PHYSICAL REVIEW B 86, 144510 (2012)

Gain, directionality, and noise in microwave SQUID amplifiers: Input-output approach

Archana Kamal,1,* John Clarke,2 and Michel H. Devoret11Departments of Applied Physics, Yale University, New Haven, Connecticut 06520, USA

2Department of Physics, University of California, Berkeley, California 94720, USA(Received 29 June 2012; published 10 October 2012)

We present a new theoretical framework to analyze microwave amplifiers based on the dc SQUID. Our analysisapplies input-output theory generalized for Josephson junction devices biased in the running state. Using thisapproach, we express the high-frequency dynamics of the SQUID as a scattering between the participating modes.This enables us to elucidate the inherently nonreciprocal nature of gain as a function of bias current and inputfrequency. This method can, in principle, accommodate an arbitrary number of Josephson harmonics generatedin the running state of the junction. We report detailed calculations taking into account the first few harmonicsthat provide simple semiquantitative results showing a degradation of gain, directionality, and noise of the deviceas a function of increasing signal frequency. We also discuss the fundamental limits on device performance andapplications of this formalism to real devices.

DOI: 10.1103/PhysRevB.86.144510 PACS number(s): 85.25.Cp, 74.50.+r, 85.25.Dq

I. INTRODUCTION

For almost half a century, the dc SQUID (superconductingquantum interference device) has enabled a broad range ofdevices, including magnetometers, gradiometers, voltmeters,susceptometers, and amplifiers.1,2 Most of these devices areused at relatively low frequencies, and all have the commonfeature of offering extremely low noise.3 The fact that thedc SQUID is potentially a quantum-limited amplifier inthe microwave regime was recognized long ago,4 but notexploited in practice until the axion dark matter experiment(ADMX), provided a powerful motivation.5 This need led tothe development of the microstrip SQUID amplifier (MSA)in which the input coil deposited on (but insulated from)the washer of a SQUID acts as a resonant microstrip.6 Suchamplifiers have achieved a noise temperature within a factorof two of the standard quantum limit.7–9 More recently, newdesigns have appeared intended to extend the frequency ofoperation to frequencies as high as 10 GHz, aimed at thereadout of superconducting qubits10 and the detection ofmicromechanical motion.11 These include incorporation of agradiometric SQUID at the end of a quarter wave resonator12

and the direct injection of the microwave signal from a quarterwave resonator into one arm of the SQUID ring.13,14

Besides having desirable properties such as high gain, widebandwidth and near quantum-limited operation, microwaveSQUID amplifiers (MWSAs)—unlike conventional Josephsonparametric amplifiers15–17—also offer an intrinsic separationof input and output channels of the signal that makes themunique among amplifiers based on Josephson tunnel junctions.This property makes them especially well suited as a pream-plifier in the measurement chain for superconducting devicesby eliminating the need for channel separation devices, suchas circulators and isolators, between the sample under test andthe first amplification stage. Although microwave SQUID am-plifiers have been successfully used experimentally, questionspertaining to their nonlinear dynamics and ultimate sensitivityas amplifiers have continued to remain challenging problems.Previous theories include quantum Langevin simulations4,18

and treatment of the SQUID as an interacting quantumpoint contact.19 The ultimate exploitation of the amplifier,

however, requires a precise quantum treatment of its behaviorat the Josephson frequency and its harmonics. Besides beingvaluable for practical considerations, such understanding mayhelp discern the cause of intrinsically nonreciprocal operationof the MWSA that has hitherto remained an open question.This concern is especially relevant to applications such as qubitreadout where the amplifier backaction may prove to be theAchilles’ heel. In this work, we develop an ab initio theoreticalframework to understand the high-frequency dynamics of theSQUID in detail. In addition to giving us crucial insights intothe amplifying mechanism of the MWSA and its nonreciprocalresponse between the input and output signal channels, thisapproach enables us to calculate the experimentally relevantquantities such as available gain, added noise and directionalityat operating frequencies of interest.

We perform our analysis in the paradigm of input-outputtheory and employ the method of harmonic balance to study thedriven dynamics of the device. The dc SQUID is biased in thevoltage regime—in contrast to the usual Josephson parametricamplifiers operated in the zero voltage state with the phaseexcursions of the Josephson junction confined to a singlecosine well—and has the dynamics of a particle samplingvarious wells of a two-dimensional tilted washboard.1 Theinput-output analysis thus needs to be generalized to take intoaccount phase running evolution in this two-dimensional po-tential. Our approach involves a self-consistent determinationof the working point of the device established by static biasparameters (the static bias current IB and external flux �ext

shown in Fig. 1) followed by a study of the rf dynamics usinga perturbative series expansion around this working point.In Sec. II, we introduce our input-output model for the dcSQUID. Using this in Sec. III, we first derive the response atzero frequency and at the Josephson oscillation frequency ωJ

in a self consistent manner. Following this, we evaluate theperturbative response at finite frequency around zero and ωJ

as a scattering matrix in the basis of relevant modes of thecircuit, which clearly elucidates the nonreciprocal dynamicsof the device. In Secs. IV and V, we calculate the figures ofmerit such as power gain, directionality and noise temperature,and identify the fundamental limits on the performance of thedevice. Section VI contains our concluding remarks.

144510-11098-0121/2012/86(14)/144510(12) ©2012 American Physical Society

ARCHANA KAMAL, JOHN CLARKE, AND MICHEL H. DEVORET PHYSICAL REVIEW B 86, 144510 (2012)

ext

IB

V1 V2

M

L

FIG. 1. Circuit schematic of a conventional MWSA. The SQUIDconsists of two Josephson junctions, arranged in a superconductingloop, with inductance L. The loop is biased using a static currentsource IB and an external flux �ext. An input voltage V1 generates anoscillating current in an input coil inductively coupled to the SQUIDthus inducing a small flux modulation δ� of the flux enclosed by theloop. For optimal flux bias [�ext = (2n + 1)�0/4] that maximizes theflux-to-voltage transfer coefficient V� ≡ (∂V2/∂�ext)IB , this causesan output voltage V2 = V�δ� to develop across the ring. Thus thedevice behaves as a low impedance voltage amplifier.

II. ANALYTICAL MODEL

A. SQUID circuit basics

The SQUID circuit considered in our analysis is shown inFig. 2(a). The dynamics of the system is modeled as a particlemoving in a two-dimensional potential of the form1

USQUID

2EJ

(ϕD,ϕC) = 1

πβL

(ϕD − ϕext

2

)2

− cos ϕD cos ϕC − IB

2I0ϕC. (1)

Here, IB is the bias current, I0 is the critical current ofeach junction, ϕext = 2π�ext/�0 represents the externallyimposed flux in the loop, βL ≡ 2LI0/�0 denotes a di-mensionless parametrization of the SQUID loop inductance,EJ ≡ I0�0/2π is the Josephson energy, and �0 ≡ h/2e isthe flux quantum. We have introduced the common, ϕC =(ϕL + ϕR)/2, and differential, ϕD = (ϕL − ϕR)/2, mode com-binations of the phases of the two junctions that form the axesof the two-dimensional orthogonal coordinate system.

To facilitate an input-output analysis of the circuit, we re-place the resistive shunts across the junction with semi-infinitetransmission lines [cf. Fig. 2(b)] of characteristic impedanceZC = R, following the Nyquist model of dissipation. Thus theshunts play the dual role of dissipation and ports (or channels)used to address the device. This allows us to switch froma standing mode representation in terms of lumped elementquantities such as voltages and currents to a propagatingwave description in terms of signal waves traveling on thetransmission lines. The amplitude of these waves is given bythe well-known input-output relation,20

Ain/outi (t) = V i

2√

ZC

∓√

ZCI i

2, i ∈ {L,R}, (2)

where V i and I i denote the voltage across the shunt resistanceand current flowing in the shunt resistance respectively. FromEq. (1), we obtain the common mode current, IC = (IL +

IR)/2, and differential mode current, ID = (IL − IR)/2, flow-ing in the shunts by identifying ϕC,D as the relevant positionvariables. The current in each mode can thus be interpreted asthe “force”21 that follows directly from Hamilton’s equationof motion as

IC,D

I0= − ∂

∂ϕC,D

(USQUID

2EJ

), (3)

which yields

ωC = ωB

2− ω0 sin ϕC cos ϕD, (4a)

ωD = ω0

πβL

(−2ϕD + ϕext) − ω0 cos ϕC sin ϕD. (4b)

Here, we have expressed the currents in equivalent fre-quency units:

ωC ≡ ICR

ϕ0and ωD ≡ IDR

ϕ0, (currents) (5)

ωB ≡ IBR

ϕ0and ω0 ≡ I0R

ϕ0, (characteristic currents) (6)

with ϕ0 = �0/(2π ). Including a capacitance across the junc-tion gives an additional term, involving a second-order deriva-tive of the common and differential mode fluxes, of the form−ω−1

B cϕC,D with c = 2πIBR2C/�0, on the right-hand

side of Eq. (4). This parametrization of capacitance, motivatedby calculational simplicity, leads to a different parametrizationof the plasma frequency, ωp ≡ (I0/ϕ0C)1/2[1 − (IB/I0)2]1/4.The more conventional parametrization with a fixed value ofcapacitance for all bias values can be implemented in a morecomprehensive calculation aided by numerical techniques.

Equation (4) represents a subtle current-phase relationshipfor the two-junction system, analogous to the first Josephsonrelation. We note that Eq. (4) can alternatively be derivedusing a first-principles Kirchoff law analysis of the circuit inFig. 2(a). Similar to the currents, we can define the commonand differential mode voltages as

ωC ≡ V C

ϕ0, ωD ≡ V D

ϕ0. (7)

Further, by the second Josephson relation, we have

〈ωC〉 = Vdc/ϕ0 = ωJ , (8)

where Vdc is the static voltage developed across the SQUIDbiased in the running state.

We note that the usual mode of operation of a dc SQUIDinvolves an input flux inductively coupled using an input trans-former of which the loop inductance forms the secondary coil(see Fig. 1). The input transformer, however, is an experimentalartifact required to ensure the impedance matching with theinput impedance of the SQUID at a desired frequency. It isnot crucial from the point of view of device characteristics,however, as it is the SQUID which provides amplification andall the relevant nonlinear dynamics of the device. In the ensuinganalysis, we do not employ a separate input port, but ratherconsider a direct input coupling through the differential modeof the ring which couples to the flux in an analogous manner[see Fig. 2(b)]. Such a scheme may also prove beneficial fora practical device to overcome the problem of low couplingat high signal frequencies, as recently shown experimentally

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VR

Ain

Aout

VL

Ain

Aout

IL IRL R

L

L

R

R

L/2 L/2

IB IB

(a)

ZC = R ZC = R

R

IBIL IR

L Rext

L/2 L/2

RVL VR

VR

Aout

VL

Ain

Aout

IL IR

L R

L

L

RL/2 L/2

ZC = R ZC = R

AinR

(b)

(c)

L R

FIG. 2. (Color online) Equivalent input-output model of the SQUID. (a) The bare SQUID loop without the input coupling circuit. As notedin Fig. 1, the circuit has two static biases—a common mode bias current IB and a differential mode external flux �ext. There is also a capacitanceC across each junction, not shown here for simplicity. (b) Equivalent SQUID circuit under Nyquist representation of shunt resistances andseparate static current biases IL

B and IRB for the left and right junction respectively. The common mode bias current IB now corresponds to

the even combination of two external bias currents −(ILB + IR

B )/2, while the external flux �ext corresponds to the differential combinationof the two current sources L(IL

B − IRB )/2. The oscillating signals are modelled as incoming and outgoing waves traveling on semi-infinite

transmission lines, representing the shunt resistances across the two junctions. (c) Effective junction representation for evaluating the signalresponse of the device. Here, we have replaced the junctions biased with a static current with effective junctions pumped by the Josephsonharmonics (represented by a “glowing” cross with a pumping wave) generated by phase running evolution in the voltage state of the junction.

using a SLUG (superconducting low-inductance undulatorygalvanometer) microwave amplifier.13,14

B. Harmonic balance treatment

Using the input-output relation of Eq. (2) with Eqs. (4) and(7), we obtain the equations

ωC,D(t) = ωC,D(t) + 2ωinC,D(t) (9)

for common and differential mode circuit quantities. Here,ωin(t) = Ain(t)

√R/ϕ0 represents the input signal drive ex-

pressed in terms of an equivalent frequency. Furthermore, it isuseful to note that Eq. (9) for the common mode quantities,in conjunction with Eq. (4a) with ϕD = 0, reduces to theequation of motion of a single resistively-shunted junction(RSJ), V C/R + I0 sin ϕC = IB + I in(t).

We employ the technique of harmonic balance and solveEq. (9) in the frequency domain, at all frequencies of interest(see Fig. 3). This is achieved by assuming two parts to thesolution for each variable of interest (ϕC and ϕD):

ϕC = ωJ t + δϕC(t), (10)

ϕD = φ0 + δϕD(t), (11)

where ωJ t and φ0 represent the average static values of thecommon and differential mode phases [cf. Eq. (8)]. The timevarying components are of the form

δϕC,D(t) = �C,D(t) + �C,D(t), (12)

where �(t) refers to the components at the Josephsonfrequency ωJ and its harmonics. The term �(t) includesthe components oscillating at the signal frequency ωm andits resultant sidebands ωn = nωJ + ωm generated by wave

m0

J 2 JJ2 J m

SC,D

FIG. 3. (Color online) Spectral density landscape of commonand differential modes of the SQUID. The tall solid arrows showthe Josephson harmonics generated internally in the running stateof the device. The small input signal frequency ωm and differentsidebands generated by mixing with Josephson harmonics are shownwith dashed arrows.

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ARCHANA KAMAL, JOHN CLARKE, AND MICHEL H. DEVORET PHYSICAL REVIEW B 86, 144510 (2012)

mixing via the nonlinearity of the SQUID:

�C,D =K∑

k=1

pC,Dk,x cos kωJ t + p

C,Dk,y sin kωJ t, (13)

�C,D =+N∑

n=−N

sC,Dn,x cos(nωJ + ωm)t

+ sC,Dn,y sin(nωJ + ωm)t. (14)

We note that the number of Josephson harmonics includedin the analysis [i.e., K in Eq. (13)] is determined by the order ofexpansion of the junction nonlinearity in δϕ. This in turn is de-termined by the bias voltage of the device set by the bias currentIB . As IB is reduced towards the critical current of the junctionI0, higher Josephson harmonics become more significant as thecharacteristics of the device become increasingly nonlinear.We can, therefore, calculate the response perturbatively byexpanding each of the coefficients p and s in Eqs. (13) and(14) as a truncated power series in the reduced bias parameter:

ε ≡ I0

IB

= ω0

ωB

. (15)

The degree of the resultant polynomial evaluation of p, s

coefficients is set by the desired order of expansion in δϕ. Asε � 0.5 (or equivalently IB > 2I0) for the SQUID to operate inthe running state at any value of flux bias,1 which is the regimeof interest for the SQUID to be operated as a voltage amplifier,it provides a convenient small parameter of choice. Equivalentexpansions in previous works19,22 have confirmed convergenceof such perturbation series methods for experimentally relevantparameters. Furthermore, this parameter serves as the effectivestrength of the different Josephson harmonics, which playa role analogous to the strong “pump” tone of conventionalparametric amplifiers.

The scalar input-output relations of Eq. (9), decomposedinto relevant temporal modes (frequency components) byperforming a harmonic series expansion of Eqs. (4) and (7),lead to complex matrix equations that reveal the various mixingprocesses performed by the nonlinear terms in Eq. (4). Thisdecomposition enables us to study independently the steadystate and the dynamic responses of the system by setting V in =0 and V in = VRF(t) respectively, where VRF(t) is an rf inputdrive at the relevant frequency. We use the former to evaluatethe system response �C,D(t) at the Josephson frequency andits harmonics and the latter to evaluate the system response�C,D(t) at the signal and sideband frequencies. In bothcases, the dimensionality of the resultant matrix equationsin the frequency domain is strictly dictated by the number oftemporal modes included in the analysis and hence by the orderof expansion, as explained earlier. Throughout our calculationswe assume βL = 1, the value found previously3 to optimizethe noise performance of the dc SQUID.

III. CALCULATION OF SQUID DYNAMICS

A. Steady state response: I-V characteristics

We first determine the working point of the SQUID bysolving for the steady state characteristics. As the zero-frequency response of the system is intrinsically related to

0.1 0.2 0.3 0.4 0.5 0.6

0.0

0.1

0.2

0.3

0.4

0.5

0.1 0.2 0.3 0.4 0.5 0.60.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7(b)

(a)

v

v

FIG. 4. (Color online) Static transfer function of the SQUIDcalculated as a function of two bias parameters ε = ω0/ωB at ϕext =π/2 for (a) strongly overdamped (C = 0) and (b) intermediatelydamped junctions (C = 1). Both plots were calculated with βL =1. The (black) triangles represent the transfer function calculatedfrom the exact numerical integration of the SQUID equations. The(green) circles correspond to the K = 1 evaluation including only theJosephson frequency [see Eq. (13)]. This first-order evaluation doesnot show any voltage modulation with flux as there is no couplingbetween the common and differential modes at this order. The (blue)squares and (red) diamonds correspond to an evaluation includingthe second (K = 2) and third Josephson (K = 3) harmonics, respec-tively. The corresponding curves represent interpolating polynomials.In both plots, the agreement of the perturbative series with the exactnumerical solution improves on including higher order correctionscorresponding to contributions of higher Josephson harmonics.

the response at the Josephson frequency through Eq. (8), wecalculate it self-consistently along with the strength of thevarious Josephson harmonics in the steady state by consideringonly the static source terms with no oscillating input drive atthe Josephson frequency and its harmonics. This yields a setof boundary conditions of the form

ω[kωJ ] − ω[kωJ ] = 0, k ∈ [0,K]. (16)

We solve this set of simultaneous equations to calculatethe strength of the various Josephson harmonics generatedinternally from the static bias due to the junction nonlinearityalong with the zero-frequency characteristics. Figure 4 showsa plot of the static transfer function υ� = ∂〈ωC〉/∂ϕext ob-tained using the perturbative series method to determine the

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GAIN, DIRECTIONALITY, AND NOISE IN MICROWAVE . . . PHYSICAL REVIEW B 86, 144510 (2012)

coefficients pC,Dk [see Eq. (13)] described in the last section.

The agreement between the exact numerical calculationand the perturbative analytical calculation improves on in-creasing the order of the perturbation series expansion byincluding mixing processes mediated by higher Josephsonharmonics. Further, from the steady state calculation for thedifferential mode, we obtain a relation for the phase anglebetween the two junctions in the ring as

φ0 = ϕext

2+ βL

K∑k�2

akεk sin ϕext, (17)

where the coefficients ak are of order unity. Thus, we see thatthe average values of both the explicit static bias parametersnamely ε (common) and ϕext (differential) participate in estab-lishing each of the implicit static biases Vdc (or equivalentlyωJ ) for the common mode and φ0 for the differential mode.The contributions arising from the bias current, as shownin Eq. (17), lead to a rolling of the static phase differencearound the SQUID loop that manifests itself as the change incurvature of the transfer function curves shown in Fig. (4).Furthermore, we note that, as indicated by the steady statecalculation, the flux dynamics of υ� evaluated using thetruncated harmonic series calculation are “slower,” that is, theyshift to higher values of bias with respect to the exact numericalresults. Nonetheless, the predicted magnitudes are comparableand hence the theory is capable of making semiquantitativepredictions in an analytically tractable manner. The majormerit of this approach over conventional methods lies in thenatural extension offered for the study of higher frequencydynamics as discussed in the following sections.

B. RF response: scattering matrix

Once we have determined the static working point for theSQUID, we can solve for its rf dynamics in the small signalregime. The aim is to calculate signal amplitudes by includingthe �C,D(t) term in our analysis and considering all the mixingprocesses mediated by the pumps �C,D(t) evaluated in thelast section, permissible by the harmonic balance of Eqs. (4),(7), and (9). This is equivalent to the representation shown inFig. 2(c), where we model the mixing of the input signal by theSQUID as a parametric interaction with different Josephsonharmonics playing the role of an effective “colored” pump. Inthe limit of a small amplitude input signal, which is the relevantlimit for most practical situations, we can then introduce alinear response description of the dynamics as an admittancematrix seen from the ports. This can be obtained from thecurrent-phase and voltage-phase relationship [see Eqs. (4) and(7)] as

−→ω = M

−→� (18)

and−→ω = M

−→� , (19)

yielding

Y = MM−1. (20)

The vectors in the equations above are defined in the basisof all signal and sideband frequencies of interest, (�C[nωJ +

ωm],�D[nωJ + ωm]), n ∈ [−N, + N ] leading to a 4(2N +1) × 4(2N + 1) admittance matrix. We further note that thematrix is block diagonal since harmonic balance leads to twodisjoint manifolds, each of which forms a closed subspace ofdimension 2(2N + 1).

From the admittance matrix of Eq. (20), we can evaluatethe scattering matrix of the SQUID using the identity

S = (U + Y )−1(U − Y ), (21)

where U represents an identity matrix of appropriate dimen-sions. Figure 5 shows the calculation for different ordersin junction nonlinearity and the relevant forward (|sCD|2)and backward scattering gain (|sDC |2). It immediately showsthe emergence of the nonreciprocal gain of the device that,unlike conventional paramps, enables a two-port operation.As the nonlinearity of the device characteristics is increasedby reducing IB towards I0 (thus increasing the expansionparameter ε), we need to include the higher Josephsonharmonics in the calculation which become significant dueto rapid running evolution of the phase of the junctions in thetwo-dimensional tilted washboard. This leads to a situationanalogous to pumping of the SQUID by an effective multitonepump of the form �(t) = ∑K

k=1 pk cos(ωJ t + φk) in both C

and D modes [see �(t) panels in Fig. 5]. The dynamics of sucha system include multi-path interference involving differentJosephson harmonics. This effect, analogous to symmetrybreaking in ratchet physics,23 implements an asymmetricfrequency conversion scheme guided by relative phases φk

of different Josephson harmonics driving the junctions.24 Thesignal in the differential mode is preferentially upconverted,coupled through higher order mixing processes into thecommon mode and then preferentially downconverted intothe common mode, yielding a net forward gain from thedifferential mode to the common mode. The reverse gainprocess from C to D is disfavored by the same reasoning,leading to the nonreciprocal operation of the SQUID amplifier.

IV. POWER GAIN OF THE SQUID

The dc SQUID operated as a two-port voltage amplifierresembles the configuration of a semiconductor, operationalamplifier (op-amp) as opposed to that of a conventionalparametric amplifier, which is a matched device (that is, theinput and output impedances are identical to the impedancesof the transmission lines or coaxial cables). In this sense,the MWSA is the magnetic dual of the rf SET (singleelectron transistor).25 The dc SQUID amplifies an input current(directly coupled as in this analysis or coupled as a flux via aninput transformer), and has a much lower impedance thanthe electromagnetic environment in which it is embedded.Conversely, the rf SET amplifies an input voltage, and has amuch higher impedance than the electromagnetic environmentin which it is embedded. The true power gain of either device,as seen from the ports, thus involves a de-embedding of thedevice characteristics.26 In the case of the SQUID, this requiresa translation from the matched (or scattering) descriptionbased on the input-output theory considered in this paper tothe op-amp or hybrid representation that is well suited fordescribing an unmatched amplifier.

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ARCHANA KAMAL, JOHN CLARKE, AND MICHEL H. DEVORET PHYSICAL REVIEW B 86, 144510 (2012)

0

0.2

0.0

0.2

0.4

0.0

0.4

1.0

0.5

0.0

0.5

1.0

0

0

m 0

m 0

m 0

C,D

(t)C

,D(t)

C,D

(t)

Jt

Jt

Jt

JJ

mm

0

2 J

J

mm

0

2 J2 J2 J

J

mm m

0

3 J

(a)

(b)

(c)

K=1, N=1

K=2, N=1

K=3, N=2

J

J

3 J

|sCD|2

|sDC|2

|sCD|2

|sDC|2

|sCD|2

|sDC|2

FIG. 5. (Color) Josephson harmonics and small signal scattering gain of the SQUID calculated using harmonic balance with expansionof the sin ϕ nonlinearity to (a) first, (b) third, and (c) fifth order, respectively. The parameters used were �ext = �0/4, βL = 1, and C = 1.Each panel shows the relevant modes of the frequency spectrum included in the calculation at that order [see Eqs. (13) and (14)]. Thedispersive mixing between various temporal modes of the system is denoted using grey arcs with the relevant Josephson harmonic acting asthe pump indicated next to them. The relative strength of the different mixing processes is indicated by the respective widths of the arcs, with thestrongest being denoted by the thickest arcs. Also shown are plots of �(t), the effective pumps in common (blue) and differential (red) modesat each order of the calculation. The box panels show the respective forward (|sCD|2) and backward (|sDC |2) scattering gains as a function ofreduced input frequency ωm/ω0 and bias parameter ε = ω0/ωB . The surface plot in (a), calculated using only the Josephson frequency, showsno asymmetry between the forward and backward gains (blue and red surface plots, respectively). The asymmetry develops on inclusion ofhigher harmonics that implement a multitone pump which is not symmetric about t = 0, as seen from the plots of �(t) in (b) and (c). As weincrease the order of calculation and include higher harmonics, the asymmetry increases and finally peaks at an optimal value of bias parameterε = 0.455.

The hybrid matrix describing a two-port amplifier is of theform27 (

V2

I1

)=

(λV Zout

Yin λ′I

)(V1

I2

), (22)

where (V1,I1) and (V2,I2) denote the voltage and currentassociated with the input and output ports, respectively. Thepower gain for such an amplifier is given by

GP = P out

P in= V 2

2 /Re[Zout]

V 21 /Re[Zin]

= λ2V

Re[Yin]Re[Zout], (23)

where λV is the voltage gain of the amplifier, Yin is the inputadmittance, and Zout is the output impedance. Equation (23)

represents the gain of an effective “matched” device account-ing for the impedance mismatch at the input and output ports.

In principle, although the calculation of quantities inEq. (22) can be performed using the scattering matrix evaluatedin Eq. (21),27 nonetheless, it is advantageous to transform to adescription that is more natural in describing the relationshipbetween standing mode current and voltage variables. We findthat an impedance matrix (Z) representation is well suited forsuch a purpose due to its rather straightforward mapping tothe standing mode quantities of Eq. (22). Using the Y matrix,derived in Eq. (20), we can write the impedance matrix Z ofthe dc SQUID as

Z = (U + Y )−1 (24)

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GAIN, DIRECTIONALITY, AND NOISE IN MICROWAVE . . . PHYSICAL REVIEW B 86, 144510 (2012)

with ⎛⎝ −→ωC

−→ωD

⎞⎠ =(

zCC zCD

zDC zDD

) ⎛⎝ −→ωC

−→ωD

⎞⎠ . (25)

Here, as before,−→ω and

−→ω are vectors defined in the space of

all signal and sideband frequencies of interest. Also U is anidentity matrix of appropriate dimensions and corresponds tothe admittance contribution of the resistive shunts across thejunctions.

The next step is to make the translation from the impedancematrix derived in the common and differential mode basisto the two-port description of Eq. (22). This requires anidentification of the correct “input” and “output” voltages andcurrents for the circuit in Fig. 2(a). As the SQUID readoutinvolves measurement of the voltage developed across it, therelevant output quantities are related to the common modequantities as V2 = V C and I2 = 2IC . The translation to theinput variables of the hybrid representation is more subtle.For this purpose we first note that, in conventional SQUIDoperation, the input flux coupled into the ring modulates thecirculating current J , which is, thus, the relevant input currentof the device. The equivalent input voltage that causes theflux modulation of the circulating current can be representedby a voltage source VJ in series with the inductance of theloop. Figure 6 summarizes the different possible two-portrepresentations of the SQUID used in this paper.

On interpreting the loop variables (VJ ,J ) described abovein terms of the differential mode voltage V D and current

I1

I1 V1 I2V2Z

V1

I2

V2Y

I1

V1 H I2V2

++++

++

VD I CV CID

VD

I C

V C

ID

V C I CVJ/2

J

VJ/2

(a)

(b)

(c)

++ ++

++ ++

FIG. 6. Different representations of a two-port network andanalog configurations for the dc SQUID. (a) Y -matrix representationdefined for closed boundary conditions Yij = dIi/dVj |Vk =j =0 forthe junctions and inductance, omitting the shunts [see Eq. (20)].(b) Z-matrix representation defined for open boundary conditionsZij = dVi/dIj |Ik =j =0 including the shunts [see Eq. (24)]. (c) (Hybrid)H-matrix or op-amp representation defined with mixed boundaryconditions [see Eq. (22)]. In effective matrices for the SQUID, thecommon mode (C) and differential mode (D) excitations of the ringplay the role of ports 1 and 2, if the SQUID is addressed using hybrids.In each panel, the quantities shown with solid arrows representthe stimulus, while those shown with dashed arrows represent thecorresponding response of the network.

ID (see Appendix A for details), we obtain the followingequivalence between the coefficients of the hybrid matrix inEq. (22) and the Z matrix of Eq. (25):

λV =(

R

iωmL

)zCD[ωm], (26)

λI =(

R

iωmL

)zDC[ωm], (27)

Zout =(

R

2

)zCC[ωm], (28)

Yin = (iωmL)−1 +(

2R

ω2L2

)zDD[ωm]. (29)

Using the above translation in Eq. (23), we find an expressionfor the power gain purely in terms of Z matrix coefficients:

GP [ωm] = |zCD[ωm]|2Re[zCC[ωm]Re[zDD[ωm]]

. (30)

Figure 7 shows the power gain of the device as a functionof bias and input frequency, calculated using Eq. (30). Itshows that power gain of the MWSA increases quadrat-ically with decreasing input signal frequency, a result

GP

(dB

)

0.00 0.02 0.04 0.06 0.08 0.100

10

20

30

40

Harmonic Balance Calculation (b)

(a)

m

Fit

0.15 0.20 0.25 0.30 0.35 0.40 0.450

5

10

15

20

25

30

35

GP

(dB

)

FIG. 7. Power gain of the MWSA calculated with K = 3, N =2, taking into account the modification of the input and outputimpedances of the device by matched loads. The parameters are�ext = �0/4, βL = 1, and C = 1. (a) Power gain versus biasparameter ε = ω0/ωB calculated for a fixed input frequency ωm =0.01 ω0. The solid curve is an interpolating polynomial of degreetwo. (b) Power gain versus input frequency ωm/ω0 calculated withbias parameter fixed at ε = 0.455, the optimum value for attainingminimum noise temperature [see Fig. 9(a)] at low frequencies (ω �ω0). The fit is of the form GP = [0.006/(ωm/ω0)2] + 2.0 measuredin linear units.

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ARCHANA KAMAL, JOHN CLARKE, AND MICHEL H. DEVORET PHYSICAL REVIEW B 86, 144510 (2012)

0.00

0.05

0.10

0.30

0.35

0.40

0.45

0

10

20

30

40

Gp

(dB

)G

p(d

B)

rev

m 0

5 dB

35 dB

Dire

ctio

nalit

y

FIG. 8. (Color) Directionality (in dB) of the MWSA as a functionof bias parameter ε = ω0/ωB and reduced input frequency ωm/ω0.The red dots represent high directionality and blue dots representlow directionality. The parameters used were the same as thosein Fig. 7.

corroborated by a simple quasistatic treatment presented inAppendix B. This result agrees well with that derived forgeneric quantum-limited linear detectors in Ref. 27 where itwas shown that power gain scales as (kBTeff/hωm)2. Here, Teff

is the effective temperature of the detector. The characteristicJosephson frequency, ω0 = 2eI0R/h = kBTeff/h, thus sets theeffective temperature of the MWSA and the scale of powergain.

The reverse power gain of the device is calculated in asimilar manner as

GrevP = I 2

1 Re[Zin]

I 22 Re[Zout]

= λ2I

Re[Yin]Re[Zout][see Eq. (22)]

= |zDC[ωm]|2Re[zCC[ωm]Re[zDD[ωm]]

. (31)

The directionality (GP − GrevP )—which is a measure of the

asymmetry between forward and reverse power gains—follows directly from the asymmetric scattering gain discussedin the previous section. Our calculation shows that it is a strongfunction of the bias ε (see Fig. 8); furthermore, the optimal biasfor maximum power gain is not the same as that for maximaldirectionality. We note that the results presented here havebeen obtained with a truncated harmonic series excluding allJosephson harmonics above 3ωJ . In the real device, the achiev-able isolation between forward (differential-to-common) andbackward (common-to-differential) gain channels may bequantitatively different due to the presence of the neglectedhigher order interferences.

V. NOISE TEMPERATURE

In this section, we evaluate the noise added by the dc SQUIDoperated as a voltage amplifier. The noise added by a systemcan be quantified by its noise temperature, TN , defined as

TN = Ahω

kB

, (32)

where A is the Caves added noise number.7 This noisetemperature corresponds to the effective input temperature ofthe amplifier obtained by referring the added noise measuredat the output to the input, and is quantified in terms ofenergy quanta per photon at the signal frequency. For aphase preserving amplifier, such as the MWSA, the minimumpossible noise temperature corresponds to half a photon ofadded noise, that is, Amin = 0.5 in the large gain limit (ingeneral, Amin = 1/2 − 1/2G).

Using the hybrid representation developed in the previoussection and Appendix A, we write the noise inequality for theMWSA as

kBTN �√

SV CV C SJJ − Re[SV CJ ]2 − Im[SV CJ ]

λV

, (33)

where SV V represents the spectral density of the voltagefluctuations at the output, SJJ represents the spectral densityof the circulating current fluctuations, and SV J is the cross-correlation between the voltage and current fluctuations.27

As in the case of power gain, we can evaluate the spectraldensities in Eq. (33) from the Z matrix of the SQUID derivedin Sec. IV. This exercise is enabled by the fact that the input-output theory treats the deterministic signal input and noiseof the system on an equal footing. Thus the linear responsedescription developed to calculate the signal gain provides astraightforward way of generalizing the theory to understandthe noise properties of the system, simply by replacing theinput current signal with a noise signal described by a spectraldensity of the form

SII [ω] = 2hωRe[Y [ω]] coth

(hω

2kBT

). (34)

Such a small-signal linear approach is valid both in the thermaland quantum regimes of noise for the SQUID as the typicalphoton energy at the frequencies of interest, the Josephsonfrequency and its first few harmonics, is much smaller thanthe energy dissipated per turn of the SQUID running phasebecause of the low Q of the Josephson oscillations.

Using the Z matrix, we can write the voltage noise spectraldensity in the common mode as

SV CV C =N∑

n=−N

∣∣zCC0n

∣∣2SICIC [nωJ + ωm]

+N∑

n=−N

∣∣zCD0n

∣∣2SIDID [nωJ + ωm]. (35)

Here, the first sum accounts for the contribution to the noisearising from the common mode signal (n = 0) and sidebandsabout the Josephson harmonics (n = ±1, ± 2) included in thecalculation; the second sum accounts for the noise generatedin the common mode output signal by the differential modesignal and sidebands, arising from coupling between C and D

modes. Similarly, we can calculate SJJ as

SJJ = 4

ω2mL2

SV DV D , (36)

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GAIN, DIRECTIONALITY, AND NOISE IN MICROWAVE . . . PHYSICAL REVIEW B 86, 144510 (2012)

by making the identification J = 2V D/(iωL). Here, as before,we calculate SV DV D from the Z matrix:

SV DV D =N∑

n=−N

∣∣zDC0n

∣∣2SICIC [nωJ + ωm]

+N∑

n=−N

∣∣zDD0n

∣∣2SIDID [nωJ + ωm]. (37)

Finally, for SV J , we have

SV CJ = −2

iωmL

(N∑

n=−N

zCC0n zDC∗

0n SICIC [nωJ + ωm]

+N∑

n=−N

zCD0n zDD∗

0n SIDID [nωJ + ωm]

). (38)

Figure 9 shows plots of the Caves noise number of thedevice, calculated in both the thermal regime [kBT hωm,where all terms in Eqs. (35)–(38) contribute equal noise

(b)

m

A= kBTN

A'= TNT

Add

ed n

oise

num

ber A'= TN(a)

A= kBTN

T

h m

h m

0.40 0.42 0.44 0.46 0.480.25

0.5

1

2

5

10

20

50

0.00 0.02 0.04 0.06 0.08 0.100

1

2

3

4

5

6

Add

ed n

oise

num

ber

FIG. 9. (Color online) Caves added noise number for the MWSA,calculated using the harmonic balance analysis with K = 3, N = 2as a function of (a) ε = ω0/ωB for ωm = 0.01 ω0 and (b) reducedinput frequency ωm/ω0 with ε = 0.455. In both plots, (round) blackmarkers show the noise number A

′ = TN/T obtained in the thermalregime kBT hωm, while (square) green markers show the noisenumber A = kBTN/hωm calculated in the quantum regime kBT �hωm. The solid curves in (a) represent interpolating polynomials. Thequantum calculation gives a minimum value for A = kBTN/hωm ≈0.5, attained at ε = 0.455, corresponding to one-half photon of addednoise [horizontal black dashed line in (a)]. The optimal value of biascurrent for minimum added noise does not coincide with that forachieving the maximum power gain [see Fig. 7(a)] or directionality(Fig. 8).

powers] and the quantum regime [kBT � hωm, where eachterm in Eqs. (35)–(38) contributes a noise power proportionalto its frequency in accordance with Eq. (34)].

There are a number of points to be highlighted. Our calcu-lation shows that, at the optimal current bias of ε = 0.455, theMWSA attains the quantum limit of added noise correspondingto half-photon at signal frequency. Moreover, the optimum biaspoint for minimum noise corresponds to the bias for maximumscattering gain [see Fig. 5(c)] rather than for the maximumpower gain [see Fig. 7(a)]. This result, previously found boththeoretically and experimentally,28 follows from the fact thatthe added noise is a property of the bare SQUID without anymatching to input and output loads. In the case of conventionalparametric amplifiers, the minimum noise indeed occurs atthe maximum scattering gain. Furthermore, the partial crosscorrelation between the output voltage noise across the SQUIDand the supercurrent noise circulating in the loop is crucial tominimizing the noise in both thermal and quantum regimes.We also note that, for sufficiently low signal frequencies, thecalculated added noise number is found to saturate at a valueslightly below the quantum limit of one half-photon at thesignal frequency. This result, we suspect, is due to the fact thatat the bias for minimum noise, the reverse gain is substantialand hence the isolation is not perfect (see Fig. 8). The quantumlimit of one half-photon is a limiting value calculated for idealdetectors with zero reverse gain and high forward gain,27 acondition which is not satisfied at the optimal noise bias in ourcalculation. Finally, our calculation shows that the minimumnoise number is achieved only when the signal frequencyis much lower than the characteristic Josephson frequencyω0 = 2πI0R/�0, and increases significantly with increasingsignal frequency.

VI. CONCLUDING REMARKS

In summary, we have developed a new method based oninput-output theory to provide a first-principles analysis of themicrowave SQUID amplifier (MWSA). In this paradigm, wetreat the SQUID biased in its running state as a parametricamplifier pumped by a combination of Josephson harmonicsgenerated internally by the motion of the phase of the junctions.This approach leads to a fully self-consistent description ofboth the static and rf dynamics of the device. The scatteringmatrix calculation shows that the nonreciprocal gain of the am-plifier arises from mixing processes involving higher Joseph-son harmonics which implement an asymmetric frequencyconversion scheme involving upconversion to and subsequentdownconversion from the Josephson frequency. We find thatthe power gain of the matched SQUID amplifier decreasesquadratically with signal frequency ωm; by comparison, thegain in the usual SQUID operation with a matched input coilscales as 1/ωm.28 However, a recently reported dc SQUIDamplifier14 using the direct coupling method considered inthis paper demonstrated that the power gain scaled as 1/ω2

m ata frequency of a few GHz and a bandwidth of several hundredMHz.

Our analysis shows that the MWSA achieves quantum-limited noise performance for optimal flux and currentbiases, and for signal frequencies significantly lower thanthe characteristic Josephson frequency ω0 = 2πI0R/�0. The

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ARCHANA KAMAL, JOHN CLARKE, AND MICHEL H. DEVORET PHYSICAL REVIEW B 86, 144510 (2012)

added noise increases significantly with increasing frequency.This problem can be alleviated by using junctions withhigher values of critical currents. With the present technologyfor niobium junctions, critical current densities of tens ofmicroamperes per square micron are readily achievable. Thistranslates into characteristic frequencies of about 100 GHz,which should be sufficient to achieve lower noise at GHzfrequencies provided hot electron effects due to dissipationin the shunts are mitigated.29 Furthermore, our analysis showsthat simultaneous optimization of gain, directionality and noiseis a delicate operation since the optimal biases for these threeproperties do not coincide. Based on our calculation, at theworking point for minimum added noise, A ≈ 0.5, power gainsof 15–18 dB and directionality of around 5–8 dB are obtained.However, higher power gains of 20–30 dB and directionalityof 10–12 dB can be realized by permitting a higher noisenumber A ≈ 5–10. Though the predicted directionality is stillmodest, it suffices to reduce the number of nonreciprocalelements (circulators, isolators) in the measurement chaintypically employed for the readout of superconducting qubits.Moreover, the noise penalty incurred with MWSAs comparesvery well to standard cryogenic amplifiers such as HEMTswhose typical noise numbers lie in the range 40–50 formicrowave frequencies.

Although the results presented in this paper are semiquan-titative, we believe that extension of the analysis to higherorders, in conjunction with numerical optimization techniques,can be a useful tool to analyze SQUID-based devices dueto rapid convergence offered by the harmonic series method.This approach would allow one to evaluate the appropriateparameters, depending on the intended application, that yieldthe best compromise between gain and noise properties.

ACKNOWLEDGMENTS

We thank O. Buisson, A. Clerk, M. Hatridge, and R. Vijayfor useful discussions. The authors also wish to thank AnandaRoy for help with the calculations in initial stages of thisproject. This research was supported by ARO under Grant No.W911NF-09-1-0514 (AK and MHD) and the US Departmentof Energy, Office of High Energy Physics, under contractnumber DE-FG02-11ER41765 (JC).

APPENDIX A: LOOP VARIABLES FOR THE DC SQUID

In this appendix, we establish the correspondence betweenthe differential mode variables that serve as the input in theanalysis of Secs. II and III and the input variables requiredfor the hybrid representation discussed in Sec. IV. The outputvariables in the two representations have a simple relationshipas explained in Sec. IV.

Figure 10 shows the two representations [see Figs. 6(b)and (c)], one in terms of differential mode quantities (V D,ID)suitable for a scattering or matched representation (since theinput and output impedances are just the transmission lineimpedance) and the other in terms of a circulating current J anda loop voltage VJ , which are the relevant input quantities forthe device in an unmatched hybrid description. In Fig. 10(a),Kirchoff’s current law gives

J = ID − IL, (A1)

RID

J

IL

L/2

(a)VD V=0

RJ

J

L/2

(b) VD

VJ//2

FIG. 10. (Color online) Equivalence between the SQUID differ-ential mode and the op-amp input variables. Here, for simplicity,we have used the symmetric version of the SQUID to divide thering along the equipotential. The circuit in (a) models the device asan impedance response function to an imposed current source ID ,as a result of which a voltage drop +V D(−V D) develops acrossthe left (right) junction. (b) Hybrid representation for the SQUID,which models the input response by introducing a differential voltagesource in the SQUID loop and recording the current that flows throughthe junction. The junction at this point is replaced with an effectivejunction pumped using the various Josephson harmonics generatedby the static bias current [also see Fig. 2(c)].

while in Fig. 10(b), from Kirchoff’s voltage law, we have

−VJ

2= V D + iωL

2J, (A2)

with J = IL.To establish the equivalence of the two representations from

the point of view of the junction, we require the voltage acrossthe junction V D and current through the junction J to beconserved (see Fig. 10). Thus, using Eq. (A1) in Eq. (A2), weobtain

VJ = −iωLID. (A3)

Similarly, it is easily seen that the circulating current J isgiven as

J = 2V D

iωL. (A4)

APPENDIX B: STATIC ANALOG CIRCUIT FOR THE SQUID

The SQUID can be thought of as a current amplifier witha current transferred from a low-impedance input port to ahigh-impedance output port. This description is analogous tothe FET dual model with the gain given by a transimpedanceinstead of a transconductance. The equivalent “current gain”of such a device (see Fig. 11) for frequencies sufficientlyclose to zero [ωm � ρinR/L = ω0/(πβL) to be precise] can

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GAIN, DIRECTIONALITY, AND NOISE IN MICROWAVE . . . PHYSICAL REVIEW B 86, 144510 (2012)

Iin Iout

V

Differential Common

RD = out( )RLin( )R Vdc

FIG. 11. Equivalent low-frequency circuit for a SQUID forcalculation of unilateral power gain. The input circuit is modelled asan effective impedance viewed by a low frequency differential modecurrent. The output circuit impedance comprises a bias-dependentresistor, denoting the dynamic impedance of the junction, thatconverts the output voltage to a corresponding output current. Thenet “transimpedance” is given by the static flux-to-voltage transferfunction of the device. The symbols ρin, out denote bias-dependentconstants of order unity.

be modeled as

Iout

Iin≈ V�L

RD

. (B1)

This leads to a power gain

GdcP =

(Iout

Iin

)2RD

Re[Zin]. (B2)

For frequencies of interest, Re[Zin] ≈ ω2mL2/(ρinR). Using

this result in Eq. (B2), we obtain the power gain

GdcP = ρin

ρout

(V�

ωm

)2

, (B3)

which can be rewritten as

GdcP ≈ ρg

(ω0

ωm

)2

, (B4)

GP

(dB

)

0.15 0.20 0.25 0.30 0.35 0.40 0.450

5

10

15

20

25

30

35

FIG. 12. (Color online) Comparison of the power gain as afunction of bias parameter ε = ω0/ωB calculated using a K =3, N = 2 calculation (solid black line) and a purely quasistaticcalculation (dashed red line) of Eq. (B3). For the quasistatic gaincalculation, V� and ρin,out were obtained from the I -V characteristicsevaluated in Sec. III A.

where ρg is a bias-dependent and frequency-independentconstant of order unity. Here, we have used the relationV

opt� = R/L = ω0/π

2 for βL = 1. Equation (B4) shows thatthe gain drops quadratically with increasing signal frequency,and that no power gain is obtained for signal frequencies closeto the plasma frequency of each junction in the SQUID. Thisfrequency dependence of the power gain is borne out by the rfanalysis employing the harmonic balance treatment, shown inFig. 7(b).

Figure 12 shows a comparison of gain as a function of biasparameter ε, calculated using quasistatic response functionsas shown in Eq. (B3) and a rf calculation at low frequenciesinvolving the third Josephson harmonic [same as that shownin Fig 7(a)]. The agreement is better for lower values ofε where high frequency components of the device are lesssignificant. The impedance matrix calculation generates extraterms due to self-summation caused by the inversion operation[cf. Eq. (24)], which leads to higher-order corrections absentfrom the purely quasistatic calculation.

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