Nonlinear Coupling in Nb/NbN Superconducting Microwave Resonators

0.45 MB·English

by  Baleegh Abdo

APS/123-QED
Nonlinear Coupling in Nb/NbN Superconducting Microwave Resonators
∗
B. Abdo, E. Segev, O. Shtempluck, and E. Buks
Microelectronics Research Center, Department of Electrical Engineering, Technion, Haifa 32000, Israel
(Dated: February 2, 2008)
5 In this experimental study we show that the coupling between Nb/NbN superconducting mi-
0 crowave resonators and their feedline can be made amplitude dependent. We employ this mech-
0 anism to tune the resonators into critical coupling condition, a preferable mode of operation for
2 a wide range of applications. Moreover we examine the dependence of critical coupling state on
otherparameterssuchastemperatureandmagneticfield. Possiblenovelapplicationsbasedonsuch
n
nonlinear coupling are briefly discussed.
a
J
PACSnumbers: 84.40.Az,85.25.-j,42.50.Dv
1
1

Recently, there has been a growing scientific interest resonatorsare presented in the insets of Figs.4 and 2 re-
]
n in nonlinear directional couplers especially in the opti- spectively. Theresonatorsweredc-magnetronsputtered,
o cal regime [1]. It has been shown in a series of publica- atroomtemperature,onsapphiresubstrateswithdimen-
c tions,thatoperatingsuchdevicesbymeansofparametric sions 34mm X 30mm X 1mm each. A gap of 0.5 mm
-
r downconversion, may exhibit some important quantum was set between the feedline and the resonators. The
p phenomena, such as generation of quadrature-squeezed thickness of the Nb sputtered film is 2200˚A, whereas the
u states [1], occurrenceofquantum Zeno andanti-Zenoef- thickness of the NbN film is 3000˚A. Following the de-
s
. fects [2], non-classical correlations [3], and quantum en- position,the Nb resonatorwaspatternedusing standard
t
a tanglement[4]. Sucheffects,ifgeneratedandengineered, photolithography process and dry etched with photo re-
m would have immediate and significant implications on a sist mask. The sputtering parameters of the NbN su-
- wide range of fields, ranging from basic science to opti- perconducting microwave resonator have been discussed
d cal communication systems, and quantum computation elsewhere [10].
n
[5],[6]. Thedependence ofthese mentionedeffectsonop-
o
timalcouplingstrengthsandcavitydetuningsarealmost The measurements of the resonators were done using
c
[ evident [1],[3]. theS parameterofavectornetworkanalyzer,whilevary-

Inthis studyweinvestigatethecouplingbetweennon- ingthe input powerorthe ambienttemperature insmall
1
linearNbN/Nbmicrowaveresonatorsandtheirfeedlines, steps. In Figs. 1,2 we show a power induced critical
v
6 asafunctionofinjectedpower,ambienttemperatureand couplingdata obtainedatthe firstmode ofthe NbN res-
3 applied magnetic field. We show that the coupling in onator. In Fig. 1 we plot the S11(ω) curves of the first
2 theseresonatorscouldbe,toalargeextent,dependenton mode in the complex plane, corresponding to three in-
1 externally applied parameters, allowing us thus to tune put powers 0.944mW, 1.072 mW, 1.14mW in the vicin-
0 the resonatorsintocriticalcouplingcondition,whichisa ity of the resonance, representing p < p , p = p , p > p
c c c
5
coupling state where the power transferredto the device statesrespectively,wherep istheinjectedpowerlevelat
0 c
is maximized and no power reflection is present. Such a whichcriticalcoupling isachieved. Thesethree trajecto-
/
t state occurs as bothimpedances ofthe device and ofthe riesshowclearlythattheNbNresonatoratitsfirstmode
a
m feedline match. has transformed fromovercoupled(strongly coupled), to
Theoretically critical coupling can be achieved by ei- undercoupled state (loosely coupled) through critically
-
d thertuningthecouplingparametersorbyalteringthein- coupled condition via input power increase only. In Fig.
n ternal/externallossesofthe device. Suchcontrolwasre- 2 (a) we show the resonance curves |S11| of the three in-
o portedinopticalsystems[7], wherecriticalcouplingwas putpowers,presentedpreviouslyinFig. 1,inthevicinity
c
obtained in microring resonator by altering the internal of the resonance. The resonances were shifted vertically
:
v lossinsidetheringbymeansofintegratedsemiconductor for clarity. In Fig. 2 (b) we present the nonlinear evo-
Xi optical amplifiers. Other critical coupling achieved in a lution of the |S11| minimum at resonance as the input
fiber ring resonator configuration may be used in opti- power is increased through the critical coupling power.
r
a calcommunicationsystems i.e. switchesandmodulators Theminimumofthegraph∼−60dBatpc =1.072mW,
[8],[9]. corresponds to critical coupling condition where there is
no power reflection. The gradual shift of the resonance
Our fabricated resonators were assembled in a stan-
frequency towards lower frequencies as the input power
dardstriplinegeometry,andwerehousedinagoldplated
is increased, is plotted in Fig. 2 (c).
Faraday package made of Oxygen Free High Conductiv-
ity (OFHC) Cupper. The layout of the Nb and NbN
In order to identify and characterize the coupling
mechanism responsible for this dependence on the drive
amplitude, we apply the following universal expression
∗Electronicaddress: [email protected] for reflection amplitude of a linear resonator near reso-
2
1
0.8
0.6 1.072mW
0.9441mW
1.14mW
0.4
S11 0.2
Imaginary −0.02
−0.4
−0.6 FIG. 3: Resonator coupling states as defined by their S11(ω)
−0.8 curves in the vicinity of the resonance ω0, (see Eq.(1)) cor-
responding for (a) undercoupled state (b) critically coupled
−1
−1 −0.5 0 0.5 1
Real S state (c) overcoupled state.
11
FIG. 1: NbN resonator first mode S11(ω) curve, drawn in
the complex plane for three input powers 0.944mW, 1.072
mW,1.14mW,with2MHzspan. Theresonatorchangesfrom magnetic field. This generalization could be justified by
overcoupled to undercoupled state as the power is increased noting that the |S11(ω)| curves measured in the vicinity
through pc = 1.072 mW. The unit circle and the origin are ofthe resonance,aretoagoodapproximationsymmetri-
also plotted on the same axis for visual comparison. calLorentzians. The different possible coupling states of
the resonatorasdescribedgenerallybyEq.(1)areshown
schematically in Fig. 3.
Byexpandingγ1 andγ2 nearcriticalcouplingdrivexc,
to lowestorder in δx=x−xc, we getγ1 =γ+g1δx and
γ2 =γ+g2δx, where γ,g1 and g2 are real constants.
Substitutingforγ1andγ2inthevicinityofxcinEq.(1)
at the corresponding resonance ω =ω0, yields
2 3
S11(ω0)=a1δx+a2(δx) +O(cid:16)(δx) (cid:17), (2)
where a1 =(g2−g1)/2γ, a2 = g12−g22 /4γ2.
(cid:0) (cid:1)
In order to evaluate the critical coupling constant γ
characterizingtheresonancepresentedearlier,weapplied
a fit to the resonance curve at p using Eq.(1) and ob-
c
tained γ = 1.3MHz. Moreover, we extracted g1 and g2
by applying a quadratic polynomial fit as in Eq.(2) to
the experimentaldata ofrealS11 atresonanceasa func-
FIG. 2: Data analysis of NbN resonator first mode. (a) tion of δp. The data and the quadratic fit are presented
|S11 | graphs representing p < pc, p = pc and p > pc cases. in Fig. 2 (d), yielding the following nonlinear coupling
The resonance curves were shifted vertically by a constant rates g1 = −0.1MHz/mW, g2 = 1.2MHz/mW. Having
offset for clarity. (b) |S11 | minimum as a function of input |g2| > |g1| and g2 > 0, implies that the critical coupling
power. The minimum of the graph is at pc. (c) Resonance in this case, is caused mainly by the increase of the dis-
frequency vs. input power. (d) Real S11 at resonance vs.
sipation coupling as the input power is increased.
power difference from critical coupling power. The resonator
In the Nb resonator, power induced critical coupling
geometry is shown in theinset of subplot (b).
was attained at 12.589GHz resonance as shown in Fig.
4. The resonator changes from undercoupled state to
overcoupled state as the power is increased through
nance [11] the critical coupling power at 6.59mW . The coef-
ficients γ,g1,g2 extracted for this case are 48.5MHz,
1.6MHz/mW, 1.5MHz/mW respectively. Implying sur-
i(ω0−ω)−(γ1−γ2)
S11(ω)= , (1) prisinglythatwhatledtocriticalcouplinginthiscase,is
i(ω0−ω)+(γ1+γ2)
the increase of the feedline-resonator coupling at higher
where γ1 and γ2 are real coupling factors associated rate than the increase of the dissipation coupling rate,
with the feedline-resonator and the resonator-reservoir since we have |g1| > |g2|, g1 > 0. Such increase in the
(dissipation) couplings respectively. However,unlike lin- feedline-resonatorcouplingcouldbe attributedtoa non-
earoscillatorswhereγ1,γ2 andω0 areconstants,herewe linear redistribution of the standing-wave voltage-mode
assumethatthese factorsdependonsomeexternallyap- insidetheresonatorrelativetothefeedlineposition,lead-
pliedparameterx,suchasinputpower,temperature,and ing to amplitude dependence of γ1.
3
(a) (b)
−10 −10
5.79 K
|S| [dB]11−−−−54320000 5.81 K5.8 K S| [dB] at resonance11−−−432000
|
−60 −50
4.339 4.34 4.341 4.342 4.343 4.344 4.345 4.346 5.7 5.75 5.8 5.85 5.9 5.95
Frequency [GHz] Temperature [K]
(c) (d)
Resonance frequency [GHz]44444.....433333.334443682464 Real S at resonance11−−0000....02442 d a qtauadratic fit
5.7 5.75 5.8 5.85 5.9 5.95 −0.1 −0.05 0 0.05 0.1
Temperature [K] δ T [K]
FIG. 4: Data analysis of Nb resonator critical coupling at FIG. 5: Data analysis of NbN second mode temperature
12.589GHz resonance. (a) |S11 | graphs representing p < pc, induced critical coupling. (a) |S11| graphs representing
p = pc and p > pc cases. The resonance curves were shifted T < Tcrit,T = Tcrit and T > Tcrit cases. The resonance
vertically bya constant offset for clarity. (b) |S11 |minimum curves were shifted vertically by a constant offset for clarity.
as a function of input power. The minimum of the graph is (b) |S11 | minimum as a function of temperature, the mini-
at pc . (c) Resonance frequency vs. input power. (d) Real mum of the graph is at Tcrit . (c) Resonance frequency vs.
S11 at resonance vs. power difference δp from critical cou- temperature. (d) Real S11 at resonance vs. temperature dif-
plingpower. Theresonatorgeometry isshown intheinsetof ference δT from critical coupling temperature.
subplot (b).
such effects in the microwave regime.
Basedonourexperimentaldemonstrationofnonlinear In addition to input power tuning, we have succeeded
coupling between superconducting stripline elements it to obtain critical coupling using other modes of oper-
is interesting to examine the feasibility of implementing ation, one of which was tuning the ambient tempera-
some of the theoretical proposals presented in Ref. [1-6] ture. To demonstrate this, we applied a constant input
in the microwave regime. For that end we consider a powertotheNbNresonatorandmeasureditssecondres-
resonator made of two stripline elements having nonlin- onanceresponseasweincreasedtheambienttemperature
ear coupling factor g between them. The resonator is in 0.01K steps. The input power applied was 0.1mW
also coupled to a feed line employed for delivering the , lower than pc of that resonance at 4.2 K, which was
input and output signals. Consider operating close to a about 0.209 mW. The resonator changed from over-
resonancehavingangularfrequencyω0 anddampingrate coupled state at T < Tcrit to undercoupled state at
γ. A detailedanalysisof this model is beyondthe scope T > Tcrit. Tcrit was found to be 5.8K. The measure-
of the present letter. However, employing dimensional- ment results of this mode of operation are summarized
ity analysis and some simplifying assumptions one finds in Fig. 5. By applying similar fitting procedures as was
that the onset of nonlinear instability and bifurcation is explained before one can obtain the following constants:
expected in such a system at input power level of order γ = 4.2MHz,g1 = −0.6MHz/K,g2 = 220MHz/K, indi-
pin ∼= γ/g. In the regime of operation when the in- cating that what led to critical coupling condition, in
NL
putpowerisclosetopin effectssuchasintermodulation this case, was the increase of the dissipation coupling as
NL
gain and quantum squeezing [1] are expected to become a result of the temperature rise.
noticeable [11]. Based on our results with the Nb res- Moreover to better understand the dual relation be-
onator, we estimate from the experimental values of γ tweentheinputpowerandtheambienttemperature,ad-
and g1 (assuming g =g1) that sucheffects canbe imple- ditional measurement was carried out on the first mode
mented with a moderate input power of order 10−2W. of the NbN resonator. We varied the temperature of
On the other hand, effects such as Zeno and anti-Zeno the resonatorasa parameterandateachgiventempera-
[2] require much stronger nonlinear coupling. We esti- ture,wescannedtheinputpowerinsearchforthecritical
mate that in the limit of zero temperature such effects coupling power p . The measurement result is shown in
c
will become noticeable only when the input power be- Fig. 6, where we see that in the case of the first mode
comes comparable to piϕn ∼= γ/ℏω0g2. Again, assuming of the NbN resonator at ∼ 2.5GHz, increasing the am-
the experimental values extracted from the data of the bient temperature decreased the input power at which
Nb resonator, one finds pin ∼= 1012W - far too high for critical coupling occurred, and thus divided the T −pin
ϕ
anypracticalimplementation. However,furtheranalysis plane into two regions representing the overcoupled and
is requiredto explore other possibilities of implementing undercoupled resonator states, approximately separated
4
by the guideline shown in the figure.
Critical coupling condition was also achieved in the
NbN resonator by means of applied magnetic field. To
show this we set a constantinput power of1.23mW (the
critical coupling power at the time of the measurement
in the absence of magnetic field was 1.259mW), and in-
creased a perpendicular magnetic field by small steps.
The results of this mode of operation, unlike power and
temperature, showed a distinct instability and frequent
transitions around the critical coupling state as can be
inferredfromFig. 7. Apossibleexplanationtothisinsta-
FIG. 6: A plot displaying the critical coupling power depen- ble behavior could be vortex penetration into the grain
dence of the first mode of the NbN resonator on ambient boundaries of our NbN film, as H in NbN could be in
c1
temperature. The critical coupling power decreases as the the order of 35mT [12].
temperature is increased.
In summary, we have succeeded to tune our supercon-
(a) (b)
0 −35 ducting microwave resonators into critical coupling con-
−20 4455..59mmTT dition using input power, ambient temperature and ap-
4467..87mmTT−40 pliedmagneticfield. Bydatafittingwehavebeenableto
−40 48.6mT
49.5mT
|S| [dB]11−−−−1186200000 55555555555660123455678901.....m.......543219876554TmmmmmmmmmmmmTTTTTTTTTTTT−−−554505 |S| [dB] at resonance11 apdaexnbliistdlncerguaidcsccsoteehundqaptunilfsiagyonnemgtthiienmteaoetdefciaovhtcmeahhlnyienicsaptamnhostees.sifnicabocIlneutexoprahalipdirnbpdegislititpicpinooaagnntrisaosiwmobnmelseeteohfeofarirsvmtehtγphisb,oegrrvc1tieaao,flrgnuiy2t--
6623..32mmTT quantum phenomena in the microwave regime.
−140 68.2mT
−60
−160
−1820.5172.5182.5192.522.5212.5222.523 −6545 50 55 60 65 70 E.B. would especially like to thank Michael L. Roukes
Frequency [GHz] Magnetic field [mT]
for supporting the early stage of this research and
for many helpful conversations and invaluable sugges-
FIG. 7: Critical coupling achieved by magnetic field at the
first mode resonance of the NbN resonator while applying tions. Very helpful conversations with Bernard Yurke
constant input power of 1.23mW. (a) |S11| vs. frequency are also gratefully acknowledged. This work was sup-
corresponding to an increased applied magnetic field. The ported by the German Israel Foundation under grant 1-
resonance curves were shifted vertically by a constant offset 2038.1114.07,the Israel Science Foundation under grant
for clarity. (b) The |S11| value at the resonance frequency. 1380021, the Deborah Foundation and Poznanski Foun-
dation.
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