Persistent Quantum Beats and Long-Distance Entanglement from Waveguide-Mediated Interactions

We study photon-photon correlations and entanglement generation in a one-dimensional waveguide coupled to two qubits with an arbitrary spatial separation. To treat the combination of nonlinear elements and 1D continuum, we develop a novel Green function method. The vacuum-mediated qubit-qubit interactions cause quantum beats to appear in the second-order correlation function. We go beyond the Markovian regime and observe that such quantum beats persist much longer than the qubit lifetime. A high degree of long-distance entanglement can be generated, increasing the potential of waveguide-QED systems for scalable quantum networking.

Figure 1

Schematic diagram of the waveguide system and single-photon transmission. (a) Two qubits (separated by $L$) interacting with the waveguide continuum. Panels (b) and (c) show color maps of the single-photon transmission probability $T$ and the phase shift $\theta$, respectively, as a function of detuning $\delta =ck-{\omega }_0$ and $2kL$. Here, we consider the lossless case ${\Gamma }^{\prime }=0$.

Figure 2

Quantum beats in the Markovian regime. The second-order photon-photon correlation function of both the transmitted (top) and reflected (bottom) fields as a function of $t$ for $k_{0}L=0$ (solid line) and $k_{0}L=\pi /2$ (dashed line). The incident weak coherent state is on resonance with the qubits: $k=k_0={\omega }_0/c$. (Parameters: ${\omega }_0=100\Gamma$ and ${\Gamma }^{\prime }=0.1\Gamma$).

Figure 3

Persistent quantum beats in the non-Markovian regime. The second-order correlation function of both the transmitted (top) and reflected (bottom) fields is plotted as a function of $t$ for $k_{0}L=25.5\pi$ (solid line) and $100.5\pi$ (dashed line). We set the incident coherent state on resonance with the qubits ($k=k_0$), ${\omega }_0=100\Gamma$ and ${\Gamma }^{\prime }=0.1\Gamma$.

Figure 4

Renormalized transition frequencies and decay rates of the symmetric ($S$) and antisymmetric ($A$) states. Panels (a)–(c) show the transition frequencies ${\omega }_S$ (thin solid line) and ${\omega }_A$ (thick solid line) obtained numerically from Eq. (7) together with ${\omega }_S$ (thin dashed line) and ${\omega }_A$ (thick dashed line) given by the Markov approximation. Panels (d)–(f) similarly show the decay rates ${\Gamma }_S$ and ${\Gamma }_A$ obtained both numerically and in the Markov approximation. (${\omega }_0=100\Gamma$ and ${\Gamma }^{\prime }=0.1\Gamma$).

Figure 5

Long-distance qubit-qubit entanglement. The steady state concurrence is plotted as a function of $k_{0}L$ for (a) $0\le k_{0}L\le 5\pi$ and (b) $95\pi \le k_{0}L\le 100\pi$. The Rabi frequencies are ${\Omega }_1=0.1\Gamma$ and ${\Omega }_2=0$. The driving laser is on resonance with the qubits. (${\omega }_0=100\Gamma$ and ${\Gamma }^{\prime }=0.1\Gamma$).