of 22
Supplementary Methods
Sa
mple fabrication
The substrate is composed of a 100nm thick low-stress SiN layer on
top of a Si wafer.
The Cooper
-Pair Box (CPB) is patterned us
ing electron-beam lithography and double-angle
evaporation of aluminum.
31
The thickness of the island and th
e ground leads are ~ 60 nm and ~
20 nm respectively. The island is coupled
to the ground leads via two small (~ 100 x 100
nm
2
) Al/AlO
x
/Al Josephson tunnel junctions, and is
arranged in a DC-SQUID configuration.
The aluminum layer used to define the nano
resonator
, and which ultimately serves as
the electrode on top of the nanoresonator, is patter
ned in the same step as the CPB. This layer
acts as an etch mask for undercutting the nanor
esonator. To protect the CPB during etching, a
layer of PMMA is spun on the sample, and a small window defining the nanoresonator is
opened using a second e-beam lithography step. The nanoresonator is then undercut in an
ECR etcher with Ar/NF3 plasma: The first step
is an anisotropic SiN etch that defines the
resonator beam; and the second is an isotropic et
ch of the underlying Si to undercut the beam.
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Sample characteristics
Nanoresonator
fundamental in-plane resonance frequency.
/ (2 )
58 MHz
NR
ωπ
=
Nanoresonator effective spring constant given by
~60
K
N
/
m
2
g
eom
NR
K
α
M
ω
=
, where
16
910
geom
M
kg
≈×
is the estimated
geometrical mass and
α
=.48 is found by assuming that the
CPB couples to the aver
age displacement of the
nanoresonator over the length of the CPB island.
Q
30,000-60,000
Typical nanoresonator qual
ity factor for the range of
T
mc
,
V
GNR
, and
V
NR
used.
Estimated CPB charging energy from spectroscopy (lower
bound) and LZ measurements (upper bound).
/
13 15 GHz
C
Eh
≈−
Estimated CPB maximum Josephson energy from
spectroscopy.
0
/
13 GHz
J
Eh
Measured nanoresonator-to-CPB capacitance.
43 aF
NR
C
=
Resonator/CPB spacing.
~300nm
d
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Derivative of nanoresonato
r/CPB capacitance estimated from
FEMLAB.
/~
NR
Cx
∂∂
40 50
p
F
/
m
Measured CPB gate capacitance.
17 aF
cpb
C
=
/(2 ) ~.3 2.3MHz
λπ
Coupling strength calculated us
ing Eq.1 and the parameters
/
14 GHz
C
Eh
=
,
60 N / m
K
=
,
50
NR
C
x
=
p
F
/
m
, and
215
NR
V
V
=
.
/(2 ) ~.5 3MHz
λπ
Coupling strength calculated using Eq. 3 and
data displayed in the inset of Fig. 1c, assuming CPB biased at
charge degeneracy and
Δ
/(
2 )
NR
ωπ
/
12 GHz
J
Eh
=
.
Circuit and measurement description
The measurement set-up is shown in Fig.S1. We use capacitive displacement
transduction and radio frequency reflectom
etry to
measure the nanomechanical frequency shift
.
Δ
/(
2 )
NR
ωπ
32
A DC potential difference
NR
GNR
VVV
=
Δ
is applied across the capacitor
C
GNR
, coupling the nanoresonator’s
motion to the charge on the actuation-detection electrode.
Upon application of
to the electrode, the respons
e of the coupled nanoresonator-
electrode system can be modeled as a series RLC circuit with impedance
)(
ω
V
RF
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()
22
()
/
NR
NR
NR
NR
Z
ω
jL
ωω ω
R
=−+
.
32
Here
and
, where
is the spacing between the nanoresonator and
electrode and
α
22
2
/(
Δ
)
NR
GNR
NR
GNR
GNR
RKd
ω
Q
α
CV
2
/
NR
NR
NR
LQR
ω
=
nm
d
GNR
100
~
GNR
is a geometric coefficient of order
unity. For typical values of coupling
voltage used in our experiment,
Δ
~5 10V
V
,
0
25.~
ZM
R
NR
>>
, where
is
the characteristic impedance of the transmission
line feeding the actuation-detection electrode.
0
40
Z
=
To overcome the impedance mismatch, we use an LC network to
transf
orm
closer to
)
(
NRNR
ω
Z
0
Z
. The LC network is formed
by a commercial, copper-wound
coil (nominally
L
T
= 5.6
μ
H) and the stray capacitance (
C
T
~ 1.0 pF) of the PC board upon
which the sample sits. The components are chosen so that
1/ (
)
LC
T
T
NR
ω
LC
ω
=≈
. Thus
when the RF drive f
requency is tuned into reso
nance with the nanoresonator and LC circuit,
, the total impedance seen looking from
the transmission line into the impedance
matching network is given by
LC
NR
ωωω
==
(
)
/ (
)
NR
T
T
NR
Z
ω
LCR
=
, where it is assumed for simplicity that
the matching network is lossless. For our
parameters, we estimate that we achieve
.
102~)(
NR
ω
Z
For perfect matching to the transmission line,
0
)(
Z
ω
Z
NR
=
, the reflection coefficient
. However, a small change
in the nanoresonator’s
frequency
leads to a correction given by
ΔΓ
0
Γ
(( )
)/(( )
) 0
NR
NR
Z
ω
ZZ
ω
Z
=− +
0
=
π
Δ
/(
2 )
NR
ω
Δ
/
NR
NR
jQ
ωω
. Thus a shift in the
nanomechanical frequency
can be monitored by tracking the shift
Δ
/(2 )
NR
ωπ
RF
r
VV
ΔΓΔ
=
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in the reflected component of the RF drive signal. This is accomplished by embedding the
nanoresonator in a phase-lock loop (Fig. S2).
Essentially, a directional coupler routes the
reflected signal to a phase-sensitive detector,
the output of which is fed into a VCO that
supplies
.
)(
ω
V
RF
The measurement is done on a dilution refri
gerator located inside of a RF-shielded
room. Four DC lines are used:
for biasing the CPB;
for controlling the coupling
between the CPB and nanoresonator;
for controlling the coupling between the
nanoresonator and detection circuit; and
cpb
V
NR
V
GNR
V
f
lux
V
(not shown) for biasing a homemade current
source to energize the solenoid for application of flux
Φ
to the CPB.
and
cpb
V
f
lux
V
are
supplied by a DAC card and routed into the shielded room using optical isolators.
and
are supplied by batteries located in the shielded room.
,
, and
are
filtered at room temperature (RT) at the input to the dilution refrigerator using commercial Pi-
filters and Cu powder filters.
NR
V
GNR
V
cpb
V
NR
V
GNR
V
33
From RT to the mixing chamber (MC), these lines are each
composed of ~ 2 meters of lossy stainless steel
coax, with two stages of Cu powder filters (at
1K and MC). The typical total attenuation from RT
to MC is measured to
be ~-100 dB at 10
GHz.
High frequency lines providi
ng radio frequency excitation
of the nanoresonator
and microwave excitation
RF
V
μ
V
of the CPB are routed into the
shielded room through DC blocks
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and into the dilution refrigerator via commerci
al high-pass filters and attenuators. They are
attenuated and thermalized at 1 K using cryogenic-compatible attenuators. At 1K,
is fed
into the coupled port (-20 dB) of
a directional coupler for the reflectometry measurements: The
Output port of the directional coupler feeds
to the actuation gate electrode of the
nanoresonator for excitation; And
the Input port of the directiona
l coupler feeds the reflected
component
through a Miteq AFS-series cryogenic
amplifier to the room temperature
electronics that compose the pha
se-lock loop (Fig. S2). Bias-tees anchored to the MC are
used to combine
RF
V
RF
V
r
V
μ
V
and
with
and
respectively. To provide additional
filtering between the MC and sample stage, all
lines are fed through lossy stainless steel coax.
RF
V
cpb
V
GNR
V
Supplementary Discussion
Model of dispersive shift incor
porating the full C
PB Hamiltonian
The spin-1/2 approximation for the CPB
Ha
miltonian is only appropriate in the
vicinity of a charge degeneracy point. Thus to model the dispersive frequency shift
of the nanoresonator over the
full range of
gate voltages
used in the
experiment (e.g. Fig. 2b) we employ the full expressions for the CPB
Δ
/(
2 )
NR
ωπ
cpb
V
34
and interaction
Hamiltonians, given respectively by
()
()
+++−
−−
=
n
J
NR
cpb
C
CPB
nnnn
E
nnnnnE
H
1
1
2
4
ˆ
2
(S1)
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and
()
(
)
nnaannn
λ
H
n
NR
cpb
⋅+−−
−=
j
ˆˆ
2
ˆ
int
h
(S2).
To account for contributions from charge states other than
n
and
n+1
, 11 adjacent charge states
are included in the model. Using Matlab, we first calculate numerically the eigenstates of the
uncoupled Hamiltonian
as a function of a
and
Φ
, where
remains as defined in the main text. The correction to the energy levels due to
and hence
the dispersive shift of the nanoresonator
are then calculated using second-order
time-independent perturbation theory. It is importa
nt to note, that in the vicinity of a charge
degeneracy point,
, use of the full CPB Hamiltonian
results in < 5% correction
to
calculated from Eq. 3 (at de
generacy the correction is
S
1%).
NR
CPB
HHH
ˆˆˆ
+=
cpb
V
NR
H
ˆ
int
ˆ
H
Δ
/(2 )
NR
ωπ
π
π
75
.
25
.
≤≤
CPB
n
Δ
/(
2 )
NR
ω
Periodicity of dispersive shift in V
cpb
and
Φ
Figure S3a displays the 2
e
-to-
e
transition in periodicity of nanoresonator
’s frequency shift
with respect to
, taken with flux biased so that
Δ
/(
2 )
NR
ω
cpb
V
0
~
J
J
EE
.
At a mixing
chamber temperature of 120 mK, the periodicity is
primarily 2e-periodic with peaks spaced by
, corresponding to a gate capacitance
Δ
18.7 mV
cpb
V
=
17.1 aF
g
C
=
. At temperatures of 40
mK and below, the quasiparticle poisoning ra
tes greatly exceed the
measurement time, and
periodicity is primarily e-periodic with peaks spaced by
Δ
910mV
cpb
V
.
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Figure S3b displays the pe
riodicity in applied flux
Φ
of
taken at a charge
degeneracy point. The average spaci
ng between the peaks is found to be
Δ
.
While there is large uncertainty in the effectiv
e area of the split-juncti
on CPB, we can estimate
the flux periodicity using the
geometric area of the CPB loop
A
~ 5
μ
m
Δ
/(
2 )
NR
ωπ
π
π
~ 200
B
μ
T
2
. This yields a flux
periodicity of
Δ
Φ
~ 1 x 10
-15
T- m
2
in reasonable agreement with th
e expected periodicity of one
flux quantum
Φ
o
= 2.07 x 10
-15
T m
2
. The background increase in
during the course of the
measurement was not typical of most magnetic
field sweeps, which were taken over a much
smaller range.
NR
ω
The x-axes of Figure 2a (main text) and Figures 3a-d (main text) were converted into
units of the flux quantum
Φ
o
by assuming that, at charge de
generacy, adjacent minima in
were separated in flux by
Φ
Δ
/(
2 )
NR
ω
o
. For the data in Figure
2 (main text), the magnetic
field sweep was applied on top of a static fiel
d of ~ 0.01 T. For the data in Figure 3 (main
text), the magnetic field sweep was applied on to
p of a static field of ~ 0.015 T.
Charge-drift correction
For a typical
vs.
vs.
Φ
“map” measurement, the flux
Φ
was held
constant while we swept
and recorded
Δ
. The flux was then incremented, and
the process repeated. For each constant flux trace,
we typically averaged for ~ sec’s. Since the
Δ
/(
2 )
NR
ωπ
cpb
V
cpb
V
/(2 )
NR
ω
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effective gate charge had a slow drift co
mponent, a post-processing correction method was
employed in order to erase the drift and to av
erage multiple maps taken over long periods of
time. Figure S4a is the uncorrected
vs.
vs.
Φ
map for Figure 2a (main text).
In this instance, we chose two charge periods in the map and fit
vs.
to two
Gaussian peaks for each value of flux
Φ
. A typical trace and fit, taken at
Φ
=-0.45 (A.U.), is
displayed in Fig. S4b. The fit gave us the po
sitions of the charge
degeneracy points and
allowed us to align traces taken at successive values of
Φ
as well as subtract offsets due to
background fluctuations in
, typically on the order of 10’
s Hz. Figure 2a (main text)
is the result of only one su
ch post-processed map.
Δ
/(
2 )
NR
ωπ
cpb
V
Δ
/(2 )
NR
ωπ
π
cpb
V
/ (2 )
NR
ω
For the spectroscopy maps in Figs.3a-d (main text) and the Landau-Zener map in Fig.
4a (m
ain text), as many as 14 maps were taken over the course of a day. Because the presence
of the microwave resonances in these maps made
it difficult to fit and find the location of the
charge degeneracy points, it was essential, for each value of
Φ
, to take one trace with
microwaves applied and then a second trace
immediately afterward without microwaves
applied. This allowed us to use the maps with
out microwaves applied to correct the charge
drift of the maps with micr
owaves applied. For better precision, a second post-processing
step was used that minimized the variance at each value of
Φ
between traces of different maps.
For reference, one of the five ‘uncorrected’ Landau-Zener maps taken at
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/ (2 )
6.50 GHz
μ
ωπ
=
that were averaged together to produ
ce Fig. 4a (main text) is displayed in
Figure S5.
Determination of qubit parameters
and
C
E
0
J
E
from spectroscopy
We can estimate the charging energy
C
E
and Josephson energy
J
E
of the qubit from
spectroscopy by plotting, for fixed
E
J
,
versus the displacement from degeneracy
/(2 )
μ
ωπ
Δ
cpb
n
at which resonance occurs (i.e.
μ
ω
E
h
=
Δ
). Figure S6 displays
vs.
for
several values of
Φ
. From the fits to
/(2 )
μ
ωπ
Δ
cpb
n
22
/(2 )
(8
Δ
)()/
μ
ccpb
j
ωπ
En
E h
=+
, we find
,
,
/
13.7 .4 GHz
c
Eh
/
12.7 .8 GHz
c
Eh
/
13.4 .2 GHz
c
Eh
=
±
for
Φ
=.1
Φ
o
, .2
Φ
o
and .35
Φ
o
respectively (all uncertainties are s.e.m.).
From the intercepts of the fits, we find
,
,
/
10.4 .5 GHz
J
Eh
/
9.5 .9 GHz
J
Eh
0 GHz
J
E
(large uncertainty) for
Φ
=.1
Φ
o
, .2
Φ
o
, and .35
Φ
o
respectively (all uncertainties are
s.e.m.). The values of maximum
Josephson energy
0
J
E
inferred from the fits and the Josephson relation are ~10-15% smaller
than what one would expect from the resonance condition
Δ
μ
J0
ω
EE
=
=
h
, which occurs
when
and
( Fig. S7b). Comparing Fig. S7a a
nd Fig. S7b, it is apparent that
this occurs for microwave frequencies in the range of
.5
cpb
n
=
Φ
0
=
/ (2 ) 12.5 13.5 GHz
μ
ωπ
=
, and thus we
estimate
.
0
/ ~ 13 GHz
J
Eh
Figure S7 also serves to demonstrate the additional resonance features that appear near
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charge degeneracy. The green-dashed lines de
note the resonance bands we expect for the
applied microwave frequency
based upon a calculation of
Δ
E
from the full
Hamiltonian Eq. S1. At the lowest microw
ave frequencies (Figs. S7a-b), the observed
-dependence of the microwave resonance appears to be consistent with the coupling
of the CPB island to an incoherent charge fluctuator.
/(2 )
μ
ωπ
)
Φ
,(
CPB
n
35
However, we obse
rved no apparent
dependence of the additional resonant features on thermal cy
cling or background electric and
magnetic fields. Additionally, we find that upon increasing
(Fig S7c-d), qualitative
agreement between the observed and expected
-dependence improves. Furthermore,
it is apparent that there are resonant features where
(e.g. Fig. 3a (main text), Fig. 3d
(main text), and Fig. S7c).
/(2 )
μ
ωπ
)
Φ
,(
CPB
n
Δ
0
NR
ω
>
Qubit linewidth versus microwave amplitude
Recording the mechanical frequency shift
/ (2 )
NR
ω
π
Δ
as a function of the microwave
frequency
(Fig. S8) we can also extract the driven qubit linewidth
/(2 )
μ
ωπ
2
21
2
/(2 )
1
/(
)
d
TT
T
γπ
π
=+Ω
(S3). As expected from
Ω
d
and Eq. S3, we find that
/(2 )
γ
π
increases with
μ
V
(Inset Fig. S8). From a fit of
/(2 )
γ
π
vs.
to Eq. S3 we find
at charge degeneracy.
2
μ
V
sec2
2
nT
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Contour plots for Landau-Zener interferogram
The contour lines overlaying the Landau-Ze
ner interferogram in Fig. 4a (m
ain text)
denote locations in
-space where the phase of the CPB wavefunction is a multiple of
2
π
, i.e.
, where
m
is an integer. These contours we
re generated via the same method
used in Ref. [
),(
μ
cpb
VV
m
πφ
2
=
36
], with
2
0
,
1
Δ
(())2
t
LR
cpb
S
t
φ
dt E n
t
φ
=
h
. For a given
and
cpb
V
μ
V
, “L” and “R”
correspond to the phase
developed while the CPB traverses
points to the left of charge
degeneracy (i.e.
< -3.66 mV) and the phase
developed while the CPB traverses points to
the right of charge degeneracy (i.e.
>-3.66 mV) respectively. The Stokes phase
φ
L
φ
cpb
V
R
φ
cpb
V
s
, is also
as defined in Ref. [36]. The parameters
used to generate the overlay were
,
,
, and
/
14.0 GHz
C
Eh
=
0
/
13.2 GHz
J
Eh
=
17.1 aF
g
C
=
/(2 ) 6.5GHz
μ
ωπ
=
. As well, a factor of 2.8 was used
to convert
μ
n
into
μ
V
, corresponding to an attenuation of 43.5 dB. Transmission
measurements of the microwave circuit made at
room temperature using a network analyzer
with a 50
Ω
input impedance yield ~50-54
dB attenuation. This is in
reasonable agreement with
value extracted from the LZ interferogram c
onsidering that the CPB gate presents an
effectively open-circuit termination to the microwave line during operation.
Supplementary Notes
References for supporting online material
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31. Fulton, T.A. & Dolan, G.J. Observation of si
ngle-electron char
ging effects in small tunnel
junctions.
Phys. Rev. Lett.
59
, 109 (1987).
32. Truitt, P.A. et al. Efficient and sensitive capacitive readout of nanomechanical resonator
arrays.
Nano Lett.
7
, 120-126 (2007).
33. Martinis, J.M, Devoret, M.H.& Clarke, J. Ex
perim
ental tests for the quantum behavior of
a macroscopic degree of freedom: The phase difference across a Josephson junction.
Phys.
Rev. B
35
, 4682-4698 (1987).
34
. Makhlin, Y., Schon, G., & Shnirman, A. Quantum-state engineering with Josephson
junction devices.
Rev. Mod. Phys.
73
, 357-400 (2001).
35. Schuster, D.I.
Cir
cuit Quantum Electrodynamics
. Ph.D. Thesis, Yale University (2007).
36
. Sillanpaa, M. et al. Continuous-time monitoring of Landau-Zener interference in a
Cooper
-pair box.
Phys. Rev. Lett.
96
, 187002 (2006).
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Supplementary Figures and Legends
Figure S1:
Schem
atic of measurement circuit
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Figur
e S2
. Phase-lock loop circuit. We use a pha
se-lock loop to measure the nanoresonator
frequency shift
. The lock-in acts as a phase dete
ctor and low-pass filter which
oscillator (VCO). The VCO output is split into
two: One for
exciting the nanoresonator
and the other REF for the refe
rence input of the lock-in.
Δ
/(
2 )
NR
ωπ
controls the voltage-controlled
RF
V
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b
a
Figure S3
. (a) 2
e
-
e
transition of the response in
. Data in (
a) taken with
and
flux bias set so that
cpb
V
7 V
NR
V
=
0
~
J
J
EE
. (b) CPB flux periodicity. Data in (b) taken near a CPB
charge degeneracy with
and
T
7 V
NR
V
=
mc
~ 120 mK.
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a
b
Figure S4
:
(a) Uncorrected data for Figure 2 (main
text). A quasiparticle switching event is
evident at
Φ
=-1.3. (b) Constant flux cross-section ta
ken at “A”, denoted by vertical black
line in (a). Black line in (b) is a fit of the cross-section “A” to
two Gaussian peaks to
determine charge-offset and background change from trace-to-trace.
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Figure S5:
One
map of
Δ
/ (2 ) vs
vs.
.
NR
cpb
μ
ωπ
V
V
for
/ (2 )
6.50 GHz
μ
ωπ
=
before charge
drift is ‘corrected’. This is one of five such
maps; the average of the five maps is plotted in
Fig.4a (main text).
V
cpb
(mV)
V
μ
(V)
-20
-15
-10
-5
0
5
0.5
1
1.5
2
-500
2.5
0
500
1000
1500
Δω
NR
/2
π
(Hz)
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Figure S6:
Plot of spectroscopic frequency
versus the value of
at which
the CPB transition energy
/(2 )
μ
ωπ
Δ
cpb
n
Δ
μ
E
ω
=
h
, for
Φ
=.1
Φ
o
(squares),
Φ
=.2
Φ
o
(circles),
Φ
=.35
Φ
o
(triangles). Data is taken with
and
T
10 V
NR
V
=
mc
~ 140 mK.
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a b
c d
Figure S7: Measurements of the nanomechanical frequency shift
Δ
versus
/(2 )
NR
ωπ
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cpb
V
and
Φ
for microwave frequencies (a)
/(2 ) 12.5GHz
μ
ωπ
=
(b) 13.5 GHz (c) 18 GHz (d)
20 GHz. Data taken at
and
T
10 V
NR
V
=
mc
= 140 mK. The experimental values of
cpb
V
and
Φ
have been converted into units of Cooper pairs and the flux quantum
Φ
o
respectively. The
maximum Josephson
0
J
E
energy occurs at
Φ
/
Φ
o
= 0.0. When
, the resonant
hyperbola overlap at
Φ
/
Φ
0
J
μ
E
ω
=
h
o
= 0.0. Thus, from (a) and (b) we estimate that
E
j0
/
h
~ 13 GHz.
The green solid lines in the pl
ots denote the expected
resonance hyperbola determined from the
qubit transition energy
Δ
E
, which we calculate from the full CPB Hamiltonian Eq. S1 using
and
.
/
14.0 GHz
c
Eh
=
/
13.2 GHz
J0
Eh
=
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