The Wigner-Lohe model for quantum synchronization and its emergent dynamics

Abstract

We present the Wigner-Lohe model for quantum synchronization which can be derived from the Schrödinger-Lohe model using the Wigner formalism. For identical one-body potentials, we provide a priori sufficient framework leading the complete synchronization, in which L 2 -distances between all wave functions tend to zero asymptotically. 1 (Pierangelo Marcati)

Systems of matrix Riccati equations, linear fractional transformations, partial integrability and synchronization

“…VII we return to the rectangular matrix system, restricted to satisfy unitarity conditions, which are of interest because they include the case of complex vector unknowns which can be regarded as quantum wavefunctions. The matrix equations describe quantum synchronization, which has been well-studied [50][51][52][53][54][55][56] , and we show in particular in Sec. VII B that partial integration formally extends to infinite-dimensional equations which constitute a nonlinear system of Schrödinger equations.…”

“…Define Ω = ωJ for some real parameter ω, then Ω is an element of the Lie algebra of SO(1, 1) as required. The evolution equations (55), in which we choose Γ = κ ∑ j λ j M j /(2N ), reduce to:…”

“…which extend the conditions for square matrices that define a which have been extensively investigated 20,[23][24][25][26][27] , and for p = 2 reduce to the Kuramoto model. For the complex case we obtain models of quantum synchronization [50][51][52][53][54][55]64 .…”

“…The system of equations can be extended to infinite dimensions in which the wavefunctions are elements of a Hilbert space in which the Hamiltonian at each node acts as a self-adjoint operator. It is known [50][51][52][53][54][55][56] for models with identical potentials that solutions exist globally, and that complete synchronization occurs under specified conditions 56 . We show that partial integration extends to this infinite-dimensional case, with a fixed number of nodes N .…”

“…with n = 1 = m the elements M i of the system are SO(1, 1) matrices which evolve according to(55), and complete synchronization occurs19 in the sense that M i M −1 j as t → ∞. This system describes the synchronization of a relativistic cluster of particles in…”