Many-particle interferometry and entanglement by path identity

We introduce a general scheme of many-particle interferometry in which two identical sources are used and “which-way information” is eliminated by making the paths of one or more particles identical (path identity). The scheme allows us to generate many-particle entangled states. We provide general forms of these states and show that they can be expressed as superpositions of various Dicke states. We illustrate cases in which the scheme produces maximally entangled two-qubit states (Bell states) and maximally three-tangled states (three-particle Greenberger-Horne-Zeilinger-class states). A striking feature of the scheme is that the entangled states can be manipulated without interacting with the entangled particles; for example, it is possible to switch between two distinct Bell states. Furthermore, each entangled state corresponds to a set of many-particle interference patterns. The visibility of these patterns and the amount of entanglement in a quantum state are connected to each other. The scheme also allows us to change the visibility and the amount of entanglement without interacting with the entangled particles and, therefore, has the potential to play an important role in quantum information science.

Figure 1

Zou-Wang-Mandel experiment. $Q$ and $Q^{\prime }$ are two identical sources emitting two photons (1,2) into beams ($b_1,b_2$) and ($b_1^{\prime },b_2^{\prime }$). The paths of photon 2 are made identical by sending beam $b_2$ through $Q^{\prime }$ and aligning it with $b_2^{\prime }$. Photon 2 is not detected. Single-photon interference is observed at detector $d$ when $b_1$ and $b_1^{\prime }$ are superposed.

Figure 2

(Case I) Two-particle interference and entanglement by one-particle path identity: (a) Schematic of the setup. $Q$ and $Q^{\prime }$ are two identical three-particle sources emitting particles (1,2,3) into beams ($b_1,b_2,b_3$) and ($b_1^{\prime },b_2^{\prime },b_3^{\prime }$). Beam $b_3$ is aligned with $b_3^{\prime }$ in such a way that it is not possible to determine the source of particle 3 if observed after $Q^{\prime }$. The phase change along $b_3$ due to propagation from $Q$ to $Q^{\prime }$ is ${\theta }_3$. Beams $b_1$ and $b_1^{\prime }$ are superposed by BS1 (beam splitter or an equivalent device) with two outputs detected at $d_1$ and $d_1^{\prime }$. The phase difference between $b_1$ and $b_1^{\prime }$ is ${\varphi }_1$. Likewise $b_2$ and $b_2^{\prime }$ are superposed by BS2 with two outputs at $d_2$ and $d_2^{\prime }$; the corresponding phase difference is ${\varphi }_2$. Particle 3 is never detected. (b) Two-particle interference patterns. Probabilities ($P_{d_1{d}_2}, P_{d_1{d}_2^{\prime }}, P_{d_1^{\prime }{d}_2}, P_{d_1^{\prime }{d}_2^{\prime }}$) of joint detection at the pairs of detectors ($d_1,d_2$), ($d_1,d_2^{\prime }$), ($d_1^{\prime },d_2$), and ($d_1^{\prime },d_2^{\prime }$) vary sinusoidally with phase ${\theta }_3$ that can only be modulated using particle 3. Interference patterns $P_{d_1{d}_2}$ and $P_{d_1^{\prime }{d}_2^{\prime }}$ are in phase (dashed line). These patterns are complementary to the patterns $P_{d_1{d}_2^{\prime }}$ and $P_{d_1^{\prime }{d}_2}$ (solid line). We have set ${\mathrm{\Phi }}^{\left(2\right)}=2n\pi , n$ being an integer. A maximum and a minimum of an interference pattern are attained for two distinct Bell states.

Figure 3

(Case II) Two-particle interference and entanglement by two-particle path identity: $Q$ and $Q^{\prime }$ are two identical four-particle sources emitting particles (1,2,3,4) into beams ($b_1,b_2,b_3,b_4$) and ($b_1^{\prime },b_2^{\prime },b_3^{\prime },b_4^{\prime }$), respectively. The beams $b_3$ and $b_4$ are aligned with $b_3^{\prime }$ and $b_4^{\prime }$, respectively; the corresponding phase changes are ${\theta }_3$ and ${\theta }_4$. Particles 3 and 4 are not detected. The rest of the notations are same as in Fig. 2. The two-particle interference patterns produced in this setup are identical to those shown in Fig. 2, except each pattern can now be modulated by both ${\theta }_3$ and ${\theta }_4$. The same Bell states are obtained under conditions strictly similar to case I.

Figure 4

The general scheme (notations are analogous to Figs. 2 and 3). Two identical $N$-particle sources emit particles ($1,2,\cdots ,N$) into beams ($b_1,b_2,\cdots ,b_N$) and ($b_1^{\prime },b_2^{\prime },\cdots ,b_N^{\prime }$), respectively. Paths of $M$ particles ($N-M+1,\cdots ,N$) are made identical by aligning the corresponding beams and these particles are not detected. Rest of the particles ($1,2,\cdots ,N-M$) produce $N-M$-particle interference patterns and entangled states when the corresponding beams are superposed.

Figure 5

Controlling the amount of entanglement. Two-particle entangled states are produced using the setup illustrated by Fig. 2. The concurrence is equal to the visibility of the two-particle interference pattern. Both concurrence and visibility are equal to the amplitude transmission coefficient of the attenuator (when there is no experimental loss).

Figure 6

Fidelity and loss of path identity. We consider the case illustrated by Fig. 2. The fidelity varies linearly with the visibility of the two-particle interference pattern. No path identity results in zero visibility and perfect path identity results in highest possible visibility.