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008

Phase sensitive amplification in a superconducting stripline resonator integrated with

a dc-SQUID

Baleegh Abdo,∗ Oren Suchoi, Eran Segev, Oleg Shtempluck, and Eyal BuksDepartment of Electrical Engineering, Technion, Haifa 32000, Israel

Miles BlencoweDepartment of Physics and Astronomy - Dartmouth College

(Dated: February 18, 2013)

We utilize a superconducting stripline resonator containing a dc-SQUID as a strong intermod-ulation amplifier exhibiting a signal gain of 25 dB and a phase modulation of 30 dB. Studyingthe system response in the time domain near the intermodulation amplification threshold reveals aunique noise-induced spikes behavior. We account for this response qualitatively via solving numer-ically the equations of motion for the integrated system. Furthermore, employing this device as aparametric amplifier yields a gain of 38 dB in the generated side-band signal.

PACS numbers: 05.45.-a, 85.25.Dq, 84.40.Dc

The field of solid-state qubits and quantum informa-tion processing has received considerable attention dur-ing the past decade [1, 2, 3, 4] and has successfullydemonstrated several milestone results to date [5, 6, 7, 8,9, 10, 11]. However, one of the fundamental challengeshindering this emerging field is noise interference, whicheither screens the output signal or leads to quantum statedecoherence [5, 6, 12, 13, 14, 15]. Hence, this may ex-plain to some extent the renewed interest exhibited re-cently by the quantum measurement community in thefield of phase-sensitive amplifiers [16, 17, 18, 19, 20, 21].This interest is driven essentially by two important prop-erties of these amplifiers: (1) Their capability to amplifyvery weak coherent signals; (2) Their ability to squeezenoise below the equilibrium level by means of employ-ing a homodyne setup and phase control. These prop-erties are expected to be highly beneficial to the areaof quantum communication [22, 23] and to the genera-tion of quantum squeezed states [16, 17, 24, 25]. Ad-ditional interest in these high-gain parametric amplifiersarises from the large body of theoretical work predict-ing photon-generation from vacuum via the dynamicalCasimir effect [26, 27], which can be achieved by em-ploying an appropriate parametric excitation mechanism[28, 29].

In this present work we study a superconductingstripline microwave resonator integrated with a dc-SQUID [30, 31, 32]. The paper is mainly devoted toa novel amplification mechanism in which the relativelylarge nonlinear inductance of the dc-SQUID is exploitedto achieve large gain in an intermodulation (IM) con-figuration. We provide theoretical evidence to supportour hypothesis that the underlying mechanism responsi-ble for the large observed gain is metastability of the dc-SQUID. In addition, at the end of the paper we demon-strate another amplification mechanism, in which the de-

∗Electronic address: baleegh@tx.technion.ac.il

pendence of the dc-SQUID inductance on external mag-netic field is exploited to achieve parametric gain [20].

The device (see Fig. 1 (a)) is implemented on a high-resistivity 34 mm×30 mm×0.5 mm Silicon wafer coatedwith a 100 nm thick layer of SiN. As a preliminary step,thick gold pads (300 nm) are realized at the periphery ofthe wafer. Subsequently, e-beam lithography is appliedin which both the macroscopic resonator and the micro-scopic Josephson gaps are written. Following a develop-ing stage, a two angle shadow evaporation of Aluminium[33] -with an intermediate stage of oxidation- implementsthe resonator as well as the two Al/AlOx/Al Josephsonjunctions comprising the dc-SQUID. The total thicknessof the Aluminium evaporated is 80 nm (40 nm at each an-gle). Finally, a lift-off process concludes the fabricationof the integrated system.

In general, IM generation is a nonlinear phenomenonwhich is associated often with the occurrence of nonlin-ear effects in the resonance curves of a superconduct-ing resonator [34, 35, 36, 37, 38]. In Fig. 2 we show atransmission measurement of the resonator response, ob-tained using a vector network analyzer at the resonancewhile sweeping the frequency upwards. The measuredresonance resides at f0 = 8.219 GHz which correspondsto the third resonance mode of the resonator having ananti-node of the rf-current waveform at the dc-SQUIDposition. Similar nonlinear effects in the transmission re-sponse have been measured at the first mode (≃ 2.766GHz) as well.

As can be seen in the figure, at excitation powers Pp be-low Pc = −60.3 dBm (at which nonlinear effects emerge)the resonance curve is linear and Lorentzian. As the in-put power is increased abrupt jumps appear at both sidesof the resonance (see black line in Fig. 2) accompanied byfrequency hysteresis loops. In addition, as one continuesto increase the input power the resonance curves becomeshallower, broader and less symmetrical. Such power de-pendency can be attributed to the nonlinearity of thedc-SQUID inside the resonator which increases consid-erably as the amplitude of the driving current becomes

2

FIG. 1: (Color) The device and IM setup. (a1) A photographof the device consisting of a circular resonator with a circum-ference of 36.4 mm and a line width of 150 µm, a dc-SQUIDintegrated into the resonator and a flux-line employed for driv-ing rf-power and flux-biasing the dc-SQUID. (a2) An electronmicrograph displaying the dc-SQUID incorporated in the res-onator and the adjacent flux-line. The area of the dc-SQUIDfabricated is 39 µm × 39 µm, while the area of each junctionis about 0.25(µm)2. The self-inductance of the dc-SQUIDloop is Ls = 1.3 · 10−10 H and its total critical Josephson cur-rent is about 2Ic = 28 µA. (a3) A micrograph of one of thedc-SQUID junctions. (b) A basic IM setup. The pump andthe signal designated by their angular frequency ωp and ωs

respectively are combined by a power combiner and fed tothe resonator. The field at the output is amplified using twoamplification stages and measured using a spectrum analyzer(path 1). A dc current was applied to the flux-line in order toflux-bias the dc-SQUID. Path (2) at the output correspondsto a homodyne setup employed in measuring IM.

comparable to the critical current.

The basic IM scheme used is schematically depicted inFig. 1 (b). The input field of the resonator is composedof two sinusoidal fields generated by external microwavesynthesizers and superimposed using a power combiner.The applied signals have unequal amplitudes. One, re-ferred to as the pump, is an intense sinusoidal field withfrequency fp, whereas the other, referred to as the sig-nal, is a small amplitude sinusoidal field with frequencyfp+δ, where δ represents the frequency offset between thetwo signals. Due to the presence of a nonlinear elementsuch as the dc-SQUID integrated into the resonator, fre-quency mixing between the pump and the signal yieldsan output idler field at frequency fp−δ. Thus the outputfield of the resonator, measured by a spectrum analyzer,consists mainly of three spectral components, the trans-

8.16

8.18

8.2

8.22

8.24

8.26

−64

−62

−60

−58

−56

−54

−52

−50

−20

−10

0

10

Frequency [GHz]Excitation Power [dBm]

|S21

| [dB

]

−1 −0.5 0 0.5 1−1

−0.9

−0.8

−0.7

−0.6

Φ/Φ0

Pum

p [a

.u.]

FIG. 2: (Color) Transmission response curves of the resonatorat its third resonance (8.219 GHz) as a function of input powerand frequency. The resonance curves exhibit nonlinear effectsat Pp ≥ Pc. The loaded quality factor for this resonance in thelinear regime is 350. The black line maps the jumps appearingat both sides of the resonance curves. The inset exhibits thedependence of the transmitted pump signal on the appliedmagnetic flux threading the dc-SQUID loop during an IMmeasurement.

mitted pump, the transmitted signal and the generatedidler. The IM amplification in the signal, idler and thenoise is obtained, as shown herein, by driving the dc-SQUID to its onset of instability via tuning the pumppower.

In Fig. 3 we show a typical IM measurement result. Inthis measurement the pump was tuned to the resonancefrequency fp = 8.219 GHz. The signal power Ps was setto −100 dBm, and its frequency offset δ to 5 kHz. As thepump power injected into the resonator is increased, thenonlinearity of the dc-SQUID increases and consequentlythe frequency mixing between the pump and the signalincreases as well. At about Pc the dc-SQUID reachesa critical point at which the idler, the signal and alsothe noise floor level (within the frequency bandwidth)undergo a large simultaneous amplification. The idlergain measured relative to the injected signal power atthe resonator input is about 13 dB (see inset of Fig. 3).The same result is obtained as well, as one calculates thepower ratio of the generated signal at the output portof the resonator to the input signal while accounting forthe losses and amplifications of the elements along eachdirection.

In order to show that this IM amplifier is also phasesensitive we applied a standard homodyne detectionscheme as schematically depicted in Fig. 1 (see path(2)), in which the phase difference between the pump andthe local oscillator (LO) at the output -having the samefrequency- can be varied. In such a scheme the pumpis down-converted to dc, whereas both signal and idler

3

FIG. 3: (Color) A spectrum power measured by a spectrumanalyzer during IM operation as a function of increasing pumppower while applying external flux Φ ≃ Φ0/4. The spectrumtaken at a constant frequency span around fp was shiftedto dc for clarity. At the vicinity of Pc the system exhibitsa large amplification. A cross-section of the measurementtaken along the pump power axis is shown in the inset. Thered line (light grey) and the blue line (dark grey) exhibit thecorresponding gain factors of the transmitted signal and theidler respectively.

are down-converted to the same frequency δ. The largestamplification is obtained when the LO phase (φLO) is ad-justed such that the signal and idler tones constructivelyinterfere. Shifting φLO by π/2 from the point of largestamplification results in destructive interference, which inturn leads to the largest de-amplification.

The IM measurement results, obtained using the ho-modyne setup while flux-biasing the dc-SQUID withabout half flux-quantum, are shown in Figs. 4 and 5. Fig-ure 4 exhibits the signal gain (blue line) versus increasingapplied pump power. As can be seen from the figure thesignal gain assumes a peak of 25 dB for Pp ≃ Pc. More-over, the response of the resonator at the pump frequency(green line) -measured simultaneously using a voltagemeter connected in parallel to the spectrum analyzer-is drawn as well for comparison. The sharp drop in theregion Pp ∼ Pc, coinciding with the amplification of thesignal, indicates that the transmitted pump is depletedand power is transferred to other frequencies.

Fig. 5 exhibits a periodic dependence of the signal gainon φLO, having a gain modulation of about 30 dB peakto peak between the amplification and de-amplificationquadratures. It is worthwhile mentioning that in thismeasurement not only the signal shows a periodic de-pendence on φLO but also the noise floor which exhibitsup to 20 dB modulation (measured at fp + 1.5δ).

Another interesting aspect of this device is revealed byexamining its response in the time domain while applyingan IM measurement using the homodyne setup shown in

−10

0

10

20

30

Sig

nal G

ain

[dB

]

−63 −62 −61 −60 −590.7

0.8

0.9

1

1.1

Res

pons

e [a

. u.]

Pump Power [dBm]

FIG. 4: (Color) A signal gain (blue line) measured as a func-tion of increasing pump power. The measurement was takenusing a homodyne scheme. The signal displays a gain of about25 dB at Pc. The corresponding dc component of the homo-dyne detector output (green line) was measured simultane-ously using a voltage meter in parallel to the spectrum ana-lyzer. In this measurement Ps = −110 dBm and δ = 5kHz.

0 0.5 1 1.5 2 2.5 3 3.5 4−15

−10

−5

0

5

10

15

20

φLO

[π]

Sig

nal G

ain

[dB

]

FIG. 5: A periodic dependence of the signal gain on the LOphase difference at the vicinity of Pc. The signal exhibits alarge amplification and de-amplification at integer and halfinteger multiples of π respectively. The phase modulationdependency shows up to 30 dB peak to peak amplitude.

Fig. 1 (see path 2). This is achieved by connecting afast oscilloscope in parallel to the spectrum analyzer atthe output. A few representative snapshots of the devicebehavior measured in the time domain are shown in Fig.6, where each subplot corresponds to a different appliedpump power. In subplot (a) the pump power is about3 dBm lower than the critical value and it displays aconstant voltage level. As the pump power is increased(subplot (b)) separated spikes start to emerge. Increasing

4

−1−0.8−0.6−0.4−0.2

(a) −63.11 dBm

−1−0.8−0.6−0.4−0.2

(b) −61.35 dBm

−1−0.8−0.6−0.4−0.2

(c) −60.45 dBm

Vol

tage

[V]

−1−0.8−0.6−0.4−0.2

(d) −60.11 dBm

−1−0.8−0.6−0.4−0.2

(e) −60.01 dBm

0 1 2 3 4 5 6 7 8 9 10

−1−0.8−0.6−0.4−0.2

(f) −59.91 dBm

Time [µ sec]

FIG. 6: Snapshots of the resonator response measured in thetime domain during an IM operation. (a)-(d) display theresonator response at pump powers lower than the criticalvalue. (e) At the critical value. (f) Exceeding the criticalvalue.

the power further causes the spikes to start bunching withone another and as a consequence forming larger groups(subplots (c) and (d)). This bunching process reaches anoptimal point at the pump power corresponding to thepeak in the IM gain (subplot (e)). Above that value, suchas the case in subplot (f), the spikes become saturatedand the gain drops.

Such behavior in the time domain reveals also the un-derlying mechanism responsible for the amplification. Asit is clear from the measurement traces the rate of spikesstrongly depends on Pp at the vicinity of Pc, thus addinga small signal at fp + δ (which is effectively equivalent toapplying an amplitude modulation of the pump at fre-quency δ), results in modulation of the rate of spikes atthe same frequency, and consequently yields a large re-sponse at the signal and the idler tones.

In an attempt to account for the unique temporal re-sponse presented in Fig. 6, we refer to the two equationsof motion governing the dc-SQUID system given by [39]

βC

··

δ1

Ic

(

Φ0

2πRJ

)2

+·

δ1

(

Φ0

2πRJ

)

= −∂U

∂δ1

+ In1

βC

··

δ2

Ic

(

Φ0

2πRJ

)2

+·

δ2

(

Φ0

2πRJ

)

= −∂U

∂δ2

+ In2 , (1)

where Φ0 is the flux quantum, RJ is the shunt resis-tance of the junctions, Ic is the critical current of eachjunction, δ1, δ2 are the gauge invariant phase differencesacross junctions 1 and 2 respectively, In1, In2 are equi-librium noise currents generated in the shunt resistorshaving -in the limit of high temperature- a spectral den-sity of SIn = 4kBT/RJ , βC is a dimensionless parameter

defined as βC ≡ 2πIcR2

JCJ/Φ0, where CJ is the junctioncapacitance and U is the potential energy of the systemwhich reads

U = −I

2(δ1 + δ2) +

2Ic

πβL

(

δ1 − δ2

2−

πΦ

Φ0

)2

− Ic (cos δ1 + cos δ2) , (2)

where Φ is the applied magnetic flux, I is the bias currentflowing through the dc-SQUID, and βL is a dimensionlessparameter defined as βL ≡ 2LsIc/Φ0. While the circu-lating current flowing in the dc-SQUID loop is given byIcirc (t) = Ic (δ1 − δ2 − 2πΦ/Φ0) /πβL.

Furthermore, in order to account for the resonator re-sponse as well, we make two simplifying assumptions:(1) We model our resonator as a series RLC tank os-cillator characterized by an angular frequency ω0 =1/

√LC = 2π ·8.219 GHz, and a characteristic impedance

Z0 =√

L/C, where L and C are the inductance and thecapacitance of the resonant circuit respectively; (2) Weneglect the mutual inductance that may exist betweenthe inductor and the dc-SQUID.

Under these simplifying assumptions one can write thefollowing equation of motion for the charge on the capac-itor denoted by q (t)

Z0q

ω0

+ Rq + ω0Z0q + Vsq + Vin + Vn = 0, (3)

where Vn is a voltage noise term having -in the limit ofhigh temperature- a spectral density of SVn

= 4RkBT ,Vin (t) = Re

(

V0e−iωpt

)

is an externally applied sinu-soidal voltage oscillating at the pump angular frequencyωp and having a constant voltage amplitude V0. Whereas,

Vsq = Φ0

(

δ1 + δ2

)

/4π designates the voltage across the

dc SQUID. Using these notations the bias current in Eq.(2) reads I = q.

Finally in order to relate the observed output signal inFig. 6 at the output of the homodyne scheme to thefast oscillating solution q, we first express the chargeon the capacitor as q (t) =

√

2~/Z0 Re[

A (t) e−iωpt]

,where A (t) is a slow envelope function, second, we em-ploy the following input-output relation [24] Vout (t) =

V0 − i√

32Z0~γ2

1A (t), where γ1 is the coupling constant

between the resonator and the feedline, in order to ob-tain the field at the output port. third, we account forthe phase shift by evaluating the following expression2VLO Re

[

Vout (t) eiφLO]

, where VLO corresponds to theamplitude of the LO.

By integrating these stochastic coupled equations ofmotion numerically, while employing device parameterswhich are relevant for our case, one finds that the ob-served temporal behavior of the system can be qualita-tively explained in terms of noise-induced jumps betweendifferent potential wells forming the potential landscapeof the dc-SQUID which is given by Eq. (2). As a con-sequence of applying the voltage Vin to the integrated

5

−0.5

0

0.5

(a) |V0|=0.000193

−0.5

0

0.5

(b) |V0|=0.0001936559

−0.5

0

0.5

(c) |V0|=0.000193663

Am

plitu

de [a

. u.]

−0.5

0

0.5

(d) |V0|=0.000193699

−0.5

0

0.5

(e) |V0|=0.000193702

0 20 40 60 80 100 120−0.5

0

0.5

(f) |V0|=0.000194

Time [nsec]

FIG. 7: A simulation result. The output field exhibits spikesin the time domain similar to the spikes shown in Fig. 6. Thespikes occur whenever the system locus jumps from one wellto another due to the presence of stochastic noise while driv-ing the system near a critical point. In this simulation runthe system was simulated over a period of 1000 time cycles ofpump oscillations (one of the constraints set on the simulationwas maintaining a reasonable computation time). The param-eters that were employed in the simulation (same as experi-ment) are ω0 = ωp = 2π · 8.219 GHz, βL = 3.99, RJ = 9.4 Ω,Φ = Φ0/2 and φLO = π. The rest of the parameters were setin order to reproduce the measurement result: γ1 = 107 Hz,R = 3.3 Ω, βC = 5.86 and Z0 = 10Ω (corresponding experi-mental values are: γ1 ≃ 2.4 · 107 Hz, Z0 ≃ 50Ω, CJ was notmeasured directly, a capacitance on the order of 0.7 pF wasassumed).

system, the potential landscape of the dc-SQUID os-cillates at the pump frequency, and its oscillation am-plitude grows with the amplitude of the incoming volt-age. However, inter-well transitions of the system becomemost dominant as the oscillation voltage reaches a criti-cal value at which the potential landscape of dc-SQUIDbecomes tilted enough -due to the current flowing in thesystem- in order to allow frequent noise-assisted escapeevents from one well to another or across several wells,which in turn cause a voltage drop to develop across thedc-SQUID and induce jumps in the circulating current.Such hopping of the system state is manifested as wellin the homodyne output field response shown in Fig. 7,which in a qualitative manner mimics successfully themain temporal features shown in Fig. 6.

It is also clear from this inter-well transitions modelthat the amplification in this case is different from theso-called Josephson bifurcation amplifier [14], as in thelatter case the system is confined to only one well and thebifurcation occurs between two oscillation states havingdifferent amplitudes.

Moreover, just as in the recent experiment by Ya-

FIG. 8: (Color) (a) A homodyne setup employed in measur-ing parametric excitation. (b) Large gain measured in a para-metric excitation experiment corresponding to the generatedthree harmonics (at δ = 5 kHz, 2δ, 3δ respectively) and thenoise spectral density at 1.5δ. The gain was measured as afunction of increasing pump power applied at 2f0+δ. The sig-nal power applied to the resonator at f0 was Ps = −80 dBm.In this measurement a maximum gain of 38 dB is achieved atthe first harmonic.

mamoto et al. [20] we have additionally employed ourdevice as a parametric amplifier. To this end, we haveused the parametric excitation scheme exhibited in Fig.8 (a) in which the pump and signal tones are applied todifferent ports. The main rf signal (pump) is applied tothe flux-line at the vicinity of twice the resonance fre-quency of the resonator 2f0 + δ (δ ≡ ∆/2π = 5 kHz) and2f0 = 16.438 GHz does not coincide with any resonanceof the device. Whereas, the signal, being several orders ofmagnitude lower than the pump, is fed to the resonatorport at f0 and its main purpose is to probe the systemresponse.

In Fig. 8 (b) we exhibit a parametric excitation mea-surement result obtained using this device at Φ ≃ 0.6Φ0.In this result there is evidence of one of the charac-teristic fingerprints of a parametric amplifier: the exis-tence of an excitation threshold above which there is anoise rise and an abrupt amplification of the harmonic atf0 + δ (which results from a nonlinear frequency mixingat the dc-SQUID). As can be seen in the figure at about−40 dBm the first harmonic generated at δ and the twohigher-order harmonics at 2δ, 3δ get amplified consider-ably up to a maximum gain of 38 dB measured at the

6

first harmonic. Also in a separate measurement result(not shown here) the first harmonic has been found todisplay ≃ 20 dB peak to peak modulation as a functionof external magnetic field.

In conclusion, in this work we have designed and fab-ricated a superconducting stripline resonator containinga dc-SQUID. We have shown that this integrated systemcan serve as a phase sensitive amplifier. We have studiedthe device using IM measurement and parametric exci-tation. In both schemes the device exhibited distinctthreshold behavior, strong noise rise and large amplifica-tion of coherent side-band signals generated due to thenonlinearity of the dc-SQUID. In addition, we have inves-tigated the system response in the time domain duringIM measurements. We have found that in the vicinity ofthe critical input power the system becomes metastableand consequently exhibits noise-activated spikes in thetransmitted power. We have shown that this kind of

behavior can be explained in terms of noise-assisted hop-ping of the system state between different potential wells.We have also demonstrated that the main features ob-served in the time domain can be qualitatively capturedby solving the equations of motion for the dc-SQUID inthe presence of rf-current bias and stochastic noise. Sucha device may be exploited under suitable conditions in avariety of intriguing applications ranging from generat-ing quantum squeezed states to parametric excitation ofzero-point fluctuations of the vacuum.

ACKNOWLEDGEMENTS

This work was supported by the Israel Science Foun-dation, the Deborah Foundation, the Poznanski Founda-tion, Russel Berrie Nanotechnology Institute, US-IsraelBinational Science Foundation and MAFAT. BA was sup-ported by the Ministry of Science, Culture and Sports.

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