Fourier analysis of many-body transition amplitudes and states

Abstract

We decompose the counting statistics of many-body interference experiments into contributions associated with distinct irreducible exchange symmetries. To do so, we perform a Fourier transform over the symmetric group $S_N$ on the collection of $N!$ many-body transition amplitudes connecting two states of a system of $N$ particles. We apply our formalism to the interference of partially distinguishable bosons and fermions and describe mechanisms responsible for completely destructive interference in many-body systems obeying specific exchange symmetries, including, but not limited to, bosons and fermions.

Figures (6)

• Figure 1: Graphical representations of the multiplication tables of the (Abelian, i.e., commutative) cyclic group of order 6 (top) and (non-Abelian) group $S_3$ (bottom). The group elements $\sigma$ are represented by points on a circle. For each group element $\tau$, the map $\sigma\mapsto \tau\circ\sigma$ is depicted by arrows connecting these points.
• Figure 2: Spectral density [Eq. \ref{['spectdens']}] of the transition amplitude function [Eq. \ref{['transamp']}] for $N=4$ particles in a Fourier interferometer [Eq. \ref{['UFT']}] with $M=4$ modes. The colour code gives $\mathop{\mathrm{Tr}}\nolimits\left[ \hat{a}(\lambda)^\dagger\hat{a}(\lambda)\right]$ for (inequivalent) pairs $\ket{\bm{i}}$ and $\ket{\bm{o}}$ of input and output states (labels along the x- and y-axes, respectively), for each symmetry type $\lambda$. Transitions to or from Pauli-forbidden states (see Sec. \ref{['sec:Pauli']}) are marked in blue. Otherwise suppressed transitions are marked in red.
• Figure 3: Spectral density [Eq. \ref{['spectdens']}] of the transition amplitude function [Eq. \ref{['transamp']}] for $N=6$ particles in a Fourier interferometer [Eq. \ref{['UFT']}] with $M=6$ modes. The colour code gives $\mathop{\mathrm{Tr}}\nolimits\left[ \hat{a}(\lambda)^\dagger\hat{a}(\lambda)\right]$ for (inequivalent) pairs $\ket{\bm{i}}$ and $\ket{\bm{o}}$ of input and output states (labels along the x- and y-axes, respectively), for $\lambda=(6)$ (top), $\lambda=(5,1)$ (middle) and $\lambda=(3,3)$ (bottom). Transitions to or from Pauli-forbidden states (see Sec. \ref{['sec:Pauli']}) are marked in blue. Otherwise suppressed transitions are marked in red.
• Figure 4: Spectral density [Eq. \ref{['spectdens']}] of the transition amplitude function [Eq. \ref{['transamp']}] for three input states $\ket{\bm{i}}$ with periodic occupations in a Fourier interferometer [Eq. \ref{['UFT']}] with $M=N=6$. The colour code gives the value of $\mathop{\mathrm{Tr}}\nolimits\left[ \hat{a}(\lambda)^\dagger\hat{a}(\lambda)\right]$ for (inequivalent) outputs $\bm{o}$ (x-axis labels), and irreps $\lambda$ (y-axis labels). Transitions to or from Pauli-forbidden states (see Sec. \ref{['sec:Pauli']}) are marked in blue. Suppressed transitions are marked in red.
• Figure 5: Left: Distribution of $N=6$ particles over $M=6$ modes, identified with the vertices of a hexagon, for the input mode list $\bm{i}= (0, 0, 0, 1, 3, 5)$. Notice the mirror symmetry across the x-axis. Middle: same for the output mode list $\bm{o}= (0, 0, 1, 2, 3, 3)$. Notice the symmetry across the y-axis. Right: Corresponding distribution in the complex plane of the many-body transition amplitudes $a(\sigma)=\braket{\bm{o}|R(\sigma)^\dagger U^{\otimes N}|\bm{i}}$, for $\sigma$ running over $S_N$, when $U$ is the Fourier unitary \ref{['UFT']}. Notice the joint mirror symmetry across the x- and y-axes, resulting in a symmetry around the origin. The radius of the blue dots is proportional to the number of particles or amplitudes, red crosses indicate unoccupied modes or absent amplitudes.