Collective phases of identical particles interfering on linear multiports

We introduce collective geometric phases of bosons and fermions interfering on a linear unitary multiport, where each phase depends on the internal states of identical particles (i.e., not affected by the multiport) and corresponds to a cycle of the symmetric group. We show that quantum interference of $N$ particles in generic pure internal states, i.e., with no pair being orthogonal, is governed by $(N-1)(N-2)/2$ independent triad phases (each involving only three particles). The deterministic distinguishability, preventing quantum interference with two or three particles, allows for the genuine $(N\ge 4)$-particle phase (interference) on a multiport: setting each particle to be deterministically distinguishable from all others except two by their internal states allows for a novel (circle-dance) interference of $N\ge 4$ particles governed by a collective $N$-particle phase, while simultaneously preventing the $R$-particle interference for $3\le R\le N-1$. The genuine $N$-particle interference manifests the $N\mathrm{th}$-order quantum correlations between identical particles at a multiport output; it does not appear in the marginal probability for a subset of the particles, e.g., it cannot be detected if at least one of the particles is lost. This means that the collective phases are not detectable by the usual “quantumness” criteria based on the second-order quantum correlations. The results can be useful for quantum computation, quantum information, and other quantum technologies with single photons.

Figure 1

(a) $N$ identical particles are impinging on a multiport with a unitary matrix $U$. The circular arrows illustrate the $N$-particle cycle $(1,2,3,...,N)$ responsible for the circle-dance interference with deterministically distinguishable particles $\alpha$ and $\beta$ for $\beta \ne \alpha \pm 1$ (mod $N$). (b) A weighted directed graph representation of particle distinguishability in panel (a) in the case of pure internal states $|{\varphi }_k\rangle , k=1,...,N$, where each particle is a vertex and a directed edge $k\to l$ has the complex weight $w(k,l)\equiv -ln\left(\langle {\varphi }_l|{\varphi }_k\rangle \right)=d_{kl}+i{\theta }_{kl}$, where $d_{kl}$ serves as the distance between the vertices (here $d_{kl}=\infty$ is indicated by the absence of such an edge) and ${\theta }_{kl}$ is the phase in the direction $k\to l$. Only two $(R\ge 3)$-particle cycles on a finite path—$1\to 2\to \cdots \to 5\to 1$ and its inverse—contribute to output probability in panel (a); therefore the latter depends on a single collective (five-particle) phase along the edges ${\theta }_{(1,2,3,4,5)}={\theta }_{12}+{\theta }_{23}+{\theta }_{34}+{\theta }_{45}+{\theta }_{51}$ (see the text).

Figure 2

The directed graph for the multiparticle phases, where the mutual phase ${\theta }_{kl}\equiv \arg \left(\langle {\varphi }_k|{\varphi }_l\rangle \right)$ weights the directed edge $k\to l$. An $(R\ge 3)$-particle collective phase corresponds to a closed oriented path on $R$ vertices. Each closed path on edges has a corresponding multiparticle phase obtained by summation of the mutual phases on the edges passed (with the minus sign, when passing an edge in the inverse direction to the indicated one). The directed paths $1\to 2\to 3\to 1, 1\to 3\to 4\to 1$, and $1\to 2\to 4\to 1$ allow expression of any closed oriented paths on the edges 1, 2, 3, 4.