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Second Quantization Functions
In this Tech Note, we present the implementation of second quantization for bosons within the QuantumFramework, utilizing a truncated Fock space. This approach enables simple manipulation and representation of quantum states, which are essential for a wide range of calculations. By leveraging the core features of the QuantumFramework, we simplify the process of working with common quantum states and operators, facilitating diverse computational tasks in fundamental quantum mechanics and quantum optics.
In[1]:=
<<Wolfram`QuantumFramework`<<Wolfram`QuantumFramework`SecondQuantization`
Core Functions
Core Functions
Fock space size and utilities
Fock space size and utilities
SetFockSpaceSize[size] | sets size as the default truncation dimension , after called all the other definitions can use this default size |
Commutator[op1,op2] | Commutator of two operators |
OperatorVariance[ψ,op] | Variance of the operator op for a state ψ |
States
States
FockState[ n | n-th Fock state with truncation dimension size |
CoherentState[size]CoherentState[size][α] | parametric state representing a non-normalized coherent state in a truncated basis of dimension size |
ThermalState[nbar,size] | thermal mixed state with average number of photons nbar in a truncated basis of dimension size |
CatState[size]CatState[size][α,ϕ] | parametric general cat state representing a superposition of coherent states in a basis of dimension size, the complex amplitude and the phase can be supplied |
Operators
Operators
AnnihilationOperator[size] | bosonic annihilation operator in a truncated Fock space with specified size |
DisplacementOperator[α,size] | phase space displacement operator of dimension size with complex displacement amplitude alpha |
SqueezeOperator[ξ,size] | squeeze operator with complex parameter xi of dimension size |
QuadratureOperators[size] | returns X 1 X 2 |
We will briefly demonstrate how to apply all these functions in Wolfram quantum framework.
Ordering Option
Ordering Option
Displacement and squeeze operators can have different definitions based on operator ordering. We include the "Ordering" option to specify the order (weak ordering, normal ordering or anti normal ordering). Theoretically in a infinite dimensional space they are all equivalent , but in a truncated space these operators have numerical error, and depending on the calculation some choice of ordering can be more accurate.
DisplacementOperator[..,"Ordering"->] | "NormalOrdering" Exp[-|α 2 | † a Exp[α † a Exp[|α 2 | † a |
SqueezeOperator[..,"Ordering"->] | With ξ= θ r Exp[- 1 2 θ †2 a † a 1 2 1 2 -θ 2 a Exp[ 1 2 2 a †2 a Exp[ 1 2 -θ 2 a † a 1 2 1 2 θ †2 a |
We will briefly demonstrate how to apply this option in Wolfram quantum framework.
Examples
Examples
Set the default size to 20:
In[4]:=
SetFockSpaceSize[20];
In[4]:=
AnnihilationOperator[]
Out[4]=
QuantumOperator
 |  |
|
Creation operator and Fock states
Define creation operator just taking the adjoint of the annihilation operator with "Dagger" property or SuperDagger:
In[5]:=
a=AnnihilationOperator[];@FockState[1]//TraditionalForm
†
a
Out[6]//TraditionalForm=
2
|2〉Coherent state
Create a symbolic coherent state of default size:
In[7]:=
CoherentState[][β]//TraditionalForm
Out[7]//TraditionalForm=
|0〉+β|1〉+|2〉+|3〉+|4〉+|5〉+|6〉+|7〉+|8〉+|9〉+|10〉+|11〉+|12〉
2
β
2
3
β
6
4
β
2
6
5
β
2
30
6
β
12
5
7
β
12
35
8
β
24
70
9
β
72
70
10
β
720
7
11
β
720
77
12
β
1440
231
+|13〉+|14〉+|15〉+|16〉+|17〉+|18〉+|19〉
13
β
1440
3003
14
β
10080
858
15
β
30240
1430
16
β
120960
1430
17
β
120960
24310
18
β
725760
12155
19
β
725760
230945
Visualize photon distribution of a coherent state with :
α=1.3+2
In[120]:=
CoherentState[25][1.3+2]["Normalize"]["ProbabilityPlot",LabelStyle6,AspectRatio1/4]
Out[120]=
Cat State
Create a cat state of amplitude and :
|α〉+|-α〉
ϕ
α=2+
ϕ=3π/4
In[115]:=
CatState[25][2+,3π/4]
Out[115]=
QuantumState
|  |
|
In[122]:=
CatState[25][2+,3π/4]["ProbabilityPlot",LabelStyle6,AspectRatio1/4]
Out[122]=
Displacement operator
Create displacement operator of size 40 with amplitude 3+ :
In[134]:=
α=3+;
In[132]:=
DisplacementOperator[α,40]
Out[132]=
QuantumOperator
| |||||||
Data not saved. Save now |
In[133]:=
DisplacementOperator[α,40][]CoherentState[40][α]
Out[133]=
True
Ordering
Verify displacement operator is unitary using "UnitaryQ" for , for this task testing with weak ordering is more accurate
α=
In[3]:=
DisplacementOperator[1.]["UnitaryQ"]
Out[3]=
False
In[4]:=
DisplacementOperator[1.,"Ordering""WeakOrdering"]["UnitaryQ"]
Out[4]=
True
Squeeze operator
Create squeeze operator of default size with complex parameter 1+:
In[12]:=
SqueezeOperator[1.+]
Out[12]=
QuantumOperator
|  |
|
OperatorVariance
Calculate the variance of the quadrature operators for the ground state:
Commutator
Quasi-probability Representations
Quasi-probability Representations
Two phase space quasi-probability representations are included Wigner and HusimiQ , which are used extensively in quantum mechanics and quantum optics. These are calculated numerically in a grid of phase space points. Glauber representation is not included because for a general state is not feasible to implement a numerical algorithm , due to the singularities of this representation.
We will briefly demonstrate how to apply all these functions in Wolfram quantum framework.
Examples
Examples
Wigner representation of the ground state
HusimiQ representation of Fock state n=4