Hierarchy Dictionary

Introduction

For hierarchical equations of motions, there are many indices the users have to deal with including the indices of the Exponent in bosonic baths, the Exponent in fermionic baths, and the ADOs formed by the hierarchy.

With the auxiliary density operators $\rho_{\textbf{j}\vert\textbf{q}}^{(m,n,p)}$, we use the following keywords :

idx : the index of the auxiliary density operators
lvl : the level $m$ ($n$) of the bosonic (fermionic) hierarchy
nvec : object Nvec which stores the number of existence for each multi-index ensemble $j$ ($q$) in vector $\textbf{j}$ ($\textbf{q}$).

Dictionary for Pure Bosonic or Fermionic Baths

An object which contains all dictionaries for pure (bosonic or fermionic) bath-ADOs hierarchy, defined as:

struct HierarchyDict <: AbstractHierarchyDict

HierarchyDict can be obtained from the field .hierarchy in M_Boson or M_Fermion, and it contains the following fields :

idx2nvec : Return the Nvec from a given index of ADO
nvec2idx : Return the index of ADO from a given Nvec
lvl2idx : Return the list of ADO-indices from a given hierarchy level
bathPtr : Records the tuple $(\alpha, k)$ for each position in Nvec, where $\alpha$ and $k$ represents the $k$-th exponential-expansion term of the $\alpha$-th bath.

# HEOMLS for bosonic baths
M::M_Boson
HDict = M.hierarchy

# HEOMLS for fermionic baths
M::M_Fermion
HDict = M.hierarchy

# obtain the nvec corresponds to 10-th ADO
nvec = HDict.idx2nvec[10]

# obtain the index of the ADO corresponds to the given nvec
nvec::Nvec
idx = HDict.nvec2idx[nvec]

# obtain a list of indices which corresponds to all ADOs in 3rd-level of hierarchy
idx_list = HDict.lvl2idx[3]

HierarchicalEOM.jl also provides a function getIndexEnsemble(nvec, bathPtr) to obtain the index of the Exponent and it's corresponding index of bath:

# HEOMLS
M::M_Boson
M::Fermion

HDict = M.hierarchy

# auxiliary density operators
ados::ADOs

for (idx, ado) in enumerate(ados)
    ado # the corresponding auxiliary density operator for idx

    # obtain the nvec corresponds to ado
    nvec = HDict.idx2nvec[idx]

    for (α, k, n) in getIndexEnsemble(nvec, HDict.bathPtr)
        α  # index of the bath
        k  # the index of the exponential-expansion term in α-th bath
        n  # the repetition number of the ensemble {α, k} in vector j (or q) in ADOs
        exponent = M.bath[α][k]  # the k-th exponential-expansion term in α-th bath

        # do some calculations you want
    end
end

Dictionary for Mixed Bosonic and Fermionic Baths

An object which contains all dictionaries for mixed (bosonic and fermionic) bath-ADOs hierarchy, defined as:

struct MixHierarchyDict <: AbstractHierarchyDict

MixHierarchyDict can be obtained from the field .hierarchy in M_Boson_Fermion, and it contains the following fields :

idx2nvec : Return the tuple (Nvec_b, Nvec_f) from a given index of ADO, where b represents boson and f represents fermion
nvec2idx : Return the index from a given tuple (Nvec_b, Nvec_f), where b represents boson and f represents fermion
Blvl2idx : Return the list of ADO-indices from a given bosonic-hierarchy level
Flvl2idx : Return the list of ADO-indices from a given fermionic-hierarchy level
bosonPtr : Records the tuple $(\alpha, k)$ for each position in Nvec_b, where $\alpha$ and $k$ represents the $k$-th exponential-expansion term of the $\alpha$-th bosonic bath.
fermionPtr : Records the tuple $(\alpha, k)$ for each position in Nvec_f, where $\alpha$ and $k$ represents the $k$-th exponential-expansion term of the $\alpha$-th fermionic bath.

# HEOMLS 
M::M_Boson_Fermion
HDict = M.hierarchy

# obtain the nvec(s) correspond to 10-th ADO
nvec_b, nvec_f = HDict.idx2nvec[10]

# obtain the index of the ADO corresponds to the given nvec
nvec_b::Nvec
nvec_f::Nvec
idx = HDict.nvec2idx[(nvec_b, nvec_f)]

# obtain a list of indices which corresponds to all ADOs in 3rd-bosonic-level of hierarchy
idx_list = HDict.Blvl2idx[3] 

# obtain a list of indices which corresponds to all ADOs in 4rd-fermionic-level of hierarchy
idx_list = HDict.Flvl2idx[4]

HierarchicalEOM.jl also provides a function getIndexEnsemble(nvec, bathPtr) to obtain the index of the Exponent and it's corresponding index of bath:

# HEOMLS
M::M_Boson_Fermion

HDict = M.hierarchy

# auxiliary density operators
ados::ADOs

for (idx, ado) in enumerate(ados)
    ado # the corresponding auxiliary density operator for idx

    # obtain the nvec(s) correspond to ado
    nvec_b, nvec_f = HDict.idx2nvec[idx]

    # bosonic bath indices
    for (β, k, n) in getIndexEnsemble(nvec_b, HDict.bosonPtr)
        β  # index of the bosonic bath
        k  # the index of the exponential-expansion term in β-th bosonic bath
        nb # the repetition number of the ensemble {β, k} in vector j in ADOs
        exponent = M.Bbath[β][k]  # the k-th exponential-expansion term in β-th bosonic bath

        # do some calculations you want
    end

    # fermionic bath indices
    for (α, h, n) in getIndexEnsemble(nvec_f, HDict.fermionPtr)
        α  # index of the fermionic bath
        h  # the index of the exponential-expansion term in α-th fermionic bath
        nf # the repetition number of the ensemble {α, h} in vector q in ADOs
        exponent = M.Fbath[α][h]  # the h-th exponential-expansion term in α-th fermionic bath

        # do some calculations you want
    end
end