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:

Journal: Development (Cambridge, England)

Article Title: A cadherin-integrin–ECM code for presomitic mesoderm fluidity

doi: 10.1242/dev.204874

Figure Lengend Snippet: Solidification of the presomitic mesoderm. (A) Illustration of the solidifying presomitic mesoderm (PSM) in a 10-somite-stage zebrafish embryo. From left to right is the sequential magnification of the posterior body. The two boxes on the right show transverse sections. PSM cells are pink, neural tube (NT) cells are cyan. The PSM is coated by fibronectin (light green) and the extracellular space is highlighted by secreted GFP (dark green). (B) 2D Simulations of PSM solidification show the fibronectin tissue cortex in light green, secreted GFP in dark green, and PSM cells in pink. As adhesion energy increases relative to kinetic energy, the packing fraction φ (i.e. the ratio of the cellular area to the tissue area) increases. (C-E) Confocal images showing localization of endogenously tagged Fibrillin 2b (in magenta) and Fibronectin1a (in green) in the tailbud. (C) The PSM is coated in an ECM of fibronectin and Fibrillin 2b. Fibrillin 2b is prominent in the notochord (NC) ECM ( n =12). (D) Transverse section of the area highlighted by the cyan dashed line in C. Fibrillin 2b localizes dorsally atop fibronectin on the PSM (white arrow). (E) Neither Fibrillin 2b nor fibronectin is assembled into a matrix within the progenitor zone (PZ). There is a fourfold increase in ECM intensity comparing 50 µm 2 regions in the mid-PSM and PZ ( n =3). (F) Cadherin 2 adhesions stabilize gradually from posterior (pink asterisk) to anterior (pink arrowhead) PSM. Cadherin 2 is tagged in tandem with the slower maturing TagRFP (maturation time 100 min) and sfGFP (maturation time 13.6 min) [TgBAC( cdh2:cdh2-tFP )]. Only TagRFP (in grayscale) is shown [ n =18; mean intensity value for TagRFP in a 30 µm 2 centered on the arrowhead area is 1.94-fold greater than the area around the asterisk, compared to sfGFP with a fold change of 1.66 (paired t -test, P =0.0063)]. Images in C,E,F are 3D projections of confocal z -stacks with anterior to the top. Scale bars: 30 µm (C,E,F); 15 µm (D). LPM, lateral plate mesoderm. See also Fig. S1 .

Article Snippet: 3D confocal projections were processed in Imaris (Oxford Instruments).

Techniques: