Selected Papers: Trend of Quantum Information

Multiphoton Transitions in a
Superconducting Flux Qubit
Shiro Saito†, Hirotaka Tanaka, Kouichi Semba, andHideaki Takayanagi
Abstract
We have observed multiphoton transitions between two macroscopic quantum-mechanical superposition states of a superconducting flux quantum bit (qubit) consisting of a superconducting loop with three
Josephson junctions. Two persistent currents circulating in opposite directions in the loop are macroscopically distinct states. Resonant peaks and dips of up to three-photon transitions were observed inspectroscopic measurements when the system was irradiated with a strong radio frequency (RF) photon
field. The widths of the multiphoton absorption dips shown to scale with the Bessel functions are inagreement with theoretical predictions derived from Bloch equations and the dressed-atom description.
From this analysis, we obtained the relaxation and dephasing time of the qubit, which are measures of
the qubit’s quality and its environment. This information should prove useful for designing and optimizing qubits in the future.

1. Introduction
Quantum state engineering has become one of themost attractive fields in modern physics. Nuclear
magnetic resonance techniques in atomic and molecular physics have been studied enthusiastically in relation to quantum computation and communication [1].
In principle, any quantum two-state system has thepotential to be applied to a quantum bit (qubit). Of themany candidates that may enable us to achieve quantum computation, superconducting qubits based on
Josephson junctions have been gaining importance.
Solid-state qubits such as semiconductor qubits or
superconducting qubits have the distinct merit of scalability due to the mature state of nanometer-scale fabrication technology. Furthermore, their coherence
times have been greatly improved in recent years [2].
There are three types of superconducting qubit. With
the first, the charge degree of freedom of Cooper pairs
is used to induce coherent quantum oscillations
between two charge states of a Cooper pair box [2],† NTT Basic Research Laboratories
Atsugi-shi, 243-0198 Japan
E-mail: s-saito@will.brl.ntt.co.jp12[3]. The second type, known as the Josephson phase
qubit, consists of a circuit with a single relatively large
Josephson junction, which is current-biased close to
its critical current [4]. The third type is the flux qubit
where three Josephson junctions are arranged in asuperconducting loop and threaded by an externallyapplied magnetic flux [5]. Several qubit operations
have been reported with this system [6]. The device
could also be prepared in a quantum superposition of
the two states carrying opposite macroscopic persistent currents [7]. From the viewpoint of macroscopic
quantum phenomena, it is interesting that macroscopically distinguishable states, such as a persistent current, form the superposition states.
The energy scale of quantum circuits containing
Josephson junctions is in the microwave regime. This
property was demonstrated in the current-voltage
characteristics of a Josephson junction under
microwave irradiation displaying the well-known
Shapiro steps [8]. They appear at voltages corresponding to integer multiples of the applied
microwave energy. With this phenomenon, the superconductor phase difference at the junction, which is amacroscopic degree of freedom, can be treated as aclassical degree of freedom moving in the Josephson
NTT Technical Review

Technologies

potential in phase space. In contrast, the quantummechanical behavior of the macroscopic phase difference was demonstrated using one-photon absorption
processes between quantized energy levels within asingle Josephson potential well [9]. Recently, Wallraff
et al. presented experimental evidence of multiphoton
absorption between quantized energy levels within asingle potential well formed by a large current-biased
Josephson junction [10]. In this paper, we report thefirst observation of multiphoton transitions between
superposition states of a macroscopically distinct
state [11], which are formed in the double-well potential system of a superconducting flux qubit [12].

tially separated but magnetically coupled (mutual
inductance M ≈ 7 pH). The area of the qubit loop is
5.1 × 5.3 µm2 and the dc-SQUID is 7.3 × 7.7 µm2. The
qubit contains two identical Josephson junctions andone that is β (=0.7) times smaller. They have junction
areas of 100 × 300 and 100 × 210 nm2, respectively.
The Josephson energy EJ = Ic/2e of the two identicaljunctions and the energy splitting ∆ at the degeneracy
point were 340 and 0.54 GHz, respectively, as estimated from spectroscopy measurements. Here, Ic is thecritical current of the junction and e is the electron
charge. The qubit system is made of aluminum and iscontacted by on-chip gold leads and pads. A 260-µmwide on-chip Au-strip line induces oscillating magnetic fields in the qubit loop. The distance between thequbit and the edge of the strip line is 20 µm. The 100nm-thick strip line and the leads and pads were manufactured on a silicon wafer with 1 µm of SiO2 by ebeam lithography and the liftoff technique. We thenfabricated the qubit system using e-beam lithography
and the standard shadow evaporation technique [13].
Two successive aluminum layers (40 and 50 nm) were
deposited on the substrate. After depositing the first

2. Experiment
2.1 Sample fabrication
Figure 1(a) shows a scanning electron microscope
image of a superconducting flux qubit system. The
inner loop with three Josephson junctions constitutes
the flux qubit and the outer loop with two junctions isa dc-SQUID (superconducting quantum interference
device) acting as a qubit state detector. They are spa-

(b)

(a)

Energyhω exReadout

(c)

1.490

1.495

1.500

1.505

1.510

Φ qubit/Φ 0

Fig. 1. (a) Scanning electron microscope image of a superconducting flux qubit system. (b) Schematic illustration of qubit
energy levels as a function of the applied flux around Φqubit / Φ0 = 1.5. The blue and red curves represent the groundand first excited states of the qubit. The upward pointing arrows indicate multiphoton absorption processes. (c)Schematic drawing of the qubit readout at around Φqubit / Φ0 = 1.5. The green curve represents the average thermal
readout between the ground (blue curve) and the first excited state (red curve) under microwave irradiation.

Vol. 2 No. 12 Dec. 2004

13

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layer, we oxidized it with 1 mTorr of pure oxygen for 2
minutes. The double evaporation direction is shifted by±17° to create Al overlaps forming Josephson junctions.
2.2 Superconducting flux qubit
By carefully designing the junction parameters [5][7], we can make the inner loop behave as an effective
quantum two-state system. In fact, the qubit readoutchanges greatly with the qubit design, ranging fromthe purely classical regime to the quantum regime[14]. It is described by the HamiltonianhHˆ qb = (ε 0σˆ z + ∆σˆ x ) , where σˆ x , z are the Pauli spin
2
operators. Two eigenstates of σ̂ z are localized states↓ corresponding to the persistent clockwise currentof the qubit and ↑ corresponding to its persistentanticlockwise current. The energy eigenstates 0 and
1 of Ĥqb show an energy anticrossing with energy
gap h∆ at the degeneracy point. An externally applied
static magnetic flux Φqubit generates the energy bias
hε 0 = IP Φ 0 Φ qubit Φ 0 − fop between the two localized states, where Φ0 = h/2e is the flux quantum,()

2
IP = IC 1 − (1 2 β ) is the persistent current of thequbit, and fop is the qubit operating point, which is ahalf integer. The effective energy gap in the biased

qubit is h∆ b = h ε 02 + ∆2 . The qubit dynamics iscontrolled by a time-dependent RF fields(t) = s cos ωex t with amplitude s and frequency ωexprovided by an on-chip RF-line. This leads to an addih
tional term in the Hamiltonian Hˆ RF (t ) = − s(t )σˆ z .
2
Figure 1(b) shows a schematic energy diagram of thequbit as a function of Φqubit/Φ0. The blue and red
curves represent the ground state 0 and the firstexcited state 1 , respectively. They show the energy
anticrossing with energy gap ∆ at the degeneracy
point Φ qubit/Φ 0 = 1.5. The arrows schematicallydescribe the processes by which the qubit absorbs the
RF photon when the effective energy gap ∆b matches
an integer multiple of the RF photon energy nhωex.
If ∆ is zero, Hˆ qb + Hˆ RF (t ) has only diagonal terms. In
this case, the RF field cannot excite the qubit from theground state to any excited state. A non-zero ∆ is therefore a prerequisite for a spectroscopic experiment. In
other words, resonant peaks and dips in the qubit signals are direct evidence for the coherent superpositionof the macroscopically distinct states and for the mul14

tiphoton transitions that occur between them.
2.3 Qubit readout
When the external flux Φqubit/Φ0 diverges slightly
from half integers, the observable difference between
the qubit states 0 and 1 is the direction of the flux
induced by the qubit circulation current. To measure
the flux, we used the current-voltage (I-V) characteristics of a dc-SQUID, which worked as a highly sensitive flux detector. The switching current, where thevoltage across the dc-SQUID jumps from zero to a
finite value in the I-V characteristics, depends on themagnetic flux piercing the dc-SQUID. We can therefore detect the qubit states by measuring the switching currents of the dc-SQUID. The switching current
was typically obtained by averaging the values of 500
measurements. Figure 1(c) shows schematic readout
curves of the qubit corresponding to 0 (blue curve)
and 1 (red curve). Because the measurements werecarried out in a dilution refrigerator at a temperatureof 30 mK and were averaged, we observed the thermal average between them (like the green curve,
which is calculated) [15]. With the microwave irradiation, we also observed resonant peaks and dips dueto the multiphoton absorption. The bias voltage needed to obtain the I-V characteristics had a triangular
waveform and was fed through large bias resistors of
1 MΩ to achieve a current-bias measurement. The
sweep rate of the current was set at 120 µA/s and thefrequency of the wave was 310 Hz. The RF line used
to apply microwaves contained three attenuators in
the input line: 30 dB at room temperature, 10 dB at
4.2 K, and 10 dB at the base temperature. We fabricated an on-chip strip line to achieve strong coupling
between the qubit and the RF line.
To achieve a good readout resolution, we chose theoperating point fop to be Φqubit/Φ0 = 1.5. Figure 2
shows the signal-to-noise (S/N) ratio ∆Iqb/Istd (redcurve) and average switching current Isw (blue curve)
as a function of the applied flux through the qubit
loop, where Istd denotes the standard deviation of the
SQUID switching currents over 150 events, which
may be considered to be the noise level in the qubit
readout. The qubit signal amplitude ∆Iqb should be
proportional to the flux derivative of Isw, which meansthat we can estimate ∆Iqb by using the signal amplitude ∆Iqb0 at Φqubit/Φ0 = 1.5 (see Fig. 3(a)). In theexperiments, the qubit signal appeared when Φqubit/Φ0
was a half-integer and the two arrows in Fig. 2 indicate the S/N ratio at Φqubit/Φ0 = 0.5 and 1.5. Hence wechose the operating point to be Φqubit/Φ0 = 1.5 toachieve a higher readout resolution.
NTT Technical Review

Selected Papers

3. Results and discussion
3.1 Experimental results
Figure 3(a) shows the qubit signal δIsw as a functionof the external flux Φqubit at Φqubit/Φ0 ≈ 1.5, which is
3
500
400
300
200

1

100
0

0
0.0

0.5

Φ qubit/Φ 0

1.0

1.5

Fig. 2. Signal-to-noise (S/N) ratio ∆ I qb / I std and theswitching current of the dc-SQUID Isw as a functionof applied flux through the qubit loop. The twoarrows show the S/N ratio at the qubit operating
points Φqubit/Φ0 = 0.5 and 1.5.

Isw (nA)∆Iqb/Istd2

derived by subtracting the sinusoidal backgroundswitching current from the SQUID switching current
Isw. It shows a change in the thermal average persistent current of the qubit [7]. We obtained a sample
temperature of 65 mK from a numerical fit to the data
by using the qubit parameters EJ and ∆, which wederived from spectroscopy measurements. Figure3(b) shows the Φqubit dependence of the qubit signal
under microwave irradiation. The distinct resonantpeaks and dips are attributed to situations where theenergy separation of the two qubit states ∆b matches
an integer multiple of the RF photon energy n ωex.
We have detected up to three resonant peaks and dipsfor various fixed RF frequencies. The half width at
half maximum (HWHMn) and the normalized amplitude An of the dips are shown in Figs. 4(a) and (b) for
various microwave amplitudes IRF for n = 1, 2, 3,
which is defined by IRF = 10PRF/20. Here, PRF (dBm) is
the microwave power at the signal generator. We
derived the HWHMn and An by fitting the Φqubit/Φ0
dependence of δIsw around the resonant dips (see Fig.
3(b)) using a Lorentzian. The An values are normal400

HWHMn (MHz)δIsw (nA)

1
∆Iqb0

0
–1

(a)

n=1

(a)

2

300

n=2
n=3

200

100

–2

0
1.2
(b)

(b)

2

1.0
3

0.8

An

δIsw (nA)

1
0

2
–1

1

0.4
2

1

0.2

–2
1.490

1.495

0.6

1.500
Φ qubit/Φ 0

1.505

1.510

Fig. 3. Applied magnetic flux dependence of the qubit signal
δ Isw. (a) Without microwave irradiation. The data
were obtained after averaging 2000 measurements.
(b) With microwave irradiation. The data wereobtained after averaging 500 measurements. The
microwave power and frequency were 5 dBm at thegenerator and 9.1 GHz, respectively.

Vol. 2 No. 12 Dec. 2004

0.0
0.0

0.4

0.8
I RF (arb. units)

1.2

1.6

Fig. 4. Half width at half maximum (a) and normalized
amplitude (b) of the resonant dips as functions of
the microwave amplitude IRF. The solid curves
represent theoretical fits obtained by Blochequations combined with the dressed-atom
approach.

15

Selected Papers

ized by the full amplitude of the dips (3 nA). The
HWHMn carries important information about thedephasing and relaxation processes caused by theinteraction with the environment. To calculate theline shape of the one-photon absorption dip, we usedphenomenological Bloch equations. To describe thestrong driving region and the n-th dip, we replaced
the single-photon Rabi frequency ω1 in the Blochequations with the corresponding multi-photon frequency derived from the theory of the dressed-atom
approach [16]. This substitution has been verified bynumerical simulations [17]. This method yields
results that are in excellent agreement with experimentally obtained ones, as seen in Fig. 4.
3.2 Bloch equations
The Bloch equations describe the dynamics of aspin-1/2 in a constant field in the z-direction and atime-dependent field perpendicular to it [18]. With
the flux qubit, the constant field represents the effective energy gap ∆b, namely the Larmor frequency
and the time-dependent field comes from the applied
microwave, which causes Rabi oscillation. To maintain the analogy between the spin-1/2 and the flux
qubit, we write the Hamiltonian in terms of the energy eigenstates [19] and obtain


h 
∆
ε
(1)
Hˆ =  ∆ b − 0 s(t )σˆ z +s(t )σˆ x  .
2 
∆b
∆b


The second term of the Hamiltonian leads to a nonadiabatic periodic variation in the Larmor frequency.
εWhen 0 s ≈ ∆ b, the Rabi frequency is decreased.
∆b
However, we can disregard this term under the usualεexperimental condition, 0 s < ∆ b. We have con∆bfirmed this by numerical simulations. If we use therotating-wave approximation, the motion of the qubit
spin σ̂ (t ) in the laboratory frame can be describedby the Bloch equationsd σˆ x (t )dtd σˆ y (t )dtd σˆ z (t )obtained in the rotating frame by setting d σ̂ i (t ) = 0 ,dtwhere i = x, y, z. Our dc-SQUID measures the flux
produced by the qubit circulating current. This meansthat the readout corresponds to the z-component of thequbit spin σ̂ (t ) . Hence we are interested in σ̂ z (t )of the steady state, which is given byω12 Tr Tφσˆ z = 1 −σ ,
 1 + ( ∆ωT )2 + ω 2 T T  0φ
1 r φ


(3)

where ∆ω ≡ ∆b –ωex. In Eq. (3), a resonant dip due to
one photon with a Lorentzian line shape appears ataround ωex ≈ ∆b. The HWHM1 and the amplitude A1of the resonant dip are
2

2

 1
T 
HWHM1 =   + ω12  r  , Tφ 
 Tφ 

(4)

ω12 Tr Tφ
A1 =σ0.
1 + ω12 Tr Tφ

3.3 Dressed-atom approach
To describe the regime of strong driving and n-photon absorption processes, we apply the dressed-atom
approach to the flux qubit. The Hamiltonian is given byhHˆ = (ε 0σˆ z + ∆σˆ x ) + hω ex a + a + gσˆ z a + a + ,2
where a and a+ are the annihilation and creation operators for a field mode with angular frequency ωex andg is a coupling constant between the qubit and thefield. The first term represents the qubit Hamiltonian()

in the basis of the localized state ↓ and ↑ . Ther→σˆ x (t ),= γ σˆ (t ) × B(t ) −x
Tφ

Hamiltonian can be explicitly diagonalized whenrσˆ y (t ),= γ σˆ (t ) × B(t ) −yTφ

exp −( + )g a + − a hω ex ↑ ↓ N and their eigen-

[
[

]

→

]

[

]

()

∆ = 0. The eigenstates are given by ↑ ↓ ; N(2)rσˆ z (t ) − σ 0→= γ σˆ (t ) × B(t ) −
.
z
dt
Tr
r
→
h
Here, γB(t ) = − (ω1 cos ω ex t, ω1 sin ω ex t, ∆ b ), σ 0
2
16

is the thermal equilibrium value of σ̂ z (t ) , Tr and Tφs∆
are the relaxation and dephasing times, and ω1 =2∆ bis the Rabi frequency of the one-photon absorption
process. The steady-state solutions of Eqs. (2) can be

[

(

)

] ()

dressed=( ) = Nhω − g 2 hω + ( − ) 1 hε ,
ex
ex
0

↑↓

energies are EN

2
where N are eigenstates of hωexa+a. For ∆ << ε0,first-order perturbation theory in the qubit Hamiltonian yields a term that is linear in ∆ and whenn ωex ≈ ∆b, the two dressed states show anticrossNTT Technical Review

Selected Papersh∆Jn (α ). Here,2
Jn(α) is the n-th order Bessel function of the first kindand α ≡ 4g N hω ex is the scaled amplitude of thedriving field. If we prepare a localized initial state,
Rabi oscillations occur with frequency ω n = ∆ Jn (α )[20].
To analyze the HWHMn and An of the resonant
dips, we substituted ωn for ω1 in Eqs. (4). We used
Tr1, Tr2, Tr3, Tφn = Tφ, and c as fitting parameters.
Here, Trn and Tφn are the relaxation and dephasing
times related to the n-th photon absorption process.
We defined a coupling constant c as cIRF = α. We
found excellent agreement between the calculated
and experimental data for both HWHMn and An (seeFig. 4). We obtained Tr1 = 31 ns, Tr2 = 84 ns, Tr3 = 430
ns, Tφ = 4.0 ns, and c = 0.55 from the fitting at a
microwave frequency of 3.8 GHz. We have obtained
similar results for ωex/2π = 9.1 and 11.4 GHz (notshown). It should be noted that the relaxation anddephasing times Trn and Tφ do not depend on the frequency of the corresponding multiphoton transition
for a pure structureless ohmic environment (see also
Eqs. (2)). This is no longer the case for a more complicated structured harmonic environment [21] as ispresent in our device. Here, the plasmon frequency ofthe dc-SQUID provides an additional energy scale ofthe environment. Nevertheless, the global physics is
captured by our simplified model.
ing due to off-diagonal coupling

4. Conclusion
In conclusion, we reported on measurements ofmacroscopic superconducting circuits that reveal
their quantum-mechanical behavior at low temperature. We identified multiphoton transition processes
in the qubit and found that the width of the n-photon
resonance scales with the n-th Bessel function with
its argument given by the ratio of the driving-fieldstrength to the frequency of the photons. We also
obtained the relaxation and dephasing times of thequbit, which are measures of the qubit quality and its
environment. This information should be usefulwhen designing and optimizing qubits in the future.
5. Acknowledgments
We thank M. Thorwart (Heinrich-Heine-Universität Düsseldorf) for theoretical analysis; M. Ueda
(NTT, Tokyo Institute of Technology), H. Nakano(NTT), J. E. Mooij (Delft University of Technology),
C. J. P. M. Harmans (Delft University of Technolo-

Vol. 2 No. 12 Dec. 2004gy), M. Grifoni (Universität Regensburg), I. Chiorescu (Michigan State University), Y. Nakamura (NECFundamental Research Laboratories), and D. Vion
(CEA Saclay) for useful discussions; and T. Kutsuzawa (NTT, Tokyo University of Science) for experimental help. This work has been supported by the
CREST project of the Japan Science and Technology
Agency (JST).
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Shiro Saito
Research Scientist, Superconducting Quantum
Physics Research Group, Physical Science Laboratory, NTT Basic Research Laboratories.
He received the B.E., M.E., and Ph.D. degrees
in applied physics from the University of Tokyo,
Tokyo in 1995, 1997, and 2000, respectively. In
2000, he joined NTT Basic Research Laboratories, Kanagawa, Japan. Since then, he has been
studying superconducting flux qubits. His current interests are improving the qubit coherencetime and achieving two-qubit entanglement insolid-state qubits. He is a member of the Japan
Society of Applied Physics (JSAP) and the Physical Society of Japan (PSJ).

Hirotaka Tanaka
Senior Research Scientist, Superconducting
Quantum Physics Research Group, Physical Science Laboratory, NTT Basic Research Laboratories.
He received the M.E. degree in electrical andelectronic engineering from Tokyo Institute ofTechnology, Tokyo in 1995. Then he joined NTT
Basic Research Laboratories, Kanagawa, Japan.
Since then, he has been engaged in the study ofquantum device physics. From 1997 to 1998, heworked in NTT Photo-electronics Laboratories
to develop nano-scale fabrication devices using
fullerene-doped electron beam resist. He studiedin Delft University of Technology in the Netherlands from 1998 to 1999. He is also a visiting lecturer at Nagoya University. He is a member of
JSAP and PSJ.

Kouichi Semba
Group Leader, Superconducting Quantum
Physics Research Group, Physical Science Laboratory, NTT Basic Research Laboratories.
He received the B.S. and M.S. degrees in science and a Ph.D. degree in applied physics fromthe University of Tokyo, Tokyo in 1983, 1985,
and 2002, respectively. In 1985, he joined NTT
Electrical Communication Laboratories, Ibaraki,
Japan. In 1987, he joined NTT Basic ResearchLaboratories, Tokyo, Japan. Since then, he has
been engaged in the study of the physics of highTc cuprate superconductors. In 2002-2003, he
was a visiting researcher at Delft University ofTechnology, the Netherlands. His current interests are achieving and controlling entanglementin superconducting qubits. He is a member of
PSJ, JSAP, and the American Physical Society(APS).

Hideaki Takayanagi
NTT R&D Fellow, Director, NTT Basic
Research Laboratories.
He received the B.S., M.S., and Ph.D. degrees
in science from the University of Tokyo, Tokyoin 1975, 1977, and 1987, respectively. In 1977,
he joined the Electrical Communication Laboratories, Nippon Telegraph and Telephone Public
Corporation (now NTT), Tokyo, Japan. Since
1985, he has been working on superconducting
quantum device physics. In 2000, he was awarded the Nissan Science Prize. He is also a visiting
professor at the University of Tokyo and Tokyo
University of Science. He is a member of PSJ,
JSAP, and APS.

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NTT Technical Review