mori projection operator  (Genovis Inc)

Journal: Communications Physics

Article Title: Renormalization and low-energy effective models in cavity and circuit quantum electrodynamics

doi: 10.1038/s42005-025-02325-5

Figure Lengend Snippet: a Schematical representation of a fluxonium qubit, defined by a Josephson junction with energy E J and a capacitance C 1 in parallel with a inductance L 1 , galvanically coupled to a L C resonator. First eigenstates of the fluxonium qubit for ϕ ext = π ( b ) and ϕ ext = 3 π /4 ( c ). In the latter, the parity symmetry is broken. In both cases, the dashed black line correspond to the potential. Comparison of the eigenvalues in the full model (solid blue line), the standard QRM (dashed green line), and the RQRM (dotted red line), as a function of the normalized coupling g / ω c , and for ϕ ext = π ( d ) and ϕ ext = 49 π /50 ( e ). As for the real atoms case, the renormalization gives better results. f Time evolution of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle i(\hat{a}-{\hat{a}}^{{\dagger} })\rangle$$\end{document} ⟨ i ( a ^ − a ^ † ) ⟩ after a π -pulse on the qubit and in the case of ϕ ext = 49 π /50. The renormalized QRM provides a better agreement with the full model. The parameters used in this Figure are: E C = q 2 /(2 C 1 ) = 2.5 GHz, E L = ( ℏ /2 q ) 2 / L 1 = 0.5 GHz, E J = 9 GHz, and ω c = 3 ω 10 , which reproduce typical experimental values for fluxonium qubits , . For the π -pulse, we used ω dr = E 10 , σ dr = 50/( E 21 − E 10 ) and t 0 = 3 σ dr (see Methods).

Article Snippet: The standard QRM is obtained by truncating the atomic Hilbert space to the two lowest energy levels, which can be formally obtained by applying the projection operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{P}}={\sum }_{n = 0}^{1} \vert n \rangle \langle n \vert$$\end{document} P ^ = ∑ n = 0 ∣ n ⟩ ⟨ n ∣ to the full Hamiltonian (see Fig. a).

Techniques: Comparison

Journal: Communications Physics

Article Title: Renormalization and low-energy effective models in cavity and circuit quantum electrodynamics

doi: 10.1038/s42005-025-02325-5

Figure Lengend Snippet: Mean square error of the eigenvalues of the first 5 excited states with respect to the full model, as a function of the gauge parameter η , for g / ω c = 0.8, m = 1, γ = 60, and ω c = 3 ω 10 . The QRM (red dashed) breaks gauge invariance, showing that the dipole gauge ( η = 1) is the most accurate. On the other hand, the RQRM \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{{{\mathcal{H}}}}}^{{\prime} (\eta )}$$\end{document} H ^ ′ ( η ) (green dotted) in Eq. is not only gauge invariant but also provides more accurate results. For completeness, we also compare these models with the gauge-preserving QRM \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{{{\mathcal{H}}}}}^{(\eta )}$$\end{document} H ^ ( η ) (blue dash-dotted) in Eq. , which, however, does not take into account the renormalization of the higher energy levels.

Article Snippet: The standard QRM is obtained by truncating the atomic Hilbert space to the two lowest energy levels, which can be formally obtained by applying the projection operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{P}}={\sum }_{n = 0}^{1} \vert n \rangle \langle n \vert$$\end{document} P ^ = ∑ n = 0 ∣ n ⟩ ⟨ n ∣ to the full Hamiltonian (see Fig. a).

Techniques: Preserving

Journal: Communications Physics

Article Title: Renormalization and low-energy effective models in cavity and circuit quantum electrodynamics

doi: 10.1038/s42005-025-02325-5

Figure Lengend Snippet: a Expectation value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\hat{a}+{\hat{a}}^{{\dagger} })}^{2}$$\end{document} ( a ^ + a ^ † ) 2 ( a ) on the third excited state of the full model (blue solid), QRM (red dashed) and the RQRM (green dash-dotted), as a function of the coupling strength g / ω c . The RQRM provides more accurate results, even for strong coupling strengths. Matrix elements of the cavity field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{a}+{\hat{a}}^{{\dagger} }$$\end{document} a ^ + a ^ † ( b ) and the atomic position operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{x}$$\end{document} x ^ ( c ) between the ground and the second excited, as a function of g / ω c . While in the panel ( b ) the renormalization improves the accuracy, in the panel c it does not. This behavior can be explained by the infidelity of the second excited state of the RQRM with respect to the QRM, as a function of g / ω c , for both the reduced density matrix of the cavity (solid light blue) and the atom (dashed orange) ( d ).

Article Snippet: The standard QRM is obtained by truncating the atomic Hilbert space to the two lowest energy levels, which can be formally obtained by applying the projection operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{P}}={\sum }_{n = 0}^{1} \vert n \rangle \langle n \vert$$\end{document} P ^ = ∑ n = 0 ∣ n ⟩ ⟨ n ∣ to the full Hamiltonian (see Fig. a).

Techniques: