Modeling of the topological waveguide

SUPPLEMENTARY INFORMATION FOR CHAPTER 5

C.1 Modeling of the topological waveguide

A p p e n d i x C

Considering the discrete translational symmetry in our system, we can rewrite the variables in terms of Fourier components as

Φ^α_n= 1

√N X

k

e^inkdΦ^α_k, i^α_n = 1

√N X

k

e^inkdi^α_k, (C.3) whereα=A,B,Nis the number of unit cells, andk = ^2πm_{N d} (m =−N/2,· · · , N/2− 1) are points in the first Brillouin zone. Equation (C.2) is written as

X

k^′

e^ink′dΦ^A_k^′ =X

k^′

e^ink′d

L0i^A_k^′ +Mvi^B_k^′+e^−ik′dMwi^B_k^′

under this transform. Multiplying the above equation withe^−inkdand summing over alln, we get a linear relation betweenΦ^α_k andi^α_k:

Φ^A_k Φ^B_k

!

= L0 Mv +Mwe^−ikd Mv +Mwe^ikd L0

! i^A_k i^B_k

! .

By calculating the inverse of this relation, the Lagrangian of the system (C.1) can be rewritten ink-space as

L=X

k

C0+Cv+Cw

2

˙Φ^A_−k˙Φ^A_k + ˙Φ^B_−k˙Φ^B_k

−Cg(k) ˙Φ^A_−k˙Φ^B_k

− L0

2 i^A_−ki^A_k +i^B_−ki^B_k

−Mg(k)i^A_−ki^B_k

=X

k

C0+C_v+C_w 2

˙Φ^A_−k˙Φ^A_k + ˙Φ^B_−k˙Φ^B_k

−Cg(k) ˙Φ^A_−k˙Φ^B_k

−

L0

2 Φ^A_−kΦ^A_k + Φ^B_−kΦ^B_k

−Mg(k)Φ^A_−kΦ^B_k L²₀−M_g(−k)M_g(k)

(C.4) where C_g(k) ≡ C_v +C_we^−ikd and M_g(k) ≡ M_v +M_we^−ikd. The node charge variablesQ^α_k ≡∂L/∂˙Φ^α_k canonically conjugate to node fluxΦ^α_k are

Q^A_k Q^B_k

!

= C0+Cv+Cw −Cg(−k)

−Cg(k) C0+Cv +Cw

! ˙Φ^A_−k

˙Φ^B_−k

! .

Note that due to the Fourier transform implemented on flux variables, the canonical charge in momentum space is related to that in real space by

Q^α_n = ∂L

∂˙Φ^α_n =X

k

∂L

∂˙Φ^α_k

∂˙Φ^α_k

∂˙Φ^α_n = 1

√N X

k

e^−inkdQ^α_k,

which is in the opposite sense of regular Fourier transform in Eq. (C.3). Also, due to the Fourier-transform properties, the constraint that Φ^α_n and Q^α_n are real

reduces to(Φ^α_k)^∗ = Φ^α_−k and(Q^α_k)^∗ =Q^α_−k. Applying the Legendre transformation H =P

k,αQ^α_k ˙Φ^α_k − L, the Hamiltonian takes the form H =X

k

CΣ(Q^A_−kQ^A_k +Q^B_−kQ^B_k) +Cg(−k)Q^A_−kQ^B_k +Cg(k)Q^B_−kQ^A_k 2C_d²(k)

+ L0(Φ^A_−kΦ^A_k + Φ^B_−kΦ^B_k)−M_g(k)Φ^A_−kΦ^B_k−M_g(−k)Φ^B_−kΦ^A_k 2L²_d(k)

, where

CΣ≡C0+Cv +Cw, C_d²(k)≡C_Σ² −Cg(−k)Cg(k), L²_d(k)≡L²₀−Mg(−k)Mg(k).

Note that C_d²(k) and L²_d(k) are real and even function in k. We impose the canonical commutation relation between real-space conjugate variables[ ˆΦ^α_n,Qˆ^β_n′] = iℏδα,βδn,n^′ to promote the flux and charge variables to quantum operators. This reduces to [ ˆΦ^α_k,Qˆ^β_k^′] = iℏδα,βδk,k^′ in the momentum space [Note that due to the Fourier transform, ( ˆΦ^α_k)^† = ˆΦ^α_−k and ( ˆQ^α_k)^† = ˆQ^α_−k, meaning flux and charge op- erators in momentum space arenon-Hermitiansince the Hermitian conjugate flips the sign ofk]. The Hamiltonian can be written as a sumHˆ = ˆH0+ ˆV, where the

“uncoupled” partHˆ0 and coupling termsVˆ are written as Hˆ0 =X

k,α

"

Qˆ^α_−kQˆ^α_k

2C₀^eff(k)+ Φˆ^α_−kΦˆ^α_k 2L^eff₀ (k)

#

, Vˆ =X

k

"

Qˆ^A_−kQˆ^B_k

2C_g^eff(k) + Φˆ^A_−kΦˆ^B_k

2L^eff_g (k) + H.c.

# , (C.5) with the effective self-capacitanceC₀^eff(k), self-inductanceL^eff₀ (k), coupling capac- itanceC_g^eff(k), and coupling inductanceL^eff_g (k)given by

C₀^eff(k) = C_d²(k) CΣ

, L^eff₀ (k) = L²_d(k) L0

, C_g^eff(k) = C_d²(k)

Cg(−k), L^eff_g (k) = −L²_d(k) Mg(k).

(C.6) The diagonal partHˆ0of the Hamiltonian can be written in a second-quantized form by introducing annihilation operators ˆak andˆbk, which are operators of the Bloch waves on A and B sublattice, respectively:

ˆ

a_k ≡ 1

√2ℏ

"

Φˆ^A_k

pZ₀^eff(k) +i q

Z₀^eff(k) ˆQ^A_−k

#

, ˆb_k ≡ 1

√2ℏ

"

Φˆ^B_k

pZ₀^eff(k) +i q

Z₀^eff(k) ˆQ^B_−k

# .

Here, Z₀^eff(k) ≡ p

L^eff₀ (k)/C₀^eff(k) is the effective impedance of the oscillator at wavevector k. Unlike the Fourier transform notation, for bosonic modes ˆak and

ˆb_k, we use the notation (ˆa_k)^† ≡ ˆa^†_k and (ˆb_k)^† ≡ ˆb^†_k. Under this definition, the commutation relation is rewritten as[ˆa_k,ˆa^†_k′] = [ˆb_k,ˆb^†_k′] = δ_k,k^′. Note that the flux and charge operators are written in terms of mode operators as

Φˆ^A_k = r

ℏZ₀^eff(k) 2

ˆ

ak+ ˆa^†_−k

, Qˆ^A_k = 1 i

s ℏ 2Z₀^eff(k)

ˆ

a−k−ˆa^†_k , Φˆ^B_k =

r

ℏZ₀^eff(k) 2

ˆbk+ ˆb^†_−k

, Qˆ^B_k = 1 i

s ℏ 2Z₀^eff(k)

ˆb−k−ˆb^†_k . The uncoupled Hamiltonian is written as

Hˆ0 =X

k

ℏω0(k) 2

ˆ

a^†_kˆak+ ˆa−kˆa^†_−k+ ˆb^†_kˆbk+ ˆb−kˆb^†_−k

, (C.7)

where the “uncoupled” oscillator frequency is given byω0(k)≡[L^eff₀ (k)C₀^eff(k)]^−1/2, which ranges between values

ω0(k= 0) =

s L0CΣ

[L²₀−(M_v+M_w)²][C_Σ² −(C_v +C_w)²], ω0

k = π

d

=

s L0CΣ

(L²₀− |Mv−Mw|²)(C_Σ² − |Cv−Cw|²). The coupling HamiltonianVˆ is rewritten as

Vˆ =−X

k

ℏg_C(k) 2

ˆ

a−kˆbk−ˆa−kˆb^†_−k−ˆa^†_kˆbk+ ˆa^†_kˆb^†_−k +ℏg_L(k)

2

ˆ

a−kˆbk+ ˆa−kˆb^†_−k+ ˆa^†_kˆbk+ ˆa^†_kˆb^†_−k +h.c.

, (C.8)

where the capacitive couplingg_C(k)and inductive couplingg_L(k)are simply written as

gC(k) = ω0(k)Cg(k) 2CΣ

, gL(k) = ω0(k)Mg(k) 2L0

, (C.9)

respectively. Note that g_C^∗(k) = gC(−k)and g_L^∗(k) = gL(−k). In the following, we discuss the diagonalization of this Hamiltonian to explain the dispersion relation and band topology.

Band structure within the rotating-wave approximation

We first consider the band structure of the system within the rotating-wave approx- imation (RWA), where we discard the counter-rotating terms ˆaˆb and ˆa^†ˆb^† in the

Hamiltonian. This assumption is known to be valid when the strength of the cou- plings |g_L(k)|, |g_C(k)| are small compared to the uncoupled oscillator frequency ω0(k). Under this approximation, the Hamiltonian in Eqs. (C.7)-(C.8) reduces to a simple form Hˆ = ℏP

k(ˆv_k)^†h(k)ˆv_k, where the single-particle kernel of the Hamiltonian is,

h(k) = ω0(k) f(k) f^∗(k) ω0(k)

!

. (C.10)

Here, vˆ_k = (ˆak,ˆbk)^T is the vector of annihilation operators at wavevector k and f(k)≡g_C(k)−g_L(k). In this case, the Hamiltonian is diagonalized to the form

Hˆ =ℏ X

k

hω+(k) ˆa^†_+,kˆa+,k+ω−(k) ˆa^†_−,kˆa−,k

i, (C.11)

where two bandsω±(k) =ω0(k)± |f(k)|symmetric with respect toω0(k)at each wavevector k appear [here, note that ˆa^†_±,k ≡ (ˆa±,k)^†]. The supermodes aˆ±,k are written as

ˆ

a±,k = ±e^−iϕ(k)aˆ_k+ ˆb_k

√2 ,

where ϕ(k) ≡ argf(k) is the phase of coupling term. The Bloch states in the single-excitation bands are written as

|ψk,±⟩= ˆa^†_±,k|0⟩= 1

√2 ±e^iϕ(k)|1k,0k⟩+|0k,1k⟩ ,

where|nk, n^′_k⟩denotes a state withn(n^′) photons in modeˆak (ˆbk).

As discussed below in App. C.2, the kernel of the Hamiltonian in Eq. (C.10) has an inversion symmetry in the sublattice unit cell which is known to result in bands with quantized Zak phase [218]. In our system the Zak phase of the two bands are evaluated as

Z =i I

B.Z.

dk 1

√2

±e^−iϕ(k) 1 ∂

∂k

"

√1 2

±e^iϕ(k) 1

!#

=−1 2

I

B.Z.

dk ∂ϕ(k)

∂k .

The Zak phase of photonic bands is determined by the behavior of f(k) in the complex plane. If the contour off(k)forkvalues in the first Brillouin zone excludes (encloses) the origin, the Zak phase is given byZ = 0(Z = π) corresponding to the trivial (topological) phase.

Band structure beyond the rotating-wave approximation

Considering all the terms in the Hamiltonian in Eqs. (C.7)-(C.8), the Hamiltonian can be written in a compact formHˆ = ^ℏ₂P

k(ˆv_k)^†h(k)ˆv_kwith a vector composed of mode operatorsˆv_k=

ˆ

ak,ˆbk,ˆa^†_−k,ˆb^†_−kT

and

h(k) =













ω0(k) f(k) 0 g(k) f^∗(k) ω0(k) g^∗(k) 0

0 g(k) ω0(k) f(k) g^∗(k) 0 f^∗(k) ω0(k)













=ω0(k)













1 ^ck−l₂ ^k 0 ^−ck₂^−lk

c^∗_k−l_k^∗

2 1 ^−c∗k₂^−l∗k 0 0 ^−ck₂^−lk 1 ^ck−l₂ ^k

−c^∗_k−l_k^∗

2 0 ^c∗k−l₂ ^∗k 1













, (C.12)

wheref(k) ≡ g_C(k)−g_L(k)as before andg(k) ≡ −g_C(k)−g_L(k). Here, l_k ≡ Mg(k)/L0andck≡Cg(k)/CΣare inductive and capacitive coupling normalized to frequency. The dispersion relation can be found by diagonalizing the kernel of the Hamiltonian in Eq. (C.12) with the Bogoliubov transformation

ˆ

w_k =S_kˆv_k, S_k = U_k V^∗_−k V_k U^∗_−k

!

(C.13) wherewˆ_k ≡(ˆa_+,k,ˆa_−,k,ˆa^†_+,−k,ˆa^†_−,−k)^T is the vector composed of supermode oper- ators andU_k,V_kare2×2matrices forming blocks in the transformationS_k. We want to findS_ksuch that(ˆv_k)^†h(k)ˆv_k = (ˆw_k)^†˜h(k)ˆw_k, whereh(k)˜ is diagonal. To preserve the commutation relations, the matrixS_k has to be symplectic, satisfying J=S_kJ(Sk)^†, withJdefined as

J =













1 0 0 0

0 1 0 0

0 0 −1 0 0 0 0 −1











 .

Due to this symplecticity, it can be shown that the matrices Jh(k) andJh(k)˜ are similar under transformationS_k. Thus, finding the eigenvalues and eigenvectors of

the coefficient matrix

m(k)≡ Jh(k) ω0(k) =













1 ^ck−l₂ ^k 0 ^−ck₂^−lk

c^∗_k−l^∗_k

2 1 ^−c∗k₂^−l∗k 0 0 ^ck+l₂ ^k −1 ^−ck₂^+lk

c^∗_k+l^∗_k

2 0 ^−c∗k₂^+l∗k −1













(C.14)

is sufficient to obtain the dispersion relation and supermodes of the system. The eigenvalues of matrixm(k)are evaluated as

± v u u

t1− lkc^∗_k+l^∗_kck

2 ±

s

1− lkc^∗_k+l^∗_kck

2

2

−(1− |lk|²)(1− |ck|²) and hence the dispersion relation of the system taking into account all terms in Hamiltonian (C.12) is

˜

ω±(k) = ˜ω0(k) v u u t1±

s

1− [L²₀−Mg(−k)Mg(k)] [C_Σ² −Cg(−k)Cg(k)]

L0CΣ− ¹₂[Mg(−k)Cg(k) +Cg(−k)Mg(k)] ² (C.15) where

˜

ω0(k)≡ω0(k) s

1− Mg(k)Cg(−k) +Mg(−k)Cg(k) 2L0CΣ

.

The two passbands range over frequencies [ω₊^min, ω₊^max] and[ω^min₋ , ω^max₋ ], where the band-edge frequencies are written as

ω^min₊ = 1

p[L0+p2(M_v −M_w)][CΣ−p2(C_v−C_w)]

ω^max₊ = 1

p[L0+p1(M_v +M_w)][CΣ−p1(C_v+C_w)], (C.16a)

ω₋^min= 1

p[L0−p1(Mv+Mw)][CΣ+p1(Cv+Cw)]

ω₋^max= 1

p[L0−p2(Mv−Mw)][CΣ+p2(Cv −Cw)]. (C.16b) Here, p1 ≡ sgn[L0(Cv +Cw)−CΣ(Mv +Mw)] and p2 ≡ sgn[L0(Cv −Cw)− CΣ(Mv−Mw)]are sign factors. In principle, the eigenvectors of the matrixm(k)in Eq. (C.14) can be analytically calculated to find the transformationS_kof the original

modes to supermodes ˆa_±,k. For the sake of brevity, we perform the calculation in the limit of vanishing mutual inductance (M_v =M_w = 0), where the matrixm(k) reduces to

m_C(k)≡













1 ck/2 0 −ck/2 c^∗_k/2 1 −c^∗_k/2 0

0 ck/2 −1 −ck/2 c^∗_k/2 0 −c^∗_k/2 −1













. (C.17)

In this case, the block matrices U_k, V_k in the transformation in Eq. (C.13) are written as

U_k = 1 2√

2

e^−iϕ(k)x+,k x+,k

−e^−iϕ(k)x−,k x−,k

!

, V_k = 1 2√

2

e^−iϕ(k)y+,k y+,k

−e^−iϕ(k)y−,k y−,k

! ,

wherex±,k =p⁴

1± |ck|+√₄ ¹

1±|ck|,y±,k =p⁴

1± |ck|−√4 ¹

1±|ck|, andϕ(k) = argck. Note that the constants are normalized by relationx²_±,k−y_±,k² = 4.

The knowledge of the transformation S_k allows us to evaluate the Zak phase of photonic bands. In the Bogoliubov transformation, the Zak phase can be evaluated as [310]

Z =i I

B.Z.

dk 1 2√

2

±e^−iϕ(k)x±,k x±,k ±e^−iϕ(k)y±,k y±,k

·J· ∂

∂k











 1 2√

2













±e^iϕ(k)x±,k

x_±,k

±e^iϕ(k)y±,k

y±,k

























=i I

B.Z.

dk 1 8

i∂ϕ(k)

∂k (x²_±,k−y_±,k² ) + ∂

∂k(x²_±,k−y²_±,k)

=−1 2

I

B.Z.

dk ∂ϕ(k)

∂k , identical to the expression within the RWA. Again, the Zak phase of photonic bands is determined by the winding off(k)around the origin in complex plane, leading toZ = 0in the trivial phase andZ =πin the topological phase.

Extraction of circuit parameters and the breakdown of the circuit model

As discussed in Fig. 5.2b, the parameters in the circuit model of the topological waveguide is found by fitting the waveguide transmission spectrum of the test structures. We find that two lowest-frequency modes inside the lower passband fail to be captured according to our model with capacitively and inductively coupled LC resonators. We believe that this is due to the broad range of frequencies (about 1.5 GHz) covered in the spectrum compared to the bare resonator frequency ∼

−π 0 π kd

Frequency (GHz) ^{Full Model}

Within RWA Final Mapping to SSH Model

6 6.5 7 7.5

Figure C.2: Band structure of the realized topological waveguide under various assumptions discussed in App. C.2. The solid lines show the dispersion relation in the upper (lower) passband, ω±(k): full model without any assumptions (red), model within RWA (blue), and the final mapping to SSH model (black) in the weak coupling limit. The dashed lines indicate the uncoupled resonator frequencyω0(k) under corresponding assumptions.

6.6GHz and the distributed nature of the coupling, which can cause our simple model based on frequency-independent lumped elements (inductor, capacitor, and mutual inductance) to break down. Such deviation is also observed in the fitting of waveguide transmission data of Device I (Fig.C.7).

C.2 Mapping of the system to the SSH model and discussion on robustness of