Floquet topological system based on frequency-modulated classical coupled harmonic oscillators

Abstract

We theoretically propose how to observe topological effects in a generic classical system of coupled harmonic oscillators, such as classical pendula or lumped-element electric circuits, whose oscillation frequency is modulated fast in time. Making use of Floquet theory in the high frequency limit, we identify a regime in which the system is accurately described by a Harper-Hofstadter model where the synthetic magnetic field can be externally tuned via the phase of the frequency-modulation of the different oscillators. We illustrate how the topologically-protected chiral edge states, as well as the Hofstadter butterfly of bulk bands, can be observed in the driven-dissipative steady state under a monochromatic drive. In analogy with the integer quantum Hall effect, we show how the topological Chern numbers of the bands can be extracted from the mean transverse shift of the steady-state oscillation amplitude distribution. Finally we discuss the regime where the analogy with the Harper-Hofstadter model breaks down.

One-Dimensional Mass-Spring Chains Supporting Elastic Waves with Non-Conventional Topology

“…Topological electronic [1], electromagnetic [2,3], and phononic crystals [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] all have demonstrated unusual topologically constrained properties. In phononic crystals and acoustic metamaterials, symmetry breaking is linked to constraints on the topological form of acoustic wave functions.…”

“…Time-reversal symmetry in intrinsic systems [4][5][6][7][8][9][10][11][12][13][14] is broken through internal resonance or symmetry breaking structural features (e.g., chirality) and without addition of energy from the outside. Energy is added to extrinsic topological systems to break time reversal symmetry [15][16][17][18][19][20]. A common example of an extrinsic approach is that of time-reversal symmetry breaking of acoustic waves by moving fluids [21][22][23][24][25][26][27][28][29].…”

Omnidirectional spin Hall effect in a Weyl spin-orbit-coupled atomic gas

“…On the other hand, when it comes to effective models, many quantum-mechanical systems with adiabatically varying parameters are naturally described in terms of Abelian gauge theories [2][3][4]. This geometric approach based on the Berry phase has paved the way to a multitude of both theoretical and experimental developments covering molecular [3][4][5], solid-state [6][7][8][9][10][11][12], photonic [13][14][15][16][17][18][19][20][21], mechanical [22,23] and electric [24,25] systems. Although the corresponding non-Abelian gauge structure in the presence of degenerate quantum states has been noticed promptly after the discovery of the Berry phase [26], a set of experimentally observed signatures of the non-Abelian geometrical phases remains limited [27,28].…”

Geometric phase and topology of elastic oscillations and vibrations in model systems: Harmonic oscillator and superlattice

“…Energy can also be added to extrinsic topological elastic systems to break time reversal symmetry. [20][21][22][23][24][25][26] For example, we have considered the externally-driven periodic spatial modulation of the stiffness of a one-dimensional elastic medium and its directed temporal evolution to break symmetry. 26 The bulk elastic states of this time-dependent super-lattice possess non-conventional topological characteristics leading to non-reciprocity in the direction of propagation of the waves.…”