of the electric field. In this paper we will extend their results and prove that cyclotron damping in electromagnetic fields still occurs.

Now we will make a brief statement about the connection and difference between the results of \[23\] and ours. First, in electric field case, Mouhot and Villani proved the existence of Landau damping under assumption of the $(\mathbf{L})$ condition, that is expressed as follows:

($\mathbf{L}$) There are constants $C_{0},\lambda,\kappa>0$ such that $|\hat{f}^{0}(\eta)|\leq C_{0}e^{-2\pi\lambda|\eta|}$ for any $\eta\in\mathbb{R}^{d};$ and for any $\xi\in\mathbb{C}$ with $0\leq \textmd{Re}\xi<\lambda,$ $$\begin{aligned}
  \inf_{k\in\mathbb{Z}^{d}}|\mathcal{L}(\xi,k)-1|\geq\kappa,
  \end{aligned}$$ where we define a function $\mathcal{L}(\xi,k)=-4\pi^{2}\int^{\infty}_{0}e^{2\pi\xi^{\ast}|k|t}\widehat{W}(k)\hat{f}^{0}(kt)|k|^{2}tdt,$ and $\xi^{\ast}$ is the complex conjugate to $\xi.$

To some extent, (0.6) of the $(\mathbf{L})$ condition is similar to the “Small Denominators" condition in KAM theory in \[1\], but is stronger. Here we will consider the condition of cyclotron damping from a totally different perspective, in detail, we will give a physical condition that we call the “$\mathbf{Physical}$ $\mathbf{Stability}$ $\mathbf{Condition}$",in short, “$\mathcal{\mathcal{\mathbf{PSC}}}$", which is stated in the following form (here we assume the background magnetic field $B_{0}$ along the $\hat{z}$ direction):

$(\mathbf{PSC})$ : for any component velocity in the $\hat{z}$ direction $v_{3}\in\mathbb{R},$ there exists some positive constant $v_{Te}$ such that if $v_{3}=\frac{\omega}{k_{3}},\omega,k$ are frequencies of time 