Entanglement generation by qubit scattering in three dimensions

A qubit (a spin-1/2 particle) prepared in the up state is scattered by local spin-flipping potentials produced by the two target qubits (two fixed spins), both prepared in the down state, to generate an entangled state in the latter when the former is found in the down state after scattering. The scattering process is analyzed in three dimensions, both to lowest order and in full order in perturbation, with an appropriate renormalization for the latter. The entanglement is evaluated in terms of the concurrence as a function of the incident and scattering angles, the size of the incident wave packet, and the detector resolution to clarify the key elements for obtaining an entanglement with high quality. The characteristics of the results are also discussed in the context of (in)distinguishability of alternative paths for a quantum particle.

Figure 1

(Color online) Qubit $X$ is scattered by qubits $A$ and $B$, and is detected by a detector with a finite resolution.

Figure 2

(Color online) Concurrence $C$ in the Born approximation for cases with a large incident wave packet, $w⪢d$, shown in polar coordinates with radius $C$ as a function of the scattering angle ${\theta }_{\text{D}}$ relative to the alignment $\mathit{d}$ of the target qubits $A$ and $B$. Parameters are: (a) $\Delta \theta =\pi /25$, $\pi /15$, $\pi /6$ with $k_{0}d=10$ (dependence on the opening angle of the detector mouth $\Delta \theta$), and (b) $k_{0}d=5$, 10, 25 with $\Delta \theta =\pi /15$ (dependence on the incident momentum $k_0$). The thin-line circles indicate $C=1$ (maximal entanglement).

Figure 3

(Color online) Concurrence $C$ when the incident wave packet is small, $w/d=0.5$, is shown for different incident angles ${\theta }_0=0°$, $60°$, $90°$ relative to the alignment $\mathit{d}$ of qubits $A$ and $B$, with $k_{0}d=10$ and $\Delta \theta =\pi /15$. The concurrence $C$ for a large incident wave packet $w/d=10$ (with the other parameters the same) is also shown (dashed curve) as a reference. In the directions indicated by the dots (where ${\theta }_{\text{D}}={\theta }_0$, i.e., in the same direction as the incident direction ${\hat{\mathit{k}}}_0$ and in its cylindrically symmetric directions with respect to $\mathit{d}$), the concurrence $C$ takes the same values in both cases $w⪢d$ and $w\lesssim d$.

Figure 4

(Color online) Resolving power of an optical device and condition (4.16). The optical device with an aperture $\Delta \theta$ can resolve the separation $d$ if the wavelength ${\lambda }_0=2\pi /k_0$ is short enough, i.e., when ${\lambda }_0\lesssim d\Delta \theta \text{sin}{\theta }_{\text{D}}$. Otherwise, it cannot distinguish two sources $A$ and $B$, the condition of which is nothing but Eq. (4.16).

Figure 5

(Color online) Two paths via $A$ and $B$ have difference in their lengths by $\left|\left({\hat{\mathit{k}}}_0-\hat{\mathit{k}}\right)\cdot \mathit{d}\right|$.

Figure 6

(Color online) (a) Concurrence $C$ and (b) yield $P$ in the full order of perturbation with a large incident wave packet, $w⪢d$, shown in polar coordinates with radii $C$ and $P$ as functions of the scattering angle ${\theta }_{\text{D}}$ relative to the alignment $\mathit{d}$ of the target qubits $A$ and $B$. Parameters are: $k_{0}d=10$, ${\theta }_0=90°$, $\Delta \theta =\pi /15$, and $m{g}_r/{\hslash }^{2}d=1$. The corresponding concurrence $C$ in the Born approximation is also shown in (a) as a reference (dashed curve).

Figure 7

(Color online) (a) Concurrence $C$ and (b) yield $P$ in the full order of perturbation with a large incident wave packet, $w⪢d$, as functions of the scattering angle ${\theta }_{\text{D}}$ and the coupling constant $g_r$. Parameters are: $k_{0}d=10$, ${\theta }_0=90°$, and $\Delta \theta =\pi /15$.

Figure 8

(Color online) Concurrence $C$ in the full order of perturbation with a large incident wave packet, $w⪢d$, as functions of the scattering angle ${\theta }_{\text{D}}$ and the incident angle ${\theta }_0$. Parameters are: (a) $k_{0}d=10$, $m{g}_r/{\hslash }^{2}d=1$; (b) $k_{0}d=10$, $m{g}_r/{\hslash }^{2}d=10$; (c) $k_{0}d=\pi$, $m{g}_r/{\hslash }^{2}d=1$; (d) $k_{0}d=\pi$, $m{g}_r/{\hslash }^{2}d=10$. $\Delta \theta =\pi /15$ for all cases.

Figure 9

(Color online) Probabilities $P$ corresponding to the concurrences in Fig. 8.

Figure 10

(Color online) Integration contours. The original contour $C_0$ and the deformed ones $C_{\pm }$.