Qubit-oscillator dynamics in the dispersive regime: analytical theory beyond rotating-wave approximation
aa r X i v : . [ qu a n t - ph ] J u l Qubit-oscillator dynamics in the dispersive regime:analytical theory beyond rotating-wave approximation
David Zueco, Georg M. Reuther, Sigmund Kohler, and Peter H¨anggi Institut f¨ur Physik, Universit¨at Augsburg, Universit¨atsstraße 1, D-86135 Augsburg, Germany Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, E-28049 Madrid, Spain (Dated: July 20, 2009)We generalize the dispersive theory of the Jaynes-Cummings model beyond the frequently em-ployed rotating-wave approximation (RWA) in the coupling between the two-level system and theresonator. For a detuning sufficiently larger than the qubit-oscillator coupling, we diagonalize thenon-RWA Hamiltonian and discuss the differences to the known RWA results. Our results extendthe regime in which dispersive qubit readout is possible. If several qubits are coupled to one res-onator, an effective qubit-qubit interaction of Ising type emerges, whereas RWA leads to isotropicinteraction. This impacts on the entanglement characteristics of the qubits.
PACS numbers: 03.67.Lx, 03.70.+k, 42.50.Hz, 42.50.Pq
I. INTRODUCTION
More than forty years ago, Jaynes and Cummings [1]introduced a fully quantum mechanical model for the in-teraction of light and matter, which are represented by asingle harmonic oscillator and a two-level system, respec-tively. Within dipole approximation for the interaction,that model is expressed by the Hamiltonian H = ~ ǫ σ z + ~ ωa † a + ~ gσ x ( a † + a ) , (1)where ~ ǫ is the level splitting of the two-level system,henceforth “qubit”, ω is the frequency of the electromag-netic field mode, and g the dipole interaction strength.The Pauli matrices σ α , α = x, y, z , refer to the two-level system, while a † and a denote the bosonic creationand annihilation operators of the electromagnetic fieldmode. This model describes a wealth of physical phe-nomena rather well and by now is a “standard model”of quantum optics. A particular experimental realiza-tion of the Hamiltonian is an atom interacting with thefield inside an optical cavity, usually referred to as cavityquantum electrodynamics (cavity QED). Correspondingexperiments have demonstrated quantum coherence be-tween light and matter manifest in phenomena such asRabi oscillations and entanglement [2, 3].Despite its simplicity, the Hamiltonian (1) cannot bediagonalized exactly and, thus, is often simplified bya rotating-wave approximation (RWA). There, one ex-presses the qubit-cavity interaction in terms of the ladderoperators σ ± = ( σ x ± iσ y ). In the interaction picturewith respect to the uncoupled Hamiltonian, the couplingoperators σ + a , σ − a † and σ − a , σ + a † oscillate with thephase factors exp[ ± i( ω − ǫ ) t ] and exp[ ± i( ω + ǫ ) t ], respec-tively. Operating at or near resonance, the cavity-qubitdetuning is small, | ǫ − ω | ≪ ǫ + ω , so that the former op-erators oscillate slowly, whereas the latter exhibit fast“counter-rotating” oscillations. If in addition, the cou-pling is sufficiently weak, g ≪ min { ǫ, ω } , one can sepa-rate time scales and replace the counter-rotating terms by their vanishing time average. Then one obtains theJaynes-Cummings Hamiltonian [1] H RWA = ~ ǫ σ z + ~ ωa † a + ~ g ( σ − a † + σ + a ) . (2)Lately, new interest in Jaynes-Cumming physics hasemerged in the solid state realm. There, one implementsartificial atoms with Cooper-pair boxes (charge qubits)[4] or superconducting loops (flux qubits) [5]. The roleof the cavity is played now by a transmission line ora SQUID depending on the architecture [6, 7], or evena nanomechanical oscillator [8]. Since the first experi-mental realizations in 2004 [4, 5], a plethora of resultshas been obtained, such as quantum-non-demolition-likereadout of a qubit state [9], the generation of Fock states[10], the observation of Berry phases [11], multi-photonresonances [7], entanglement between two qubits insideone cavity [12, 13], and the demonstration of a two-qubitalgorithm [14].These experiments have in common that they operatein the strong coupling limit, that is, the coupling g islarger than the linewidth of the resonator. On the otherhand, g is typically two orders of magnitude less thanthe qubit and resonator frequencies. In this scenario theJaynes-Cummings model (2) has been shown to describethe experiments faithfully.Of practical interest is the dispersive limit, in whichthe qubit and the resonator are far detuned compared tothe coupling strength, g ≪| ǫ − ω | [6, 15]. In this regimea non-demolition type measurement of the qubit can beperformed by probing the resonator [4, 16]. Moreover,it is possible to simulate quantum spin chains with twoor more qubits that are dispersively coupled to one res-onator [12, 14, 17]. The complementary architectureof two cavities dispersively coupled to one qubit allowsbuilding a quantum switch [18]. All these ideas havebeen developed from the RWA model (2) in the dispersivelimit or from according generalizations to many qubits ormany oscillators. Thus, these theories are restricted tothe range g ≪ | ǫ − ω | ≪ ǫ + ω, (3)where the first inequality refers to the dispersive limit,while the second one has been used to derive the RWAHamiltonian (2) from the original model (1).In recent experiments, efforts are made to reach aneven stronger qubit-cavity coupling g . Thus, it will even-tually be no longer possible to fulfill both inequalities(3) [19]. In particular, when trying to operate in thedispersive limit, the second inequality may be violated,so that RWA is no longer applicable. Non-RWA effectsof the model Hamiltonian (1) have already been studiedin the complementary adiabatic limits ǫ ≪ ω and ω ≪ ǫ [20, 21, 22]. Furthermore, Van Vleck perturbation theoryhas been used in the resonant and close-to-resonant cases[23]. Finally, polaron transformation [24], cluster meth-ods [25], wave-packet approach [26] and even generalizedRWA approximations [27] have been considered.Motivated by the importance of the dispersive regimeand in view of the experimental tendency towardsstronger qubit-oscillator coupling, we present in this worka dispersive theory beyond RWA, so that the second con-dition in Eq. (3) can be dropped. This means that ourapproach is valid under the less stringent condition g ≪ | ǫ − ω | , (4)which implies that the detuning is not necessarily smallerthan ǫ and ω . In order to set the stage, we briefly reviewin Sec. II the dispersive theory within RWA. In Sec. III,we derive a dispersive theory for Hamiltonian (1) beyondRWA, which we generalize in Sec. IV to the presence ofseveral qubits. II. DISPERSIVE THEORY WITHIN RWA
The dispersive limit is characterized by a large detun-ing ∆ = ǫ − ω as compared to the qubit-oscillator coupling g . Thus, λ = g ∆ (5)represents a small parameter, while the RWA Hamilto-nian (2) is valid for | ǫ − ω | ≪ ǫ + ω . Then it is convenientto separate the coupling term from the RWA Hamilto-nian, i.e., to write H RWA = H + ~ gX + with the contri-butions H = ~ ǫ σ z + ~ ωa † a , (6) X ± = σ − a † ± σ + a . (7)Applying the unitary transformation, D RWA = e λX − (8)one obtains for the transformed Hamiltonian H disp = D † RWA H RWA D RWA to second order in λ : H disp = H RWA + λ [ H RWA , X − ] + λ [[ H RWA , X − ] , X − ], which can be eval-uated to read H RWA , disp = ~ ǫ σ z + ~ g σ z + (cid:18) ~ ω + ~ g ∆ σ z (cid:19) a † a . (9)The physical interpretation of (9) is that the oscillatorfrequency is shifted as ω → ω ± g / ∆ , (10)where the sign depends on the state of the qubit. Ifone now probes the resonator with a microwave signalat its bare resonance frequency ω , the phase of the re-flected signal possesses a shift that depends on the qubitstate. This allows one to measure the low-frequency dy-namics of the qubit [6]. Since, according to Eq. (9), thequbit Hamiltonian ( ~ ǫ/ σ z commutes with the disper-sive coupling ( ~ g / ∆) σ z a † a , this constitutes a quantumnon-demolition measurement of the qubit, which has al-ready been implemented experimentally [4, 22, 28, 29].In turn, the qubit energy splitting is shifted dependingon the mean photon number n = h a † a i . Accordingly, onecan also measure the mean photon number, and even per-form a full quantum state tomography of the oscillatorstate [30]. Note also that besides the condition of λ be-ing small, the perturbational result (9) is accurate only ifthe mean photon number h n i does not exceed the criticalvalue n crit = 1 / λ . For larger photon numbers, higherpowers of the number operator a † a must be taken into ac-count [31, 32] . Henceforth, we restrict ourselves to theso-called linear dispersive regime, in which the photonnumber is clearly below the critical value n crit . III. DISPERSIVE REGIME BEYOND RWA
It is now our aim to treat the original Hamiltonian (1)in the dispersive limit accordingly, i.e., to derive an ex-pression that corresponds to Eq. (9) but is valid in thefull dispersive regime defined by inequality (4). Goingbeyond RWA, we have to keep the counter-rotating cou-pling terms Y ± = σ + a † ± σ − a , (11)which are relevant if either of the relations g ≪ min { ǫ, ω } or | ǫ − ω | ≪ ǫ + ω is violated. Separating again thequbit-oscillator coupling from the bare terms, we rewriteHamiltonian (1) as H = H + ~ gX + + ~ gY + , (12)which differs from H RWA by the last term. It will turn outthat a unitary transformation corresponding to Eq. (8)is achieved by the operator D = e λX − + λY − . (13)Here we have introduced the parameter λ = gǫ + ω = g ǫ − ∆ , (14)which obviously fulfills the relation λ < λ , since ǫ and ω are positive. Thus, whenever λ is small, λ is small aswell. Nevertheless, under condition (4), λ and λ may beof the same order.Proceeding as in Sec. II, we define the dispersiveHamiltonian H disp = D † HD . Using the commutation re-lations [ Y + , Y − ] = σ z (2 a † a + 1) −
1, [ ǫ/ σ z + ωa † a, Y − ] = − ( ǫ + ω ) Y + and [ Y ± , X ∓ ] = σ z { a + ( a † ) } , we obtainthe expression H disp = ~ ǫ σ z + ~ ωa † a + ~ g (cid:18)
1∆ + 12 ǫ − ∆ (cid:19) σ z ( a † + a ) , (15)which is valid up to second order in the dimensionlesscoupling parameters λ and λ .As compared to the RWA result (9), we find two dif-ferences: First, the prefactor of the coupling has a con-tribution that obviously stems from λ . Second and moreimportantly, the coupling is no longer proportional to thenumber operator a † a , but rather to ( a † + a ) . Thus, theoperator Y ± has turned into the counter-rotating con-tributions a and ( a † ) . For this reason, the dispersiveHamiltonian (15) is not diagonal in the eigenbasis of theuncoupled Hamiltonian H .Nevertheless, it is possible to interpret the result asa qubit-state dependent frequency shift by the followingreasoning. Let us interpret ~ ωa † a as the Hamiltonian ofa particle with unit mass in the potential ω x , where x = p ~ / ω ( a † + a ). Then the qubit-oscillator couplingin Eq. (15) modifies the potential curvature ω , such thatthe oscillator frequency undergoes a shift according to ω → ω = ω s ± g ω (cid:18)
1∆ + 12 ǫ − ∆ (cid:19) . (16)Again the sign depends on the qubit state. To be con-sistent with the second-order approximation in g , wehave to expand also the square root to that order. Thiscomplies with the experimentally interesting parameterregime where g < ω . We finally obtain ω = ω ± g (cid:18)
1∆ + 12 ǫ − ∆ (cid:19) . (17)As for the RWA Hamiltonian, we find that the qubit stateshifts the resonance frequency of the oscillator. This re-sult is not only of appealing simplicity, but also has arather important consequence: Dispersive readout is pos-sible even when the qubit-oscillator coupling is so strongthat condition (3) cannot be fulfilled, that is, when theRWA result is not valid.For a quantitative analysis of our analytical findings,we compare the frequency shifts (10) and (17) with nu-merical results. In doing so, we diagonalize the Hamil-tonian (1) in the subspace of the qubit state |↓i , where σ z |↓i = −|↓i . The results are depicted in Fig. 1. − − δ ω / ω ǫ [ ω ] g / ω = g / ω = g / ω = − − δ ω / ω g / ω = g / ω = g / ω = FIG. 1: Oscillator frequency shift as function or the qubitsplitting ǫ = ω + ∆ for the spin state |↓i obtained (a) withinRWA, Eq. (10), and (b) beyond RWA, Eq. (17). The linesmark the analytical results, while the symbols refer to thenumerically obtained splitting between the ground state andthe first excited state in the subspace of the qubit state |↓i . For a qubit splitting ǫ close to the cavity frequency ω ,i.e., outside the dispersive regime, the analytically ob-tained frequency shifts diverge. This behavior is cer-tainly expected for an expansion in g/ ∆. For a rela-tively small coupling g/ω . . g/ω & .
05, the predictions from RWA exhibit clear differ-ences. The general tendency is that RWA overestimatesthe frequency shift for blue detuning ∆ = ǫ − ω <
0, whileit predicts a too small shift for red detuning.The data shown in panel (b) demonstrates that thetreatment beyond RWA yields the correct frequency shiftin the entire dispersive regime, i.e., whenever the detun-ing significantly exceeds the coupling, | ∆ | ≫ g . Thus,as long as the coupling remains much smaller than theoscillator frequency, g ≪ ω , it is always possible to tunethe qubit splitting ǫ into a regime in which 4 is fulfilled.Moreover, the excellent quantitative agreement of our an-alytical result (17) with the numerically exact solution in-dicates the feasibility to determine g from measurementsin the strong-coupling limit [19].A particular limit is ∆ → − ω , which corresponds to avanishing qubit splitting, ǫ →
0. In this case it is obvi-ous from Hamiltonian (1) that the coupling to the qubitmerely entails a linear displacement of the oscillator coor-dinate, while the oscillator frequency remains unaffected.This limit is perfectly reproduced by our non-RWA result(17), irrespective of the coupling strength. The RWAresult, by contrast, predicts a spurious frequency shift,indicating the failure of RWA.
IV. SEVERAL QUBITS IN A CAVITY
An experimentally relevant generalization of the model(1) is the case of several qubits coupling to the sameoscillator. The corresponding Hamiltonian reads [33] H = ~ X j ǫ j σ zj + ~ ωa † a + ~ X j g j σ xj ( a † + a ) , (18)where the index j labels the qubits. As for the one-qubit case, the rotating wave-approximation is frequentlyapplied and yields the Tavis-Cummings Hamiltonian [34] H = ~ X ǫ j σ zj + ~ ωa † a + ~ X g j X j + , (19)where X j ± = σ − j a † j ± σ + j a j , cf. Eq. (6). A. Dispersive theory within RWA
We obtain for each qubit the dimensionless couplingparameter λ j = g j / ( ǫ j − ω ). The dispersive limit isnow determined by | λ j | ≪ j . Effective decou-pling of the qubits and the cavity to second order is thenachieved via a transformation with the unitary operatorexp( − P j λ j X j − ), cf. Eq. (8). The resulting dispersiveHamiltonian reads [33] H disp = ~ ωa † a + ~ X j (cid:18) ǫ j + g ∆ j (cid:19) σ zi + X g ∆ j a † aσ jz + X j>k J jk ( σ − j σ + k + σ + j σ − k ) . (20)Remarkably, the oscillator entails an effective couplingbetween the qubits with the strength J jk = g j g k (cid:18) j + 1∆ k (cid:19) , (21)which has already been observed experimentally [12]. Ithas been proposed to employ this interaction for buildingqubit networks [17] and for generating qubit-qubit entan-glement [35, 36, 37]. Moreover, quantum tomography ofa two-qubit state has been implemented by probing thecavity at its bare resonance frequency [38]. In this sce-nario the oscillator frequency exhibits a shift depending on a collective coordinate of all qubits. Consequently,the cavity response experiences a phase shift from theingoing signal, which in turn contains information aboutthat collective qubit coordinate. B. Dispersive theory beyond RWA
As in Sec. III for the one-qubit case, we now extend thedispersive theory of the Tavis-Cummings model beyondRWA, taking into account the counter-rotating terms ofthe Hamiltonian (18). In analogy to transformation (13),we employ the ansatz D = e P λ j X j − + λ j Y j − , (22)where Y j − = σ − j a − σ + j a † and λ j = g/ (2 ǫ − ∆ j ). Follow-ing the lines of Sec. III, i.e. expanding the transformedHamiltonian to second order in λ and λ , we obtain thedispersive Hamiltonian H disp = D † HD = ~ ωa † a + ~ X j ǫ j σ zi + 12 X j g j (cid:18) j + 12 ǫ − ∆ j (cid:19) ( a † + a ) σ zj + X j>k J jk σ xj σ xk . (23)We have introduced the modified coupling strength J jk = g i g k (cid:18) j + 1∆ k − ǫ − ∆ j − ǫ − ∆ k (cid:19) , (24)which describes the effective interaction between qubits j and k , and represents the extension of Eq. (21) be-yond RWA. The dispersive shifts of the qubit and cav-ity frequencies, given by the second and third term ofEq. (23), are equally modified as compared to the RWAresult (20).Interestingly enough, the effective qubit-qubit interac-tion in Eq. (23) is of the Ising type σ xj σ xk , whereas RWApredicts the isotropic XY interaction σ + j σ − k + σ − j σ + k , seeEq. (20). Thus, the treatment beyond RWA predicts aqualitatively different effective model and not merely arenormalization of parameters. The Ising term even per-sists in the limit 1 / ∆ j ≫ / (2 ǫ − ∆ j ). Nevertheless, onecan recover the RWA Hamiltonian (20) by writing theinteraction term as σ xj σ xk = σ + j σ − k + σ + j σ + k + h . c . andperforming a RWA for the Ising coupling. This corre-sponds to discarding small-weighted, rapidly oscillatingterms of the type J jk σ + j σ + k + h . c . The difference between the effective models (20) and(23) has some physically relevant consequences. First,in contrast to the RWA result (20), Hamiltonian (23)does not conserve the number of qubit excitations, whichwill affect the design of two-qubit gates [33]. Moreover,both models possess different spectra, which influencesentanglement creation. For instance, the ground state ofthe Hamiltonian (23) for two degenerate qubits ( ǫ = ǫ )coupled to one cavity is | i [ |↓↓i − ( J/ ǫ ) |↑↑i ] and thusexhibits qubit-qubit entanglement. By contrast, the cor-responding ground state of the multi-qubit RWA Hamil-tonian (20) is the product state | i|↓↓i . For the caseof the Hamiltonian (23), thermal qubit-qubit entangle-ment will consequently be present at zero temperatureand even at thermal equilibrium [39, 40]. V. SUMMARY
We have generalized the dispersive theory for a qubitcoupled to a harmonic oscillator to the case of fardetuning. In this limit, it is no longer possible totreat the qubit-oscillator interaction Hamiltonian withinthe rotating-wave approximation. Therefore, previousderivations need some refinement. It has turned out thatdiagonalizing the Hamiltonian analytically up to secondorder in the coupling constant is possible as well be-yond RWA. The central result is that as within RWA,the oscillator experiences a shift of its resonance fre-quency, the sign of the shift depending on the qubit state.In this respect, the difference between both approachesseems to be merely quantitative. Nevertheless, our resultimplies an important fact for currently devised qubit- oscillator experiments with ultra-strong cavity-qubit cou-pling: Dispersive qubit readout is possible as well in thatregime. The comparison with numerical results has con-firmed that our approach is quantitatively satisfactory inthe whole dispersive regime.The corresponding treatment of many qubits coupledto the same oscillator is equally possible. In such archi-tectures, the oscillator mediates an effective qubit-qubitinteraction which may be used for gate operations andentanglement creation. We have revealed that the formof the effective interaction depends on whether or notone employs RWA. While RWA predicts an isotropic XYinteraction, the inclusion of the counter-rotating termsyields an interaction of Ising type. This difference im-pacts on various proposed entanglement creation proto-cols as soon as they operate in the far-detuned dispersiveregime.
Acknowledgments
We would like to thank Michele Campisi, Frank Deppe,Johannes Hausinger, Matteo Mariantoni and EnriqueSolano for discussions. We gratefully acknowledge finan-cial support by the German Excellence Initiative via the“Nanosystems Initiative Munich (NIM)”. This work hasbeen supported by DFG through SFB 484 and SFB 631. [1] E. T. Jaynes and F. W. Cummings, Proc. IEEE , 89(1963).[2] J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod.Phys. , 565 (2001).[3] H. Walther, B. T. H. Varcoe, B.-G. Englert, andT. Becker, Rep. Prog. Phys. , 1325 (2006).[4] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S.Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J.Schoelkopf, Nature (London) , 162 (2004).[5] I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C. J.P. M. Harmans, and J. E. Mooij, Nature (London) ,159 (2004).[6] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, andR. J. Schoelkopf, Phys. Rev. A , 062320 (2004).[7] F. Deppe, M. Mariantoni, E. P. Menzel, A. Marx,S. Saito, K. Kakuyanagi, H. Tanaka, T. Meno, K. Semba,H. Takayanagi, et al., Nature Phys. , 686 (2008).[8] M. LaHaye, J. Suh, P.M.Echternach, K.C.Schwab, andM. Roukes, Nature (London) , 960 (2009).[9] A. L. S., Saito, T. Picot, P. C. de Groot, C. J. P. M.Harmans, and J. E. Mooij, Nature Phys. , 119 (2007).[10] M. Hofheinz, E. M. Weig, M. Ansmann, R. C. Bialczak,E. Lucero, M. Neeley, A. D. O’Connell, H. Wang, J. M.Martinis, and A. N. Cleland, Nature (London) , 310(2008).[11] P. J. Leek, J. M. Fink, A. Blais, R. Bianchetti, M. G¨oppl,J. M. Gambetta, D. I. Schuster, L. Frunzio, R. J.Schoelkopf, and A. Wallraff, Science , 1889 (2007).[12] J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A.Houck, A. Wallraff, et al., Nature (London) , 443(2007).[13] M. A. Sillanp¨a¨a, J. I. Park, and R. W. Simmonds, Nature(London) , 438 (2007).[14] L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop,B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frun-zio, S. M. Girvin, et al., ArXiv: 0903.2020 [cond-mat].[15] Y.-X. Liu, L. F. Wei, and F. Nori, Phys. Rev. A ,033818 (2005).[16] A. Lupascu, E. F. C. Driessen, L. Roschier, C. J. P. M.Harmans, and J. E. Mooij, Phys. Rev. Lett. , 127003(2006).[17] F. Helmer, M. Mariantoni, A. G. Fowler, J. von Delft,E. Solano, and F. Marquardt, EPL , 50007 (2009).[18] M. Mariantoni, F. Deppe, A. Marx, R. Gross, F. K. Wil-helm, and E. Solano, Phys. Rev. B , 104508 (2008).[19] J. Bourassa, J. M. Gambetta, A. A. Abduma-likov, O. Astafiev, Y. Nakamura, and A. Blais,ArXiv:0906.1383 [cond-mat].[20] E. K. Irish and K. Schwab, Phys. Rev. B , 1553111(2003).[21] E. K. Irish, J. Gea-Banacloche, I. Martin, and K. C.Schwab, Phys. Rev. B , 195410 (2005).[22] G. Johansson, L. Tornberg, and C. M. Wilson, Phys. Rev.B , 100504(R) (2006).[23] J. Hausinger and M. Grifoni, New J. Phys. , 115015(2008).[24] P. Neu and R. J. Silbey, Phys. Rev. A , 5323 (1996). [25] R. F. Bishop and C. Emary, J. Phys. A , 5635 (2001).[26] J. Larson, Phys. Scr. , 146 (2007).[27] E. K. Irish, Phys. Rev. Lett. , 259901 (2007).[28] M. Grajcar, A. Izmalkov, E. Il’ichev, T. Wagner,N. Oukhanski, U. H¨ubner, T. May, I. Zhilyaev, H. E.Hoenig, Ya. S. Greenberg, et al., Phys. Rev. B ,060501(R) (2004).[29] M. Sillanp¨a¨a, T. Lehtinen, A. Paila, Y. Makhlin, andP. Hakonen, Phys. Rev. Lett. , 187002 (2006).[30] M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz,N. Katz, E. Lucero, A. O’Connell, H. Wang, A. N. Cle-land, and J. M. Martinis, Nature Phys. , 523 (2008).[31] M. Boissonneault, J. M. Gambetta, and A. Blais, Phys.Rev. A , 060305 (2008).[32] M. Boissonneault, J. M. Gambetta, and A. Blais, Phys.Rev. A , 013819 (2009).[33] A. Blais, J. Gambetta, A. Wallraff, D. I. Schuster, S. M.Girvin, M. H. Devoret, and R. J. Schoelkopf, Phys. Rev.A , 032329 (2007). [34] M. Tavis and F. W. Cummings, Phys. Rev. , 379(1968).[35] F. Helmer and F. Marquardt, Phys. Rev. A , 052328(2009).[36] C. L. Hutchison, J. M. Gambetta, A. Blais, and F. K.Wilhelm, ArXiv:0812.0218 [cond-mat].[37] L. S. Bishop, L. Tornberg, D. Price, E. Ginossar, A. Nun-nenkamp, A. A. Houck, J. M. Gambetta, J. Koch, G. Jo-hansson, S. M. Girvin, et al., ArXiv:0902.0324 [cond-mat].[38] S. Filipp, P. Maurer, P. J. Leek, M. Baur, R. Bianchetti,J. M. Fink, M. G¨oppl, L. Steffen, J. M. Gambetta,A. Blais, et al., Phys. Rev. Lett. , 200402 (2009).[39] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev.Mod. Phys. , 517 (2008).[40] G. L. Kamta and A. F. Starace, Phys. Rev. Lett.88