A V-shape superconducting artificial atom based on two inductively coupled transmons

1.9 MB·English

by  É. Dumur

A V-shape superconducting artificial atom based on two inductively coupled
transmons
E´. Dumur,1 B. Ku¨ng,1 A.K. Feofanov∗,1 T. Weissl,1 N. Roch,1 C. Naud,1 W. Guichard,1 and O. Buisson1
1Institut N´eel, CNRS–Universit´e Joseph Fourier, BP 166, 38042 Grenoble-cedex 9, France
(Dated: May 8, 2015)
Circuit quantum electrodynamics systems are typically built from resonators and two-level arti-
ficial atoms, but the use of multi-level artificial atoms instead can enable promising applications in
quantum technology. Here we present an implementation of a Josephson junction circuit dedicated
to operate as a V-shape artificial atom. Based on a concept of two internal degrees of freedom,
the device consists of two transmon qubits coupled by an inductance. The Josephson nonlinearity
5 introduces a strong diagonal coupling between the two degrees of freedom that finds applications
1 in quantum non-demolition readout schemes, and in the realization of microwave cross-Kerr media
0 based on superconducting circuits.
2
PACSnumbers: 85.25.Cp,03.67.Lx,45.50.Pq
y
a
M
Both in scientific and technological interest, the elec- a single degree of freedom is challenging, but when us-
tromagnetic coupling of two-level systems and light has ing two degrees of freedom the separated |g(cid:105) ↔ |e(cid:105) and
7
been the source of a great number of studies on quan- |g(cid:105)↔|a(cid:105) transitions naturally occur. For quantum mea-

] tum systems [1]. Replacing two-level by multi-level sys- surements,thisoffersthepossibilitytocouplethereadout
h
tems offers possibilities that go beyond the addition of transition |g(cid:105) ↔ |a(cid:105) to the outside world, while keeping
p
complexity. In experiments focusing on light and single the qubit transition |g(cid:105) ↔ |e(cid:105) decoupled from it, and in
-
t photons, multi-level systems are at the origin of effects particular protected from decay induced by the Purcell
n
a like electromagnetically induced transparency (EIT) [2] effect[12,13]. Selectivecouplingisalsoaprerequisitefor
u or the generation of entangled photon pairs [3]. In ex- photon interaction schemes between spatially separated
q
periments focusing on two-level systems, advanced tools modes in which the multi-level device plays the role of a
[
suchassidebandcooling[4]andstatemeasurement[5,6] cross-Kerr medium [14–16]. V-shape properties were al-

2 become accessible when incorporating ancillary levels. readymentionnedinfluxonium[17],phasequbit[18]and,
v Inspired by these quantum experiments with natural tunablecouplingqubit[12,19,20]but,toourknowledge,
2
atoms and ions, an adaptation to the field of super- notyetverifiedexperimentally. Inthispaper,wepresent
9
8 conducting circuits is an obvious line of research that a demonstration of a V-shape superconducting artificial
4 has been followed using systems derived from qubits [7]. atom verifying the ensemble of properties listed previ-
0 In these systems with a single degree of freedom, selec- ously.
1. tionrulesareabsentorfavorladder-shapelevelschemes. Our system is based on two inductively coupled trans-
0 Sideband cooling [8] and EIT [9] have for instance been mons and the two degrees of freedom are given by two
5 realized using flux qubits. Related effects have also been normal modes of the circuit [21]. Their frequencies can
1
studied in transmons and phase qubits [10, 11]. be freely chosen by design. The characteristics of the
:
v two modes predestine them to play the roles of a logi-
In comparison, only few theoretical studies have ad-
i cal qubit and an ancillary qubit (or simply: qubit and
X dressed the V-shape level scheme [12, 13] despite its
ancilla). The qubit part is played by the low-frequency
r very successful application for quantum measurements
a in trapped ions [5] and nitrogen–vacancy centers in di- modewhichshowsgoodcoherencepropertiesandalarge
anharmonicity, as it is equivalent to the well-established
amond [6]. In this context, a V-shape system is under-
transmon. In a circuit quantum electrodynamics archi-
stood as a qubit with good coherence properties formed
tecture [22], this mode couples to the photon field in a
by a ground and excited state |g(cid:105) and |e(cid:105), and an ancil-
nearby resonator via its electric dipole moment. The
lary level |a(cid:105) coupled to the ground state. The transi-
ancilla part is played by the high-frequency mode mag-
tion between |a(cid:105) and |e(cid:105) should be suppressed. Finally,
netically coupled to the resonator.
transitions to higher states should be well out of reso-
Interestingly the Josephson nonlinearity is at the ori-
nance with the two principle transitions |g(cid:105) ↔ |e(cid:105) and
ginoftheV-shapeproperty. UsuallytheJosephsonnon-
|g(cid:105) ↔ |a(cid:105). Combining these properties in a system with
linearity produces a strong anharmonicity in the low-
frequency mode of a superconducting circuit. This pre-
vents from contamination by higher energy states and
∗Presentaddress: E´colePolytechniqueF´ed´eraledeLausanne,1015 thus reduces the quantum dynamics to those of a two-
Lausanne,Switzerland level system [27]. In our device, the Josephson nonlin-
2
(a) (b) L. The magnitude of L is comparable to the Joseph-
son inductance L = Φ /(2πI ). The device shown in
J 0 c
Fig. 1(a,b) is fabricated from thin-film aluminium on a
high-resistivity silicon substrate. After patterning the
larger parts of the structure by electron beam lithogra-
phy and wet etching, the qubit structure as well as the
center conductor of a coplanar-waveguide resonator are
fabricated by lift-off using the controlled-undercut tech-
nique [24]. The linear inductance is realized in the form
ofachainoftwelvelargeJosephsonjunctions[17]ofcrit-
(c) (d)
ical current I(cid:48) (cid:29)I such that L=12×Φ /(2πI(cid:48)).
c c 0 c
The simplified diagram of our circuit is shown in
Fig. 1(d). The currents I and I through the two small
1 2
junctionsrepresentthetwodegreesoffreedominthecir-
cuit. The circuit exhibits two modes: a symmetric one
correspondingtoanin-phaseoscillationofI andI ,and
1 2
an antisymmetric one corresponding to an oscillation of
I and I in anti-phase. The symmetric (or qubit) mode
1 2
corresponds to the plasma oscillation of the supercon-
FIG. 1: (Color) (a) False-colored scanning electron micro- ductingquantuminterferencedevice(SQUID)formedby
graphofthesample. TheV-shapequbitcircuit(blue)iscou- the two junctions in a superconducting loop. Its electric
pledtotheresonator(red)viaasharedinductor. (b)Magni-
dipolemomentpointsinlineofthejunctions,asindicated
fied view of the core part showing parts of the chain, a small
in the diagram, whereas its magnetic dipole moment is
junction, and parts of the capacitor (from top to bottom).
zero. Theantisymmetric(orancilla)modeisusuallynot
(c) Energy level diagram of the V-shape qubit. Solid levels
show the energy levels of the coupled perturbative Hamilto- accessible in a SQUID due to its elevated frequency. In
nian H in Eq. (1). Dashed levels show the energy levels of our device however, the large inductance L ensures that
the same Hamiltonian (up to an energy offset) without the the frequency of this mode falls within the measurement
coupling term−(cid:126)g σ(qb)σ(a)/2. Thanks to theenergy shifts bandwidth. Its magnetic dipole moment points out of
zz z z
governed by the coupling strength gzz, the lowest three lev- the circuit plane, whereas its electric dipole moment is
elscanbeaddressedwithoutpopulatingthefourthlevel,and zero. These two orthogonal dipoles enable selective cou-
thus the experimental circuit realizes a V-shape level scheme
plingbetweenthequbitandtheancilla,openingtheway
(thickarrows)(d)Equivalentqubitcircuitdiagramconsisting
to novel circuit architecture possibilities [13, 15, 16].
of two capacitances C, two Josephson junctions with critical
The SQUID flux bias Φ controls the mode energies
current Ic, and an inductor L. The current oscillations and b
the magnetic dipole moment µ associated with the ancilla and their mutual coupling. An optimal point for the op-
modeareindicatedwithredsymbols,thecurrentoscillations erationasaV-shapedeviceisreachedatthe“sweetspot”
and the electric dipole moment p associated with the qubit Φ =0,whichweassumeforthefollowingtheoreticalde-
b
mode are indicated with green symbols. The loop is biased scription [21]. Both modes are anharmonic and we may
with a magnetic flux Φ .
b consider them as two-level systems with transition ener-
√ √
gies (cid:126)ω ≈ 2E E and (cid:126)ω ≈ 2E E (cid:112)1+2L /L.
qb J C a J C J
Remarkably, the two modes are coupled by a σ σ term,
z z
earity generates in addition a cross-anharmonicity effect whereas other coupling terms are absent at zero flux due
between the two modes which induces a diagonal cou- tosymmetryreasons. Thisfollowsfromtheperturbative
pling[23]. Thiscouplingleadstoafrequencyshiftofone treatment of the full Hamiltonian of the circuit depicted
modeofmorethan100MHzconditionalontheexcitation in Fig. 1(d). For the purpose of this paper, we will thus
state of the other mode. Thus this cross-anharmonicity describe the system by the simplified Hamiltonian
preventsfromcontaminationbythefourthstateinwhich
thereisoneexcitationineachmode. Inthatwaythesys- H=(cid:126)ωqbσz(qb)/2+(cid:126)ωaσz(a)/2−(cid:126)gzzσz(qb)σz(a)/2, (1)
tem dynamics are reduced to those of a V-shape system.
where σ(qb) (σ(a)) are Pauli matrices of the qubit (an-
z z
Thetransmoncircuitthatformsthebasisofourcircuit
cilla). The cross-anharmonicity is expressed as
consists of a small Josephson junction with critical cur-
rent I that is shunted by an interdigital capacitance C. E
c (cid:126)g = C . (2)
We introduce the associated Josephson and charging en- zz 8(cid:112)1+2L /L
J
ergies as E = Φ I /(2π) and E = (2e)2/(2C), where
J 0 c C
Φ = h/(2e) is the magnetic flux quantum. We cou- In Fig. 1(c), we show an energy level diagram of our
0
ple two identical transmons by integrating their Joseph- system to clarify the role of the coupling term. The
son junctions into a loop with a large linear inductance eigenstates of the uncoupled Hamiltonian (cid:126)ω σ(qb)/2+
qb z
3
(cid:126)ω σ(a)/2 are shown as dashed levels. The solid levels
a z
show the eigenenergies E of the coupled Hamiltonian
i,j
given in Eq. (1) with eigenstates |i,j(cid:105), where i(j)=g, e
denotes the qubit (ancilla) state. The transition energy
E −E is detuned from E −E by an amount
e,g g,g e,e g,e
2(cid:126)g . Equally, the transition energy E −E is de-
zz g,e g,g
tuned from E − E by 2(cid:126)g . As long as (cid:126)g is
e,e e,g zz zz
large enough, it will allow for V-shape system dynamics
in which only the states |g,g(cid:105) = |g(cid:105), |e,g(cid:105) = |e(cid:105), and
|g,e(cid:105)=|a(cid:105) play a role. In particular, g must be signifi-
zz
cantlylargerthanthelinewidthsofthequbitandancilla
transitions.
In the following, we describe the measurements per-
formed to determine the mode energies and to demon-
stratethecross-anharmonicityinourdevice. Oursample
isplacedinadilutionrefrigeratoratatemperatureofap-
FIG. 2: (Color) Spectroscopy of the V-shape artificial atom.
proximately 30mK. The quantum circuit is coupled to
The gray scale encodes the transmission of a readout tone
a coplanar-waveguide resonator through an inductance
close to the resonator frequency ω /(2π) in the presence of
r
shared by the qubit loop and the resonator [9] as well a spectroscopy tone whose frequency f is swept. The mag-
s
as through stray capacitances. By placing the circuit at netic field is varied along the horizontal axis and converted
the grounded end of a quarter-wave resonator, we can to flux Φb through the SQUID loop. The gray lines repre-
achieve the interesting configuration in which the induc- sent the transition of the ancilla in the top graph and the
transition of the qubit in the bottom graph as illustrated in
tivecouplingbetweenancillaandresonatorismaximized
the insets. Dashed lines show numerical model calculations
whereas the capacitive coupling between qubit and res-
of these transition energies. The small discrepancy on the
onator is eliminated at all frequencies. This allows for
ancillaspectroscopybetweenexperimentandpredictionclose
a fast qubit readout protocol free of decay induced by to Φ /Φ ≈ ±1/2 may be explained by taking into account
b 0
the Purcell effect [13]. In our device, this configuration a 35 % asymetrical critical current in the two coupled trans-
is realized within geometrical constraints, leading to a mons.
coupling between qubit and resonator small enough to
suppress Purcell decay but large enough to allow excit-
ing the qubit with a microwave tone sent through the flux, but for the purpose of characterization we study
resonator. the full flux dependence. We vary the external magnetic
At its open end, the resonator is capacitively coupled field with a small coil around the sample. In the data in
to a microwave transmission line through which we mea- Fig.2wedistinguishtwodarklinescorrespondingtothe
sure the transmission of a readout tone at frequency qubit transition (cid:126)ωqb+(cid:126)gzz =Ee,g−Eg,g (bottom) and
fro close to the resonator frequency ωr/2π ≈ 7.2GHz. the ancilla transition (cid:126)ωa +(cid:126)gzz = Eg,e −Eg,g, respec-
Through the presence of a SQUID in the center conduc- tively. As a function of flux, the frequency ωqb of the
tor [25], ω is tunable by magnetic field over a range of qubit mode varies more strongly on a relative scale than
r
∼ 150MHz. The input signal is attenuated by 20dB at that of the ancilla mode, ωa. The ancilla mode involves
4.2K and by 40dB at base temperature and the output principally the elements L and C that are insensitive to
signalpassesthroughalow-noiseamplifierat4.2K. The flux. In contrast, the qubit mode does not involve L and
sample is protected from amplifier noise by two circula- itsfrequencyisexpectedtodropstronglyasthefluxbias
tors and a low-pass filter. A second sample consisting of reaches Φb =Φ0/2.
a resonator at 7.7GHz and a V-shape device are present InordertocomparethecircuitmodelinFig.1(d)with
onthesamechip. Thankstothewell-separatedresonator experiment, we performed numerical calculations of the
frequencies, we can independently measure the two sam- spectrumofitsfullHamiltonianderivedinRef.[21]. The
ples using the same transmission line. model depends on the three circuit parameters Ic, C,
We performed two-tone spectroscopy to map out the and L which we take as fitting parameters. The numer-
energydiagramoftheartificialatomshowninFig.2. We ical calculation consists of a solution of the discretized
send a spectroscopy tone at frequency f through the Schr¨odingerequationusingtheKwantcode[26]. There-
s
transmission line while measuring transmission at f . sults of these calculations are shown as dashed lines in
ro
Boththequbitandtheancillaaredispersivelycoupledto Fig. 2. The fit to the experimental data yields the pa-
theresonator. Theexcitationoftheirtransitionsleadsto rameters Ic =8.19nA, C =39.7fF, and L=0.192×LJ.
ashiftinω ofafewMHz, andconsequentlytoachange Due to mode anharmonicity, the transition from the
r
in transmission at f [22]. first to the second excited state of the qubit is detuned
ro
Asmentioned,thededicatedoperationpointisatzero from ω by −0.30GHz, as measured via two-photon ex-
qb
4
(a) pulse sequence is sketched in Fig. 3(b). In the top trace
(b)
P = -20 dBm in Fig. 3(a), we observe a single dip corresponding to
the qubit transition |g,g(cid:105) → |e,g(cid:105) indicated in the level
-10dBm
scheme in panel (c). Towards the bottom, a second dip
-7 dBm (c) emerges next to the first one. As indicated in the level
schemeinpanel(d),theseconddipiscausedbythepop-
-4 dBm
ulationoftheexcitedancillastate|g,e(cid:105)bythecondition-
-1 dBm ing pulse (red arrow). The transition |g,e(cid:105)→|e,e(cid:105) thus
becomes available. The depth of the two dips is ideally
(d)
2 dBm
proportional to the occupation of the states |g,g(cid:105) and
5 dBm |g,e(cid:105), theirfrequenciesareseparatedbygzz/π. Thispic-
ture corresponds well to the observation, and we can ex-
tractacross-anharmonicityofg /π =115MHz(dashed
zz
lines).
(e) (f)
Weperformedtwofurthermeasurementtotestthere-
producibility and consistency of this result. Firstly, we
performed a measurement on the other, nominally iden-
tical V-shape device on the same chip, secondly, we in-
terchanged the role of the qubit and the ancilla. We
could obtain a better overall data quality when measur-
ingincontinuousratherthanpulsedmode,meaningthat
FIG. 3: (Color) (a) Measurement of the cross-anharmonicity a measurement tone, a spectroscopy tone, and a condi-
g based on a sequence involving a conditioning pulse at
zz tioning tone were applied simultaneously. In Fig. 3(e),
f = ω /(2π) to populate the ancilla, a spectroscopy pulse
c a we plot the first measurement showing a spectroscopy
at f to scan over the qubit transition, and a readout pulse
s around the qubit frequency, with the conditioning tone
at f , cf. the scheme in (b). From the topmost to the low-
ro
est curve, we increase the power of the conditioning pulse at the ancilla frequency turned off (top trace), or turned
in steps. Below the qubit dip present for all powers at on (bottom trace). The result is largely consistent with
ω /(2π)≈3.67GHz, a second dip emerges for high powers. that in Fig. 3(a), with a slightly smaller peak separa-
qb
Thepeakseparationisameasureofthecross-anharmonicity. tion of about 110MHz. In Fig. 3(f), we plot a mea-
Thesituationswithoutandwiththefirstpulsearerepresented
surement with inverted roles of qubit and ancilla. We
inthediagramsin(c)and(d). (e)Controlmeasurementofthe
sweep the spectroscopy frequency f around the ancilla
peak separation on the twin sample on the same chip. This s
frequency, and excite the qubit with the conditioning
measurementisperformedwithcontinuoustones,withatone
atfc =ωa/(2π)turnedoffforthetoptrace,andturnedonfor tone at ωqb/(2π). We observe a quantitatively similar
the bottom trace. The result is consistent with that shown behavior to the previous measurements with a peak sep-
in (a). (f) Second control measurement with inverted roles aration of about 110MHz. The experimental value of
of qubit and ancilla, i.e., the spectroscopy tone frequency is g /π compares well to the theoretical value 144MHz
zz
sweptaroundtheancillafrequency,whereastheconditioning
obtained from Eq. (2), corroborating the validity of our
tone is resonant with the qubit at f =ω /(2π).
c qb circuit model.
With this result we demonstrate the nature and
strength of the cross-anharmonicity in our device. On
citation into that state at large microwave power. We thisbasiswecanqualifyourcircuitasaV-shapesystem.
performed measurements of Rabi oscillations and qubit The measurement of the cross-anharmonicity and its in-
relaxation to estimate a coherence time of T2,Rabi = terpretation are based on a solid theoretical model ver-
0.5µs and a lifetime T1 =0.6µs of the qubit. ified by spectroscopy. Our circuit furthermore exhibits
Inordertoobservethecross-anharmonicityg atzero the good coherence properties required for the use in
zz
flux, we choose the experimental approach of detecting quantum experiments. Integration in existing setups is
changes in the qubit transition frequency depending on straightforward thanks to a device technology proven in
the presence or absence of a microwave excitation of the a wide range of circuit quantum electrodynamics exper-
ancilla. In essence, we perform a spectroscopy, in which iments. Among the applications, we highlight fast qubit
we apply a spectroscopy pulse (frequency f , duration readouttechniques[13]andcross-Kerrinteractionsatthe
s
80ns), followed by a readout pulse (frequency f , dura- few-photon level [15]. Thanks to its electrical and mag-
ro
tion 250ns). Before these two pulses, we apply a condi- netic dipole allowing a selective coupling of the ancilla
tioningpulseofduration10nsandfrequencyf resonant and logical qubit, our circuit also fulfilled the require-
c
withtheancillatransition|g,g(cid:105)→|g,e(cid:105). Inthemeasure- ments of the single-photon transistor proposed by Ref.
ment shown in Fig. 3(a), the power of the conditioning [16].
pulse is gradually increased from top to bottom. The The authors thank M. Hofheinz, B. Huard, Y. Kubo,
5
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