Driven dynamics and rotary echo of a qubit tunably coupled to a harmonic oscillator

0.99 MB·English

by  Simon Gustavsson

Driven dynamics and rotary echo of a qubit tunably coupled to a harmonic oscillator
S. Gustavsson1, J. Bylander1, F. Yan2, P. Forn-D´ıaz1,†, V. Bolkhovsky3, D. Braje3, G. Fitch3, K.
Harrabi4,(cid:53), D. Lennon3, J. Miloshi3, P. Murphy3, R. Slattery3, S. Spector3, B. Turek3,∗, T. Weir3,
P.B. Welander3, F. Yoshihara4, D.G. Cory5,6, Y. Nakamura4,7,‡, T.P. Orlando1, and W.D. Oliver1,3
1Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2Department of Nuclear Science and Engineering, MIT, Cambridge, MA 02139, USA
3MIT Lincoln Laboratory, 244 Wood Street, Lexington, MA 02420, USA
4The Institute of Physical and Chemical Research (RIKEN), Wako, Saitama 351-0198, Japan
5Institute for Quantum Computing and Department of Chemistry, University of Waterloo, Ontario, Canada
6The Perimeter Institute for Theoretical Physics, Ontario, Canada
2 7Green Innovation Research Laboratories, NEC Corporation, Tsukuba, Ibaraki 305-8501, Japan
1 †Present address: Norman Bridge Laboratory of Physics,
0 California Institute of Technology, Pasadena, California 91125, USA
2 (cid:53)Present address: Physics Department, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia
∗Present address: The Johns Hopkins University Applied Physics Laboratory,
n
11100 Johns Hopkins Road, Laurel, MD 20723, USA
a
J ‡Present address: Dept. of Applied Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

0
Wehaveinvestigatedthedrivendynamicsofasuperconductingfluxqubitthatistunablycoupled
3
to a microwave resonator. We find that the qubit experiences an oscillating field mediated by off-

resonantdrivingoftheresonator,leadingtostrongmodificationsofthequbitRabifrequency. This
]
n opens an additional noise channel, and we find that low-frequency noise in the coupling parameter
o causes a reduction of the coherence time during driven evolution. The noise can be mitigated with
c the rotary-echo pulse sequence, which, for driven systems, is analogous to the Hahn-echo sequence.
-
r
p
u Circuit quantum electrodynamics implemented with cuit. Thediabaticstatesofthequbitcorrespondtoclock-
s superconducting artificial atoms and microwave res- wise or counterclockwise persistent currents in the qubit
.
t onatorshasemergedasaframeworkforstudyingon-chip loop[bluearrowinFig.1(a)],withenergiescontrolledby
a
light-matter interactions [1–3]. It has enabled a range the flux in the loop. The resonator mode of interest is
m
of experiments including lasing [4], the creation [5–7] theSQUIDplasmamode,depictedbythetworedarrows
- anddetection[8]ofarbitraryFockstates,andmicrowave inFig.1(a). TheSQUIDservesdualpurposes: itactsas
d
n photon-correlationmeasurements[9,10]. Microwaveres- a tunable coupler between the qubit and the resonator,
o onators also provide a means to couple distant qubits and it is also used as a sensitive magnetometer for qubit
c [11, 12] and, in this role, have been used to implement read-out [22].
[
quantumalgorithmsinsuperconductingcircuits[13]and We have investigated two devices with similar lay-
1 to develop quantum computer architectures [14]. How- outs but slightly different parameters, both made of alu-
v ever,thecouplingofaqubittoaresonatoralsoinfluences minum. Device A was designed and fabricated at MIT
1 the qubit coherence, for example by modifying its relax- LincolnLaboratoryanddeviceBwasdesignedandfabri-
4 ation rate through the Purcell effect [15]. catedatNEC.Figure1(b)showsaspectroscopymeasure-
3
In this work, we study an additional consequence of ment of device A versus applied flux, with the qubit flux
6
. the resonator by investigating the driven dynamics and detuningΦqb definedasΦqb =Φ+Φ√0/2andΦ0 =h/2e.
1
thedephasingofafluxqubit[16]thatistunablycoupled The qubit frequency follows f = ∆2+ε2, where the
0 qb
to a harmonic oscillator [17–20]. We find that the res- tunnel coupling ∆=2.6GHz is fixed by fabrication and
2
1 onator mediates an indirect driving field that interferes the energy detuning ε = 2IPΦqb/h is controlled by the
: with the direct drive set by the qubit-antenna coupling, applied flux Φ (IP is the persistent current in the qubit
v
thereby modifying both the amplitude and the phase of loop). Theresonatorfrequencyf isaround2.3GHzand
i r
X the net driving field. The tunable coupling allows the depends only weakly on Φ and I . In addition, there
qb b
r indirect driving to be switched off, but it also opens an are features visible at frequencies corresponding to the
a additional channel for noise to couple into the system. sum of the qubit and resonator frequencies, illustrating
Fluctuations in the coupling parameter translate into ef- the coherent coupling between the two systems [3, 23].
fective driving-field amplitude noise, which reduces the The system is described by the Jaynes-Cummings
qubit coherence during driven evolution. We show that Hamiltonian [1, 24, 25]
thequbitdephasingduetoamplitudenoise(whetherdue
to tunable coupling or otherwise) can be mitigated by a H/h=−∆σ − εσ +f (cid:18)a†a+ 1(cid:19)+ g1 (cid:0)a+a†(cid:1)σ .
rotary echo [21], a pulse sequence originally developed 2 x 2 z r 2 2 z
for nuclear magnetic resonance.
(1)
Thedevice, showninFig.1(a), consistsofafluxqubit Here, aanda† arethecreation/annihilationoperatorsof
and a SQUID embedded in a two-mode LC resonant cir- the resonator field and g is the dipole coupling between
1
2
(a) Ib (b)6 fqb+fr (Hz]a)100 DToattaal drive εtmowtal (by) 0.6
L SCIIQrmwUIDL C L Frequency [GHz]4 fqb (fqb+fr)/2 bi frequency f [MRabi24680000 DDririveec tv diari vrees εodmnirweacttor Switching probabilit0000....3636 IIbb== -7200 n nAA
2 fr Ra 0 Ib* 0.3 Ib= 170 nA
Qubit f /2 -200 0 200 0 50 100
qb
I [nA] Time [ns]
Imw -5 0 5 (c) b (d)
antenna 40
Flux detuning Φqb [mΦ0] z] 200
H
(c) (d) 2 M 30 A]
ΦΦ [m]qb000..23 DDeevviiccee AB I∂ [MHz/nA]b 01 DDeevviiccee ABg=0 -24200001Coupling g [M mwε Drive [direct12000 D3rive freq4uency [5GHz] 6mwI [nr1000 D3rive freq4uency [5GHz] 6
0.1 ε/-1 1 H
∂ at I=I* z
b b -40]
0 -2 FIG. 2. (a) Rabi frequency of qubit A, measured vs Ib at
-400 -200 0 200 400 -400 -200 0 200 400 Φ = 0. The driving field seen by the qubit contains two
qb
Bias current I [nA] Bias current I [nA]
b b components: one is due to direct coupling to the antenna,
the other is due to the coupling mediated by the resonator.
FIG. 1. (a) Circuit diagram of the qubit and the oscillator. (b) Rabi traces for a few of the data points in panel (a).
The qubit state is encoded in currents circulating clockwise Themicrowavesintheantennahavethesameamplitudeand
or counterclockwise in the qubit loop (blue arrow), while the frequency for all traces. (c) Direct coupling between the an-
mode of the harmonic oscillator is shown by the red arrows. tenna and the qubit, extracted from measurements similar
(b) Spectrum for device A, showing the qubit and the har- to the one shown in panel (a). The coupling depends only
monic oscillator. In addition, the two-photon qubit (fqb/2) weaklyonfrequency. (d)Microwavecurrentintheresonator,
andthequbit+resonator(fqb+fr)transitionsarevisible. (c) inducedbyafixedmicrowaveamplitudeintheantenna. The
Flux induced in the qubit loop by the dc bias current Ib. black line is a fit to the square root of a Lorentzian, describ-
The black lines are parabolic fits. (d) First-order coupling ing the oscillation amplitude of a harmonic oscillator with
between the qubit and the ground state of the harmonic os- f =2.3GHz and Q=100.
r
cillator,showingthatthecouplingistunablebyadjustingI .
b
The coupling is zero at I =I∗, which is slightly offset from
b b
I =0 due to fabricated junction asymmetry. The derivative
b
∂ε/∂I is calculated from the curves in panel (c). The qubit display similar behavior, with the bias current gener-
b
parameters are: IP = 175nA for device A and IP = 180nA ating a parabolic shift in Φqb [28, 29]. Since Φqb con-
for device B. The resonators have quality factors Q ≈ 100. trols the qubit energy detuning ε, the first-order qubit-
The right-hand axis is calculated using f = 2.2GHz and
r resonator coupling strength is determined by the deriva-
C =2C =14pF for both samples.
eff tive ∂ε/∂I = (2I /h)(∂Φ /∂I ). The bare coupling
b P qb b
coefficient between qubit and resonator is then g =
1
(cid:112)
(∂ε/∂I )δI , where δI = 2π2hf3C is the rms am-
b 0 0 r eff
the qubit and the resonator. In this work, we do not plitude of the vacuum fluctuations and C = 2C is the
eff
consider higher-order coupling parameters [24–26]. totalcapacitanceoftheresonantcircuit[25]. Thederiva-
The coupling g1 is mediated by the SQUID. When tive ∂ε/∂Ib and the coupling g1 are plotted in Fig. 1(d),
the two SQUID junctions are symmetric, the current of determined at dc from the measured relation between
the resonator mode splits equally into the two SQUID Φqb andIb showninFig.1(c). Notethatinbothdevices
arms, and therefore no net flux is induced into the qubit g1 can be tuned over a range of a few tens of MHz, and
loop. Thequbitisthuseffectivelydecoupledfromtheres- that the coupling is turned off at Ib = Ib∗. The device
onator. However, in the presence of a magnetic field, ap- parameters are given in the figure caption.
plying a dc bias current Ib creates an asymmetric phase Having determined the coupling coefficients, we turn
dropoverthetwoSQUIDjunctions. Thiscausestheres- toanalyzinghowthepresenceoftheresonatorinfluences
onator current to be slightly larger in one of the arms, the qubit’s driven dynamics. Figure 2(a) shows the ex-
which will produce a flux in the qubit loop. The cou- tracted Rabi frequency f of qubit A as a function
Rabi
pling to the resonator can thus be controlled in situ by of I , measured at f = 2.6GHz. We find that f
b qb Rabi
changing Ib [27]. changes by a factor of five over the range of the mea-
Figure 1(c) shows the flux induced into the qubit loop surement, which is surprising since both the amplitude
as a function of the dc bias current I for the two de- and the frequency of the microwave current Imw in
b antenna
vices. Given the similarity in the design, both samples theantennaarekeptconstant. Thedatapointswereob-
3
shseotfeaihanadwicoTrnehrwmoreeendsxdoteoanahbxinmtineypcaaplnfiiFoapnlctsieiodentcgsiiii.nnarlotletg2nafhc,(dteRRtboto)araadr..tbebrmsiiiAtWvuoahtsilrskenteacsesgiedcclseeoleaoaunsgfrftreefiieFtonfonuhirnetglFerhsld.ayiatiqgc2fftou.yc(etadb1arhpl()eieiteob,bcnia)rtqmnw,yautvetaienbaedt(ldginεiutzaese=ettweirshendoad0euos)flftsfl.booduIuyirxbdcxivsotad,ehnarenaeet-- (Frequency [GHz]a) 468 fqb/2fqb fqb-ffrr (bbi frequency f [MHz]Rabi)15000 IFIaamminntwwtt,ee nnEnnaaq==s 18. 4.(12.2 ,μ 3μA)A
2 a
tuning the qubit frequency (fqb = 2.6GHz) is relatively R 0
-5 0 5 -500 0 500
close to the resonator frequency (f = 2.3GHz). We
r Flux detuning Φ [mΦ] I [nA]
therefore expect the microwave drive in the antenna to qb 0 b
off-resonantlyinduceamicrowavecurrentImw intheres-
r FIG. 3. (a) Spectrum of device B. The spectral line at
onator,whichisproportionaltothesquarerootoftheav-
(cid:112) 2GHz is the resonator, whereas the qubit tunnel coupling is
erage photon population, (cid:104)n(cid:105). By setting I (cid:54)=I∗, the
b b ∆=5.4GHz. (b)RabifrequencyvsbiascurrentIb,measured
coupling between the resonator and the qubit is turned at f =5.4GHz and Φ =0 and for two different microwave
qb
on, and the resonator current Imw will start driving the drive amplitudes Imw . Similar to device A, the Rabi fre-
r antenna
qubit. To describe this indirect driving, we treat the quencydependsstronglyonI ,andscaleslinearlywithdrive
b
resonator classically and write the qubit Hamiltonian in amplitude. The black lines are fits to Eqs. (3,4), using the
Eq. (1) as same coupling parameters for both sets of data. Note that
the range of I in Fig. 3(b) is several times larger than in
b
H /h=−1/2(cid:2)∆σ +(cid:2)εdc+εmwcos(2πft)(cid:3)σ (cid:3). (2) Fig. 2(a).
qb x z
Here, the drive amplitude εmw experienced by the qubit
becomes a combination of the drive εmw , due to di- Fig. 2(d) is the frequency response of a harmonic oscil-
direct
rect coupling between antenna and qubit, and the drive lator with f = 2.3GHz and Q = 100, with amplitude
r
(∂ε/∂I )Imw mediated by the resonator. We get: normalized to match the data.
b r
To further investigate how the presence of the res-
(cid:115)
(cid:20) ∂ε (cid:21)2 (cid:20) ∂ε (cid:21)2 onator affects the qubit dynamics at large detunings, we
εmw= εmw +cosθ Imw + sinθ Imw , (3)
direct ∂I r ∂I r performed measurements on device B. Figure 3(a) shows
b b
a spectrum of that device, where the qubit and the res-
onatormode(f =2GHz)areclearlyvisible. Thisdevice
where θ ≡ θ −θ is the phase difference between the r
d r
hasalargertunnelcoupling(∆=5.4GHz),whichallows
direct drive and the drive mediated by the resonator.
TheRabifrequencyduetothedriveεmw dependsonthe us to operate the qubit at large frequency detuning from
the resonator while still staying at ε = 0, where the
qubit’s quantization axis, which changes with the static dc
qubit, to first order, is insensitive to flux noise [28, 29].
energy detuning ε :
dc
Thequbit-resonatordetuningcorrespondstoseveralhun-
εmw ∆ dred linewidths of the resonator, which is the regime of
f = . (4)
Rabi 2 (cid:112)ε2 +∆2 most interest for quantum information processing [12].
dc
Figure 3(b) shows the Rabi frequency vs bias cur-
Fitting the data in Fig. 2(a) to Eqs. (3,4) allows us to rent Ib of device B, measured at f = 5.4GHz and
extract the parameters εmw , Imw and θ. The different for two different values of the microwave drive current
direct r
drive contributions are plotted together with the data in Imw . Similarly to Fig. 2(a), the Rabi frequency
antenna
Fig.2(a). ThedirectdriveisindependentofIb,whilethe clearly changes with Ib, but the dependence is weaker
drive Imw(∂ε/∂I ) mediated by the resonator increases than in Fig. 2(a) because of the larger frequency de-
r b
linearlywith|Ib|,whichoriginatesfromthelineardepen- tuning. Note that fRabi scales linearly with Iamnwtenna
dence of g1 shown in Fig. 1(d). The minimum in Rabi for all values of Ib. By fitting the data to Eqs. (3,4),
frequency occurs at a value of I slightly shifted from we find εmw /Imw = 6.4MHz/µA, Imw/Imw =
b direct antenna r antenna
the point I∗ where g = 0. This offset appears because 2.4nA/µA and θ = −155◦. The large phase differ-
b 1
of the phase difference θ between the two drive compo- ence θ for device B causes the minimum in fRabi to shift
nents. The fit gives θ =−75◦, which is consistent with a away from the point close to Ib = 0 where the coupling
resonator driven above its resonance frequency. g1 = 0 [see Fig. 1(d)]. We attribute the large phase
Figures 2(c) and 2(d) show how the two drive com- shift to influences from a second resonant mode, which
ponents depend on microwave frequency, measured by is formed by the two L and the two C in the outer loop
changing the static flux detuning Φ to increase the of Fig. 1(a) [18, 30]. For sample B, this mode resonates
qb
qubit frequency [see Fig. 1(b)]. The direct drive only de- around 5GHz.
pendsweaklyonfrequency(duetocablelosses),whereas The results of Figs. 2 and 3 show that the microwave
thedrivemediatedbytheresonatordropssharplyasthe signal mediated by the resonator plays a significant role
qubit-resonator detuning increases. The black curve in when driving the qubit, appearing already at moderate
4
qubit-resonatorcouplingg andpersistingevenwhenthe (a) Rotary echo (b)15
1
thFweigroe. s1ay(lsdlto)ew]m,ssbtuahtreeitcfoacuropmdlieenstguwnteiotdh.baeTdthurearnwdebedsaicgoknff: i[tnghv1ees=ptiagra0atmeind- amplitude00..21 1R1//aeeb ddieeccaayy:: 93..37 μμss μme T [s]e10 4/3 T1 Rotary echo
eptreorviudseesdatowacyonftorrollotwh-efrceoquupenlicnygn(Iobiseintoouerntseertutph)e asylsso- Osc. cay ti 5 Rabi
de Fit, Eq. (6)
tem. Consider the relation between the fRabi and Ib in 00 5 10 15 1/e 00 50 100
Fig.3(b): fluctuationsδI nearI =0willcausefluctua-
b b Time t [μs] Rabi frequency f [MHz]
tionsintheamplitudeofthedrivefieldseenbythequbit, (c) p (d) Rabi
which will lead to dephasing during driven evolution. pe t tp
o p
To quantify the dephasing, we linearize the relation el0 0
between the Rabi frequency and Ib close to Ib = 0 as Env Rabi Time Rotary echo Time
f = f [1 + rδI ], where f = f (I = 0) and r =
0 b 0 Rabi b
(∂f /∂I )/f = −1.28(µA)−1 is given by Eqs (3,4) FIG. 4. (a) Decay envelopes of the Rabi and rotary-echo
Rabi b 0 sequences for device B, measured with f =65MHz. The
or from Fig. 3(b). We model the fluctuations δI as nor- Rabi
b solid lines are fits to Eq. (6). (b) Decay times for Rabi and
mallydistributed,withstandarddeviationσ . Assuming
I rotary echo, extracted from fits similar to the ones shown in
thenoisetobequasi-static,wherethevalueofδI iscon-
b panel(a). Thedashedlineshowstheupperlimitsetbyqubit
stantduringasingletrialbutdiffersfromruntorun[31], energyrelaxation. Thedottedlinemarksthepositionforthe
we find that the Rabi oscillations decay as decayenvelopeshowninpanel(a). (c,d)Schematicdiagrams
describingthetwopulsesequencesin(a)and(b). Forrotary
(cid:90) e−δIb2/(2σI2) cos(2πf [1+rδI ]t) dδI = echo, the phase of the microwaves is rotated by 180◦ during
(cid:112) 0 b b
2πσ2 the second half of the sequence.
I
=e−2(πf0rσIt)2cos(2πf t). (5)
0
In addition, the qubit energy relaxation time T1 gives Hahn-echo experiment [33]. Similarly, the fluctuations
anexponentialcontributiontotheRabidecay,withtime in drive field that cause decay of the Rabi oscillations
constant 4T1/3 given by the Bloch equations. The Rabi in Fig. 4(b) can be mitigated with the rotary-echo pulse
decay also depends on the flux noise at the Rabi fre- sequence[21],depictedinFig.4(c),whichfordrivensys-
quency, but this contribution can be disregarded when temsisanalogoustotheHahn-echosequence. Byshifting
operating the qubit at εdc = 0, where the qubit is in- thephaseofthedriveby180◦afteratimetp/2,anyaddi-
sensitive to first-order flux noise [32]. The total decay tional rotations, acquired due to slow fluctuations in the
envelope f(t) of the Rabi oscillations becomes driveamplitudeduringthefirsthalfofthesequence,will
√ cancel out during the reversed rotations in the second
f(t)=e−4T31te−(t/Tϕ)2, with Tϕ =1/( 2πf0rσI). (6) half of the sequence.
Note that the Gaussian decay constant T due to the The blue squares in Fig. 4(a) shows the decay of the
ϕ
effective amplitude fluctuations is inversely proportional rotary-echosequence,measuredforfRabi =65MHz. The
to f , the average Rabi frequency. This is a consequence rotary-echo data shows a clear improvement compared
0
ofhavingnoiseinthecouplingbetweenthequbitandthe to the Rabi decay for the same parameters [red circles in
antenna; the effective amplitude fluctuations seen by the Fig.4(a)]. Wefittherotary-echodatatoEq.(6)andplot
qubit will scale with the drive amplitude. the extracted decay times together with the results from
The red circles in Fig. 4(a) show the envelope of Rabi Rabi measurements in Fig. 4(b). The rotary-echo signal
oscillations measured for f = 65MHz, together with outperforms the Rabi decay over the full range of Rabi
Rabi
afittoEq.(6). ThequbitenergyrelaxationT =11.7µs frequencies, and reaches the upper limit set by qubit re-
1
is known from separate experiments [32], leaving Tϕ = laxation(4T1/3)atlowfrequencies. Forintermediatefre-
4.3µs as a fitting parameter. In Fig. 4(b), we plot the quencies,therotary-echodecaytimesareslightlyshorter
Rabi decay time versus fRabi, extracted from envelopes than 4T1/3; we attribute the reduced coherence times to
similartoFig.4(a). Tocaptureboththeexponentialand fluctuations in Ib that occur on time scales comparable
the Gaussian decay, we plot the time T for the envelope to the length of the pulse sequence. Noise at frequen-
e
to decrease by a factor 1/e. For the lowest Rabi fre- cies around 1/tp will not be refocused by the reversed
quency, the decay time is within 25% of the upper limit drive pulse, since the rotary-echo sequence has similar
set by qubit relaxation, but it decreases with f , as filteringpropertiesastheHahn-echo[21]. Atthehighest
Rabi
expected from Eq. (6). The black solid line shows a fit driveamplitudes(fRabi >100MHz),weobserveastrong
to Eq. (6), giving a value of σ =0.8nA for the noise in increase in decoherence, probably due to heating. The
I
I . The effective fluctuations in the drive amplitude are indirect driving can also be reduced by driving the qubit
b
rσ =0.06%. We cannot rule out that part of that noise with two antennas with different amplitudes and phases
I
may be caused by instrument imperfections. [34], but it requires a more complicated setup.
Dephasing due to low-frequency fluctuations of the To conclude, we have investigated interference effects
qubit frequency is routinely reduced by performing a occurring when driving a qubit that is tunably coupled
5
to a harmonic oscillator. Although the addition of a properdecouplingprotocols. Inanalogywithmulti-pulse
coupling control parameter opens up an extra channel Hahn-echo experiments, we expect the incorporation of
for dephasing, we show that its influence is reversible additional rotary echos to further improve the coherence
with dynamical decoupling techniques. The results are times [32].
relevant for any type of qubit that is tunably coupled We thank M. Gouker, X. Jin, and M. Neeley for help-
to a resonator, and they show that despite engineering ful discussions and technical support. K.H. gratefully
limitations, imperfections can be reversed by applying acknowledgesthesupportofRIKENandKFUPM(DSR
FT100009).
[1] A. Blais, R. Huang, A. Wallraff, S. Girvin, and [17] M. S. Allman, F. Altomare, J. D. Whittaker, K. Cicak,
R. Schoelkopf, Phys. Rev. A 69, 062320 (2004). D. Li, A. Sirois, J. Strong, J. D. Teufel, and R. W. Sim-
[2] A.Wallraff,D.Schuster,A.Blais,L.Frunzio,R.Huang, monds, Phys. Rev. Lett. 104, 177004 (2010).
J.Majer,S.Kumar,S.Girvin,andR.Schoelkopf,Nature [18] A.Fedorov,A.Feofanov,P.Macha,P.Forn-D´ıaz,C.Har-
431, 162 (2004). mans,andJ.Mooij,Phys.Rev.Lett.105,60503(2010).
[3] I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C. J. [19] S. Srinivasan, A. Hoffman, J. Gambetta, and A. Houck,
P.M.Harmans,andJ.E.Mooij,Nature431,159(2004). Phys. Rev. Lett. 106, 083601 (2011).
[4] O.Astafiev,K.Inomata,A.O.Niskanen,T.Yamamoto, [20] J. Gambetta, A. Houck, and A. Blais, Phys. Rev. Lett.
Y.A.Pashkin,Y.Nakamura,andJ.S.Tsai,Nature449, 106, 030502 (2011).
588 (2007). [21] I. Solomon, Phys. Rev. Lett. 2, 301 (1959).
[5] A. A. Houck, D. I. Schuster, J. M. Gambetta, J. A. [22] I. Chiorescu, Y. Nakamura, C. J. P. M. Harmans, and
Schreier, B.R.Johnson, J.M.Chow, L.Frunzio, J.Ma- J. E. Mooij, Science 299, 1869 (2003).
jer,M.H.Devoret,S.M.Girvin,etal.,Nature449,328 [23] M. Goorden, M. Thorwart, and M. Grifoni, Phys. Rev.
(2007). Lett. 93, 267005 (2004).
[6] M. Hofheinz, E. M. Weig, M. Ansmann, R. C. Bialczak, [24] P. Bertet, I. Chiorescu, C. J. P. M. Harmans, and J. E.
E. Lucero, M. Neeley, A. D. O’Connell, H. Wang, J. M. Mooij, arXiv (2005), cond-mat/0507290v1.
Martinis, and A. N. Cleland, Nature 454, 310 (2008). [25] P. Bertet, C. Harmans, and J. Mooij, Phys. Rev. B 73,
[7] M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, 064512 (2006).
E.Lucero,M.Neeley,A.D.O’Connell,D.Sank,J.Wen- [26] A. Hoffman, S. Srinivasan, S. Schmidt, L. Spietz, J. Au-
ner, J. M. Martinis, et al., Nature 459, 546 (2009). mentado,H.Tu¨reci,andA.Houck,Phys.Rev.Lett.107,
[8] D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, 053602 (2011).
J.M.Gambetta,A.Blais,L.Frunzio,J.Majer,B.John- [27] T.Hime,P.A.Reichardt,B.L.T.Plourde,T.L.Robert-
son, M. H. Devoret, et al., Nature 445, 515 (2007). son,C.-E.Wu,A.V.Ustinov,andJ.Clarke,Science314,
[9] D. Bozyigit, C. Lang, L. Steffen, J. M. Fink, C. Eich- 1427 (2006).
ler, M. Baur, R. Bianchetti, P. J. Leek, S. Filipp, M. P. [28] P. Bertet, I. Chiorescu, G. Burkard, K. Semba, C. J.
da Silva, et al., Nat. Phys. 5, 1 (2010). P.M.Harmans,D.P.DiVincenzo,andJ.E.Mooij,Phys.
[10] F. Mallet, M. Castellanos-Beltran, H. Ku, S. Glancy, Rev. Lett. 95, 257002 (2005).
E. Knill, K. Irwin, G. Hilton, L. Vale, and K. Lehnert, [29] F.Yoshihara,K.Harrabi,A.O.Niskanen,Y.Nakamura,
Phys. Rev. Lett. 106, 220502 (2011). and J. S. Tsai, Phys. Rev. Lett. 97, 167001 (2006).
[11] M.A.Sillanp¨a¨a,J.I.Park,andR.W.Simmonds,Nature [30] J. Johansson, S. Saito, T. Meno, H. Nakano, M. Ueda,
449, 438 (2007). K. Semba, and H. Takayanagi, Phys. Rev. Lett. 96,
[12] J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. 127006 (2006).
Johnson,J.A.Schreier,L.Frunzio,D.I.Schuster,A.A. [31] G. Ithier, E. Collin, P. Joyez, P. Meeson, D. Vion,
Houck, A. Wallraff, et al., Nature 449, 443 (2007). D. Esteve, F. Chiarello, A. Shnirman, Y. Makhlin, and
[13] L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, J. Schriefl, Phys. Rev. B 72, 134519 (2005).
B.R.Johnson,D.I.Schuster,J.Majer,A.Blais,L.Frun- [32] J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara,
zio, S. M. Girvin, et al., Nature 460, 240 (2009). K. Harrabi, G. Fitch, D. G. Cory, Y. Nakamura, J.-S.
[14] M. Mariantoni, H. Wang, T. Yamamoto, M. Neeley, Tsai, and W. D. Oliver, Nat. Phys. 7, 565 (2011).
R.C.Bialczak, Y.Chen,M.Lenander,E.Lucero,A.D. [33] E. Hahn, Physical Review 80, 580 (1950).
O’Connell, D. Sank, et al., Science (2011). [34] P.C.deGroot,J.Lisenfeld,R.N.Schouten,S.Ashhab,
[15] A.A.Houck,J.A.Schreier,B.R.Johnson,J.M.Chow, A.Lupa¸scu,C.J.P.M.Harmans,andJ.E.Mooij,Nat.
J. Koch, J. M. Gambetta, D. I. Schuster, L. Frunzio, Phys. 6, 763 (2010).
M. H. Devoret, S. M. Girvin, et al., Phys. Rev. Lett. [35] J. M. Chow, J. M. Gambetta, L. Tornberg, J. Koch,
101, 080502 (2008). L. S. Bishop, A. A. Houck, B. R. Johnson, L. Frunzio,
[16] J. Mooij, T. Orlando, L. Levitov, L. Tian, van der S.M.Girvin,andR.J.Schoelkopf,Phys.Rev.Lett.102,
Wal CH, and S. Lloyd, Science 285, 1036 (1999). 090502 (2009).