Optimal design for the Josephson mixerJ.-D. Pillet,1, 2 E. Flurin,1 F. Mallet,1, a) and B. Huard11)Laboratoire Pierre Aigrain, Ecole Normale Supérieure-PSL Research University, CNRS,Université Pierre et Marie Curie-Sorbonne Universités, Université Paris Diderot-Sorbonne Paris Cité,24 rue Lhomond, 75231 Paris Cedex 05, France2)Collège de France, 11 place Marcelin Berthelot, 75005 Paris, France

(Dated: 27 March 2015)

We present an optimal design in terms of gain, bandwidth and dynamical range for the Josephson mixer, thesuperconducting circuit performing three-wave mixing at microwave frequencies. In a compact all lumped-element based circuit with galvanically coupled ports, we demonstrate non degenerate amplification for mi-crowave signals over a bandwidth up to 50 MHz for a power gain of 20 dB. The quantum efficiency of themixer is shown to be about 70% and its dynamical range reaches 5 quanta per inverse dynamical bandwidth.

PACS numbers: 85.25.Cp, 84.30.Le, 84.40.Dc, 85.25.-j, 42.65.Hw, 42.50.Lc

Analog processing of microwave signals has recentlyentered the quantum regime owing to the developmentsof superconducting circuits. Quantum limited amplifiersthat are based on the non-linearity provided by Joseph-son junctions have been developed in various designs1–19.Non-degenerate three-wave mixing, a key operation, isrealized by the Josephson ring modulator (JRM), whichis a ring of four identical Josephson junctions20,21. Thiselement is at the core of several tools able to generateand manipulate quantum microwave modes such as phasepreserving amplifiers22–24, non-local entanglement gener-ators25, frequency converters26, quantum memories27 orcirculators28,29. In all previous implementations of theJosephson mixer, the JRM was embedded at the cross-ing of two distributed or lumped resonators, which putsa constraint on bandwidth and dynamical range that canbe detrimental to quantum operations. In this letter, wediscuss the origin of this constraint and how to optimizethe figures of merit of the Josephson mixer. These ideasare put in practice on an experiment in which a phasepreserving amplifier is solely built out of a JRM that isshunted with lumped plate capacitors. Compared to pre-vious implementations, we report an order of magnitudeincrease of its dynamical bandwidth, up to 50 MHz at apower gain of 20 dB while keeping the dynamical rangeas high as 5 photons per inverse bandwidth.

In a Josephson mixer, the JRM couples three inde-pendent fluxes ϕa, ϕb and ϕp (Fig. 1a), through the low-est order coupling term Hmix = −EJ sin(Φ/4ϕ0)ϕaϕbϕp,where EJ is the Josephson energy of each junction,ϕ0 = ~/2e the reduced flux quantum and Φ the fluxthreading the ring20,21. Three wave mixing occurs byembedding the ring in resonant circuits (Fig. 1b) so that

each flux can be expressed as ϕk ∝ (k̂ + k̂†) where k̂ isthe canonical annihilation operator of a microwave modeof characteristic impedance Zk and resonance frequencyfk. Although the Josephson mixer can be used in vari-ous ways20–22,24–28, we will focus here on the amplifica-

a)corresponding author: francois.mallet@lpa.ens.fr

tion regime in order to describe the figures of merit on aconcrete case. The signals sent towards a and b modesare amplified in reflection in a phase preserving mannerby driving the mode p out of resonance at the frequencyfa+fb

20. Three main specifications matter in analog pro-cessing of quantum microwave signals. First, the powergain G of the amplifier needs to be large enough so thatthe quantum noise at the input of the amplifier dominatesall other noise sources on the detection setup. Typically20 dB is enough if a cryogenic HEMT is used as a secondstage of amplification30. Second, the time correlations ofthe quantum signals should be dominated by the systemof interest and not by the Josephson mixer. This requiresto have as large a dynamical frequency bandwidth Γ aspossible. Finally, the maximum input power Pmaxin thatdoes not affect the gain by more than 1 dB needs to belarge enough to avoid any limitation on the amplitude ofthe quantum signals.

Optimizing these three parameters for a practical am-plifier has been at the center of recent experimentalworks in various geometries12–14,16. Recently JosephsonParametric Amplifiers have reached up to 20 dB gainover a 100 MHz bandwidth in an all lumped-elementbased design14,16 while 5 dB gain over 2 GHz band-width has been reported in a TiN traveling wave para-metric amplifiers13. The constraints on the parametersof the Josephson mixer are similar in origin to those ofits degenerate cousin, the Josephson Parametric Ampli-fier31, but with some differences21. First, there is anupper bound on the energy that is stored in the p pumpmode, originating from the small flux ϕp assumption inthe three-wave mixing term Hmix. Therefore, in orderto allow pump powers to reach the onset of parametricoscillations, at which large gain G develops, one has toensure that21

paQapbQb > Ξ, (1)

where Ξ is a number depending on the exact geometry ofthe mixer. Ξ = 8 will be used in the following32. In thisexpression, Qk is the quality factor of mode k, defined asQk = fk/Γk where Γk is the resonance bandwidth. Theparticipation ratio pk of the JRM Josephson junctions in

2

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FIG. 1. (a) Flux quadratures of the eigenmodes of theJosephson Ring Modulator. (b) Original mixer design wherethe JRM is placed at the crossing of two resonators of differ-ent frequencies. Two lines represent the spatial profiles of thevoltage in the λ/2 resonators. (c) Ideal design where everyinductive parts originate from the Josephson junctions. (d)Equivalent circuit of the differential modes a and b when thepump is turned off. A parasitic inductance Lex in series withthe JRM is considered. (e) The two shaded regions, corre-sponding to Lex = 1 nH and Lex = 50 pH, indicate the combi-nations of quality factor Q and Josephson junction inductanceLJ that verify the inequalities (1) and (2) for a power gainG = 20 dB, a saturating input power Pmaxin = −110 dBm andoperating frequency f = 7 GHz. For a given extra inductanceLex and saturating power P

maxin , there is a minimum quality

factor which limits the bandwidth of the Josephson mixer anda maximum inductance for the Josephson junction. f) Max-imal allowed value of the resonator extra inductance Lmaxexwithin the constraints (1) and (2) as a function of saturatingpower Pmaxin plotted for various quality factors Q.

the k mode quantifies the ratio of the total energy in thismode that is actually stored across the JRM junctions.For a serial coupling, it is given by pk = LJ/(LJ + L

kex)

where Lkex is the effective extra inductance of the res-onator defining the k mode and LJ = ϕ

20/EJ is the induc-

tance of a single junction of the JRM (Fig. 1d). Second,there is an upper bound on the power spectral densityof the amplified signals coming from the the small fluxϕa,b assumption in the three-wave mixing term Hmix.Indeed, neither the input signal power Pin, nor the vac-uum noise should be amplified beyond a fraction of theJosephson energy EJ . For large gain, this condition can

be approximated as

pkG (Pin/(2πΓk) + hf) Ξ′ < EJ , (2)

where Ξ′ is a number of order 1 and EJ is the Josephsonenergy EJ = ϕ

20/LJ . The two above constraints (1) and

(2) indicate that increasing both Pmaxin and Γ for a givengain G requires to increase the participation ratios pk andthe Josephson junction energy EJ . However, these twofigures are related since pk is a decreasing function of EJ .To minimize the detrimental influence of an increasingEJ on pk, it seems that the optimal design is when allthe inductive parts of the resonators originate from theJosephson junction themselves so that Lex = 0 (Fig. 1c).

It is enlightening to represent graphically these con-straints in the parameter phase spaces. In figure 1e,shaded areas delimitate, for two different values of Lex,the allowed values of the quality factor Q and Josephsoninductance LJ for a typical quantum limited amplifieroperating at a frequency f = 7 GHz with a 20 dB powergain, a saturating input power Pmaxin = −110 dBm. Notethat for the sake of clarity the two a and b modes havebeen set to identical parameters. As can be seen in thefigure, lower Q (larger bandwidth) can be obtained onlyby lowering Lex. Conversely, Fig. 1f shows the maximalallowed extra inductance Lmaxex as a function of saturatingpower Pmaxin for several desired quality factors Q. Fromthese curves, one can deduce which maximal extra induc-tance Lmaxex can be used for a given bandwidth. If Lexcomes from the geometrical inductance of some wires,their length is of the order of Lex/µ0, which is repre-sented on the right axis of Fig. 1f. From these considera-tions, one also determines the maximal spatial extensionof a Josephson mixer to ensure a given bandwidth.

Our implementation of the optimal design (Fig. 1c) ispresented in Fig. 2. The a and b mode resonators arecomposed of the JRM that is shunted by a cross of platecapacitors. The circuit is fabricated on a 500 µm thick Sichip covered with a 300 nm layer of SiO2 on top. In a firststep, a Ti(5nm)/Al(30nm) common counter electrode forall the plate capacitors is fabricated using standard e-beam lithography (dark yellow in Fig. 2b). It spreadsall over the surface underneath the rest of the circuit,except for a hole in the center and a thin stripe (brownin Fig. 2b) allowing to flux bias the circuit without theconstraints imposed by the Meissner effect. The wholechip is then covered with 200 nm of amorphous dielec-tric silicon nitride by plasma-enhanced chemical vapordeposition. Finally, the second metallic plate of the ca-pacitors (area 285 µm × 86 µm for the a resonator and140 µm × 86 µm for the b resonator) and the Joseph-son junctions (area 4.2 µm × 1 µm) are fabricated bydouble angle deposition of 100 nm and 120 nm of alu-minum with an intermediate oxydation step (Fig. 2c).The circuit is then placed in a copper box enclosed in aCryoperm magnetic shielding box anchored at base tem-perature of a dilution refrigerator (Tdil ' 50 mK). Two180o hybrid couplers address separately the differential aand b modes through their ∆ ports as well as the pump

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m5 μ

Capacitor Junction

Si3N4

SiO2

AlAl2O3

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FIG. 2. (a) Simplified schematic of the experimental setup.Differential a and b modes of the Josephson mixer are ad-dressed in reflection through two 180o hybrid couplers. Allinput lines are filtered and attenuated (partially shown). Out-put signals are separated from input signals by a directionalcoupler and amplified by a low noise HEMT amplifier at 4K.(b) Optical microscope picture of the device showing the pla-nar capacitors (right) and the Josephson junction ring (left).(c) Side view of the device. The thickness of the differentlayers is given in the main text.

mode c through one of the Σ ports. The resonators havecharacteristic impedances smaller than 10 Ω and are gal-vanically connected to the 50 Ω ports of the device so asto maximize bandwidth, only limited by impedance mis-match. Owing to the large coupling between the differen-tial modes and the input/output ports, the gain is sensi-tive to the frequency dependence of the impedance16. Inorder to probe the characteristics of the Josephson mixeralone without carefully engineering the impedance of theenvironment, we connect a 6 dB attenuator on the ∆ports of the hybrid couplers. A coil allows to control theflux threading the JRM loop to tune the mixing term inHmix via its current Icoil.

The effect of the flux bias can also be seen (Fig. 3a)on the resonance frequencies fa and fb which depend inan hysteretic manner of the flux with a period 4φ0

20.This hysteretic behavior could be removed by insertingadditional inductances in the JRM in order to extendthe static bandwidth of the amplifier24. However, thiscomes at the expense of lowering participation ratios,which we aim at maximizing, and becomes less usefulwith a large dynamical bandwidth. In this device, weobserve a frequency dependence on Icoil, which is notperfectly periodic. This observed non-linear dependenceof Φext on Icoil may originate form vortex dynamics inthe large superconducting capacitor plate that is buriedunder the silicon nitride. For each resonance frequency,it was possible to measure the quality factor (inset of

5.45 5.5 5.5505

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FIG. 3. (a) Measured resonant frequencies of resonators a andb as a function of increasing current Icoil in the coil generat-ing the flux bias Φext of the JRM when the pump is turnedoff. The insets show the measured quality factors Qk as afunction of resonance frequency. Dashed lines correspond tothe predicted Qk using the parameters given in the text. Thecorresponding bandwidths for resonators a and b at the fluxbias φ1 are respectively Γa = 365 MHz. and Γb = 200 MHz.(b) Gain in reflection at the flux bias φ1 indicated as a linein (a). The color bar encodes the pump power referred to theparametric oscillation threshold. The pump frequency is setto 12.26 GHz.

Fig. 3a). This dependence can lead to a quantitativemodel describing the Josephson mixer. In this detailedmodel, based on Fig. 1d, a stray inductance Lstray isconsidered in series with the capacitor Cres. Using firstfull 3D microwave simulations of the whole device, it waspossible to estimate the geometrical electrical parametersof Cares = 3 pF, L

astray +L

aex = 130 pH and C

bres = 6 pF,

Lbstray + Lbex = 85 pH. Then, by fitting LJ = 90 pH, one

gets f(fit)a = 6.95 GHz and f

(fit)b = 5.7 GHz, which are

close to the measured resonance frequencies at Φext = 0(Fig. 3a). From there, one can estimate the participationratios to be pa = 25% and pb = 35%. Note that simi-lar values for the participation ratios can be obtained byfitting directly the flux dependence of the resonance fre-quency (see supplementary material of Ref.27). Finally,one can fit the stray inductances to best recover the curveQ(f) and find Lastray = 75 pH, L

bstray = 51 pH.

The Josephson mixer can be used as an amplifier bysetting the flux φ1 slightly lower than 2φ0 = h/e and driv-ing the c pump mode at the frequency fp = 12.26 GHz,which is close to fa + fb

20. The gain was measured inreflection on both amplifier ports as a function of fre-quency (Fig. 3b) for various values of the pump powerPp at the fixed flux φ1. As the pump power rises towards

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FIG. 4. (a) Efficiency η of the amplifier as a function of thegain measured on port a. The inset shows a comparison be-tween experiments (orange points) and numerical simulation(continuous line) of the dependence of the resonant frequencyfa on input power Pin. This allows an in situ calibration ofthe input power at the level of the amplifier and thus deducethe attenuation of the input and output line connecting theamplifier to the instruments. The error bars on the efficiencycorresponds to an error of ±0.3 dB on this calibration. (b)Measured gain G on the a port as a function of input powerPin at 6.76 GHz for various pump powers (black line: pump isOFF, from yellow to red: increasing pump powers -8,-5,-4,-3,-2 dB). The dashed vertical line indicates the 1dB compressionpoint at 20 dB.

the parametric oscillation threshold, the gain increaseson both ports up to 30 dB. Conversely, the operatingbandwidth decreases. This curve demonstrates a band-width of 50 MHz at 20 dB, which is an order of magni-tude higher than using previous implementations of theJosephson mixer22–24.

The amplifier added noise is evaluated by amplifica-tion of zero point fluctuations in a and b modes. A spec-trum analyzer measures the spectral power density com-ing from aout and bout as a function of Pp. No signal issent into ain and bin. The difference between spectraldensities while the amplifier is ON and OFF is given by

SON − SOFF = GLNA × hfG(Sadd + 1), (3)

where G is the gain of the amplifier and GLNA is the totalgain of the output lines. Here the modes are assumed tobe in the vacuum state and G� 1. In this case the addednoise Sadd can be related to the quantum efficiency η ofthe mixer25 by Sadd = (1−η)/2η. Determination of η re-quires thus separate measurements of the amplifier gainG, as those of Fig. 3, and fine calibrations of the attenu-ations and gains of the input and output lines connecting

the mixer to the detectors. We obtain this calibration bymeasuring the shift in frequency as a function of powersent into ain and bin. It provides, by comparison withnumerical calculations of the circuit shown in Fig. 1 (d),a precise calibration of the attenuation of the input linebetween ain and the mixer. The gain GLNA between theJosephson mixer and aout is then deduced from the to-tal transmission between ain and aout, the pump beingturned off. Note that we also observe a clear frequencyshift in the gain measurements of Fig. 3 (b) while chang-ing pump power due to higher order cross-Kerr33 termsthat are proportional to Ppa

†a and Ppb†b. Figure 4(a)

presents the measured η as a function G. It indicates anefficiency of 0.7, up to 33 dB of gain above which theamplifier enters the parametric oscillation regime whereη is near 0.2.

The last important specification of an amplifier is itsdynamical range. It is characterized by the 1 dB com-pression point Pmaxin of the amplifier. In Josephson para-metric devices, such as the Josephson mixer, this sat-uration can be caused either by depletion of the pump(i.e. the gain is so large that the pump cannot refillquickly enough to feed the amplifier), or by reaching alarge enough number of photons such that higher ordernon-linearities cannot be neglected. Using the calibrationof the input power Pin on port b above, we used a vectornetwork analyzer to measure the output power Pout as afunction of input power Pin (Fig. 4b) at 6.76 GHz (cen-ter frequency when G = 20 dB) for various pump powersfollowing Ref.21. At low input powers Pin < −120 dBm,the gain goes from 0 to 25 dB for increasing pump powerPp without depending on Pin. For the pump power corre-sponding to G = 20 dB at low input power, the amplifierbehaves linearly for low power until it reaches the 1 dBcompression point at −112 dBm at the JRM input. At6.76 GHz and for a bandwidth of 50 MHz, this powercorresponds to 4.5 photons per bandwidth. Above thisthreshold the gain drops and finally saturates. This dy-namical range is large enough not only for performingqubit readout23 but also for amplifying vacuum squeezedstates25.

In conclusion, we have discussed an optimal and com-pact design for the Josephson mixer and applied theseprinciples to demonstrate phase preserving quantum lim-ited amplification. The resulting device operates withgains reaching 30 dB within 0.4 photons of the quan-tum limit of noise and a saturation power of −112 dBm,which is promising for analog information processing ofquantum signals, directional amplification and on-chipcirculators29. These specifications do not hinder the dy-namical bandwidth of the mixer, which reaches 50 MHzat G = 20 dB. Such device is suited for fast operationon superconducting qubit, which are necessary to theimprovement of the efficiency of quantum feedback36,37,multiplexing several qubits35 or more generally quantumerror correction schemes.

5

ACKNOWLEDGMENTS

We thank Michel Devoret for enlightening discussions.Nanofabrication has been made within the consortiumSalle Blanche Paris Centre. This work was supported bythe program ANR-12-JCJC-TIQS of Agence Nationalepour la Recherche. JDP acknowledges financial supportfrom Michel Devoret.

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