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

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 quantum vacuum of the system, since it becomes a novel entangled ground state $|G\rangle$ 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., $\langle G|a^{\dagger}a|G\rangle\neq 0$, where $a^{\dagger}(a)$ is the creation (annihilation) operator of the resonator mode, the ground-state photons are actually virtual (tightly bound to the artificial atom Munoz2018NC ) and cannot be directly detected. Theoretically, it was proposed to employ non-adiabatic modulations vacanti2012photon , sudden turn-off of the qubit–resonator interaction garziano2013switching , or a spontaneous decay mechanism of multi-level systems stassi2013spontaneous to convert these virtual photons into real ones (similar to the dynamical Casimir effect johansson2009prl ; wilson2011nature ), so as to generate radiation out of the resonator. However, these are still experimentally challenging.

In the standard model of particle physics, the $W^{\pm}$ and $Z$ weak gauge bosons obtain mass via the Higgs mechanism, in which the electroweak gauge symmetry $\rm{SU(2)\times U(1)}$ 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 Liu2005PRL . In this parity-symmetry case, the quantum-vacuum expectation value of the resonator field is zero, $\langle G|(a+a^{\dagger})|G\rangle=0$, where $|G\rangle$ is the qubit–resonator 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 $\langle G|(a+a^{\dagger})|G\rangle\neq 0$ garziano2014vacuum . 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 $\langle G|(a+a^{\dagger})|G\rangle$ 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.

Similar to the three-junction flux qubit Mooij1999Science , the five-junction flux qubit has both clockwise and counterclockwise persistent-current states. Away from the optimal point $\Phi_{\rm ext}=(n+\frac{1}{2})\Phi_{0}$, where $\Phi_{\rm ext}$ is the external flux threading the loop of the flux qubit, $\Phi_{0}=h/2e$ is the superconducting flux quantum, and $n$ is an integer, these two persistent-current states have an energy difference $\varepsilon=2I_{p}\delta\Phi_{\rm ext}$, depending on the maximum persistent current $I_{p}$ and the flux bias $\delta\Phi_{\rm ext}\equiv\Phi_{\rm ext}-(n+\frac{1}{2})\Phi_{0}$. Also, there is a barrier between these two persistent-current states, which removes their degeneracy at the optimal point by opening an energy gap $\Delta$. In the basis of eigenstates, the Hamiltonian of the flux qubit can be written as (setting $\hbar=1$) $H_{q}=\omega_{q}\sigma_{z}/2$, where $\omega_{q}=\sqrt{\Delta^{2}+\varepsilon^{2}}$ is the transition frequency of the qubit and $\sigma_{z}$ 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 yoshihara2017superconducting : $H_{r}=\omega_{r}a^{\dagger}a$, where $\omega_{r}$ is the resonance frequency of the resonator mode. This $\omega_{r}$ is V-shaped versus $\delta\Phi_{\rm ext}$ around the optimal point yoshihara2017superconducting because the inductance across the qubit loop, as part of the total inductance of the lumped-element resonator, depends approximately linearly on $|\delta\Phi_{\rm ext}|$. The large JJ shared by the flux qubit and the lumped-element resonator acts as an effective inductance to produce an interaction between them, $H_{\rm int}=g[\cos\theta\,\sigma_{z}-\sin\theta\,\sigma_{x}](a^{{\dagger}}+a)$, where $\tan\theta=\Delta/\varepsilon$, and $g=MI_{p}I_{r}$ is the coupling strength, with $M\approx L_{c}$ being the mutual inductance and $I_{r}=\sqrt{\omega_{r}/2(L_{0}+L_{c})}$ 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 $H_{\rm int}$, and the Hamiltonian of the qubit–resonator system is written as

i.e., the generalized quantum Rabi model Nori2017PhyRep .

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 $\omega_{p}/2\pi=4.8$ 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 $|G\rangle\equiv|0\rangle_{s}$ to the first- and second-excited states, $|1\rangle_{s}$ and $|2\rangle_{s}$ of the qubit–resonator system, respectively, i.e., $\omega_{01}$ and $\omega_{02}$ (see the solid fitting curves in Figs. 2c and 2b). Around $\omega_{p}/2\pi=11.9$ GHz, we observe the transition from the ground state $|G\rangle$ to the third-excited state $|3\rangle_{s}$ (Fig. 2a), with the solid fitting curves corresponding to $\omega_{03}$. 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 qubit–resonator 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 john1990prl ; goban2014nc . 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 qubit–resonator 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 john1990prl .

By fitting the transition frequencies $\omega_{01}$, $\omega_{02}$ and $\omega_{03}$ with experimental results in Fig. 2, we can derive the parameters of the generalized Rabi model in Eq. (1), which are $I_{p}=245~{}{\rm nA}$ and $\Delta/2\pi=15.0~{}{\rm GHz}$ for the flux qubit, $\omega_{r}/2\pi=4.82~{}{\rm GHz}$ for the lumped-element resonator, and $g/2\pi=4.55~{}{\rm GHz}$ for the qubit–resonator coupling. Here the obtained resonance frequency $\omega_{r}/2\pi=4.82~{}{\rm GHz}$ is the value when the external flux bias is at the optimal point $\delta\Phi_{\rm ext}=0$. In our qubit–resonator system, we achieve $g/\omega_{r}\approx 0.944$, indicating that it indeed reaches the deep-strong coupling regime $g/\omega_{r}\sim 1$.

When $\delta\Phi_{\rm ext}=0$ ($\theta=\pi/2$), the deep-strongly coupled system in Eq. (1) reduces to the standard quantum Rabi model. Instead of a trivial (product) ground state $|g,0\rangle$ in the Jaynes-Cummings model, it has a quantum vacuum (i.e., entangled ground state) $|G\rangle$, with $\langle G|a^{{\dagger}}a|G\rangle\neq 0$. This standard Rabi model has a well-defined parity symmetry, characterized by $\sigma_{z}e^{i\pi a^{{\dagger}}a}$, which ensures that the ground state is a superposition of all states with an even number of excitations kockum2019ultrastrong . For this quantum vacuum, $\langle G|(a+a^{\dagger})|G\rangle$=0. When $\delta\Phi_{\rm ext}\neq 0$, $H_{s}$ in Eq. (1) has an extra longitudinal coupling term proportional to $\sigma_{z}$. It breaks the parity symmetry of the model, and hence both even and odd numbers of excitations are allowed in the new ground state garziano2014vacuum as well as in the excited states of the coupled system. Now, both $\langle G|a^{{\dagger}}a|G\rangle\neq 0$ and $\langle G|(a+a^{\dagger})|G\rangle\neq 0$.

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 $\omega_{\rm CPW}/2\pi=3.554$ GHz and the coupling strength between the waveguide resonator and the Xmon qubit is $g_{X}/2\pi=28~{}$MHz. Then, we obtain the transition frequency $\omega_{X}/2\pi=5.181~{}{\rm GHz}$ of the Xmon qubit and its anharmonicity $A/2\pi=-0.16~{}{\rm GHz}$. With these parameters as well as the relations $\omega_{X}=\sqrt{8E_{c}E_{J}}-E_{c}$ and $A=-E_{c}$, we have $E_{c}/2\pi=0.16$ GHz and $E_{J}/2\pi=20.97$ GHz. Because the lumped-element resonator couples to the Xmon, it induces an offset charge to the Josephson junction, leading $H_{X}$ to $\tilde{H}_{X}=4E_{c}(n-n_{R})^{2}-E_{J}\cos\varphi$, where $n_{R}=i\frac{g^{\prime}}{8E_{c}}(a-a^{{\dagger}})$, and $g^{\prime}\approx g_{X}$ (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 $H_{\mathrm{tot}}=H_{s}+\tilde{H}_{X}$. 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 $|G\rangle$ at the temperature $\sim$30 mK.

Note that in the dispersive regime, where the Xmon–resonator coupling rate is much lower than the corresponding detuning, the energy transitions of the Xmon are almost unaffected by the interaction (i.e., flux-bias-insensitive) 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 $\omega_{X}$ of the Xmon qubit, and the resonance frequency of the two-photon transition is $\frac{1}{2}\omega_{X}$. In our chip, $\frac{1}{2}\omega_{X}$ is designed to be well separated from both $\omega_{01}$ and $\frac{1}{2}\omega_{02}$ 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 $\delta\Phi_{\rm ext}=0$, 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 $v=\langle G|(a+a^{\dagger})|G\rangle\neq 0$, is able to break the parity symmetry of the Xmon artificial atom, without however inducing any $\delta\Phi_{\rm ext}$ 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 $\cot{\theta}=\varepsilon/\Delta\simeq 10^{-2}$, is able to activate two-photon 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 ($\gtrsim 20$ 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 $b=\sum_{n=0}^{3}\sqrt{n+1}\,|n\rangle\langle n+1|$ is the annihilation operator for the Xmon, and $H_{X}^{(4)}=\sum_{n=0}^{3}\varepsilon_{n}|n\rangle\langle n|$ 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 $\langle P(\omega_{d})\rangle=\text{Tr}[-i(b-b^{\dagger})\rho(\omega_{d})]$, with $\rho$ being the density operator of the system. The latter can be calculated using the master equation approach in the dressed picture ridolfo2012photon (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 $|G\rangle$ at a temperature of $\sim$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 $\omega_{d}\simeq\omega_{X}/2$. 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 $\varepsilon/\Delta$, quantifying the degree of symmetry breaking induced by the flux offset on the flux qubit, is very small ($\varepsilon/\Delta\simeq 10^{-2}$) at $\delta\Phi_{\rm ext}=0.1$ m$\Phi_{0}$, 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 $g/\omega_{r}$ 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 $w=\langle G|(a-a^{\dagger})|G\rangle\neq 0$, the Xmon–resonator interaction in Eq. (2) could be approximated as $\sim-g^{\prime}w(b-b^{\dagger})$. 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 $v\neq 0$. 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 $v=\langle G|(a+a^{\dagger})|G\rangle\neq 0$. In the present case, the symmetry breaking of the qubit–resonator 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 $H_{\rm tot}$, the two-photon transition rate is proportional to the product $|Y_{0,1}Y_{1,2}|$, where $Y_{i,j}=\langle E_{i}|-i(b-b^{\dagger})|E_{j}\rangle$, with $|E_{j}\rangle$ eigenvectors of $H_{\rm tot}$ sorted from the lower to the higher corresponding energy levels. Thus, with $|E_{0}\rangle$ being the ground state of the whole interacting system, we identify $|E_{1}\rangle$ as the first excited level of $H_{s}$ (slightly dressed by the interaction with the Xmon) and $|E_{2}\rangle$ the corresponding first excited dressed level of $H_{X}^{(4)}$. 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 $\delta\Phi_{\rm ext}$ (see Supplementary Fig. 4) and it is zero for $\delta\Phi_{\rm ext}=0$, due to the parity symmetry. This explains the onset of the parity-symmetry breaking felt by the Xmon.

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 deep-strong coupling regimes and can be used as a tool to explore the coherence properties of quantum vacua in these exotic hybrid quantum systems hwang2015quantum ; liu2017universal ; xie2014anisotropic , and the occurrence of superradiant phase transitions in Dicke-like systems nataf2010vacuum .