Dissipative nonequilibrium synchronization of topological edge states via self-oscillation

The interplay of synchronization and topological band structures with symmetry-protected midgap states under the influence of driving and dissipation is largely unexplored. Here, we consider a trimer chain of electron shuttles, each consisting of a harmonic oscillator coupled to a quantum dot positioned between two electronic leads. Each shuttle is subject to thermal dissipation and undergoes a bifurcation toward self-oscillation with a stable limit cycle if driven by a bias voltage between the leads. By mechanically coupling the oscillators together, we observe synchronized motion at the ends of the chain, which can be explained using a linear stability analysis. Because of the inversion symmetry of the trimer chain, these synchronized states are topologically protected against local disorder. Furthermore, with current experimental feasibility, the synchronized motion can be observed by measuring the dot occupation of each shuttle. Our results open another avenue to enhance the robustness of synchronized motion by exploiting topology.

Figure 1

Dissipative nonequilibrium system with topological edge states: An open chain of $N$ coupled systems splits into trimers with units $A,B$, and $C$. Each system consists of an electron shuttle, that is a quantum dot (QD) hosted by a nanomechanical oscillator positioned between electronic leads. The QD is tunnel coupled to the leads, source (S) and drain (D), in a nonlinear fashion such that electron tunneling is enhanced in the proximity of the lead (indicated by the thickness of the arrows). The same bias voltage $V$ is applied to each shuttle and induces an electrostatic field $\alpha V$ between the leads, which in turn forces the charged shuttle to move toward the drain. Each shuttle is coupled to its nearest neighbors via the mechanical coupling of the oscillators to build a chain. Trimerization of the chain is then achieved by assigning different frequencies to nanomechanical oscillators of each unit, e.g., by varying oscillator length.

Figure 2

(a) Frequencies ${\omega }_l$ of units $A$ (pink), $B$ (yellow), and $C$ (blue) as function of the global phase $\varphi$. (b) Frequency spectrum of $\mathit{\omega }$ [see Eq. (3)] for $N=24$ and $\mathrm{\Delta }/g=1$. At the crossings in the spectrum ($\varphi =2\pi /3$ and $\varphi =5\pi /3$) the Hamiltonian is inversion-symmetric with topologically protected edge states. [(c)–(e)] Corresponding eigenstates of the pink and blue branch in panel (bb) for different values of $\varphi$ indicating bulk states ($\varphi =3\pi /2$), edge states located at one side of the chain ($\varphi =\pi /2$), or inversion symmetric edge states located at both sides of the chain ($\varphi =2\pi /3$). The behavior of the orange states is similar (not shown).

Figure 3

Synchronization of a trimer chain of electron shuttles for $N=24$. [(a)–(c)] Oscillating position $\delta x_l\left(t\right)$ of the shuttles, [(d)–(f)] corresponding dot occupation $\delta q_l\left(t\right)$ at steady state after initial relaxation. For $\varphi =\pi /2$ (top figures), only the right end of the chain performs synchronized motion while the left end is at rest. Consequently, only the dot occupation on the right end of the chain oscillates in time. For $\varphi =2\pi /3$ (middle figures) the inversion symmetry is preserved and both edges of the chain are synchronized with an inversion symmetric initial state. For $\varphi =0.7\pi$ also, both edges oscillate; however, each side of the chain oscillates with its own frequency, such that only shuttles on either side of the chain are synchronized. Parameters: $\beta V=150.0,\alpha \lambda =0.06,\mathrm{\Delta }/g=1.0,\mathrm{\Gamma }/\gamma =1.0$.

Figure 4

Linear stability analysis: Real (a) and imaginary part (b) of eivenvalues $z_i$ of the Jacobian $\mathbf{J}\left({\overline{\mathbf{X}}}_{\text{fix}}\right)$ evaluated at the fixed point. As the global phase $\varphi$ is varied, collective edge states are excited if $\text{Re}\left(z_i\right)>0$. The corresponding oscillation frequency is given by $\text{Im}\left(z_i\right)$. In the light gray shaded area, only one edge state is excited (blue right and pink left), whereas in the dark gray shaded area, both edge states are excited in general with different oscillation frequencies. At $\varphi =2\pi /3$, the oscillation frequencies cross in panel (b), such that both edge states are synchronized. Inset: Real part of $z_i$ for a single shuttle as function of the oscillator frequency. Parameters: $\beta V=150.0,\alpha \lambda =0.06,\mathrm{\Delta }/g=1.0,\mathrm{\Gamma }/\gamma =1.0$.

Figure 5

(a) Frequency spectrum of the trimer chain of $N=24$ harmonic oscillators for $\varphi =2\pi /3$ [vertical line in Fig. 4] with increasing amount of disorder $r$ for 30 realizations of disorder. The edge state (blue) is topologically protected against the disorder for large values of $r$. [(b), (c)] $\text{Re}\left(z_i\right)$ and $\text{Im}\left(z_i\right)$ of the Jacobian of the linear stability analysis with increasing amount of disorder. The topologically protected edge state located at both ends of the chain persists for large values of $r$; however, additional bulk states with different frequencies may become linearly unstable. [(d), (e)] Oscillating position $\delta x_l\left(t\right)$ of the shuttles for one realization of disorder with $r/g=0.1$ and $r/g=0.4$, respectively, and inversion symmetric initial states. Parameters: $\beta V=150.0,\alpha \lambda =0.06,\mathrm{\Delta }/g=1.0,\mathrm{\Gamma }/\gamma =1.0$.