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Journal of Physics A Mathematical and Theoretical

Adiabatic approximation for the Rabi model with broken inversion symmetry

Li-Tuo Shen, Zhen‐Biao Yang, Huai‐Zhi Wu

Published Dec 23, 2016• 1 citation
Publisher

Abstract

Abstract We study the properties and behavior of the Rabi model with broken inversion symmetry. Using an adiabatic approximation approach, we explore the high-frequency qubit and oscillator regimes, and obtain analytical solutions for the qubit–oscillator system. We demonstrate that, due to broken inversion symmetry, the positions of two potentials and zero-point energies in the oscillators become asymmetric and have a quadratic dependence on the mean dipole moments within the high-frequency oscillator regime. Furthermore, we find that there is a critical point above which the qubit–oscillator system becomes unstable, and the position of this critical point has a quadratic dependence on the mean dipole moments within the high-frequency qubit regime. Finally, we verify this critical point based on the method of semiclassical approximation.

DOI
10.1088/1751-8121/50/4/044002

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We study the effect of inversion symmetry breaking on properties of the ground state of a qubit-oscillator system within the ultrastrong-coupling regime. We obtain the solution of the ground state through the approximate analytical approach, which, under the near-resonance condition, agrees well with the numerical simulation. We demonstrate that, due to the inversion symmetry breaking, the ground state becomes asymmetric and the qubit-oscillator entanglement entropy strongly depends on the mean dipole moments. Furthermore, we find that the attainable maximum entanglement entropy undergoes a sharp change across a critical point and the minimum value of the Wigner function has a quadratic dependence on the mean dipole moments.
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A generalized quantum Rabi Hamiltonian with both one- and two-photon terms has emerged in the circuit quantum electrodynamics system for a decade. The usual parity symmetry is broken naturally in the simultaneous presence of both couplings, which complicates analytical treatments, even in the rotating-wave approximations. In this paper, we propose an adiabatic approximation to this generic model by using Bogoliubov operators and obtain a very concise analytical solution for both eigenvalues and eigenstates. Although the adiabatic approximation is only exact in the vanishing limit of the qubit frequency, the results for some physical observables nevertheless agree well with the numerical ones in a wide parameter regime. In the rotating-wave approximations, we also derive an analytical eigensolution. Two dominant Rabi frequencies are found in the Rabi oscillations of this generalized model. We also apply the present analytical theory to the vacuum Rabi splitting. It is found that some different phenomena emerge just because of the presence of the additional two-photon coupling term.
The Rabi model, a two-level atom coupled to a harmonic oscillator, can undergo a second-order quantum phase transition (QPT) [M.-J. Hwang et al., Phys. Rev. Lett. 115, 180404 (2015)]. Here we show that the Rabi QPT accompanies critical behavior in the higher-energy excited states, i.e., the excited-state QPT (ESQPT). We derive analytic expressions for the semiclassical density of states, which show a logarithmic divergence at a critical energy eigenvalue in the broken symmetry (superradiant) phase. Moreover, we find that the logarithmic singularities in the density of states lead to singularities in the relevant observables in the system such as photon number and atomic polarization. We corroborate our analytical semiclassical prediction of the ESQPT in the Rabi model with its numerically exact quantum mechanical solution.
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Abstract The asymmetric quantum Rabi model with broken parity invariance shows spectral degeneracies in the integer case, that is when the asymmetry parameter equals an integer multiple of half the oscillator frequency, thus hinting at a hidden symmetry and accompanying integrability of the model. We study the expectation values of spin observables for each eigenstate and observe characteristic differences between the integer and noninteger cases for the asymptotics in the deep strong coupling regime, which can be understood from a perturbative expansion in the qubit splitting. We also construct a parent Hamiltonian whose exact eigenstates possess the same symmetries as the perturbative eigenstates of the asymmetric quantum Rabi model in the integer case.