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arXiv:2008.07708 · Scientific Reports

Hamiltonian of a flux qubit-LC oscillator circuit in the deep–strong-coupling regime

F. Yoshihara, Sahel Ashhab, T. Fuse, Motoaki Bamba, Kouichi Semba

Published Apr 26, 2022• 15 citations

Abstract

We derive the Hamiltonian of a superconducting circuit that comprises a single-Josephson-junction flux qubit inductively coupled to an LC oscillator, and we compare the derived circuit Hamiltonian with the quantum Rabi Hamiltonian, which describes a two-level system coupled to a harmonic oscillator. We show that there is a simple, intuitive correspondence between the circuit Hamiltonian and the quantum Rabi Hamiltonian. While there is an overall shift of the entire spectrum, the energy level structure of the circuit Hamiltonian up to the seventh excited states can still be fitted well by the quantum Rabi Hamiltonian even in the case where the coupling strength is larger than the frequencies of the qubit and the oscillator, i.e., when the qubit-oscillator circuit is in the deep-strong-coupling regime. We also show that although the circuit Hamiltonian can be transformed via a unitary transformation to a Hamiltonian containing a capacitive coupling term, the resulting circuit Hamiltonian cannot be approximated by the variant of the quantum Rabi Hamiltonian that is obtained using an analogous procedure for mapping the circuit variables onto Pauli and harmonic oscillator operators, even for relatively weak coupling. This difference between the flux and charge gauges follows from the properties of the qubit Hamiltonian eigenstates.

DOI: 10.1038/s41598-022-10203-1

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We report on experimentally measured light shifts of superconducting flux qubits deep-strongly coupled to LC oscillators, where the coupling constants are comparable to the qubit and oscillator resonance frequencies. By using two-tone spectroscopy, the energies of the six lowest levels of each circuit are determined. We find huge Lamb shifts that exceed 90% of the bare qubit frequencies and inversions of the qubits' ground and excited states when there are a finite number of photons in the oscillator. Our experimental results agree with theoretical predictions based on the quantum Rabi model.

We have observed the coherent exchange of a single energy quantum between a flux qubit and a superconducting LC circuit acting as a quantum harmonic oscillator. The exchange of an energy quantum is known as the vacuum Rabi oscillation: the qubit is oscillating between the excited state and the ground state and the oscillator between the vacuum state and the first excited state. We also show that we can detect the state of the oscillator with the qubit and thereby obtained evidence of level quantization of the LC circuit. Our results support the idea of using oscillators as couplers of solid-state qubits.

We study the system where a superconducting flux qubit is capacitively coupled to an LC resonator. In three devices with different coupling capacitance, the magnitude of the dispersive shift is enhanced by the third level of the qubit and quantitatively agrees with the theory. We show by numerical calculation that the capacitive coupling plays an essential role for the enhancement in the dispersive shift. We investigate the coherence properties in two of these devices, which are in the strong-dispersive regime, and show that the qubit energy relaxation is currently not limited by the coupling. We also observe the discrete ac-Stark effect, a hallmark of the strong-dispersive regime, in accordance with the theory.

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We have analyzed and proposed coupling mechanisms between Three Josephson Junction Flux Qubits (3JJQ). For this, we have developed a numerical method to extract the effective Hamiltonian of a system of coupled qubits via the Schrieffer-Wolff transformation (SWT). We then give a comprehensive introduction to the 3JJQ, and study it analytically by approximating its potential with a Harmonic well. With a clear understanding of the 3JJQs, we use the SWT to gain intuition about their effective dipolar interaction with the electromagnetic field, and use that intuition to propose and study analytically and numerically the capacitive coupling of two 3JJQs via a non-tunable capacitor, and the inductive coupling of two 3JJQs via a tunable Josephson Junction (dc-SQUID), showing that we are able to reproduce non-stoquastic Hamiltonians in the strong-coupling regime.

Recent measurement on an LC resonator magnetically coupled to a superconducting qubit [ Phys. Rev. Lett. 105 (2010) 237001 ] shows that the system operates in the ultra-strong coupling regime and crosses the limit of validity for the rotating-wave approximation of the Jaynes-Cummings model. By using extended bosonic coherent states, we solve the Jaynes—Cummings model exactly without using the rotating-wave approximation. Our numerically exact results for the spectrum of the flux qubit coupled to the LC resonator are fully consistent with the experimental observations. The smallest Bloch—Siegert shift obtained is consistent with that observed in this experiment. In addition, the Bloch—Siegert shifts in arbitrary level transitions and for arbitrary coupling constants are predicted.