Quantum entanglement creation based on quantum scattering in one-dimensional waveguides

We propose two heralded schemes for generating multipartite Greenberger-Horne-Zeilinger (GHZ) and W states. They are realized by photon scattering in one-dimensional waveguides, and the qubits are encoded in the degenerate ground states of the quantum emitters. In our schemes, errors caused by the system imperfections and faulty photon scattering can be turned into detectable photon loss, which is advantageous for quantum information science. That is, our schemes for creating GHZ and W states work in a heralded way. Moreover, the generation of the two entangled states can be theoretically generalized to many-qubit cases in a straightforward way. Our calculations show that both the success probabilities and fidelities of our two schemes are high with current technique on manipulating quantum dots in slow-light photonic crystal waveguides.

Figure 1

The structure for a photon mirror, where a two-level quantum emitter (the black dot) is coupled to a 1D waveguide (the cylinder). An input photon wave packet (black) from the left scatters with the emitter, resulting in a transmitted part (green) and a reflected part (blue). (b) A heralded setup for implementing a $Z$ gate on an emitter confined in a 1D waveguide. PBS denotes a polarizing beam splitter, which transmits the horizontal polarization photon $|H\rangle$ and reflects the vertical polarization photon $|V\rangle$. The black lines represent free space photon paths, $M$ denotes a fully reflected mirror, and BS is a 50:50 beam splitter. The inset gives four-level structure and optical transitions of the quantum emitter, e.g., a negatively charged quantum dot in experiment. In detail, the ground state $|\downarrow \rangle$ ($|\uparrow \rangle$) is the electron-spin state with $J_z=-\frac{1}{2}$ ($J_z=\frac{1}{2}$), and $|\uparrow \downarrow \Downarrow \rangle$ ($|\uparrow \downarrow \Uparrow \rangle$) is the trion state of a negatively charged exciton with $J_z=-\frac{3}{2}$ ($J_z=\frac{3}{2}$).

Figure 2

Schematic setup for implementing the four-emitter GHZ state. ${\text{HWP}}_i$ ($i=1,2$) represents a half-wave plate oriented at ${45}^{\circ }$ to realize the conversion of the photon polarization $|H\rangle \leftrightarrow |V\rangle$, and ${\text{HWP}}_3$ denotes a half-wave plate oriented at $22.5^{\circ }$ to apply a Hadamard operation on the polarizations of the flying photon, i.e., $|H\rangle \to \left(\right|H\rangle +|V\rangle )/\sqrt{2}$ or $|V\rangle \to \left(\right|H\rangle -|V\rangle )/\sqrt{2}$ [98, 99]. $D_i$ ($i=1,2,3$) is a single-photon detector.

Figure 3

Schematic setup for realizing the four-emitter W state. ${\text{HWP}}_i$ ($i=1,2$) denotes a half-wave plate oriented at $22.5^{\circ }$ to apply a Hadamard operation on the polarization states, i.e., $|H\rangle \to \left(\right|H\rangle +|V\rangle )/\sqrt{2}$ or $|V\rangle \to \left(\right|H\rangle -|V\rangle )/\sqrt{2}$.

Figure 4

The success probabilities of the scattering configuration shown in Fig. 1 (black solid line) and the setup for creating the GHZ or W state (blue dashed line) vs (a) the frequency detuning $\delta /{\mathrm{\Gamma }}_{{}_{1\mathrm{D}}}$ and (b) the Purcell factor $P$. Parameters: (a) $P=100$ and (b) $\delta =0$.

Figure 5

The fidelities of the setups for creating the GHZ state (black solid line) and W state (blue dashed line) as a function of (a) the frequency detuning $\delta /{\mathrm{\Gamma }}_{{}_{1\mathrm{D}}}$ and (b) the Purcell factor $P$. Parameters: (a) $P=100$ and (b) $\delta =0$.

Figure 6

The fidelities of the setups for creating the (a) GHZ state and (b) W state as a function of $\delta /{\mathrm{\Gamma }}_{{}_{1\mathrm{D}}}$ for $\eta =0$ (black solid lines), $\eta =0.05{\mathrm{\Gamma }}_{{}_{1\mathrm{D}}}$ (red dashed lines), and $\eta =0.1{\mathrm{\Gamma }}_{{}_{1\mathrm{D}}}$ (blue dotted lines). (a, b) $P=100$.