Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Second Quantization Functions
This tech note introduces the QuantumFramework implementation of bosonic second quantization on a truncated Fock space. The truncation provides a finite-dimensional representation that is convenient for computation while retaining the structure of common states and operators. The examples below focus on practical workflows for building states, applying operators, and visualizing results in quantum mechanics and quantum optics.
Install and load the QuantumFramework paclet and the second-quantization subpackage (skip the install step if it is already available):
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PacletInstall["https://www.wolfr.am/DevWQCF",ForceVersionInstallTrue]
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<<Wolfram`QuantumFramework`<<Wolfram`QuantumFramework`SecondQuantization`
Bosonic states and operators
Fock space size and utilities
Fock space size and utilities
SetFockSpaceSize | Sets size as the default truncation dimension , 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 normalized coherent state in a basis of dimension size , the complex amplitude is a parameter |
ThermalState | Thermal mixed state with the 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 order |
DisplacementOperator | Phase space displacement operator of a single mode with complex amplitude alpha , basis dimension size and qudit order order |
SqueezeOperator | Squeeze operator of a single mode with complex parameter xi , basis dimension size and qudit order order |
QuadratureOperators | X 1 X 2 |
PhaseShiftOperator | Phase space rotation operator with rotation angle θ, basis dimension size and qudit order order |
BeamSplitterOperator | Two mode beam-splitter operator, θ indicates the reflectivity, ϕ the relative phase in a basis of dimension size and qudit order order |
In the sections below, we demonstrate how to use these functions in the QuantumFramework.
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 |
Displacement operator ordering
Displacement operator ordering
By default, when "Ordering" is not specified, normal ordering is used. The formulas for the three orderings 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 for the three orderings 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 |
Coherence and correlation functions
Coherence and correlation functions
G2Coherence | Second-order coherence (2) g |
G1Correlation r 1 t 1 r 2 t 2 | First-order correlation function (1) G x 1 x 2 r 1 t 1 r 2 t 2 |
Basic usage
Setting the space size
Setting the space size
The global variable stores the default truncation size (dimension) of the Fock space. It is used by all second-quantization functions when the size parameter is not explicitly specified.
• Default value: 16
• Use to change the default.
• Use to reset to 16.
$FockSize
• Default value: 16
• Use
SetFockSpaceSize[n]
• Use
SetFockSpaceSize[]
Note: Larger truncation sizes improve accuracy for high photon-number states, but increase computational cost.
Show the default truncation size:
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$FockSize
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16
Set the truncation size to 20:
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SetFockSpaceSize[20];
Show new truncation size:
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$FockSize
Out[5]=
20
Set the default size again:
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SetFockSpaceSize[];
Show the current truncation size:
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$FockSize
Out[7]=
16
Fock States
Fock States
Create a single-mode Fock state with occupation number 5 using the default truncation size:
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FockState[5]
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QuantumState
 |  |
|
Create a single-mode Fock state with occupation number 5 using truncation size 10:
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FockState[5,10]
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QuantumState
|  |
|
Create a two-mode Fock state with occupation numbers 2 and 12:
Verify the shorthand string representation matches the two-mode state:
This shorthand does not work for numbers ≥ 10 because the digits are interpreted as separate qudits:
Annihilation and creation operators
Annihilation and creation operators
Create a single-mode annihilation operator with the default truncation size:
Create an annihilation operator with truncation size 8 (an 8×8 matrix) using the default mode order {1}:
Create an annihilation operator acting on mode 2 using the default truncation size:
Create an annihilation operator with size 5 acting on mode 3:
Apply the annihilation operator to a Fock state:
Define the creation operator by taking the adjoint (SuperDagger) of the annihilation operator, then apply it to a Fock state:
Or use the "Dagger" property directly:
Coherent state
Coherent state
Create a symbolic coherent state with the default truncation size:
Cat State
Cat State
Photon distribution of the same state:
Thermal state
Thermal state
Define a parametric thermal state in terms of the mean photon number:
Plot the entropy as a function of the mean photon number:
Displacement operator
Displacement operator
Create a displacement operator with truncation size 40 and amplitude 3+:
Verify that displacing the vacuum produces a coherent state:
Specify the mode (order) of the operator with the default size, mode 2 in this case:
Phase shift operator
Phase shift operator
Matrix form of the phase-shift operator for truncation size 10:
Effect of phase shift on a single-mode symbolic state:
Effect of phase shift on a two-mode superposition state:
Phase shift operator coincides with the "Z" gate for qudits when the angle is 2π/n (up to conjugation):
Squeeze operator
Squeeze operator
Create a squeeze operator with complex parameter 1+ using the default size:
Plot the non-zero amplitudes of the state:
Beam-Splitter Operator
Beam-Splitter Operator
Transform the state |1,1〉 with the symmetric beam splitter:
Specify the mode order (where the operator acts) and the truncation size:
BeamSplitter operator: Method option
BeamSplitter operator: Method option
BeamSplitterOperator supports two computation methods via the Method option. The default is MatrixExp which uses direct matrix exponentiation; Recurrence uses an efficient recurrence relation for the beam splitter matrix elements. Performance can vary depending on whether the angle arguments are numeric or exact/symbolic.
Use the "Recurrence" method with exact parameters:
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:
G2Coherence (Second-Order Coherence)
G2Coherence (Second-Order Coherence)
Improve the result increasing the truncation size:
It is not exactly 2 because of truncation: the thermal distribution has some weight at photon numbers beyond the cutoff.
Consider the mixed state (1-ϵ) |0〉 〈0| + ϵ |2〉 〈2|:
Extreme bunching when ϵ 0:
G1Correlation (first - order correlation)
G1Correlation (first - order correlation)
The first-order correlation is defined for some state |ψ〉 as
Quadrature Operators
Quadrature Operators
Or alternatively because the operator is Hermitian:
Mode Order (Multimode Systems)
Mode Order (Multimode Systems)
Single-mode operator with explicit order:
Verify the property:
Two-mode beam splitter acting on modes 1 and 3:
Displacement operator on mode 3:
Note: When using mode order, all operators in a calculation should use consistent mode assignments. The order parameter does NOT change the Fock space dimension—it only labels which mode the operator acts on.
Error Messages
Error Messages
The following error messages may be generated by second-quantization functions:
Example: FockState index clipping
Ordering option
Ordering option
Displacement and squeeze operators can be defined with different orderings. Use the "Ordering" option to choose "Weak", "Normal", or "Antinormal". In the infinite-dimensional limit these are equivalent, but in a truncated space the numerical error depends on the ordering, so one choice may be more accurate for a given calculation.
Repeat the check using weak ordering:
Amplitudes obtained from a known analytic formula:
Compute absolute errors:
Show the results:
Applications: States and operators
Applications: States and operators
Mean number of photons
Mean number of photons
Define the annihilation operator and create a random pure state:
Compute the mean number of photons using QuantumMeasurementOperator:
Number operator is Hermitian:
Using the probabilities of the diagonal of the density matrix:
Heisenberg evolution of annihilation operator
Heisenberg evolution of annihilation operator
Define the Hamiltonian:
Construct the evolution super-operator:
Evolve the operator:
Compare the result to the known result of quantum mechanics:
Quasi-probability Representations
Two phase-space quasi-probability representations are included: Wigner and HusimiQ. They are used extensively in quantum mechanics and quantum optics. Each is computed numerically on a grid of phase-space points and returns an interpolating function. Glauber representation is not included because, for a general state, a stable numerical algorithm is not feasible due to highly singular behavior.
In the sections below, we demonstrate how to use these functions in the QuantumFramework.
Examples
Examples
Ground state Wigner Representation:
Ground state Wigner Representation:
Compute the Wigner representation on the square region [-4, 4] × [-4, 4]:
Visualize the function in 3D:
HusimiQ Representation of a Fock state
HusimiQ Representation of a Fock state
Visualize the density plot of the representation:
Quasi-probability representations: Applications and options
Applications
Applications
Expectation values
Expectation values
Compute the Wigner function for the Fock state |2〉:
Phase space integral (normalization):
Marginal distributions
Marginal distributions
Visualize momentum distribution:
Visualize position distribution:
Squeezed vacuum Wigner Representation
Squeezed vacuum Wigner Representation
Plot the function, the nonclassical quadrature behavior of squeezed states is visible:
Options
Options
Scaling of the representation for ground state
Scaling of the representation for ground state
Set the Gaussian scaling parameter to 0.5:
Visualize the result:
Using GridSize
Using GridSize
Compute the Wigner function value at the origin for the Fock state |2〉 using a grid size of 140:
A grid size of 20 is insufficient for this region and yields noticeable numerical error:
The exact value from the analytic expression is:
Compare absolute errors:
Advanced examples
Decay of a coherent state
Decay of a coherent state
Set the decay rate γ:
Define the Hamiltonian including photon loss:
State evolution, we can do it symbolically:
Plot the mean photon number as a function of time:
Plot the fidelity with respect to the analytic expression:
Show the results:
Optical balance
Optical balance
Consider the following master equation, which balances a driving field Hamiltonian and single-photon loss:
Set the drive strength g:
Random initial state:
Set up the master equation:
Initial state:
Evolved state:
Photon number expectations and fidelity:
Plot both quantities versus time:
Jaynes-Cummings Model: Calculating the unitary
Jaynes-Cummings Model: Calculating the unitary
Define the basis and the relevant operators:
Define the atomic operators and the annihilation operator in second mode:
Going to the interaction picture at resonance, we get a time-independent Hamiltonian and from it we get the unitary operator to evolve states:
Jaynes-Cummings Model: Evolving states
Jaynes-Cummings Model: Evolving states
Field in a Fock State
Field in a Fock State
Rabi oscillations when the field is initially in a Fock state
Find the probability of the atom being in the excited state, using the projector as a measurement operator:
Plot the result:
Calculate the evolution for the states |g〉 ⊗ |n〉, varying the number state n:
Coherent state: Collapse and revivals
Coherent state: Collapse and revivals
Atom "thermalization" (atom interacting with a thermal state field)
Atom "thermalization" (atom interacting with a thermal state field)
Reduced density matrix of the atom:
Plot two components of the matrix:
Atom gets "thermalized"; the time-average of the coherences → 0:
Jaynes-Cummings with dissipation
Jaynes-Cummings with dissipation
Definition of the basis and the necessary operators:
Parameter setting:
Hamiltonian definition:
Jump operators:
Helper definition to get the probability of finding the atom in the excited state in a time window:
Damped Rabi oscillations:
Define the states:
Define the master equation super-operator:
Calculate the probabilities of the evolved states:
Plot the results:
Driven-Dissipative Kerr Nonlinear Oscillator
Driven-Dissipative Kerr Nonlinear Oscillator
The Hamiltonian is:
Define the Hamiltonian:
We take the vacuum as the initial state:
Wigner function snapshots:
We see the steady state resembles a coherent state up to distortion that depends of the parameters, it is known that evolution in Kerr nonlinear oscillators can produce coherent state manifold convergence
Taking the root with less energy: