Unexpectedly allowed transition in two inductively coupled transmons

1.3 MB·

by  Étienne Dumur

JOURNALOFLATEXCLASSFILES,VOL.13,NO.9,SEPTEMBER2014 1
Unexpectedly allowed transition in
two inductively coupled transmons
E´tienne Dumur, Bruno Ku¨ng, Alexey Feofanov, Thomas Weißl, Yuriy Krupko, Nicolas Roch, Ce´cile Naud,
Wiebke Guichard, and Olivier Buisson
Universite´ Grenoble Alpes, Institut NEEL, F-38000 Grenoble, France
CNRS, Institut NEEL, F-38000 Grenoble, France
6
1 We present experimental results in which the unexpected zero-two transition of a circuit composed of two inductively coupled
0 transmons is observed. This transition shows an unusual magnetic flux dependence with a clear disappearance at zero magnetic
2 flux. In a transmon qubit the symmetry of the wave functions prevents this transition to occur due to selection rule. In our circuit
the Josephson effect introduces strong couplings between the two normal modes of the artificial atom. This leads to a coherent
n superposition of states from the two modes enabling such transitions to occur.
a
J
Index Terms—Josephson junctions, Cavity resonator
0
2

l] I. INTRODUCTION (b) (a)
l
a
h CIRCUIT Quantum ElectroDynamics (CQED) has re-
- vealed itself to be an extensive platform to address both
s
e fundamental quantum mechanics issues [1], [2] and applied
m technological interests [3], [4]. In CQED, circuit elements as
Josephson junctions (JJ’s) and microwave resonators can be
.
t
a used as elementary components to engineer quantum systems
m such as qubits and photon cavities. Nowadays the most com-
- monly qubit used is the transmon qubit which consists of
d a small Josephson junction shunted by a capacitance [5]. It
n
is described as an anharmonic oscillator with one degree of (c)
o
freedom. Only the two first levels of the transmon mode have
c
[ to be considered to realise a qubit. Due to the symmetry of

the wave functions, direct transition between the ground state
1
and the second excited state of a transmon type qubit is a
v
3 forbidden transition [14].
6 In this article we present the study of a quantum system
2
based on two inductively coupled transmons. This circuit
5
exhibits two degrees of freedom with a V-shape energy level
0
. diagram [6]. Such an artificial atom presents a strong interest
1
for fast qubit readout [7], cross-Kerr interaction [8], [9], and
0 Fig. 1. (a) False-colored scanning electron micrograph of the sample. The
6 single photon transistor [10]. The two degrees of freedom feedline(lightgreen)throughwhichtransmissionismeasurediscapacitively
1 are given by the two normal modes of the circuit [11] coupled to a quarter-wave resonator (red). Local DC flux bias lines (purple
andcyan)allowtoapplylocalmagneticfield.Theyellowlineisamicrowave
: whichcorrespondtoasymmetricandanantisymmetricmode.
v excitationlinecapacitivelycoupledtotheartificialatom.(b)Magnifiedview
i Interestingly the symmetric mode is equivalent to the well- oftheartificialatom.Theartificialatom(blue)iscapacitivelyandinductively
X established transmon mode. The Josephson nonlinearity pro- coupledtomicrowaveresonator(red).ThechainofJJisvisibleattheright
r duces an anharmonicity in the two modes of the artificial side of the SQUID loop. (c) Equivalent circuit diagram composed of two
a capacitorsofcapacitanceC,twoJJ’sofcriticalcurrentIc andaninductorof
atom. We observed by spectroscopy an unexpected transition inductanceL.Thesymmetricandantisymmetricmodesaredepictedasgreen
between the ground state and the second excited state at andredarrows,respectively.TheloopisbiasedwithamagneticfluxΦb.
nonzero magnetic flux in the symmetric mode. Moreover this
transition becomes forbidden close to zero magnetic flux.
ascoupledfluxqubits[12],[13],fluxonium[14],[15]andflux
Theseobservationsarediscussedandexplainedthroughparity
qubit coupled to a resonator [16].
effects and the non-linear coupling of the two modes which
leadstoacoherentsuperpositionofstatesofthesymmetricand
antisymmetric mode. Selection rules and symmetry breaking II. SAMPLEPRESENTATION
has been predicted and observed in other qubits system such Our artificial atom is composed of two identical transmons
integrated into a loop of large inductance. The transmon
Correspondingauthor:E.Dumur(email:[email protected]). consists of a small JJ of critical current Ic shunted by an
JOURNALOFLATEXCLASSFILES,VOL.13,NO.9,SEPTEMBER2014 2
interdigital capacitor. The capacitance of the JJ in parallel of
the capacitor is denoted C. The dynamics of the transmon is Hz] 150
M
given by the ratio of its Josephson energy EJ = Φ0Ic/(2π) [ 120
on its Cooper-pair charging energy EC = (2e)2/(2C), with 2π) 90
Φ = h/(2e) is the magnetic flux quantum. The linear (
0 /2 60
inductance of the loop L is comparable to the Josephson 2
ω
inductance of the transmons LJ = Φ0/(2πIc). The resulting z] 3
circuit is shown Fig. 1(a). H 2
G
The device shown in Fig. 1(a) and Fig. 1(b) is fabricated [ 1
from thin-film aluminium on a high resistive silicon substrate. π) 0
(2 1
Coarse structures are patterned by electron beam lithography /1 −2
and wet etched. Fine structure such as the artificial atom and ω2 −3
− 1/2 1/4 0 1/4 1/2
thecentrallineofcoplanar-waveguideresonatorarefabricated − −
by lift-off using the Bridge free fabrication technique [17]. In Φb/Φ0
order to reach a linear inductance of the loop large enough to
be comparable to the Josephson inductance, we incorporated Fig. 2. Dependence of the non-linear coupling terms on a magnetic flux.
in the loop a chain of 12 large JJ’s [15] of critical current Typical values for ω22/(2π) are some hundreds of megahertz whereas
I(cid:48) (cid:29)I such that L=12×Φ /(2πI(cid:48)). ω21/(2π)arearoundfewgigahertz.Thetheoreticalpredictionsarerealised
c c 0 c from circuit parameters extracted from the fits performed on data presented
A simplified diagram of our artificial atom is shown in inFig.3.
Fig. 1(c). The system presents two modes of oscillations: a
symmetric one corresponding to an in-phase oscillation of
to the usual transmon mode. The last line shows non-linear
the supercurrent across the two JJ and an antisymmetric one
interactionbetweentheseoscillators.Duetothelargecoupling
corresponding to an oscillation out-of-phase. The symmetric
inductanceoftheloopthesecouplingtermshaveanimportant
mode can be seen as an electrical dipole pointing in line of
effectonthedynamicsofthesystem.Inthefollowing,wewill
the JJ’s with an average phase x = (ϕ + ϕ )/2 where
s 1 2
define |n ,n (cid:105)=|n (cid:105)|n (cid:105) as the eigenstates of the uncoupled
ϕ are the phase difference across the two JJ’s. A conjugate s a s a
1|2 Hamiltonianwheren ∈Nindexestheenergylevelsofeach
charge p = (q + q )/2 is associated to this oscillating s|a
x 1 2
mode. The eigenstates of the full system will be denoted |ψ (cid:105)
mode with q = C(Φ /(2π))ϕ˙ is the conjugate charge k
1|2 0 1|2 with k ∈N indexes the energy level
of each JJ. The antisymmetric mode is usually not accessible
The non-linear coupling term ω has been studied in
in conventional SQUID due to its high frequency. In our 21
Ref. [11] to demonstrate at a certain flux bias a coherent
artificial atom, the large inductance of the loop L ensures
frequency conversion of one excitation in the antisymmetric
that the frequency of the antisymmetric mode falls within our
mode in two excitation in the symmetric one. In Ref. [6],
measurementbandwidth.Thatmodecanbeseenasamagnetic
authorsusedthetermω torealiseatzerofluxbiasaso-called
dipolepointingoutoftheartificialatomrelatedtooscillations 22
V-shape energy diagram. This property emerges from a cross-
ofphasedifferencex =(ϕ −ϕ )/2withaconjugatecharge
a 1 2
anharmonicity between levels of the two modes. In Fig. 2,
p =(q −q )/2.
y 1 2
weshowthecalculatedmagneticfluxdependenceofcoupling
The artificial atom composed of two inductively coupled
terms. We see that ω exhibit a odd parity in respect to
transmons has already been theoretically explored in greater 21
magnetic flux while ω an even one. This parity is related to
detail in Ref. [11]. We remind here the Hamiltonian of the 22
the parity order of the Taylor expansion of the coupling term.
systemwritteninthebaseofthesymmetricandantisymmetric
We also notice the order of magnitude difference between the
mode expanded to the fourth order by Taylor expansion
two couplings term.
1 (cid:16) (cid:17)
H(cid:98) = 2(cid:126)ωs p(cid:98)2s +x(cid:98)s2 −(cid:126)ωsδsx(cid:98)s4
III. EXPERIMENTALRESULTS
1 (cid:16) (cid:17)
+ 2(cid:126)ωa p(cid:98)2a +x(cid:98)a2 −(cid:126)ωaσax(cid:98)a3−(cid:126)ωaδax(cid:98)a4 Our sample is placed in a wet dilution refrigerator with
+(cid:126)ω x 2x +(cid:126)ω x 2x 2, (1) a base temperature of 30mK. As shown in Fig. 1(b), the
21(cid:98)s (cid:98)a 22(cid:98)s (cid:98)a
artificial atom is coupled to a coplanar-waveguide resonator
where x and p are the reduced position and momentum through a shared inductance and a stray capacitance. These
(cid:99)s|a (cid:98)s|a
quantumoperatorsofsymmetricandantisymmetricmodesuch couplings ensure that the two modes of the artificial atom
√ √
asx =(a +a† )/ 2andp =i(a −a† )/ 2.Thefirst will be effectively coupled to the resonator. By placing the
(cid:99)s|a (cid:98)s|a (cid:98)s|a (cid:98)s|a (cid:98)s|a (cid:98)s|a
two lines of Eq. 1 correspond to two independent anharmonic circuit close to the grounded end of the resonator, we achieve
oscillators corresponding to the symmetric and antisymmetric aconfigurationinwhichthetwocouplingbecomecomparable.
mode of angular frequency ω and ω respectively. The anhar- The resonant frequency of the quarter-wave resonator is made
s a
monicity is written as dimensionless factors and is denoted σ tunablebyintegratingaSQUIDinitscentralline.Theresonant
a
and δ for the corrections at the third and fourth order. The frequency can be tuned over a range of about 150MHz by
s|a
correctiontermscomefromthenon-linearityoftheJosephson changing the flux threading the SQUID loop. The readout
effect.WenotethatthefirstlineisequaltotheHamiltonianof of the artificial atom transitions is performed by standard
the transmon qubit. The symmetric mode is therefore similar dispersive state measurement. The input signal is attenuated
JOURNALOFLATEXCLASSFILES,VOL.13,NO.9,SEPTEMBER2014 3
parameters I =8.19nA, C =39.7fF, and L=0.192×L .
c J
Surprisingly in our experiments the second transition of the
symmetricmodeisclearlyvisible.Thisresultwasnotexpected
since such transitions have never been observed in transmon
qubits. Moreover, close to zero magnetic field, we observe in
our experiment a disappearance of the second transition peak
(seetheinsetinFig.3).Thenwehavetoanswertwoquestions:
why are we able to directly measure the peak of the second
level of the symmetric mode and why does the peak vanish at
zero magnetic flux ?
A simple way to know whether a transition is forbidden
is to look at the parity of the initial and final state as
well as of the coupling operator. In a first time we will
consider the uncoupled system. In this case, the transition
probability between the ground |0 (cid:105)|0 (cid:105) and the excited state
s a
|2 (cid:105)|0 (cid:105) is given by P ∝ |(cid:104)ψ |Ω x +Ω y |ψ (cid:105)|2 where
s a 0→2 2 s(cid:98)s a(cid:98)a 0
Ω and Ω are the amplitude of coupling between microwave
s a
field and the symmetric and antisymmetric mode [18]. From
previous equation we obtain P ∝ |Ω (cid:104)2 |x |0 (cid:105)(cid:104)0 |0 (cid:105)+
0→2 s s (cid:98)s s a a
Ω (cid:104)0 |x |0 (cid:105)(cid:104)2 |0 (cid:105)|2.Withthefirsttermweretrievetheusual
a a (cid:98)a a s s
results observed in transmon qubits. Indeed the two states,
|0 (cid:105) and |2 (cid:105), have the same parity and the coupling term is
s s
odd.Consequently(cid:104)2 |x |0 (cid:105)iszeroforsymmetryreason.The
s (cid:98)s s
second term, which does not exist in transmon type qubits,
is due to the coupling between the microwave field and the
antisymmetric mode. Nevertheless due to the orthogonality of
Fig. 3. Spectroscopy of the artificial atom as function of frequency and theeigenstates,thistermisstrictlyzero.Thetransitionisthen
magnetic field. From top to bottom, panels show the first transition of the forbidden for the uncoupled system.
antisymmetric mode, the second transition of the symmetric mode, and the
To explain why the transition is observed, we need to
first transition of the symmetric mode. The inset presents a zoom in the
second transition of the symmetric mode for Φb/Φ0 close to 0.For each consider the full Hamiltonian with its non linear coupling
measuredfrequencysweepwesubtractedameasurementoffset.Themagnetic terms. The calculation of the eigenstates of the full Hamil-
fieldisconvertedtofluxΦb throughtheSQUIDloopoftheartificialatom. tonian |ψ (cid:105) is a hard problem and in our work we only
Dashed line shows numerical model calculations of these transitions. The k
small discrepancy on the antisymmetric spectroscopy between experimental considerthecorrectedeigenstatesatthefirstorderbyquantum
data and theoretical prediction close to Φb/Φ0 ≈ ±1/2 may be explained perturbationtheory.Thecompleteexpressionofthefirstorder
bytakingintoaccounta35%asymmetricalcriticalcurrentbetweenthetwo
corrected eigenstates our system is given in Ref. [19]. We
JJ’s.
observed that the corrected eigenstates become contaminated
by states of higher and lower energy, and more importantly,
by 20dB at 4.2K and by 40dB at base temperature before theymixstatesofthesymmetricandantisymmetricmodedue
passing through a feedline (Fig. 1(a)). tonon-linearcouplings.Byusingthecorrectedeigenstateswe
Duetothecapacitivecouplingbetweenthefeedlineandthe derive the transition probability1 P which is proportional
0→2
quarter-wave resonator, the output signal carries information to |(ω Ω /(8ω −4ω )|2.
21 a s a
aboutthequantumstateoftheartificialatom.Next,thesignal Due to the non-linear coupling term ω , the transition
21
is amplified by a High Electron Mobility Transistor amplifier from the ground state to the second excited state is allowed.
thermalised at 4.2K. The sample is protected from noise Moreover, the disappearance of the transition at zero flux can
coming from the amplifier by two isolators and a low-pass be explained by the magnetic flux dependence of ω which
21
filter. goes to zero at zero flux, see Fig. 2. We also notice that the
In the three panels presented in Fig. 3, we see three transitionisinducedbytheΩ y couplingoperator.Indeedwe
a(cid:98)a
resonances of the artificial atom which depend on magnetic have to keep in mind that at Φ (cid:54)=0, the different eigenstates
b
field. These curves correspond to the first and second tran- of the system are a linear combination of symmetric and
sition of the symmetric mode and the first transition of the antisymmetric states.
antisymmetric mode. As a function of flux, the transitions of
the symmetric mode vary more strongly on a relative scale
IV. CONCLUSION
than that of the antisymmetric one. The antisymmetric mode
In this article we have presented an unexpected allowed
involves the two JJ’s as well as the linear inductance L which
transition between the ground and the second excited state
is insensitive to flux. In contrast, the symmetric mode only
involves the JJ’s and so its transition frequencies are expected
1The result only presents first order term, the higher order terms are
to drop to to zero as the magnetic flux tends to Φ ≈ Φ /2.
b 0 neglected. However all terms, even those which are neglected here, exhibit
From fitting (see Ref. [6]) we obtain the following circuit thesamebehaviouratzeroflux,theydroptozero.
JOURNALOFLATEXCLASSFILES,VOL.13,NO.9,SEPTEMBER2014 4
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