High Saturation Power Josephson Parametric Amplifier with GHz Bandwidth
aa r X i v : . [ phy s i c s . i n s - d e t ] O c t High Saturation Power Josephson Parametric Amplifier with GHz Bandwidth
O. Naaman, a) D. G. Ferguson, and R. J. Epstein Northrop Grumman Systems Corp., Baltimore, Maryland 21240, USA (Dated: 22 November 2017)
We present design and simulation of a Josephson parametric amplifier with bandwidth exceeding 1.6 GHz, andwith high saturation power approaching -90 dBm at a gain of 22.8 dB. An improvement by a factor of roughly50 in bandwidth over the state of the art is achieved by using well-established impedance matching techniques.An improvement by a factor of roughly 100 in saturation power over the state of the art is achieved byimplementing the Josephson nonlinear element as an array of rf-SQUIDs with a total of 40 junctions. WRSpicesimulations of the circuit are in excellent agreement with the calculated gain and saturation characteristics.Josephson parametric amplifiers have been in extensiveuse over the past few years, providing quantum limitednoise performance at gains exceeding 20 dB, and enablinghigh fidelity qubit readout, squeezed microwave fieldgeneration, weak measurement, and feedback control. However, state-of-the-art devices of the eponymous JPA and Josephson Parametric Converter (JPC) types sufferfrom either narrow bandwidth ∼
10 MHz, or low satu-ration power ∼ −
110 dBm, or in many cases from both.Traveling-wave parametric amplifier architectures canachieve large bandwidths of several GHz at the cost ofhigh junction counts, typically exceeding 2000, with onlymodest improvement in saturation power. Here, we de-scribe a flux-pumped JPA-type three-wave mixing am-plifier with over 20 dB gain, in which we implement well-established impedance matching techniques to achieveover 1.6 GHz bandwidth, and a recently developed junc-tion array design to achieve high saturation power ap-proaching -90 dBm. Overall, this work represents 50-foldimprovement over the state of the art in bandwidth and100-fold improvement saturation power, in a circuit withless than 100 junctions. We compare the calculated am-plifier response to Spice simulations of the full nonlinearcircuit.Both JPA and JPC, and their variants, are built withresonant structures embedding Josephson junctions orSQUIDs, which serve as the nonlinear active elements inthe amplifier. Traditionally, their design has been drivenby the principle that the loaded quality factor of the res-onated nonlinearity must be relatively high in order toachieve high power gains. As a result, most Josephsonparametric amplifiers are extremely narrow-band, and aconsiderable effort has been directed into making theircenter frequency tunable to enable their practical usein the lab. This concept has been challenged recentlyby the work of Mutus et al. and Roy et al. , whohave shown that JPAs have been traditionally operatingfar from their maximum possible gain-bandwidth prod-uct, and that high gains and wide-band operation canbe achieved simultaneously by improving the amplifierimpedance match to the 50 Ω environment.The pumped nonlinearity in a parametric amplifier a) [email protected] presents the signal port with an effective negative re-sistance, giving rise to reflection gain. At the cen-ter of the amplifier band, the gain of a device withan effective signal resistance R sq < G / =( −| R sq | − Z ) / ( −| R sq | + Z ) = 1 / Γ p , where Γ p is thereflection coefficient of an identical circuit having a pas-sive, positive resistance load | R sq | . Therefore, the prob-lem of designing the gain of a parametric amplifier can bemapped onto the problem of impedance-matching a pas-sive load of equal magnitude. This also implies that thegain-bandwidth product of a parametric amplifier is onlylimited by the Bode-Fano theorem. If the pumped non-linearity in a JPA has an effective admittance Y sq ( ω ),then the bandwidth ∆ ω and the power gain G are relatedvia ∆ ω × ln G / ≤ − πω | Re { Y sq ( ω ) } | /Im { Y sq ( ω ) } ∼ πω L a / | R sq | , where L a is the linear inductance of thejunction (or SQUID) that depends on the flux bias, Φ dc ,and ω is the amplifier center frequency. For typical pa-rameters, a 7.5 GHz amplifier with 25 dB of gain, a linearinductance L a = 90 pH, and an effective negative resis-tance of R sq = 1 /Re { Y sq ( ω ) } = -10 Ω, could have amaximum bandwidth of about 3.5 GHz. With a three-pole physically realizable matching network, one can the-oretically achieve up to 60% of that bandwidth. In what follows, we will use the so-called pumpistormodel to obtain an expression for Y sq ( ω ), and design athree-pole bandpass network to match the effective loadto 50 Ω. Figure 1(a) shows the overall topology of theresulting circuit, whose gain characetristics are shown inFig. 2. We will outline a design procedure that is quitegeneral to Josephson amplifiers, however, we will concen-trate on a particular nonlinear element shown in Figure1(b) —a parallel arrangement of two rf-SQUID arrays, which we have previously demonstrated to be capable ofcarrying up to -53 dBm of power, and should thereforeallow amplification of up to -90 dBm input signals with20 dB of gain without saturation. The relatively highjunction I c in the array can support higher mode cur-rents than a typical JPA, and the low-inductance shuntof each of the junctions eliminates phase slips that arelikely to occur otherwise when the amplifier is biased andpumped.The design of the matching network follows Ref. 15.We choose to implement a three-pole coupled-resonatorbandpass network; larger number of poles gives only FIG. 1. (a) Schematic of the amplifier circuit including aband-pass matching network embedding the pumped Joseph-son nonlinearity, represented by Y sq . The signal port is shownon the right and labeled Z . Y ext is the admittance seenby the nonlinearity out through the matching network. (b)Schematic of the particular implementation of the active el-ement in our amplifier, built with two rf-SQUID arrays inparallel. Each array has N sections that contain a junctionwith critical current I c shunted by inductors L and L . Thepump is coupled inductively to the loop and the self induc-tance of the coupling transformer, L m is treated as a parasiticin our calculation. marginal improvement in bandwidth per pole. The setof filter prototype coefficients { g i } that describe the net-work, which were calculated specifically to accommodatenegative-resistance loads, are tabulated in Refs. 15 and20 for specified gain and ripple characteristics. Once thecenter frequency of the network (which we take to coin-cide with half the pump frequency ω = ω p /
2) is spec-ified, as well as its desired bandwidth, gain, and ripple,the only remaining free parameter in the design is thecapacitance shunting the Josephson element. All otherparameters, particularly the dc flux bias to the Joseph-son element and the amplitude of the pump tone, will beconstrained and can be calculated from the design equa-tions.The first pole of the network, resonator Z in Fig.1(a), is comprised of the Josephson element’s linear in-ductance L a (Φ dc ) in parallel with a shunt capacitance C . The choices for C and ω determine both the re-quired flux bias Φ dc operating point such that L a (Φ dc ) =1 /ω C , as well as the impedance of the resonator, Z = p L a (Φ dc ) /C . The other two poles of the net-work are built with passive LC resonators whose fre-quencies are ω , and impedances are Z = p L p /C and Z = p L p /C (the values of the resonator shunt ca-pacitances will be modified below from C ... to C p ... ).The resonators are coupled via admittance inverters J ij whose values are calculated from { g i } according to J = w/ √ Z Z g g , J = w/ √ Z Z g g , and J = p w/ ( Z Z g g ), where w is the fractional bandwith, Z = 50 Ω, and with the additional constraint w × | R sq | Z = g . (1)We choose the impedance of the passive resonator Z to satisfy Z = wZ /g g ; this allows us to eliminatethe last inverter J . The impedance Z can be chosenarbitrarily, and we set Z = √ Z Z . We can now imple-ment all admittance inverters as capacitive pi-sections with C ij = J ij /ω to obtain the circuit shown in Fig.1(a), where C p = C − C , C p = C − C − C and C p = C − C , and all component values are determinedby the above. The admittance seen by the nonlinear el-ement looking through the matching network out to the50 Ω environment, Y ext ( ω ) in Fig. 1(a), can be evaluatedat the center of the band: Y ext ( ω ) = jω C + (cid:18) J J (cid:19) Z . (2)The matching network design is now complete, andwithout assuming anything about the particular form ofthe Josephson nonlinearity, it is quite general and canbe used to broadband amplifiers based on dc-SQUIDs,Josephson dipole elements, and rf-SQUID arrays (Fig.1(b)) alike. However, we still need to find the optimalpump amplitude Φ ac and calculate the gain profile, whichrequire knowledge of the admittance Y sq . We find thisadmittance by use of the pumpistor model of Ref. 18 fora flux-pumped nonlinearity in 3-wave mixing operation,which we can write as Y sq ( ω s ) = 1 /jω s L a + 1 /jω s ( L b + L c ), where ω s is the signal frequency, L a = L T (Φ dc ) isthe linear inductance of the Josephson element at theoperating point, and L b = − L T (Φ dc ) L ′ T (Φ dc ) ac (3) L c = 4 iω i L T (Φ dc ) Y ∗ ext ( ω i ) L ′ T (Φ dc ) ac , (4)where L ′ T (Φ dc ) is the flux derivative of the inductanceevaluated at the operating point, and ω i = ω p − ω s is the idler frequency. If the amplifier is built using asimple dc-SQUID with a total critical current I c , then L T (Φ) = ¯ h/ eI c cos ( π Φ / Φ ). Our nonlinear element,shown in Fig. 1(b), is constructed from two arrays ofrf-SQUIDs, each of the array’s N stages composed of ajunction with critical current I c shunted by linear induc-tors L and L . In this case we have for the two arraysin parallel L T ( δ (Φ dc )) = N L + L ) L J + L L cos δ L J + (4 L + L ) cos δ , (5) L ′ T ( δ (Φ dc )) = (2 L + L ) L J π sin δ [( L + L ) L J + L L cos δ ] [ L J + (4 L + L ) cos δ ] , (6)where L J = ¯ h/ eI c , Φ is the flux quantum, and δ (Φ dc )is given implicitely by (cid:18) L + 1 L (cid:19) δ + 1 L J sin δ = π Φ dc N Φ (cid:18) L + 2 L (cid:19) . (7)Using Eqs. (2)-(7), we can find the pump amplitude Φ ac for which R sq = 1 /Re { Y sq ( ω ) } satisfies the constraintin Eq. (1) at the center of the band.We now have all circuit element values, the dc fluxoperating point and the optimal amplitude of the pump.To calculate the gain profile of the amplifier, we have toevaluate, at each signal frequency ω s , the admittance atthe idler frequency Y ext ( ω p − ω s ), and from it calculatethe admittance of the pumped nonlinearity Y sq ( ω s ). Wethen calculate the impedance of the whole amplifier asseen from the 50 Ω signal port to the right of Fig. 1(a), Z amp ( ω s ), and the gain in dB is given by G ( ω s ) = 20 × log (cid:12)(cid:12)(cid:12)(cid:12) Z amp ( ω s ) − Z Z amp ( ω s ) + Z (cid:12)(cid:12)(cid:12)(cid:12) . (8)As a concrete example, we design an amplifier with acenter frequency of 7.5 GHz. We target a design witha gain of 25 dB and with 0.5 dB gain ripple. From ta-bles in Refs. 15,20, we find the coefficients g = 0 . g = 0 . g = 0 . g = 0 . w = 0 .
25 for the fractional bandwidth parameter, and aninitial shunt capacitance of C = 5 . I c = 35 µ A, and the array inductors are L = 1 .
45 pHand L = 3 .
52 pH. The loop enclosing the two arrays iscoupled to the pump via a 10 pH transformer with a selfinductance of L m = 20 pH. Following the above proce-dure we find the dc flux operating point Φ dc = 0 .
26 Φ per junction (total of 10.38 Φ over the entire loop of2 N = 40 junctions), and the pump amplitude that gives R sq = − .
15 Ω at ω and satisfies Eq. (1) is found tobe Φ ac = 0 .
073 Φ per junction for a total of 2.93 Φ over the entire 40-junction loop. With reference to Fig.1(a), the coupling capacitors evaluate to C = 1 .
19 pFand C = 0 . C p = 4 .
01 pF, C p = C p = 85 fF. The passive res-onator inductances are L p = 254 pH, and L p = 769pH.The solid curve in Figure 2 shows the resulting gaincharacteristics of the amplifier as calculated using Eq.(8) and the pumpistor model with Eq. (3)-(7). We seethat the amplifier has a maximum gain of 22.8 dB, a 3dB bandwidth of 1.63 GHz, and gain ripple of 0.74 dB.We have additionally simulated the full nonlinear circuitin WRSpice in the time domain using transient analy-sis. We extracted the reflection coefficient of the circuitfrom the voltage waveform at the port of the amplifer by frequency (GHz) ga i n ( d B ) pumpistorspice FIG. 2. Amplifier gain as a function of signal frequency, ascalculated using the pumpistor model and the circuit param-eters given in the text (solid), and as simulated in WRSpice(circles). numerical I-Q demodulation, allowing us to separate thereflected waves from the incident signal. The results ofthese simulations are shown as circles in Fig. 2, and are inexcellent agreement with the calculated response. In thesimulation, we found that the optimal pump amplitudeand dc flux were both higher than estimated within thepumpistor model by roughly 20% and 30% respectively,which is not surprising as the pumpistor is a linearizedmodel while the Spice simulation captures the full non-linearity of the circuit.In Figure 3, we use the Spice simulation to character-ize the large-signal response of the circuit and estimateits 1 dB compression power, independent of any theoryfor the saturation mechanism. The simulated data areshown in the figure as circles for a signal frequency of 7.3GHz, and we obtain a saturation power of P sat = − . The loaded pump power P load available todrive the amplification process is P load = P av − P loss ,where P av is the available power from the pump port,and P loss = G load (cid:0) P in + w ¯ hω (cid:1) . Here the last term isthe input quantum noise power over the amplifier band, P in is the input signal power, and G load is the loadedgain evaluated with P load drive. The available and loadedpump powers relate to the respective pump amplitudes -120 -110 -100 -90 -80 P in (dBm) G a i n ( d B ) depletionspice FIG. 3. Loaded gain as a function of input signal power at ω s / π = 7 . Φ ac such that P av = ω p Φ ac,av / L T and similarly for theloaded amplitude. The solid curve in Fig. 3 is a plot of G load and is calculated by solving self consistently for theloaded gain,Φ ac,load = Φ ac,av − L T ω p G load (cid:0) P in + w ¯ hω (cid:1) . (9)The pump depletion model gives P sat = −
91 dBm, andis in reasonable agreement with the simulated data. Werecognize that other saturation mechanism exist, and while Fig. 3 is not sufficient evidence that pump de-pletion dominates, the figure does suggest that compet-ing mechanisms contribute up to the same order in thistype of amplifier.To summarize, we have shown how to design animpedance matching network to embed Josephson para-metric amplifiers and achieve broadband performance tomeet prescribed gain, bandwidth, and ripple specifica-tions. We additionally introduced a Josephson elementcomposed of rf-SQUID arrays to achieve high saturationpowers. Calculated gain and saturation curves agree wellwith simulated data that capture the full nonlinear cir-cuit. Similar techniques can be readily implemented tobroadband amplifiers based on the JPC architecture, butwill require separate designs for the spatially and spec-trally distinct signal and idler modes.We thank D. Dawson and A. Marakov for technicalassistance. R. Vijay, D. H. Slichter, and I. Siddiqi, “Observationof quantum jumps in a superconducting artificial atom,”Phys. Rev. Lett. , 110502 (2011). E. Jeffrey, D. Sank, J. Y. Mutus, T. C. White, J. Kelly,R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth,A. Megrant, P. J. J. O’Malley, C. Neill, P. Roushan,A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis,“Fast accurate state measurement with superconducting qubits,”Phys. Rev. Lett. , 190504 (2014). B. Abdo, K. Sliwa, S. Shankar, M. Hatridge, L. Frunzio,R. Schoelkopf, and M. Devoret, “Josephson directional am-plifier for quantum measurement of superconducting circuits,”Phys. Rev. Lett. , 167701 (2014). K. W. Murch, S. J. Weber, K. M. Beck, E. Ginossar, and I. Sid-diqi, “Reduction of the radiative decay of atomic coherence insqueezed vacuum,” Nature , 62 (2013). K. W. Murch, S. J. Weber, C. Macklin, and I. Siddiqi, “Observ-ing single quantum trajectories of a superconducting quantumbit,” Nature , 211 (2013). R. Vijay, C. Macklin, D. H. Slichter, S. J. Weber, K. W. Murch,R. Naik, A. N. Korotkov, and I. Siddiqi, “Stabilizing rabi os-cillations in a superconducting qubit using quantum feedback,”Nature , 77 (2012). J. Y. Mutus, T. C. White, E. Jeffrey, D. Sank, R. Barends,J. Bochmann, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth,J. Kelly, A. Megrant, C. Neill, P. J. J. O’Malley, P. Roushan,A. Vainsencher, J. Wenner, I. Siddiqi, R. Vijay, A. N.Cleland, and J. M. Martinis, “Design and characteriza-tion of a lumped element single-ended superconducting mi-crowave parametric amplifier with on-chip flux bias line,”Appl. Phys. Lett. , 122602 (2013). B. Abdo, J. M. Chavez-Garcia, M. Brink, G. Keefe,and J. M. Chow, “Time-multiplexed amplification in ahybrid-less and coil-less josephson parametric converter,”Appl. Phys. Lett. , 082601 (2017). C. Macklin, K. O’Brien, D. Hover, M. E. Schwartz,V. Bolkhovsky, X. Zhang, W. D. Oliver, and I. Siddiqi, “Anear–quantum-limited josephson traveling-wave parametric am-plifier,” Science , 307 (2015). T. C. White, J. Y. Mutus, I.-C. Hoi, R. Barends, B. Camp-bell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey,J. Kelly, A. Megrant, C. Neill, P. J. J. O’Malley, P. Roushan,D. Sank, A. Vainsencher, J. Wenner, S. Chaudhuri, J. Gao,and J. M. Martinis, “Traveling wave parametric amplifier withjosephson junctions using minimal resonator phase matching,”Appl. Phys. Lett. , 242601 (2015). O. Naaman, J. A. Strong, D. G. Ferguson, J. Egan, N. Bailey,and R. T. Hinkey, “Josephson junction microwave modulatorsfor qubit control,” J. Appl. Phys. , 073904 (2017). N. Roch, E. Flurin, F. Nguyen, P. Morfin, P. Campagne-Ibarcq,M. H. Devoret, and B. Huard, “Widely tunable, nondegeneratethree-wave mixing microwave device operating near the quantumlimit,” Phys. Rev. Lett. , 147701 (2012). J. Y. Mutus, T. C. White, R. Barends, Y. Chen, Z. Chen,B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant,C. Neill, P. J. J. O’Malley, P. Roushan, D. Sank, A. Vainsencher,J. Wenner, K. M. Sundqvist, A. N. Cleland, and J. M. Marti-nis, “Strong environmental coupling in a josephson parametricamplifier,” Appl. Phys. Lett. , 263513 (2014). T. Roy, S. Kundu, M. Chand, A. M. Vadiraj, A. Ranadive,N. Nehra, M. P. Patankar, J. Aumentado, A. A. Clerk,and R. Vijay, “Broadband parametric amplification withimpedance engineering: Beyond the gain-bandwidth product,”Appl. Phys. Lett. , 262601 (2015). G. Matthaei, L. Young, and E. M. T. Jones,
Microwave Filters,Impedance-Matching Networks, and Coupling Structures (ArtechHouse, 1980). R. M. Fano, “Theoretical limitations on the broadband matchingof arbitrary impedances,” Journal of the Franklin Institute ,57 (1950). E. Kuh and J. Patterson, “Design theory of optimum negative-resistance amplifiers,” Proceedings of the IRE , 1043 (1961). K. M. Sundqvist and P. Delsing, “Negative-resistance models forparametrically flux-pumped superconducting quantum interfer-ence devices,” EPJ Quantum Technology , 6 (2014). W. Zhang, W. Huang, M. Gershenson, and M. Bell, “Joseph-son metamaterial with a widely tunable positive/negative kerrconstant,” arXiv preprint arXiv:1707.06834 (2017). W. J. Getsinger, “Prototypes for usein broadbanding reflection amplifiers,”IEEE Transactions on Microwave Theory and Techniques , 486 (1963). D. Pozar,
Microwave Engineering (Wiley, 2004). N. E. Frattini, U. Vool, S. Shankar, A. Narla, K. M. Sliwa, and M. H. Devoret, “3-wave mixing josephson dipole element,”Appl. Phys. Lett. , 222603 (2017). S. R. Whiteley, “Josephson junctions in spice3,” IEEE Trans.Mag. , 2902 (1991). B. Abdo, A. Kamal, and M. Devoret, “Nondegener- ate three-wave mixing with the josephson ring modulator,”Phys. Rev. B , 014508 (2013).25