text: *Theoretical framework:*We consider the case of a two-level 
  atomic beam interacting with a single-mode Ramsey cavity 
  of separated-oscillating-ﬁeld resonators with the cavity mode 
  linewidth is much wider than the atomic gain linewidth. Thus 
  we call it bad-cavity Ramsey laser. All atoms are pumped 
  onto the upper lasing state**a**before entering the ﬁrst cavity 
  of seperated ﬁeld, and the lower lasing state is**b**. We assume 
  all the atoms have the same velocities υ, that means what we 
  consider here is a homogeneous laser system. And for the 
  sake of simplicity, we consider the two-standing waves linear 
  optical Ramsey conﬁguration with a grid as spatial selector 
  [20, 21]. Our treatment can be extended to other conﬁgura- 
  tions as in [22–24]. The length of each oscillating part is*l*, 
  and the length of the free drift region is*L*. The corresponding 
  Hamiltonian is 
  
  [ω*j*
  *a*(*t*)σ*j* *a*+ ω*j*
  *b*(*t*)σ*j*
  *b*] 
  *H*= ~ωˆ*a*† ˆ*a*+ ~ 
  
  X*j*
  Γ*j*(*t*)(ˆ*a*† ˆσ*j*
  − 
  + ~*g*
  X*j*