Ultrastrong waveguide QED with giant atoms

Sergi Terradas-Briansó, Carlos A. González-Gutiérrez, Franco Nori, Luis Martín-Moreno, and David Zueco

Phys. Rev. A 106, 063717 – Published 22 December 2022

Abstract

Quantum optics with giant emitters has shown a new route for the observation and manipulation of non-Markovian properties in waveguide QED. In this paper we extend the theory of giant atoms, hitherto restricted to the perturbative light-matter regime, to deal with the ultrastrong-coupling regime. Using static and dynamical polaron methods, we address the low-energy subspace of a giant atom coupled to an Ohmic waveguide beyond the standard rotating-wave approximation. We analyze the equilibrium properties of the system by computing the atomic frequency renormalization as a function of the coupling characterizing the localization-delocalization quantum phase transition for a giant atom. We show that virtual photons dressing the ground state are nonexponentially localized around the contact points but decay as a power law. The dynamics of an initially excited giant atom is studied, pointing out the effects of ultrastrong coupling on the Lamb shift and the spontaneous emission decay rate. Finally, we comment on the existence of the so-called oscillating bound states beyond the rotating-wave approximation.

Received 16 June 2022
Accepted 4 November 2022

DOI:https://doi.org/10.1103/PhysRevA.106.063717

Physics Subject Headings (PhySH)

Circuit quantum electrodynamics Light-matter interaction Quantum optics Quantum optics with artificial atoms

Atomic, Molecular & Optical

Authors & Affiliations

Sergi Terradas-Briansó ¹, Carlos A. González-Gutiérrez ¹, Franco Nori ^2,3,4, Luis Martín-Moreno ¹, and David Zueco ¹

¹Instituto de Nanociencia y Materiales de Aragón, CSIC, Universidad de Zaragoza, Zaragoza 50009, Spain
²Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
³Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
⁴RIKEN Center for Quantum Computing, 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan

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Vol. 106, Iss. 6 — December 2022

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Images

Figure 1
(a) Pictorial illustration of the giant emitter with three connection points. (b) Schematic of a circuit-QED implementation of a giant atom coupled to an Ohmic waveguide with three connection points. (c) Dispersion relations for the discrete and continuous models for the Ohmic waveguide. The group velocity for the waveguide is $v_{g} = c = 1$ throughout this work. (d) Spectral function $J (ω)$ for a continuous dispersion relation $ω_{k} = v_{g} | k |$ shown by solid lines and the corresponding results using the dispersion relation from Eq. (3) for $N_{c} = 1$ (squares), $N_{c} = 2$ (triangles), $N_{c} = 3$ (circles), and $N_{c} = 10$ (diamonds). The spacing between coupling points is $x = 5 δ x$ and the coupling strength $α = 0.1$ . The cutoff frequency is $ω_{c} = 3$ and the number of modes is $N = 300$ for both plots (c) and (d).
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Figure 2
(a) Renormalization of the two-level system energy with the coupling strength $α$ and the distance between coupling points $x$ for $N_{c} = 3$ . (b) Set of specific values for the distance $x$ between connections showing the phase transition as $α$ increases. For limiting cases we have analytical expressions; for intermediate distances, the transition is more abrupt.
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Figure 3
(a) Ground-state photons for a giant atom with five connection points ( $N_{c} = 5$ ) with coupling strength $α / N_{c} = 1 / 5$ , in this case $Δ_{r} \neq 0$ . The inset focuses on the profile of the photonic clouds around the rightmost coupling point. A fitting into a power-law decay $x^{- a}$ , plotted as a black dashed line, leads to the exponent $a ≃ 2.96$ . (b) Ground-state photons for $α / N_{c} = 2 / 5$ , corresponding to $Δ_{r} = 0$ . The decay also fits into a power law, in this case with $a ≃ 1.09$ . For both plots, the distance between the closest couplings is $x = 20 δ x$ , the cutoff frequency is $ω_{c} = 3$ , $Δ = 1$ , and the number of modes $N = 15 001$ .
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Figure 4
(a) Normalized effective decay rate and (b) Lamb shift for a giant emitter with three coupling points ( $N_{c} = 3$ ) and bare qubit frequency $Δ = 1$ as a function of the spacing between connections. Dashed lines indicate the behavior for $α = 0.01$ (RWA) and solid lines $α = 0.16$ (beyond the RWA). (c) Full dependence of the effective decay rate on the coupling strength and the distance between coupling points.
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Figure 5
Evolution of the excited-state probability of a giant emitter with three coupling points $N_{c} = 3$ for three different distances between contact points and fixed coupling strength $α = 0.8$ , $Δ = 1$ , $ω_{c} = 6$ , and $N = 3000$ . The expected equilibrium probabilities $P_{e}^{GS}$ are illustrated with horizontal dotted lines.
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Figure 6
(a) Time evolution of the excited-state population $P_{e}$ with increasing coupling $α$ . Black horizontal lines point to the two cases highlighted in the other plots of the figure. (b) Plot of the spontaneous emission into an oscillating bound state in the RWA for $α ≃ 0.118$ (solid line) and for an increased coupling of $α ≃ 0.557$ , within the USC (dashed line). The field emitted into the waveguide for these cases is plotted in (d) and (c), respectively. These simulations are carried out for a giant atom with three connections $N_{c} = 3$ with the parameters $ζ = 186 δ x$ , $ω_{c} = 3$ , $Δ = 1$ , and number of modes $N = 4001$ .
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