optical atomic clock. Our results also show that a subnatural
linewidth spectroscopy, superior to any other available su bnat-
ural spectroscopy technique at present [3–10], can be reach ed
by this kind of laser, if a suitable atomic level structure is cho-
sen. Thus, this method can provide an e ﬀective subnatural
spectroscopy, and the possibilities for the new optical clo ck
scheme [15] and atom interferometers [2].
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 cavitymode
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 statea 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 line ar
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 isl,
and the length of the free drift region is L . The corresponding
Hamiltonian is
H = ℏωˆa †ˆa +ℏ
∑
j
[ ωj
a ( t) σj
a +ωj
b ( t) σj
b ]
+ ℏg
∑
j
Γj ( t)(ˆa † ˆσj
−e −i⃗k ·⃗r j + ˆσj
+ˆae i⃗k ·⃗r j ) , (1)
where ˆ a , ˆa † are the annihilation and creation operators of the
ﬁeld mode inside the cavity, with the frequency ω, σj
a =
( |a ⟩⟨a |) j and σj
b = ( |b ⟩⟨b |) j are the projection operators for the
jth atom corresponding to the upper and lower lasing levels,