arXiv:2003.03090 · Physical Review A

Highly degenerate photonic waveguide structures for holonomic computation

Julien Pinske, Lucas Teuber, Stefan Scheel

Published Jun 5, 2020• 16 citations

Abstract

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 $M$-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 $\mathrm{U}(3)$ 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.

DOI: 10.1103/physreva.101.062314

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arXiv:2003.03090 · Physical Review A

Highly degenerate photonic waveguide structures for holonomic computation

Julien Pinske, Lucas Teuber, Stefan Scheel

Published Jun 5, 2020• 16 citations

arXiv HTML PDF

Abstract

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

M

-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

\mathrm{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.

DOI: 10.1103/physreva.101.062314

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Carlo Danieli, Valentina Brosco, Claudio Conti +1 more

Controlled and multi-controlled quantum gates, whose action on a target qubit depends on the state of multiple control qubits, represent a fundamental logical building block for complex quantum algorithms. We propose a scheme for realizing this class of gates based on non-Abelian holonomies in modulated photonic waveguide networks. Our approach relies on linear photonic cicuits formed by two star networks coupled via a two-path circuit. A star network with M peripheral waveguides coupled to a single central site, or M-pod, naturally generalizes the tripod structure used in non-Abelian Thouless pumping and stimulated Raman adiabatic passage (STIRAP). In the present work, we first analyze the minimal case of two connected tripods and design adiabatic driving loops that implement single-qubit, CNOT, and SWAP gates. We then show how extending the approach to larger M-pod structures enables the realization of multiply controlled operations, which we exemplify by designing Toffoli and the OR gate on two coupled pentapods. Finally, we demonstrate that networks of connected tripods can implement the Deutsch quantum query algorithm.Controlled and multi-controlled quantum gates, whose action on a target qubit depends on the state of multiple control qubits, represent a fundamental logical building block for complex quantum algorithms. We propose a scheme for realizing this class of gates based on non-Abelian holonomies in modulated photonic waveguide networks. Our approach relies on linear photonic cicuits formed by two star networks coupled via a two-path circuit. A star network with M peripheral waveguides coupled to a single central site, or M-pod, naturally generalizes the tripod structure used in non-Abelian Thouless pumping and stimulated Raman adiabatic passage (STIRAP). In the present work, we first analyze the minimal case of two connected tripods and design adiabatic driving loops that implement single-qubit, CNOT, and SWAP gates. We then show how extending the approach to larger M-pod structures enables the realization of multiply controlled operations, which we exemplify by designing Toffoli and the OR gate on two coupled pentapods. Finally, we demonstrate that networks of connected tripods can implement the Deutsch quantum query algorithm.

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Holonomic quantum computation (HQC) realizes quantum gates through non-Abelian geometric phases, providing an experimentally accessible approach to quantum control. While the nonadiabatic HQC framework has been extensively developed for three-level $Λ$ systems encoding qubits, its systematic extension to higher-dimensional qudits remains largely unexplored. In this work, we generalize nonadiabatic HQC to a $d$ -pod configuration, where a single excited state is coupled to $d$ ground states, the latter forming the computational subspace. This scheme enables universal holonomic single- and two-qudit gates using only optical or microwave pulses on trapped atoms or ions, offering an efficient route to implement a discrete universal gate set with minimal pulse coordination. As an explicit example, we analyze in detail the qutrit ( $d=3$ ) case, demonstrating compact realizations of single- and two-qutrit holonomic gates, each gate requiring at most two loops in the Grassmannian generated by at most three pulses.Holonomic quantum computation (HQC) realizes quantum gates through non-Abelian geometric phases, providing an experimentally accessible approach to quantum control. While the nonadiabatic HQC framework has been extensively developed for three-level $Λ$ systems encoding qubits, its systematic extension to higher-dimensional qudits remains largely unexplored. In this work, we generalize nonadiabatic HQC to a $d$-pod configuration, where a single excited state is coupled to $d$ ground states, the latter forming the computational subspace. This scheme enables universal holonomic single- and two-qudit gates using only optical or microwave pulses on trapped atoms or ions, offering an efficient route to implement a discrete universal gate set with minimal pulse coordination. As an explicit example, we analyze in detail the qutrit ($d=3$) case, demonstrating compact realizations of single- and two-qutrit holonomic gates, each gate requiring at most two loops in the Grassmannian generated by at most three pulses.

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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 $J=0$ and $J=1$ , 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 $N$ 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 $J$ 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.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 $J=0$ and $J=1$, 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 $N$ 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 $J$ 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.

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Julien Pinske, Stefan Scheel

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.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.

arXiv:2105.04859Physical Review AJan 5, 20224 citations

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

M

\mathrm{U}(3)

Λ

systems encoding qubits, its systematic extension to higher-dimensional qudits remains largely unexplored. In this work, we generalize nonadiabatic HQC to a

d

-pod configuration, where a single excited state is coupled to

d

ground states, the latter forming the computational subspace. This scheme enables universal holonomic single- and two-qudit gates using only optical or microwave pulses on trapped atoms or ions, offering an efficient route to implement a discrete universal gate set with minimal pulse coordination. As an explicit example, we analyze in detail the qutrit (

d=3

) case, demonstrating compact realizations of single- and two-qutrit holonomic gates, each gate requiring at most two loops in the Grassmannian generated by at most three pulses.

J=0

and

J=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

N

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

J

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.

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.