2
with frequency ωj
a and ωj
b , and σj
− = ( |b ⟩⟨a |) j is the “spin-
ﬂip” operator for the jth atom, with its adjoint σj
+ = ( |a ⟩⟨b |) j .
The coupling constant g is given by g = µ√ω/2 ℏǫ0 V , where
µ is the magnitude of the atomic dipole moment, and V is the
e ﬀective volume of the cavity.
In order to denote the ﬁnite-time interaction between the
atoms and Ramsey separated ﬁeld, we introduce the function
Γj ( t) = Θ( t −t j ) −Θ( t −t j −τ) +Θ( t −t j −τ−T ) −Θ( t −t j −2 τ−T ) ,
(2)
where Θ( t) is the Heaviside step function [ Θ( t) = 1 for t > 0,
Θ( t) = 1 /2 for t = 0, and Θ( t) = 0 for t < 0]. T is the free
drift time of the atoms, and τ is the interacting time between
the atom and one cavity.
By the standard way [25], we can get the Heisenberg-
Langevin equations of the motion for the single-atom and
ﬁled operators. By introducing the macroscopic atomic oper -
ator, M ( t) = −i ∑
j Γj ( t) σj
−( t), N a ( t) = ∑
j Γj ( t) σj
aa ( t), N b ( t) =∑
j Γj ( t) σj
bb ( t), the dynamic equations for the ﬁeld and macro-
scopic atomic operators yield
˙a ( t) = −κ
2 a ( t) +gM ( t) +F κ( t) , (3)
˙N a ( t) = R (1 −A 0 + A 1 −A 2 ) −( γa +γ′
a ) N a ( t)
− g [ M †( t) a ( t) +a †( t) M ( t)] +F a ( t) , (4)
˙N b ( t) = − R ( B 0 − B 1 + B 2 ) −γb N b ( t) +γ′
a N a ( t)
+ g [ a †( t) M ( t) + M †( t) a ( t)] +F b ( t) , (5)
˙M ( t) = − R ( C 0 −C 1 +C 2 ) −γab M ( t)
+ g [ N a ( t) −N b ( t)] a ( t) +F M ( t) , (6)
where the macroscopic noise operators are deﬁned as
F a ( t) =
∑
j
˙Γj ( t) σj
a ( t) −R (1 −A 0 + A 1 −A 2 ) +
∑
j
Γj ( t) f j
a ( t) ,
F b ( t) =
∑
j
˙Γj ( t) σj
b ( t) +R ( B 0 − B 1 + B 2 ) +
∑
j
Γj ( t) f j
b ( t) ,
F M ( t) = −i
∑
j
˙Γj ( t) ˜σj
−( t) +R ( C 0 −C 1 +C 2 ) −i
∑
j
Γj ( t) f j
σ( t) ,
with A 0 =
⣨
σj
a ( t j +τ)
⟩
q , A 1 =
⣨
σj
a ( t j +τ+T )
⟩
q ,
A 2 =
⣨
σj
a ( t j +2 τ+T )
⟩
q , B 0 =
⣨
σj
b ( t j +τ)
⟩
q ,
B 1 =
⣨
σj
b ( t j +τ+T )
⟩
q , B 2 =
⣨
σj
b ( t j +2 τ+T )
⟩
q ,
C 0 =
⣨
−iσj
−( t j +τ)
⟩
q , C 1 =
⣨
−iσj
−( t j +τ+T )
⟩
q ,
C 2 =
⣨
−iσj
−( t j +2 τ+T )
⟩
q . R is the mean pumping
rate, which is deﬁned in [26]. It is very easy to check that the
average values of the above Langevin forces are all zero.
By using the above deﬁnitions of the noise operators, we
ﬁnd the correlation functions of macroscopic noise forces c an
be generally written in the form
⟨F k ( t) F l ( t′) ⟩
= D (0)
kl δ( t −t′) + D (1)
kl δ( t −t′ −τ)
+ D (2)
kl δ( t −t′ +τ) + D (3)
kl δ( t −t′ −τ−T )
+ D (4)
kl δ( t −t′ +τ+T ) +D (5)
kl δ( t −t′ −2 τ−T )
+ D (6)
kl δ( t −t′ +2 τ+T ) +D (7)
kl δ( t −t′ −T )
+ D (8)
kl δ( t −t′ +T ) , (7)
where D ( i)
kl ( k ,l = a ,b ,M ,M †; i = 0 ,1 ,2) are the quantum dif-
fusion coe ﬃcients.
c-number correlation functions: By choosing some partic-
ular ordering for products of atomic and ﬁeld operators, one
could derive the c-number stochastic Langevin equations from
the quantum Langevin equations derived above, and all of the
dynamic equations for c-number stochastic variables are th e
same as in [26]. The di ﬀerences are from the correlation func-
tions. On the other hand, we convert the quantum noise oper-
ators into the c-number noise variables˜F k ( t)( k = a ,b ,M ,M †),
whose correlation functions are expressed as
⣨˜F k ( t) ˜F k ( t′)
⟩
= ˜D (0)
kl δ( t −t′) + ˜D (1)
kl δ( t −t′ −τ)
+ ˜D (2)
kl δ( t −t′ +τ) + ˜D (3)
kl δ( t −t′ −τ−T )
+ ˜D (4)
kl δ( t −t′ +τ+T ) + ˜D (5)
kl δ( t −t′ −2 τ−T )
+ ˜D (6)
kl δ( t −t′ +2 τ+T ) + ˜D (7)
kl δ( t −t′ −T )
+ ˜D (8)
kl δ( t −t′ +T ) , (8)
where ˜D ( i)
kl are the c-number Langevin di ﬀusion coe ﬃcients,
related to quantum Langevin di ﬀusion coe ﬃcients D ( i)
kl as in
[27].
Steady-state solutions: The steady-state solutions for the
mean values of the ﬁeld and atomic variables for laser op-
eration are obtained by dropping the noise terms of the c-
number Langevin equations and setting the time derivatives
equal to zero. The analytical solutions are very complex, and
one could numerically solve the steady-state equations. In this