Probing higher-order transitions through scattering of microwave photons in the ultrastrong-coupling regime of circuit QED

Guan-Ting Chen, Po-Chen Kuo, Huan-Yu Ku, Guang-Yin Chen, and Yueh-Nan Chen

Phys. Rev. A 98, 043803 – Published 2 October 2018

Abstract

Higher-order transitions can occur in the ultrastrong-coupling regime of circuit QED through virtual processes governed by the counter-rotating interactions. We propose a feasible way to probe higher-order transitions through the scattering of propagating microwave photons incident on the hybrid qubit-cavity system. The line shapes in the scattering spectra can indicate the coherent interaction between the qubits and the cavity, and the higher-order transitions can be identified in the population spectra. We further find that if the coupling strengths between the two qubits and the cavity are tuned to be asymmetric, the dark antisymmetric state with the Fano line shape can also be detected from the variations in the scattering spectra.

Received 22 May 2018

DOI:https://doi.org/10.1103/PhysRevA.98.043803

Physics Subject Headings (PhySH)

Cavity quantum electrodynamics Light-matter interaction Quantum optics Scattering theory

Atomic, Molecular & Optical

Authors & Affiliations

Guan-Ting Chen¹, Po-Chen Kuo¹, Huan-Yu Ku¹, Guang-Yin Chen^2,*, and Yueh-Nan Chen^1,3,†

¹Department of Physics, National Cheng Kung University, Tainan 701, Taiwan
²Department of Physics, National Chung Hsing University, Taichung 402, Taiwan
³Physics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan

^*gychen@phys.nchu.edu.tw
^†yuehnan@mail.ncku.edu.tw

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Vol. 98, Iss. 4 — October 2018

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Images

Figure 1
Schematic diagram of two flux qubits coupled to a resonator in a transmission-line waveguide. A microwave photon incident from the left is coherently scattered by the qubit-cavity system.
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Figure 2
(a) The figure shows the energy levels for the lowest energy eigenstates of the qubit-cavity system as a function of the normalized cavity frequency $ω_{c} / ω_{q}$ , using the parameters $λ_{1} = λ_{2} = 0.1 ω_{q}$ and $θ = π / 6$ , where $ω_{q}$ is the qubit frequency as a reference point. In the inset we can observe the anticrossing between $| g g 10 〉$ and $| e e 00 〉$ , adopted from [33]. (b) The reflection spectra $R$ of the incident microwave photons in a one-dimensional waveguide for the coupling strengths: $λ_{1} = λ_{2} = 0 ω_{q}$ (blue-solid curve) and $λ_{1} = λ_{2} = 0.1 ω_{q}$ (red-dashed curve). (c) The reflection spectra $R$ with the atomic and the cavity losses. The coupling strengths are set to be $λ_{1} = λ_{2} = 0.5 ω_{q}$ (green-dot-dashed curve), and the corresponding loss rates are $γ = κ = 3 \times 10^{- 5} ω_{q}$ [33].
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Figure 3
The populations of the energy eigenstates with the atomic and the cavity losses. The parameters $λ_{1}$ , $λ_{2}$ , $g$ , $Γ = \frac{4 g^{2}}{v}$ (the decay rate into the microwave photon modes), and $ω_{c}$ are normalized to the qubit frequency $ω_{q}$ . We set $λ_{1} = λ_{2} = 0.5 ω_{q}$ , $g = 0.01 ω_{q}$ , $Γ = 0.05 ω_{q}$ , and $ω_{c} = 1.5 ω_{q}$ [51, 55]. The black-solid, red-dashed, green-dot-dashed, and blue-dotted curves represent the states $| g g 10 〉$ , $| e e 00 〉$ , $(| g e 00 〉 + | e g 00 〉) / \sqrt{2}$ , and $(| g e 00 〉 - | e g 00 〉) / \sqrt{2}$ , respectively. We can observe the populations of $| e e 00 〉$ and $| g g 10 〉$ exchange at the normalized frequency near 1.5.
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Figure 4
(a) and (b) Density plots of the reflection spectra $R$ of the incident microwave photons as functions of $ω_{/} ω_{q}$ and $ω_{c} / ω_{q}$ . The bright regions indicate the high reflection probability around 1. The coupling strengths between the qubits and the cavity are chosen as $λ_{1} = λ_{2} = 0.1 ω_{q}$ in (a) and $λ_{1} = 0.1 ω_{q}, λ_{2} = 0.2 ω_{q}$ in (b).
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Figure 5
The reflection spectra $R$ for the asymmetric coupling strengths between the two qubits and the cavity as a function of the normalized microwave photon frequency without the losses (a) and with the losses (c). The populations of the dark antisymmetric state $(| g e 00 〉 - | e g 00 〉) / \sqrt{2}$ as a function of the normalized microwave photon frequency without the losses (b) and with the losses (d). The red-solid, blue-dashed, and green-dot-dashed curves in (a) and (b) denote $(λ_{1} = 0.1 ω_{q}, λ_{2} = 0.15 ω_{q})$ , $(λ_{1} = 0.1 ω_{q}, λ_{2} = 0.2 ω_{q})$ , and $(λ_{1} = 0.1 ω_{q}, λ_{2} = 0.25 ω_{q})$ , respectively. The red-solid, blue-dashed, and green-dot-dashed curves in (c) and (d) denote $(λ_{1} = 0.5 ω_{q}, λ_{2} = 0.55 ω_{q})$ , $(λ_{1} = 0.5 ω_{q}, λ_{2} = 0.6 ω_{q})$ , and $(λ_{1} = 0.5 ω_{q}, λ_{2} = 0.65 ω_{q})$ , respectively.
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