Pseudo-Giant Atoms in Quantum Optics

• Pseudo-giant atoms are quantum optical emitters coupled at multiple, well-separated points, breaking the standard dipole approximation.
• They exhibit phase-dependent interference that modulates spontaneous emission, Lamb shifts, and supports bound states in the continuum.
• Tailored coupling geometries enable dynamic switching between protected and strongly radiative regimes, with applications in quantum state transfer and decoherence-free operations.

A pseudo-giant atom is a quantum optical emitter—typically an engineered superconducting qubit or atom—coupled to a one-dimensional continuum, such as a waveguide or transmission line, not at a single spatial location but rather via multiple, well-separated discrete points. This nonlocal coupling breaks the standard dipole (pointlike) approximation, leading to fundamentally altered radiative properties including phase-dependent interference in spontaneous emission, modified Lamb shifts, non-Markovian memory effects, formation of dark states and bound states in the continuum, and the ability to dynamically switch between coherent protection and strong radiative emission. The term “pseudo-giant atom” denotes an engineered structure where the physical emitter remains spatially compact, but the interaction with the field is distributed over several coupling ports to synthesize all dynamical features of a true, physically giant atom (Kannan et al., 2019).

1. Theoretical Framework and Model Hamiltonians

The core Hamiltonian for a pseudo-giant atom subject to the rotating-wave and continuum-mode approximations is: $$H = H_{\mathrm{qubit}} + H_{\mathrm{waveguide}} + H_{\mathrm{int}},$$ with

$$H_{\mathrm{qubit}} = \hbar\frac{\omega_q}{2}\sigma_z, \qquad H_{\mathrm{waveguide}} = \sum_k \hbar\omega_k a_k^\dagger a_k,$$

$$H_{\mathrm{int}} = \hbar\sum_{j=1}^N \sum_k \left[ g_j e^{i k x_j} a_k \sigma_+ + g_j^* e^{-i k x_j} a_k^\dagger \sigma_- \right].$$

Here, the qubit transition frequency is $\omega_q$, $a_k$ annihilates a continuum mode at $k$, and $g_j$ is the complex coupling at point $x_j$. The crucial phase factor $e^{i k x_j}$ embeds the delay or relative phase from photon propagation between spatially separated coupling points (Kannan et al., 2019, Du et al., 2021, Kockum, 2019, Andersson et al., 2018).

Passing to the continuum, the effective coupling function is

$G(\omega) = \sum_{j=1}^N g_j e^{i \omega x_j / v}$

such that the atom couples with a single, frequency-dependent “form factor” to the field continuum.

2. Interference, Frequency-dependent Decay, and Lamb Shift

The emission and decoherence properties of a pseudo-giant atom arise from interference between emission paths at different coupling points:

• Decay rate (Fermi’s Golden Rule): $\Gamma(\omega_q) = 2\pi |G(\omega_q)|^2$
• Lamb shift: $$\Delta(\omega_q) = P \int_0^\infty d\omega\, [|G(\omega)|^2 / (\omega_q - \omega)]$$
• Decoherence-free points: Occur when $G(\omega_q) = 0$, corresponding to perfect destructive interference at the emission frequency. For $N=2$ with $g_1 = g_2 = g$ and separation $\Delta x$,

$$G(\omega_q) = g[1 + e^{i\omega_q \Delta x/v}] = 0 \iff \omega_q \Delta x / v = (2n+1)\pi$$

realizing a “dark” qubit (Kannan et al., 2019, Andersson et al., 2018, Kockum, 2019, Du et al., 2021).

By tuning the inter-tap spacing $\Delta x$ and the transition frequency $\omega_q$, one can realize a continuous range from strongly radiative to fully protected configurations in situ. For multi-level atoms or multi-point couplings $(N>2)$, the spectral structure of decay rates and Lamb shifts can be engineered in a highly nontrivial, comb-like fashion (Kockum, 2019, Du et al., 2021).

3. Non-Markovian Dynamics and Bound States

When the time delay between coupling points is non-negligible on the scale of atomic decay ($\gamma T \gtrsim 1$), the evolution becomes non-Markovian. The delay-differential equation for the excited-state amplitude is: $$\dot{c}_e(t) = -\gamma\,c_e(t) - \sum_{j\neq k} \gamma_{jk}\,c_e(t - T_{jk})\,\Theta(t-T_{jk})$$ where $T_{jk}$ denotes propagation times between points $x_j$ and $x_k$ (Andersson et al., 2018, Guo et al., 2016, Xu et al., 2023).

Consequences include:

• Nonexponential decay: The atom’s excited-state population decays in a sequence of revival pulses or with subexponential envelopes, analytically captured via series expansions or the solution structure of transcendental characteristic equations, often involving the multivalued Lambert $W$ function (Andersson et al., 2018, Guo et al., 2016).
• Bound states in the continuum (BICs): For certain geometries and coupling strengths, persistent atom–field oscillations between the emitter and a localized, cavity-free wave packet are established, yielding extremely long-lived subradiant modes (Guo et al., 2019, Xu et al., 2023, Longhi, 2020, Xiao et al., 2021).
• Critical non-Markovianity parameter: In the multi-leg regime with equal coupling and spacing, the criterion for observable non-Markovian (multiple perfect-reflection) features is $\gamma_{\text{tot}} T > 6/(1+1/N)$ (Xu et al., 2023).

4. Multi-qubit Effects, Braided Geometries, and Decoherence-free Subspaces

Pseudo-giant atom architectures permit extension to several spatially distributed qubits or multi-port coupling of one or more transmons:

• Braided geometries: In two-qubit systems with interleaved coupling points (e.g., $x_1^a < x_1^b < x_2^a < x_2^b$), both qubits can be rendered “dark” at a shared frequency while maintaining a finite mediated exchange interaction, enabling decoherence-free, high-fidelity gates (Kannan et al., 2019).
• Phase-tunable interactions: Exchange, cross-damping, and collective Lamb shifts are all functions of the accumulated inter-tap phase. The ability to adjust $\omega_q$ dynamically enables switching between protected and dissipative regimes (Kannan et al., 2019, Kockum, 2019).
• Entanglement and nonclassical emission: Phase-dependent decay channels allow delayed entanglement onset in coupled emitters and robust antibunching of emitted photons, as seen in both microwave and optical domain giant-atom emulators (Chen et al., 2023, Xiao et al., 2021).

5. Experimental Realizations and Device Architectures

Pseudo-giant atoms have been implemented in several platforms:

• Superconducting transmons coupled to coplanar waveguides (CPW): MIT–Lincoln Lab devices with N=2 or 3 taps spaced millimeters apart, enabling $T_1 \gg 30\,\mu s$ at decoherence-free points and selective two-qubit gate activation. Example: two-tap device with $\Delta x \approx 20$ mm, maximal individual point strength $\gamma_0 / 2\pi \approx 2$ MHz (Kannan et al., 2019).
• Surface acoustic wave (SAW) devices: Qubits coupled via multi-finger interdigital transducers, exploiting the slow SAW velocity to achieve delays $T$ up to hundreds of ns (e.g., $L=50–550\,\mu$m, $T \sim 19–190$ ns, $\gamma T \approx 0.8$–7) (Andersson et al., 2018, Guo et al., 2019).
• Optical photonics and photonic crystal waveguides: Synthetic giant atoms via coupled Rydberg atomic arrays or multi-mode photonic crystal platforms, with programmable coupling locations and detunings (Chen et al., 2023).
• Metamaterial slow-light waveguides and multimode cavity networks: Emulation of nonlocal coupling via tunable circuit nodes or multi-mode capacitive connections (Kockum, 2019).

A summary table of device types and key parameters:

Platform	Coupling Points	Typical Spacing	Achievable Regime
CPW+Transmon	N=2–3	mm–cm	Decoherence-free, in-situ
SAW+IDT	N=2–100	50–550 μm	Non-Markovian, BICs
Photonic crystal+Rydberg	N=2	λ~optical	Optical giant-atom
Metawaveguide/multimode	N=2–N	λ–multiple λ	Flexible interference

6. Functionalities: Photon Control, Frequency Conversion, and Quantum Simulation

Pseudo-giant atoms permit dynamical protocols inaccessible in pointlike settings:

• Catch and release of flying qubits: Non-Markovian bound states in multi-point coupled transmon loops function as photonic or phononic “tweezers,” allowing on-demand trapping and emission of propagating pulses without traditional resonators (Xu et al., 2023).
• Frequency conversion and Sagnac enhancement: Phase-tunable interference with giant-atom geometries produces efficient, even unit-efficiency, single-photon frequency conversion, further boosted by Sagnac interferometry at coupling points (Du et al., 2021).
• Programmable line-shape engineering: Arbitrary spectral response shaping via $N$-point coupling design, implementing Ramsey-like interference, mode filtering, and quantum memory operations (Du et al., 2021, Longhi, 2020).
• Quantum Zeno and anti-Zeno effects: Measurement-modified decay rates can be sharply modulated in pseudo-giant atoms via their size parameter, controllable Lamb shift, and engineered coupling spectrum (Zhang et al., 2022).

7. Design Principles, Regimes, and Outlook

The central design variable is the set $\{g_j, x_j\}$, determining the “form factor” $G(\omega)$ and thus all spectral, temporal, and correlation properties:

• Large $N$ and/or long delays $T$ highly non-Markovian, supporting robust dark states and slow decay ($t^{-1/2}$ envelope).
• Geometry (inter-leg spacing, tap location, circuit phase) precisely tunes between protected (zero decay) and radiative (maximal emission) operation.
• Pseudo-giant implementation reduces the need for physically large atoms, allowing chip-scale devices to exhibit “giant” atom physics.
• Applications include decoherence-free subspaces, robust quantum state transfer, quantum memories, nonclassical photon sources, and high-fidelity gate engineering (Kannan et al., 2019, Kockum, 2019, Chen et al., 2023, Xu et al., 2023).

Pseudo-giant atoms thus bridge quantum optics, circuit QED, and quantum information with tailored nonlocal coupling, providing a versatile platform for exploration of waveguide QED, non-Markovian feedback, and protected quantum operations at both microwave and optical wavelengths.