ABSTRACT (original article): In this work we develop an open quantum system view of the parametric approximation, which allows us to obtain systematic perturbative corrections to it. We consider the Jaynes-Cummings model with dissipation, assuming that the field is in the regime close to the parametric approximation with depletion. We obtain non-unitary corrections to the parametric approximation and additional dynamical Lamb-shift contributions to it. For high detuning, these nonunitary corrections appear to be non-Markovian before depletion. We also show that even after depletion, initial non-Markovian behavior contributes to the dynamics via laser-induced polishing of the density matrix. CITATION (original article): Artem Karasev, Alexander Teretenkov (2025), Parametric approximation as an open-quantum-system problem, Phys. Rev. A 111, 052203. https://doi.org/10.1103/PhysRevA.111.052203

1. Perturbative expansion

Suppose we have two weakly interacting subsystems A and B, so the density matrix of full system is factorized
ρ=
ρ
A
⊗
ρ
B
. The total evolution is governed by equation

t
ρ=
ℒ
0
+λℒ
. Generator
ℒ
0
describes fee dynamics and
ℒ
defines interaction, proportional to a small constant
λ
.Subsystem
A
is a system of interest and we aim to reduce degrees of freedom of the subsystem
B
. To do so, we use standard projector

ρ
=
tr
B
(ρ)⊗
fixed
ρ
B
. Using this projector we ca write down perturbative expansion for the generator of the reduced evolution.
d
dt
ρ
A
K(t)
ρ
A

∞
∑
n0
n
λ
K
n
(t)
ρ
A
(
1
)
K
n
(t)
n
∑
q0
q
(-1)
∑
q
∑
j0
k
j
n,
k
j
0

O
k
0
-1
O
0
O
k
1
-1
O
0
...
-1
O
0
O
k
q
-1
O
0
,
(
2
)
where
O
k
(t)
t
∫
t
0
d
t
1
…
t
k-1
∫
t
0
d
t
k
P
L
0
t
e
L(
t
1
)…L(
t
k
)P
(
3
)
O
0
(t)P
L
0
t
e
P
(
4
)
We call

n
(t)
cumulants and

k
(t)
moments.

2. Model and vectorization

We consider dissipative Jaynes Cummings model, so
L
0
-i[
ω
A
+
σ
-
σ
+
ω
c
†
b
b,· ]+γb·
†
b
-
1
2
{
†
b
b,·}
​​
L-i[g
+
σ
b+
g
-
σ
†
b
,· ]
​​
P

B
(· )⊗|z〉〈z|
(
5
)
With two-level birth/annihilation operators
{
-
σ
,
+
σ
}=
I
A
, bosonic operators
[b,
†
b
]=
I
B
, and coherent state
bz〉=zz〉
.
For convenience of calculations we will work with a vectorized form of generators
L
0
-i
ω
A
+
σ
l
-
σ
l
+i
ω
A
+
σ
r
-
σ
r
-i
ω
c
†
b
l
b
l
+i
ω
c
†
b
r
b
r
​​+γ
b
l
b
r
-
1
2
†
b
l
b
l
-
1
2
†
b
r
b
r
(
6
)
L-ig
+
σ
l
b
l
-i
g
-
σ
l
†
b
l
+i
g
+
σ
r
b
r
+ig
-
σ
r
†
b
r
,
(
7
)
where
b
l
≡b⊗I
,
b
r
≡I⊗b
,
-
σ
l
≡
-
σ
⊗I
,
-
σ
r
≡I⊗
-
σ
etc.
Projector takes form
P|z
〉
l
|
z
〉
r
〈1|
,
(
8
)
where is such that
〈1||ψ〉
l
|φ〉
r
〈φ|ψ〉
.
(
9
)
and
〈1|
X
l
Y
r
|ψ〉
l
|φ〉
r
〈φ|
T
Y
X|ψ〉.
(
10
)
The moments

k
will contain averages, due to action of
P
on subsystem
B
. For example
1|
k

†
b
l

m
b
l
z
l

z
〉
r

k
z
m
z
(
11
)

3. Details

Using our notebook one can calculate moments

k
applying proposed vectorization technique. Combinatorial formulae (2) helps in collecting this moments in cumulants. After taking averages, only thing left to do is to calculate repeated integrals
t
∫
t
0
d
t
1
…
t
k-1
∫
t
0
d
t
k
L(
t
1
)…L(
t
k
)
,
(
12
)
where
L(t)-ig
+
σ
l
B
l
(t)-i
g
-
σ
l
†
B
l
(t)+i
g
+
σ
r
B
r
(t)
​​
+ig
-
σ
r
†
B
r
(t)
(
13
)
and
B
l
(t)≡
-Γt
e
b
l
,
(
14
)
B
r
(t)≡
-
Γ
t
e
b
r
,
(
15
)
†
B
l
(t)≡
Γt
e
†
b
l
+
-
Γ
t
e
(1-
γt
e
)
b
r
,
(
16
)
†
B
r
(t)≡
Γ
t
e
†
b
r
+
-Γt
e
(1-
γt
e
)
b
l
,
(
17
)
where
Γ=-i(
ω
A
-
ω
c
)+
1
2
γ
.
We use precalculated integrals and after insert (13) in precatculated constructions.

Auxiliary functions

Normal forms and averages with respect to coherent state

In[]:=
Unprotect[NonCommutativeMultiply];​​A_**(B_+C_):=A**B+A**C;​​(B_+C_)**A_:=B**A+C**A;​​A_**c_?NumberQ:=cA;​​c_?NumberQ**A_:=cA;​​A_**(B_c_?NumberQ):=cA**B;​​(A_c_?NumberQ)**B_:=cA**B;​​A_**(B_c_Rational):=cA**B;​​(A_c_Rational)**B_:=cA**B;​​A_**(B_c_Power):=cA**B;​​(A_c_Power)**B_:=cA**B;​​NormalForm[nonComm_]:=ExpandAll[nonComm//.b**bcbc**b+1]/.{NonCommutativeMultiplyTimes,bz,bc->zc};​​(*References:*)​​(*https://mathematica.stackexchange.com/a/240282*)​​(*JamesF.Feagin'sQuantumMethodswithMathematica15.1CommutatorAlgebra*)​​​​OrderFunction[x_]:=Join[Reverse[Select[x,#==br||#brc&]]/.{brcb,brbc},Select[x,#==bl||#blc&]/.{blb,blcbc}];​​Average[list_]:=NormalForm[NonCommutativeMultiply@@OrderFunction[list]];
Tests
In[]:=
VerificationTest[OrderFunction[{bl,br,blc,brc,bl,br,blc}],{bc,b,bc,b,bc,b,bc},TestID"OrderFunction"]//Print;​​VerificationTest[NormalForm[b**bc**b],z+
2
z
zc,TestID"NormalForm"]//Print;​​VerificationTestAverage[{bl,br,blc,brc,bl}],z+3
2
z
zc+
3
z
2
zc
,TestID"Average"//Print;
TestObject
Outcome: Success
Test ID:
OrderFunction

TestObject
Outcome: Success
Test ID:
NormalForm

TestObject
Outcome: Success
Test ID:
Average


Vectorized Lindbladian

CircleTimes[x_,y_]:=KroneckerProduct[x,y]​​
σ
+
=
0
1
0
0
;
σ
-
=
0
0
1
0
;II=IdentityMatrix[2];​​L[i_]:=-g
σ
+
⊗IIBl[i]-gc
σ
-
⊗IIBlc[i]+gcII⊗
σ
+
Br[i]+gII⊗
σ
-
Brc[i];​​​​VerificationTestL[1].L[2],
-ggcBl[1]Blc[2]-ggcBr[1]Brc[2]
0
0
ggcBl[2]Br[1]+ggcBl[1]Br[2]
0
-ggcBl[1]Blc[2]-ggcBr[2]Brc[1]
2
g
Bl[2]Brc[1]+
2
g
Bl[1]Brc[2]
0
0
2
gc
Blc[2]Br[1]+
2
gc
Blc[1]Br[2]
-ggcBl[2]Blc[1]-ggcBr[1]Brc[2]
0
ggcBlc[2]Brc[1]+ggcBlc[1]Brc[2]
0
0
-ggcBl[2]Blc[1]-ggcBr[2]Brc[1]
,TestID"L[i]"//Print;​​
TestObject
Outcome: Success
Test ID:
L[i]


Calculation of “moments”
(
O
i
)

O
0

In[]:=
O0=MatrixExp[(-ωA(
σ
+
.
σ
-
)⊗II+ωAII⊗(
σ
+
.
σ
-
))t];​​O0//MatrixForm//Print;
1
0
0
0
0
-tωA

0
0
0
0
tωA

0
0
0
0
1

O
1

In[]:=
O1=O0.Integrate[#,{
t
1
,0,t}]&@EvaluateExpandL[1]/.Bl[i_]
-
1
2
γ-Δω
t
i

bl[i],Br[i_]
-
1
2
γ+Δω
t
i

br[i],Blc[i_]
1
2
γ-Δω
t
i

blc[i]+
-
1
2
γ+Δω
t
i

(1-
γ
t
i

)br[i],Brc[i_]
1
2
γ+Δω
t
i

brc[i]+
-
1
2
γ-Δω
t
i

(1-
γ
t
i

)bl[i]/.x_[i_]Average[Normal@SparseArray[{ix}]];​​O1//MatrixForm
Precalculation of integrals
Precalculation of integrals
Precalculation of integrals
Precalculation all degenerate cases

Calculation of cumulants in terms of moments

Tests for lower cumulants
Calculation for higher cumulants
See full output here: https://www.wolframcloud.com/obj/37a69740-f2e9-4e24-aba9-2af78fc3a6b8

CITE THIS NOTEBOOK

Parametric approximation as an open-quantum-system problem​
by Artem Karasev & Alexander Teretenkov
Wolfram Community, STAFF PICKS, June 27, 2025
​https://community.wolfram.com/groups/-/m/t/3487051