Department of Applied Physics, Waseda University, Okubo 3-4-1, Shinjuku-ku, Tokyo, 169-8555, Japan.
Department of Advanced Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo, 169-8555, Japan.
Sci Rep. 2023 Jul 13;13(1):11340. doi: 10.1038/s41598-023-36547-w.
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 ([Formula: see text] 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 [Formula: see text] for the coupling between the FQ and the modes in the CPWR, where the coupling grows as [Formula: see text] for [Formula: see text] and decreases as [Formula: see text] for [Formula: see text]. Here [Formula: see text] is the frequency of the nth 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 [Formula: see text] 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.
我们报告了在多模电路量子电动力学（QED）系统中处于深强耦合（DSC） regime 下的极其大的兰姆移动的实验和理论结果，其中qubit-谐振器耦合强度可与 qubit 和谐振器频率相媲美或更大。该系统包括超导磁通量子比特（FQ）和四分之一波长共面波导谐振器（[Formula: see text] CPWR），它们通过共享边缘上的约瑟夫森结进行感应耦合，以实现 DSC regime。光谱在 CPWR 的基模频率周围进行，并且通过单模量子 Rabi 哈密顿量对光谱进行拟合，以获得系统参数。由于 qubit 也与谐振器中的许多更高模式耦合，因此单模拟合不能提供裸 qubit 能量，而是包含来自所有其他模式的重整化的值。我们推导出多模谐振器系统中兰姆移动的理论公式。如先前研究所示，FQ 与 CPWR 中的模式之间的耦合存在截止频率[Formula: see text]，其中耦合随[Formula: see text]增长，随[Formula: see text]减小。这里[Formula: see text]是第 n 个模式的频率。截止效应发生是因为 qubit 充当了谐振器中电流的障碍物，它抑制了在 qubit 位置处的谐振器中高于[Formula: see text]的模式的电流，导致耦合强度降低。使用我们观察到的光谱和理论公式，我们估计基本模式的兰姆移动为 82.3%，所有模式的总兰姆移动为 96.5%。这一结果表明，与 CPWR 中的大量模式的耦合产生了极其大的兰姆移动，但不会将 qubit 能量抑制为零，这在没有高频截止的情况下会发生。
