Strong Coupling Cavity QED with Gate-Defined Double Quantum Dots Enabled by a High Impedance Resonator

5.5 MB·

by  Anna Stockklauser

Strong Coupling Cavity QED with Gate-Defined Double Quantum Dots
Enabled by a High Impedance Resonator
A. Stockklauser,∗ P. Scarlino,∗ J. V. Koski, S. Gasparinetti, C. K. Andersen,
C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff
Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland
(Dated: January 13, 2017)
The strong coupling limit of cavity quantum electrodynamics (QED) implies the capability of a
matter-like quantum system to coherently transform an individual excitation into a single photon
withinaresonantstructure. Thisnotonlyenablesessentialprocessesrequiredforquantuminforma-
7 tion processing but also allows for fundamental studies of matter-light interaction. In this work we
1 demonstrate strong coupling between the charge degree of freedom in a gate-defined GaAs double
0 quantum dot (DQD) and a frequency-tunable high impedance resonator realized using an array of
2 superconducting quantum interference devices (SQUIDs). In the resonant regime, we resolve the

n vacuumRabimodesplittingofsize2g/2π=238MHzataresonatorlinewidthκ/2π=12MHzand
a a DQD charge qubit dephasing rate of γ2/2π = 80MHz extracted independently from microwave
J spectroscopy in the dispersive regime. Our measurements indicate a viable path towards using
2 circuit based cavity QED for quantum information processing in semiconductor nano-structures.
1

] In the strong coupling limit, cavity QED realizes the ing the resonator impedance Zr beyond the typical 50Ω
l
l coherentexchangeofasinglequantumofenergybetween of a standard coplanar waveguide. We have realized a
a
anonlinearquantumsystemwithtwoormoreenergylev- frequency-tunable microwave resonator with impedance
h
(cid:112)
- els, e.g. a qubit, and a single mode of a high quality cav- Zr = Lr/Cr ∼ 1.8kΩ using the large inductance
s
ity capable of storing individual photons [1]. The distin- L ∼ 50nH of a SQUID array [26–28] combined with
e r
m guishingfeatureofstrongcouplingisacoherentcoupling a small stray capacitance Cr ∼ 15fF. Its resonance fre-
rate g, determined by the product of the dipole moment quency and thus also its impedance is tunable by apply-
.
t ofthemulti-levelsystemandthevacuumfieldofthecav- ingasmallmagneticfieldusingamm-sizedcoilmounted
a
m ity, which exceeds both the cavity mode linewidth κ, de- on the sample holder. The frequency-tunability of the
termining the photon life time, and the qubit linewidth resonatorisparticularlyusefulinthiscontextasitallows
-
d γ = γ /2+γ , set by its energy relaxation and pure for the systematic study of its interaction with semicon-
2 1 ϕ
n
dephasing rates, γ and γ , respectively. ductor nano-structures without changing their electrical
1 ϕ
o
The strong coupling limit of Cavity QED has been bias conditions.
c
[ reached with a multitude of physical systems including Theresonator,withasmallfootprintof300×120µm2

alkaliatoms[2],Rydbergatoms[3],superconductingcir- (Fig. 1a,b), is fabricated using standard electron-beam
1
v cuits[4,5]andopticaltransitionsinsemiconductorquan- lithography and shadow evaporation of aluminum (Al)
3 tum dots [6, 7]. Of particular interest is the use of this onto a GaAs heterostructure. The embedded two-
3 concept in quantum information processing with super- dimensional electron gas (2DEG) has been etched away
4 condcuting circuits where it is known as circuit QED everywherebutinasmallmesaregionhostingtheDQD.
3
[4, 8, 9]. Thearray,composedof32SQUIDs(Fig.1d),isgrounded
0
. Motivated by the ability to suppress the spontaneous at one end and terminated in a small island at the other
1
emission of qubits beyond the free space limit [10], to end to which a single coplanar drive line is capacitively
0
7 perform quantum non-demolition (QND) qubit read-out coupled. A gate line extends from the island and forms
1 [11,12],tocoupledistantqubitsthroughmicrowavepho- one of the plunger gates of the double quantum dot (or-
: tons coherently [13, 14] and to convert quantum infor- ange) (Fig. 1c).
v
i mation stored in stationary qubits to photons [15, 16], The double quantum dot is formed in the mesa struc-
X
research towards reaching the strong coupling limit of ture using gold (Au) top gates (yellow in Fig. 1a,b,c)
r cavityQEDispursuedforthechargeandspindegreesof controllingthetunnelcouplingoftheDQDtothesource
a
freedom in semiconductor nano-structures [17–22]. Re- anddrainleads(blue)aswellastheinter-dottunnelcou-
cently, inparallelwiththeworkdiscussedhere, indepen- plingt. Theleftandrightsidegates(LSG,RSG)control
dent efforts to reach this goal have come to fruition with the on-site electrostatic energies of each of the two dots,
gate defined DQDs in silicon [23] and carbon nanotubes whiletheplungergatesarenotbiasedintheexperiment.
[24]. An additional gate and pair of leads can be configured
The essence of our approach to reach the strong cou- as a quantum point contact for charge detection. The
pling limit with individual electronic charges in GaAs microwave response of the system is probed in reflection
DQDs is rooted in the enhancement of the electric com- (Fig. 1e) using standard circuit QED heterodyne detec-
√
ponentofthevacuumfluctuations∝ Z [25]byincreas- tion techniques [4, 18].
r
2
(a) (b)
500 µm
50 µm
(c) (d) (e) in
500 nm 2 µm
50 Ω out
LSG RSG
C
PG
S D
FIG. 1. Sample and simplified circuit diagram. (a) False-color optical micrograph of a representative device indicating the
substrate(darkgray),thesuperconductingstructures(lightgray),thegoldtopgates(yellow)formingtheDQDanditssource
and drain leads and contacts (blue). (b) Optical micrograph displaying a SQUID array resonator (light gray) and its coupling
gate to the DQD and the DQD biasing structures (yellow). (c) Electron micrograph of the DQD showing its electrostatic top
gates (yellow) and the plunger gate coupled to the resonator (orange). (d) Electron micrograph of three SQUID loops (dark
grey) in the array deposited on the etched GaAs heterostructure (light gray). (e) Circuit diagram schematically displaying
the DQD (source contact labeled S, drain contact labeled D, and coupling capacitance C to the resonator) and essential
PG
componentsinthemicrowavedetectionchain(circulator,amplifier)usedforperformingreflectancemeasurementsofthedevice.
Boxes with crosses and rectangles indicate Josephson and normal tunnel junctions, respectively.
We show that the resonance frequency of the SQUID berofchargesineachdottobeoftheorderof10electrons
array resonator can be tuned from a maximum value of [18, 30].
ν ∼ 6.0GHz to well below 4.5GHz (which is the lower To explore their mutual coupling, we first fix the
r
cut-offfrequencyofourdetectionelectronics)inmeasure- SQUID array resonance frequency to ν =5.03GHz and
r
ments of its reflectance |S (ν )| as a function of applied set the tunnel coupling of the DQD to 2t ∼ 4.13GHz <
11 p
magneticfluxΦ andprobefrequencyν (Fig.2a). From ν . This ensures that tuning the difference energy δ
m p r
this data we extract the characteristic circuit parame- between the charge states in the right and left quan-
tersoftheresonatorandfindthatitsimpedancechanges tum dot results in a resonance (ν = ν ) between the
q r
from Z ∼ 1.3kΩ to 1.8kΩ in this frequency range. charge qubit transition frequency ν and the resonator
r q
(cid:112)
With the DQD well detuned from the resonator biased at δ =± (ν (Φ ))2−(2t)2 [31].
± r m
at νr =5.02GHz, we determine its internal loss rate, its Varyingthedetuningδ(alongthedashedlineindicated
externalcouplingratetotheinputlineandthetotalline in Fig. 2b) by applying appropriately chosen voltages to
width (κint,κext,κ)/(2π)∼(10.0,2.3,12.3)MHz [29]. the two side gates we observe the dispersive (i.e. non-
We configure the double quantum dot and determine resonant) interaction between the DQD and the res-
its characteristic properties by extracting the amplitude onator in a probe-frequency-dependent reflectance mea-
andphasechangeofacoherenttonereflectedofftheres- surement of the resonator (Fig. 3a). As a function of
onatoratfrequencyν usingameasurementofthereflec- δ,thereflectancespectrum|S (ν )|showscharacteristic
p 11 p
tion coefficient S (ν ) in response to changes of the po- shifts in the dispersive regime (ν (cid:29)ν or ν (cid:28)ν ) and
11 p q r q r
tentialsappliedtothegateelectrodesformingthedouble indications of an avoided crossing at δ ∼ ±2.86GHz
±
quantum dot. Using this by now well-established tech- at resonance (ν = ν ) which we analyze in more detail
q r
nique [17–19], we record characteristic hexagonal charge below.
stability diagrams (Fig. 2b) from which we extract the We first extract the frequency ν˜ of the resonator, as
r
DQDchargingenergyof580GHzandestimatethenum- renormalizedbyitsinteractionwiththeDQD,byfittinga
3
Lorentzianlinetothereflectancespectrumateachvalue of two Lorentzian lines yields a splitting of 2g/2π ∼
of δ. When varying δ, the experimentally extracted shift 238MHz, with an effective linewidth of 93MHz. The
∆ν =ν˜ −ν reachesupto∼100MHzclosetoresonance vacuum Rabi mode splitting is found to be in good
r r r
(blue dots, Fig. 3b). The measured values of ∆ν are in agreement with the spectrum evaluated from the master
r
excellentagreementwiththeresultsofamasterequation equation simulation (red solid line) with the parameters
simulation (solid line) analyzed in the same way find- (g ,γb,γb)/(2π)=(155,35,63)MHz, which is consistent
0 1 ϕ
ing the parameters (g ,γb,γb)/(2π) = (155,35,63)MHz with the analysis of the dispersive frequency shift dis-
0 1 ϕ
while keeping the bare resonator linewidth κ fixed at its cussed above. We note that the small amplitude of the
independently determined value stated above. In the signalinreflectionisadirectconsequenceofthefactthat
Jaynes-Cummings model we use to describe the cou-
pled system, both the coupling rate and the decoher-
ence rates depend on the mixing angle θ. The effec-
(a)
tive coupling strength g is given by g = g sinθ, where
0 0.4 0.6 0.8 1.0
(cid:112)
sinθ = 2t/ (2t)2+δ2, while the decay and decoher-
|S | (d)
ence rates are given by γ = sin2θγb + cos2θγb and 11
1 ϕ 1
bqrftflrpbγierhaeϕyunluocasmebettmoymwiien=hdvtotyiaaeetderwnbvhretclsecresiihoititlnendoshhwqκeg2νpiawuz.elqssθneaaiiWγ∼gdttnaaϕLbiwntltohshuofνiono+afierrmcnceκw˜wfittansaoehndniitoltnprezdotlfh2iplotarryt,rθfeeetnhoxγhss4tleaooac1hbeldcnleri.ervnhgDluaeleeUieeertsndQroeonsgtssrnDtisohlonirparannagetteecphesagoflchoweterrtteanoehricireiardaanetngmslntatedhencslepriiraeqcsnenraγ,mpeuntoec2ilbosbttdeaw.riie(cid:29)yoiotmsiss.entfesiahrnDttκwelnhic.qetzioeorthuteetfhuNdeeabdnnspteaebiheecarasfddyyeer--- (bGHzFrequency,)pν()4445555......789123. 238 MHz●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●93 MHz●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Tuning the DQD into resonance with the resonator 0.96 0.98 1.
(ν = ν ), indicated by arrows in Fig. 3a, we observe 100 |S |
ar(deflqcalesechateradnvrcgaerceuseupnmeclitnRreua)mboifomftohtdeheespsrpeeclsitotrtnuianmtgort(ob(lFauiegs.udpo3etdsr))p.oisnAittiohfinet (cMHz)r-()Δν1000●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●|S111|.●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●1●●●●●●●●●●●●●●●●●●●1●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●(●●●●●●●●●●●●●●●●●●e●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●)●●●●●●●●●●●●●●●●●●●●●●●●
●●●●
0.8 ●●
(a) 0.4 0.6 0.8 1.0 (b) 52 54 56 58 60 MHz)60 ●●●● ● ●●●
(40 ● 0.6 ●
|S161|. ϕ(°) ~2πκ/20●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●
Hz) V) 0 0.4 ●●
G 5.5 m 970 5 0 5 4.8 5. 5.2
( (
ncy,pν5. e,VLSG -980 - Detuning,δ(GHz) Frequency,νp(GHz)
eque 4.5 oltag - δ
Fr V FIG. 3. Dispersive and strong resonant interaction. (a) Res-
990
onator reflectance |S | as a function of probe frequency ν
0. 0.1 0.2 0.3 - 1000 990 andDQDdetuningδ.11Resonance(ν =ν )occurringatδ ips
q r ±
- - indicated by arrows. (b) Extracted resonator frequency shift
Magneticflux, m 0 Voltage,VRSG mV
∆ν (dots)and(c)linewidthκ˜ (dots)vs.DQDdetuningδ in
Φ /Φ ( ) r
comparison to results of a master equation simulation (line)
FIG. 2. Characterization of the SQUID array resonator and for (g ,γb,γb)/(2π) = (155,35,63)MHz. (d) Measured res-
0 1 ϕ
double quantum dot. (a) Reflectance spectrum |S | of the onator reflectance |S | (dots) vs. probe frequency ν at res-
11 11 p
resonator as a function of probe frequency ν and applied onance (ν = ν ) displaying a strong coupling vacuum Rabi
p q r
magneticfluxΦ /Φ . (b)Hexagonalchargestabilitydiagram modesplitting. Thesolidlineistheresultofthemasterequa-
m 0
oftheDQDdetectedinthephaseφofthemicrowavetoneat tion simulation, the dashed line is a fit to a superposition of
frequency ν reflected of the resonator close to its resonance twoLorentzianlines. (e)Resonatorreflectancespectrum|S |
p 11
frequency ν as a function of the applied side gate voltages with a Lorentzian fit (dashed line) in the dispersive regime
r
V . vs. probe frequency ν .
RSG,LSG p
4
((aa)) (b) (c)
0.4 0.6 0.8 1.0 1.2 50 55 60 65 70 75
24.5
|S11| A(mV) mV)24.
(
A 23.5
6.0 6.0
3.9 4. 4.1 4.2 4.3 4.4
(d)
z) z) Frequency, s GHz
GH 5.5 GH 5.5 ν ( )
pν( sν( 2z)0.04
Frequency, 5.0 Frequency, 5.0 (e)2GHqδν(000...000123 ●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ● ● ●
P
4.5 4.5 n, 0.4
4.0 4.0 Populatio 00.2. ●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ● ● ●
6 4 2 0 2 4 6 6 4 2 0 2 4 6 0. 0.02 0.04 0.06 0.08
- - - - - -
Detuning, GHz Detuning, GHz Ps nW
δ( ) δ( ) ( )
FIG.4. DQDchargequbitspectroscopy. (a)Resonatorreflectancespectra|S |asafunctionofprobefrequencyν andDQD
11 p
detuningδ forν (Φ )≈{4.5,5.0,5.5,5.9}GHz. Redpointsindicateresonance(ν =ν )extractedformthedata. Thedashed
r m q r
line indicates the calculated transition frequency of the charge qubit. (b) Amplitude A of fixed frequency measurement tone
ν =5.947GHzreflectedfromtheresonatorvs.qubitspectroscopyfrequencyν andqubitdetuningδ. Thedashedlineindicates
p s
the expected qubit resonance frequency for 2t = 4.13GHz. (c) Qubit line shapes A(ν ) (dots) measured at δ = 0 (arrows in
s
b) for for drive strengths P = {−5,−10,−15}dBm at the generator and fits to Lorentzian lines (on a linear background)
s
extracting the linewidth δν . Probe frequency ν = 5.022GHz, probe power P = −35dBm at the generator. (d) Extracted
q p p
qubitlinewidthδν2(bluedots)vs.spectroscopydrivepowerP withlinearfit(redsolidline). (e)Saturationofqubitpopulation
q s
with spectroscopy drive power P .
s
the qubit decoherence rate γ is significantly larger than qubit is excited from its ground state |g(cid:105) to a mixture
2
the resonator decay rate κ, an observation that is also between ground and excited state |e(cid:105). This mixed state
reproduced in the theoretical analysis of the data. changes the resonance frequency ν˜ of the resonator by
r
dispersive coupling resulting in a detectable change of
Furthermore, we analyze the spectroscopic properties
the amplitude A (and also phase φ , not shown) of the
oftheDQDchargequbitintwocomplementarymeasure- r
microwave tone reflected at frequency ν (Fig. 4b). This
ments. First, we make use of the frequency tunability of p
techniquehasbeenpioneeredforsuperconductingqubits
the high impedance SQUID array resonator by applying
[11, 12] where it is widely used. Varying both the qubit
asmallmagneticfluxΦ toitsSQUIDloopsandkeeping
m detuning δ and the spectroscopy frequency ν we map
theDQDchargequbitatafixedtunnelcoupling2t. Ata s
out the spectrum of the qubit (dashed line, Fig. 4b) and
setoffrequencies{ν (Φ )},weobserveresonatorspectra
r m determine its tunnel coupling 2t=4.13GHz .
characteristic for its dispersive and resonant interaction
with the qubit (Fig. 4a). The resonances (ν = ν ) oc-
q r Usingthistechniquewearenotonlyabletoaccurately
curring at δ for the set of values {ν (Φ )} (red data
± r m determinethetransitionfrequencyν oftheDQDcharge
points) are in good agreement with the expected depen- q
qubitbutalsoitslineshape,shownforthreedrivepowers
dence of the qubit energy levels on δ, see dashed line in
P in Fig. 4c. The observed line shape depends on the
Fig. 4a. We note that at each resonance (ν = ν (Φ )) s
q r m qubit intrinsic linewidth, as set by its dephasing time
anavoidedcrossingdisplayingavacuumRabimodesplit-
T(cid:63), and on the strength of the applied microwave drive
ting is observed. 2
P whichbroadensthelineproportionaltoitsamplitude.
s
We also perform qubit spectroscopy by probing the In the limit of weak driving (P → 0), the spectroscopic
s
amplitudeandphaseoftheresonatorreflectanceatfixed linewidth δν ∼ 80MHz is determined by the dephasing
q
measurement frequency ν = 5.947GHz while applying time T(cid:63) =12.5ns of the DQD qubit as extracted from a
p 2
an additional spectroscopy microwave tone at frequency linear fit to the data in Fig. 4d. This is consistent with
ν to the resonator. When the spectroscopy tone is res- thepreviouslyextractedvaluesofγ . Increasingthedrive
s 2
onant with the qubit transition frequency (ν = ν ) the strengthP weobservethequbittransitionandthusalso
s q s
5
the resonator response to approach saturation (Fig. 4e). (2008).
The data presented in this manuscript indicates that [10] A.A.Houck,J.A.Schreier,B.R.Johnson,J.M.Chow,
thestrongcouplinglimitofasemiconductorchargequbit J. Koch, J. M. Gambetta, D. I. Schuster, L. Frunzio,
formed in a double quantum dot coupled to a microwave M.H.Devoret,S.M.Girvin, andR.J.Schoelkopf,Phys.
Rev. Lett. 101, 080502 (2008).
photon has been realized. This is achieved by the use
[11] D. I. Schuster, A. Wallraff, A. Blais, L. Frunzio, R.-S.
of a high impedance SQUID array resonator increasing
Huang, J. Majer, S. M. Girvin, and R. J. Schoelkopf,
the coupling strength by a factor of 6 relative to cou-
Phys. Rev. Lett. 94, 123602 (2005).
pling schemes using conventional 50Ω resonators. This [12] A.Wallraff,D.I.Schuster,A.Blais,L.Frunzio,J.Majer,
approachisuniversallyapplicabletoanycircuitQEDap- S.M.Girvin, andR.J.Schoelkopf,Phys.Rev.Lett.95,
plicationstrivingtomaximizethecouplingtothecharge 060501 (2005).
degree of freedom. The realization of strong coupling in [13] J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R.
Johnson,J.A.Schreier,L.Frunzio,D.I.Schuster,A.A.
thissemiconductorcircuitQEDdevicealsoenabledusto
Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M.
performspectroscopyoftheDQDqubitinthedispersive
Girvin, and R. J. Schoelkopf, Nature 449, 443 (2007).
regimetoevaluateitslineshapeindependenceonthemi-
[14] M.A.Sillanpa¨¨a,J.I.Park, andR.W.Simmonds,Nature
crowavedrivepower, indicatingthepossibilityoftempo- 449, 438 (2007).
rally resolving the charge dynamics. These results carry [15] A.Houck,D.Schuster,J.Gambetta,J.Schreier,B.John-
promisetofurtheradvancequantuminformationprocess- son,J.Chow,L.Frunzio,J.Majer,M.Devoret,S.Girvin,
ingeffortsbasedonsemiconductorchargeandspinqubits and R. Schoelkopf, Nature 449, 328 (2007).
[16] C. Eichler, C. Lang, J. M. Fink, J. Govenius, S. Filipp,
using circuit QED approaches, e.g. to perform quantum
and A. Wallraff, Phys. Rev. Lett. 109, 240501 (2012).
non-demolition (QND) readout and to realize coupling
[17] M. R. Delbecq, V. Schmitt, F. D. Parmentier, N. Roch,
between distant qubits through microwave photons.
J. J. Viennot, G. F`eve, B. Huard, C. Mora, A. Cottet,
WeacknowledgecontributionsbyMicheleCollodo,An- and T. Kontos, Phys. Rev. Lett. 107, 256804 (2011).
dreasLandig,VilleMaisiandAntonPotoˇcnik. Wethank [18] T.Frey,P.J.Leek,M.Beck,A.Blais,T.Ihn,K.Ensslin,
AlexandreBlaisforvaluablefeedbackonthemanuscript. and A. Wallraff, Phys. Rev. Lett. 108, 046807 (2012).
This work was supported by the Swiss National Science [19] K.D.Petersson,L.W.McFaul,M.D.Schroer,M.Jung,
J.M.Taylor,A.A.Houck, andJ.R.Petta,Nature490,
Foundation through the National Center of Competence
380 (2012).
in Research (NCCR) Quantum Science and Technology
[20] H. Toida, T. Nakajima, and S. Komiyama, Phys. Rev.
and by ETH Zurich.
Lett. 110, 066802 (2013).
[21] A. Wallraff, A. Stockklauser, T. Ihn, J. R. Petta, and
A. Blais, Phys. Rev. Lett. 111, 249701 (2013).
[22] J.J.Viennot,M.C.Dartiailh,A.Cottet, andT.Kontos,
Science 349, 408 (2015).
∗ These authors contributed equally to this work. [23] X. Mi, J. V. Cady, D. M. Zajac, P. W. Deelman, and
[1] S.HarocheandJ.-M.Raimond,Exploring the Quantum: J. R. Petta, Science (2016).
Atoms, Cavities, and Photons (OxfordUniversity Press, [24] L. E. Bruhat, T. Cubaynes, J. J. Viennot, M. C. Dar-
New York, USA, 2006). tiailh, M. M. Desjardins, A. Cottet, and T. Kontos,
[2] R. J. Thompson, G. Rempe, and H. J. Kimble, Phys. arXiv:1612.05214 (2016).
Rev. Lett. 68, 1132 (1992). [25] M. Devoret, S. Girvin, and R. Schoelkopf, Ann. Phys.
[3] M.Brune,F.Schmidt-Kaler,A.Maali,J.Dreyer,E.Ha- 16, 767 (2007).
gley, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. [26] M. A. Castellanos-Beltran and K. W. Lehnert, Appl.
76, 1800 (1996). Phys. Lett. 91, 083509 (2007).
[4] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. [27] N. A. Masluk, I. M. Pop, A. Kamal, Z. K. Minev, and
Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. M. H. Devoret, Phys. Rev. Lett. 109 (2012).
Schoelkopf, Nature 431, 162 (2004). [28] C. Altimiras, O. Parlavecchio, P. Joyez, D. Vion,
[5] I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C. J. P. Roche, D. Esteve, and F. Portier, Appl. Phys. Lett.
P.M.Harmans, andJ.E.Mooij,Nature431,159(2004). 103, 2601 (2013).
[6] T.Yoshie,A.Scherer,J.Hendrickson,G.Khitrova,H.M. [29] M. Go¨ppl, A. Fragner, M. Baur, R. Bianchetti, S. Fil-
Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. ipp, J. M. Fink, P. J. Leek, G. Puebla, L. Steffen, and
Deppe, Nature 432, 200 (2004). A. Wallraff, J. Appl. Phys. 104, 113904 (2008).
[7] J. P. Reithmaier, G. Sek, A. Loffler, C. Hofmann, [30] J. Basset, D.-D. Jarausch, A. Stockklauser, T. Frey,
S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Ku- C. Reichl, W. Wegscheider, T. M. Ihn, K. Ensslin, and
lakovskii, T. L. Reinecke, and A. Forchel, Nature 432, A. Wallraff, Phys. Rev. B 88, 125312 (2013).
197 (2004). [31] The conversion factor between the gate voltages and en-
[8] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and ergydetuningδ hasbeenestimatedfromtheDQDspec-
R. J. Schoelkopf, Phys. Rev. A 69, 062320 (2004). troscopy measurements reported in Fig. 4.
[9] R. J. Schoelkopf and S. M. Girvin, Nature 451, 664