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Boson-fermion complementarity in a linear interferometer

Michael G. Jabbour, Nicolas J. Cerf

Abstract

Bosonic and fermionic statistics are well known to give rise to antinomic behaviors, most notably boson bunching vs. fermion antibunching. Here, we establish a fundamental relation that combines bosonic and fermionic multiparticle interferences in an arbitrary linear interferometer. The bosonic and fermionic transition probabilities appear together in a same equation which constrains their values, hence expressing a boson-fermion complementarity that is independent of the details of the interaction. For two particles in any interferometer, for example, it implies that the average of the bosonic and fermionic probabilities must coincide with the probability obeyed by classical particles. Incidentally, this fundamental relation also provides a heretofore unknown mathematical identity connecting the squared moduli of the permanent and determinant of arbitrary complex matrices.

Boson-fermion complementarity in a linear interferometer

Abstract

Bosonic and fermionic statistics are well known to give rise to antinomic behaviors, most notably boson bunching vs. fermion antibunching. Here, we establish a fundamental relation that combines bosonic and fermionic multiparticle interferences in an arbitrary linear interferometer. The bosonic and fermionic transition probabilities appear together in a same equation which constrains their values, hence expressing a boson-fermion complementarity that is independent of the details of the interaction. For two particles in any interferometer, for example, it implies that the average of the bosonic and fermionic probabilities must coincide with the probability obeyed by classical particles. Incidentally, this fundamental relation also provides a heretofore unknown mathematical identity connecting the squared moduli of the permanent and determinant of arbitrary complex matrices.
Paper Structure (5 sections, 5 theorems, 109 equations, 1 figure)

This paper contains 5 sections, 5 theorems, 109 equations, 1 figure.

Table of Contents

  1. Appendix
  2. Generating function for the transition probabilities
  3. Recurrence relation for the transition probabilities
  4. General relations for matrices
  5. Further physical applications of Theorem \ref{['theo:RecNmain']}

Key Result

Lemma 1

Let $\boldsymbol{x}, \boldsymbol{z} \in [0,1)^N$. The generating function $g(\boldsymbol{x},\boldsymbol{z})$ of the bosonic transition probabilities via a linear interferometer given by the unitary matrix $U$ satisfies where $X = \mathrm{diag}(\boldsymbol{x})$ and $Z = \mathrm{diag}(\boldsymbol{z})$.

Figures (1)

  • Figure 1: Examples of the components of the fundamental relation \ref{['eq:theo:RecNmain']} for an arbitrary interferometer with four modes ($N=4$) and at most one particle per mode. We consider different numbers of particles: (a) $|\boldsymbol{i}| = |\boldsymbol{n}| = 1$, (b) $|\boldsymbol{i}| = |\boldsymbol{n}| = 2$ and (c) $|\boldsymbol{i}| = |\boldsymbol{n}| = 3$. The full (open) circles represent bosons (fermions). The blue and red components correspond, respectively, to the terms with positive and negative signs in Eq. \ref{['eq:theo:RecNmain']}. The dashed vertical lines at the inputs and outputs of each interferometer separate different input and output patterns (rather than multiple particles per input or output mode); the summation over all input and output patterns is implicit.

Theorems & Definitions (13)

  • Lemma 1
  • proof : Proof of Lemma \ref{['lem:GenFunc']}
  • Lemma 2
  • proof : Proof of Lemma \ref{['lem:RecNmain2']}
  • Theorem 1: Boson--fermion complementarity
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof : Proof of Equation \ref{['eq:MacMahon']}
  • ...and 3 more