• To: mathgroup at smc.vnet.net
• Subject: [mg109714] need your help!
• From: Ben <berihut at gmail.com>
• Date: Thu, 13 May 2010 07:24:37 -0400 (EDT)

```Dear Sir,

My name is Berihu Teklu. Currently I'm doing a research about
propagation of entanglement in channel described by structured
reservoirs. I'm working with Mathematica (version 6.0) .
And now I have a problem, I try to plot an equation at the end  in the
attached mathematica file (which is in my case Entanglement of
Formation). And even after hours of working I don't get any plot.

This is thus to kindly request, if you could write me any comments on
my attached mathematica notebooks.

In[1]:= s[\[Omega]_, t_] =
FullSimplify[
Integrate[
Sin[\[Omega] \[Tau]] Sin[\[Omega]0 \[Tau]], {\[Tau], 0, t}]]

Out[1]= (\[Omega]0 Cos[t \[Omega]0] Sin[t \[Omega]] - \[Omega] Cos[
t \[Omega]] Sin[t \[Omega]0])/(\[Omega]^2 - \[Omega]0^2)

In[2]:= c[\[Omega]_, t_] =
FullSimplify[
Integrate[
Cos[\[Omega] \[Tau]] Cos[\[Omega]0 \[Tau]], {\[Tau], 0, t}]]
d[\[Omega]_, t_] =
FullSimplify[
Integrate[
Cos[\[Omega] \[Tau]] Sin[\[Omega]0 \[Tau]], {\[Tau], 0, t}]]

Out[2]= (\[Omega] Cos[t \[Omega]0] Sin[t \[Omega]] - \[Omega]0 Cos[
t \[Omega]] Sin[t \[Omega]0])/(\[Omega]^2 - \[Omega]0^2)

Out[3]= (-\[Omega]0 + \[Omega]0 Cos[t \[Omega]] Cos[
t \[Omega]0] + \[Omega] Sin[t \[Omega]] Sin[
t \[Omega]0])/(\[Omega]^2 - \[Omega]0^2)

Jpbg[\[Omega]_] = If[\[Omega] > 0, 1/Sqrt[\[Omega]], 0];

J1[\[Alpha]_, \[Omega]0_, t_] =
FullSimplify[
Integrate[ \[Alpha]^2/
Sqrt[\[Omega]]*(\[Omega] Cos[t \[Omega]0] Sin[
t \[Omega]] - \[Omega]0 Cos[t \[Omega]] Sin[
t \[Omega]0])/(\[Omega]^2 - \[Omega]0^2), {\[Omega],
0, \[Infinity]},
Assumptions -> {\[Alpha] > 0, \[Omega]0 > 0, t > 0}], {\[Alpha] >
0, \[Omega]0 > 0, \[Omega] > 0, t > 0}];

\[CapitalDelta][\[Alpha]_, \[Omega]0_, t_] :=
Re[Simplify[J1[\[Alpha], \[Omega]0, t]]];

In[7]:= \[CapitalDelta][0.1, 0.5, 0.1, 5.]

Out[7]= 0.199423

J2[\[Alpha]_, \[Omega]0_] =
FullSimplify[
Integrate[ \[Alpha]^2/
Sqrt[\[Omega]]*-\[Omega]0/(\[Omega]^2 - \[Omega]0^2), {\[Omega],
0, \[Infinity]},
Assumptions -> {\[Alpha] > 0, \[Omega]0 > 0, \[Lambda] >
0}], {\[Alpha] > 0, \[Omega]0 > 0}];
J3[\[Alpha]_, \[Omega]0_, t_] =
FullSimplify[
Integrate[ \[Alpha]^2/
Sqrt[\[Omega]]*(\[Omega]0 Cos[t \[Omega]] Cos[
t \[Omega]0])/(\[Omega]^2 - \[Omega]0^2), {\[Omega],
0, \[Infinity]},
Assumptions -> {\[Alpha] > 0, \[Omega]0 > 0, t > 0}], {\[Alpha] >
0, \[Omega]0 > 0, t > 0}];

During evaluation of In[8]:= Integrate::idiv: Integral of 1/((\
\[Lambda]^2+\[Omega]^2) (\[Omega]^2-\[Omega]0^2)) does not converge \
on {0,\[Infinity]}. >>

J4[\[Alpha]_, \[Omega]0_, t_] =
FullSimplify[
Integrate[ \[Alpha]^2/
Sqrt[\[Omega]]*(\[Omega] Sin[t \[Omega]] Sin[
t \[Omega]0])/(\[Omega]^2 - \[Omega]0^2), {\[Omega],
0, \[Infinity]},
Assumptions -> {\[Alpha] > 0, \[Omega]0 > 0, t > 0}], {\[Alpha] >
0, \[Omega]0 > 0, t > 0}];

\[CapitalPi][\[Alpha]_, \[Omega]0_, t_] :=
Re[Simplify[
J2[\[Alpha], \[Omega]0] + J3[\[Alpha], \[Omega]0, t] +
J4[\[Alpha], \[Omega]0, t]]];

In[12]:= \[CapitalPi][0.1, 0.5, 0.1, 5.]

Out[12]= 0.42693

J5[\[Alpha]_, \[Omega]0_, \[Lambda]_, t_] =
FullSimplify[
Integrate[ \[Alpha]^2/
Sqrt[\[Omega]]*(\[Omega]0 Cos[t \[Omega]0] Sin[
t \[Omega]] - \[Omega] Cos[t \[Omega]] Sin[
t \[Omega]0])/(\[Omega]^2 - \[Omega]0^2), {\[Omega],
0, \[Infinity]},
Assumptions -> {\[Alpha] > 0, \[Omega]0 > 0, \[Lambda] > 0,
t > 0}], {\[Alpha] > 0, \[Omega]0 > 0, \[Lambda] > 0, t > 0}];

\[Gamma][\[Alpha]_, \[Omega]0_, t_] :=
Re[Simplify[J5[\[Alpha], \[Omega]0, t]]];

In[15]:= \[Gamma][0.1, 0.5, 0.1, 5.]

Out[15]= 0.168173

\[CapitalGamma][\[Alpha]_, \[Omega]0_, t_] =
FullSimplify[
Integrate[2*\[Gamma][\[Alpha], \[Omega]0, s], {s, 0, t},
Assumptions -> {\[Alpha] > 0, \[Omega]0 > 0}], {\[Alpha] >
0, \[Omega]0 > 0}];

Subscript[\[CapitalDelta], \[CapitalGamma]][\[Alpha]_, \[Omega]0_,
t_] = FullSimplify[
Integrate[J1[\[Alpha], \[Omega]0, s], {s, 0, t},
Assumptions -> {\[Alpha] > 0, \[Omega]0 > 0}], {\[Alpha] >
0, \[Omega]0 > 0}];

In[18]:= Subscript[\[CapitalDelta], \[CapitalGamma]][0.1, 0.5, 0.1, \
5.]

Out[18]= 1.04639

\[CapitalDelta]co[\[Alpha]_, \[Omega]0_, t_] =
Re[FullSimplify[
Integrate[
J1[\[Alpha], \[Omega]0, s]*Cos[2 \[Omega]0 (t - s)], {s, 0, t},
Assumptions -> {\[Alpha] > 0, \[Omega]0 > 0, t > 0}], {\[Alpha] >
0, \[Omega]0 > 0, t > 0}]];

\[CapitalDelta]si[\[Alpha]_, \[Omega]0_, t_] =
Re[FullSimplify[
Integrate[
J1[\[Alpha], \[Omega]0, s]*Sin[2 \[Omega]0 (t - s)], {s, 0, t},
Assumptions -> {\[Alpha] > 0, \[Omega]0 > 0, \[Lambda] > 0,
t > 0}], {\[Alpha] > 0, \[Omega]0 > 0, t > 0}]];

\[CapitalPi]co[\[Alpha]_, \[Omega]0_, t_] =
Re[FullSimplify[
Integrate[(J2[\[Alpha], \[Omega]0, s] +
J3[\[Alpha], \[Omega]0, s] + J4[\[Alpha], \[Omega]0, s])*
Cos[2 \[Omega]0 (t - s)], {s, 0, t},
Assumptions -> {\[Alpha] > 0, \[Omega]0 > 0, t > 0}], {\[Alpha] >
0, \[Omega]0 > 0, \[Lambda] > 0, t > 0}]];

\[CapitalPi]si[\[Alpha]_, \[Omega]0_, t_] =
Re[FullSimplify[
Integrate[(J2[\[Alpha], \[Omega]0, s] +
J3[\[Alpha], \[Omega]0, s] + J4[\[Alpha], \[Omega]0, s])*
Sin[2 \[Omega]0 (t - s)], {s, 0, t},
Assumptions -> {\[Alpha] > 0, \[Omega]0 > 0, t > 0}], {\[Alpha] >
0, \[Omega]0 > 0, t > 0}]];

Out[22]= \$Aborted

\[CapitalDelta]co[0.1, 0.5, 0.1, 5.]

-0.198356 - 1.08996*10^-17 \[ImaginaryI]

\[CapitalDelta]si[0.1, 0.5, 0.1, 5.]

0.404226+ 0. \[ImaginaryI]

\[CapitalPi]co[0.1, 0.5, 0.1, 5.]

0.148022- 5.29091*10^-17 \[ImaginaryI]

\[CapitalPi]si[0.1, 0.5, 0.1, 5.]

0.54091- 5.42101*10^-17 \[ImaginaryI]

Clear[A0]

A0[r_] = {{Cosh[2 r], 0}, {0, Cosh[2 r]}};
B0 = {{b, 0}, {0, b}};

Clear[At]

At[\[Alpha]_, \[Omega]0_, t_, r_] :=
Re[A0[r] + {{Subscript[\[CapitalDelta], \[CapitalGamma]][\[Alpha], \
\[Omega]0,
t] + (\[CapitalDelta]co[\[Alpha], \[Omega]0,
t] - \[CapitalDelta]si[\[Alpha], \[Omega]0,
t]), -(\[CapitalDelta]si[\[Alpha], \[Omega]0,
t] - \[CapitalPi]co[\[Alpha], \[Omega]0,
t])}, {-(\[CapitalDelta]si[\[Alpha], \[Omega]0,
t] - \[CapitalPi]co[\[Alpha], \[Omega]0, t]),
Subscript[\[CapitalDelta], \[CapitalGamma]][\[Alpha], \[Omega]0,
t] - (\[CapitalDelta]co[\[Alpha], \[Omega]0,
t] - \[CapitalDelta]si[\[Alpha], \[Omega]0, t])}}];

MatrixForm[ArrayFlatten[At[0.1, 0.5, 0.1, 5., 1.]]]

\!\(\*
TagBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{"4.206008542870848`",
RowBox[{"-", "0.25620413800854397`"}]},
{
RowBox[{"-", "0.25620413800854397`"}], "5.411172395695137`"}
},
GridBoxAlignment->{
"Columns" -> {{Left}}, "ColumnsIndexed" -> {},
"Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.7]},
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset[0.2], {
Offset[0.4]},
Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
Function[BoxForm`e\$,
MatrixForm[BoxForm`e\$]]]\)

Ct[\[Omega]0_, t_, r_] =
ArrayFlatten[{{Sinh[2 r]/2  Cos [2 \[Omega]0 t],
Sinh[2 r]/2 Sin [2 \[Omega]0 t]}, {Sinh[2 r]/2  Cos [
2 \[Omega]0 t], -Sinh[2 r]/2  Cos [2 \[Omega]0 t]}}];

MatrixForm[ArrayFlatten[Ct[0.1, 5., 1.]]]

\[Sigma]t[\[Alpha]_, \[Omega]0_, t_, r_] =
ArrayFlatten[{At[\[Alpha], \[Omega]0, t, r],
Ct[\[Omega]0, t, r]}, {Transpose[Ct[\[Omega]0, t, r]],
At[\[Alpha], \[Omega]0, t, r]}];
MatrixForm[ArrayFlatten[\[Sigma]t[0.1, 0.5, 5., 1.]]]

Subscript[I, 1] =
Simplify[Det[
At[\[Alpha], \[Omega]0, t, r]], {\[Alpha] > 0, \[Omega]0 > 0,
t > 0, r > 0}]

Subscript[I, 3] =
Simplify[Det[Ct[\[Omega]0, t, r]], {\[Omega]0 > 0, t > 0, r > 0}]

Subscript[I, 4] =
Simplify[Det[\[Sigma]t[\[Alpha], \[Omega]0, t, r]], {\[Alpha] >
0, \[Omega]0 > 0, t > 0, r > 0}];

Subscript[C, +] =
Simplify[Sqrt[(
Subscript[I, 1]^2 + Subscript[I, 3]^2 - Subscript[I, 4] +
Sqrt[(Subscript[I, 1]^2 + Subscript[I, 3]^2 - Subscript[I,
4])^2 - (2 Subscript[I, 1]*Subscript[I, 3])^2        ])/(
2 Subscript[I, 1])]];
Subscript[C, -] =
Simplify[Sqrt[(
Subscript[I, 1]^2 + Subscript[I, 3]^2 - Subscript[I, 4] -
Sqrt[(Subscript[I, 1]^2 + Subscript[I, 3]^2 - Subscript[I,
4])^2 - (2 Subscript[I, 1]*Subscript[I, 3])^2        ])/(
2 Subscript[I, 1])]];

Subscript[a, n] = Simplify[Sqrt[Subscript[I, 1]]];

Subscript[\[Kappa], -] =
Simplify[
Sqrt[(Subscript[a, n] - Subscript[C, +])*(Subscript[a, n] -
Subscript[C, -])]];

Subscript[x, m] =
Simplify[(Subscript[\[Kappa], -]^2 + 1/4)/(
2 Subscript[\[Kappa], -])];

Subscript[E, F][\[Alpha]_, \[Omega]0_, \[Lambda]_, t_, r_] :=
Re[(Subscript[x, m] + 1/2) ln [
Subscript[x, m] + 1/2] - (Subscript[x, m] - 1/2)  ln [
Subscript[x, m] - 1/2]];

Subscript[E, F][0.1, 0.5, 5., 1.]

Plot[Subscript[E, F][0.1, 0.5, t, 1.],
{t, 0, 5}]

```

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