Projected dipole moments of individual two-level defects extracted using circuit quantum electrodynamics

0.6 MB·English

by  Bahman Sarabi

Projected dipole moments of individual two-level defects extracted using circuit
quantum electrodynamics
B. Sarabi,1,2 A. N. Ramanayaka,1,2 A. L. Burin,3 F. C. Wellstood,2,4 and K. D. Osborn1,4
1Laboratory for Physical Sciences, College Park, MD 20740, USA
2Department of Physics, University of Maryland, College Park, MD 20742, USA
3Department of Chemistry, Tulane University, New Orleans, LA 70118, USA
4Joint Quantum Institute, University of Maryland, College Park, MD 20742, USA
(Dated: April 6, 2016)
Material-basedtwo-levelsystems(TLSs),appearingasdefectsinlow-temperaturedevicesinclud-
6 ing superconducting qubits and photon detectors, are difficult to characterize. In this study we
1
apply a uniform dc-electric field across a film to tune the energies of TLSs within. The film is
0
embedded in a superconducting resonator such that it forms a circuit quantum electrodynamical
2
(cQED)system. TheenergyofindividualTLSsisobservedasafunctionoftheknowntuningfield.

r By studying TLSs for which we can determine the tunneling energy, the actual p , dipole moments
z
p
projectedalongtheuniformfielddirection,areindividuallyobtained. Adistributioniscreatedwith
A
60 p . We describe the distribution using a model with two dipole moment magnitudes, and a fit
z
yields the corresponding values p = p = 2.8±0.2 Debye and p = p = 8.3±0.4 Debye. For a
5 1 2
strong-coupledTLSthevacuum-Rabisplittingcanbeobtainedwithp andtunnelingenergy. This
z
allows a measurement of the circuit’s zero-point electric field fluctuations, in a method that does
]
n not need the electric-field volume.
o
c PACSnumbers: 03.67.-a,03.67.Pp,33.15.Kr,66.35.+a
-
r
p
Dielectrictwo-levelsystems(TLSs)haveattractedthe sult only a lower bound of p is determined from those
u z
s attention of the quantum computing community ever measurements (∆0/E ≤1). However, in studies of many
t. since they were identified as a major source of decoher- individual TLSs in the barrier of a JJ, the distribution
a
ence in superconducting qubits [1]. Subsequent studies of observations is consistent with a single TLS moment
m
foundthatTLSswerealsoaperformance-limitingsource magnitude p [1, 15], assuming a standard distribution
-
of noise in photon detectors used for astronomy [2, 3]. of ∆ [16, 17]. Here only the average projected moment
d 0 √
n This motivation has led to quantum characterization of (cid:104)pz(cid:105)≈p/ 3isdeterminedfromthemodelfit,andhasan
o TLSs in the tunneling barrier of superconducting qubits accuracy limitation caused by the aforementioned vari-
c [1, 4] and both noise and loss characterization in high able barrier thickness.
[
quality superconducting resonator circuits. Thus TLSs TLSs have been recently studied with an applied field

3 are found as defects in various dielectric structures: de- which tilts the TLS potential energies. A strain tun-
v positedinsulatingfilms[1,5,6],JosephsonJunction(JJ) ing study with individual TLS observations [18] veri-
5
tunneling barriers [1, 7, 8], imperfect interfaces between fied agreement with the standard model which employs
6
8 superconductingfilmswithcrystallinesubstrates[9],and double-well potentials [16, 17]. Further work revealed
5 the native oxides on materials [10]. Recent modeling TLS-TLS interactions [19]. Finally, a swept electric-field
0 has predicted possible structures and values for the TLS study measured nonequilibrium microwave loss which
1. dipole moment [11–13]. While these TLSs are generally was explained by Landau-Zener dynamics and a single
0 knowntobechargedatomicconfigurationsthatspatially representative dipole moment (magnitude) [20, 21].
5 tunnel,theirmicroscopicstructureandelementalcompo- Inourdevice,TLSsarecoupledtoaLCresonatorand
1
sition are generally unknown. tuned with a uniform and known dc electric field E .
: bias
v In a qubit made from an anharmonic oscillator the in- This tuning distinguishes it from an earlier study of the
i
X teraction energy is observed as a spectroscopic splitting vacuum Rabi splitting and Glauber state made with a
inthequbitstate. However, thisquantityisnotonlyde- TLS [22]. TLSs are chosen for analysis only near an
r
a pendent on circuit element parameters, but also on the observedenergyminima, whichrevealsthetunnelingen-
thickness of the tunneling barrier [1]. The barrier has ergies ∆ . In our experiment, p ≡ p|cosθ| is measured
0 z
a thickness variation of 1-2 nm [14], such that this lack foreachindividualTLSandenabledbythatof∆ . This
0
of accuracy is also present in the electric field amplitude leaves the relationship to the dipole moment magnitude
and the measured TLS dipole moment. In general, indi- p uncertain only by the angle θ between the dipole and
vidual measurements of TLSs within JJs find the tran- the known dc-field direction. 60 p are individually ex-
z
sition dipole moment p = p ∆ /E, where p ≡ |p | tracted. The distribution created from these values is fit
tr z 0 z z
is the absolute value of the dipole moment projected in to a model with two dipole moment magnitudes, p = p
1
the field direction, and the TLS tunneling energy over and p = p , and thus is related to earlier studies using
2
the total energy ∆ /E is generally unknown. As a re- two loss tangent sources [23, 24]. While p can be de-
0 tr
2
to allow many TLSs to be observed in a small range of
bias voltage V . The microwave input power was gen-
bias
erally set such that the photon occupancy on resonance
is n¯ (cid:39) 0.4. This photon occupancy allows sufficient
max
signal to noise, and in a similar device is found to re-
solve nearly degenerate resonator-TLS energy levels in
the presence of other weaker coupled TLSs [29].
The micro-V device was measured at a temperature
T ≤25 mK in three data sets from separate cool-downs.
In each data set the bias voltage V was varied by 60
bias
mVacrossthedielectriccorrespondingtoanelectricfield
FIG.1. (a)SchematicofLCresonatorcoupledtoTLSswithin
range of 240 kV/m within it. The bias line was low-pass
the capacitors C1-C4 of the electric field controlled device.
filteredatroomtemperatureandonthe3Kelvinstageto
The voltage V induces a uniform dc-electric field across
bias
the capacitors. (b) Optical image of the fabricated tunable reducenoiseinducedtotheTLSenergies. Inaddition,a3
device. Aluminum appears light and the sapphire substrate dBmicrowaveattenuatorwasinstalledonthe0.7Kplate
appears black. to thermalize the center conductor of the corresponding
coaxial cable. Finally, a Cu-powder low-pass filter was
used at the base temperature stage of the refrigerator to
termined in general from dc-tuned measurements, some
absorb Cooper-pair breaking photons.
TLSs also have strong-coupling such that the vacuum
Figure 2(a) shows the transmission spectroscopy re-
Rabi splitting is observed with the corresponding cou-
sults from one of the data sets. The wide dark re-
pling frequency g = p E /¯h. By combining a mea-
tr RMS giongenerallycorrespondstothebroadcircuitresonance
surement of p , ∆ and g we get a complete measure-
z 0 ω /2π, and a large number of fine features are caused by
ment of the uniform zero-point electric field fluctuations c
near-resonance interactions with individual TLSs. Us-
E ofthedevice. Aswillbediscussedbelow,ourfilm
RMS ing a first analysis which neglects vacuum splitting ef-
containsTLSsrelatedtothosefromJJbarriers,material
fects, we interpret the fine features as the TLS ener-
interfaces,contamination,anddielectricsingeneral,such
gies. In agreement with the standard model for energy
thattheaccuratemomentscangenerallybeextractedby
Eq. 1, some TLSs exhibit a minimum in their energy
this type of device.
(at ¯hω (cid:39) ∆ ) versus E . The energy is also sym-
Inthisexperimentweuseasuperconductingresonator c 0 bias
metric about the minimum in the same quantity, as ex-
withbridgearrangementoffourparallel-platecapacitors
pected from the referenced equation. Since E is di-
shown in Fig. 1. The capacitors allow a uniform dc elec- bias
rectly applied, a fit to the data, as shown in Fig. 2(b),
tricbiasE totheTLSswithin,creatingthetransition
bias produces a fairly direct determination of the individual
energy
projected dipole moments p . Each p is extracted with
z z
(cid:113) a precision estimated at approximately 1 percent (see
E = ∆2+(∆+2p E )2. (1)
0 z bias below). ∆ = −p V /d when the energy is at a mini-
z bias 0
Here, ∆ is the energy difference between the potential mum (E = ∆0). At this bias, degnerate wells (DW) are
well minima. The tunneling dipole can alternatively be created in the TLS double well potential (as well as the
described with a distance between the wells 2p/q for a even-amplitude superposition of the single well states).
tunneling charge q. The TLS distribution from the stan- Fig. 3 shows the distribution of 60 pz extracted us-
dardmodelisdn2 ∝d∆ d∆/∆ [16,17,25–27]. Through ing data from three separate cool downs. The result-
0 0
symmetry the bias lead at voltage Vbias is isolated from ing distribution of pz can be interpreted to have two
the ac resonance. The capacitors are formed with two peaks, and suggests that we compare it to the standard
aluminum films and a d =125 nm thick layer of silicon amorphousmodelofTLSscontainingtwodipole-moment
0
nitride (SiNx) in between. Since the bridge capacitance magnitudes, p1 and p2. In accordance with the random-
contains two capacitors in series, the TLSs within expe- orientation defect model, we write the TLS number den-
riencethebiasfieldEbias =Vbias/(2d0). Thecapacitance sity d3n=dpzd∆d∆0(P1Θ(p1−pz)/(p1∆0)+P2Θ(p2−
and a quadrupole spiral inductor produce a resonance at pz)/(p2∆0))intermsoftheprojecteddipolemomentpz,
6.0 GHz. The SiNx dielectric volume within the capac- where Θ is the Heaviside step function and Pi is the ma-
itors V = 78 µm3 is micron-scaled (micro-V) to allow terial constant for pi, and i = 1,2 [28]. The range of
strong enough coupling for individual TLS observations. dc-biascreatesanobservablesetofDWswithadistribu-
The dielectric used in this study has a loss tangent of tion
tanδ = 7.8×10−4 as obtained from the p data below dn κ p δV P Θ(p −p ) P Θ(p −p )
0 z DW = z bias( 1 1 z + 2 2 z ).
[28]. The silicon nitride is deposited with PECVD[6]. dp ω d p p
z c 0 1 2
Individual TLS have also been observed using films with (2)
smallertanδ [22],butthefilmforthisstudywasselected Here the experimental ranges are δ∆ = h¯κ and δ∆ =
0 0
3
FIG. 2. (a) False-color plot of transmission |S |=|V /V | vs. frequency f and bias voltage V . Data is taken at T =25
21 out in bias
mKandn¯ (cid:39)0.4. Lightcopperandblackcorrespondto|S |=0.56and|S |=0.40,respectively. TLSsareobservedwith
max 21 21
minimainenergy(thetunnelingenergy∆ )nearlydegeneratewiththecircuitphotonenergy¯hω . (b)Fitswithenergymodel
0 c
(Eq. 1) to data in panel (a) are shown which find the projected moment p and tunneling energy ∆ of 30 TLSs.
z 0
25 bye and p = 8.4 Debye, as shown in the figure, where
2
circle markers and the dashed line indicate bin heights.
p p
1 2 The fit to the data is consistent with the model if the
20
bin to the right of 2.8 Debye has some counts from the
p distribution while the bin to the left of 2.8 Debye has
1
15 lower counts than the fit value due to Poisson statisti-
N cal fluctuations. Using P1 and P2 from the initial fit,
Monte Carlo analysis was performed using different in-
10
put values for p and p . Simulation datasets drew the
1 2
statistically correct number of p in simulation distribu-
z
5
tions. Fits with optimization for Poisson statistics [30]
were then used. These fits created simulation-extracted
0 p1 andp2 whichgenerallydifferfrominputvaluesdueto
0 2 4 6 8 10 Poisson sampling error. The difference was used to find
p (Debye) the probable range of p and p for a two-moment ma-
z 1 2
terial. Hundreds of simulated datasets were used to find
the range of moments consistent with the observed data.
FIG. 3. Distribution of p as determined from fits similar
z Theanalysisgivesp =2.8±0.2Debyeandp =8.3±0.4
to Fig. 2 (b). An initial fit to the model with two dipole 1 2
moment magnitudes p is shown in circle markers and dashed Debyeforthemomentsofthemodel,wheretherangein-
lines. The two-sawtooth pattern follows from the technique dicates a single standard deviation. Thus we find our
andastandardmodelofrandomTLSorientations. Themost data is consistent with a model of two dipole moments,
likely ranges for moments magnitudes, p1 = 2.8±0.2 Debye and that with the use of this model the standard devia-
and p = 8.3±0.4 Debye, are shown by the markers with
2 tions of dipole moments are ≤7%.
horizontal error bars.
To the knowledge of the authors, this experiment
marks the first study of a single sample from which two
TLS moment magnitudes were studied. In applications
p δV /d ,whereκisthecouplingratebetweentheres-
z bias 0 with this material we expect that p is the more impor-
1
onator and transmission line. We see from theory that
tant moment for material characterization since TLSs in
for each p there should be a number of degenerate-well
i thispartofthedistributionaremuchmorenumerous. In
TLS in a bin N, where N ∝p . This feature is a conse-
z analuminabarrierofaJJqubitaTLSdipolemomentof
quence of how larger p will be swept further in energy
z 3Debyewasfoundifoneassumesthebarrierthicknessof
for a fixed domain in δV .
bias 2 nm [1] (TLSs detected in tunneling barriers are depen-
An initial fit of Eq. 2 to the data yields p = 2.9 De- dent on an unknown barrier thickness). In δ -averaged
1 0
4
measurements of bulk SiO with OH− correlated TLSs
2
themomentmagnitudeextractedis3.3Debye[31]. p of
1
our nitride film is similar and, from SIMS analysis, our
filmsarealsoknowntocontainsomeOandH.Although
our value of p is large, it is within the range of values
2
reported in other measurements of silicon nitride [20].
WebelievethattheDWextractiontechniquecreatesthe
potentialforquantitativeanalysisonsamples,wheredif-
ferent dipole distribution models could be tested with
statistical analysis.
Furthermore,sinceourmethodusesaninsulating(cur-
rent blocking) layer for TLS, it can be extended for use
FIG. 4. (a) False-color plot of measured |S | vs. frequency
to characterize various materials which are important to 21
f and bias voltage V for one of the most strongly coupled
quantum-sensitive devices. Any insulating film can be bias
TLSs. Theresonator-TLScouplinginducesavoidedcrossings
tested using this technique, and this includes films of
dependent on the coupling g. Data is taken at T = 24 mK
alumina, the material commonly used in JJ tunneling and a maximum photon occupancy of n¯ (cid:39) 0.4. Light
max
barriers. In addition, the technique can accommodate copperandblackcorrespondto|S |=0.60and|S |=0.43,
21 21
multiple layers of TLS-laden films, such that it could respectively. (b)Optimumfittopanel(a)datausingacQED-
test surface oxides, states at a superconductor/insulator based model (see main text). White dashed curves show the
extractedlow-powereigenenergiesE andE ofthesystem.
boundary, process contamination at the same boundary, + −
g is extracted along with p and ∆ , without the electric
the thickness dependence of TLSs in films, etc. Finally, z 0
volume.
our technique could test the dependence of p on an in-
dependent variable, such as hydrogen concentration at a
host material or interface. fluctuation field E without using the geometrically
RMS
WenextanalyzeoneofthemoststronglycoupledTLS estimated electric-field volume. From the optimum fit
to the resonator, where Fig. 4(a) shows a close-up of the parameters we calculate E = h¯gE/p ∆ = 25 V/m
RMS z 0
DW spectra. Here we clearly see the minima energy of (where∆ /E (cid:39)1). Ontheotherhand,thedesignparam-
0
(cid:112)
the TLS with the vacuum Rabi splitting. The Jaynes- eters allow us to calculate E = ¯h ω /(2(cid:15) (cid:15) ¯hV) =
RMS c r 0
Cummings eigenenergies for this system are 21 V/m (using (cid:15) = 6.5 [6] and the bridge capacitors’
r
dielectric volume V), which is 16% smaller than the fit
1 (cid:112)
E = ¯h(ω +ω )±¯h g2+(δ/2)2, (3) determined E . Since V was not underestimated and
± 2 c TLS RMS
p isveryaccurate,thedifferenceisbelievedtobecaused
z
where ω = E/¯h (Eq. 1), and δ represents the TLS- by the uncertainty in the measured g.
TLS
resonator detuning. Figure 4(b) shows the least squares In conclusion, a resonator was studied which accu-
MonteCarlo(LSM)fittothespectrausingthetheoryof ratelyappliesauniformdc-electricfieldtoTLSsinafilm.
theirlineshapesRef. [22,29]. Theeigenenergiesfromthe ThefieldallowsTLSenergiestobeanalyzedasafunction
fit determined p = 6.0 Debye and coupling g/2π = 753 ofaknownfieldandthedirectextractionof60projected
z
kHzareshownaswhitedashedcurves. Thefitalsoyields moments p . An analysis reveals a good fit to a model
z
the coherence time T = 2/γ = 313 ns, related to with two representative dipole moment magnitudes for a
2 TLS
the spectral width. Here, γ is the TLS relaxation silicon nitride film, p =2.8±0.2 and p =8.3±0.4 De-
TLS 1 2
rate which limits coherence because the temperature is bye. Thep ofournitridefilmwascomparabletothatof
1
lowandstrongcouplingtotheenvironment,throughthe OH− found in bulk silicon oxide. With additional data
cavity, provides enhanced relaxation. For this TLS the this method should allow one to search for additional
eigenenergy exhibits a visible splitting (g) such that this dipole magnitudes (TLS types). Since this study uses a
fit yields a more accurate p than extracted from the depositedinsulatingfilmcontainingTLSs,itcouldbeex-
z
previousmethod,e.g.,themethodofFig. 2(b). However, tendedtoanalyzeothermaterialsandstructures(surface
the difference in p caused by even the strongest cQED oxides, dielectric-superconductor interface states, fabri-
z
coupling is small; p is only different by 1% between the cation contamination, interdigital capacitors, etc.). This
z
twomethods. Thisshowsanexampleofhowp extracted shouldallowtheclassificationofTLSsbydipolemoment
z
earlier(forFig. 3)usingaweak-couplingassumptionare and the optimization of quantum computing or photon
only weakly perturbed. Since g/κ = 1.7 is larger than 1 detectordevices. ThecQEDmeasurementalsoalloweda
we are in the strong coupling regime. However, the TLS completemeasurementofthezero-pointfluctuationelec-
relaxation is large in this case, g/γ = 0.74, showing tric field. This demonstrates how strong-coupling cQED
TLS
thatthestrongcouplinglimitdoesnotrelyongoodTLS can be achieved with TLSs in a practical energy-tunable
coherence [32]. circuit such that many measurements which are avail-
This analysis also gives a measurement of the RMS able for superconducting qubits should now be available
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