When quantum systems evolve through Hilbert space, they acquire geometric phases [1]. For many applications, this can be expressed by a line integral over an (Abelian) vector potential, which leads to a scalar phase. However, systems with degenerate energy levels experience matrix-valued geometric phases [2], known as Non-Abelian holonomies. While Non-Abelian structures naturally arise in Yang Mills theories, they have also proven to be cruicial for topological quantum computation [3]. In our work we use an integrated photonic structure to realize a non-Abelian phase utilizing an Abelian gauge field, namely electro-magnetic wavepackets.
Abstract When a quantum system undergoes slow changes, the evolution of its state depends only on the corresponding trajectory in Hilbert space. This phenomenon, known as quantum holonomy, brings to light the geometric aspects of quantum theory. Depending on the number of degrees of freedom involved, these purely geometric entities can be scalar or belong to a matrix-valued symmetry group. In their various forms, holonomies are vital elements in the description of the fundamental forces in particle physics as well as theories beyond the standard model such as loop quantum gravity or topological quantum field theory. Yet, implementing matrix-valued holonomies thus far has proven challenging, being further complicated by the difficulties involved in identifying suitable dark states for their construction in bosonic systems. Here we develop a representation of holonomic theory founded on the Heisenberg picture and leverage these insights for the experimental realization of a three-dimensional quantum holonomy. Its non-Abelian geometric phase is implemented via the judicious manipulation of bosonic modes constructed from indistinguishable photons and obeys the U(3) symmetry relevant to the strong interaction. Our findings could enable the experimental study of higher-dimensional non-Abelian gauge symmetries and the exploration of exotic physics on a photonic chip.
Implementing quantum gates as non-Abelian holonomies, a class of topologically protected unitary operators, is a particularly promising paradigm for the design of intrinsically stable quantum computers [1]. In contrast to dynamic phases, the geometric phase accumulated by a quantum system propagating through a Hilbert space
Hdepends exclusively on its path. In general, geometric phases can exhibit arbitrary dimensionality. Wilczek and Zee introduced the idea of multi-dimensional, non-Abelian geometric phases - so called holonomies [2]. Anandan later dropped the requirement of adiabaticity to create holonomies, that are truly time-independent [3]. Non-adiabatic holonomies rely on a subspace
Hgeoof the Hilbert-space that is spanned by states
{∣Φk⟩}kthat fulfill
(Φk∣H^∣Φj⟩=0, where
H^is the system's Hamiltonian. Restricting the propagation to
Hgeoensures parallel transport and, thus, a purely geometric phase (see Fig. 1a) [4], [5]. Quantum optics constitutes a particularly versatile platform for quantum information processing, and in particular for the construction of non-adiabatic holonomic quantum computers: In addition to integration and miniaturization provided by the platform, the bosonic nature of photons also conveniently allows for multiple excitations of the same mode, readily expanding
Hgeoand enabling the synthesis of holonomies from higher symmetry groups
U(N)as larger and more capable computational units [6], [7].
In 1984 Berry pointed out that abelian gauge structures exist naturally in slowly varying quantum system. Since then, a great deal of interest both theoretically and experimentally has been devoted to the interpretation, generalization and detection of geometric phases. In particular, relevant for the present work, it has been shown that a non-Abelian gauge structure emerges if a set of quantum states remains degenerate as the Hamiltonian varies. Non-Abelian phases have been originally investigated in Nuclear Quadrupole Resonance. More recently, they have found a renewed interest within the field of Quantum Computation, since it has been proved that quantum gates can be also implemented by geometrical means using both Abelian and non-Abelian phases. In this novel geometric approach, known as Holonomic Quantum Computation (HQC), information is encoded in a degenerate eigenspace of a parametric family of Hamiltonians and the computational network of unitary quantum gates is realized by driving adiabatically the Hamiltonian parameters along loops in a control manifold. By properly designing such loops, the non trivial curvature of the geometry of the control manifold gives rise to unitary transformations, i.e. holonomies, that implement desired quantum gates. Until now only one scheme has been proposed to realize HQC in trapped ions by means of pulsed coherent excitations. On the other hand, it would be of great interest to have an implementation for solid-state systems, in view of their scalability and integrability. In particular, the impressing development of nanotechnologies with the consequent control of artificial two-level system in superconducting devices stimulates a more detailed investigation of geometric interference in mesoscopic systems. Here we show that non-Abelian phases can appear in the quantum charge dynamics of Josephson networks. Besides their possible detection, we shall see how they may be employed both in adiabatic charge pumping and as an implementation of HQC with Josephson junctions. 2. Josephson network for non abelian phases