Resolving Fock states near the Kerr-free point of a superconducting resonator Yong Lu12Marina Kudra1 Timo Hillmann1 Jiaying Yang13 Hangxi Li1 Fernando Quijandr a14 and Per Delsing1y

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Resolving Fock states near the Kerr-free point of a superconducting resonator

Yong Lu1,2,∗Marina Kudra1, Timo Hillmann1, Jiaying Yang1,3,

Hangxi Li1, Fernando Quijandr´ıa 1,4, and Per Delsing1†

1Microtechnology and Nanoscience, Chalmers University of Technology, SE-412 96, G¨oteborg, Sweden

23.Physikalisches Institut, University of Stuttgart,70569 Stuttgart, Germany

3Ericsson research, Ericsson AB, SE-164 83, Stockholm, Sweden

4Quantum Machines Unit, Okinawa Institute of Science and

Technology Graduate University, Onnason, Okinawa 904-0495, Japan

We have designed a tunable nonlinear resonator terminated by a SNAIL (Superconducting

Nonlinear Asymmetric Inductive eLement). Such a device possesses a sweet spot in which the

external magnetic ﬂux allows to suppress the Kerr interaction. We have excited photons near this

Kerr-free point and characterized the device using a transmon qubit. The excitation spectrum of the

qubit allows to observe photon-number-dependent frequency shifts about nine times larger than the

qubit linewidth. Our study demonstrates a compact integrated platform for continuous-variable

quantum processing that combines large couplings, considerable relaxation times and excellent

control over the photon mode structure in the microwave domain.

Encoding quantum information in the inﬁnite Hilbert

space of a harmonic oscillator is a promising avenue for

quantum computing. Recently, signiﬁcant progress has

been made by using three-dimensional (3D) microwave

cavities [1–6]. Thanks to the strong-dispersive coupling,

quantum states such as cat states [7,8], GKP states [9]

and the cubic phase state [10] have been engineered.

However, currently, the scalability and connectivity is

diﬃcult, limited by the size of the cavity. Another

simpler method is to use coplanar microwave resonators

where resonators and qubits can be fabricated together

in a single chip [11–13]. The drawback is the shorter

relaxation time of coplanar resonators compared to 3D

cavities.

Both 2D and 3D microwave resonators as well as

acoustic resonators [14,15] host linear modes. Therefore,

an ancillary qubit is customarily used to introduce

a nonlinearity for state preparation and operation.

However, the limited coherence of the ancillary qubit and

the imperfect operations on it will decrease the ﬁdelity

of the actual states [10,16,17]. To avoid operations

on ancillary qubits, a Superconducting QUantum

Interference Device (SQUID) can be used to terminate a

coplanar resonator. This not only provides the tunability

of the mode frequency by changing the external magnetic

ﬂux through the loop [18–23], it also induces a suﬃciently

nonlinearity. Using this nonlinearity, experiments in

waveguide quantum electrodynamics have demonstrated

the generation of entangled microwave photons by the

parametrical pumping of a symmetrical SQUID loop [24,

25]. Non-Gaussian states, regarded as a resource for

quantum computing, have also been realized [26–28].

Therefore, state preparation and operations can be

implemented in a nonlinear resonator, even without

ancillary qubits. However, in those experiments, the

∗e-mail:kdluyong@outlook.com

†e-mail:per.delsing@chalmers.se

generated states can not be stored for a long time since

the resonators are directly coupled to the waveguides.

Theoretically, it has been shown that pulsed operations

on a novel tunable nonlinear resonator can be used

to achieve a universal gate set for continuous-variable

(CV) quantum computation [29]. The proposed device

is similar to a parametric ampliﬁer where the Josephson

junction or SQUID loop is replaced by an asymmetric

Josephson device known as the SNAIL (Superconducting

Nonlinear Asymmetric Inductive eLement) [30,31], is

applied. Using a SNAIL or an asymmetrically threaded

SQUID loop [32,33], it is possible to realize three-wave

mixing free of residual Kerr interactions by biasing the

element at a certain external magnetic ﬂux. In this work,

we refer to this ﬂux sweet spot as the Kerr-free point.

Therefore, it is meaningful to investigate a SNAIL-

terminated resonator in circuit quantum electrodynamics

(cQED) where the nonlinear resonator, decoupled to the

waveguide, can be used for state preparation and storage.

Strong dispersive coupling between oscillators and qubits

was achieved a decade ago for photons [11] and recently

for phonons [34–36], in linear resonators. Here, we

observed the very well-resolved photon-number splitting

up to the 9th-photon Fock state in a SNAIL-terminated

resonator coupled to a qubit. Our nonlinear resonator

has a considerable relaxation time up to T1= 20 µs under

a few-photon drive, limited by (Two-Level Systems)

TLSs. Our study opens the door to implement, operate

and store quantum states on this scalable platform in

the future. Moreover, compared to a linear resonator,

our resonator has a non-negligible dephasing rate from

its high sensivity to the magnetic ﬂux noise due to the

SNAIL loop.

Characterization of the SNAIL-terminated

resonator. In order to characterize the parameters

for the SNAIL-terminated resonator directly, we

ﬁrst fabricated a SNAIL-terminated λ/2 resonator

capacitively coupled to a coplanar transmission line

[not shown], which is the same as the nonlinear

resonator in Fig. 1. By measuring the transmission

arXiv:2210.09718v1 [quant-ph] 18 Oct 2022

Feedline

Readout resonator

Qubit Charge line

SNAIL-terminated resonator

SNAIL

Flux line

JJ 10um

10um

200um

FIG. 1. A scanning optical micrograph of the measured sample.A superconducting qubit with a cross-shaped island (green) and

a single Josephson junction (JJ, light blue in the middle inset), capacitively coupled to both a coplanar read-out resonator (red) and

the nonlinear resonator (blue). The nonlinear resonator is formed by a linear coplanar resonator terminated by a superconducting

nonlinear asymmetric inductive element (SNAIL) that has three big Josephson junctions and one small junction (orange in right

inset). The charge line (pink) is used to operate the qubit and drive the nonlinear resonator. The qubit state read-out is implemented

through the transmission of the feedline. The on-chip ﬂux line is not used in this work, instead, a superconducting coil on the top

of the chip [not shown] is used to generate the external magnect ﬂux Φext through the SNAIL. See more fabrication details and the

measurement setup for this device in Supplementary.

Qubit

Nonlinear resonator

a b

FIG. 2. Spectroscopy for the qubit frequency and the SNAIL-

terminated resonator frequency. a, The qubit frequency

vs. the external magnetic ﬂux through the SNAIL. b, The

SNAIL-terminated resonator frequency vs. the external magnetic

ﬂux through the SNAIL. The nonlinear resonator is dispersively

coupled to the qubit as shown in Fig. 1.S(ρee)∝ρee is the

measured signal from the read-out pulse of the qubit. ωpis the

probe frequency.

coeﬃcient through a vector network analyzer similar

to measuring conventional resonators [37–40], at the

10 mK stage of a dilution refrigerator, we can extract

the nonlinear resonator frequency at diﬀerent external

magnetic ﬂuxes [see Supplementary]. For our SNAIL-

terminated resonator, the inductive energy can be

written as [29–31]

USNAIL(φ) = −βEJcos(φ)−3EJcos φext −φ

3,(1)

where βis the ratio of the Josephson energies of the

small and the big junctions of the SNAIL, φis the

TABLE I. SNAIL-terminated resonator parameters at zero

magnetic ﬂux and near the Kerr-free point. The values are

extracted from a transmission coeﬃcient measurement on a

SNAIL-terminated resonator coupled to a transmission line.

We have γs= 1/Ts=ωs/Qswhere γs,Ts,ωsand Qs

are the resonator intrinsic decoherence rate, coherence time,

frequency and the internal Q value, respectively. The intrinsic

decoherence rate is given by γs=Γ1

2+ Γφwhere the intrinsic

relaxation rate Γ1=1

T1with the lifetime T1and the pure

dephasing rate Γφ=1

Tφwith the pure dephasing time Tφ.

Φext/Φ0ωs/2π Qsγs/2π Ts

GHz kHz µs

0 5.14 2.23 ×10523 6.92

0.386 4.31 3.86 ×104112 1.42

superconducting phase across the small junction, φext =

2πΦext/Φ0is the reduced external magnetic ﬂux, and

EJ, related to the Josephson inductance LJ, is the

Josephson energy of the big junctions in the SNAIL.

Upon quantization, the Hamiltonian for the SNAIL-

terminated resonator becomes

Hs=~ωs+g3(a+a†)3+g4(a+a†)4,(2)

where g3(g4) is the couplings for the three (four)-

wave mixing coupling strength, and ωsis the resonator

frequency which follows the relation [29]

ωstan π

ωs

ωr0 =Zcc2

3LJ

,(3)

where ωr0 describes the bare resonance frequency of

the resonator without the SNAIL, Zc= 58.7 Ω for

the characteristic impedance of the resonator, and c2

is a numerically determinable coeﬃcient for the linear

coupling whose speciﬁc value depends on βand φext in

our case.

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ResolvingFockstatesneartheKerr-freepointofasuperconductingresonatorYongLu1;2,MarinaKudra1,TimoHillmann1,JiayingYang1;3,HangxiLi1,FernandoQuijandra1;4,andPerDelsing1y1MicrotechnologyandNanoscience,ChalmersUniversityofTechnology,SE-41296,Goteborg,Sweden23.PhysikalischesInstitut,UniversityofStuttga...

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Resolving Fock states near the Kerr-free point of a superconducting resonator Yong Lu12Marina Kudra1 Timo Hillmann1 Jiaying Yang13 Hangxi Li1 Fernando Quijandr a14 and Per Delsing1y

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