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arXiv:1303.3367 · Physical Review A

Ground state of a resonant two-qubit cavity system in the ultrastrong-coupling regime

Kelvin M. C. Lee, C. K. Law

Published Jul 8, 2013• 21 citations
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Abstract

We examine a generalized Rabi model formed by two identical qubits interacting with a common oscillator mode. In the near-resonance configuration where the oscillator frequency is close to the transition frequency of the qubit, we determine the ground state of the model approximately by using a variational method and a transformation method. Both methods are shown to approximate the ground state accurately for a range of coupling strengths in the ultrastrong-coupling regime. In addition, the accuracy of the approximation can be further improved by using perturbation theory. We also characterize the ground-state entanglement between the two qubits by calculating the negativity.

DOI
10.1103/physreva.88.015802

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We study the Rabi model composed of three qubits coupled to a harmonic oscillator without involving the rotating-wave approximation. We show that the ground state of the three-qubit Rabi model can be analytically treated by using the transformation method, and the transformed ground state agrees well with the exactly numerical simulation under a wide range of qubit-oscillator coupling strengths for different detunings. We use the pairwise entanglement to characterize the ground-state entanglement between any two qubits and show that it has an approximately quadratic dependence on the qubit-oscillator coupling strength. Interestingly, we find that there is no qubit-qubit entanglement for the ground state if the qubit-oscillator coupling strength is large enough.
We study the system involving mutual interaction between three qubits and an oscillator within the ultrastrong coupling regime. We apply the adiabatic approximation approach to explore two extreme regimes: (i) the oscillator's frequency is far larger than each qubit's frequency and (ii) the qubit's frequency is far larger than the oscillator's frequency, and analyze the energy-level spectrum and the ground-state property of the qubit-oscillator system under the conditions of various system parameters. For the energy-level spectrum, we concentrate on studying the degeneracy in low-energy levels. For the ground state, we focus on its nonclassical properties that are necessary for preparing the nonclassical states. We show that the minimum qubit-oscillator coupling strength needed for generating the nonclassical states of the Schr\"odinger-cat type in the oscillator is just one half of that in the Rabi model. We find that the qubit-qubit entanglement in the ground state vanishes if the qubit-oscillator coupling strength is strong enough, for which the entropy of three qubits remains larger than 1. We also observe the phase-transition-like behavior in the regime where the qubit's frequency is far larger than the oscillator's frequency.
We study the ground state of the two-qubit squeezed Rabi model. Two special transformations are found to diagonalize the system Hamiltonian when each qubit’s frequency is close to the field frequency, where both the squeezing and counterrotating-wave interactions are removed, leading to an effective integrable Hamiltonian. The analytical ground state is determined and matches with numerical solutions well for a range of squeezing strengths and qubit-field detunings in the ultrastrong-coupling regime. We demonstrate that the ground state exhibits a first-order quantum phase transition at a phase boundary linearly induced by the squeezed light. We characterize the two-qubit negativity analytically and find that its two-qubit entanglement increases with the increasing squeezing strength nonlinearly. The average photon numbers of the field mode and variances of position and momentum quadratures are also analyzed and discovered to have a nonlinear relation with the squeezing strength. Finally, we discuss the experimental scheme and realization possibility of the predicted results.
We revisit the analytic solution of the two-qubit quantum Rabi model with inhomogeneous coupling and transition frequencies using a displaced oscillator basis. This approach enables us to apply the same truncation rules and techniques adopted in the Rabi model to the two qubits system. The derived analytical spectra match perfectly with the numerical solutions in the parameter regime where the qubits’ transition frequencies are far off-resonance with the field frequency and the interaction strengths reach the ultrastrong coupling regime. We further explore the dynamical behavior of the two qubits as well as the evolution of entanglement. The analytical methods provide unexpectedly accurate results in describing the dynamics of the two qubits in the present experimentally accessible coupling regime. The time evolutions of the probability for the qubits show that the collapse-revival phenomena emerge, survive and finally disappear when one coupling strength increases from weak to strong coupling regimes and the other coupling strength is well into the ultrastrong coupling regime. The inhomogeneous coupling system exhibits new dynamics, which are different from the homogeneous coupling case.
We present an analytical method for the two-qubit quantum Rabi model. While still operating in the frame of the generalized rotating-wave approximation (GRWA), our method further embraces the idea of introducing variational parameters. The optimal value of the variational parameter is determined by minimizing the energy function of the ground state. Comparing with numerically exact results, we show that our method evidently improves the accuracy of the conventional GRWA in calculating fundamental physical quantities, such as energy spectra, mean photon number, and dynamics. Interestingly, the accuracy of our method allows us to reproduce the asymptotic behavior of a mean photon number in a large frequency ratio for the ground state and investigate the quasiperiodical structure of the time evolution, which the GRWA is incapable of predicting. The applicable parameter ranges cover the ultrastrong-coupling regime, which will be helpful to recent experiments.