Traveling Wave Parametric Amplifier based on a chain of Coupled Asymmetric SQUIDs
aa r X i v : . [ c ond - m a t . s up r- c on ] S e p Traveling Wave Parametric Amplifier based on a chain of Coupled AsymmetricSQUIDs
M. T. Bell
1, 2 and A. Samolov Engineering Department, University of Massachusetts Boston, Boston, Massachusetts 02125 Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854 (Dated: August 5, 2018)A traveling wave parametric amplifier (TWPA) composed of a transmission line made up of achain of coupled asymmetric superconducting quantum interference devices (SQUIDs) is proposed.The unique nature of this transmission line is that its nonlinearity can be tuned with an externalmagnetic flux and can even change sign. This feature of the transmission line can be used to performphase matching in a degenerate four-wave mixing process which can be utilized for parametricamplification of a weak signal in the presence of a strong pump. Numerical simulations of theTWPA design have shown that with tuning, phase matching can be achieved and an exponentialgain as a function of the transmission line length can be realized. The flexibility of the proposeddesign can realize: compact TWPAs with less than 211 unit cells, signal gains greater than 20 dB,3 dB bandwidth greater than 5.4 GHz, and saturation powers up to -98 dBm. This amplifier designis well suited for multiplexed readout of quantum circuits or astronomical detectors in a compactconfiguration which can foster on-chip implementations.
PACS numbers: 85.25.Am, 85.25.Cp, 84.30.Le
I. INTRODUCTION
Over the past decade Josephson parametric amplifiershave proven essential in experiments studying quantumjumps , tracking quantum trajectories and real-timemonitoring and feed-back control of quantum bits(qubits). In parametric amplifiers high gain is achievedwhen the signal to be amplified interacts with a non-linear medium for as long as possible. State-of-the-artJosephson parametric amplifiers utilize resonant circuitsto increase the interaction time of the signal with the non-linear medium . As a consequence the instantaneousbandwidth and maximum input power allowed are sig-nificantly reduced. Such limitations have renewed inter-est in superconducting traveling wave parametric ampli-fiers (TWPA) which achieve long interaction timeswith the nonlinear medium by extending the electricallength over a long transmission line instead of multiplebounces in a resonant circuit, as a result TWPAs do notsuffer from the same bandwidth and dynamic range lim-itations that cavity based amplifiers do. The main chal-lenges in TWPA designs is that optimum gain is achievedwhen phase matching conditions are met. Supercon-ducting TWPAs have been investigated by many groups,thus far taking one of two approaches, either utilizing atransmission line composed of a series array of Joseph-son junctions or a transmission line utilizing thenonlinear kinetic inductance of a narrow superconduct-ing wire . These investigations have revealed theneed for engineered dispersion to be introduced into thetransmission line to facilitate phase matching .Designs which utilize periodic loading and the ad-dition of resonant elements to facilitate phase match-ing have shown promise, however at the expense of in-creased complexity, higher tolerances, and longer propa-gation lengths (2 cm − . The amplifica-tion process is the most efficient when the total phasemismatch is close to zero. However due to the nonlin-earity of the transmission line a strong pump modifiesphase matching through self and cross phase modulationresulting in a phase mismatch. The linear dispersion ofthe transmission line along with spectral separation ofthe signal and pump angular frequencies can be usedto compensate for the nonlinear phase mismatch. Theunique feature of the proposed TWPA is that the linearand nonlinear dispersion can be tuned with Φ, and thenonlinearity can even change sign. By adjusting Φ for agiven pump power, phase matching can be achieved. II. TWPA DESIGN
The design of the proposed TWPA is shown in Figure1(a). Each cell of the transmission line is an asymmet-ric SQUID with a single ”small” Josephson junction withcritical current I js and capacitance C js in one arm andtwo ”large” Josephson junctions with critical current I jl and capacitance C jl in the other arm. -1 0 1-707-1 0 1-707 (d) I/I j s ( ) / =0 / =0.3 / =0.5r = 6/ =0.5 I/I j s ( ) r = 4 r = 5 r = 6 (c) ’ FIG. 1. (color online). TWPA based on a chain of asymmet-rically coupled SQUIDs (a) Circuit schematic of the proposedTWPA. Each unit cell of the TWPA is threaded with a mag-netic flux Φ and has a parasitic capacitance to ground C gnd .The coupled SQUID transmission line can realistically be im-plemented in a coplanar transmission line geometry. The ge-ometrical inductance of the transmission line can be madenegligible in comparison to the Josephson inductance of thecoupled asymmetric SQUIDs making up the line. (b) Unitcell of the TWPA. (c) and (d) Current phase relation of aunit cell for various ratios r and Φ / Φ . Adjacent cells to one another are coupled via the largeJosephson junctions. A feature of this arrangement isthat for an even number of asymmetrical SQUIDs inthe chain, the Josephson energy E j ( φ ) remains an evenfunction of the phase difference φ across the chain forany value of Φ . The proposed TWPA requires a longtransmission line with many unit cells, this allows to ne-glect boundary effects and focus on translationally in-variant solutions. At arbitrary Φ the transmission lineremains symmetric under the translation by two cells.The defined unit cell of the transmission line is com-posed of two large and two small junctions (Figure 1(b)).Each unit cell is of length a and has a capacitance toground of C gnd . The ”backbone” of the unit cell ismade up of ”large” Josephson junctions highlighted inred, which are designed to have Josephson energies, E jl ,two orders of magnitude larger than the charging energy, E cl = e / (2 C jl ), of the junction. In this case quan-tum fluctuations of the phase across individual junctionsare small ∝ exp[ − (8 E jl /E cl ) / ] and thus a classicaldescription of the system can be used. The phases onthe two large junctions for each unit cell are α and α ′ ,and the total phase across the unit cell is ϕ = α + α ′ .The Josephson energy E j ( ϕ ) of the unit cell is minimizedwhen α = α ′ , and a gauge was chosen such that an ex-ternal magnetic flux would induce phases 2 π Φ / Φ on the small junctions. An expansion of the current phase rela-tion of the current flowing through the backbone of theunit cell around ϕ = 0 is given by I ( ϕ ) = I js h r (cid:16) π ΦΦ (cid:17)i ϕ − I js h r
48 + 13 cos (cid:16) π ΦΦ (cid:17)i ϕ , (1)where Φ = h/ e is the flux quantum, h is Planck’s con-stant, e is the electron charge, and r = I jl /I js . ForΦ / Φ = 0 . r = 4 where the linearterm in I ( ϕ ) goes to zero and the cubic term dominates.At this point the inductance L ∝ (cid:0) dI ( ϕ ) /dϕ (cid:1) − of theunit cell is the largest. This is the unique feature of thesuperinductor which is similar by design to the pro-posed TWPA. As for the proposed TWPA a large induc-tance of the unit cell is not desired, however tunabilityof the nonlinearity of I ( ϕ ) is. Fig. 1(c) and 1(d) showthe current phase relation of a unit cell of the TWPA forvarious r and Φ. Fig. 1(c) and equation (1) show thatthe nonlinearity at I ( ϕ ≈
0) is always positive at fullfrustration Φ / Φ = 0 . r <
16. Fig.1(d) and equa-tion (1) show that for certain r values (example r = 6)by adjusting Φ the nonlinearity can be tuned over a widerange, and can even change sign from negative to posi-tive. By tuning the nonlinearity it is possible to optimizethe parametric amplification efficiency of the FWM pro-cess.Using equation (1) a nonlinear wave equation was de-rived to describe the node flux ϕ n shown in Fig. 1(a)along the length of the transmission line . Assumingconstant C js , C jl , C gnd , I jl , and I js along the lengthof the transmission line, no dissipation, and the contin-uum approximation for a wave-type excitation ( λ ≫ a )the following wave equation is derived a L h r (cid:16) π ΦΦ (cid:17)i ∂ ϕ∂z + a C js (cid:16) r (cid:17) ∂ ϕ∂t ∂z − C gnd ∂ ϕ∂t − γ ∂∂z h(cid:16) ∂φ∂z (cid:17) i = 0 , (2)where L = ϕ /I js and ϕ = Φ / (2 π ). The first threeterms of the wave equation represents the linear contribu-tions to the dispersion on the transmission line due to thedistributed inductances and capacitances and how theycan be tuned with r and Φ. The fourth term describesthe nonlinearity and how the nonlinear coupling coeffi-cient, γ = (cid:2) a / ( ϕ L ) (cid:3)(cid:2) ( r/
48) + (1 /
3) cos (cid:0) π Φ / Φ (cid:1)(cid:3) canbe tuned with Φ. The solution to equation (2) is assumedto be four traveling waves, where in the degenerate casethe two pump angular frequencies ω p are equal, a sig-nal ω s , and a generated idler tone ω i = 2 ω p − ω s . Bytaking the slowly varying envelope and undepleted pumpapproximations a set of coupled mode equations is de-rived to describe the propagation of the signal and idlertraveling waves: ∂a s ∂z − i γk p k i k s (2 k p − k i ) a ∗ i | A p | ω s C gnd e iκz = 0 , (3) ∂a i ∂z − i γk p k s k i (2 k p − k s ) a ∗ s | A p | ω i C gnd e iκz = 0 , (4)where a s and a i are the complex signal and idler ampli-tudes, k m is the wave vectors of the pump, signal, andidler ( m = { p, s, i } ) (see Appendix A), ∆ k = k s + k i − k p is the phase mismatch due to the linear dispersion in thetransmission line, the total phase mismatch including selfand cross phase modulation is κ = − ∆ k + 2 α p − α s − α i ,and | A p | is the undepleted pump amplitude. The terms α m ∝ γ | A p | , are the self and cross phase modulation ofthe wave vectors per unit length.Equations (3) and (4) are similar to well establishedfiber parametric amplifier theory and have the fol-lowing solution to describe the power gain of the signalin the presence of a strong pump with zero initial idleramplitude G s = | cosh( gz ) − i ( κ/ g ) sinh( gz ) | where theexponential gain factor is g = s(cid:18) k s k i (2 k p − k s )(2 k p − k i ) ω p k p ω i ω s (cid:19) α p − (cid:18) κ (cid:19) . (5)At nearly phase matching conditions κ ≈ g is posi-tive and real and the signal gain has an exponential de-pendence on the length of the transmission line G s = | e gz / | . For small pump amplitudes | A p | ∼ = | A s | and ω s ∼ = ω p phase matching is not a concern. For largerpump amplitudes the parametric amplification processlosses phase matching through self-phase modulation ofthe pump, when this happens g becomes small and κ is large, the gain scales quadratically as a function ofTWPA length.The phase mismatch due to the large pump amplitudecan be compensated with the linear dispersion along thetransmission line if α p and ∆ k are of opposite sign, κ = − α p ω p k p (cid:18) k s ω s + k i ω i − k p ω p (cid:19) − ∆ k. (6) III. NUMERICAL SIMULATIONS
From this point on numerical results are presented for arealizable set of parameters for the proposed TWPA. Foreach unit cell r = 6, C gnd = 50 fF, C js = 50 fF, C jl = rC js , I js = 1 µ A, and I jl = rI js . In choosing C gnd andthe inductance of the large junctions L jl = Φ / ( rI js )which ultimately sets the characteristic impedance of thetransmission line special attention was made to achievean impedance near 50 Ω over the tunable range of theTWPA in order to maintain compatibility with commer-cial electronics. A realizable unit cell size based on our -3 -3 p ( a - ) k p ( a - ) / FIG. 2. (color online) Pump tone wave vector and pump self-phase modulation per unit length a as a function of magneticflux Φ / Φ . The inset shows the change in sign of α p versusΦ / Φ . fabrication process is a = 8 µ m . The traveling wavesused in the numerical results are a pump tone with an-gular frequency ω p / (2 π ) = 6 . −
76 dBmwhich is equivalent to I prms ≈ . µ A , the signal angu-lar frequency ω s was varied in most cases and the idlerangular frequency is ω i = 2 ω p − ω s with initial signaland idler power levels 80 dB and 160 dB lower than thepump power respectively.Shown in Figure 2 is the dependence of k p and α p onΦ / Φ . From the inset in Figure 2 it can be seen that γ and as a result α p changes sign from positive to negativefor Φ > . and more importantly is of opposite signto ∆ k ≥ ω p and ω s it is possibleto utilize ∆ k which increases with Φ in the transmissionline to compensate the phase mismatch due to self-phasemodulation of the pump.Fig 3(a) shows numerical simulations of the signalgain as a function of signal frequency for the proposedTWPA with a transmission line length of 600 a . For amagnetic flux tuning of Φ / Φ = 0 .
45 and pump power −
76 dBm, Fig. 3(a) (red line) there are two regions ω s / (2 π ) = 3 . . κ = 0 can be achieved, and for comparison the phasemismatch dependence on signal frequency is shown inFig. 3(b) (red line). For κ ≈ g is real andlarge, the gain depends exponentially on the TWPAlength Fig. 3(c) (solid red line). When the phase mis-match is the largest at ω s / (2 π ) = 6 . g is smalland κ is large, the gain depends quadratically on thelength of the TWPA as shown in Fig. 3(c) (dashed redline). Under the phase matching conditions there existregions of exponential gain and quadratic gain dependingon signal frequency, the 3 dB bandwidth of the TWPA ∼ . κ ≈ S i gna l G a i n ( d B ) TWPA Length (a) (c) (b) S i gna l G a i n ( d B ) (a) ( a - ) Signal Frequency (GHz)
FIG. 3. (color online) Calculated gain of the proposed TWPA.In all three panels the color red and blue represent flux tun-ings (pump powers) of Φ / Φ = 0 .
45 ( −
76 dBm) and Φ / Φ =0 . −
73 dBm) respectively. (a) Signal gain in dB as a func-tion of signal frequency. (b) Phase mismatch as a function ofsignal frequency. (c) Dependence of the signal gain as a func-tion of transmission line length in units a . Solid red and bluelines correspond to ω s / (2 π ) = 9 . . ω s / (2 π ) =6 . κ ≈
0. According to equation (5) the most optimal gain doesnot necessarily occur at perfect phase matching κ = 0 dueto the pre-factor to α p . At significant κ the gain dependsquadratically on the length shown in panel (c) dashed linesand solid blue line. For the flux tuning of Φ / Φ = 0 . at ω s / (2 π ) = 3 . . / Φ = 0 . −
70 dBm, κ is large Fig. 3(b) (blue line) and only a quadratic gaindependence is possible. Since the signal gain increasesquadratically for all frequencies a relatively flat gain char-acteristic of the amplifier can be achieved with a sig-nal gain of 23 dB over a 3 dB bandwidth greater than5 . P B ( d B m ) / Leng t h ( a ) -140 -120 -100 -800510152025 S i gna l G a i n ( d B ) Signal Power (dBm)
FIG. 4. (color online) Calculated 1 dB compression pointand minimum TWPA length as a function of magnetic fluxΦ / Φ to maintain a signal gain of 20 dB. As Φ / Φ variesfrom 0 .
35 to 0 . −
68 dBmto −
76 dBm to maintain phase matching conditions betweena signal and pump tone at angular frequencies ω s / (2 π ) =6 GHz and ω p / (2 π ) = 9 GHz respectively. The inset showsthe signal gain as a function of signal power for Φ / Φ = 0 . a , the 1 dB compression point occurs at −
98 dBm. the coupled mode equations (see Appendix B) withoutthe un-depleted pump approximation and taking into ac-count self- and cross-phase modulation are solved to de-termine the real amplitude and phase mismatch as a func-tion of z along the length of the transmission line. Figure4 (inset) shows how the signal gain decreases with the sig-nal power due to pump depletion effects, for Φ / Φ = 0 . a . The signal gain andthe phase mismatch depend on magnetic flux through α p ∝ γk p | A p | and k m . For each Φ / Φ the pump poweris varied over the range −
76 dBm ( I prms ≈ . I jl ) to −
68 dBm ( I prms ≈ . I jl ) to maintain phase matchingat ω s / (2 π ) = 6 GHz with a ω p / (2 π ) = 9 GHz pump. Foreach Φ / Φ the minimum length of the TWPA to achievea signal gain of 20 dB is shown in Fig. 4 (red line). When α p and k m decrease with decreasing Φ / Φ a strongerpump is required to maintain phase matching conditionswhich results in a larger P dB up to −
98 dBm, smaller γ ,and a longer transmission line to maintain a signal gain of20 dB. When α p and k m increases with Φ / Φ a smallerpump power is required to achieve phase matching, P dB decreases, γ increases resulting in a shorter transmissionline with a minimum length of 211 a to achieve a signalgain of 20 dB. IV. SUMMARY
In conclusion a TWPA design based on a chain of cou-pled asymmetric SQUIDs has been presented. The pro-posed design allows for great flexibility where a magneticflux can be used to tune the nonlinearity of the trans-mission line to achieve phase matching conditions in afour-wave mixing process. Numerical simulations haveshown that the proposed amplifier can achieve gains of23 dB with a 3 dB bandwidth of greater than 5 . . −
98 dBm with 341 unitcells. The proposed amplifier is ideally suited for mul-tiplexed readout of quantum bits or kinetic inductancebased astronomical detectors.
ACKNOWLEDGMENTS
We would like to thank M. Gershenson for helpful dis-cussions. This work was supported in part by Solid StateScientific Corporation (US Army Small Business Tech-nology Transfer Program).
Appendix A: Analytical Approximation
In this section we derive an analytical solution to thecoupled nonlinear wave equations used to describe theinteraction between the signal and pump traveling wavespropagating along the transmission line composed of achain of coupled asymmetric SQUIDs. The energy phaserelation of a unit cell Figure 1(a) is E J ( ϕ ) = − E jl cos (cid:16) ϕ (cid:17) − E js cos (cid:16) ϕ − π ΦΦ (cid:17) − E js cos (cid:16) ϕ + 2 π ΦΦ (cid:17) , (A1)Expanding ∂E J ( ϕ ) /∂ϕ | ϕ =0 gives the approximate cur-rent phase relation describing the current I n ( ϕ ) flowingthrough the backbone of unit cell n Eq. (1). UtilizingKirchhoff’s current law I n − ( ϕ ) − C gnd d ϕ/dt = I n ( ϕ )and neglecting the effects of dissipation and assuminga sufficiently long wavelength λ ≫ a of the signal andpump the following wave equation is derived for a posi-tion z along the transmission line a L h r (cid:16) π ΦΦ (cid:17)i ∂ ϕ∂z + a C js (cid:16) r (cid:17) ∂ ϕ∂t ∂z − C gnd ∂ ϕ∂t − γ ∂∂z h(cid:16) ∂ϕ∂z (cid:17) i = 0 , (A2)where L = ϕ /I js , ϕ = Φ / π , r = I jl /I js , and γ = (cid:2) a / ( ϕ L ) (cid:3)(cid:2) ( r/
48) + (1 /
3) cos (cid:0) π Φ / Φ (cid:1)(cid:3) . The so-lution to Eq.(A2) is assumed to be a superposition ofa pump, signal, and idler traveling waves propagating along the transmission line of the form ϕ ( z, t ) = 12 h A p ( z ) e i ( k p z − ω p t ) + A s ( z ) e i ( k s z − ω s t ) + A i ( z ) e i ( k i z − ω i t ) + c.c i , (A3)where c.c. denotes complex conjugate, A m is the complexamplitudes, k m is the wave vectors, and ω m is the angularfrequencies of the pump, signal, and idler ( m = { p, s, i } ).A degenerate four-wave mixing process is considered un-der the following frequency matching condition ω s + ω i =2 ω p . Eq.(A3) is substituted into Eq.(A2) and assuming aslowly varying envelope of the propagating waves where | ∂ A m /∂z | ≪ | k m ∂A m /∂z | and | ∂A m /∂z | ≪ | k m A m | and a uniform transmission line where C gnd , C js and k m are constant, a set of coupled mode equations which de-scribes the propagation of the pump, signal, and idlerwaves along the transmission line is determined: ∂A p ∂z − iα p A p = 0 , (A4) ∂A s ∂z − iα s A s − i γk p k i k s (2 k p − k i ) A ∗ i A p ω s C gnd e − i ∆ kz = 0 , (A5) ∂A i ∂z − iα i A i − i γk p k s k i (2 k p − k s ) A ∗ s A p ω i C gnd e − i ∆ kz = 0 , (A6)where a large pump amplitude relative to the signal andidler amplitudes was assumed, decoupling the pump, andthe quadratic terms in A s,i were neglected, ∆ k = k s + k i − k p is the phase mismatch due to linear dispersion,and α m is the self-phase modulation per unit length a : α s = 3 γk s k p | A p | C gnd ω s ,α i = 3 γk i k p | A p | C gnd ω i ,α p = 3 γk p | A p | C gnd ω p , (A7)where | A p | is the initial pump amplitude. The lineardispersion relation for this transmission line is k m = ω m p LC gnd a q(cid:2) r + 2 cos (cid:0) π ΦΦ (cid:1)(cid:3) − ω m LC js (cid:0) r + 2 (cid:1) . (A8)Assuming an un-depleted pump amplitude and the fol-lowing substitutions A p ( z ) = A p e iα p z solution to Eq.(A4), A s ( z ) = a s ( z ) e iα s z , and A i ( z ) = a i ( z ) e iα i z into(A5) and (A6) to obtain: ∂a s ∂z − i γk p k i k s (2 k p − k i ) a ∗ i | A p | ω s C gnd e iκz = 0 , (A9) ∂a i ∂z − i γk p k s k i (2 k p − k s ) a ∗ s | A p | ω i C gnd e iκz = 0 , (A10)where κ = − ∆ k + 2 α p − α s − α i is the total phase mis-match. Equations (A9) and (A10) are similar to wellestablished fiber parametric amplifier theory and havethe following solution to describe the amplitude of thesignal along the length of the transmission line assumingzero initial idler amplitude: a s ( z ) = a s h cosh( gz ) − iκ g sinh( gz ) i e iκz/ (A11)A similar solution to (A11) exists for the idler amplitude.The exponential gain factor is g = s(cid:18) k s k i (2 k p − k s )(2 k p − k i ) ω p k p ω i ω s (cid:19) α p − (cid:18) κ (cid:19) . (A12)The signal power gain can be determined from Eq. A11 G s = (cid:12)(cid:12)(cid:12) cosh( gz ) − iκ g sinh( gz ) (cid:12)(cid:12)(cid:12) . (A13)For the proposed chain of asymmetric SQUIDs the phasemismatch due to linear dispersion is always real and non-negative for r > r . Using the frequency matching con-dition and Eq. A8 we show that∆ k = c (cid:18) ω s p − c ω s + ω i p − c ω i − ω p q − c ω p (cid:19) , where c = p LC gnd a q r + 2 cos(2 π ΦΦ ) and c = LC js (cid:0) r + 2 (cid:1)(cid:0) r + 2 cos (cid:0) π ΦΦ (cid:1)(cid:1) ,∆ k ≈ c c ∆ ω ω p ≥ , where ∆ ω ≡ ω s − ω p = ω p − ω i from the frequency match-ing condition. We also show that the sign of the first termin Eq.(6) is only dependent on α p for r > r , k m ω m = c ω m (cid:0) − c ω m (cid:1) ≈ c ω m (cid:16) c ω m (cid:17) (A14) k s ω s + k i ω i − k p ω p ≈ c ω p (cid:16) c ∆ ω + 3 c ω p (cid:17) > . Appendix B: Numerical Analysis
Below the solutions to the coupled mode equationswhich govern the degenerate four wave mixing processbetween the signal, idler, and pump tones and pump de-pletion effects are presented. Plugging Eq. A3 into Eq.A2 and making the slowly varying envelope approxima-tion the following coupled mode equations are derived: ∂A p ∂z − i γk p ω p C gnd h k p (cid:16) k p | A p | + 2 k s | A s | + 2 k i | A i | (cid:17) A p +2 k s k i (cid:16) k s + k i − k p (cid:17) A ∗ p A s A i e i ∆ kz i = 0 ,∂A s ∂z − i γk s ω s C gnd h k s (cid:16) k p | A p | + k s | A s | + 2 k i | A i | (cid:17) A s + k p k i (2 k p − k i ) k s A ∗ i A p e − i ∆ kz i = 0 ,∂A i ∂z − i γk i ω i C gnd h k i (cid:16) k p | A p | + 2 k s | A s | + k i | A i | (cid:17) A i + k p k s (2 k p − k s ) k i A ∗ s A p e − i ∆ kz i = 0 . The coupled complex differential equations are solvedby converting the complex amplitudes to A m = B m ( z ) e iθ m ( z ) and finding the solutions for the real ampli-tudes B p ( z ), B s ( z ), B i ( z ), and the total phase mismatch θ s ( z )+ θ i ( z ) − θ p ( z )+∆ k using the Runge-Kutta method. Appendix C: Parameter Variations
In this section we determine how variations in Joseph-son junction parameters and tuning magnetic flux gradi-ents effect the operation of the proposed TWPA. Bothtypes of variations manifest themselves as a position de-pendent alteration to both the linear and nonlinear dis-persion from unit cell to unit cell which ultimately effectsthe phase matching and the signal gain in the TWPA.Since the transmission line of the TWPA is made upof coupled asymmetric SQUIDs variations in one unitcell through mutual coupling, effects the superconduct-ing phase distribution in neighboring unit cells.We model the variations in Josephson junction param-eters as a normal distribution with mean critical currents I js and I jl for small and large Josephson junctions re-spectively, and a relative standard deviation σ JJ whichapplies to both small and large junctions. The proposedTWPA was modeled by minimizing the total Josephsonenergy of a 200 unit cell chain by adjusting the phases onall the large junctions. A gauge was chosen such that theexternal magnetic flux would induce phases 2 π Φ / Φ onthe small junctions. The constraint of the minimizationprocedure was that the sum of the phase drops on allof the large junctions would be equal to the total phaseacross the chain. From the minimization a position de-pendent ∆ k m and ∆ γ from unit cell to unit cell was de-termined, vectorized, and introduced into the numericalanalysis of the coupled mode equations. S i gna l G a i n ( d B ) JJ Standard Deviation JJ (%)r = 5f p = 9 GHz/ = 0.5 (a)(b) r = 6f p = 9 GHz S i gna l G a i n ( d B ) JJ Standard Deviation JJ (%)/ = 0.5/ = 0.45 FIG. A1. Simulated signal gain as a function of the relativestandard deviation of the critical currents of both the smalland large Josephson junctions which make up the TWPA.Simulations were performed at Φ / Φ = 0 .
45 (blue circles)and Φ / Φ = 0 . ω p / (2 π ) = 9 GHz, ω s / (2 π ) =6 GHz, TWPA length 200a, and r = 5 and r = 6 for Panel (a)and (b) respectively. The error bars represent the variationin signal gain due to the stochastic solution of the coupledmode equations to a normal distribution of junction parame-ters along the length of the TWPA. Numerical simulations of the TWPA which take intoaccount a normal distribution in Josephson junction pa-rameters were performed on the shortest 200 a TWPAwith a ω p / (2 π ) = 9 GHz and pump power of −
76 dBm.Shown in FIG. A1(a) and FIG. A1(b) is the signal gainat ω s / (2 π ) = 6 GHz) where κ ≈ σ JJ for different r andΦ. For each data point 50 numerical simulations wereperformed and the error bars represent the spread insignal gain due to the stochastic nature of the solu-tion to the coupled mode equations to a normal distri-bution in junction parameters. FIG. A1(a) shows for r = 5 and Φ / Φ = 0 . S i gna l G a i n ( d B ) m per TWPA unit cell JJ f p = 9 GHz / = 0.5/ = 0.45 FIG. A2. Simulated signal gain as a function of magneticflux gradient per unit cell across a TWPA composed of 200unit cells, for magnetic flux tunings of Φ / Φ = 0 .
45 (bluecircles) and Φ / Φ = 0 . σ JJ = 2 . p = 3 GHzf p = 5 GHz S i gna l G a i n ( d B ) Signal Frequency (GHz)f p = 9 GHzr = 6/ = 0.45 FIG. A3. Numerical simulation of the signal gain as afunction of signal frequency for pump frequencies ω p / (2 π ) =3 GHz (green line), 5 GHz (red line), and 9 GHz (blue line)respectively, pump power of −
76 dBm, r = 6, TWPA lengthof 600 a and Φ / Φ = 0 . to σ JJ = 1%, where the signal gain drops by 1 dB. FIG.A1(b) shows for r = 6 and Φ / Φ = 0 .
5, the tolerance in-creases to σ JJ = 3% which can be realized with presentday fabrication technology where the on-chip variationin junction parameters follows a normal distribution and σ = 2 . − . . When the magnetic flux is tunedto Φ / Φ = 0 .
45 the nonlinearity is decreased, however,the tolerated σ JJ increases to 4 . r or large Φ) and the tolerance of the TWPA to a variationin junction parameters.Numerical simulations of the signal gain as a func-tion of magnetic flux gradient per unit cell across theTWPA is shown in FIG. A2 for tuning magnetic fluxes ofΦ / Φ = 0 .
45 (blue circles) and Φ / Φ = 0 . σ JJ = 2 . / Φ = 0 . per unit cell. When the magnetic flux is tunedto Φ / Φ = 0 .
45 where the TWPA is more sensitive tomagnetic flux variations, the signal gain drops by 1 dBat 0 . per unit cell. In our measurement setup withmoderate magnetic shielding we have observed magnetic flux gradients of 24 µ Gauss /µ m which is equiva-lent to 0 . per unit cell of the TWPA. Appendix D: Different Pumping Conditions
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