IQ Mixer Calibration for Superconducting Circuits

IQ Mixer Calibration for Superconducting CircuitsS. W. Jolin,1, a) R. Borgani,1 M. O. Tholén,1, 2 D. Forchheimer,1, 2 and D. B. Haviland11)Nanostructure Physics, KTH Royal Institute of Technology, Stockholm, Sweden2)Intermodulation Products AB, Segersta, Sweden

(Dated: 18 August 2020)

An important device for modulation and frequency translation in the field of circuit quantum electrodynamics is theIQ mixer, an analog component for which calibration is necessary to achieve optimal performance. In this paper,we introduce techniques originally developed for wireless communication applications to calibrate upconversion anddownconversion mixers. A Kalman filter together with a controllable carrier frequency offset calibrates both mixerswithout removing them from the embedding measurement infrastructure. These techniques can be embedded into roomtemperature control electronics and they will find widespread use as circuit QED devices continue to grow in size andcomplexity.

I. INTRODUCTION

The superconducting qubit processors are continuouslygrowing in scale and complexity, placing increasing de-mands on the performance of room temperature electron-ics. An essential component is the IQ mixer. In circuitquantum electrodynamics (QED)1, IQ mixers are used to up-convert control signals from classical electronics and down-convert signals emitted from the quantum circuit. In wire-less radiofrequency (RF) and microwave systems, IQ mix-ers are particularly useful for Quadrature Amplitude Mod-ulation (QAM) schemes, commonly used for cost-efficientand high-data-rate systems2–6. Applications include point-to-point communication2 and various radar technologies7,8.

IQ mixers are analog components, which by their very na-ture require calibration. The error we calibrate for in thispaper is the so-called mixer imbalance. For upconversion,the imbalance manifests as the generation of a spurious sig-nal called the image, accompanying every desired signal. Fordownconversion imbalance prevents unambiguous separationof the signal and image sidebands in a heterodyne detectionscheme (or equivalently, a low-IF receiver setup). In eithercase, calibration of the IQ mixer allows for conversion to asingle sideband of the carrier frequency and suppression of itsimage, thus leading to more efficient use of frequency space.This efficiency becomes critical, for example, when imple-menting high-fidelity qubit gates9 for quantum computing andapplications where multiple qubits are controlled and mea-sured on the same signal line. Such frequency-domain mul-tiplexing of signals is also highly desirable in modern com-munication technology10.

Technological progress in direct synthesis and sampling ofRF signals is rapidly progressing11,12, but until this technol-ogy covers the entire frequency band used in circuit QED,IQ mixers are an unfortunate necessity. However we canuse various digital techniques to compensate for their im-perfections. This is occasionally referred to as the dirty RFparadigm, where imperfect, or dirty, RF devices are improvedthrough digital means13. In this paper, we combine ideas orig-inally developed for orthogonal frequency-division multiplex-

a)Electronic mail: shanjo@kth.se

0 20 40 60 80 100 120 140time (minutes)

80

75

70

65

60

55

ILR

(dB)

FIG. 1. The image leakage ratio (ILR) of an upconversion mixer asit changes over time, due to drifts in the mixer parameters. Similarphenomena will also occur at the downconversion stage. In eithercase, regular re-calibration is necessary for optimal performance.

ing (OFDM) applications to calibrate mixers in a circuit-QEDsetting.

The ideal calibration method should avoid any additionalexternal equipment such as RF sources and spectrum analyz-ers, and it should calibrate both mixers simultaneously. Inaddition it should also permit frequent re-calibration of themixers to compensate for drift, which can be substantial. Forexample, in Fig. 1 we plot the image leakage ratio (ILR), de-fined as the ratio of power at the image band to the power atthe signal band, vs. time. Initially the image is suppressed, butgradually increases over time by 20 dB. We attribute this driftto changes in the mixer’s temperature. This strongly suggeststhat regular re-calibration of mixers is necessary in order tomaintain optimal performance of superconducting qubit pro-cessors. For frequent re-calibration to be feasible, the bothmixers must be calibrated in situ. In this paper, we introducea novel technique that achieves these goals.

The paper is organized as follows: in section II we in-troduce the simplified IQ mixer model for upconversion anddownconversion and a model for mixer imbalance. In section

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IQ Mixer Calibration for Superconducting Circuits 2

+

−

cosΩ𝑡

𝑟(𝑡)

𝑧𝐼(𝑡)

𝑧𝑄(𝑡)

I port

Q port

RF port𝐺up sin(Ω𝑡 + 𝜙up)

LO port

𝑟(𝑡)

−𝐺down sin(Ω𝑡 + 𝜙down)

cosΩ𝑡

RF port

Q port

I port𝑧𝐼(𝑡)

𝑧𝑄(𝑡)

LO port

(b)

(a)

FIG. 2. a) A schematic upconversion circuit with phase φ and am-plitude G imbalances indicated in red font. b) A schematic downcon-version circuit with imbalances indicated in red font.

III we discuss the digital algorithms for calibration. In sectionIV we apply these techniques to a typical experimental circuit-QED setup with low-intermediate frequency (IF) transmissionand receiver stage. In addition, we introduce our method tocalibrate both upconversion and downconversion mixers si-multaneously. In the final section V, we conclude with a briefsummary.

II. MODELLING THE IQ MIXER

An ideal IQ mixer should behave as a multiplier of signalsin the time domain. One of the signals is often a pure tone athigh frequency, called a carrier wave, and the mixer is usedto convert or translate the frequency components of the othertone, either up or down in frequency. Starting with upconver-sion, mixing in the time domain can be described as6,14

r(t) = ℜz(t)e jΩt, (1)

where r(t) is the signal at the RF output port, Ω is the fre-quency of the carrier applied to the local-oscillator (LO) inputport and z(t) = zI(t) + jzQ(t) denotes the signals applied tothe I and Q input ports (see Fig. 2 (a) ).

In the frequency domain this description is well illustrated

in matrix form(c[ω]

c∗[−ω]

)=

12

(1 00 1

)(z[Ω+ω]z∗[Ω−ω]

), (2)

where r[ω] = c[ω]+c∗[−ω]. Upconversion mixing will there-fore create two sidebands at Ω±ω . In this paper, we refer tothe upper sideband Ω+ω as the signal and the lower sidebandΩ−ω as the image.

The image is generally undesirable as it crowds the limitedbandwidth, otherwise needed for multiplexing, with unwantedsignals. This frequency crowding is particularly detrimentalto applications in wireless local-area networks (WLAN)5,13,as well to the fidelity of frequency-multiplexed control andreadout of superconducting qubits15,16. The image can becancelled by injecting the appropriate signals into the I andQ ports of the IQ mixer, such that zI(t)+ jzQ(t) = x(t)e jωt ,where x(t) is real.

However, this cancellation is negatively affected by imper-fections in the mixer. We will characterize these imperfectionsas amplitude Gup and phase φup imbalance, stemming from anasymmetry in the I and Q branches of the mixer. The imbal-ance modifies Eq. (1) according to

r(t) = ℜz(t)[cos(Ωt)+ jGup sin

(Ωt +φup

)], (3)

thus Eq. (2) becomes(c[ω]

c∗[−ω]

)=

14

(J∗up KupK∗up Jup

)(z[Ω+ω]z∗[Ω−ω]

), (4)

where we define the complex quantities Jup = 1+Gupe− jφup

and Kup = 1−Gupe jφup . We regain the perfect mixer relationsby setting Gup = 1 and φup = 0. Due to the presence of the off-diagonal terms, injecting z(t) = x(t)e jωt will no longer cancelthe image completely. However, if the phase and amplitudeimbalances were known (or equivalently, Jup and Kup), an ap-propriate z(t) could be generated to cancel the image.

The ideal behavior of a downconversion mixer is very simi-lar to upconversion. In the time domain the output at the I andQ ports are given by10,17

z(t) = zI(t)+ jzQ(t) = LPFr(t)e− jΩt, (5)

where LPF denotes a low-pass filter(see Fig. 2 (b)). The out-put at the I and Q ports contain both signal and image, whichwe write as z(t) = Zsig(t)+Zimg(t). In the frequency domainwe have(

Zsig[ω]Z∗img[−ω]

)=

12

(1 00 1

)(Ysig[ω +Ω]

Y ∗img[−ω +Ω]

). (6)

where Ysig[ω +Ω] and Yimg[ω +Ω] is the signal and imagerespectively entering the RF input port.

Amplitude and phase imbalance are introduced in an anal-ogous manner. The time domain expression is modified ac-cording to

z(t) = LPFr(t) [cos(Ωt)− jGdown sin(Ωt +φdown)], (7)

IQ Mixer Calibration for Superconducting Circuits 3

and the analogous frequency-domain expression is10,13,17–19(Zsig[ω]

Z∗img[−ω]

)=

14

(Jdown KdownK∗down J∗down

)(Ysig[ω +Ω]

Y ∗img[−ω +Ω]

). (8)

The definitions for Jdown and Kdown are analogous to the up-conversion case. Since the matrix is no longer diagonal, im-balance will cause downconversion from both signal and im-age sidebands of the carrier.

Further simplification is made by decomposing the matrixin Eq. (8) into10(

Zsig[ω]Z∗img[−ω]

)=

12

(1 kqk∗q 1

)︸ ︷︷ ︸

leakage matrix

(Jdown 0

0 J∗down

)︸ ︷︷ ︸

scaling matrix

(Ysig[ω +Ω]

Y ∗img[−ω +Ω]

),

(9)

where kq is the ratio

kq =Kdown

J∗down=

1−Gdowne jφdown

1+Gdowne jφdown. (10)

III. CORRECTING THE IQ MIXER

In this section we take a look at how to calibrate the IQ mix-ers at both the upconversion and downconversion stage. Westart by discussing the upconversion mixer calibration in sub-section III A and then proceed to discuss the downconversioncalibration in subsection III B.

A. Upconversion stage

For upconversion, the signals zI and zQ are typically gen-erated with a digital source, allowing for fine tuning of theiramplitude and phase so as to cancel the image. We refer tothis process as digital pre-distortion. The quality of the can-cellation is measured by the image leakage ratio (ILR) definedas,

ILR =power at image frequencypower at signal frequency

. (11)

To correct for this drift one can introduce feedback to adjustthe amplitude and phase of the digital pre-distortion signals atthe I and Q ports, so as to continuously minimize the ILR. Theupconversion circuit schematic in Fig. 2 (a) is extended to in-clude the digital source, depicted in Fig. 3. The amplitudeand phase imbalances of the upconversion mixer are replacedby the alternative imbalance parameters α = Gup cosφup andβ = Gup sinφup. In the digital pre-distortion stage these pa-rameters are denoted as α and β . The ILR can then be ex-pressed as14

ILR(α, β ) =(α− α)2 +(β − β )2

(α + α)2 +(β − β )2, (12)

DAC

DAC

𝑧𝐼(𝑡)

𝑧𝑄(𝑡)

𝛼 sinΩ𝑡

𝛽

ො𝛼

መ𝛽

Digital pre-distortion IQ mixer for upconversion

+

−+

+

+

−

cos Ω𝑡

𝑟(𝑡)

FIG. 3. Schematic for the upconversion circuit. The amplitude andphase imbalances α and β are compensated by digitally distortingthe IF input as indicated in the figure. See Ref.14 for details.

revealing that when α = α and β = β we have optimal imagesuppression. In the absence of any digital pre-distortion, i.e.α = 1 and β = 0, the ILR then depends on the intrinsic imbal-ances of the mixer, which vanishes for the perfectly balancedmixer α = 1 and β = 0.

We can use semi-analytical methods6 or use various min-imization techniques such as the Nelder-Mead method20 toidentify parameters α and β . In this work we apply an itera-tive algorithm from Ref.14.

The cost function to minimize is the quantity:

C(αi, β j) = 4α2i ILRi, j (13)

where the subscripts i and j track the iteration steps and ILRi, jis the measured image leakage ratio with the applied pre-distortion parameters αi and β j.

The updated distortion parameter αi is calculated accordingto

αi =12

(αi−2 + αi−1−

C(αi−2, βi−2)−C(αi−1, βi−2)

αi−2− αi−1

),

(14)

while the updated βi is calculated according to

βi =12

(βi−2 + βi−1−

C(αi−1, βi−2)−C(αi−1, βi−2)

βi−2− βi−1

).

(15)

We initialize the algorithm by choosing α0, α1≈ 1, β0, β1≈0, and proceed as follows:

1. Distort signals by α0 and β0, measure C(α0, β0).

2. Distort signals by α1 and β0, measure C(α1, β0).

3. Distort signals by α1 and β1, measure C(α1, β1). Com-pute α2.

4. Distort signals by α2 and β1, measure C(α2, β1). Com-pute β2.

IQ Mixer Calibration for Superconducting Circuits 4

5. Distort signals by α2 and β2, measure C(α2, β2). Com-pute α3.

6. Distort signals by α3 and β2, measure C(α3, β2).Compute β3.

...

The iteration is continued until a desired threshold on theILRi, j is satisfied.

B. Downconversion stage

Given an observed signal Zsig[ω] and idler Z∗img[−ω] af-ter downconversion, we want to reconstruct the actual signalYsig[ω +Ω] and idler Y ∗img[−ω +Ω] before entering the imbal-anced IQ mixer. This is possible according to Eq. (9), if wemanage to determine kq and Jdown. For simplicity, the scalingmatrix is henceforth neglected since it does not contribute tothe leakage between sidebands. This leaves us with only onecomplex parameter kq.

Before proceeding, we simplify our notation by introducingYm =Ysig[ω +Ω] and Y ∗−m =Y ∗img[−ω +Ω]. The subscript−mdenotes the images while m denotes signals. Analogous nota-tion will also be used for Zsig[ω] and Z∗img[−ω]. The currentbest estimate of kq is denoted kq. With kq we reconstruct theundistorted signal and image according to(

YmY ∗−m

)=

(1 −kq−k∗q 1

)(Zm

Z∗−m

), (16)

neglecting the complex scaling factor.We use a method developed by Windisch and

Fettweis13,18,21 to estimate kq. If we assume 〈YmY−m〉= 0, it ispossible to estimate the mixer imbalance by first calculating:

kp =〈ZmZ−m〉⟨∣∣Zm +Z∗−m

∣∣2⟩ , (17)

where 〈...〉 indicates an average. Then kq is related to kp ac-cording to:

kq =1−a− jb1+a+ jb

, (18)

b =−2ℑ[kp], (19)

a =√

1−b2−4ℜ

kp, (20)

Consequentially, the amplitude and phase imbalances areGdown =

√a2 +b2 and φdown = arctan(b/a) respectively.

Equations (17) - (20) are the core of the downconversion cal-ibration procedure. Because the downconversion mixer alsodrifts in time (see Fig. 1), feedback is also needed to updatekq. The procedure is therefore enhanced with a Kalman filter.

Kalman filters are used to determine state vari-ables/parameters (stationary or dynamical) from noisyobservations. The filter is employed in such diverse

applications as navigation22 and multifrequency atomicforce-microscopy23. Other applications include coolingof nanoparticles through feedback24, and state estimationin atomic magnetometry25 and optomechanics26,27. For adetailed theoretical discussion, we refer the interested readerto Ref.10,21. To set up a discrete-time Kalman filter, we needa process equation and a measurement equation22.

The process equation relates the sought after state variablesat two different points in time. In this case, the state variableis kq and our process equation is

kq(i+1) = kq(i)+ vp(i). (21)

The argument i denotes ordered, finite intervals of time thatwe henceforth refer to as time frames. The IQ mixer statevariable kq(i) is assumed to evolve in time according to equa-tion (21), in which the process noise vp(i) is Gaussian withzero mean and variance σ2

p(i). We attribute the origin of thisnoise term to various fluctuations in the analog electronics andtemperature.

The measurement equation describes how our estimatedvalue of the state variable, which is determined from noisymeasurement data, is related to the actual or true value of thestate variable

kq(i) = kq(i)+ vm(i). (22)

The measurement noise vm(i) is Gaussian with zero mean andvariance σ2

m(i).The Kalman filter provides the estimated kq at different time

frames. The filtered estimate kq(i|Hi), with variance σ2m(i), is

the best estimate (in the least-square sense) of the state kq atthe current time frame i, based on a history of observationsHi up to and including frame i. The predicted estimate kq(i+1|Hi), with variance σ2

m(i+ 1, i), is the best estimate for thefuture frame i+1 based on the past history of observations.

The Kalman filter is initialized at i= 0 such that kq(1|H0) =

0 and its variance σ2m(1,0) = ∞. Other values of kq(1|H0)

with smaller σ2m(1,0) are possible (for example, if we have

values from a previous calibration run). For i > 1 we do thefollowing10:

1. Record n = 1,2, ...,N samples of Znk (i), Zn

−k(i) and re-construct Y n

k (i), Y n−k(i) using the predicted kq(i|Hi−1)

according to Eq. (16). Thereafter, estimate the vari-ances, σ2

k (i) and σ2−k(i) from the Y n

k (i) and Y n−k(i) data.

2. Simultaneously, use Znk (i), Zn

−k(i) to compute kq(i) ac-cording to Eq. (17). Here kq(i) is the current timeframe’s noisy estimate. The error variance of this quan-tity is approximately

σ2q (i)≈

[N

(1+

σ2k

σ2−k

)(1+

σ2−k

σ2k

)]−1

. (23)

3. Update the variance of the filtered estimate as

1σ2

m(i)=

1σ2

m(i, i−1)+

1σ2

q (i). (24)

IQ Mixer Calibration for Superconducting Circuits 5

0 20 40 60 80 100time frame

70

65

60

55

50

45

40

35

ILR

(dB)

FIG. 4. Reconstructed ILR through the downconversion mixer usinga Kalman filter as a function of time frames. We initialize the algo-rithm by assuming a perfect mixer Gdown = 1 and φdown = 0, withnear-zero variance. After 100 iterations, the ILR becomes limited bythe noise floor and we extract the filtered parameters Gdown = 0.961and φdown = 0.96.

4. Update the predicted estimate

kq(i+1|Hi) = σ2m(i)

(kq(i|Hi−1)

σ2m(i, i−1)

+kq(i)σ2

q (i)

). (25)

5. Compute the variance of the predicted estimate as

σ2m(i+1, i) = σ

2m(i)+σ

2p(i). (26)

6. Finally, update the filtered estimate

kq(i|Hi) = kq(i+1|Hi). (27)

Notice that the drift σ2p(i) is not updated. This needs to be

specified from the start and its magnitude (compared to σ2m(i))

will determine how little weight is given to past in contrast topresent observations.

We test the filter on a system consisting only of a downcon-version mixer driven by a pure RF tone corresponding to thesignal. The Kalman filter is initialized with the erroneous as-sumption that the mixer is ideal with near-zero variance. Theincoming RF signal is integrated for 4 ms and the Kalmanfilter algorithm is used to acquire an estimate of the filteredparameter kq(i|Hi). The image and signal sidebands are re-constructed according to Eq. (16), whose quality can be as-sessed by plotting the ILR at each iteration step, see Fig. 4.There is a steady decrease in the ILR until the noise floor isreached, indicating a successful calibration of the downcon-version mixer.

Spectrum analyzer

Vivace

PC

DUT

FIG. 5. Schematic setup for mixer calibration. The output RF signalis split with a Wilkinson power divider. One branch leads to a fullyequipped cryogenic low-noise microwave reflection setup inside adilution refrigerator. The other branch is monitored by a spectrumanalyzer, to prove our procedure is working as intended.

IV. COMPLETE CALIBRATION OF CONVERSION CHAIN

The upconversion and downconversion mixers are individ-ually straightforward to calibrate, as previously shown. How-ever, many experimental setups consists of both upconversionand downconversion stages, each with imbalances that needsto be corrected. Calibrating both mixers in situ introduces an-other problem not addressed so far: how does one go aboutdistinguishing the error of one mixer from another? We in-troduce and demonstrate two new approaches that resolvesthis issue and permits automated calibration of the entire fre-quency conversion chain.

These ideas are tested on a fairly typical experimental setupfor research in circuit QED, shown in Fig. 5. The digitalsource used to generate the I and Q signals is the Vivace plat-form from Intermodulation Products AB28. This same instru-ment also coherently demodulates the downconverted signals.With multiple output and input ports to drive and measure atthe I and Q ports of both up- and downconversion mixers,and with its ability to coherently modulate and demodulateup to 32 frequencies simultaneously, Vivace greatly simpli-fies our measurement setup. The output is upconverted witha IQ0318M mixer from Marki Microwave and split with apower divider into two branches. One branch is connectedto a spectrum analyzer. The spectrum analyzer is not requiredand its sole purpose is to act as an independent observer of thecalibration procedure. Meanwhile, the other branch is con-nected to a dilution refrigerator, wired with low-noise am-plifiers, attenuators, circulators and isolators. The signal isfinally downconverted with another IQ0318M mixer, beforebeing digitally demodulated from intermediate frequency (IF)to baseband with Vivace.

The ILR produced by the imbalanced upconversion mixer

IQ Mixer Calibration for Superconducting Circuits 6

Calibration complete

distort output with

Calculate

Update Update

update

calculate ILR measure signals

& images

Kalman ilter

Downconversion stage

Upconversion stage

RF signal

After several iterations of Kalman ilteringIf ILR < threshold

with Eq. 14 with Eq. 15

FIG. 6. Flow chart graphically summarises the calibration loop. The upconversion calibration alternately updates α and β using an estimatedvalue of C(n,m) obtained from the downconversion stage. A Kalman filter is used to calibrate the downconversion stage to ensure a correctvalue of C(n,m) is obtained.

can be measured by both the spectrum analyzer and the imbal-anced downconversion stage. If the downconversion calibra-tion works ideally, the ILR measured at the downconversionstage (henceforth referred to as the downconversion ILR) willbe identical to the ILR measured with the spectrum analyzer(referred to as the upconversion ILR).

The calibration loop is graphically summarized in Fig. 6.It can be divided into two sections, corresponding to the up-and downconversion stages. The loop principally follows thealgorithm in section III A. The new addition is the Kalmanfilter (see section III B) at the downconversion stage. TheKalman filter is run for several iterations (in our case, the num-ber of iterations ranged from 20-100) to calibrate the down-conversion mixer, which is necessary to acquire an accuratevalue for C(αi, β j) (see Eq. (13)). This in turn suggests thenext αi or βi through Eq. (14) and (15). However, the im-balanced upconversion-mixer will generate correlated signalsand images, violating the assumption 〈YmY−m〉 = 0 necessaryfor downconversion-mixer calibration.

We present two methods to circumvent this problem. Thefirst method involves calibrating with white noise at the sig-nal and image bands. While calibrating the downconversionmixer, we turn off the upconversion-mixer output, and samplenoise instead. For the sake of clarity, we explicitly list the firstfew steps of the procedure. First initialize by choosing α1 andα2 close to 1, β1 and β2 close to 0, then:

1. Turn off the RF upconversion-mixer output (by turningoff the IF signal).

2. Run the Kalman filter (according to section III B) bysampling incoming noise in the signal and image bands.This results in a best kq.

3. Turn on the RF upconversion-mixer output. Distort sig-nals by α1 and β1. Measure both signal and image side-bands at the downconversion stage.

4. Proceed to reconstruct the undistorted sidebands ac-cording to Eq. (16), using the recently acquired kq. Theratio of powers is the downconversion ILR. Use it tocalculate the quantity C(α1, β1), as defined in Eq. (13).

5. Turn off the RF upconversion-mixer output.

6. Run the Kalman filter (according to section III B) bysampling incoming noise in the signal and image bands.This results in a new best kq.

7. Turn on the RF upconversion-mixer output. Distort sig-nals by α2 and β1. Measure both signal and image side-bands at the downconversion stage.

8. Proceed to reconstruct the undistorted sidebandsaccording to equation (16), using the recently updatedkq. The ratio of powers is the downconversion ILR.Use it to calculate the quantity C(α2, β1).

...

Notice how we follow the algorithm given in section III A,with the addition of turning off the upconversion mixer outputand calibrating the downconversion mixer before every newestimate of C(n,m). The initialization of the Kalman filterwould be either a best guess or using the best estimate valuesfrom the previous Kalman filter loop.

The second method to separate the calibration of upconver-sion and downconversion relies on two different carrier tones,

IQ Mixer Calibration for Superconducting Circuits 7

RF input

Downconvertedoutput

With CFOWithout CFO

𝑓 𝑓

𝑓 𝑓2𝛿𝑓

𝛿𝑓

0 0

FIG. 7. Downconverting with an LO detuned by δ f , referred to asa carrier frequency offset (CFO), spectrally separates the two side-bands at downconversion by 2δ f . As a consequence, the downcon-version stage effectively measures two signal-image pairs. If we callthe red and blue arrows signals, then their respective images containonly noise. Each signal-image pair is therefore uncorrelated and thedownconversion calibration is valid.

detuned by a carrier frequency offset (CFO). Referring to Fig.7, a CFO will prevent the incoming signal (blue) and image(red) sidebands from being superimposed upon downconver-sion. Thus from the perspective of the downconversion stage,the blue and red arrows are two incoming signals, with theirrespective image bands containing only noise. The Kalmanfilter is run for each pair separately (essentially two filtersrunning in parallel) and generates two values kred

q and kblueq .

The red and blue signals are distorted by the imbalances inthe downconversion mixer, and are corrected by inserting kred

q

and kblueq into Eq. (16)(

YredY ∗−,red

)=

(1 −kq,red

−k∗q,red 1

)(Zred

Z∗−,red

), (28)(

YblueY ∗−,blue

)=

(1 −kq,blue

−k∗q,blue 1

)(Zblue

Z∗−,blue

). (29)

The ILR used to calibrate the upconversion mixer is then cal-culated as ILR = |Yred|/|Yblue|. Using a CFO, a proper estima-tion of the ILR necessitates measuring four frequency bands(two signal-image pairs), instead of the two (one signal-imagepair). The Vivace platform is well suited to this task as itsmultifrequency lock-in can measure up to 32 bands. It isalso worth noting that kred

q and kblueq are frequency dependent

and the relative difference is expected to vary as a function ofCFO. In our case, we used a CFO of 20kHz resulting in typi-cal magnitude ratios of |kred

q |/|kblueq | ≈ 1.007 and phase ratios

of ∠kredq /∠kblue

q ≈ 0.988.Fig. 8 plots the upconversion ILR, as measured by the spec-

trum analyzer, vs. the downconversion ILR, as measured byVivace. Successive iterations of the calibration are plotted aspoints, connected with dashed lines, starting in the upper rightwith large ILR, and proceeding toward the lower left with re-duced ILR. The colors represent different calibration scenar-

80 60 40 20 0 20 40Downconversion ILR (dB)

80

60

40

20

0

Upco

nver

sion

ILR

(dB)

idealwith noise, 2s integration timewith noise, 10s integration timewith CFOwithout CFO

FIG. 8. Comparing the upconversion and downconversion ILR.Ideally the two quantities should be equal, as indicated by the blackdashed line. Major deviations are due to faulty calibration. Smalldeviations are attributed to frequency dependent attenuation in thecryogenic electronics. Differences between the various methods areexplained in the main text.

0 24 48 72 96 120time (s)

70

60

50

40

30

20

ILR

(dB)

downconversion ILRupconversion ILR

FIG. 9. Calibration with CFO as a function of iteration number.Calibration time is limited by the spectrum analyzer used to monitorthe upconversion ILR. If we remove this monitor, -70 dB ILR canbe reached within a couple of seconds. The imbalance parametersare α = 0.923 and β =−0.0327, corresponding to Gup = 0.961 andφup =−2.03.

ios. The dashed diagonal line represents the ideal case. Noticethat attempts to calibrate both up- and downconversion simul-taneously while 〈YmY−m〉 6= 0 (red), fail to produce the idealresult, as expected.

Two calibrations using only noise (blue and orange) are de-picted, the difference being the amount of time spent samplingnoise (2s and 10s respectively). The accuracy is decent forlarge ILR, but suffers as we approach -40 dB.

The calibration improves significantly if we use a signal and

IQ Mixer Calibration for Superconducting Circuits 8

a CFO (green). We can accurately reconstruct the ILR downto -70 dB. Notice the offset of roughly 3 dB from the diagonal,corresponding to an overestimate of the ILR at the downcon-version stage. We attribute this to an unequal attenuation ofthe sidebands in the cryogenic electronics.

Finally, Fig. 9 depicts calibration with CFO of two imbal-anced mixers as a function of time. Each point correspondsto a complete loop around the chart in Fig. 6. The down-conversion ILR is successfully suppressed to less than -60 dBwith our setup, showing that we can calibrate both up- anddownconversion mixers simultaneously. The offset from theupconversion ILR is attributed to frequency-dependent atten-uation in the cryogenic electronics.

V. CONCLUSION

Current IQ-mixer components for circuit QED experimentsare analog devices that, due to their analog nature, come withimperfections and imbalances. These imbalances can be mit-igated by digital pre-distortion techniques. In this paper, wehave presented several such techniques to calibrate IQ mixers,both for up- and downconversion. These techniques, origi-nally developed for OFDM applications, are shown to be suit-able for circuit-QED applications. By careful combination ofup- and downconversion techniques, it is possible to calibrateboth stages simultaneously. In addition, we show that it is pos-sible to calibrate the mixers in situ, provided the frequency-dependent attenuation is not too large. We believe this willpave the way for more efficient mixer-calibration proceduresas superconducting-qubit processors continue to increase insize and complexity in the near future.

ACKNOWLEDGMENTS

We acknowledge the financial support from the Knut andAlice Wallenberg Foundation through the Wallenberg Centerfor Quantum Technology (WACQT). We also acknowledgePer Delsing and Simone Gasparinetti for helpful comments.

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