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arXiv:1506.01903 · International Journal on Perceptive and Cognitive Computing

Analytical Solution of The Two-Qubit Quantum Rabi Model

D. A. M. Abo-Kahla, M. Abdel-Aty, Salah A. Aly Ahmed

Published May 18, 2016• 1 citation
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Abstract

In this paper, an analytical solution of the two-qubit Rabi model for the general case is presented.Furthermore, a comparison between the information entropies and the Von Neumann entropy S(ρ A ) is given forsome special values of the qubit-photon coupling constants for the two qubits and the detuning parameters. Itis demonstrated that oscillations of the occupation probabilities ρ11 , ρ22 , ρ33 and ρ44 are equivalent to the case ofthe spontaneous emission. The occupation probability ρ11 reaches the case of sudden death, when the detuningparameter ∆2 equals zero.

DOI
10.31436/ijpcc.v2i1.18

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In this paper, an analytical solution of the two-qubit Rabi model for the general case is presented.Furthermore, a comparison between the information entropies and the Von Neumann entropy S(ρ A ) is given forsome special values of the qubit-photon coupling constants for the two qubits and the detuning parameters. Itis demonstrated that oscillations of the occupation probabilities ρ11 , ρ22 , ρ33 and ρ44 are equivalent to the case ofthe spontaneous emission. The occupation probability ρ11 reaches the case of sudden death, when the detuningparameter ∆2 equals zero.
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 studied the two-qubit quantum Rabi model and found its dark state solutions with at most N photons. One peculiar case presents when N=3N=3N=3, which has constant eigenenergy in the whole coupling regime and leads to level crossings within the same parity subspace. We also discovered asymptotic solutions with at most N=2i+3N=2i+3N=2i+3 (i=1,2,3,… )(i=1,2,3,\dots)(i=1,2,3,…) photons, and constant eigenenergy NℏωN\hbar ωNℏω when coupling ggg becomes much larger than photon frequency ωωω. Although generally all photon number states are involved in the two-qubit quantum Rabi model, such NNN-photon solutions exist and may have applications in quantum information processing with ultrastrong couplings.
We consider quantum information entropy phenomenon for multi-qubit Rabi system. By introducing different measurements schemes, we establish the relation between information entropy approach and Von Neumann entropy. It is shown that the information entropy is more sensitive to the time development than the Von Neumann entropy. Furthermore, the suggested protocol exhibits excellent scaling of relevant characteristics, with respect to population dynamics, such that more accurate dynamical results may be obtained using information entropy due to variation of the frequency detuning and the coupling constant.
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We study the two-qubit Rabi model in the most general case where the qubits are different from each other. The spectrum of the system in the ultrastrong-coupling regime is shown to converge to two forced oscillator chains by perturbation theory. An even and odd decomposition of the Hilbert space allows us to calculate the spectra in any given parameter regime; the cases studied confirm our perturbation theory prediction in the ultrastrong-coupling regime and point to crossings in the spectra within each parity subspace in the moderate-coupling regime. The normal modes of the system are calculated by two different methods, the first a linear algebra approach via the parity bases that delivers a four-term recurrence relation for the amplitudes of the proper states and, the second, via Bargmann representation for the field that delivers five-term recurrence relations. Finally, we show some examples of the time evolution of the mean photon number, population inversion, von Neuman entropy and Wootters concurrence under the two-qubit quantum Rabi Hamiltonian by taking advantage of the parity decomposition.
Two-qubit system is the foundation of constructing the universal quantum gate. We have studied the two-qubit Rabi model for the general case and its generalizations with dipole, XXX and XYZ Heisenberg qubit-qubit interactions, which are commonly used in quantum computation. Their solutions are presented analytically with eigenstates can be obtained in terms of extended coherent states or Fock states and applied to the construction of the ultrafast two-qubit quantum gate in circuit QED and quantum state storage and transfer. Some novel kinds of quasi-exact solutions are found for specific sets of parameters, causing level crossings within the same parity subspace, which do not appear in the regular spectrum, indicating its non-integrability. These eigenstates are very interesting for quantum computing and single photon experiments because they are formed by just a few Fock states, and in some cases, with at most one photon. They are also easy to prepare, since they exist for all qubit-photon coupling values with constant eigenenergy and the qubit energy splittings can be fine tuned in the experiment in contrast to the coupling.