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• W4384200911 abstract "Abstract We report experimental and theoretical results on the extremely large Lamb shift in a multimode circuit quantum electrodynamics (QED) system in the deep-strong coupling (DSC) regime, where the qubit-resonator coupling strength is comparable to or larger than the qubit and resonator frequencies. The system comprises a superconducting flux qubit (FQ) and a quarter-wavelength coplanar waveguide resonator ( $$lambda /4$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> CPWR) that are coupled inductively through a shared edge that contains a Josephson junction to achieve the DSC regime. Spectroscopy is performed around the frequency of the fundamental mode of the CPWR, and the spectrum is fitted by the single-mode quantum Rabi Hamiltonian to obtain the system parameters. Since the qubit is also coupled to a large number of higher modes in the resonator, the single-mode fitting does not provide the bare qubit energy but a value that incorporates the renormalization from all the other modes. We derive theoretical formulas for the Lamb shift in the multimode resonator system. As shown in previous studies, there is a cut-off frequency $$omega _{textrm{cutoff}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>ω</mml:mi> <mml:mtext>cutoff</mml:mtext> </mml:msub> </mml:math> for the coupling between the FQ and the modes in the CPWR, where the coupling grows as $$sqrt{omega _n}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msqrt> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:msqrt> </mml:math> for $$omega _n/omega _{textrm{cutoff}}ll 1$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mtext>cutoff</mml:mtext> </mml:msub> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and decreases as $$1/sqrt{omega _n}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msqrt> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:msqrt> </mml:mrow> </mml:math> for $$omega _n/omega _{textrm{cutoff}}gg 1$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mtext>cutoff</mml:mtext> </mml:msub> <mml:mo>≫</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . Here $$omega _n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> is the frequency of the n th mode. The cut-off effect occurs because the qubit acts as an obstacle for the current in the resonator, which suppresses the current of the modes above $$omega _{textrm{cutoff}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>ω</mml:mi> <mml:mtext>cutoff</mml:mtext> </mml:msub> </mml:math> at the location of the qubit and results in a reduced coupling strength. Using our observed spectrum and theoretical formulas, we estimate that the Lamb shift from the fundamental mode is 82.3% and the total Lamb shift from all the modes is 96.5%. This result illustrates that the coupling to the large number of modes in a CPWR yields an extremely large Lamb shift but does not suppress the qubit energy to zero, which would happen in the absence of a high-frequency cut-off." @default.
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• W4384200911 date "2023-07-13" @default.
• W4384200911 modified "2023-09-30" @default.
• W4384200911 title "Extremely large Lamb shift in a deep-strongly coupled circuit QED system with a multimode resonator" @default.
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• W4384200911 doi "https://doi.org/10.1038/s41598-023-36547-w" @default.
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