Circuit quantum electrodynamic model of dissipative-dispersive Josephson traveling-wave parametric ampliers Yongjie Yuan Michael HaiderJohannes A. Russer Peter Russer and Christian Jirauschek

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Circuit quantum electrodynamic model of dissipative-dispersive

Josephson traveling-wave parametric ampliﬁers

Yongjie Yuan, Michael Haider,∗Johannes A. Russer, Peter Russer, and Christian Jirauschek

TUM School of Computation, Information and Technology,

Technical University of Munich, Hans-Piloty-Str. 1, 85748 Garching, Germany

(Dated: October 19, 2022)

We present a quantum mechanical model for a four-wave mixing Josephson traveling-wave para-

metric ampliﬁer including substrate losses and associated thermal ﬂuctuations. Under the assump-

tion of a strong undepleted classical pump tone, we derive an analytic solution for the bosonic

annihilation operator of the weak signal photon ﬁeld using temporal equations of motion in a ref-

erence timeframe, including chromatic dispersion. From this result, we can predict the asymmetric

gain spectrum of a Josephson traveling-wave parametric ampliﬁer due to non-zero substrate losses.

We also predict the equivalent added input noise including quantum ﬂuctuations as well as thermal

noise contributions. Our results are in excellent agreement with recently published experimental

data.

I. INTRODUCTION

In conventional low-noise microwave ampliﬁers, am-

pliﬁcation is achieved by modulating the channel of a

high-electron-mobility transistor (HEMT). A character-

istic measure for the ampliﬁer’s performance is the noise

temperature, which is deﬁned as the equivalent temper-

ature of a resistor that would produce the same level of

Johnson-Nyquist noise [1, 2]. The added noise power de-

pends on the channel resistance and easily exceeds the

energy of a few tens of microwave photons, even when

the circuit is cooled down to cryogenic temperatures [3].

In superconducting quantum computing where the qubit

state is probed by ultra-low-power microwave signals,

however, quantum-limited [4] noise performance is key

for high-ﬁdelity single-shot readouts of quantum infor-

mation [5]. This and the limited cooling power budget in

the lowest temperature stage of dilution refrigerators ren-

der the use of traditional solid-state ampliﬁers impossible

in the context of low-power dispersive qubit readout.

For the detection and ampliﬁcation of single-photon-

level microwave signals, a diﬀerent type of low-noise am-

pliﬁer is required where the gain does not originate from

modulating a dissipative channel. Parametric ampliﬁca-

tion is accomplished by nonlinear mixing of the input

signal with a strong coherent pump ﬁeld. The wave-

mixing interaction is non-dissipative and thus achieves

superior noise performance [6]. Superconducting para-

metric ampliﬁers based on the nonlinear kinetic induc-

tance of Josephson junctions [7] approach the quantum

noise limit [4] with very little power dissipation. Hence,

they are used as ﬁrst-stage ampliﬁers in the readout of su-

perconducting qubits [8–10]. In recent years, research on

parametric ampliﬁers gained a lot of momentum due to

the growing interest in superconducting quantum com-

puting. The ﬁrst experimental evidence for the feasi-

bility of parametric ampliﬁcation with Josephson junc-

∗michael.haider@tum.de

tions was already obtained in 1967 [11]. The ﬁrst the-

oretical study was given in [12]. General energy rela-

tions for frequency conversion in nonlinear reactances

have been found in [13], which are needed to describe

DC-pumped parametric ampliﬁcation [14]. An exhaus-

tive review of diﬀerent microwave parametric ampliﬁer

designs in the context of quantum information experi-

ments is given in [15]. In a typical architecture, a sin-

gle Josephson junction is coupled to a resonator in order

to increase the gain of the ampliﬁer by increasing the

interaction time of the signal and pump modes. The

microwave resonator, however, limits the bandwidth of

the ampliﬁer to the resonator bandwidth. The ampliﬁer

resonator is operated in reﬂection mode, where a bulky

microwave circulator needs to be used to separate the re-

spective input and output waves [16]. Both limitations,

the limited bandwidth as well as the need for a circu-

lator, can be overcome by using a traveling-wave type

architecture as proposed by [17], where Josephson junc-

tions are periodically embedded in a microwave transmis-

sion line. Josephson traveling-wave parametric ampliﬁers

(JTWPAs) achieve large gain by increasing the interac-

tion time of the signal and pump modes through a large

propagation distance [18, 19]. The interaction along the

nonlinear transmission line shows a strong phase sensitiv-

ity. Thus, optimum parametric gain can only be achieved

if the ampliﬁcation process is phase-matched by careful

dispersion engineering [20–22].

In [23], the Josephson-embedded nonlinear transmis-

sion line is described by a classical nonlinear wave equa-

tion. The operation principle was modeled using coupled-

mode equations (CMEs), allowing to straightforwardly

calculate the ampliﬁer’s gain spectrum. Single-photon

applications at ultra-low temperatures, however, neces-

sitate a quantum mechanical treatment of the device.

Quantum mechanical models for Josephson circuits can

be obtained in the framework of quantum circuit the-

ory [24–26], where the term mesoscopic physics has been

coined for systems which contain a large number of elec-

tons and yet show distinct quantum features [27].

Quantum models for lumped-element Josephson para-

arXiv:2210.10032v1 [quant-ph] 18 Oct 2022

metric ampliﬁers (JPAs) have been investigated in [14,

28, 29]. Dissipation and thermal noise in the JPA were

introduced phenomenologically in [30] by coupling the

inner degrees of freedom to a bath. A Hamiltonian de-

scription of a JTWPA using continuous-mode operators

was derived in [31]. A similar discrete-mode mesoscopic

Hamiltonian was presented in [32], which serves as a

starting point for the investigations of the present paper.

For the derivation of the CMEs in [20, 23] and the quan-

tum descriptions in [31, 32] non-dissipative parametric

ampliﬁcation was assumed, i.e. all losses were neglected.

In this work, we extend the Hamiltonian framework

from [31, 32] in order to treat noise and dissipation,

mainly arising from substrate losses along the transmis-

sion line since the transmission line conductor itself is

in a superconducting state. Noise and dissipation are

included in the Hamiltonian by phenomenologically cou-

pling the signal and idler modes to a bath, consisting of

an inﬁnite number of quantum harmonic oscillators [33].

The coupling constants depend on the substrate material

and can be represented by diﬀerent coupling models [34].

Having a dissipative model for a Josephson traveling-

wave parametric ampliﬁer at hand enables studying the

frequency-dependent attenuation and noise performance

close to the quantum limit. Fluctuations and dissipation

in quantum traveling-wave parametric ampliﬁers were

studied in [35] using input-output theory as well as a

distributed loss model. In the following, we present an

analytic solution for the photon ﬁeld annihilation oper-

ator in a Josephson-embedded transmission line includ-

ing substrate losses and additional thermal noise. From

there we calculate expressions for the ampliﬁer gain and

the added input noise which both match experimental re-

sults [36]. We predict an added input noise equivalent to

approximately 1.3 excess photons, which is in excellent

agreement with recent experimental observations [36].

First, we introduce a circuit model of a dissipative-

dispersive Josephson junction-embedded transmission

line in Section II. We then discuss the traveling-wave

mode quantization in Section III and introduce the

JTWPA dispersion relation along with a reference time-

frame in Section IV. In Section V we introduce our four-

wave mixing Hamiltonian. Afterwards, the Heisenberg

equations of motion are derived within the moving refer-

ence timeframe in Section VI. The resulting equations of

motion for the photon ﬁeld are then solved in order to get

an analytic expression for the signal mode annihilation

operator under a strong classical pump approximation

in Section VII. Next, we present analytic results for the

gain proﬁle and the temporal dynamics of an exemplary

JTWPA structure from the literature [20, 32] including

substrate losses in Section VIII. In Section IX we derive

an analytic expression for the number of added noise pho-

tons due to thermal ﬂuctuations. We conclude the section

with a discussion on the equivalent added input noise for

an experimental structure [36], where our theoretical pre-

dictions are found to be in excellent agreement with their

observations.

C0∆z

vnvn+1

∆z

Figure 1. Unit cell of a JTWPA. The resonant phase match-

ing circuit and a resistive representation of the bath are high-

lighted in blue and orange color, respectively. The bath adds

noise to the system via the current source, and dissipates en-

ergy through the resistor.

II. QUANTUM MECHANICAL TREATMENT

OF A DISSIPATIVE JTWPA

Consider a superconducting transmission line with pe-

riodically embedded identical Josephson junction load-

ings. The distance between the Josephson nonlinearities

is small compared to the wavelength under consideration,

which allows for a continuum treatment of the Josephson

embedded transmission line [23]. A circuit representation

of the unit cell of such a stucture is given in Figure 1. As

the transmission line itself is in a superconducting state,

losses and thermal ﬂuctuations only occur due to sub-

strate imperfections, which are included in terms of the

resistor and the associated noise current source in the

circuit model in Figure 1, highlighted in orange color. In

order to derive a quantum model of the continuous dis-

sipative nonlinear transmission line, the resistor and the

associated noise current source are modeled in terms of

a distributed Markovian heat-bath, consisting of an inﬁ-

nite number of harmonic oscillators with thermal initial

occupations. It has been shown in [20] that the gain and

bandwith of Josephson traveling-wave parametric ampli-

ﬁers can be signiﬁcantly improved by dispersion engi-

neering using resonant phase-matching (RPM). Resonant

phase-matching can be included using periodically em-

bedded LC resonators, which are capacitively coupled to

the transmission line. The resonators tailor the disper-

sion relation such that the total phase mismatch along

the transmission line remains small over a large band-

width. In Figure 1, and throughout the rest of this pa-

per, C0is the ground capacitance per unit length of the

transmission line, LJ,0and CJare the linear inductance

and intrinsic capacitance of the Josephson junctions, ∆z

is the length of a single unit cell, Ccis the RPM cou-

pling capacitance, and Crand Lrare the RPM resonator

capacitance and inductance. Note that the transmission

line inductance has been neglected, as it can be easily

included into the linear Josephson inductance.

III. QUANTIZATION OF DISCRETE AND

CONTINUOUS MODES

A discrete-mode mesoscopic Hamiltonian for a non-

dissipative Josephson-embedded transmission line has

been derived in [32]. For the quantum mechanical model

in this work we take a similar approach, where we how-

ever consider a continuous mode spectrum in order to de-

rive a consistent noise model in terms of a noise spectral

density [37, 38]. We use canonical quantization of right-

moving traveling-wave modes [27], where the magnetic

ﬂux through the Josephson element ∆ΦJand the conju-

gate charge Qin the ground capacitance per unit length

C0are used as canonical pair of variables, deﬁned over a

continuous spatial argument z. Assuming a monochro-

matic right-traveling wave propagating through a trans-

mission line, we can describe the voltage operator by

V(z, t) = ˆc(z) eik(ω)z−iωt + H.c. , (1)

where ˆc(z) is the amplitude operator, ωis the angular

frequency of the propagating wave, k(ω) is the associated

wave number, and H.c.denotes the Hermitian conjugate.

A right-traveling wave where the voltage is deﬁned over

a continuous frequency spectrum, on the other hand, can

be decomposed into monochromatic plane wave contri-

butions in terms of the inverse Fourier transform

V(z, t) = 1

√2π

∞

−∞

ˆcω0eik(ω0)z−iω0tdω0,(2)

where the spectral Fourier coeﬃcient operator ˆcω0can be

obtained by means of the Fourier transform of the voltage

operator

ˆcω0(z) = 1

√2π

∞

−∞

V(z, t) e−ik(ω0)z+iω0tdt . (3)

Hence, monochromatic right-traveling wave amplitude

operators can be represented by the spectral coeﬃcient

operators

ˆcω0=1

√2π

∞

−∞

ˆc(z) ei[k(ω)−k(ω0)]ze−i(ω−ω0)tdt

=√2πˆc(z)δ(ω0−ω).(4)

In the following, relation (4) will be used to trans-

late continuous frequency amplitude operators to their

monochromatic limits.

IV. DISPERSION AND REFERENCE

TIMEFRAME

Traveling-wave amplitudes inside a dispersionless non-

linear transmission line exhibit space-time translation in-

variance [39], i.e.

A(z, t) = A0, t −z

v=A(z−vt, 0) ,(5)

with the spatial argument zand the frequency indepen-

dent propagation velocity v. Thus, the evolution of a

corresponding wave amplitude operator ˆ

A(z, t) can be

described by either a spatial or a temporal dependence

A(z, t)→ˆ

A(z)⇐⇒ ˆ

At=z

v,(6)

where again only right-propagating waves are taken into

account. In case of a dispersive transmission line, the

phase velocity vph(ω) = ω/k(ω) is frequency-dependent,

where k(ω) represents the dispersion relation. As each

mode now travels at a diﬀerent velocity, distinct fre-

quency components arrive at a certain location xat dif-

ferent times. In other words, when looking at a cer-

tain location xalong the transmission line, each mode

travels within its own frequency-dependent timeframe.

The corresponding timeframes t(ω) are considered to

be frequency-dependent functions which are given by

t(ω) = z/vph(ω). Therefore, the system operators can

be described similar to the dispersionless case, by a

frequency-dependent time argument

ˆaω(z)→ˆaωt(ω) = z

vph(ω).(7)

This way, the spatial dependence of the system annihi-

lation operator ˆaωcan be expressed by a temporal de-

pendence which permits the use of Heisenberg’s equa-

tions of motion for calculating the temporal evolution

of the system. However, since the Heisenberg equations

for diﬀerent modes are given by means of a frequency-

dependent, and thus mode-dependent timeframe t(ω), it

is diﬃcult to ﬁnd analytic solutions. In order to circum-

vent this problem, we introduce a frequency-independent

reference velocity vr=p∆z/LJ,0C0with a correspond-

ing reference timeframe tr=z/vrwhich is associated

with a wave propagating in a dispersionless ideal trans-

mission line with capacitance C0and inductance LJ,0per

unit length, respectively. Accordingly, we can deﬁne the

translation between the reference and phase timeframes

by ∂t(ω)/∂tr=pΛ(ω). The dimensionless dispersion

factor Λ(ω) is given by

Λ(ω) =

1 + q1 + 1

ω2R2C02∆z2

2(1 −ω2LJ,0CJ)≈1

1−ω2LJ,0CJ

,(8)

and may include dispersion due to the intrinsic substrate

resistance R, the shunt capacitance C0, as well as the

Josephson capacitance CJ. However, the contributions of

Rand C0can usually be neglected. Note that, diﬀerent

from [32], the nonlinearity of the Josephson inductance is

not explicitly taken into account for deriving the disper-

sion factor Λ(ω), where we only use the linear Josephson

inductance LJ,0. Ignoring substrate losses, however, (8)

exactly matches the dispersion relation in [20].

V. FOUR-WAVE MIXING HAMILTONIAN FOR

A JTWPA

In order to provide a consistent investigation of ther-

mal noise, we construct the system Hamiltonian in terms

of operators with a continuous mode spectrum. Hence,

we use [27]

V=1

√2π

∞

0s~ω

2C0vph(ω)hˆaωeik(ω)z−iωt + H.c.idω ,

where C0is the ground capacitance per unit length and

vph(ω) represents the frequency-dependent phase veloc-

ity. Accordingly, the magnetic ﬂux through the Joseph-

son element can be expressed by the operator

∆ˆ

ΦJ=1

√2π

∞

k(ω)∆z

ω×

×s~ω

2C0vph(ω)hˆaωeik(ω)z−iωt + H.c.idω .

(9)

The Hamiltonian of a lossless JTWPA can then be ob-

tained by integrating the Hamiltonian density, i.e. the

energy stored in the ground capacitors and in the linear

and nonlinear inductances of the Josephson junctions per

unit length, over the entire device length x[32]

HJTWPA =1

2∆z2

0("∆z

LJ,0

∆ˆ

ΦJ−∆z

12LJ,0ϕ2

∆ˆ

Φ3

+CJ∆z∂2∆ˆ

ΦJ

∂t2#∆ˆ

ΦJ+1

C0ˆ

Q2)dz ,

(10)

where ϕ0=~/(2e) is the reduced magnetic ﬂux quan-

tum, ~is the reduced Planck’s quantum of action and e

is the elementary charge. The quadratic terms in (10)

contribute to the linear part of the system Hamiltonian,

while the fourth-order term describes the nonlinearity

which is responsible for the four-wave-mixing ampliﬁ-

cation process. The continuous mode Hamiltonian de-

rived in (10) agrees with the Hamiltonian in equation (9)

of [31].

The input mode spectrum can be decomposed into

weak input signal components ˆaω, together with a strong

pump tone ˆaΩpat a pump frequency of Ωp. Inserting this

decomposition into the Josephson ﬂux operator ∆ΦJ(9)

and then inserting the result into the Hamiltonian (10),

we obtain the lossless system Hamiltonian in terms of the

continuous mode operators ˆaωand ˆaΩp

HJTWPA =

∞

~Ωpˆa†

ΩpˆaΩpdΩp+

∞

~ωˆa†

ωˆaωdω

−~2∆z3

64π2C02L3

J,0I2

∞

k(˜

Ωp)k(˜

Ω0

p)k(Ωp)k(Ω0

p) e−i(˜

Ω0

p−˜

Ωp+Ω0

p−Ωp)t

q˜

Ωp˜

Ω0

pΩpΩ0

pvph(˜

Ωp)vph(˜

Ω0

p)vph(Ωp)vph(Ω0

ˆa†

Ωpˆa˜

Ω0

pˆa†

ΩpˆaΩ0

p×

ei[k(˜

Ω0

p)−k(˜

Ωp)+k(Ω0

p)−k(Ωp)]zdzd˜

Ωpd˜

Ω0

pdΩpdΩ0

−~2∆z3

16π2C02L3

J,0I2

∞

k(ω)k(ω0)k(Ωp)k(Ω0

p) e−i(ω0−ω+Ω0

p−Ωp)t

qωω0ΩpΩ0

pvph(ω)vph(ω0)vph(Ωp)vph(Ω0

ˆa†

ωˆaω0ˆa†

ΩpˆaΩ0

p×

ei[k(ω0)−k(ω)+k(Ω0

p)−k(Ωp)]zdzdωdω0dΩpdΩ0

−~2∆z3

32π2C02L3

J,0I2

∞





k(ω)k(ω0)k(Ωp)k(Ω0

p) e−i(−ω−ω0+Ωp+Ω0

p)t

qωω0ΩpΩ0

pvph(ω)vph(ω0)vph(Ωp)vph(Ω0

ˆa†

ωˆa†

ω0ˆaΩpˆaΩ0

p×

ei[−k(ω)−k(ω0)+k(Ωp)+k(Ω0

p)]z+ H.c.

dzdωdω0dΩpdΩ0

(11)

where we have dropped the non-resonant and fast rotat-

ing terms. The ﬁrst line in equation (11) describes the

propagation of the free photon ﬁelds. The second and

third lines describe self-phase modulation and the fol-

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Circuitquantumelectrodynamicmodelofdissipative-dispersiveJosephsontraveling-waveparametricampliersYongjieYuan,MichaelHaider,JohannesA.Russer,PeterRusser,andChristianJirauschekTUMSchoolofComputation,InformationandTechnology,TechnicalUniversityofMunich,Hans-Piloty-Str.1,85748Garching,Germany(Dated:O...

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Circuit quantum electrodynamic model of dissipative-dispersive Josephson traveling-wave parametric ampliers Yongjie Yuan Michael HaiderJohannes A. Russer Peter Russer and Christian Jirauschek

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