Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Second Quantization Functions
Details and options | |
Quasi Probability Representations and Examples | |
Advanced Examples | |
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.
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<<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 |
OperatorVariance | Variance of the operator op for a state ψ |
States
States
FockState n | n-th Fock state in a basis of dimension size |
FockState n 1 n 2 | Multimode Fock state with occupation numbers n i |
CoherentState CoherentState | Parametric state representing a non-normalized coherent state in a basis of dimension size , the complex amplitude is a parameter |
ThermalState | Thermal mixed state with average number of photons nbar in a 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 are parameters |
Operators
Operators
AnnihilationOperator | Bosonic annihilation operator in a truncated Fock spacewith basis dimension size and qudit order ord |
DisplacementOperator | Phase space displacement operator of a single mode with complex amplitude alpha , basis dimension size and qudit order ord |
SqueezeOperator | Squeeze operator of a single mode with complex parameter xi , basis dimension size and qudit order ord |
QuadratureOperators | X 1 X 2 |
PhaseShiftOperator | Phase space rotation operator with rotation angle θ, basis dimension size and qudit order ord |
BeamSplitterOperator | Two mode beam-splitter operator, θ indicates the reflectivity, ϕ the relative phase in a basis of dimension size and qudit order ord |
We will briefly demonstrate how to apply all these functions in Wolfram quantum framework.
Details and options
Operators Mathematical Definitions
Operators Mathematical Definitions
DisplacementOperator | Exp[α † a * α |
SqueezeOperator | Exp 1 2 2 a †2 a |
PhaseShiftOperator | Exp[ θ † a |
BeamSplitterOperator | Expθ ϕ a 1 † a 2 - ϕ † a 1 a 2 |
QuadratureOperators | X 1 1 2 † a X 2 1 2 † a |
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.
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:
DisplacementOperator | st can be "Normal", "Weak" or "Antinormal".For "Normal", one gets: Exp[-|α 2 | † a Exp[α † a Exp[|α 2 | † a |
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:
SqueezeOperator | st 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 |
Basic Examples
Basic Examples
Setting the space size
Setting the space size
Set the default size to 20:
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SetFockSpaceSize[20];
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AnnihilationOperator[]
Out[4]=
QuantumOperator
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Calling it with no arguments, sets again the default size when loading the paclet which is 16:
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SetFockSpaceSize[];
In[18]:=
AnnihilationOperator[]
Out[18]=
QuantumOperator
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Annihilation operator
Annihilation operator
Define an annihilation operator with a given size and uses default order (1):
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AnnihilationOperator[8]
Out[21]=
QuantumOperator
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Define an annihilation operator to the second mode with default truncation size:
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AnnihilationOperator[{2}]
Out[19]=
QuantumOperator
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Define an annihilation operator with dimension 5 in the third order (or mode):
In[20]:=
AnnihilationOperator[5,{3}]
Out[20]=
QuantumOperator
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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:
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a=AnnihilationOperator[];@FockState[1]//TraditionalForm
†
a
Multimode Fock State
Multimode Fock State
Create a two mode Fock state with occupation numbers 2 and 12:
Apply the annihilation operator to both modes , state|1,11〉 is expected up to a global constant:
Equivalent using QuantumState with strings arguments, but QuantumState with string arguments will not work if one number is bigger than 9( it will interpret it as extra qudits):
Coherent state
Coherent state
Symbolic coherent state:
Cat State
Cat State
Photon distribution of the same state:
Displacement operator
Displacement operator
Create displacement operator of size 40 with amplitude 3+ :
Create a coherent state displacing the vacuum:
Specify the order(mode) of the operator with default size, second mode in this case:
Phase shift operator
Phase shift operator
Specify the basis size and order:
Phase shift operator coincides with the "Z" gate for qudits when the angle is 2π/n (up to conjugation):
Squeeze operator
Squeeze operator
Create squeeze operator of default size with complex parameter 1+:
Plot the non-zero amplitudes of the state:
Beam-Splitter Operator
Beam-Splitter Operator
Transform the state |1,1〉 using the symmetric beam-splitter:
Specify the order (where the operator acts) and the basis dimension:
OperatorVariance
OperatorVariance
Calculate the variance of the quadrature operators for the ground state:
A Fock state has a definite number of photons so the variance of the number operator is zero:
Quadrature Operators
Quadrature Operators
Ordering option
Ordering option
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
Define the Hamiltonian:
Superoperator that allows evolving another operator:
Evolve the operator:
Compare the result:
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. They are calculated numerically in a grid of phase space points and an interpolating function is returned. 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:
In the region from -4 to 4:
Fock state HusimiQ Representation
Fock state HusimiQ Representation
Applications
Applications
Expectation values
Expectation values
We will use the Fock state |2〉:
Phase space integral:
Squeezed vacuum Wigner Representation
Squeezed vacuum Wigner Representation
This exhibits graphically 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
Using GridSize
Using GridSize
Calculating the value of the Wigner function of Fock state |2〉 at the origin:
However, "GridSize" of 20 is not enough for this region and has a lot of numerical error:
The exact value from analytical expression is:
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
Define the basis and the relevant operators:
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
Find the probability of the atom being in the excited state:
Plot the result:
2. Collapse and revivals when the initial field state is a coherent state
Define the initial state and evolve it (we use a random α):
3. Atom "thermalization" (atom interacting with a thermal state field)
Reduced density matrix of the atom:
Plot two components:
Atom gets "thermalized", time average of the coherences → 0:
Jaynes-Cummings with dissipation
Jaynes-Cummings with dissipation
Hamiltonian definition:
Jump operators:
Helper definition to get the probability of finding the atom in the excited state:
Damped Rabi oscillations: