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International Journal of Theoretical Physics

The Phase Transition in Two-Photon Rabi Model Under Mean Field Approximation

Xian-Lei Hu

Published Aug 21, 2019• 7 citations
Publisher
DOI
10.1007/s10773-019-04246-w

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We use the mean field approximation method to study the quantum phase transitions of the Jaynes-Cummings lattice model and the Rabi lattice model. The effective Hamiltonians are obtained for the JC and Rabi model including the Kerr nonlinear term. Numerically we diagonalized the Hamiltonian matrix and calculated the superfluidity order parameter and the two-photon correlation function by solving the iteration equations. We have explored the Mott insulating-superfluid quantum phase transition, the bunching-antibunching behavior of light, and the effect of Kerr nonlinear term on the quantum phase transition and photon statistical characteristics. Our results show that in the JC lattice model, by increasing J, a quantum phase transition takes place and the system is driven to a superfluid phase. The phase boundaries of the Mott lobes are N-dependent. However the photon will always be in a bunching statistical behavior irrelevant of the coupling strength between the two-level atom and the phonton and the nonlinear Kerr effect. In the Rabi lattice model, the anti-rotating wave term breaks Mott-lobe structure of the phase diagram and the increase of the two-level atom and photon interaction strength g and the photon transition strength J between the lattices drive the system from the Mott insulating phase to the superfluid phase. The photon statistical behavior changes from the bunching to the antibunching one when considering the anti-rotating wave term, which is important in the strongly coupled systems. Most interestingly, the increase of the Kerr nonlinear coefficient will inhibit the Mott insulating phase-superfluid phase transition, but favor the superfluid phase and the transition from the bunching to anti-bunching statistics.
Embedding quantum dot circuits into microwave cavities has emerged as a novel platform for controlling photon emission statistics by electrical means. With such a circuit version of the Rabi model, we reveal previously undefined quantum phase transitions in electrically driven lasing regimes, which do not require deep strong light-matter couplings. For one-photon interaction, the scaling analysis indicates that the system undergoes a continuous phase transition from thermal to coherent photon emissions. Going beyond this, a discontinuous quantum phase transition from superbunched to coherent states in two-photon processes, accompanied by the bistability within a mean-field theory, is predicted. Both the order of phase transitions and the critical electron-photon coupling can be easily controlled by an electric field, while the tunneling current can be used as a fingerprint of such transitions. Our prediction, along with its extension to multiphoton processes, represents a key step towards accessing lasing phase transitions.
A two-interacting-qubit quantum Rabi-like model with vanishing transverse fields on the qubit pair is studied. Independently of the coupling regime, this model can be exactly and unitarily reduced to two independent single-spin quantum Rabi models, where the spin-spin coupling plays the role of the transverse field. This transformation and the analytical treatment of the single-spin quantum Rabi model provide the key to prove the integrability of our model. The existence of different first-order quantum phase transitions, characterized by discontinuous two-spin magnetization, mean photon number, and concurrence, is brought to light.