Bloch Redfield master equation implementation

CHANGELOG.md

@@ -8,6 +8,7 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 ## [Unreleased](https://github.com/qutip/QuantumToolbox.jl/tree/main)
 
 - Introduce `Lanczos` solver for `spectrum`. ([#476])
+- Add Bloch-Redfield master equation solver. ([#473])
 
 ## [v0.31.1]
 Release date: 2025-05-16
@@ -230,4 +231,5 @@ Release date: 2024-11-13
 [#455]: https://github.com/qutip/QuantumToolbox.jl/issues/455
 [#456]: https://github.com/qutip/QuantumToolbox.jl/issues/456
 [#460]: https://github.com/qutip/QuantumToolbox.jl/issues/460
+[#473]: https://github.com/qutip/QuantumToolbox.jl/issues/473
 [#476]: https://github.com/qutip/QuantumToolbox.jl/issues/476

docs/src/resources/api.md

@@ -208,6 +208,9 @@ smesolve
 dfd_mesolve
 liouvillian
 liouvillian_generalized
+bloch_redfield_tensor
+brterm
+brmesolve
 ```
 
 ### [Steady State Solvers](@id doc-API:Steady-State-Solvers)

src/QuantumToolbox.jl

@@ -106,6 +106,7 @@ include("time_evolution/callback_helpers/mcsolve_callback_helpers.jl")
 include("time_evolution/callback_helpers/ssesolve_callback_helpers.jl")
 include("time_evolution/callback_helpers/smesolve_callback_helpers.jl")
 include("time_evolution/mesolve.jl")
+include("time_evolution/brmesolve.jl")
 include("time_evolution/lr_mesolve.jl")
 include("time_evolution/sesolve.jl")
 include("time_evolution/mcsolve.jl")

src/time_evolution/brmesolve.jl

@@ -0,0 +1,196 @@
+export bloch_redfield_tensor, brterm, brmesolve
+
+@doc raw"""
+    bloch_redfield_tensor(
+        H::QuantumObject{Operator},
+        a_ops::Union{AbstractVector, Tuple},
+        c_ops::Union{AbstractVector, Tuple, Nothing}=nothing;
+        sec_cutoff::Real=0.1,
+        fock_basis::Union{Val,Bool}=Val(false)
+    )
+
+Calculates the Bloch-Redfield tensor ([`SuperOperator`](@ref)) for a system given a set of operators and corresponding spectral functions that describes the system's coupling to its environment.
+
+## Arguments
+
+- `H`: The system Hamiltonian. Must be an [`Operator`](@ref)
+- `a_ops`: Nested list with each element is a `Tuple` of operator-function pairs `(a_op, spectra)`, and the coupling [`Operator`](@ref) `a_op` must be hermitian with corresponding `spectra` being a `Function` of transition energy
+- `c_ops`: List of collapse operators corresponding to Lindblad dissipators
+- `sec_cutoff`: Cutoff for secular approximation. Use `-1` if secular approximation is not used when evaluating bath-coupling terms.
+- `fock_basis`: Whether to return the tensor in the input (fock) basis or the diagonalized (eigen) basis.
+
+## Return
+
+The return depends on `fock_basis`.
+
+- `fock_basis=Val(true)`: return the Bloch-Redfield tensor (in the fock basis) only.
+- `fock_basis=Val(false)`: return the Bloch-Redfield tensor (in the eigen basis) along with the transformation matrix from eigen to fock basis.
+"""
+function bloch_redfield_tensor(
+    H::QuantumObject{Operator},
+    a_ops::Union{AbstractVector,Tuple},
+    c_ops::Union{AbstractVector,Tuple,Nothing} = nothing;
+    sec_cutoff::Real = 0.1,
+    fock_basis::Union{Val,Bool} = Val(false),
+)
+    rst = eigenstates(H)
+    U = QuantumObject(rst.vectors, Operator(), H.dimensions)
+    sec_cutoff = float(sec_cutoff)
+
+    # in fock basis
+    R0 = liouvillian(H, c_ops)
+
+    # set fock_basis=Val(false) and change basis together at the end
+    R1 = 0
+    isempty(a_ops) || (R1 += mapreduce(x -> _brterm(rst, x[1], x[2], sec_cutoff, Val(false)), +, a_ops))
+
+    SU = sprepost(U, U') # transformation matrix from eigen basis back to fock basis
+    if getVal(fock_basis)
+        return R0 + SU * R1 * SU'
+    else
+        return SU' * R0 * SU + R1, U
+    end
+end
+
+@doc raw"""
+    brterm(
+        H::QuantumObject{Operator},
+        a_op::QuantumObject{Operator}, 
+        spectra::Function;
+        sec_cutoff::Real=0.1, 
+        fock_basis::Union{Bool, Val}=Val(false)
+    )
+
+Calculates the contribution of one coupling operator to the Bloch-Redfield tensor.
+
+## Argument
+
+- `H`: The system Hamiltonian. Must be an [`Operator`](@ref)
+- `a_op`: The operator coupling to the environment. Must be hermitian.
+- `spectra`: The corresponding environment spectra as a `Function` of transition energy.
+- `sec_cutoff`: Cutoff for secular approximation. Use `-1` if secular approximation is not used when evaluating bath-coupling terms.
+- `fock_basis`: Whether to return the tensor in the input (fock) basis or the diagonalized (eigen) basis.
+
+## Return
+
+The return depends on `fock_basis`.
+
+- `fock_basis=Val(true)`: return the Bloch-Redfield term (in the fock basis) only.
+- `fock_basis=Val(false)`: return the Bloch-Redfield term (in the eigen basis) along with the transformation matrix from eigen to fock basis.
+"""
+function brterm(
+    H::QuantumObject{Operator},
+    a_op::QuantumObject{Operator},
+    spectra::Function;
+    sec_cutoff::Real = 0.1,
+    fock_basis::Union{Bool,Val} = Val(false),
+)
+    rst = eigenstates(H)
+    term = _brterm(rst, a_op, spectra, sec_cutoff, makeVal(fock_basis))
+    if getVal(fock_basis)
+        return term
+    else
+        return term, Qobj(rst.vectors, Operator(), rst.dimensions)
+    end
+end
+
+function _brterm(
+    rst::EigsolveResult,
+    a_op::T,
+    spectra::F,
+    sec_cutoff::Real,
+    fock_basis::Union{Val{true},Val{false}},
+) where {T<:QuantumObject{Operator},F<:Function}
+    _check_br_spectra(spectra)
+
+    U = rst.vectors
+    Id = I(prod(rst.dimensions))
+
+    skew = @. rst.values - rst.values' |> real
+    spectrum = spectra.(skew)
+
+    A_mat = U' * a_op.data * U
+
+    ac_term = (A_mat .* spectrum) * A_mat
+    bd_term = A_mat * (A_mat .* trans(spectrum))
+
+    if sec_cutoff != -1
+        m_cut = similar(skew)
+        map!(x -> abs(x) < sec_cutoff, m_cut, skew)
+        ac_term .*= m_cut
+        bd_term .*= m_cut
+
+        vec_skew = vec(skew)
+        M_cut = @. abs(vec_skew - vec_skew') < sec_cutoff
+    end
+
+    out =
+        0.5 * (
+            + _sprepost(A_mat .* trans(spectrum), A_mat) + _sprepost(A_mat, A_mat .* spectrum) - _spost(ac_term, Id) -
+            _spre(bd_term, Id)
+        )
+
+    (sec_cutoff != -1) && (out .*= M_cut)
+
+    if getVal(fock_basis)
+        SU = _sprepost(U, U')
+        return QuantumObject(SU * out * SU', SuperOperator(), rst.dimensions)
+    else
+        return QuantumObject(out, SuperOperator(), rst.dimensions)
+    end
+end
+
+@doc raw"""
+    brmesolve(
+        H::QuantumObject{Operator}, 
+        ψ0::QuantumObject,
+        tlist::AbstractVector, 
+        a_ops::Union{Nothing, AbstractVector, Tuple}, 
+        c_ops::Union{Nothing, AbstractVector, Tuple}=nothing;
+        sec_cutoff::Real=0.1,
+        e_ops::Union{Nothing, AbstractVector}=nothing,
+        kwargs...,
+    )
+
+Solves for the dynamics of a system using the Bloch-Redfield master equation, given an input Hamiltonian, Hermitian bath-coupling terms and their associated spectral functions, as well as possible Lindblad collapse operators.
+
+## Arguments
+
+- `H`: The system Hamiltonian. Must be an [`Operator`](@ref)
+- `ψ0`:  Initial state of the system $|\psi(0)\rangle$. It can be either a [`Ket`](@ref), [`Operator`](@ref) or [`OperatorKet`](@ref).
+- `tlist`: List of times at which to save either the state or the expectation values of the system.
+- `a_ops`: Nested list with each element is a `Tuple` of operator-function pairs `(a_op, spectra)`, and the coupling [`Operator`](@ref) `a_op` must be hermitian with corresponding `spectra` being a `Function` of transition energy
+- `c_ops`: List of collapse operators corresponding to Lindblad dissipators
+- `sec_cutoff`: Cutoff for secular approximation. Use `-1` if secular approximation is not used when evaluating bath-coupling terms.
+- `e_ops`: List of operators for which to calculate expectation values. It can be either a `Vector` or a `Tuple`.
+- `kwargs`: Keyword arguments for [`mesolve`](@ref).
+
+## Notes
+
+- This function will automatically generate the [`bloch_redfield_tensor`](@ref) and solve the time evolution with [`mesolve`](@ref).
+
+# Returns
+
+- `sol::TimeEvolutionSol`: The solution of the time evolution. See also [`TimeEvolutionSol`](@ref)
+"""
+function brmesolve(
+    H::QuantumObject{Operator},
+    ψ0::QuantumObject,
+    tlist::AbstractVector,
+    a_ops::Union{Nothing,AbstractVector,Tuple},
+    c_ops::Union{Nothing,AbstractVector,Tuple} = nothing;
+    sec_cutoff::Real = 0.1,
+    e_ops::Union{Nothing,AbstractVector} = nothing,
+    kwargs...,
+)
+    R = bloch_redfield_tensor(H, a_ops, c_ops; sec_cutoff = sec_cutoff, fock_basis = Val(true))
+
+    return mesolve(R, ψ0, tlist, nothing; e_ops = e_ops, kwargs...)
+end
+
+function _check_br_spectra(f::Function)
+    meths = methods(f, [Real])
+    length(meths.ms) == 0 &&
+        throw(ArgumentError("The following function must accept one argument: `$(meths.mt.name)(ω)` with ω<:Real"))
+    return nothing
+end

test/core-test/brmesolve.jl

@@ -0,0 +1,115 @@
+@testitem "Bloch-Redfield tensor sec_cutoff" begin
+    N = 5
+    H = num(N)
+    a = destroy(N)
+    A_op = a+a'
+    spectra(x) = (x>0) * 0.5
+    for sec_cutoff in [0, 0.1, 1, 3, -1]
+        R = bloch_redfield_tensor(H, [(A_op, spectra)], [a^2], sec_cutoff = sec_cutoff, fock_basis = true)
+        R_eig, evecs = bloch_redfield_tensor(H, [(A_op, spectra)], [a^2], sec_cutoff = sec_cutoff, fock_basis = false)
+        @test isa(R, QuantumObject)
+        @test isa(R_eig, QuantumObject)
+        @test isa(evecs, QuantumObject)
+
+        state = rand_dm(N) |> mat2vec
+        fock_computed = R * state
+        eig_computed = (sprepost(evecs, evecs') * R_eig * sprepost(evecs', evecs)) * state
+        @test isapprox(fock_computed, eig_computed, atol = 1e-15)
+    end
+end
+
+@testitem "Compare brterm and Lindblad" begin
+    N = 5
+    H = num(N)
+    a = destroy(N) + destroy(N)^2/2
+    A_op = a+a'
+    spectra(x) = x>0
+
+    # this test applies for limited cutoff
+    lindblad = lindblad_dissipator(a)
+    computation = brterm(H, A_op, spectra, sec_cutoff = 1.5, fock_basis = true)
+    @test isapprox(lindblad, computation, atol = 1e-15)
+end
+
+@testitem "brterm basis" begin
+    N = 5
+    H = num(N)
+    a = destroy(N) + destroy(N)^2/2
+    A_op = a+a'
+    spectra(x) = x>0
+    for sec_cutoff in [0, 0.1, 1, 3, -1]
+        R = brterm(H, A_op, spectra, sec_cutoff = sec_cutoff, fock_basis = true)
+        R_eig, evecs = brterm(H, A_op, spectra, sec_cutoff = sec_cutoff, fock_basis = false)
+        @test isa(R, QuantumObject)
+        @test isa(R_eig, QuantumObject)
+        @test isa(evecs, QuantumObject)
+
+        state = rand_dm(N) |> mat2vec
+        fock_computed = R * state
+        eig_computed = (sprepost(evecs, evecs') * R_eig * sprepost(evecs', evecs)) * state
+        @test isapprox(fock_computed, eig_computed, atol = 1e-15)
+    end;
+end
+
+@testitem "brterm sprectra function" begin
+    f(x) = exp(x)/10
+    function g(x)
+        nbar = n_thermal(abs(x), 1)
+        if x > 0
+            return nbar
+        elseif x < 0
+            return 1 + nbar
+        else
+            return 0.0
+        end
+    end
+
+    spectra_list = [
+        x -> (x>0),  # positive frequency filter
+        x -> one(x), # no filter
+        f, # smooth filter
+        g, # thermal field
+    ]
+
+    N = 5
+    H = num(N)
+    a = destroy(N) + destroy(N)^2/2
+    A_op = a+a'
+    for spectra in spectra_list
+        R = brterm(H, A_op, spectra, sec_cutoff = 0.1, fock_basis = true)
+        R_eig, evecs = brterm(H, A_op, spectra, sec_cutoff = 0.1, fock_basis = false)
+        @test isa(R, QuantumObject)
+        @test isa(R_eig, QuantumObject)
+        @test isa(evecs, QuantumObject)
+
+        state = rand_dm(N) |> mat2vec
+        fock_computed = R * state
+        eig_computed = (sprepost(evecs, evecs') * R_eig * sprepost(evecs', evecs)) * state
+        @test isapprox(fock_computed, eig_computed, atol = 1e-15)
+    end
+end
+
+@testitem "simple qubit system" begin
+    pauli_vectors = [sigmax(), sigmay(), sigmaz()]
+    γ = 0.25
+    spectra(x) = γ * (x>=0)
+    _m_c_op = √γ * sigmam()
+    _z_c_op = √γ * sigmaz()
+    _x_a_op = (sigmax(), spectra)
+
+    arg_sets = [[[_m_c_op], [], [_x_a_op]], [[_m_c_op], [_m_c_op], []], [[_m_c_op, _z_c_op], [_z_c_op], [_x_a_op]]]
+
+    δ = 0
+    ϵ = 0.5 * 2π
+    e_ops = pauli_vectors
+    H = δ * 0.5 * sigmax() + ϵ * 0.5 * sigmaz()
+    ψ0 = unit(2basis(2, 0) + basis(2, 1))
+    times = LinRange(0, 10, 100)
+
+    for (me_c_ops, brme_c_ops, brme_a_ops) in arg_sets
+        me = mesolve(H, ψ0, times, me_c_ops, e_ops = e_ops, progress_bar = Val(false))
+        brme = brmesolve(H, ψ0, times, brme_a_ops, brme_c_ops, e_ops = e_ops, progress_bar = Val(false))
+
+        @test all(me.expect .== brme.expect)
+    end
+end;