Counting statistics of many-particle quantum walks

We study quantum walks of many noninteracting particles on a beam splitter array as a paradigmatic testing ground for the competition of single- and many-particle interference in a multimode system. We derive a general expression for multimode particle-number correlation functions, valid for bosons and fermions, and infer pronounced signatures of many-particle interferences in the counting statistics.

Figure 1

Beam splitter array with each BS described by the unitary matrix $U$, Eq. (1). Particles start in modes with label $(0)$, propagate from top to bottom through the adjacent rows of BS, and are eventually detected in modes with label ($n$), on exit.

Figure 2

Single-mode counting statistics $P^{(25)}$($k$) for mode $i=25$ after $n=6$ steps on a logarithmic scale. Bosonic bunching enhances $P^{(25)}$($k$) for bosons by several orders of magnitude (blue circles) as compared to the result for distinguishable particles (red squares). The inset shows the distribution for small $k$.

Figure 3

Joint counting statistics for two different pairs of modes: Particle distribution $P_m^{(20)}(\Delta k=k_i-k_j)$ among two modes after $n=20$ steps of $N=8$ particles, for lattice size $2L=50$, conditioned on finding $m=4$ or 8 particles.)a) Joint probability distribution of the symmetric (with respect to the lattice center) modes $i=19$, $j=32$ to find $m=4$ particles (dashed; distinguishable: red squares, bosons: blue circles) and $m=8$ particles (solid; distinguishable: red squares, bosons: blue circles). (b) Same as a), for asymmetric modes $i=18$, $j=32$. (c) $P_{m=N}^{(20)}(\Delta k=k_i-k_j)$ of modes $i=19$, $j=32$ for $N=4$ (orange diamonds), $N=6$ (green squares) and $N=8$ (black circles).

Figure 4

Two-mode statistics as in Fig. 3, averaged over all pairs of modes, after (a) $n=19$ steps and (b) $n=20$ steps. The color code is as in Fig. 3. Note that while the distribution is essentially invariant under changes of $n$ for distinguishable particles, the correlations fluctuate with $n$, in the bosonic case.