HEOMLS Matrix for Master Equation

HierarchicalEOM.jl allows users to further add the Lindbladian (superoperator) on any types (AbstractHEOMLSMatrix) of HEOM Liouvillian superoperator $\hat{\mathcal{M}}$. The Lindbladian describes the dissipative interaction between the system and extra environment.

This method is more efficient than the full HEOM when some of the baths are weakly coupled to the system since it does not require any extra ADOs space to describe the dynamics and hence reduces the size of $\hat{\mathcal{M}}$.

Bosonic Dissipative Env.

If the system is weakly coupled to an extra bosonic environment, the explicit form of the Lindbladian is given by

\[\hat{\mathcal{D}}(J)\Big[\cdot\Big]=J\left[\cdot\right]J^\dagger-\frac{1}{2}\Big[J^\dagger J, \cdot\Big]_+,\]

where $J\equiv \sqrt{\gamma}V$ is the jump operator, $V$ describes the dissipative part (operator) of the dynamics, $\gamma$ represents a non-negative damping rate and $[\cdot, \cdot]_+$ stands for anti-commutator.

One can add the Lindbladian $\hat{\mathcal{D}}$ of bosonic environment to the HEOM Liouvillian superoperator $\hat{\mathcal{M}}$ by calling

addBosonDissipator(M::AbstractHEOMLSMatrix, jumpOP) with the parameters:

• M::AbstractHEOMLSMatrix : the matrix given from HEOM model
• jumpOP::AbstractVector : The list of collapse (jump) operators $\{J_i\}_i$ to add. Defaults to empty vector [].

Finally, the function returns a new $\hat{\mathcal{M}}$ with the same type:

M0::AbstractHEOMLSMatrix
J = [J1, J2, ..., Jn]  # jump operators

M1 = addBosonDissipator(M0, J)

Fermionic Dissipative Env.

If the fermionic system is weakly coupled to an extra fermionic environment, the explicit form of the Lindbladian acting on EVEN-parity operators is given by

\[\hat{\mathcal{D}}_{\textrm{even}}(J)\Big[\cdot\Big]=J\left[\cdot\right]J^\dagger-\frac{1}{2}\Big[J^\dagger J, \cdot\Big]_+,\]

where $J\equiv \sqrt{\gamma}V$ is the jump operator, $V$ describes the dissipative part (operator) of the dynamics, $\gamma$ represents a non-negative damping rate and $[\cdot, \cdot]_+$ stands for anti-commutator.

For acting on ODD-parity operators, the explicit form of the Lindbladian is given by

\[\hat{\mathcal{D}}_{\textrm{odd}}(J)\Big[\cdot\Big]=-J\left[\cdot\right]J^\dagger-\frac{1}{2}\Big[J^\dagger J, \cdot\Big]_+,\]

One can add the Lindbladian $\hat{\mathcal{D}}$ of fermionic environment to the HEOM Liouvillian superoperator $\hat{\mathcal{M}}$ by calling

addFermionDissipator(M::AbstractHEOMLSMatrix, jumpOP) with the parameters:

• M::AbstractHEOMLSMatrix : the matrix given from HEOM model
• jumpOP::AbstractVector : The list of collapse (jump) operators $\{J_i\}_i$ to add. Defaults to empty vector [].

Finally, the function returns a new $\hat{\mathcal{M}}$ with the same type and parity:

M0_even::AbstractHEOMLSMatrix # constructed with EVEN-parity
M0_odd::AbstractHEOMLSMatrix  # constructed with  ODD-parity
J = [J1, J2, ..., Jn]  # jump operators

M1_even = addFermionDissipator(M0_even, J)
M1_odd  = addFermionDissipator(M0_odd,  J)