Probing the symmetry breaking of a light–matter system by an ancillary qubit

Hybrid quantum systems in the ultrastrong, and even more in the deep-strong, coupling regimes can exhibit exotic physical phenomena and promise new applications in quantum technologies. In these nonperturbative regimes, a qubit–resonator system has an entangled quantum vacuum with a nonzero average photon number in the resonator, where the photons are virtual and cannot be directly detected. The vacuum field, however, is able to induce the symmetry breaking of a dispersively coupled probe qubit. We experimentally observe the parity symmetry breaking of an ancillary Xmon artificial atom induced by the field of a lumped-element superconducting resonator deep-strongly coupled with a flux qubit. This result opens a way to experimentally explore the novel quantum-vacuum effects emerging in the deep-strong coupling regime.

In the nonperturbative ultrastrong-coupling regime, the qubit-resonator system can be described by a quantum Rabi model.It is particularly interesting to harness controllable physical parameters to tune the quan-tum vacuum of the system, since it becomes a novel entangled ground state |G rather than the trivial product ground state of the Jaynes-Cummings model.In such an exotic quantum vacuum, while the average photon number in the resonator is nonzero, i.e., G|a † a|G = 0, where a † (a) is the creation (annihilation) operator of the resonator mode, the ground-state photons are actually virtual (tightly bound to the artificial atom [41]) and cannot be directly detected.Theoretically, it was proposed to employ non-adiabatic modulations [37], sudden turn-off of the qubit-resonator interaction [38], or a spontaneous decay mechanism of multi-level systems [39] to convert these virtual photons into real ones (similar to the dynamical Casimir effect [42,43]), so as to generate radiation out of the resonator.However, these are still experimentally challenging.
In the standard model of particle physics, the W ± and Z weak gauge bosons obtain mass via the Higgs mechanism, in which the electroweak gauge symmetry SU(2) × U(1) is broken due to the interaction with a symmetry-broken vacuum field (the Higgs field) displaying a nonzero vacuum expectation value.In our experiment, we observe the parity symmetry breaking of a probe superconducting circuit (Xmon) dispersively coupled to a qubit-resonator system in the deep-strong coupling regime.This effect, although rather different (in our case the broken symmetry is discrete and it is not spontaneous), shares some interesting analogies with the Higgs mechanism.At the optimal point, both the flux qubit and the qubit-resonator system have a well-defined parity symmetry [44].In this parity-symmetry case, the quantum-vacuum expectation value of the resonator field is zero, G|(a + a † )|G = 0, where |G is the qubitresonator ground state.With the external flux tuned away from the optimal point, parity-symmetry breaking is induced in the flux qubit and, in the presence of a very strong qubit-resonator coupling, it also significantly affects the resonator vacuum, giving rise to G|(a + a † )|G = 0 [35].In our experiment, the achieved qubit-resonator system is in the deep-strong coupling regime, so the quantum-vacuum state is very different and, when away from the optimal point, this can produce a sizable nonzero value of G|(a+a † )|G as well as observable symmetry breaking effects.Indeed, as demonstrated in our experiment, the qubit-resonator system is able to break the parity selection rule of the Xmon dispersively coupled to the resonator, thus enabling forbidden transitions.We should emphasize that the similarity between the Higgs mechanism and our observation only comes from two key features: (i) the symmetry-broken vacuum has a nonzero expectation value, and (ii) the field with a nonzero expectation value can induce symmetry breaking in another quantum system, in the absence of real excitations of the field.Of course, a system composed by just two qubits and a lumped-element resonator cannot fully reproduce the far more complex Higgs model.

Deep-strongly coupled qubit-resonator circuit
The system is composed of a five-junction flux qubit deep-strongly coupled to a superconducting lumpedelement resonator via a common Josephson junction (JJ) (Fig. 1).In addition, we use an Xmon as a quantum detector, which is capacitively coupled to the lumped-element resonator on the left and to a coplanarwaveguide resonator on the right.The whole device is placed in a dilution refrigerator cooled down to a temperature of ∼30 mK.
Similar to the three-junction flux qubit [2], the fivejunction flux qubit has both clockwise and counterclockwise persistent-current states.Away from the optimal point Φ ext = (n + 1  2 )Φ 0 , where Φ ext is the external flux threading the loop of the flux qubit, Φ 0 = h/2e is the superconducting flux quantum, and n is an integer, these two persistent-current states have an energy difference ε = 2I p δΦ ext , depending on the maximum persistent current I p and the flux bias δΦ ext ≡ Φ ext − (n + 1  2 )Φ 0 .Also, there is a barrier between these two persistent-current states, which removes their degeneracy at the optimal point by opening an energy gap ∆.In the basis of eigenstates, the Hamiltonian of the flux qubit can be written as (setting = 1) H q = ω q σ z /2, where The lumpedelement resonator is composed of two identical large "fishbone" interdigitated capacitors and a center conductor in between.The flux qubit consists of three identical larger JJs and a smaller JJ reduced by a factor of 0.42 in area.To enhance the coupling between the flux qubit and the lumped-element resonator, an even larger JJ (with its area doubled) is added in the qubit loop and shared with the center conductor.Extending to the left (right) is another section of 50 Ω coplanar waveguide which couples to the input signal line (the Xmon qubit).b, Zoom-in optical image of the area denoted by the green rectangular box in a. c, Circuit diagram of the device (cf.Supplementary Fig. 1).
the transition frequency of the qubit and σ z is a Pauli operator.The quantum two-level system is a good model for the flux qubit because of its relatively large anharmonicity.
Compared to the coplanar-waveguide resonator, the lumped-element resonator has the advantage of only a single resonator mode [24]: H r = ω r a † a, where ω r is the resonance frequency of the resonator mode.This ω r is V-shaped versus δΦ ext around the optimal point [24] because the inductance across the qubit loop, as part of the total inductance of the lumped-element resonator, depends approximately linearly on |δΦ ext |.The large JJ shared by the flux qubit and the lumped-element resonator acts as an effective inductance to produce an inter- )Φ0 (which is a more stable flux bias point than Φext = 1 2 Φ0 in our system).The solid blue curves in a-c are the fitted transition frequencies between the ground state to the third-, second-and first-excited states of the qubitresonator system, respectively (i.e., ω03, ω02, and ω01).In a, the additional transitions indicated by the dashed red curves correspond to sideband transitions (assisted by the Xmon levels) in the system.Source data are provided as a Source Data file.
action between them, H int = g[cos θ σ z − sin θ σ x ](a † + a), where tan θ = ∆/ε, and g = M I p I r is the coupling strength, with M ≈ L c being the mutual inductance and I r = ω r /2(L 0 + L c ) the vacuum fluctuation current along the center conductor of the lumped-element resonator, where L c is the inductance of the large JJ and L 0 is the geometry inductance.When the qubit-resonator coupling is in the ultrastrong or deep-strong regime, one cannot apply the rotating-wave approximation (RWA) to H int , and the Hamiltonian of the qubit-resonator system is written as i.e., the generalized quantum Rabi model [15].We can extract the parameters in H s by fitting the reflection spectra of the qubit-resonator system, as measured by applying a probe tone to the system.Around ω p /2π = 4.8 GHz (near the bare frequency of the lumped-element resonator) and 5.6 GHz, clear transitions are observed; of which the corresponding frequencies are found to be consistent with the transition frequencies from the ground state |G ≡ |0 s to the first-and second-excited states, |1 s and |2 s of the qubit-resonator system, respectively, i.e., ω 01 and ω 02 (see the solid fitting curves in Figs.2c and 2b).Around ω p /2π = 11.9GHz, we observe the transition from the ground state |G to the third-excited state |3 s (Fig. 2a), with the solid fitting curves corresponding to ω 03 .Moreover, similar to those in Ref. 26, additional transitions are observed in Fig. 2a, which are attributed to the sideband transitions (the dashed curves in Fig. 2a) involving the Xmon levels as well, see Supplementary Information.Near 5.6 GHz and 11.9 GHz, the transmission background of the probe tone changes abruptly, forming two band edges (see Supplementary Fig. 2).The qubitresonator system coupled to the band edges in Figs.2a and 2b is analogous to the case of an atom coupled to a band edge in a photonic crystal waveguide [45,46].The abrupt changes in the transmission background originate from the wire bonding and filters.Near the band edge, a photon emitted by the atom (in our case it is the qubitresonator system) is Bragg reflected and reabsorbed, resulting in the emergence of spectrally resolvable polariton states (similar to the vacuum Rabi splitting), which will disappear away from the band edge [45].
By fitting the transition frequencies ω 01 , ω 02 and ω 03 with experimental results in Fig. 2, we can derive the parameters of the generalized Rabi model in Eq. ( 1), which are I p = 245 nA and ∆/2π = 15.0GHz for the flux qubit, ω r /2π = 4.82 GHz for the lumpedelement resonator, and g/2π = 4.55 GHz for the qubitresonator coupling.Here the obtained resonance frequency ω r /2π = 4.82 GHz is the value when the external flux bias is at the optimal point δΦ ext = 0.In our qubitresonator system, we achieve g/ω r ≈ 0.944, indicating that it indeed reaches the deep-strong coupling regime g/ω r ∼ 1.
When δΦ ext = 0 (θ = π/2), the deep-strongly coupled system in Eq. ( 1) reduces to the standard quantum Rabi model.Instead of a trivial (product) ground state |g, 0 in the Jaynes-Cummings model, it has a quantum vacuum (i.e., entangled ground state) |G , with G|a † a|G = 0.This standard Rabi model has a well-defined parity symmetry, characterized by σ z e iπa † a , which ensures that the ground state is a superposition of all states with an even number of excitations [28].For this quantum vacuum, G|(a + a † )|G =0.When δΦ ext = 0, H s in Eq. ( 1) has an extra longitudinal coupling term proportional to σ z .It breaks the parity symmetry of the model, and hence both even and odd numbers of excitations are allowed in the new ground state [35] as well as in the excited states of the coupled system.Now, both G|a † a|G = 0 and G|(a + a † )|G = 0.

Detection of the induced symmetry breaking
Below we harness an Xmon [7] to detect the symmetry breaking of the lumped-element resonator.The Xmon is both largely detuned and weakly coupled to it via a small capacitor (cf.Fig. 1a).In such a dispersive regime, the effect of the Xmon on the qubit-resonator system is greatly reduced.The Xmon can be modeled by the Hamiltonian H X = 4E c n 2 − E J cos ϕ, where E c is the single-electron charging energy of the JJ, E J is the Josephson coupling energy, n = −i∂/∂ϕ, and ϕ is the phase drop across the JJ.In the Xmon, the metallic cross and the ground metal provides the JJ with a large shunt capacitor to reduce its sensitivity to the charge noise [5,6].
The Xmon's parameters can be determined with the dispersive readout technique by coupling the Xmon to a coplanar-waveguide resonator (see Fig. 1c and Supplementary Fig. 1).The resonance frequency of this waveguide resonator is measured to be ω CPW /2π = 3.554 GHz and the coupling strength between the waveguide resonator and the Xmon qubit is g X /2π = 28 MHz.Then, we obtain the transition frequency ω X /2π = 5.181 GHz of the Xmon qubit and its anharmonicity A/2π = −0.16GHz.With these parameters as well as the relations ω X = √ 8E c E J − E c and A = −E c , we have E c /2π = 0.16 GHz and E J /2π = 20.97GHz.Because the lumped-element resonator couples to the Xmon, it induces an offset charge to the Josephson junction, leading ), and g ≈ g X (by a symmetric design) is the coupling strength between the lumped-element resonator and the Xmon qubit.
The total Hamiltonian of the deep-strongly coupled qubit-resonator system plus the Xmon can be expressed as H tot = H s + HX .Owing to the large transition frequency between the ground and first-excited states, the deep-strongly coupled qubit-resonator system nearly stays in the ground state |G at the temperature ∼30 mK.
Note that in the dispersive regime, where the Xmonresonator coupling rate is much lower than the corresponding detuning, the energy transitions of the Xmon are almost unaffected by the interaction (i.e., flux-biasinsensitive) and can be easily identified with standard spectroscopic techniques.We also observe that, neglecting the interaction between the resonator and the flux qubit, or considering a flux qubit at the optimal point, the Xmon-resonator system displays parity symmetry.On the contrary, when the qubit is brought out of the optimal point, the very strong qubit-resonator coupling strength can induce a symmetry breaking of the Xmon, even for moderate Xmon-resonator coupling strengths (see Supplementary Information).
In Figs.3a and 3b, we show the single-and two-photon transitions between the lowest two levels of the Xmon using two-tone spectroscopy.Here the resonance frequency of the single-photon transition corresponds to the transition frequency ω X of the Xmon qubit, and the resonance �� ext (m� 0 ) frequency of the two-photon transition is 1  2 ω X .In our chip, 1  2 ω X is designed to be well separated from both ω 01 and 1  2 ω 02 of the deep-strongly coupled qubit-resonator system to avoid any unwanted transitions.The drive power (−65 dBm) applied at the local drive port for exciting the two-photon transition is much stronger than that for the single-photon transition (−120 dBm).In Fig. 3b, the signal of the two-photon transition is found to disappear at the optimal point δΦ ext = 0, evidencing that the well-defined parity symmetry of the standard Rabi model preserves the parity selection rule of the Xmon.When deviating from the optimal point, the parity-symmetry breaking in H s , in addition to producing a nonzero vacuum expectation value v = G|(a + a † )|G = 0, is able to break the parity symmetry of the Xmon artificial atom, without however inducing any δΦ ext dependent Lamb shift.Figure 3b shows that even a very small deviation from the parity symmetry point of the system Hamiltonian, which can be quantified by the adimensional parameter cot θ = ε/∆ 10 −2 , is able to activate twophoton transitions in the Xmon, in agreement with the theoretical calculations in Fig. 3d.
These results can be described by adopting a simplified model for the Xmon using only its four lowest energy levels.Considering the large detunings ( 20 GHz), the effects from higher levels are negligible.With the Xmon now approximated as a four-level system (qudit), the total Hamiltonian of the qubit-resonator system plus the Xmon can be written as where b = is the annihilation operator for the Xmon, and H (4) X = 3 n=0 ε n |n n| is the bare Xmon energy (H X ), projected into the reduced four-dimensional Hilbert space.We can evaluate the single-and two-photon absorptions under coherent drive of the Xmon by studying its effective polarization , with ρ being the density operator of the system.The latter can be calculated using the master equation approach in the dressed picture [47] (see Supplementary Information).The simulated results are shown in Figs.3c and 3d, which are in a good agreement with the experimental observations.This demonstrates the symmetry breaking of a quantum system which is coupled to the vacuum of another quantum system displaying symmetry breaking.Note that the deep-strongly coupled qubit-resonator system nearly stays in the ground state |G at a temperature of ∼30 mK, and in the present case of dispersive coupling with a largely detuned Xmon.Moreover, no real excitations of this system are coherently generated by the coherent drive at ω d ω X /2.We also point out that here the symmetry breaking is not spontaneous but due to the parity symmetry breaking of H s in Eq. ( 2), induced by the presence of a flux offset applied to the flux qubit.However, the adimensional parameter ε/∆, quantifying the degree of symmetry breaking induced by the flux offset on the flux qubit, is very small (ε/∆ 10 −2 ) at δΦ ext = 0.1 mΦ 0 , when the Xmon two-photon transitions start to be observed (see Fig. 3b) and it does not affect the transition frequency of the Xmon.According to the additional calculations shown in Supplementary Fig. 6, the two-photon signals of the Xmon disappear if the effective coupling g/ω r between the flux qubit and the LC resonator is reduced to 0.6.This provides evidence that the observed induced symmetry breaking of the Xmon is a unique feature in the near deep-strong coupling regime.
We observe that the interaction-induced symmetry breaking mechanism detected here is more complex with respect to the Higgs mechanism and to that described in Ref. 35.In these two cases, the effect is directly induced by the vacuum expectation value of the field.For example, for w = G|(a − a † )|G = 0, the Xmonresonator interaction in Eq. ( 2) could be approximated as ∼ −g w(b−b † ).It can be shown that this term directly determines the symmetry breaking of the probe qubit in Ref. 35.However, in the present case, it turns out that w = 0, since the inductive coupling between the resonator and the flux qubit determines w = 0 and v = 0. Nonetheless, a full quantum analysis (see Supplementary Information) shows that in such a case (w = 0) as well, the Xmon can undergo symmetry breaking, when interacting with a field with no real excitations and displaying symmetry breaking.Using a probe qubit which is inductively coupled to the resonator would give rise to a symmetry breaking mechanism directly determined by the nonzero vacuum expectation value v = G|(a + a † )|G = 0.In the present case, the symmetry breaking of the qubitresonator system determines a nonzero matrix element entering the Xmon two-photon transition rate; thus enabling two-photon transitions in the Xmon.Considering the eigenstates of the total Hamiltonian H tot , the two-photon transition rate is proportional to the prod- eigenvectors of H tot sorted from the lower to the higher corresponding energy levels.Thus, with |E 0 being the ground state of the whole interacting system, we identify |E 1 as the first excited level of H s (slightly dressed by the interaction with the Xmon) and |E 2 the corresponding first excited dressed level of H (4) X .It turns out that Y 0,1 as well as Y 0,2 are nonzero and almost constant in the interval of flux offset reported here, while Y 1,2 is very well approximated by a linear function of δΦ ext (see Supplementary Fig. 4) and it is zero for δΦ ext = 0, due to the parity symmetry.This explains the onset of the parity-symmetry breaking felt by the Xmon.

Discussion
One interesting future possibility is that the current experimental method could be used to characterise the spontaneous vacuum symmetry breaking in the Dicke model (i.e., equilibrium superradiant phase transition) when more flux qubits are integrated in the lumpedelement resonator and operated at the optimal point simultaneously.If an equilibrium superradiant phase transition occurs, the small gap in the two-photon spectra of the Xmon in Fig. 3b will disappear; which means that two-photon transitions could be observed even for ε/∆ 0. Note that in the present case, with a resonator interacting very strongly with only one flux qubit, we observe these parity-forbidden transitions for values ε/∆ 1 (specifically ε/∆ 0.01).We also point out that, considering a setup with the capacitively coupled Xmon replaced by a galvanically coupled artificial atom (e.g. a flux qubit), the measured rate of parityforbidden one-or two-photon transitions, would provide a direct measurement of the vacuum field expectation value G|(a + a † )|G , with a rate proportional to its square modulus [35].
In conclusion, we have experimentally probed the symmetry breaking of a lumped-element resonator, by observing the activation of two-photon transitions in a probe artificial atom (Xmon).The latter is dispersively coupled to the resonator and probed in the absence of any real coherent excitation of the resonator field; which however displays a nonzero vacuum expectation value, as confirmed by theoretical calculations.The violation of the Xmon parity selection rule comes from virtual paths enabled by its interaction with an electromagnetic resonator whose parity symmetry is significantly broken by the deep-strong light-matter interaction with a flux qubit.The experimental results are in very good agreement with our theoretical analysis.The proposed setting offers a novel way to explore quantum-vacuum effects emerging in the light-matter ultra-strong and deepstrong coupling regimes and can be used as a tool to explore the coherence properties of quantum vacua in these exotic hybrid quantum systems [48][49][50], and the occurrence of superradiant phase transitions in Dicke-like systems [36].

Methods
The experimental setup is shown in Supplementary Fig. 1.
The superconducting lumped-element and coplanar-waveguide resonators are fabricated by patterning a niobium thin film of thickness 50 nm deposited on a 10×3 mm 2 silicon chip via electron beam lithography.The flux qubit is also fabricated on the silicon substrate in the middle of the center conductor of the lumpedelement resonator by using both electron beam lithography and double-angle evaporation of aluminium.An external magnetic field generated by a magnetic coil surrounding the device is applied to tune the magnetic flux threading through the qubit loop.The Josephson junction in the Xmon is connected to the cross-shaped capacitor at one end and fabricated using separate steps of electron beam lithography and double-angle evaporation.Reflection spectra of the deep-strongly coupled qubitresonator system at the frequency ω p of the probe tone are measured with a vector network analyser (VNA).Another microwave signal at frequency ω d is further applied at the local drive port of the Xmon for two-tone spectroscopy measurements.The input signals are attenuated and filtered at various temperature stages before finally reaching the sample.Also, two isolators and a low-pass filter (LPF) are used to protect the sample from the amplifier's noise.Owing to the parity symmetry, two-photon transitions in the bare Xmon artificial atom are forbidden.Thus, an observation of the two-photon transition represents a signature of induced breaking of the selection rules in the Xmon coupled to the qubit-resonator system.Below we show that the two-photon transition rate can be directly calculated using second-order perturbation theory.
Let us consider a single-tone external drive exciting the Xmon, where with |E j being the eigenstates of the Hamiltonian (2), which are ordered so that j < k if ω j < ω k .
We choose system's ground state |E 0 as the initial state.The first-excited state of the system corresponds to the dressed first-excited state of the DSC qubit-resonator system.The second-excited state of the system corresponds to the dressed first-excited state of the Xmon.The transition rate from |E 0 to |E 2 can be written as where ω j,k = ω k − ω j , and Here, the matrix elements of the perturbation potential are The losses of the system components can be included by converting the Dirac delta function in Supplementary Eq. ( 7) into a Lorentzian: The definition of the loss rates as Γ 2,0 can be found in the next section.The two-photon Xmon power spectrum can be obtained as where ρ is system's density operator.It can be written as where Y j,k = E j |Y + |E k .In comparision with Y 0,2 , the matrix element Y 1,2 is almost negligible.

Density matrix approach
Here we derive the two-photon emission rate of the three-component system described in the previous section, by using a perturbative master equation approach.We start from the master equation for the coupled system in the basis of its eigenstates (dressed-states approach), including the external drive (that is the perturbation): where H 0 = n ω n |E n E n | is the diagonalized Hamiltonian of Supplementary Eq. ( 2).The external driving field is described by the time-dependent interaction and L[ρ] is the dissipator expressed in the eigenstates of the DSC system (see, e.g., Ref. 2).Applying the rotatingwave approximation, we can write We will also consider an expansion of the density matrix in terms of powers of the drive amplitude, so that we can separate each contribution, i.e.
Moreover, since the drive can be expanded in Fourier components as (actually, we will use here only a single-tone drive) the same can be done with ρ (n) , leading to the expression where ω k = ±k ω d , with k being an integer number.
Considering the steady state, plugging Supplementary Eq. ( 18) into the master equation, we obtain a set of closed equations separated in driving orders and Fourier components, that at the lowest order looks like: which leads to the thermal density operator with the partition function Z = j exp − ωj k B T .The next two sets of equations for the first-and second-order density matrices can be formally derived from the master equation, and they have the following forms: (1)  ωj , H (1) The solutions of the above expressions can be derived sequentially, by plugging the already found solution for the previous order.In this way we find for the first-and second-order steady-state density matrices: ωj , ρ The decay rates Γ n,m are the sum of all the lossy channels involved in the |E m → |E n transition and its expression is shown below.For a nonzero temperature reservoir, one has to take care of the modified decay rates for the generic transition |E j → |E i (with j > i).
For the sake of simplicity, we assume that both the artificial atoms and the lumped-element resonator are thermalized at the same temperature.Thus, the rates can be calculated via the Fermi's golden-rule.At zerotemperature, we obtain Γ 0 i,j = γ where ni,j is the thermal noise that affects the |E j → |E i transition, i.e., ni,j = 1/[exp( ω i,j /k B T ) − 1].In the following theoretical analysis, corroborated by the standard working temperature of the DSC system (T ∼ 30 mK), we can safely ignore the enhancement induced rates, by choosing the relevant ni,j = 0.
In our calculations, we stop the perturbative development up to the second order, as the essential physics is fully caught.In particular, we can obtain the single-and two-photon absorption by looking at the polarization of the Xmon as a function of the drive frequency, which can be defined as In this formula, we can apply further simplifications by (i) considering the case of a ground state environment (T = 0) and (ii) dropping the negligible terms of P (ω d ) , finding that the leading terms that drive the onset of the sought effect yield 2,0 represent a constant background as they are far-detuned resonances, and thus they can be ignored.Furthermore, we also find that |Y 2,0 | 1, as it represents the transition matrix element of the weakly coupled Xmon, from its ground to the firstexcited level.Following our perturbative approach, we find that (ω d − ω 0,1 + iΓ 0,1 /2)(2ω d − ω 0,2 + iΓ 0,2 /2) .
(25) Finally, as the detected field has frequency ω d ∼ 1  2 ω 2,0 , the non-resonant part of the denominator can be expanded in series, leading to the final expression:

Results
Below we show the numerical results achieved with parameters extrapolated by the experimental data.Namely, we use ω r /2π = 4.82 GHz, ∆/2π = 15 GHz, g/2π = 4.55 GHz, g /2π ≈ g X /2π = 28 MHz, and loss rates for the flux qubit, lumped-element resonator and Xmon are chosen to be γ (q) /2π = γ (a) /2π = γ (b) /2π = 2 MHz.Also, we consider a thermalized DSC system at a temperature of 30 mK.Our model is able to reproduce the experimental results with a very good agreement.In the simulations, we use a quite strong external field exciting the Xmon, in order to produce a two-photon absorption, and its value in terms of linewidth is Ω = 200×γ (b) .Such a value is large enough to ensure the onset of the second-order processes.While for the direct excitation of the single-photon absorption, we reduce Ω by a factor 2 × 10 6 , in doing so we can get the same order of the amplitudes of the single-and two-photon signals, which is consistent with the experimental observation.the interaction of the two-photon resonance with other spectral lines.
Symmetry breaking of the quantum vacuum makes the flux-bias-insensitive Xmon experience a virtual path to its two-photon resonance.It is not possible if the Xmon is coupled with a normal product-state vacuum (when g/ω r 1), or when the vacuum symmetry is preserved (δΦ ext = 0), in contrast to the cases of flux-tunable qubits, where the two-photon transition becomes allowed naturally when there is a finite external flux bias.

FIG. 1 .
FIG.1.Device.a, Optical image of the device.The lumpedelement resonator is composed of two identical large "fishbone" interdigitated capacitors and a center conductor in between.The flux qubit consists of three identical larger JJs and a smaller JJ reduced by a factor of 0.42 in area.To enhance the coupling between the flux qubit and the lumped-element resonator, an even larger JJ (with its area doubled) is added in the qubit loop and shared with the center conductor.Extending to the left (right) is another section of 50 Ω coplanar waveguide which couples to the input signal line (the Xmon qubit).b, Zoom-in optical image of the area denoted by the green rectangular box in a. c, Circuit diagram of the device (cf.Supplementary Fig.1).

FIG. 2 .
FIG. 2. Reflection spectra.Reflection spectra of the deep-strongly coupled qubit-resonator system versus the external flux bias δΦext and the probe frequency ωp around Φext = (3 + 12 )Φ0 (which is a more stable flux bias point than Φext = 1 2 Φ0 in our system).The solid blue curves in a-c are the fitted transition frequencies between the ground state to the third-, second-and first-excited states of the qubitresonator system, respectively (i.e., ω03, ω02, and ω01).In a, the additional transitions indicated by the dashed red curves correspond to sideband transitions (assisted by the Xmon levels) in the system.Source data are provided as a Source Data file.

FIG. 3 .
FIG. 3. Excitation spectra.Excitation spectra of the Xmon qubit versus the external flux bias δΦext and the drive frequency ω d around Φext = (3 + 1 2 )Φ0.The frequency of the probe tone is fixed at 3.554 GHz, in resonance with the λ/2 mode of the coplanar-waveguide resonator.a and b show the experimental results, corresponding to the single-and twophoton transitions of the Xmon qubit with frequencies ωX and 1 2 ωX , respectively.c and d show the simulated results.The theoretical calculations display the changes in the amplitude of the Xmon polarization | P (ω d ) |. Loss rates for the flux qubit, lumped-element resonator and Xmon are chosen to be γ (q) /2π = γ (a) /2π = γ (b) /2π = 2 MHz, for simplicity and to be consistent with the observed linewidth in a. e (f ), Cross sections along the excitation spectra in a (b) and c (d) when ω d /2π = 5.181 GHz (2.5905 GHz).Source data are provided as a Source Data file.

1 .
Schematic of the experimental setup.Figure1ain the main text is the area denoted by the blue rectangular box.RT: room temperature.Cryo: cryogenic.LPF: low pass filter.

d
Supplementary Fig. 2. a(c) and b(d) are bare (normalized) reflection spectra without the fitting curves of Figs.2a and 2b in the main text.Source data are provided as a Source Data file.

3 .
Caculated excitation spectra of the total system (ω0,n).The parameters used in the caculation are the same as in the main text.The dressed excitation energy levels of the deep-strongly coupled qubit-resonator system ω0n (Xmon ω0n,X ) are in blue (magenta).The black levels correspond to multiple excitation levels ω0n + ω0m,X .The red dashed double-arrowed lines indicate the observed sideband transitions in Fig.2aof the main text.

SupplementaryFig. 4 .
Calculated most relevant terms in the two-photon absorption rate as a function of the external flux bias δΦext.The detected signal is approximately proportional to the product of these quantities.The calculated Y1,2 term goes to zero for δΦext → 0, when the parity selection rule for the dressed Xmon is restored.The Y0,2 is not shown in the figure as it is almost unity in the whole range investigated here.

Figure 3
Figure 3 in the main text shows the comparison between experimental (top) and theoretical (middle) results calculated for the reported parameters.The left panel shows the weak-excitation, two-tone spectroscopy of the dressed Xmon (|E 0 → |E 2 ) transition, while in the right panel, a strong-excitation spectroscopy shows the twophoton resonance at half of the Xmon (|E 0 → |E 2 ) transition.Both the experimental and theoretical signals are displayed as a function of the drive frequency ω d and of the external flux bias, δΦ ext .For zero flux bias δΦ ext = 0, the two-photon resonance disappears as the parity symmetry is completely restored.The theoretical calculations display the changes in the amplitude of the Xmon polarization | P (ω d ) |.