the josephson bifurcation - q-lab - yale university

Invited Review Article: The Josephson bifurcation amplifierR. Vijay,1 M. H. Devoret,2 and I. Siddiqi11Quantum Nanoelectronics Laboratory, Department of Physics, University of California,Berkeley, California 94720, USA2Department of Applied Physics and Department of Physics, Yale University,New Haven, Connecticut 06520-8284, USA

Received 28 April 2009; accepted 17 August 2009; published online 17 November 2009

We review the theory, fabrication, and implementation of the Josephson bifurcation amplifier JBA.At the core of the JBA is a nonlinear oscillator based on a reactively shunted Josephson junction. Aweak input signal to the amplifier couples to the junction critical current I0 and results in adispersive shift in the resonator plasma frequency p. This shift is enhanced by biasing the junctionwith a sufficiently strong microwave current Irf to access the nonlinear regime where p varies withIrf. For a drive frequency d such that =2Q1−d /p3, the oscillator enters the bistableregime where two nondissipative dynamical states OL and OH, which differ in amplitude and phase,can exist. The sharp I0 dependent transition from OL to OH forms the basis for a sensitive digitalthreshold amplifier. In the vicinity of the bistable regime 3, analog amplification ofcontinuous signals is also possible. We present experimental data characterizing amplifierperformance and discuss two specific applications—the readout of superconducting qubits digitalmode and dispersive microwave magnetometry analog mode. © 2009 American Institute ofPhysics. doi:10.1063/1.3224703

I. INTRODUCTION

Electrical amplifiers are ubiquitous in experimentalphysics. They raise the energy of a signal coming from ameasurement to a level sufficient to exceed the noise of therecording and processing electronics. Amplifiers are basedon nonlinear elements such as the transistor for instance.Typically, such components are dissipative and exhibit achange in their dynamic two terminal resistance in responseto an input signal applied to a third terminal, such as a gate.The superconducting tunnel junction1 is a unique two termi-nal device that can be operated as either a nonlinear resistoror inductor. A number of different circuit elements, such as asuperconducting island or loop, can serve as the third inputterminal to couple an external signal, such as an electriccharge or magnetic flux, respectively, to the properties of thejunction, in particular the critical current I0. Small variationsin I0 can significantly change the transport properties of thejunction under appropriate bias conditions, giving rise to asensitive amplifier with near quantum limited noiseperformance.2–4

Let us review the basic properties of Josephson tunneljunctions. A Josephson tunnel junction is formed by separat-ing two superconducting electrodes with a barrier such as athin oxide layer. The current It and voltage Vt of thejunction can be expressed in terms of t, the gauge-invariant phase difference, as It= I0 sin t and Vt=0d /dt, where the parameter I0 is the junction critical cur-rent and 0= /2e is the reduced flux quantum. These equa-tions parameterize a nonlinear inductor with inductance LJ

=0 / I0 cos, which for 1 or equivalently It / I01reduces to LJ0=0 / I0. This inductance is by constructionshunted in parallel by the geometric capacitance CJ of the

junction, thereby forming an electrical oscillator. The dy-namics of this oscillator can be described by the motion of aphase particle with coordinate in an anharmonic, sinusoidalpotential U=−0I0 cos. The natural frequency of smalloscillations in this potential, the plasma frequency,5,6 is p0

=1 /LJ0CJ. In general, due to the nonlinear nature of theJosephson inductance, the natural frequency of the oscillatorwill be a function of the oscillation amplitude.

In a Josephson amplifier, an external signal typicallymodulates the critical current I0 of the junction, changingboth the height of the sinusoidal potential U and the plasmafrequency p. Two measurement protocols to detect varia-tions in I0 are illustrated in Fig. 1. In the dissipative approacha, the height of the potential barrier UI0 , Idc is deter-mined by biasing the junction with a current Idc I0 anddetecting the onset of a finite resistance.7 In this scheme, thephase particle has sufficient energy to escape the potentialwell and the phase difference advances linearly in time,generating a finite dc voltage. Switching to the finite voltagestate of the junction can be avoided by using a dispersivemeasurement technique to determine pI0 , Irf as shown inFig. 1b. This involves applying an alternating currentIrf sindt and measuring the time varying voltage acrossthe junction at the drive frequency d.8–10 If Irf I0, the junc-tion remains in the superconducting state, and the phase par-ticle oscillates in only one well of the sinusoidal potentialwith no dc voltage generated. However, the amplitude andphase of the oscillating junction voltage Vt=VJ cosdt+J vary with pI0 , Irf. The Josephson bifurcation ampli-fier JBA is a dispersive detector that exploits the anharmo-nicity of the junction oscillator for enhanced measurementsensitivity. In this oscillator, the plasma frequency pI0 , Irf

REVIEW OF SCIENTIFIC INSTRUMENTS 80, 111101 2009

0034-6748/2009/8011/111101/17/$25.00 © 2009 American Institute of Physics80, 111101-1

Downloaded 05 Oct 2010 to 128.36.14.215. Redistribution subject to AIP license or copyright; see http://rsi.aip.org/about/rights_and_permissions

http://dx.doi.org/10.1063/1.3224703

http://dx.doi.org/10.1063/1.3224703

varies not only with the critical current I0 but also with thedrive amplitude Irf. When biased with a sufficiently strongcurrent Irf, the plasma frequency shift for a small modulationin I0 is significantly increased9,11 as compared to the casewith lower drive current when the oscillator is in the linearregime.

The dependence of the plasma frequency on drive am-plitude is depicted in Fig. 2 where the steady state amplitudeVJ and the phase J of the junction voltage are plotted as afunction of d and Irf see Sec. II for theory. The resonantresponse sharpens and shifts to lower frequency as the driveamplitude is increased. At higher drive powers, the systembifurcates from a single valued to a bistable regime. Whenthe drive frequency d solid vertical line is sufficiently de-tuned from the small oscillation plasma frequency p0, thesystem can have two possible oscillation states OL and OH,

which differ in amplitude and phase. The dashed lines on theresponse curves indicate unstable solutions. For a bias pointsuch as the one indicated by the black dot in the figure, asmall decrease in I0 will cause the oscillator to evolve fromOL to OH, thus realizing a sensitive threshold detector for I0.The smallest variation in I0 that can be resolved is limited bythermal or quantum fluctuations, which broaden this transi-tion. In this digital mode of operation, small changes in thecritical current translate into changes in the occupation prob-ability of the two dynamical states.

The circuit schematic of the JBA is shown in Fig. 3. Thetunnel junction is shunted in parallel by an additional purelyreactive impedance Z, which in the simplest case is acapacitor with CSCJ. This reactive impedance allows oneto tune the resonant frequency and the quality factor of theresonator. The junction can also be embedded in a transmis-sion line cavity to form a nonlinear transmission lineresonator.12,13 We will consider the capacitively shunted casein detail in this review though a brief description of the trans-mission line version will be provided in Sec. II. The inputsignal couples to the junction I0. Examples of input circuitsused to couple weak electrical signals are shown in the inset.In a, an external signal is inductively coupled to a super-conducting quantum interference device SQUID loop,which modulates I0. In b, a superconducting single electrontransistor S-SET is used to couple signals capacitively. Amicrowave generator with Z0=50 output impedancedrives a current Irftsindt via the circulator C, whichallows microwave signals to flow in one direction. The signalreflected from the resonator is the amplifier output and iscoupled to a cryogenic semiconductor amplifier through theadjacent port of the circulator. This signal is further amplifiedand demodulated at room temperature using quadrature mix-ers to determine the signal amplitude and phase.

The idea of amplifying signals using a bifurcation isquite common and observed in electrical, optical, chemical,and biological systems see Ref. 14 and references therein.

(a)

(b)

dc

ac

U

δ

U

δ

I(t) = Idc = 0.6 I0

I(t) = Irf sin (ωd t)

∆U(I0,Idc)

ωp (I0,Irf)

FIG. 1. Color online Dissipative a and dispersive b techniques to de-termine I0. In a, we directly sample the barrier height UI0 , Idc by bias-ing with a current It= Idc I0. By measuring the escape rate out of the zerovoltage state, one can infer the height of the barrier. In b, we drive thesystem with an ac excitation It= Irf sindt and measure the response atthe drive frequency to determine the plasma frequency pI0 , Irf.

-0.1-0.2 0 0.1 0.2

-0.1-0.2 0 0.1 0.21 - ωd / ωp0

1 - ωd / ωp0

V J/2πϕ

0ωp

0.05

0.1

0.15

0.2

0.250.010.020.030.040.050.06

Irf/I0

0.010.020.030.040.050.06

Irf/I0

-0.4

-0.2

0.0

0.2

0.4

φ J/π

OH

OH

OL

OL

Q=20

Q=20

FIG. 2. Color Computed steady state junction a voltage amplitude VJ

and b phase J of the JBA when driven with an ac excitation It= Irf sindt. The different traces correspond to different drive amplitudesIrf. Solutions are plotted as a function of normalized drive frequency for anoscillator with Q=20. OL and OH denote the low amplitude phase laggingand the high amplitude phase leading oscillation states, respectively. Theblack dot indicates a typical bias point where the system is highly sensitiveto changes in I0.

DRIVE

I0

Cs

Cs

Z(ω)

INPUT

OUTPUT

FOLLOWERAMPLIFIER

JBA

INPUTCIRCUIT

SQUIDLOOP JBA

INPUTCIRCUIT S-SET JBA

Irfsin(ωdt+φ(I0,Irf))

Irfsin(ωdt)

R=50Ω

C

C

C

(a)

(b)

FIG. 3. Schematic diagram of the JBA. A junction with critical current I0

and a parametrically coupled input is driven by a rf pump signal, whichprovides the power for amplification. When biased in the vicinity of thedynamical bifurcation point, the phase of the reflected signal dependssensitively on the input signal. The circulator C is used to couple signals inand out of the resonator. It also suppresses the noise of the following am-plifier reaching the oscillator and helps ensure that the fluctuations felt bythe oscillator correspond to the noise of a 50 resistor at the bath tempera-ture T. Inset: examples of parametric input coupling circuits using a SQUIDand S-SET.

111101-2 Vijay, Devoret, and Siddiqi Rev. Sci. Instrum. 80, 111101 2009


Recently, such nonlinear behavior has been observed in na-nomechanical resonators and is being extensively studied forapplications in parametric sensing.15 A biological example ofbifurcation amplification in nature is the human ear. The co-chlea in our ear is biased close to a Hopf bifurcation,16 whichleads to several remarkable properties in our hearing such ascompression of dynamic range, infinitely sharp tuning at zeroinput, and generation of combination tones. The ear is essen-tially a nonlinear amplifier, i.e., the response depends quitestrongly on the strength of the input signal. Researchers inrobotics and medical sciences are currently in the process ofconstructing a hearing sensor, which can mimic the nonlinearproperties of the cochlea.17 In superconducting devices, theJosephson parametric amplifier18 also exploits proximity to abifurcation. These devices are capable of achieving largegain with near quantum limited noise and can show quantumeffects such as squeezing of noise below the standard quan-tum limit.19 Note that the JBA can also be operated as aconventional parametric amplifier by combining a weak in-put signal along with the drive signal. When biased in thevicinity of the bifurcation points, the system can exhibitparametric gain20 like in recently developed amplifiers basedon multi-junction rather than single junction circuits em-bedded in microwave cavities21,22

In this review, we will detail the theory of operation ofthe JBA in Sec. II, focusing in particular on the case wherethe shunting impedance Z is a simple capacitor. In Sec.III, we discuss device design and fabrication. The cryogenicmeasurement setup is detailed in Sec. IV. In Sec. V, measure-ments characterizing the stand alone amplifier are presented.In Sec. VI, we discuss two particular applications of theJBA—measurements of the quantronium qubit and highspeed magnetometry. We present concluding remarks in Sec.VII.

II. THEORY

We study bifurcation amplification in two types of cir-cuits. The first one is a lumped element resonator where weshunt the Josephson junction with a capacitor to form thenonlinear resonator. This circuit is shown in Fig. 4a. Here,I0 is the critical current of the junction, CJ is the intrinsic

junction capacitance, CSCJ is the shunt capacitance to tunethe resonant frequency, and R is the real part of the currentsource impedance, which provides damping. In the secondtechnique, we embed a Josephson junction in a half-wavetransmission line resonator as shown in Fig. 4b. If we makea single mode approximation for the transmission line reso-nator near its resonant frequency, one arrives at the equiva-lent LCR circuit shown in Fig. 4c. The values of the effec-tive circuit elements are given by the following equations

Veff = Z00CinVd, Reff = Z02R0

2Cout2 , 1

Leff = Z0/20, Ceff = 2/ Z00 ,

where Z0 is the characteristic impedance of the transmissionline, CinCout are the input and output coupling capacitors,0 is the bare resonant frequency of the cavity set by itslength and speed of microwaves in the transmission line andR=50 as before is the characteristic impedance of themicrowave feed lines. The transmission line version, whichhas been dubbed the cavity bifurcation amplifier CBA, es-sentially works on the same principle as the lumped elementJBA though it offers some practical advantages such as easeof fabrication, absence of dielectric layers, possibility ofmultiplexing, etc. In this section, we will begin by establish-ing the connection between the JBA and the CBA, but wewill restrict the detailed description to that of the JBA. Formore details on the CBA, see Ref. 23.

The classical differential equations describing the dy-namics of a Josephson junction oscillator driven with a rfcurrent for the circuits depicted in Figs. 4a and 4c aregiven below,

CS0d2t

dt2 +0

R

dtdt

+ I0 sint = Irf cosdt

+ INt , 2

Leff +0

I01 − q2/I0

2d2qtdt2 + Reff

dqtdt

+q

Ceff

= Vrf cosdt + VNt . 3

In Eq. 2, is the gauge-invariant phase difference acrossthe junction, d is the drive frequency, and 0= /2e is thereduced flux quantum. The noise current INt models thethermal noise of the source impedance R. In Eq. 3, q is thecharge on the capacitor Ceff and VNt is the effective thermalnoise of the impedance Reff. Under appropriate driving con-ditions, this nonlinear oscillator can have two steady drivenstates differing in amplitude and phase.9,24 Even though Eqs.2 and 3 look different, especially with respect to the lo-cation of the nonlinear term, they reduce to the same effec-tive equation. If we assume harmonic solutions for t orqt at the drive frequency d of the form yteidt+c.c., wecan show that the equation of motion for the dimensionlessslowly varying complex amplitude u corresponding toEqs. 2 and 3 can be reduced to

du

d= −

u

− iuu2 − 1 − i + fN , 4

where

LeffCeffReff I0Veff

R R

Cin CoutI0Vd

Z0,ω0 Z0,ω0

λ/4 λ/4

CSI(t) R I0CJ

(a)

(b)

(c)

FIG. 4. a Circuit diagram of the JBA. b Circuit diagram of the CBA witha half-wave transmission line cavity embedded with a Josephson junction inthe middle. c The equivalent LCR circuit for the CBA.



= 2Q1 − d/p0 5

is the reduced drive detuning, p0 is the resonant frequencyfor small oscillations, Q is the quality factor, = p0−dt isthe dimensionless slow time, is the reduced drive power,and fN is the reduced noise term. Equation 4 has beenderived under the rotating wave approximation with onlyleading order nonlinear terms being retained in Eqs. 2 and3. The expressions for p0, Q, u, and for the JBA and theCBA are given in Table I, where =1+LeffI0 /0 /Q. Soapart from different expressions for the reduced variables inthe terms of circuit parameters, the nonlinear dynamics ofJBA and CBA oscillators converges to the universal form Eq.4 for large Q. Hence, from now on, we will restrict ourdiscussion to the JBA.

The scaled steady state solutions of Eq. 4 du /d=0for the JBA are plotted in Fig. 2 for different excitation am-plitude Irf. These curves are computed with the noise termfN set to zero. The different curves correspond to differentdrive strength. For Irf I0, the response is linear with a char-acteristic Lorentzian line shape. In this regime, the phaseparticle only samples small values of so that the sinusoidalpotential is well approximated by a parabola, which de-scribes harmonic behavior. For increasing drive amplitude,the higher order nonlinear terms contribute and reduce theplasma frequency pI0 , Irf, but the response is still singlevalued. For sufficiently strong drive, the oscillator bifurcatesand can occupy one of two metastable states OL and OH.These states differ significantly in oscillation amplitude andphase. The dashed lines in Fig. 2 indicate the unstable solu-tions.

To illustrate how a small variation in critical current canbe detected, the phase response of the oscillator for two dif-ferent values of I0 is plotted in Fig. 5 for three different drivecurrents corresponding to the linear response regime, thesingle-valued nonlinear regime, and the bifurcation regimedescribed above. An input signal modulates the critical cur-rent and shifts the plasma frequency, shown in Fig. 5 as adisplacement of the response curves. To operate the JBA, adrive current at fixed frequency d is applied. An alternativetechnique where the drive amplitude is kept fixed but thefrequency is swept has been recently presented in Ref. 25. Ashift in p is detected as a shift in phase of the reflected

signal output, as described in Fig. 3. In the linear regimeshown in Fig. 5a, the maximum phase shift and thus themaximum gain are achieved by driving the oscillator atd=p0 as indicated by the solid line. Sensitivity can beimproved by increasing the quality factor Q=RCsp0 of theoscillator but at the direct cost of reducing the amplifierbandwidth. Another approach to improve sensitivity, whichis at the heart of the JBA, is to increase the drive current toaccess the nonlinear regimes shown in Figs. 5b and 5c. InFig. 5b, the phase response is sharper on the lower fre-quency part of the response curve, and a greater sensitivitycan be obtained by driving at a frequency d red detunedfrom p0, as indicated by the dashed line. For yet strongerdrives, the response curve transitions sharply between thetwo branches, as shown in Fig. 5c. If the drive frequencyd is chosen in the vicinity of this bifurcation point, then theoscillator will occupy one oscillation state or the other de-pending on the value of I0. By adjusting the drive currentamplitude and frequency, the phase difference in the re-flected signal between these states can approach 180°, real-izing a very sensitive threshold detector for small variationsin I0, which does not rely on switching to the dissipativefinite voltage state.

In the linear and the single-valued nonlinear regime, theminimum critical current variation that can be resolved andthus the ultimate sensitivity of the amplifier depend on thesmallest phase or amplitude shift, analog quantities, that canbe resolved by the cryogenic microwave measurement elec-tronics in a given time interval.26,27 In the bifurcation regime,the reflected signal phase is a digital quantity with a suffi-ciently large difference between the two oscillation statesthat it can be fully resolved within practical integration times100 ns. The amplifier sensitivity in the bifurcation re-gime is thus set by the minimal critical current variation,which causes the oscillator to switch from one dynamicalstate to the other. This is determined by the dominant fluc-tuations, thermal or quantum, in the system. Sensitivity isthus not limited by microwave electronics, an importantpractical advantage.

For a given drive frequency d such that =2RCSp0

−d3, the systems exhibits bistability within a certainrange of drive amplitude such that IB

− Irf IB+. Here IB

− and

TABLE I. Variables used in Eqs. 2–4 expressed in terms of circuitparameters.

Parameter JBA CBA

p0 I0

0Cs

1

Leff +0/I0Ceff

Q RCsp0 Z0

2Reff

u p02

4dp0 −dy 1

22

d

I0y

p0

6 Irf2

64I02d

3p0 −d3

Vrf2

02d

2 1

223

0

180

-180

φ(deg)

ωd ωp0

(c)

ωd=ωp0

(a)

ωd ωp0

(b)

FIG. 5. Color Principle of operation of the JBA. The curves in all threepanels are the reflected signal phase response for three different drive pow-ers. The red right and blue left curves correspond to two different valuesof the small oscillation plasma frequency p0. The value of p0 for the bluecurve is indicated by the vertical solid line. Panel a is for the smallest drivepower and corresponds to linear regime. For a larger drive power in b, theJBA oscillator is in the single-valued nonlinear regime. When biased withsufficient power to access the bistable regime c, the phase response exhib-its a sharp transition. Thus, the maximum phase shift for a given change inp0 increases with drive power a–c and is achieved at the bias pointswith increasing detuning as indicated by the vertical dashed lines.



IB+ are called the lower and upper bifurcation currents, re-

spectively. Figure 6 shows a plot of the steady state solutionsof Eq. 4 as a function of reduced drive power for differentvalues of reduced drive detuning. Multivalued solutions existonly for 3. The bifurcation points are the turning pointsof these curves where the slope du2 /d=. The reducedbifurcation points are indicated B

IB2 for the curve

with =3.0. The saddle-node bifurcation is found in the vi-cinity of the upper bifurcation current IB

+, which is given by

IB+ =

16

33I0

3/21 +9

16Q221/2, 6

where =1−d /p0 is the dimensionless detuning of thedrive. The above formula has been computed by retainingonly one nonlinear term in the expansion of sin using themethod outlined in Ref. 24 and is given here in the limit ofhigh quality factor Q 10 such that =2Q1. The JBA,when operated as a digital threshold detector, is biased nearthis point by ramping the current. Hence, in what follows,whenever we mention bifurcation point/current, we mean theupper bifurcation point/current.

There are two important quantities that go into determin-ing the sensitivity of the JBA to changes in critical currentwhen operated as a digital threshold detector. The first quan-tity is the shift in the bifurcation current IB

+ with critical cur-rent I0. This quantity can be written as follows:

IB+

I0=

IB+

I0

3

4. 7

We can infer from the above equation that the fractionalvariation in the bifurcation current IB

+ / IB+ is always larger

than the fractional variation in critical current I0 / I0 since3 /41. The second quantity that determines the sensitivityof the JBA is the intensity of fluctuations at a given tempera-ture T. These fluctuations induce switching between themetastable states of the JBA, which gives a finite width tothe transition from the low OL to the high OH amplitudestate. In the classical regime kBTp0, this dynamicalswitching can be described by an Arrhenius law in which therate of escape from an effective cubic metapotential

escrf = a/2 exp− Uesc

rf /kBT 8

is written as the product of an attempt frequency a /2 anda Boltzmann factor, which contains the barrier height Uesc

rf ,and the system temperature T. In this case, the effective bar-rier height24 is

Uescrf = Uesc

rf 1 − Irf/IB23/2, 9a

Uescrf =

64

930I0 , 9b

where the coefficient Uescrf is given to lowest order in 1 / Q.

Similarly, the attempt frequency in the metapotential is givenby

a = a01 − Irf/IB21/2, 10a

a0 = 4Qp2/33 , 10b

where we have assumed that the JBA is biased sufficientlyclose to the bifurcation point resulting in an overdampedmotion in the rotating frame. We should point out that forbias points sufficiently far from the bifurcation point, themotion in the rotating frame is underdamped and Eq. 10ahas a different power law28 but Eq. 9a remains valid.

We now introduce the idea of a switching probabilitycurve or an “S-curve.” If the JBA is biased by ramping thecurrent Irf to a value slightly lower than IB

+, there is a finiteprobability that it will make a transition from the low ampli-tude state to the high amplitude state in a given time tw. Thisis called the switching probability Pswitch at that bias point.Since we know the escape rate esc

rf from the low amplitudestate to the high amplitude state of the JBA, we can definethe probability of switching in a given time tw,

PswitchIrf,I0,tw = 1 − exp− twescrf Irf,I0 , 11

where PswitchIrf , I0 has been defined as function of twoparameters—the drive amplitude Irf and the critical currentI0—as we are interested in the variation in Pswitch with re-spect to these two parameters. The S-curve is essentially theplot of Pswitch versus Irf for a given I0 and varies smoothlyfrom zero to one Fig. 7. The finite width of this S-curve isset by the intensity of fluctuations and hence the temperature.Clearly, a change in the value of critical current will shift theS-curve, resulting in a change in Pswitch. This is the principleof detection in the digital mode. The switching probability ismeasured by sending many pulses to the JBA and recordingthe resulting oscillation state of the JBA for each pulse. Theshift in S-curves is essentially related to the shift in the bi-furcation current and can be inferred from Eq. 7,

Ishift =IB

+

I0I0 =

IB+

I0

3

4I0. 12

The computation of the S-curve width is subtle. Since Eq. 8and hence Eq. 11 is only valid for small escape ratesUesc

rf kBT, it essentially breaks down near the top of theS-curve where the switching rates are very high. So we com-pute the width of the S-curve in terms of the drive amplitudeusing a different procedure. The upper boundary of theS-curve is clearly set by IB

+ since the escape barrier vanishesat the bifurcation point. For the lower boundary we use the

1 2 3 4 5 6

1

0

2

3

4

Ω3βΒ

Ω|u|2

Ω = 1.5 , 3 , 2.0 , 2.5 , 3.0

+Ω3βΒ-

P~

FIG. 6. Color Steady state solutions of Eq. 4 for increasing left to rightvalues of reduced detuning . The magnitude squared of reduced oscillation

amplitude is plotted as a function of reduced drive power P=3. The x andy axes have been scaled to clearly depict the variation with . We observemultivalued solutions for 3. The turning points of the curves for 3 are the bifurcation points B

and are indicated for the curve with=3.0.



point when Uescrf kBT since Pswitch starts becoming non-

zero around that point. The resulting width is given by

Iwidth = IB+ kBT

Uescrf 2/3

, 13

where Uescrf is given by Eq. 9b. Note that this definition of

Iwidth is not strictly identical to that depicted in Fig. 7 wherewe qualitatively illustrate the idea of the width of an S-curve.We get a unit change in Pswitch when the shift in the S-curveis about the same as its width. Combining the above twoequations, we can compute the change I0 in critical cur-rent that can be resolved in a single pulsed measurement orthe critical current resolution as

I0 =32/3

4I0

1/3 kBT

02/3

1/31 − −2. 14

We note that the critical current resolution depends quiteweakly on most parameters, the strongest being the tempera-ture T2/3. The dependence on I0 is quite weak I0

1/3, whichmeans that there is flexibility in choosing the value of junc-tion critical current. If the total time taken for one pulsedmeasurement is tpulse set by Q and noise temperature of themicrowave amplification chain, then we can express thecritical current resolution I0 in terms of a critical currentsensitivity per unit bandwidth as

SI0

1/2 =32/3

4I0

1/3 kBT

02/3

1/31 − −2tpulse. 15

An important point to be noted is that the above analysis hasbeen done in the limit of large Q and small escape rates esc

rf

in order to simplify calculations and see trends clearly. Of-ten, many of these approximations become too crude in realexperiments, especially for small Q20, and the only wayto obtain accurate predictions is with full numerical simula-tions of Eq. 2. Nevertheless, the above formulae serve as aguide to the phenomena and provide quick estimates. Formore details on the calculation, see Ref. 20. Furthermore, wehave neglected intrinsic fluctuations in the junction critical

current as these have a well studied intensity and 1 / f fre-quency spectrum29 which diminishes their contribution atmicrowave frequencies.

In the quantum regime kBTp0, the above analysisremains valid if we replace T with an effective temperatureTeff=d /2kB. This result has been predicted theoreticallyusing the idea of quasienergies in the rotating frame for bothunderdamped30,31 and overdamped motion in the rotatingframe.32 Note that this effective temperature is different fromthe saturation temperature seen in the escape to the voltagestate in a dc current-biased junction T=d /7.2kB, whichis determined by a tunneling rate at low temperatures.33 Onecan also arrive at these results by treating the JBA as a para-metric amplifier when biased near the bifurcation point. Theparametric amplification process converts the virtual zeropoint quantum fluctuations into real classical fluctuations.The effective temperature of this classical bath tends to Teff

=d /2kB as the bath temperature T→0. For operatingpoints close to the bifurcation point, one can compute the fullexpression for the effective temperature34 as

Teff =d

2kBcoth d

2kBT , 16

and we note that this expression tends to the correct limits asT→0,. This expression is identical to the one obtained inRef. 32, if we ignore effects of dephasing due to oscillatorfrequency modulation. This is justified since the only sourceof dephasing in JBA circuits would be due to 1 / f fluctuationsin the critical current of the junction or the dielectric constantof the capacitor, and they are quite small in the frequencyrange relevant for escape dynamics.

III. DESIGN AND FABRICATION

In this section we will briefly discuss the techniques forfabricating the JBA and some design considerations. In thefirst step, a metallic underlayer—either a normal metal Auor Cu or a superconductor Nb—is deposited on a siliconsubstrate to form one plate of the shunting capacitor. This isfollowed by deposition of a 200 nm insulating SiN layer byplasma-enhanced chemical vapor deposition at 400 °C. Thislayer is the dielectric material in the capacitor. Using e-beamlithography and double-angle shadow mask evaporation, wethen fabricate the top capacitor plates along with a fewsquare micron sized Al /AlOx /Al tunnel junction. The topcapacitor plates along with the metallic underlayer form twocapacitors in series. Figure 8 shows an optical image of theon-chip capacitor along with the schematic profile of theground plane multilayer used for Cu based capacitors. Theother metallic layers Ti and Cr were employed for protect-ing the Cu layer during the deposition of the dielectric layer.It also ensured that the Cu layer would adhere properly to thesilicon substrate. For Au and Nb underlayers, SiN was di-rectly deposited on top. Also shown in the bottom right cor-ner is a scanning electron microscrope SEM image of atypical Josephson tunnel junction. The critical current of thejunction was in the range of I0=1–2 A corresponding to aJosephson inductance in the range of LJ=0.3–0.15 nH. Byvarying both the dielectric layer thickness and the pad area,

1

0Bias amplitude (Irf)

P switch

∆Ishift

∆Iwidth

twIrf

FIG. 7. Color online Switching probability Pswitch of the JBA when ener-gized with a pulse of duration tw and amplitude Irf. The curves indicate theprobability of switching, as a function of Irf from the low amplitude to thehigh amplitude state when biased near the upper bifurcation point IB

+. Thered left and blue right curves correspond to smaller and larger junctioncritical currents, respectively. The shift Ishift depends on the change in I0

and the relative detuning . The width Iwidth depends on the intensity offluctuations in the oscillator.



the total shunt capacitance CS was varied between 16 and 40pF. The plasma frequency of the JBA was kept in the rangeof 1–2 GHz. Several samples were fabricated and their pa-rameters are listed in Table II. For more details on fabricationtechniques, including the transmission line version of theJBA, see Ref. 23.

We will now discuss some important design consider-ations to ensure that a well controlled bifurcation can beobserved. In particular, we will discuss the microwave char-acteristics of the shunting capacitor, which reduces the bareplasma frequency of the Josephson junction. A more gener-alized discussion on the requirements of the shunting imped-ance for observing a bifurcation can be found in Ref. 12. Theshunting capacitor has to be implemented carefully to ensurethat it behaves as a capacitor at the relevant frequencies withminimal parasitic resistance and inductance. At higher fre-quencies 5 GHz, it is much easier and practical to usedistributed circuit elements such as transmission line resona-

tors. This was one of the driving factors for the developmentof the CBA, which uses transmission line resonators with anembedded Josephson junction to implement the nonlinearoscillator.35

The first generation of our microwave capacitors weremade using a thin 100 nm Au ground plane. This was fol-lowed by Nb ground planes 50 and 200 nm and then thick1 m Cu. Imperfect screening currents in the capacitorplates due to the finite conductivity Au and Cu and thick-ness of the ground planes result in a stray inductance LSand a stray resistance RS. Figure 8b depicts a model thataccounts for the stray inductance and resistance. The samplesmade with superconducting Nb ground planes did not haveany stray resistance, while the data for samples with Au andCu ground planes lead to values of RS=0.8 and 0.02 ,respectively. The lowest parasitic inductance LS=0.026 nHwas obtained with the 1 m Cu ground planes, while theother samples had a value in the range of 0.15–0.34 nH. Thepresence of stray resistance results in dissipation in the os-cillator, and the phase shift in the reflected signal upon cross-ing the resonance is much less than the 360° expected for alossless oscillator. In the high power regime, the sampleswith large stray resistance made a transition into a chaoticstate before encountering the bistable regime. This chaoticstate is also observed in well behaved samples at higherpowers black region in Fig. 13, but there is a large enoughrange of powers where the bistable regime is observed. Thestray inductance has a similar effect as it increases the lin-earity of the oscillator, thereby pushing the onset of the non-linear regime to higher drive powers and closer to the chaoticstate. To summarize, the stray parameters LS and RS reduceand in the worst case eliminate the available phase spacewhere one can observe the bistable regime of the nonlinearoscillator. For the JBA with Q20, one should aim toachieve LS /LJ10 and RS /Z010−3 for optimum results.

IV. EXPERIMENTAL SETUP

The JBA chip is placed on a low-loss microwave circuit-board r=10 and is wire-bonded to the end of a coplanarstripline, which is soldered to a coaxial launcher affixed tothe side wall of the copper sample box. We anchor the rfleak-tight sample box to the cold stage of a dilution refrig-erator with base temperature of 15 mK. Some preliminarydata were also obtained in a 3He cryostat with a base tem-perature of 270 mK. The measurement setup including thevalues of filters and attenuators is shown in Fig. 9.

For frequency domain measurements, microwave excita-tion signals are generated by a HP 8722D vector networkanalyzer and are coupled to the sample via the 13 dB sideport of a directional coupler after passing through cryogenicattenuators. The reflected microwave signal passes throughthe direct port of the coupler and is amplified first using acryogenic 1.20–1.85 GHz high electron mobility transistorHEMT amplifier with noise temperature TN=4 K beforereturning to the network analyzer. The isolators on the returnline allow microwave signals to propagate only in one direc-tion and suppress the HEMT amplifier input noise irradiatingthe sample by 60 dB. To suppress noise outside the band of

CSI(t)RS

R LJ

LS

CJ

(a)

(b)

SiNSiN

SiN

FIG. 8. Color Optical image of the microfabricated on-chip capacitor isshown in the left panel. The top Al layer forms two capacitors in series withthe Cu underlayer as the common electrode. Top right panel indicates theprofile of the different layers in the ground plane. The additional layers Tiand Cr sandwiching the Cu layer are used to protect the Cu layer during thedeposition of SiN. These additional layers are not required for Nb groundplanes. The bottom right panel shows a SEM image of a typical Josephsonjunction, which is shunted by the capacitor. b Circuit schematic of the JBAchip including stray elements. The junction inductance LJ is shunted witha capacitor CS including the stray inductance LS and stray resistance RS.CJCS is the intrinsic junction capacitance. The stray inductance and resis-tance arise due to imperfect screening currents and the finite conductivity ofthe ground planes.

TABLE II. Sample parameters from Ref. 11. LJ=0 / I0 and p0 are mea-sured values. CS and LS are fit values to the data. Samples 1 and 2 have a100 nm thick Au underlayer, sample 3 has a 50 nm thick Nb underlayer,samples 4 and 6 have a 1 m thick Cu underlayer, and sample 5 has a 200nm thick Nb underlayer.

SampleLJ

nHp0 /2 GHz

CS

pFLS

nHRS

1 0.28 1.18 391 0.200.02 0.82 0.18 1.25 304 0.340.04 0.83 0.32 1.64 161 0.270.02 0.04 0.38 1.81 191 0.030.02 0.025 0.40 1.54 191 0.150.02 0.06 0.28 1.80 271 0.010.02 0.0



the isolators 1.20–1.85 GHz, we use lossy transmission linefilters.23,36 The attenuators on the input line carry out a simi-lar function by reducing the noise from higher temperaturestages. These components ensure that the intensity of thefluctuations at the sample correspond to the temperature ofthe cold stage to which it is anchored. This is important formaximizing the sensitivity of the JBA.

For the time domain measurements, we used a TektronixAWG 520 to create the pulse envelopes. These envelopeswere combined with cw microwave signals from a synthe-sized microwave generator using analog mixers. The re-flected signal was then mixed down to a lower frequency anddigitized using a 1 GS/s digitizer from Acqiris. The phasewas then determined by fitting this demodulated signal insoftware. Alternatively, we also used an Analog Devicesphase detector AD 8302 for directly converting the re-flected signal phase into a voltage. This was the preferredmethod when the JBA was being used as a digital thresholddetector and enabled rapid determination of the oscillationstate with no software processing overhead.

For ultralow noise, simultaneous dc and rf measurementson the same JBA sample, we developed a fully differentialmeasurement scheme as shown in Fig. 10. I-V characteristicand low frequency switching current measurements7 are of-ten required to extract system parameters such as the criticalcurrent and the effective temperature.33 Nonequilibriumnoise is effectively rejected in low frequency measurementsby using twisted-pair cables with fully differential signals.Radio frequency signals on the other hand are traditionallysingle ended and are susceptible to common mode noisewhen combining low and high frequency measurements. Toremedy this problem, we used a 180° hybrid coupler to cre-ate two out of phase rf signals, which were then combinedwith the dc signals using two bias tees to create a fully dif-

ferential excitation. This setup was used only for ultralownoise measurements such as the escape temperature measure-ments in the JBA described at the end of Sec. V B.

V. RESULTS

A. Frequency domain measurements

Frequency domain measurements are used to character-ize the nonlinear oscillator and extract circuit parameters.The linear plasma resonance is located by sweeping the ex-citation frequency d and measuring the reflection coeffi-cient Rd= Zd−Z0 / Zd+Z0. Here Z0=50 isthe characteristic impedance of our transmission lines, andZd is the impedance seen at the sample holder plane. Foran ideal LC resonator without intrinsic dissipation, we expecta phase shift =dp0

−dp0=2 . The excitation

power was kept low enough to access the linear regime ofthe oscillator. In Fig. 11, we present the reflected signal

300 K

4.2 K

0.8 K

0.05 K

0.012 K

VECTORNETWORKANALYZER

COMPUTER

IQ MIXER/PHASE

DEMODULATOR

DIGITIZERARB CW

IF

RF

LO

Iout(t) sin (ωdt+φ)Iin(t) sin (ωdt)

COAXIALCABLETYPE

SS/SPCWSS/SSNb/NbCu/SPCW

-10 dB

FIG. 9. Color Cryogenic microwave measurement setup. The rf excitationline is attenuated and filtered with reactive components. The signal returnline has isolators and lossy filters. Typical values of components along withthe type of coaxial cables used are indicated.

BIASTEE

BIASTEE

Σ +∆ −

+Vdc/2

-Vdc/2

RFSOURCE

50 Ω

Vrf

+Vrf / √2

−Vrf / √2

HYBRIDRb

Rb

SUPERCONDUCTINGALUMINUM BOX

DCBIASING

RFBIASING

RF (IN)FILTERCHAIN

DCFILTERCHAIN

RF (OUT)FILTERCHAIN

C

PHASEDETECTORAMP

PRE-AMP

Vdc

Idc

DCOUTPUT

(a)

(b)

CIRCULATORS

BIASTEE

LOSSYLOW PASSFILTERS

REFLECTIVELOW PASSFILTERS

ALUMINUM BOX

SAMPLE BOX

dcIN/OUT

rfIN/OUT

FIG. 10. Color Schematic a and optical image b of the combined dif-ferential rf and dc biasing scheme. Shielded twisted-pair wires implement afour wire dc I-V measurement. A 180° hybrid coupler is used to split the rfsignal into two out of phase components, creating the differential rf drive.The rf and dc signals are combined using a bias tee before they reach thedevice inside blue rectangle. A superconducting Al box green rectangle isused to shield the device from low frequency magnetic fields since thetwisted pairs have to be separated before they can be combined with the rfsignals. This differential biasing scheme rejects noise over a wide frequencyrange.



phase as a function of excitation frequency for sample 5.The point where =0 is the linear-regime plasma frequency.For sample 5, p0 /2 =1.54 GHz.

The precise frequency and critical current dependence ofthe reflected signal phase of our samples can be explained bythe lumped element model shown in Fig. 8b. The plasmafrequency in the linear regime is determined by the totalinductance LJ+LS and capacitance CS and is given by thefollowing relation:

1

p02

= CSLJ + LS =0CS

I0+ CSLS. 17

A plot of 2 /p02 versus 1 / I0=LJ /0 is shown in Fig. 12for samples 1, 2, 4, and 5. As the critical current is decreasedby applying a magnetic field in the plane of the junction, itsinductance increases, and the plasma frequency is reduced.For each sample, a linear fit to the data of Fig. 12 yields thevalues of CS and LS see Table II. The fit values for CS agreewell with simple estimates made from the sample geometry.

For samples with a thin underlayer 1–3, a stray induc-tance in the range LS=0.20–0.34 nH was observed. Forsamples 4 and 5 with a significantly thicker underlayer, LS

was reduced to 0.026 and 0.15 nH, respectively. This behav-ior is consistent with the calculated screening properties of

our thin films. Using LS and CS we can accurately predict theobserved resonant line shape solid line in Fig. 11 in whichRS0. Samples with a normal underlayer yielded finite val-ues of RS.

We now discuss measurements of the power dependenceof the plasma resonance. The reflected signal phase as afunction of frequency for increasing power for sample 5 ispresented in the lower panel of Fig. 13 as a two dimensionalcolor plot. Each row is a single frequency sweep, similar toFig. 11. For small excitation power, we recover the linearplasma resonance at 1.54 GHz, indicated in yellow, corre-sponding to =0. As the incident power is increased beyond115 dBm, the plasma frequency decreases as explained inSec. II. For powers greater than 105 dBm, there is anabrupt transition from the phase leading green to phaselagging red state. This marks the transition from the lowamplitude state to the high amplitude state. One can crossthis transition by sweeping the power at fixed frequency alsodashed line. For powers in excess of 90 dBm, we en-counter a new dynamical regime black region in Fig. 13where appears to diffuse between the wells of the cosinepotential. This was confirmed by the presence of an unam-biguous audio frequency ac resistance in the black regionsee Ref. 20. In the lower panel of Fig. 13 inset, we illus-trate the sequence of dynamical transitions by plotting as afunction of incident power at d /2 =1.375 GHz. For P

φ(deg)

-180

-90

0

90

180

1.81.71.61.51.41.3

Csfit= 19 pF

Lsfit = 0.15 nH

LJ = 0.40 nH

CS

LS

LJI(t)

ωd/2π (GHz)

50Ω

I(t)=Irf sin(ωd t)

FIG. 11. Color online Normalized reflected signal phase as a function ofexcitation frequency d /2 for sample 5. The open circles are measureddata for LJ=0.40 nH. The solid line is calculated from the equivalent circuitmodel shown in the inset. The magnitude of the reflected signal is unitywithin experimental uncertainty.

1.5

1.0

0.5

0.02.52.01.51.00.50.0

1/ I0 (µA-1)

1245

[2π/ωp0]2(GHz-2)

FIG. 12. Color online Inverse squared of the plasma frequency 2 /p02

as a function of the inverse critical current 1 / I0 for samples 1, 2, 4, and 5.Solid lines are linear fits to the data corresponding to the model in Fig. 11with a single best fit line drawn for samples 1 and 2, which nominally differonly in I0.

1.81.61.41.2

-120

-110

-100

-90

P(dBm) -180

-90090180

-120 -110 -100 -90

A

BC

P (dBm )

EXPERIMENT

1.81.61.41.2

-120

-110

-100

-90

P(dBm)

00

1800

-1800

THEORY

ωd/2π (GHz)

ωd/2π (GHz)

φ(deg)

FIG. 13. Color Normalized reflected signal phase as a function of ex-citation frequency d /2 and excitation power P is shown for sample 5.Experimental data are shown in the bottom panel, while the top panel isthe result of numerical simulations. A vertical slice taken at d /2 =1.375 GHz dashed line shows the abrupt transition between two oscilla-tion states of the system.



−102 dBm, the phase is independent of power and oscil-lates in a single well in the harmonic-like phase leadingstate letter A. For −102 dBmP−90 dBm, the phaseevolves with power and still remains within the same wellbut oscillates in the anharmonic phase lagging state letterB. Finally, for P−90 dBm, the average phase of the re-flected signal saturates to 180°, corresponding to a capaci-tive short circuit. This last value is expected if hops ran-domly between wells, the effect of which is to neutralize theJosephson inductance.

To explain the complete frequency and power depen-dence of the transitions shown in the lower panel of Fig. 13,we numerically solved the full circuit model shown in Fig.8b, including the exact sinusoidal junction current-phaserelation. The result of this calculation is shown in the upperpanel of Fig. 13. It correctly predicts the variation in theplasma frequency with excitation power and the boundariesof the phase diffusion region. The agreement between theoryand experiment is remarkable in view of the simplicity ofthis model, which uses only measured parameters.

B. Time domain measurements

The measurements described so far characterized thetime averaged response of the Josephson oscillator undercontinuous microwave excitation. We will now describe theexperiments that probed the dynamics of the JBA at shorttime scales 10 ns under pulsed microwave excitation.Data from sample 6 are presented in this section.

We first characterized in detail the switching associatedwith the OL→OH transition near the bifurcation point IB

+. Weexcited the oscillator with two different readout pulse proto-cols. In the first protocol, the drive current was ramped fromzero to its maximum value in 40 ns and was then held con-stant for 40 ns before returning to zero. Only the final 20 nsof the constant drive period were used to determine the os-cillation phase with the first 20 ns allotted for settling ofthe phase. Histograms taken with a 10 MHz acquisition rateare shown in Fig. 14. In the upper panel, the two peakscorresponding to states OL and OH can be easily resolvedwith a small overlap of 10−2. The finite width of each peak isdue to the output noise and is consistent with the system

noise temperature of our microwave electronics. In this firstmethod, the latching property of the system has not beenexploited. In the second protocol for the readout pulse, weagain ramp for 40 ns and allow a settling time of 20 ns, butwe then reduce the drive current by 20% and measure thereflected signal for 300 ns. In that latter period, whateverstate was reached at the end of the initial 60 ns period is“latched” and this time is used to improve the signal/noiseratio of the reflected phase measurement. As shown in thelower panel of Fig. 14, the two peaks are now fully sepa-rated, with an overlap of 610−5, allowing a determinationof the state OH probability with an accuracy better than 10−3.This second protocol is well suited to precise time-resolvedmeasurements of I0 or for applications where a low noisefollower amplifier is unavailable. At the end of the pulse, theamplitude is ramped back to zero in 10–20 ns. The ramp upand ramp down times depend on the quality factor of theresonator, and we ensure that this process is adiabatic andthere is no ringing in the oscillator. This entire process can berepeated almost immediately since there is no heating due tothe absence of on-chip dissipation leading to very fast rep-etition rates 10 MHz.

We then measured the switching probability curves orS-curves, PswitchIrf, for different values of I0 to evaluate thesensitivity of the JBA. Using the readout protocol and thediscrimination threshold shown in Fig. 14, we obtain theswitching probability curves shown in Fig. 15. Defining thediscrimination power d as the maximum difference betweentwo S-curves, which differ in I0, we find that at T=280 mK, d=57% for I0 / I0=1%. The switching prob-ability curves should shift according to IB / IB / I0 / I0=3 /4 Eq. 7, which for our case takes the value of 6.1. InFig. 15, the curves are shifted by 6%, which agrees well withthis prediction.

As discussed in Sec. II, the width of the S-curves de-pends on the effective intensity of fluctuations in the JBAoscillator. A well designed experimental setup will ensurethat this intensity of fluctuations is set by the thermal/quantum fluctuations corresponding to the operating tem-perature of the JBA. It is possible to determine the effectivetemperature characterizing these fluctuations by measuring

2

1

0

counts(10)4

9

6

3

0

12080400-40phase φ (deg)

STATEOL

STATEOH

FIG. 14. Histograms of the reflected signal phase at Irf / I0=0.145. The upperhistogram contains 1.6106 counts with a measurement time m=20 ns.The lower panel, taken with the latching technique, has 1.5105 countswith a measurement time m=300 ns. The dashed line represents the dis-crimination threshold between the OL and OH states.

Irf / I0

P switch

T=280 mK

1.0

0.8

0.6

0.4

0.2

0.00.1100.1000.0900.080

ηd

FIG. 15. Color online Switching probability curves at T=280 mK as afunction of the normalized drive current Irf / I0. The discrimination power d

is the maximum difference between the two curves. The two curves differ byapproximately 1% in I0, and the curve corresponding to the higher criticalcurrent lies at higher values of Irf / I0.



the escape rate from state OL to OH as a function of thevarious bias parameters. The classical theory of escape wasbriefly described in Sec. II. In the quantum regime Tp0 /kB, the effective intensity of fluctuations is given bysetting T=Teff=p0 /2kB.30,31 The full expression for the es-cape temperature34 is given by Eq. 16.

We will now describe the procedure for measuring theescape rate from the low amplitude state of the JBA. TheJBA was biased with a long trapezoidal pulse at frequencyd with amplitude Irf. The pulse ramp time used was 40 ns.The total pulse length was 1 ms. The phase of the reflectedpulse is analyzed and the time i

switch at which the JBAmakes a transition from low to high amplitude state is re-corded. The experiment is repeated typically for N=105

times. We then construct the probability PL of the JBAbeing the low amplitude state as

PL = 1 −1

Ni=1

N − iswitch , 18

where

= 0 0

=1 0 19

is the Heaviside unit step function. The probability PLdecays exponentially with a decay constant given by the es-cape rate esc

rf ,

PL = exp− escrf . 20

So by fitting an exponential to PL, we can determineesc

rf Irf for different bias amplitudes Irf. The results areshown in Fig. 16 where we have plotted the reduced escape

rate escrf Irf= loga /2 esc

rf 2/3 given by

escrf Irf = Uesc

rf

kBTescrf 2/31 −

Irf2

IB2 21

as a function of Ief2 / IB

2 for selected values of bath tempera-tures. Data are shown for two samples with different reso-nant frequencies in panels a and b.

We observe straight lines with different slopes for differ-

ent bath temperatures. The slope slope and intercept int ofthe line are given by

slope = − Uescrf

kBTescrf 2/3 1

IB2 , 22a

int = Uescrf

kBTescrf 2/3

. 22b

From these two quantities we can determine the bifurcationcurrent IB and escape temperature Tesc

rf using the followingformulae:

IB = − int/slope1/2, 23a

Tescrf =

Uescrf

kBint3/2

. 23b

This confirms that the escape rate follows the behavior givenby Eqs. 8 and 9. We can then extract the escape tempera-

ture Tescrf for each operating point. Note that in the experi-

ment, we control Irf by varying the pulse amplitude Arf of

the rf generator. Since IrfArf, we can plot escrf as a function

of Arf2 and extract the bifurcation amplitude AB IB using

Eq. 23a. Once we have AB we can plot the data as a func-tion of Arf

2 /AB2 , which is the same as Irf

2 / IB2 . When plotted this

way, the straight lines should intersect escrf =0 at Irf

2 / IB2 =1 at

all temperatures since the bifurcation current IB and henceAB does not depend on temperature. The data for each tem-perature in Fig. 16 have been plotted after normalizing thex-axis with the value of IB AB extracted at the lowest tem-perature. In practice, we always observe small differences inthe extracted values of AB at different temperatures as isevident from the data. This could imply that the data start todeviate from the prediction of Eq. 8. However we foundthat this variation was predominantly due to the slow in-crease in the attenuation of the input coaxial lines due toslowly decreasing levels of liquid Helium in the dewar of thedilution refrigerator. We observed that the extracted values ofAB always increased with time even when the temperaturewas kept constant and reverts back when the liquid heliumlevel is restored. The data shown in Fig. 16 take about 2 daysto acquire since one has to wait for the system to thermalizeat each operating temperature. During such long periods, the

8

6

4

2

0

1.00.80.60.4

12 mK50 mK75 mK125 mK200 mK

I0 =1.14 µA IB =0.1 µA

I0 = 3.71 µA IB = 0.4 µA

ωd /2π =1.525 GHz

ωd /2π =4.450 GHz

13 mK75 mK125 mK175 mK200 mK

8

6

4

2

0

1.00.80.60.4

(a)

(b)

Γesc(Irf)

∼rf

Γesc(Irf)

∼rf

Irf / IB2 2

Irf / IB2 2

FIG. 16. Color The reduced escape rate escrf Irf as a function of Irf

2 / IB2 at

different bath temperatures decreasing from left to right. Data are shownfor two samples with a p /2 =1.670 GHz and d /2 =1.525 GHz andb p /2 =4.690 GHz and d /2 =4.450 GHz. The solid lines arestraight line fits to the data. The slope and intercept of these fits are used toextract the escape temperature Tesc

rf and the bifurcation current IB+.



liquid helium level can change appreciably. We also verifiedthat the extracted values of AB remained constant when noisewas added to the system to artificially elevate the tempera-ture. All these checks were done to ensure that our experi-ment follows the prediction of Eq. 8 and we can extractmeaningful results for the value of Tesc

rf .In order to extract Tesc

rf from the data, we need the valueof Uesc

rf . Data from numerical simulations of the JBA circuityielded a value for Uesc

rf which was 10%–20% larger thanthose obtained from Eq. 9b. This would lead to a 10%–20%smaller value for Tesc

rf , and we observed this discrepancy inthe experiments. In order to overcome this problem, we usedthe following procedure to extract Uesc

rf from the data itself.At the highest temperature point, we assumed Tesc

rf =T sincethat is the expected result in the classical regime.24 We can

then use Eq. 23b to extract Uescrf =kBTesc

rf int3/2. This extracted

value of Uescrf is then used for all the lower temperature

points. This is valid provided we can ensure that the intensityof fluctuations in the JBA really correspond to the thermalfluctuations at that temperature. We verified this by biasingthe junction with a dc current and measuring the escape ratefrom the superconducting to the normal state of the junction.We used the procedure described in Ref. 33 to measure theescape rates esc

dc Idc and extract the dc escape temperatureTesc

dc and found that the escape temperature varied linearlywith bath temperature at high temperatures. For our sampleparameters, the relation Tesc

dc =T should be satisfied down tothe lowest temperature of our fridge 12 mK. We only ob-served a small discrepancy at the lowest temperatures, whichwe attribute to improperly thermalized filters and unfilterednoise in the dual dcrf, Fig. 10 biasing configuration. Theblue and red arrows in Fig. 17 indicate the smallest value ofTesc

dc . Nevertheless, the dc escape temperature data validateour normalization procedure described above.

Figure 17 plots Tescrf extracted from the data shown in

Fig. 16 as a function of bath temperature T. The solid linesare a plot of Eq. 16 with the corresponding values of d

used in the measurement. The agreement between theory andexperiment is excellent for both the samples with differentresonant frequencies. Note that this is not a fit to the data.

The only scaling of data performed, as explained above, isthe extraction of Uesc

rf from the highest temperature data pointfor each sample. The results for Tesc

rf not only agree withtheory for the lowest temperature point, but the functionaldependence on temperature as we cross over from the clas-sical kBTd to quantum regime kBTd is also wellreproduced. This verifies that the sensitivity of the JBA whenoperated as a digital threshold detector is only limited byquantum fluctuations.

VI. APPLICATIONS

A. Qubit readout

Superconducting quantum bits qubits are electronic cir-cuits with discrete energy levels and are thus “artificialatoms.”37–41 As discussed in Sec. II, the transition from thelow amplitude state to the high amplitude state of the JBAdepends sensitively on the critical current of the junction.This can be used to readout the state of superconductingqubits. We engineer the circuit such that the two quantumstates of the qubit differ sufficiently in critical current toinduce switching in the JBA when it is biased in the bistableregime. This approach has been applied to charge-phase,42

flux,43 and transmon qubits44 and is currently being exploredin phase45 qubits as well. We detail measurements of a quant-ronium qubit39 using the readout protocol depicted in Fig.18. The qubit and the JBA are phase coupled via the largejunction. The desired qubit state is prepared by applying theappropriate microwave pulses to the write port. The readoutoperation is performed by applying microwave pulses to theread port, which energizes the JBA oscillator. Biasing theJBA near its bifurcation point encodes the qubit state in theoscillation state of the JBA. The latter is ascertained by ana-lyzing the reflected signal phase. Repeated measurements arethen used to obtain the probability of switching from the lowamplitude to the high amplitude state. The measurement isarranged so that the switching probability is close to zerowhen the qubit is in its ground state, while the switchingprobability is close to one when the qubit is in its first ex-

10

2

4

6

100

2

T esc(mK)

T (mK)10 2 4 6 8100 2

ωd /2π = 1.525 GHz

ωd /2π = 4.450 GHz

8rf

FIG. 17. Color online Tescrf vs T obtained from the data in Fig. 16. The solid

lines are a plot of Eq. 16 for d /2 =1.525 GHz and d /2 =4.450 GHz, respectively. Data show excellent agreement with the theoret-ical prediction. The dashed line is the classical dependence Tesc

rf =T. Thearrow indicates the lowest escape temperature measured in the dc escapemeasurements for the corresponding sample.

DRIVE

OUTPUT AMPC

|1⟩

|0⟩

GATE

WRITEPORT

READPORT

QUANTRONIUM+JBA

FIG. 18. Color Schematic of the qubit measurement setup. The quantro-nium qubit is comprised of two small Josephson junctions and a large junc-tion in a loop. This latter junction is shunted by a capacitor and forms thenonlinear oscillator of the JBA. The qubit state is manipulated by sendingpulses to the gate write port, while readout operation is performed bysending a pulse to the nonlinear resonator via the read port. The circulatorC is used to separate the incident and reflected signals. The phase of thereflected signal carries information about the qubit state.



cited state. Ideally, we would like the switching probabilityto be zero for qubit ground state and one for qubit excitedstate.

Neglecting the dissipation induced in the transmissionlines, the total Hamiltonian of the quantronium coupled to a

JBA resonator is Ht= Hboxt+ Hrest Ref. 42 with

Hboxt = 4ECN −1

2+

CgUt2e

2

− EJ cos

2cos ,

24

Hrest =q2

2Cs− EJ

R cos − 0It .

This Hamiltonian has been written supposing that the asym-metry between the two small junctions is zero and the dcvalues of the offset gate charge and loop flux have beenadjusted to operate at the degeneracy point, i.e., CgU /2e

=1 /2 and zero flux in the loop.39 Here, N and q /2e are the

momenta conjugate to the generalized positions and , re-spectively. The control parameters Ut=Urftcos rft andIt= Irftsin dt are analogous to electromagnetic probefields in an atomic system and induce a charge excitation ofthe write port and a phase excitation of the read port, respec-tively. If we keep these two lowest states in the Hilbert space

of Hbox Ref. 46 and we express Hres in terms of the photoncreation and annihilation operators, we obtain an effectiveHamiltonian

Heff =2CgUt

eEC X −

EJ

2 Z + p01 + ! Za†a

− 1 +!

4 Za + a†4 − fa + a†It , 25

where

p0 = EJR

02CS

, 26a

! =EJ

4EJR , 26b

=EC

R

12=

1

12

e2

2CS, 26c

f = 02ECR

EJR 1/4

. 26d

The photon annihilation operator a is related to by

=a + a†

EJR/2EC

R1/4 , 27

which represents the decomposition of the gauge-invariantphase difference into annihilation and creation operators ofthe “plasma” mode whose bare frequency is p0. The opera-tors X and Z are the Pauli spin operators, and EC

R

=e2 / 2CS is the single electron charging energy of the read-out junction. In this effective Hamiltonian, the expansion of

cos is carried out only to the first anharmonic term, which

describes the nonlinear resonator dynamics with sufficientaccuracy for a bifurcation readout.

We now describe the role of each term in Eq. 25. Thefirst term describes the influence on the qubit of the chargeport drive, which is used to manipulate its state. The secondterm describes the free evolution of the qubit at the Larmorfrequency 01=EJ /. We have supposed here that the ratioEJ /EC is sufficiently small that corrections to the Larmorfrequency involving EC are small. To model the behavior ofqubit samples with an appreciable EJ /EC ratio, we wouldkeep higher order terms, yielding renormalized values of thecoefficients in Eq. 25. The third term describes the domi-nant coupling between the qubit and the resonator and resultsin the qubit state affecting the resonant frequency of the reso-nator. Note that this term commutes with the Hamiltonian ofthe qubit when U=0, offering the possibility of quantumnondemolition measurements.47 The fourth term is the con-sequence of the nonlinearity of the resonator and leads to adecrease in the frequency of the resonator when its photonpopulation increases as discussed earlier in Sec. II. Finally,the fifth term describes the excitation of the resonator by thedrive current applied through the read port.

The coherence properties of different qubit samples weremeasured and typical results are shown in Fig. 19. Panel a

0.8

0.6

0.4

0.25004003002001000

a)

P switch

τ (ns)

0.7

0.6

0.5

0.4

0.31086420

tw (µs)

0.8

0.6

0.4

0.25004003002001000

P switch

∆t (ns)

1.0

0.8

0.6

0.4

0.2

0.010610410210098

IRF (nA)

P switch

P switch

max

|0>|1>48%

a) RABI OSCILLATIONS

c) RAMSEY OSCILLATIONS

b) EXCITED STATE LIFETIME

d) S-CURVES

FIG. 19. Color online Summary of qubit coherence measurements. TheLarmor frequency of the qubit is 9.513 GHz with EJ /EC=2.7. a Rabioscillations as a function of the duration of a square pulse applied on the

gate. Solid green curve is an exponentially decaying sinusoidal fit with T2

=1.6 s. b Decay of the excited state, prepared by applying a pulse, asa function of the waiting time tw between the preparation and readout pulses.Solid green curve is an exponential fit with a decay constant T1=3.2 s.The dashed black line indicates the value of Pswitch in the absence of a pulse. c Ramsey fringes obtained with two /2 pulses separated by thetime interval t. The pulse frequency was detuned from the Larmor fre-quency by 20 MHz. The green curve is an exponentially decaying sinusoidalfit. The decay time T2

is 320 ns. d Switching probability as a function ofdrive current amplitude for ground and excited qubit states. The verticaldotted line is the value of the drive current at which the maximal differencein Pswitch is observed. The solid line connects the observed data points in the0 state, and the dashed line is a copy of the solid line horizontally shiftedto overlap the 1 state data at low values of Pswitch.



shows the Rabi oscillation data. The Rabi decay time T2 wasfound to be in the range 0.8–1.7 s depending on thesample and precise biasing conditions. A linear dependenceof the Rabi oscillation frequency "Rabi with the microwavedrive amplitude Urf

max was observed, in agreement with thetheory of driven two level quantum systems. Panel b showsthe decay of the excited state lifetime T1 with typical life-times being in the range of 1–5 s. The values of T1 ob-tained with our dispersive readout are comparable with theresults of Vion et al.39 but are significantly shorter than thevalues expected from coupling to a well thermalized 50 microwave environment shunting the qubit. The loss mecha-nisms giving rise to the observed energy relaxation are notunderstood at this time. Panel c shows the Ramsey oscilla-tion data, which allow one to measure the decay time ofqubit phase coherence during free evolution of the qubitstate. Typical Ramsey decay times observed were T2

300 ns. The Ramsey fringes decay time T2 has a compo-nent due to energy relaxation and one due to pure dephasing:1 /T2=1 / 2T1+1 /T, where T represents pure dephasing.In our measurements, T2 is usually dominated by puredephasing, which is due to fluctuations in the qubit transitionfrequency originating from 1 / f offset charge noise. Recentqubit measurements using the cavity version of the JBAshow that the dephasing times are compatible with the mag-nitude of the typical 1 / f offset charge noise seen in thesesystems.13 Immunity to 1 / f charge noise can be achieved byincreasing the EJ /EC ratio in these qubits, and we observedsome improvement in the pure dephasing time for suchsamples. This strategy is implemented in new qubit imple-mentations, which use very large EJ /EC ratios to almosteliminate the gate charge dependence of the transitionfrequency.41,48

Panel d shows the S-curves corresponding to the qubitbeing in the ground and excited states. The open circles inblue and red correspond to data for the qubit ground andexcited states, respectively, while the solid black line is thebest fit through the ground state data. The dashed black lineis a shifted version of the solid black line to match the ex-cited state data for low switching probabilities. This wasdone to indicate the small difference in the shape of theexcited state S-curve resulting in the reduction in readoutcontrast. The observed contrast for this data is about 15%–30% smaller than expected. In a set of experiments describedin Ref. 20, we used two readout pulses in succession to de-termine that a 15%–30% loss of qubit population occurseven before the resonator is energized to its operating point.We attribute this loss to spurious on-chip defects.49 As pho-tons are injected into the resonator, the effective qubit fre-quency is lowered due to a Stark shift via the phase port.50

When the Stark shifted frequency coincides with the fre-quency of an on-chip defect, a relaxation of the qubit canoccur. Typically, the qubit frequency samples 200–300 MHzbefore the state of the qubit is registered by the readout, andthree to four spurious resonances are encountered in thisrange. Presently, this phenomenon seems to be limiting thequbit measurement fidelity. The detailed back-action of theJBA on the qubit is still not fully understood and is a topic ofcurrent research.51

B. High speed magnetometry

As discussed in Sec. I, one can replace the single junc-tion in the JBA with a two junction SQUID to make a sen-sitive flux detector. Magnetic flux coupled to the SQUIDloop modifies its effective critical current and hence can bedetected using the JBA principle. There are several modes inwhich such a magnetometer can be operated. Biasing in thehysteretic regime permits a digital readout, similar to thequbit measurements described above, and could potentiallybe used for spin state measurements of magnetic molecules.It is also possible to bias in the vicinity of the bifurcationpoint where the response is nonhysteretic, but the nonlinear-ity boosts sensitivity. Such a device can detect fast fluxchanges, which can be used to study the dynamics ofnanomagnets.52

It is important to distinguish a JBA magnetometer from aconventional dc SQUID device and other rf biased SQUIDs.In the dc SQUID, the magnetometer is biased near the resis-tive transition where the static voltage is used to register fluxcoupled to the SQUID loop. A shunt resistor is used to sup-press hysteresis in the current-voltage characteristic. Suchdevices can exhibit bandwidth in excess of 100 MHz with#0 sensitivity53 but at the expense of dissipation in andaround the SQUID. The JBA magnetometer is an inductivedevice in which the magnitude of the phase excursion isalways less than , thus avoiding the dissipation and back-action associated with switching into the voltage state. Assuch, flux sensitivity alone is not sufficient to compare thesetwo types of magnetometers as their measurement back-action is very different. Radio frequency biased SQUIDsRef. 54 do not generate a static voltage but typically dohave phase excursions beyond a single well of the washboardpotential and the associated back-action. SQUIDs have alsobeen operated in the inductive mode,55,56 typically where thephase excursion is very small and the anharmonicity of theSQUID potential is not sampled. In the JBA magnetometer,we discuss a device where the quality factor is kept low forenhanced bandwidth and the SQUID is biased near the bifur-cation point for enhanced sensitivity.

We built a rf magnetometer by embedding a tunnel junc-tion SQUID in a ! /4 coplanar stripline resonator. The prin-ciple of operation is exactly the same as the lumped elementversion with a slight difference in the effective circuit.13

Figure 20a shows a schematic of this circuit, and im-ages of the magnetometer are shown in Fig. 20b. The de-vice was fabricated using standard e-beam lithography withdevice parameters chosen to yield a resonant frequency of1.35 GHz when the flux through the SQUID loop is zero.The quality factor Q of the resonator was around 800. Asuperconducting coil was used to apply a dc magnetic flux tobias the SQUID. An on-chip coplanar waveguide transmis-sion line with a short circuit termination was used to couplehigh frequency signals to calibrate the device. The samplebox along with the bias coil was enclosed in a superconduct-ing aluminum box to shield from magnetic noise. The micro-wave measurement setup was similar to the one shown inFig. 9 but had additional lines to bias the coil and the on-chipflux line.

The reflected signal phase as a function of the bias flux



and drive frequency is shown in the inset of Fig. 21. Asexpected, we observe the periodic dependence of the reso-nance frequency yellow color corresponding to zero phaseas a function of bias flux. The reflected signal phase is plot-ted as a function of drive power and frequency in Fig. 21 andcharacterizes the nonlinear response of the resonator. Thisplot is similar to Fig. 13, but here the power was swept inboth directions for a given frequency. The different sweepdirections are interlaced in the plot. The sweep direction onlyaffects the hysteretic regime as observed in the striped regionin the top left part. One can clearly see the upper and lowerbifurcation points. For magnetometry, the device was oper-ated in the linear and the single-valued nonlinear regime byadjusting the microwave power. Small changes in input fluxare converted into small changes in the phase of the micro-

wave signal reflected from the resonator, forming the basis ofdetection. In this technique, the noise temperature of the mi-crowave electronics sets the phase resolution and ultimatelydetermines the magnetometer sensitivity. In the linear re-gime, the dominant noise contribution comes from the coldHEMT amplifier assembly which has a system noise tem-perature of about 10 K.

In order to characterize the performance of this device asa flux sensor, we carried out the following experiment. Thedc coil was used to bias the SQUID at a sensitive point. Thefast flux line was then used to send a small calibrated ac fluxsignal at a frequency of 10 Hz. A vector network analyzerwas used to determine the oscillations in the reflected signalphase, which was then transferred to a computer. These os-cillations were then Fourier-transformed and the signal tonoise ratio SNR of the resulting peak at 10 Hz was deter-mined. Since we know the magnitude of the ac flux signal,we can extract the value of the smallest flux signal, whichcan be resolved with unity SNR in a 1 Hz bandwidth. We callthis number the effective flux noise in our system. Figure 22shows a plot of the effective flux noise as a function ofmicrowave drive power for three different flux bias pointsdifferent sample than one in Fig. 21. One expects the SNRto improve when larger microwave power is used to probethe resonator, resulting in a lower effective flux noise. Thedashed lines indicate the improvement expected if the reso-nator was linear. The advantage of the JBA principle isclearly visible as the nonlinearity sets in at higher drive pow-ers see Figs. 5a and 5b. Under these bias conditions theJBA exhibits significant parametric gain, effectively lower-ing the system noise temperature, and results in the lowestobserved flux noise of 1 #0 /Hz for this device. Thisdemonstrates that a microwave readout of a SQUID can leadto a fast sensitive flux detector.

One should note that it is possible to operate the magne-tometer in the digital mode by biasing the oscillator in thebistable region as was done for the qubit readout. In thismode, the magnetometer is essentially a 1-bit flux sampler.However for measuring continuous changes in flux, it is amore cumbersome mode of operation involving pulse prepa-

λ / 4 resonator input fluxΦ

squid

(a)

(b)

200 µm10 µm

FIG. 20. Color online a Schematic of a ! /4 resonator terminated with anunshunted two junction SQUID. The input flux applied via a coil modulatesthe critical current of the SQUID, resulting in a shift in the resonant fre-quency. b Optical image of Al coplanar stripline resonator terminated withan unshunted SQUID, shown in the SEM image on the right.

1.2875 1.2900 1.2925 1.2950

-130

-125

-120

P(dBm)

ωd/2π (GHz)

Reflected phase

-180o 180o0

-0.5-1.0 0.0

1.36

1.32

1.28

Φ / Φ0

ωd/2

π(GHz)

FIG. 21. Color Reflected signal phase color as a function of drive powerand frequency. This plot is similar to the one in Fig. 13, but here the datawere obtained by sweeping power in both directions at a given frequency.Alternate sweep directions are interlaced, allowing one to see the hysteresisin the driven response striped region in the top left corner. The upper andlower bifurcation boundaries are clearly visible. Inset: reflected signal phasecolor as a function of drive frequency and flux bias for a drive power of140 dBm. The linear resonant frequency shows the expected periodicdependence on flux bias.

4

8

10-4

10-5

10-6

2

4

8

2

4

8

-140 -135 -130 -125 -120

ac flux modulation frequency = 10 Hz

Effectivefluxnoise(Φ

0/Hz1/2)

P (dBm)

Φ = 0.18 Φ0

Φ = 0.25 Φ0

Φ = 0.32 Φ0

FIG. 22. Color Effective flux noise of the magnetometer as a function ofdrive power. This quantity is the smallest change in the flux one can detectin a 1 Hz bandwidth. Data are shown for a different sample than the one inFig. 21. The dotted curves from top to bottom correspond to data for threeincreasing values of dc flux bias points. Lowest effective flux noise obtainedis 1 #0 /Hz. The dashed lines indicate the expected reduction in fluxnoise with power for a comparable linear resonator. Biasing in the nonlinearregime yields an order of magnitude reduction in flux noise.



ration and processing and one would prefer the analog modewhere only a continuous microwave signal is required. Wetested the magnetometer in the digital mode data not shownas well and obtained similar effective flux noise perfor-mance.

We will conclude this section with some remarks aboutoptimizing the performance of such a magnetometer. Asmentioned earlier, the system noise temperature of the am-plification chain determines the lowest flux noise for a givenset of SQUID parameters. In principle, operating the JBA athigh parametric gain points should result in a noise tempera-ture approaching the standard quantum limit. We are pres-ently exploring this possibility. The choice of the qualityfactor Q of the resonator is another important criterion foroptimizing effective flux noise. A higher Q will give you alarger phase shift per unit change in flux but at the cost ofreduced bandwidth and possibly lower operating powers,which can even make the effective flux noise worse. As thedata suggest, accessing the nonlinearity of the SQUID givesyou enhanced performance even for a low Q device. How-ever this also comes at the cost of reduced bandwidth datanot shown since gain-bandwidth for such systems is fixed.Another parameter that influences the effective flux noise isthe drive power, and in principle one can improve the per-formance by increasing it. In practice there is always a maxi-mum operable limit for a given device due to its finite dy-namic range of operation, e.g., entering the bistable regimein the JBA. One could also trade-off nonlinearity for higheroperating powers by reducing the coupling of the SQUIDnonlinear device to the resonator linear device,56 but thiswill predictably decrease the phase shift per unit change influx. To summarize, quality factor, nonlinearity, operatingpower, and noise temperature are related to each other in acomplex way, and we are presently exploring strategies tooptimize for minimum effective flux noise and maximumbandwidth.

VII. CONCLUSIONS

The Josephson Bifurcation Amplifier JBA exploits thebifurcation dynamics of the microwave-driven plasma oscil-lation of a Josephson junction. When biased near the bifur-cation point, the driven voltage response varies strongly withthe plasma frequency which is modulated by a weak externalsignal parametrically coupled to the critical current of thejunction, thus giving rise to amplification. The JosephsonBifurcation Amplifier is capable of both analog and digitaldispersive measurement with variable gain.

We have demonstrated experimentaly the principle andoperation of this novel amplifier by observing the Josephsonplasma oscillation transition between the two dynamicalstates predicted for a driven nonlinear system. Using differ-ent samples, we have shown that this dynamical phenom-enon is stable, reproducible, and can be precisely controlled,thus opening the possibility for practical applications such asamplification. A signal coupled to the critical current of thejunction can be detected by monitoring the changes in thedynamical state of the nonlinear oscillator.

This approach was used to develop a nonlinear disper-

sive readout for superconducting qubits by coupling aCooper-pair box with the JBA. In order to perform a readout,the resonator is rf-energized to a level where its oscillationstate now acts as a sensitive pointer of the qubit state. Thistechnique does not generate any dissipation on-chip since theresonator is only damped by circuitry outside the chip, i.e., a50 transmission line with a matched circulator and ampli-fier, and enables a high-fidelity qubit readout with a mega-hertz repetition rate. We have measured Rabi oscillations andRamsey fringes with sufficient speed that real time filteringto correct for drifts in the charge and flux bias becomes pos-sible. Also, several successive readouts may be performedwithin the energy relaxation time of the qubit T1. Thisgives valuable information on the readout-induced interac-tion between the qubit and its environment and accounts forthe observed contrast.

We also exploited the JBA principle to construct a highspeed magnetometer. This was done by incorporating a tun-nel junction SQUID in a coplanar stripline resonator operat-ing at 1.3 GHz. Using the nonlinearity of the SQUID, wewere able to achieve an effective flux noise of about1 #0 /Hz. Further improvements are possible by optimiz-ing the flux to critical current transfer function of theSQUID. This method can be extended to high speed magne-tometry of single molecule magnets by replacing the tunneljunction SQUIDs with nano-SQUIDs which use nanocon-strictions as the Josephson elements.

The advantage of the JBA over other dissipative ampli-fication techniques such as the dc SQUIDs resides in itsminimal back-action. Since there is no on-chip dissipation,the only source of back-action is the matched isolator load,which is efficiently thermalized at the bath temperature. Animportant point is that in the JBA, only fluctuations from theload that occur in a narrow band centered about the plasmafrequency contribute to the back-action, whereas in the dcSQUID noise from many high frequency bands is also sig-nificant. Finally, the bifurcation amplifier does not sufferfrom quasiparticle generation associated with hystereticSQUIDs Ref. 57 and dc current-biased junctions58 whichswitch into the voltage state. Long quasiparticle recombina-tion times at low temperatures limit the acquisition rate ofthese devices, while the recombination process itself pro-duces excess noise for adjacent circuitry.59

We would like to thank M. Metcalfe, C. Rigetti, E.Boaknin, and V. Manucharyan for their contribution to theJBA and CBA experiments. We would like to acknowledgeM. Hatridge for his contribution to the magnetometry experi-ments. Funding from the following sources is acknowledged:AFOSR under Grant No. FA9550-08-1-0104 R.V. and I.S.,ONR under Grant No. N00014-07-1-0774 I.S., NSA underARO Contract No. W911NF-05-1-0365, NSF under GrantNo. DMR-0653377 M.H.D., the Keck FoundationM.H.D., the Agence Nationale de la Recherche M.H.D.and College de France M.H.D..

1 B. D. Josephson, Rev. Mod. Phys. 36, 216 1964.2 A. B. Zorin, Phys. Rev. Lett. 76, 4408 1996.3 R. J. Schoelkopf, P. Wahlgren, A. A. Kozhevnikov, P. Delsing, and D. E.Prober, Science 280, 1238 1998.



http://dx.doi.org/10.1103/RevModPhys.36.216

http://dx.doi.org/10.1103/PhysRevLett.76.4408

http://dx.doi.org/10.1126/science.280.5367.1238

4 M. Muck, C. Welzel, and J. Clarke, Appl. Phys. Lett. 82, 3266 2003.5 A. J. Dahm, A. Denenstein, T. F. Finnegan, D. N. Langenberg, and D. J.Scalapino, Phys. Rev. Lett. 20, 859 1968.

6 T. Holst and J. B. Hansen, Physica B 165, 1649 1990.7 T. A. Fulton and L. N. Dunkleberger, Phys. Rev. B 9, 4760 1974.8 E. Il’ichev, T. Wagner, L. Fritzsch, J. Kunert, V. Schultze, T. May, H. E.Hoenig, H. G. Meyer, M. Grajcar, D. Born, W. Krech, M. V. Fistul, and A.M. Zagoskin, Appl. Phys. Lett. 80, 4184 2002.

9 I. Siddiqi, R. Vijay, F. Pierre, C. M. Wilson, M. Metcalfe, C. Rigetti, L.Frunzio, and M. H. Devoret, Phys. Rev. Lett. 93, 207002 2004.

10 A. Lupascu, C. J. M. Verwijs, R. N. Schouten, C. J. P. M. Harmans, and J.E. Mooij, Phys. Rev. Lett. 93, 177006 2004.

11 I. Siddiqi, R. Vijay, F. Pierre, C. M. Wilson, L. Frunzio, M. Metcalfe, C.Rigetti, R. J. Schoelkopf, M. H. Devoret, D. Vion, and D. Esteve, Phys.Rev. Lett. 94, 027005 2005.

12 V. E. Manucharyan, E. Boaknin, M. Metcalfe, R. Vijay, I. Siddiqi, and M.Devoret, Phys. Rev. B 76, 014524 2007.

13 M. Metcalfe, E. Boaknin, V. Manucharyan, R. Vijay, I. Siddiqi, C. Rigetti,L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, Phys. Rev. B 76, 1745162007.

14 K. Wiesenfeld and B. McNamara, Phys. Rev. Lett. 55, 13 1985.15 J. S. Aldridge and A. N. Cleland, Phys. Rev. Lett. 94, 156403 2005.16 V. M. Eguíluz, M. Ospeck, Y. Choe, A. J. Hudspeth, and M. O. Magnasco,

Phys. Rev. Lett. 84, 5232 2000.17 S. Martignoli, J.-J. van der Vyver, A. Kern, Y. Uwate, and R. Stoop, Appl.

Phys. Lett. 91, 064108 2007.18 B. Yurke, L. R. Corruccini, P. G. Kaminsky, L. W. Rupp, A. D. Smith, A.

H. Silver, R. W. Simon, and E. A. Whittaker, Phys. Rev. A 39, 25191989.

19 R. Movshovich, B. Yurke, P. G. Kaminsky, A. D. Smith, A. H. Silver, R.W. Simon, and M. V. Schneider, Phys. Rev. Lett. 65, 1419 1990.

20 R. Vijay, Ph.D. thesis, Yale University, 2008. Available online at http://qulab.eng.yale.edu/archives.htm.

21 M. A. Castellanos-Beltran and K. W. Lehnert, Appl. Phys. Lett. 91,083509 2007.

22 N. Bergeal, R. Vijay, V. E. Manucharyan, I. Siddiqi, R. J. Schoelkopf, S.M. Girvin, M. H. Devoret, arXiv:0805.3452v1

23 M. Metcalfe, Ph.D. thesis, Yale University, 2008. Available online athttp://qulab.eng.yale.edu/archives.htm.

24 M. I. Dykman and M. A. Krivoglaz, Physica A 104, 480 1980.25 O. Naaman, J. Aumentado, L. Friedland, J. S. Wurtele, and I. Siddiqi,

Phys. Rev. Lett. 101, 117005 2008.26 A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer, M. H. Devoret,

S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. Lett. 95, 060501 2005.27 M. Hatridge, R. Vijay, J. J. Lee, J. Clarke, and I. Siddiqi unpublished.28 M. I. Dykman, I. B. Schwartz, and M. Shapiro, Phys. Rev. E 72, 021102

2005.29 D. J. Van Harlingen, T. L. Robertson, B. L. T. Plourde, P. A. Reichardt, T.

A. Crane, and J. Clarke, Phys. Rev. B 70, 064517 2004.30 M. I. Dykman and V. N. Smelyansky, Zh. Eksp. Teor. Fiz. Pis’ma Red. 94,

61 1988.31 M. Marthaler and M. I. Dykman, Phys. Rev. A 73, 042108 2006.32 M. I. Dykman, Phys. Rev. E 75, 011101 2007.33 J. M. Martinis, M. H. Devoret, and J. Clarke, Phys. Rev. B 35, 4682

1987.34 R. Vijay, I. Siddiqi, P. Hyafil, M. Metcalfe, L. Frunzio, and M. H. Devoret

unpublished.35 E. Boaknin, V. E. Manucharyan, S. Fissette, M. Metcalfe, L. Frunzio, R.

Vijay, I. Siddiqi, A. Wallraff, R. J. Schoelkopf, and M. Devoret,arXiv:cond-mat/0702445 2007.

36 D. F. Santavicca and D. E. Prober, Meas. Sci. Technol. 19, 087001 2008.37 Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, Nature London 398, 786

1999.38 J. E. Mooij, T. P. Orlando, L. Levitov, L. Tian, C. H. van der Wal, and S.

Lloyd, Science 285, 1036 1999.39 D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve,

and M. H. Devoret, Science 296, 886 2002.40 J. M. Martinis, S. Nam, J. Aumentado, and C. Urbina, Phys. Rev. Lett. 89,

117901 2002.41 J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A.

Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 76,042319 2007.

42 I. Siddiqi, R. Vijay, M. Metcalfe, E. Boaknin, L. Frunzio, R. J. Schoe-lkopf, and M. H. Devoret, Phys. Rev. B 73, 054510 2006.

43 A. Lupascu, E. F. C. Driessen, L. Roschier, C. J. P. M. Harmans, and J. E.Mooij, Phys. Rev. Lett. 96, 127003 2006.

44 F. Mallet, F. R. Ong, A. Palacios-Laloy, F. Nguyen, P. Bertet, D. Vion, andD. Esteve, Nat. Phys. 5, 791 2009.

45 F. Wellstood private communication.46 G. Ithier, E. Collin, P. Joyez, P. J. Meeson, D. Vion, D. Esteve, F.

Chiarello, A. Shnirman, Y. Makhlin, J. Schriefl, and G. Schön, Phys. Rev.B 72, 134519 2005.

47 N. Boulant, G. Ithier, P. Meeson, F. Nguyen, D. Vion, D. Esteve, I. Sid-diqi, R. Vijay, C. Rigetti, F. Pierre, and M. Devoret, Phys. Rev. B 76,014525 2007.

48 J. A. Schreier, A. A. Houck, J. Koch, D. I. Schuster, B. R. Johnson, J. M.Chow, J. M. Gambetta, J. Majer, L. Frunzio, M. H. Devoret, S. M. Girvin,and R. J. Schoelkopf, Phys. Rev. B 77, 180502 2008.

49 R. W. Simmonds, K. M. Lang, D. A. Hite, S. Nam, D. P. Pappas, and J. M.Martinis, Phys. Rev. Lett. 93, 077003 2004.

50 D. I. Schuster, A. Wallraff, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S.M. Girvin, and R. J. Schoelkopf, Phys. Rev. Lett. 94, 123602 2005.

51 H. Nakano, S. Saito, K. Semba, and H. Takayanagi, Phys. Rev. Lett. 102,257003 2009.

52 S. D. Bader, Rev. Mod. Phys. 78, 1 2006.53 D. Drung, J. Beyer, M. Peters, J.-H. Storm, and T. Schurig, IEEE Trans.

Appl. Supercond. 19, 772 2009.54 J. E. Zimmerman, P. Thiene, and J. T. Harding, J. Appl. Phys. 41, 1572

1970.55 K. K. Likharev, Dynamics of Josephson Junctions and Circuits Gordon

and Breach, New York, 1985, Chap. 14, p. 490.56 J. A. B. Mates, G. C. Hilton, K. D. Irwin, L. R. Vale, and K. W. Lehnert,

Appl. Phys. Lett. 92, 023514 2008.57 I. Chiorescu, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij, Science

299, 1869 2003.58 A. Cottet, Ph.D. thesis, Université Paris VI, 2002.59 J. Mannik and J. E. Lukens, Phys. Rev. Lett. 92, 057004 2004.



http://dx.doi.org/10.1063/1.1572970


http://dx.doi.org/10.1016/S0921-4526(09)80410-9

http://dx.doi.org/10.1103/PhysRevB.9.4760

http://dx.doi.org/10.1063/1.1481988










http://dx.doi.org/10.1063/1.2768204

http://dx.doi.org/10.1063/1.2768204

http://dx.doi.org/10.1103/PhysRevA.39.2519


http://qulab.eng.yale.edu/archives.htm


http://dx.doi.org/10.1063/1.2773988


http://dx.doi.org/10.1016/0378-4371(80)90010-2



http://dx.doi.org/10.1103/PhysRevE.72.021102



http://dx.doi.org/10.1103/PhysRevE.75.011101


http://dx.doi.org/10.1088/0957-0233/19/8/087001

http://dx.doi.org/10.1038/19718

http://dx.doi.org/10.1126/science.285.5430.1036

http://dx.doi.org/10.1126/science.1069372












http://dx.doi.org/10.1103/RevModPhys.78.1

http://dx.doi.org/10.1109/TASC.2009.2017887

http://dx.doi.org/10.1109/TASC.2009.2017887

http://dx.doi.org/10.1063/1.1659074

http://dx.doi.org/10.1063/1.2803852

http://dx.doi.org/10.1126/science.1081045


the josephson bifurcation - q-lab - yale university

Documents