Abstract
Understanding the interaction between light and matter is very relevant for fundamental studies of quantum electrodynamics and for the development of quantum technologies. The quantum Rabi model captures the physics of a single atom interacting with a single photon at all regimes of coupling strength. We report the spectroscopic observation of a resonant transition that breaks a selection rule in the quantum Rabi model, implemented using an LC resonator and an artificial atom, a superconducting qubit. The eigenstates of the system consist of a superposition of bare qubitresonator states with a relative sign. When the qubitresonator coupling strength is negligible compared to their own frequencies, the matrix element between excited eigenstates of different sign is very small in presence of a resonator drive, establishing a signpreserving selection rule. Here, our qubitresonator system operates in the ultrastrong coupling regime, where the coupling strength is 10% of the resonator frequency, allowing signchanging transitions to be activated and, therefore, detected. This work shows that signchanging transitions are an unambiguous, distinctive signature of systems operating in the ultrastrong coupling regime of the quantum Rabi model. These results pave the way to further studies of signpreserving selection rules in multiqubit and multiphoton models.
Introduction
The quantum Rabi model describes the dipolar interaction between a twolevel system, a qubit of frequency ω_{q}, and a quantized electromagnetic field mode, an oscillator of frequency ω_{r}^{1,2}. The strength of the interaction g between qubit and oscillator defines different regimes of coupling, reflecting the dominating energy scales in the dynamics of the system. The limit of coupling strength g much smaller than qubit/resonator frequencies is the wellknown JaynesCummings (JC) model^{3}, where the rotatingwave approximation (RWA) holds. The ultrastrong coupling (USC) regime is defined by g/ω_{r} ≥ 0.1. Such condition implies that counterrotating terms in the interaction Hamiltonian cannot be neglected. The USC regime presents interesting features such as a nontrivial ground state with photonic excitations^{4}, the generation of qubitoscillator nonclassical states^{5} and ultrafast gates for quantum computation^{6}. The intricate groundstate structure has motivated various proposals for quantum simulation^{7,8}. Despite its fundamental interest and applications, the experimental study of the USC regime is challenging. In this sense, superconducting quantum circuits^{9} are suitable systems to achieve large coupling strengths, despite some intrinsic limitations for most superconducting qubit types^{10}. In fact, recent ideas have emerged^{11,12,13} on how to reach the USC regime using superconducting flux qubits. The latter can attain a large galvanic coupling to resonators^{14} and have a huge anharmonicity, typically larger than , both of which are crucial to reach g/ω_{r} > 0.1. USC in an open system has recently been reported using a superconducting flux qubit^{15}.
In the JC model , the energylevel structure consists of a manifold of dressedstate doublets each containing n excitations n, ±〉^{3}. Each doublet n〉 is labeled with a quantum number ± corresponding to the relative sign in the superposition of uncoupled qubitresonator states {e, n − 1〉, g, n〉}:
The mixing angle is defined as . In the case of resonator driving represented by the operator H_{d} ~ (a + a^{†}), transitions between dressed states of different sign, , are much less favored than those preserving it, n, ±〉_{JC} ↔ n ± 1, ±〉_{JC}. For USC rates, g/ω_{r} ~ 0.1, one can still effectively describe the spectrum of the system using dressedstate doublets with correspondingly larger transition matrix elements and corrections of order g/(ω_{r} + ω_{q}) to manifolds with total number of excitations n ± 2^{16,17,18,19}. The considerations on signchanging transitions in the USC regime do not apply. Thus for coupling strength much smaller than qubit/resonator frequencies, one can establish a selection rule for the case of driven resonator H_{d} ~ (a + a^{†}). In the USC regime, due to the different nature of the system eigenstates, the selection rule may be broken. For qubit driving H_{d} ~ σ_{x}, the selection rules are slightly different than the resonator driving case, as detailed in the results section.
Observations of signpreserving transitions have been reported for superconducting transmon qubits coupled to transmission line resonators with g/ω_{r} ~ 0.02^{20,21}. To date, only a few experimental systems have reached the USC regime. In superconducting circuits, artificial atoms have been ultrastrongly coupled to resonators^{14,22,23,24} and transmission lines^{15}, where new physics beyond the RWA were reported. Polaritons in quantum wells have been ultrastrongly coupled to microcavities, causing large frequency shifts due to enhanced nonlinearities^{25}. Finally, electrons in a twodimensional electron gas undergoing cyclotron transitions ω_{c} were coupled to metamaterial cavities where the coupling strength achieved was g/ω_{c} ~ 0.58^{26}. None of the above cited experiments reported observations or was able to provide evidence of signchanging transitions.
On the theory side, despite the considerable amount of recent progress on studying the physics of the USC regime^{1,7,27,28,29,30,31,32}, only recently transitions between dressed states have been carefully analyzed^{33}.
Here, we report the spectroscopy of a qubitresonator system in the USC regime of the quantum Rabi model. We use the fact that our system is at finite temperature to excite several transitions between dressed states. One of the transitions is shown to break the signpreserving selection rule, demonstrating another paradigmatic example of the unique physics occurring in the USC regime. Using a superconducting flux qubit coupled to a lumpedelement LC resonator we are able to reproduce the quantum Rabi model of a qubit coupled to a single oscillator^{1}. In our experimental system, the qubit and the resonator have no higher energy levels within the relevant frequency range, which is advantageous for observing the breakdown of the signselection rule.
Results
Theoretical model
Our experiment operates in the regime of g/ω_{r} ~ 0.1, which will be referred to as the quantum BlochSiegert (QBS) regime. The eigenstates of the system in this regime are analogous to the doublet structure of dressed states of the JC model (see the methods section for more details):
With respect to the JC model, a correction of order λ = g/(ω_{q} + ω_{r}) appears connecting to states containing n ± 2 number of excitations due to the effect of the counterrotating terms. The mixing angle is here defined as
which is different from the JC in a photonnumber dependent detuning δ_{n} ≡ δ + 2nω_{BS} and a photonnumber dependent coupling strength . δ ≡ ω_{q} − ω_{r} is the detuning in the JC model, ω_{BS} ≡ g^{2}/(ω_{r} + ω_{q}) is the BlochSiegert shift that originates from the counterrotating terms. The existence of a selection rule for transitions between dressed states of different sign depends on the relative magnitude of matrix elements for signchanging/preserving transitions. The matrix elements between dressed states for resonatortype driving H_{d}/ = A_{r}(a + a^{†}) cos(ω_{d}t) can be calculated to be
Clearly, the signpreserving transitions are always possible while the signchanging transitions are less likely to be excited. For vanishing qubitresonator coupling g and δ > 0 (δ < 0), φ_{n} → 0 (φ_{n} → π), so sin(φ_{n}/2) → 0 (sin(φ_{n}/2) → 1), cos(φ_{n}/2) → 1 (cos(φ_{n}/2) → 0). This implies that signchanging transitions disappear when the coupling strength is much smaller than qubit/resonator frequencies, establishing a selection rule for this type of transitions.
For single qubit driving H_{d}/ = A_{q}σ_{x} cos(ω_{d}t) the transition matrix elements between dressed states read
For δ < 0, as is the case in our experiment, as g → 0, implying a signpreserving rule. However, the other transitions do not follow the same rule since as g → 0. Clearly for singlequbit driving, the selection rule considerations of driven resonator do not apply. In the rest of this work we only discuss the transitions 1, −〉 ↔ 2, −〉 and 1, −〉 ↔ 2, +〉 that follow the signpreserving rule both for resonatordriven as well as qubitdriven transitions. The case of singlequbit σ_{z} driving is described in the Methods. Therefore all our conclusions on broken selection rules still hold. Throughout the rest of the text we will drop the tilde to refer to the dressed states in the USC regime simply as n, ±〉.
Analyzing spectra in the USC regime requires accounting for the outofequilibrium quantum dynamics featuring the coloured nature of dissipative baths. One possibility is to project the master equation in the dressedstate basis n, ±〉^{19}. Another possibility is the secondorder timeconvolutionless projection operator method (TCPOM)^{34}, which has been proven useful to correctly describe the open system dynamics in the USC regime^{18,35}. Here, we follow the latter approach where the master equation reads
with , , , and the operators are defined as
We consider thermal baths at temperature T for all dissipative channels, where the relaxation coefficients depend on the spectral density of the baths d_{k}(ω) and the systembath coupling strength α_{k}(ω). Moreover, the system Hamiltonian includes the qubitresonator interaction as well as the presence of classical microwave fields, that is
The qubit mixing angle θ_{q} is defined as tan θ_{q} = Δ/ε, with Δ being the tunnel coupling between the qubit eigenstates and ε = 2I_{p}(Φ − Φ_{0}/2) the magnetic energy of the qubit (see Methods). The outofequilibrium dynamics assumes a perturbative treatment of driving mechanisms which is valid for amplitudes , ω_{r}, g^{36}, as is the case throughout this work. In all numerical simulations we use up to 6 Fock states in the resonator basis which is sufficient for g/ω_{r} ~ 0.1.
Signpreserving transition
In order to explore dressedstate transitions, we use a superconducting flux qubit galvanically coupled to an LC resonator. The results presented here are obtained with the same device where the BlochSiegert was first reported^{14}. All relevant experimental details can be found there. The resonator has frequency ω_{r}/2π = 8.13 GHz, while the qubit has persistent current I_{p} = 500 nA and a minimal splitting of Δ/2π = 4.2 GHz, so the detuning is δ = Δ − ω_{r} = −2π × 3.93 GHz at the symmetry point of the qubit. The coupling strength is g/2π = 0.82 GHz. The latter implies , so the system operates in the USC regime. The qubit and the resonator are thermally anchored to the mixing chamber stage of a dilution refrigerator at 20 mK. The qubit state is read out using a DCSQUID magnetometer. An external superconducting coil is used to set the magnetic flux in the qubit loop near Φ = Φ_{0}/2. At this flux bias point the effects of the counterrotating terms of the Hamiltonian are strongest (equation (16) with sin θ_{q} = 1).
A numerical diagonalization of the Hamiltonian in equation (16) leads to the spectrum shown in Fig. 1(a). The solid lines correspond to transitions from the ground state, while the dashed lines are transitions from the first excited state. The intermediate dashed transition 1, −〉 ↔ 2, −〉 (see Fig. 1(b)) around 8 GHz is the one studied in detail in this section and corresponds to a signpreserving transition. The highestenergy, dashed transition 1, −〉 ↔ 2, +〉 is studied in the next section and is the first transition in ascending energy to break the signpreserving selection rule, given our choice of parameters. At the qubit symmetry point Φ = Φ_{0}/2 only transitions connecting states of different parity are allowed^{37}. Therefore, the red sideband 1, −〉 ↔ 1, +〉 and blue sideband 0, g〉 ↔ 2, −〉 are forbidden at that point. Outside the symmetry point all transitions are permitted due to the asymmetry in the flux qubit potential^{38,39}. At the qubit symmetry point Φ = Φ_{0}/2, the system eigenstates correspond to those in equations (3) and (4) and the selection rules introduced in the previous section are therefore expected.
Figure 2(a) shows the spectrum of the system at low power near the resonator frequency ω_{r}/2π = 8.13 GHz. The solid and dashed curves are a fit of equation (16). The dotted line is the JC model with the same fitted parameters. The difference between the JC and Rabi models is the BlochSiegert shift ω_{BS}^{14}, with a maximum of 55 MHz at Φ = Φ_{0}/2. A negative (positive) shift of the frequency corresponds to a resonator (qubit)like dressed state transition. Figure 2(b) shows a spectroscopy trace at Φ = Φ_{0}/2. The measurements are performed following a similar protocol as the one developed previously for a tunablegap flux qubit^{40} (see also Methods). Following from Fig. 1, we find that the higher frequency resonance in Fig. 2(b) corresponds to the transition 0, g〉 ↔ 1, +〉, which given the detuning δ/2π = −3.93 GHz is mostly a resonatorlike excited dressed state. The lower frequency resonance corresponds to 1, −〉 ↔ 2, −〉, therefore it is a signpreserving transition. The finite temperature in our system enables the excitation of this transition. The signal from this resonance is weak given the small amount of population of qubitlike state 1, −〉 in thermal equilibrium. The fact that we can observe this transition at low driving strength indicates a large matrix element as predicted by equations (6, 7, 8, 9, 10, 11, 12, 13). The low signaltonoise ratio leads to spurious peaks and dips which do not represent additional transitions, as verified by the spectrum in Fig. 1. The geometry of our system is such that qubit and resonator driving amplitudes have a comparable value, (see Methods), therefore both systems are simultaneously driven in all measurements presented.
The resonances in this spectrum look broad due to the low relaxation time of this particular device, mostly due parasitic capacitance to the SQUID readout circuitry. Other flux qubit experiments have shown much better performance, even with SQUID detectors^{41,42}. There is a clear asymmetry between the two resonances in Fig. 2(b). This difference allows us to approximately calibrate the electronic temperature of the system. Solving the master equation iteratively to reach steadystate is a lengthy process, making it impractical to run a fit of the observed spectrum. Instead, the unknown parameters are swept in order to find the optimal set that leads to a calculated spectrum closest to the measurements. We plot the expectation value of the qubit current operator , where is the steadystate solution of the TCPOM master equation, equation (14) using the eigenstates from equation (16). More details can be found in the Methods. The optimal set of parameters is , , . For the temperature, we plot T = 90 mK (blue asterisks) and T = 150 mK (red circles) to indicate the possible range of temperatures. Other experiments using superconducting qubits have observed comparable finite electronic temperatures^{43}.
The results of the simulation clearly indicate an asymmetry between the two resonances near 8 GHz. A global scaling factor and an offset have been applied to the numerical solution to match the amplitude of the experimental resonances. The scaling factor gives the transduction between average persistent current in the qubit to SQUID switching probability. In the experiment, the offset value is chosen to be 50% probability of the DCSQUID to switch to the running state where a finite voltage is generated^{42}. The differences between simulation and experiment for the resonance at 8.02 GHz is most likely due to the low signaltonoise, leading to an imprecise determination of the temperature of the system.
Broken signpreserving selection rule
In this section, we perform spectroscopy of the qubitresonator system with driving amplitude approximately 5 times larger than the one used in Fig. 2. The enhanced driving strength enables additional transitions to occur, both one and twophoton transitions. The highpower spectrum of the system is shown in Fig. 3(a), showing a rich spectral structure. For frequencies below 8 GHz, multiphoton processes can be identified. The frequency of each multiphoton transition follows the proper scaling , as was already reported for a transmon qubit coupled to a resonator^{44}. θ_{q} is the qubit mixing angle defined below equation (16). The resonance at 8.25 GHz from Fig. 2 has been powerbroadened, overlapping with the signpreserving transition 1, −〉 ↔ 2, −〉 at 8.02 GHz. The large signal generated by this resonance may be related to a buildup of the population in the resonator due to the strong drive^{45}.
Above 8 GHz, there are two distinctive resonances near . The transition increasing in frequency away from the symmetry point corresponds to the blue sideband, 0, g〉 ↔ 2, −〉. The reason it vanishes at Φ = Φ_{0}/2 is due to parity selection rules^{37}. The other weaker resonance corresponds to the transition 1, −〉 ↔ 2, +〉 and is, therefore, a signchanging transition that breaks the signpreserving selection rule explained in the introductory section. To our knowledge, this is the first observation of this kind of transitions between excited states of the quantum Rabi model. The small signal from this resonance evidences the difference in magnitude of the transition matrix elements as compared to the one for 1, −〉 ↔ 2, −〉 given the difference in driving amplitudes used to acquire the data in Figs 2 and 3.
The spectroscopic sign of any resonance detected by the SQUID magnetometer relates to the magnetic field generated by the qubit in steadystate when an external drive is present. The transition 1, −〉 ↔ 2, +〉 increases the switching probability above the reference at 50%, compared to all other resonances where the switching probability decreases below 50%. This implies that in steadystate the qubit is more polarized to the ground state than in thermal equilibrium, contrary to the other resonances where the qubit ends up with more population in the excited state. The resonance 1, −〉 ↔ 2, +〉 therefore cools the qubit down by transferring its excess thermal energy to the resonator, since the state 2, +〉 is a resonatorlike excited dressed state. A spectroscopy trace at the symmetry point can be observed in Fig. 3(b). The switching probability at 12.3 GHz increases with respect to the background set at P_{sw} = 0% (referenced to 50%).
Discussion
In Fig. 4(a,b) we calculate the matrix element for transitions T_{1−〉↔2−〉}, T_{1−〉↔2+〉}, respectively, as function of normalized coupling strength g/ω_{r} for driving the resonator . The blue solid line is calculated numerically from equation (16), the greendashed line corresponds to the quantum BlochSiegert (QBS) regime equations (7), (8), (11) and (12) and the reddotted line is the JaynesCummings model calculated from the states in equations (1) and (2). In Fig. 4(c,d) we also plot the matrix elements T_{1−〉↔2−〉}, T_{1−〉↔2+〉} for the case of singlequbit driving H_{d} ~ σ_{x}. The difference in matrix elements for sign changing/preserving transitions is very clear for both types of driving. Figure 4(b,d) show that in our experiment the brokensign transition is mostly driven by the direct qubit drive. As shown in Fig. 4(e), the relative weight T_{1−〉↔2−〉}/T_{1−〉↔2+〉} decreases with increasing g/ω_{r} for both types of driving since we consider finite detuning δ/2π = −3.91 GHz. If we had the resonance condition ω_{r} = ω_{q} = 2π × 8.13 GHz, the relative weight of the matrix elements for resonator (qubit) driving would reach T_{1,−〉↔2,−〉}/T_{1,−〉↔2,+〉}≈7(1) at g/ω_{r} = 0.1, instead of 234(4.8) as shown in Fig. 4(e). The dashed vertical line in Fig. 4(a–e) marks the operating point of our device. The observation of the signchanging transition 1, −〉 ↔ 2, +〉 in this experiment can therefore be attributed to the system operating in the USC regime.
In Fig. 5 we compare the measured spectrum near 12 GHz (Fig. 5(a)) with a simulation (Fig. 5(b)) with the TCPOM master equation (14) using larger drive amplitude, A_{qb}/2π = A_{r}/2π = 50 MHz, keeping the rest of terms equal as those found in the previous section, with T = 100 mK. The increased driving amplitude used here is approximately a factor of 5 larger compared to the amplitude used for the simulations in Fig. 2(b), closely matching the ratio of driving amplitudes used in the measurements of Fig. 3 relative to those in Fig. 2. The results are plotted in Fig. 5(b) for the probability to find the system excited, , where ρ is the density matrix of the coupled system calculated with the master equation in steadystate. The position of the main two resonances appearing in the experiment in Fig. 5(a) are clearly reproduced up to ~−1 mΦ_{0}, particularly at the symmetry point Φ = Φ_{0}/2 where only the signchanging transition 1, −〉 ↔ 2, +〉 is excited. Beyond −1 mΦ_{0}, the experimental signal in Fig. 5(a) disappears due to an increase of the dephasing rate of the qubit away from the symmetry point^{41,46,47} which is not taken into account in the numerical simulations.
Conclusions
We report the observation of transitions between excited states of a superconducting flux qubitresonator system in the ultrastrong coupling regime. The strength of the coupling combined with initial thermal population of the qubit permits the observation of a transition between excited states that was never detected before. Such transition changes the relative sign in the superposition of bare qubitresonator states in the dressedstate level structure. We developed a theoretical model based on the time convolutionless projection operator method that reproduces all transitions observed spectroscopically. Our work, therefore, verifies the existence of signchanging transitions in the ultrastrong coupling regime of the quantum Rabi model, despite their weak strength. Until now, evidence for superconducting qubitresonator systems to operate in the USC regime was found as spectral deviations from the JaynesCummings model^{14,22,23}. We instead put forward the observation of signchanging transitions in the quantum Rabi model as a direct signature of any physical system in the USC regime for resonatortype driving. Our work can be extended to multiqubit^{33} or multiphoton^{48} quantum Rabi models where forbidden transitions play a key role in understanding and manipulating the energylevel structure of the system.
Methods
Quantum BlochSiegert Hamiltonian and eigenenergies
For a qubit coupled to a resonator the Hamiltonian that describes the full system dynamics is the quantum Rabi model^{1}. When g/ω_{r}, the rotatingwave approximation (RWA) holds and the JaynesCummings (JC) model is obtained^{3}. In the regime g/ω_{r} ~ g/ω_{q} ~ 0.1 the RWA is no longer valid but a perturbative treatment still permits deriving the Hamiltonian of the system analytically^{16,17}, as well as a model on dissipation^{18,19,36}.
The BlochSiegert Hamiltonian derives from the quantum Rabi Hamiltonian via perturbative treatment of the parameter ^{16,17,19}. In this perturbative regime, the counterrotating terms are treated as offresonant fields and the quantum Rabi model can be transformed in an effective dispersive picture. The resulting Hamiltonian is:
where the BlochSiegert shift ω_{BS} is defined as
with photondependent coupling strength
Here is the flux qubit splitting with ε = 2I_{p}(Φ − Φ_{0}/2) being the magnetic energy of the qubit in its truncated twostate Hilbert space. The qubit mixing angle θ_{q} is defined as tan θ_{q} = Δ/ε.
Projecting the Hamiltonian on the product state Ψ〉 = i, n〉, i = {e, g} has a boxdiagonal representation that simplifies the full diagonalization:
For box n, the following eigenvalues are found:
with ground state
where n ≥ 1 represents the number of total excitations in the uncoupled basis. All levels are shifted by ω_{BS} as compared to the JC model. In addition, both the effective detuning between levels δ + 2nω_{BS} as well as the effective coupling constant (second term in the square root of equation (22)) depend on the number of excitations. The latter is the secondorder correction in this dispersive picture. Using the following definitions:
the eigenenergies take the form
exactly analogous to the JC model^{9}. Following this analogy the eigenstates of the system can be expressed similarly to the eigenstates of the JC model, with corrections to states with different total number of excitations, as
where the mixing angle is defined as
Equations (28) and (29) show that the energy eigenstates of the BlochSiegert Hamiltonian correspond to doublets of superpositions exactly as with the JaynesCummings model, with coefficients having a different dependence on the number of photons n and corrections of order λ = g/(ω_{r} + ω_{q}) to other states with different number of excitations. Therefore the selection rule considerations between transitions that change the sign quantum number explained in the introduction break down in the USC regime.
Dissipation dynamics
Here, the parameters of the qubitresonator system described in the main text are derived using numerical simulations. Besides the temperature of the bath T, there are a set of unknown parameters that need to be determined: the qubit relaxation and decoherence rates Γ_{1}, Γ_{2}, the resonator decay rate Γ_{r} and the amplitudes of the external fluxes A_{qb} and A_{r} driving the qubit and the resonator, respectively. Estimates of the mutual inductance between qubit/resonator and microwave line permit setting A_{qb} = A_{r} by geometry (see Fig. 6). We also assume that qubit decoherence is governed by relaxation alone Γ_{2} = Γ_{1}/2 since this device shows very short T_{1} ~ 10 ns times near the symmetry point Φ = Φ_{0}/2.
A technique to characterize the temperature of a transmon qubit by driving Rabi oscillations was already developed^{43}, but due to the short relaxation time of our qubit this technique is not suitable. We would like to emphasize that the coherence of the flux qubit used in this experiment is limited by spontaneous emission to the readout circuitry of the SQUID detector^{42}. Using proper filtering^{49} and a more symmetric circuit design^{40} would allow better quantitative study of the spectroscopic resonances. Other experiments have shown much longer values of qubit lifetime at the symmetry point^{41,46,47}, exceeding 10 μs.
We run a numerical simulation of the master equation presented in the theory section and compare the steadystate solution to the spectroscopy measurements. The numerical simulation of the master equation (Equation (14) in the main article) is performed by a RungeKutta method considering a Fock space of up to 6 Fock states in the resonator, sufficient for the relative coupling strength g/ω_{r} = 0.1. The operators and are numerically built by means of projecting them into the eigenstates of the qubitresonator Hamiltonian, , where
For instance, the operator can be represented as
Notice that taking into account Equation (16) from the main article in the above expression will lead to integrals of the form
where . The second term in the above equation stands for the Cauchy principal value. The latter leads to small Lamb shifts that we have neglected in our numerical simulation.
An example of the dynamical evolution governed by the master equation can be seen in Fig. 7. The plots show the probability to populate the lowest four energy levels from Fig. 1 in the main text as function of applied driving time. The parameters used to obtain each figure are ρ(0) = 1, −〉〈1, −, T = 90 mK, Γ_{r}/2π = 0.1 MHz, Γ_{qb}/2π = 15 MHz, ω_{D}/2π = 4.12 GHz, A_{qb} = A_{r} = 90 MHz (12 MHz) Fig. 7(a,b). The value of the driving frequency ω_{D} is resonant with the transition 0, g〉 ↔ 1, −〉. Figure 7(a) features small oscillations due to a driving amplitude stronger than the qubit decay rate. Figure 7(b) shows a smooth behavior given that the dissipative mechanisms surpass the driving microwave fields.
In order to compare the numerical simulations with the actual data we need to take into account that the DCSQUID detector is sensitive to the flux generated by the qubit, Φ_{qb} = M_{SQ−qb}〈I_{circ}〉, where M_{SQ−qb} is the SQUIDqubit mutual inductance and 〈I_{circ}〉 the expectation value of the circulating current operator of the qubit. The current operator has the form with the phase operator across the α junction. Using the twolevel approximation of the flux qubit, calculated in the energy basis of the qubit, with I_{p} being the persistent current. In this case it is easy to show that for a pure state 〈σ_{z}〉 = ±cos θ_{q}, with θ_{q} defined in the previous section. The sign depends on whether the qubit is in the ground or the excited state, respectively. For a mixed state , .
Measurement protocol
To perform measurements at the symmetry point we use a protocol developed in previous experiments^{40}. The qubit is initiated in an external bias flux Φ_{i} = −2.4 mΦ_{0} where it produces a net magnetic field. Then a flux square pulse is applied to the qubit via the microwave line, followed by a long microwave burst. The length of the burst is sufficient to bring the system in steadystate for all resonances. At the end of the burst the flux pulse is ramped down until the qubit is brought back to the initial point. The rise time of the pulse (~2 ns) is slow enough that the LandauZener tunneling probability is below 1% when the qubit is brought across the avoidedlevel crossing with the resonator, so the qubit population is transferred adiabatically from the symmetry point to the readout point.
Selection rule for longitudinal driving
A longitudinal singlequbit driving characterized by a σ_{z} operator will not break the parity symmetry of the quantum Rabi model (QRM) at the qubit symmetry point. Correspondingly, the driving may induce transitions between states of the same parity subspace. For instance, transitions will be allowed between states to or to . Such transitions are activated by a proper choice of the resonance condition, with the signal amplitude being much smaller than the energy differences between higherlevel states in the QRM. This assures the validity of the rotatingwave approximation. Away from the symmetry point, the QRM breaks its parity symmetry and consequently its selection rule associated with cavity and/or qubit driving.
More explicitly, let us consider the QRM with a flux qubit at its symmetry point driven by a microwave field along the longitudinal axis σ_{z}. The total Hamiltonian reads:
where . The Hamiltonian (34) can be written in terms of the eigenstates of H_{QRM}, that is, H_{QRM}j〉 = ω_{j}j〉. Notice that the basis contain states that belong to the parity subspace +1 and states that belong to the parity subspace −1. The Hamiltonian (34) can be rewritten as
where . As previously mentioned, the longitudinal driving acting upon the qubit does not break the parity symmetry characterized by the parity operator , as clearly σ_{z} commutes with . The consequence is that the matrix elements Z_{jk} are different from zero only for j = k and for states that belong to the same parity subspace.
If we go to an interaction representation with respect to H_{QRM}, the Hamiltonian (35) reads
where we explicitly assume that for k > j, ω_{k} > ω_{j}. From the above Hamiltonian it is clear that if we satisfy the condition , then timedependent corrections to the QRM eigenstates might be neglected. Moreover, if the condition is satisfied, so that the rotatingwave approximation holds, it might be possible to activate transitions between states belonging to the same parity subspace under the resonance condition ω_{D} = ω_{kj}.
Additional Information
How to cite this article: FornDíaz, P. et al. Broken selection rule in the quantum Rabi model. Sci. Rep. 6, 26720; doi: 10.1038/srep26720 (2016).
Change history
19 October 2016
A correction has been published and is appended to both the HTML and PDF versions of this paper. The error has not been fixed in the paper.
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Acknowledgements
We would like to acknowledge Jürgen Lisenfeld for his contributions to the measurements and Thomas Picot and Johannes Fink for fruitful discussions. P. F.D. acknowledges funding from NSERC, Canada Foundation for Innovation, Ontario Ministry of Research and Innovation and Industry Canada. E. S. would like to acknowledge funding from Spanish MINECO grants FIS201236673C0302 and FIS201569983P, Basque Government grant IT47210, UPV/EHU grant UFI 11/55, PROMISCE and SCALEQIT EU projects. G. R. acknowledges the support from the Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT, Chile) under grant 1150653.
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Affiliations
Institute for Quantum Computing, Department of Physics and Astronomy, and Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, N2L 3G1, Canada
 P. FornDíaz
Kavli Institute of Nanoscience, Delft University of Technology, Post Office Box 5046, 2600 GA Delft, The Netherlands
 P. FornDíaz
 , C. J. P. M. Harmans
 & J. E. Mooij
Departamento de Física, Universidad de Santiago de Chile (USACH), Avenida Ecuador 3493, 9170124, Santiago, Chile
 G. Romero
Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, E48080 Bilbao, Spain
 E. Solano
IKERBASQUE, Basque Foundation for Science, María Díaz de Haro 3, E48013 Bilbao, Spain
 E. Solano
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Contributions
P.F.D., C.J.P.M.H. and J.E.M. designed the experiment. P.F.D. fabricated the device and carried out the measurements. P.F.D., G.R. and E.S. analyzed the results theoretically. G.R. developed the numerical analysis. P.F.D. wrote the manuscript. All authors reviewed the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to P. FornDíaz.
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