Flow-equation approach to quantum systems driven by an amplitude-modulated time-periodic force
Author
Viktor Novičenko
Title
Flow-equation approach to quantum systems driven by an amplitude-modulated time-periodic force
Description
Notebook realize expansion of the Hamiltonian in terms of the inverse frequency
Category
Academic Articles & Supplements
Keywords
high frequency expansions, periodic hamiltonians, perturbation theory
URL
http://www.notebookarchive.org/2022-01-biomdb4/
DOI
https://notebookarchive.org/2022-01-biomdb4
Date Added
2022-01-25
Date Last Modified
2022-01-25
File Size
129.01 kilobytes
Supplements
Rights
Redistribution rights reserved
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This file contains supplementary data for V. Novičenko, G. Žlabys and E. Anisimovas, “Flow-equation approach to quantum systems driven by an amplitude-modulated time-periodic force,” Phys. Rev. A 105, 012203 (2022). https://doi.org/10.1103/PhysRevA.105.012203
Flow-equation approach to quantum systems driven by an amplitude-modulated time-periodic force
Flow-equation approach to quantum systems driven by an amplitude-modulated time-periodic force
Viktor Novičenko
In[]:=
calculating order:1
calculating order:2
calculating order:3
calculating order:4
calculating order:5
Simplifying...
Solutions:
h
1
g(t)(cos(ϕ)(6(t)-7g(t)(t)+7+8)+4Δg(t)sin(ϕ)(t))
2
′
g
′′
g
2
Δ
2
g(t)
4
g(t)
′
g
64
4
ω
1
256
5
ω
′
g
2
′
g
′′
g
2
Δ
2
g(t)
4
g(t)
2
g(t)
(3)
g
2
′
g
′′
g
2
Δ
2
g(t)
4
g(t)
2
g(t)
′
g
16
3
ω
3
g(t)
4
2
ω
h
2
g(t)(4Δg(t)cos(ϕ)(t)-sin(ϕ)(6(t)-7g(t)(t)+7+8))
′
g
2
′
g
′′
g
2
Δ
2
g(t)
4
g(t)
64
4
ω
1
256
5
ω
2
′
g
′′
g
2
Δ
2
g(t)
4
g(t)
2
g(t)
(3)
g
′
g
2
′
g
′′
g
2
Δ
2
g(t)
4
g(t)
2
g(t)
′
g
16
3
ω
3
g(t)
4
2
ω
h
3
Δ
2
2
′
g
′′
g
2
Δ
2
g(t)
16
3
ω
Δ(-3(t)+3g(t)(t)-2+)
2
′
g
′′
g
2
Δ
2
g(t)
4
g(t)
32
4
ω
1
256
5
ω
2
g(t)
4
Δ
2
′
g
2
Δ
2
′
g
(4)
g
2
Δ
′′
g
(3)
g
′
g
3
g(t)
′′
g
2
′′
g
2
Δ
4
g(t)
6
g(t)
Δ
2
g(t)
4
2
ω
2
g(t)
2ω
In[]:=
(*Runthiscelltoobtaindtheexpansionforthemicromotion,thatistheoperatorSsuchthatU=exp(-iS)asinEq.(25).Theobtainedcoefficientsd_jmultiplythegeneratorsG_j.*)resArray=ArrayReshape[results,{fourierCompNo+1,basisNo}];resArray=Expand[ϵ*resArray/.ω1/ϵ]/.(ϵ^pow_/;pow>magnusTermNo)0;Do[fH[i]=resArray[[i+1]];,{i,0,fourierCompNo}]Fix=Now;A=Sum[(fH[n]*Exp[I*n*ω*t]-ComplexExpand[Conjugate[fH[n]],complexParameters]*Exp[-I*n*ω*t]),{n,1,fourierCompNo}];Print["Now we have the generator ... "];Print["doing 1 ... ",TimeObject[Fix]];Print["Integrating ... ",Now-Fix];Ω[1]=Integrate[A,{s,0,s1}]/.s1s;magnusRes=Ω[1];Do[Print["doing ",n," ... ",Now-Fix];S[n,1]=Expand[Comm[Ω[n-1],A]]/.(ϵ^pow_/;pow>magnusTermNo)0;Do[S[n,j]=Expand[Sum[Comm[Ω[m],S[n-m,j-1]],{m,1,n-j}]]/.(ϵ^pow_/;pow>magnusTermNo)0;,{j,2,n-1}];Print["Integrating ... ",Now-Fix];Ω[n]=Sum[BernoulliB[j]/j!*Integrate[S[n,j],{s,0,s1}],{j,1,n-1}]/.s1s;magnusRes=magnusRes+Ω[n];,{n,2,magnusTermNo}]Print["proceeding to limits ... ",Now-Fix];(*cutoffhigherorders*)magnusLim=Limit[magnusRes,s+Infinity];Print["collecting ... ",Now-Fix];resMagnusCol=Collect[magnusLim*(-I),ϵ,FullSimplify]/.ϵ1/ω;Do[Print[TraditionalForm[i]],{i,resMagnusCol}];Print["All done ... ",Now-Fix];
S
Position[resMagnusCol,i][[1,1]]
Now we have the generator ...
doing 1 ...
16:44:08GMT+3.
Integrating ...
0.159036
s
doing 2 ...
2.90144
min
Integrating ...
3.46966
min
doing 3 ...
14.0077
min
Integrating ...
17.3104
min
doing 4 ...
33.6349
min
Integrating ...
40.5531
min
doing 5 ...
40.7784
min
Integrating ...
47.6657
min
proceeding to limits ...
47.6694
min
collecting ...
53.0224
min
S
1
′
g
′′
g
2
Δ
3
g(t)
3
ω
′
g
′′
g
2
g(t)
′
g
(3)
g
3
Δ
3
g(t)
4
ω
3
Δ
′
g
2
Δ
′′
g
(3)
g
(4)
g
4
Δ
2
′
g
2
g(t)
′
g
′′
g
2
Δ
3
g(t)
5
g(t)
5
ω
′
g
2
ω
S
2
′
g
′′
g
2
Δ
3
g(t)
3
ω
2
Δ
′
g
′′
g
(3)
g
2
g(t)
′
g
3
Δ
3
g(t)
4
ω
3
Δ
′
g
2
Δ
′′
g
(3)
g
(4)
g
4
Δ
2
′
g
2
g(t)
′
g
′′
g
2
Δ
3
g(t)
5
g(t)
5
ω
′
g
2
ω
S
3
′
g
′′
g
2
′
g
2
Δ
2
g(t)
4
g(t)
4
ω
′
g
′′
g
(3)
g
′
g
′
g
′′
g
3
g(t)
′
g
3
Δ
2
g(t)
4
g(t)
5
ω
′
g
3
ω
2
g(t)
2
ω
All done ...
53.271
min
Cite this as: Viktor Novičenko, "Flow-equation approach to quantum systems driven by an amplitude-modulated time-periodic force" from the Notebook Archive (2022), https://notebookarchive.org/2022-01-biomdb4
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