Waveguide QED in a dispersive photonic channel

WAVEGUIDE QUANTUM ELECTRODYNAMICS

2.3 Waveguide QED in a dispersive photonic channel

While the original concept of waveguide QED revolved around coupling quantum emitters to a waveguide inside a transmission band, a new paradigm to induce long-range photon-mediated interactions between quantum emitters inside a photonic bandgap was proposed in Ref. [125]. In this scheme, photonic structures are engineered to host a bandgap where the propagation of photons are prohibited. Tun- ing the emitters’ frequencies inside the bandgap, the emitters cannot radiate to the waveguide channel while the coherent interaction between distant emitters are still allowed by means of exchange of virtual photons. Compared to the case when the emitters are tuned inside the transmission band discussed in Sec. 2.2, the bandgap regime offers a practical direction to achieve strong and long-range coupling between emitters without suffering from significant dissipation to the photonic band.

In this section, I introduce the basic concepts of this direction to achieve a scalable quantum many-body system with long-range connectivity.

Emitter-photon bound states

When the transition frequency of a quantum emitter is tuned inside a photonic bandgap of an electromagnetic structure, the spontaneous emission of the emitter is forbidden due to the absence of resonant photonic modes to absorb the emitted

a

b

−π/d 0 π/d k ω

ωc

ω0

ωb BG

Figure 2.7: Emitter-photon bound state. a, Illustration of a quantum emitter (green circle) coupled to a 1D dispersive waveguide channel (gray) represented as a periodic electromagnetic structure with lattice constant 𝑑. The emitter’s frequency is tuned inside the bandgap of the waveguide, inducing a emitter-photon bound state exponentially localized at a length scale𝜉, shaded in blue.b,An example of photonic bandstructure of a dispersive waveguide. The transmission band (bandgap) is shaded in red (gray). Tuning the bare transition frequency𝜔₀of the emitter (green arrow) below the frequency𝜔_𝑐of band-edge, the emitter-photon bound state exists at a frequency𝜔_𝑏lower than that of the emitter (blue arrow).

photons. Instead, the emitted photon is scattered back to the original emitter, resulting in a coupled eigenstate of photonic modes of the electromagnetic structure and the emitter. This is known as theemitter-photon bound state3, first predicted by Sajeev John and Jian Wang in 1990 [126] in the context of Anderson localization of light [127]. In the emitter-photon bound state, the emitter is dressed with a photonic tail exponentially localized with respect to the emitter, gaining a spatial extent. This length scale of the emitter-photon bound state can be adjusted by controlling the detuning of the emitter from the edge of the photonic band.

The Hamiltonian of a quantum emitter coupled to such 1D photonic structure, a dispersive waveguide (an example illustrated in Fig. 2.7), can be written as

𝐻 = ℏ𝜔₀|𝑒ih𝑒| +Õ

𝑘

ℏ𝜔_𝑘𝑎ˆ^†

𝑘𝑎ˆ_𝑘 +Õ

𝑘

ℏ

𝑔_𝑘𝑎ˆ^†

𝑘|𝑔ih𝑒| +𝑔^∗

𝑘𝑎ˆ_𝑘|𝑒ih𝑔|

, (2.14) where 𝜔₀ is the transition frequency of the emitter, ˆ𝑎_𝑘 ( ˆ𝑎^†

𝑘) is the annihilation (creation) operator of photonic mode at wavevector𝑘 satisfying the canonical commutation relation[𝑎ˆ_𝑘,𝑎ˆ^†

𝑘⁰] =𝛿_{𝑘 , 𝑘}0, and𝑔_𝑘is the momentum-space coupling between the emitter and the waveguide photons. Here, |𝑔i and |𝑒idenote the ground state and the excited state of the emitter, respectively. Limiting our analysis to the single-

3Also known as the atom (qubit)-photon bound state if an atom (a qubit) plays the role of a two- level emitter.

excitation manifold, we look for a emitter-photon bound state of the following form

|𝜙_𝑏i=cos𝜃|{0}i|𝑒i +sin𝜃 Õ

𝑘

𝑐_𝑘𝑎ˆ^†

𝑘|{0}i|𝑔i (2.15) that satisfies the time-independent Schrödinger equation ˆ𝐻|𝜙_𝑏i = ℏ𝜔_𝑏|𝜙_𝑏i. Here,

|{0}i represents the vacuum state of photon modes and the coefficients 𝑐_𝑘 are normalized byÍ

𝑘 |𝑐_𝑘|²=1. Writing out the algebraically independent terms in the equation we get

𝜔₀cos𝜃+ Õ

𝑘

𝑔^∗

𝑘𝑐_𝑘

sin𝜃 =𝜔_𝑏cos𝜃 (2.16a) 𝜔_𝑘𝑐_𝑘sin𝜃+𝑔_𝑘cos𝜃 =𝜔_𝑏𝑐_𝑘sin𝜃 (2.16b) From equating the coefficients in Eq. (2.16b), we obtain the probability amplitudes at wavevector 𝑘to be

𝑐_𝑘 = 𝑔_𝑘 (𝜔_𝑏−𝜔_𝑘)tan𝜃

. (2.17)

Substituting the Eq. (2.17) into Eq. (2.16a), we get a transcendental equation for evaluating the energy of the emitter-photon bound state

𝜔_𝑏=𝜔₀+Õ

𝑘

|𝑔_𝑘|² 𝜔_𝑏−𝜔_𝑘

(2.18a) subjected to the normalization condition

tan²𝜃 =Õ

𝑘

|𝑔_𝑘|²

(𝜔_𝑏−𝜔_𝑘)². (2.18b)

For a generic 1D waveguide with a quadratic dispersion relation, a simple analytical relation to describe the emitter-photon bound state can be derived. We specifically consider the dispersion relation of the form

𝜔_𝑘 =𝜔_𝑐+𝛼(𝑘− 𝑘₀)², (2.19) where𝜔_𝑐 is the frequency of the band-edge,𝛼 > 0 is the curvature of the photonic band, and𝑘₀is the wavevector at which the band-edge occurs.

Assuming that the coupling of emitter and the waveguide locally occurs at position 𝑥 = 𝑥₀ with the strength 𝑔, the emitter-photon interaction Hamiltonian takes the form ˆ𝐻_int = 𝑔(𝑎ˆ^†

x0|𝑔ih𝑒| +𝑎ˆ_x

0|𝑒ih𝑔|) in terms of real-space annihilation operator ˆ

𝑎_𝑥

0 = ^√¹

𝑁

𝑘𝑒^{𝑖 𝑘 𝑥}⁰𝑎ˆ_𝑘 where 𝑁 is the number of modes inside the band. This is

translated into the last term in Eq. (2.14) with momentum-space coupling given by 𝑔_𝑘 =𝑔 𝑒^{−𝑖 𝑘 𝑥}⁰/√

𝑁. In this case, the Eqs. (2.18a)-(2.18b) are simplified into4 𝜔₀−𝜔_𝑏 = 𝑔²𝑑

𝛼(𝜔_𝑐−𝜔_𝑏) (2.20a)

and

tan²𝜃 = 𝑔²𝑑 4p

𝛼(𝜔_𝑐−𝜔_𝑏)³

= 𝜔₀−𝜔_𝑏

2(𝜔_𝑐−𝜔_𝑏). (2.20b) Here,𝑑is the shortest length scale (lattice constant) of the waveguide that determines the first Brillouin zone. It can be seen from Eq. (2.20a) that the emitter-photon bound state inside the bandgap (𝜔_𝑏 < 𝜔_𝑐) has a frequency lower than the bare emitter frequency (𝜔_𝑏 < 𝜔₀) due to the negative Lamb shift from hybridization with photonic modes at higher frequencies. Also, combining Eqs. (2.20a)-(2.20b), it can be shown that the photonic component of the bound state becomes

sin²𝜃 =

1+4√ 𝛼 𝑔²𝑑

(𝜔_𝑐−𝜔_𝑏)^3/2 −1

. (2.21)

This means that the emitter-photon bound state becomes more photon (emitter)-like as the frequency get closer to (farther from) the band-edge frequency𝜔_𝑐.

One can also evaluate the real-space coefficients𝑐_𝑥 = ^√¹

𝑁

𝑘𝑒^{𝑖 𝑘 𝑥}𝑐_𝑘 of photonic part of the wavefunction|𝜙_𝑏iusing Eq. (2.17)5:

𝑐_𝑥 =− 𝑔 𝑑 p2𝛼(𝜔₀−𝜔_𝑏)

𝑒^{𝑖 𝑘}⁰^(𝑥−𝑥⁰⁾𝑒^{−|𝑥−𝑥}⁰^|/𝜉 (2.22) where

𝜉 =

r 𝛼 𝜔_𝑐−𝜔_𝑏

(2.23)

4In the derivation, the summationÍ

𝑘 over the first Brillouin zone was replaced with the integral

𝑑 2𝜋

∫

𝑑 𝑘 in the thermodynamic limit (𝑁→ ∞) whose upper and lower limits are extended from±𝜋 to±∞assuming that the integration performed at high|𝑘|values are negligible. Also, the integral identities

∫ d𝑥 𝑎²+𝑥²

= 1

𝑎tan⁻¹𝑥 𝑎

∫ d𝑥 𝑎²+𝑥²

= 1 2𝑎³

𝑎𝑥 𝑎²+𝑥²

+ 1

2𝑎³tan⁻¹𝑥 𝑎

are employed.

5Here, the integral identity

∫ ∞

−∞

d𝑥 𝑒^{𝑖 𝑎 𝑥}

𝑥²+1 =𝜋 𝑒^{− |𝑎}^| is used.

−π/d 0 π/d k ω

ωc

BG (i)

(ii)

Figure 2.8: Waveguide-mediated interactions between emitter-photon bound states.

When two quantum emitters are tuned inside the bandgap, the spatial overlap of correspond- ing emitter-photon bound states induces emitter-emitter interaction. The range of interaction depends on the localization length𝜉which is long when the emitters are tuned close to the band-edge frequency𝜔_𝑐 (i, green arrow), while the interaction is short-ranged deep inside the bandgap (ii, blue arrow).

is the localization length. Equations (2.22)-(2.23) directly shows a few important properties of the photonic component of the bound state. First, the magnitude of 𝑐_𝑥 exponentially decays at a length scale 𝜉 with respect to the location 𝑥 = 𝑥₀ of the emitter. The localization length 𝜉 characterizes the effective spatial extent of the emitter-photon bound state. Second, the probability amplitude 𝑐_𝑥 collects an additional factor𝑒^{𝑖 𝑘}⁰⁽^𝑥⁻^𝑥⁰⁾ associated with propagation with wavevector 𝑘 = 𝑘₀ of the band-edge. Also, the localization length is inversely proportional to the detuning of bound state from the band-edge, i.e.,𝜉 ∝ (𝜔_𝑐−𝜔_𝑏)^−1/2, diverging as𝜔_𝑏 → 𝜔_𝑐. This means that the spatial extent of the emitter-photon bound state can take a wide range of values depending on the frequency tuning of the emitter inside the bandgap.

Waveguide-mediated interactions between emitter-photon bound states

The spatially extended nature of the emitter-photon bound state has been viewed as a method to induce effective interactions between emitters not long since the first theoretical investigation of emitter-photon bound states [128, 129]. When two resonant emitters are tuned inside the bandgap of a common photonic structure, it was shown that the photonic band off-resonantly mediates exchange interaction between the emitters in a way similar to how each emitter is dressed by photonic modes to form the emitter-bound state (self-interaction). Such interaction takes a special form that falls off exponentially with the distance between the emitters, whose length scale𝜉 can be tuned by adjusting the frequency of the emitters inside

the bandgap (see Fig. 2.8).

The Hamiltonian of two quantum emitters (labeled by𝑗 =1,2) coupled to a common 1D dispersive waveguide is given by

ˆ 𝐻 =Õ

𝑗

ℏ𝜔⁽^𝑗)

0 |𝑒ih𝑒|𝑗+Õ

𝑘

ℏ𝜔_𝑘𝑎ˆ^†

𝑘𝑎ˆ_𝑘 +Õ

𝑗 , 𝑘

ℏh 𝑔⁽^𝑗)

𝑘 𝑎ˆ^†

𝑘|𝑔ih𝑒|𝑗 +𝑔⁽^𝑗)

𝑘

∗𝑎ˆ_𝑘|𝑒ih𝑔|𝑗

i , (2.24) where𝜔⁽^𝑗)

0 is the frequency of the emitter 𝑗, ˆ𝑎_𝑘 ( ˆ𝑎^†

𝑘) is the annihilation (creation) operator of photonic mode at wavevector 𝑘 satisfying the canonical commutation relation [𝑎ˆ_𝑘,𝑎ˆ^†

𝑘⁰] = 𝛿_{𝑘 , 𝑘}0, and 𝑔⁽^𝑗)

𝑘 is the momentum-space coupling between the emitter 𝑗 and the waveguide. Here, |𝑔i𝑗 and |𝑒i𝑗 denote the ground state and the excited state of the emitter 𝑗, respectively.

Again, we look for a emitter-photon bound state in the single-excitation manifold of the following form

|𝜙_𝑏i=cos𝜃|{0}i h

𝑐_𝑞⁽¹⁾|𝑒i₁|𝑔i₂+𝑐_𝑞⁽²⁾|𝑔i₁|𝑒i₂i +sin𝜃

𝑘

𝑐_𝑘𝑎ˆ^†

𝑘|{0}i|𝑔i₁|𝑔i₂ (2.25) that satisfies the time-independent Schrödinger equation ˆ𝐻|𝜙_𝑏i = ℏ𝜔_𝑏|𝜙_𝑏i. Here,

|{0}i represents the vacuum state of photon modes and the coefficients 𝑐_𝑘 are normalized byÍ

𝑘|𝑐_𝑘|²=1 and𝑐_𝑞 byÍ

𝑗 |𝑐⁽

𝑗)

𝑞 |²=1. Writing out the algebraically independent terms in the equation we get

𝜔⁽

𝑗) 0 𝑐⁽

𝑗)

𝑞 cos𝜃+ Õ

𝑘

𝑔⁽

𝑗) 𝑘

∗

𝑐_𝑘

sin𝜃 =𝜔_𝑏𝑐⁽

𝑗)

𝑞 cos𝜃 , (2.26a) Õ

𝑗

𝑔⁽^𝑗)

𝑘 𝑐⁽_𝑞^𝑗)

cos𝜃+𝜔_𝑘𝑐_𝑘sin𝜃 =𝜔_𝑏𝑐_𝑘sin𝜃 . (2.26b) Equation (2.26b) can be simplified into an expression for coefficients𝑐_𝑘of photonic modes:

𝑐_𝑘 = Í

𝑗𝑔⁽

𝑗) 𝑘

𝑐⁽

𝑗) 𝑞

(𝜔_𝑏−𝜔_𝑘)tan𝜃

. (2.27)

Substituting Eq. (2.27) into Eq. (2.26a), we obtain an eigenequation for probability amplitudes𝑐⁽

𝑗)

𝑞 of quantum emitters given by 𝜔_𝑏

𝑐⁽_𝑞¹⁾ 𝑐⁽_𝑞²⁾

= 𝜔⁽¹⁾

0 +𝐽₁₁ 𝐽₁₂ 𝐽₂₁ 𝜔⁽²⁾

0 +𝐽₂₂

! 𝑐⁽_𝑞¹⁾ 𝑐⁽_𝑞²⁾

, (2.28)

where

𝐽_{𝑖 𝑗} =Õ

𝑘

𝑔^(𝑖)

𝑘

∗

𝑔⁽

𝑗) 𝑘

𝜔_𝑏−𝜔_𝑘

. (2.29)

The matrix 𝐽_{𝑖 𝑗} is Hermitian and represents the effective interaction between the emitters mediated by photons of the transmission band. The diagonal terms (𝑖= 𝑗) correspond to self-interaction of a quantum emitter with itself, also known as the Lamb shift identical to Eq. (2.18a).

Considering again a 1D waveguide with a quadratic dispersion relation described in Eq. (2.19) where each emitter 𝑗 is coupled at position𝑥 = 𝑥_𝑗 of the waveguide (resulting in momentum-space coupling𝑔⁽^𝑗)

𝑘 =𝑔 𝑒^{−𝑖 𝑘 𝑥}^𝑗/√

𝑁), we can readily evaluate the sum over 𝑘 in the thermodynamic limit. Following procedures similar to the derivation of Eq. (2.22), Equation (2.29) reduces to

𝐽_{𝑖 𝑗} =− 𝑔²𝑑 2p

𝛼(𝜔_𝑐−𝜔_𝑏)

𝑒^{𝑖 𝑘}⁰^(𝑥^𝑖^−𝑥^𝑗⁾𝑒^−|𝑥^𝑖^−𝑥^𝑗^|/𝜉, (2.30) where 𝜉 is the localization length defined in (2.23). It can be easily seen that the interaction between emitters mediated by the dispersive waveguide in Eq. (2.30) follows the spatial shape of emitter-photon bound state in Eq. (2.22), collecting a phase factor 𝑒^{𝑖 𝑘}⁰^Δ^𝑥 and an attenuation factor 𝑒^−|Δ^𝑥^|/^𝜉 associated with suppressed propagation of a photon inside the bandgap along the displacement Δ𝑥 = 𝑥_𝑖 −𝑥_𝑗 between the emitters.

Dalam dokumen Superconducting Circuits (Halaman 33-39)