Decoherence in dc SQUID phase qubits

Abstract

We report measurements of Rabi oscillations and spectroscopic coherence times in an Al/AlOx/Al and three Nb/AlOx/Nb dc SQUID phase qubits. One junction of the SQUID acts as a phase qubit and the other junction acts as a current-controlled nonlinear isolating inductor, allowing us to change the coupling to the current bias leads in situ by an order of magnitude. We found that for the Al qubit a spectroscopic coherence time T2* varied from 3 to 7 ns and the decay envelope of Rabi oscillations had a time constant T' = 25 ns on average at 80 mK. The three Nb devices also showed T2* in the range of 4 to 6 ns, but T' was 9 to 15 ns, just about 1/2 the value we found in the Al device. For all the devices, the time constants were roughly independent of the isolation from the bias lines, implying that noise and dissipation from the bias leads were not the principal sources of dephasing and inhomogeneous broadening. Comment: 5 figures. Accepted to Physical Review B

Full text

Page 1

1
Decoherence in dc SQUID phase qubits

Hanhee Paik, S. K. Dutta, R. M. Lewis, T. A. Palomaki, B. K. Cooper, R. C. Ramos,† H. Xu,**
A. J. Dragt, J. R. Anderson, C. J. Lobb, and F. C. Wellstood
Center for Nanophysics and Advanced Materials, and Joint Quantum Institute
Department of Physics, University of Maryland, College Park MD 20742-4111

Abstract
We report measurements of Rabi oscillations and spectroscopic coherence times in an
Al/AlOx/Al and three Nb/AlOx/Nb dc SQUID phase qubits. One junction of the SQUID acts as a
phase qubit and the other junction acts as a current-controlled nonlinear isolating inductor,
allowing us to change the coupling to the current bias leads in situ by an order of magnitude. We
found that for the Al qubit a spectroscopic coherence time
*
2 T varied from 3 to 7 ns and the
decay envelope of Rabi oscillations had a time constant ' T = 25 ns on average at 80 mK. The
three Nb devices also showed
*
2 T in the range of 4 to 6 ns, but ' T was 9 to 15 ns, just about 1/2
the value we found in the Al device. For all the devices, the time constants were roughly
independent of the isolation from the bias lines, implying that noise and dissipation from the bias
leads were not the principal sources of dephasing and inhomogeneous broadening.


Present Address: ** School of Applied Physics, Cornell University, Ithaca NY 14853

† Department of Physics, Drexel University, Philadelphia PA 19104
PACS numbers: 03.67.Lx, 03.65.Yz, 85.25.Dq

Page 2

2
As the size of a physical system increases beyond the atomic scale, the behavior typically
crosses over from the quantum to the classical limit.1 The scale at which this crossover occurs is
not, however, a fundamental one. In principle, a large object that is well isolated from
environmental influences can display quantum mechanical nature. Superconducting circuits
containing Josephson junctions provide good examples of quantum behavior at larger length
scales.2 Despite the coupling to two-level systems and dielectric layers,3,4 junctions that are tens
of micrometers on a side and which interact with other junctions hundreds of micrometers away
have shown superposition of quantum states as well as entangled quantum states.5-7 Although the
junctions were cooled to millikelvin temperatures, it is quite remarkable that in many
experiments, the junctions’ behavior was monitored by room-temperature amplifiers that were
attached through low-pass-filtered meter-length wire leads and still exhibited isolation from
room temperature noise. Such large-scale connections to a quantum system from a noisy
environment can only be tolerated if the quantum system is sufficiently decoupled from the leads,
but not so decoupled that measurements become impossible.
In this paper, we have measured Rabi oscillations and spectroscopic coherence times in
order to examine the effects of varying the coupling between a quantum system and noise from
the bias leads. Our quantum system is a dc SQUID phase qubit8 made with Al/AlOx/Al or
Nb/AlOx/Nb junctions. In a dc SQUID phase qubit [see Fig. 1(a)], one junction 1J is a phase
qubit9 and the rest of the SQUID circuit (fixed inductance
1 L , isolation junction 2J
with
parallel capacitance
2
C , and parasitic inductance
2 L ) serves as an on-chip inductive isolation
network providing variable isolation from the bias leads.
When the applied flux Φa is held constant, fluctuations I
∆ in the bias current cause small
changes
1I
∆ in the current
1I flowing through the qubit junction 1J . It is useful to define the

Page 3

3
current noise power isolation factor
()2
1
∆I
∆IrI≡
. If the frequency f of the current
fluctuations is much less than the resonance frequency of either junction, then
2
2J2
2J21
2
2J2
2J211J
I
LL
LL
+
L
LL
LLLL
r






++
≈






+
+++
=
,
(1)
where
1J1
LL >>
, as appropriate for our devices. Here

2
2
2
02
o

=
2J
II
π 2
Φ
L
−
(2)
is the Josephson inductance of the isolation junction,10
02
I is the critical current of the isolation
junction,
2I is the current going through the isolation junction, and
1J L
is the Josephson
inductance of the qubit junction. We neglect the mutual inductance between the arms of the
SQUID.
Equation (1) implies that the current noise power from the bias leads is reduced by a
factor of
Ir through the inductive isolation network before it reaches the qubit junction and the
effective impedance that the bias leads present to the qubit junction is stepped up by a factor of
Ir . If the current on the leads has a noise power spectral density
)f(SI
, then the current noise
power spectral density
)f(S1J
that reaches the qubit junction 1J is
2
J221
J22
I
I
I
J1
LLL
L
+
L
+
)f(S
r
)f(S
)f(S






+
≈=
. (3)
We can vary
Ir and thus
)f(S1J
in situ because
2J L
can be changed by varying the current
2I
through the isolation junction [see Eq. (2)]. Good isolation can be achieved by choosing
22J1
LLL
+ >>
and the best isolation occurs at
0I2=
, where
2J L
is a minimum. The choice

Page 4

4
1J1
LL >>
also ensures that the qubit and isolation junctions are decoupled by the relatively large
loop inductance. In this limit the qubit junction behaves like a single Josephson junction qubit.11
Figure 1(b) shows schematically the corresponding potential energy and energy levels of the 1-D
tilted washboard potential experienced by the qubit junction.12
A simple characterization of the effects of noise on the qubit is given by measuring the
spectroscopic coherence time,

f
1
∆
T
*
2
π
≡
(4)
where f ∆ is the full-width in frequency at the half-maximum of the 0 to 1 spectroscopic
resonance peak, measured in the low-power limit. Dephasing,13 inhomogeneous broadening,14
dissipation,15 and power broadening14 all contribute to the spectroscopic width of resonant
transitions.16 Dissipation is the loss of energy by the qubit with timescale
1 T ; dephasing refers to
the loss of phase coherence caused by noise at frequencies comparable or faster than
1 T/1
; in
contrast, inhomogeneous broadening is due to slow variations in the energy level spacing.
Which effect dominates depends on the nature of the measurement and the frequency range of
the noise.16,17
If
)f(S1J
is constant below a cut-off frequency
1c
T/1f <<
and inhomogeneous
broadening dominates decoherence, then
*
2 T
is given by16

1
1
01
I
cI
2/1
I
1
1
01
I
J1
*
2
f
(0)fS 2π
1.65


r
f
σ 2π
1.65


T
−−



∂
∂
⋅=



∂
∂
⋅=
. (5)
Here
cIIc1J1J
f )0(Sr2 f )0(S2
ππσ
==
is the rms current noise in
1I due to
)f(S1J
and
01
f
is the 0 to 1 transition frequency. Equation (5) shows that
*
2 T will depend on
1I because12

Page 5

5

4/1
2
01
1
p 01
I
I
1ff














−≈
. (6)
From Eq. (6), we find







≈
∂
∂
2
01
1
3
01
4
p
1
01
I
I
I
f
f
2
1
f
, (7)
where
1o 01p
C2If
Φπ
=
is the unbiased qubit junction's plasma frequency. We note that for
typical biasing conditions
1I is nearly equal to
01
I and in this limit
1 01
If
∂∂
varies rapidly as a
function of
1I or
01
f . From Eqs. (5) and (7), we expect that
*
2 T would vary as
3
01
f and scale
with
Ir if
)f(S1J
is the principal noise source in our qubit.
In contrast, if
)f(S1J
has a cutoff frequency
1c
T/1f >>
, the effect is to produce
dephasing with
*
2 T given by16
2
1
01
I
I
I
2
1
*
2
f
r
)0(S
π
T2
1
T
1






∂
∂
+=
, (8)
where
1 T is the energy relaxation time. In the case of noise dominated decoherence,
*
2 T will
behave as
6
01Ifr
, which varies even more rapidly than was the case for Eq. (5).
Additional information about current noise and dissipation from the leads can be obtained
by examining Rabi oscillations. The decay time constant ' T
of the envelope of Rabi oscillations
is sensitive to noise at the Rabi frequency, while the shape of the envelope is affected by
inhomogeneous broadening from low frequency noise.17,18 When both dephasing and dissipation
are present, the Rabi decay constant T′ is related to the energy relaxation time
1 T and the
coherence time
2 T by

Page 6

6
21
T2
1
T2
1
T
1
+=
′
(9)
when there is zero detuning.19,20,21 Here
2 T corresponds to the conventional definition of the
coherence time used in NMR and appears in the Bloch equations where
12
T2T =
if decoherence
is solely due to dissipation.21
Since T′ and
1 T can be measured separately, Eq. (9) allows us to obtain information on
2 T even in a system where it is difficult to perform a clean spin-echo measurement. If current
noise in the leads is the dominant source of decoherence, we expect T′ to scale with
)(
RII
fSr

where
π
2
Ω
/f
RR=
is the Rabi frequency,17 while if dissipation associated with the lead
impedance is the dominant source of decoherence, we expect
1 T ,
12
T2T =
and
3/T4 ' T
1
=
to
scale with
0IZr
, where
0
Z is the impedance of the leads at the transition frequency.
To examine the contribution of the leads to decoherence, we measured four dc SQUID
phase qubits. Device AL1 [see Fig. 1(c)] was made in our laboratory using photolithography
followed by double-angle evaporation of approximately 50 nm thick Al films on an oxidized Si
substrate. The 40 µm x 2 µm Al/AlOx/Al qubit junction had a zero-field critical current of
01
I =
21.28 µA and the device had a single-turn square loop with a 3 µm-width line and sides 300 µm
long. Devices NB1, NB2, and NBG were made by Hypres, Inc., from Nb/AlOx/Nb trilayers.
Devices NB1 [see Fig. 1(d)] and NB2 had similar layouts consisting of a 6-turn SQUID loop that
formed the isolation inductance
1 L . Device NBG was configured as a gradiometer, with two 6-
turn loops in series, wound oppositely to make the device relatively insensitive to uniform
external magnetic fields.22 In device NB1 the trilayer had a nominal critical current density of
100 A/cm2 while for the other two niobium devices the critical current density was nominally 30

Page 7

7
A/cm2. For NB1 only, we suppressed the critical currents of the qubit and isolation junctions by
applying a small magnetic field in the plane of the junctions so that
01
I = 34.4 µA for the 10 µm
x 10 µm qubit junction; the initial value was
01
I = 108 µA. The devices were measured in two
separate dilution refrigerators using similar detection electronics, microwave filters, and wiring.
Devices AL1 and NBG were measured on an Oxford Instruments Kelvinox 25 at a base
temperature of 80 mK, while NB1 and NB2 were measured on an Oxford Instruments model 200
at a base temperature of 25 mK. Each refrigerator was enclosed in an rf-shielded room and the
devices were shielded against low-frequency magnetic noise by means of a superconducting
aluminum sample box and a room-temperature mu-metal cylinder.
For each device, we first measured the current-flux switching characteristics. We found
the inductance parameters and rough estimates for the critical current of each junction (see Table
I) by fitting the complete
Φ
−
I
curve to the expected characteristics of an asymmetric dc
SQUID.23 We then measured the transition spectrum and Rabi oscillations. For these
measurements we simultaneously ramped the current and flux in the appropriate ratio so as to
increase the current through the qubit junction linearly with time while keeping the current
through the isolation junction fixed.8,24 Because of the shape of the qubit’s washboard potential,
higher energy states are more likely to escape via tunneling to the voltage state than lower energy
states. As the current through the qubit junction increases, the tilt of the washboard potential
increases, decreasing the barrier height and causing the tunneling rates for all levels to increase.
We recorded the time at which the switching voltage occurred and from this found the current at
which the device tunneled. Repeating this sequence of order 105 times yields a histogram of
switching events as a function of current, which we then use to construct a total escape rate Γ
versus current
1I .25 Since the relatively large loop inductances and critical currents (see Table I)

Page 8

8
allowed for multiple possible levels of trapped flux in the loop, we used a flux shaking method 24,
26 to initialize the SQUIDs into a desired flux state before each measurement was made.27
To vary the isolation from the leads, we used two techniques. For the Nb devices, the
measurement was done on each flux state that corresponds to a different but reproducible amount
of current circulating in the loop, causing different
2I and
Ir values for the same current
1I in
the qubit junction. For device AL1, we first used flux shaking to initialize the SQUID into the
zero flux state, corresponding to no circulating current in the loop. We next applied a small static
offset flux to the SQUID to induce circulating current in the loop, thereby driving current
through the isolation junction to set
Ir .
Figure 2(a) shows the total escape rate Γ versus
1I in device AL1 measured at 80 mK
for isolation
Ir = 1000 (solid curve) and
Ir = 200 (dots). We did not apply microwaves for either
of these "background" escape rate curves. The
Ir = 200 curve shows an overall increase in the
escape rate compared to the
Ir = 1000 curve, as expected if high-frequency noise was present on
the bias leads. The broad peaks at 21.02 µA and 21.07 µA in the
Ir = 200 curve varied in size and
location depending on the isolation factor. In separate experiments, we found that these
anomalous peaks in the background escape rate were due to noise-induced populations in the 2
and 3 states caused when the 0 to 2 or 1 to 3 transition frequency of the qubit matched
the 0 to 1 resonance frequency of the isolation junction.11,28,29 We note that the total escape
rate is given by
∑
i
=
iiΓρΓ
where
i ρ and
i Γ are the normalized occupation probability and
escape rate from level i. Since the escape rates increase by two to three orders of magnitude for
each successive level, very small populations in the excited states are detectable. For example, in

Page 9

9
Fig. 2(a) at
1I = 21.07 µA we estimate that the probability of occupying 2 increases by only
about 50 parts per million when
Ir changes from 1000 to 200.
We next measured the total escape rate Γ versus
1I while applying microwaves for
Ir =
1000 [see Fig. 2(b)]. For all the devices, clear resonant peaks were found in the 6-8 GHz range
and the dependence of the 0 to 1 transition on current was in good agreement with that
expected for a single Josephson junction. We fit the data to the expected spectrum of a single
Josephson junction to determine the qubit critical current and capacitance (see Table I). With the
power set low enough that power broadening was not apparent, we measured f ∆ for the 0 to
1 transition and then applied Eq. (4) to obtain
*
2 T .
Figure 3(a) shows
*
2 T versus
Ir for device AL1 measured at 80 mK and 7.45 GHz. We
find that
*
2 T varies between 3 ns and 7 ns in an apparently random fashion as
Ir varies by an
order of magnitude. We note that
*
2 T showed neither the linear, nor the square root dependence
on
Ir , predicted by Eqs. (5) and (8). For comparison, Fig. 3(b) shows
*
2 T versus
Ir for device
NB1 measured at 7.2, 7.3, 7.4, and 7.5 GHz at 25 mK. It shows an apparent random variation
between about 3 and 6 ns. Devices NB2 and NBG were measured at fixed isolations of
Ir = 2500
and
Ir = 2300, respectively, and showed
*
2 T values that were about the same as for NB1 (see
Table I). Furthermore, none of the four devices showed the strong systematic dependence of
*
2 T
on
1I or
01
f predicted by Eqs. (5) or (8) [see Fig. 3(b), for example].22
Finally, we measured Rabi oscillations in the four devices by applying microwave current
at the 0 to 1 transition frequency and then monitoring Γ as a function of time from when
the microwaves started. Figure 4 shows Rabi oscillations in the escape rate for device AL1 at 80

Page 10

10
mK with a 7 GHz drive for
Ir = 1000 [Fig. 4(a)] and for
Ir = 200 [Fig. 4(b)]. The solid curves
are χ2−fits to our phenomenological model for the oscillations
[]
{}
)e (1g)t (t
R
cose1gg
Γ
000
)/Tt -(t
20
)/T't -(t
10
−−
−+−−+=
Ω
. (10)
The fitting parameter T′ gives the decay time constant of Rabi oscillations,
1 g sets the
amplitude of the oscillations, and t0 is the time when the microwave power was turned on. The
parameter
0 T and the
)e (1g
00)/Tt -(t
2
−
−
term account for the rise time of the microwave pulse, and
emulate the effect of increased occupancy in higher levels such as 2 at high microwave power.
The parameter
0 g
combined with
)e (1g
00)/Tt -(t
2
−
−
accounts for the initial background escape
rate, and includes contributions from 0 , 1 , and higher levels caused by noise or thermal
excitation. We found that including
0 g and
2
g significantly improved the quality of the fits but
had little effect on the estimated Rabi decay time T′.
Comparing Figs. 4(a) and 4(b), we note the increased escape rate at t= 0 for
Ir = 200, as
expected from an increase in noise-induced transitions to an excited state due to a decrease in the
isolation. However, fitting Eq. (10) to the data yields T′= 33 ns for the
Ir = 200 curve and T′=
28.2 ns for the
Ir = 1000 curve. We also measured Rabi oscillations for
Ir = 1000 at six different
Rabi frequencies (from 111 MHz to 188 MHz) and for
Ir = 200 at twelve different Rabi
frequencies (from 33 MHz to 122 MHz). The range of T′was 20 ns to 28 ns for
Ir = 1000 and 20
ns to 33 ns for
Ir = 200. Thus the decay time constant of Rabi oscillations did not scale with
Ir
or
Ir .
If the decoherence were entirely due to dissipation, we would expect the coherence time
12
T2T =
, and the Rabi decay time
3/T4 ' T
1
=
. To test whether our qubits are dissipation

Page 11

11
limited, we found
1 T by measuring a series of background escape rates at temperatures from 80
mK to 200 mK in the maximally isolated case (
Ir = 1000). We noted the size and location of the
shoulder of each thermal escape rate to estimate
1 T .30 In device AL1, this procedure yielded
≈
1 T
20 ns, a value for which
≈=
3/T4 ' T
1
27 ns.
We also observed that T′was independent of the isolation factor
Ir in the Nb devices.
For example, Fig. 5 shows Rabi oscillations in NB1 at 25 mK with a 7.6 GHz microwave drive
for
Ir = 1300 [Fig. 5(a)] and
Ir = 450 [Fig. 5(b)]. Fitting to Eq. (10) yields
≈
' T
12 ns for
Ir =
1300 and
≈
' T
15 ns for
Ir = 450. We found T′from 9 to 15 ns for
Ir in the range of 50 to 1300.
As with device AL1, no apparent systematic dependence of T′on the isolation was shown in
NB1.
However, there was one significant difference in behavior between NB1 and AL1. From
1 T measurements, we found
≈
1 T
14 ns for NB1. If the decoherence were entirely due to
dissipation, we would expect
≈=
3/T4 ' T
1
19 ns, which manifestly disagrees with the observed
T′ data for this device. Other Nb devices such as NB2 and NBG showed qualitative behavior
and quantitative results that were very similar to device NB1 (see Table I). This disagreement
means that an additional dephasing mechanism is present beyond that due to dissipation.
Table I summarizes the parameters and main results for all four devices. The fact that T′
and
*
2 T did not depend systematically on the isolation from the leads implies that neither current
noise from the leads nor dissipation in the leads is the main source of decoherence in these
devices, even though we observe clear noise-induced transitions in the escape rate that vary with
the isolation. We also note that
1 T was in the 15 to 20 ns range for all four devices, but the
aluminum qubit AL1 showed a substantially longer Rabi decay time T′than the Nb devices.

Page 12

12
Another possible source of decoherence is local 1/f flux noise of unknown origin that has
been found in other SQUIDs at millikelvin temperatures.31,32 Decoherence from such a source
would be largely independent of
Ir but would depend on
1 01
If
∂∂
, which in turn depends
strongly on
01
f or
1I . Our data do not support such a dependence. Also, T′ and
*
2 T for the
gradiometer NBG were very comparable to those for magnetometers NB1 and NB2, but shorter
than for magnetometer AL1.22 This strongly suggests that spatially uniform flux noise was not
responsible for the short coherence times in our dc SQUID phase qubits.
Simmonds et al. and Martinis et al.3,4 have pointed out that the likely source of
decoherence in phase qubits is spurious two-level charge fluctuators that reside in the substrate
or dielectric layers. The fact that T′for device AL1 was two times longer than for the three Nb
devices is suggestive of a materials related effect. AL1 had a thermally grown AlOx tunnel
barrier, native oxide on the exposed metal surfaces, and the thermally grown SiO2 layer on a Si
substrate but no wiring insulation layer. In contrast, the Nb devices had all of the above plus
sputtered SiO2 insulation layers between the wiring layers. While we have not seen clear
spurious resonant splittings (down to a resolution of about 10 MHz) in spectroscopic data on
AL1, we have identified small apparent splittings of about 5-10 MHz in NB1.33
In conclusion, we have measured the spectroscopic coherence time
*
2 T and the time
constant T′ for the decay of Rabi oscillations in four dc SQUID phase qubits with variable
coupling to the leads. From these measurements we can determine the impact of the leads on
noise and decoherence in the dc SQUID phase qubits. We found that varying the isolation from
the leads produced no systematic effect on either
*
2 T or T′, and that with comparable isolation,
the aluminum device had a coherence time
2 T that was two to three times longer than that of the
Nb devices. This implies that the leads are not the dominant source for decoherence in these

Page 13

13
devices. Instead, our data are consistent with a local, materials related source of decoherence.
This work was funded by the NSF through the QuBIC Program (grant number:
EIA0323261), the NSA through the Laboratory for Physical Sciences, and the state of Maryland
through the CNAM, formerly the Center for Superconductivity Research. We acknowledge
many helpful conversations with A. J. Przybysz, H. Kwon, K. Mitra, F. W. Strauch, P. R.
Johnson, B. Palmer, M. Manheimer, M. Mandelberg, J. M. Martinis and R. W. Simmonds.

Page 14

14
References
[1] A. J. Leggett, Prog. Theor. Phys. Suppl. 69, 80 (1980).
[2] J. Q. You and F. Nori, Phys. Today 58, 42 (2005).
[3] R. W. Simmonds, K. M. Lang, D. A. Hite, S. Nam, D. P. Pappas, and J. M. Martinis, Phys.
Rev. Lett. 93, 077003 (2004).
[4] J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K.
Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev. Lett. 95, 210503
(2005).
[5] A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R.
Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, Science 300, 1548 (2003).
[6] M. Steffen, M. Ansmann, R. C. Bialczak, N. Katz, E. Lucero, R. McDermott, M. Neeley, E.
M. Weig, A. N. Cleland, and John M. Martinis, Science 313, 1423 (2006).
[7] M. A. Sillanpää, J. I. Park, and Raymond W. Simmonds, Nature 449, 438 (2007).
[8] J. M. Martinis, S. Nam, J. Aumentado, and C. Urbina, Phys. Rev. Lett. 89, 117901 (2002).
[9] R. C. Ramos, M. A. Gubrud, A. J. Berkley, J. R. Anderson, C. J. Lobb, and F. C. Wellstood,
IEEE Trans. Appl. Supercond. 11, 998 (2001).
[10] M. Tinkham, Introduction to Superconductivity, 2nd Ed. (McGraw-Hill, New York, 1996).
[11] K. Mitra, F. W. Strauch, C. J. Lobb, J. R. Anderson, and F. C. Wellstood, submitted to Phys.
Rev. B. (2008)
[12] J. M. Martinis, M. H. Devoret, and J. Clarke, Phys. Rev. B 35, 4682 (1987).
[13] Y. Makhlin, G. Schon, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001).
[14] L. Allen and J. H. Eberly, Optical Resonances and Two-level Atoms, New Ed. (Dover,
Mineola, NY, 1987).

Page 15

15
[15] U. Weiss, Quantum Dissipative Systems, 2nd Ed. (World Scientific, River Edge, NJ, 1999).
[16] H. Xu, A. J. Berkley, R. C. Ramos, M. A. Gubrud, P. R. Johnson, F. W. Strauch, A. J. Dragt,
J. R. Anderson, C. J. Lobb, and F. C. Wellstood, Phys. Rev. B 71, 064512 (2005).
[17] J. M. Martinis, S. Nam, J. Aumentado, K. M. Lang, and C. Urbina, Phys. Rev. B 67,
094510 (2003).
[18] D. J. Van Harlingen, T. L. Robertson, B. L. T. Plourde, P. A. Reichardt, T. A. Crane, and J.
Clarke, Phys. Rev. B. 70, 064517 (2004).
[19] H. C. Torrey, Phys. Rev. 76, 1059 (1949).
[20] R. A. Smith, Proc. Roy. Soc. London A362, 1(1978).
[21] A. Abragam, The Principles of Nuclear Magnetism (Clarendon, Oxford, 1961).
[22] H. Paik, B. K. Cooper, S. K. Dutta, R. M. Lewis, R. C. Ramos, T. A. Palomaki, A. J.
Przybysz, A. J. Dragt, J. R. Anderson, C. J. Lobb, and F. C. Wellstood, IEEE Trans. Appl.
Supercond. 17, 120 (2007).
[23] W. Tsang and T. Van Duzer, J. Appl. Phys. 46, 4573 (1975).
[24] T. A. Palomaki, S. K. Dutta, H. Paik, H. Xu, J. Matthews, R. M. Lewis, R. C. Ramos, K.
Mitra, P. R. Johnson, F. W. Strauch, A. J. Dragt, C. J. Lobb, J. R. Anderson, and F. C.
Wellstood, Phys. Rev. B 73, 014520 (2006).
[25] T. A. Fulton and L. N. Dunkleberger, Phys. Rev. B 9, 4760 (1974).
[26] V. Lefevre-Seguin, E. Turlot, C. Urbina, D. Esteve, and M. H. Devoret, Phys. Rev. B 46,
5507 (1992).
[27] C. D. Tesche and J. Clarke, J. Low Temp. Phys. 29, 301 (1977).
[28] T. A. Palomaki, S. K. Dutta, R. M. Lewis, H. Paik, K. Mitra, B. K. Cooper, A. J. Przybysz,
A. J. Dragt, J. R. Anderson, C. J. Lobb, and F. C. Wellstood, IEEE Trans. Appl. Supercond.