Totally Destructive Many-Particle Interference

In a general, multimode scattering setup, we show how the permutation symmetry of a many-particle input state determines those scattering unitaries that exhibit strictly suppressed many-particle transition events. We formulate purely algebraic suppression laws that identify these events and show that the many-particle interference at their origin is robust under weak disorder and imperfect indistinguishability of the interfering particles. Finally, we demonstrate that all suppression laws so far described in the literature are embedded in the general framework that we here introduce.

Figure 1

Initial and final eigenvalue distribution for $N=5$ particles injected into $n=8$ modes, with input and output mode occupation lists $\overrightarrow{r}$ and $\overrightarrow{s}$, respectively. The input $\overrightarrow{r}$ is invariant under the permutation $\pi =(123)(456)(78)$, with eigenvalue matrix $D=\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)$, ${\lambda }_1={\lambda }_2={\lambda }_3=1$, ${\lambda }_4={\lambda }_5=e^{i(2\pi /3)}$, ${\lambda }_6={\lambda }_7=e^{i(4\pi /3)}$, and ${\lambda }_8=-1$. The mode assignment list on output, $\overrightarrow{d}(\overrightarrow{s})=(2,2,4,5,6)$, then defines the associated final eigenvalue distribution by mapping $\{{\lambda }_1,\dots ,{\lambda }_n\}$ onto $\mathrm{\Lambda }(\overrightarrow{s})=\{{\lambda }_{d_1(\overrightarrow{s})},\dots ,{\lambda }_{d_N(\overrightarrow{s})}\}=\{1,1,e^{i(2\pi /3)},e^{i(2\pi /3)},e^{i(4\pi /3)}\}$. The initial eigenvalue distribution ${\mathrm{\Lambda }}_{\mathrm{ini}}$, defined by $\overrightarrow{r}$ together with the cycle lengths of $\pi$, is a subset of $\{{\lambda }_1,\dots ,{\lambda }_n\}$: ${\mathrm{\Lambda }}_{\mathrm{ini}}=\{e^{i(2\pi /3)},e^{i(4\pi /3)},1,-1,1\}$ (see main text for the definition of ${\mathrm{\Lambda }}_{\mathrm{ini}}$).

Figure 2

Numerical evaluation of mean transmission probabilities $⟨P_{\mathrm{B}/\mathrm{F}/\mathrm{D}}(\overrightarrow{r},\overrightarrow{s},U)⟩$ for the example illustrated in Fig. 1, with bosonic ($B$) and fermionic ($F$) event probabilities depicted by blue bars in panels (a) and (b), respectively. Yellow bars represent event probabilities for distinguishable ($D$) particles, renormalized on the set of singly occupied modes in (b). Data were generated by averaging over 10 000 randomly chosen eigenbases of the permutation operator $\mathcal{P}$, for all output configurations of bosons, fermions, and distinguishable particles, respectively. Forbidden output events are subdivided in classes (I, II, III), where suppression is due to single-particle dynamics (SPD) in classes (I, II), and due to many-particle dynamics (MPD) in (III). All suppressed events of classes (II, III) are predicted by our suppression laws (SL) (4), (5), while those SPD-suppressed events in (I) need some special consideration, see [37]. From altogether 792 bosonic and 56 fermionic transmission events, the subsets of those with nonvanishing detection probabilities are collected in set (IV), in increasing order.