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

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.

Figure 1

(a) Spectrum of the two-qubit two-mode quantum Rabi model with ${\omega }_1={\omega }_2=\omega$, ${\mathrm{\Delta }}_1=0.9\omega$, ${\mathrm{\Delta }}_2=0.1\omega$, $g_{11}=g_{12}=g_{21}=g_{22}=g$. (b) Spectrum of the three-qubit two-mode quantum Rabi model with ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2={\mathrm{\Delta }}_3={\omega }_1={\omega }_2=\omega$, $g=g_{11}=g_{12}+g_{13}=g_{21}$, $g_{12}=g_{22}=g_{13}=g_{23}$. Red lines correspond to even parity while blue lines to odd parity.

Figure 2

Adiabatic evolution of the special dark state $|{\psi }_{2+}⟩$ of Eq. (7) from $|0_M\uparrow \uparrow ⟩$ to $|W_M{\psi }_B⟩$ for the two-qubit $M$-mode quantum Rabi model, where $|{\psi }_B⟩=(1/\sqrt{2})(|\downarrow \uparrow ⟩-|\uparrow \downarrow ⟩)$, and $|W_M⟩$ is the normalized prototype $W$ state with all $g_i$’s equal. (a) Adiabatic trajectory used to vary the parameters for $M=2$. $g_1=g_2=g$. (b) Fidelity $F_M=|⟨W_M{\psi }_B|\psi (T)⟩{|}^2$ when $T$ is fixed to $100{\omega }^{-1}$ for mode number $M$. (c) Population of different states during the adiabatic process for $M=2$. (d) Interaction time needed to reach $F_M>0.99$ for mode $M$ in unit of ${\omega }^{-1}$.

Figure 3

(a) Schematic setup for the generation and release of the $W$ state: Two SQs are capacitively coupled to $M$ CWRs. Each CWR is connected to a TL through a variable coupler $C$, such that the photon emission rate into the TL is controllable. (b) A superconducting circuit design for the two-qubit two-mode Rabi model [56].

Figure 4

Numerical simulation for the catch and release of the prototype three-mode $W$ state $|W_3⟩=(1/\sqrt{3})(|100⟩+|010⟩+|001⟩)$ (left panel) and the four-mode $W$ state $|W_4^{\prime }⟩=(1/\sqrt{10})(|1000⟩+\sqrt{2}|0100⟩+\sqrt{3}|0010⟩+2|0001⟩)$ (right panel). (a) and (b) Population of different states inside resonators. (c) and (d) Emission rates into transmission lines.

Figure 5

Numerical simulation for the catch and release of the perfect three-mode $W$ state $|W_3^{\prime }⟩=\frac{1}{2}(|100⟩+|010⟩+\sqrt{2}|001⟩)$ for quantum teleportation. (a) Population of different states inside resonators. (b) Emission rates into transmission lines.