Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
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 | sets size as the default truncation dimension , after called all the other definitions can use this default size |
Commutator | Commutator of two operators |
OperatorVariance | Variance of the operator op for a state ψ |
States
States
FockState n | n-th Fock state with truncation dimension size |
CoherentState CoherentState | parametric state representing a non-normalized coherent state in a truncated basis of dimension size |
ThermalState | thermal mixed state with average number of photons nbar in a truncated basis of dimension size |
CatState CatState | 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
bosonic annihilation operator in a truncated Fock space with specified size | |
phase space displacement operator of dimension size with complex displacement amplitude alpha | |
squeeze operator with complex parameter xi of dimension 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 , normal or anti normal ). 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.
Basic Examples
Basic Examples
Setting the space size
Setting the space size
Set the default size to 20:
In[4]:=
SetFockSpaceSize[20];
In[4]:=
AnnihilationOperator[]
Out[4]=
QuantumOperator
 |  |
|
Creation operator and Fock states
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[];
In[5]:=
†
a
Out[6]//TraditionalForm=
2
|2〉Coherent state
Coherent state
Symbolic coherent state:
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
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
Cat State
Create a cat state (|α〉+|-α〉 of amplitude and :
α
ϕ
)
α=2+
ϕ=3π/4
In[115]:=
CatState[25][2+,3π/4]
Out[115]=
QuantumState
|  |
|
Photon distribution of the same state:
In[122]:=
CatState[25][2+,3π/4]["ProbabilityPlot",LabelStyle6,AspectRatio1/4]
Out[122]=
Displacement operator
Displacement operator
Create displacement operator of size 40 with amplitude 3+ :
In[3]:=
DisplacementOperator[3+,40]
Out[3]=
QuantumOperator
| |||||||
Data not saved. Save now |
Create a coherent state displacing the vacuum:
In[4]:=
DisplacementOperator[α,40][]CoherentState[40][α]
Out[4]=
True
Squeeze operator
Squeeze operator
Create squeeze operator of default size with complex parameter 1+:
In[12]:=
SqueezeOperator[1.+]
Out[12]=
QuantumOperator
|  |
|
OperatorVariance
OperatorVariance
Calculate the variance of the quadrature operators for the ground state:
In[18]:=
OperatorVariance[FockState[0],#]&/@QuadratureOperators[]
Out[18]=
,
1
4
1
4
A Fock state has a definite number of photons so the variance of the number operator is zero:
In[5]:=
a=AnnihilationOperator[];OperatorVariance[FockState[2],@a]
†
a
Out[6]=
0
Commutator
Commutator
Compute the commutator of quadrature operators, it is 1 the error in the last term is expected behaviour due to truncation:
2
In[6]:=
2
 | QuantumOperator |
Out[6]//TraditionalForm=
10|19〉〈19|
Applications
Applications
Mean number of photons
Mean number of photons
Using QuantumMeasurementOperator:
Using the probabilities in the diagonal of the density matrix:
Heisenberg evolution of annihilation operator
Heisenberg evolution of annihilation operator
Options
Options
Displacement operator ordering
Displacement operator ordering
By default when "Ordering" is not specified normal ordering is used, the formulas of the three orders are shown below:
Squeeze operator ordering
Squeeze operator ordering
By default when "Ordering" is not specified normal ordering is used, the formulas of the three orders are shown below:
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.
Basic Examples
Basic Examples
Ground state Wigner Representation:
Ground state Wigner Representation:
Using a grid of phase space points from -4 to 4:
Fock state HusimiQ Representation
Fock state HusimiQ Representation
Applications
Applications
Interpolation of a Representation
Interpolation of a Representation
Phase space integral:
Squeezed vacuum Wigner Representation
Squeezed vacuum Wigner Representation
This exhibits the non classical behavior of squeezed states regarding the quadratures
Options
Options
Scaling of the representation for ground state
Scaling of the representation for ground state
Advanced examples
Decay of a coherent state
Decay of a coherent state
Hamiltonian definition:
State evolution:
Saving mean number of photons and fidelity respect to an analytical expression:
Plotting the results:
Optical balance
Optical balance
Considering the following master equation, we look for balance between a driving field Hamiltonian and the single photon loss mechanism:
Random initial state:
Hamiltonian definition associated with the master equation:
Jaynes-Cummings Model
Jaynes-Cummings Model
Going to the interaction picture at resonance, we get a time independent Hamiltonian:
Now we try different initial states
1. Rabi oscillations when the field is initially in a Fock state
2. Collapse and revivals when the initial field state is a coherent state
3. Atom "thermalization" (atom interacting with a thermal state field)
Atom gets "thermalized", time average of the coherences → 0:
Jaynes-Cummings with dissipation
Jaynes-Cummings with dissipation
Hamiltonian definition:
Jump operators:
Damped Rabi oscillations: