Nonlinear dynamical topological phases in Cooper-pair box array Motohiko Ezawa Department of Applied Physics University of Tokyo Hongo 7-3-1 113-8656 Japan

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Nonlinear dynamical topological phases in Cooper-pair box array

Motohiko Ezawa

Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan

The topological property of a system is a static property in general. For instance, the topological edge state is

observed by measuring the local density of states. In this work we propose a system whose topological property

is only revealed by dynamics. As a concrete example, we explore a nonlinear dynamical topological phase

transition revealed by a quench dynamics of a Cooper-pair box array connected with capacitors. It is described

by coupled nonlinear differential equations due to the Josephson effect. It is trivial as far as the static system

is concerned. However, the wave propagation induced by the quench dynamics demonstrates a rich topological

phase diagram in terms of the strength of the input.

Topological physics is one of the most extensively studied

ﬁelds in condensed-matter physics[1, 2]. The notion of topol-

ogy is also applicable to artiﬁcial topological systems such as

photonic[3–8], acoustic[9–13], mechanical[14–21] and elec-

tric circuit[22–27] systems. New feature of these artiﬁcial

topological systems are that nonlinearity is naturally intro-

duced as in photonics[28–36], mechanics[37–39] and electric

circuits[40–42]. In particular, electric circuits simulate almost

all topological phases. Active topological electric circuits[43]

is realized by using nonlinear Chua’s diode circuit. Topo-

logical Toda lattice is realized by using variable capacitance

diodes[42].

Recently, the development of superconducting qubits is

very rapid as a method for quantum computation[44, 45]. The

transmon qubit is a successful example of a superconducting

qubit[46, 47] based on a Cooper-pair box (CPB)[44, 48, 49]

made of the Josephson junction and a capacitor, as shown in

Fig.1(a).

In this work we explore a nonlinear topological phase tran-

sition in a CPB array connected with capacitors as in Fig.1(d),

where alternating capacitances are arranged for the array to

acquire a structure akin to that of the one-dimensional Su-

Schrieffer-Heeger (SSH) chain. The SSH model is the well-

known topological system, where the topological phase is

signaled by the emergence of zero-energy topological edge

states. They are detected by the measurement of the local den-

sity of states. However, the CPB array reveals an unfamiliar

feature in topological physics, where the system is trivial as

far as the static system is concerned.

To reveal the topological structure, we evoke a dynamical

property. We analyze a quench dynamics by giving a ﬂux to

the left-end CPB and studying its time evolution. We have

found four types of propagations. (i) There are standing waves

mainly at the left-end CPB and weakly at a few adjacent odd-

number CPBs. Additionally, there are propagating waves into

the bulk. (ii) There are only propagating waves spreading into

other CPBs. (iii) The standing wave is trapped strictly at the

left-end CPB. (iv) The coupled standing waves are trapped to

double CPBs at the left-end. These four modes correspond to

the topological phase, the trivial phase, the trapped phase and

the dimer phase. We determine the phase diagram with the aid

of a phase indicator deﬁned by the saturated amplitude. The

present model presents a system whose topological property

is only revealed by a dynamical method.

Model: A CPB is an electric circuit made of a Josephson

(a1)

(c)

(b)(a2)

SC SC

capacitor

insulator

C’

C C

AC’

C’

AC’

C’

FIG. 1. Illustration of (a) a Cooper-pair box, (b) double Cooper-pair

boxes connected via a capacitor and (c) a dimerized Cooper-pair box

array.

junction and a capacitor with capacitance C, as illustrated in

Fig.1(a). We investigate a CPB array connected by capacitors

shown in Fig.1(c). The Lagrangian is given by

L(Φ,˙

Φ) = X

nCn

2˙

Φ2

n+IcΦ0

2πcos 2πΦn

Φ0

2(˙

Φn−˙

Φn+1)2,(1)

where Icis the critical current of the Josephson junction,

Φ0≡h/2eis the unit ﬂux, Cnis the capacitance in the n-

th CPB, and Φnis a magnetic ﬂux across the n-th Josephson

junction. The ﬁrst line describes the n-th CPB, and the sec-

ond line describes the coupling between the n-th and (n+1)-th

CPBs via the capacitance C0

n. It is summarized as

L(Φ,˙

Φ) = 1

n,m

Mnm ˙

Φn˙

Φm+X

IcΦ0

2πcos 2πΦn

Φ0

,(2)

where

Mnm = [Cn+C0

n+C0

n−1]δnm −[C0

nδm,n+1 +C0

mδn,m+1],

(3)

with C0

0= 0. The Euler-Lagrange equation reads

Mnm ¨

Φm+Icsin 2πΦn

Φ0

= 0.(4)

The canonical conjugate of the ﬂux Φn=∂L/∂ ˙

Φnis the

charge acquired in the capacitor, and given by

Qn=Mnm ˙

Φm.(5)

arXiv:2210.00496v1 [cond-mat.mes-hall] 2 Oct 2022

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NonlineardynamicaltopologicalphasesinCooper-pairboxarrayMotohikoEzawaDepartmentofAppliedPhysics,UniversityofTokyo,Hongo7-3-1,113-8656,JapanThetopologicalpropertyofasystemisastaticpropertyingeneral.Forinstance,thetopologicaledgestateisobservedbymeasuringthelocaldensityofstates.Inthisworkweproposeasys...

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Nonlinear dynamical topological phases in Cooper-pair box array Motohiko Ezawa Department of Applied Physics University of Tokyo Hongo 7-3-1 113-8656 Japan

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