WAVEGUIDE QUANTUM ELECTRODYNAMICS
2.2 Waveguide QED in the bandgap regime
assumes that a quantum emitter couples to the waveguide at multiple points, thus forming a decoherence-free state on its own when the coupling points are chosen wisely. Braiding or nesting giant atoms such that their waveguide sections overlap can induce coherent interaction between the giant atoms [151].
Another volume of research studying the passband regime is chiral waveguide QED [152], which can be physically implemented by, e.g., quantum dots [153]. In this case, instead of emitting into both left and right propagating modes of the waveguide, the emitter only emits into one of the direction or exhibiting asymmetric couplinggL,j andgR,j in Eqs.2.10-2.11. This area gives rise to, e.g., photon routers in quantum network applications, as well as the driven-dissipative preparation of non-trivial many-body states.
-π/d 0 π/d ωe
ωq
ω
k
d ξ
|{0}〉|g〉
|{0}〉|e〉 gk
|{0}〉|g〉
|ϕb〉
a b
c
Figure 2.5: A single emitter-photon bound state. a, Schematics of a quan- tum emitter (red ball) coupled to a dispersion-engineered waveguide (blue wavy structure) with unit cell sized. The coupling results in an exponentially localized emitter-photon bound state (red shade) with localization length ξ. b, A quadratic dispersion relation in the first Brillouin zone withk0 = 0. The passband is shaded blue above the bandedge frequencyωe, and the bandgap below it is shaded in green.
An example frequency of a quantum emitter in the bandgap is indicated by the red arrow. c, Level diagrams of the quantum emitter in the bandgap and the waveguide modes. The left part shows the bare emitter and waveguide modes, as well as the couplinggkbetween them. The right part shows the dressed level diagram with the bound state|ϕb⟩exhibiting a lower energy (solid line) than that of the bare emitter (dashed line). Panelsa-bis inspired by [133].
photon bound state, first discussed by John and Wang in 1990 [156]. Mathematically, this picture is still captured by the Dicke Hamiltonian in Eq. 2.2 with the emitter frequency ωq outside of possible ωk’s. In the following, we ignore the spurious decayΓ′for simplicity.
To describe the single-excitation emitter-photon bound state|ϕb⟩, we start with the superposition of single-excitation basis without loss of generosity
|ϕb⟩= cosθ|{0}⟩|e⟩+ sinθX
k
cka†k|{0}⟩|g⟩, (2.21)
where |{0}⟩ is the vacuum state for the photonic modes, |g⟩ (|e⟩) is the ground (excited) state of the quantum emitter, θ parameterizes the emitter or total photon population and ck is the coefficient for the propagating photon mode with wave vectork satisfyingP
k|ck|2 = 1. To solve for the parameters, we note that|ϕb⟩is an eigenstate of the Hamiltonian in Eq.2.2 H|ϕb⟩ = ℏωb|ϕb⟩with ωb the angular frequency of the bound state. Assuming RWA, the eigenstate equation yields [120,
131,156,157]
ck= gk
(ωb−ωk) tanθ (2.22) ωb =ωq+X
k
|gk|2
ωb−ωk (2.23)
tan2θ=X
k
|gk|2
(ωb−ωk)2. (2.24)
Up to now, the result has been general with no assumption about the waveguide or its dispersion relation. Moving forward, we consider the generic quadratic dispersion relation (Fig.2.5b) at the bandedgeωk =ωe+α(k−k0)2, whereωeis the bandedge frequency,α is the band curvature, andk0 is the corresponding wave vector at the bandedge. This is suitable to approximate bandedge dispersion with the quadratic term as the leading order, such as the rectangular waveguide example in Eq. 2.1.
For the derivation below, we assume the bandgap is below the passband withα >0 and the emitter frequencyωq < ωe. The opposite case is left as an exercise for the readers. Assuming the quadratic dispersion relation and uniform coupling to each modek after changing the summation to integral1[120,133],
ωq−ωb = g2d 2p
α(ωe−ωb) (2.27)
tan2θ = g2d 4p
α(ωe−ωb)3 = ωq−ωb
2(ωe−ωb). (2.28) From Eq.2.27, we can see that the bound state frequency is pushed below the bare emitter frequency because of the Lamb shift (Fig. 2.5c). The shape of the bound state can be deduced by the photonic mode coefficient in the real space, resulting in an exponentially localized shape (Fig.2.5a)
cx = 1
√N X
k
eikxck =− gd
p2α(ωq−ωb)eik0(x−x0)e−|x−x0|/ξ, (2.29)
1This means
1 N
X
k
→ d 2π
Z
dk, (2.25)
whereNis the number of unit cells,dis the size of a unit cell in the waveguide exhibiting translational symmetry, and the integration covers the entire first Brillouin zone. This change from summation to integral is valid whenN → ∞, the thermodynamic limit. For example, the summation in Eq.2.23 becomes
d 2πg2
Z dk ωb−ωk
, (2.26)
whereg2=N gk2.
where the localization length is given by ξ =
r α ωe−ωb
. (2.30)
This localization length is consistent with the result from the imaginary part of the wave vector at frequencyωb
ωb =ωe+α(k−k0)2, k =k0+ i
ξ. (2.31)
The properties of the bound state is determined by the detuning between the bound state and the bandedge. Decreasing the detuning results in (i) a growing photonic population in the bound state from Eq. 2.28, i.e., the bound state becoming less emitter-like and more photon-like; and (ii) the spatial extend of the bound state becoming more delocalized shown in Eq. 2.30. This tunability of the bound state properties serves as the foundation of the tunable exchange interaction discussed in the following subsection.
Exchange interaction between two bound states
Intuitively, when the two bound states spatially overlap with each other, there will be interaction between them with the interaction strength and range determined by the bound state properties. A more rigorous picture is that the virtual photons in the passband modes mediates the interactions between the two emitters, similar to the formalism in Sec.2.1. The difference lies in the emitter frequency relative to the passband. Now let us make it more quantitative by starting from the Hamilto- nian in Eq. 2.12 that still holds in this case. Additionally, the expression for the single-excitation eigenstate of the two-emitter interacting Hamiltonian can be found analytically using a similar method as the one for a single bound state. We can directly extract the interaction between the two bound states from the eigenstate equation.
Let us rewrite the Hamiltonian Eq.2.12in a clearer way H = X
j=1,2
ℏωq,j|e⟩⟨e|j +X
k
ℏωka†kak+X
j,k
ℏ
hgk,ja†k|g⟩⟨e|j +h.c.i
. (2.32) The two-emitter-photon bound state in the single-excitation manifold is described by
|ϕb⟩= cosθ|{0}⟩(cq,1|e⟩1|g⟩2+cq,2|g⟩1|e⟩2) + sinθX
k
cka†k|{0}⟩|g⟩1|g⟩2, (2.33)
where the additional parameters cq,j represents the relative population of the two emitters, satisfyingP
j|cq,j|2 = 1. Solving the eigenstate equationH|ϕb⟩=ℏωb|ϕb⟩ gives [131]
ωb
cq,1 cq,2
!
= ωq,1+J1,1 J1,2 J2,1 ωq,2 +J2,2
! cq,1 cq,2
!
, (2.34)
where
Jj,l =X
k
gk,j∗ gk,l ωb −ωk
. (2.35)
For j = l, Jj,j is the single-emitter Lamb shift the same as the one in Eq. 2.23.
The cross-emitter termJ1,2 represents the exchange interaction between the bound states, exhibiting the form of virtual-photon-mediated interaction g1g2/∆. Using similar techniques as deriving Eq. 2.29, we arrive at the expression by assuming
|gk,j|=|gk,l|
J1,2 =− g2d 2p
α(ωe−ωb)eik0(x1−x2)e−|x1−x2|/ξ, (2.36) where the localization length is the same as the one in Eq. 2.30. Although the single-emitter eigenstate equation always has a bound state solution, the existence of two bound states in the two-emitter eigenstate equation is not guaranteed. Specif- ically, because of the interaction between the single-emitter bound states, one of the hybridized two-emitter bound state may be pushed into the passband frequency and become delocalized [157].
-π/d 0 π/d ωe
ωq
ω
k
b
ωq’
a
Figure 2.6: Two quantum emitters in the bandgap frequency. a, Schematics of two emitter-photon bound states. The upper (lower) one shows a higher (lower) emitter frequency ωq (ω′q), resulting in larger (smaller) population in the photonic envelope and a larger (smaller) spatial extend of the bound states. Consequently, the interaction between the bound states exhibits larger (smaller) amplitude and a longer (shorter) range. b, Dispersion relation showing the bandedge frequencyωe
and the different quantum emitter frequencies. This figure is adapted from [133].
The bound state interaction inherits the tunability of the bound state, exhibiting a larger amplitude and more extended range when the emitter-bandedge detuning is
smaller (Fig.2.6). Although exponentially localized interaction is not considered long range in the thermodynamic limit, the flexibility to tune the interaction range to cover the entire physical device [158] acts as the foundation of long-range connec- tivity in waveguide-based quantum architecture in the bandgap regime (Chapter6).
This tunable interaction also gives access to a wide range of many-body Hamiltoni- ans [159]. Moreover, with local control or additional levels, the interaction profile can be engineered to exhibit beyond exponential decay, emulating, e.g., power-law decay or even designed connectivity [158,160] (see also Sec.7.2).
Beyond the single-excitation manifold, bound states consisting of a quantum emitter and multiple binding photons have also been explored showing intriguing properties in quantum optics and many-body physics [157, 161, 162]. When the quantum emitter is extended beyond a two-level system, such as the superconducting transmon qubit, higher-excitation manifold have been studied using spectroscopic tools [163].
Lastly, the dispersion relation can be engineered to exhibit, e.g., multiple bands or topological features, where the quadratic dispersion assumption no longer holds [131,164] (an example is discussed in detail in Chapter5).
C h a p t e r 3