Chiral route to spontaneous entanglement generation

We study the spontaneous entanglement generation between two qubits chirally coupled to a waveguide. The maximum achievable concurrence is demonstrated to increase by a factor of $4/e\sim 1.5$ as compared to the non-chiral coupling situation. The proposed entanglement scheme is shown to be robust against variation of the qubit properties such as detuning and separation, which are critical in the nonchiral case. This result relaxes the restrictive requirements of the nonchiral situation, paving the way toward a realistic implementation. Our results demonstrate the potential of chiral waveguides for quantum entanglement protocols.

Figure 1

System under study. Two qubits of frequency ${\omega }_0$ and separated by a distance $d$ are placed in the vicinities of a waveguide. The energies ${\gamma }_{j\alpha }$ quantify the chiral coupling of qubit $j$ to the photonic propagating mode $\alpha (=L,R)$. In the same fashion, the decay rate into the 3D environment and other lossy modes is described by a coupling constant ${\mathrm{\Gamma }}_j$.

Figure 2

Time evolution of the concurrence, Eq. (6), for different qubit-qubit separations $d$, in the nonchiral (a) and chiral (b) cases. (c) Dependence of $C_{\text{max}}$ on the separation between the qubits, $d$. The condition ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2$ is chosen because it optimizes the maximum achievable concurrence $C_{\text{max}}$ (see the main text). In all three panels, the ideal case ${\beta }_j=1$ is considered.

Figure 3

Maximum achievable concurrence in the ideal case $\left(\beta =1\right)$, vs the directionalities of each qubit, ${\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2$, for the left qubit initially excited, and a separation $d={\lambda }_0$. The blue line displays the nonchiral value $C_{\text{max}}=0.5$.

Figure 4

Maximum concurrence $C_{\text{max}}$ as a function of the various system parameters $({\mathrm{\Delta }}_j=0.9,{\beta }_j=0.98)$. (a) Effect of the detuning between the transition frequencies of the qubits, $\delta$. (b) Dependence on the total coupling to the waveguide, $\gamma$. (c) Dependence on the qubit-qubit separation, $d$, for different values of $\gamma$. All the coupling constants are normalized to ${\omega }_0$.