Probing the Tavis-Cummings level splitting with intermediate-scale superconducting circuits
Ping Yang, Jan David Brehm, Juha Leppäkangas, Lingzhen Guo, Michael Marthaler, Isabella Boventer, Alexander Stehli, Tim Wolz, Alexey V. Ustinov, Martin Weides
aa r X i v : . [ qu a n t - ph ] J un Probing the Tavis-Cummings level splitting with intermediate-scale superconductingcircuits
Ping Yang, Jan David Brehm, Juha Lepp¨akangas,
1, 2
Lingzhen Guo, Michael Marthaler,
2, 4, 5
Isabella Boventer,
1, 6
Alexander Stehli, Tim Wolz, Alexey V. Ustinov,
1, 7 and Martin Weides
1, 8, ∗ Institute of Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany HQS Quantum Simulations GmbH, 76131 Karlsruhe, Germany Max Planck Institute for the Science of Light, 91058 Erlangen, Germany Institute for Theoretical Condensed Matter physics,Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany Theoretische Physik, Universit¨at des Saarlandes, 66123 Saarbr¨ucken, Germany Institute of Physics, University Mainz, 55128 Mainz, Germany Russian Quantum Center, National University of Science and Technology MISIS, Moscow 119049, Russia James Watt School of Engineering, University of Glasgow, Glasgow G12 8QQ, United Kingdom (Dated: June 9, 2020)We demonstrate the local control of up to eight two-level systems interacting strongly with amicrowave cavity. Following calibration, the frequency of each individual two-level system (qubit)is tunable without influencing the others. Bringing the qubits one by one on resonance with thecavity, we observe the collective coupling strength of the qubit ensemble. The splitting scales up withthe square root of the number of the qubits, which is the hallmark of the Tavis-Cummings model.The local control circuitry causes a bypass shunting the resonator, and a Fano interference in themicrowave readout, whose contribution can be calibrated away to recover the pure cavity spectrum.The simulator’s attainable size of dressed states with up to five qubits is limited by reduced signalvisibility, and -if uncalibrated- by off-resonance shifts of sub-components. Our work demonstratescontrol and readout of quantum coherent mesoscopic multi-qubit system of intermediate scale underconditions of noise.
I. INTRODUCTION
Most of today’s quantum information systems rely onan interplay between an artificial atom and a resonatormode used for readout [1]. In absence of dissipation,its dynamics is well described by the Jaynes-Cummingsmodel [2]. For N atoms interacting with one resonatorTavis and Cummings predicted a √ N enhancement ofthe effective coupling strength at degeneracy, leading toa level repulsion with a frequency gap 2 g √ N , where g isthe coupling strength of one artificial atom to the res-onator [3]. After early experimental realizations withtrapped ions [4], the √ N -enhancement has been demon-strated with three locally tunable superconducting trans-mon qubits [5], followed by eight qubits in a globallycontrolled ensemble [6], and a comparable number oftransmons [7].Novel applications have been proposed involving morethan one controllable two-level system coupled to a sin-gle resonator. These include bus systems realizing a tun-able long-range interaction between distant qubits [8–10] and a quantum von Neumann architecture [11]. Thecollective interaction also creates multi-qubit entangle-ment [12] and provides protection against radiation de-cay [13]. This versatility supports the use of Tavis-Cummings systems in future quantum simulators andcomputers. An analog quantum simulation [14, 15] of a ∗ [email protected] Dicke model [16] (generalized Tavis-Cummmings model)would provide a direct access to eigenenergies and tran-sient dynamics of light-matter interaction in the ultra-strong coupling regime [17, 18].Experimentally, the increase of circuit complexity is agrowing challenge for system control, e.g., due to cross-talk or circuit topology. Global control of a large numberof qubits is adversely affected by disorder in the ensem-ble, such as local flux offsets, which can be mitigated bylocal controls. Ideally, a residual finite interaction be-tween the qubits themselves and cross-talk between fluxlines can be calibrated away.In this work, we increase the circuit complexity to studythe Tavis-Cummings circuit consisting of a superconduct-ing microwave resonator interacting with up to eight in-dividually frequency-controllable transmon qubits. It isa well suited platform to study both desired and parasiticeffects occurring in scaled-up quantum circuits. We showa calibration method allowing for local qubit control of alleight qubits, and demonstrate its adequate analog quan-tum simulation of the Tavis-Cummings system by mea-suring the √ N coupling enhancement as the hallmarksignature. Our circuit complexity is positioned betweenwell-understood few qubit-resonator systems and theirscaled up versions constructed to achieve quantum ad-vantage. The experiment contains key properties such asdecoherence, local control and crosstalk, reactive and dis-sipative background, and higher qubit levels, all of whichare subtle features of any near-term physical quantumsimulator. (b) flux bias qubitsHEMT+40dB+20dBRoomtemperature 4 K 20 mKVNA microwavegenerator IaIh qubit 1 qubit 8-20dB resonator-20dB-20dB-50dB
50 50 powercombiner currentdivider FIG. 1. (a) Optical micrograph of the chip bonded to the sam-ple box. The meander-structure coplanar resonator is coupledat each end to four transmon qubits. U-shaped leads carryDC current to control the local magnetic flux. The enlargedimage (red rectangle) shows two cavity-embedded transmonqubits and their flux bias lines. (b) Schematic of the measure-ment setup. The tone generated by the VNA is attenuatedat different temperature stages of the refrigerator to reachsingle-photon regime and to lower the thermal noise at thequantum chip.
II. MULTI-QUBIT CHIP AND SETUP
The quantum chip studied in this work contains acoplanar waveguide half-wavelength resonator with fourtransmon qubits capacitively coupled to each end of theresonator, as shown in Fig. 1 (a), for a maximal couplingstrength to each qubit. Each qubit frequency is individ-ually controlled by a local DC flux bias. The theoreticaldescription is given by the Tavis-Cummings model [3]with Hamiltonianˆ H/ ~ = ω r ˆ a † ˆ a + X i ω i σ z i + X i g i (ˆ a † ˆ σ − i +ˆ a ˆ σ + i ) . (1)Here ω r is the pure resonator frequency, ω i the qubit i frequency, g i their coupling strength, and ˆ σ + , − are thespin raising and lowering operators. The two-level sys-tems are realized by transmon qubits [19], each includingtwo Josephson junctions in a SQUID geometry enablingthe local tunability by the applied magnetic fluxes.Our experimental setup is illustrated in Fig. 1 (b), furtherdetails, including the sample are given in the Appendix.The microwave drive tone of a vector network analyzer (VNA) is attenuated along the signal chain by 120 dB intotal. The power reaching the chip is −
137 dBm, wherethe average photon number on resonance is estimated tobe h n i ≈ . III. TAKING LOCAL CONTROL
Being designed with local flux control of each qubit,the cross-talk between qubits and non-corresponding fluxcontrol lines is small, but not negligible, due to residualon-chip coupling, parasitic coupling in the DC wiring orwithin the DC current sources. For multi-qubit circuitswith flux-tunable components careful calibration of thelinear cross-talk is part of the experiment [23, 24]. Thisensures true single-qubit control with DC current com-pensation routines on all the flux control lines.Using one global readout, the calibration is fast, repro-ducible and -to some extent- scalable. Single-tone mea-surement of the resonator without exact qubit identifi-cation is sufficient to build the 8 × ∆Φ ...∆Φ = M a M b · · · M h ... ... . . . ... M a M b · · · M h ∆ I a ...∆ I h , (2) (c) (d)(a) (b)I a (mA) I b ( m A ) f r e q u e n c y ( G H z ) power (dBm) I a (mA)f VNA (GHz) | S | f VNA (GHz) | S | I a (mA) cor
10 100 1000 I b ( m A ) c o r FIG. 2. (a) Power scan with all qubits far detuned (log-scale).The photon-dressed resonator frequency changes from low tohigh powers in transmission height and frequency. (b) Un-calibrated single-flux scan after background removal. Onlyone qubit is tuned through the cavity resonance frequency,all the other qubits are far detuned. The resonator frequencybefore background subtraction shows weakening of signal andtransformation between peaks and dips [20]. The horizontalblack dashed line indicates the frequency point chosen to dothe flux calibration. (c) Example for uncalibrated two-coilsweeps. The red lines are the fitted slopes which give theratio between two mutual inductances. (d) Repeated mea-surement after calibration. The absence of a tilt indicatesgood isolation between the pair of flux lines. where 1 , . . . , a, . . . , h indicate theflux bias. For instance, ∆Φ is the flux variation throughthe first qubit, ∆ I b is the change of the DC current run-ning through the second bias line, and M h is the mutualinductance between the first qubit and the eighth fluxbias line. Changing the current in one flux line does notonly tune the frequency of its adjacent qubit, but alsomay bias other qubits. For calibration, a frequency closeto the anti-crossing [see black dashed line in Fig. 2 (b)]is chosen. By observing the change in transmission whilesweeping two bias currents, the mutual inductance ma-trix element is obtained. This value is used for the com-pensation currents to counteract the induced bias fluxesto effectively keep all other qubits at their frequencies.The corrected current after calibration I cor i is employed,rather than the absolute value of current I i . Almost no-tilt indicates a good flux calibration, as seen in Fig. 2(d). For more details see Appendix. ( c ) n = = = (a) -9 -7 -5 f r e q u e n c y ( G H z ) I a (mA) cor (b) -0.31 -0.21 FIG. 3. Comparison between measurement and simulation ofone transmon tuned through the cavity frequency. The plot-ted transmission amplitude is in log-scale. (a) Measured dataof the anti-crossing. The black lines correspond to excitationsfrom the ground state and cause the vacuum Rabi splitting.The other colored lines correspond to higher-level transitionsand are identified in (c). (b) Master equation simulation byQuTiP [25] for a three-level artificial atom interacting witha resonator which has an average thermal photon populationof 0.1 photons. (c) Energy-diagram of the first two excitationmanifolds (schematic, not to scale) of the dressed system.
IV. INDIVIDUAL QUBIT SPECTROSCOPY
Before probing the full Tavis-Cummings model, we de-termine the coupling strengths g i of each qubit from theminimal level-splitting, while parking all other qubitsat their maximum frequencies. This level splitting iseffectively described by eigenenergies E ± / ~ = ω i + ω r ±√ ∆ +4 g i , with qubit frequency ω i and ∆ = ω i − ω r . Atdegeneracy (∆ = 0), the frequency difference E R / ~ isgiven by the vacuum Rabi splitting ( E + − E − ) / ~ = 2 g i ,i.e. the minimum distance between the major splittings(black dashed lines) as shown in Fig. 3 (a). The mea-sured coupling strengths g i , see Table I, indicate a goodagreement between the designed and observed values.The major splitting on resonance between one qubitand one resonator is described well by the Jaynes-Cummings model. For detailed understanding of all fea-tures away from the Rabi splitting, the transmon has tobe considered as a multi-level anharmonic oscillator, with | g i , | e i , | f i denoting the first three uncoupled eigenstatesrespectively, and a Hamiltonian:ˆ H / ~ = ω r ˆ a † ˆ a + X j =g , e , f ω j | j ih j | + (cid:0) ˆ a † +ˆ a (cid:1) X i,j =g , e , f g ij | i ih j | , (3)in the base of {| g , i , | e , i , | f , i , | g , i , · · · , | f , n i} ,and eigenenergies ω g , ω e and ω f . Only single-photontransitions between adjacent levels of the first three lev-els of transmon are taken into consideration, since two-photon transitions require much higher drive powers [26].The analysis based on the (two-level) Jaynes-Cummingsmodel, considered before, included transitions indicatedby the two black arrows in Fig. 3 (c). Including the thirdtransmon state, additional transitions appear betweenthe higher manifolds. All these six transitions are visual-ized in Fig. 3 (c) and plotted together with the measured designed qubit 1 qubit 2 qubit 3 qubit 4 qubit 5 qubit 6 qubit 7 qubit 8 ω max / π (GHz) .
11 7 . ± .
05 7 . ± .
05 7 . ± .
04 11 . ± . . ± . . ± .
08 10 . ± .
08 12 . ± . g/ π (MHz) . . ± . . ± . . ± . ± ± ± . ± . ± g i and maximal energy-level splittings ω max i of each transmon qubit. Qubits 1-6 were used inthe experiments probing the Tavis-Cummings level splitting. T times were estimated to vary between 50-80 ns and dephasingwas limited by qubit decay, T ≈ T / γ ≈ π × . anti-crossing in Fig. 3 (a), next to the numerical masterequation simulation using QuTiP [25] shown in Fig. 3 (b).We obtain a good agreement between the measured dataand the model. This demonstrates detailed understand-ing of resonances appearing in our spectroscopy, and con-firms that the additional features do not correspond totwo-photon transitions (requring higher drive powers),but to single-photon transitions starting from the first,thermally excited, manifold. In combination with thelow anharmonicity they can also cause additional vac-uum Rabi splittings. V. MULTI-QUBIT SPECTROSCOPY AND √ N SCALING OF THE COUPLING
By bringing the transmons one by one on resonancewith the cavity, we demonstrate the local control of mul-tiple qubits and are able to measure the collective cou-pling. The theoretical vacuum Rabi splitting generalizesto E R N / ~ = p ∆ +4 N g , assuming identical couplings g i = g . When the N qubits are exactly on resonance (i.e.∆ = 0), the splitting is 2 g √ N [5]. Already for one qubitbeing slightly detuned, the measured splitting increases.Furthermore, considering g i being different for each qubit(even though relatively small), the Rabi splitting is givenby E R N / ~ = 2 pP i g i at resonance.Fig. 4 shows the transmission spectra revealing thecollective vacuum Rabi splitting. As a prerequisite,one qubit has been tuned into resonance, leading toan avoided level crossing as in Fig. 3 (a). In Fig. 4(a), this qubit is kept on resonance, while the sec-ond qubit is tuned in. N = 2 qubits (and one res-onator mode) have N +1 = 3 single-excitation eigen-states. One of the states is dark (no photons in theresonator), the excitation being shared only betweenthe qubits. Similarly, we bring more qubits on reso-nance in (b) to (d). On resonance a bright doublet ap-pears, corresponding to collective qubits-photon superpo-sition states, | N ±i = √ | g , .., g , i± √ N ( | e , .., g , i + ... + | g , .., e , i ). These eigenfrequencies are separated by thecollective coupling 2 g √ N . The other N − dark state N=2 N=3 (b)I c (mA)I b (mA) (a) f r e q u e n c y ( G H z ) cor cor N=4(c) -10 -9 f r e q u e n c y ( G H z ) N=5(d) -9 -8 I d (mA) cor I e (mA) cor FIG. 4. Multiple qubits on resonance with the cavity. Trans-mission amplitude in the log-scale for two (a) to five (d) qubitsinteracting resonantly with the cavity. The white dashed linesare fitting curves used to extract the collective coupling, seeAppendix. The red dashed lines indicate the splittings, withthe red triangle marking their centres (namely the resonatorfrequency). The black dashed lines show the resonator fre-quency when tuning only qubit 1 in resonance. and tunable, although we manage to bring a maximumof six qubits on, or close, to resonance. For up to fivequbits a good agreement to the prediction of the TavisCummings model is obtained. A drift of the ensemblefrequency, ∆ f , appears due to effective dispersive shifts P i = N +1 g i / ∆ i from off-resonance qubits. The induceddrift relative to the measured splitting is however mi-nor as | ∆ f / g √ N | ≪
1. The effective splitting and res-onator drift ∆ f are close the theoretical expectations,as shown in Fig. 4 (e). The tuning precision is mainlylimited by the steep flux dependence at the resonatorfrequency, in particular for high frequency qubits. Forinstance, a change of flux ∆Φ = 0 . (correspondingto 100 . µA in bias current) on qubit 5 results in a shift of130 MHz in qubit frequency. The large flux susceptibilityrenders them sensitive to fluctuations in either the biascurrent (from the current source or picked up in the wirechain towards the qubits) or the magnetic backgroundfield. The qubit’s larger linewidth reduces their signalstrength, and therefore limits the tuning precision.A central limiting factor in our experiment is thesignal strength measured on-resonance in both traces, | S | ∝ p N γ N / ( κ N + γ N ). We define it here by cou- ( M H z ) (a)theoryexperimentf expected f -1000.20.6 number of qubits (b) li n e w i d t h b r o a d e n i n g ( M H z ) linewidth broadening GND probabilitysignal strength s i g n a l s t r e n g t h ( a . u . ) , G N D p r o b a b ili t y FIG. 5. Multiple qubits on resonance with the cavity. (a)Comparison between theoretical (average measured individ-ual coupling g avg = P i g i /N ) and experimental vacuum Rabisplittings for N qubits. Up to N = 5 calibration for all qubitshas been applied, the N = 6 data (grey area) is from uncali-brated measurement (red points). Using the same reference,the rhombuses show the measured and expected shift of centerfrequencies ∆ f . The pure resonator frequency is calibratedto 0 MHz. (b) Linewidth broadening κ , probability p thatthe system is in the ground excitation-manifold, and the sig-nal strength | S | as function of total number of on-resonancequbits N . pling to the transmission line γ , the linewidth broad-ening κ , and the probability for being in the groundexcitation-manifold p (i.e., not being thermally excitedout of the probed manifold). These values depend onthe bias points, here labelled simply by the amount ofqubits brought on resonance N . The off-resonance cou-pling is γ ≈ π × . N > γ N> = γ / N = 1. Here the degrading effect of elevated temper-ature and fast qubit decay becomes relevant. This isbecause for N >
0, excitation manifolds appear wherethe splitting cannot be observed. The system can escapein these manifolds using thermal fluctuations. Conse- quently, the probability p drops below 1, see Fig. 5(b).We have estimated that the temperature at different biaspoints varies between 130 mK and 175 mK [20], and ismaximal for N = 1. Furthermore, the broadening κ in-creases at N = 1, since an on-resonance qubit sharesphotons with the cavity, and thereby can also dissipatethem (here with a decay rate much higher than γ ). Thiseffect is not additive for N >
1, since at collective reso-nance it is the average decay rate of all the qubits broughton-resonance, which defines the dissipation rate. As a re-sult, the signal strength decays more slowly for
N > N = 0 and N = 1. However, when increas-ing N >
1, another effect comes into play, that insteadtends to reduce the broadening κ : the number of qubitsthat cause broadening of the effective cavity frequencythrough off-resonance hopping (by inducing fluctuationsto the dispersive shift of the cavity [20]) reduces. How-ever, the signal peak becomes narrower and more difficultto detect it from the overall noise. This behavior of thesignal can also be reproduced by master-equation simu-lations for cavity coupled to eight qubits [20].The actual limit of maximal observable ensemble sizedepends on the qubit parameters, chosen bias points andprobing tone strength. Using the calibration scheme,the signal vanishes for six qubits simultaneously on-resonance, and in another cooldown using no calibra-tion and different probing power, the signal disappearsfor seven qubits. Thermal leakage out of the Tavis-Cummings subspace, decay of superconducting qubits,and variations between cooldowns need to be suppressedfor coherent control of larger N . VI. CONCLUSIONS
We have demonstrated the enhancement of the col-lective coupling between a harmonic oscillator and lo-cally tunable two-level systems. The set of up to sixcollectively coupled qubits is one of the largest ensembleand the one with largest collective coupling demonstrat-ing the Tavis-Cummings splitting in circuit QED to thebest of our knowledge. After the submission we becameaware of a related work showing up to N = 10, but ap-proximately 5 times less ensemble coupling strength [27]than our work. The system was realized by a supercon-ducting coplanar resonator coupled to eight frequency-controllable transmons. Our experiment showed that thismoderately scaled circuit can be well controlled even inthe presence of parasitic effects like background trans-mission, dissipation, flux control crosstalk, low anhar-monicity and elevated sample temperatures, all of whichare likely subtle features of near-term physical quantumsimulators. A method was presented to calibrate for thecrosstalk between the qubits and non-neighbouring fluxcoils using a single, shared readout resonator, allowing forprecise individual qubit control. The spectroscopic mea-surement on the collective interaction confirmed that inthis system the collective coupling strength scales with √ N .Increasing the collective coupling opens up the path forfurther research [18] such as ultra-strong coupling be-tween two modes, ground-state squeezing, and superra-diant emission. VII. ACKNOWLEDGEMENT
The authors thank Lucas Radtke for experimental,and the China Scholarship Council (CSC), Studiens-tiftung des deutschen Volkes, the European ResearchCouncil (ERC-648011), the DFG-Center for FunctionalNanostructures National Service Laboratory, HelmholtzIVF grant ’Scalable solid state quantum computing’,DFG project INST 121384/138-1 FUGG, Russian Sci-ence Foundation (16-12-00095), and Ministry of Educa-tion and Science of Russian Federation (K2-2017-081) forgrant support.
Appendix A: Wiring
The chip is mounted in an aluminium sample box, andwire bonded to input and output microwave lines. Theconstant (DC) currents for flux bias control of the qubitsare provided via bonds to a printed circuit board. Thesample is located inside a cryoperm magnetic shield andcooled down by a dilution refrigerator to around 20 mK.A microwave generator is employed when multi-photontransitions are probed dispersively. The signal comingout from the sample goes through two circulatorsand is amplified by a high-electron-mobility transistor(HEMT) at 4 . . Appendix B: Sample fabrication
The sample is patterned in a single step by electron-beam lithography, followed by double angle aluminium deposition (total 80 nm) on the intrinsic silicon substrate.The size of the Josephson junction is 100 ×
100 nm with acritical current of 40 . . Appendix C: Calibration
For an arbitrary selection of two flux lines, the sig-nal traces as shown in Fig.1(b) are not always orthog-onal to each other due to finite crosstalk. The slopesof the traces correspond to the mutual inductance ma-trix elements normalized to the self inductance of thecorresponding flux lines and qubits. M xy is the mutualinductance between the x qubit and the y flux bias line. This is a consequencefrom Eq.2 in the main text. We extract the slopes fromlinear fits to the data traces. To obtain the full mutualinductance matrix we repeat this measurement scheme28 times for all combinations of flux lines. M b M a M c M a M d M a M e M a M f M a M g M a M h M a M a M b M c M b M d M b M e M b M f M b M g M b M h M b M a M c M b M c M d M c M e M c M f M c M g M c M h M c M a M d M b M d M c M d M e M d M f M d M g M d M h M d M a M e M b M e M c M e M d M e M f M e M g M e M h M e M a M f M b M f M c M f M d M f M e M f M g M f M h M f M a M g M b M g M c M g M d M g M e M g M f M g M h M g M a M h M b M h M c M h M d M h M e M h M f M h M g M h . (C1)The compensation scheme of the crosstalk is basedon counter-currents which are applied to all other qubitcoils, while only one qubit is effectively tuned. Thecounter-currents cancel out the flux in the non-tunedqubits, which therefore stay at a fixed frequency. Toobtain the necessary compensation currents a 7 variablelinear equation set has to be solved. For example, to tunequbit 1, this function set needs to be solved: M a M b ∆ I a +∆ I b + M c M b ∆ I c + M d M b ∆ I d + M e M b ∆ I e + M f M b ∆ I f + M g M b ∆ I g + M h M b ∆ I h = 0 M a M c ∆ I a + M b M c ∆ I b +∆ I c + M d M c ∆ I d + M e M c ∆ I e + M f M c ∆ I f + M g M c ∆ I g + M h M c ∆ I h = 0 M a M d ∆ I a + M b M d ∆ I b + M c M d ∆ I c +∆ I d + M e M d ∆ I e + M f M d ∆ I f + M g M d ∆ I g + M h M d ∆ I h = 0 M a M e ∆ I a + M b M e ∆ I b + M c M e ∆ I c + M d M e ∆ I d +∆ I e + M f M e ∆ I f + M g M e ∆ I g + M h M e ∆ I h = 0 M a M f ∆ I a + M b M f ∆ I b + M c M f ∆ I c + M d M f ∆ I d + M e M f ∆ I e +∆ I f + M g M f ∆ I g + M h M f ∆ I h = 0 M a M g ∆ I a + M b M g ∆ I b + M c M g ∆ I c + M d M g ∆ I d + M e M g ∆ I e + M f M g ∆ I f +∆ I g + M h M g ∆ I h = 0 M a M h ∆ I a + M b M h ∆ I b + M c M h ∆ I c + M d M h ∆ I d + M e M h ∆ I e + M f M h ∆ I f + M g M h ∆ I g +∆ I h = 0 , (C2)where ∆ I b , ∆ I c , · · · , ∆ I g , ∆ I h are the 7 variables. Tosolve the equation set the relation between these variablesand ∆ I a has to be computed to apply the compensationcurrents for I a . In other words, with matrix C1, weare able to calibrate out the cross-talk between all thecoils. The variation in current ∆ I is used, rather thanthe absolute value of current I . Fig.2(d) shows the resultafter calibration of (c). Almost no-tilt indicates there isno residual cross-talk between these two flux bias lines. Appendix D: subtraction of background fromtransmission data
Boundary conditions between the cavity and transmis-sion lines in the presence of a background transmissionare [20]ˆ a out ( t ) = √ κ c ˆ a ( t ) − iǫ ˆ a in ( t ) − iǫ iǫ ˆ b in ( t ) (D1)ˆ b out ( t ) = √ κ c ˆ a ( t ) − iǫ ˆ b in ( t ) − iǫ iǫ ˆ a in ( t ) . (D2)Here operators ˆ a in / out describe propagating modes on oneside of the two-sided cavity and ˆ b in / out on the other side.The cavity mode is described by the operator ˆ a and thebackground coupling by parameter ǫ . We consider here aweak background coupling, i.e. | ǫ | ≪
1. Assuming thatwe have measured the output h ˆ b out ( t ) i , and there is noinput from side b , and we know ǫ , then the cavity fieldcan be deduced from Eq. (D2), √ κ c h ˆ a ( t ) i = h ˆ b out ( t ) i + 2 iǫ iǫ h ˆ a in ( t ) i . (D3)Here κ c is the effective coupling between the dressed res-onator and the transmission line. Since the cavity equa-tion of motion depends only weakly on ǫ [20], it followsthat this solution is (up to a constant front factor) alsothe solution for an output without the presence of a back-ground. The data before and after background removing I a (mA)f VNA (GHz) | S | f r e q u e n c y ( G H z ) -5 0 5 10 (b)I a (mA)f VNA (GHz) | S | (a) FIG. 6. Single-flux scan. (a) Original data without back-ground substraction. The two insets show the shape of theresonator when the flux is 1 mA and 6 mA. (b) The data afterbackground substraction (i.e. Fig.2 b). The two insets showthe shape of the resonator when the flux is 1 mA and 6 mA. is shown in Fig. 6. For the original data before back-ground removing, a transformation between Fano-shapedpeaks and dips is observed. Fig. 6 b)(i.e. Fig.2 b) showthe result after background extraction, in which Fanoresonances do not appear.
Appendix E: Multi-photon transitions
The multi-photon transition [26] of qubit 7 of the 8-qubit chip in the power spectrum is shown in Fig. 7. Allof the qubits are tuned to their maximum frequencies.The VNA is set to the single-photon power and observesthe dispersive shift of the resonator while driving qubit7 separately by a microwave generator. With low driv-ing power, only the fundamental transition is visible. Themulti-photon transitions from ground state to higher lev-els are visible while increasing the power. We determine ω , max / π = 10 . ± .
08 GHz and an anharmonicity 410 ± E c / π ~ = 462 MHz). For trans-mon qubit, E c /h = e / hC total , and is approximately FIG. 7. Multi-photon transition experiment. (a) Measuredqubit frequencies of qubit 7 with increased drive power. Atlow power, only the fundamental single photon transitionfrom ground state to first excited state is visible. While in-creasing power, multi-photon transition are observable, andthe higher the power, the more transitions show up. Thetransmitted amplitude is in log-scale. (b) Illustration for themulti-photon transition among the eigen-energy levels of thequbit. the anharmonicity [19]. E J max /h = 34 . ± . Appendix F: Extended Jaynes-Cummings model
In order to explain all features visible in our mea-surements with one qubit on resonance, we extend theJaynes-Cummings Model to the case where an anhar-monic three level atom is interacting with a bosonic res-onator mode. Hamiltonian H L in Eq.3 in the main texthas a block diagonal from, and each block is associatedwith a fixed conserved number of total excitations in thesystem consisted by the resonator and 1 qubit. When thetotal excitation is 0, H L = 0, with basis vector | g, i .When the total excitation is 1,ˆ H L = (cid:18) ω r g ge g ge ω e (cid:19) , (F1)with basis vectors {| g, i , | e, i} . And when the totalexcitation is 2,ˆ H L = ω r √ g ge √ g ge ω r + ω e g ef g ef ω f , (F2)with basis vectors {| g, i , | e, i , | f, i} . Diagonalizationof the Hamiltonians in Eq. F1 and Eq. F2 yields theeigenenergies of the first two excitation manifolds of thesystem that is indicated in Fig.3 c. Appendix G: Fitting the splitting
Consider a single two-level qubit couples to a res-onator, the Hamiltonian is the same as Eq. F1. Theeigenvalues of this Hamiltonian are E ± ~ = ω r + ω e ± q g ge +( ω r − ω e ) (G1)In the vicinity of the strong coupling to the resonatorrange, the relation between qubit energy and the appliedflux bias current is simplified to a linear function ω e ( I ) =2 π ( aI + b ). By substitution into Eq. G1, one gets thefitting function for a single qubit interacting with theresonator. f ± ( I ) = f r + aI + b ± r g ge π ) +( f r − aI − b ) . (G2)For multiple qubit case, treating them as an ensemble(ens), the effective total coupling strength is enhanced.In order to obtain the value, the multiple-qubit anticross-ing is fitted with the following formula: f ( I ) ens + = f r + aI + b r g ge π ) +( f r − aI − b ) ,f ( I ) ens − = f r + a ( I + I shift )+ b − f shift − r g ge π ) +[ f r − a ( I + I shift ) − b ] . (G3)Eq. G3 has the same form as Eq. G2 but the lowerbranch of the anticrossing has two more degrees of free-dom ( I shift , f shift ) which shift its position comparedto the single qubit anticrossing. The effective couplingstrength is extracted by the minimum distance betweenthese two branches (i.e. the ensemble and the resonatorare exactly on-resonance). g ens ( I )2 π = f ( f r − ba − I shift ) ens + − f ( f r − ba − I shift ) ens − Appendix H: Analysis of signal strength
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