Nonlinear amplifier and frequency shifter using a tunable periodic drive

We consider the superposition of a weak and a strong force acting on an overdamped particle moving on either an asymmetric-periodic or a double-well potential. The velocity of the particle has only harmonics $\left(n{\omega }_1\right)$ of the strong force, when the particle either oscillates near a minimum or runs away from it. Near a threshold drive (bistable point) separating these two dynamical regimes, the weak force drastically changes the velocity spectrum, greatly amplifying the mixing harmonics. This effect can be used either to amplify or to shift the frequency of a weak signal and can be observed in a wide variety of systems, including domain walls in a ferromagnet, SQUIDs, and tiny particles in a ratchet potential.

Figure 1

Spatially asymmetric ratchet (a) and double-well (b) potentials used here to obtain the amplification of a weak signal. A particle moving in these potentials exhibits bistable dynamics, needed for the signal amplification studied here.

Figure 2

(Color online) (a) The Fourier spectrum of the velocity of a particle in a ratchet potential, shown in Fig. 1a, for the driving amplitude $a=7$, which is away from the bistable point, the response is given by the strong drive and its harmonics (the weak signal plays no role here). (b) Same as in (a) but with $a=6.72$ (which is close to a bistable point between confined and running solutions) and frequency ${\omega }_1=2\pi$. The weak signal amplitude is chosen to be $ϵ=0.03$ and ${\omega }_2=\sqrt{5}{\omega }_1$. It is remarkable that even though the weak signal is $\lesssim 0.5\%$ of the strong force, the produced harmonic amplitudes are of the same order as the main drive. (c) Same as in (a) for commensurate frequencies, ${\omega }_2=(9∕5){\omega }_1=2.25{\omega }_1$ and $a=6.72$, i.e., close to the bistable point. Even though the weak signal frequency ${\omega }_2=(9∕4){\omega }_1$ is close to the incommensurate frequency in (b), the dense set of weak peaks disappears for the commensurate case (c). The labels (“Incommensurate”) “Commensurate” in the figures refer to the (in)commensurate ratio ${\omega }_2∕{\omega }_1$. The incommensurate case has fractal spectral structure.

Figure 3

The Fourier spectrum of the velocity of a particle in a double-well potential [shown in Fig. 1b] with frequencies as in Fig. 2b and $a=11.8,ϵ=0.4$. In order to achieve amplification of a weak signal, the driving amplitude $a$ is chosen to be close to the bistable point [located between the regions (i) where the particle motion is confined within one well, and (ii) where the motion occurs between the wells].

Figure 4

The motion of the particle, on the ratchet potential [Fig. 1a] and the double-well potential [Fig. 1b], is shown in (a) and (b), respectively, for the parameters used in Fig. 2b and Fig. 3. The running solution $y\left(\tau \right)$ for the particle on the ratchet produces a net transport (shown by the arrow). The particle in the double-well potential jumps in (b) between two potential wells.

Figure 5

Amplified harmonic ${\widehat{M}}_{3,-1}\left(ϵ\right)=\widehat{M}(3{\omega }_1-{\omega }_2)$ of the response $\widehat{M}\left(\omega \right)$ versus the amplitude $ϵ$ of a weak input signal for $a=6.72$ (near bistable point), ${\omega }_1={\omega }_2∕\sqrt{5}=2\pi$, for the ratchet potential shown in Fig. 1a. Circles correspond to numerical results and the solid line is a fit using our derived result ${\widehat{M}}_{n,m}\left(ϵ\right)=\alpha \arccos [(a_c-a)∕ϵ]$, here with $a-a_c=0.025$ and $\alpha =0.36$. The region for the effective amplification of the weak signal is located between the two dashed lines.

Figure 6

(Color online) Gradual suppression of the amplified signal (particle velocity) by white noise for the ratchet potential shown in Fig. 1a. Here, $a=6.72,{\omega }_1={\omega }_2∕\sqrt{5}=2\pi$, and $ϵ=0.06$. For noise intensity $\kappa ={10}^{-3}$ [panel (a)], a weak trace of the input weak signal is barely visible (marked by red vertical arrows); for $\kappa ={10}^{-4}$ [panel (b)] the amplified peaks are well resolved.

Figure 7

(Color online) (a) A schematic view of the asymmetric SQUID with two Josephson junctions on the left branch (both of them described by the gauge-invariant phase ${\varphi }_l\equiv \varphi$) and one Josephson junction on the right branch (with phase ${\varphi }_r$). (b) 3D sketch of a triangular-patterned magnetic film, indicating the height $l\left(x\right)$ and width $d$. The magnetic domain-wall (shown by the short vertical dashed magenta line) separates regions having different orientations of the magnetization (shown by the up and down arrows). This wall can be driven through the patterned magnetic sample by applying a time-dependent external magnetic field $H_e\left(t\right)$.