Derivation from Cavity Quantum Electrodynamics Maxwell–Bloch equations

beginning jaynes-cummings hamiltonian under coherent drive

h
=

ω

c



a

†


a
+

ω

a



σ

†


σ
+
i
g
(

a

†


σ
−
a

σ

†


)
+
i
j
(

a

†



e

−
i

ω

l


t


−
a

e

i

ω

l


t


)


{\displaystyle h=\omega _{c}a^{\dagger }a+\omega _{a}\sigma ^{\dagger }\sigma +ig(a^{\dagger }\sigma -a\sigma ^{\dagger })+ij(a^{\dagger }e^{-i\omega _{l}t}-ae^{i\omega _{l}t})}

where



a


{\displaystyle a}

lowering operator cavity field, ,



σ
=


1
2



(

σ

x


−
i

σ

y


)



{\displaystyle \sigma ={\frac {1}{2}}\left(\sigma _{x}-i\sigma _{y}\right)}

atomic lowering operator written combination of pauli matrices. time dependence can removed transforming wavefunction according




|

ψ
⟩
→

e

−
i

ω

l


t

(

a

†


a
+

σ

†


σ
)



⁡

|

ψ
⟩


{\displaystyle |\psi \rangle \rightarrow \operatorname {e} ^{-i\omega _{l}t\left(a^{\dagger }a+\sigma ^{\dagger }\sigma \right)}|\psi \rangle }

, leading transformed hamiltonian

h
=

Δ

c



a

†


a
+

Δ

a



σ

†


σ
+
i
g
(

a

†


σ
−
a

σ

†


)
+
i
j
(

a

†


−
a
)


{\displaystyle h=\delta _{c}a^{\dagger }a+\delta _{a}\sigma ^{\dagger }\sigma +ig(a^{\dagger }\sigma -a\sigma ^{\dagger })+ij(a^{\dagger }-a)}

where




Δ

i


=

ω

i


−

ω

l




{\displaystyle \delta _{i}=\omega _{i}-\omega _{l}}

. stands now, hamiltonian has 4 terms. first 2 self energy of atom (or other 2 level system) , field. third term energy conserving interaction term allowing cavity , atom exchange population , coherence. these 3 terms alone give rise jaynes-cummings ladder of dressed states, , associated anharmonicity in energy spectrum. last term models coupling between cavity mode , classical field, i.e. laser. drive strength



j


{\displaystyle j}

given in terms of power transmitted through empty two-sided cavity



j
=


2
p
(

Δ

c


2


+

κ

2


)

/

(

ω

c


κ
)




{\displaystyle j={\sqrt {2p(\delta _{c}^{2}+\kappa ^{2})/(\omega _{c}\kappa )}}}

,



2
κ


{\displaystyle 2\kappa }

cavity linewidth. brings light crucial point concerning role of dissipation in operation of laser or other cqed device; dissipation means system (coupled atom/cavity) interacts environment. end, dissipation included framing problem in terms of master equation, last 2 terms in lindblad form

ρ
˙



=
−
i
[
h
,
ρ
]
+
2
κ

(
a
ρ

a

†


−


1
2



(

a

†


a
ρ
+
ρ

a

†


a
)

)

+
2
γ

(
σ
ρ

σ

†


−


1
2



(

σ

†


σ
ρ
+
ρ

σ

†


σ
)

)



{\displaystyle {\dot {\rho }}=-i[h,\rho ]+2\kappa \left(a\rho a^{\dagger }-{\frac {1}{2}}\left(a^{\dagger }a\rho +\rho a^{\dagger }a\right)\right)+2\gamma \left(\sigma \rho \sigma ^{\dagger }-{\frac {1}{2}}\left(\sigma ^{\dagger }\sigma \rho +\rho \sigma ^{\dagger }\sigma \right)\right)}

the equations of motion expectation values of operators can derived master equation formulas



⟨
o
⟩
=
tr
⁡

(
o
ρ
)



{\displaystyle \langle o\rangle =\operatorname {tr} \left(o\rho \right)}

,



⟨



o
˙



⟩
=
tr
⁡

(
o



ρ
˙



)



{\displaystyle \langle {\dot {o}}\rangle =\operatorname {tr} \left(o{\dot {\rho }}\right)}

. equations of motion



⟨
a
⟩


{\displaystyle \langle a\rangle }

,



⟨
σ
⟩


{\displaystyle \langle \sigma \rangle }

, ,



⟨

σ

z


⟩


{\displaystyle \langle \sigma _{z}\rangle }

, cavity field, atomic ground state population, , atomic inversion respectively, are

d

d
t



⟨
a
⟩
=
i

(
−

Δ

c


⟨
a
⟩
−
i
g
⟨
σ
⟩
−
i
j
)

−
κ
⟨
a
⟩


{\displaystyle {\frac {d}{dt}}\langle a\rangle =i\left(-\delta _{c}\langle a\rangle -ig\langle \sigma \rangle -ij\right)-\kappa \langle a\rangle }








d

d
t



⟨
σ
⟩
=
i

(
−

Δ

a


⟨
σ
⟩
−
i
g
⟨
a

σ

z


⟩
)

−
γ
⟨
σ
⟩


{\displaystyle {\frac {d}{dt}}\langle \sigma \rangle =i\left(-\delta _{a}\langle \sigma \rangle -ig\langle a\sigma _{z}\rangle \right)-\gamma \langle \sigma \rangle }








d

d
t



⟨

σ

z


⟩
=
−
2
g

(
⟨

a

†


σ
⟩
+
⟨
a

σ

†


⟩
)

−
2
γ
⟨

σ

z


⟩
−
2
γ


{\displaystyle {\frac {d}{dt}}\langle \sigma _{z}\rangle =-2g\left(\langle a^{\dagger }\sigma \rangle +\langle a\sigma ^{\dagger }\rangle \right)-2\gamma \langle \sigma _{z}\rangle -2\gamma }

at point, have produced 3 of infinite ladder of coupled equations. can seen third equation, higher order correlations necessary. differential equation time evolution of



⟨

a

†


σ
⟩


{\displaystyle \langle a^{\dagger }\sigma \rangle }

contain expectation values of higher order products of operators, leading infinite set of coupled equations. heuristically make approximation expectation value of product of operators equal product of expectation values of individual operators. akin assuming operators uncorrelated, , approximation in classical limit. turns out resulting equations give correct qualitative behavior in single excitation regime. additionally, simplify equations make following replacements

⟨
a
⟩
=
(
γ

/



2


g
)
x


{\displaystyle \langle a\rangle =(\gamma /{\sqrt {2}}g)x}






⟨
σ
⟩
=
−
p

/



2




{\displaystyle \langle \sigma \rangle =-p/{\sqrt {2}}}






⟨

σ

z


⟩
=
−
d


{\displaystyle \langle \sigma _{z}\rangle =-d}






Θ
=

Δ

c



/

κ


{\displaystyle \theta =\delta _{c}/\kappa }






c
=

g

2



/

2
κ
γ


{\displaystyle c=g^{2}/2\kappa \gamma }






y
=


2


g
j

/

κ
γ


{\displaystyle y={\sqrt {2}}gj/\kappa \gamma }






Δ
=

Δ

a



/

γ


{\displaystyle \delta =\delta _{a}/\gamma }

and maxwell–bloch equations can written in final form

x
˙



=
κ

(
−
2
c
p
+
y
−
(
i
Θ
+
1
)
x
)



{\displaystyle {\dot {x}}=\kappa \left(-2cp+y-(i\theta +1)x\right)}









p
˙



=
γ

(
−
(
1
+
i
Δ
)
p
+
x
d
)



{\displaystyle {\dot {p}}=\gamma \left(-(1+i\delta )p+xd\right)}









d
˙



=
γ

(
2
(
1
−
d
)
−
(

x

∗


p
+
x

p

∗


)
)



{\displaystyle {\dot {d}}=\gamma \left(2(1-d)-(x^{*}p+xp^{*})\right)}