Circuit Quantum Electrodynamics:

Coherent Coupling of a Single Photon to a Cooper Pair Box

A. Wallraff, D. Schuster, A. Blais, L. Frunzio, R.-S. Huang,∗ J. Majer, S. Kumar, S. M. Girvin, and R. J. SchoelkopfDepartments of Applied Physics and Physics, Yale University, New Haven, CT 06520

(Dated: July 13, 2004, accepted for publication in Nature (London))

Under appropriate conditions, superconducting electronic circuits behave quantum mechanically,with properties that can be designed and controlled at will. We have realized an experiment in whicha superconducting two-level system, playing the role of an artificial atom, is strongly coupled to asingle photon stored in an on-chip cavity. We show that the atom-photon coupling in this circuitcan be made strong enough for coherent effects to dominate over dissipation, even in a solid stateenvironment. This new regime of matter light interaction in a circuit can be exploited for quantuminformation processing and quantum communication. It may also lead to new approaches for singlephoton generation and detection.

The interaction of matter and light is one of the fun-damental processes occurring in nature. One of its mostelementary forms is realized when a single atom interactswith a single photon. Reaching this regime has been amajor focus of research in atomic physics and quantumoptics1 for more than a decade and has created the field ofcavity quantum electrodynamics2 (CQED). In this articlewe show for the first time that this regime can be realizedin a solid state system, where we have experimentallyobserved the coherent interaction of a superconductingtwo-level system with a single microwave photon. Thisnew paradigm of circuit quantum electrodynamics mayopen many new possibilities for studying the strong in-teraction of light and matter.

In atomic cavity QED an isolated atom with electricdipole moment d interacts with the vacuum state electricfield E0 of a cavity. The quantum nature of the electro-magnetic field gives rise to coherent oscillations of a singleexcitation between the atom and the cavity at the vac-uum Rabi frequency νRabi = 2dE0/h, which can be ob-served experimentally when νRabi exceeds the rates of re-laxation and decoherence of both the atom and the field.This effect has been observed both in the time domain us-ing large dipole moment Rydberg atoms in 3D microwavecavities3 and spectroscopically using alkali atoms in verysmall optical cavities with large vacuum fields4,5.

Our experimental implementation of cavity quantumelectrodynamics in a circuit consists of a fully electricallycontrollable superconducting quantum two-level system,the Cooper pair box6, coupled to a single mode of thequantized radiation field in an on-chip cavity formed bya superconducting transmission line resonator. Coherentquantum effects have been recently observed in severalsuperconducting circuits7–12, making these systems wellsuited for use as quantum bits (qubits) for quantum infor-mation processing13,14. Of these superconducting qubits,the Cooper pair box is especially well suited for cavityQED because of its large effective electric dipole momentd, which can be made more than 104 times larger thanin an alkali atom in the ground state and still 10 timeslarger than in a typical Rydberg atom. In addition, thesmall size of the quasi one-dimensional transmission line

cavity used in our experiments generates a large vacuumelectric field E0. As suggested in our earlier theoreti-cal study15, the simultaneous combination of large dipolemoment and large field strength in our implementationis ideal for reaching the strong coupling limit of cavityQED in a circuit. Other solid state analogs of strongcoupling cavity QED have been envisaged by many au-thors in superconducting16–23, semiconducting24,25 andeven micro-mechanical systems26,27. First steps to-wards realizing such a regime have been made forsemiconductors24,28,29. Our experiments constitute thefirst experimental observation of strong coupling cavityQED with a single artificial atom and a single photon ina solid state system.

I. THE CAVITY QED CIRCUIT

The on-chip cavity is realized as a distributed transmis-sion line resonator patterned into a thin superconductingfilm deposited on the surface of a silicon chip. The quasi-one-dimensional coplanar waveguide resonator30 consistsof a narrow center conductor of length l and two nearbylateral ground planes, see Fig. 1a. Its full wave resonanceoccurs at a frequency of νr = 6.04GHz. Close to its res-onance frequency ωr = 2πνr = 1/

√LC the resonator can

be modelled as a parallel combination of a capacitor Cand an inductor L (a resistor R can be omitted since theinternal losses are small). This simple resonant circuitbehaves as a harmonic oscillator described by the Hamil-tonian Hr = hωr(a

†a + 1/2), where a† (a) is the creation(annihilation) operator for a single photon in the electro-magnetic field and 〈a†a〉 = 〈n〉 = n is the average photonnumber. Cooling such a resonator down to temperatureswell below T ? = hωr/kB ≈ 300mK, the average pho-ton number n can be made very small. At our operatingtemperatures of T < 100mK the thermal occupancy isn < 0.06, so the resonator is near its quantum groundstate. The vacuum fluctuation energy stored in the res-onator in the ground state gives rise to an effective rmsvoltage of Vrms =

√hωr/2C ≈ 1µV on its center conduc-

tor. The magnitude of the electric field between the cen-

2

50 µm

5 µm

1 mmA

B C

FIG. 1: Integrated circuit for cavity QED. a The superconducting niobium coplanar waveguide resonator is fabricated on anoxidized 10 × 3 mm2 silicon chip using optical lithography. The width of the center conductor is w = 10 µm separated fromthe lateral ground planes extending to the edges of the chip by a gap of width d = 5 µm resulting in a wave impedance of thestructure of Z = 50 Ω being optimally matched to conventional microwave components. The length of the meandering resonatoris l = 24 mm. It is coupled by a capacitor at each end of the resonator (see b) to an input and output feed line, fanning out tothe edge of the chip and keeping the impedance constant. b The capacitive coupling to the input and output lines and hencethe coupled quality factor Q is controlled by adjusting the length and separation of the finger capacitors formed in the centerconductor. c False color electron micrograph of a Cooper pair box (blue) fabricated onto the silicon substrate (green) into thegap between the center conductor (top) and the ground plane (bottom) of a resonator (beige) using electron beam lithographyand double angle evaporation of aluminum. The Josephson tunnel junctions are formed at the overlap between the long thinisland parallel to the center conductor and the fingers extending from the much larger reservoir coupled to the ground plane.

ter conductor and the ground plane in the vacuum stateis then a remarkable Erms ≈ 0.2V/m, which is some 100times larger than in atomic microwave cavity QED3. Thelarge vacuum field strength resulting from the extremelysmall effective mode volume (∼ 10−6 cubic wavelengths)of the resonator is clearly advantageous for cavity QEDnot only in solid state but also in atomic systems31. Inour experiments this vacuum field is coupled capacitivelyto a superconducting qubit. At the full wave resonancethe electric field has antinodes at both ends and at thecenter of the resonator and its magnitude is maximal inthe gap between the center conductor and the groundplane where we place our qubit.

The resonator is coupled via two coupling capacitorsCin/out, one at each end, see Fig. 1b, to the input andoutput transmission lines through which its microwavetransmission can be probed, see Fig. 2a. The predomi-nant source of dissipation is the loss of photons from theresonator through these input and output ports at a rateκ = ωr/Q, where Q is the (loaded) quality factor of the

resonator. The internal (uncoupled) loss of the resonatoris negligible (Qint ≈ 106). Thus, the average photon life-time in the resonator Tr = 1/κ exceeds 100 ns even formoderate quality factors of Q ≈ 104.

Our choice of artificial atom is a mesoscopic supercon-ducting electrical circuit, the Cooper pair box6 (CPB),the quantum properties of which have been clearlydemonstrated7,16,32. The Cooper pair box consists ofa several micron long and sub-micron wide supercon-ducting island which is coupled via two sub-micron sizeJosephson tunnel junctions to a much larger supercon-ducting reservoir. We have fabricated such a structureinto the gap between the center conductor and the groundplane in the center section of the coplanar waveguide res-onator, see Fig. 1c.

The Cooper Pair box is a tunable two-state system de-scribed by the Hamiltonian Ha = −1/2 (Eel σx + EJ σz),where Eel is the electrostatic energy and EJ is theJosephson energy of the circuit. The electrostatic en-ergy Eel = 4EC (1 − ng) is proportional to the charg-

3

ing energy EC = e2/2CΣ, where CΣ is the total capac-itance of the island. Eel can be tuned by a gate chargeng = VgC

?g/e, where Vg is the voltage applied to the input

port of the resonator and C?g the effective dc-capacitance

between the input port of the resonator and the islandof the Cooper pair box. The Josephson energy EJ =EJ,max cos (πΦb) is proportional to EJ,max = h∆/8e2RJ ,where ∆ is the superconducting gap and RJ is the tun-nel junction resistance. Using an external coil, EJ canbe tuned by applying a flux bias Φb = Φ/Φ0 to the loopformed by the two tunnel junctions, the island and thereservoir. We denote the ground state of this system as|↓〉 and the excited state as |↑〉, see Fig. 2d. The energylevel separation between these two states is then given

by Ea = hωa =√

E2J,max cos2(πΦb) + 16E2

C (1 − ng)2.

The Hamiltonian Ha of the Cooper pair box can bereadily engineered by choosing the parameters EJ,max

and EC during fabrication and by tuning the gate chargeng and the flux bias Φb in situ during experiment. Thequantum state of the Cooper pair box can be manip-ulated using microwave pulses7,33. Coherence in theCooper pair box is limited both by relaxation from theexcited state |↑〉 into the ground state |↓〉 at a rate γ1

corresponding to a lifetime of T1 = 1/γ1 and by fluctu-ations in the level separation giving rise to dephasing ata rate γϕ corresponding to a dephasing time Tϕ = 1/γϕ.The total decoherence rate γ = γ1/2 + γϕ has contri-butions from the coupling to fluctuations in the electro-magnetic environment32,34 but is possibly dominated bynon-radiative processes. Relevant sources of dephasinginclude coupling to low frequency fluctuations of chargesin the substrate or tunnel junction barrier and couplingto flux motion in the superconducting films.

II. THE QUBIT-CAVITY COUPLING

The coupling between the radiation field stored in theresonator and the Cooper pair box is realized through thecoupling capacitance Cg. A voltage Vrms on the centerconductor of the resonator changes the energy of an elec-tron on the island of the Cooper pair box by an amounthg = eVrmsCg/CΣ, where g is the coupling strength. Theenergy hg is the dipole interaction energy dE0 betweenthe qubit and the vacuum field in the resonator. We haveshown15 that the Hamiltonian describing this coupledsystem has the form HJC = Hr + Ha + hg(a†σ− + aσ+),where σ+ (σ−) creates (annihilates) an excitation in theCooper pair box. The Jaynes-Cummings Hamiltonian,HJC, describing our coupled circuit is well known fromcavity QED. It describes the coherent exchange of energybetween a quantized electromagnetic field and a quantumtwo-level system at a rate g/2π, which is observable if g ismuch larger than the decoherence rates γ and κ. This sit-uation is achieved in our experiments and is referred to asthe strong coupling limit (g > [γ, κ]) of cavity QED35. Ifsuch a system is prepared with the electromagnetic field

in its ground state |0〉, the two-level system in its ex-cited state |↑〉 and the two constituents are in resonanceωa = ωr, it oscillates between states |0, ↑〉 and |1, ↓〉 atthe vacuum Rabi frequency νRabi = g/π. In this situa-tion the eigenstates of the coupled system are symmetricand antisymmetric superpositions of a single photon ina resonator and an excitation in the Cooper pair box|±〉 = (|0, ↑〉 ± |1, ↓〉) /

√2 with energies E± = h (ωr ± g).

Although the eigenstates |±〉 are entangled, their entan-gled character is not addressed in our current CQED ex-periment. In our experiment, it is the energies E± of thecoherently coupled system which are probed spectroscop-ically.

In the non-resonant (‘dispersive’) case the qubit is de-tuned by an amount ∆ = ωa − ωr from the cavity res-onance. The strong coupling between the field in theresonator and the Cooper pair box can be used to per-form a quantum nondemolition (QND) measurement ofthe quantum state of the Cooper pair box. For |∆| g,diagonalization of the coupled quantum system leads tothe effective Hamiltonian15

H ≈ h

(ωr +

g2

∆σz

)a†a +

1

2h

(ωa +

g2

∆

)σz ,

where the terms proportional to a†a determine the res-onator properties and the terms proportional to σz de-termine the circuit properties. It is easy to see that thetransition frequency of the resonator ωr ± g2/∆ is con-ditioned by the qubit state σz = ±1 and depends on thecoupling strength g and the detuning ∆. Thus by mea-suring the transition frequency of the resonator the qubitstate can be determined. Similarly, the level separationin the qubit ∆Ea = h

(ωa + 2a†a g2/∆ + g2/∆

)is condi-

tioned by the number of photons in the resonator. Theterm 2 a†a g2/∆ linear in the photon number n is knownas the ac-Stark shift and the term g2/∆ as the Lamb shiftas in atomic physics. All terms in this Hamiltonian, withthe exception of the Lamb shift, are clearly identified inthe results of our circuit QED experiments.

III. THE MEASUREMENT TECHNIQUE

The properties of this strongly coupled circuit QEDsystem are determined by probing the level separationof the resonator states spectroscopically15. The ampli-tude T and phase φ of a microwave probe beam of powerPRF transmitted through the resonator are measured ver-sus probe frequency ωRF. A simplified schematic of thebasic microwave circuit is shown in Fig. 2a. In thissetup the Cooper pair box acts as a circuit element thathas an effective capacitance which is dependent on itsσz eigenstate, the coupling strength g, and its detuning∆. It is connected through Cg in parallel with the res-onator. This variable capacitance changes the resonancefrequency of the resonator which can be probed by mea-suring its microwave transmission spectrum. The trans-mission T 2 and phase φ of the resonator for a far detuned

4

B

A

C D

ni = 0

EaEJ

-2 +2

−2 −1 0 1 2

gate charge, ng

0

1

2

3

4

En

erg

y,

EêE

c2

νr

δνr

6.042 6.044 6.046

frequency,νRF @GHzD

−20

−10

0

tra

nsm

issio

n,

T@d

BD

6.042 6.044 6.046

frequency , νRF @GHz D

−90

−45

0

45

90

ph

ase

,φ@d

eg

.D

ωRF ωLOω

circulatorbias Tattenuator RF amp mixer

transmissionamplitude and phase

IFRF

Cin

L CJ,EJ

Cout

resonator CPB

100 mK 1 K RTRTRTRT 20 mK

Vg

V

Cg LO

Φ

C

FIG. 2: Measurement scheme, resonator and Cooper pair box. a The resonator with effective inductance L and capacitance Ccoupled through the capacitor Cg to the Cooper pair box with junction capacitance CJ and Josephson energy EJ forms thecircuit QED system which is coupled through Cin/out to the input/ouput ports. The value of EJ is controllable by the magneticflux Φ. The input microwaves at frequencies ω and ωRF are added to the gate voltage Vg using a bias-tee. After the transmittedsignal at ωRF is amplified using a cryogenic high electron mobility (HEMT) amplifier and mixed with the local oscillator atωLO, its amplitude and phase are determined. The circulator and the attenuator prevent leakage of thermal radiation into theresonator. The temperature of individual components is indicated. b Measured transmission power spectrum of the resonator(blue dots), the full line width δνr at half maximum and the center frequency νr are indicated. The solid red line is fit to aLorentzian with Q = νr/δνr ≈ 104. c Measured transmission phase φ (blue dots) with fit (red line). In panels b and c thedashed lines are theory curves shifted by ±δνr with respect to the data. d Energy level diagram of a Cooper pair box. Theelectrostatic energy (ni − ng)2e2/2CΣ is indicated for ni = 0 (solid black line), −2 (dotted line) and +2 (dashed line) excesselectrons forming Cooper pairs on the island. CΣ is the total capacitance of the island given by the sum of the capacitances CJ

of the two tunnel junctions, the coupling capacitance Cg to the center conductor of the resonator and any stray capacitances.In such a system ni is well determined if EC/kB is much larger than the bath temperature T and if the quantum fluctuationsinduced by the tunnelling of electrons through the junctions are small. In the absence of Josephson tunnelling the states withni and ni + 2 electrons on the island are degenerate at ng = 1. The Josephson coupling mediated by the weak link formed bythe tunnel junctions between the superconducting island and the reservoir lifts this degeneracy and opens up a gap proportionalto the Josephson energy EJ . A ground state band |↓〉 and an excited state band |↑〉 are formed with a gate charge and fluxbias dependent energy level separation of Ea.

qubit (g2/∆κ 1), i.e. when the qubit is effectively de-coupled from the resonator, are shown in Figs. 2b andc. In this case, the transmission is a Lorentzian line ofwidth δνr = νr/Q at the resonance frequency νr of thebare resonator. The transmission phase φ displays a stepof π at νr with a width determined by Q. The expectedtransmission spectrum in both phase and amplitude fora frequency shift ±g2/∆ of one resonator line width areshown by dashed lines in Fig. 2b and c. Such small shiftsin the resonator frequency are sensitively measured in ourexperiments using probe beam powers PRF which control-lably populate the resonator with average photon num-bers from n ≈ 103 down to the sub photon level n 1.It is worth noting that in this architecture both the res-onator and the qubit can be controlled and measuredusing capacitive and inductive coupling only, i.e. without

attaching any dc-connections to either system.

IV. THE DISPERSIVE REGIME

In the dispersive regime, the qubit is detuned from theresonator by an amount ∆ and thus induces a shift g2/∆in the resonance frequency which we measure as a phaseshift tan−1

(2g2/κ∆

)of the transmitted microwave at

a fixed probe frequency ωRF. The detuning ∆ is con-trolled in situ by adjusting the gate charge ng and theflux bias Φb of the qubit. For a Cooper pair box withJosephson energy EJ,max/h > νr two different cases canbe identified. In the first case, for bias fluxes such thatEJ(Φb)/h > νr, the qubit does not come into resonancewith the resonator for any value of gate charge ng, see

5

−2 −1 0 1 2

0

5

10

15

20

ν@G

HzD

νa@G

HzD

A

C

B

D

−2 −1 0 1 2

gate charge , ng

−1

− 0.5

0.

0.5

1.

flu

xb

ias

,Φ

b

1 0 1 2

gate voltage, V g @V D

35

40

45

50

ma

gn

eti

cfi

eld

,B@µ

TD

νrνa

Φb = 0.80

Φb = 0.35

δ = νa − νr

−2 −1 0 1 2

gate charge , nggate charge , ng

−40

−30

−20

−10

0

10

20

ph

ase

shif

t,φ@d

egD

−1 0 1 2

gate voltage, V g @V D

−2 −1 0 1 2

ng

−1

−0.5

0

0.5

1

Φb

0

10

20

5

0

0.5

1

a

FIG. 3: Strong coupling circuit QED in the dispersive regime. a Calculated level separation νa = ωa/2π = Ea/h betweenground |↓〉 and excited state |↑〉 of qubit for two values of flux bias Φb = 0.8 (orange line) and Φb = 0.35 (green line). Theresonator frequency νr = ωr/2π is shown by a blue line. Resonance occurs at νa = νr symmetrically around degeneracy ng = 1,see red arrows. The detuning ∆/2π = δ = νa − νr is indicated. b Measured phase shift φ of the transmitted microwave forvalues of Φb in a. Green curve is offset by −25 deg for visibility. c Calculated qubit level separation νa versus bias parametersng and Φb. The resonator frequency νr is indicated by the blue plane. At the intersection, also indicated by the red curvein the lower right hand quadrant, resonance between the qubit and the resonator occurs (δ = 0). For qubit states below theresonator plane the detuning is δ < 0, above δ > 0. d Density plot of measured phase shift φ versus ng and Φb. Light colorsindicate positive φ (δ > 0), dark colors negative φ (δ < 0). The red line is a fit of the data to the resonance condition νa = νr.In c and d, the line cuts presented in a and b are indicated by the orange and the green line, respectively. The microwaveprobe power PRF used to acquire the data is adjusted such that the maximum intra resonator photon number n at νr is about10 for g2/∆κ 1. The calibration of the photon number has been performed in situ by measuring the ac-Stark shift of thequbit levels37.

Fig. 3a. As a result, the measured phase shift φ is max-imum for the smallest detuning ∆ at ng = 1 and getssmaller as the detuning ∆ increases, see Fig. 3b. More-over, φ is periodic in ng with a period of 2e, as ex-pected. In the second case, for values of Φb resultingin EJ/h < νr, the qubit goes through resonance with theresonator at two values of ng. Thus, the phase shift φ is

largest as the qubit approaches resonance (∆ → 0) at thepoints indicated by red arrows, see Fig. 3b. As the qubitgoes through resonance, the phase shift φ changes signas ∆ changes sign. This behavior is in perfect agreementwith our predictions based on the analysis of the circuitQED Hamiltonian in the dispersive regime.

In Fig. 3c the qubit level separation νa = Ea/h is plot-

6

ted versus the bias parameters ng and Φb. The qubitis in resonance with the resonator (νa = νr) at the setof parameters [ng,Φb] also indicated by the red curve inone quadrant of the plot. The measured phase shift φ isplotted versus both ng and Φb in Fig. 3d. We observethe expected periodicity in flux bias Φb with one fluxquantum Φ0. The set of parameters [ng,Φb] for whichthe resonance condition is met is marked by a suddensign change in φ. From this set, the Josephson energyEJ,max and the charging energy EC of the qubit can bedetermined in a fit to the resonance condition. Using themeasured resonance frequency νr = 6.044GHz, we ex-tract the qubit parameters EJ,max = 8.0 (±0.1)GHz andEC = 5.2 (±0.1)GHz.

This set of data clearly demonstrates that the prop-erties of the qubit in this strongly coupled system canbe determined in a transmission measurement of the res-onator and that full in situ control over the qubit pa-rameters is achieved. We note that in the dispersiveregime this new read-out scheme for the Cooper pair boxis most sensitive at charge degeneracy (ng = 1), wherethe qubit is to first order decoupled from 1/f fluctuationsin its charge environment, which minimizes dephasing.This property is advantageous for quantum control of thequbit at ng = 1, a point where traditional electrometry,using an SET for example32, is unable to distinguish thequbit states. Moreover, we note that this dispersive mea-surement scheme performs a quantum non-demolitionread-out of the qubit state15 which is the complementof the atomic microwave cQED measurement in whichthe state of the photon field is inferred non-destructivelyfrom the phase shift in the state of atoms sent throughthe cavity3,36.

V. THE RESONANT REGIME

Making use of the full control over the qubit Hamil-tonian, we tune the flux bias Φb such that the qubit isat ng = 1 and in resonance with the resonator. Initially,the resonator and the qubit are cooled into their com-bined ground state |0, ↓〉, see inset in Fig. 4b. Due to thecoupling, the first excited states become a doublet. Weprobe the energy splitting of this doublet spectroscop-ically using a microwave probe beam which populatesthe resonator with much less than one photon on aver-age. The intra resonator photon number is calibratedmeasuring the ac-Stark shift of the qubit level separa-tion in the dispersive case when probing the resonatorat its maximum transmission37. The resonator transmis-sion T 2 is first measured for large detuning ∆ with aprobe beam populating the resonator with a maximumof n ≈ 1 at resonance, see Fig. 4a. From the Lorentzianline the photon decay rate of the resonator is determinedas κ/2π = 0.8MHz. The probe beam power is subse-quently reduced by 5 dB and the transmission spectrumT 2 is measured in resonance (∆ = 0), see Fig. 4b. Weclearly observe two well resolved spectral lines separated

by the vacuum Rabi frequency νRabi ≈ 11.6MHz. Theindividual lines have a width determined by the averageof the photon decay rate κ and the qubit decay rate γ.The data is in excellent agreement with the transmis-sion spectrum numerically calculated from the Jaynes-Cummings Hamiltonian characterizing the qubit decayand dephasing by the single parameter γ/2π = 0.7MHz.

In fact, the transmission spectrum shown in Fig. 4bis highly sensitive to the thermal photon number in thecavity. Due to the anharmonicity of the coupled atom-cavity system in the resonant case, an increased thermalphoton number would reduce transmission and give riseto additional peaks in the spectrum due to transitions be-tween higher excited doublets38. The transmission spec-trum calculated for a thermal photon number of n = 0.5,see light blue curve in Fig.4b, is clearly incompatible withour experimental data. The measured transmission spec-trum, however, is consistent with the expected thermalphoton number of n <∼ 0.06 (T < 100mK), see red curvein Fig.4b, indicating that the coupled system has in factcooled to near its ground state.

Similarly, multiphoton transitions between the groundstate and higher excited doublets can occur at higherprobe beam powers. Such transitions also probe the non-linearity of the cavity QED system. In our experimentthese additional transitions are not resolved individually,but they do lead to a broadening of vacuum Rabi peakswith power. Eventually, at high drive powers the trans-mission spectrum becomes single peaked again as transi-tions between states high up in the dressed states ladderare driven.

We also observe the anti-crossing between the singlephoton resonator state and the first excited qubit stateas the qubit is brought into resonance with the resonatorat ng = 1 when tuning the gate charge and weakly prob-ing the transmission spectrum, see Fig. 4c. For smalldeviations from ng = 1 the vacuum Rabi peaks evolvefrom a state with equal weight in the photon and qubit,as shown in Fig. 4b, to predominantly photon states forng 1 or ng 1. Throughout the full range of gatecharge the observed peak positions agree well with cal-culations considering the qubit with level separation νa,a single photon in the resonator with frequency νr and acoupling strength of g/2π, see solid lines in Fig. 4c.

We have performed the same measurement for a differ-ent value of flux bias Φb such that Ea/h < νr at ng = 1.The corresponding plot of the transmission amplitudeshows two anti-crossings as the qubit is brought fromlarge positive detuning ∆ through resonance and largenegative detuning once again through resonance to posi-tive detuning, see Fig. 4d. Again, the peak positions arein agreement with the theoretical prediction.

VI. DISCUSSION

We have for the first time observed strong couplingcavity quantum electrodynamics at the level of a single

7

0.98 1. 1.02 1.04

gate charge , ng

6.02

6.03

6.04

6.05

6.06

6.07

fre

qu

en

cy,ν

RF@G

HzD

−0.14 −0.12 −0.1 −0.08 −0.06

gate voltage, V g g@V D

6.02 6.03 6.04 6.05 6.06 6.07

frequency , νRF @GHz D

0

0.02

0.04

0.06

0.08

0.1

tra

nsm

issi

on

,TêT

022

0

0.2

0.4

0.6

0.8

1A

B

C

datafit

datafit

0.95 1. 1.05

gate charge , ng

6.03

6.04

6.05

6.06

fre

qu

en

cy,ν

RF@G

HzD

−0.6 −0.55 0.5

gate voltage, V @V D

D

1

,1 + ,0

1

1

ωr - g2/∆

∆

ωr + g2/∆

0

12g

2g 2

2

0

1

0

1

0

1

2

)/ 2(

,1 - ,0 )/ 2( ,1 - ,0 )/ 2(

FIG. 4: Vacuum Rabi mode splitting. a Measured transmission T 2 (blue line) versus microwave probe frequency νRF for largedetuning (g2/∆κ 1) and fit to Lorentzian (dashed red line). The peak transmission amplitude is normalized to unity. Theinset shows the dispersive dressed states level diagram. b Measured transmission spectrum for the resonant case ∆ = 0 atng = 1 (blue line) showing the Vacuum Rabi mode splitting compared to numerically calculated transmission spectra (red andlight blue lines) for thermal photon numbers of n = 0.06 and 0.5, respectively. The dashed line is the calculated transmission forg = 0 and κ/2π = 0.8 MHz. The inset shows the resonant dressed states level diagram. c Resonator transmission amplitude Tplotted versus probe frequency νRF and gate charge ng for ∆ = 0 at ng = 1. Dashed lines are uncoupled qubit level separationνa and resonator resonance frequency νr. Solid lines are level separations found from exact diagonalization of HJC. Spectrumshown in b corresponds to line cut along red arrows. d As in c, but for EJ/h < νr. The dominant character of the correspondingeigenstates is indicated.

qubit and a single photon in an all solid state system.We have demonstrated that the characteristic parame-ters of our novel circuit QED architecture can be chosenat will during fabrication and can be fully controlled insitu during experiment. This system will open new possi-bilities to perform quantum optics experiments in solids.It also provides a novel architecture for quantum controland quantum measurement of electrical circuits and willfind application in superconducting quantum computa-tion, for example as a single-shot QND read-out of qubitsand for coupling of qubits over centimeter distances us-ing the resonator as a quantum bus. This architecturemay readily be used to control, couple and read-out twoqubits and can be scaled up to more complex circuits15.Non-radiative contributions to decoherence in solid statequbits can be investigated in this system by using theresonator to decouple the qubit from fluctuations in itselectromagnetic environment. Finally, our circuit QEDsetup may potentially be used for generation and detec-tion of single microwave photons in devices that could be

applied in quantum communication.

Acknowledgments

We thank John Teufel, Ben Turek and Julie Wyattfor their contributions to the project and are grateful toPeter Day, David DeMille, Michel Devoret, Sandy Wein-reb and Jonas Zmuidzinas for numerous conversations.This work was supported in part by the National SecurityAgency (NSA) and Advanced Research and DevelopmentActivity (ARDA) under Army Research Office (ARO)contract number DAAD19-02-1-0045, the NSF ITR pro-gram under grant number DMR-0325580, the NSF un-der grant number DMR-0342157, the David and LucilePackard Foundation, the W. M. Keck Foundation, andthe Natural Science and Engineering Research Councilof Canda (NSERC).

∗ Department of Physics, Indiana University, Bloomington,IN 47405

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