Observation of Autler–Townes effect in a dispersively dressed Jaynes–Cummings system

The system studied in this experiment comprises a transmon [13], which can be thought of as a multi-level artificial 'atom' coupled to a single harmonic mode of a superconducting resonator. The coupled transmon–resonator system can be modelled to a good approximation [28] by a Jaynes–Cummings Hamiltonian [29] generalized to a multi-level atom [16, 30, 31]

$$\begin{equation}\fl H_{\rm JC} = \hbar {\omega}_{\rm{r}} (a^{\dagger} a ) + \hbar\!\!\!\!\!\! \sum_{j = \{g,e,f\ldots\}}\!\!\!\!\!\!{\omega}_j |j\rangle \langle j| + \hbar \!\!\!\!\!\! \sum_{j = \{g,e,f\ldots\}}\!\!\!\!\!\!g_{j,j\!+\!1} (a^{\dagger} |j\rangle \langle j\!+\!1| + a |j\!+\!1\rangle \langle j|)\,. \end{equation} \tag{ 1 }$$

Here ω_r is the bare resonator frequency, a^† (a) is the creation (annihilation) operator for the resonator mode, the transmon states |j〉 are labelled {g,e,f,...} and g_j,j+1 is the coupling strength of the $|j\rangle \leftrightarrow |j+1\rangle$ transition of the transmon with the resonator mode, assuming only coupling to quasi-resonance transitions. This Hamiltonian can be approximately diagonalized in the dispersive limit [11, 31, 32], Δ_j,j+1 ≡ ω_j,j+1 − ω_r ≫ g_j,j+1, where ω_j,j+1 ≡ ω_j+1 − ω_j is the frequency of the $|j\rangle \leftrightarrow |j+1\rangle$ transition. We can then write the diagonal Hamiltonian as a perturbative expansion in the small parameters λ_j,j+1 ≡ g_j,j+1/Δ_j,j+1 ≪ 1. In the context of our experiment, where the Rabi frequency of the transmon excitation is much smaller than the anharmonicity, we truncate the transmon to a two level system with ground-state |g〉 and first-excited state |e〉. To compensate for the finite anharmonicity of the transmon we include perturbative shifts to the energy levels of the system due to the second-excited state |f〉 [28]. The dispersively diagonalized Hamiltonian up to second order in λ_j,j+1 is [11, 28]

$$\begin{equation} \widetilde{H}^{(2)}_{\rm JC} \approx \hbar {\widetilde{\omega}_{\rm{r}}} (a^{\dagger} a ) + \frac{\hbar {\widetilde{\omega}_{ge}}}{2} \sigma_z + \hbar \chi (a^{\dagger} a) \sigma_z, \end{equation} \tag{ 2 }$$

where χ ≃ χ_ge − χ_ef/2 is the effective dispersive shift of the resonator due to the transmon levels, χ_j,j+1 ≡ g²_j,j+1/Δ_j,j+1 are the partial couplings, $\widetilde {\omega }_{\rm {r}} \simeq \omega _{\rm {r}} - \chi _{ef}/2$ is the dressed resonator frequency, $\widetilde {\omega }_{ge} \simeq \omega _{ge}+\chi _{ge}$ is the dressed qubit transition frequency and σ_z is the z-Pauli spin operator for the qubit.

The last term in the Hamiltonian represents the dispersive shift of the resonator frequency by an amount ±χ depending on the qubit state. It also produces an ac Stark shift of the qubit frequency by an amount 2χn due to n = 〈n|a^†a|n〉 photons in the resonator [11]. In the strong dispersive limit the dispersive shift χ ≫ {Γ,κ}, where Γ is the spectral line width of the qubit transition and κ is the resonator linewidth. The ac Stark shift due to each constituent Fock state |n〉 results in a separate peak in the qubit spectrum. The stationary distribution w_n of the Fock states, can then be well approximated by the relative areas under the individual photon-number peaks, with the average number of photons given by

$$\begin{equation} \bar{n} = \frac{\sum_{n} \! w_n n }{\sum_{n} \! w_n }. \end{equation} \tag{ 3 }$$

For a finite temperature, and in the absence of resonator driving, w_n is just the usual thermal distribution [18]. When the resonator is driven coherently from zero temperature [33], w_n approaches a Poisson distribution $w^{\mathrm { coh}}_n = {\mathrm { e}}^{-\bar {n}}(\bar {n})^n / n!$.

We now introduce a general drive Hamiltonian with independent 'coupler' and 'probe' tones in the basis of dressed resonator–qubit states. In our experiment the coupler tone with frequency ω_c drives the 'resonator-like' dressed transitions $|\widetilde {g,n}\rangle \leftrightarrow |\widetilde {g,n+1}\rangle$ and $|\widetilde {e,n}\rangle \leftrightarrow |\widetilde {e,n+1}\rangle$. The probe tone with frequency ω_p drives the 'qubit-like' transitions $|\widetilde {g,n}\rangle \leftrightarrow |\widetilde {e,n}\rangle$ in the Jaynes–Cummings ladder. The drive Hamiltonian in the rotating wave approximation (RWA) is

$$\begin{eqnarray} &&\fl H_{\rm drive} = \sum^{\infty}_{n=0} \left[\vphantom{\sum_{k=\{g,e\}} \frac{\hbar \Omega_{\rm{c}}}{2}} \frac{\hbar \Omega_{\rm{p}}}{2}(|\widetilde{g,n}\rangle \langle\widetilde{e,n}| {\rm e}^{{\rm i}\omega_{\rm{p}} t} + |\widetilde{e,n}\rangle \langle\widetilde{g,n}| {\rm e}^{-{\rm i}\omega_{\rm{p}} t}) \right. \nonumber\\ &&+\left. \sum_{k=\{g,e\}} \frac{\hbar \Omega_{\rm{c}}}{2}(|\widetilde{k,n}\rangle \langle\widetilde{k,n+1}| {\rm e}^{{\rm i}\omega_{\rm{c}} t} + |\widetilde{k,n+1}\rangle \langle\widetilde{k,n}|{\rm e}^{-{\rm i}\omega_{\rm{c}} t})\right] , \end{eqnarray} \tag{ 4 }$$

where Ω_p and Ω_c are the amplitudes of the probe and coupler tones [16], $|\widetilde {g,n}\rangle \langle \widetilde {e,n}|\,$ is a 'qubit-like' lowering operator, and $|\widetilde {k,n}\rangle \langle \widetilde {k,n+1}|\,$, with k∈{g,e}, being a 'resonator-like' lowering operator. For the 'resonator-like' transitions, taking into account higher order Kerr-type nonlinearities, only a few terms survive in the summation (4) and all others can be neglected in the RWA.

The drive Hamiltonian, (4), neglecting corrections of the order λ²_j,j+1 in the dispersive approximation, can be written as

$$\begin{equation} H_{\rm drive} \simeq \frac{\hbar \Omega_{\rm{c}}}{2}(a {\rm e}^{{\rm i}\omega_{\rm{c}} t}+ a^{\dagger} {\rm e}^{-{\rm i}\omega_{\rm{c}} t})+ \frac{\hbar \Omega_{\rm{p}}}{2}(\sigma^-{\rm e}^{{\rm i}\omega_{\rm{p}} t}+ \sigma^+ {\rm e}^{-{\rm i}\omega_{\rm{p}} t}). \end{equation} \tag{ 5 }$$

To simulate the system we use a density matrix formulation. We take into account Kerr-type nonlinearities by including terms up to fourth order in λ_j,j+1:

$$\begin{equation} H^{(4)}_{\rm{Kerr}} \simeq \hbar \zeta (a^{\dagger} a)^2 \sigma_z + \hbar \zeta' (a^{\dagger} a)^2, \end{equation} \tag{ 6 }$$

where ζ ≈ (χ_efλ²_ef − 2χ_geλ²_ge + 7χ_efλ²_ge/4 − 5χ_geλ²_ef/4) is the resonator–qubit cross-Kerr coefficient and ζ' ≈ (χ_ge − χ_ef)(λ²_ge + λ²_ef) is the resonator self-Kerr coefficient [16, 30]. Transforming the total Hamiltonian into the rotating frame of both the drives [28], we obtain a time-independent Hamiltonian

$$\begin{equation} \fl H_{\rm{tot}} \approx \hbar \widetilde{\Delta}_{\rm{c}} (a^{\dagger} a)+ \frac{\hbar \widetilde{\Delta}_{\rm{p}}}{2} \sigma_z + \hbar \chi (a^{\dagger} a)\sigma_z + H^{(4)}_{\rm{Kerr}}+ \frac{\hbar \Omega_{\rm{c}}}{2}(a+a^{\dagger}) + \frac{\hbar \Omega_{\rm{p}}}{2}(\sigma^+ + \sigma^-), \end{equation} \tag{ 7 }$$

where $\widetilde {\Delta }_{\rm {c}} = \widetilde {\omega }_{\rm {r}} - \omega _{\rm {c}}$ and $\widetilde {\Delta }_{\rm {p}} = \widetilde {\omega }_{ge}- \omega _{\rm {p}}$ are the detunings of the dressed resonator and spectroscopy tones.

Dephasing and relaxation of the qubit and losses in the resonator are accounted for in the master equation in the Markovian approximation [28, 34]. For the resonator, we model the dissipation of photons as a decay rate κ₋. For the transmon, we model relaxation through a decay rate Γ₋ and pure dephasing through the dephasing rate γ_ϕ in the master equation. In our experiment, the finite population in the qubit excited state |e〉 and the resonator n = 1 Fock state due to finite temperature are taken into account by including excitation rates κ₊ for the resonator and Γ₊ for the transmon, where $\kappa _+/\kappa _- = \Gamma _+/\Gamma _- \simeq \exp ({-\hbar \omega _{\rm {r}}/k_{\mathrm {B}} T})$, giving us an estimate for the effective temperature of the system. The master equation for the density matrix ρ can then be written as [28]

$$\begin{equation} \fl \dot{\rho} = -\frac{\rm i}{\hbar} [H_{\rm{tot}} , \rho] + \kappa_- \mathcal{D}[a]\rho+\kappa_+ \mathcal{D}[a^{\dagger}]\rho+ \Gamma_- \mathcal{D}[\sigma^-]\rho+\Gamma_+\mathcal{D}[\sigma^+]\rho + \frac{\gamma_{\phi}}{2} \mathcal{D}[\sigma_z]\rho, \end{equation} \tag{ 8 }$$

where the super-operator $\mathcal {D[{A}]{\rho }} \equiv {A\rho A^{\dagger} - (A^{\dagger} A \rho + \rho A^{\dagger} A})/2$. We have neglected terms of the order λ²_j,j+1 for each of the operators in the dissipation part of the master equation (8) after making the dispersive approximation. Photon-number induced dephasing [31] has been neglected in this simple model since the average number of photons in the resonator is very small ($\bar {n} \approx 1$).

The parameters χ,Ω_c,Ω_p,ζ,ζ',κ₋,κ₊,Γ₋,Γ₊ and γ_ϕ are measured experimentally (section 3.3) and input into the simulation. We then solve (8) for the steady state solution $\dot {\rho } = 0$. In our experiment, the qubit state-projective read-out signal is proportional to Tr[ρσ_z] in the steady state.

When the transition between two quantum levels is driven strongly with a resonant drive field, the resulting 'dressed' system can be equivalently viewed as two split levels, with splitting equal to the Rabi frequency of the drive field (figure 1). This splitting can be observed spectroscopically by probing transitions to a third level in the system, which comprises the Autler–Townes effect [19]. In this section, we take a closer look at (7) to understand the mechanism of the Autler–Townes splitting of the n = 1 photon peak observed in our experiment. The Kerr-type nonlinearities are neglected in the context of this simple model.

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We begin by truncating the set of basis states to the four lowest dressed levels $|\widetilde {g,0}\rangle$, $|\widetilde {e,0}\rangle$, $|\widetilde {g,1}\rangle$ and $|\widetilde {e,1}\rangle$ (figure 1). In the experiment, the $|\widetilde {e,0}\rangle \leftrightarrow |\widetilde {e,1}\rangle$ transition is driven by a strong 'coupler' field with strength Ω_c and detuning $(\widetilde {\omega }_{\rm {r}} +\chi ) - \omega _{\rm {c}} \equiv \widetilde {\Delta }_{\rm {c}}+\chi$, and the $|\widetilde {g,1}\rangle \leftrightarrow |\widetilde {e,1}\rangle$ is 'probed' by a weak field of strength Ω_p and detuning $(\widetilde {\omega }_{ge} + 2\chi ) - \omega _{\rm {p}} \equiv \widetilde {\Delta }_{\rm {p}} +2\chi$ (figure 1).

In this truncated basis, the Hamiltonian can be represented by the matrix

$$\begin{equation} H^{(2)}_{\rm 4-levels} \approx \hbar \left( \begin{array}{@{}cccc@{}} -\widetilde{\Delta}_{\rm{p}}/2 & 0 & 0 & 0\\ 0 & \widetilde{\Delta}_{\rm{p}}/2 & 0 & \Omega_{\rm{c}}/2\\ 0 & 0 & \widetilde{\Delta}_{\rm{c}}-\widetilde{\Delta}_{\rm{p}}/2-\chi & \Omega_{\rm{p}}/2\\ 0 & \Omega_{\rm{c}}/2 & \Omega_{\rm{p}}/2 & \widetilde{\Delta}_{\rm{c}} + \widetilde{\Delta}_{\rm{p}}/2+\chi\\ \end{array} \right), \end{equation} \tag{ 9 }$$

where we have assumed the RWA and excluded transitions detuned from the drives. When the probe and coupler drives are exactly resonant with the respective transitions, the detunings obey $\tilde {\Delta }_{\rm {p}}+2\chi = 0$ and $\tilde {\Delta }_{\rm {c}}+\chi = 0$. Under this condition the eigenvalues of the Hamiltonian in (9) show a splitting equal to

$$\begin{equation} \delta = (\Omega^2_{\rm{c}} + \Omega^2_{\rm{p}})^{1/2}, \end{equation} \tag{ 10 }$$

and for Ω_p ≪ Ω_c, this gives an Autler–Townes splitting linear in the coupler Rabi frequency δ ≃ Ω_c ∼ (P_rf)^1/2, where P_rf is the drive power of the coupler tone. This simple model for the splitting offers good quantitative agreement with the experiment (section 4.2).

Figure 2 shows the transmon [13] coupled to a superconducting lumped-element resonator [35], which is in turn coupled to a coplanar waveguide transmission line used for excitation and measurement. The transmon is formed from two Josephson junctions (junction area ≈100 × 100 nm²) shunted by an interdigitated capacitor with finger widths of 10 μm, lengths of 70 μm and separation between fingers of 10 μm. The junctions are connected to form a superconducting loop with a nominal loop area of 4 × 4.5 μm². The Josephson junction loop is placed close to a shorted current bias line to finely tune the critical current of the parallel junctions and hence the transition frequency of the qubit.

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The device was fabricated by depositing a 100 nm thin film of aluminium by thermal evaporation on a commercial c-plane sapphire substrate. The resonator, transmission line, and on-chip flux bias line were patterned using photolithography and etched with a standard aluminium etchant. The transmon was subsequently patterned using electron-beam lithography and the Al/AlO_x/Al tunnel junctions were formed by double-angle evaporation with an intermediate oxygen exposure step.

The device was mounted in a hermetically sealed copper box and attached to the mixing chamber of an Oxford Kelvinox 100 dilution refrigerator with a base temperature of 20 mK. To isolate the device from thermal noise at higher temperatures, the input microwave line to the device has a 10 dB attenuator mounted at 4 K, 20 dB at 0.7 K and 30 dB at 20 mK on the mixing chamber. On the output (i.e. read-out) microwave line, two 18 dB isolators with bandwidths from 4 to 8 GHz are placed in series at 20 mK. The output microwave signal then goes through a 3 dB attenuator and a high electron mobility transistor amplifier at 4 K before being amplified further at room temperature and mixed down to an intermediate frequency (IF) of 10 MHz.

Three microwave sources are used in the measurement: probe, coupler and read-out. The read-out and probe tones are pulsed on and off, while the coupler tone is applied continuously for the duration of the measurement. To measure the excited-state population of the transmon, a high power (i.e. 50 dB larger than the power of the coupler tone) at a frequency corresponding to the bare cavity is applied [17]. This read-out relies on the Jaynes–Cummings nonlinearity of the coupled resonator–qubit system [30, 36]. When the transmon is in the |g〉 state a large transmissivity is observed and when the transmon is in an excited state (|e〉, |f〉, etc) a small transmissivity is observed.

For spectroscopic measurements, a probe pulse is first applied for a duration of 5 μs at an amplitude just large enough to saturate the qubit transition without causing large power broadening. A read-out pulse of duration 5 μs is applied 20 ns after turning off the probe. The mixed down IF signal is digitized using a data acquisition card and the in-phase and quadrature components are demodulated before being recorded.

From spectroscopic and time-domain measurements the main system parameters were determined. The resonator has a bare resonant frequency ω_r/2π = 5.464 GHz, internal quality factor Q_I = 190 000 and a loaded quality factor Q_L ≡ ω_r/κ = 18 000 with κ = 2π(300) kHz. The normal-state resistance of the two Josephson junctions in parallel yielded a maximum Josephson energy E_J,max/h ≈ 25 GHz and the transmon has a Coulomb charging energy of E_c/h = 250 MHz. This gives a maximum ground-to-first excited state transition frequency, $\omega _{ge,{\mathrm { max}}}/2\pi \simeq (\sqrt { 8E_{\rm {J,max}} E_{\rm {c}}} - E_{\rm {c}})/h = 7.1\,{\mathrm { GHz}}$ for the qubit. The qubit transition frequency was tuned using a combination of an external superconducting magnet and the on-chip flux bias to $\widetilde {\omega }_{ge}/2\pi = 4.982\,{\mathrm { GHz}}$, which corresponds to a detuning of Δ_ge/2π = −482 MHz from the resonator. From spectroscopic measurements of the qubit, we determined the effective dispersive shift χ/2π = −4.65 MHz. From the measurement of the bare and dressed resonator frequencies, and the effective dispersive shift χ, we determine χ_ge/2π = −10 MHz and χ_ef/2π = −10.7 MHz, respectively, (2). From the definitions of χ_ge and χ_ef, we determine g_ge/2π = 70 MHz and g_ef/2π = 89 MHz. The resonator self-Kerr coefficient ζ/2π = −85 kHz and the transmon–resonator cross-Kerr coefficient ζ'/2π = −23 kHz are then determined from ζ ≈ (χ_efλ²_ef − 2χ_geλ²_ge + 7χ_efλ²_ge/4 − 5χ_geλ²_ef/4) and ζ' ≈ (χ_ge − χ_ef)(λ²_ge + λ²_ef) [16, 30].

Time-domain coherence measurements performed at this resonator–qubit (Δ_ge/2π = −482 MHz) detuning revealed a qubit relaxation time T₁ = 1/(Γ₋ + Γ₊) = 1.6 μs and a Rabi decay time T' = 1.6 μs for the first excited state of the qubit. From these measurements, the pure dephasing rate is estimated at γ_ϕ ≡ 1/T_ϕ ≈ 2 × 10⁵ s⁻¹.

Figure 3(a) shows spectroscopy of the transmon with no drive field applied to the resonator. The spectrum shows the dressed qubit ground-to-first excited state transition at $\widetilde {\omega }_{ge}/2\pi = 4.982\,{\mathrm { GHz}}$. The smaller spectroscopic peak detuned by −9.3 MHz at $(\widetilde {\omega }_{ge}+2\chi )/2\pi =4.973\,{\mathrm { GHz}}$ is due to one $\widetilde {\omega }_{\rm {r}} -\chi /2\pi =5.474\,{\mathrm { GHz}}$ photon occasionally being present in the resonator [18] from thermal excitations. From the relative areas under the two spectral peaks, we estimate a fractional thermal population of n_th = 0.1 photons, corresponding to a temperature of about 120 mK for the resonator. This effective temperature is much higher than the base temperature ∼20 mK of the dilution refrigerator, possibly due to a leakage of infrared photons [7, 8].

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Upon driving the resonator with a coupler tone at ω_c/2π = 5.474 GHz, which is resonant with the transition $|\widetilde {g,0}\rangle \leftrightarrow |\widetilde {g,1}\rangle$ (figure 1), we increase the mean occupancy of the resonator from its equilibrium value and create a coherent state. Since the coherent state is a superposition of Fock states, the qubit spectrum has multiple peaks, one for each Fock state. When applying a power of P_rf = 1.25 aW at the dressed resonator frequency (figure 3(b)), an increase in the height of the $\widetilde {\omega }_{ge}+2\chi$ peak is observed and a spectral peak at $\widetilde {\omega }_{ge}+4\chi$ begins to appear. The peak at $\widetilde {\omega }_{ge}$ is still the largest. Figure 3(c) shows the spectrum for an applied resonator drive power of 25 aW. In this case, more spectral peaks are observed and the $\widetilde {\omega }_{ge}+2\chi$ is the largest.

We can calculate the average number of photons ($\bar {n}$) by fitting the spectral peak associated with each Fock state and using the relative peak areas to weight each Fock state (3). The relative weights are also found to follow a Poisson distribution once the resonator is pumped into a coherent state [14]. The inset of figure 3 shows the average number of photons versus the applied power in the weak driving limit. For very weak driving P_rf < 0.1 aW, the thermal photon population n_th = 0.1 is the dominant contribution to $\bar {n}$. Above an applied power of P_rf > 0.1 aW $\bar {n}$ monotonically increases. For small applied powers in this region, $\bar {n} = (2 Q_{\rm {C}}/Q_{\rm {L}})P_{\rm {rf}}/(\hbar \tilde {\omega }_{\rm {r}}\kappa _-)$ where the first term renormalizes the power applied to the transmission line to the power stored in the resonator and Q_C is the quality factor due to external coupling. Using this linear relation and the excess photon number population from the applied coupler tone in figure 3(b) an attenuation of α = 65 dB is calculated for the input microwave line. The red curve in the inset is a model for $\bar {n}$ consisting of a contribution from a power independent thermal contribution plus a coherent state population with an applied linear power dependence. As can be seen in this figure, we see a strong deviation from linear behaviour. It is important to note that the average occupancy of the resonator $\bar {n}$ is, in general, expected to be a nonlinear function of P_rf [16]. A detailed analysis of the full nonlinear relation is beyond the scope of this paper.

Figure 4 shows two-tone spectroscopy of the qubit as we vary the frequency of the coupler tone. In this plot, the vertical bands at frequencies ω_s/2π = 4.982 and 4.973 GHz are just the n = 0 and 1 photon peaks as shown in figure 3. The prominent diagonal band seen between the n = 0 and 1 photon bands in this figure corresponds to a two-photon 'blue' sideband transition from the $|\widetilde {g,0}\rangle$ to $|\widetilde {e,1}\rangle$ [37]; the sum of the frequencies (ω_p,ω_c) of the drive photons along this diagonal band is equal to the corresponding transition frequency. In general, a diagonal band with slope −1/n should appear when the detunings of the drives satisfy $\widetilde {\Delta }_{\rm {p}}-n\widetilde {\Delta }_{\rm {c}} = n\chi$ corresponding to the sideband transition $\widetilde {|g,n-1}\rangle \leftrightarrow \widetilde {|e,n}\rangle$. For example, the sideband transition $|\widetilde {g,0}\rangle \leftrightarrow |\widetilde {e,1}\rangle$ level appears as a band of slope −1 and the $|\widetilde {g,1}\rangle \leftrightarrow |\widetilde {e,2}\rangle$ transition appears as a faintly visible band of slope −1/2 in figure 4 (red arrow).

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Another feature of the data in figure 4 (red box) is a small splitting in the thermal n = 1 photon peak when the frequencies of the probe and coupler tones are ($\omega _{\rm {p}},\omega _{\rm {c}}) \simeq (\widetilde {\omega }_{ge}+2\chi ,\widetilde {\omega }_{\rm {r}}+\chi$). We examine this splitting more closely in this section. Here we follow the convention of section 2.3 and refer to the strengths of the probe and coupler tones in terms their Rabi frequencies Ω_p and Ω_c. The Rabi oscillations at zero detuning on the $|\widetilde {g,0}\rangle \to |\widetilde {e,1}\rangle$ transition were used to quantify Ω_p. For the coupler tone, the following relation between Ω_c and P_rf was used [16]:

$$\begin{equation} \frac{\Omega_{\rm c}}{2\pi} = \left( \frac{P_{\rm rf}}{2Q_{\rm c}h}\right)^{1/2}. \end{equation} \tag{ 11 }$$

Figures 5(a)–(c) show measurements of the splitting in the thermal n = 1 photon peak as we vary the strength of the coupler tone while keeping the strength of the probe tone fixed Ω_p/2π ≃ 0.3 MHz. Figure 5(a) shows that when we apply a coupler tone with strength Ω_c/2π ≃ 1.3 MHz, we observe a splitting in the thermal n = 1 photon peak with a splitting size that is almost equal to Ω_c. In figure 5(b), we lower the strength of the coupler to Ω_c/2π ≃ 1 MHz and we notice a corresponding decrease in the splitting size to ∼1 MHz. Upon further lowering the strength of the coupler to Ω_c/2π ≃ 0.6 MHz, the splitting decreases in size to ∼0.6 MHz.

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The splitting size can be understood based on an Autler–Townes mechanism (section 2.3) shown in figure 1. To begin with, the $|\widetilde {g,1}\rangle$ level is populated due to thermal excitation of photons in the resonator [7, 8]. Subsequently the probe and coupler tones are applied on resonance with the transitions $|\widetilde {g,1}\rangle \leftrightarrow |\widetilde {e,1}\rangle$ and $|\widetilde {e,0}\rangle \leftrightarrow |\widetilde {e,1}\rangle$ (see section 2.3), respectively. In the presence of the two drive fields, the $|\widetilde {e,1}\rangle$ level splits into a pair of levels separated by δ = (Ω²_p + Ω²_c)^1/2 (10). In the limit Ω_c ≫ Ω_p this splitting is almost equal to Ω_c and is observed spectroscopically upon probing the $|\widetilde {g,1}\rangle \leftrightarrow |\widetilde {e,1}\rangle$ transition.

The simple model in (9) also predicts the overall 'shape' of the splitting in the (ω_p,ω_c) plane in figures 5(a)–(c). When $\tilde{\Delta}_{\rm{p}} + 2\chi\!=\!0$ and $\tilde{\Delta}_{\rm{c}}\!+\!\chi \neq 0$, only the $|\widetilde{g,1}\rangle \leftrightarrow |\widetilde {e,1}\rangle$ transition is resonantly driven, while the influence of level $|\widetilde {e,0}\rangle$ is energy suppressed. This explains the vertical band corresponding to the n = 1 photon peak at 4.973 GHz. When $\tilde{\Delta}_{\rm{p}}+2\chi\!=\!\tilde{\Delta}_{\rm {c}} +\chi \neq 0$, the difference of the drive frequencies corresponds to the $|\widetilde {e,0}\rangle \leftrightarrow |\widetilde {g,1}\rangle$ transition. This two-photon 'red' sideband transition [37] explains the slope +1 of the splitting in the figure 5.

Figures 5(d)–(f) show steady state simulations of the system–bath master equation at the steady state (section 2.2) in the region around the thermal n = 1 photon peak when driving the coupler with a strength of Ω_c/2π = 1.3 MHz (d), 1 MHz (e) and 0.6 MHz (f). The parameters χ,Ω_c,Ω_p,ζ,ζ',κ₋,κ₊,Γ₋,Γ₊ and γ_ϕ in the master equation were determined from independent experiments (section 3.3) and equation (8) was then solved for ρ for the steady state solution. Here we plot Tr(ρ.σ_z) to simulate the qubit state-projective read-out in our experiment [17, 30, 36]. We included the two lowest qubit levels |g〉,|e〉, the ten lowest resonator levels, the resonator self-Kerr and qubit–resonator cross-Kerr terms in the simulation. The population in the n = 1 Fock state of the resonator and the |e〉 state of the transmon due to finite temperature were taken into account through the excitation rates κ₊ ≈ κ₋/10, Γ₊ ≈ Γ₋/10 in the system master equation (8).

Finally, figure 6 shows the experimentally measured splitting versus the Rabi frequency of the coupler. The Rabi frequency of the coupler Ω_c was calibrated using (11). The root-mean-square voltage V_rf = (Z₀P_rf)^1/2 is calculated by assuming Z₀ = 50 Ω for the impedance of the transmission line. The red curve in the figure 6 is the splitting size δ/2π ≃ (Ω²_c + Ω²_p)^1/2/2π predicted by the simple model (10) for a fixed Ω_p/2π ≃ 0.3 MHz and it agrees well with the experimental data. Here we see that the observed Autler–Townes splitting is nearly linear in the amplitude of the coupler drive voltage V_rf and is almost equal to the Rabi frequency of the coupler field, as expected from (10).

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We note that our simple model also predicts an Autler–Townes splitting of the n = 0 photon qubit peak when the probe and coupler frequencies are $(\omega _{\rm {p}}, \omega _{\rm {c}}) \simeq (\tilde {\omega }_{ge}, \tilde {\omega }_{\rm {r}}+\chi )$. We see a hint of this splitting in figure 4 but it is not fully resolved because of the line-width of the qubit $|\widetilde {g,0}\rangle \leftrightarrow |\widetilde {e,0}\rangle$ transition. The Autler–Townes mechanism for the splitting in the n = 0 photon peak involves the $|\widetilde {g,0}\rangle ,\, |\widetilde {e,0}\rangle ,\, |\widetilde {e,1}\rangle$ levels in figure 1. The three-level subsystem is operated in a 'ladder' configuration in this case, with the $|\widetilde {e,0}\rangle \leftrightarrow |\widetilde {e,1}\rangle$ transition driven strongly with a 'coupler' tone and the $|\widetilde {g,0}\rangle \leftrightarrow |\widetilde {e,0}\rangle$ transition weakly 'probed'.