Quantum Theory of Fractional Topological Pumping of Lattice Solitons

One of the hallmarks of topological systems is the robust quantization of particle transport. It is the origin of the integer-valued quantum Hall conductivity and a potential tool for quantum information technology. Recent experiments on topological pumps constructed by using arrays of photonic waveguides and described by the (lattice-translational invariant) Aubry-André-Harper model, have demonstrated both integer and fractional transport of lattice solitons. In these systems, a background medium mediates interactions between photons via a Kerr nonlinearity and leads to the formation of self-bound multiphoton states. Upon increasing the interaction strength, a sequence of transitions was observed from a phase with integer transport in a pump cycle through different phases of fractional transport to a phase with no transport. We here present a quantum description of topological pumps of self-bound many-particle states in terms of an effective Hamiltonian of their center-of-mass (c.m.) motion, which allows one to introduce an effective band structure <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>E</a:mi> <a:mi>μ</a:mi> </a:msub> <a:mo stretchy="false">(</a:mo> <a:mi>K</a:mi> <a:mo stretchy="false">)</a:mo> </a:math> with <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>K</e:mi> </e:math> being the c.m. momentum and to classify topological phases in terms of generalized symmetries. We provide an explicit analytic expression of the effective Hamiltonian for few particles in the strong interaction limit and present numerical results in the more general case. We identify a topological invariant, an effective single-particle Chern number, which fully governs the soliton transport. Increasing the interaction strength in the Aubry-André-Harper model leads to a successive merging of c.m. bands, which is the origin of the observed sequence of topological phase transitions and also the potential breakdown of topological quantization for some interaction strength.