Ellipse errors

In[]:=
Manipulate​​Module​​{dx=1/4Exp[-2r],dy=1/4Exp[2r]},​​GraphicsEdgeForm[Thick],Green,​​Rotate[Disk[{0,0},{dx,dy}],θ1/2],​​Black,Dashed,​​Line[Outer[Times,{-dx,dx},{Cos[θ1/2],Sin[θ1/2]}]],​​Line[Outer[Times,{-dy,dy},{-Sin[θ1/2],Cos[θ1/2]}]],​​TextFramed
2
(ΔX1)
==14(
2
Cosh[r]
+
2
Sinh[r]
-2Sinh[r]Cosh[r]Cos[θ1]),​​FrameStyle->Directive[Red,Thick,Dashing[0,0,0]],{0.8,0.85},TextFramed
2
(ΔX2)
==14(
2
Cosh[r]
+
2
Sinh[r]
+2Sinh[r]Cosh[r]Cos[θ1]),​​FrameStyle->Directive[Red,Thick,Dashing[0,0,0]],{0.8,0.55},TextFramed
2
(ΔX1)
2
(ΔX2)
==116(
2
(
2
Cosh[r]
+
2
Sinh[r]
)
-
2
(2Sinh[r]Cosh[r]Cos[θ1])
),FrameStyle->Directive[Red,Thick,Dashing[0,0,0]],{0.75,-0.75}​​,​​Axes->True,​​ImageSize->Medium,​​AxesLabel->{"X1","X2"},​​PlotRange->{{-1,1},{-1,1}},​​PlotLabel->"Quadratures of a squeezed vacuum state |ξ〉"​​,{{r,0,"Modulus r"},0,1},{{θ1,0,"Argument θ"},0,2π},SaveDefinitions->True
Out[]=
​
Modulus r
Argument θ

Poll Question #2: Single photon interference (non balanced BS)

Actually this one was not worded properly, so i apologize if it was confusing. If BOTH beam splitter are changed to non balanced, the probabilities will never reach zero BUT only at one of the output ports. The case i was referring to is when only one of the beam-splitter is changed to non balanced.

Both non balanced

In[]:=
SetFockSpaceSize[3];
In[]:=
a5=a;​​a6=QuantumOperator[a,{2}];
We now use a dielectric beam splitter (non balanced):
In[]:=
nonbalancedBS=GeneralBSArcCos[
3/10
],0,0,0;
Compute the operator of the arrangement:
In[]:=
MZ=nonbalancedBS@QuantumOperator["P"[ϕ]]@nonbalancedBS;
Relating the final and initial operators (Heisenberg-like evolution):
In[]:=
{a1,a2}=Inverse@MZ["Matrix"].{a5,a6};
In[]:=
probs=(a1["Dagger"]@FockState[{0,0}])["Probability"]
Out[]=
01
2Im[ϕ]

2
Abs-
21
10
+
1
10
21
-Conjugate[ϕ]


2Im[ϕ]

2
Abs
3
10
+
7
10
-Conjugate[ϕ]


+
2Im[ϕ]

2
Abs-
21
10
+
1
10
21
-Conjugate[ϕ]


,10
2Im[ϕ]

2
Abs
3
10
+
7
10
-Conjugate[ϕ]


2Im[ϕ]

2
Abs
3
10
+
7
10
-Conjugate[ϕ]


+
2Im[ϕ]

2
Abs-
21
10
+
1
10
21
-Conjugate[ϕ]



In[]:=
simplifiedProbs=FullSimplify[probs//Values,Element[ϕ,Reals]]//ComplexExpand//TrigReduce
Out[]=
-
21
50
(-1+Cos[ϕ]),
1
50
(29+21Cos[ϕ])
In[]:=
Plot[Evaluate@MapThread[Callout[#1,#2,Right]&,{simplifiedProbs,{"D1","D2"}}],{ϕ,0,6Pi},​​AxesLabel->{"ϕ","Probability"},​​PlotLegends->{"Output 5","Output 6"},PlotRange->{0,1}]
Out[]=
Output 5
Output 6
In[]:=
Total[simplifiedProbs]//Simplify
Out[]=
1

Only one non balanced

In[]:=
balancedBS=GeneralBS[π/4,0,0,0];​​MZ=nonbalancedBS@QuantumOperator["P"[ϕ]]@balancedBS;​​{a1,a2}=Inverse@MZ["Matrix"].{a5,a6};
In[]:=
probs=(a1["Dagger"]@FockState[{0,0}])["Probability"]
Out[]=
01
2Im[ϕ]

2
Abs-
7
5
2
+
1
2
3
5
-Conjugate[ϕ]


2Im[ϕ]

2
Abs-
7
5
2
+
1
2
3
5
-Conjugate[ϕ]


+
2Im[ϕ]

2
Abs
3
5
2
+
1
2
7
5
-Conjugate[ϕ]


,10
2Im[ϕ]

2
Abs
3
5
2
+
1
2
7
5
-Conjugate[ϕ]


2Im[ϕ]

2
Abs-
7
5
2
+
1
2
3
5
-Conjugate[ϕ]


+
2Im[ϕ]

2
Abs
3
5
2
+
1
2
7
5
-Conjugate[ϕ]



In[]:=
simplifiedProbs=FullSimplify[probs//Values,Element[ϕ,Reals]]//ComplexExpand//TrigReduce
Out[]=

1
10
5-
21
Cos[ϕ],
1
10
5+
21
Cos[ϕ]
In[]:=
Plot[Evaluate@MapThread[Callout[#1,#2,Right]&,{simplifiedProbs,{"D1","D2"}}],{ϕ,0,6Pi},​​AxesLabel->{"ϕ","Probability"},​​PlotLegends->{"Output 5","Output 6"},PlotRange->{0,1}]
Out[]=
Output 5
Output 6
For this case the minimum is
1
10
(5
-
21
) that is not zero, that will change depending on how asymmetric is the non balanced beam-splitter.
In[]:=
Minimize[#,ϕ][[1]]&/@simplifiedProbs

Comments and Questions

Question:

How are quasiprobability functions calculated? Using the fast Fourier transform or with the series given as an exercise in Peter Knight’s book?”

Answer:

I answered during the class, but i wanted to say it more explicitly. The Wigner function is calculated using Clenshaw recurrence to evaluate Laguerre polynomials in a faster way. The Husimi Q function does not use any particular algorithm, it is just based on the definition shown on the class but with some numerical improvements regarding speed.