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arXiv:2105.04859 · Physical Review A

Symmetry-protected non-Abelian geometric phases in optical waveguides with nonorthogonal modes

Julien Pinske, Stefan Scheel

Published Jan 5, 2022• 4 citations
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Abstract

The generation of non-Abelian geometric phases from a system of evanescently coupled waveguides is considered within the framework of nonorthogonal coupled-mode theory. Here, we study an experimentally feasible tripod arrangement of waveguides that contain dark states from which a nontrivial $U(2)$ mixing can be obtained by means of an adiabatic parameter variation. We investigate the influence of higher-order contributions beyond nearest-neighbor coupling as well as self-coupling on the stability of a non-Abelian $U(3)$ phase generated from an optical tetrapod setup. Our results indicate that, despite the mode nonorthogonality, the symmetry of dark states protects the geometric evolution of light from distortion.

DOI
10.1103/physreva.105.013507

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The generation of non-Abelian geometric phases from a system of evanescently coupled waveguides is considered within the framework of nonorthogonal coupled-mode theory. Here, we study an experimentally feasible tripod arrangement of waveguides that contain dark states from which a nontrivial U(2)U(2)U(2) mixing can be obtained by means of an adiabatic parameter variation. We investigate the influence of higher-order contributions beyond nearest-neighbor coupling as well as self-coupling on the stability of a non-Abelian U(3)U(3)U(3) phase generated from an optical tetrapod setup. Our results indicate that, despite the mode nonorthogonality, the symmetry of dark states protects the geometric evolution of light from distortion.
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We investigate an all-out optical setup allowing for generation of non-Abelian geometric phases on its large degenerate eigenspaces. The proposal has the form of an MMM-pod system and can be implemented in terms of integrated photonic waveguide structures. We show that by injecting a larger number of photons into the optical setup, the degeneracy of eigenspaces scales rapidly. After studying the spectral properties of our system for the general case, we show how arbitrary U(3)\mathrm{U}(3)U(3) transformations can be generated on the dark subspace of an optical tripod filled with two photons. Moreover, a degeneracy in the bright subspaces of the system, absent in any atomic analog, allows for the generation of universal single-qubit manipulations. Finally, we address the complexity issue of holonomic computation. Particularly, we show how two-qubit and three-qubit states can be implemented on a photonic tripod, where a natural multipartite structure is inherited from the spatial mode structure of the waveguides.
A popular quantum system, in which the Pancharatnam-Berry non-Abelian geometric phase has been generated and exploited, is the atomic tripod system. It is conveniently created by linking a single atomic state with three other states by three electromagnetic fields. Such a linkage pattern naturally emerges between the magnetic sublevels of two atomic levels with angular momenta J=0J=0J=0 and J=1J=1J=1, although tripod implementations between other suitable sublevels are also used. Here we go beyond the limitation of a tripod system and show that it is possible to generate the non-Abelian geometric phase in a quantum system composed of NNN lower and $N\ensuremath{-}2$ upper sublevels. The theoretical instrument is the Morris-Shore transformation which reveals the existence of two uncoupled (dark) states composed of the lower sublevels only. A possible physical implementation is the atomic transition $J\ensuremath{\leftrightarrow}J\ensuremath{-}1$, with JJJ arbitrary, which is driven, as in the case of tripod system, by three electromagnetic fields of different polarizations. This generalization considerably broadens the range of systems that can be used to generate a geometric phase, with the the same experimental complexity as in the tripod system. Specific calculations of the non-Abelian geometric phase are presented for $J=\frac{3}{2}\ensuremath{\leftrightarrow}J=\frac{1}{2}$ and $J=2\ensuremath{\leftrightarrow}J=1$ systems. A method for measuring the geometric phase is proposed.
It is known that the dynamics and geometric phase of a quantum system could be mapped mathematically into a classical system of coupled oscillators without loss of physics. In this work, we show that this quantum-classical mapping can also be used to simulate non-Abelian dynamics and phase of the quantum system with energy degeneracy by interpreting the quantum degeneracy as the resonance between the eigenfrequencies of the classical oscillators. The quantum evolution and Wilczek-Zee phase in the degenerate subspace can also be described by the classical non-Abelian evolution and a non-Abelian geometric angle in the resonant subspace. By studying the adiabatic evolution in this classical non-Abelian case, our results can also be used to generalize the concept of averaging principle and Hannay's angle into the classical system with resonant eigenfrequencies. The quantum-classical mapping of a tripod-scheme Hamiltonian with two degenerate dark states is employed to illustrate the classical non-Abelian dynamics and geometric angles.
To what extent can coupling to a dissipative environment hinder the emergence of a geometric phase? And does dissipation undermine its possibly non-Abelian nature? Here, the authors show that non-Abelian geometric phase survives the presence of an environment, albeit modified. The dissipative coupling is manifest in the non-Abelian dynamics. Moreover, it gives rise to non-Abelian geometric dephasing, which may suppress or enhance coherent contributions, depending on the order of adiabatic operations.