Superconducting qubit–oscillator circuit beyond the ultrastrong-coupling regime

Abstract

The interaction between an atom and the electromagnetic field inside a cavity 1-6 has played a crucial role in developing our understanding of light-matter interaction, and is central to various quantum technologies, including lasers and many quantum computing architectures. Superconducting qubits 7,8 have allowed the realization of strong 9,10 and ultrastrong 11-13 coupling between artificial atoms and cavities. If the coupling strength g becomes as large as the atomic and cavity frequencies (∆ and ω o , respectively), the energy eigenstates including the ground state are predicted to be highly entangled 14 . There has been an ongoing debate 15-17 over whether it is fundamentally possible to realize this regime in realistic physical systems. By inductively coupling a flux qubit and an LC oscillator via Josephson junctions, we have realized circuits with g/ω o ranging from 0.72 to 1.34 and g/∆ 1. Using spectroscopy measurements, we have observed unconventional transition spectra that are characteristic of this new regime. Our results provide a basis for ground-state-based entangled pair generation and open a new direction of research on strongly correlated light-matter states in circuit quantum electrodynamics.We begin by describing the Hamiltonian of each component in the qubit-oscillator circuit, which comprises a superconducting flux qubit and an LC oscillator inductively coupled to each other by sharing a tunable inductance L c , as shown in the circuit diagram in Fig. 1a.The Hamiltonian of the flux qubit can be written in the basis of two states with persistent currents flowing in opposite directions around the qubit loop 18 , |L q and |R q , as H q = − (∆σ x + εσ z )/2, where ∆ and ε = 2I p 0 (n φq − n φq0 ) are the tunnel splitting and the energy bias between |L q and |R q , I p is the maximum persistent current, and σ x, z are Pauli matrices. Here, n φq is the normalized flux bias through the qubit loop in units of the superconducting flux quantum, 0 = h/2e, and n φq0 = 0.5 + k q , where k q is the integer that minimizes |n φq − n φq0 |. The macroscopic nature of the persistent-current states enables strong coupling to other circuit elements. Another important feature of the flux qubit is its strong anharmonicity: the two lowest energy levels are well isolated from the higher levels.The Hamiltonian of the LC oscillator can be written asC is the resonance frequency, L 0 is the inductance of the superconducting lead, L qc ( L c ) is the inductance across the qubit and coupler (see Supplementary Section 2), C is the capacitance, andâ (â † ) is the oscillator's annihilation (creation) operator. Figure 1b shows a laser microscope image of the lumped-element LC oscillator, where L 0 is designed to be as small as possible to maximize the zeropoint fluctuations in the currentand hence achieve strong coupling to the flux qubit, while C is adjusted so as to achieve a desired value of ω o . The freedom of choosing L 0 for large I zpf is one of the advantages of lumped-element LC oscillators over coplanar-waveguide ...

Discrete Time-Crystalline Order in Cavity and Circuit QED Systems

“…This model is relevant to cavity QED systems based on cold atoms [53][54][55][56] and circuit QED systems based on superconducting qubits [57][58][59][60][61][62][63]. As schematically illustrated in Fig.…”

Fate of photon blockade in the deep strong-coupling regime

“…achieve a coupling strength that is comparable or even larger than the cavity frequency [18][19][20][21][22][23][24][25]. In this so-called ultrastrong coupling regime, the rotating-wave approximation on which the Jaynes-Cummings model is based is no longer valid and thus the counter-rotating terms cannot be neglected.…”

“…In the latter case, an output-photon statistics that is significantly different from the standard scenario based on the JC model is predicted. The effect of counter-rotating terms on the output-photon statistics and the photon blockade when the coupling strength is of the same order of the cavity frequency [25,[36][37][38], however, has remained largely unexplored.…”

Approaching ultrastrong coupling in transmon circuit QED using a high-impedance resonator

“…Pioneered by the experiments of Refs. [12,13], experiments in the DSC regime have now been achieved with flux qubits coupled to resonators [14] as well as an electromagnetic continuum [15]. Additionally, the U/DSC coupling regime of the Rabi model was the subject of recent analog quantum simulations [16,17].…”