All-Phononic Digital Transistor on the Basis of Gap-Soliton Dynamics in an Anharmonic Oscillator Ladder

A conceptual mechanism of amplification of phonons by phonons on the basis of a nonlinear band-gap transmission (supratransmission) phenomenon is presented. As an example, a system of weakly coupled chains of anharmonic oscillators is considered. One (source) chain is driven harmonically by a boundary with a frequency located in the upper band close to the band edge of the ladder system. Amplification happens when a second (gate) chain is driven by a small signal in the counterphase and with the same frequency as the first chain. If the total driving of both chains overcomes the band-gap transmission threshold, the large amplitude band-gap soliton emerges and the amplification scenario is realized. The mechanism is interpreted as the nonlinear superposition of evanescent and propagating nonlinear modes manifesting in a single or double soliton generation working in band-gap or bandpass regimes, respectively. The results could be straightforwardly generalized for all-optical or all-magnonic contexts and have all the promise of logic gate operations.

Figure 1

(a) The conceptual scheme of a phonon transistor with indications of source and gate signal forms and supply places. Amplified signal is monitored at the Drain port. (b) Space-time dynamics of gap-soliton creation and propagation in the lower chain, where, in the range $1<n<150$, one has a linear chain (green balls in the upper graph), and for $150<n<250$, soliton propagation occurs in a nonlinear chain (blue balls). The upper chain is coupled with the lower one in the range $150<n<200$. (c) Energies of the input gate signal (green curve) and the amplified gap-soliton output Drain signal (blue curve) monitored in the same lower chain.

Figure 2

Red and blue curves represent linear dispersion relations of the antisymmetric ${\mathrm{\Omega }}_1(p)$ and symmetric ${\mathrm{\Omega }}_2(p)$ branches, respectively. Dashed horizontal lines represent three different cases ($\mathrm{\Omega }=1.90$, $\mathrm{\Omega }=2.10$, and $\mathrm{\Omega }=2.18$) which lead to the different kind of nonlinear wave transport in the system. Inset shows the schematics of the system of the FPU ladder.

Figure 3

Numerical simulations on the FPU ladder (1) with the boundary driving according to (10): the upper chain (upper graph) is driven with a frequency $\mathrm{\Omega }=1.9$ and the lower chain (lower graph) is kept pinned. Inset shows how the ultimate left $n=0$ balls of the ladder are driven in time.

Figure 4

Same as in Fig. 3, but now, the upper chain is driven by a frequency $\mathrm{\Omega }=2.1$ and the lower chain is again pinned.

Figure 5

The amplification scenario for the FPU ladder. In the upper chain (upper panel), we provide the continuous driving of the left end with an amplitude just below the threshold (12) and the band-gap frequency $\mathrm{\Omega }=2.18$, while the left end of the lower chain (lower panel) is perturbed by a small amplitude signal with the same carrier frequency. Inset shows how the ultimate left $n=0$ balls of the ladder are driven in time.