Band structure analysis - SUPPLEMENTARY INFORMATION FOR CHAPTER 5

SUPPLEMENTARY INFORMATION FOR CHAPTER 5

D.1 Band structure analysis

A p p e n d i x D

where 𝑀 = 2𝑁 +1 is the total number of periods in the waveguide. Using the Fourier relation we find the Lagrangian in 𝑘-space as

𝐿 =Õ

𝜅

2(𝐶₀+𝐶_g) | ¤Φ^a_𝜅|²−

1−𝑒^−𝑖²^𝜋(^𝜅/𝑀)

2 |Φ^a_𝜅|² 2𝐿₀ + 1

2(𝐶_g+𝐶_r) | ¤Φ^b_𝜅|²− |Φ^b_𝜅|² 2𝐿_r

−𝐶_g

Φ¤^b_𝜅Φ¤^a_−𝜅+ ¤Φ^b_−𝜅Φ¤^a_𝜅 2

To proceed further, we need to find the canonical node charges which are defined as𝑄^a^,^b

𝜅 = ^{𝜕 𝐿}

𝜕Φ¤^a𝜅^,^b

, and subsequently derive the Hamiltonian of the system by using a Legendre transformation. Doing so we find

𝐻 =Õ

𝜅

𝑄^a

𝜅𝑄^a_−𝜅 2𝐶⁰

1−𝑒⁻^𝑖²^𝜋⁽^𝜅^/^𝑀⁾

2Φ^a_𝜅Φ^a_−𝜅 2𝐿₀

+ 𝑄^b

𝜅𝑄^b_−𝜅 2𝐶⁰_r

+ Φ^b_𝜅Φ^b_−𝜅 2𝐿_r

+ 𝑄^a

𝜅𝑄^b_−𝜅+𝑄^a_−𝜅𝑄^b

𝜅

2𝐶_g⁰

Here, we have defined the following quantities 𝐶⁰

𝐶_g𝐶_r+𝐶_g𝐶₀+𝐶₀𝐶_r 𝐶_g+𝐶_r

, 𝐶⁰

r =

𝐶_g𝐶_r+𝐶_g𝐶₀+𝐶₀𝐶_r 𝐶_g+𝐶₀

, 𝐶⁰

𝐶_g𝐶_r+𝐶_g𝐶₀+𝐶₀𝐶_r 𝐶_g

The canonical commutation relation[Φ^𝑖_𝜅, 𝑄

𝑗

−𝜅⁰] =𝑖ℏ𝛿_{𝑖, 𝑗}𝛿_𝜅,𝜅0 allows us to define the following annihilation operators as a function of charge and flux operators

ˆ 𝑎_𝜅 =

s 𝐶⁰

0Ω𝑘

2ℏ

Φ^a_𝜅+ 𝑖 𝐶⁰

0Ω𝑘

𝑄^a

𝜅

, 𝑏ˆ_𝜅 = r

𝐶⁰

r𝜔₀ 2ℏ

Φ^b_𝜅+ 𝑖 𝐶_r⁰𝜔₀

𝑄^b

𝜅

. (D.3)

Here, we have defined the resonance frequency for each mode as Ω𝑘 =

4sin²(𝑘 𝑑/2) 𝐿₀𝐶⁰

, 𝜔₀= 1 p

𝐿_r𝐶_r⁰

, (D.4)

where 𝑘 = 2𝜋 𝜅/(𝑀 𝑑) is the wavenumber. It is evident that Ω𝑘 has the expected dispersion relation of a discrete periodic transmission line and𝜔₀ is the resonance frequency of the loaded microwave resonators. Using the above definitions for ˆ𝑎_𝜅,𝑏ˆ_𝜅

ˆ 𝐻 = ℏ

2 Õ

𝑘

Ω𝑘

ˆ 𝑎^†

𝑘𝑎ˆ_𝑘+𝑎ˆ_−𝑘𝑎ˆ^†

−𝑘

+𝜔₀

𝑏ˆ^†

𝑘

𝑏_𝑘+𝑏ˆ_−𝑘𝑏ˆ^†

−𝑘

−𝑔_𝑘

𝑏ˆ_−𝑘 −𝑏ˆ^†

𝑘 𝑎ˆ_𝑘 −𝑎ˆ^†

−𝑘

−𝑔_𝑘

ˆ 𝑎^†

𝑘 −𝑎ˆ_−𝑘 𝑏ˆ^†

−𝑘 −𝑏ˆ_𝑘

, (D.5)

along with the coupling coefficient 𝑔_𝑘 =

p𝐶⁰

0𝐶⁰_r 2𝐶_g⁰

𝜔₀Ω𝑘 =

𝐶_g

√ 𝜔₀Ω𝑘

(𝐶₀+𝐶_g) (𝐶_r+𝐶_g)

. (D.6)

An alternative structure for coupling microwave resonators is depicted in the bottom panel of Fig. D.1. In this geometry, the coupling is controlled by the inductive element𝐿_g. Repeating the analysis above for this case, we find

Ω𝑘 = s

4sin²(𝑘 𝑑/2) 𝐶₀𝐿⁰

, 𝜔₀= 1 p

𝐶_r𝐿⁰

, 𝑔_𝑘 = p𝐿⁰

0𝐿⁰

2𝐿⁰

𝜔₀Ω𝑘. (D.7) We have defined the modified inductance values as

𝐿⁰

𝐿_g𝐿_r+𝐿_g𝐿₀+𝐿₀𝐿_r 𝐿_g+𝐿_r

, 𝐿⁰

r =

𝐿_g𝐿_r+𝐿_g𝐿₀+𝐿₀𝐿_r 𝐿_g+𝐿₀

, 𝐿⁰

g =

𝐿_g𝐿_r+𝐿_g𝐿₀+𝐿₀𝐿_r 𝐿_g

Band structure calculation with RWA

Using the rotating wave approximation, the Hamiltonian in Eq. (D.5) can be simplified to

𝐻 = ℏÕ

𝑘

Ω𝑘𝑎ˆ^†

𝑘𝑎ˆ_𝑘 +𝜔₀𝑏ˆ^†

𝑘

ˆ 𝑏_𝑘+𝑔_𝑘

𝑏ˆ^†

𝑘𝑎ˆ_𝑘 +𝑎ˆ^†

𝑘

ˆ 𝑏_𝑘

. (D.8)

Note that this approximation is applicable only when the coupling is sufficiently weak, 𝑔_𝑘 min(𝜔₀,Ω𝑘), and the detuning is sufficiently small |𝜔₀ − Ω𝑘| (𝜔₀+Ω𝑘). AssumingΩ𝑘 and𝜔₀ are of the same order, this condition is satisfied when𝐶_g 2p

(𝐶₀𝐶_r).

The simplified Hamiltonian can be written in the compact form ˆ

𝐻 = ℏÕ

𝑘

x^†_𝑘H𝑘x𝑘, (D.9)

where

H𝑘 =

Ω𝑘 𝑔_𝑘 𝑔_𝑘 𝜔₀

, x𝑘 =

ˆ 𝑎_𝑘

ˆ 𝑏_𝑘

. (D.10)

We desire to transform the Hamiltonian to a diagonalized form H˜𝑘 =

𝜔₊_{, 𝑘} 0 0 𝜔₋_{, 𝑘}

. (D.11)

It is straightforward to use the eigenvalue decomposition to find𝜔_±_{, 𝑘} as 𝜔_{±, 𝑘} = 1

(Ω𝑘 +𝜔₀) ± q

(Ω𝑘−𝜔₀)²+4𝑔²

𝑘

, (D.12)

along with the corresponding eigenstates|±, 𝑘i =𝛼ˆ_±_{, 𝑘}|0i, where ˆ

𝛼_{±, 𝑘} = (𝜔_{±, 𝑘}−𝜔₀) q

(𝜔_{±, 𝑘}−𝜔₀)²+𝑔²

𝑘

𝑎_𝑘+ 𝑔_𝑘 q

(𝜔_{±, 𝑘}−𝜔₀)²+𝑔²

𝑘

𝑏ˆ_𝑘. (D.13)

Band structure calculation beyond RWA

The exact Hamiltonian in Eq. (D.5) can be written in the compact form ˆ

𝐻 = ℏ 2

𝑘

x^†_𝑘H𝑘x𝑘, (D.14)

where

H^𝑘 =







Ω𝑘 0 𝑔_𝑘 −𝑔_𝑘 0 Ω𝑘 −𝑔_𝑘 𝑔_𝑘 𝑔_𝑘 −𝑔_𝑘 𝜔₀ 0

−𝑔_𝑘 𝑔_𝑘 0 𝜔₀







, x^𝑘 =





 ˆ 𝑎_𝑘 ˆ 𝑎^†

−𝑘

𝑏ˆ_𝑘 𝑏ˆ^†

−𝑘







. (D.15)

To find the eigenstates of the system, we can use a linear transform to map the state vector ˜x𝑘 =S𝑘x𝑘 such thatx^†_𝑘H𝑘x𝑘 = ˜x^†_𝑘H˜𝑘˜x𝑘 with the transformed diagonal Hamiltonian matrix

H˜𝑘 =







𝜔_{+, 𝑘} 0 0 0

0 𝜔_{+, 𝑘} 0 0

0 0 𝜔₋_{, 𝑘} 0 0 0 0 𝜔₋_{, 𝑘}.







(D.16)

In order to preserve the canonical commutation relations, the matrix S𝑘 has to be symplectic, i.e. J = S𝑘JS^†_𝑘, with the matrix J = diag(1,−1,1,−1). A linear transformation (such asS^𝑘) that diagonalizes a set of quadratically coupled boson fields while preserving their canonical commutation relations is often referred to as a Bogoliubov-Valatin transformation. While it is generally difficult to find the transform matrixS𝑘, it is easy to find the eigenvalues of the diagonalized Hamiltonian by exploiting some of the properties ofS𝑘. Note that sinceJ=S𝑘JS^†_𝑘, the matrices J ˜H𝑘 andJH𝑘share the same set of eigenvalues. The eigenvalues ofJ ˜H𝑘are the two frequencies𝜔_{±, 𝑘}, and thus we have

𝜔²

±, 𝑘 = 1 2

Ω²_𝑘 +𝜔²

± r

Ω²

𝑘−𝜔²

+16𝜔₀Ω𝑘𝑔²

𝑘

. (D.17)

Circuit theory derivation of the band structure

Consider the pair of equations that describe the propagation of a monochromatic electromagnetic wave of the form 𝑣(𝑥 , 𝑡) = 𝑉(𝑥)𝑒^{−𝑖 𝑘 𝑥}𝑒^𝑖𝜔𝑡 (along with the corresponding current relation) inside a transmission line

d d𝑥

𝑉(𝑥) =−𝑍(𝜔)𝐼(𝑥), d d𝑥

𝐼(𝑥) =−𝑌(𝜔)𝑉(𝑥). (D.18)

Here,𝑍(𝜔)and𝑌(𝜔)are frequency dependent impedance and admittance functions that model the linear response of the series and parallel portions of a transmission line with length𝑑. It is straightforward to check that the solutions to these equation satisfy 𝑘(𝜔) = 𝑛𝜔/𝑐 = p

−𝑍(𝜔)𝑌(𝜔)/𝑑. For a loss-less waveguide and in the absence of dispersion we have𝑍(𝜔) =𝑖𝜔 𝐿₀and𝑌(𝜔)=𝑖𝜔𝐶₀, and thus we find the familiar dispersion relation𝑘(𝜔) =𝜔

√

𝐿₀𝐶₀/𝑑. Nevertheless, the pair of equations above remain valid for arbitrary impedance and admittance functions 𝑍(𝜔) and 𝑌(𝜔), provided that the dimension of the model circuit remains much smaller than the wavelength under consideration. In this model, a real and negative quantity for the product𝑍𝑌 results in an imaginary wavenumber and subsequently creates a stop band in the dispersion relation. This situation can be achieved by periodically loading a transmission line with an array of resonators [343, 344]. Assuming a unit length of𝑑 we find

𝑘²= 𝜔

𝑐 2

𝑛²

1+ 2𝑐𝛾_e 𝑛𝑑

1 𝜔²

0−𝜔²

. (D.19)

Here,𝜔₀is the resonance frequency, and𝛾_eis the external coupling decay rate of an individual resonator in the array. For moderate values of gap-midgap ratio (Δ/𝜔_𝑚), the frequency gap can be found as

Δ = 𝑐 𝑛𝑑

𝛾_e 𝜔₀

, (D.20)

and𝜔_𝑚 =𝜔₀+Δ/2. We have defined the gap as the range of frequencies where the wavenumber is imaginary.

Although a microwave resonator can be realized by using a two-element LC-circuit, the three-element circuits in Fig. D.1 provide an additional degree of freedom which enables setting the coupling𝛾_e independent of the resonance frequency𝜔₀. Using circuit theory, it is straightforward to show

𝜔₀ = 1

𝐿_r(𝐶_r+𝐶_g)

, 𝛾_e = 𝑍₀ 2𝐿_r

𝐶_g 𝐶_r+𝐶_g

. (D.21)

Here, 𝑍₀ is the characteristic impedance of the unloaded waveguide. It is easy to check that for small values of 𝐶_g/𝐶_r, the resonance frequency is only a weak function of𝐶_g. As a result, it is possible to adjust the coupling rate𝛾_eby setting the capacitor𝐶_gwhile keeping the resonance frequency almost constant. Fig. D.1 also depicts an alternative strategy for coupling microwave resonators to the waveguide.

In this design, the inductive element 𝐿_g is used to set the coupling in a “current

divider" geometry. We provide experimental results for implementation of bandgap waveguide based on both designs in the next section.

While the “continuum" model described above provides a heuristic explanation for formation of bandgap in a waveguide loaded with resonators, its results remains valid as far as 𝑘 2𝜋/𝑑. To avoid this approximation, we can use the transfer matrix method to find the exact dispersion relation for a system with discrete periodic symmetry [180]. In this case, Equation (D.19) is modified to

cos(𝑘 𝑑) =1− 𝜔 𝑐

2𝑛²𝑑²

2 − 𝑛𝑑 𝛾_e 𝑐

𝜔² 𝜔²

0−𝜔²

. (D.22)

Note that this relation still requires𝑑to be much smaller than the wavelength of the unloaded waveguide𝜆=2𝜋 𝑐/(𝑛𝜔).

Dispersion and group index near the band-edges

Equation (D.17) can be reversed to find the wavenumber𝑘as a function of frequency.

Assuming, a linear dispersion relation of the form𝑘 =𝑛Ω𝑘/𝑐for the bare waveguide we find

𝑘 = 𝑛𝜔 𝑐

𝜔²−𝜔²_𝑐+

𝜔²−𝜔²_𝑐−

. (D.23)

Here, 𝜔_𝑐+ = 𝜔₀ and 𝜔_𝑐− = 𝜔₀ q

1−4𝑔²

𝑘/(Ω𝑘𝜔₀) are the upper and lower cut- off frequencies, respectively. The quantity 𝑔²

𝑘/(Ω𝑘𝜔₀) is a unit-less parameter quantifying the size of the bandgap and is independent of the wavenumber𝑘. The dispersion relation can be written in simpler forms by expanding the wavenumber in the vicinity of the two band-edges

𝑘 =











𝑛𝜔_𝑐−

𝑐

q Δ

−𝛿₋ for𝜔 ≈𝜔_𝑐−,

𝑛𝜔𝑐+

𝑐

𝛿₊

Δ for𝜔 ≈𝜔_𝑐+.

(D.24) Here, Δ = 𝜔_𝑐+ −𝜔_𝑐− is the frequency span of the bandgap and 𝛿_± =𝜔 −𝜔_𝑐± are the detunings from the band-edges.

The form of the dispersion relation Eq. (D.17) suggests that the maxima of the group index happens near the band-edges. Having the wavenumber, we can evaluate the group velocity𝑣_𝑔 =𝜕 𝜔/𝜕 𝑘 and find the group index𝑛_𝑔=𝑐/𝑣_𝑔as

𝑛_𝑔 =













𝑛𝜔𝑐−

√

√ Δ

−4(𝛿₋−𝑖 𝛾𝑖)³ for𝜔≈ 𝜔_𝑐−,

𝑛𝜔_𝑐+

√

4Δ(𝛿+−𝑖 𝛾𝑖) for𝜔≈ 𝜔_𝑐₊.

(D.25)

Note that we have replaced 𝛿_± with 𝛿_± −𝑖 𝛾_𝑖 to account for finite internal quality factor of the resonators in the structure.

Coupling a Josephson junction qubit to a metamaterial waveguide

We consider the coupling of a Josephson junction qubit to the metamaterial waveguide. Assuming rotating wave approximation (valid for weak coupling 𝑓_𝑘 𝜔_𝑘, 𝜔_q), the Hamiltonian of this system can be written as

𝐻 = ℏÕ

𝑘

𝜔_𝑘𝑎ˆ^†

𝑘𝑎ˆ_𝑘 + 𝜔_q

2 𝜎ˆ_𝑧+ 𝑓_𝑘

ˆ 𝑎^†

𝑘𝜎ˆ⁻+𝑎ˆ_𝑘𝜎ˆ⁺

(D.26) Here 𝑓_𝑘 is the coupling factor of the qubit to the waveguide photons, and𝜔_𝑘 =𝜔_±_{, 𝑘}, where the plus or minus sign is chosen such that the qubit frequency𝜔_qlies within the band. Without loss of generality, we assume 𝑓_𝑘 to be a real number. The Heisenberg equations of motions for the qubit and the photon operators can be written as

𝜕

𝜕 𝑡 ˆ

𝑎_𝑘 =−𝑖𝜔_𝑘𝑎ˆ_𝑘 −𝑖 𝑓_𝑘𝜎ˆ⁻ (D.27)

𝜕

𝜕 𝑡𝜎ˆ⁻ =−𝑖𝜔_q𝜎ˆ⁻−𝑖 Õ

𝑘

𝑓_𝑘𝑎ˆ_𝑘 (D.28)

The equation for ˆ𝑎_𝑘 can be formally integrated and substituted in the equation for ˆ

𝜎⁻ to find

𝜕

𝜕 𝑡𝜎ˆ⁻ =−𝑖𝜔_q𝜎ˆ⁻−𝑖 Õ

𝑘

𝑓_𝑘𝑒^−𝑖𝜔^𝑘^(𝑡−𝑡⁰⁾𝑎ˆ_𝑘(𝑡₀)

−Õ

𝑘

𝑓²

𝑘

∫ ^𝑡

𝑡₀

𝑒⁻^𝑖⁽^𝜔^𝑘^{) (}^𝑡⁻^𝜏⁾𝜎ˆ⁻(𝜏)d𝜏 . (D.29) We now use the Markov approximation to write ˆ𝜎⁻(𝜏) ≈𝜎ˆ⁻(𝑡)𝑒⁻^𝑖⁽^𝜔^q^{) (}^𝜏⁻^𝑡⁾, and thus

𝜕

𝜕 𝑡 ˆ

𝜎⁻ =−𝑖𝜔_q𝜎ˆ⁻−𝑖 Õ

𝑘

𝑓_𝑘𝑒⁻^𝑖𝜔^𝑘⁽^𝑡⁻^𝑡⁰⁾𝑎ˆ_𝑘(𝑡₀)

−Õ

𝑘

𝑓²

𝑘

∫ 𝑡 𝑡₀

𝑒⁻^𝑖⁽^𝜔^𝑘⁻^𝜔^q^{) (}^𝑡⁻^𝜏⁾d𝜏

𝜎⁻(𝑡). (D.30) Considering the generic equation of motion for a linearly decaying qubit,(𝜕/𝜕 𝑡)𝜎ˆ⁻ =

−𝑖𝜔_q𝜎ˆ⁻− (𝛾/2)𝜎ˆ⁻, we can identify real part of the last term in the equation above as the decay rate due to radiation of the qubit into the waveguide. We can extend the integral’s bound to approximately evaluate this term as𝛾 ≈ 2𝜋Í

𝑘 𝑓²

𝑘𝛿(𝜔_𝑘 −𝜔_q).

Band Frequency (GHz) 𝑔/2𝜋(MHz) 𝑄_e(×10³) 𝑄_i(×10³)

Lower 4.2131 15.3 49.47 74.99

Lower 4.6012 19.74 35.09 76.25

Lower 4.7395 18.14 43.58 75

Lower 4.8044 16.53 94.17 75.59

Lower 4.8373 14.03 152.06 73.77

Lower 4.856 9.73 455.8 76.47

Lower 4.8654 4.48 2100 72

Upper 6.6768 39.44 15.74 68.06

Upper 7.309 58.06 12.02 70.44

Table D.1: Measured resonance parameters for metamaterial waveguide. The values are measured for the waveguide of Figs. 5.2-5.4. The resonances are measured in reflection from the input 50-Ω CPW port. The qubit-resonance coupling, 𝑔, is inferred from the anti-crossing observed as the qubit is tuned through each waveguide resonance.

Assuming the coupling rate 𝑓_𝑘 is a smooth function of the𝑘-vector, we can evaluate this some in the continuum limit as

𝛾 =2𝜋 Õ

𝑘

𝑓²

𝑘𝛿(𝜔_𝑘 −𝜔_q) ≈ 𝑀 𝑑

∫

d𝑘 𝑓²

𝑘𝛿(𝜔_𝑘−𝜔_q)

=𝐿

∫ d𝜔

𝜕 𝑘

𝜕 𝜔

𝑓²

𝑘𝛿(𝜔_𝑘 −𝜔_q) = 𝐿 𝑐

𝑓(𝜔_q)²𝑛_g(𝜔_q).

It is evident that reducing the group velocity increases the radiation decay rate of the qubit. A similar analysis can be applied to find the decay rate of a linear cavity with resonance frequency of𝜔₀(i.e. a harmonic oscillator) that has been coupled to the waveguide with coupling constant𝑔(𝜔). In this case we find

𝛾 = 𝐿 𝑐

𝑔(𝜔₀)²𝑛_g(𝜔₀), 𝑄_e =𝜔₀/𝛾 = 𝜔₀𝑐 𝐿

1 𝑔(𝜔₀)²𝑛_g(𝜔₀)

Dalam dokumen Superconducting Circuits (Halaman 177-184)