Dynamical decoupling and dynamical isolation in temporally modulated coupled pendulums 寸 GRAZIA SALERNO and IACOPO CARUSOTTO INO-CNR BEC Center and Dipartimento di Fisica,Universita di Trento,via Sommarive 14,38123 Povo,Italy 5.v quantum mechanics.we antic ate dynamical localization and dyn Introduction.- Dynamical localization is a surpris ate synhe tic gauge field for photons [25,26] ing combi quence quantum mec s applied o par- dy of dynam This effect was first observed as renormalization of the cal this text.The dynamic stabilization of the inverted pendulum when its pivot point is made to oscillate in space is a well elebrated example of nor anical effect stem [2 fal in a stro nside system closely related effect is the coherent destruction of tunnel- and periodic ally varied in time In analogy to 导 made c dict quant dable material y between the pendulums tions a and electronic decoherence.the robust coherence effect,where exchange of ene is suppressed.When the pendulums are driven by an ex- ernal force,we an a nove Further studies of dynamical The system and the theoretical model.- The system were 113,14h of a d induced superfluid to Mott-insulator transition [15]. mi=pI (1) scheme 1--m[1+h(t)x1+k(x2-x1)-11(2) he to gen m2=2 (3) sites 16,17]and then fields for atoms 118-231 Corres ndingly to these advances in 2=-mu1+(t切2+k(1-x2)-2. omicphyic erve dyes light in t从ex,。aiables indicate theo he penc thom the um pos avegu arXiv:1401.3978v2 [physics.class-ph] 16 Apr 2014 epl draft Dynamical decoupling and dynamical isolation in temporally modulated coupled pendulums GRAZIA SALERNO and IACOPO CARUSOTTO INO-CNR BEC Center and Dipartimento di Fisica, Universit`a di Trento, via Sommarive 14, 38123 Povo, Italy PACS 45.20.D- – Newtonian mechanics PACS 46.40.Ff – Resonance, damping, and dynamic stability PACS 03.65.Vf – Phases: geometric; dynamic or topological Abstract –We theoretically study the dynamics of a pair of coupled pendulums subject to a peri￾odic temporal modulation of their oscillation frequency. Inspired from analogous developments in quantum mechanics, we anticipate dynamical localization and dynamical isolation effects, as well as the occurrence of non-trivial coupling phases. Perspectives in the direction of studying synthetic gauge fields in a classical mechanics context are outlined. Introduction. – Dynamical localization is a surpris￾ing consequence of quantum mechanics applied to par￾ticles subject to a strong time-dependent external force. This effect was first observed as renormalization of the magnetic response of an atom illuminated by a strong rf field [1]. In a solid state context, dynamical localization was proposed in [2–6] as a dramatic suppression of the d.c. conductivity of a metal in a strong a.c. field. Another closely related effect is the coherent destruction of tunnel￾ing in a double-well geometry, first predicted in [7,8] and extensively compared to dynamical localization in [10]. While the experimental study of these effects in solids is made difficult by the unavoidable material imperfec￾tions and electronic decoherence, the robust coherence and the clean periodic potential experienced by atomic matter waves in temporally modulated optical lattices has allowed for a clear observation of Bloch band sup￾pression in a new atomic physics context [9]. Further studies of dynamical matter wave localization effects were reported in [11, 12] using Bose-condensed atomic samples. Following the proposal [13, 14], this research line culminated in the observation of a dynamically￾induced superfluid to Mott-insulator transition [15]. Very exciting further developments of these ideas aim at using more complex modulation schemes to gen￾erate non-trivial hopping phases between the lattice sites [16, 17] and then synthetic gauge fields for neutral atoms [18–23]. Correspondingly to these advances in atomic physics, the same ideas are being explored in pho￾tonics to observe dynamical localization of light in cou￾pled optical waveguides [24] and, very recently, to gener￾ate synthetic gauge field for photons [25, 26]. In this Letter, we report a theoretical study of dynami￾cal localization phenomena in a classical mechanics con￾text. The dynamic stabilization of the inverted pendulum when its pivot point is made to oscillate in space is a well celebrated example of non-trivial mechanical effect stem￾ming from a temporal modulation of the system parame￾ters [27]. Here we consider a system of two coupled pen￾dulums, whose oscillation frequencies are independently and periodically varied in time. In analogy to the coher￾ent destruction of tunneling of a quantum particle in a double-well potential, we predict a dynamical decoupling effect, where exchange of energy between the pendulums is suppressed. When the pendulums are driven by an ex￾ternal force, we anticipate a novel dynamic isolation effect, where the temporal modulation effectively decouples the system from the external force. The system and the theoretical model. – The system of two identical coupled pendulums is modelled as a pair of coupled harmonic oscillators of equal masses m follow￾ing the motion equations: m x˙ 1 = p1 (1) p˙1 = −mω2 0 [1 + ν1(t)] x1 + k(x2 − x1) − ξ1x˙ 1 (2) m x˙ 2 = p2 (3) p˙2 = −mω2 0 [1 + ν2(t)] x2 + k(x1 − x2) − ξ2x˙ 2 . (4) The x1,2 variables indicate the spatial displacement of the pendulums from the equilibrium position. The lin￾earised form of the motion equations is legitimate in p-1