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Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Measurement with indefinite causal order and the Sagnac interferometer

Stephen Mark Barnett, Sarah Croke, Sonja Franke‐Arnold

Published Dec 24, 2024• 1 citation
Publisher

Abstract

It has been shown that measurements involving indefinite causal order can be superior to those in which a sequence of operations occurs in a specified order. In optics, such measurements are realized naturally in a Sagnac interferometer. We show that such an arrangement can measure the solid angle (on the Poincaré sphere) enclosed by a sequence of unitary transformations of the polarization. This is the Pancharatnam-Berry phase. Extension from the classical or single-photon treatment to a fully quantized treatment allows the analysis of the interferometer for arbitrary quantum states of light.This article is part of the theme issue 'The quantum theory of light'.

DOI
10.1098/rsta.2024.0447

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It has been shown that measurements involving indefinite causal order can be superior to those in which a sequence of operations occurs in a specified order. In optics, such measurements are realized naturally in a Sagnac interferometer. We show that such an arrangement can measure the solid angle (on the Poincaré sphere) enclosed by a sequence of unitary transformations of the polarization. This is the Pancharatnam-Berry phase. Extension from the classical or single-photon treatment to a fully quantized treatment allows the analysis of the interferometer for arbitrary quantum states of light.This article is part of the theme issue 'The quantum theory of light'.
Sagnac interferometers are important tools for precision measurements. Cancellation of common-path noise ensures a high degree of stability against external perturbations, while their sensitivity to rotations provides the basis of laser gyroscopes. Here we highlight another, less explored feature of Sagnac interferometers: the fact that optical elements are traversed in reverse order for clockwise and counterclockwise propagating light components. This provides the opportunity to explore a classical realization of indefinite causal order, where the order of events within a sequence is not fixed. We explore this concept to identify the geometric phase associated with two non-commuting polarization operations with a single measurement. This idea may have applications for the rapid determination of polarization manipulations or optical activity within chiral media, but foremost it provides a geometric illustration of indefinite causal order.
We report on theoretical calculations and experimental observations of Pancharatnam's phase originating from arbitrary SU(2) transformations applied to polarization states of light. We have implemented polarimetric and interferometric methods, which allow us to cover the full Poincar\'e sphere. As a distinctive feature, our interferometric array is robust against mechanical and thermal disturbances, showing that the polarimetric method is not inherently superior over the interferometric one, as previously assumed. Our strategy effectively amounts to feeding an interferometer with two copropagating beams that are orthogonally polarized with respect to each other. It can be applied to different types of standard arrays, such as a Michelson, a Sagnac, or a Mach-Zehnder interferometer. We exhibit the versatility of our arrangement by performing measurements of Pancharatnam's phases and fringe visibilities that closely fit the theoretical predictions. Our approach can be easily extended to deal with mixed states and to study decoherence effects.
We study two types of topological phases, i.e., Aharonov-Bohm phases, by means of optical interferometry. The first arises from a cycle of redirections of the photon spin and is closely related to the Aharonov-Anandan phase. The second arises from a cycle of polarization changes on the Poincare sphere and was discovered by Pancharatnam. Both of these phases, and their combined effects, have been investigated by a nonplanar Mach-Zehnder interferometer. The signs of the fringe shifts arising from these topological effects have been determined and agree with theory.
Classically, the causal order of two timelike separated events A and B is fixed-either A before B or B before A. This is no longer true in quantum theory, where it is possible to encounter superpositions of causal orders. The quantum switch is one of the most prominent processes with indefinite causal order. Optical realizations of the quantum switch have been successfully implemented in experiments, but some argue that this merely simulates a process with indefinite causal order and that a superposition of spacetime metrics is required for a true realization. Here, we provide a relativistic definition of causal order between operationally defined events that defines a meaningful observable in both the general relativistic and quantum mechanical sense. Importantly, this observable does not distinguish between the indefinite causal order implemented on an optical bench and the gravitational quantum switch, a gedanken experiment where the indefinite causal order is achieved by a quantum superposition of gravitational fields. Therefore, our results support the thesis that the optical quantum switch is just as much a realization of indefinite causal order as its gravitational counterpart, which makes use of the quantum mechanical behavior of spacetime.
In our paper we consider the non-classical behaviour of both the Hermitian (observable) Stokes parameters of light and the phase difference of two modes that describe the quantum polarization states of optical field. To characterize the degree of polarization of light we introduce a new quantity taking into account the quantum properties of different quantum states of two orthogonally polarized modes. The problem of determination of the phase difference in two modes of optical field for the quantum polarization states of light is discussed. To describe in general such a quantum field we introduce two pairs of the phase operators: the phase angles for the Stokes parameters of light in a three-dimensional picture of the Poincaré sphere. We also consider a special type of the eight-port polarization interferometer (polarimeter) for simultaneous homodyne detection of both the Stokes parameters of light and the polarization phase operators and their fluctuations as well. Using an anisotropic (spatioperiodic) Kerr-like nonlinear medium associated with the polarization interferometer we could generate and also observe the polarization-squeezed phase states of light. The fluctuations in the phase difference between two orthogonally polarized modes for these non-classical states are less than the fluctuations for light in coherent state.