Auxiliary Density Operators

Introduction

The auxiliary density operators (ADOs) $\rho_{\textbf{j}\vert\textbf{q}}^{(m,n,p)}(t)$ encode environmental effects related to different exponential terms (Exponent) present in the Bosonic Bath and Fermionic Bath correlation functions and provide an iterative description of high-order system-baths memory effects.

In $\rho_{\textbf{j}\vert\textbf{q}}^{(m,n,p)}(t)$, the tuple $(m, n, p)$ represents the $m$th-level-bosonic-and-$n$th-level-fermionic ADO with parity $p$, and $\textbf{j}$ ($\textbf{q}$) denotes a vector $[j_m,\cdots,j_1]$ ($[q_n,\cdots,q_1]$) where each $j$ ($q$) represents a specific multi-index ensemble $\{\beta, l\}$ ($\{\alpha, h\}$) with

• $\beta$ : denotes the index of bosonic bath
• $\alpha$ : denotes the index of fermionic bath
• $l$ : denotes the index of exponent in the bosonic bath
• $h$ : denotes the index of exponent in the fermionic bath

In HierarchicalEOM.jl, we express all the auxiliary density operators into a single column vector and store it in the object defined as :

struct ADOs,

which is usually obtained after solving the time evolution or stationary state by a given HEOM Liouvillian superoperator Matrix.

Fields

The fields of the structure ADOs are as follows:

• data : the vectorized auxiliary density operators
• dims : the dimension list of the coupling operator (should be equal to the system dims).
• N : the number of auxiliary density operators
• parity: the parity label

One obtain the value of each fields as follows:

# usually obtained after solving time evolution or stationary state
ados::ADOs

ados.data
ados.dims
ados.N
ados.parity

Reduced Density Operator

In order to obtain the system reduced density operator in the type of QuantumObject, just simply call getRho

ados::ADOs
ρ = getRho(ados)

High-Level Auxiliary Density Operators

Although we express all the auxiliary density operators in the vector ADOs.data, we still make the ADOs like a list where accessing each element would return a specific auxiliary density operator in matrix type.

In order to obtain the auxiliary density operator in the type of QuantumObject with a specific index i, just simply call getADO

ados::ADOs
i::Int

ρ   = getADO(ados, 1) # the first element will always be the reduced density operator
ado = getADO(ados, i) # the i-th auxiliary density operator

Also, ADOs supports all the element-wise methods (functions) :

Supports length(::ADOs) which returns the total number of auxiliary density operators (same as ADOs.N) :

ados::ADOs
length(ados)

Supports bracket operation [] which is similar to access the element of a list :

ados::ADOs

# all the following returned ADO will be in matrix form
ados[1]    # returns the first auxiliary density operator (which is always the reduced density operator)
ados[10]   # returns the 10-th auxiliary density operator
ados[3:10] # returns a list of auxiliary density operators from index 3 to 10
ados[end]  # returns the last auxiliary density operator

Supports iteration (for-loop) process :

ados::ADOs

for ado in ados  # iteration
    ado # each auxiliary density operator in matrix form
end

Expectation Value

Given an observable $A$ and ADOs $\rho^{(m,n,p)}_{\textbf{j} \vert \textbf{q}}$, one can calculate the expectation value by

\[\langle A \rangle = \textrm{Tr}\left[A \rho^{(0,0,p)}_{ \vert }\right],\]

where, $m=n=0$ represents the reduced density operator.

One can directly calculate the expectation values using the function QuantumToolbox.expect:

A::QuantumObject # observable

# with a single ADOs
ados::ADOs
E = expect(A, ados)

# with a list contains many ADOs
ados_list::Vector{ADOs}
Elist = expect(A, ados_list)

Here, Elist contains the expectation values corresponding to the ados_list.