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Order-Invariant Two-Photon Quantum Correlations in PT-Symmetric Interferometers

Tom A. W. Wolterink, Matthias Heinrich, Stefan Scheel, Alexander Szameit

Published Jan 1, 2023• 0 citations
Publisher

Abstract

We identify sequences of concatenated two-mode linear optical transformations whose two-photon behavior is invariant under reversal of the order. We experimentally verify this systematic behavior in parity-time-symmetric complex interferometer arrangements of varying composition.
DOI
10.1364/cleo_fs.2023.fm4a.5

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Multiphoton correlations in linear photonic quantum networks are governed by matrix permanents. Yet, surprisingly few systematic properties of these crucial algebraic objects are known. As such, predicting the overall multiphoton behavior of a network from its individual building blocks typically defies intuition. In this work, we identify sequences of concatenated two-mode linear optical transformations whose two-photon behavior is invariant under reversal of the order. We experimentally verify this systematic behavior in parity-time-symmetric complex interferometer arrangements of varying compositions. Our results underline new ways in which quantum correlations may be preserved in counterintuitive ways, even in small-scale non-Hermitian networks.
Quantum correlations in linear optical networks are a crucial resource for quantum information processing. Correlations between multiple photons are described by the permanent of the transmission matrix of the optical network [1]. Yet, despite their central role, very few properties of permanents are known to exist and the calculation of matrix permanents remains a computationally hard problem, which, at large scale, has sparked the field of boson sampling. However, the lack of tangible properties also affects small-scale networks, as it hinders intuitive prediction of the behavior of composite systems from their components. This especially holds for non-Hermitian systems that incorporate losses, which are known to drastically alter quantum correlations even in networks of just two modes, such as a lossy beam splitter [2], [3]. Here, we systematically identify sequences of two-mode systems that perform distinct transformations, whereas their permanents remain invariant under reversal of the entire sequence's order. We experimentally verify this property by probing the two-photon correlations in parity-time (PT)-symmetric interferometers of wildly different composition.
With their conceptual description of Parity-Time symmetry in 1998 [1], Bender and Boettcher inaugurated a new field of studies with fascinating implications on topology [2] and exceptional point dynamics [3], to name a few. Typically, PT-symmetric Hamiltonians exhibit real eigenvalues below a certain threshold, whereas above, the PT-symmetry is spontaneously broken as the eigenvalue spectrum becomes complex. Entirely passive configurations based solely on differential losses [7] allow the physics of PT symmetry to be carried over to a quantum-optical context [6] and explored therein [4]. Here we study the ways in which the degree of loss systematically impacts the characteristics of quantum correlations between interfering photons by analyzing the two-photon correlations in PT-symmetric directional couplers.
The average quantum physicist on the street would say that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under combined matrix transposition and complex conjugation) in order to guarantee that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian H = p 2 + ix 3 , which is obviously not Dirac Hermitian, has a positive real discrete spectrum and generates unitary time evolution, and thus it defines a fully consistent and physical quantum theory. Evidently, the axiom of Dirac Hermiticity is too restrictive. While H = p 2 + ix 3 is not Dirac Hermitian, it is PT symmetric; that is, invariant under combined parity P (space reflection) and time reversal T . The quantum mechanics defined by a PT -symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics is extended into the complex domain, new kinds of theories having strange and remarkable properties emerge. In the past few years, some of these properties have been verified in laboratory experiments. A particularly interesting PT-symmetric Hamiltonian is H = p 2 − x 4 , which contains an upside-down potential. This potential is discussed in detail, and it is explained in intuitive as well as in rigorous terms why the energy levels of this potential are real, positive, and discrete. Applications of PT -symmetry in quantum field theory are also discussed.