Nonadiabatic geometric quantum computation with optimal control on superconducting circuits
NNonadiabatic geometric quantum computation with optimal control on superconducting circuits
Jing Xu, Sai Li, Tao Chen, and Zheng-Yuan Xue ∗ Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, and School of Physicsand Telecommunication Engineering, South China Normal University, Guangzhou 510006, China (Dated: April 23, 2020)Quantum gates, which are the essential building blocks of quantum computers, are very fragile. Thus, torealize robust quantum gates with high fidelity is the ultimate goal of quantum manipulation. Here, we propose anonadiabatic geometric quantum computation scheme on superconducting circuits to engineer arbitrary quantumgates, which share both the robust merit of geometric phases and the capacity to combine with the optimalcontrol technique to further enhance the gate robustness. Specifically, in our proposal, arbitrary geometricsingle-qubit gates can be realized on a transmon qubit, by a resonant microwave field driving, with both theamplitude and phase of the driving being time-dependent. Meanwhile, nontrivial two-qubit geometric gates canbe implemented by two capacitively coupled transmon qubits, with one of the transmon qubits’ frequency beingmodulated to obtain effective resonant coupling between them. Therefore, our scheme provides a promisingstep towards fault-tolerant solid-state quantum computation.
Recently, constructing the quantum computer based on thequantum mechanical theory, is highly desired, to deal withhard problems. However, quantum systems will inevitablyinteract with their surrounding environment. On the otherhand, the precise quantum manipulation of a quantum systemis limited by the precision of controlling the driving fields.Thus, fast and robust quantum manipulation is highly desired.To construct a fault-tolerant quantum computer, topologicalquantum computation strategy is one of the most exciting ad-vances [1], however, the realization of an elementary quan-tum gate there is still an experimental difficulty currently. No-tably, geometric phases [2–4] possess the intrinsic characterof noise-resilience against certain local noises, and thus cannaturally be used to construct robust quantum gates for con-structing a fault-tolerant quantum computer.Previously, geometric quantum computation (GQC) hasbeen proposed based on adiabatic evolutions with bothAbelian [5] and non-Abelian geometric phases [6, 7]. How-ever, the adiabatic evolution requires long running time suchthat a quantum state can be ruined by the decoherence ef-fect. To overcome this problem, GQC with nonadiabatic evo-lutions has been proposed to achieve high-fidelity quantumgates based on both Abelian [8–11] with experimental demon-strations [12–16] and non-Abelian geometric phases [17, 18].Unfortunately, the existence of systematic errors will devas-tate the advantage of the robustness of geometric quantumgates [19, 20]. Recently, theoretical [21, 22] and experimentalworks [23] have been proposed to further enhance the robust-ness of nonadiabatic non-Abelian geometric quantum gatesagainst the control errors, based on three-level systems, bycombining the gate operations with optimal control technique(OCT) [11, 24–26]. However, compared to the non-Abeliancase, quantum gates induced from Abelian geometric phasesbased on two levels are easier to be realized, and the requiredtwo-qubit interaction is experimentally accessible.Therefore, we here propose a fast GQC scheme that can becompatible with OCT on superconducting circuits, to furtherimprove the robustness of the implemented quantum gatesagainst control errors of the driving fields. Superconducting circuits [27–30] have shown the unique merits of the large-scale integrability and flexibility of operations, and thus aretreated as one of the promising platforms for the physical im-plementation of scalable quantum computation. Meanwhile, asuperconducting transmon device [29] can be easily addressedto be a two-level system, i.e., the ground and first excitedstates {| (cid:105) , | (cid:105)} , which can serve as a qubit and operated bya resonant driving microwave field. Thus, arbitrary geometricsingle-qubit gates in our scheme can be accurately achievedafter canceling the leakage to the higher excited states, mainlythe second excited state | (cid:105) , by combing with the DRAG cor-rection [31–33]. In addition, a recent experiment [15] showsthat the time-dependent effective resonant coupling can be in-duced in a two coupled superconducting qubits system, andthus our nontrivial geometric two-qubit control-phase gatescan also be resonantly realized [34, 35] in a simple experi-mental setup. Furthermore, by combining with OCT, the ro-bustness of the implemented geometric gates against the staticsystematic error can be greatly enhanced.We first consider the construction of arbitrary single-qubitgeometric gates in the computation basis S = {| (cid:105) , | (cid:105)} . Fora driving Hamiltonian H d ( t ) in the S , assuming ¯ h = 1 here-after, its dynamic evolution is governed by the time-dependentSchr¨odinger equation of i ∂∂t | ψ ( t ) (cid:105) = H d ( t ) | ψ ( t ) (cid:105) , (1)where | ψ ( t ) (cid:105) = e − i f ( t )2 (cid:20) cos χ ( t )2 e − i β ( t )2 | (cid:105) + sin χ ( t )2 e i β ( t )2 | (cid:105) (cid:21) can be generally defined [25] by two time-dependent angles χ ( t ) and β ( t ) , and a parameterized phase f ( t ) . Meanwhile,the orthogonal evolution state | ψ ⊥ ( t ) (cid:105) = e i f ( t )2 (cid:20) − sin χ ( t )2 e − i β ( t )2 | (cid:105) + cos χ ( t )2 e i β ( t )2 | (cid:105) (cid:21) of | ψ ( t ) (cid:105) will also satisfy Eq. (1). By modulating the pa-rameters of the driving field, the system undergoes a cyclic a r X i v : . [ qu a n t - ph ] A p r (a) (b) γ FIG. 1. Illustration of our single-qubit geometric quantum gates. (a)The qubit states are resonantly driven to realize arbitrary single-qubitgates, while the driving field will also simultaneously introduce un-wanted dispersive transitions to the higher energy states. (b) Geomet-ric illustration of the proposed single-qubit gate on a Bloch sphere. evolution, and the initial state | ψ (0) (cid:105) ( | ψ ⊥ (0) (cid:105) ) can acquire aglobal phase γ = [ f (0) − f ( τ )] / − γ ) at the final time τ ,which consists of a dynamical phase of γ D = − (cid:90) τ (cid:104) ψ ( t ) | H d ( t ) | ψ ( t ) (cid:105) dt = 12 (cid:90) τ ˙ β ( t ) sin χ ( t )cos χ ( t ) dt, and a geometric phase of γ G = i (cid:90) τ (cid:104) ˜ ψ ( t ) | ˙˜ ψ ( t ) (cid:105) dt = 12 (cid:90) τ ˙ β ( t ) cos χ ( t ) dt, where | ˜ ψ ( t ) (cid:105) = e if ( t ) / | ψ ( t ) (cid:105) . Therefore, by canceling thedynamical phase, i.e., γ D = 0 , in the global phase at the endof the cyclic evolution, we will obtain a pure geometric evolu-tion. Since the evolution here is not governed by the adiabaticcondition, the geometric phases will be induced in a fasterway than that of the adiabatic schemes [4]. Especially, whenthe phase in Hamiltonian is a constant, our scheme will reduceto the conventional nonadiabatic schemes [10, 11]. Then, thefinal geometric evolution operator in S is U ( τ ) = e iγ | ψ (0) (cid:105)(cid:104) ψ (0) | + e − iγ | ψ ⊥ (0) (cid:105)(cid:104) ψ ⊥ (0) | = e iγ(cid:126)n · (cid:126)σ , (2)where χ = χ (0) , β = β (0) . It is a rotation around the axisof (cid:126)n = (sin χ cos φ , sin χ sin φ , cos χ ) by an angle γ ,from which arbitrary single-qubit gates can be induced.We now proceed to implement scheme on superconductingcircuits. As shown in Fig. 1(a), the two lowest levels | (cid:105) and | (cid:105) of a single transmon qubit can be resonantly coupled bya microwave field with time-dependent amplitude Ω( t ) andphase φ ( t ) . Into the interaction picture, neglecting the high-order oscillating terms by the rotating-wave approximation.Then, the Hamiltonian of the system in S is H d ( t ) = 12 (cid:18) t ) e iφ ( t ) Ω( t ) e − iφ ( t ) (cid:19) . (3)In the following, to let H d ( t ) fulfil Eq. (1) at any moment, weobtain the relation of the parameters as ˙ f ( t ) = − ˙ β ( t )cos χ ( t ) , (4a) ˙ χ ( t ) = − Ω( t ) sin[ β ( t ) + φ ( t )] , (4b) ˙ β ( t ) = − Ω( t ) cot χ ( t ) cos [ β ( t ) + φ ( t )] . (4c)Thus, we find that a target evolution path of the evolution state | ψ ( t ) (cid:105) ( | ψ ⊥ ( t ) (cid:105) ) can be determined by designing the parame-ters Ω( t ) and φ ( t ) of the microwave field. Here, we constructa single-loop evolution path by defining the evolution param-eters χ ( t ) and β ( t ) to fulfill a cyclic evolution, as illustratedin Fig. 1(b). Thus we can inversely determine the parameters Ω( t ) and φ ( t ) , i.e., Ω( t ) = − ˙ χ ( t )sin ( β ( t ) + φ ( t )) ,φ ( t ) = arctan (cid:18) ˙ χ ( t )˙ β ( t ) cot χ ( t ) (cid:19) − β ( t ) . (5)In addition, we also need to ensure the accumulated dynamicalphases are zero at the end of cyclic evolution to achieve puregeometric operations.Specifically, we consider rotations around X and Z axes astwo typical examples in detail. Firstly, to realize the geometricrotation operators around the X axis, we divide a single-loopevolution path into four equal parts, which aims to cancel dy-namical phases at the end of cyclic evolution. The shape of χ j ( t ) and initial values of β j ( t ) in each part are t ∈ [0 , τ /
4] : χ ( t ) = π [1 + sin (2 πt/τ )] / ,β (0) = 0 ,t ∈ [ τ / , τ /
2] : χ ( t ) = π [1 + sin (2 πt/τ )] / ,β ( τ /
4) = β ( τ / − γ,t ∈ [ τ / , τ /
4] : χ ( t ) = π [1 − sin (2 πt/τ )] / ,β ( τ /
2) = β ( τ / ,t ∈ [3 τ / , τ ] : χ ( t ) = π [1 − sin (2 πt/τ )] / ,β (3 τ /
4) = β (3 τ /
4) + γ, (6)where the shape of β j ( t ) is set as β j ( t ) = − (cid:82) ˙ f j ( t ) cos χ j ( t ) dt for the j th part with f j ( t ) =cos 2 χ j ( t ) / . Therefore, we can obtain the shape of Ω( t ) and φ ( t ) in the different evolution parts according toEq. (5). Meanwhile, in this setting, the dynamical phase iszero and the geometric phase is γ G = γ due to the saltationof β ( t ) at the moment of t = τ / and t = 3 τ / . In this way,the geometric rotation operations e iγσ x can be realized.Similarly, to realize the geometric rotation operators aroundthe Z axis, we divide a single-loop evolution path into onlytwo equal parts. The shape of χ j ( t ) and initial values of β j ( t ) in the each parts are t ∈ [0 , τ /
2] : χ ( t ) = π sin ( πt/τ ) ,β (0) = 0 ,t ∈ [ τ / , τ ] : χ ( t ) = π sin ( πt/τ ) ,β ( τ /
2) = β ( τ / − γ, (7) t/ τ |0 (cid:2) |1 (cid:2) F T t/ τ |0 (cid:2) |1 (cid:2) F N t/ τ -2-10 Ω / Ω max φ / π t/ τ -2-10 Ω / Ω max φ / π ( c )(a)( b ) (d) FIG. 2. Implementation of single-qubit geometric gates and their per-formance. The shapes of Ω( t ) and φ ( t ) for the NOT and Phase gatesare shown in (a) and (b), respectively. The qubit-state population andthe state-fidelity dynamics of the NOT and Phase gate operations areshown in (c) and (d), respectively. where the shape of β j ( t ) is set to be β j ( t ) = − (cid:82) ˙ f j ( t ) cos χ j ( t ) dt for the j th part with f j ( t ) = [2 χ j ( t ) − sin 2 χ j ( t )] / . The shape of Ω( t ) and φ ( t ) can also be ob-tained in the different evolution parts according to Eq. (5).By only the saltation of β ( t ) at the moment of t = τ / ,the dynamical phase at the end of the cyclic evolution can beeliminated and the pure geometric phase can be accumulated.Therefore, the geometric rotations e iγσ z can be realized.Due to the weak anharmonicity of the transmon qubit, here,the DRAG correction [31–33] is also introduced to suppressthe leakage error beyond the qubit basis. Considering all ef-fects of decoherence and dominant counter-rotating terms, weuse the Lindblad master equation ˙ ρ = i [ ρ , H d ( t ) + H leak ( t )] + [Γ L ( σ ) + Γ L ( σ )] , (8)with H leak ( t ) = − α | (cid:105)(cid:104) | + (cid:104) √ t ) e iφ ( t ) | (cid:105)(cid:104) | + H . c . (cid:105) , (9)to evaluate the performance of the implemented single-qubitgates, where ρ is the density matrix of the considered systemand L ( A ) = A ρ A † − A † A ρ / − ρ A † A / is the Lind-bladian of the operator A with σ = | (cid:105)(cid:104) | + √ | (cid:105)(cid:104) | and σ = | (cid:105)(cid:104) | + 2 | (cid:105)(cid:104) | , and Γ and Γ are the decay and de-phasing rates of the transmon qubit, respectively. We considerthe case of Γ = Γ = Γ = 2 π × kHz, which is easily ac-cessible with current experimental technologies. The anhar-monicity of the transmon is set to be α = 2 π × MHz, andthe maximum amplitude Ω max = 2 π × MHz. We next takethe NOT ( N ) and Phase ( T ) gates as two typical examples,which can be realized by setting χ (0) = π/ , γ = π/ and χ (0) = 0 , γ = − π/ with the same β (0) = 0 , respectively.Assuming the maximum amplitude Ω max = 2 π × MHz, the cyclic evolution time τ is about 102 ns for the NOT gate and125 ns for the Phase gate. The corresponding shapes of Ω( t ) and φ ( t ) for the NOT and Phase gates are shown in Figs. 2(a)and 2(b), respectively. Assuming the initial states of quantumsystem are | Φ(0) (cid:105) N = | (cid:105) and | Φ(0) (cid:105) T = ( | (cid:105) + | (cid:105) ) / √ forthe cases of the NOT and Phase gates, the obtained fidelitiesare as high as F N = 99 . and F T = 99 . , as shownin Figs. 2(c) and 2(d), respectively. In addition, we find thegate fidelities of can reach F GN = 99 . and F GT = 99 . ,respectively.Here, we proceed to design the evolution path by combiningit with OCT [24–26], to further enhance the robustness of ourscheme against systematic error. Specifically, in the case ofthe geometric Z rotations, we consider the existence of thestatic systematic error, i.e. Ω( t ) → (1 + (cid:15) )Ω( t ) . Then, theHamiltonian will be H (cid:15) ( t ) = (1 + (cid:15) ) Ω( t )2 e iφ ( t ) | (cid:105)(cid:104) | + H . c .. (10)Due to the symmetry of the evolution path of the consideredgeometric rotations, we take the first path [0 , τ / to evaluatethe gate robustness, which can be calculated by the perturba-tion theory with probability amplitude P defined as P = |(cid:104) ψ ( τ / | ψ (cid:15) ( τ / (cid:105)| = 1 + ˜ O + ˜ O + · · · , (11)where | ψ (cid:15) ( τ / (cid:105) is the state with the systematic error, and ˜ O n denotes the term of the perturbation at the n th order. Up to thesecond order, it is calculated to be ˜ O = 0 , ˜ O = − (cid:15) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) χ τ χ e − if sin χdχ (cid:12)(cid:12)(cid:12)(cid:12) . (12)Defining f ( χ ) = η [2 χ − sin (2 χ )] , β (0) = 0 and β ( τ /
2) = β ( τ / − γ , resulting in ˜ O = − (cid:15) sin ηπ/ (2 η ) , we canensure ˜ O = 0 for P = 1 by setting η to a non-zero integer,which directly demonstrates that the designed evolution pathis combined with OCT. It is important to note that when η =0 , ˜ O = − π (cid:15) / , the current implementation will reduce tothe previous non-adiabatic schemes [8–11].For a typical example, we simulate the Phase gate under theeffect of the systematic error (cid:15) Ω max in the range of π × [ − , MHz with Ω max = 2 π × MHz. In Fig. 3(a), we plot thegate fidelity as a function of the systematic error (cid:15) Ω max for thecases of η = 0 and η = 1 without decoherence with the gatetime τ being 405 ns and 98 ns, respectively. we find that therobustness of our geometric gates in the case of η = 1 can besignificant improved comparing with the case of η = 0 (pre-vious implementations). Meanwhile, as shown in Figs. 3(b)and 3(c), considering both the systematic error and the deco-herence effects, our implementation still has the advantage ofimproving robustness in a certain decoherence range.We now turn to the implementation of nontrivial two-qubitgeometric gates based on two capacitively coupled transmons[36–38], which are respectively labeled by transmons A andB with qubit frequency ω A , B and anharmonicity α A , B . How-ever, the frequency difference ∆ = ω A − ω B and coupling (a) (b) (c) FIG. 3. Single-qubit gate performance without ( η = 0 ) and with ( η = 1 ) optimization. (a) The gate fidelity of the Phase gate with different η under the systematic error (cid:15) without decoherence. The gate fidelity of the Phase gate without and with optimization are shown in (b) and (c),respectively, under both the systematic error (cid:15) Ω max and a uniform decoherence rate Γ . ′ i e (b) A B (a)(c)
FIG. 4. Illustration of the implementation of the two-qubit geometricgates. (a) Two capacitively coupled transmon qubits configurationfor non-trivial two-qubit gates, where the frequency of qubit A is acmodulated to induce effective resonant interaction between the twoqubits. (b) The coupling structure for the states of the two transmonqubits. (c) The gate fidelity of a nontrivial geometric control-phasegate with γ (cid:48) = π/ . strength g between this two transmons A and B are usuallyfixed and can not be adjustable. Profitably, in a recent ex-perimental setup [15], as illustrated in in Fig. 4(a), time-dependent tunable coupling interaction can be realized by in-troducing a qubit-frequency driving ζ ( ε ( t )) on transmon A,which can be experimentally induced by adding a longitudi-nal field ε ( t ) = ζ − ( ˙ F ( t )) , where F ( t ) = λ ( t ) sin[ νt + ϕ ( t )] with ν and ϕ ( t ) being the frequency and phase of the longi-tudinal field, respectively. Then, in the interaction picture, theeffective Hamiltonian is H t ( t ) = g [ | (cid:105) AB (cid:104) | e i ∆ t + √ | (cid:105) AB (cid:104) | e i (∆+ α B ) t + √ | (cid:105) AB (cid:104) | e i (∆ − α A ) t ] e − iλ ( t ) sin[ νt + ϕ ( t )] + H.c. . (13)The corresponding coupling configuration of these two cou-pled transmons is shown in Fig. 4(b). We consider the case of the resonant interaction in the subspace {| (cid:105) AB , | (cid:105) AB } by choosing the driving frequency ν = ∆ − α A with g (cid:28){ ν, ∆ − ν, ∆ + α B − ν } , and then using Jacobi-Anger identityand neglecting the high-order oscillating terms, the obtainedeffective Hamiltonian can be reduced to H ( t ) = 12 (cid:18) g (cid:48) ( t ) e iϕ ( t ) g (cid:48) ( t ) e − iϕ ( t ) (cid:19) , (14)in the two-qubit subspace {| (cid:105) AB , | (cid:105) AB } , where g (cid:48) ( t ) =2 √ gJ ( λ ( t )) is effective time-dependent coupling strengthbetween transmon qubits A and B, with J ( λ ( t )) being theBessel function of the first kind. Similar to the single-qubit case in Eq. (3), within the two-qubit subspace {| (cid:105) AB , | (cid:105) AB , | (cid:105) AB , | (cid:105) AB } , we can also use the effec-tive Hamiltonian H ( t ) to acquire a pure geometric phase e iγ (cid:48) condition on two-qubit state of | (cid:105) AB by a cyclic evolution.The resulting geometric control-phase gates is U ( γ (cid:48) ) = diag (1 , , , exp( iγ (cid:48) )) . (15)Here, we also use the Lindblad master equation to evaluatethe nontrivial two-qubit geometric control-phase gates with γ (cid:48) = π/ as a typical example. We set the parameters of thetransmon qubits as ∆ = 2 π × MHz, α A = 2 π × MHz, α B = 2 π × MHz and g = 2 π × MHz, and the drivingfrequency ν = ∆ − α A = 2 π × MHz. Furthermore, we fixthe evolution time τ (cid:48) for the two-qubit gate to be ns undera corresponding coupling strength of g (cid:48) max = 2 π × MHz,and the form of the auxiliary parameters χ ( t ) , β ( t ) being thesame as that of the single-qubit case for the geometric rota-tion operators around Z axis. In this way, the shape of g (cid:48) ( t ) and ϕ ( t ) can finally be determined. In addition, we can nu-merically define λ ( t ) /ν = J − [ g (cid:48) ( t ) / (2 √ g )] , and then usethe original Hamiltonian H t ( t ) to faithfully verify our pro-posal. As shown in Fig. 4(c), we can get the gate fidelity F G = 99 . . Finally, comparing the two-qubit Hamilto-nian H ( t ) with the single-qubit Hamiltonian H d ( t ) , one findthat they are in the same form, both with the tunable couplingstrength and phase. Thus, the OCT presented in the single-qubit case can be directly incorporated in the two-qubit case.In summary, we have proposed a general method to con-struct fast universal GQC. Then, we physically implement ourproposal on superconducting circuits, where arbitrary single-qubit gates are realized by resonant driving on a transmonqubit with a microwave field, and nontrivial two-qubit gatescan be implemented by ac driving on one of the transmonqubits, which leads to effectively resonant coupling betweenthem. Finally, our scheme can combine with OCT to fur-ther enhance the gate robustness against the static system-atic error. We note that our proposal can be expanded toa two-dimensional capacitively coupled lattice of transmonqubits, and thus provides a promising step towards fault-tolerant quantum computation on superconducting circuits.This work was supported by the National Natural ScienceFoundation of China (Grant No. 11874156). ∗ [email protected][1] C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. D. Sarma,Rev. Mod. Phys. , 1083 (2008).[2] M. V. Berry, Proc. R. Soc. Lond., Ser. A , 45 (1984).[3] F. Wilczek and A. Zee, Phys. Rev. Lett. , 2111 (1984).[4] Y. Aharonov and J. Anandan, Phys. Rev. Lett. , 1593 (1987).[5] J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, Nature(London) , 869 (2000).[6] P. Zanardi and M. Rasetti, Phys. Lett. A , 94 (1999).[7] L. M. Duan, J. I. Cirac, and P. Zoller, Science , 1695 (2001).[8] X. B. Wang and M. Keiji, Phys. Rev. Lett. , 097901 (2001).[9] S. L. Zhu and Z. D. Wang, Phys. Rev. Lett. , 097902 (2002).[10] P. Z. Zhao, X. D. Cui, G. F. Xu, E. Sj¨oqvist, and D. M. Tong,Phys. Rev. A , 052316 (2017).[11] T. Chen and Z.-Y. Xue, Phys. Rev. Appl. , 054051 (2018). [12] G. Falci, R. Fazio, G. M. Palma, J. Siewert, and V. Vedral, Na-ture (London) , 355 (2000).[13] D. Leibfried et al ., Nature (London) , 412 (2003).[14] J. Du, P. Zou, and Z. D. Wang, Phys. Rev. A , 020302(R)(2006).[15] J. Chu et al ., arXiv:1906.02992.[16] Y. Xu et al ., arXiv:1910.12271.[17] E. Sj¨oqvist, D. M. Tong, L. Mauritz Andersson, B. Hessmo, M.Johansson, and K. Singh, New J. Phys. , 103035 (2012).[18] G. F. Xu, J. Zhang, D. M. Tong, E. Sj¨oqvist, and L. C. Kwek,Phys. Rev. Lett. , 170501 (2012).[19] S. B. Zheng, C. P. Yang, and F. Nori, Phys. Rev. A , 032313(2016).[20] J. Jing, C.-H. Lam, and L.-A. Wu, Phys. Rev. A , 012334(2017).[21] B.-J. Liu, X.-K. Song, Z.-Y. Xue, X. Wang, and M.-H. Yung,Phys. Rev. Lett. , 100501 (2019).[22] S. Li, T. Chen, and Z.-Y. Xue, Adv. Quantum Technol. ,2000001 (2020).[23] T. Yan et al ., Phys. Rev. Lett. , 080501 (2019).[24] A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, New J.Phys. , 093040 (2012).[25] D. Daems, A. Ruschhaupt, D. Sugny and S. Gu´erin, Phys. Rev.Lett. , 050404 (2013).[26] S. J. Glaser et al ., Eur. Phys. J. D , 279 (2015).[27] J. Clarke and F. K. Wilhelm, Nature , 1031 (2008).[28] J. Q. You and F. Nori, Nature , 589 (2011).[29] M. H. Devoret and R. J. Schoelkopf, Science , 1169 (2013).[30] G. Wendin, Rep. Prog. Phys. , 106001 (2017).[31] J. M. Gambetta, F. Motzoi, S. T. Merkel, and F. K. Wilhelm,Phys. Rev. A , 012308 (2011).[32] T. H. Wang , Z. X. Zhang, L. Xiang, Z. H. Gong, J. L. Wu, andY. Yin, Sci. China: Phys. Mech. AStro. , 047411 (2018).[33] T. H. Wang et al ., New J. Phys. , 065003 (2018).[34] F. W. Strauch, P. R. Johnson, A. J. Dragt, C. J. Lobb, J. R. An-derson, and F. C. Wellstood, Phy. Rev. Lett. , 167005 (2003).[35] L. DiCarlo et al ., Nature , 574 (2010).[36] M. Reagor et al ., Sci. Adv. , eaao3603 (2018).[37] S. A. Caldwell et al ., Phys. Rev. Appl. , 034050 (2018).[38] X. Li et al ., Phys. Rev. Appl.10