3
˜N bss = R τ
2
[
1 − C 0 −C 1 +C 2
g τ
√ κ
R ( B 0 − B 1 + B 2 )
]
.
A detailed analysis about the stability of the steady-state can
be found such as in [28]. In this paper, we assume the steady-
state solution is stable.
Laser linwidth: Suppose the quantum ﬂuctuation is small,
the evolution of the ﬂuctuations can be obtained by making a
linearization of the c-number Langevin equations around th e
steady-state solution. Then the measured spectra of ﬁeld ﬂu c-
tuations will be directly related to these quantities. By Fo urier
transformations of the linearized equation, we get the ampl i-
tude and phase quadrature components δX ( ω) and δY ( ω) [26].
Well above threshold, one can neglect the amplitude ﬂuctu-
ations, and the linewidth inside the cavity is related to the
phase-diﬀusion coe ﬃcient [25]. For small ﬂuctuation of laser
phase, the spectrum of phase ﬂuctuations is simply related t o
the spectrum of the phase quadrature component of the ﬁeld
ﬂuctuations, namely,
( δϕ2 ) ω = 1
I 0
( δY 2 ) ω.
In the region γab ≪ T −1 ≪ τ−1 ≪ κ/2, as in the recently
proposed active optical clock [15] with atomic beam. The
phase quadrature component of the ﬁeld ﬂuctuations can be
expressed as
( δϕ2 ) ω
≈ ( κ/2 +γab ) 2
I 0 ω2 [( κ/2 +γab ) 2 +ω2 ]
g 2
4( κ/2 +γab ) 2 {4 γab ˜N ass
+ 2 R [( A 0 + B 0 ) +( A 2 + B 2 )]
+ Rp [( C 0 −C ∗
0 ) 2 +( C 1 −C ∗
1 ) 2 +( C 2 −C ∗
2 ) 2 ] }. (9)
Since the time τ and T is much shorter than the time scale
of the atomic dampings, we can neglect the dampings when
calculateA i , B i , C i . By using
A 0 = cos 2
( ΩR
2 τ
)
, A 1 = cos 2
( ΩR
2 τ
)
,
A 2 = 1 −sin 2 ( ΩR τ) cos 2
( ∆2
2 T
)
, B 0 = sin 2
( ΩR
2 τ
)
,
B 1 = sin 2
( ΩR
2 τ
)
, B 2 = sin 2 ( ΩR τ) cos 2
( ∆2 T
2
)
,
( C 0 −C ∗
0 ) 2 = 0 ,( C 1 −C ∗
1 ) 2 = −sin 2 ( ΩR τ) sin 2 ( ∆2 T ) ,
( C 2 −C ∗
2 ) 2 = −sin 2 ( ΩR τ) sin 2 ( ∆2 T ) ,
we get
( δϕ2 ) ω = ( κ/2 +γab ) 2
ω2 [( κ/2 +γab ) 2 +ω2 )]
γ2
ab
( κ/2 +γab ) 2 {D S T
+ D Ram [2 − p sin 2 ( ΩR τ) sin 2 ( ∆2 T )] }, (10)
where ΩR is the Rabi frequency on resonance,
D S T =g 2 ˜N ass /I 0 γab , D Ram = g 2 R /2 I 0 γ2
ab , and
∆2 = ω − ( ωa 2 − ωb 2 ) presents the detuning in the free
drift region. p is a parameter, which characterizes the pump-
ing statistics: a Poissonian excitation statistics corres ponds to
p = 0 , and for a regular statistics we have p = 1.
Then the linewidth of Ramsey laser with bad cavity is given
by
D = γ2
ab
( κ/2 +γab ) 2 {D S T + D Ram [2 − p sin 2 ( ΩR τ) sin 2 ( ∆2 T )] }.
(11)
Since D S T /D Ram ≪ 1 in our situation, and in the case of max-
imal photon number, the steady state value of ˜N ass is about
R τ/2. Then we get the
D ≈ 2 g 2
κ [2 − p sin 2 ( ΩR τ) sin 2 ( ∆2 T )] . (12)
From the expression above, we ﬁnd that the pumping statis-
tic can inﬂuence the linewidth. For regular injection (p = 1),
the linewidth is the narrowest, while for Poissonian inject ion
( p = 0), the linewidth is the broadest. But even for regular
injection, the linewidth is larger than the case of one cavit y.
That means the mechanism of separated-ﬁeld does not play
the role in reducing the linewidth as in the conventional opt i-
cal Ramsey method, which is counter-intuitive. However, th e
separated ﬁelds are indispensable for any phase detection l ike
atom interferometry. The details about the method of active
atom interferometry will appear elsewhere.
Our method of Ramsey laser is suitable for any atoms with
metastable energy level, as an example, we choose the tran-
sition from the metastable state 4s 4 p 3 P 1 to the ground state
4 s 2 1 S 0 of 40 Ca to check the striking feature of this laser: sub-
natural linewidth. As mentioned in [29], the corresponding
natural linewidth of the metastable state 4s 4 p 3 P 1 is 320Hz.
As in the recently proposed active optical clock with atomic
beam [15], the velocity of the atoms in thermal atomic beam is
about 500m/s, and the length of the interaction region is about
1mm, then the time for the atom to traverse each coherent-