Resonantly phase-matched Josephson junction traveling wave parametric amplifier
RResonantly phase-matched Josephson junction traveling wave parametric amplifier
Kevin O’Brien, Chris Macklin, Irfan Siddiqi, and Xiang Zhang
1, 3, ∗ Nanoscale Science and Engineering Center, University of California, Berkeley, California 94720, USA Quantum Nanoelectronics Laboratory, Department of Physics,University of California, Berkeley, California 94720, USA Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
We develop a technique to overcome phase-mismatch in Josephson-junction traveling wave para-metric amplifiers in order to achieve high gain over a broad bandwidth. Using “resonant phasematching,” we design a compact superconducting device consisting of a transmission line with sub-wavelength resonant inclusions that simultaneously achieves a gain of 20 dB, an instantaneousbandwidth of 3 GHz, and a saturation power of -98 dBm. Such an amplifier is well-suited to cryo-genic broadband microwave measurements such as the multiplexed readout of quantum coherentcircuits based on superconducting, semiconducting, or nano-mechanical elements as well as tradi-tional astronomical detectors.
Josephson parametric amplifiers[1–5] routinely ap-proach quantum-noise-limited performance [6–9], and arecurrently used in sensitive experiments requiring high-fidelity detection of single-photon-level microwave sig-nals, such as the readout and feedback control of su-perconducting quantum bits [10–17], and magnetometrywith the promise of single-spin resolution [3]. To obtaina large parametric gain, the interaction time with thematerial nonlinearity—the order-unity nonlinear induc-tance of the Josephson junction—must be maximized.Current Josephson parametric devices increase the in-teraction time by coupling the junction to a resonantcavity albeit at the expense of instantaneous bandwidth.In contrast, traveling wave parametric amplifiers[18–21](TWPAs) achieve long interaction times by utilizinglong propagation lengths rather than employing multi-ple bounces in a cavity, thereby avoiding the inherentgain-bandwidth tradeoff associated with cavity based de-vices. A major challenge in the design of TWPAs, how-ever, is that optimum parametric gain is only achievedwhen the amplification process is phase matched. TW-PAs based on Josephson junctions have been investigatedtheoretically[22–25] and experimentally[26–28], but havenot demonstrated sufficient gain, in part due to phase-matching limitations, to replace existing semiconduc-tor amplifier technology. TWPAs based on the weakernonlinear kinetic inductance of thin titanium nitridewires and phase matched through periodic loading havealso been demonstrated[29, 30], but require significantlylonger propagation lengths and higher pump powers toachieve comparable gain. In this paper, we show thatby adding a resonant element into the transmission line,phase matching and exponential gain can be achievedover a broad bandwidth.The proposed traveling wave parametric amplifier con-sists of a Josephson-junction-loaded transmission line(Fig. 1a) with a capacitively-coupled parallel LC res-onator shunt to allow phase matching. The LC res-onator shunt (colored red in Fig. 1a,b) creates a stopband (Fig. 1c, red) in the otherwise approximately lin- ear dispersion relation (Fig. 1c, black dashed). In thepresence of a strong co-propagating pump wave, a weaksignal propagating in the TWPA is amplified through afour wave mixing interaction. Four wave mixing in theweak pump limit is perfectly phase-matched for a lineardispersion; however, a strong pump modifies the phasevelocities through self and cross phase modulation, gener-ating a phase mismatch and preventing exponential gain.We compensate this phase mismatch by tuning the pumpfrequency near the pole of the LC resonator. In a dissi-pationless system such as a superconducting circuit, aresonant element opens a stop band (inset of Fig. 1c), inwhich the wave vector is purely imaginary, surrounded byregions in which the wave vector is purely real and variesfrom 0 to π/a where a is the size of the unit cell. Thewavevector of the pump can be set to arbitrary values byvarying the frequency with respect to the resonance inorder to eliminate the phase mismatch.We now calculate the value of the phase mismatch andthe expected device performance when phase matchingis achieved. We use a first principles model for the non-linear dynamics in the Josephson junction transmissionline[24, 25] which has been validated by experiments[28].By making the ansatz that the solutions are travelingwaves, taking the slowly varying envelope approximation,and neglecting pump depletion, we obtain a set of cou-pled wave equations which describe the energy exchangebetween the pump, signal, and idler in the undepletedpump approximation (with the derivation in Eqs. 5-27in the Appendix): ∂a s ∂x − iκ s a ∗ i e i (∆ k L +2 α p − α s − α i ) x = 0 (1) ∂a i ∂x − iκ i a ∗ s e i (∆ k L +2 α p − α s − α i ) x = 0 (2)where a s and a i are the signal and idler amplitudes,∆ k L = 2 k p − k s − k i is the phase mismatch in the lowpump power limit, and the coupling factors α p , α s , and α i represent the change in the wave vector of the pump,signal, and idler due to self and cross phase modula- a r X i v : . [ c ond - m a t . s up r- c on ] J un FIG. 1. Resonantly phase-matched traveling wave parametricamplifier (a) Signal photons are amplified through a nonlin-ear interaction with a strong pump as they propagate alongthe 2000 unit cell transmission line with a lattice period of a =10 µ m. (b) In each unit cell a Josephson junction, a non-linear inductor, is capacitively coupled to an LC resonator.The circuit parameters are: C j =329 fF, L =100 pH, C =39 fF, C c =10 fF, C r =7.036 pF, L r =100 pH, I =3.29 µ A (c) The LC circuit opens a stop band (red) in the dispersion relationof the TWPA (black dashed) whose frequency depends on thecircuit parameters. In the inset, we plot the pump frequencyto phase-match a pump current of 0.3 I (blue), 0.5 I (purple),and 0.7 I (green), where I is the junction critical current tion induced by the pump. The coupling factors de-pend on the circuit parameters (see Eqs. 19, 20, and21 in the Appendix) and scale quadratically with thepump current. Maximum parametric gain is achievedwhen the exponential terms are constant: the phase mis-match, ∆ k = ∆ k L + 2 α p − α s − α i , must then be zero.The coupled wave equations (1), (2) are similar to thecoupled amplitude equations for an optical parametricamplifier[31] and have the solution: a s ( x ) = a s (0) (cid:18) cosh gx − i ∆ k g sinh gx (cid:19) e i ∆ kx/ (3)with the gain coefficient g = (cid:112) κ s κ ∗ i − (∆ k/ . Forzero initial idler amplitude and perfect phase matching,this leads to exponential gain, a s ( x ) ≈ a s (0) e gx /
2. Forpoor phase matching g is imaginary and the gain scalesquadratically with length rather than exponentially.Without resonant phase matching, the parametric am-plification is phase matched at zero pump power, but (cid:31) (cid:31) FIG. 2. Gain of the resonantly phase matched traveling waveparametric amplifier (RPM TWPA). (a) The gain as a func-tion of signal frequency in dB with RPM (purple) and without(black dashed) for a pump current of 0.5 I and a pump fre-quency of 5.97 GHz. (b) The phase mismatch with (purple)and without (black dashed) RPM. (c) The peak gain as afunction of pump current without RPM (black dashed) andwith RPM for three different pump frequencies, which phasematch the parametric amplification for pump currents of 0.3 I (red), 0.5 I (purple), and 0.7 I (green). (d) The phasemismatch as a function of pump current. The dots mark thepump current where the parametric amplification is perfectlyphase matched. rapidly loses phase matching as the pump power in-creases. Neglecting dispersion and frequency dependentimpedances, the exact expression for the phase mismatchcan be simplified to yield ∆ k ≈ k p − k s − k i − k p κ , where κ = a k p | Z char | L ω p (cid:16) I p I (cid:17) . The nonlinear process creates apump power dependent phase mismatch which can becompensated by increasing the pump wave vector.In Fig. 2, we show the increase in gain due to resonantphase matching for the device described in Fig 1. Reso-nant phase matching increases the gain by more than oneorder of magnitude from 10 dB to 21 dB (Fig. 2a) for apump current of half the junction critical current and apump frequency, 5.97 GHz, on the lower frequency tailof the resonance as shown in the inset of Fig. 1c. The in-crease in the pump wave vector due to the resonance com-pensates the phase mismatch from cross and self phasemodulation (Fig. 2b, black dashed) leading to perfectphase matching near the pump frequency (Fig. 2b, pur-ple). For higher pump currents, the benefits are evenmore pronounced: the RPM TWPA achieves 50 dB ofgain (compared to 15 dB for the TWPA) with a pumpcurrent of 0.7 I (Fig. 2c). Achieving 50 dB of gain overa 3 GHz bandwidth would require a larger junction crit-ical current than used here to prevent gain saturation byvacuum photons. By varying the pump frequency rela-tive to the resonance, the parametric amplification canbe phase matched for arbitrary pump currents (Fig. 2d).Due to this ability to tune the pump phase mismatchover the entire range of possible wavevectors, this tech-nique is highly flexible and can accommodate a varietyof pumping conditions.We now examine the scaling relations for the gain inorder to obtain the optimum gain through engineeringthe linear and nonlinear properties of the transmissionline. Simplifying the expression for the gain by assum-ing perfect phase matching and neglecting the effects ofthe resonant element and the junction resonance on thedispersion we find that the exponential gain coefficientis directly proportional to the wave vector g ∝ k p I p /I .Full expressions for the wave vector and characteristicimpedance are given by Eq. 31 and Eq. 32. Thus, for afixed pump strength relative to the junction critical cur-rent, the gain coefficient is proportional to the electricallength. In other words, a larger wave vector and thusslower light leads to a larger effective nonlinearity dueto the higher energy density; this effect is well known inphotonic crystals[32]. For convenient integration withcommercial electronics the characteristic impedance isdesigned to be Z char ≈ (cid:112) L/ ( C + C c ) ≈
50Ω which fixesthe ratio of the inductance and capacitance. The wavevector is proportional to the product of the capacitanceand inductance k ≈ ω/a (cid:112) L ( C + C c ). Increasing boththe capacitance and inductance or decreasing the unitcell size are effective strategies for increasing the gainper unit length while maintaining impedance matchingfor a 50 ohm load. The capacitors and inductors takea finite amount of space which constrains the minimumsize of the unit cell. The current design represents a tradeoff between unit cell size and component values which isconvenient to fabricate.Next we consider the dynamic range of the amplifier.The upper limit of the dynamic range of a parametric am-plifier is given by pump depletion: the pump transfers en-ergy to the signal and idler which reduces the parametricgain. To investigate this regime, we solve for the coupledwave equations without the undepleted pump approxi-mation, resulting in four coupled nonlinear differential (cid:31)(cid:30)(cid:29) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) Μ FIG. 3. Effect of pump depletion on dynamic range. (a) Thegain as a function of input signal current (normalized to thepump current) for a small signal gain of 10, 15, and 20 dBobtained with a pump current of 0 . I and device lengthsof 1150, 1530, and 1900 unit cells. The approximation forthe gain depletion (dashed lines) from Eq. 4 is in excellentagreement with the result obtained by solving the full nonlin-ear dynamics (solid lines). (b) P dB , the input signal powerwhere the gain decreases by 1 dB, as a function of junctioncritical current I with the pump current fixed at 0 . I . Theblack dashed line corresponds to the device considered in thisarticle which has a gain compression point of P dB =-87,-93,and -98 dBm for a gain of 10, 15, and 20 dB for a junctioncritical current of I = 3 . µ A. equations (Eqs. 47-50 in the Appendix). We solve theseequations by transforming them to real differential equa-tions and expressing them as a Jacobi elliptic integral[33].The gain as a function of input signal power calculatedfrom Eqs. 47-50 (solid lines in Fig. 3) is in excellentagreement with the approximate yet general solution forpump depletion (dashed lines in Fig. 3) in a four photonparametric amplifier[34]: G = G G I s /I p (4)where G is the small signal gain in linear units and I s and I p are the input signal and pump currents.From Eq. 4, the gain compression point is approximately P dB = P p / (2 G ). Thus, the threshold for gain satura-tion is independent of the specific device configurationand depends only on the small signal gain and the pumppower. For the considered device, the gain as a functionof input signal current is plotted for three values of thesmall signal gain in Fig. 3a. The signal current at whichthe gain drops by 1 dB is marked on the curves of Fig. 3a.For the considered device with a pump current of 1 . µA the signal power where the gain decreases by 1 dB is − −
93, and −
98 dBm for a small signal gain of 10, 15, and20 dB, respectively. These gain compression points areconsistent with the approximate relation with the pumppower of -69 dBm. The dynamic range of the TWPA issignificantly higher than a cavity based Josephson para-metric amplifier with the same junction critical currentsince the lack of a cavity enables a higher pump currentbefore the Josephson junction is saturated.To further increase the threshold for gain saturation,the junction critical current can be scaled up, as seen inFig. 3b. However, increasing the junction critical currentdecreases the inductance which reduces the wave vector;leading to a weaker nonlinearity. One potential solutionis to use N Josephson junctions each with a critical cur-rent I /N in series as a superinductor[35]. The 1 dBpower then scales as √ N leading to a larger dynamicrange at the expense of more complex fabrication.We now estimate the potential for pump distortion dueto third harmonic or higher harmonic generation. Thiswas suggested as a mechanism leading to self-steepeningand instabilities in parametric amplification[36, 37]. Wefind that the third harmonic is poorly phased matchedand less than 0 .
1% of the pump is converted to thethird harmonic. Due to the junction resonance at1 / √ LC J ≈ π · . a th = (1 − e ix ∆ k ) κ / ∆ k where ∆ k = ∆ k L + 3 κ − κ and∆ k L = 3 k p − k th . The couplings are defined in Eq. 39-42and are different from the couplings for parametric am-plification. The third harmonic amplitude oscillates withdevice length and pump power (Fig 5c,d in the Appendix)and has a maximum amplitude given by coupling κ / ∆ k .For a pump current of one half the junction critical cur-rent, the third harmonic power is 3 orders of magnitudeweaker than the pump power. As seen in the above anal-ysis, the third harmonic is too weak to cause significantpump depletion for the system under consideration.Resonant phase matching has an important advan-tage over dispersion engineering through periodic loadingwhich has been used to phase match TWPAs based on theweaker nonlinear kinetic inductance[29, 30]. A disadvan-tage of periodic loading is the potential phase matchingof backward parametric amplification. Dispersion engi-neering through periodic loading opens a photonic bandgap (Fig. 4) near the pump frequency and through bandbending changes the pump wave propagation constant tophase match forward parametric amplification. Periodicloading creates an effective momentum inversely propor-tional to the periodicity of the loading G = 2 π/ Λ where λ λ δ FIG. 4. Band structure of resonantly phase matched and pe-riodically loaded TWPAs. The parametric amplifier withouta (a, black only) resonant element, (a, black and red) with aresonant element, and (b) with periodic loading: every 37thunit cell has a slightly different capacitance and inductance(green). (c) The wave vector as a function of frequency forthe TWPA (black dashed), RPM TWPA (red), and TWPAwith periodic loading (green). The yellow shaded region in-dicates the photonic band gap due to the periodic loading.The main difference between the photonic band gap and theresonator is the edge of the Brillouin zone. For the resonantelement, the zone boundary is at π/a while the other period-ically loaded transmission line has zone boundary at π/ (37 a )which is determined by the period of the loading. The ef-fective momentum due to the periodic loading is close to k p ,which may phase match backward parametric amplification. Λ is the periodicity of the loading and G is the reciprocallattice vector. The periodicity is chosen so that the stopband is at G/ ≈ k p . In such a periodic system the phasematching relation needs only to be satisfied up to an inte-ger multiple of the reciprocal lattice vector[38]. As can beseen from the phase matching relation, the effective mo-mentum from the lattice phase-matches the parametricamplification process for a backward propagating signal∆ k L,b = 2 k p + k s + k i + nG ≈ n = −
2. Under thiscondition, any backward propagating photons present inthe system will be amplified, leading to gain ripples anda reduced threshold for parametric oscillations. Due toimperfect impedance matching over the operating bandof the amplifier, a weak standing wave condition will beset up in the nonlinear transmission line due to the re-turn loss at the output and input. If the return loss indB at the transition from nonlinear line to linear lineis R , then the magnitude of the standing signal will beof order 2 R . However, the signal experiences some gain(in dB) in the forward and reverse directions, G f and G r . If G f + G r + 2 R approaches unity, the device be-comes a parametric oscillator. The proposed resonantphase matching technique phase matches only the for-ward parametric amplification process, so the maximumgain before the onset of parametric oscillations may behigher than in a device utilizing periodic loading.In conclusion, we have developed a traveling waveparametric amplifier which is phase matched by sub-wavelength resonant elements and achieves 20 dB of gain,3 GHz of bandwidth, and a saturation power ( P dB ) of-98 dBm. This device is well suited to multiplexed read-out of quantum bits and astronomical detectors. Apply-ing metamaterial design techniques to nonlinear super-conducting systems may yield a number of useful devicesfor circuit quantum electrodynamics such as backwardparametric amplifiers or mirror-less optical parametricoscillators[39]. ACKNOWLEDGMENTS
The authors wish to acknowledge L. Friedland and O.Yaakobi for prior theory work and N. Antler for use-ful discussions. This research was supported in part bya Multidisciplinary University Research Initiative fromthe Air Force Office of Scientic Research (AFOSR MURIAward No. FA9550-12-1-0488), the Army Research Of-fice (ARO) under grant W911NF-14-1-0078 and the Of-fice of the Director of National Intelligence (ODNI), Intel-ligence Advanced Research Projects Activity (IARPA),through the Army Research Office. All statements offact, opinion or conclusions contained herein are those ofthe authors and should not be construed as representingthe official views or policies of IARPA, the ODNI or theUS government. ∗ [email protected][1] M. A. Castellanos-Beltran and K. W. Lehnert, Appl.Phys. Lett. , 083509 (2007).[2] N. Bergeal, F. Schackert, M. Metcalfe, R. Vijay, V. E.Manucharyan, L. Frunzio, D. E. Prober, R. J. Schoelkopf,S. M. Girvin, and M. H. Devoret, Nature (London) ,64 (2010).[3] M. Hatridge, R. Vijay, D. H. Slichter, J. Clarke, andI. Siddiqi, Phys. Rev. B , 134501 (2011).[4] N. Roch, E. Flurin, F. Nguyen, P. Morfin, P. Campagne-Ibarcq, M. H. Devoret, and B. Huard, Physical ReviewLetters , 147701 (2012).[5] C. Eichler, Y. Salathe, J. Mlynek, S. Schmidt, andA. Wallraff, arXiv:1404.4643 [quant-ph] (2014).[6] W. H. Louisell, A. Yariv, and A. E. Siegman, Phys. Rev. , 1646 (1961). [7] E. M. Levenson-Falk, R. Vijay, N. Antler, and I. Siddiqi,Supercond. Sci. Tech. , 055015 (2013).[8] M. A. Castellanos-Beltran, K. D. Irwin, G. C. Hilton,L. R. Vale, and K. W. Lehnert, Nat. Phys. , 929 (2008).[9] F. Mallet, M. A. Castellanos-Beltran, H. S. Ku,S. Glancy, E. Knill, K. D. Irwin, G. C. Hilton, L. R.Vale, and K. W. Lehnert, Phys. Rev. Lett. , 220502(2011).[10] K. W. Murch, S. J. Weber, C. Macklin, and I. Siddiqi,Nature (London) , 211 (2013).[11] R. Vijay, C. Macklin, D. H. Slichter, S. J. Weber, K. W.Murch, R. Naik, A. N. Korotkov, and I. Siddiqi, Nature , 77 (2012).[12] J. E. Johnson, C. Macklin, D. H. Slichter, R. Vijay, E. B.Weingarten, J. Clarke, and I. Siddiqi, Physical ReviewLetters , 050506 (2012).[13] R. Vijay, D. H. Slichter, and I. Siddiqi, Physical ReviewLetters , 110502 (2011).[14] M. Hatridge, S. Shankar, M. Mirrahimi, F. Schackert,K. Geerlings, T. Brecht, K. M. Sliwa, B. Abdo, L. Frun-zio, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret,Science , 178 (2013).[15] P. Campagne-Ibarcq, E. Flurin, N. Roch, D. Darson,P. Morfin, M. Mirrahimi, M. H. Devoret, F. Mallet, andB. Huard, Physical Review X , 021008 (2013).[16] D. Riste, M. Dukalski, C. A. Watson, G. de Lange,M. J. Tiggelman, Y. M. Blanter, K. W. Lehnert, R. N.Schouten, and L. DiCarlo, Nature , 350 (2013).[17] D. Rist`e, J. G. van Leeuwen, H. S. Ku, K. W. Lehnert,and L. DiCarlo, Physical Review Letters , 050507(2012).[18] A. L. Cullen, Nature (London) , 332 (1958).[19] A. L. Cullen, Institution of Electrical Engineers Papers ,101 (1959).[20] A. S¨orenssen, Appl. Sci. Res., Section B , 463 (1962).[21] P. K. Tien, J. Appl. Phys. , 1347 (1958).[22] M. J. Feldman, P. T. Parrish, and R. Y. Chiao, J. Appl.Phys. , 4031 (1975).[23] M. Sweeny and R. Mahler, IEEE Trans. Magn. , 654(1985).[24] O. Yaakobi, L. Friedland, C. Macklin, and I. Siddiqi,Phys. Rev. B , 144301 (2013).[25] O. Yaakobi, L. Friedland, C. Macklin, and I. Siddiqi,Phys. Rev. B , 219904 (2013).[26] S. Wahlsten, S. Rudner, and T. Claeson, Appl. Phys.Lett. , 298 (1977).[27] B. Yurke, M. L. Roukes, R. Movshovich, and A. N.Pargellis, Appl. Phys. Lett. , 3078 (1996).[28] C. Macklin et al. , In preparation.[29] B. Ho Eom, P. K. Day, H. G. LeDuc, and J. Zmuidzinas,Nat. Phys. , 623 (2012).[30] C. Bockstiegel, J. Gao, M. R. Vissers, M. Sandberg,S. Chaudhuri, A. Sanders, L. R. Vale, K. D. Irwin, andD. P. Pappas, J. Low Temp. Phys. , 1 (2013).[31] J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S.Pershan, Phys. Rev. , 1918 (1962).[32] M. Solja˘ci´c, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen,and J. D. Joannopoulos, J. Opt. Soc. Am. B , 2052(2002).[33] Y. Chen, J. Opt. Soc. Am. B , 1986 (1989).[34] P. Kylemark, H. Sunnerud, M. Karlsson, and P. A. An-drekson, J. Lightwave Technol. , 3471 (2006).[35] N. A. Masluk, I. M. Pop, A. Kamal, Z. K. Minev, andM. H. Devoret, Phys. Rev. Lett. , 137002 (2012). [36] R. Landauer, J. Appl. Phys. , 479 (1960).[37] R. Landauer, IBM J. Res. Dev. , 391 (1960).[38] N. Bloembergen and A. J. Sievers, Appl. Phys. Lett. ,483 (1970).[39] A. K. Popov, S. A. Myslivets, and V. M. Shalaev, Opt.Lett. , 1165 (2009). APPENDIX 1: PARAMETRIC AMPLIFICATION
Here we derive the coupled wave equations for a travel-ing wave parametric amplifier. The nonlinear wave equa-tion is (Eq. 22 from Ref. [24]): C ∂ φ∂t − a L ∂ φ∂x − C j a ∂ φ∂x ∂t = a I L ∂ φ∂x (cid:18) ∂φ∂x (cid:19) (5)We take the ansatz that the solutions will be forwardpropagating waves of the form: φ = 12 [ A p ( x ) e i ( k p x + ω p t ) + A s ( x ) e i ( k s x + ω s t ) + A i ( x ) e i ( k i x + ω i t ) + c.c ] (6)Where A m is the slowly varying amplitude, k m is thewave vector, and ω m is the angular frequency. We substi-tute the above expression into the nonlinear wave equa-tion then make the following approximations:1. Neglect the second derivatives of the slowly vary-ing amplitudes using the slowly varying envelopeapproximation: (cid:12)(cid:12)(cid:12) ∂ A m ∂x (cid:12)(cid:12)(cid:12) (cid:28) (cid:12)(cid:12) k m ∂A m ∂x (cid:12)(cid:12) .2. Neglect the first derivatives of the slowly vary-ing amplitudes on the right side of the nonlinearwave equation (ie, in the nonlinear polarizability): (cid:12)(cid:12) ∂A m ∂x (cid:12)(cid:12) (cid:28) | k m A m | .Considering only the left side of Eq. 5 and separating outthe terms that oscillate at the pump, signal, and idlerfrequencies we get the following equation: (cid:34) a e i ( tω m + k m x ) k m L − C e i ( tω m + k m x ) ω m − a C j e i ( tω m + k m x ) k m ω m (cid:35) A m ( x )+ (cid:34) ia C j e i ( tω m + k m x ) k m ω m − ia e i ( tω m + k m x ) k m L (cid:35) ∂A m ( x ) ∂x (7)where m = p, s, i . Defining the wave vector as k m = ω m √ C La √ − C j Lω m , Eq. 7 simplifies to: − iC ω m k m e i( tω m + k m x ) ∂A m ( x ) ∂x (8) Now we consider the nonlinear component (the right sideof Eq. 5). The propagation equation for the pump is: ∂A p ∂x − ia k p C I L ω p A p A ∗ p = 0 (9)where we have neglected the terms proportional to theamplitudes of the signal and idler as they are muchsmaller than the pump field. The propagation equa-tion for the signal and idler, neglecting terms which arequadratic in the signal and idler amplitudes: ∂A s ∂x − i a k p k s C I L ω s A p A ∗ p A s − i a k p (2 k p − k i ) k s k i C I L ω s A p A ∗ i e i∆ k L x = 0 (10) ∂A i ∂x − i a k p k i C I L ω i A p A ∗ p A i − i a k p (2 k p − k s ) k s k i C I L ω i A p A ∗ s e i∆ k L x = 0 (11)Now we solve for the pump propagation, assuming noloss, and obtain: A p ( x ) = A p, e i a k pApA ∗ p C I L ω p x (12)We substitute the solution for the pump field (Eq. 12)into Eqs. 10 and 11: A p ( x ) = A p, e iα p x (13) ∂A s ∂x − iα s A s − iκ s A ∗ i e i (∆ k L +2 α p ) x = 0 (14) ∂A i ∂x − iα i A i − iκ i A ∗ s e i (∆ k L +2 α p ) x = 0 (15)where the couplings are defined as: α s = 2 κk s a LC ω s κ s = κ (2 k p − k i ) k s k i a LC ω s (16) α i = 2 κk i a LC ω i κ i = κ (2 k p − k s ) k s k i a LC ω i (17) α p = κk p a LC ω p κ = a k p A p, A ∗ p, I L (18)To generalize these equations for arbitrary circuits, wemake the substitution C = 1 / ( iωZ ) and express thepump amplitude in terms of the characteristic impedanceand pump current: A p, = I p Z char /ω p . The couplingsare now: α s = 2 κk s a iZ ( ω s ) Lω s κ s = κ (2 k p − k i ) k s k i iZ ( ω s ) a Lω s (19) α i = 2 κk i a iZ ( ω i ) Lω i κ i = κ (2 k p − k s ) k s k i iZ ( ω i ) a Lω i (20) α p = κk p a iZ ( ω p ) Lω p κ = a k p | Z char | L ω p (cid:18) I p I (cid:19) (21)We solve the coupled amplitude equations (Eqs. 14 and15) by making the substitutions A s = a s e iα s x and A i = a i e iα i x to obtain: ∂a s ∂x − iκ s a ∗ i e i (∆ k L +2 α p − α s − α i ) x = 0 (22) ∂a i ∂x − iκ i a ∗ s e i (∆ k L +2 α p − α s − α i ) x = 0 (23)These equations are analogous to the coupled amplitudeequations for an optical parametric amplifier, which havethe following solution[31]: a s ( x ) = (cid:34) a s (0) (cid:18) cosh gx − i ∆ k g sinh gx (cid:19) + iκ s g a ∗ i (0) sinh gx (cid:35) e i ∆ kx/ (24) a i ( x ) = (cid:34) a i (0) (cid:18) cosh gx − i ∆ k g sinh gx (cid:19) + iκ i g a ∗ s (0) sinh gx (cid:35) e i ∆ kx/ (25)where ∆ k and g are defined as:∆ k = ∆ k L + 2 α p − α s − α i = 2 k p − k s − k i + 2 α p − α s − α i (26) g = (cid:113) κ s κ ∗ i − (∆ k/ (27) APPENDIX 2: LINEAR PROPERTIES
For the considered circuit topology (Fig. 1b) theABCD matrix is: (cid:18) − Z − /Z Z /Z (cid:19) where Z = Z L || Z C j = (cid:18) iωL + iωC j (cid:19) − (28) Z = Z C || Z res (29) Z res = Z C c + Z C r || Z L r = 1 − ( C c + C r ) L r ω iωC c (1 − C r L r ω ) (30)The wavevector and characteristic impedance in terms ofthe ABCD matrix elements are: k = cos − A + B Z char = ( A − D ) + (cid:112) ( A + D + 2)( A + D − C (32) APPENDIX 3: THIRD HARMONICGENERATION
The general procedure applies to third harmonic gen-eration as well. We make the undepleted pump approxi-mation which means that we assume the third harmonicis always much weaker than the pump. Depending onthe phase matching, this may not be the case in the ex-periment. In the undepleted pump approximation, thewave equations are: ∂A p ∂x − ia k p C I L ω p A p A ∗ p = 0 (33) ∂A th ∂x − i a k p k th C I L ω th A p A ∗ p A th + i a k p k th C I L ω th A p e i ∆ k L x = 0 (34)Using the same procedure of solving for the pump field A p ( x ) = A p, e i a k pApA ∗ p C I L ω p x (35) ∂A th ∂x − i a k p k th C I L ω th A p, A ∗ p, A th + i a k p k th C I L ω th A p, e i (cid:18) ∆ k L +3 a k pApA ∗ p C I L ω p (cid:19) x = 0 (36)and rewriting it in terms of the couplings A p ( x ) = A p, e iκ x (37) ∂A th ∂x − iκ A th + iκ e i(∆ k L +3 κ ) x = 0 (38) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:30) FIG. 5. Third harmonic. (a) The wave vector with the reso-nance (red), without the resonance (black dashed), and witha linearized dispersion (gold) equal to ωa (cid:112) L ( C + C c ) whichneglects the junction resonance. (b) The phase mismatch cor-responding to the dispersion relations in (a) for a pump of 5.97GHz. As expected, the linearized resonance is perfectly phasematched in the limit of zero pump power (gold). The more re-alistic dispersions with the resonance (red) and without (blackdashed) have a large phase mismatch which gets larger as thepump power increases. (c) The relative third harmonic powerincreases with the pump power, but is negligible with (red)and without (black) the resonance.(d) The third harmonicpower of the linearized resonance is significantly larger thanin the realistic system. with the couplings defined as: κ = a k p | Z char | L ω p (cid:18) I p I (cid:19) (39) κ = κk p iZ ( ω p ) a Lω p (40) κ = 2 κk th iZ ( ω th ) a Lω th (41) κ = κk p k th iZ ( ω th ) a Lω th (cid:18) | Z char | I p ω p I (cid:19) (42)We solve this making the substitutions a th = A th e iκ x and obtain: ∂a th ∂x + iκ e i(∆ k L +3 κ − κ ) x = 0 (43)which has the solution: a th = (1 − e ix (∆ k L +3 κ − κ ) ) κ ∆ k L + 3 κ − κ (44)= (1 − e ix ∆ k ) κ ∆ k (45)when the initial amplitude is zero , where ∆ k = ∆ k L +3 κ − κ and ∆ k L = 3 k p − k th . In Fig. 5 we plot the dispersion relation, phase mismatch, and the third har-monic power for the case with resonant phase matching,without, and with a fictitious linearized dispersion re-lation (equivalent to neglecting the junction resonance).The third harmonic is too weak to cause significant pumpdepletion for the system under consideration. APPENDIX 4: PARAMETRIC AMPLIFICATIONINCLUDING PUMP DEPLETION
We now consider the general solution including pumpdepletion effects and non-degenerate pumps. We nowlook for a traveling wave solution with two pump waves,a signal and an idler. φ = 12 [ A ( x ) e i ( k x + ω t ) + A ( x ) e i ( k x + ω t ) + A ( x ) e i ( k x + ω t ) + A ( x ) e i ( k x + ω t ) + c.c ] (46)where A and A are the pumps and A and A are theidler and signal. Plugging this ansatz into the nonlinearwave equation (Eq. 5) and making the slowly varyingenvelope approximation, we obtain the following coupledamplitude equations: dA dx − iκ A ∗ A A e − i ∆ kx − iA (cid:88) m =1 α m A m A ∗ m = 0(47) dA dx − iκ A ∗ A A e − i ∆ kx − iA (cid:88) m =1 α m A m A ∗ m = 0(48) dA dx − iκ A A A ∗ e i ∆ kx − iA (cid:88) m =1 α m A m A ∗ m = 0(49) dA dx − iκ A A A ∗ e i ∆ kx − iA (cid:88) m =1 α m A m A ∗ m = 0(50)where ∆ k L = k + k − k − k is the phase mismatch inthe weak field limit and the couplings are defined by: κ n = a k k k k ( k n − ε n ∆ k L )8 C I j L ω n (51) α nm = a k n k m (2 − δ nm )16 C I j L ω n (52)where the coupling constant κ n describes the four wavemixing process, α nm describes self and cross phase mod-ulation, ε = ε = 1 and ε = ε = − δ nmnm