API

API

Quantum-objects

Types

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.

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.

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.

Abstract base class for Bra and Ket states.

The state vector class stores the coefficients of an abstract state in respect to a certain basis. These coefficients are stored in the data field and the basis is defined in the basis field.

Bra(b::Basis[, data])

Bra state defined by coefficients in respect to the basis.

Ket(b::Basis[, data])

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

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.

DenseOperator(b1[, b2, data])

Dense array implementation of Operator.

The matrix consisting of complex floats is stored in the data field.

SparseOperator(b1[, b2, data])

Sparse array implementation of Operator.

The matrix is stored as the julia built-in type SparseMatrixCSC in the data field.

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.

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.

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.

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.

\[A_{bl_1,bl_2} = S_{(bl_1,bl_2) ↔ (br_1,br_2)} B_{br_1,br_2} \\ A_{br_1,br_2} = B_{bl_1,bl_2} S_{(bl_1,bl_2) ↔ (br_1,br_2)}\]
DenseSuperOperator(b1[, b2, data])

SuperOperator stored as dense matrix.

SparseSuperOperator(b1[, b2, data])

SuperOperator stored as sparse matrix.

Functions

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

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

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.

identityoperator(a::Basis[, b::Basis])

Return an identityoperator in the given bases.

diagonaloperator(b::Basis)

Create a diagonal operator of type SparseOperator.

randoperator(b1[, b2])

Calculate a random unnormalized dense operator.

spre(op)

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

For operators $A$, $B$ the relation

\[ \mathrm{spre}(A) B = A B\]

holds. op can be a dense or a sparse operator.

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

For operators $A$, $B$ the relation

\[ \mathrm{spost}(A) B = B A\]

holds. op can be a dense or a sparse operator.

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

\[S ρ = -\frac{i}{ħ} [H, ρ] + \sum_i J_i ρ J_i^† - \frac{1}{2} J_i^† J_i ρ - \frac{1}{2} ρ J_i^† J_i\]

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.
samebases(a, b)

Test if two objects have the same bases.

check_samebases(a, b)

Throw an IncompatibleBases error if the objects don't have the same bases.

multiplicable(a, b)

Check if two objects are multiplicable.

check_multiplicable(a, b)

Throw an IncompatibleBases error if the objects are not multiplicable.

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.

dagger(x)

Hermitian conjugate.

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.

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

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

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.

projector(a::Ket, b::Bra)

Projection operator $|a⟩⟨b|$.

projector(a::Ket)

Projection operator $|a⟩⟨a|$.

projector(a::Bra)

Projection operator $|a⟩⟨a|$.

dm(a::StateVector)

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

LinearAlgebra.normMethod.
norm(x::StateVector)

Norm of the given bra or ket state.

LinearAlgebra.trFunction.
tr(x::AbstractOperator)

Trace of the given operator.

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.

normalize(x::StateVector)

Return the normalized state so that norm(x) is one.

normalize(op)

Return the normalized operator so that its tr(op) is one.

normalize!(x::StateVector)

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

normalize!(op)

In-place normalization of the given operator so that its tr(x) is one.

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.

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

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

Tensor product of operators 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.

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.expMethod.
exp(op::AbstractOperator)

Operator exponential.

Base.expMethod.
exp(op::DenseSuperOperator)

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

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.

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.

dense(op::AbstractOperator)

Convert an arbitrary Operator into a DenseOperator.

sparse(op::AbstractOperator)

Convert an arbitrary operator into a SparseOperator.

Exceptions

Exception that should be raised for an illegal algebraic operation.

Quantum systems

Fock

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.

number(b::FockBasis)

Number operator for the specified Fock space.

destroy(b::FockBasis)

Annihilation operator for the specified Fock space.

create(b::FockBasis)

Creation operator for the specified Fock space.

displace(b::FockBasis, alpha)

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

fockstate(b::FockBasis, n)

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

coherentstate(b::FockBasis, alpha)

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

Phase space

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 $α = \frac{1}{\sqrt{2}}(x + i y)$.

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 $α = \frac{1}{\sqrt{2}}(x + i y)$.

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.

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.

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

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

NLevelBasis(N)

Basis for a system consisting of N states.

transition(b::NLevelBasis, to::Int, from::Int)

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

nlevelstate(b::NLevelBasis, n::Int)

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

Spin

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.

sigmax(b::SpinBasis)

Pauli $σ_x$ operator for the given Spin basis.

sigmay(b::SpinBasis)

Pauli $σ_y$ operator for the given Spin basis.

sigmaz(b::SpinBasis)

Pauli $σ_z$ operator for the given Spin basis.

sigmap(b::SpinBasis)

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

sigmam(b::SpinBasis)

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

spinup(b::SpinBasis)

Spin up state for the given Spin basis.

spindown(b::SpinBasis)

Spin down state for the given Spin basis.

Particle

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 $dp=(p_{max}-p_{min})/N$.

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 $dx=(x_{max}-x_{min})/N$.

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.

samplepoints(b::PositionBasis)

x values of the real space basis.

samplepoints(b::MomentumBasis)

p values of the momentum basis.

Base.positionMethod.
position(b::PositionBasis)

Position operator in real space.

Base.positionMethod.
position(b:MomentumBasis)

Position operator in momentum space.

momentum(b::PositionBasis)

Momentum operator in real space.

momentum(b:MomentumBasis)

Momentum operator in momentum space.

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

Create a Gaussian state around x0 andp0 with width sigma.

In real space the gaussian state is defined as

\[\Psi(x) = \frac{\sqrt{\Delta x}}{\pi^{1/4}\sqrt{\sigma}} e^{i p_0 (x-\frac{x_0}{2}) - \frac{(x-x_0)^2}{2 \sigma^2}}\]

and is connected to the momentum space definition

\[\Psi(p) = \frac{\sqrt{\sigma} \sqrt{\Delta x}}{\pi^{1/4}} e^{-i x_0 (p-\frac{p_0}{2}) - \frac{1}{2}(p-p_0)^2 \sigma^2}\]

via a Fourier-transformation

\[\Psi(p) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ipx}\Psi(x) \mathrm{d}x\]

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 factora $\sqrt{Δx}$ and $\sqrt{Δp}$ are used so that $Ψx_i = \sqrt{Δ x} Ψ(x_i)$ and $Ψp_i = \sqrt{Δ x} Ψ(p_i)$ so that the resulting Ket state is normalized.

FFTOperator

Abstract type for all implementations of FFT operators.

FFTOperators

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

FFTKets

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

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

\[ψ(x_i) = \sum_{n=0}^N ⟨x_i|n⟩ ψ_n\]

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.:

\[⟨x_i|n⟩ = π^{-\frac{1}{4}} \frac{e^{-\frac{1}{2}\left(\frac{x}{x_0}\right)^2}}{\sqrt{x_0}} \frac{1}{\sqrt{2^n n!}} H_n\left(\frac{x}{x_0}\right)\]

Subspace bases

SubspaceBasis(basisstates)

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

orthonormalize(b::SubspaceBasis)

Orthonormalize the basis states of the given SubspaceBasis

A modified Gram-Schmidt process is used.

projector(b1, b2)

Projection operator between subspaces and superspaces or between two subspaces.

Many-body

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.

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.

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.

number(b::ManyBodyBasis, index)

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

number(b::ManyBodyBasis)

Total particle number operator.

destroy(b::ManyBodyBasis, index)

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

create(b::ManyBodyBasis, index)

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

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

\[X = \sum_{ij} a_i^† a_j ⟨u_i| x | u_j⟩\]

and for the interaction case by

\[X = \sum_{ijkl} a_i^† a_j^† a_k a_l ⟨u_i|⟨u_j| x |u_k⟩|u_l⟩\]

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.

onebodyexpect(op, state)

Expectation value of the one-body operator op in respect to the many-body state.

Metrics

tracenorm(rho)

Trace norm of rho.

It is defined as

\[T(ρ) = Tr\{\sqrt{ρ^† ρ}\}.\]

Depending if rho is hermitian either tracenorm_h or tracenorm_nh is called.

tracenorm_h(rho)

Trace norm of rho.

It uses the identity

\[T(ρ) = Tr\{\sqrt{ρ^† ρ}\} = \sum_i |λ_i|\]

where $λ_i$ are the eigenvalues of rho.

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

\[ T(ρ) = Tr\{\sqrt{ρ^† ρ}\} = \sum_i σ_i\]

where $σ_i$ are the singular values of rho.

tracedistance(rho, sigma)

Trace distance between rho and sigma.

It is defined as

\[T(ρ,σ) = \frac{1}{2} Tr\{\sqrt{(ρ - σ)^† (ρ - σ)}\}.\]

It calls tracenorm which in turn either uses tracenorm_h or tracenorm_nh depending if $ρ-σ$ is hermitian or not.

tracedistance_h(rho, sigma)

Trace distance between rho and sigma.

It uses the identity

\[T(ρ,σ) = \frac{1}{2} Tr\{\sqrt{(ρ - σ)^† (ρ - σ)}\} = \frac{1}{2} \sum_i |λ_i|\]

where $λ_i$ are the eigenvalues of rho - sigma.

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

\[ T(ρ,σ) = \frac{1}{2} Tr\{\sqrt{(ρ - σ)^† (ρ - σ)}\} = \frac{1}{2} \sum_i σ_i\]

where $σ_i$ are the singular values of rho - sigma.

entropy_vn(rho)

Von Neumann entropy of a density matrix.

The Von Neumann entropy of a density operator is defined as

\[S(ρ) = -Tr(ρ \log(ρ)) = -\sum_n λ_n\log(λ_n)\]

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.
fidelity(rho, sigma)

Fidelity of two density operators.

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

\[F(ρ, σ) = Tr\left(\sqrt{\sqrt{ρ}σ\sqrt{ρ}}\right),\]

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

ptranspose(rho, index)

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

PPT(rho, index)

Peres-Horodecki criterion of partial transpose.

negativity(rho, index)

Negativity of rho with respect to subsystem index.

The negativity of a density matrix ρ is defined as

\[N(ρ) = \|ρᵀ\|,\]

where ρᵀ is the partial transpose.

logarithmic_negativity(rho, index)

The logarithmic negativity of a density matrix ρ is defined as

\[N(ρ) = \log₂\|ρᵀ\|,\]

where ρᵀ is the partial transpose.

Time-evolution

Schroedinger

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.
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

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.
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.

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:

\[H_{nh} = H - \frac{i}{2} \sum_k J^†_k J_k\]

Further information can be found at master.

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:

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.
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.

\[H_{nh} = H - \frac{i}{2} \sum_k J^†_k J_k\]

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.

Monte Carlo wave 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.
  • kwargs...: Further arguments are passed on to the ode solver.
mcwf_h(tspan, rho0, Hnh, J; <keyword arguments>)

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

For more information see: mcwf

mcwf_nh(tspan, rho0, Hnh, J; <keyword arguments>)

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

\[H_{nh} = H - \frac{i}{2} \sum_k J^†_k J_k\]

For more information see: mcwf

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.
  • kwargs...: Further arguments are passed on to the ode solver.
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

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.

Spectral analysis

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.

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.

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

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.
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.
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

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.
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

\[S(ω, t) = 2\Re\left\{\int_0^{∞} dτ e^{-iωτ}⟨A^†(t+τ)A(t)⟩\right\}\]

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.

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.
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.

Stochastics

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.
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.
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

\[H_{det} = H_0 + H_{nl},\]

where

\[H_{nl} = iCe^{-i\theta} \langle X_\theta \rangle - \frac{i}{2} C^\dagger C,\]

and

\[H_s = iCe^{-i\theta}.\]
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.
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.
semiclassical.schroedinger_stochastic(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.
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

randstate(basis)

Calculate a random normalized ket state.

thermalstate(H,T)

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

coherentthermalstate(basis::FockBasis,H,T,alpha)

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

phase_average(rho)

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

passive_state(rho,IncreasingEigenenergies::Bool=true)

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

Printing

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.