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arXiv:1312.7610

Regular and exceptional spectra of the two-qubit quantum Rabi model

Jie Peng, Zhongzhou Ren, Daniel Braak, Guangjie Guo, Ju Guo-Xing, Zhang, Xin, Xiaoyong Guo

Published Dec 30, 2013• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1312.7610

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•Jul 7, 2018•2 citations
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.
We have studied the two-qubit quantum Rabi model in the asymmetric case and its generalizations with dipole and Heisenberg-type qubit-qubit interactions. The solutions are obtained analytically with eigenstates given in terms of the extended coherent states or photon number states. For identical qubit-photon couplings, a novel type of quasi-exact solution which exists for all coupling values with constant eigenenergy is found, leading to level crossings within the same parity subspace even for non-identical qubits. In contrast to the quasi-exact eigenstates of the single-qubit model, these exceptional eigenstates are formed by just a few Fock states (photon number bounded from above at one or three), and the condition for them depends only on a fine-tuning of the qubit level splittings with respect to the photon energy, but not on the coupling to the photon field if the qubit-qubit interaction is not taken into consideration. This makes them excellent candidates for direct experimental observation within circuit quantum electrodynamics (QED) and application in single photon experiments. Besides, exceptional solutions with finite photon numbers N are also found.
In this report, we have studied exceptional spectra of the two-qubit quantum Rabi model in two situations. Firstly, an exceptional spectra is achieved in resonant condition, in which the frequencies of two qubit and photon field satisfy resonant relation. With transformed rotating-wave approximation (TRWA) method, the Rabi model can be mapped into the solvable formation in qubit-photon Fock states space. Based on the quantum supperconducting circuits experiment setup, the best value range of the system parameters are discussed. In the resonant station working window, the energy spectrums are calculated. Secondly, a special quasi-exact solution of two qubits Rabi model in reservoir is achieved. The algebraic structure of Hamiltonian is analyzed in the photon number space, a closed quasi-exact eigenstates space is found, and the quasi-exact solution can be clearly found from the algebraic structure of Hamiltonian. The results could be used to test quantum Rabi model.
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.
Abstract There are special solutions, the well-known dark states, to the even-qubit Dicke models, which are the products of two-qubit singlets and a Fock state, where the qubits are decoupled from the photon field. These spin singlets can be used to store quantum correlations since they preserve entanglement even under dissipation, driving and dipole-dipole interactions. One of the features for these ‘dark states’ is that their eigenenergies are independent of the qubit-photon coupling strength. Here we have obtained a novel kind of exceptional solution—‘dark-like states’ for the multi-qubit and multi-photon Rabi models, whose eigenenergies are also constant in the whole coupling regime. But unlike the ‘dark states’, the qubits and photon field are coupled. Furthermore, the photon numbers are bounded from above commonly at 1, which is different from the one-qubit case. The existence conditions of these exceptional solutions are simpler than the exact isolated solutions, and may be fine tuned in experiments. While solutions to the single-qubit and multi-photon Rabi model exist only if the photon number M ⩽ 2 and the coupling strength is below a certain critical value, the dark-like eigenstates for the multi-qubit and multi-photon Rabi model exist regardless of these constraints. In view of these properties of the dark-like states, they may find similar applications as ‘dark states’ in quantum information.
In this work, the anisotropic quantum Rabi model with different coupling strengths of the rotating-wave and counter-rotating wave terms is studied by using two kinds of extended coherent states (ECS). By the first kind of ECS, we can derive a so-called GGG-function, by which both the regular and exceptional solutions can be given. The exceptional solution are just corresponding to the crossing points of two energy levels with different parities, so is doubly degenerate. By the second kind of ECS, a general scheme for the eigensolutions is derived analytically in a unified way. The zero-order approximation is just the adiabatic approximation, and the first-order approximation is actually a generalized rotating-wave approximation. The algebraic formulae for the eigensolutions are given explicitly in two approximations. The generalized rotating-wave approximations work well in a wide range of two different coupling strengths and the qubit detunings.
The analytically exact solutions to the mixed quantum Rabi model including both one- and two-photon terms are found by using Bogoliubov operators. Transcendental functions in terms of 4×44 \times 44×4 determinants responsible for the exact solutions are derived. These so-called GGG-functions with pole structure can be reduced to the previous ones in the unmixed models. The zeros of GGG-functions reproduce completely the regular spectra. The exceptional eigenvalues can be also obtained by another transcendental function. From the pole structure, we can derive two energy limits when the two-photon coupling strength tends to the collapse point 1/21/21/2. Surprisingly, all energy levels only collapse to the lower one, which diverges negatively. All the level crossings in the unmixed models are relaxed to avoided crossings in the present mixed model due to absence of parity symmetry. Very interestingly, the effective one-photon coupling is enhanced in the presence of the two-photon coupling, which paves a highly efficient and economic way to access the deep-strong coupling regime even though the original one-photon coupling is weak.