Dynamics of a driven damped particle in the presence of a magnetic field: Asymmetric splitting of the output signal

Abstract

In this paper, we have investigated the dynamics of a damped harmonic oscillator in the presence of an electromagnetic field. Magnetic field may induce asymmetric splitting of the spectrum of the output signal with two peaks in the case of a driven damped two-dimensional harmonic oscillator. One more additional peak may appear for the three-dimensional case. At the same time, one may observe an antiresonance phenomenon even for the driven damped cyclotron motion where the system with the purely non-conservative force fields is driven by an electric field. Finally, our calculation exhibits how the magnetic field can modulate the frequency of a harmonic oscillator, the phase difference (between the input and the output signals) and the efficiency like quantity of the energy storing process, respectively. Thus, the present study might be applicable in areas related to refractive index, the barrier crossing dynamics and autonomous stochastic resonance, respectively.

Appendices

Appendix A: Definition of relevant quantities which appear in eqs (28) and (29)

$A_1, A_2, B_1, B_2, C_1$ and $C_2$ which appear in eqs (28) and (29) are defined as

$$\begin{aligned}&A_{1}=(H_{c}H_{x}-H_{z}H_{2})(H_{b}H_{c}+H_{3}H_{6})\\&\qquad \quad +(H_{c}H_{y}+H_{z}H_{3})(H_{c}H_{1}+H_{2}H_{6}),\\&A_{2}=(H_{c}H_{x}^{\prime }-H_{z}^{\prime }H_{2})(H_{b}H_{c}+H_{3}H_{6})\\&\qquad \quad +(H_{c}H_{y}^{\prime }+H_{z}^{\prime }H_{3})(H_{c}H_{1}+H_{2}H_{6}),\\&B_{1}=(H_{c}H_{y}+H_{z}H_{3})(H_{a}H_{c}+H_{2}H_{5})\\&\qquad \quad -(H_{c}H_{x}-H_{z}H_{2})(H_{c}H_{4}-H_{3}H_{5}),\\&B_{2}=(H_{c}H_{y}^{\prime }+H_{z}^{\prime }H_{3})(H_{a}H_{c}+H_{2}H_{5})\\&\qquad \quad -(H_{c}H_{x}^{\prime }-H_{z}^{\prime }H_{2})(H_{c}H_{4}-H_{3}H_{5}),\\&C_{1}=(H_{b}H_{z}-H_{y}H_{6})(H_{a}H_{b}+H_{1}H_{4})\\&\qquad \quad +(H_{b}H_{x}+H_{y}H_{1})(H_{b}H_{5}+H_{4}H_{6}),\\&C_{2}=(H_{b}H_{z}^{\prime }-H_{y}^{\prime }H_{6})(H_{a}H_{b}+H_{1}H_{4})\\&\qquad \quad +(H_{b}H_{x}^{\prime }+H_{y}H_{1})(H_{b}H_{5}+H_{4}H_{6}),\\&D_{1}=(H_{a}H_{c}+H_{2}H_{5})(H_{b}H_{c}+H_{3}H_{6})\\&\qquad \quad +(H_{c}H_{1}+H_{2}H_{6})(H_{c}H_{4}-H_{3}H_{5}),\\&D_{2}=(H_{a}H_{b}+H_{1}H_{4})(H_{b}H_{c}+H_{3}H_{6})\\&\qquad \quad +(H_{b}H_{5}+H_{4}H_{6})(H_{b}H_{2}-H_{1}H_{3}) \end{aligned}$$

with

$$\begin{aligned}&H_{a}=\big (\omega _{x}^{2}-\omega _{E}^{2}\big )^{2}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )+\Big \{\gamma ^{2}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )-\Omega _{y}^{2}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\\&\qquad -\Omega _{z}^{2}\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\Big \}\omega _{E}^{2},\\&H_{b}=\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )^{2}\big (\omega _{z}^{2}-\omega _{E}^{2}\big )+\Big \{\gamma ^{2}\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\big (\omega _{x}^{2}-\omega _{E}^{2}\big )-\Omega _{z}^{2}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\\&\qquad -\Omega _{x}^{2}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\Big \}\omega _{E}^{2},\\&H_{c}=\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )^{2}+\Big \{\gamma ^{2}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )-\Omega _{x}^{2}\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\\&\qquad -\Omega _{y}^{2}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\Big \}\omega _{E}^{2},\\&H_{x}=\frac{q}{m}\Big \{ E_{0x}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\cos \phi _{x}-E_{0x}\gamma \omega _{E}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\sin \phi _{x}\\&\qquad +E_{0y}\Omega _{z}\omega _{E}\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\sin \phi _{y}-E_{0z}\Omega _{y}\omega _{E}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\sin \phi _{z}\Big \}\\&H_{y}=\frac{q}{m}\Big \{ E_{0y}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\cos \phi _{y}-E_{0x}\Omega _{z}\omega _{E}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\sin \phi _{x}\\&\qquad -E_{0y}\gamma \omega _{E}\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\sin \phi _{y}+E_{0z}\Omega _{x}\omega _{E}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\sin \phi _{z}\Big \},\end{aligned}$$

$$\begin{aligned}&H_{z}=\frac{q}{m}\Big \{ E_{0z}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\cos \phi _{z}+E_{0x}\Omega _{y}\omega _{E}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\sin \phi _{x}\\&\qquad -E_{0y}\Omega _{x}\omega _{E}\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\sin \phi _{y}-E_{0z}\gamma \omega _{E}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\sin \phi _{z}\Big \},\\&H_{x}^{\prime }=\frac{q}{m}\Big \{ E_{0x}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\sin \phi _{x}+E_{0x}\gamma \omega _{E}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\cos \phi _{x}\\&\qquad -E_{0y}\Omega _{z}\omega _{E}\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\cos \phi _{y}+E_{0z}\Omega _{y}\omega _{E}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\cos \phi _{z}\Big \},\\&H_{y}^{\prime }=\frac{q}{m}\Big \{ E_{0y}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\sin \phi _{y}+E_{0x}\Omega _{z}\omega _{E}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\cos \phi _{x}\\&\qquad +E_{0y}\gamma \omega _{E}\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\cos \phi _{y}-E_{0z}\Omega _{x}\omega _{E}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\cos \phi _{z}\Big \},\\&H_{z}^{\prime }=\frac{q}{m}\Big \{ E_{0z}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\sin \phi _{z}-E_{0x}\Omega _{y}\omega _{E}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\cos \phi _{x}\\&\qquad +E_{0y}\Omega _{x}\omega _{E}\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\cos \phi _{y}+E_{0z}\gamma \omega _{E}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\cos \phi _{z}\Big \},\\&H_{1}=\Big \{\gamma \Omega _{z}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )+\gamma \Omega _{z}\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\big (\omega _{x}^{2}-\omega _{E}^{2}\big )-\Omega _{x}\Omega _{y}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\Big \}\omega _{E}^{2},\\&H_{2}=\Big \{\gamma \Omega _{y}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )+\gamma \Omega _{y}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )+\Omega _{z}\Omega _{x}\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\Big \}\omega _{E}^{2},\\&H_{3}=\Big \{\gamma \Omega _{x}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )+\gamma \Omega _{x}\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\big (\omega _{x}^{2}-\omega _{E}^{2}\big )-\Omega _{y}\Omega _{z}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\Big \}\omega _{E}^{2},\\&H_{4}=\Big \{\gamma \Omega _{z}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )+\gamma \Omega _{z}\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\big (\omega _{x}^{2}-\omega _{E}^{2}\big )+\Omega _{x}\Omega _{y}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\Big \}\omega _{E}^{2},\\&H_{5}=\Big \{\gamma \Omega _{y}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )+\gamma \Omega _{y}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )-\Omega _{z}\Omega _{x}\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\Big \}\omega _{E}^{2} \end{aligned}$$

and

$$\begin{aligned}&H_{6}=\Big \{\gamma \Omega _{x}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\\&\qquad +\gamma \Omega _{x}\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\\&\qquad +\Omega _{y}\Omega _{z}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big (\omega _{z}^{2}-\omega _{E}^{2}\big )\Big \}\omega _{E}^{2}. \end{aligned}$$

For the condition, $\omega _{x}=\omega _{y}=\omega _{z}=\omega$ and $\Omega _{x}=\Omega _{y}=\Omega _{z}=\Omega$ the above relations become

$$\begin{aligned}&A_{1}=(H_{x}H_{0}-H_{z}H_{2})(H_{0}^{2}+H_{1}H_{2})\\&\qquad +(H_{y}H_{0}+H_{z}H_{1})(H_{2}^{2}+H_{0}H_{1}),\\&A_{2}=(H_{x}^{\prime }H_{0}-H_{z}^{\prime }H_{2})(H_{0}^{2}+H_{1}H_{2})\\&\qquad +(H_{y}^{\prime }H_{0}+H_{z}^{\prime }H_{1})(H_{2}^{2}+H_{0}H_{1}),\\&B_{1}=(H_{y}H_{0}+H_{z}H_{1})(H_{0}^{2}+H_{1}H_{2})\\&\qquad +(H_{x}H_{0}-H_{z}H_{2})(H_{1}^{2}-H_{0}H_{2}),\\&B_{2}=(H_{y}^{\prime }H_{0}+H_{z}^{\prime }H_{1})(H_{0}^{2}+H_{1}H_{2})\\&\qquad +(H_{x}^{\prime }H_{0}-H_{z}^{\prime }H_{2})(H_{1}^{2}-H_{0}H_{2}),\\&C_{1}=(H_{z}H_{0}+H_{x}H_{1})(H_{0}^{2}+H_{1}H_{2})\\&\qquad +(H_{y}H_{0}-H_{x}H_{2})(H_{1}^{2}-H_{0}H_{2}), \end{aligned}$$

$$\begin{aligned}&C_{2}=(H_{z}^{\prime }H_{0}+H_{x}^{\prime }H_{1})(H_{0}^{2}+H_{1}H_{2})\\&\qquad +(H_{y}^{\prime }H_{0}-H_{x}^{\prime }H_{2})(H_{1}^{2}-H_{0}H_{2}) \end{aligned}$$

and

$$\begin{aligned}&D_{1}=D_{2}=D=(H_{0}^{2}+H_{1}H_{2})^{2}\\&-(H_{1}^{2}-H_{0}H_{2})(H_{2}^{2}+H_{0}H_{1}) \end{aligned}$$

with

$$\begin{aligned}&H_{0}=\big (\omega ^{2}-\omega _{E}^{2}\big )^{2}-\big (2\Omega ^{2}-\gamma ^{2}\big )\omega _{E}^{2},\\&H_{1}=\Omega \omega _{E}^{2}\big (2\gamma -\Omega \big ),\\&H_{2}=\Omega \omega _{E}^{2}\big (2\gamma +\Omega \big ),\\&H_{x}=\frac{q}{m}E_{0x}\big (\omega ^{2}-\omega _{E}^{2}\big ),\\&H_{y}=\frac{q}{m}E_{0y}\big (\omega ^{2}-\omega _{E}^{2}\big ),\\&H_{z}=\frac{q}{m}E_{0z}\big (\omega ^{2}-\omega _{E}^{2}\big ),\\&H_{x}^{\prime }=\frac{q}{m}\omega _{E}\big (\gamma E_{0x}-\Omega E_{0y}+\Omega E_{0z}\big ),\\&H_{y}^{\prime }=\frac{q}{m}\omega _{E}\big (\gamma E_{0y}-\Omega E_{0z}+\Omega E_{0x}\big ) \end{aligned}$$

and

$$\begin{aligned} H_{z}^{\prime }=\frac{q}{m}\omega _{E}\big (\gamma E_{0z}-\Omega E_{0x}+\Omega E_{0y}\big ). \end{aligned}$$

Appendix B: Definition of the relevant quantities which appear in eq. (34)

$H_0, H_1, H_2, H_3$ and $H_4$ which appear in eq. (34) are defined as

$$\begin{aligned}&H_{0}=H_{0x}H_{0y}+\gamma ^{2}\Omega ^{2}\omega _{E}^{4}\big (\omega _{x}^{2}+\omega _{y}^{2}-2\omega _{E}^{2}\big )^{2},\\&H_{1}=\frac{q}{m}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big \{ E_{0x}H_{0y}\\&\quad +E_{0y}\gamma \Omega \omega _{E}^{2}\big (\omega _{x}^{2}+\omega _{y}^{2}-2\omega _{E}^{2}\big )\big \},\\&H_{2}=\frac{q}{m}\omega _{E}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big \{ E_{0x}H_{0y}\gamma -E_{0y}H_{0x}\Omega \\&\quad +\gamma \Omega \omega _{E}^{2}\big (\omega _{x}^{2}+\omega _{y}^{2}-2\omega _{E}^{2}\big )\big (E_{0y}\gamma +E_{0x}\Omega \big )\big \},\\&H_{3}=\frac{q}{m}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\big \{ E_{0y}H_{0x}\\&\quad -E_{0x}\gamma \Omega \omega _{E}^{2}\big (\omega _{x}^{2}+\omega _{y}^{2}-2\omega _{E}^{2}\big )\big \} \end{aligned}$$

and

$$\begin{aligned}&H_{4}=\frac{q}{m}\omega _{E}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\big \{E_{0y}H_{0x}\gamma +E_{0x}H_{0y}\Omega \\&\qquad +\gamma \Omega \omega _{E}^{2}\big (\omega _{x}^{2}+\omega _{y}^{2}-2\omega _{E}^{2}\big )\big (E_{0y}\Omega -E_{0x}\gamma \big )\big \} \end{aligned}$$

with

$$\begin{aligned}&H_{0x}=\big (\omega _{x}^{2}-\omega _{E}^{2}\big )^{2}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )\\&\qquad +\gamma ^{2}\omega _{E}^{2}\big (\omega _{y}^{2}-\omega _{E}^{2}\big )-\Omega ^{2}\omega _{E}^{2}\big (\omega _{x}^{2}-\omega _{E}^{2}\big ) \end{aligned}$$

and

$$\begin{aligned}&H_{0y}=\big (\omega _{y}^{2}-\omega _{E}^{2}\big )^{2}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )\\&\qquad +\gamma ^{2}\omega _{E}^{2}\big (\omega _{x}^{2}-\omega _{E}^{2}\big )-\Omega ^{2}\omega _{E}^{2}\big (\omega _{y}^{2}-\omega _{E}^{2}\big ). \end{aligned}$$