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arXiv:1801.07019 · Physical Review A

Totally destructive interference for permutation-symmetric many-particle states

Christoph Dittel, Gabriel Dufour, Mattia Walschaers, Gregor Weihs, Andreas Buchleitner, Robert Keil

Published Jun 15, 2018• 30 citations
arXivHTMLPDF

Abstract

Interference is at the heart of quantum mechanics and is most intriguing in the complexity of multiple interfering particles. Here fundamental symmetry considerations are used to develop a general formalism for understanding the totally destructive interference of many bosons and fermions.
DOI
10.1103/physreva.97.062116

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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.
Abstract Particle-exchange symmetry shapes the counting statistics of many-body systems by giving rise to multiple alternative evolutions whose transition amplitudes interfere. Here, we apply a Fourier transform over the symmetric group S N to the collection of N ! many-body transition amplitudes connecting two states of an N -particle system. This provides a decomposition of the counting statistics of many-body interference experiments into contributions associated with distinct irreducible exchange symmetries. As a first application of our formalism, we consider bosons or fermions which are rendered partially distinguishable from one another through unobserved degrees of freedom, and whose state is also subjected to a Fourier analysis. We thereby obtain a systematic description of partial distinguishability, which is both an important experimental noise source and a possible handle for the control of many-body interference. As a second application, we identify mechanisms responsible for completely destructive interference in many-body systems with other exchange symmetries than those of bosons or fermions. For an interferometer implementing the discrete Fourier transform unitary, we predict many instances of such suppressed transitions.
The Pauli exclusion principle (PEP) can be considered from two aspects. First, it asserts that particles that have half-integer spin (fermions) are described by antisymmetric wave functions, and particles that have integer spin (bosons) are described by symmetric wave functions. It is called spin-statistics connection (SSC). The physical reasons why SSC exists are still unknown. On the other hand, PEP is not reduced to SSC and can be consider from another aspect, according to it, the permutation symmetry of the total wave function can be only of two types: symmetric or antisymmetric. They both belong to one-dimensional representations of the permutation group, while other types of permutation symmetry are forbidden. However, the solution of the Schrödinger equation may have any permutation symmetry. We analyze this second aspect of PEP and demonstrate that proofs of PEP in some wide-spread textbooks on quantum mechanics, basing on the indistinguishability principle, are incorrect. The indistinguishability principle is insensitive to the permutation symmetry of wave function. So, it cannot be used as a criterion for the PEP verification. However, as follows from our analysis of possible scenarios, the permission of states with permutation symmetry more general than symmetric and antisymmetric leads to contradictions with the concepts of particle identity and their independence. Thus, the existence in our Nature particles only in symmetric and antisymmetric permutation states is not accidental, since all symmetry options for the total wave function, except the antisymmetric and symmetric, cannot be realized. From this an important conclusion follows, we may not expect that in future some unknown elementary particles that are not fermions or bosons can be discovered.