Coherent dynamics of a Josephson charge qubit
T. Duty,
*
D. Gunnarsson, K. Bladh, and P. Delsing
Microtechnology and Nanoscience, MC2, Chalmers University of Technology, S-412 96 Go
¨
teborg, Sweden
共Received 10 May 2003; revised manuscript received 31 October 2003; published 5 April 2004兲
We have fabricated a Josephson charge qubit by capacitively coupling a single-Cooper-pair box 共SCB兲 to an
electrometer based upon a single-electron transistor 共SET兲 configured for radio-frequency readout 共rf-SET兲.
Charge quantization of 2e is observed and microwave spectroscopy is used to extract the Josephson and
charging energies of the box. We perform coherent manipulation of the SCB by using very fast dc pulses and
observe quantum oscillations in time of the charge that persist to ⯝10 ns. The observed contrast of the
oscillations is high and agrees with that expected from the finite E
J
/E
C
ratio and finite rise time of the dc
pulses. In addition, we are able to demonstrate nearly 100% initial charge state polarization. We also present
a method to determine the relaxation time T
1
when it is shorter than the measurement time T
meas
.
DOI: 10.1103/PhysRevB.69.140503 PACS number共s兲: 85.25.Hv, 74.40.⫹k, 85.35.Gv
Although a large number of physical systems have been
suggested as potential implementations of qubits, solid-state
systems are attractive in that they offer a realistic possibility
of scaling to a large number of interacting qubits. Recently
there has been considerable experimental progress using su-
perconducting microelectronic circuits to construct artificial
two-level systems. A variety of relative Josephson and Cou-
lomb energy scales have been used to construct qubits based
upon a single-Cooper-pair box
1,2
and flux qubits based upon
a three-junction loop.
3,4
Coherence times of the order of
0.5
s have been achieved for a single-Cooper-pair box
qubit.
2
Rabi oscillations between energy levels of a single
large tunnel junction have also been observed.
5,6
Despite the
encouraging results, one aspect that is not well understood
concerns the contrast of the oscillations, which in all previ-
ously reported experiments is smaller than expected.
The experimental systems reported so far can also be dis-
tinguished by the readout method and the manner of per-
forming single-qubit rotations. The first demonstration of co-
herent control of a single-Cooper-pair box
1
共SCB兲 employed
a weakly coupled probe junction to determine the charge on
the island. In the more recent experiment reported by Vion
et al.,
2
the SCB was incorporated into a loop containing a
large tunnel junction, for which the switching current de-
pends on the state of the SCB. Switching current measure-
ments of superconducting quantum-interference devices
共SQUID’s兲 have also been used for flux and phase-type
qubits.
3–6
Nakamura et al.
1
performed single-qubit rotations
by applying very fast dc pulses to a gate lead in order to
quickly move the SCB into and away from the charge degen-
eracy point. This technique produces qubit rotations with an
operation time that can be of the order of the natural oscil-
lation period. Other experiments utilize microwave pulses to
perform NMR-like rotations of the qubit.
2,4–6
The latter ap-
proach requires less stringent microwave engineering, since
rf rotations can be accomplished with pulses that are more
than an order of magnitude slower in rise time and duration
than the natural oscillation period. By using fast pulses, how-
ever, single-qubit rotations can be performed ⬇20 times
faster than with rf rotations.
In this Rapid Communication, we report measurements
made on a SCB-type qubit with very fast dc pulses used to
effect the qubit rotations, as in Nakamura et al.
1
For our
qubit, however, the readout system consists of a single-
electron transistor 共SET兲 capacitively coupled to the SCB
and configured for radio-frequency readout 共rf-SET兲.Byin-
corporating a SET into a tank circuit, rf-SET electrometers
can be made that are both fast
7
and sensitive,
8
and are well
suited for SCB-qubit readout.
9,10
One advantage of using rf-
SET readout is that it can easily be turned on and off, al-
though for the measurements reported here it is operated in a
continuous way.
10
More significantly, the rf-SET is funda-
mentally different from qubit readout based on switching
currents in that it involves a weak measurement of the charge
in contrast to a yes or a no answer. This could be a significant
advantage for readout of multiple-qubit systems. For ex-
ample, a switching current measurement on one qubit would
also directly affect the other qubits due to its strong interac-
tion. Furthermore, switching current measurements are inher-
ently stochastic in time—an important consideration if one is
interested in measuring time correlations.
Electron-beam lithography and double-angle shadow
evaporation of aluminum films onto an oxidized silicon sub-
strate were used to fabricate the combined SCB-SET system
共see Fig. 1兲.
11
The SCB box consists of a low-capacitance
superconducting island connected to a superconducting res-
ervoir by two parallel junctions that define a low-inductance
SQUID loop. An additional gate lead placed near the island
is used to change the electrostatic potential of the island by a
gate voltage V
g
through a gate capacitance C
g
. By adjusting
the magnetic flux ⌽ through the loop, the effective Joseph-
son energy is tunable as E
J
⫽ E
J
max
兩
cos(
⌽/⌽
0
)
兩
, where ⌽
0
is
the magnetic-flux quantum (h/2e). Coupling of the SCB to
the SET was accomplished by extending part of the SET
island to the proximity of the box island and resulted in a
weak dimensionless coupling, C
I
/C
⌺
of 0.5–4 % for
samples with slightly different geometries and total SCB ca-
pacitances C
⌺
. C
I
is the capacitance between the SCB and
SET islands.
The samples were placed at the mixing chamber of a di-
lution refrigerator with a base temperature of ⬇20 mK. An
external superconducting magnet was used to produce a large
field parallel to the plane of the samples and a small super-
conducting coil close to the sample was used to produce a
RAPID COMMUNICATIONS
PHYSICAL REVIEW B 69, 140503共R兲共2004兲
0163-1829/2004/69共14兲/140503共4兲/$22.50 ©2004 The American Physical Society69 140503-1
field perpendicular to the SQUID loop. This allowed us to
independently suppress the superconducting gap of the alu-
minum film, and change the effective Josephson energy of
the Cooper-pair box. All control lines were filtered by a com-
bination of low-pass and stainless steel and copper-powder
filters. To present sharp pulses to the SCB gate, a high-
frequency coaxial line was used having 10 dB attenuation at
4.2 K and 20 dB at 1.5 K. The tank circuit had a resonant
frequency of 380 MHz and the rf-SET electrometer was bi-
ased near a feature in the IV vs V
g
SET
landscape known as the
double Josephson quasiparticle peak,
12
such that the bias cur-
rent through the SET was typically 200–300 pA. Under
these conditions, the electrometer sensitivity for the com-
bined SCB-SET system was found to be 50⫻ 10
⫺ 6
e/
冑
Hz
with a bandwidth of 10 MHz.
To make an artificial two-level system, the energy scales
of the SCB must be chosen so that k
B
T⬍ E
C
, E
J
⬎ ⌬, where
E
C
is the energy cost required to add a single electron to the
island and is set by total capacitance of the island E
C
⫽ e
2
/2C
⌺
.If⌬ is large enough compared to E
C
, then the
ground states will be even parity states that differ only by the
average number of Cooper pairs on the island. The effective
Hamiltonian of the box, including the Josephson coupling, is
given by
H⫽ 4E
C
兺
n
共
n⫺ n
g
兲
2
兩
n
典具
n
兩
⫺
E
J
2
兺
n
共
兩
n⫹ 1
典具
n
兩
⫹
兩
n
典具
n⫹ 1
兩
兲
, 共1兲
where we define gate charge as n
g
⫽ C
g
V
g
/2e⫺ n
0
, and n
0
is
the offset charge due to stray charges near the box. Figure
2共a兲 shows the ground-state and first excited-state energy
bands calculated using the Hamiltonian in Eq. 共1兲.Onac-
count of the 2e periodicity of the system, we limit the dis-
cussion to gate charge 0⭐n
g
⭐1.
In thermodynamic equilibrium, the actual quantity that
determines the island parity is the even-odd free energy dif-
ference ⌬
˜
(T), which differs from ⌬ due to entropic
considerations.
13
Measurements were made for several dif-
ferent samples with Coulomb energies E
C
/k
B
(E
C
/h) rang-
ing from 0.43 to 1.65 K 共9–34 GHz兲 and E
J
/E
C
ratios of 0.4
to 2.25. Coloumb staircases were measured by slowly ramp-
ing the box gate charge 共at a rate of ⬃40e/s), and measuring
the power reflected from the rf-SET tank circuit. Examples
of the measured box charge Q
box
(n
g
) is shown in Fig. 2共b兲.
Our samples show 2e periodicity when E
c
/k
B
⬍ 1 K, well
below the gap ⌬/k
B
⫽ 2.2 K. For samples with larger E
c
, the
FIG. 1. 共a兲 Scanning electron micrograph of a sample. The de-
vice consists of a Cooper-pair box and SET electrometer and was
fabricated from an aluminum 共lighter regions兲 evaporated onto an
oxidized Si substrate 共darker regions兲. 共b兲 Circuit diagram of box
and electrometer.
FIG. 2. 共a兲 Ground-state and first excited-state energies vs n
g
for
E
J
/E
C
⫽ 0.7 共solid line兲 and E
J
⫽ 0 共dotted line兲. 共b兲 Coulomb stair-
cases, Q
box
versus n
g
for a sample with E
c
/h⫽ 9.25⫾ 0.10 GHz
and E
J
max
/h⫽ 20.24⫾ 0.10 GHz. In 共i兲 a staircase taken from a
sample under microwave radiation with a frequency of 35 GHz is
shown. In 共ii兲, we show data where E
J
/h has been suppressed to
11.4 GHz by applying a small magnetic field through the SQUID
loop. The dotted lines in 共i兲 and 共ii兲 show the calculated Q
box
vs n
g
for the respective Josephson energies. 共iii兲 Measured Q
box
vs n
g
共solid line兲 when a fast pulse train is applied to the SCB gate having
an amplitude 0.88e, width ⌬t⫽ 130 ps, and repetition time T
R
⫽ 130 ns 共here E
J
/h⫽ 4 GHz). The dashed line is the measured
Q
box
without the pulse train. 共c兲 Avoided level crossing shown by
the positions of the spectral peaks and dips, ⌬n
g
⫽ (n
g
dip
⫺ n
g
peak
)/2, for varying microwave frequency. 共d兲 Josephson energy
E
J
vs applied flux ⌽/⌽
0
through the SQUID loop determined from
spectroscopic data 共circles兲 and coherent oscillation data 共crosses兲.
This sample had E
c
/h⫽ 17.45⫾ 0.13 GHz and E
J
max
/h⫽ 15.5
⫾ 0.2 GHz.
RAPID COMMUNICATIONS
T. DUTY, D. GUNNARSSON, K. BLADH, AND P. DELSING PHYSICAL REVIEW B 69, 140503共R兲共2004兲
140503-2
staircases acquire an extra step around n
g
⫽ 0.5 due to poi-
soning by nonequilibrium quasiparticles.
The characteristic energies E
C
and E
J
max
of the SCB can
be determined directly using microwave spectroscopy.
10
When monochromatic microwaves are applied to the SCB
gate, resonant peaks and dips occur in the measured Q
box
vs
n
g
staircase when the microwave photon energy matches
with the energy-level splitting
01
关see Fig. 2共b兲兴. As the
microwave frequency is reduced, the positions of these peaks
exhibit an avoided level crossing. Within the two-level ap-
proximation, the positions of the single-photon resonances
depend on the applied frequency
as ⌬n
g
⫽
冑
h
2
2
⫺ E
J
2
/8E
C
, where we used ⌬n
g
⬅(n
g
dip
⫺ n
g
peak
)/2.
An example of such an avoided level crossing is shown in
Fig. 2共c兲 where we have used least-squares fits to this equa-
tion to estimate the Coulomb and Josephson energies.
Measurements of time-resolved, coherent oscillations of
the charge on the SCB are made by slowly ramping the SCB
gate charge n
g
as before, while applying fast 共nonadiabatic兲
rectangular dc-voltage pulses of amplitude a in the form of a
pulse train to the SCB gate. When n
g
is far away from the
charge degeneracy and a such that n
g
⫹ a places the system
at the charge degeneracy, the leading and trailing edges of
each pulse act as successive Hadamard transformations. In
between, the system coherently oscillates between charge
eigenstates for a time ⌬t. The trailing edge of the pulse
returns the gate charge to n
g
with the system in a superposi-
tion of ground and excited states corresponding to the phase
of the oscillation acquired during ⌬t. After the pulse, the
excited-state component decays with a relaxation time T
1
.
The pulses are separated by a repetition time T
R
, typically
greater than T
1
.
Since the staircase is acquired by ramping n
g
on a time
scale much slower than T
R
, one measures an enhanced
charge ⌬Q
box
⫽ Q
box
on
⫺ Q
box
off
that is time averaged over the
pulse train and proportional to the probability of finding the
system in the excited state. Peaks in ⌬Q
box
vs n
g
will be
located at gate charges n
g
peak
such that approximately a half-
integer number of oscillations occur at gate charge n
g
peak
⫹ a during the time ⌬t. Figure 2共b兲共bottom curve兲 shows a
measured staircase using such a pulse train. When one mea-
sures such staircases for varying ⌬t, then time-domain oscil-
lations are evident in the ⌬t cross sections of ⌬Q
box
vs n
g
.
This is shown in Fig. 4共a兲, where we plot the time evolution
near the charge degeneracy point. As T
R
is increased, the
pulse-induced charge contributes less to the time average. A
simple calculation which ignores coherence effects remain-
ing after a time T
R
leads to the dependence
⌬Q
box
共
T
R
兲
⫽ 2n
0
T
1
T
R
1⫺ e
⫺ T
R
/T
1
1⫹ e
⫺ T
R
/T
1
, 共2兲
where n
0
is the initial peak amplitude. One can estimate T
1
at the readout gate voltage by measuring the dependence of
the peak amplitude on T
R
and fitting the data to this formula.
Using such a pulse-train allows a measurement of the
excited-state probability even when the relaxation time T
1
is
shorter than the measurement time. Population averaging and
mixing limit the maximum observed amplitude of the ⌬Q
box
oscillations to 1e rather than 2e, and this occurs only for an
ideal rectangular pulse, short repetition times, and vanishing
E
J
/E
C
ratio. We have numerically integrated the Schro
¨
-
dinger equation using pulses with a finite rise time to study
the various factors affecting both the contrast of the oscilla-
tions and the degree of charge state polarization. The results
are best understood when plotted on the Bloch sphere as
shown in Fig. 3. The dash-dotted lines show the evolution in
time of the state vector starting from an initial condition
共upward pointing arrows兲 and evolving through a maximum
polarization of the charge state 共downward arrows兲. The ini-
tial state differs from pure S
z
due to a finite E
J
/E
C
. Figure
3共a兲 shows evolution for a pulse that takes the system to the
charge degeneracy. One finds that a finite pulse rise time
mimics a higher E
J
/E
C
ratio and reduces the oscillation con-
trast. Nonetheless, Fig. 3共b兲 shows that by pulsing past the
charge degeneracy, one can achieve nearly 100% polariza-
tion.
Measurements of the T
R
dependence of ⌬Q
box
are shown
in Fig. 4共b兲, where we have used an appropriate pulse am-
plitude and duration to achieve nearly 100% initial polariza-
tion. We find T
1
⫽ 108⫾ 20 ns at the readout point n
g
⯝0.25
using a least-squares fit to Eq. 共2兲. The data of Fig. 4共a兲 show
an oscillation contrast of ⬇0.55e. Taking into account the
experimentally determined E
J
/E
C
and T
1
/T
R
, along with
the measured rise time
␦
t⯝35 ps of our pulse generator 共An-
ritsu MP1763C兲, we find good agreement between the ob-
served and expected initial oscillation contrast. Numerical
solutions of Schro
¨
dinger equation show that the reduction of
contrast due to the pulse rise time depends strongly on E
J
.
This was checked experimentally by varying E
J
and found to
be in agreement with the calculations. A fit of the data of Fig.
4共a兲 to an exponentially decaying sinusoid gives an oscilla-
tion frequency
0
⫽ 3.6 GHz and decoherence time of 2.9 ns.
Using fast dc pulse trains we were able to observe oscil-
lations in time out to 10 ns. The dependence of decoherence
time on gate charge at the top of the pulse is shown in Fig.
4共c兲 and has a sharp maximum at the charge degeneracy.
This tells us that low-frequency charge noise is the dominant
source of dephasing in our qubit. As discussed and demon-
strated by Vion et al.,
2
dephasing due to charge fluctuations
is minimum at the charge degeneracy. The coherence time
FIG. 3. 共a兲 Bloch sphere showing evolution about the charge
degeneracy for E
J
/h⫽ 6 GHz, E
C
/h⫽ 9 GHz, and a pulse rise
time of 35 ps. 共b兲 Evolution for a pulse that takes the system past
the charge degeneracy and achieves 100% charge state polarization
共downward pointing arrow兲.
RAPID COMMUNICATIONS
COHERENT DYNAMICS OF A JOSEPHSON CHARGE QUBIT PHYSICAL REVIEW B 69, 140503共R兲共2004兲
140503-3
found by Vion et al. is much larger than that found in our
measurements although their initial oscillation amplitude is
much smaller. If we extrapolate the gate charge dependence
of T
1
共not shown兲 to the charge degeneracy, we expect T
1
⬃10 ns. This implies that the decoherence time we measure
at the charge degeneracy is dominated by the short relaxation
time rather than pure dephasing.
Our measured T
1
was also found to be independent of
SET bias current in the subgap regime. Previously, a consid-
erably larger T
1
was measured in a similar sample using a
spectroscopic technique,
10
and found to agree with the theo-
retical T
1
⯝1
s expected from quantum fluctuations of a
50 ⍀ environment. The reduced T
1
observed here could be
attributed to several factors. One is coupling to the relatively
unfiltered high-frequency coaxial line, in addition to un-
wanted coupling to other metal traces and the microwave
environment of our sample. In practice it is difficult to esti-
mate the actual real part of the impedance on these leads at
such high transition frequencies. Another possibility con-
cerns quasiparticles. Our samples were fabricated without
quasiparticle traps and despite the observed static 2e period-
icity, there could be considerable nonequilibrium quasiparti-
cle dynamics occurring on these time scales. Finally, it may
be that the use of fast dc pulses leaves the bath of charge
fluctuators in a configuration having additional channels for
relaxtion of the qubit. Clearly what is needed is T
1
measure-
ments using a variety of techniques—rf rotations, spectros-
copy, as well as fast dc pulse—on the same sample.
In conclusion, we have fabricated and measured a solid-
state qubit based upon a SCB combined with a rf-SET read-
out system. Due to a relatively short T
1
, continuous mea-
surement of the SCB was employed. Fast dc pulses were
used to coherently manipulate the qubit and time-coherent
oscillations of the charge were observed. The initial contrast
of the oscillations is relatively large and quantitatively un-
derstood as due to finite E
J
/E
C
combined with the finite
pulse rise time. Furthermore, nearly 100% charge state po-
larization can be achieved. The oscillations show a maxi-
mum decoherence time at the charge degeneracy which indi-
cates that charge fluctuations dominate the dephasing rate.
We would like to acknowledge fruitful discussions with
R. Schoelkopf, K. Lehnert, A. Wallraff, G. Johansson,
A. Ka
¨
ck, G. Wendin, and Y. Nakamura. The samples were
made at the MC2 clean room. The work was supported by
the Swedish SSF and VR, by the Wallenberg and Go
¨
ran
Gustafsson foundations, and by the EU under the IST-
SQUBIT programme.
*
Electronic address: tim@mc2.chalmers.se
1
Y. Nakamura, Y.A. Pashkin, and J.S. Tsai, Nature 共London兲 398,
786 共1999兲.
2
D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina,
D. Esteve, and M.H. Devoret, Science 296, 886 共2002兲.
3
C.H. van der Wal et al., Science 290, 773 共2000兲.
4
I. Chiorescu, Y. Nakamura, C.J.P.M. Harmans, and J.E. Mooij,
Science 299, 1869 共2003兲.
5
Y. Yu, S. Han, X. Chu, S.-I. Chu, and Y. Wang, Science 296, 889
共2002兲.
6
J.M. Martinis, S. Nam, J. Aumentado, and C. Urbina, Phys. Rev.
Lett. 89, 117901 共2002兲.
7
R.J. Schoelkopf, P. Walgren, A.A. Kozhevnikov, P. Delsing, and
D.E. Prober, Science 280, 1238 共1998兲.
8
A. Aassime, D. Gunnarsson, K. Bladh, and P. Delsing, Appl.
Phys. Lett. 79, 4031 共2001兲.
9
A. Aassime, G. Johansson, G. Wendin, R.J. Schoelkopf, and P.
Delsing, Phys. Rev. Lett. 86, 3376 共2001兲.
10
K.W. Lehnert, K. Bladh, L.F. Spietz, D. Gunnarsson, D.I.
Schuster, P. Delsing, and R.J. Schoelkopf, Phys. Rev. Lett. 90,
027002 共2003兲.
11
K. Bladh et al., Phys. Scr. T102, 167 共2002兲.
12
A.A. Clerk, S.M. Girvin, A.K. Nguyen, and A.D. Stone, Phys.
Rev. Lett. 89, 176804 共2002兲.
13
P. Lafarge, P. Joyez, D. Esteve, C. Urbina, and M.H. Devoret,
Nature 共London兲 365, 6445 共1993兲.
FIG. 4. 共a兲⌬Q
box
vs ⌬t. The solid line is a least-squares fit to
an exponentially decaying sinusoid and gives a decay time T
2
⯝2.9 ns and frequency
0
⫽ 3.6 GHz. 共b兲 Dependence of the peak
height ⌬Q
box
on repetition time T
R
for ⬃100% initial polarization.
The solid line is a least-squares fit to Eq. 共2兲 giving T
1
⫽ 108
⫾ 20 ns. 共c兲 Decoherence time T
2
vs gate charge n
g
.
RAPID COMMUNICATIONS
T. DUTY, D. GUNNARSSON, K. BLADH, AND P. DELSING PHYSICAL REVIEW B 69, 140503共R兲共2004兲
140503-4
