We have measured noise in thin-film superconducting coplanar waveguide
resonators. This noise appears entirely as phase noise, equivalent to a jitter
of the resonance frequency. In contrast, amplitude fluctuations are not
observed at the sensitivity of our measurement. The ratio between the noise
power in the phase and amplitude directions is large, in excess of 30 dB. These
results have important implications for resonant readouts of various devices
such as detectors, amplifiers, and qubits. We suggest that the phase noise is
due to two-level systems in dielectric materials.Comment: 4 pages, 3 figures, accepted for publication in Applied Physics
  Letter

Benjamin A. Mazin

Henry G. LeDuc

Jiansong Gao

Jonas Zmuidzinas

Peter K. Day

Wilhelm F. K.

English

arXiv

Crossref

Noise properties of superconducting coplanar waveguide microwave resonators

The authors have measured noise in thin-film superconducting coplanar waveguide resonators. This noise appears entirely as phase noise, equivalent to a jitter of the resonance frequency. In contrast, amplitude fluctuations are not observed at the sensitivity of their measurement. The ratio between the noise power in the phase and amplitude directions is large, in excess of 30 dB. These results have important implications for resonant readouts of various devices such as detectors, amplifiers, and qubits. They suggest that the phase noise is due to two-level systems in dielectric materials

Gao, Jiansong

Zmuidzinas, Jonas

Mazin, Benjamin A.

LeDuc, Henry G.

Day, Peter K.

Caltech Authors - Main

Noise properties of superconducting coplanar waveguide
microwave resonators
Jiansong Gaoa and Jonas Zmuidzinas
Physics Department, California Institute of Technology, Pasadena, California 91125
Benjamin A. Mazin, Henry G. LeDuc, and Peter K. Day
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109
Received 20 September 2006; accepted 1 February 2007; published online 8 March 2007
The authors have measured noise in thin-film superconducting coplanar waveguide resonators. This
noise appears entirely as phase noise, equivalent to a jitter of the resonance frequency. In contrast,
amplitude fluctuations are not observed at the sensitivity of their measurement. The ratio between
the noise power in the phase and amplitude directions is large, in excess of 30 dB. These results
have important implications for resonant readouts of various devices such as detectors, amplifiers,
and qubits. They suggest that the phase noise is due to two-level systems in dielectric materials.
© 2007 American Institute of Physics. DOI: 10.1063/1.2711770
Thin-film superconducting microwave resonators are of
interest for a number of applications, including the multi-
plexed readout of single electron transistors,1 microwave
kinetic inductance detectors MKIDs,2,3 normal metal-
insulator-superconductor tunnel junction detectors,4 super-
conducting quantum interference devices,5,6 and qubits.7,8
The device to be measured presents a variable dissipative or
reactive load to the resonator, influencing the resonator qual-
ity factor Qr or frequency fr, respectively. Changes to both
Qr and fr may be determined simultaneously by sensing the
amplitude and phase of a microwave probe signal.2 While
several early demonstrations used hand-assembled lumped-
element circuits,1,4,5 frequency-domain multiplexing of large
arrays generally will require compact microlithographed
high-Qr resonators.
1 Such resonators are also needed for
strong coupling to charge qubits.7 Noise in microlitho-
graphed resonators has been observed2,3 and can be a limit-
ing factor for device performance but is not well understood.
In this letter, we report measurements of resonator noise,
show how the noise spectra separate into amplitude and
phase components, and discuss the physical origin of the
noise.
We studied quarter-wavelength coplanar waveguide
CPW resonators2 Fig. 1a with center strip widths w of
0.6–6 m and gaps g between the center strip and ground
planes of 0.4–4 m, and with impedances Z050 . Reso-
nator lengths of 3–7 mm produce resonance frequencies fr
between 4 and 10 GHz. Frequency multiplexed arrays of up
to 100 resonators are coupled to a single CPW feedline.
The CPW circuits are patterned from a film of either Al
Tc=1.2 K or Nb Tc=9.2 K on a crystalline substrate, ei-
ther sapphire, Si, or Ge. The surfaces of the semiconductor
substrates are not intentionally oxidized, although a native
oxide due to air exposure is expected to be present.
A microwave synthesizer at frequency f is used to excite
a resonator. The transmitted signal is amplified with a cryo-
genic high electron mobility transistor HEMT amplifier and
is compared to the original signal using an IQ mixer, whose
output voltages I and Q are proportional to the in-phase and
quadrature amplitudes of the transmitted signal2,3 see Fig. 2
inset. As f is varied, the output = I ,QT the superscript T
represents the transpose traces out a resonance circle Fig.
1b. With f fixed,  is seen to fluctuate about its mean, and
the fluctuations t= It ,QtT are digitized for noise
analysis, typically over a 10 s interval using a sample rate of
250 kHz.
The fluctuations t are observed to be primarily in the
direction tangent to the resonance circle, while the fluctua-
tions in the orthogonal direction are small. These two direc-
tions correspond to fluctuations in the phase and amplitude
of the resonator’s electric field E, respectively. This observa-
tion can be quantified by studying the spectral-domain noise
covariance matrix S defined by
† = S − , S
=  SII SIQ
SIQ
*  SQQ
	 , 1
where  is the Fourier transform of the time-domain
data, the dagger represents the Hermitian conjugate, SII
and SQQ are the autopower spectra, and SIQ is the
aElectronic mail: jiansong@caltech.edu
FIG. 1. Color online a Schematic illustration not to scale of the reso-
nator and feedline geometry; See Day et al. Ref. 2 for the equivalent
circuit. Black represents the superconducting film and white represents bare
substrate. The coupler relies on the mutual capacitance between the center
strips of the feedline and resonator CPW lines and may be considered to be
a lumped element C=1/ frZ0
8Qc, where Qc is the coupling-limited
quality factor. b Resonance circle of a 200 nm Nb on Si resonator at
120 mK solid line, quasiparticle trajectory calculated from Mattis-Bardeen
theory Ref. 10 dashed line. For this figure, the readout point = I ,Q is
located at the resonance frequency fr. c Noise ellipse magnified by a
factor of 30. Other parameters are fr=4.35 GHz, Qr=3.5105 coupling
limited, w=5 m, g=1 m, readout power Pr−84 dBm, and internal
power Pint−30 dBm.
APPLIED PHYSICS LETTERS 90, 102507 2007
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cross-power spectrum. The matrix S is Hermitian and may
be diagonalized using a unitary transformation; however, we
find that the imaginary part of SIQ is negligible and that an
ordinary rotation applied to the real part Re S gives al-
most identical results. The eigenvectors and eigenvalues are
calculated at every frequency ,
OTRe SO = Saa 0
0 Sbb
	 , 2
where O= va ,vb is an orthogonal rotation matrix.
We use Saa and va to denote the larger eigenvalue and
its eigenvector.
A typical pair of spectra Saa and Sbb shown in
Fig. 2, along with the rotation angle , defined as the
angle between va and the I axis. Three remarkable fea-
tures are found for all noise data. First,  is independent
of  within the resonator bandwidth the rms scatter is 	

0.4° per 10 Hz frequency bin from 1 Hz to 5 kHz, which
means that only two special directions, va and vb, diagonal-
ize S. Equation 2 shows that Saa and Sbb are the
noise spectra projected into these two constant directions.
Second, va is always tangent to the IQ resonance circle while
vb is always normal to the circle, even when f is detuned
from fr. Third, Saa is well above Sbb see Fig. 2.
The character of the noise can be clearly visualized by plot-
ting a noise ellipse, defined by TC−1=1, where
C=1
2Re Sd is the covariance matrix for I and Q fil-
tered for the corresponding bandpass we use 1=1 Hz and
2=1 kHz. The major axis of the noise ellipse is always in
the phase direction, and the ratio of the two axes is always
very large Fig. 1c.
Figure 2 also shows that the amplitude noise spectrum is
flat except for a 1/ knee at low frequency contributed by
the electronics. The amplitude noise is independent of
whether the synthesizer is tuned on or off the resonance and
is consistent with the noise temperature of the HEMT ampli-
fier. The phase noise spectrum9 has a 1/ slope below 10 Hz,
typically a −1/2 slope above 10 Hz, and a roll off at the
resonator bandwidth fr /2Qr. The phase noise is well above
the HEMT noise, usually by two or three orders of magni-
tude in rad2 /Hz at low frequencies. It is well in excess of
the synthesizer phase noise contribution or the readout sys-
tem noise.2,3
Quasiparticle fluctuations in the superconductor can be
securely ruled out as the source of the measured noise by
considering the direction in the IQ plane that would corre-
spond to a change in quasiparticle density nqp. Both the real
and inductive parts of the complex conductivity 	 respond
linearly to nqp, 	=	1− i	2, resulting in a trajectory that
is always at a nonzero angle =tan−1	1 /	2 to the reso-
nance circle, as indicated by the dashed lines in Figs. 1b
and 1c. Mattis-Bardeen10 calculations yield 7° for Nb
below 1 K, so quasiparticle fluctuations are strongly ex-
cluded since 	. Furthermore,  is measured experimen-
tally by examining the response to x-ray, optical/UV, or sub-
millimeter photons and is typically 15°.11
The phase noise depends on the microwave power inside
the resonator Pint, the materials used for the resonator, and
the operating temperature. The power dependence for vari-
ous material combinations is shown in Fig. 3; all follow the
scaling Saa Pint
−1/2. For comparison, amplifier phase noise
is a multiplicative effect that would give a constant noise
level independent of Pint, while the amplifier noise tempera-
ture is an additive effect that would produce a 1/ Pint depen-
dence. Sapphire substrates generally give lower phase noise
than Si or Ge. However, the Nb/Si device showed low noise
comparable with Al/sapphire, suggesting that the etching or
interface chemistry, which is different for Nb and Al, may
play a role. Two Al/Si resonators with very different Al
thicknesses and kinetic inductance fractions12 fall onto the
dashed equal-noise scaling line, strongly suggesting that the
superconductor is not responsible for the phase noise.3 Fur-
thermore, the noise of a Nb/Si resonator decreased by a
factor of 10 when warmed from 0.2 to 1 K, stronger evi-
dence against superconductor noise since Nb has Tc=9.2 K
and its properties change very little for TTc. More detail
on the temperature dependence will be published separately.
The evidence leads us to suggest that the noise is caused
by fluctuating two-level systems TLSs in the dielectric
materials—either the bulk substrate or its exposed surface,
the interface layers between the metal films and the substrate,
or any oxide layers on the metal surfaces. Models assuming
a collection of TLS with a wide range of excitation energies
E and relaxation rates have long been used to explain the low
temperature physical properties of noncrystalline solids.13–15
TLSs are also found in crystalline materials16,17 but at lower
densities. Fluctuations due to TLS are of interest
theoretically18,19 and have been observed in dielectrics, either
as telegraph or 1/ noise, using tunnel junctions,20 single
electron transistors,21 and atomic force microscopes.22 Other
examples include telegraph noise in the resistance of metallic
FIG. 2. Noise spectra in the phase Saa, solid line and amplitude Sbb,
dashed line directions, and the rotation angle , dotted line. The noise
data are from the same Nb/Si resonator under the same condition as in
Fig. 1. The inset shows the diagram of the homodyne readout system.
FIG. 3. Power and material dependence of the phase noise at =1 kHz. To
compare resonators with different fr and Qr, phase noise is converted to
fractional frequency noise, calculated by Sfr / fr
2=Saa /4Qr
2. All the
resonators have w=3 m and g=2 m and are measured around 120 mK.
102507-2 Gao et al. Appl. Phys. Lett. 90, 102507 2007
Downloaded 09 Mar 2007 to 131.215.225.9. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
nanoconstrictions23 and qubit dephasing effects.8
Recent experiments7,24 have illustrated the reactive load-
ing effect that occurs when a single TLS a qubit is coupled
to a microwave resonator. For weak coupling g or large de-
tuning f = E /h− fr, gf , this reactive loading causes the
resonator frequency to shift by ±g2 /f depending on the
quantum state of the TLS. Thermal fluctuations—absorption
or emission of thermal phonons by a collection of TLS—
could therefore cause phase noise, equivalent to a fluctuating
dielectric constant r , t. For this model one expects the
noise to vanish exponentially for Thfr /2k as the TLSs
settle into their ground states, while the increase in TLS-
phonon transition rates might explain the observed noise de-
crease at high temperatures. However, at present we do not
have a model which quantitatively explains the data. Alter-
natively, TLS frequency fluctuations Et /h produced by
TLS-TLS interactions15,25 and observed in single molecule
optical fluorescence experiments26,27 should give phase noise
with a power law rather than exponential decrease at low
temperatures.
The Pint
−1/2 noise scaling is indicative of TLS saturation;
otherwise, the phase noise would be independent of Pint as
expected for dielectric constant fluctuations. Similarly,
power-independent thermal fluctuations of the TLS dielectric
polarization P19 are ruled out since this would give additive
noise with equal amplitude and phase components and scal-
ing as 1/ Pint. TLS saturation effects are well known;
8,15 in-
deed, fr and Qr show anomalous temperature and power
dependence28 for TTc. Saturation of a single TLS depends
on its frequency detuning f , its position r in the spatially
varying resonator field Er, and the orientation and strength
of its dipole moment d. The growth of the saturation zone in
this parameter space with increasing Pint may explain the
observed Pint
−1/2 noise scaling.
The microwave fields in our resonators are sufficiently
strong to cause TLS saturation. In the Bloch equation frame-
work, saturation for zero detuning occurs when 2T1T21,
where =d ·E / is the Rabi frequency and T1
−1, T2
−1 are the
usual energy relaxation and dephasing rates.29 The distribu-
tion of T1
−1 is extremely broad,15 extending above and below
the range of noise frequencies we observe, because T1
−1
0
2 /E2 is controlled by the tunnel coupling 0 and is there-
fore exponentially sensitive to the potential barrier.15,29 For
silica, T1
−11 MHz E /4 GHz3 0 /E2 cothE /2kT. The
dipole moment d is proportional to 0 /E,
15 so 2T1 is inde-
pendent of 0. Meanwhile, T2
−1 scales as 1−0
2 /E2 Ref. 15
and 29 and has a relatively narrow distribution; for silica,
T2
−110 MHz T /200 mK1.5.30 Saturation is therefore
mostly independent of 0 and occurs when E ·d / d exceeds
a critical value Ecfr ,T. Indeed, the shape of the observed
phase noise spectrum does not change significantly with
Pint until the superconductors become nonlinear.
3 For
silica, Ec4 GHz,200 mK31 V m−1 and corresponds to
Pint−75 dBm which is significantly below the power levels
we use see Fig. 3. We also find Ec7.2 GHz,25mK
10 V m−1, which is appropriate for the experimental con-
ditions of Martinis et al.8 and is similar to their observed
onset of TLS saturation.
In summary, we find that lithographed resonators display
−1/2 noise directed purely in the phase direction which
varies with readout power, temperature, and the substrate
material. These results are important for the device
optimization—devices relying on resistive loading should be
able to avoid this noise source. For MKIDs, the sin  ampli-
tude component of the signal is already available when using
an IQ readout, and an optimally weighted phase/amplitude
measurement can substantially improve the sensitivity at low
frequencies where the phase noise is larger.
The authors thank Sunil Golwala, Kent Irwin, Andrew
Lange, Konrad Lehnert, John Martinis, and Harvey Moseley
for useful discussions. This work was supported in part by
the NASA Science Mission Directorate, JPL, Gordon and
Betty Moore Foundation, and Alex Lidow, a Caltech Trustee.
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commonly used to represent the phase noise of active oscillators or fre-
quency sources. Our noise spectra refer to a passive resonator driven at
fixed frequency; in this case, a resonance frequency shift fr causes a
proportional phase shift =2Qrfr / fr in the resonator field. In contrast, if
we locked an active oscillator to our passive resonator, the oscillator’s
output phase shift relative to a perfect clock would now be the time inte-
gral of 2frt, and we would therefore observe an extra 1 /2 factor in
the oscillator’s phase noise spectrum at low frequencies.
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Zmuidzinas, and F. A. Harrison, Appl. Phys. Lett. 89, 222507 2006.
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102507-3 Gao et al. Appl. Phys. Lett. 90, 102507 2007
Downloaded 09 Mar 2007 to 131.215.225.9. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp


Noise properties of superconducting coplanar waveguide
microwave resonators
Jiansong Gaoa and Jonas Zmuidzinas
Physics Department, California Institute of Technology, Pasadena, California 91125
Benjamin A. Mazin, Henry G. LeDuc, and Peter K. Day
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109
Received 20 September 2006; accepted 1 February 2007; published online 8 March 2007
The authors have measured noise in thin-film superconducting coplanar waveguide resonators. This
noise appears entirely as phase noise, equivalent to a jitter of the resonance frequency. In contrast,
amplitude fluctuations are not observed at the sensitivity of their measurement. The ratio between
the noise power in the phase and amplitude directions is large, in excess of 30 dB. These results
have important implications for resonant readouts of various devices such as detectors, amplifiers,
and qubits. They suggest that the phase noise is due to two-level systems in dielectric materials.
© 2007 American Institute of Physics. DOI: 10.1063/1.2711770
Thin-film superconducting microwave resonators are of
interest for a number of applications, including the multi-
plexed readout of single electron transistors,1 microwave
kinetic inductance detectors MKIDs,2,3 normal metal-
insulator-superconductor tunnel junction detectors,4 super-
conducting quantum interference devices,5,6 and qubits.7,8
The device to be measured presents a variable dissipative or
reactive load to the resonator, influencing the resonator qual-
ity factor Qr or frequency fr, respectively. Changes to both
Qr and fr may be determined simultaneously by sensing the
amplitude and phase of a microwave probe signal.2 While
several early demonstrations used hand-assembled lumped-
element circuits,1,4,5 frequency-domain multiplexing of large
arrays generally will require compact microlithographed
high-Qr resonators.1 Such resonators are also needed for
strong coupling to charge qubits.7 Noise in microlitho-
graphed resonators has been observed2,3 and can be a limit-
ing factor for device performance but is not well understood.
In this letter, we report measurements of resonator noise,
show how the noise spectra separate into amplitude and
phase components, and discuss the physical origin of the
noise.
We studied quarter-wavelength coplanar waveguide
CPW resonators2 Fig. 1a with center strip widths w of
0.6–6 m and gaps g between the center strip and ground
planes of 0.4–4 m, and with impedances Z050 . Reso-
nator lengths of 3–7 mm produce resonance frequencies fr
between 4 and 10 GHz. Frequency multiplexed arrays of up
to 100 resonators are coupled to a single CPW feedline.
The CPW circuits are patterned from a film of either Al
Tc=1.2 K or Nb Tc=9.2 K on a crystalline substrate, ei-
ther sapphire, Si, or Ge. The surfaces of the semiconductor
substrates are not intentionally oxidized, although a native
oxide due to air exposure is expected to be present.
A microwave synthesizer at frequency f is used to excite
a resonator. The transmitted signal is amplified with a cryo-
genic high electron mobility transistor HEMT amplifier and
is compared to the original signal using an IQ mixer, whose
output voltages I and Q are proportional to the in-phase and
quadrature amplitudes of the transmitted signal2,3 see Fig. 2
inset. As f is varied, the output = I ,QT the superscript T
represents the transpose traces out a resonance circle Fig.
1b. With f fixed,  is seen to fluctuate about its mean, and
the fluctuations t= It ,QtT are digitized for noise
analysis, typically over a 10 s interval using a sample rate of
250 kHz.
The fluctuations t are observed to be primarily in the
direction tangent to the resonance circle, while the fluctua-
tions in the orthogonal direction are small. These two direc-
tions correspond to fluctuations in the phase and amplitude
of the resonator’s electric field E, respectively. This observa-
tion can be quantified by studying the spectral-domain noise
covariance matrix S defined by
† = S − , S
=  SII SIQSIQ*  SQQ 	 , 1
where  is the Fourier transform of the time-domain
data, the dagger represents the Hermitian conjugate, SII
and SQQ are the autopower spectra, and SIQ is the
aElectronic mail: jiansong@caltech.edu
FIG. 1. Color online a Schematic illustration not to scale of the reso-
nator and feedline geometry; See Day et al. Ref. 2 for the equivalent
circuit. Black represents the superconducting film and white represents bare
substrate. The coupler relies on the mutual capacitance between the center
strips of the feedline and resonator CPW lines and may be considered to be
a lumped element C=1/ frZ0
8Qc, where Qc is the coupling-limited
quality factor. b Resonance circle of a 200 nm Nb on Si resonator at
120 mK solid line, quasiparticle trajectory calculated from Mattis-Bardeen
theory Ref. 10 dashed line. For this figure, the readout point = I ,Q is
located at the resonance frequency fr. c Noise ellipse magnified by a
factor of 30. Other parameters are fr=4.35 GHz, Qr=3.5105 coupling
limited, w=5 m, g=1 m, readout power Pr−84 dBm, and internal
power Pint−30 dBm.
APPLIED PHYSICS LETTERS 90, 102507 2007
0003-6951/2007/9010/102507/3/$23.00 © 2007 American Institute of Physics90, 102507-1
Downloaded 09 Mar 2007 to 131.215.225.9. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
cross-power spectrum. The matrix S is Hermitian and may
be diagonalized using a unitary transformation; however, we
find that the imaginary part of SIQ is negligible and that an
ordinary rotation applied to the real part Re S gives al-
most identical results. The eigenvectors and eigenvalues are
calculated at every frequency ,
OTRe SO = Saa 00 Sbb 	 , 2
where O= va ,vb is an orthogonal rotation matrix.
We use Saa and va to denote the larger eigenvalue and
its eigenvector.
A typical pair of spectra Saa and Sbb shown in
Fig. 2, along with the rotation angle , defined as the
angle between va and the I axis. Three remarkable fea-
tures are found for all noise data. First,  is independent
of  within the resonator bandwidth the rms scatter is 	

0.4° per 10 Hz frequency bin from 1 Hz to 5 kHz, which
means that only two special directions, va and vb, diagonal-
ize S. Equation 2 shows that Saa and Sbb are the
noise spectra projected into these two constant directions.
Second, va is always tangent to the IQ resonance circle while
vb is always normal to the circle, even when f is detuned
from fr. Third, Saa is well above Sbb see Fig. 2.
The character of the noise can be clearly visualized by plot-
ting a noise ellipse, defined by TC−1=1, where
C=1
2Re Sd is the covariance matrix for I and Q fil-
tered for the corresponding bandpass we use 1=1 Hz and
2=1 kHz. The major axis of the noise ellipse is always in
the phase direction, and the ratio of the two axes is always
very large Fig. 1c.
Figure 2 also shows that the amplitude noise spectrum is
flat except for a 1/ knee at low frequency contributed by
the electronics. The amplitude noise is independent of
whether the synthesizer is tuned on or off the resonance and
is consistent with the noise temperature of the HEMT ampli-
fier. The phase noise spectrum9 has a 1/ slope below 10 Hz,
typically a −1/2 slope above 10 Hz, and a roll off at the
resonator bandwidth fr /2Qr. The phase noise is well above
the HEMT noise, usually by two or three orders of magni-
tude in rad2 /Hz at low frequencies. It is well in excess of
the synthesizer phase noise contribution or the readout sys-
tem noise.2,3
Quasiparticle fluctuations in the superconductor can be
securely ruled out as the source of the measured noise by
considering the direction in the IQ plane that would corre-
spond to a change in quasiparticle density nqp. Both the real
and inductive parts of the complex conductivity 	 respond
linearly to nqp, 	=	1− i	2, resulting in a trajectory that
is always at a nonzero angle =tan−1	1 /	2 to the reso-
nance circle, as indicated by the dashed lines in Figs. 1b
and 1c. Mattis-Bardeen10 calculations yield 7° for Nb
below 1 K, so quasiparticle fluctuations are strongly ex-
cluded since 	. Furthermore,  is measured experimen-
tally by examining the response to x-ray, optical/UV, or sub-
millimeter photons and is typically 15°.11
The phase noise depends on the microwave power inside
the resonator Pint, the materials used for the resonator, and
the operating temperature. The power dependence for vari-
ous material combinations is shown in Fig. 3; all follow the
scaling SaaPint
−1/2
. For comparison, amplifier phase noise
is a multiplicative effect that would give a constant noise
level independent of Pint, while the amplifier noise tempera-
ture is an additive effect that would produce a 1/ Pint depen-
dence. Sapphire substrates generally give lower phase noise
than Si or Ge. However, the Nb/Si device showed low noise
comparable with Al/sapphire, suggesting that the etching or
interface chemistry, which is different for Nb and Al, may
play a role. Two Al/Si resonators with very different Al
thicknesses and kinetic inductance fractions12 fall onto the
dashed equal-noise scaling line, strongly suggesting that the
superconductor is not responsible for the phase noise.3 Fur-
thermore, the noise of a Nb/Si resonator decreased by a
factor of 10 when warmed from 0.2 to 1 K, stronger evi-
dence against superconductor noise since Nb has Tc=9.2 K
and its properties change very little for TTc. More detail
on the temperature dependence will be published separately.
The evidence leads us to suggest that the noise is caused
by fluctuating two-level systems TLSs in the dielectric
materials—either the bulk substrate or its exposed surface,
the interface layers between the metal films and the substrate,
or any oxide layers on the metal surfaces. Models assuming
a collection of TLS with a wide range of excitation energies
E and relaxation rates have long been used to explain the low
temperature physical properties of noncrystalline solids.13–15
TLSs are also found in crystalline materials16,17 but at lower
densities. Fluctuations due to TLS are of interest
theoretically18,19 and have been observed in dielectrics, either
as telegraph or 1/ noise, using tunnel junctions,20 single
electron transistors,21 and atomic force microscopes.22 Other
examples include telegraph noise in the resistance of metallic
FIG. 2. Noise spectra in the phase Saa, solid line and amplitude Sbb,
dashed line directions, and the rotation angle , dotted line. The noise
data are from the same Nb/Si resonator under the same condition as in
Fig. 1. The inset shows the diagram of the homodyne readout system.
FIG. 3. Power and material dependence of the phase noise at =1 kHz. To
compare resonators with different fr and Qr, phase noise is converted to
fractional frequency noise, calculated by Sfr / fr2=Saa /4Qr2. All the
resonators have w=3 m and g=2 m and are measured around 120 mK.
102507-2 Gao et al. Appl. Phys. Lett. 90, 102507 2007
Downloaded 09 Mar 2007 to 131.215.225.9. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
nanoconstrictions23 and qubit dephasing effects.8
Recent experiments7,24 have illustrated the reactive load-
ing effect that occurs when a single TLS a qubit is coupled
to a microwave resonator. For weak coupling g or large de-
tuning f = E /h− fr, gf , this reactive loading causes the
resonator frequency to shift by ±g2 /f depending on the
quantum state of the TLS. Thermal fluctuations—absorption
or emission of thermal phonons by a collection of TLS—
could therefore cause phase noise, equivalent to a fluctuating
dielectric constant r , t. For this model one expects the
noise to vanish exponentially for Thfr /2k as the TLSs
settle into their ground states, while the increase in TLS-
phonon transition rates might explain the observed noise de-
crease at high temperatures. However, at present we do not
have a model which quantitatively explains the data. Alter-
natively, TLS frequency fluctuations Et /h produced by
TLS-TLS interactions15,25 and observed in single molecule
optical fluorescence experiments26,27 should give phase noise
with a power law rather than exponential decrease at low
temperatures.
The Pint
−1/2 noise scaling is indicative of TLS saturation;
otherwise, the phase noise would be independent of Pint as
expected for dielectric constant fluctuations. Similarly,
power-independent thermal fluctuations of the TLS dielectric
polarization P19 are ruled out since this would give additive
noise with equal amplitude and phase components and scal-
ing as 1/ Pint. TLS saturation effects are well known;8,15 in-
deed, fr and Qr show anomalous temperature and power
dependence28 for TTc. Saturation of a single TLS depends
on its frequency detuning f , its position r in the spatially
varying resonator field Er, and the orientation and strength
of its dipole moment d. The growth of the saturation zone in
this parameter space with increasing Pint may explain the
observed Pint
−1/2 noise scaling.
The microwave fields in our resonators are sufficiently
strong to cause TLS saturation. In the Bloch equation frame-
work, saturation for zero detuning occurs when 2T1T21,
where =d ·E / is the Rabi frequency and T1
−1
, T2
−1 are the
usual energy relaxation and dephasing rates.29 The distribu-
tion of T1
−1 is extremely broad,15 extending above and below
the range of noise frequencies we observe, because T1
−1
0
2 /E2 is controlled by the tunnel coupling 0 and is there-
fore exponentially sensitive to the potential barrier.15,29 For
silica, T1
−11 MHz E /4 GHz3 0 /E2 cothE /2kT. The
dipole moment d is proportional to 0 /E,15 so 2T1 is inde-
pendent of 0. Meanwhile, T2
−1 scales as 1−0
2 /E2 Ref. 15
and 29 and has a relatively narrow distribution; for silica,
T2
−110 MHz T /200 mK1.5.30 Saturation is therefore
mostly independent of 0 and occurs when E ·d / d exceeds
a critical value Ecfr ,T. Indeed, the shape of the observed
phase noise spectrum does not change significantly with
Pint until the superconductors become nonlinear.3 For
silica, Ec4 GHz,200 mK31 V m−1 and corresponds to
Pint−75 dBm which is significantly below the power levels
we use see Fig. 3. We also find Ec7.2 GHz,25mK
10 V m−1, which is appropriate for the experimental con-
ditions of Martinis et al.8 and is similar to their observed
onset of TLS saturation.
In summary, we find that lithographed resonators display
−1/2 noise directed purely in the phase direction which
varies with readout power, temperature, and the substrate
material. These results are important for the device
optimization—devices relying on resistive loading should be
able to avoid this noise source. For MKIDs, the sin  ampli-
tude component of the signal is already available when using
an IQ readout, and an optimally weighted phase/amplitude
measurement can substantially improve the sensitivity at low
frequencies where the phase noise is larger.
The authors thank Sunil Golwala, Kent Irwin, Andrew
Lange, Konrad Lehnert, John Martinis, and Harvey Moseley
for useful discussions. This work was supported in part by
the NASA Science Mission Directorate, JPL, Gordon and
Betty Moore Foundation, and Alex Lidow, a Caltech Trustee.
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Noise Properties of Superconducting Coplanar Waveguide Microwave Resonators

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