Coherent frequency conversion in a superconducting artificial atom with two internal degrees of freedom

1.4 MB·English

by  Florent Lecocq

Coherent frequency conversion in a superconducting artificial atom with two internal
degrees of freedom
F. Lecocq, I. M. Pop, I. Matei, E. Dumur, A. K. Feofanov, C. Naud, W. Guichard, O. Buisson
Institut N´eel, C.N.R.S.- Universit´e Joseph Fourier, BP 166, 38042 Grenoble-cedex 9, France
(Dated: January 20, 2012)
By adding a large inductance in a dc-SQUID phase qubit loop, one decouples the junctions’
dynamics and creates a superconducting artificial atom with two internal degrees of freedom. In
2 addition to the usual symmetric plasma mode (s-mode) which gives rise to the phase qubit, an
1 anti-symmetric mode (a-mode) appears. These two modes can be described by two anharmonic
0 oscillators with eigenstates |nsiand|naiforthesanda-mode,respectively. Weshowthatastrong
2 nonlinear coupling between the modes leads to a large energy splitting between states |0s,1ai and
n |2s,0ai. Finally, coherent frequency conversion is observed via free oscillations between the states
a |0s,1ai and |2s,0ai.
J
PACSnumbers:
9
1

In atomic physics the presence of multiple degrees of ducting loop of large inductance L interrupted by two
]
l freedom (DoF) constitutes a precious resource for the identicalJosephsonjunctionswithcriticalcurrentIc and
l
a development of quantum mechanics experiments. In capacitance C, operated at zero current bias (see Fig.1).
h trapped ions or nitrogen-vacancy centers, the V-shaped Aswewillseeinthefollowing,thepresenceofalargeloop
-
s energyspectrumenablesveryhighfidelityreadoutofthe inductance modifies dractically the quantum dynamics
e statesencodedinthefirsttwolevelsbyusingthefluores- of this system. The two phase differences φ1 and φ2
m
cencepropertiesofthe transitionto the thirdlevel[1,2]. across the two junctions correspond to the two degrees
. In quantum optics, the multiple DoF of the optically of freedom of this circuit, which lead to two oscillating
t
a active crystals have lead to many quantum effects such modes: the symmetric (s-) and the anti-symmetric (a-)
m
as CoherentPopulationTrapping (CPT) and the associ- plasma modes [23]. The s-mode correspondsto the well-
- ated effect of ElectromagneticallyInduced Transparency known in-phase plasma oscillation of the two junctions
d
n (EIT)[3],spontaneousemissioncancelationviaquantum with the average phase xs = (φ1 +φ2)/2, oscillating at
o interference[4,5]orgenerationofentangledphotonpairs acharacteristicfrequency givenbythe plasmafrequency
c [6]. In the field of superconducting qubits, experimental of the dc-SQUID, ωs. The a-mode is an opposite-phase
[ p
effortshavemainlyfocusedontwo-levelsystems[7,8]and plasma mode related to oscillations of the phase differ-

1 multilevel systems [9–13] with a single DoF. Supercon- ence xa = (φ1 − φ2)/2, producing circulating current
v ducting artificial atoms currently need additionnal DoF oscillations at frequency ωa. In previous experiments
p
5
inordertorealizeΛ,V,N ordiamond-shapedenergylev- [9, 11, 22] , the loop inductance L was small compared
8
0 els and therefore to perform new quantum experiments to the Josephson inductance LJ = Φ0/(2πIc). There-
4 [14, 15]. Only recently a superconducting device with fore the two junctions were strongly coupled and the
. V-shapeenergyspectrumwasexperimentallyconsidered dynamics of the phase difference x was neglected and
1 a
[16]andathreeDoFsuperconductingringwasdeveloped fixed by the applied flux. The quantum behavior of
0
2 for parametric amplification [17]. the circuit was described by the s-mode only, showing
1 aone-dimensionalmotionoftheaveragephasex . Here-
In this letter we present an artificial atom with two s
v: internal DoF which constitutes a basic block for the re- after we will consider a circuit with a large inductance
Xi alization of V- or Diamond-shaped energy levels. In this (L ≥ LJ) that decouples the phase dynamics of the two
junctions. This large inductance lowers the frequency of
systemwebenefitfromthenaturalnonlinearcouplingbe-
r thea-modeandthedynamicsofthesystembecomesfully
a tween the two DoF to observe a coherent frequency con-
two-dimensional. The a-modewaspreviouslyintroduced
versionprocessinthetimedomain. Contrarytoprevious
to discuss the thermal and quantum escape of a current-
frequency conversion proposals and implementations in
biaseddc-SQUID[24,25]butitsdynamicswasneverob-
solid state devices, we observe this process in the strong
served. Wepresentmeasurementsofthe fullspectrumof
coupling limit, where multiples oscillations can be seen
thisartificialatom,independentcoherentcontrolofboth
before losing coherence [18, 19], and without any exter-
modes and finally we exploit the strong nonlinear cou-
nal coupling device or additionnal source of power [20].
plingbetweenthetwoDoFtoobserveatimeresolvedup
In addition this system could be used for triggered and
anddownfrequencyconversionofthesystemexcitations.
high-efficiency generation of entangled pairs of photons,
a key component for quantum information [21].
Theelectronicpropertiesofasymmetricandinductive
Thecircuitisacamelbackphasequbit[22]withalarge dc-SQUIDatzero-currentbiascanbe describedbyafic-
loop inductance, i.e a dc-SQUID build by a supercon- titious particle of mass m = 2C(Φ0/2π)2 moving inside
2
tively the symmetric and antisymmetric plasma modes
(a) 1µm
[26, 27]. C describes the coupling between the two
If(t) oscillators. s,aH = ~ωα (pˆ2 +xˆ2)/2−σ xˆ3 +δ xˆ4 ,
α p α α α α α α
where α = s,a with (ωs(cid:2))2 = (2E /m)cosxmin an(cid:3)d
p J a
(ωa)2 =(ωs)2+(4E /m)(L /L). The operators xˆ and
p p J J α
pˆ arethe reduced positionandmomentum operatorsin
α
both directions. The coupling term has a very simple
expression at zero current bias:
Cs,a =~g21xˆ2sxˆa+~g22xˆ2sxˆ2a (2)
with~g21 =−EJ(~/m)23(ωps ωpa)−1sinxmain and~g22 =
−E (~/m)2(ωsωa)−1cosxmipn/2. In the following, we
3µm I J p p a
b willdefine|n ,n i≡|n i|n iastheeigenstatesofuncou-
s a s a
(b) (d) pledhamiltonianHs+Ha,where|nαiindexestheenergy
levels of each mode. The potentials associated with the
4,0 s-modeanda-modearedepictedinthefigureFig.1band
ergy 2s0 1s3s s a 2s,1a 0s,2a Fig.1c for a workingpoint (Ib =0,Φb =0.37Φ0). At the
n s 3,0 same bias point the complete spectrum of the system is
E s a 1,1
Xs, symmetric mode s a presented in Fig.1d.
(c) de 2s,0a The complete aluminum device is fabricated using an
o 0,1
m s a angle evaporation technique without suspended bridges
gy 2a 1 3a metric 1s,0a qubit [w2e8r]e, acnonddiutctisedprineseantdeidlutiinonFirge.f1raig.erTathoermatea4s0urmemKenutss-
ner 0a a ym 0s,0a ingastandardexperimentalconfiguration,previouslyde-
E S
Xa, anti-symmetric mode Anti-symmetric mode scribed in Ref.[22].
The readout of the circuit is performed using switch-
FIG. 1: Description of the device. (a) A micrograph of ing current techniques. For spectroscopy measurements
the aluminium circuit. The two small squares are the two we apply a microwave pulse field, through the current
Josephson junctions (enlarged in the top right inset, 10µm2 bias line (see Fig 1a), followed by the readout nanosec-
area, Ic =713 nA and C =510 fF) decoupled by a large in- ondfluxpulsethatproducesaselectiveescapedepending
ductiveloop(L=629pH).ThewidthofthetwoSQUIDarms
on the quantum state of the circuit [9, 22]. The energy
were adjusted to reduce the inductance asymmetry to about
10%. Very narrow current bias lines, with a large 15 nH- spectrum versus current bias at Φb = 0.48Φ0 and ver-
inductance, isolate the quantum circuit from the dissipative sus flux bias at Ib = 0 are plotted in Fig.2a and Fig.2b
environment at high frequencies . The symmetric and anti- respectively. In the following we will denote να as the
nm
symmetricoscillationmodesareillustratedbyblueandredar- transition frequency between the states |n i and |m i,
α α
rows,respectively. (b)and(c)Potentialsofthesanda-mode with the other mode in the ground state. The first tran-
respectively,forthebiasworkingpoint(Ib=0,Φb =0.37Φ0). sition frequency in Fig.2 is the one of the camelback
(d) Schematic energy level diagram indexed by the quan-
phase qubit, νs . With a maximum frequency at zero-
tumexcitationnumberofthetwomodes|ns,naiatthesame 01
current bias, the system is at an optimal working point
working point. Climbing each vertical ladder one increases
the excitation number of the s-mode, keeping the excitation with respect to current fluctuations [22]. At higher fre-
number of the a-mode constant. The two first levels, |0s,0ai quency, the second transition of the s-mode is observed
and |1s,0ai,realize a camelback phase qubit. with ν0s2 ≈ 2ν0s1. In the flux biased spectrum the third
transition νs is also visible. An additional transition
03
is observed at about 14.6 GHz, with a very weak cur-
a two-dimensional potential :
rent dependence (Fig.2a) but a finite flux dependence
(Fig.2b). It corresponds to the first transition of the a-
L πΦ
U(xs,xa)=2EJ(cid:20)−cosxscosxa+ LJ(xa− Φ0b)2(cid:21) (1) mode, ν0a1. The s-modetransitionfrequenciesdropwhen
Φb/Φ0 approaches 0.7 which is consistent with the crit-
where EJ =Φ0Ic/(2π). Hereafter we will consider the ical flux Φc/Φ0 = 1/2 + L/(2πLJ) = 0.717 for which
particle trapped in a local minimum (xms in,xmain) veri- ωps → 0 and xmain → π/2. On the contrary ωpa remains
fying xms in = 0 and xmain+(L/2LJ)sinxmain = πΦb/Φ0. finitewhenΦb →Φc withωpa → (4EJ/m)(LJ/L). One
Byexpandingthepotentialatthisminimumuptofourth alsoobservesalargelevelanti-crpossingofabout700MHz
order, its quantum dynamics is given by the Hamil- betweenthetwotransitionsνs andνa . Additionallyno
02 01
tonian H = H + H + C where H and H are level anti-crossing is measurable between νs and νa .
s a s,a s a 03 01
anharmonic oscillator Hamiltonians describing respec- TheexperimentalenergyspectruminFig.2canbeper-
3
18(a) νs (b) νs02 νs03 froersptohneditnrgandseitpiohnassinνg0s1tiamnedsνa0ar1ereesstpimecattivedelyt.o bTeheT2sco≈r-
16 02 160 ns and T2a ≈ 180 ns. At the working point (Ib =
νa
GHz]14 νa01 01 0th,Φebs-=an0d.37aΦ-m0)odReabbyi-laikpeploysicnigllamtiiocnroswaarveeppeorwfoerrmaetdtohne
ncy [12 resonance frequencies ν0s1 and ν0a1 respectively. Rabi-
ue decay time is measured in the two-levellimit with about
q
Fre10 νs 170 ns and 50 ns for s- and a-mode. Relaxation times
8 01 T1s = 200 ns and T1a = 74 ns are extracted from the ex-
νs ponential population decay of the excited levels |1 ,0 i
01 s a
6 and |0 ,1 i. The measured coherence times of the a-
s a
−200 0 200 0.3 0.4 0.5 0.6 modearemuchsmallerthanexpectedfromtheminimum
Bias current, I [nA] Bias flux, Φ /Φ
b b 0 linewidth. The origin of this additional decoherence will
be discussed later. Nevertheless, the coherence time is
FIG. 2: Energy spectrum. Escape probability Pesc versus sufficientlylargeforindependentcoherentcontrolofeach
frequency as a function of current bias (a) and flux bias (b)
mode.
measured at Φb = 0.48Φ0 and Ib = 0 respectively. Pesc is
enhanced when the frequency matches a resonant transition Oneofthe opportunitiesgivenbythe richspectrumof
of the circuit. The microwave amplitude was tuned to keep thistwoDoFartificialatomistheobservationofacoher-
the resonance peak amplitude at 10%. Dark and bright blue entfrequencyconversionprocessusingthexˆ2xˆ coupling
s a
scalecorrespondtohighandsmallPesc. Thereddashedlines ofEq.2. Thepulsesequence,similartootherstatesswap-
are the transition frequencies deduced from the spectrum of
ping experiment [29, 30], is presentedin (see Fig.3a). At
the full hamiltonian with C = 510 fF (see text). The green
t = 0, the system is prepared in the state |0 ,1 i, at
diamond is the initial working point for the measurement of s a
coherent free oscillations between the two modes, presented the initial working point Φb = 0.37Φ0 (green diamond
in Fig.3,and thegreen dottedsquare is thearea wherethese inFig.2b). Immediately after,a non-adiabaticflux pulse
oscillations take place. [32] brings the system to the working point defined by
Φ , close to the degeneracy point (νs ≈ νa ). Af-
int 02 01
ter the free evolution of the quantum state during the
fectly fitted by deriving the spectrum of the full Hamil- time ∆t , we measure the escape probility P . Fig.3b
int esc
tonian described above. As all the other parameters of presents Pesc as a function of ∆tint for Φint = 0.515Φ0.
the device can be extractedfrom switching-currentmea- The observed oscillations have a 815MHz-characteristic
surements [22], the only free parameter is the junction frequency (inset of Fig.3b) that matches precisely the
capacitance. FromthefitweobtainC =510fF,whichis theoritical frequency splitting at this flux bias (red ar-
consistentwith the 10µm2 junction area. Numericalcal- row). In Fig.3c, we present these oscillations as func-
culations are used to describe the energy dependence in tion of Φ . Their frequency varies with Φ , showing
int int
Fig.2a in orderto take into accountthe currentbias and a typical “chevron” pattern. In the inset of Fig.3c, the
the inductance asymmetryas wellas additionalcoupling oscillationfrequency versus Φ is compared to theoret-
int
terms. The model describes well the level anti-crossing ical predictions. The good agreement between theory
between the quantum states |2 ,0 i and |0 ,1 i, which andexperimentisastrikingconfirmationofthe observa-
s a s a
is given by the nonlinear coupling term ~g21xˆ2sxˆa. The tionofswappingbetweenthequantumstates|0s,1aiand
coupling strength g21/2π strongly depends on the work- |2s,0ai. Instead of the well known linear coupling xˆsxˆa
ing point. It is predicted to change from zero at Φ =0, between two oscillators, which corresponds to a coher-
b
to 700 MHz at the anti-crossing, up to about 1200 MHz ent exchange of single excitations between the two sys-
atΦ ≈0.65closeto the criticalline. The couplingterm tems, here the coupling xˆ2xˆ is non-linear and produces
b s a
xˆ2xˆ mainlycouplesthestates|0 ,1 iand|2 ,0 i. Start- a coherent exchange of a single excitation of the a-mode
s a s a s a
ing from these uncoupled states at Φ =0, they become with a double excitation of the s-mode, i.e. a coherent
b
a maximally entangled state at the resonance condition, frequency conversion. Starting from the state |0 ,1 i,
s a
νa = νs . Close to Φ = Φ the states are still en- an excitation pair |2 ,0 i is then deterministically pro-
01 02 b c s a
tangled because g21 diverges. In this device the second duced in about a single nanosecond at the degeneracy
nonlinearcouplingterm, xˆ2sxˆ2a,isoneorderofmagnitude point Φint =0.537Φ0.
smaller with g22/2π ≈ 50 MHz. Nevertheless the shift We now discuss the coherent properties and measure-
of transition frequencies that it induces must be taken ment contrast in our device. The unexpectedly short
into account to fit the experimental spectrum. Finally coherence time of the a-mode can be explained by the
the absence of xˆ3sxˆa in the coupling term explains why coupling to spurious two-level systems (TLS) [31]. With
no level anti-crossing between ν0s3 and ν0a1 is observed. a junction area of 10µm2 our device suffers from a large
From spectroscopic measurements, we obtain mini- TLS density of about 12 TLS/GHz (barely visible in
mumresonancewidthsasnarrowas4MHzand3.5 MHz Fig.2). Therefore it is very difficult to operate the a-
4
(a) Preparation Interaction + ing the system at Φb = 0 will lift this limitation since
2,0 _ it is an optimal point with respect to flux noise. The
s a 0s,1a 2s,0a 0s,1a smalloscillationamplitude in Fig.3has two additionnals
1s,0a 1s,0a origins. First the duration tπ of the π-pulse applied for
nt preparation of the state |0s,1ai has to fulfill the condi-
urre 0s,0a 0s,0a nt tion t−π1 <ν1a2−ν0a1 to avoid multilevel dynamics. Since
Flux & C p-pulse D@int easureme νttπi1am2=e−oν8f00a1thn=es.1a-8HmMoowdHeezvewirnhtitcπhheisipmropefslietehnsetaeoxsrptdreeorrnimgtherenetdr,euwlcatexioafintxioeodnf
Fb=Fint m the |0s,1ai occupancy after the π-pulse, to about 40%.
The second origin of the small contrast raises the ques-
Time
tion of the readout contrast between the states |0 ,1 i
s a
(b)
0.4 and |2 ,0 i. In absence of coupling the escape probabil-
s a
bility0.3 Φint =0.515Φ0 T [a.u]00..48 itthyeocfotnhterasrtya,teth|e0se,sncaaipsehporuolbdanboiltitbyeosfenthsietisvteatteon|na.,0Oni
a F s a
prob0.2 F 00 0.5 1 1.5 is very sensitive to ns. This difference should lead to a
pe Frequency [GHz] strong contrast between the states |0s,1ai and |2s,0ai.
ca0.1 However, since the coupling strength is very large close
s
E to the critical lines, |0 ,1 i and |2 ,0 i are still entan-
s a s a
0
0 5 10 15 20 25 30 gledwhentheescapeoccurs. Thisleadstoanadditionnal
Interaction time ∆t [ns]
int reductionofthecontrastbetweenthetwostatesthatpre-
(c) 0.05 0.1 0.15 0.2 vents the extraction of the theoriticaly high efficiency of
the frequency conversion[27].
30 z]
H
G1.5
ns] 25 cy [ 1 Inconclusion,wehavedesignedandstudiedanewtype
∆me, t [int1250 Frequen0.050.5 0.54 0.58 oTerfhisneugpspetwreccootrnaudnmuhcaotrifnmtgohanisrictdifieocvsiciacillelaaittsoowmrsewlcloidtuhepstlcewrdiobviendiatebrnynoacnlolDinnsoeiFadr-.
on ti Φint/Φ0 coupling terms. Coherent manipulation of the two DoF
acti 10 is demonstrated and the strong nonlinear coupling al-
er
Int 5 lows the observation of a coherent frequency conversion
betweenthetwointernalDoF.Ifinsertedinamicrowave
0 0.5 0.52 0.54 0.56 0.58 cavity this process should enable parametric amplifica-
Interaction flux, Φ /Φ tion or generation of correlated microwave photons. Fi-
int 0
nally by reducing the critical current of the junctions,
keeping the ratio L /L constant, one can increase the
FIG. 3: Free coherent oscillations between states J
|0s,1ai and |2s,0ai produced by a nonlinear coupling. strength of the coupling term g22/2π up to 500 MHz
(a) Schematic pulse sequence. The energy diagram, with- providingastrongdispersivefrequencyshiftproportional
out coupling in blue/red and with coupling in black, is rep- to the population of each mode. This should allow the
resented for both the preparation and interaction steps.(b) realization of C-NOT quantum gate inside the two DoF
Escape probability Pesc versus interaction time ∆tint. The of this artificial atom or enable an efficient readout of
insetpresentstheFouriertransform oftheseoscillations with
the camelback phase qubit state by probing the transi-
a clear peak at 815 MHz. The red arrow indicates the theo- tionfrequencyofthea-mode. Consideringthelonglistof
retically expectedfrequency. (c)Pesc versusinteractiontime
unique and interesting features, this new artificial atom
∆tint fordifferentinteractionfluxΦint closetotheresonance
conditionbetweenν0s2 andν0a1. Forclaritythedataisnumer- with two DoF and a V-shaped energy level structure is
icallyprocessedusing200MHzhigh-passfilter(dashedlinein a valuable addition to the growing set of building blocks
inset of (b)). Inset : oscillation frequency as function of flux. for superconducting quantum electronics.
The dashed red line shows the theoretical predictions.
We acknowledge C. Hoarau for his help on the elec-
modeinafrequencywindowfreeofTLSsinceνa isonly tronic setup. We thank P. Milman and M. H. Devoret
01
slightly flux dependent. However this is not a real is- for fruitful discussions. We acknowledge the technical
sue as it can be solved easily by reducing the junction support of the PTA facility in CEA Grenoble and of the
area [33]. The minimum linewidth of both a-mode and Nanofabfacility in CNRS Grenoble. This workwas sup-
s-mode, and therefore their coherence times, are limited ported by the European SOLID projects, by the French
in our experiment by low frequency flux noise. Operat- ANR ”QUANTJO”and by the Nanoscience Foundation.
5
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