# API

## Quantum-objects

### Types

- General basis types. Specialized bases can be found in the section API: Quantum-systems.

`QuantumOpticsBase.Basis`

— Type.Abstract base class for all specialized bases.

The Basis class is meant to specify a basis of the Hilbert space of the studied system. Besides basis specific information all subclasses must implement a shape variable which indicates the dimension of the used Hilbert space. For a spin-1/2 Hilbert space this would be the vector `Int[2]`

. A system composed of two spins would then have a shape vector `Int[2 2]`

.

Composite systems can be defined with help of the `CompositeBasis`

class.

`QuantumOpticsBase.GenericBasis`

— Type.`GenericBasis(N)`

A general purpose basis of dimension N.

Should only be used rarely since it defeats the purpose of checking that the bases of state vectors and operators are correct for algebraic operations. The preferred way is to specify special bases for different systems.

`QuantumOpticsBase.CompositeBasis`

— Type.`CompositeBasis(b1, b2...)`

Basis for composite Hilbert spaces.

Stores the subbases in a vector and creates the shape vector directly from the shape vectors of these subbases. Instead of creating a CompositeBasis directly `tensor(b1, b2...)`

or `b1 ⊗ b2 ⊗ …`

can be used.

- States

`QuantumOpticsBase.StateVector`

— Type.`QuantumOpticsBase.Bra`

— Type.`Bra(b::Basis[, data])`

Bra state defined by coefficients in respect to the basis.

`QuantumOpticsBase.Ket`

— Type.`Ket(b::Basis[, data])`

Ket state defined by coefficients in respect to the given basis.

- General purpose operators. A few more specialized operators are implemented in API: Quantum-systems.

Abstract base class for all operators.

All deriving operator classes have to define the fields `basis_l`

and `basis_r`

defining the left and right side bases.

For fast time evolution also at least the function `gemv!(alpha, op::AbstractOperator, x::Ket, beta, result::Ket)`

should be implemented. Many other generic multiplication functions can be defined in terms of this function and are provided automatically.

`QuantumOpticsBase.DataOperator`

— Type.Abstract type for operators with a data field.

This is an abstract type for operators that have a direct matrix representation stored in their `.data`

field.

`QuantumOpticsBase.DenseOperator`

— Type.`DenseOperator(b1[, b2, data])`

Dense array implementation of Operator.

The matrix consisting of complex floats is stored in the `data`

field.

`QuantumOpticsBase.SparseOperator`

— Type.`SparseOperator(b1[, b2, data])`

Sparse array implementation of Operator.

The matrix is stored as the julia built-in type `SparseMatrixCSC`

in the `data`

field.

`QuantumOpticsBase.LazyTensor`

— Type.`LazyTensor(b1[, b2], indices, operators[, factor=1])`

Lazy implementation of a tensor product of operators.

The suboperators are stored in the `operators`

field. The `indices`

field specifies in which subsystem the corresponding operator lives. Additionally, a complex factor is stored in the `factor`

field which allows for fast multiplication with numbers.

`QuantumOpticsBase.LazySum`

— Type.`LazySum([factors,] operators)`

Lazy evaluation of sums of operators.

All operators have to be given in respect to the same bases. The field `factors`

accounts for an additional multiplicative factor for each operator stored in the `operators`

field.

`QuantumOpticsBase.LazyProduct`

— Type.```
LazyProduct(operators[, factor=1])
LazyProduct(op1, op2...)
```

Lazy evaluation of products of operators.

The factors of the product are stored in the `operators`

field. Additionally a complex factor is stored in the `factor`

field which allows for fast multiplication with numbers.

- Super operators:

`QuantumOpticsBase.SuperOperator`

— Type.Base class for all super operator classes.

Super operators are bijective mappings from operators given in one specific basis to operators, possibly given in respect to another, different basis. To embed super operators in an algebraic framework they are defined with a left hand basis `basis_l`

and a right hand basis `basis_r`

where each of them again consists of a left and right hand basis.

`DenseSuperOperator(b1[, b2, data])`

SuperOperator stored as dense matrix.

`SparseSuperOperator(b1[, b2, data])`

SuperOperator stored as sparse matrix.

### Functions

- Functions to generate general states, operators and super-operators

`QuantumOpticsBase.basisstate`

— Function.`basisstate(b, index)`

Basis vector specified by `index`

as ket state.

For a composite system `index`

can be a vector which then creates a tensor product state $|i_1⟩⊗|i_2⟩⊗…⊗|i_n⟩$ of the corresponding basis states.

`basisstate(b::ManyBodyBasis, occupation::Vector{Int})`

Return a ket state where the system is in the state specified by the given occupation numbers.

`QuantumOpticsBase.identityoperator`

— Function.`identityoperator(a::Basis[, b::Basis])`

Return an identityoperator in the given bases.

`QuantumOpticsBase.diagonaloperator`

— Function.`diagonaloperator(b::Basis)`

Create a diagonal operator of type `SparseOperator`

.

`QuantumOpticsBase.randoperator`

— Function.`randoperator(b1[, b2])`

Calculate a random unnormalized dense operator.

`QuantumOpticsBase.spre`

— Function.`spre(op)`

Create a super-operator equivalent for right side operator multiplication.

For operators $A$, $B$ the relation

holds. `op`

can be a dense or a sparse operator.

`QuantumOpticsBase.spost`

— Function.Create a super-operator equivalent for left side operator multiplication.

For operators $A$, $B$ the relation

holds. `op`

can be a dense or a sparse operator.

`QuantumOpticsBase.liouvillian`

— Function.`liouvillian(H, J; rates, Jdagger)`

Create a super-operator equivalent to the master equation so that $\dot ρ = S ρ$.

The super-operator $S$ is defined by

**Arguments**

`H`

: Hamiltonian.`J`

: Vector containing the jump operators.`rates`

: Vector or matrix specifying the coefficients for the jump operators.`Jdagger`

: Vector containing the hermitian conjugates of the jump operators. If they are not given they are calculated automatically.

- As far as it makes sense the same functions are implemented for bases, states, operators and superoperators.

`QuantumOpticsBase.samebases`

— Function.`samebases(a, b)`

Test if two objects have the same bases.

`QuantumOpticsBase.check_samebases`

— Function.`check_samebases(a, b)`

Throw an `IncompatibleBases`

error if the objects don't have the same bases.

`QuantumOpticsBase.multiplicable`

— Function.`multiplicable(a, b)`

Check if two objects are multiplicable.

`QuantumOpticsBase.check_multiplicable`

— Function.`check_multiplicable(a, b)`

Throw an `IncompatibleBases`

error if the objects are not multiplicable.

`QuantumOpticsBase.basis`

— Function.`basis(a)`

Return the basis of an object.

If it's ambiguous, e.g. if an operator has a different left and right basis, an `IncompatibleBases`

error is thrown.

`QuantumOpticsBase.dagger`

— Function.`dagger(x)`

Hermitian conjugate.

`QuantumOpticsBase.tensor`

— Function.`tensor(x, y, z...)`

Tensor product of the given objects. Alternatively, the unicode symbol ⊗ (\otimes) can be used.

`tensor(x::Basis, y::Basis, z::Basis...)`

Create a `CompositeBasis`

from the given bases.

Any given CompositeBasis is expanded so that the resulting CompositeBasis never contains another CompositeBasis.

`tensor(x::Ket, y::Ket, z::Ket...)`

Tensor product $|x⟩⊗|y⟩⊗|z⟩⊗…$ of the given states.

`tensor(x::AbstractOperator, y::AbstractOperator, z::AbstractOperator...)`

Tensor product $\hat{x}⊗\hat{y}⊗\hat{z}⊗…$ of the given operators.

`tensor(x::Ket, y::Bra)`

Outer product $|x⟩⟨y|$ of the given states.

`QuantumOpticsBase.projector`

— Method.`projector(a::Ket, b::Bra)`

Projection operator $|a⟩⟨b|$.

`QuantumOpticsBase.projector`

— Method.`projector(a::Ket)`

Projection operator $|a⟩⟨a|$.

`QuantumOpticsBase.projector`

— Method.`projector(a::Bra)`

Projection operator $|a⟩⟨a|$.

`QuantumOpticsBase.dm`

— Function.`dm(a::StateVector)`

Create density matrix $|a⟩⟨a|$. Same as `projector(a)`

.

`LinearAlgebra.norm`

— Method.`norm(x::StateVector)`

Norm of the given bra or ket state.

`LinearAlgebra.tr`

— Function.`tr(x::AbstractOperator)`

Trace of the given operator.

`QuantumOpticsBase.ptrace`

— Function.`ptrace(a, indices)`

Partial trace of the given basis, state or operator.

The `indices`

argument, which can be a single integer or a vector of integers, specifies which subsystems are traced out. The number of indices has to be smaller than the number of subsystems, i.e. it is not allowed to perform a full trace.

`LinearAlgebra.normalize`

— Method.`normalize(x::StateVector)`

Return the normalized state so that `norm(x)`

is one.

`LinearAlgebra.normalize`

— Method.`normalize(op)`

Return the normalized operator so that its `tr(op)`

is one.

`LinearAlgebra.normalize!`

— Method.`normalize!(x::StateVector)`

In-place normalization of the given bra or ket so that `norm(x)`

is one.

`LinearAlgebra.normalize!`

— Method.`normalize!(op)`

In-place normalization of the given operator so that its `tr(x)`

is one.

`QuantumOpticsBase.expect`

— Function.`expect(op, state)`

Expectation value of the given operator `op`

for the specified `state`

.

`state`

can either be a (density) operator or a ket.

`expect(index, op, state)`

If an `index`

is given, it assumes that `op`

is defined in the subsystem specified by this number.

`QuantumOpticsBase.variance`

— Function.`variance(op, state)`

Variance of the given operator `op`

for the specified `state`

.

`state`

can either be a (density) operator or a ket.

`variance(index, op, state)`

If an `index`

is given, it assumes that `op`

is defined in the subsystem specified by this number

`QuantumOpticsBase.embed`

— Function.`embed(basis1[, basis2], indices::Vector, operators::Vector)`

Tensor product of operators where missing indices are filled up with identity operators.

`embed(basis1[, basis2], indices::Vector, operators::Vector)`

Embed operator acting on a joint Hilbert space where missing indices are filled up with identity operators.

`embed(basis1[, basis2], operators::Dict)`

`operators`

is a dictionary `Dict{Vector{Int}, AbstractOperator}`

. The integer vector specifies in which subsystems the corresponding operator is defined.

`QuantumOpticsBase.permutesystems`

— Function.`permutesystems(a, perm)`

Change the ordering of the subsystems of the given object.

For a permutation vector `[2,1,3]`

and a given object with basis `[b1, b2, b3]`

this function results in `[b2, b1, b3]`

.

`Base.exp`

— Method.`exp(op::AbstractOperator)`

Operator exponential.

`Base.exp`

— Method.`exp(op::DenseSuperOperator)`

Operator exponential which can for example used to calculate time evolutions.

`QuantumOpticsBase.gemv!`

— Function.`gemv!(alpha, a, b, beta, result)`

Fast in-place multiplication of operators with state vectors. It implements the relation `result = beta*result + alpha*a*b`

. Here, `alpha`

and `beta`

are complex numbers, while `result`

and either `a`

or `b`

are state vectors while the other one can be of any operator type.

`QuantumOpticsBase.gemm!`

— Function.`gemm!(alpha, a, b, beta, result)`

Fast in-place multiplication of of operators with DenseOperators. It implements the relation `result = beta*result + alpha*a*b`

. Here, `alpha`

and `beta`

are complex numbers, while `result`

and either `a`

or `b`

are dense operators while the other one can be of any operator type.

- Conversion of operators

`QuantumOpticsBase.dense`

— Function.`dense(op::AbstractOperator)`

Convert an arbitrary Operator into a `DenseOperator`

.

`SparseArrays.sparse`

— Method.`sparse(op::AbstractOperator)`

Convert an arbitrary operator into a `SparseOperator`

.

### Exceptions

Exception that should be raised for an illegal algebraic operation.

## Quantum systems

### Fock

`QuantumOpticsBase.FockBasis`

— Type.`FockBasis(N)`

Basis for a Fock space where `N`

specifies a cutoff, i.e. what the highest included fock state is. Note that the dimension of this basis then is N+1.

`QuantumOpticsBase.number`

— Method.`number(b::FockBasis)`

Number operator for the specified Fock space.

`QuantumOpticsBase.destroy`

— Method.`destroy(b::FockBasis)`

Annihilation operator for the specified Fock space.

`QuantumOpticsBase.create`

— Method.`create(b::FockBasis)`

Creation operator for the specified Fock space.

`QuantumOpticsBase.displace`

— Function.`displace(b::FockBasis, alpha)`

Displacement operator $D(α)$ for the specified Fock space.

`QuantumOpticsBase.fockstate`

— Function.`fockstate(b::FockBasis, n)`

Fock state $|n⟩$ for the specified Fock space.

`QuantumOpticsBase.coherentstate`

— Function.`coherentstate(b::FockBasis, alpha)`

Coherent state $|α⟩$ for the specified Fock space.

`QuantumOpticsBase.coherentstate!`

— Function.`coherentstate!(ket::Ket, b::FockBasis, alpha)`

Inplace creation of coherent state $|α⟩$ for the specified Fock space.

### Phase space

`QuantumOptics.qfunc`

— Function.```
qfunc(a, α)
qfunc(a, x, y)
qfunc(a, xvec, yvec)
```

Husimi Q representation $⟨α|ρ|α⟩/π$ for the given state or operator `a`

. The function can either be evaluated on one point α or on a grid specified by the vectors `xvec`

and `yvec`

. Note that conversion from `x`

and `y`

to `α`

is done via the relation $α = \frac{1}{\sqrt{2}}(x + i y)$.

`QuantumOptics.wigner`

— Function.```
wigner(a, α)
wigner(a, x, y)
wigner(a, xvec, yvec)
```

Wigner function for the given state or operator `a`

. The function can either be evaluated on one point α or on a grid specified by the vectors `xvec`

and `yvec`

. Note that conversion from `x`

and `y`

to `α`

is done via the relation $α = \frac{1}{\sqrt{2}}(x + i y)$.

`QuantumOptics.coherentspinstate`

— Function.`coherentspinstate(b::SpinBasis, θ::Real, ϕ::Real)`

A coherent spin state |θ,ϕ⟩ is analogous to the coherent state of the linear harmonic oscillator. Coherent spin states represent a collection of identical two-level systems and can be described by two angles θ and ϕ (although this parametrization is not unique), similarly to a qubit on the Bloch sphere.

`QuantumOptics.qfuncsu2`

— Function.```
qfuncsu2(ket,Ntheta;Nphi=2Ntheta)
qfuncsu2(rho,Ntheta;Nphi=2Ntheta)
```

Husimi Q SU(2) representation $⟨θ,ϕ|ρ|θ,ϕ⟩/π$ for the given state.

The function calculates the SU(2) Husimi representation of a state on the generalised bloch sphere (0 < θ < π and 0 < ϕ < 2 π) with a given resolution `(Ntheta, Nphi)`

.

```
qfuncsu2(rho,θ,ϕ)
qfuncsu2(ket,θ,ϕ)
```

This version calculates the Husimi Q SU(2) function at a position given by θ and ϕ.

`QuantumOptics.wignersu2`

— Function.```
wignersu2(ket,Ntheta;Nphi=2Ntheta)
wignersu2(rho,Ntheta;Nphi=2Ntheta)
```

Wigner SU(2) representation for the given state with a resolution `(Ntheta, Nphi)`

.

The function calculates the SU(2) Wigner representation of a state on the generalised bloch sphere (0 < θ < π and 0 < ϕ < 2 π) with a given resolution by decomposing the state into the basis of spherical harmonics.

```
wignersu2(rho,θ,ϕ)
wignersu2(ket,θ,ϕ)
```

This version calculates the Wigner SU(2) function at a position given by θ and ϕ

`QuantumOptics.ylm`

— Function.`ylm(l::Integer,m::Integer,theta::Real,phi::Real)`

Spherical harmonics Y(l,m)(θ,ϕ) where l ∈ N, m = -l,-l+1,...,l-1,l, θ ∈ [0,π], and ϕ ∈ [0,2π).

This function calculates the value of Y(l,m) spherical harmonic at position θ and ϕ.

### N-level

`QuantumOpticsBase.NLevelBasis`

— Type.`NLevelBasis(N)`

Basis for a system consisting of N states.

`QuantumOpticsBase.transition`

— Method.`transition(b::NLevelBasis, to::Int, from::Int)`

Transition operator $|\mathrm{to}⟩⟨\mathrm{from}|$.

`QuantumOpticsBase.nlevelstate`

— Function.`nlevelstate(b::NLevelBasis, n::Int)`

State where the system is completely in the n-th level.

### Spin

`QuantumOpticsBase.SpinBasis`

— Type.`SpinBasis(n)`

Basis for spin-n particles.

The basis can be created for arbitrary spinnumbers by using a rational number, e.g. `SpinBasis(3//2)`

. The Pauli operators are defined for all possible spin numbers.

`QuantumOpticsBase.sigmax`

— Function.`sigmax(b::SpinBasis)`

Pauli $σ_x$ operator for the given Spin basis.

`QuantumOpticsBase.sigmay`

— Function.`sigmay(b::SpinBasis)`

Pauli $σ_y$ operator for the given Spin basis.

`QuantumOpticsBase.sigmaz`

— Function.`sigmaz(b::SpinBasis)`

Pauli $σ_z$ operator for the given Spin basis.

`QuantumOpticsBase.sigmap`

— Function.`sigmap(b::SpinBasis)`

Raising operator $σ_+$ for the given Spin basis.

`QuantumOpticsBase.sigmam`

— Function.`sigmam(b::SpinBasis)`

Lowering operator $σ_-$ for the given Spin basis.

`QuantumOpticsBase.spinup`

— Function.`spinup(b::SpinBasis)`

Spin up state for the given Spin basis.

`QuantumOpticsBase.spindown`

— Function.`spindown(b::SpinBasis)`

Spin down state for the given Spin basis.

### Particle

`QuantumOpticsBase.PositionBasis`

— Type.```
PositionBasis(xmin, xmax, Npoints)
PositionBasis(b::MomentumBasis)
```

Basis for a particle in real space.

For simplicity periodic boundaries are assumed which means that the rightmost point defined by `xmax`

is not included in the basis but is defined to be the same as `xmin`

.

When a `MomentumBasis`

is given as argument the exact values of $x_{min}$ and $x_{max}$ are due to the periodic boundary conditions more or less arbitrary and are chosen to be $-\pi/dp$ and $\pi/dp$ with $dp=(p_{max}-p_{min})/N$.

`QuantumOpticsBase.MomentumBasis`

— Type.```
MomentumBasis(pmin, pmax, Npoints)
MomentumBasis(b::PositionBasis)
```

Basis for a particle in momentum space.

For simplicity periodic boundaries are assumed which means that `pmax`

is not included in the basis but is defined to be the same as `pmin`

.

When a `PositionBasis`

is given as argument the exact values of $p_{min}$ and $p_{max}$ are due to the periodic boundary conditions more or less arbitrary and are chosen to be $-\pi/dx$ and $\pi/dx$ with $dx=(x_{max}-x_{min})/N$.

`QuantumOpticsBase.spacing`

— Function.`spacing(b::PositionBasis)`

Difference between two adjacent points of the real space basis.

`spacing(b::MomentumBasis)`

Momentum difference between two adjacent points of the momentum basis.

`QuantumOpticsBase.samplepoints`

— Function.`samplepoints(b::PositionBasis)`

x values of the real space basis.

`samplepoints(b::MomentumBasis)`

p values of the momentum basis.

`Base.position`

— Method.`position(b::PositionBasis)`

Position operator in real space.

`Base.position`

— Method.`position(b:MomentumBasis)`

Position operator in momentum space.

`QuantumOpticsBase.momentum`

— Method.`momentum(b::PositionBasis)`

Momentum operator in real space.

`QuantumOpticsBase.momentum`

— Method.`momentum(b:MomentumBasis)`

Momentum operator in momentum space.

`QuantumOpticsBase.potentialoperator`

— Function.`potentialoperator(b::PositionBasis, V(x))`

Operator representing a potential $V(x)$ in real space.

`potentialoperator(b::MomentumBasis, V(x))`

Operator representing a potential $V(x)$ in momentum space.

`potentialoperator(b::CompositeBasis, V(x, y, z, ...))`

Operator representing a potential $V$ in more than one dimension.

**Arguments**

`b`

: Composite basis consisting purely either of`PositionBasis`

or`MomentumBasis`

. Note, that calling this with a composite basis in momentum space might consume a large amount of memory.`V`

: Function describing the potential. ATTENTION: The number of arguments accepted by`V`

must match the spatial dimension. Furthermore, the order of the arguments has to match that of the order of the tensor product of bases (e.g. if`b=bx⊗by⊗bz`

, then`V(x,y,z)`

).

`QuantumOpticsBase.gaussianstate`

— Function.```
gaussianstate(b::PositionBasis, x0, p0, sigma)
gaussianstate(b::MomentumBasis, x0, p0, sigma)
```

Create a Gaussian state around `x0`

and`p0`

with width `sigma`

.

In real space the gaussian state is defined as

and is connected to the momentum space definition

via a Fourier-transformation

The state has the properties

- $⟨p⟩ = p_0$
- $⟨x⟩ = x_0$
- $\mathrm{Var}(x) = \frac{σ^2}{2}$
- $\mathrm{Var}(p) = \frac{1}{2 σ^2}$

Due to the numerically necessary discretization additional scaling factors $\sqrt{Δx}$ and $\sqrt{Δp}$ are used so that $\langle x_i|Ψ\rangle = \sqrt{Δ x} Ψ(x_i)$ and $\langle p_i|Ψ\rangle = \sqrt{Δ p} Ψ(p_i)$ so that the resulting Ket state is normalized.

`QuantumOpticsBase.FFTOperator`

— Type.`FFTOperator`

Abstract type for all implementations of FFT operators.

`QuantumOpticsBase.FFTOperators`

— Type.`FFTOperators`

Operator performing a fast fourier transformation when multiplied with a state that is a Ket or an Operator.

`QuantumOpticsBase.FFTKets`

— Type.`FFTKets`

Operator that can only perform fast fourier transformations on Kets. This is much more memory efficient when only working with Kets.

`QuantumOpticsBase.transform`

— Function.```
transform(b1::MomentumBasis, b2::PositionBasis)
transform(b1::PositionBasis, b2::MomentumBasis)
```

Transformation operator between position basis and momentum basis.

`transform(b1::CompositeBasis, b2::CompositeBasis)`

Transformation operator between two composite bases. Each of the bases has to contain bases of type PositionBasis and the other one a corresponding MomentumBasis.

```
transform(b1::PositionBasis, b2::FockBasis; x0=1)
transform(b1::FockBasis, b2::PositionBasis; x0=1)
```

Transformation operator between position basis and fock basis.

The coefficients are connected via the relation

where $⟨x_i|n⟩$ is the value of the n-th eigenstate of a particle in a harmonic trap potential at position $x$, i.e.:

### Subspace bases

`QuantumOpticsBase.SubspaceBasis`

— Type.`SubspaceBasis(basisstates)`

A basis describing a subspace embedded a higher dimensional Hilbert space.

`QuantumOpticsBase.orthonormalize`

— Function.`orthonormalize(b::SubspaceBasis)`

Orthonormalize the basis states of the given `SubspaceBasis`

A modified Gram-Schmidt process is used.

`QuantumOpticsBase.projector`

— Method.`projector(b1, b2)`

Projection operator between subspaces and superspaces or between two subspaces.

### Many-body

`QuantumOpticsBase.ManyBodyBasis`

— Type.`ManyBodyBasis(b, occupations)`

Basis for a many body system.

The basis has to know the associated one-body basis `b`

and which occupation states should be included. The occupations_hash is used to speed up checking if two many-body bases are equal.

`QuantumOpticsBase.fermionstates`

— Function.```
fermionstates(Nmodes, Nparticles)
fermionstates(b, Nparticles)
```

Generate all fermionic occupation states for N-particles in M-modes. `Nparticles`

can be a vector to define a Hilbert space with variable particle number.

`QuantumOpticsBase.bosonstates`

— Function.```
bosonstates(Nmodes, Nparticles)
bosonstates(b, Nparticles)
```

Generate all bosonic occupation states for N-particles in M-modes. `Nparticles`

can be a vector to define a Hilbert space with variable particle number.

`QuantumOpticsBase.number`

— Method.`number(b::ManyBodyBasis, index)`

Particle number operator for the i-th mode of the many-body basis `b`

.

`QuantumOpticsBase.number`

— Method.`number(b::ManyBodyBasis)`

Total particle number operator.

`QuantumOpticsBase.destroy`

— Method.`destroy(b::ManyBodyBasis, index)`

Annihilation operator for the i-th mode of the many-body basis `b`

.

`QuantumOpticsBase.create`

— Method.`create(b::ManyBodyBasis, index)`

Creation operator for the i-th mode of the many-body basis `b`

.

`QuantumOpticsBase.transition`

— Method.`transition(b::ManyBodyBasis, to::Int, from::Int)`

Operator $|\mathrm{to}⟩⟨\mathrm{from}|$ transferring particles between modes.

`QuantumOpticsBase.manybodyoperator`

— Function.`manybodyoperator(b::ManyBodyBasis, op)`

Create the many-body operator from the given one-body operator `op`

.

The given operator can either be a one-body operator or a two-body interaction. Higher order interactions are at the moment not implemented.

The mathematical formalism for the one-body case is described by

and for the interaction case by

where $X$ is the N-particle operator, $x$ is the one-body operator and $|u⟩$ are the one-body states associated to the different modes of the N-particle basis.

`QuantumOpticsBase.onebodyexpect`

— Function.`onebodyexpect(op, state)`

Expectation value of the one-body operator `op`

in respect to the many-body `state`

.

## Metrics

`QuantumOpticsBase.tracenorm`

— Function.`tracenorm(rho)`

Trace norm of `rho`

.

It is defined as

Depending if `rho`

is hermitian either `tracenorm_h`

or `tracenorm_nh`

is called.

`QuantumOpticsBase.tracenorm_h`

— Function.`tracenorm_h(rho)`

Trace norm of `rho`

.

It uses the identity

where $λ_i$ are the eigenvalues of `rho`

.

`QuantumOpticsBase.tracenorm_nh`

— Function.`tracenorm_nh(rho)`

Trace norm of `rho`

.

Note that in this case `rho`

doesn't have to be represented by a square matrix (i.e. it can have different left-hand and right-hand bases).

It uses the identity

where $σ_i$ are the singular values of `rho`

.

`QuantumOpticsBase.tracedistance`

— Function.`tracedistance(rho, sigma)`

Trace distance between `rho`

and `sigma`

.

It is defined as

It calls `tracenorm`

which in turn either uses `tracenorm_h`

or `tracenorm_nh`

depending if $ρ-σ$ is hermitian or not.

`QuantumOpticsBase.tracedistance_h`

— Function.`tracedistance_h(rho, sigma)`

Trace distance between `rho`

and `sigma`

.

It uses the identity

where $λ_i$ are the eigenvalues of `rho`

- `sigma`

.

`QuantumOpticsBase.tracedistance_nh`

— Function.`tracedistance_nh(rho, sigma)`

Trace distance between `rho`

and `sigma`

.

Note that in this case `rho`

and `sigma`

don't have to be represented by square matrices (i.e. they can have different left-hand and right-hand bases).

It uses the identity

where $σ_i$ are the singular values of `rho`

- `sigma`

.

`QuantumOpticsBase.entropy_vn`

— Function.`entropy_vn(rho)`

Von Neumann entropy of a density matrix.

The Von Neumann entropy of a density operator is defined as

where $λ_n$ are the eigenvalues of the density matrix $ρ$, $\log$ is the natural logarithm and $0\log(0) ≡ 0$.

**Arguments**

`rho`

: Density operator of which to calculate Von Neumann entropy.`tol=1e-15`

: Tolerance for rounding errors in the computed eigenvalues.

`QuantumOpticsBase.fidelity`

— Function.`fidelity(rho, sigma)`

Fidelity of two density operators.

The fidelity of two density operators $ρ$ and $σ$ is defined by

where $\sqrt{ρ}=\sum_n\sqrt{λ_n}|ψ⟩⟨ψ|$.

`QuantumOpticsBase.ptranspose`

— Function.`ptranspose(rho, index)`

Partial transpose of rho with respect to subspace specified by index.

`QuantumOpticsBase.PPT`

— Function.`PPT(rho, index)`

Peres-Horodecki criterion of partial transpose.

`QuantumOpticsBase.negativity`

— Function.`negativity(rho, index)`

Negativity of rho with respect to subsystem index.

The negativity of a density matrix ρ is defined as

where `ρᵀ`

is the partial transpose.

`QuantumOpticsBase.logarithmic_negativity`

— Function.`logarithmic_negativity(rho, index)`

The logarithmic negativity of a density matrix ρ is defined as

where `ρᵀ`

is the partial transpose.

## Time-evolution

### Schroedinger

`QuantumOptics.timeevolution.schroedinger`

— Function.`timeevolution.schroedinger(tspan, psi0, H; fout)`

Integrate Schroedinger equation.

**Arguments**

`tspan`

: Vector specifying the points of time for which output should be displayed.`psi0`

: Initial state vector (can be a bra or a ket).`H`

: Arbitrary operator specifying the Hamiltonian.`fout=nothing`

: If given, this function`fout(t, psi)`

is called every time an output should be displayed. ATTENTION: The state`psi`

is neither normalized nor permanent! It is still in use by the ode solver and therefore must not be changed.

`QuantumOptics.timeevolution.schroedinger_dynamic`

— Function.`timeevolution.schroedinger_dynamic(tspan, psi0, f; fout)`

Integrate time-dependent Schroedinger equation.

**Arguments**

`tspan`

: Vector specifying the points of time for which output should be displayed.`psi0`

: Initial state vector (can be a bra or a ket).`f`

: Function`f(t, psi) -> H`

returning the time and or state dependent Hamiltonian.`fout=nothing`

: If given, this function`fout(t, psi)`

is called every time an output should be displayed. ATTENTION: The state`psi`

is neither normalized nor permanent! It is still in use by the ode solver and therefore must not be changed.

### Master

`QuantumOptics.timeevolution.master`

— Function.`timeevolution.master(tspan, rho0, H, J; <keyword arguments>)`

Time-evolution according to a master equation.

There are two implementations for integrating the master equation:

`master_h`

: Usual formulation of the master equation.`master_nh`

: Variant with non-hermitian Hamiltonian.

For dense arguments the `master`

function calculates the non-hermitian Hamiltonian and then calls master_nh which is slightly faster.

**Arguments**

`tspan`

: Vector specifying the points of time for which output should be displayed.`rho0`

: Initial density operator. Can also be a state vector which is automatically converted into a density operator.`H`

: Arbitrary operator specifying the Hamiltonian.`J`

: Vector containing all jump operators which can be of any arbitrary operator type.`rates=nothing`

: Vector or matrix specifying the coefficients (decay rates) for the jump operators. If nothing is specified all rates are assumed to be 1.`Jdagger=dagger.(J)`

: Vector containing the hermitian conjugates of the jump operators. If they are not given they are calculated automatically.`fout=nothing`

: If given, this function`fout(t, rho)`

is called every time an output should be displayed. ATTENTION: The given state rho is not permanent! It is still in use by the ode solver and therefore must not be changed.`kwargs...`

: Further arguments are passed on to the ode solver.

`QuantumOptics.timeevolution.master_h`

— Function.`timeevolution.master_h(tspan, rho0, H, J; <keyword arguments>)`

Integrate the master equation with dmaster_h as derivative function.

Further information can be found at `master`

.

`QuantumOptics.timeevolution.master_nh`

— Function.`timeevolution.master_nh(tspan, rho0, H, J; <keyword arguments>)`

Integrate the master equation with dmaster_nh as derivative function.

In this case the given Hamiltonian is assumed to be the non-hermitian version:

Further information can be found at `master`

.

`QuantumOptics.timeevolution.master_dynamic`

— Function.`timeevolution.master_dynamic(tspan, rho0, f; <keyword arguments>)`

Time-evolution according to a master equation with a dynamic Hamiltonian and J.

There are two implementations for integrating the master equation with dynamic operators:

`master_dynamic`

: Usual formulation of the master equation.`master_nh_dynamic`

: Variant with non-hermitian Hamiltonian.

**Arguments**

`tspan`

: Vector specifying the points of time for which output should be displayed.`rho0`

: Initial density operator. Can also be a state vector which is automatically converted into a density operator.`f`

: Function`f(t, rho) -> (H, J, Jdagger)`

or`f(t, rho) -> (H, J, Jdagger, rates)`

`rates=nothing`

: Vector or matrix specifying the coefficients (decay rates) for the jump operators. If nothing is specified all rates are assumed to be 1.`fout=nothing`

: If given, this function`fout(t, rho)`

is called every time an output should be displayed. ATTENTION: The given state rho is not permanent! It is still in use by the ode solver and therefore must not be changed.`kwargs...`

: Further arguments are passed on to the ode solver.

`QuantumOptics.timeevolution.master_nh_dynamic`

— Function.`timeevolution.master_dynamic(tspan, rho0, f; <keyword arguments>)`

Time-evolution according to a master equation with a dynamic non-hermitian Hamiltonian and J.

In this case the given Hamiltonian is assumed to be the non-hermitian version.

The given function can either be of the form `f(t, rho) -> (Hnh, Hnhdagger, J, Jdagger)`

or `f(t, rho) -> (Hnh, Hnhdagger, J, Jdagger, rates)`

For further information look at `master_dynamic`

.

`@skiptimechecks`

Macro to skip checks during time-dependent problems. Useful for `timeevolution.master_dynamic`

and similar functions.

### Monte Carlo wave function

`QuantumOptics.timeevolution.mcwf`

— Function.`mcwf(tspan, psi0, H, J; <keyword arguments>)`

Integrate the master equation using the MCWF method.

There are two implementations for integrating the non-hermitian schroedinger equation:

`mcwf_h`

: Usual formulation with Hamiltonian + jump operators

separately.

`mcwf_nh`

: Variant with non-hermitian Hamiltonian.

The `mcwf`

function takes a normal Hamiltonian, calculates the non-hermitian Hamiltonian and then calls `mcwf_nh`

which is slightly faster.

**Arguments**

`tspan`

: Vector specifying the points of time for which output should

be displayed.

`psi0`

: Initial state vector.`H`

: Arbitrary Operator specifying the Hamiltonian.`J`

: Vector containing all jump operators which can be of any arbitrary

operator type.

`seed=rand()`

: Seed used for the random number generator.`rates=ones()`

: Vector of decay rates.`fout`

: If given, this function`fout(t, psi)`

is called every time an

output should be displayed. ATTENTION: The state `psi`

is neither normalized nor permanent! It is still in use by the ode solve and therefore must not be changed.

`Jdagger=dagger.(J)`

: Vector containing the hermitian conjugates of the jump

operators. If they are not given they are calculated automatically.

`display_beforeevent=false`

:`fout`

is called before every jump.`display_afterevent=false`

:`fout`

is called after every jump.`display_jumps=false`

: If set to true, an additional list of times and indices

is returned. These correspond to the times at which a jump occured and the index of the jump operators with which the jump occured, respectively.

`kwargs...`

: Further arguments are passed on to the ode solver.

`QuantumOptics.timeevolution.mcwf_h`

— Function.`mcwf_h(tspan, rho0, Hnh, J; <keyword arguments>)`

Calculate MCWF trajectory where the Hamiltonian is given in hermitian form.

For more information see: `mcwf`

`QuantumOptics.timeevolution.mcwf_nh`

— Function.`mcwf_nh(tspan, rho0, Hnh, J; <keyword arguments>)`

Calculate MCWF trajectory where the Hamiltonian is given in non-hermitian form.

For more information see: `mcwf`

`QuantumOptics.timeevolution.mcwf_dynamic`

— Function.`mcwf_dynamic(tspan, psi0, f; <keyword arguments>)`

Integrate the master equation using the MCWF method with dynamic Hamiltonian and Jump operators.

The `mcwf`

function takes a normal Hamiltonian, calculates the non-hermitian Hamiltonian and then calls `mcwf_nh`

which is slightly faster.

**Arguments**

`tspan`

: Vector specifying the points of time for which output should

be displayed.

`psi0`

: Initial state vector.`f`

: Function`f(t, psi) -> (H, J, Jdagger)`

or`f(t, psi) -> (H, J, Jdagger, rates)`

that returns the time-dependent Hamiltonian and Jump operators.`seed=rand()`

: Seed used for the random number generator.`rates=ones()`

: Vector of decay rates.`fout`

: If given, this function`fout(t, psi)`

is called every time an

output should be displayed. ATTENTION: The state `psi`

is neither normalized nor permanent! It is still in use by the ode solve and therefore must not be changed.

`display_beforeevent=false`

:`fout`

is called before every jump.`display_afterevent=false`

:`fout`

is called after every jump.`display_jumps=false`

: If set to true, an additional list of times and indices

is returned. These correspond to the times at which a jump occured and the index of the jump operators with which the jump occured, respectively.

`kwargs...`

: Further arguments are passed on to the ode solver.

`QuantumOptics.timeevolution.mcwf_nh_dynamic`

— Function.`mcwf_nh_dynamic(tspan, rho0, f; <keyword arguments>)`

Calculate MCWF trajectory where the dynamic Hamiltonian is given in non-hermitian form.

For more information see: `mcwf_dynamic`

`QuantumOptics.timeevolution.diagonaljumps`

— Function.`diagonaljumps(rates, J)`

Diagonalize jump operators.

The given matrix `rates`

of decay rates is diagonalized and the corresponding set of jump operators is calculated.

**Arguments**

`rates`

: Matrix of decay rates.`J`

: Vector of jump operators.

### Bloch-Redfield master

`bloch_redfield_tensor(H, a_ops; J=[], use_secular=true, secular_cutoff=0.1)`

Create the super-operator for the Bloch-Redfield master equation such that $\dot ρ = R ρ$ based on the QuTiP implementation.

See QuTiP's documentation (http://qutip.org/docs/latest/guide/dynamics/dynamics-bloch-redfield.html) for more information and a brief derivation.

**Arguments**

`H`

: Hamiltonian.`a_ops`

: Nested list of [interaction operator, callback function] pairs for the Bloch-Redfield type processes where the callback function describes the environment spectrum for the corresponding interaction operator. The spectral functions must take the angular frequency as their only argument.`J=[]`

: Vector containing the jump operators for the Linblad type processes (optional).`use_secular=true`

: Specify whether or not to use the secular approximation.`secular_cutoff=0.1`

: Cutoff to allow a degree of partial secularization. Terms are discarded if they are greater than (dw_min * secular cutoff) where dw_min is the smallest (non-zero) difference between any two eigenenergies of H. This argument is only taken into account if use_secular=true.

`timeevolution.master_bloch_redfield(tspan, rho0, R, H; <keyword arguments>)`

Time-evolution according to a Bloch-Redfield master equation.

**Arguments**

`tspan`

: Vector specifying the points of time for which output should be displayed.`rho0`

: Initial density operator. Can also be a state vector which is automatically converted into a density operator.`H`

: Arbitrary operator specifying the Hamiltonian.`R`

: Bloch-Redfield tensor describing the time-evolution $\dot ρ = R ρ$ (see timeevolution.bloch_redfield_tensor).`fout=nothing`

: If given, this function`fout(t, rho)`

is called every time an output should be displayed. ATTENTION: The given state rho is not permanent! It is still in use by the ode solver and therefore must not be changed.`kwargs...`

: Further arguments are passed on to the ode solver.

## Spectral analysis

`QuantumOptics.eigenstates`

— Function.`eigenstates(op::AbstractOperator[, n::Int; warning=true])`

Calculate the lowest n eigenvalues and their corresponding eigenstates.

This is just a thin wrapper around julia's `eigen`

and `eigs`

functions. Which of them is used depends on the type of the given operator. If more control about the way the calculation is done is needed, use the functions directly. More details can be found at [http://docs.julialang.org/en/stable/stdlib/linalg/].

NOTE: Especially for small systems full diagonalization with Julia's `eigen`

function is often more desirable. You can convert a sparse operator `A`

to a dense one using `dense(A)`

.

If the given operator is non-hermitian a warning is given. This behavior can be turned off using the keyword `warning=false`

.

For sparse operators by default it only returns the 6 lowest eigenvalues.

`QuantumOptics.eigenenergies`

— Function.`eigenenergies(op::AbstractOperator[, n::Int; warning=true])`

Calculate the lowest n eigenvalues.

This is just a thin wrapper around julia's `eigvals`

. If more control about the way the calculation is done is needed, use the function directly. More details can be found at [http://docs.julialang.org/en/stable/stdlib/linalg/].

If the given operator is non-hermitian a warning is given. This behavior can be turned off using the keyword `warning=false`

.

For sparse operators by default it only returns the 6 lowest eigenvalues.

`QuantumOptics.simdiag`

— Function.`simdiag(ops; atol, rtol)`

Simultaneously diagonalize commuting Hermitian operators specified in `ops`

.

This is done by diagonalizing the sum of the operators. The eigenvalues are computed by $a = ⟨ψ|A|ψ⟩$ and it is checked whether the eigenvectors fulfill the equation $A|ψ⟩ = a|ψ⟩$.

**Arguments**

`ops`

: Vector of sparse or dense operators.`atol=1e-14`

: kwarg of Base.isapprox specifying the tolerance of the approximate check`rtol=1e-14`

: kwarg of Base.isapprox specifying the tolerance of the approximate check

**Returns**

`evals_sorted`

: Vector containing all vectors of the eigenvalues sorted by the eigenvalues of the first operator.`v`

: Common eigenvectors.

## Steady-states

`QuantumOptics.steadystate.master`

— Function.`steadystate.master(H, J; <keyword arguments>)`

Calculate steady state using long time master equation evolution.

**Arguments**

`H`

: Arbitrary operator specifying the Hamiltonian.`J`

: Vector containing all jump operators which can be of any arbitrary operator type.`rho0=dm(basisstate(b))`

: Initial density operator. If not given the $|0⟩⟨0|$ state in respect to the choosen basis is used.`tol=1e-3`

: Tracedistance used as termination criterion.`hmin=1e-7`

: Minimal time step used in the time evolution.`rates=ones(N)`

: Vector or matrix specifying the coefficients for the jump operators.`Jdagger=dagger.(Jdagger)`

: Vector containing the hermitian conjugates of the jump operators. If they are not given they are calculated automatically.`fout=nothing`

: If given this function`fout(t, rho)`

is called every time an output should be displayed. To limit copying to a minimum the given density operator`rho`

is further used and therefore must not be changed.`kwargs...`

: Further arguments are passed on to the ode solver.

`QuantumOptics.steadystate.eigenvector`

— Function.```
steadystate.eigenvector(L)
steadystate.eigenvector(H, J)
```

Find steady state by calculating the eigenstate with eigenvalue 0 of the Liouvillian matrix `L`

, if it exists.

**Keyword arguments:**

`tol = 1e-9`

: Check`abs(eigenvalue) < tol`

to determine zero eigenvalue.`nev = 2`

: Number of calculated eigenvalues. If`nev > 1`

it is checked if there is only one eigenvalue with real part 0. No checks for`nev = 1`

: use if faster or for avoiding convergence errors of`eigs`

. Changing`nev`

thus only makes sense when using SparseSuperOperator.`which = :LR`

: Find eigenvalues with largest real part. Keyword for`eigs`

function (ineffective for DenseSuperOperator).`kwargs...`

: Keyword arguments for the Julia`eigen`

or`eigs`

function.

`QuantumOptics.steadystate.liouvillianspectrum`

— Function.```
steadystate.liouvillianspectrum(L)
steadystate.liouvillianspectrum(H, J)
```

Calculate eigenspectrum of the Liouvillian matrix `L`

. The eigenvalues and -states are sorted according to the absolute value of the eigenvalues.

**Keyword arguments:**

`nev = min(10, length(L.basis_r[1])*length(L.basis_r[2]))`

: Number of eigenvalues.`which = :LR`

: Find eigenvalues with largest real part. Keyword for`eigs`

function (ineffective for DenseSuperOperator).`kwargs...`

: Keyword arguments for the Julia`eigen`

or`eigens`

function.

## Time correlations

`QuantumOptics.timecorrelations.correlation`

— Function.`timecorrelations.correlation([tspan, ]rho0, H, J, op1, op2; <keyword arguments>)`

Calculate two time correlation values $⟨A(t)B(0)⟩$.

The calculation is done by multiplying the initial density operator with $B$ performing a time evolution according to a master equation and then calculating the expectation value $\mathrm{Tr} \{A ρ\}$

Without the `tspan`

argument the points in time are chosen automatically from the ode solver and the final time is determined by the steady state termination criterion specified in `steadystate.master`

.

**Arguments**

`tspan`

: Points of time at which the correlation should be calculated.`rho0`

: Initial density operator.`H`

: Operator specifying the Hamiltonian.`J`

: Vector of jump operators.`op1`

: Operator at time`t`

.`op2`

: Operator at time`t=0`

.`rates=ones(N)`

: Vector or matrix specifying the coefficients (decay rates) for the jump operators.`Jdagger=dagger.(J)`

: Vector containing the hermitian conjugates of the jump`kwargs...`

: Further arguments are passed on to the ode solver.

`QuantumOptics.timecorrelations.spectrum`

— Function.`timecorrelations.spectrum([omega_samplepoints,] H, J, op; <keyword arguments>)`

Calculate spectrum as Fourier transform of a correlation function

This is done with the Wiener-Khinchin theorem

The argument `omega_samplepoints`

gives the list of frequencies where $S(ω)$ is caclulated. A corresponding list of times is calculated internally by means of a inverse discrete frequency fourier transform. If not given, the steady-state is computed before calculating the auto-correlation function.

Without the `omega_samplepoints`

arguments the frequencies are chosen automatically.

**Arguments**

`omega_samplepoints`

: List of frequency points at which the spectrum is calculated.`H`

: Operator specifying the Hamiltonian.`J`

: Vector of jump operators.`op`

: Operator for which the auto-correlation function is calculated.`rho0`

: Initial density operator.`tol=1e-4`

: Tracedistance used as termination criterion.`rates=ones(N)`

: Vector or matrix specifying the coefficients for the jump operators.`Jdagger=dagger.(J)`

: Vector containing the hermitian conjugates of the jump operators. If they are not given they are calculated automatically.`kwargs...`

: Further arguments are passed on to the ode solver.

`timecorrelations.correlation2spectrum(tspan, corr; normalize_spec)`

Calculate spectrum as Fourier transform of a correlation function with a given correlation function.

**Arguments**

`tspan`

: List of time points corresponding to the correlation function.`corr`

: Correlation function of which the Fourier transform is to be calculated.`normalize_spec`

: Specify if spectrum should be normalized to its maximum.

## Semi-classical

Semi-classical state.

It consists of a quantum part, which is either a `Ket`

or a `DenseOperator`

and a classical part that is specified as a complex vector of arbitrary length.

`QuantumOptics.semiclassical.schroedinger_dynamic`

— Function.`semiclassical.schroedinger_dynamic(tspan, state0, fquantum, fclassical[; fout, ...])`

Integrate time-dependent Schrödinger equation coupled to a classical system.

**Arguments**

`tspan`

: Vector specifying the points of time for which the output should be displayed.`psi0`

: Initial semi-classical state`semiclassical.State`

.`fquantum`

: Function`f(t, psi, u) -> H`

returning the time and or state dependent Hamiltonian.`fclassical`

: Function`f(t, psi, u, du)`

calculating the possibly time and state dependent derivative of the classical equations and storing it in the vector`du`

.`fout=nothing`

: If given, this function`fout(t, state)`

is called every time an output should be displayed. ATTENTION: The given state is neither normalized nor permanent!`kwargs...`

: Further arguments are passed on to the ode solver.

`QuantumOptics.semiclassical.master_dynamic`

— Function.`semiclassical.master_dynamic(tspan, state0, fquantum, fclassical; <keyword arguments>)`

Integrate time-dependent master equation coupled to a classical system.

**Arguments**

`tspan`

: Vector specifying the points of time for which output should be displayed.`rho0`

: Initial semi-classical state`semiclassical.State`

.`fquantum`

: Function`f(t, rho, u) -> (H, J, Jdagger)`

returning the time and/or state dependent Hamiltonian and Jump operators.`fclassical`

: Function`f(t, rho, u, du)`

calculating the possibly time and state dependent derivative of the classical equations and storing it in the complex vector`du`

.`fout=nothing`

: If given, this function`fout(t, state)`

is called every time an output should be displayed. ATTENTION: The given state is not permanent!`kwargs...`

: Further arguments are passed on to the ode solver.

`QuantumOptics.semiclassical.mcwf_dynamic`

— Function.`semiclassical.mcwf_dynamic(tspan, psi0, fquantum, fclassical, fjump_classical; <keyword arguments>)`

Calculate MCWF trajectories coupled to a classical system.

**Arguments**

`tspan`

: Vector specifying the points of time for which output should be displayed.`psi0`

: Initial semi-classical state`semiclassical.State`

featuring a`Ket`

(@ref).`fquantum`

: Function`f(t, psi, u) -> (H, J, Jdagger)`

returning the time and/or state dependent Hamiltonian and Jump operators.`fclassical`

: Function`f(t, psi, u, du)`

calculating the possibly time and state dependent derivative of the classical equations and storing it in the complex vector`du`

.`fjump_classical`

: Function`f(t, psi, u, i)`

performing a classical jump when a quantum jump of the i-th jump operator occurs.`fout=nothing`

: If given, this function`fout(t, state)`

is called every time an output should be displayed. ATTENTION: The given state is not permanent!`display_beforeevent`

: Choose whether or not an additional point should be saved before a jump occurs. Default is false.`display_afterevent`

: Choose whether or not an additional point should be saved after a jump occurs. Default is false.`display_jumps=false`

: If set to true, an additional list of times and indices is returned. These correspond to the times at which a jump occured and the index of the jump operators with which the jump occured, respectively.`kwargs...`

: Further arguments are passed on to the ode solver.

## Stochastics

`QuantumOptics.stochastic.schroedinger`

— Function.`stochastic.schroedinger(tspan, state0, H, Hs[; fout, ...])`

Integrate stochastic Schrödinger equation.

**Arguments**

`tspan`

: Vector specifying the points of time for which the output should be displayed.`psi0`

: Initial state as Ket.`H`

: Deterministic part of the Hamiltonian.`Hs`

: Stochastic part(s) of the Hamiltonian (either an operator or a vector of operators).`fout=nothing`

: If given, this function`fout(t, state)`

is called every time an output should be displayed. ATTENTION: The given state is neither normalized nor permanent!`normalize_state=false`

: Specify whether or not to normalize the state after each time step taken by the solver.`kwargs...`

: Further arguments are passed on to the ode solver.

`QuantumOptics.stochastic.schroedinger_dynamic`

— Function.`stochastic.schroedinger_dynamic(tspan, state0, fdeterm, fstoch[; fout, ...])`

Integrate stochastic Schrödinger equation with dynamic Hamiltonian.

**Arguments**

`tspan`

: Vector specifying the points of time for which the output should be displayed.`psi0`

: Initial state.`fdeterm`

: Function`f(t, psi, u) -> H`

returning the deterministic (time- or state-dependent) part of the Hamiltonian.`fstoch`

: Function`f(t, psi, u, du) -> Hs`

returning a vector that contains the stochastic terms of the Hamiltonian.`fout=nothing`

: If given, this function`fout(t, state)`

is called every time an output should be displayed. ATTENTION: The given state is neither normalized nor permanent!`noise_processes=0`

: Number of distinct white-noise processes in the equation. This number has to be equal to the total number of noise operators returned by`fstoch`

. If unset, the number is calculated automatically from the function output. NOTE: Set this number if you want to avoid an initial calculation of the function output!`normalize_state=false`

: Specify whether or not to normalize the state after each time step taken by the solver.`kwargs...`

: Further arguments are passed on to the ode solver.

`QuantumOptics.stochastic.homodyne_carmichael`

— Function.`stochastic.homodyne_carmichael(H0, C, theta)`

Helper function that defines the functions needed to compute homodyne detection trajectories according to Carmichael with `stochastic.schroedinger_dynamic`

.

**Arguments**

`H0`

: The deterministic, time-independent system Hamiltonian.`C`

: Collapse operator (or vector of operators) of the detected output channel(s).`theta`

: The phase difference between the local oscillator and the signal field. Defines the operator of the measured quadrature as $X_\theta = C e^{-i\theta} + C^\dagger e^{i\theta}$. Needs to be a vector of the same length as`C`

if`C`

is a vector.`normalize_expect=true`

: Specifiy whether or not to normalize the state vector when the expectation value in the nonlinear term is calculated. NOTE: should only be set to`false`

if the state is guaranteed to be normalized, e.g. by setting`normalize_state=true`

in`stochastic.schroedinger_dynamic`

.

Returns `(fdeterm, fstoch)`

, where `fdeterm(t, psi) -> H`

and `fstoch(t, psi) -> Hs`

are functions returning the deterministic and stochastic part of the Hamiltonian required for calling `stochastic.schroedinger_dynamic`

.

The deterministic and stochastic parts of the Hamiltonian are constructed as

where

and

`QuantumOptics.stochastic.master`

— Function.`stochastic.master(tspan, rho0, H, J, C; <keyword arguments>)`

Time-evolution according to a stochastic master equation.

For dense arguments the `master`

function calculates the non-hermitian Hamiltonian and then calls master_nh which is slightly faster.

**Arguments**

`tspan`

: Vector specifying the points of time for which output should be displayed.`rho0`

: Initial density operator. Can also be a state vector which is automatically converted into a density operator.`H`

: Deterministic part of the Hamiltonian.`J`

: Vector containing all deterministic jump operators which can be of any arbitrary operator type.`C`

: Vector containing the stochastic operators for a superoperator of the form`C[i]*rho + rho*Cdagger[i]`

.`rates=nothing`

: Vector or matrix specifying the coefficients (decay rates) for the jump operators. If nothing is specified all rates are assumed to be 1.`Jdagger=dagger.(J)`

: Vector containing the hermitian conjugates of the jump operators. If they are not given they are calculated automatically.`fout=nothing`

: If given, this function`fout(t, rho)`

is called every time an output should be displayed. ATTENTION: The given state rho is not permanent! It is still in use by the ode solver and therefore must not be changed.`kwargs...`

: Further arguments are passed on to the ode solver.

`QuantumOptics.stochastic.master_dynamic`

— Function.`stochastic.master_dynamic(tspan, rho0, fdeterm, fstoch; <keyword arguments>)`

Time-evolution according to a stochastic master equation with a dynamic Hamiltonian and J.

**Arguments**

`tspan`

: Vector specifying the points of time for which output should be displayed.`rho0`

: Initial density operator. Can also be a state vector which is automatically converted into a density operator.`fdeterm`

: Function`f(t, rho) -> (H, J, Jdagger)`

or`f(t, rho) -> (H, J, Jdagger, rates)`

giving the deterministic part of the master equation.`fstoch`

: Function`f(t, rho) -> (C, Cdagger)`

giving the stochastic superoperator of the form`C[i]*rho + rho*Cdagger[i]`

.`rates=nothing`

: Vector or matrix specifying the coefficients (decay rates) for the jump operators. If nothing is specified all rates are assumed to be 1.`fout=nothing`

: If given, this function`fout(t, rho)`

is called every time an output should be displayed. ATTENTION: The given state rho is not permanent! It is still in use by the ode solver and therefore must not be changed.`noise_processes=0`

: Number of distinct white-noise processes in the equation. This number has to be equal to the total number of noise operators returned by`fstoch`

and all optional functions. If unset, the number is calculated automatically from the function outputs. NOTE: Set this number if you want to avoid an initial calculation of function outputs!`kwargs...`

: Further arguments are passed on to the ode solver.

`stochastic.schroedinger_semiclassical(tspan, state0, fquantum, fclassical[; fout, ...])`

Integrate time-dependent Schrödinger equation coupled to a classical system.

**Arguments**

`tspan`

: Vector specifying the points of time for which the output should be displayed.`state0`

: Initial semi-classical state`semiclassical.State`

.`fquantum`

: Function`f(t, psi, u) -> H`

returning the time and or state dependent Hamiltonian.`fclassical`

: Function`f(t, psi, u, du)`

calculating the possibly time and state dependent derivative of the classical equations and storing it in the vector`du`

.`fstoch_quantum=nothing`

: Function`f(t, psi, u) -> Hs`

that returns a vector of operators corresponding to the stochastic terms of the Hamiltonian. NOTE: Either this function or`fstoch_classical`

has to be defined.`fstoch_classical=nothing`

: Function`f(t, psi, u, du)`

that calculates the stochastic terms of the derivative`du`

. NOTE: Either this function or`fstoch_quantum`

has to be defined.`fout=nothing`

: If given, this function`fout(t, state)`

is called every time an output should be displayed. ATTENTION: The given state is neither normalized nor permanent!`noise_processes=0`

: Number of distinct quantum noise processes in the equation. This number has to be equal to the total number of noise operators returned by`fstoch`

. If unset, the number is calculated automatically from the function output. NOTE: Set this number if you want to avoid an initial calculation of the function output!`noise_prototype_classical=nothing`

: The equivalent of the optional argument`noise_rate_prototype`

in`StochasticDiffEq`

for the classical stochastic function`fstoch_classical`

only. Must be set for non-diagonal classical noise or combinations of quantum and classical noise. See the documentation for details.`normalize_state=false`

: Specify whether or not to normalize the state after each time step taken by the solver.`kwargs...`

: Further arguments are passed on to the ode solver.

`QuantumOptics.stochastic.master_semiclassical`

— Function.`stochastic.master_semiclassical(tspan, rho0, H, Hs, J; <keyword arguments>)`

Time-evolution according to a stochastic master equation. For dense arguments the `master`

function calculates the non-hermitian Hamiltonian and then calls master_nh which is slightly faster.

**Arguments**

`tspan`

: Vector specifying the points of time for which output should be displayed.`rho0`

: Initial density operator. Can also be a state vector which is automatically converted into a density operator.`fquantum`

: Function`f(t, rho, u) -> (H, J, Jdagger)`

or`f(t, rho, u) -> (H, J, Jdagger, rates)`

giving the deterministic part of the master equation.`fclassical`

: Function`f(t, rho, u, du)`

that calculates the classical derivatives`du`

.`fstoch_quantum=nothing`

: Function`f(t, rho, u) -> C, Cdagger`

that returns the stochastic operator for the superoperator of the form`C[i]*rho + rho*Cdagger[i]`

.`fstoch_classical=nothing`

: Function`f(t, rho, u, du)`

that calculates the stochastic terms of the derivative`du`

.`rates=nothing`

: Vector or matrix specifying the coefficients (decay rates) for the jump operators. If nothing is specified all rates are assumed to be 1.`fout=nothing`

: If given, this function`fout(t, rho)`

is called every time an output should be displayed. ATTENTION: The given state rho is not permanent! It is still in use by the ode solver and therefore must not be changed.`noise_processes=0`

: Number of distinct quantum noise processes in the equation. This number has to be equal to the total number of noise operators returned by`fstoch`

. If unset, the number is calculated automatically from the function output. NOTE: Set this number if you want to avoid an initial calculation of the function output!`noise_prototype_classical=nothing`

: The equivalent of the optional argument`noise_rate_prototype`

in`StochasticDiffEq`

for the classical stochastic function`fstoch_classical`

only. Must be set for non-diagonal classical noise or combinations of quantum and classical noise. See the documentation for details.`kwargs...`

: Further arguments are passed on to the ode solver.

## State definitions

`QuantumOpticsBase.randstate`

— Function.`randstate(basis)`

Calculate a random normalized ket state.

`QuantumOpticsBase.thermalstate`

— Function.`thermalstate(H,T)`

Thermal state $exp(-H/T)/Tr[exp(-H/T)]$.

`QuantumOpticsBase.coherentthermalstate`

— Function.`coherentthermalstate(basis::FockBasis,H,T,alpha)`

Coherent thermal state $D(α)exp(-H/T)/Tr[exp(-H/T)]D^†(α)$.

`QuantumOpticsBase.phase_average`

— Function.`phase_average(rho)`

Returns the phase-average of $ρ$ containing only the diagonal elements.

`QuantumOpticsBase.passive_state`

— Function.`passive_state(rho,IncreasingEigenenergies::Bool=true)`

Passive state $π$ of $ρ$. IncreasingEigenenergies=true means that higher indices correspond to higher energies.

## Pauli

`QuantumOpticsBase.PauliBasis`

— Type.`PauliBasis(num_qubits::Int)`

Basis for an N-qubit space where `num_qubits`

specifies the number of qubits. The dimension of the basis is 2²ᴺ.

`Base class for Pauli transfer matrix classes.`

`DensePauliTransferMatrix(B1, B2, data)`

DensePauliTransferMatrix stored as a dense matrix.

`QuantumOpticsBase.ChiMatrix`

— Type.`Base class for χ (process) matrix classes.`

`QuantumOpticsBase.DenseChiMatrix`

— Type.`DenseChiMatrix(b, b, data)`

DenseChiMatrix stored as a dense matrix.

## Printing

`QuantumOpticsBase.set_printing`

— Function.`QuantumOptics.set_printing(; standard_order, rounding_tol)`

Set options for REPL output.

**Arguments**

`standard_order=false`

: For performance reasons, the order of the tensor product is inverted, i.e.`tensor(a, b)=kron(b, a)`

. When changing this to`true`

, the output shown in the REPL will exhibit the correct order.`rounding_tol=1e-17`

: Tolerance for floating point errors shown in the output.