Heralded Bell State of Dissipative Qubits Using Classical Light in a Waveguide

Maximally entangled two-qubit states (Bell states) are of central importance in quantum technologies. We show that heralded generation of a maximally entangled state of two intrinsically open qubits can be realized in a one-dimensional (1D) system through strong coherent driving and continuous monitoring. In contrast to the natural idea that dissipation leads to decoherence and so destroys quantum effects, continuous measurement and strong interference in our 1D system generate a pure state with perfect quantum correlation between the two open qubits. Though the steady state is a trivial product state that has zero coherence or concurrence, we show that, with carefully tuned parameters, a Bell state can be generated in the system’s quantum jump trajectories, heralded by a reflected photon. Surprisingly, this maximally entangled state survives the strong coherent state input—a classical state that overwhelms the system. This simple method to generate maximally entangled states using classical coherent light and photon detection may, since our qubits are in a 1D continuum, find application as a building block of quantum networks.

Figure 1

Schematic of two qubits coupled to a 1D waveguide. We have a right-going coherent state as input from the left end. Transmitted and reflected photons are measured using photon counting detection at the right and left end, respectively.

Figure 2

(a) Energy level diagram for $kL=\pi /2$. Red and blue arrows represent left jumps and driving, respectively, $|\pm i⟩\equiv (|ge⟩\pm i|eg⟩)/\sqrt{2}$, $J_{\mathrm{L}}^{-}$ is the left jump operator, and $c^{\pm }$ comes from the right jump operator $J_{\mathrm{R}}^{-}$. The effective qubit-qubit interaction $H_{\mathrm{qq}}$ is suppressed by the strong driving. (b) Second order correlation function for the reflected photons calculated from input-output theory. (Parameters: $kL=\pi /2$, $\alpha =100$.)

Figure 3

Example trajectories of entanglement (first row) and populations (second row) for (a) perfect photon detection and (b) lossy photon detection with efficiency ${\eta }_{\mathrm{i}=\mathrm{L},\mathrm{R}}=\eta =0.95$. The entanglement for pure states in (a) and mixed states in (b) is quantified using the von Neumann entropy $S$ and the entanglement of formation $S_F$, respectively. The times at which quantum jumps occur are marked with blue triangles. Longer trajectories, from $\mathrm{\Gamma }\tau =0$ to 20, are shown in the Supplemental Material [41]. (Parameters: $kL=\pi /2$, $\alpha =100$, qubits initially in the ground state $|gg⟩$.)