Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
QuantumMob`Q3mini`
WickSimulate  |  |
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Details and Options
Examples
(3)
Basic Examples
(3)
Choose how many fermion modes to consider.
In[1]:=
$n=4;
Although unnecessary, consider a specific set of fermion modes for conversion between the Wick and conventional representations.
In[2]:=
Let[Fermion,c]cc=c[Range@$n]
Out[2]=
{,,,}
c
1
c
2
c
3
c
4
Choose a initial state with half filling.
In[3]:=
in=WickState[{1,0},$n]
Out[3]=
WickState
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In[4]:=
ExpressionFor[in,cc]//Chop
Out[4]=
1
c
1
0
c
2
1
c
3
0
c
4
Construct the single-particle Hamiltonian matrix.
In[5]:=
SeedRandom[373];
In[6]:=
ham=RandomWickHermitian[$n];ham//ArrayShort
Out[6]//MatrixForm=
0 | 0.175429 | -0.12261 | 0.0940521 | … |
-0.175429 | 0 | 0.835042 | -0.577971 | … |
0.12261 | -0.835042 | 0 | 1.18605 | … |
-0.0940521 | 0.577971 | -1.18605 | 0 | … |
… | … | … | … | … |
Take quantum jump operators.
In[7]:=
$N=2;jmp=RandomWickJump[$N,$n]
Out[7]=
WickJump
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In[8]:=
WickElements[jmp,cc]//TableForm
Out[8]//TableForm=
(-0.953177+2.02279) c 1 c 2 c 3 c 4 † c 1 † c 2 † c 3 † c 4 |
(-2.21583-0.00918327) c 1 c 2 c 3 c 4 † c 1 † c 2 † c 3 † c 4 |
Now, perform the Monte Carlo simulation.
In[9]:=
EchoTiming[data=WickSimulate[in,ham,jmp,{$T=0.5,dt=0.01},"Samples"100];]
⌚
7.40636
Check the size of the data.
In[10]:=
Dimensions[data]
Out[10]=
{100,51}
Check a few state in the first trajectory.
In[11]:=
data〚1,;;3〛
Out[11]=
WickState
,WickState
,WickState
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Let us now examine various entanglement properties. Take the first half of the fermion modes as a subsystem.
In[1]:=
kk=Range[$n/2]
Out[1]=
{1,2}
Calculate the entanglement entropy (EE) between the subsystem and the rest.
In[2]:=
EchoTiming[ee=WickEntanglementEntropy[data,kk]];
⌚
0.596549
Plot the entanglement entropy averaged over the quantum trajectories. Here, γ refers to the damping rate, which equals to 1 in our unit system.
In[3]:=
ListLinePlot[Mean[ee],DataRange{0,$T},PlotMarkers{Automatic,12},FrameLabel{"γt","〈EE (, /2)〉"}]
L
L
Out[3]=
Calculate the mutual information (MI) between the subsystem and the rest.
In[4]:=
EchoTiming[mi=WickMutualInformation[data,kk]];
⌚
2.20996
Plot the entanglement entropy averaged over the quantum trajectories. Here, γ refers to the damping rate, which equals to 1 in our unit system.
In[5]:=
ListLinePlot[Mean[Re@mi],DataRange{0,$T},PlotMarkers{Automatic,12},FrameLabel{"γt","〈MI (L, L/2)〉"}]
Out[5]=
Calculate the logarithmic entanglement negativity (LN) between the subsystem and the rest.
Plot the entanglement entropy averaged over the quantum trajectories. Here, γ refers to the damping rate, which equals to 1 in our unit system.
Finally, calculate the logarithmic entanglement negativity (LN) in the mixed states (density matrices).
Plot the entanglement entropy averaged over the quantum trajectories. Here, γ refers to the damping rate, which equals to 1 in our unit system.