Experimental realizations of waveguide QED

WAVEGUIDE QUANTUM ELECTRODYNAMICS

2.4 Experimental realizations of waveguide QED

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 𝑥𝑗}/√

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 𝑘0(𝑥𝑖−𝑥𝑗)}𝑒^{−|𝑥𝑖−𝑥𝑗|/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 𝑘0Δ𝑥} and an attenuation factor 𝑒^{−|Δ𝑥|/𝜉} associated with suppressed propagation of a photon inside the bandgap along the displacement Δ𝑥 = 𝑥_𝑖 −𝑥_𝑗 between the emitters.

1 mm

200 μm

a b c

Figure 2.9: Examples of experimental platforms for waveguide QED. a, Atoms are interfaced with tapered optical nanofiber (top) or an alligator photonic crystal waveguide (bottom). b, A quantum dot is interfaced with a photonic crystal waveguide. c, Super- conducting qubits are interfaced with a coplanar waveguide transmission line (top) or a machined 3D waveguide (bottom). The panel a is adapted from Refs. [133, 134]; b is adapted from Ref. [132];cis adapted from Refs. [108, 135].

Quantum Emitter Waveguide 𝑃_1D 𝑁 Ref.

Atom Nanofiber ∼10⁻²–10⁻¹ ∼10³ [136]

PhC waveguide ^∼10

−1–10⁰ < 5 [137]

Quantum dot ^∼10

1–10² 1 [132]

Plasmonic waveguide ∼10⁰–10¹ 1 [138]

SC qubit On-chip TL ^∼10² ∼10 [108]

3D waveguide ∼10² <5 [135]

Table 2.1: Experimental platforms for studying waveguide QED. State-of-the-art ex- periments in waveguide QED, performed in various combinations of quantum emitters and waveguides, are summarized. Here, typical values of the highest achieved Purcell factor 𝑃_1Dand the number of resonant quantum emitters𝑁are compared between platforms. PhC:

photonic crystal, SC: superconducting, TL: transmission line.

scattering cross section𝜎₀=3𝜆²

0/(2𝜋)at best comparable to the focused mode area 𝐴 ∼ 𝜆²

0 (diffraction limit). Also, this technique cannot be extended to the case of many atoms since the beam waist diverges rapidly under tight focusing.

For these reasons, alternative approaches to achieve strong atom-photon interactions were investigated using guided modes of 1D photonic structures which realizes small mode area. Optical nanofibers with radius of few hundred nanometers were first used to couple atoms to the evanescent tail of guided nanofiber mode [139]. Trapped atoms near the nanofiber, it was shown that 𝑁 ∼ 10³ atoms could be coupled to the photonic mode [140]. However, suboptimal mode overlap between atoms and photons at the evanescent tail result in small single-atom Purcell factor. To overcome this challenge, a novel direction to employ engineered nanophotonic waveguides was

pursued. In particular, a Purcell factor on the order of unity was achieved for the first time with atoms by interfacing with a photonic crystal waveguide engineered to have the highest mode intensity at the atoms’ position [137, 141]. Also, a similar structure was utilized to study atom-atom interactions inside the bandgap regime discussed in Sec. 2.3, especially near the band-edge [142]. An outstanding challenge in this approach is to trap the atoms near the nanophotonic structures [134, 143], which has been difficult due to strong attractive forces at the dielectric surface [144].

Quantum dots

There has been numerous investigations to interface quantum dots with nanophotonic structures [132] such as a photonic crystal waveguide or a plasmonic waveguide [138], realizing single-emitter Purcell factors on the order of 10–100. However, most of the studies to date were limited to the case of a single quantum dot with photons being conceived as resources for quantum applications [145].

Superconducting qubits

Superconducting circuits offer a promising platform to study waveguide QED in the microwave domain due to strong emitter-photon coupling and a wide variety of mi- crowave photonic structures that could be fabricated on chip with high flexibility. In superconducting circuits, the strong sub-wavelength confinement of electromagnetic modes in transverse direction and large transition dipole moment of superconducting qubits themselves result in high Purcell factors. The first notable waveguide QED experiment in superconducting circuits was performed by Astafiev and colleagues [146] where they have used a superconducting flux qubit coupled to a transmission line on chip. The resonance fluorescence showed strong extinction in transmission spectrum of 94 %, indicating a Purcell factor beyond unity𝑃_1D > 1. Experiments involving multiple qubits [147] were also investigated, demonstrating strong coop- erative interactions between qubits mediated by the waveguide channel [108]. There has also been new efforts to utilize machined 3D microwave waveguides together with multiple qubit chips placed along the waveguide [135, 148].

The current limitations of superconducting qubit platform is related to the scalability.

The wavelength at microwave frequencies (few GHz) is on the order of a few centimeters and therefore channeling qubits with a microwave waveguide requires a large footprint on chip or a large 3D enclosure. Therefore, a compact microwave structure for waveguide QED must be envisioned [109] in order to increase the effective size of the system. Also, additional wiring for individual addressing of

qubits is necessary to harness the full power of superconducting qubits, which makes the scale-up more challenging. Nevertheless, with the technology developed for state-of-the-art quantum processors [6, 76], realizing a waveguide QED system with𝑁 ∼ 10²superconducting qubits is expected to be feasible.

C h a p t e r 3