Role of Coherence and Degeneracies in Quantum Synchronisation

Abstract

Progress on the study of synchronisation in quantum systems has been largely driven by specific examples which resulted in several examples of frequency entrainment as well as mutual synchronisation. Here we study quantum synchronisation by utilising Liouville space perturbation theory. We begin by clarifying the role of centers, symmetries and oscillating coherences in the context of quantum synchronisation. We then analyse the eigenspectrum of the Liouville superoperator generating the dynamics of the quantum system and determine the conditions under which synchronisation arises. We apply our framework to derive a powerful relationship between energy conservation, degeneracies and synchronisation in quantum systems. Finally, we demonstrate our approach by analysing two mutually coupled thermal machines and the close relationship between synchronisation and thermodynamic quantities.

Observation of quantum phase-synchronization in a nuclear spin-system

“…where L represents the Liouvillian superoperator describing the open system dynamics and |ρ is the vectorised form of density matrix in Liouville space. The mathematical form of L can be obtained by applying the transformation BρC → C * ⊗ B|ρ to the master equation where |ρ is obtained by vertically stacking the columns of density matrix [15,[40][41][42].…”

“…The Liouvillian superoperator L can be decomposed into two parts as shown in Eq. (15). The first part L 0 defines the dynamics of system in absence of external drive (V=0) and is given by…”

“…Platforms that exhibit quantized dynamics have inspired theoretical studies in systems such as trapped ions [2,3], superconducting circuits [4], atomic ensembles [5], optomechanical systems [6][7][8][9] and nanomechanical systems [10,11]. Theoretical studies have shown fundamental implications of synchronization to other fields such as entanglement generation [12,13], thermodynamics [14,15], quantum networks [16] and continuous time crystals [17].…”

Steady-State Tunable Entanglement Thermal Machine Using Quantum Dots

“…Here L is the Liouvillian superoperator [55,57,58,95] and |ρ is the vector representation of matrix ρ. Eigenvector |ρ is obtained by stacking the columns of density matrix ρ while L is obtained by the related transformation AρB → B * ⊗ A|ρ [55,57]. Following the given transformation, mathematical form of L is given by…”

“…We also show broad stability regions of control parameters where the steady states are entangled. Unlike previous approaches towards controllable thermal machines, our proposal relies only on steady state currents (dc steady state) and voltages analyzed through Liouville eigenspectrum techniques [55][56][57][58] and quantum transport theory.…”