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arXiv:2102.11129 · Physical Review Letters

One-Photon Solutions to the Multiqubit Multimode Quantum Rabi Model for Fast W -State Generation

Jie Peng, Jun‐Cong Zheng, Jing Yu, Pinghua Tang, G. Alvarado Barrios, Jianxin Zhong, E. Solano, F. Albarrán-Arriagada, Lucas Lamata

Published Jul 22, 2021• 35 citations
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Abstract

General solutions to the quantum Rabi model involve subspaces with an unbounded number of photons. However, for the multiqubit multimode case, we find special solutions with at most one photon for an arbitrary number of qubits and photon modes. Such solutions exist for arbitrary single qubit-photon coupling strength with constant eigenenergy, while still being qubit-photon entangled states. Taking advantage of their peculiarities and the reach of the ultrastrong coupling regime, we propose an adiabatic scheme for the fast and deterministic generation of a two-qubit Bell state and arbitrary single-photon multimode W states with nonadiabatic error less than 1%. Finally, we propose a superconducting circuit design to catch and release the W states, and shows the experimental feasibility of the multimode multiqubit quantum Rabi model.

DOI
10.1103/physrevlett.127.043604

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General solutions to the quantum Rabi model involve subspaces with an unbounded number of photons. However, for the multiqubit multimode case, we find special solutions with at most one photon for an arbitrary number of qubits and photon modes. Such solutions exist for arbitrary single qubit-photon coupling strength with constant eigenenergy, while still being qubit-photon entangled states. Taking advantage of their peculiarities and the reach of the ultrastrong coupling regime, we propose an adiabatic scheme for the fast and deterministic generation of a two-qubit Bell state and arbitrary single-photon multimode W states with nonadiabatic error less than 1%. Finally, we propose a superconducting circuit design to catch and release the W states, and shows the experimental feasibility of the multimode multiqubit 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.
We have obtained the solutions of the multimode quantum Rabi model when all modes have identical frequencies ωωω, including dark states ∣ϕK⟩|ϕ_K\rangle∣ϕK​⟩ with at least KKK (K=1,2,3,…)(K=1,2,3,\ldots)(K=1,2,3,…) photons. Extended to the multiqubit case, they lie close to another dark state ∣ψ⟩\vert ψ\rangle∣ψ⟩ with at most one photon in the spectrum. Taking advantages of such solutions, we find a linear and symmetry-protected adiabatic passage through ∣ψ⟩\vert ψ\rangle∣ψ⟩ to fast generate arbitrary single-photon MMM-mode WWW states ∣WM⟩\vert W_M\rangle∣WM​⟩ with exactly the same speed. The effective minimum energy gap during the adiabatic evolution is further enlarged to 0.63ω0.63ω0.63ω when Stark shifts are included, such that arbitrary ∣WM⟩\vert W_M\rangle∣WM​⟩ can be ultrafast generated in 1.55×2πω−11.55\times 2πω^{-1}1.55×2πω−1 with fidelity 99%99\%99%, indepedent of MMM. This work reveals the existence of linear ultrafast adiabatic passages in light-matter systems.
We propose a superconducting circuit to implement a two-photon quantum Rabi model in a solid-state device, where a qubit and a resonator are coupled by a two-photon interaction. We analyze the input-output relations for this circuit in the strong-coupling regime and find that fundamental quantum-optical phenomena are qualitatively modified. For instance, two-photon interactions are shown to yield a single- or two-photon blockade when a pumping field is either applied to the cavity mode or to the qubit, respectively. In addition, we derive an effective Hamiltonian for perturbative ultrastrong two-photon couplings in the dispersive regime, where two-photon interactions introduce a qubit-state-dependent Kerr term. Finally, we analyze the spectral collapse of the multiqubit two-photon quantum Rabi model and find a scaling of the critical coupling with the number of qubits. Using realistic parameters with the circuit proposed, three qubits are sufficient to reach the collapse point.
We have found the special dark state solutions of the anisotropic two-qubit quantum Rabi model (QRM), which has at most one photon, and constant eigenenergy in the whole coupling regime. Accordingly, we propose a scheme to deterministically generate two kinds of the two-qubit Bell states through adiabatic evolution along the dark states. With the assistance of the Stark shift, the generation time can be reduced to subnanosecond scales, proportional to the reverse of the resonator frequency, with fidelity reaching 99%. Furthermore, the other two kinds of Bell states can also be ultrafast generated.
We propose a unified and deterministic scheme to generate arbitrary single-photon multimode WWW states in circuit QED. A three-level system (qutrit) is driven by a pump-laser pulse and coupled to NNN spatially separated resonators. The coupling strength for each spatial mode gi{g}_{i}gi​ totally decide the generated single-photon NNN-mode WWW state $|{W}_{N}\ensuremath{\rangle}=\frac{1}{A}{\ensuremath{\sum}}_{i=1}^{N}{g}_{i}|{0}_{1}{0}_{2}\ensuremath{\cdots}{1}_{i}{0}_{i+1}\ensuremath{\cdots}{0}_{N}\ensuremath{\rangle}$, so arbitrary $|{W}_{N}\ensuremath{\rangle}$ can be generated inside resonators by adjusting gi{g}_{i}gi​ and NNN. Moreover, such WWW states can be fast emitted in symmetric temporal waveform with a probability reaching 98.998.9%98.9. The generation (or emission) time and fidelity can both be the same for arbitrary $|{W}_{N}\ensuremath{\rangle}$, independent of mode numbers. Finally, the time evolution process is easy to control since only the pump pulse is time-dependent.
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.