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Baker-Hausdorff Lemma
Expands expr | |
Represents the non-commutative multiplication, and also denoted a**b**… |
Functions to be used in the demonstration of the lemma.
In[81]:=
Needs["QuantumMob`Q3`"]
Bosons
In[82]:=
Let[Real,ϕ]
Consider two species of Bosons.
In[83]:=
Let[Boson,a,b]
Let us start with a so-called single-model squeezing operator, U:
In[84]:=
A=(z*a[1]**a[1]-Conjugate[z]*Dagger[a[1]**a[1]])/2U=Exp[A]
Out[84]=
1
2
a
1
a
1
*
z
†
a
1
†
a
1
Out[85]=
1
2
a
1
a
1
*
z
†
a
1
†
a
1
In the following, Multiply is already smart enough to expand the first two cases. However, the last example of almost the same complexity is not properly manipulated.
In[86]:=
ex1=U**a[1]**Dagger[U]ex2=U**Dagger[a[1]]**Dagger[U]ex3=U**Dagger[a[1]]**a[1]**Dagger[U]
Out[86]=
Out[87]=
Out[88]=
In[89]:=
ex4=Elaborate[ex3]//ExpToTrig//Simplifyex4a=ex4//CauchySimplify
Out[89]=
1
2
z
*
z
z
*
z
z
*
z
z
*
z
†
a
1
a
1
a
1
a
1
z
*
z
*
z
†
a
1
†
a
1
z
*
z
Out[90]=
Cosh[2z]++
†
a
1
a
1
Cosh[z]z+Sinh[z]
a
1
a
1
*
z
†
a
1
†
a
1
z
2
Sinh[z]
In[91]:=
ex5=LieExp[A,Q[a[1]]]//CauchySimplifyex4a-ex5//CauchySimplify
Out[91]=
-2z
2
(-1+)
2z
4z
†
a
1
a
1
4z
a
1
a
1
*
z
†
a
1
†
a
1
4z
Out[92]=
0
Let us check the above result. It should be equal to the following
In[93]:=
ex1a=Elaborate[ex1]ex2a=Elaborate[ex2]ex5=ex2a**ex1a//ExpToTrig//Simplifyex5a=ex5//CauchySimplifyex6=LieExp[A,Q@a[1]]//CauchySimplify
Out[93]=
1
2
-
z
*
z
2
z
*
z
a
1
-
z
*
z
2
z
*
z
*
z
†
a
1
2
z
Out[94]=
-
z
*
z
2
z
*
z
z
a
1
2
*
z
1
2
-
z
*
z
2
z
*
z
†
a
1
Out[95]=
1
2
z
*
z
z
*
z
z
*
z
z
*
z
†
a
1
a
1
a
1
a
1
z
*
z
*
z
†
a
1
†
a
1
z
*
z
Out[96]=
Cosh[2z]++
†
a
1
a
1
Cosh[z]z+Sinh[z]
a
1
a
1
*
z
†
a
1
†
a
1
z
2
Sinh[z]
Out[97]=
-2z
2
(-1+)
2z
4z
†
a
1
a
1
4z
a
1
a
1
*
z
†
a
1
†
a
1
4z
In[98]:=
ex4a-ex5a//Elaborate[#]&//CauchySimplifyex5a-ex6//CauchySimplify
Out[98]=
0
Out[99]=
0
Fermions
Here are some more examples:
In[100]:=
b3=I*ϕ*(Dagger[a[1]]**a[1]+2Dagger[a[1]]**a[2]+2Dagger[a[2]**a[1]])tmp1=Exp[b3]**Dagger[a[1]]**a[2]**Exp[-b3]tmp2=Exp[b3]**Dagger[a[3]]**a[2]**Exp[-b3]tmp3=Exp[b3]**Dagger[a[2]]**a[2]**Exp[-b3]
Out[100]=
ϕ+2+2
†
a
1
a
1
†
a
1
a
2
†
a
1
†
a
2
Out[101]=
Out[102]=
Out[103]=
In[104]:=
Elaborate[tmp1]Elaborate[tmp2]Elaborate[tmp3]
Out[104]=
ϕ
†
a
1
a
2
ϕ
ϕ
†
a
1
†
a
1
Out[105]=
(2-2)+
ϕ
†
a
1
†
a
3
†
a
3
a
2
Out[106]=
2(-1+)-4+(2-2)+
ϕ
†
a
1
a
2
2
(-1+)
ϕ
†
a
1
†
a
1
ϕ
†
a
1
†
a
2
†
a
2
a
2
More Examples Related to Spins
In[107]:=
Let[Boson,a,b]Let[Fermion,c,d]Let[Complex,z]Let[Grassmann,g]
In[111]:=
Let[Real,ϕ]
In[116]:=
Sx=FockSpin[c,1];Ux=Exp[I*ϕ*Sx];ex1=Ux**Dagger[c[Up]]**c[Up]**Dagger[Ux]ex2=Ux**c[Up]**Dagger[Ux]
Out[118]=
Out[119]=
ϕ+
1
2
†
c
↓
c
↑
1
2
†
c
↑
c
↓
c
↑
-ϕ+
1
2
†
c
↓
c
↑
1
2
†
c
↑
c
↓
In[120]:=
ex1a=Elaborate[ex1]//Simplify
Out[120]=
2
Cos
ϕ
2
†
c
↑
c
↑
†
c
↓
c
↓
2
Sin
ϕ
2
1
2
†
c
↓
c
↑
†
c
↑
c
↓
If the above result is correct, then it should be equal to Dagger[ex2]**ex2:
More Examples with Bilinear Expressions
Bosonic Bilinear Expressions
Bosonic Bilinear Expressions
Fermionic Bilinear Expressions
Fermionic Bilinear Expressions
Bilinear Expression in Majorana Fermions
Bilinear Expression in Majorana Fermions