Intermodulation Distortion in a Josephson Traveling Wave Parametric Amplier Ants Remm1Sebastian Krinner1Nathan Lacroix1Christoph Hellings1Francois Swiadek1Graham Norris1Christopher Eichler1and Andreas Wallra1 2

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Intermodulation Distortion in a Josephson Traveling Wave Parametric Ampliﬁer

Ants Remm,1Sebastian Krinner,1Nathan Lacroix,1Christoph Hellings,1Francois

Swiadek,1Graham Norris,1Christopher Eichler,1and Andreas Wallraﬀ1, 2

1Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland

2Quantum Center, ETH Zurich, CH-8093 Zurich, Switzerland

(Dated: October 11, 2022)

Josephson traveling wave parametric ampliﬁers enable the ampliﬁcation of weak microwave signals

close to the quantum limit with large bandwidth, which has a broad range of applications in super-

conducting quantum computing and in the operation of single-photon detectors. While the large

bandwidth allows for their use in frequency-multiplexed detection architectures, an increased num-

ber of readout tones per ampliﬁer puts more stringent requirements on the dynamic range to avoid

saturation. Here, we characterize the undesired mixing processes between the diﬀerent frequency-

multiplexed tones applied to a Josephson traveling wave parametric ampliﬁer, a phenomenon also

known as intermodulation distortion. The eﬀect becomes particularly signiﬁcant when the ampli-

ﬁer is operated close to its saturation power. Furthermore, we demonstrate that intermodulation

distortion can lead to signiﬁcant crosstalk and reduction of ﬁdelity for multiplexed readout of su-

perconducting qubits. We suggest using large detunings between the pump and signal frequencies

to mitigate crosstalk. Our work provides insights into the limitations of current Josephson traveling

wave parametric ampliﬁers and highlights the importance of performing further research on these

devices.

I. INTRODUCTION

Ampliﬁcation of weak microwave signals is essential for

many applications, including readout of superconduct-

ing qubits [1–4] and quantum dot devices [5–7], and ra-

dio astronomy [8, 9]. State-of-the-art low-noise ampliﬁers

in the microwave domain [10–15] approach the quantum

limit in noise performance [16] by operating at millikelvin

temperatures and using parametric pumping of a nonlin-

ear circuit made of Josephson junctions or high kinetic

inductance elements. While parametric ampliﬁers based

on nonlinear resonators have typical bandwidths on the

order of tens of MHz limited by the gain and the res-

onator linewidth [17, 18], traveling wave parametric am-

pliﬁers (TWPAs) [10, 14, 19–23] can have much higher

bandwidths of up to several GHz. The high bandwidth

enables a high degree of frequency-multiplexing, for ex-

ample for qubit readout [7, 24] and single-photon detec-

tors [9]. Multiplexed use of hardware resources [24–30] is

essential for the operation of large quantum devices.

So far material losses and the generation of signal side-

bands have been identiﬁed as the main sources of excess

noise in TWPAs above the quantum limit [14, 22, 31],

characterized by the intrinsic quantum eﬃciency. In ad-

dition to adding as little noise as possible at the signal fre-

quency, broadband ampliﬁers should not generate spuri-

ous tones due to intermodulation of the inputs [32, 33], in

particular when they are used in frequency-multiplexed

applications. However, due to ampliﬁer nonlinearities,

intermodulation distortion is unavoidable and constitutes

a well-known phenomenon in classical ampliﬁers [34]. In-

termodulation products can lead to crosstalk between

the ampliﬁed signals if any of their frequencies overlap

with one of the signals. The probability of such collisions

increases with increasing degree of frequency multiplex-

ing because the number of intermodulation products in-

creases.

In this work we characterize the intermodulation dis-

tortion of a resonantly phase-matched traveling wave

parametric ampliﬁer [14] (Section II). We identify inter-

modulation products of order up to ﬁve in the output

spectrum and characterize their power and frequency de-

pendence on the input signals. We then show that the

frequency collision of an intermodulation product with a

readout signal can lead to signiﬁcant crosstalk and reduc-

tion of readout ﬁdelity (Section III). Finally, we discuss

strategies to mitigate these errors by the choice of pump

and signal frequency and of power levels (Section IV).

We ﬁnd that intermodulation distortion, if not ac-

counted for, can lead to signiﬁcant crosstalk and read-

out errors in frequency-multiplexed architectures, which

can be detrimental for applications that rely on fast,

high-ﬁdelity readout, such as quantum error correc-

tion [35–38]. Therefore, continued development of cryo-

genic microwave ampliﬁers is needed to allow for fast low-

crosstalk multiplexed readout of many qubits.

II. INTERMODULATION DISTORTION

CHARACTERIZATION

The TWPA investigated in this study is a nonlin-

ear transmission line, formed by a series of Josephson

junctions and capacitors to ground, which is shunted to

ground by capacitively coupled LC resonators at 8.1 GHz

to achieve phase matching [14], see Fig. 1 (a). The non-

linearity of the Josephson junctions allows for four-and-

higher-wave mixing processes between the tones travers-

ing the TWPA. In the presence of a strong pump tone,

the mixing processes lead to desired phase-insensitive

ampliﬁcation of weak tones [ω1,ω2in Fig. 1 (b)], but also

to sideband generation by absorbing or emitting pump

arXiv:2210.04799v1 [quant-ph] 10 Oct 2022

FIG. 1. (a) The circuit of a resonantly phase-matched TWPA.

(b) Examples of tones propagating along the length of the

TWPA and their amplitudes (thickness of the lines, not to

scale) in the case of frequency-multiplexed readout at ω1

and ω2with the pump tone at ωp. In particular, a tone at

ω2+ωp−ω1is created, which is an example of the intermod-

ulation products studied in this work. (c) The measured gain

G(black) and signal-to-noise ratio improvement (red) spectra

of the device under study.

photons [31] and to the mixing of multiple signals [32],

known as intermodulation distortion. An example of in-

termodulation distortion is the creation of a tone (inter-

modulation product) at frequency ω2+ωp−ω1at the

output of the TWPA, as shown in Fig. 1 (b).

In our experimental setup the TWPA is mounted at

the base temperature stage of a dilution cryostat op-

erated at 10 mK. The signals, generated at room tem-

perature, are up-converted to microwave frequencies us-

ing an IQ-mixer. The output of the TWPA is am-

pliﬁed by a cryogenic high-electron mobility transistor

(HEMT) ampliﬁer and by room-temperature ampliﬁers.

Finally, the signals are down-converted using an IQ-mixer

with a local oscillator at 6.9 GHz and digitized, see Ap-

pendix A for details. The power −62 dBm and frequency

ωp/2π= 7.92 GHz of the pump tone are chosen to max-

imize the signal-to-noise ratio of signals applied between

6.7 GHz and 7.6 GHz used for qubit readout. The power

levels at the input of the TWPA are calculated based on

room-temperature measurements and the attenuation of

the components within the cryostat. We achieve a mean

gain of G= 18.4 dB and a signal-to-noise ratio rise of

13.0 dB relative to not pumping the TWPA, in which

case the noise is dominated by the HEMT ampliﬁer, see

Fig. 1 (c).

To observe the intermodulation products that we want

to characterize, we apply signal tones at frequencies ω1

and ω2to the input of the TWPA through a directional

coupler which is also used for applying the pump tone.

The signal power level at the input of the TWPA is ap-

proximately −102 dBm. We acquire 2.275 µs long time

traces and multiply them with a Blackman-Harris win-

FIG. 2. (a) Measured output power spectra when sweeping

the frequency of a single tone ω1in the presence of a second

signal tone ω2. Various intermodulation products are high-

lighted in color according to their signal order, see text for

details. (b) Line cut of the data in (a) at ω1indicated by the

dashed line. (c) Dependence of the power of the intermodula-

tion products Pon the applied signal power p1. The power of

the two signal tones is swept together. The gray lines indicate

the mean power level for products of signal order Os= 1, 2,

and 3 according to Eq. (1), and the arrows indicate the 1 dB

compression power p1dB and the intermodulation intercept

points pIP for Os= 2 and 3 tones.

dow [39] before taking the Fourier transform to avoid

windowing eﬀects. By ﬁxing ω2/2π= 7.1924 GHz and

sweeping ω1/2πfrom 6.9 GHz to 7.8 GHz, we record the

spectra shown in Fig. 2 (a). In addition to the signals

at ω1and ω2(green lines), we observe intermodulation

products at frequencies ω=npωp+Pi=1,2niωiwith inte-

gers npand ni. We can classify the intermodulation prod-

ucts according to their total order Ot=|np|+Pi|ni|.

Four-wave mixing processes lead to products of total or-

der Ot= 3 while cascaded four-wave mixing processes

and higher-order mixing processes can lead to products

with odd order Ot= 5 and higher. Of the intermodu-

lation products that lie within the acquisition band, we

observe all 17 products with total order Ot= 3 or 5 and

several products with Ot= 7 above the noise ﬂoor of

−160 dBm (at the output of the TWPA). Potential in-

termodulation products of even total order fall outside

the acquisition bandwidth in our setup. The presence

of all the intermodulation products implies that there

are no selection rules determining which intermodulation

products can be created, other than the parity of Ot.

While the allowed total orders are determined by the

order of the mixing process, the power level of the in-

termodulation product is mostly determined by the sig-

nal order Os=Pi|ni|, as we will see below. The in-

termodulation products of Os= 1, 2, 3, and ≥4 are

highlighted in Fig. 2 (a) in green, red, blue, and yellow,

respectively. Each intermodulation product also appears

mirrored around the local oscillator frequency due to im-

perfections of the frequency down-conversion process.

Next, we investigate the power of the intermodulation

products P. We ﬁx the frequency ω1/2π= 7.5551 GHz,

as indicated by the linecut in Fig. 2 (b), and vary the

input powers p1and p2= 0.5p1of the two signals, chosen

such that the output powers are similar. We record the

power Pof the intermodulation products of Os≤3, see

Fig. 2 (c), and ﬁnd that they follow power laws with the

signal order Osas the exponent as long as the ampliﬁer is

not saturated. Comparing the powers of diﬀerent Os= 3

products, we ﬁnd that they can be of similar magnitude

even for diﬀerent Otvalues. The observed power-laws

together with the data from independent power sweeps

of the two tones (see Appendix C) motivate an empirical

model of the output power

P=GpIP Y

ipi

pIP |ni|

.(1)

The model is parametrized by the mean gain Gand the

intermodulation distortion intercept point pIP [34], i.e.,

the input power level at which the extrapolated inter-

modulation product power equals the signal power. Us-

ing the average gain of the two signals G= 17.2(13) dB,

we calculate pIP for each intermodulation product ac-

cording to Eq. (1) at input power p1=−106 dBm, which

is signiﬁcantly below the saturation power. We ﬁnd a

mean second order intercept point pIP2 =−91(3) dBm

(for Os= 2) and a mean third order intercept point

pIP3 =−88(3) dBm (for Os= 3), see dashed gray lines in

Fig. 2 (c) calculated according to Eq. (1) and the mean

pIP values. The uncertainties indicate one standard devi-

ation of the spread over diﬀerent intermodulation prod-

ucts. The pIP values, visualized as the intercepts of the

gray dashed lines with the solid gray line (mean signal

power) in Fig. 2 (c), are close to the 1 dB compression

power p1dB =−96.7(23) dBm (see Appendix D). The

power diﬀerences between intermodulation products of

the same signal order might be due to diﬀerences in the

conversion rates or due to the frequency-dependence of

the gain.

We can use a simple model to describe the relation be-

tween the 1 dB compression power p1dB and the third or-

FIG. 3. (a) Change in the signal gain Gand noise power

Sas a function of applied power pΣand number of signal

tones N, compared to the single-signal (N= 1), low-power

(pΣ=−126 dBm) values. (b) Change in measurement ef-

ﬁciency ηrelative to the single-signal, low-power value ηref .

The intersection points of the gray lines indicate the 1 dB gain

loss in panel (a) and 1 dB eﬃciency loss in panel (b).

der intermodulation intercept power pIP3. In the lowest-

order series expansion that can explain four-wave mix-

ing, we write the output voltage of the ampliﬁer as

Vout =√GVin1−kV 2

in, for input voltage Vin and a co-

eﬃcient kwhich determines both saturation and inter-

modulation properties of the ampliﬁer. From this model,

we ﬁnd pIP3/p1dB = 9.6 dB [34], similar to the observed

pIP3/p1dB = 9(4) dB. We therefore expect that the in-

termodulation distortion intercept powers increase if the

1 dB compression power of the ampliﬁer is increased.

III. IMPLICATIONS FOR MULTIPLEXED

READOUT

We assess the impact of intermodulation distortion on

the performance of frequency-multiplexed qutrit readout

using the device presented in Krinner et al. [30]. Specif-

ically, we study how frequency multiplexing aﬀects the

signal-to-noise ratio, and how intermodulation products

can lead to crosstalk and increased readout errors.

To investigate the performance of the TWPA in the

presence of multiple input tones, we apply 31 dif-

ferent subsets of ﬁve frequency components ωi/2π=

{7.5551,7.1924,7.3725,6.979,6.76076}iGHz that could

be used for multiplexed readout of ﬁve qubits. We scale

the power of the all the applied tones by a common factor

and record time traces. We calculate the signal gain as

Gi=|hAii|2/piand noise as Si=|Ai|2−|hAii|2, where

Aiis the integrated amplitude of the timetrace, down-

converted from frequency ωi,piis the applied power,

and the averaging is done over 210 acquired time traces.

To ﬁnd the average signal gain and noise for a given de-

gree of multiplexing N, we average Giand Siover all

the frequency components iand subsets of Nfrequency

components that include that component i. We compare

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IntermodulationDistortioninaJosephsonTravelingWaveParametricAmplierAntsRemm,1SebastianKrinner,1NathanLacroix,1ChristophHellings,1FrancoisSwiadek,1GrahamNorris,1ChristopherEichler,1andAndreasWallra1,21DepartmentofPhysics,ETHZurich,CH-8093Zurich,Switzerland2QuantumCenter,ETHZurich,CH-8093Zurich,Swit...

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Intermodulation Distortion in a Josephson Traveling Wave Parametric Amplier Ants Remm1Sebastian Krinner1Nathan Lacroix1Christoph Hellings1Francois Swiadek1Graham Norris1Christopher Eichler1and Andreas Wallra1 2

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