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Scientific Reports

Fractional topological phase measurement with a hyperentangled photon source

A. A. Matoso, R. A. Ribeiro, Luis E. Oxman, Antonio Zelaquett Khoury, Sebastião Pádua

Published Jan 24, 2019• 4 citations
Publisher

Abstract

Pairs of photons simultaneously entangled in their path and polarization degrees of freedom are used to measure the topological phase acquired by bipartite entangled states. Conditional phase local unitary operations having the polarization degree of freedom as the control variable are applied. Qudits of arbitrary dimensions are encoded on the photons transverse positions while polarization entanglement is used as an auxiliary resource for quantum interference measurements. With this scheme the fractional phases predicted for dimensions d = 2, 3 and 4 could be measured with visibilities for the interference curves beyond the limit allowed for classical sources, which is expected for a source of quantum correlated photons. The strategy of perform a quantum interferometry experiment with photons entangled in an auxiliary degree of freedom and apply unitary local operations conditioned to this auxiliary variable shows an increase to the signal to noise ratio, simplifies alignment and can be used in different applications. This offers an interesting perspective for the efficient implementation of phase gates in quantum computing with hyperentangled photon sources in polarization and path degrees of freedom. Furthermore, one can conjecture whether the measured phase can serve as a dimensionality identifier of the Hilbert space dimension for an unknown state preparation.

DOI
10.1038/s41598-018-37344-6

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Pairs of photons simultaneously entangled in their path and polarization degrees of freedom are used to measure the topological phase acquired by bipartite entangled states. Conditional phase local unitary operations having the polarization degree of freedom as the control variable are applied. Qudits of arbitrary dimensions are encoded on the photons transverse positions while polarization entanglement is used as an auxiliary resource for quantum interference measurements. With this scheme the fractional phases predicted for dimensions d = 2, 3 and 4 could be measured with visibilities for the interference curves beyond the limit allowed for classical sources, which is expected for a source of quantum correlated photons. The strategy of perform a quantum interferometry experiment with photons entangled in an auxiliary degree of freedom and apply unitary local operations conditioned to this auxiliary variable shows an increase to the signal to noise ratio, simplifies alignment and can be used in different applications. This offers an interesting perspective for the efficient implementation of phase gates in quantum computing with hyperentangled photon sources in polarization and path degrees of freedom. Furthermore, one can conjecture whether the measured phase can serve as a dimensionality identifier of the Hilbert space dimension for an unknown state preparation.
In this work we performed two experiments to observe the fractional topological phases in two qudits photonic systems. The qudits are encoded in the transverse position degree of freedom of photon pairs generated by the spontaneous parametric down conversion after being transmitted through multiple slits. The photons in this phenomenon are quantum correlated in momentum and energy. Initially we review the theory about the origin of the phases acquired by the evolving quantum states of one and two qubits, where we identify the dynamic, the geometric and the topological phases. Then we examine the generalization of topological phases for two qudits, and discuss its relation with the concept of connectedness of a space. Next we show a way to measure the topological phase, considering that it must appear as a relative phase between two interfering quantum states. To realize the measurement we build an optical interferometer with a spatial light modulator inside it to implement the necessary unitary operations. The polarization is used as an auxiliary degree of freedom for the operations to act conditionally. The results are close to the theoretical predictions, showing the dependence of the phase with the Hilbert space dimension of the state. In a second setup we measure the topological phase for qudits without using a longitudinal interferometer. A new source of polarization entangled photon pairs is used in this new setup. The entanglement in the auxiliary degree of freedom, polarization, is used as a resource to eliminate the interferometer, propitiating a significant improvement in the visibility of the measured interference curves, and a large increase in the signal to noise ratio. In this new setup, we are also capable of realize the unitary operations in both parts of the two qubit system.
In this work we propose two experimental setups to observe fractional topological phases in qudits. The degree of freedom in which the qudits are encoded is the transverse position of photon pairs generated by spontaneous parametric down-conversion after being transmitted through multiple slits. The photons generated in this phenomenonare quantum correlated in energy and momenta. Initially we discuss the basic theory about topological phases, the context in which they are fractional, and its relation with entanglement. Then, we see how local unitary operations can be applied in this degree of freedom with the aid of spatial light modulators that introduce a programmable phase prole on the wavefront of the photons. Thus, the desired fractional phases canbe observed through polarization-controlled two-photon interference. The roles played by entanglement and Hilbert space dimension are revealed by the behavior of the interference fringes as the unitary operations are continuously changed. These ideas are supported by analytical and numerical calculations of the expected interference fringes. We also show how the so called Hong-Ou-Mandel eect is present in the proposed interferometer, which requires simultaneity on the recombination of the photons generated to allow for the interference
Geometrical and topological phases play a fundamental role in quantum theory. Geometric phases have been proposed as a tool for implementing unitary gates for quantum computation. A fractional topological phase has been recently discovered for bipartite systems. The dimension of the Hilbert space determines the topological phase of entangled qudits under local unitary operations. Here we investigate fractional topological phases acquired by photonic entangled qudits. Photon pairs prepared as spatial qudits are operated inside a Sagnac interferometer and the two-photon interference pattern reveals the topological phase as fringes shifts when local operations are performed. Dimensions d=2d=2d=2, 3, and 4 were tested, showing the expected theoretical values.
We characterize a source of photon pairs simultaneously entangled in polarization and in the transverse spatial degree of freedom, with the spatial measurements performed in intermediate planes of the fractional Fourier transform. The source consists of two crossed type-I crystals in a non-collinear geometry, such that the transverse variables carry information about the crystal of origin. Detection in the crystal plane reveals this information completely and destroys the polarization entanglement, so that a scheme using the near and far fields as conjugate spatial bases cannot support a parallel polarization channel. In intermediate planes both spatial bases combine position and momentum, such that the which-crystal information is partially erased, and the polarization entanglement survives in all four combinations of spatial bases. Applying Bell and Einstein-Podolsky-Rosen criteria to the same photon pairs certifies both degrees of freedom simultaneously. We evaluate quantum bit error rates and asymptotic secret key rates in each degree of freedom, and find that the source delivers two simultaneously usable channels per photon pair.
In this work, we demonstrate theoretically via Kraus maps that the fractional topological phase in qudits is robust to phase noise. In our proposal, dephasing noise is inserted in an optical setup designed for measuring the fractional topological phase on photonic qudits encoded in path variables after local unitary operations are applied. Qudit states can be prepared with photon pairs generated by spontaneous parametric down-conversion crossing a multiple-slit array. Polarization as an additional degree of freedom and a two-photon longitudinal interferometer are the basic ingredients of the proposed setup. Although the visibility of the interference pattern decreases when noise is added, our calculation shows that the topological phase is preserved. This result stimulates further studies for the use of fractional topological phase in quantum gates due to its robustness in environments where phase noise is present.