Topological photonics on superconducting quantum circuits with parametric couplings

Zheng-Yuan Xue1, 2, ∗ and Yong Hu3, †

1Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, and School of Physicsand Telecommunication Engineering, South China Normal University, Guangzhou 510006, China

2Guangdong-Hong Kong Joint Laboratory of Quantum Matter, and Frontier Research Institute for Physics,South China Normal University, Guangzhou 510006, China

3School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China(Dated: November 7, 2021)

Topological phases of matter is an exotic phenomena in modern condense matter physics, which has attractedmuch attention due to the unique boundary states and transport properties. Recently, this topological concept inelectronic materials has been exploited in many other fields of physics. Motivated by designing and controllingthe behavior of electromagnetic waves, in optical, microwave, and sound frequencies, topological photonicsemerges as a rapid growing research field. Due to the flexibility and diversity of superconducting quantumcircuits system, it is an promising platform to realize exotic topological phases of matter and to probe and ex-plore topologically-protected effects in new ways. Here, we review theoretical and experimental advances oftopological photonics on superconducting quantum circuits via the experimentally demonstrated parametric tun-able coupling techniques, including using of the superconducting transmission line resonator, superconductingqubits, and the coupled system of them. On superconducting circuits, the flexible interactions and intrinsic non-linearity making topological photonics in this system not only a simple photonic analog of topological effectsfor novel devices, but also a realm of exotic but less-explored fundamental physics.

Keywords: superconducting quantum circuit, topological photonics, parametric coupling

I. INTRODUCTION

Topology considers the geometric properties of objects thatcan be preserved under continuous deformations, such asstretching and bending, and quantities remaining invariant un-der such continuous deformations are denoted as topologicalinvariants [1–3]. For instance, a two-dimensional (2D) closedsurface embedded in 3D space has the topological invariantgenus which counts the number of the holes within the sur-face. This topological invariant does not respond to small per-turbations as long as the holes are not created by tearing orremoved by gluing. Objects which look very different can beequivalent in the sense of topology, and objects with differenttopologies can be classified into different equivalent classesby their topological invariants. For instance, a sphere anda spoon are equivalent because both of their genus are zero,while a torus and a coffee cup are also topological equivalentas their genus are one.

Historically, the topological concept combined with con-densed matter physics through the discovery of the quantumHall effect (QHE) [4], in which the quantized Hall conduc-tance of 2D electronic materials can be interpreted by thetopological Chern number of the filled Landau levels [5]. Thisexotic phase of matters cannot be explained by the conven-tional spontaneous symmetry-breaking mechanism. Instead,it should be understood within the framework of the newly-developed topological band theory (TBT) [6–9]. A directconsequence of a nontrivial topological band structure is theemergence of edge state modes (ESMs), i.e. the bulk-edgecorrespondence [10–13]. The existence of the ESMs can be

∗ zyxue83@163.com† yonghu@hust.edu.cn

explained by the following simple argument [14]: we put twoinsulators with different topological band structures togetherto share a boundary, and away from the boundary the two in-sulators extend to infinity. Then, these two insulators can notbe connected trivially at the boundary region, and thus a topo-logical phase transition has to happen somewhere when theband gaps are closed and reopened, i.e. there must have ESMscrossing the band gap. The presence of the gapless ESMs canthus be treated as an unambiguous signature of the topologi-cal non-trivial band structure of the bulk. The gapless spectraof the ESMs are topologically protected as their existence isguaranteed by the difference of the topologies of the bulk ma-terials on the two sides, and their propagation is robust againstperturbations in the materials including disorder and defects.

As TBT mainly considers the band topologies of Hermitianelectronic systems in the single-particle picture, the obtainedresults can be directly applied to interaction-free bosonic sys-tems, leading to the emerging research field of topologicalphotonics [15, 16]. During the past few years, there are nu-merous theoretical and experimental explorations of the pho-tonic analogue of the QHE in various artificial photonic meta-materials [17–20]. The motivation lies in both the sides ofexperimental application and fundamental physics. On inte-grated optical circuits, the back-reflection induced by disor-der and defects is a major hindrance of efficient informationtransmission. Therefore, it is expected that the unidirectionalESMs, if synthesized, can be used to transmit electromag-netic waves without the back-reflection even in the presenceof large disorder. This closely mimics the topologically pro-tected quantized Hall conductance of 2D electronic materials,and such ideal transport may offer novel functionalities forphotonic systems with topological immunity to fabrication er-rors or environmental changes. On the other hand, topolog-ical photonics also brings new physics of fundamental im-portance. As the band topology of spinless particles remains

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trivial as long as the time-reversal symmetry is preserved, ef-fective magnetic fields have to be synthesized for the charge-neutral photons in the metamaterials [19–21], which is repre-sented by the nontrivial hopping phases on the lattice throughPeierel’s substitution. In condensed matter physics, the elec-trons’ magnetic lengths of the the current-achievable strongmagnetic field are still far more larger than the lattice spac-ing [22, 23]. Meanwhile, based on the fabrication and ma-nipulation of artificial photonic metamaterials, it is expectedto obtain arbitrarily large Aharonov-Bohm phase for hoppingphotons in a loop containing only few unit-cells, indicatingthat the synthetic magnetic field for photons can be several or-ders larger than those for electronic materials. In addition, itshould be noticed that the photonic systems is naturally non-equilibrium and bosonic, exhibiting neither chemical potentialnor fermionic statistics. This is in stark contrast to the con-ventional electronic materials. Therefore, from the above twopoints of view, topological photonics is not only the simplephotonic analog of known topological phases of matter, butalso a realm of exotic but less-explored fundamental physics.

Here, we focus on reviewing experimental and theoreticaladvances of topological photonics on superconducting quan-tum circuits (SQC) [24–29], including using of the supercon-ducting qubits, superconducting transmission line resonator(TLR) and the coupled system of them. SQC can be re-garded as the realization of atom-photon interaction [30] inon-chip Josephson junction based mesoscopic electronic cir-cuit. It uses the superconducting TLRs [31, 32] to replacethe standing-wave optical cavities and the superconductingqubits [24, 33] to substitute the atoms. Historically, this on-chip superconducting architecture is proposed as a candidateplatform for quantum computation. While borrowed a vari-ety of ideas from atomic cavity QED [30] in its early devel-opment, SQC has now become an independent research fielddue to its unprecedented advantages including the strong non-linearity at the single photon level, the long coherence times,the flexibility in circuit design, the detailed control of atom-photon interaction at the quantum level, and the scalabilitybased on current microelectronic technology [34–36]. Re-cently, it has been realized that the above mentioned meritsare also essential for the synthesization of photonic metama-terial in the microwave regime and the quantum simulation ofvarious complicated many-body effects. For the purpose ofquantum simulation, individual SQC elements such as super-conducting TLRs or qubits play the role of the bosons and/orfermions in lattice models, and the inter-site coupling is estab-lished through the connection of the SQC elements via variouskinds of couplers. From this point of view, a coupler whichcan support both tunable coupling strength and hopping phaseis highly desirable. The former can be exploited to the inves-tigation of the competition between the inter-site hopping andthe on-site Hubbard interaction [37–39], while the latter, be-ing equivalent to the artificial gauge field for the photons onthe lattice [16, 40, 41], is required to realize non-trivial topo-logical effect, which is our central topic here.

The research of synthesizing gauge fields in artificial mod-els was originated in the ultracold atomic systems [41–43],while SQC system takes the naturally advantages of indi-

vidual addressing and in situ tunability of circuit parameters[26, 34, 35]. The other distinct merits of quantum simulationtopological phases with SQC are the following aspects. First,the topological state of matters are highly related with the ge-ometry of the lattice. Meanwhile, by taking the advantage ofarbitrary wiring, SQC can be used to implement lattice config-urations which are difficult for other simulators. Second, theeffective strong interaction between photons can be synthe-sized in a tunable way via many mechanisms, including theelectromagnetically induced transparency [37, 44], Jaynes-Cummings-Hubbard nonlinearity [45], as well as nonlinearJosephson coupling [46–48]. This can be understood by thefact that a superconducting qubit can be treated as a nonlinearresonator in the microwave regime [49], and thus the topolog-ical photonics on SQC can be directly extended to the non-linear regime, i.e., nonlinear topological photonics [50]. Inparticular, the Hamiltonian of a qubit chain can be describedby the Bose-Hubbard model, where the nonlinearities can nat-urally introduce photonic interactions [51], and thus interac-tion induced topological photonics can be possible. As thenumber of lattice sites grows, the quantum dynamics quicklybecome intractable even for classical super-computers. There-fore, combining the strong and tunable photonic interactionwith the synthetic gauge fields make the SQC system promis-ing in realizing bosonic fractional QHE [52, 53] and nontrivialtopological ESMs for photons in the microwave regime [54].Such method can offer facilities in the study of the compe-tition effect between synthetic gauge fields, photon hopping,and Hubbard repulsion. The third point is that, as the controlof superconducting devices and their interaction is now veryprecise and diverse, it is possible to engineer the gain and lossof a system. In this sense, SQC can be exploited to investigatethe non-Hermitian induced topological phases which exhibitdrastic difference from their Hermitian counterpart.

II. INTRODUCTION TO TBT

In this section, we briefly review the basic concepts of TBT.Currently there have already been many excellent reviews andtutorials on this subject, see, e.g., Ref. [8] for the pedagogi-cal tutorial of TBT and Ref. [16] for the summary of recentadvances in topological photonics.

A. The topological invariant

Most topological invariants being involved in this reviewcan be understood in the framework of degree of map [3]: Fora smooth map between two closed oriented manifolds withthe same dimension, an integer can be defined to characterizethe algebraic times an image point is covered. To illustrate thisconcept, let us take the winding number w of a one-dimension(1D) closed loop M parameterized by φ ∈ [0, 2π) as an ex-ample, see Section II C for the two-dimension (2D) case. Asshown in Fig. 1, we consider the radical map f from M tothe 1D unit circle S1 parameterized by the azimuthal angle θ.The rule of f is that we move each point of M radically un-

3

(a) (b) (c)

q2

q2

w=1 w=2 w=0

q1 q1 q11 1

p1

2p2p 2

3

p4 ooo

+ -p+

-p

+p ++p

FIG. 1. Winding number w of 1D closed loops, which counts thealgebraic times an image point on S1 is radically mapped, or equiv-alently the times a loop wraps around the origin. For the loopssketched, w takes values (a) w = 1, (b) w = 0, and (c) w = 2,and exhibit topological robustness against the small variation of theloops.

til it strikes S1. Since the variation of f can be equivalentlyrealized by keeping the radical rule but changing the shape ofM , the degree of map of f in this situation is recognized asthe winding number of of the loop M , which is the Jacobianintegration of f over M ,

w =1

2π

∫ φ=2π

φ=0

dθ

dφdφ ∈ Z. (1)

Here the 1/2π factor indicates the length of the target mani-fold S1. Clearly,wmeasures the times that the target manifoldS1 is covered by the image of M under f . There is anotherequivalent discrete form of w. For the loop in Fig. 1(a), let uschoose a point q1 ∈ S1 and count its inverse images p1, p2,and p3. In this counting, every inverse image contribute +1 or−1, depending on the direction, i.e. the sign of the Jacobian off , at that inverse image. Therefore, p1 and p3 contribute +1each and p2 contributes −1, resulting an overall w = 1 (seealso Figs. 1(b) and (c) for thew = 0 andw = 2 cases). Noticethat w does not depend on the choice of the image point: wecan select another image point q2, and its single inverse imagep4 also results in an overall +1 counting.

The topological nature of w is manifested by the fact thatit dose not alter under the slight deformation of M as long asthe deformation does not hit the origin where the radical rule isill-defined. We can check the other loops in Fig. 1(b) and (c),which have winding numbers 0 and 2, respectively. It is obvi-ous that we can not change their winding numbers until we letsome of the point on the loop cross the origin. Such topolog-ical robustness is intuitive: as shown in Eq. (1), w is a con-tinuous and integer-valued functional of M . Therefore, underthe continuous deformation of M , w can not change continu-ously, but has to either remain invariant or change abruptly.

B. TBT in 1D: SSH Model

A prominent example of closed manifolds in physics is thefirst Brillouin zone of periodic lattices, and perhaps the mosttypical and simple model exhibiting nontrivial band topologyis the 1D Su-Schrieffer-Heeger (SSH) model describing poly-acetylene chain formed by bonded CH groups [55]. As shown

in Fig. 2, the SSH model describes electrons hopping on achain with two sublattices and staggered hopping constants.The Hamiltonian takes the form

HSSH =∑n

[gAa

†n,Aan,B + gBa

†n+1,AanB

]+ H.c., (2)

where an,α (a†n,α) is the annihilation (creation) operator ofthe αth sublattice in the nth unit-cell, and gA, gB denotes theintra- and inter-unit-cell hopping strength, respectively. In thereciprocal space, HSSH has the form

HSSH(k) = −d(k) · σ, k ∈ [0, 2π] , (3)

with d(k) = − (t1 + t2 cos k, t2 sin k, 0) and σ =(σx, σy, σz) with σx,y,z being the Pauli matrices. Thus, d(k)defines a closed loop in the x− y plane, and the Bloch vectorn(k) = d(k)/ ‖d(k)‖ of the lower band eigenstate |u−(k)〉 isexactly the radical image of d(k). The SSH model can thus beclassified into two topologically distinct regions |gA| < |gB |and |gA| > |gB |, corresponding to w = 1 and w = 0, respec-tively. The variation of gA and gB can not change w as longas the band gap 2 |d(k)| is nonzero, that is, d(k) does not hitthe origin in the variation. In this sense we say the topologyof the Bloch band is protected by the band gap. Meanwhile,at the critical value |gA| = |gB |, there exists a point on d(k)touching the origin where the band gap is closed and n(k) isill-defined. That is to say, if we want to change the topol-ogy of the Bloch band, we need to first close the band gap atsomewhere and then reopen it.

Like any realistic materials, the SSH chain does not onlyhave a bulk part, but also has to have boundaries, i. e. twoends.The topological winding number of the Bloch band man-ifests itself through the existence and absence of localizedzero-energy edge modes at these two ends. This embod-ies the important bulk-edge correspondence principle in TBT[10, 11]. We can understand this issue from the two dimerizedlimits g1 = 1, g2 = 0 and g1 = 0, g2 = 1 (see Fig. 2). In bothsituation, ths SSH chain falls into a sequence of disconnecteddimers. The difference is that each dimer is shared by theneighboring unit-cell in the topological nontrivial case, leav-ing two isolated, unpaired sites on the two ends and thus local-ized zero-energy edge modes. Moving away from this limit,the wavefunctions of the edge modes are still exponentially lo-calized because the zero of the energy is in the bulk band gap.Therefore, the edge modes will only disappear when the bandgap is closed and reopened. In this situation the topologicalphase transition occurs and the chain goes into the topologicaltrivial region.

C. TBT in 2D

While the above 1D example is illustrative, historicallythe combination of the topology concept and the Bloch bandtheory first brought fruitful results in 2D electronic systems,opening up the rapid growing research field topological in-sulators [8]. The bulk of topological insulators behaves asan insulator while electrons can move along the edge without

4

g1=1

g1=0 g2=1

g2=0

FIG. 2. Sketch of the SSH model in the two topological trivial (a)and non-trivial (b) dimerization limits.

dissipation or back-scattering even in the presence of defect.The first example of topological insulators is the QHE [4] of2D electrons in a uniform external magnetic field. It provedout later that the quantized Hall conductance has its root inthe nontrivial topology of the bulk band structure in recipro-cal space [5]. A filled energy band below Fermi surface con-tributes a quantized portion of transverse conductance

σh =e2

2π~Ch, (4)

where h is the band index and Ch is the corresponding topo-logical Chern number. In particular, Ch has the form

Ch =1

2π

∫FBZ

d2kFhxy(k), (5)

where

Fhxy(k) = i

[⟨∂uh(k)

∂kx

∣∣∣∣∂uh(k)

∂ky

⟩− (x↔ y)

](6)

is the Berry curvature at k with uh(k) being the Bloch func-tion of momentum k in the hth band. Such topological in-variant is insensitive to the sample sizes, compositions, de-fects, and local perturbation, and can only be changed throughthe closing and reopening of the band gaps. Later, it is notedthat the emergence of nonzero Chern number is related to thebreaking of time-reversal symmetry, and a translational invari-ant Haldane model with zero net magnetic field is proposed[56].

The 2D Chern number can also be understood in the frame-work of degree of map. For a two-band system with Hamil-tonian H2D = −d(k) · σ, the Berry curvature of the lowestband is calculated as [57]

Fxy(k) =1

2 |d(k)|3d(k) ·

[∂d(k)

∂kx× ∂d(k)

∂ky

]=

1

2n(k) ·

[∂n(k)

∂kx× ∂n(k)

∂ky

], (7)

that is, Fxy(k) is the Jacobian of the following map

k = [k1, k2]

⇒ d(k) = [dx(k), dy(k), dz(k)]

⇒ n(k) =d(k)

‖d(k)‖, (8)

(a)

(b)

a

f

(N , N )x y

(1, 1)

(1, N )y

(N , 1)xx

y

FIG. 3. Sketch of the Laughlin cylinder. The Laughlin cylinderin (a) can be regarded as a plane material wrapped along one direc-tion (here along the y direction). Parallel to the cylinder, a magneticflux α is inserted. The modulation of α alters the spectrum of theESM. Through this modulation, the topological winding number ofthe ESMs can be extracted. Adapted from Ref. [60].

from the first Brillouin zone T 2 to S2. In this sense, the inte-gration in Eq. (5) counts the times T 2 wraps around S2 underthis map. The degeneracy point d = 0 is equivalent to a ficti-tious monopole with topological charge 1/2. In this sense, thed(k)/

[2 |d(k)|3

]factor represents the field generated by the

monopole, the ∂d(k)∂kx

× ∂d(k)∂ky

factor counts the infinitesimalarea on the two-dimensional surface d(k), and the quantizedvalue of the integration of Fxy(k) coincides exactly with thecelebrated Gauss’ law in electromagnetics.

D. Topological photonics

Here, topological photonics means simulating the topolog-ical phases on superconducting quantum circuits. However,the transfer from electronics to photonics also imposes sev-eral challenges in theory and experiments. An obvious andimportant issue is how to measure the topological invariants.For electronic systems, the topological invariant of the bandsis measured through the quantized Hall conductance in trans-port experiments. This method is nevertheless not applica-ble in photonic systems due to the absence of the fermionicstatistics. In the optics community, this problem is overcomethrough the realization of the gedanken adiabatic pumpingprocess proposed by R. B. Laughlin [14, 58–60]. This methodtakes the advantage of photonic platforms that the ESMs canbe manipulated and visualized in a way much better than inconventional electronic materials. We consider a 2D lattice

5

with a uniform perpendicular magnetic field φ, as shown inFig. 3, where periodic boundary condition in the y-directionbut open edges on the sample in the x-direction are placed.Due to the periodic boundary condition in the y-direction, athought experiment can be devised in which the sample iswrapped in the y-direction into a cylinder, with x being par-allel to the axis of the cylinder. Therefore, the uniform mag-netic flux φ becomes radial to the cylinder and ESMs emergeat the two edges. Through the cylinder and parallel to the x-axis, a magnetic flux α is inserted. The flux α can shift themomentum of the ESMs. When one flux quanta is threadedthrough the Laughlin cylinder exactly, the ESM spectrum re-turn to its original form with an integer number of ESMs beingtransferred. This integer is the winding number of the ESMs,which is equivalent to the Chern number of the bulk bands.

III. BASICS OF SUPERCONDUCTING QUANTUMCIRCUITS

In this section we briefly introduce the basic concepts ofSQC. For the purpose of our review, we put special emphasison the 2D superconducting TLR and superconducting trans-mon qubit and the parametric coupling mechanism based onthese SQC elements.

A. Superconducting TLR

A TLR consists of a coplanar waveguide formed by a centerconductor separated on both sides from a ground plane. Thisplanar structure can be regarded as a 2D realization of conven-tional coaxial cable [31]. The characteristic electromagneticparameter of the coplanar waveguide are its characteristicimpedance Zr =

√l0/c0 and the speed of light in the waveg-

uide v0 = 1/√l0c0 , with c0 and l0 being the capacitance to

ground and the inductance l0 per unit length. Typical valuesof these parameters are Zr ∼ 50Ω and v0 ∼ 1.3 × 108m/s[32]. In the continuum limit, the Hamiltonian density in thebulk of the wave guide has the form

HTLR =1

2c0Q(x, t)2 +

1

2l0[∂xΦ(x, t)]

2, (9)

where Q(x, t) is the charge density distribution functionon the waveguide and the generalized flux Φ(x, t) =∫ t−∞ dt′V (x, t′) is the canonical conjugate of Q(x, t) withV (x, t) being the voltage to ground on the center conductor.By using Hamilton’s equations, the wave equation describingthe surface plasma propagation along the transmission line,

v20

∂2Φ(x, t)

∂x2− ∂2Φ(x, t)

∂t2= 0, (10)

can be derived.A resonator mode, or a cavity mode, is further establish

from Eq. (9) by imposing boundary conditions at the two endpoints of the waveguide, which can be achieved by microfabri-cating a gap in the center conductor. A TLR has a fundamental

frequency ω0 = πv0/d and harmonics at ωm = mω0 with dbeing the length of the waveguide. Therefore, we get a single-mode λ/2 TLR resonator if we focus only on the lowest modeof the coplanar waveguide. The similar derivation can also beapplied to the shorted boundary condition considered in thefollowing section where the two ends are grounded. It can benoticed that such coplanar waveguide geometry is flexible anda large range of eigenfrequency ω0 can be achieved. TypicalSQC experiments use TLR resonators with ω0 ∈ [5, 15]GHz,which is much larger than the thermal temperature in the diluterefrigerator (∼ 20mK) such that thermal excitation is sup-pressed, and much smaller than the superconducting energygap of aluminum or niobium such that the unwanted quasipar-ticle excitation can be safely neglected. In addition, in experi-ments it is always advantageous to maximize the internal qual-ity factor of the TLR mode. With current level of technology,the loss of the TLR mode due to its coupling to uncontrolleddegrees of freedom in the environment can now be effectivelysuppressed, and internal Q-factor in the range [105, 106] cannow be routinely achieved [28, 29].

B. Superconducting transmon qubit

The described 2D TLR is a linear circuit. However, nonlin-earity is essential to establish two-level qubits in which quan-tum information is encoded and manipulated. Meanwhile,superconductivity allows the introduction of non-dissipativenonlinearity in mesoscopic electrical circuits as Josephsonjunction can be regarded as a nonlinear inductance. For twosuperconducting metals separated by a thin insulating barrier,a dissipationless supercurrent could flow between them, de-scribed by the relation [61]

I = Ic sinϕ, (11)

with Ic being the critical current of the Josephson junction,and ϕ is the phase difference between the superconductingmetals on the two sides of the junction (see Fig. 4(a)). More-over, the time dependence of ϕ is related to the voltage acrossthe junction as

dϕ

dt=

2π

Φ0V, (12)

with Φ0 = π~/2 being the flux quantum. Here ϕ can beassociated with the previously introduced generalized flux asϕ(t) = 2πΦ(t)/Φ0 = 2π

∫dt′V (t′) /Φ0. Therefore, we can

regard the Josephson junction as a nonlinear inductance

LJ(Φ) =

(∂I

∂Φ

)−1

=Φ0

2πIc

1

cos (2πΦ/Φ0). (13)

With the nonlinear Josephson junction inductance, we canbuild a nonlinear LC resonator by replacing the inductanceof a linear LC oscillator with a Josephson junction. In this sit-uation, the energy levels of the nonlinear LC resonator are nolonger equidistant, and we can get an effective two level qubitif we restrict our attention to the lowest two energy levels.

6

More explicitly, we consider the circuit diagram of a trans-mon qubit depicted in Fig. 4(b), which consists of a Josephsonjunction shunted by a large capacitance C. Since the energystored in the Josephson junction can be written as

E =

∫IV dt = −EJ cos

(2π

Φ0Φ

)(14)

with EJ = Φ0Ic/2π being the Josephson energy of the junc-tion, the quantized Hamiltonian of the capacitively shuntedJosephson junction therefore reads

HTransmon =Q2

2C− EJ cos

(2π

Φ0Φ

)= 4ECn

2 − EJ cosϕ, (15)

where C denotes the total capacitance around the transmonqubit, EC = e2/2C is the charging energy, and n = Q/2eis the canonical momentum of ϕ describing the excess num-ber of Cooper pairs on the metallic island. The spectrum ofHTransmon depends on the ratio EJ/EC . It has been shownthat, in the range EJ/EC ∈ [20, 80], the energy levels ofHTransmon are insensitive to the perturbation from the low-frequency 1/f background charge noise [33]. In this situa-tion HTransmon describes a virtual particle with generalizedcoordinate ϕ, generalized momentum n, and effective mass(8EC)−1, moving in an anharmonic potential −EJ cosϕ. Inturn, HTransmon can have an anharmonic oscillator form

HTransmon ≈ ~ωqb†b−EC2b†b†bb, (16)

where b(b†) is the annihilation(creation) operator of the an-harmonic oscillator, and ~ωq = E1 − E0 =

√8ECEJ − EC

denotes the energy spacing of the lowest two levels (En be-ing the eigenenery of the Fock state |n〉). The anharmonicityof the resonator is described by the level difference −EC =(E2 − E1) − (E1 − E1). Such anharmonicity factor sets theupper speed limit of manipulating the lowest two levels.

In addition, while the critical current Ic depends on thejunction size and the material parameters, we can have tun-able effective critical current by replacing the single junctionby a superconducting quantum interference device (SQUID)loop formed by two identical junctions and penetrated byan external flux Φext, as shown in Fig. 4(c). In this sit-uation the SQUID can be regarded as a Josephson junctionwith tunable critical current Ic(Φext) = 2Ic cos(πΦext/Φ0)with Φ0 = π~/2 being the flux quanta. Such tunability ori-gins from the Aharonov-Bohm interference of the supercur-rent flowing across the two junctions. Therefore, replacingthe single junction by a SQUID loop yields directly a trans-mon qubit with flux-tunable eigenfrequency. Moreover, as theflux in the SQUID loop can be tuned with a very high fre-quency in practice, this additional control knob results in theparametric driving mechanism which we will discuss in thesubsequent sections.

a

b

b ca

S I Sex

FIG. 4. Schematic plot of (a) a Josephson junction consistingof a weak link between two spuerconducting islands, (b) a trans-mon qubit formed by a Josephson junction with maximal Joseph-son coupling energy EJ shunted by a large capacitance, and (c) afrequency-tunable transmon qubit, with the single junction replacedby a SQUID penetrated by external magnetic flux Φext.

C. Static couplings

The described TLR resonators and transmon qubits can beconnected to form a scaled lattice, and their individual modescan be used to play the role of localized Wannier modes of thelattice. To perform meaningful quantum simulation, one hasto induce the coupling between the connected SQC elements.An intuitive idea is to connect them directly by coupling ca-pacitances or inductances. Since the voltages and currents onthe SQC elements can be represented by the mode creationand annihilation operators as forms like V = VZPF (a + a†)and I = IZPF (ia− ia†) (ZPF for zero-point fluctuation), thisdirect coupling mechanism results in exchange type coupling∼ CcouplingV1V2 = g(a†b + b†a) between SQC elements.However, it should be notice that the resulting coupling con-stant can not be tuned either in amplitude or in phase. Actu-ally, this kind of coupling results in only static and completelyreal coupling constants. This is particularly not desirable be-cause complex coupling constants between SQC elements isneeded to break the time reversal symmetry and obtain theconsequent nontrivial topology. Also, a method exploitingthe double electromagnetically induced transparency scheme[62, 63] to realize effective magnetic fields for polaritons ina 2D cavity lattice has been proposed. However, the usingof multi-level artificial atoms there are still challenging withcurrent experimental technologies.

IV. PARAMETRIC COUPLINGS

As the static coupling is not tunable, we now introducean alternative method based on the mechanism of parametricfrequency conversion (PFC), which can induce coupling be-tween lattice sites with controllable strengths and phases, anddiscuss a variety of its consequent topological effects. Thismethod is inspired by the laser assisted tunneling techniqueinvented in atomic lattices [64]. With appropriate modulatingfrequency imposed on either the couplers between SQC ele-ments or the on-site eigenfrequencies of the SQC elements,effective parametric coupling between SQC elements can beinduced, where a hopping phase accompanies during the hop-ping process, and from the hopping phases, an effective mag-netic field on the microwave photons can be induced.

7

A. Parametric couplings among TLRs

We first consider how to establish the tunable photonichopping between the lattice sites by using the PFC method[65–68]. The essential physics can be illustrated intuitivelythrough a toy model of two coupled cavities, assuming ~ ≡ 1hereafter, with a Hamiltonian of

HTC = H0 +H12ac (t), (17)

with

H0 =

2∑i=1

ωia†iai, (18)

H12ac (t) = g12(t)[a†1 + a1][a†2 + a2], (19)

where ai/a†i and ωi are the annihilation/creation operators

and the eigenfrequencies of the two cavities, respectively.In addition, the form of H12

ac (t) roots from the inductivecurrent-current coupling of two TLRs [68, 69], and the time-dependent coupling constant g12(t) corresponds to the acmodulation of the coupling SQUID between the TLRs.

Our aim is to implement a controllable inter-TLR photonhopping process. As the two cavities are far off-resonant, therequired photon hopping can hardly be achieved if g12(t) isstatic. Meanwhile, the effective 1↔ 2 photon hopping can beimplemented by modulating g12(t) dynamically as

g12(t) = 2J cos[(ω1 − ω2)t− θ12]

= Jei[(ω1−ω2)t−θ12] + h.c.

. (20)

Explicitly, under the assumption of (ω2 +ω1) |ω2−ω1| J , the effective Hamiltonian in the interaction picture withrespect to U0 = exp−iH0t takes the form

H12eff = U†0H

12ac (t)U0

= g12(t)[a†1eiω1t + a1e

−iω1t][a†2eiω2t + a2e

−iω2t]

= g12(t)[a†1a†2ei(ω1+ω2)t + a†1a2e

i(ω1−ω2)t + h.c.]

≈ g12(t)[a†1a2ei(ω1−ω2)t + h.c.]

= Jei[(ω1−ω2)t−θ12] + h.c.

[a†1a2e

i(ω1−ω2)t + h.c.]

= Ja†1a2eiθ12 + ei[2(ω1−ω2)t−θ12] + h.c.≈ Ja†1a2e

iθ12 + h.c., (21)

with J and θ12 being the effective hopping rate and the hop-ping phase, respectively. Physically speaking, we can imag-ine that there is a photon initially placed in the 1st cavity. Asg12(t) carries energy quanta filling the gap between the twocavity modes, the photon can absorb the needed energy fromthe oscillating coupling strength g12(t), convert its frequencyto ω2, and finally jump into the 2nd cavity. During this hop-ping, the initial phase of g12(t) is adopted by the photon. Thishopping process can be further described in a rigorous way byusing the rotating wave approximation: in the rotating frameofH0,H12

ac (t) reduces to the form in Eq. (21), with the fast os-cillating a†1a

†2 + h.c. term neglected. The essential advantage

of the described PFC method is that both the effective hop-ping strength J and the hopping phase θ12 can be controlledon-demand by the modulating pulse g12(t). In particular, theimplementation of the hopping phase θ12 is important as weare using this method to synthesize artificial gauge fields.

From the above derivation, it becomes clear that the essen-tial of realizing this parametric coupling formalism is to real-ize the tunable coupling constant g12(t). Here we rememberthat a SQUID can be regarded as a Josephson junction withtunable critical current controlled by the external bias flux.If we can couple the currents from two SQC elements by aSQUID, and if the SQUID is working in the linear region, wecan have a tunable coupling between these two SQC elementsbecause in this situation the coupling SQUID can be regardedas a tunable inductance. In the next section we come back tothe explicit circuit realization of this idea.

B. Parametric tunable coupling among qubits

In most quantum information processing tasks, tunable cou-pling between qubits is of particular importance. However,this tunability is usually achieved at the cost of introducingadditional decoherence or circuit complexity [70–76]. Alter-natively, parametric modulation of qubits’ frequencies can beused to realize this tunability [77–79] without coupling de-vices, and thus simplifies the circuit.

For the parametrically tunable coupling, we here first re-view the 1D chain case [78]. This tunability method can alsobe directly applicable to the time-dependent tuning [79] and2D [80] cases. The system Hamiltonian for N 1D capacitivelycoupled transmon qubits [33, 81] is

H =

N−1∑j=0

ωj2σzj +

N−1∑j=1

gjσxj−1σ

xj , (22)

where σz,xj are the Pauli operators on the jth qubit Qj withtransition frequency ωj , and gj is the static coupling strengthbetween qubits Qj−1 and Qj , which is fixed. The couplingstrength can be fully tuned via modulating the qubits Qj with1 ≤ j ≤ N − 1 so that the frequencies of the qubits are oscil-lating as

ωj = ωoj + εj sin(νjt+ ϕj), (23)

where ωoj is the fixed operating qubit-frequency, εj , νj , andϕj are the modulation amplitude, frequency, and phase, re-spectively. When ∆j = ωoj − ωo(j−1) equals to νj (−νj)for odd (even) j, in the interaction picture and ignoring thehigher-order oscillating terms, we get a chain of qubits andthe interaction among then is the nearest-neighbor resonantXY coupling i.e.,

HI =

N−1∑j=1

g′jσ+j−1σ

−j + H.c., (24)

where σ± = (σx ± σy)/2 and the effective tunable coupling

8

10

8

6

4

2

g 1'/2π

(MH

z)

12010080604020ε1/2π (MHz)

6

5

4

3

2

1

g 2'/2π

(MH

z)

14012010080604020ε2/2π (MHz)

220200180160ν 2

/2π

(MH

z)

4003002001000t (ns)

220200180160ν 2

/2π

(MH

z)

4003002001000t (ns)

14012010080ν 1

/2π

(MH

z)

4003002001000t (ns)

14012010080ν 1

/2π

(MH

z)4003002001000

t (ns)

(b)

ε1/2π= 30 MHz

ε1/2π= 80 MHz

FIG. 5. Experimental parametric tunable coupling g′1 between Q0

andQ1 as a function of ε1, whereQ0 is operated at a fixed frequencyω0 and initially prepared in its first excited state |e〉, while Q1 is fluxbiased in a time-dependent way so that its frequency is oscillating inthe form as in Eq. (23) and initially prepared in |g〉. Coherent excita-tion oscillation between Q0 and Q1 as a function of ν1 and time t atfixed ε1 produces a chevron pattern, from which the correspondingg′1 (dots) can be extracted, i.e., the frequency of the full-scale oscilla-tion (ν1 = ∆1) pattern is the corresponding g′1. Two typical chevronpatterns are presented in the inset, with blue and red correspondingto |g〉 and |e〉 ofQ1, respectively. The red line is fitted using Eq. (25)with g1/2π ≈ 19 MHz. Adapted from Ref. [78].

strengths are

g′j = gjJ1(ηj)×

ei(ϕ1+π/2), j = 1;J0(ηj−1)e−i(ϕj−π/2), j is even;J0(ηj−1)ei(ϕj+π/2), j is odd and 6= 1,

(25)

with ηj = εj/νj , Jm(ηj) being the mth Bessel function ofthe first kind. In this way, the tunability of g′j is achieved bychanging the amplitude εj of the modulation to tune ηj , asexperimental verified in Fig. 5 for a two-qubit case.

For tuning the coupling strength via an auxiliary device, thecurrent wisdom is to capacitively coupled to two weakly ca-pacitively coupled computational superconducting qubits viaa tunable coupler [72], which is actually a superconductingqubit, and thus this architecture is easy to scale up. In thisscenario, the qubit-qubit coupling will attribute to two paths,one is the original fixed weak capacitively coupling, the otheris induced from the stronger large detuned qubit-coupler in-teraction, which can be adjustable by tuning the frequency ofthe coupler. In this way, the total qubit-qubit coupling is stillin the form as that of Eq. (24), and it can also be fully tuned ina continuous way, even from positive to negative values [73].Alternatively, the cross-Kerr ZZ-interaction can be obtainedfor two qubits with large qubit-frequency difference, and, inthis case, controlled-phase gates can be implemented [74, 75].As the total interaction is adjusted from large detuned qubit-coupler interaction, the unwanted qubit interactions can alsobe completely turned off, and thus high-fidelity quantum op-erations can be possible.

V. TBT WITH TLR

In this section, we present the idea of using PFC betweenTLRs to establish SQC lattice and observe the consequent

ci

eq

ai

eq

bi

eq

Measu

remen

t

dev

ice1

Mea

sure

men

t

dev

ice

2

Measurementdevice 3

(a) (b)

1 2 3 4

0.4

0.5

0.6

0.7

0.8

0.9

1

IS (μA)

Lo

cali

zati

on

mode 1

mode 2

mode 3

(c)

0

200

400

0 5 10 15 200

200

400

x (mm)

0

200

400

|f(x

)|2

(m−

1/2

)

Is=1 Aµ Is=2 Aµ Is=3 Aµ

TLR 3

mo

de 2

mod

e 3

TLR 2

mo

de 1

TLR 1

° ( )aa

tdF + F

° ( )bb

tdF + F ° ( )cc

tdF + F

ω1

ω2

ω3

ω4ω2 ω4

ω1

(d)

ω2 ω1

ω3

ω2

(t)ex

F

FIG. 6. (a) Schematic plot of the three-TLR necklace coupled bygrounding SQUIDs. The loop of each SQUID is penetrated by astatic bias Φ and an ac modulating pulse δΦ. (b), (c) The localizationproperty of the eigenmodes. In (c), the normalized mode functionsof the lowest eigenmodes of the necklace versus the critical currentIS of the grounding SQUIDs are depicted. The three panels of (c)describe the mode functions of the eigenmodes 1, 2, and 3 from topto bottom. In (b), we quantify the localization property of the mtheigenmodes by the energy stored in the mth TLR versus the totaleigenenergy of the mth mode. Adapted from Ref. [82].

topological effects.

A. TBT in quasi 1D

We then illustrate explicitly the implementation of the de-scribed PFC method in SQC system. In particular, we con-sider how to synthesize artificial gauge field for the microwavephotons with this method, and a variety of the consequentnovel topological effects. Our first work in this direction isto investigate the chiral photon flow effect in a TLR necklace[82], which has been demonstrated experimentally by usingthree coupler coupled superconducting transmon qubits [76].As shown in Fig. 6(a), the proposed circuit consists of threeTLRs coupled by three grounding SQUIDs with very large ef-fective Josephson coupling energies and can be described by

H0 =

3∑m=1

ωma†mam, (26)

where am are the annihilation operators of the λ/2 modes ofthe three TLRs and ωm are the corresponding eigenfrequen-cies. Here the parameters are specified as (ω1, ω2, ω3) =(ω0, ω0 − ∆, ω0 + 2∆) with ω0/2π ∈ [10, 15] GHz and∆/2π ∈ [1, 2] GHz. Such configuration is for the followingapplication of the PFC scheme and can be achieved in exper-iment through the length selection of the TLRs [65–67, 83].

9

0

0.5

1

0

0.5

1

Ene

rgy

(uni

t of

ω1/2

π)

0 10 20 30 40 50 60 70 800

0.5

1

Time (ns)

TLR 1 TLR 2 TLR 3

FIG. 7. Chiral photon flow in the TLR necklace. We have consideredthree situations θΣ = π/2, π, and 3π/2, and depict the correspond-ing results in the three panels from top to bottom. In addition, we setthe homogeneous coupling strength g/2π = 20 MHz and decay rateκ/2π = 250 kHz. Adapted from Ref. [82].

The mode structure of the lattice can be explained in an in-tuitive way: from the point of view of a particular TLR, itsgrounding SQUIDs act as shortcuts of its neighbors and it-self, as most of the currents from the neighboring TLRs willflow directly to the ground through their common groundingSQUIDs without crossing each other. The separated and lo-calized modes for the TLR lattice can then be approximatedby the individual λ/2 modes of the TLRs, as the small induc-tances of the grounding SQUIDs impose the grounding nodesat the edges of the TLRs.

We turn to the issue of establishing the inter-TLR cou-pling through the described PFC method. First we take the1↔ 2 coupling as an example. Physically speaking, the com-mon grounding SQUID can be regarded as a tunable mutualinductance between these two TLRs, because currents fromthese two TLRs flow to the ground through the same ground-ing SQUID. Therefore, we add an ac flux driving δΦ12(t) =∆Φ12 cos[(ω1 − ω2)t − θ12] to modulate the effective induc-tance of the 1↔ 2 SQUID, such that the parametric couplingHamiltonian

Hint,12 = 2J cos[(ω1 − ω2)t− θ12]a†1a2 + h.c., (27)

can be induced with J being the coupling strength propor-tional to ∆Φ12. In the rotating frame of H0 in Eq. (26), aneffective 1↔ 2 hopping process described by

Hint,12 = Jeiθ12a†1a2 + h.c., (28)

can then be obtained, which exactly reproduced Eq. (21).Moreover, we can add similar pumping pulses on the othertwo SQUIDs and finally get the effective Hamiltonian

HI = J[eiθ12a†1a2 + eiθ23a†2a3 + eiθ31a†3a1

]+ H.c.. (29)

Here the hopping strength J and the three hopping phases θ12,θ23, and θ31 can be modulated by the amplitudes and the initial

phases of the modulating pulses, respectively. In this step, thevector potential A(x) can manifest its presence through thePeierls substitution

θij =

∫ i

j

A(x) · dx, (30)

and the loop summation of the hopping phases in turn has thephysical meaning of the synthetic magnetic flux:

θΣ =

∮A(x) · dx =

∫∫B(x) · dS. (31)

The synthetic magnetic field leads to the chiral photon flowin this necklace, which is the photonic counterpart of theLorentz circulation of electrons in an external magnetic field.We assume initially a photon is populated in the 1st mode andnumerically simulate its subsequent time evolution by usingthe master equation approach. Results related with the threesituations θΣ = π/2, π and 3π/2 are plotted in Fig. 7. Fromthe first panel corresponding to θΣ = π/2, we can observethe clear temporal phase delay of the energy population in thethree TLRs. Such pattern implies that the photon flow on thelattice is unidirectional, first from TLR 1 to TLR 2 and thenfrom TLR 2 to TLR 3. This chiral character is a consequenceof the time-reversal symmetry breaking in this necklace. Herewe should emphasize that the chiral character of the photonflow survives although we have chosen the cavity decay rateκ much stronger than reported experimental data during thisnumerical simulation, Similarly, in the third panel which cor-responds to the opposite magnetic field θΣ = 3π/2, the chiralphoton flow is in the opposite direction. Meanwhile, in thesecond panel corresponding to the trivial case θΣ = π, theenergy transfer exhibit symmetric patterns.

Meanwhile, we can notice that our discussions up to noware mainly restricted to the synthesization of and the effectinduced by Abelian background gauge field [84]. On theother hand, non-Abelian gauge field, in particular the spin-orbital coupling (SOC) mechanism [85], has already playedimportant roles in the physics of topological quantum mat-ters [8]. In recent year, synthetic SOC has been experimen-tally implemented in artificial systems including ultra-coldatoms [86], exciton-polariton microcavities [87], and coupledpendula chains [88]. Therefore, curious questions arise thatwhether the proposed PFC method can be extended to synthe-size non-Abelian gauge in the SQC architecture, and whetherthe synthetic non-Abelian gauge field would bring any newphysics that can be observed in the near future.

These two issues have been addressed in our recent work inRef. [89], where we derived explicitly that the PFC methodcan be extended to synthesize U(2) non-Abelian gauge fieldsin coupled TLR lattices, as shown in Fig. 8. The de-tailed scheme is similar to the described PFC scheme forthe Abelian gauge field. For the lattice sites, we merelyneed to incorporate the λ modes of the TLRs into consid-eration such that the λ/2 and the λ modes of the individ-ual TLRs play the role of the pseudo-spin components. Asthe lattice sites become multi-component, the linking vari-ables Ur′,r = exp

[i∫ r′

rdx · A(x)

]become matrix-valued as

10

B

CC

B

AAA

B

C

A

(a) (c)

(b) (d)

A

B

C B

C

AAA

C

B

FIG. 8. Implementation of a quasi 1D AB caging model. (a) Illustra-tion of the model in a periodic rhombic lattice. (b) The extension of(a) to a two component non-Abelian case and its implementation withTLR lattice, where the TLRs with three different lengths are con-nected at the common ends by the grounded coupling SQUIDs. (c)Illustration of the effective [n, α, σ1]⇔ [m,β, σ2] coupling inducedby the PFC via the ac modulation of the coupling SQUID. (d) Spec-trum and coupling of two neighboring TLR sites, with δ = ωβ1−ωα1

being the frequency difference of the neighboring [n, α] and [m,β]TLRs. Adapted from Ref. [89].

the non-Abelian background potential A (x) is matrix-valued.The establishment of A (x) in turn becomes the establishmentof each of the hopping branch defined by the matrix elementsof Ur′,r. Thus, the generalization of the PFC method to thesynthesization of the non-Abelian gauge field is straightfor-ward: for TLRs coupled by common SQUIDs, we only needto guarantee that their eigenfrequencies are far off resonant,see Fig. 8(d), such that a multi-tone modulation can be ap-plied to manipulate each hopping branch in the linking vari-ables independently.

The physics of Aharonov-Bohm (AB) caging in a periodic1D rhombic lattice is sketched in Fig. 8(a) [90–92], with itsTLR lattice realization shown in Fig. 8(b). Here each latticesite consists of N (pseudo)spin modes. When exposed to anU(N) background gauge field A, the Hamiltonian of the lat-tice takes the form

H = −J∑

〈nκ,mβ〉

κ†nUnκ,mββm, (32)

where κn = [κn,1, κn,2, · · · , κn,N ]T with κ = A,B,C is thevector of the annihilation operators of the κth site in the nthunit-cell, and Unκ,mβ = exp

[i∫ [n,κ]

[m,β]dx · A(x)

]. We then

define the non-Abelian AB caging by the nilpotency of inter-ference matrix

I =1

2(U2U1 + U4U3) , (33)

with U1, U2, U3, and U4 being the rightward link variableslabeled in Fig. 8(a). For the Abelian situation N = 1,AB caging coincides with the π magnetic flux penetration ineach loop of the lattice. In this situation, a particle initiallypopulated in the site [n,A] cannot move to sites further than[n ± 1, A] due to the destructive interference of the two up

and down paths shown in Fig. 8(a). For the non-Abelian sit-uation N > 1, the matrix feature of A offers much more richphysics: A matrix-formed nonzero interference matrix I canbe nilpotent. To illustrate this idea, let us take an U(2) designas the minimal realization of the proposed non-Abelian ABcaging. The link variables are set as

U1 = U4 =

[1 00 1

], U2 =

[0 11 0

], U3 =

[0 1−1 0

], (34)

such that a nilpotent

I =1

2(U2U1 + U4U3) =

[0 10 0

], (35)

can be achieved. That is, an excitation in the [n,A, 1] modecan move only to the [n + 1, A, 2] mode in a rightward way,but then it can move to neither the [n+2, A] nor the [n−1, A]sites. Similarly, an excitation in the [n,A, 2] mode can moveto the [n−1, A, 1] mode, but it cannot arrive the [n−2, A] and[n + 1, A] sites. This is due to the fact that the non-Abelianfeature of the synthetic gauge field leads to exotic asymmetricspatial configuration of excitations, which is determined byboth the nilpotent power I and the initialization of the photon.These exotic new features stem from the non-Abelian natureof A and have no Abelian analog.

B. TBT in 2D

With the developed technique of synthesizing gauge fieldsby the PFC method, we further consider the quantum sim-ulation of integer QHE and the consequent measurement ofinteger topological quantum number in a 2D photonic lat-tice [60]. For this aim, we propose a square lattice con-sisting of TLRs with four different frequencies, placed in asquare lattice form and grounded by SQUIDs at their com-mon ends, as shown in Fig. 9(a). Similar to Ref. [82], theeigenfrequencies of the four kinds of TLRs are specified as[ω0, ω0 + ∆, ω0 + 3∆, ω0 + 4∆], respectively. In each of thegrounding SQUIDs, a developed three-tone PFC pulses is ap-plied, such that the hopping strength J and hopping phase ofevery branch can be independently controlled. Such in situtunability can in turn lead to the synthesization of the follow-ing artificial gauge field for microwave photons on the lattice

A = [0, Ay(x), 0] ,

B = Bez =

[0, 0,

∂

∂xAy(x)

]. (36)

For the aim of measuring the integer topological invariants ofthis system, a nontrivial ring geometry is further endowed tothe TLR lattice, that is, we consider anNx×Ny square latticewith an nx × ny vacancy at its middle, as shown in Fig. 9(b).An uniform magnetic flux φ in each plaquette of the latticeand an extra magnetic flux α at the central vacancy can bepenetrated through the careful setting of the hopping phasesof the hopping branches. This configuration closely mimicsthe previously discussed Laughlin cylinder shown in Fig. 3.

11

xN

yNa

xnyn

f

J0EextH

1 2

34

pumping

measure

pκr

in

out

(a)

(b) (c)

1 2

34

FIG. 9. (a) Sketch of the TLR square lattice simulator of IQHE.The four TLRs building the lattice are labelled by the four differentcolors and the grounding SQUIDs are labelled by the black dots.The effective hopping between the TLRs is established through thePFC modulation of the SQUIDs. (b) Geometric configuration of theproposed lattice. (c) Chiral property of the ESMs located at the innerand outer edges. The lattice sketched in (b) can be regarded as thegluing of a simply-connected plane by its two sides (the upper panel).Such gluing process can intuitively interpret a variety of oppositeproperties of the inner and the outer ESMs, with the opposite chiralproperty being the prime example (the arrows in the lower panel).Adapted from Ref. [60].

Following the bulk-edge correspondence principle [14], theESMs of the lattice provides an efficient handle to probe thetopological invariants of the bulk band. Explicitly speaking,the ESMs located in the band gap between the hth and the (h+1)th bands can be characterized by the topological windingnumber wh, which can be related to the Chern number Ch ofthe hth band as

Ch = wh − wh−1. (37)

Therefore, the measurement of the Chern numbers Ch of thebulk bands is equivalent to the measurement of the windingnumbers wh of the ESMs. For the rational magnetic flux situ-ation φ/2π = p/q with p, q being co-prime integers, the topo-logical winding numbers wh of the ESMs are given by theDiophantine equation

h = shq + whp, |wh| ≤ q/2, (38)

with h being the gap index and wh, sh being integers.Since the spatial configuration of the proposed lattice is

equivalent to the Laughlin cylinder shown in Fig. 3, we canrealize the adiabatic Laughlin pumping of the ESMs throughcontrolling the central vacancy flux α. By increasing αmono-tonically from 0 to 2π, we can observe that the spectrum of thelattice varies and returns finally to its original form, with aninteger number of ESMs been transferred. This integer is ex-actly the winding number th of the ESMs. In Figs. 10(a) and

-:0:-:0:

kp

-:0:-:0:

-1.6 -1.5 -1.4 -1.3 -1.2

-:0:

-:0:-:0:

kp

-:0:

0.5

1

1.5

2

2.5

-:0:

1.2 1.3 1.4 1.5 1.6

-:0:

-1.75 -1.5 -1.25 -1.0

1.0 1.25 1.5 1.75 2.0

-6 -5 -4 -3

+MP=T3 4 5 6

(a) (b) (c)

(d) (e)-2.0

(f)

nMP

FIG. 10. Laughlin pumping in the proposed square lattice. Here theuniform magnetic field is set as φ/2π = 1/4 for (a), (c), (d), and (f),and φ/2π = 1/5 for (b) and (e). In each of the subfigures, the panelscorrespond to α/2π = 0, 1/4, 1/2, 3/4 and 1 from top to button,respectively. For (a), (b), (d), (e), the pumped ESMs are located atthe outside edge of the lattice, while for (c) and (f) the pumped ESMsare located at the inner side edge. Adapted from Ref. [60].

10(d), the ESMs spectrum versus α is numerically simulatedfor p/q = 1/4. The guidance of the dashed white lines andthe solid yellow arrows clearly indicates the movements of theESM peaks in the 1st and 3rd gaps, which is in agreement withthe calculated topological winding numbers w1 = −w3 = 1.A similar situation φ/2π = 1/5 is also calculated and shownin Figs. 10(b) and 10(e), where we observe that the ESMs inthe 2nd and 3rd gaps cross two peaks with opposite movingdirections during the whole pumping process. This result is inaccordance with the calculated w2 = −w3 = 2. In addition,Figs. 10(c) and 10(f) indicate that the ESMs in the inner andouter edges take the opposite moving directions during thepumping process. In summary, this measurement scheme ofthe topological invariants can be summarized as follow: dur-ing the Laughlin pumping process, we only need to follow themovement of the ESM peaks, as ESMs in the hth gap willmove by |wh| peaks, with the moving direction determined bythe sign of wh. From the obtained wh, we can directly calcu-late the Chern numbers Ch following the relation Eq. (37).

The extension of the discussed chiral photon flow on thenecklace is the coherent chiral photon flow dynamics of theESM on the lattice. Such investigation can offer an intuitiveinsight into the chiral property of the ESMs. We assume thatwe initialize the lattice in its ground state and then add a driv-ing pulse on an edge site rp with the pumping frequency ΩSP

being an ESM eigenfrequency. Then we numerically simu-late the time evolution of the lattice and sketch several screen-shots of the photon flow dynamics in Fig. 11, from which wecan observe the unidirectional photon flows around the edgeand the directions of the chiral photon flow are determined byboth ΩSP and rp. If we choose rp on the outside edge, wecan find out that the chiral photon flow is clockwise /counter-clockwise if we choose ΩSP in the 1st/3rd gap (Figs. 11(a) and

12

1

12

241

12

241

12

241

12

24

0

0.3

0.6

0.9

1.2

1 12 24

1

12

241 12 24 1 12 24 1 12 24

(e)

(d)

(c)

(b)

(a)

hayrari

FIG. 11. Chiral photon flow dynamics on the TLR lattice. Thepumping sites are selected as r = (1, 13) for (a), (c), and (e), andr = (9, 13) for (b) and (d). In addition, we set the imposed syntheticmagnetic field as φ/2π = 1/4 and α/2π = 0; ΩSP/J is chosenas [−1.76, −1.97, 1.47, 1.97, −1.75] for (a)–(e), respectively. Thetimes of the panels from left to right are arithmetic progressions withthe first terms J1/2π = [6, 1, 6, 1, 15]J−1 and the common dif-ferences ∆J/2π = [13, 5, 13, 5, 2.5]J−1 for (a)–(e), respectively.In the calculation of (e) we have incorporated the effect of latticedisorder and defect. Adapted from Ref. [60].

11(c)). The different chirals of the ESMs in different energygaps can be understood by regarding the chiral photon flow asa macroscopic rotating spin [20]: if we place such a rotatingspin in a magnetic field B, its energy will split according toits spinning direction. Moreover, from Figs. 11(a) and 11(b)(also Figs. 11(c) and 11(d)) we can observe that the inner andouter ESMs in the same gap have opposite chiral properties.This can be explained by the spatial topology of the proposedlattice. As shown in the upper panel of Fig. 9(c), if we tear thelattice apart, we can get a simply-connected plane with onlyone connected edge. Meanwhile, the inverse of the tearing,that is, the gluing of the two sides marked by dashed lines,cancels the ESM flow (marked by the arrows) on the gluedsides, and results in finally two closed circulations located atthe inner and outer edges with opposite circulating directions.As each of the hopping branches can be tuned in a site re-solved way by the proposed PFC formalism, we can expectthe future experimental demonstration of this “tearing-and-gluing” process.

The influence of disorder and defect on the chiral flow isalso calculated. In this numerical simulation, we add Gaussiandistributed diagonal and off-diagonal disorder with σ(δωr) =

σ(Jr′r) = 0.05J . In addition, we place a 2 × 2 hindrance onthe upper outer edge with δω(12−13,23−24)/J = 30. In thissituation, the topological robustness of the ESMs is clearlyverified by the survival of the chiral photon flow displayed inFig. 11(e).

C. Extension: Parametric amplification

A direct extension of the proposed PFC method is the para-metric amplification (PA) process. Compared with the de-scribed synthetic gauge field issue, this process is unique forphotonic systems. We come back to Eq. (20). Now we mod-ulate g12(t) by the summation tone of the two resonators:

g12(t) = 2J cos[(ω1 + ω2)t]. (39)

Then we can get an effective nondegenerate parametric am-plification Hamiltonian

H12eff = eiH0tH12

ac (t)e−iH0t

≈ g12a†1a†2 + h.c., (40)

which takes the similar form of the fermionic p-wave pairing[93]. From this point of view, the road map of simulatingone dimensional fermionic topological superconductor mod-els is clear: The lattice sites are built by hard-core boson,e.g. TLRs with strong nonlinearity or superconducting trans-mon qubits such that the fermionic statistics can be inducedby the Jordan-Wigner transformation [54, 94], while the hop-ping and pairing terms between the lattice sites are induced bythe PFC and parametric amplification modulation of the cou-pling grounding SQUIDs, respectively. With further modifieddispersive amplification modulation method, this architecturecan be even exploited to simulate spinful fermionic models.Actually, this idea has already been illustrated in Ref. [95],where we proposed a 1D hard-core boson lattice built by su-perconducting transmon qubit as a faithful simulator of thefollowing fermionic time-reversal invariant DIII model [96]

HDIII = −µ(c†j,εcj,ε − 1)

−∑jε

(gc†j,εcj+1,ε + i∆cj+1,εcj,ε + h.c.), (41)

where c†j,ε is the creation operator of the spin-ε fermion on thejth site, g, µ, ∆ denote the real-value hopping strength, thechemical potential, and the amplitude of p-wave pairing pa-rameter. The nontrivial coupling between the lattice sites isinduced through the inductive connection between transmonqubits [33, 97–99]. In particular, we developed a general-ized parametric amplification approach to facilitate the exoticU(1) gauge factor emerged in the Jordan-Wigner bosoniza-tion. Furthermore, we proposed that the odd-parity Kramersdoublet ground states can be used as the basis of topologicalqubits, and with these topological bases the universal topolog-ical quantum gates can be implemented in principle.

13

VI. TBT WITH SUPERCONDUCTING QUBITS

A. TBT in 1D

With the above tunable coupling among qubits, we are read-ily in the position to simulate topological quantum phases onsuperconducting quantum circuits. The first example is the1D SSH model with the Hamiltonian of Eq. (2), introducedin Section II B and it can be realized by parametrically tuningthe inter-qubit coupling strength in Eq. (24) in the requireddimerized form. Note that, although the Hamiltonian in Eq.(24) is of the bosonic nature, in the single excitation subspace,its topology is equivalent to that of the SSH model, which hastwo different topological phases with different winding num-bers. Specifically, when the qubit couplings are tuned intogA < gB (gA > gB) configuration, with gA ≡ g′2j+1 andgB ≡ g′2j+2 in Eq. (24), the winding number will be 0 (1), andthen the system is in a topologically trivial (nontrivial) state.Then, by monitoring the quantum dynamics of a single-qubitexcitation in the chain, the associated topological quantitiescan be inferred [100].

B. TBT in 2D

The gauge effect is essential in investigating novel phe-nomena in modern physics, e.g., for exploring exotic quan-tum many-body physics. In this subsection, we review twoschemes to synthesize gauge field by introducing ac modula-tion on superconducting circuits, and the motivation is to ob-tain an extremely strong effective magnetic field for bosons,which is hard in conventional solid-state systems. Whilethese pioneering examples are associated with the chiral ef-fect, novel quantum many-body properties induced by the in-terplay of the synthetic gauge field, bosonic hopping, and in-teraction of bosons are also deserve further exploration.

We consider a two-leg superconducting circuits consistingof superconducting transmon qubits, as shown in Fig. 12(a).Extending the parametric coupling in Eq. (24) to this two-legcase, the considered model Hamiltonian here is [80]

H =∑νj

(tνje

iϕνj a†ν(j−1)aνj + H.c.)

+∑j

(tje

iϕBj a†Aj aBj + H.c.), (42)

where j denotes the number of the rung, ν ∈ A,B la-bels the leg, a†νj (aνj) is the creation (annihilation) operatorfor the jth site on the νth leg, ϕνj = (−1)

j+1ϕνj + π/2,

tνj = gνjJ0(ηνj−1)J1(ηνj), and tj = gjJ0(ηAj)J1(ηBj)with gνj is the original static hopping strength between thenearest-neighbor sites along the leg ν, gj is the static interlegcoupling strength at the rung j. In this two-leg lattice model,the non-zero magnetic flux case supports a chiral current fol-low within the lattice, which is plotted [80] in Fig. 12. Forsimplicity, we have set ηνj = η and gνj = gj = g, andthus tνj = tj = t0 = gJ0(η)J1(η). Meanwhile, setting

FIG. 12. The chiral currents for a two-leg quantum circuit. (a)Schematic diagram of the two-leg lattice model with an syntheticmagnetic flux per plaquette. The blue and red solid circles representtransmon qubits at the A and B legs. The chiral currents betweenneighboring sites in the single-excitation case for (b) ϕ = 0.1π and(c) ϕ = 0.9π with N = 50, where the thicknesses of the arrowsrepresent their strengths and the sizes of the circles denote their localdensities. The ground-state chiral current as a function of ϕ for (d)single- and (e) two-excitation cases. The solid line in (d) is plottedfrom the analytical result. Reproduced from Ref. [80].

ϕAj = −ϕBj = ϕ/2, the synthetic magnetic flux will beϕj = ϕ. The chiral currents between any neighboring sitesare plotted for (b) ϕ = 0.1π and (c) ϕ = 0.9π cases withN = 50. While in Fig. 12(d), the ground-state chiral currentis plotted as a function of ϕ, where as the increasing of ϕ,the current increases to a maximal value at the critical pointϕc and then decreases. This indicates a quantum phase transi-tion appears, i.e., from the Meissner phase to the vortex phase[101], which survivals in a finite size system, and thus is read-ily observable in current experimental setups. In Fig. 12(e),the ground-state chiral current for the two-excitation case isplotted, where shows similar properties as that of the single-excitation case.

VII. TBT WITH DRESSED STATE SYSTEMS

The above discussed quantum simulation examples on su-perconducting circuits are limited to the spinless cases, as it isdifficult to engineer the spin degree of freedom in either TLRor qubit systems. To simulate spinful systems, one needs to

14

a

G

,1+,1

,2+,2

... b

Δ-gA-gB

2gA

2gB

ωrA

ωrB

FIG. 13. Polariton states and the two-polariton transitions. (a) Theeigen energy spectrum of the JC coupled circuit QED system. Underapproximate parameters of a two-tone driven field, the lowest threelevels, of the V-configuration, can be addressed to realize single-polariton-qubit manipulations. (b) The transitions of two-polaritonsystems with ∆ > gA + gB .

introduce more degrees of freedom. One of the possible plat-form is the TLR-qubit coupled system, a typical circuit QEDsetup [27], and the dressed-state or polariton of which canmimic the spin degree of freedom [102]. In this platform, thesynthetic polaritonic SOC and Zeeman field can be inducedsimultaneously with fully anin situ tunability [103], and thusit provides a flexible platform to explore topological physics.

A. The Dressed-state qubits

We begin with implementing the dressed-state on a conven-tional circuit QED setup, where a superconducting qubit of thetransmon type is capacitively coupled to a 1D TLR. The inter-action Hamiltonian is in the Jaynes-Cummings (JC) form of[27]

HJC = ωa|1〉〈1|+ ωra†a+ g(aσ+ + a†σ−), (43)

where ωa and ωr are the frequencies of qubit and cavity, re-spectively; |n〉r labels the Fock state of the TLR, the groundstate of Hamiltonian in Eq. (43) is |G〉 = |g〉|0〉r ≡ |g, 0〉withits eigenvalue is set to beEG = 0. As shown in Fig. 13(a), forn ≥ 1, due to the TLR-qubit interaction, the two-fold eigen-states for a certain n are

|−, n〉 = cosαn|g, n〉 − sinαn|e, n− 1〉, (44a)|+, n〉 = sinαn|g, n〉+ cosαn|e, n− 1〉, (44b)

and corresponding eigenvalues are

En,± = nωr +1

2

(δ ±

√δ2 + 4ng2

), (45)

where tan(2αn) = 2g√n/δ with the detuning δ = ωr − ωa.

Note that the transition frequencies among eigenstates are dif-ferent, and thus selective transition between two target statesmay be achieved. The three lowest eigenstates are |G〉, |−, 1〉,and |+, 1〉, which form a V-type artificial atom, as shown inFig. 13(a). For the sake of simplicity, we denote the polari-tonic states |+, 1〉 and |−, 1〉 as | ↑〉 and | ↓〉 in the following,

0 0.1 0.2 0.3 0.4 0.50

0.5

1

t( Sμ )

P

| ↑〉| ↓〉| ↑〉| ↓〉

(a)

D

Lab oratory Frame Rotating Frame

0.9979

AB BA

B

B

A

A

A B

J J

(b) (c)

FIG. 14. Implementation of 1D spin-1/2 lattice models on a super-conducting quantum circuit. (a) The “spin-1/2” lattice with A- (red)and B-type (blue) of unit cells, arranged alternately. The zoom-infigure details the equivalent superconducting circuits of the unit. (b)The resonant spin-preserved and detuned spin-flipped couplings ofdifferent spin states. Since the alternative arrangement, two sets ofdriving, JAB and JBA, are needed to preserve the translation symme-try. (c) In the rotating frame, and the levels and designed hoppingsof both A- and B-types of the unit cells can be treated as identical.Adapted from Ref. [103].

mimic a spin 1/2 system, or a spin qubit. Compared with con-ventional superconducting qubits, this combined definition ofqubit has better stability against low-frequency noises [104].

B. SOC for polariton spin

The described parametric coupling scheme for the mi-crowave TLRs can also be used here with the inclusion ofa transmon qubit in each TLR. As the dressed qubits are ofthe half-TLR plus half-transmon nature, the effective hoppingamong them can be induced by only coupling the photoniccomponent. Here, we present an example with two dressedqubits case, i.e., A- and B-type TLR-qubit coupled system,which are different as they have different eigen-frequencies(ωAr and ωBr ) and coupling strengths (gA and gB) in the JCHamiltonian of Eq. (43). Setting δ = 0. The energy spec-trum and their transitions are shown in Fig. 13(b) whenωBr − ωAr = ∆ > gA + gB . The transitions can be inducedwith the parametric photonic coupling, and the addressing ofeach transition is obtained when the frequency of the cou-pling strengths between TLRs, see Eq. (20), meet the reso-nant condition of a target transition [102]. Meanwhile, theenergy difference of the four different hoppings are 2gA or2gB , which can be set to be much larger than the effectivehopping strength between the two polariton states, to ensurethe selective frequency addressing of each transition. There-fore, the fully tunable spin-preserved hopping and SOC termscan be induced in a same setup, and thus this element can

15

naturally be scaled up into different types of lattices to studyvirous SOC topological states.

C. TBT in 1D

We firstly proceed to present a spin-1/2 chain with ad-justable SOC and Zeeman field that can be simulated with theabove JC lattice [103]. We set the A- and B-types units to bearranged in an alternate way, as shown in Fig. 14(a). Then, theenergy differences of the four hopping are also set to be muchlarger than the effective hopping strength to ensure the selec-tive frequency addressing of the target transitions. Moreover,a detuning 2m to the spin-flipped transition tunes, as shownin Fig. 14(b), can induce a tunable spin splitting m for eachcell, in the rotating frame, as shown in Fig. 14(c). In this way,neglecting the fast oscillating terms, we obtain

H ′JC =N∑l

mSzl +

N−1∑l=1

∑ε,ε

(t0,lεεeiϕlεε c†l,εcl+1,ε + h.c.

),

(46)where Sz

l = | ↑〉l〈↑ | − | ↓〉l〈↓ |, c†l,ε = |ε〉l 〈G| is the cre-ation operator for spin-ε polariton state at the lth unit cell, andt0,lεε is the effective hopping strength. Therefore, the Hamil-tonian in Eq. (46) simulates a 1D tight-binding lattice modelconsists of spin-1/2 particles, with m, t0,lεε and ϕlεε beingthe corresponding effective Zeeman energy, coupling strengthand phase, which can respectively be tuned via the deliber-ately chosen of the frequencies, amplitudes and phases of theexternal ac driving field. Note that although we used two typesof unit cells, in an approximate rotating frame and by adjust-ing the two types of coupling parameters JAB and JBA, thetranslation symmetry in the model can still be preserved, andthus different sites can then be treat as identical.

Our first example is a 1D spin-1/2 lattice model whichcan support non-Abelian quantum operations, the consideredHamiltonian is [105]

H =∑l

tz(c†l,↑cl+1,↑ − c†l,↓cl+1,↓) + H.c.

+∑l

hz(c†l,↑cl,↑ − c

†l,↓cl,↓)

−∑l

i∆0(c†l,↑cl+1,↓ − c†l+1,↑cl,↓) + H.c.,

(47)

which relates to Eq. (46) according to the following: t0,lεε =tz , ϕlεε = 0; m = hz; t0,lεε = ∆0 and ϕlεε = −π/2. ThisHamiltonian has a chiral symmetry and the topological invari-ant is related to the non-Abelian charge, and thus obeys thenon-Abelian statistics. In Fig. 15(a), the eigenenergies of achain with 16 lattices is plotted, where two zero-energy ESMsappear, in the middle of the energy gap, which localize at dif-ferent edges. Moreover, a topological quantum phase tran-sition critical value is hz = ±2tz , where the gap closes oropens. The phase diagram of the 1D chain model with arbi-trary ∆0 is plotted in Fig. 15(b), which indicates that a finitesize system is sufficient to demonstrate the non-Abelian na-

0

0.3 11 8 16 24 32-2.5

0

2.5(a)

16

0

0.7(d)(c)

O1 O1

O2 O2

E(t

0)

1.5

0.5

-0.5

-1.5 -0.5

-1-1

(b)1

0.5

81 124 1681 124

-0.3

0.3

0

t z(t

0)

hz(t0)

0.5

-0.5-0.7

FIG. 15. Numerical results for a 1D lattice with 16 unit-cells. (a)Spectrum of eigenenergies in terms of an energy unit t0 with tz/t0 =1, ∆0/t0 = 0.99, and hz/t0 = 0.3, where two ESMs appears.(b) The Phase diagram of the chain with arbitrary ∆0. The yellowand dark blue regions denote the topologically nontrivial and trivialcases with invariant 1 and 0, respectively. The four red circles A, B,C, and D indicate the parameters for demonstrating the non-Abeliantransformations. (c) and (d) are the plot of the left and right ESMsof the 1D chain, respectively, where the solid (dashed) lines plot theamplitude of the | ↑〉 components (| ↓〉 components divided by i)of the corresponding wave functions, with the same parameters as in(a). Adapted from Ref. [105].

ture of the quantum operations with the prescribed parame-ters. Specifically, the circles A, B, C, and D in 15(b) indicatethe parameters for this demonstration. The red arrow repre-sents that O1 is executed before O2 follows; the blue arrowmeans that O1 is executed after O2, see Ref. [105] for details.In Fig. 15 (c) and (d), the two ESMs are plotted with the sameparameters as in (a).

Our second example is to simulate a 3D nodal-loop modelHamiltonian with our 1D system with the other 2D being theparametric dimensions [103]. The implementation is furthersimplified by making a unitary transformation to the simu-lated Hamiltonian in Eq. (46), i.e., V =

∑Nl=1(−i)l−1Il with

Il = |↑〉l〈↑| + |↓〉l〈↓|. When appropriately set the parame-ters of the JC Hamiltonian, the target coupling can be imple-mented using driving with only two tunes [103]. Then, thetransformed Hamiltonian is

Hnod =V †HoriV

=

N∑l=1

m′(ky, kz)Szl +

N−1∑l=1

∑α,α′

it′0c†l,↑cl+1,↑ (48)

− it′0c†l,↓cl+1,↓ + it′0c

†l,↑cl+1,↓ − it′0c

†l,↓cl+1,↑ + h.c.,

where m′(ky, kz) = M + 2d(cos ky + cos kz) with M beingthe effective Zeeman splitting energy and d is the effectivehopping energy along the y and z directions. This accom-plishes the quantum simulation of the topological nodal-loopsemimetals, and the phase diagram is plotted in Fig. 16, wherethe topological features can be detected using the chiral dis-

16

-6 -4 -2 0 2 4 60

0.5

1

1.5

M=t'0

d=t' 0

A

B

C

FIG. 16. The quantum phase diagram of a 3D nodal-loop semimetalmodel with the associated configurations of the winding number,with each color denotes a different phase. The yellow and greenregions are topological nontrivial and one nodal loop will appear inthe ky-kz plane; while two nodal loops will appear for the light blueregion. Their winding number configurations for points A (-2.5, 0.5),B (0, 1) and C (2.5, 0.5) of these three regions are plotted in the threecorresponding insets, where the red (blue) regions of the inserts rep-resent the winding number beingw = 1 (w = 0). The red region is atopological gapped phase with w = 1, but without nodal loops. Thetwo blue regions are both topological trivial gapped phases. Adaptedfrom Ref. [103].

placement operator method, see Ref. [103] for details.

D. TBT in 2D

The simulation of 1D lattice Hamiltonian in Eq. (46) can beextended to a 2D square lattice case, where quantum spin Halleffect can be simulated [106]. Then, we consider a 2D squarelattice model with the Hamiltonian of [107]

H = −t0∑m,n

(c†m+1,ne

iθxcm,n + c†m,n+1eiθycm,n + H.c.

)+∑m,n

χm,nc†m,ncm,n, (49)

where t0 is the neighboring hopping strength; cm,n =(cm,n,↑, cm,n,↓)

T is a 2-component operator for a lattice site(x = ma, y = nb) with a and b are the lattice spacings andm and n being integers; θx = 2πjyσz and θy = 2πkσx with(j, k) being parameters determined by the synthetic magneticflux and spin mixing; χm,n = (−1)nχ is the on-site poten-tial, which is staggered in y-direction. The Hamiltonian inEq. (49) reserves the time reversal symmetry, so it belongs tothe Z2 class topological phase, and it can simulate the quan-tum spin Hall phase, see Fig. 17 for numerical results of thisquantum simulation, and Ref. [106] for details.

0 1/2 1 3/2 2-3-2-10123

kx/:

E/t 0

0 1.0 2.0-0.1

-0.05

6/t0

-

20 40

10203040

x

y

2 4 6

2

4

6

x

y

0.20.40.60.8

(b) 0.10.05

0

(a)

(c) (d)

I

II

FIG. 17. Simulation of the quantum spin Hall effect. (a) The energyband for χ = 0 and k = 0. (b) The phase diagram with Fermienergy in the range of [t0, 2t0], which consists of two phases, i.e., (I)quantum spin Hall phase and (II) the metal phase. The distributionsof the ESMs’ wave functions for (c) 42×42 and (d) 6×6 lattices.Adapted from Ref. [106].

VIII. EXPERIMENTAL REALIZATIONS OF TBT

The essential advantage of the descirbed parametric cou-pling formalisms is that they can establish the coupling be-tween SQC elements in a time- and site- resolved way andoffer consequently unprecedented flexibility to the quantumsimulation of various many-body models. In recent yearsthere have been many experimental progresses in this direc-tion, and it is the purpose of this section to briefly introducethem.

For the formalism of coupling superconducting transmis-sion line resonators by grounding SQUIDs, the first experi-ment in this direction is reported in Ref. [76], where threesuperconducting qubits are connected to form a closed loopgeometry. The coupling between the three qubits is estab-lished by grounding inductive couplers with tunable induc-tance, and the synthetic magnetic field penetrated in the loopis obtained from the oscillation phase of the temporal modu-lation of the three couplers which compensate the frequencymismatch between the qubits. From this point of view, thistechnique falls within the parametric coupling mechanism wehave introduced in Sec. III.

Due to the long coherence time currently achieved by SQC,experiments in Ref. [76] are performed by first preparing theloop in a suitable one- or two-photon Fock state and then fol-lowing its dynamics without any pump during the evolution.As illustrated in Fig. 18, the existence of the synthetic mag-netic field is manifested by the directional circulation of pho-tons in the loop with the circulation direction determined sim-ply by the synthetic magnetic field. When the strong nonlin-earity of the superconducting qubits is taken into considera-tion, two interacting photons would circulate in the directionopposite to that of the single photon. This can be explained bythe additional π phase induced by the Jordan-Wigner transfor-mation. In addition, a few-body interacting ground state can

17

FIG. 18. Experimental realization of artificial magnetic field in aparametric coupled qubit necklace. The circuit and the circulationdynamics of the single- and two-excitation are sketched in the rightof (a), and the time evolution of the excitation probability in the threequbits for the single- and two-excitation at magnetic flux −π/2 aredepicted in the left of (a) and in (b), respectively. Adapted from [76].

also be prepared by adiabatically tuning up the synthetic mag-netic field to the desired value. The same platform has beenfurther exploited to observe the Hofstadter butterfly of one-photon states and the localization effects in the two-photonsector [51].

On the other hand, with the tunable coupling among qubitsin Eq. (24) and its demonstration [78], we are readily in theposition to simulate topological quantum phases on supercon-ducting quantum circuits. The first example is the 1D SSHmodel with the Hamiltonian of Eq. (2), introduced in Sec-tion II B and experimentally demonstrated in Ref. [108] byparametrically tuning the inter-qubit coupling strength in Eq.(24) in the required dimerized form. Note that, although theHamiltonian in Eq. (24) is of the bosonic nature, in the singleexcitation subspace, its topology is equivalent to that of theSSH model, which has two different topological phases withdifferent winding numbers. Specifically, when the qubit cou-plings are tuned into gA < gB (gA > gB) configuration, withgA ≡ g′2j+1 and gB ≡ g′2j+2 in Eq. (24), the winding numberwill be 0 (1), and then the system is in a topologically trivial(nontrivial) state. By monitoring the quantum dynamics of asingle-qubit excitation in the chain, the associated topologi-cal winding number can be inferred, an experimental verifica-tion of which for a four qubits case is presented in Fig. 19.The measured value of the topologically trivial configurationis very close to the ideal one while the deviation from the ideavalue 1 in the topologically nontrivial case is mainly due tothe decoherence effect of the qubits. Similarly, one can alsodetect the topological magnon ESMs and defect states [108].

IX. CONCLUSION AND PROSPECTS

In summary, in this review we have illustrated the applica-tion of the parametric coupling method in investigating topo-logical photonics in SQC system. Such method is a very

00.20.40.60.8

1Pe

0250

500750

1000 Qubit 2

3

time (ns)

00.20.40.60.8

1eP

0250

500750

time (ns) 1000 01

32

5M 1M 5MTopo trivial(a)

(c) (d)

b1a1 b2a2

1000500time (ns)

Topo nontrivial

𝑃𝑑

(b)

(e)

(f)

1M 5M 1Mb1a1 b2a2

0.01.02.0

10005000

𝑃𝑑

-1.0-2.0

2.01.0 0.0

-1.0-2.0 0

time (ns)

Topo trivial

Topo nontrivial

10 Qubit

FIG. 19. Measuring the topological winding number in a four qubitschain with (a) gA = 5 MHz and gB = 1 MHz and (b) gA = 1 MHzand gB = 5 MHz, for the topologically trivial and nontrivial cases,respectively. (c), (d) The four qubits excitation (P ej−1 = |e〉j−1〈e|)dynamics with an initial state |gegg〉 for the two prescribed couplingconfigurations, where dots represent the experimental data whilesolid lines are plotted by numerically simulations from Hamiltonianin Eq. (24). (e), (f) The time-dependent average of the chiral dis-placement operator P d = (P e0 − P e2 ) + 2(P e1 − P e3 ) for the twocases. In the long time limit, the winding number equals twice ofthe time-averaged P d [100], and thus can be extracted. Dots are ex-perimental record, red dashed lines are from numerical simulationswith the black horizontal lines are the corresponding oscillation cen-ters, i.e., the time-averaged P d. Both experiments agree well withthe numerical simulation and theoretical prediction. The measuredtopological winding numbers are 0.030 and 0.718 for the two cases,respectively. Adapted from Ref. [108].

powerful tool in the sense that it can synthesize artificialgauge field for the microwave photons with fully in situ tun-ability. Using the adiabatic pumping process, the simulatedtopologically-protected effects can be probed and further ex-plored in new ways.

Despite the rapid progress in recent years, it is our feelingthat we are still in the beginning rather than the end of this re-search direction. Due to the flexibility of SQC system, we canexpect that the PFC architecture allows the further incorpora-tions of many other mechanisms, including on-site Hubbardinteraction, disorder, and non-Hermicity [109], and their in-terplay will definitely bring us into the realm of more fruitfulphysics. The first of our perspectives is that we should pay

18

extensive attention to the advances in the technical innova-tion of SQC which may have potential application in quantumsimulation. In particular, the reduction of the fabrication er-ror of SQC elements can be exploited to suppress or introducethe mechanism of disorder in a controllable way. In addition,we can notice that the lattice models we discuss in this re-view are typically 1D or 2D. We expect that the advance oftechnology can overcome such tyranny of planar connectiv-ity which places limit on the achievable graph configurationof the lattice models. Another expectation comes from thenotation that the discussion in this review are still limited inthe single particle region. Based on the fact that the syntheticgauge field has been implemented and the nonlinearity can beincorporated into SQC in a variety of manners, an immediatechallenge in this field is to realize the bosonic analogues of thefractional QHE. Our third perspective comes from the fact thatthe decoherence of SQC can now be suppressed to a very lowlevel. This technique can be used to incorporate and manipu-late the gain and loss of the system. From this point of view,SQC can become a versatile platform of investigating non-Hermitian topology [109]. Currently reported experiments ofnon-Hermitian topology focus mainly on the single-particle

non-Hermitian skin effect in 1D lattices induced by asymmet-ric hopping. We then expect SQC can play an important rolein this direction by pushing the research into the region ofhigher dimension by taking into the account of the competi-tion of various mechanisms including interaction, gauge field,and disorder. In addition to fundamental interests, the novelproperties in topological photonics are also promising in fu-ture applications. Similar to the electronic systems, a firstpossible application of this newly discovered topological de-gree of freedom should be the invention of disorder-immunedevices for high-speed information transfer and processing.

ACKNOWLEDGMENTS

We thanks Dr J. Liu for proofreading of the manuscript.This work was supported by the National Natural Science

Foundation of China (No. 11874156 and No. 11774114),and the Science and Technology Program of Guangzhou (No.2019050001).

[1] M. A. Armstrong, Basic topology, Springer, 1st edition, 1978.[2] M. Nakahara, Geometry, Topology and Physics, Institute of

Physics Publishing, Bristol, 2nd edition, 2003.[3] T. Frankel, The Geometry of Physics: An Introduction, Cam-

bridge University Press, Cambridge, 3rd edition, 2012.[4] K. v. Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 1980, 45

494.[5] D. J. Thouless, M. Kohmoto, M. P. Nightingale, M. den Nijs,

Phys. Rev. Lett. 1982, 49 405.[6] M. Z. Hasan, C. L. Kane, Rev. Mod. Phys. 2010, 82 3045.[7] X.-L. Qi, S.-C. Zhang, Rev. Mod. Phys. 2011, 83 1057.[8] A. B. Bernevig, T. L. Hughes, Topological Insulators and

Topological Superconductor, Princeton University Press,Princeton and Oxford, 1st edition, 2013.

[9] A. Bansil, H. Lin, T. Das, Rev. Mod. Phys. 2016, 88 021004.[10] Y. Hatsugai, Phys. Rev. Lett. 1993, 71 3697.[11] Y. Hatsugai, Phys. Rev. B 1993, 48 11851.[12] X.-L. Qi, Y.-S. Wu, S.-C. Zhang, Phys. Rev. B 2006, 74

045125.[13] O. Shtanko, L. Levitov, Proc. Natl. Acad. Sci. USA 2018, 115

5908[14] R. B. Laughlin, Phys. Rev. B 1981, 23 5632.[15] L. Lu, J. D. Joannopoulos, M. Soljacic, Nat. Photon. 2014, 8

821.[16] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi,

L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg,I. Carusotto, Rev. Mod. Phys. 2019, 91 015006.

[17] F. D. M. Haldane, S. Raghu, Phys. Rev. Lett. 2008, 100013904.

[18] Z. Wang, Y. Chong, J. D. Joannopoulos, M. Soljacic, Nature2009, 461 772.

[19] K. Fang, Z. Yu, S. Fan, Nat. Photon. 2012, 6 782.[20] M. Hafezi, S. Mittal, J. Fan, A. Migdall, J. M. Taylor, Nat.

Photon. 2013, 7 1001.[21] M. Hafezi, E. A. Demler, M. D. Lukin, J. M. Taylor, Nat.

Phys. 2011, 7 907.[22] M. O. Goerbig, arXiv e-prints 2009, arXiv:0909.1998.[23] E. Akkermans, G. Montambaux, Mesoscopic physics of elec-

trons and photons, Cambridge University Press, Cambridge,2011.

[24] Y. Makhlin, G. Schon, A. Shnirman, Rev. Mod. Phys. 2001,73 357.

[25] J. Clarke, F. K. Wilhelm, Nature 2008, 453 1031.[26] J. Q. You, F. Nori, Nature 2011, 474 589.[27] M. H. Devoret, R. J. Schoelkopf, Science 2013, 339 1169.[28] X. Gu, A. F. Kockum, A. Miranowicz, Y.-X. Liu, F. Nori,

Phys. Rep. 2017, 718-719 1 .[29] A. Blais, A. L. Grimsmo, S. M. Girvin, A. Wallraff, Rev. Mod.

Phys. 2021, 93 025005 .[30] J. M. Raimond, M. Brune, S. Haroche, Rev. Mod. Phys. 2001,

73 565.[31] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, R. J.

Schoelkopf, Phys. Rev. A 2004, 69 062320.[32] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang,

J. Majer, S. Kumar, S. M. Girvin, R. J. Schoelkopf, Nature2004, 431 162.

[33] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schus-ter, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, R. J.Schoelkopf, Phys. Rev. A 2007, 76 042319.

[34] A. A. Houck, H. E. Tureci, J. Koch, Nat. Phys. 2012, 8 292.[35] S. Schmidt, J. Koch, Ann. Phys. 2013, 525 395.[36] I. Carusotto, A. A. Houck, A. J. Kollar, P. Roushan, D. I.

Schuster, J. Simon, Nat. Phys. 2020, 16 268.[37] M. J. Hartmann, F. G. S. L. Brandao, M. B. Plenio, Nat. Phys.

2006, 2 849.[38] I. Carusotto, D. Gerace, H. E. Tureci, S. De Liberato, C. Ciuti,

A. Imamoglu, Phys. Rev. Lett. 2009, 103 033601.[39] I. Carusotto, C. Ciuti, Rev. Mod. Phys. 2013, 85 299.[40] M. Aidelsburger, S. Nascimbene, N. Goldman, Comp. Rend.

Phys. 2018, 19 394.

19

[41] N. Goldman, G. Juzeliunas, Pohberg, I. B. Spielman, Rep.Prog. Phys. 2014, 77 126401.

[42] V. Galitski, I. B. Spielman, Nature 2013, 494 49.[43] N. R. Cooper, J. Dalibard, I. B. Spielman, Rev. Mod. Phys.

2019, 91 015005.[44] Y. Hu, G.-Q. Ge, S. Chen, X.-F. Yang, Y.-L. Chen, Phys. Rev.

A 2011, 84 012329.[45] A. D. Greentree, C. Tahan, J. H. Cole, L. C. L. Hollenberg,

Nat. Phys. 2006, 2 856.[46] J. Bourassa, F. Beaudoin, J. M. Gambetta, A. Blais, Phys. Rev.

A 2012, 86 013814.[47] J. Jin, D. Rossini, R. Fazio, M. Leib, M. J. Hartmann, Phys.

Rev. Lett. 2013, 110 163605.[48] M. Leib, F. Deppe, A. Marx, R. Gross, M. Hartmann, New J.

Phys. 2012, 14, 7 075024.[49] C. Noh, D. G. Angelakis, Rep. Prog. Phys. 2016, 80 016401.[50] D. Smirnova, D. Leykam, Y. Chong, Y. Kivshar, Appl. Phys.

Rev. 2020, 7 021306.[51] P. Roushan et al., Science 2017, 358 1175.[52] R. O. Umucalılar, I. Carusotto, Phys. Rev. Lett. 2012, 108

206809.[53] A. L. C. Hayward, A. M. Martin, A. D. Greentree, Phys. Rev.

Lett. 2012, 108 223602.[54] C.-E. Bardyn, A. Imamoglu, Phys. Rev. Lett. 2012, 109

253606.[55] W. P. Su, J. R. Schrieffer, A. J. Heeger, Phys. Rev. Lett. 1979,

42 1698.[56] F. D. M. Haldane, Phys. Rev. Lett. 1988, 61 2015.[57] X.-L. Qi, T. L. Hughes, S.-C. Zhang, Phys. Rev. B 2008, 78

195424.[58] M. Hafezi, Phys. Rev. Lett. 2014, 112 210405.[59] S. Mittal, S. Ganeshan, J. Fan, A. Vaezi, M. Hafezi, Nat. Pho-

ton. 2016, 10 180.[60] Y.-P. Wang, W.-L. Yang, Y. Hu, Z.-Y. Xue, Y. Wu, npj Quan-

tum Inf. 2016, 2 16015.[61] M. Tinkham, Introduction to Superconductivity, Dover Publi-

cations, Mineola, 2nd edition, 2013.[62] J. Cho, D. G. Angelakis, S. Bose, Phys. Rev. Lett. 2008, 101

246809.[63] W. L. Yang, Z.-Q. Yin, Z. X. Chen, S.-P. Kou, M. Feng, C. H.

Oh, Phys. Rev. A 2012, 86 012307.[64] D. Jaksch, P. Zoller, New J. Phys. 2003, 5 56.[65] E. Zakka-Bajjani, F. Nguyen, M. Lee, L. R. Vale, R. W. Sim-

monds, J. Aumentado, Nat. Phys. 2011, 7 599.[66] A. J. Sirois, M. A. Castellanos-Beltran, M. P. DeFeo, L. Ran-

zani, F. Lecocq, R. W. Simmonds, J. D. Teufel, J. Aumentado,Appl. Phys. Lett. 2015, 106 172603.

[67] F. Nguyen, E. Zakka-Bajjani, R. W. Simmonds, J. Aumentado,Phys. Rev. Lett. 2012, 108 163602.

[68] F. Wulschner et al., EPJ Quantum Tech. 2016, 3 10.[69] B. Peropadre, D. Zueco, F. Wulschner, F. Deppe, A. Marx,

R. Gross, J. J. Garcia-Ripoll, Phys. Rev. B 2013, 87 134504.[70] Y.-x. Liu, L. F. Wei, J. S. Tsai, F. Nori, Phys. Rev. Lett. 2006,

96 067003.[71] A. O. Niskanen, K. Harrabi, F. Yoshihara, Y. Nakamura,

S. Lloyd, J. S. Tsai, Science 2007, 316 723.[72] F. Yan, P. Krantz, Y. Sung, M. Kjaergaard, D. L. Campbell,

T. P. Orlando, S. Gustavsson, W. D. Oliver, Phys. Rev. Appl.2018, 10 054062.

[73] X. Li, T. Cai, H. Yan, Z. Wang, X. Pan, Y. Ma, W. Cai, J. Han,Z. Hua, X. Han, Y. Wu, H. Zhang, H. Wang, Y. Song, L. Duan,L. Sun, Phys. Rev. Appl. 2020, 14 024070.

[74] M. C. Collodo, J. Herrmann, N. Lacroix, C. K. Andersen,A. Remm, S. Lazar, J.-C. Besse, T. Walter, A. Wallraff,

C. Eichler, Phys. Rev. Lett. 2020, 125 240502.[75] Y. Xu, J. Chu, J. Yuan, J. Qiu, Y. Zhou, L. Zhang, X. Tan,

Y. Yu, S. Liu, J. Li, F. Yan, D. Yu, Phys. Rev. Lett. 2020, 125240503.

[76] P. Roushan et al., Nat. Phys. 2017, 13 146.[77] M. Reagor et al., Sci. Adv. 2018, 4 eaao3603.[78] X. Li, Y. Ma, J. Han, T. Chen, Y. Xu, W. Cai, H. Wang,

Y. Song, Z.-Y. Xue, Z.-q. Yin, L. Sun, Phys. Rev. Appl. 2018,10 054009.

[79] J. Chu, D. Li, X. Yang, S. Song, Z. Han, Z. Yang, Y. Dong,W. Zheng, Z. Wang, X. Yu, D. Lan, X. Tan, Y. Yu, Phys. Rev.Appl. 2020, 13 064012.

[80] X. Guan, Y. Feng, Z.-Y. Xue, G. Chen, S. Jia, Phys. Rev. A2020, 102 032610.

[81] J. Q. You, X. Hu, S. Ashhab, F. Nori, Phys. Rev. B 2007, 75140515.

[82] Y.-P. Wang, W. Wang, Z.-Y. Xue, W.-L. Yang, Y. Hu, Y. Wu,Sci. Rep. 2015, 5 8352.

[83] M. S. Allman, J. D. Whittaker, M. Castellanos-Beltran, K. Ci-cak, F. da Silva, M. P. DeFeo, F. Lecocq, A. Sirois, J. D.Teufel, J. Aumentado, R. W. Simmonds, Phys. Rev. Lett. 2014,112 123601.

[84] J. B. Kogut, Rev. Mod. Phys. 1983, 55 775.[85] V. Galitski, I. B. Spielman, Nature 2013, 494 49.[86] Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji,

Y. Deng, S. Chen, X.-J. Liu, J.-W. Pan, Science 2016, 354 83.[87] V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin,

A. Lemaıtre, H. Tercas, A. Nalitov, M. Abbarchi, E. Galopin,I. Sagnes, J. Bloch, G. Malpuech, A. Amo, Phys. Rev. X 2015,5 011034.

[88] G. Salerno, A. Berardo, T. Ozawa, H. M. Price, L. Taxis, N. M.Pugno, I. Carusotto, New J. Phys. 2017, 19 055001.

[89] S. Li, Z.-Y. Xue, M. Gong, Y. Hu, Phys. Rev. A 2020, 102023524.

[90] S. Longhi, Opt. Lett. 2014, 39 5892.[91] S. Mukherjee, M. Di Liberto, P. Ohberg, R. R. Thomson,

N. Goldman, Phys. Rev. Lett. 2018, 121 075502.[92] M. Di Liberto, S. Mukherjee, N. Goldman, Phys. Rev. A 2019,

100 043829.[93] A. Y. Kitaev, Phys. Usp. 2001, 44 131.[94] I. Carusotto, D. Gerace, H. E. Tureci, S. De Liberato, C. Ciuti,

A. Imamoglu, Phys. Rev. Lett. 2009, 103 033601.[95] Y. Hu, Y. X. Zhao, Z.-Y. Xue, Z. D. Wang, npj Quantum Inf.

2017, 3 8.[96] Y. X. Zhao, Z. D. Wang, Phys. Rev. B 2014, 90 115158.[97] J. A. Schreier, A. A. Houck, J. Koch, D. I. Schuster, B. R.

Johnson, J. M. Chow, J. M. Gambetta, J. Majer, L. Frunzio,M. H. Devoret, S. M. Girvin, R. J. Schoelkopf, Phys. Rev. B2008, 77 180502.

[98] Y. Chen et al., Phys. Rev. Lett. 2014, 113 220502.[99] M. R. Geller, E. Donate, Y. Chen, M. T. Fang, N. Leung,

C. Neill, P. Roushan, J. M. Martinis, Phys. Rev. A 2015, 92012320.

[100] F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccir-illo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato,L. Marrucci, M. Lewenstein, P. Massignan, Nat. Commun.2017, 8 15516.

[101] M. Piraud, F. Heidrich-Meisner, I. P. McCulloch,S. Greschner, T. Vekua, U. Schollwock, Phys. Rev. B2015, 91 140406.

[102] Z.-Y. Xue, F.-L. Gu, Z.-P. Hong, Z.-H. Yang, D.-W. Zhang,Y. Hu, J. Q. You, Phys. Rev. Appl. 2017, 7 054022.

[103] F.-L. Gu, J. Liu, F. Mei, S. Jia, D.-W. Zhang, Z.-Y. Xue, npjQuantum Inf. 2019, 5 36.

20

[104] Z.-Y. Xue, W.-C. Yu, L.-N. Yang, Y. Hu, Eur. Phys. J. D 2015,69.

[105] J.-Y. Cao, J. Liu, L. B. Shao, Z.-Y. Xue, Phys. Rev. A 2020,101 022313.

[106] J. Liu, J.-Y. Cao, G. Chen, Z.-Y. Xue, Quantum Engin. 2021,3 e61.

[107] N. Goldman, I. Satija, P. Nikolic, A. Bermudez, M. A. Martin-

Delgado, M. Lewenstein, I. B. Spielman, Phys. Rev. Lett.2010, 105 255302.

[108] W. Cai, J. Han, F. Mei, Y. Xu, Y. Ma, X. Li, H. Wang, Y. P.Song, Z.-Y. Xue, Z.-q. Yin, S. Jia, L. Sun, Phys. Rev. Lett.2019, 123 080501.

[109] Y. Ashida, Z. Gong, M. Ueda, arXiv e-prints 2020,arXiv:2006.01837.