1}{2L_{{\rm q}}}\Bigl(\frac{\Phi_0}{2\pi}\Bigr)^2 \delta^2 \end{equation} \tag{ A.13 }$$

$$\begin{equation} \hphantom{U_{\rm l}} = \frac{E_{{\rm J}}(1+2\alpha)}{2\alpha \beta_L} \delta_{{\rm t}}^2. \end{equation} \tag{ A.14 }$$

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The kinetic energy of the JJs is

$$\begin{equation} T = \frac{C_{{\rm J}}}{2}\Bigl(\frac{\Phi_0}{2\pi}\Bigr)^2[\dot{\delta_1}^2+\dot{\delta_2}^2+\alpha \dot{\delta_3}^2], \end{equation} \tag{ A.15 }$$

where C_J is the capacitance of the larger JJ. Using equations (A.8) and (A.10)

$$\begin{equation} T = \frac{C_{{\rm J}}}{2}\Bigl(\frac{\Phi_0}{2\pi}\Bigr)^2\Bigl[2\dot{\delta_{{\rm a}}}^2+ 2(1+2\alpha)\dot{\delta_{{\rm s}}}^2+\frac{1+2\alpha}{\alpha} \dot{\delta_{{\rm t}}}^2\Bigr]. \end{equation} \tag{ A.16 }$$

Note that there are no cross terms such as $\dot {\delta _{{\rm a}}}\dot {\delta _{{\rm s}}}$ thanks to the variable transformation. The Lagrangian of the system is

$$\begin{equation} \mathcal{L}_{{\rm q}} = T - U_{{\rm J}} - U_{{\rm l}} \end{equation} \tag{ A.17 }$$

and the canonical momentum is

$$\begin{equation} p_i = \frac{\partial \mathcal{L}_{{\rm q}}}{\partial \dot{\delta_i}}, \end{equation} \tag{ A.18 }$$

where i = a, s and t. Finally, the Hamiltonian is given by

$$\begin{equation} \mathcal{H}_{{\rm q}} = \sum_i p_i \dot{\delta_i} - \mathcal{L}_{{\rm q}} \end{equation} \tag{ A.19 }$$

$$\begin{equation} \hphantom{\mathcal{H}_{{\rm q}}} = \frac{2E_{{\rm c}}}{\hbar^2}\Bigl( p_{{\rm a}}^2 + \frac{1}{1+2\alpha}p_{{\rm s}}^2 + \frac{2\alpha}{1+2\alpha}p_{{\rm t}}^2 \Bigr) + U_{{\rm J}} + U_{{\rm l}}, \end{equation} \tag{ A.20 }$$

where E_c is defined for a single charge, namely E_c = e²/(2C_J). This Hamiltonian can be regarded as a 3JJ flux qubit without the loop inductance interacting with a harmonic oscillator. The harmonic oscillator part (third and fifth terms in equation(A.20)) can be transformed as

$$\begin{equation} \frac{2E_{{\rm c}}}{\hbar^2}\frac{2\alpha}{1+2\alpha}p_{{\rm t}}^2 + \frac{E_{{\rm J}}(1+2\alpha)}{2\alpha \beta_L} \delta_{{\rm t}}^2 = \frac{p_{{\rm t}}^2}{2m_{{\rm t}}} + \frac{1}{2}m_{{\rm t}}\omega_{{\rm t}}^2\delta_{{\rm t}}^2, \end{equation} \tag{ A.21 }$$

$$\begin{equation} m_{{\rm t}} = \frac{\hbar^2}{8E_{{\rm c}}}\frac{1+2\alpha}{\alpha}, \end{equation} \tag{ A.22 }$$

$$\begin{equation} \omega_{{\rm t}} = \frac{2E_{{\rm c}}}{\hbar}\sqrt{\frac{2E_{{\rm J}}}{\beta_L E_{{\rm c}}}}. \end{equation} \tag{ A.23 }$$

As in [28], we choose for our basis functions the product states

$$\begin{equation} |\phi_{klm} \rangle = |\phi_k^{{\rm a}} \rangle |\phi_l^{{\rm s}} \rangle |\phi_m^{{\rm t}} \rangle, \end{equation} \tag{ A.24 }$$

where the first two states are plane waves

$$\begin{equation} |\phi_k^{{\rm a}} \rangle = (2\pi)^{-1/2} \mathrm{e}^{-\mathrm{i}k\delta_{{\rm a}}}, \end{equation} \tag{ A.25 }$$

$$\begin{equation} |\phi_l^{{\rm s}} \rangle = (2\pi)^{-1/2} \mathrm{e}^{-\mathrm{i}l\delta_{{\rm s}}} \end{equation} \tag{ A.26 }$$

and the third state is a harmonic oscillator wave function in δ_t

$$\begin{equation} |\phi_m^{{\rm t}} \rangle = \Bigl[ \frac{m_{{\rm t}}\omega_{{\rm t}}}{2^{2m}\pi \hbar (m!)^2} \Bigr]^{1/4} H_m\Bigl[ \sqrt{\frac{m_{{\rm t}}\omega_{{\rm t}}}{\hbar}} \delta_{{\rm t}} \Bigr] \exp\Bigl[ \frac{-m_{{\rm t}}\omega_{{\rm t}}\delta_{{\rm t}}^2}{2\hbar} \Bigr], \end{equation} \tag{ A.27 }$$

where H_m[δ] is the mth Hermite polynomial. The wavefunction of the Hamiltonian is given by

$$\begin{equation} |\phi \rangle = \sum_{(k,l) \in \vec{g}} \sum_m C(k,l,m)|\phi_{klm} \rangle, \end{equation} \tag{ A.28 }$$

where the first sum of k and l is taken for the reciprocal lattice of U_J(k,l) [29]. The qubit potential energy has the following translational symmetry with respect to δ_a and δ_s

$$\begin{equation} \delta_{{\rm a}} \rightarrow \delta_{{\rm a}} + 2\pi, \end{equation} \tag{ A.29 }$$

$$\begin{equation} \delta_{{\rm s}} \rightarrow \delta_{{\rm s}} + 2\pi, \end{equation} \tag{ A.30 }$$

$$\begin{equation} \delta_{{\rm a}},~\delta_{{\rm s}} \rightarrow \delta_{{\rm a}} + \pi,~\delta_{{\rm s}}+\pi. \end{equation} \tag{ A.31 }$$

Thus, its symmetry forms a two-dimensional face-centered lattice as shown in figure A.2(a). It is not a simple square lattice, which is due to the variable transformation. The corresponding reciprocal lattice is also a face-centered lattice shown in figure A.2(b). Thus, the sum for (k,l) in equation (A.28) must be taken for this lattice, and how many points we should take depends on what accuracy we need. Using $p_i = -\mathrm {i}\hbar \frac {\partial }{\partial \delta _i}$, the Schrödinger equation $\mathcal {H}|\phi _{klm} \rangle = \epsilon |\phi _{klm} \rangle$ is written as

$$\begin{eqnarray}&&\fl \sum\nolimits{{\hbox{$\!'$}}} \Bigl[ 2E_{{\rm c}} k^2 + \frac{2E_{{\rm c}}}{1+2\alpha}l^2 + m\hbar \omega_{{\rm t}} \Bigr] C(k,l,m) |\phi_{klm} \rangle - \sum\nolimits{{\hbox{$\!'$}}} U_{{\rm J}} C(k,l,m) |\phi_{klm} \rangle \nonumber\\ &&= \epsilon \sum\nolimits{{\hbox{$\!'$}}} C(k,l,m) |\phi_{klm} \rangle, \end{eqnarray} \tag{ A.32 }$$

where $\sum'$ means $\sum_{(k,l) \in \vec{g}} \sum_m$. By multiplying 〈ϕ_k'l'm'| on both sides, we obtain the following central equation:

$$\begin{eqnarray} &&\fl \left[ 2E_{{\rm c}} {k'}^2 + \frac{2E_{{\rm c}}}{1+2\alpha}{l'}^2 + {m'}\hbar \omega_{{\rm t}}\right] C(k',l',m') - \frac{E_{{\rm J}}}{2}\sum_m \left[ C(k'-1,l'-1,m) \langle \phi_{m'}^{{\rm t}} |\mathrm{e}^{\mathrm{i}\delta_{{\rm t}}}|\phi_m^{{\rm t}} \rangle \right.\nonumber\\ &&+ \left.C(k'+1,l'-1,m) \langle \phi_{m'}^{{\rm t}} |\mathrm{e}^{\mathrm{i}\delta_{{\rm t}}}|\phi_m^{{\rm t}} \rangle+ C(k'-1,l'+1,m) \langle \phi_{m'}^{{\rm t}} |\mathrm{e}^{-\mathrm{i}\delta_{{\rm t}}}|\phi_m^{{\rm t}} \rangle \right.\nonumber\\ &&\left.+ C(k'+1,l'+1,m) \langle \phi_{m'}^{{\rm t}} |\mathrm{e}^{-\mathrm{i}\delta_{{\rm t}}}|\phi_m^{{\rm t}} \rangle\right]- \alpha \frac{E_{{\rm J}}}{2} \sum_m \left[ C(k',l'-2,m) \mathrm{e}^{\mathrm{i}2\pi f}\right.\nonumber\\ &&\left. \times \langle \phi_{m'}^{{\rm t}} |\mathrm{e}^{-\mathrm{i}\delta_{{\rm t}}/\alpha}|\phi_m^{{\rm t}} \rangle + C(k',l'+2,m) \mathrm{e}^{-\mathrm{i}2\pi f}\langle \phi_{m'}^{{\rm t}} |\mathrm{e}^{\mathrm{i}\delta_{{\rm t}}/\alpha}|\phi_m^{{\rm t}} \rangle \right]= \epsilon C(k',l',m').\nonumber\\ \end{eqnarray} \tag{ A.33 }$$

To solve the above equation, we use the following formula [30]:

$$\begin{eqnarray} &&\fl \langle n |\mathrm{e}^{c\hat{x}}|m \rangle =(m!n!)^{-1/2}\mathrm{e}^{\frac{\hbar c^2}{4m_{{\rm t}}\omega_{{\rm t}}}} \sum_{j=0}^{{\rm min}(m,n)} j!\left( \begin{array}{c} m \\ j \end{array} \right) \left( \begin{array}{c} n \\ j \end{array} \right) \Bigl( \frac{\hbar}{2m_{{\rm t}} \omega_{{\rm t}}} \Bigr)^{(m+n-2j)/2}c^{m+n-2j}, \end{eqnarray} \tag{ A.34 }$$

where |m〉 is the mth eigenfunction of the harmonic oscillator and c is a constant which can be a complex number. For example, to calculate 〈ϕ^t_m'|e^iδ_t|ϕ^t_m〉, we use the above formula with

$$\begin{equation} c = i \end{equation} \tag{ A.35 }$$

$$\begin{equation} \frac{\hbar}{2m_{{\rm t}}\omega_{{\rm t}}} = \frac{2\alpha}{1+2\alpha}\sqrt{\frac{\beta_L E_{{\rm c}}}{2 E_{{\rm J}}}}. \end{equation} \tag{ A.36 }$$

By numerically solving the central equation (A.33) for (k',l') in the subset of the reciprocal lattice and m' from 0 to a certain number, we obtain the energy band diagram for a flux qubit with non-negligible loop inductance.

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Next, we consider a flux qubit inductively coupled to an LC resonator as shown in figure 1(b). The Hamiltonian consists of three parts, namely,

$$\begin{equation} \mathcal{H}_{L} = \mathcal{H}_{{\rm q}}^l + \mathcal{H}_{{\rm r}}^l + \mathcal{H}_{{\rm c}}^l. \end{equation} \tag{ B.1 }$$

$\mathcal {H}_{{\rm q}}^l$ is already shown in equation (A.20). $\mathcal {H}_{{\rm r}}^l$ is simply given by

$$\begin{equation} \mathcal{H}_{{\rm r}}^l = E_{{\rm r}} \Bigl( a^{\dagger} a + \frac{1}{2} \Bigr), \end{equation} \tag{ B.2 }$$

where $E_{{\rm r}} = \hbar /\sqrt {L_{{\rm r}}C_{{\rm r}}}$, and a (a^†) is the annihilation (creation) operator of the photons in the resonator. $\mathcal {H}_{{\rm c}}^l$ is given by

$$\begin{equation} \mathcal{H}_{{\rm c}}^l = \frac{M}{L_{{\rm r}}L_{{\rm q}}}\Bigl( \frac{\Phi_0}{2\pi} \Bigr)^2 \delta_{{\rm r}} \delta. \end{equation} \tag{ B.3 }$$

Using equations (A.3), (A.5) and

$$\begin{equation} \frac{\delta_{{\rm r}}}{L_{{\rm r}}} = \Bigl( \frac{2\pi}{\Phi_0} \Bigr) \sqrt{\frac{E_{{\rm r}}}{2L_{{\rm r}}}} (\hat{a}^{\dagger} + \hat{a}), \end{equation} \tag{ B.4 }$$

we obtain

$$\begin{equation} \mathcal{H}_{{\rm c}}^{l} = -M\frac{E_{{\rm J}}}{\beta_L} \Bigl( \frac{2\pi}{\Phi_0} \Bigr) \sqrt{\frac{E_{{\rm r}}}{2L_{{\rm r}}}}(\hat{a}^{\dagger} + \hat{a}) \delta_{{\rm t}} \end{equation} \tag{ B.5 }$$

$$\begin{equation} \hphantom{\mathcal{H}_{{\rm c}}^{l}} = -\frac{\pi}{\beta_L}\frac{E_{{\rm J}}}{E_{{M}}}\sqrt{E_{{\rm r}}E_{L{\rm r}}}(\hat{a}^{\dagger} + \hat{a}) \delta_{{\rm t}}, \end{equation} \tag{ B.6 }$$

where we defined

$$\begin{equation} E_{M} \equiv \frac{\Phi_0^2}{2M}, \end{equation} \tag{ B.7 }$$

$$\begin{equation} E_{L{{\rm r}}} \equiv \frac{\Phi_0^2}{2L_{{\rm r}}}. \end{equation} \tag{ B.8 }$$

The matrix elements 〈ϕ_klm|δ_t|ϕ_k'l'm'〉 are given by

$$\begin{equation} \langle \phi_{klm} | \delta_{{\rm t}} | \phi_{k'l'm'} \rangle = \delta_{k,k'}\delta_{l,l'} \langle \phi_{m}^t | \delta_{{\rm t}} | \phi_{m'}^t \rangle. \end{equation} \tag{ B.9 }$$

And the last term is given by

$$\begin{equation} \langle \phi_{m}^t | \delta_{{\rm t}} | \phi_{m'}^t \rangle = \left\{\begin{array}{@{}ll@{}}\sqrt{m'+1}\sqrt{\frac{2\alpha}{1+2\alpha}} \Bigl( \frac{\beta_L E_{{\rm c}}}{2E_{{\rm J}}} \Bigr)^{1/4} & (m=m'+1),\\ \sqrt{m'}\sqrt{\frac{2\alpha}{1+2\alpha}} \Bigl( \frac{\beta_L E_{{\rm c}}}{2E_{{\rm J}}} \Bigr)^{1/4} & (m=m'-1),\\ 0 & (m \neq m' \pm 1). \end{array}\right. \end{equation} \tag{ B.10 }$$

In the calculation of figure 5(d), 121 reciprocal lattice points are used for k and l, while m is taken from 0 to 2. The number of bases used for the resonator state is 5.

In this appendix, we show the results of the numerical calculation for the inductively coupled system of the qubit and the resonator, but with different parameters from those used in figure 5. Figures C.1(a)–(c) show the flux bias dependence of the excitation energy from the ground state, the coupling strength |g_ij| and the dispersive shift χ_ij, respectively. Here, we adjusted the qubit parameter α in such a way that ω₀₁ is almost equal to that for the capacitive coupling case (figures 5(a)). Also, the mutual inductance M is adjusted so that |g₀₁| becomes almost equal to that for the capacitive coupling case (figure 5(b)). Similar to the results shown in figures 5(e) and (f), |g₁₂|/|g₀₁| and χ_total are much smaller than those for the capacitive coupling case. We can also adjust the qubit parameters (E_c and α) such that both ω₀₁ and ω₁₂ are similar to those for the capacitive coupling case. As shown in figures C.1(d)–(f), the overall behavior is the same, which indicates that the smaller |g₁₂|/|g₀₁| is not due to the larger anharmonicity.

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In this appendix, we analyze the microwave response of the present qubit–resonator system. Figure D.1 is a schematic of the theoretical model, which consists of a superconducting qubit, a resonator and waveguides. The lowest two levels of the qubit (|0〉 and |1〉) are relevant here in essence. However, to account for the dispersive level shift due to the second excited state |2〉 [13], we incorporate this level in our model. We refer to the readout line coupled to the resonator as waveguide 1, and the qubit control line as waveguide 2. We apply a monochromatic field E(t) = Ee^−iω_dt to drive the qubit through waveguide 2. By defining v as the microwave velocity in the waveguides, the Hamiltonian of the system is

$$\begin{equation} \fl {\cal H}(t) = {\cal H}_{{\rm sys}}(t)+{\cal H}_{{\rm damp}}, \end{equation} \tag{ D.1 }$$

$$\begin{equation} \fl {\cal H}_{{\rm sys}}(t)/\hbar = \sum_{j} \bar{\omega}_j \sigma_{jj} +\bar{\omega}_{{\rm r}} a^{\dagger}a +\sum_{i,j}g_{ij}(\sigma_{ji}a+a^{\dagger}\sigma_{ij})+\sum_{i,j}\sqrt{\gamma_{ij}}[E(t)\sigma_{ji}+E^*(t)\sigma_{ij}], \end{equation} \tag{ D.2 }$$

$$\begin{eqnarray} &&\fl {\cal H}_{{\rm damp}}/\hbar = \int \mathrm{d}k \left[ vkb_k^{\dagger}b_k +\sqrt{v\kappa/2\pi}(a^{\dagger}b_k+b_k^{\dagger}a) \right]\nonumber\\ &&+\int \mathrm{d}k \left[ vkc_k^{\dagger}c_k +\textstyle{\sum_{i,j}}\sqrt{v\gamma_{ij}/2\pi} (\sigma_{ji}c_k+c_k^{\dagger}\sigma_{ij}) \right], \end{eqnarray} \tag{ D.3 }$$

where σ_ij = |i〉〈j| is the transition operator of the qubit, a is the annihilation operator of the resonator and b_k (c_k) is the annihilation operator of the photon propagating in waveguide 1 (2) with wavenumber k. $\bar {\omega }_j$ is the bare eigenenergy of the jth qubit level with $\bar {\omega }_0=0$ (the eigenenergy of $\mathcal {H}_{{\rm q}}^{\mathrm {c}}$, equation (2)), $\bar {\omega }_{{\rm r}}$ is the bare resonator frequency ($\hbar \bar {\omega }_{{\rm r}}=E_{{\rm r}}\sqrt {Y/X}$ (see equation (3))), g_ij is the qubit–resonator coupling, κ is the resonator decay rate and γ_ij is the qubit decay rate for the |j〉 → |i〉 transition. From the parity selection rule and the rotating-wave approximation, g_ij and γ_ij are non-zero only for (i,j) = (0,1) and (1, 2).

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By switching to a frame rotating at the drive frequency ω_d, the Hamiltonian becomes static. $\cal {H}_{{\rm sys}}$ is then given by

$$\begin{eqnarray} &&\fl {\cal H}_{{\rm sys}}/\hbar=\sum_{j}(\bar{\omega}_j-j\omega_{{\rm d}})\sigma_{jj} + (\bar{\omega}_{{\rm r}}-\omega_{{\rm d}})a^{\dagger}a + \sum_{i,j}\left[ g_{ij}(\sigma_{ji}a+a^{\dagger}\sigma_{ij})+\sqrt{\gamma_{ij}}(E\sigma_{ij}+E^*\sigma_{ij}) \right].\nonumber\\ \end{eqnarray} \tag{ D.4 }$$

$\cal {H}_{{\rm damp}}$ remains unchanged except that the photon frequency is measured from ω_d. We define the dressed states of the qubit–resonator system as the eigenstates of $\cal {H}_{{\rm sys}}$. We denote them by $|\tilde {j}\rangle$ and their energies by $\tilde {\omega }_j$ (j = 1,2,...) from the lowest. $|\tilde {j}\rangle$ corresponds to the states shown in figures 9(b)–(d), or the states in figure 9(a) if E = 0. In the dressed-state basis, the Hamiltonian of the overall system is rewritten as

$$\begin{equation} \cal{H} = \cal{H}_{{\rm sys}}+\cal{H}_{{\rm damp}}, \end{equation} \tag{ D.5 }$$

$$\begin{equation} {\cal H}_{{\rm sys}}/\hbar = \sum_j \tilde{\omega}_j\tilde{\sigma}_{jj}, \end{equation} \tag{ D.6 }$$

$$\begin{eqnarray} &&\fl {\cal H}_{{\rm damp}}/\hbar = \int \mathrm{d}k \left[ vkb_k^{\dagger}b_k +\textstyle{\sum_{i,j}}\sqrt{v\tilde{\kappa}_{ij}/2\pi} (\tilde{\sigma}_{ij}b_k+b_k^{\dagger}\tilde{\sigma}_{ji}) \right]\nonumber\\ &&+ \int \mathrm{d}k \left[ vkc_k^{\dagger}c_k +\textstyle{\sum_{i,j}}\sqrt{v\tilde{\gamma}_{ij}/2\pi} (\tilde{\sigma}_{ij}c_k+c_k^{\dagger}\tilde{\sigma}_{ji}) \right], \end{eqnarray} \tag{ D.7 }$$

where $\tilde {\sigma }_{ij}=|\tilde {i}\rangle \langle \tilde {j}|$, $\tilde {\kappa }_{ij}$ ($\tilde {\gamma }_{ij}$) is the radiative decay rate into waveguide 1 (2) for the $|\tilde {j}\rangle \to |\tilde {i}\rangle$ transition. They are, respectively, given by

$$\begin{equation} \tilde{\kappa}_{ij} = \kappa|\langle\tilde{j}|a^{\dagger}|\tilde{i}\rangle|^2, \end{equation} \tag{ D.8 }$$

$$\begin{equation} \tilde{\gamma}_{ij} = \left|\textstyle{\sum_{m,n}\sqrt{\gamma_{mn}}\langle\tilde{j}|\sigma_{mn}|\tilde{i}\rangle }\right|^2. \end{equation} \tag{ D.9 }$$

We work in the dressed-state basis to analyze the microwave response of the system to a probe field applied from waveguide 1. From equation (D.5), the Heisenberg equation for $\tilde {\sigma }_{ij}$ is

$$\begin{eqnarray} &&\fl \frac{\mathrm{d}}{\mathrm{d}t}\tilde{\sigma}_{ij}= \sum_{m,n} \left[ \eta_{ijmn}\tilde{\sigma}_{mn} +\xi^{\kappa}_{ijmn}\tilde{\sigma}_{mn}b_{{\rm in}}(t) +\xi^{\kappa}_{jinm}b_{{\rm in}}^{\dagger}(t)\tilde{\sigma}_{mn} +\xi^{\gamma}_{ijmn}\tilde{\sigma}_{mn}c_{{\rm in}}(t) +\xi^{\gamma}_{jinm}c_{{\rm in}}^{\dagger}(t)\tilde{\sigma}_{mn} \right],\nonumber\\ \end{eqnarray} \tag{ D.10 }$$

where b_in (c_in) is the input field operator for waveguide 1 (2). By denoting $S_{\mu }=\sum _{i,j}\sqrt {\tilde {\mu }_{ij}}\tilde {\sigma }_{ji}$ (μ = κ,γ), the coefficients η_ijmn and ξ^μ_ijmn are given by

$$\begin{equation} \eta_{ijmn} = \mathrm{i}(\tilde{\omega}_i-\tilde{\omega}_j)\delta_{im}\delta_{jn} -\sum_{\mu=\kappa,\gamma} \langle \tilde{m}| [\tilde{\sigma}_{ij},S_{\mu}^{\dagger}]S_{\mu}+S_{\mu}^{\dagger}[S_{\mu},\tilde{\sigma}_{ij}] |\tilde{n}\rangle/2, \end{equation} \tag{ D.11 }$$

$$\begin{equation} \xi^{\mu}_{ijmn}= \mathrm{i}\sqrt{v}\langle\tilde{m}| [S_{\mu}^{\dagger},\tilde{\sigma}_{ij}] |\tilde{n}\rangle. \end{equation} \tag{ D.12 }$$

We apply a probe field with amplitude F and frequency ω_p from waveguide 1, while we apply no probe field from waveguide 2. Therefore, 〈b_in(t)〉 = Fe^{−i(ω_p−ω_d)t} and 〈c _in(t)〉 = 0. Note that the probe frequency ω_p is measured from the drive frequency ω_d in the rotating frame. The stationary solution of the equation (D.10) is then written as

$$\begin{equation} \langle\tilde{\sigma}_{ij}\rangle = \sum_{m=0,\pm1,\ldots} A^m_{ij}\mathrm{e}^{\mathrm{i}m(\omega_{{\rm p}}-\omega_{{\rm d}})t}. \end{equation} \tag{ D.13 }$$

Substituting equation (D.13) into equation (D.10) and solving the linear simultaneous equations together with the condition that $\mathrm {Tr}(\tilde {\sigma })=1$, i.e. $\sum _j A^m_{jj}=\delta _{m0}$, we numerically determine A^m_ij. Reliable numerical results are obtained by setting −2 ⩽ m ⩽ 2 and 1 ⩽ i,j ⩽ 8. The output field amplitude 〈b_out(t)〉 for waveguide 1 is related to the input one by 〈b_out(t)〉 = 〈b_in(t)〉 − i〈S_κ(t)〉. The reflected field contains high-order harmonics due to the nonlinear effect. However, in the reflectivity measurement, only the field at the fundamental frequency (m = −1) is relevant. The reflection coefficient is defined by Γ = 〈b_out(t)〉^m=−1/〈b_in(t)〉^m=−1. Therefore,

$$\begin{equation} \Gamma=1-\mathrm{i}\sum_{i,j}\sqrt{\tilde{\kappa}_{ij}}A_{ji}^{-1}/F. \end{equation} \tag{ D.14 }$$

In the numerical simulation, we have chosen the bare parameters of the qubit–cavity system ($\bar {\omega }_1$, $\bar {\omega }_2$, g₀₁ and g₁₂) to reproduce the following experimentally measured parameters, ω₀₁/2π = 5.35 GHz, ω_r/2π = 10.27 GHz and 2χ/2π = 71 MHz. We have set κ/2π = 15 MHz as stated in the main text. We employed the qubit radiative decay rate into waveguide 2 as γ₁₀, which is estimated to be γ₁₀/2π = 88 Hz from the Rabi oscillation measurement. We determined F based on the experimental parameter P_p. γ₂₁ has almost no effect on the numerical results.