Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Wolfram`QuantumFramework`
QuantumEvolve  |
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Details and Options
Examples
(10)
Basic Examples
(4)
Evolve a symbolic quantum state by a time-independent Hamiltonian, with independent variable :
t
In[1]:=
ψf=
["X"],
[{α,β}],t
 | QuantumEvolve |
 | QuantumOperator |
| QuantumState |
Out[1]=
QuantumState
 |  |
|
Start with register state, without specifying any independent variable (note will be treated as formal parameter):
t
In[2]:=
 | QuantumEvolve |
| QuantumOperator |
Out[2]=
cos(t.)|0〉-sin(t.)|1〉
In[3]:=
U=
["X"],None,t
| QuantumEvolve |
| QuantumOperator |
Out[3]=
Cos[t]|0〉〈0|-Sin[t]|0〉〈1|-Sin[t]|1〉〈0|+Cos[t]|1〉〈1|
Evolve by above unitary:
|0〉
In[4]:=
U[]
Out[4]=
cos(t)|0〉-sin(t)|1〉
Define a time-dependent Hamiltonian:
In[1]:=
h=Cos[α]
["Z"]+Sin[α]Cos[ωt]
["X"]+Sin[ωt]
["Y"]
ω
1
2
| QuantumOperator |
ω
1
2
 | QuantumOperator |
| QuantumOperator |
Out[1]=
1
2
ω
1
1
2
ω
1
1
2
ω
1
1
2
ω
1
Define a symbolic initial state:
In[2]:=
ψi=
[{Cos[α/2],Sin[α/2]}]
| QuantumState |
Out[2]=
QuantumState
|  |
|
Evolve the initial state with the Hamiltonian:
In[3]:=
ψf=
[h,ψi,t];
 | QuantumEvolve |
Show the final state vector, with :
λ=
-+2ωCos[α]-
2
ω
ω
1
2
ω
1
In[4]:=
FullSimplify[Normal[ψf["StateVector"]]]/.
_
λ,1_
1λOut[4]=
CosCosh+,SinCosh-
-tω
1
2
α
2
tλ
2
Sinh(ω-)
tλ
2
ω
1
λ
tω
2
α
2
tλ
2
Sinh(ω+)
tλ
2
ω
1
λ
Define a time-dependent Hamiltonian:
In[1]:=
hamiltonian=ω0
["JZ"]+2ωpCos[ω0t]
["JX"];
 | QuantumOperator |
| QuantumOperator |
Set numerical values for Hamiltonian parameters:
In[2]:=
ω0=2π;ωp=π/10;tf=2π/ωp;
Find the final state :
In[3]:=
r1=
[hamiltonian,{t,0,tf}];
| QuantumEvolve |
Plot the evolution of Bloch vector components:
In[4]:=
Plot[Evaluate@Re@r1[t]["BlochVector"],{t,0,tf},PlotRangeAll,PlotLegends{"","",""}]
r
x
r
y
r
z
Out[4]=
Include relaxation dynamics with raising and lowering spin angular momentum operators, in a Lindblad dynamical equation:
In[5]:=
γ=1/20;r2=
hamiltonian,
["J+"],
["J-"],{t,0,2tf}
| QuantumEvolve |
γ
| QuantumOperator |
| QuantumOperator |
Out[5]=
QuantumState
|  |
|
Plot the evolution of Bloch vector components:
In[6]:=
Plot[Evaluate@Re@r2[t]["BlochVector"],{t,0,2tf},PlotRangeAll,PlotLegends{"","",""}]
r
x
r
y
r
z
Out[6]=
T
1
T
2
In[1]:=
Ls=
["J+"],
["J-"],
["JZ"];
1
T1
| QuantumOperator |
1
T1
| QuantumOperator |
1
T2
| QuantumOperator |
Show the density matrix:
In[2]:=
$Assumptions=T1>0&&T2>0&&Ω0>0;
In[3]:=
FullSimplify/@Normal
Ω0
["JZ"],Ls,
[Array[ρ,{2,2}]],t["Computational"]["DensityMatrix"]//MatrixForm
| QuantumEvolve |
| QuantumOperator |
| QuantumState |
Out[3]//MatrixForm=
1 2 - 2t T1 | t- 1 T1 1 2T2 |
t- 1 T1 1 2T2 | 1 2 - 2t T1 |