Nonlinear Targeted Energy Transfer and Macroscopic Analogue of the Quantum Landau-Zener Effect in Coupled Granular Chains

Abstract

Resonance is the main mechanism for energy propagation in spatially periodic linear/nonlinear systems. For the case of two weakly coupled identical Hamiltonian oscillators in resonance, any amount of energy imparted to one of the oscillators gets transferred back and forth between these oscillators with a frequency proportional to the coupling.

Appendix

In this appendix, we develop the approximate expression of the slow flow modulation equation shown in Eq. (11.9). Rewriting Eq. (11.8), which shows the slow/fast partition of the dynamics of the system:

$$ \begin{aligned} & \frac{{\partial \psi_{n0}^{x} }}{{\partial \tau_{0} }} - i\psi_{n0}^{x} = 0\, \Rightarrow \,\psi_{n0}^{x} = \varphi_{no}^{x} \left( {\tau_{1} } \right)\exp \left( {i\tau_{0} } \right) \\ & \frac{{\partial \psi_{n0}^{y} }}{{\partial \tau_{0} }} - i\psi_{n0}^{y} = 0\, \Rightarrow \,\psi_{n0}^{y} = \varphi_{no}^{y} \left( {\tau_{1} } \right)\exp \left( {i\tau_{0} } \right) \\ \end{aligned} $$

(11.37)

Proceeding to the $O(\varepsilon^{3/2} )$ approximation, we derive the following system:

$$ \begin{aligned} \frac{{\partial \psi_{n0}^{x} }}{{\partial \tau_{1} }} + \frac{{\partial \psi_{n1}^{x} }}{{\partial \tau_{0} }} - i\psi_{n1}^{x} & = \alpha \left\{ {\frac{{\psi_{{\left( {n - 1} \right)0}}^{x} - \psi_{{\left( {n - 1} \right)0}}^{x*} - \psi_{\left( n \right)0}^{x} + \psi_{\left( n \right)0}^{x*} }}{2i}} \right\}_{ + }^{3/2} \\&-\, \alpha \left\{ {\frac{{\psi_{\left( n \right)0}^{x} - \psi_{\left( n \right)0}^{x*} - \psi_{{\left( {n + 1} \right)0}}^{x} + \psi_{{\left( {n + 1} \right)0}}^{x*} }}{2i}} \right\}_{ + }^{3/2} - i\lambda \left[ {\psi_{\left( n \right)0}^{y} - \psi_{\left( n \right)0}^{y*} } \right] \\ \frac{{\partial \psi_{n0}^{y} }}{{\partial \tau_{1} }} + \frac{{\partial \psi_{n1}^{y} }}{{\partial \tau_{0} }} - i\psi_{n1}^{y} & = \alpha \left\{ {\frac{{\psi_{{\left( {n - 1} \right)0}}^{y} - \psi_{{\left( {n - 1} \right)0}}^{y*} - \psi_{\left( n \right)0}^{y} + \psi_{\left( n \right)0}^{y*} }}{2i}} \right\}_{ + }^{3/2} \\ &-\, \alpha \left\{ {\frac{{\psi_{\left( n \right)0}^{y} - \psi_{\left( n \right)0}^{y*} - \psi_{{\left( {n + 1} \right)0}}^{y} + \psi_{{\left( {n + 1} \right)0}}^{y*} }}{2i}} \right\}_{ + }^{3/2} - i\lambda \left[ {\psi_{\left( n \right)0}^{x} - \psi_{\left( n \right)0}^{x*} } \right] \\ \end{aligned} $$

(11.38)

Introducing (11.37) into (11.38) yields the following:

$$ \begin{aligned} & \frac{{\partial \left[ {\varphi_{n0}^{x} \exp \left( {i\tau_{0} } \right)} \right]}}{{\partial \tau_{1} }} + \frac{{\partial \psi_{n1}^{x} }}{{\partial \tau_{0} }} - i\psi_{n1}^{x} \\ & \quad = \alpha \left\{ {\frac{{\varphi_{{\left( {n - 1} \right)0}}^{x} \exp \left( {i\tau_{0} } \right) - \varphi_{{\left( {n - 1} \right)0}}^{x*} \exp \left( { - i\tau_{0} } \right) - \varphi_{\left( n \right)0}^{x} \exp \left( {i\tau_{0} } \right) + \varphi_{\left( n \right)0}^{x*} \exp \left( { - i\tau_{0} } \right)}}{2i}} \right\}_{ + }^{3/2} \\ & \quad -\, \alpha \left\{ {\frac{{\varphi_{\left( n \right)0}^{x} \exp \left( {i\tau_{0} } \right) - \varphi_{\left( n \right)0}^{x*} \exp \left( { - i\tau_{0} } \right) - \varphi_{{\left( {n + 1} \right)0}}^{x} \exp \left( {i\tau_{0} } \right) + \varphi_{{\left( {n + 1} \right)0}}^{x*} \exp \left( { - i\tau_{0} } \right)}}{2i}} \right\}_{ + }^{3/2} \\ & \quad - \,i\lambda \left[ {\phi_{n0}^{y} \exp \left( {i\tau_{0} } \right) - \phi_{n0}^{y*} \exp \left( { - i\tau_{0} } \right)} \right] \\ \end{aligned} $$

$$ \begin{aligned} & \frac{{\partial \left[ {\varphi_{n0}^{y} \exp \left( {i\tau_{0} } \right)} \right]}}{{\partial \tau_{1} }} + \frac{{\partial \left[ {\psi_{n1}^{y} } \right]}}{{\partial \tau_{0} }} - i\psi_{n1}^{y} \\ & \quad = \alpha \left\{ {\frac{{\varphi_{{\left( {n - 1} \right)0}}^{y} \exp \left( {i\tau_{0} } \right) - \varphi_{{\left( {n - 1} \right)0}}^{y*} \exp \left( { - i\tau_{0} } \right) - \varphi_{\left( n \right)0}^{y} \exp \left( {i\tau_{0} } \right) + \varphi_{\left( n \right)0}^{y*} \exp \left( { - i\tau_{0} } \right)}}{2i}} \right\}_{ + }^{3/2} \\ & \quad -\, \alpha \left\{ {\frac{{\varphi_{\left( n \right)0}^{y} \exp \left( {i\tau_{0} } \right) - \varphi_{\left( n \right)0}^{y*} \exp \left( { - i\tau_{0} } \right) - \varphi_{{\left( {n + 1} \right)0}}^{y} \exp \left( {i\tau_{0} } \right) + \varphi_{{\left( {n + 1} \right)0}}^{y*} \exp \left( { - i\tau_{0} } \right)}}{2i}} \right\}_{ + }^{3/2} \\ & \quad -\, i\lambda \left[ {\phi_{n0}^{x} \exp \left( {i\tau_{0} } \right) - \phi_{n0}^{x*} \exp \left( { - i\tau_{0} } \right)} \right] \\ \end{aligned} $$

(11.39)

Upon imposing solvability conditions in (11.39), yields the following slow flow, i.e., the system of modulation equations in the slow timescale governing the (slow) evolutions of the complex envelopes in (11.37):

$$ \begin{aligned} \frac{{\partial \varphi_{n0}^{x} }}{{\partial \tau_{1} }} & = \alpha \left\{ {\frac{{\varphi_{{\left( {n - 1} \right)0}}^{x} \exp \left( {i\tau_{0} } \right) - \varphi_{{\left( {n - 1} \right)0}}^{x*} \exp \left( { - i\tau_{0} } \right) - \varphi_{\left( n \right)0}^{x} \exp \left( {i\tau_{0} } \right) + \varphi_{\left( n \right)0}^{x*} \exp \left( { - i\tau_{0} } \right)}}{2i}} \right\}_{ + }^{3/2} \\ & \times\,\exp ( - i\tau_{0} ) \\ & -\, \alpha \left\{ {\frac{{\varphi_{\left( n \right)0}^{x} \exp \left( {i\tau_{0} } \right) - \varphi_{\left( n \right)0}^{x*} \exp \left( { - i\tau_{0} } \right) - \varphi_{{\left( {n + 1} \right)0}}^{x} \exp \left( {i\tau_{0} } \right) + \varphi_{{\left( {n + 1} \right)0}}^{x*} \exp \left( { - i\tau_{0} } \right)}}{2i}} \right\}_{ + }^{3/2} \\ & \times\,\exp ( - i\tau_{0} ) - i\lambda \left[ {\phi_{n0}^{y} \exp \left( {i\tau_{0} } \right) - \phi_{n0}^{y*} \exp \left( { - i\tau_{0} } \right)} \right]\exp ( - i\tau_{0} ) \\ \end{aligned} $$

$$ \begin{aligned} \frac{{\partial \varphi_{n0}^{y} }}{{\partial \tau_{1} }} & = a\left\{ {\frac{{\varphi_{{\left( {n - 1} \right)0}}^{y} \exp \left( {i\tau_{0} } \right) - \varphi_{{\left( {n - 1} \right)0}}^{y*} \exp \left( { - i\tau_{0} } \right) - \varphi_{\left( n \right)0}^{y}\exp \left( {i\tau_{0} } \right) + \varphi_{\left( n \right)0}^{y*} \exp \left( { - i\tau_{0} } \right)}}{2i}} \right\}_{ + }^{3/2} \\ &\times\, \exp ( - i\tau_{0} )\\ &-\, \alpha \left\{ {\frac{{\varphi_{\left( n \right)0}^{y} \exp \left( {i\tau_{0} } \right) - \varphi_{\left( n \right)0}^{y*} \exp \left( { - i\tau_{0} } \right) - \varphi_{{\left( {n + 1} \right)0}}^{y} \exp \left( {i\tau_{0} } \right) + \varphi_{{\left( {n + 1} \right)0}}^{y*} \exp \left( { - i\tau_{0} } \right)}}{2i}} \right\}_{ + }^{3/2} \\ &\times\,\exp ( - i\tau_{0} ) - i\lambda \left[ {\phi_{n0}^{x} \exp \left( {i\tau_{0} } \right) - \phi_{n0}^{x*} \exp \left( { - i\tau_{0} } \right)} \right]\exp ( - i\tau_{0} ) \\ \end{aligned} $$

(11.40)

To evaluate the non-smooth terms in (11.40), we follow (Starosvetsky et al. 2012) and introduce the Fourier expansions:

$$ \begin{aligned} F_{{\left( {n - 1} \right)0}}^{x} & \equiv \varphi_{{\left( {n - 1} \right)0}}^{x} - \varphi_{n0}^{x} = \left| {F_{{\left( {n - 1} \right)0}}^{x} } \right|\exp \left( {i\theta_{n - 1}^{x} } \right),\quad F_{\left( n \right)0}^{x} \equiv \varphi_{n0}^{x} - \varphi_{{\left( {n + 1} \right)0}}^{x} = \left| {F_{\left( n \right)0}^{x} } \right|\exp \left( {i\theta_{n}^{x} } \right) \\ F_{{\left( {n - 1} \right)0}}^{y} & \equiv \varphi_{{\left( {n - 1} \right)0}}^{y} - \varphi_{n0}^{y} = \left| {F_{{\left( {n - 1} \right)0}}^{y} } \right|\exp \left( {i\theta_{n - 1}^{y} } \right),\quad F_{\left( n \right)0}^{y} \equiv \varphi_{n0}^{y} - \varphi_{{\left( {n + 1} \right)0}}^{y} = \left| {F_{\left( n \right)0}^{y} } \right|\exp \left( {i\theta_{n}^{y} } \right) \\ \end{aligned} $$

(11.41)

which when substituted into the non-smooth terms on the right-hand sides of the slow flow (11.40) lead to the following expressions:

$$ \begin{aligned} \left\{ {\frac{{F_{{\left( {n - 1} \right)0}}^{x} \exp \left( {i\tau_{0} } \right) - F_{(n - 1)0}^{x*} \exp \left( { - i\tau_{0} } \right)}}{2i}} \right\}_{ + }^{3/2} &= \left| {F_{{\left( {n - 1} \right)0}}^{x} } \right|^{3/2} \left\{ {\sin (\tau_{0} + \theta_{n - 1}^{x} )} \right\}_{ + }^{3/2} \\ &= \left| {F_{{\left( {n - 1} \right)0}}^{x} } \right|^{3/2} \left\{ {\cos\Phi_{n - 1}^{x} } \right\}_{ + }^{3/2} \\ \left\{ {\frac{{F_{\left( n \right)0}^{x} \exp \left( {i\tau_{0} } \right) - F_{(n)0}^{x*} \exp \left( { - i\tau_{0} } \right)}}{2j}i} \right\}_{ + }^{3/2} &= \left| {F_{\left( n \right)0}^{x} } \right|^{3/2} \left\{ {\sin (\tau_{0} + \theta_{n}^{x} )} \right\}_{ + }^{3/2} \\ &= \left| {F_{\left( n \right)0}^{x} } \right|^{3/2} \left\{ {\cos\Phi_{n}^{x} } \right\}_{ + }^{3/2} \\ \left\{ {\frac{{F_{{\left( {n - 1} \right)0}}^{y} \exp \left( {i\tau_{0} } \right) - F_{(n - 1)0}^{y*} \exp \left( { - i\tau_{0} } \right)}}{2i}} \right\}_{ + }^{3/2} &= \left| {F_{{\left( {n - 1} \right)0}}^{y} } \right|^{3/2} \left\{ {\sin (\tau_{0} + \theta_{n - 1}^{y} )} \right\}_{ + }^{3/2} \\ &= \left| {F_{{\left( {n - 1} \right)0}}^{y} } \right|^{3/2} \left\{ {\cos\Phi_{n - 1}^{y} } \right\}_{ + }^{3/2} \\ \left\{ {\frac{{F_{\left( n \right)0}^{y} \exp \left( {i\tau_{0} } \right) - F_{(n)0}^{y*} \exp \left( { - i\tau_{0} } \right)}}{2i}} \right\}_{ + }^{3/2} &= \left| {F_{\left( n \right)0}^{y} } \right|^{3/2} \left\{ {\sin (\tau_{0} + \theta_{n}^{y} )} \right\}_{ + }^{3/2} \\ &= \left| {F_{\left( n \right)0}^{y} } \right|^{3/2} \left\{ {\cos\Phi_{n}^{y} } \right\}_{ + }^{3/2} \\ \end{aligned} $$

(11.42)

where $\Phi_{k}^{x,y} + \frac{\pi }{2} = \tau_{0} + \theta_{k}^{x,y}$. Substituting (11.42) into (11.40) and performing averaging with respect to the fast timescale $\tau_{0}$ yields the following averaged slow flow equations:

$$ \begin{aligned} \frac{{\partial \varphi_{n0}^{x} }}{{\partial \tau_{1} }} & = \left( {\alpha /2\pi } \right)\left\{ {\left| {F_{{\left( {n - 1} \right)0}}^{x} } \right|^{3/2} \exp \left[ {i\left( {\theta_{n - 1}^{x} - \pi /2} \right)} \right]\,\oint\limits_{2\pi } {\left\{ {\cos\Phi_{n - 1}^{x} } \right\}_{ + }^{3/2} \exp ( - i\Phi_{n - 1}^{x} )} \,\text{d}\Phi_{n - 1}^{x} } \right. \\ & \quad - \,\left. {\left| {F_{\left( n \right)0}^{x} } \right|^{3/2} \exp \left[ {i\left( {\theta_{n}^{x} - \pi /2} \right)} \right]\,\oint\limits_{2\pi } {\left\{ {\cos\Phi_{n}^{x} } \right\}_{ + }^{3/2} \exp ( - i\Phi_{n}^{x} )} \,\text{d}\Phi_{n}^{x} } \right\} \\ & \quad + \,\left( {\lambda /\pi } \right)\exp ( - i\pi /2)\,\oint\limits_{2\pi } {\Phi_{n0}^{y} \left\{ {\cos \left( {\Phi_{n}^{y} - \theta_{n}^{y} } \right)} \right\}\exp \left\{ { - i\left( {\Phi_{n}^{y} - \theta_{n}^{y} } \right)} \right\}} \,\text{d}\left( {\Phi_{n}^{y} - \theta_{n}^{y} } \right) \\ \frac{{\partial \varphi_{n0}^{y} }}{{\partial \tau_{1} }} & = \left( {\alpha /2\pi } \right)\left\{ {\left| {F_{{\left( {n - 1} \right)0}}^{y} } \right|^{3/2} \exp \left[ {i\left( {\theta_{n - 1}^{y} - \pi /2} \right)} \right]\,\oint\limits_{2\pi } {\left\{ {\cos\Phi_{n - 1}^{y} } \right\}_{ + }^{3/2} \exp ( - i\Phi_{n - 1}^{y} )} \,\text{d}\Phi_{n - 1}^{y} } \right. \\ & \quad - \,\left. {\left| {F_{\left( n \right)0}^{y} } \right|^{3/2} \exp \left[ {i\left( {\theta_{n}^{y} - \pi /2} \right)} \right]\,\oint\limits_{2\pi } {\left\{ {\cos\Phi_{n}^{y} } \right\}_{ + }^{3/2} \exp ( - i\Phi_{n}^{y} )} \,\text{d}\Phi_{n}^{y} } \right\} \\ & \quad + \,\left( {\lambda /\pi } \right)\exp ( - i\pi /2)\,\oint\limits_{2\pi } {\Phi_{n0}^{x} \left\{ {\cos \left( {\Phi_{n}^{x} - \theta_{n}^{x} } \right)} \right\}\exp \left\{ { - i\left( {\Phi_{n}^{x} - \theta_{n}^{x} } \right)} \right\}} \,\text{d}\left( {\Phi_{n}^{x} - \theta_{n}^{x} } \right) \\ \end{aligned} $$

(11.43)

Using Fourier expansion, retaining only the leading-order harmonics, and rearranging terms yields the following averaged slow flow system:

$$ \begin{aligned} & i\frac{{\partial \varphi_{n0}^{x} }}{{\partial \tau_{1} }} = \tilde{\alpha }\left( {F_{{\left( {n - 1} \right)0}}^{x} \left| {F_{{\left( {n - 1} \right)0}}^{x} } \right|^{1/2} - F_{\left( n \right)0}^{x} \left| {F_{\left( n \right)0}^{x} } \right|^{1/2} } \right) + \tilde{\lambda }\varphi_{n0}^{y} \\ & i\frac{{\partial \varphi_{n0}^{y} }}{{\partial \tau_{1} }} = \tilde{\alpha }\left( {F_{{\left( {n - 1} \right)0}}^{y} \left| {F_{{\left( {n - 1} \right)0}}^{y} } \right|^{1/2} - F_{\left( n \right)0}^{y} \left| {F_{\left( n \right)0}^{y} } \right|^{1/2} } \right) + \tilde{\lambda }\varphi_{n0}^{x} \\ \end{aligned} $$

(11.44)