Bath State Ansatze

The second-order QME contains the full density matrix $\hat{W}^{(I)}(\tau) = \hat{\rho}^{(I)}(\tau) \oplus \hat{w}^{(I)}(\tau)$ inside the memory integral. To avoid propagating the full system+bath state, we can replace the relative bath part (RBP) $\hat{w}^{(I)}(\tau)$ with an ansatz – an approximate form that is cheaper to evaluate. Different ansatze offer different accuracy/cost trade-offs. The derivations follow Section 3.3 of [Her23].

Constant ansatz

The simplest approximation freezes the bath at its initial equilibrium state:

\[\hat{w}_{ab}^{(I)}(t) = \hat{w}_{eq}\]

for all electronic states $a, b$ and all times $t$. This is justified when the bath is infinite and initially in equilibrium – any effect of the system on the bath is negligible. For finite systems, this is accurate only for short times (typically $t < 400$ fs for weak coupling).

In the package: QME_sI_ansatz(W0, tspan, agg; ansatz=:const_sch) or :const_int.

Linear ansatz (L1)

For short times, a Taylor expansion of the evolution operators yields a first-order correction:

\[\hat{w}(t) = \hat{w}_{eq} - \frac{i}{\hbar}[\hat{H}, \hat{w}_{eq}]\,t.\]

In the interaction picture:

\[\hat{w}_{ab}^{(I)}(t) = \langle a|U_0^\dagger(t) \bigl[\hat{w}_{eq} - \tfrac{i}{\hbar}[\hat{H}, \hat{w}_{eq}]\,t\bigr] U_0(t)|b\rangle.\]

This is more accurate than the constant ansatz for short times but the polynomial correction diverges for long simulations.

In the package: ansatz=:linear_sch.

Piecewise-linear ansatz (L2)

To extend the validity of the linear correction, the evolution is broken into $k$ steps of length $t/k$:

\[\hat{w}^{(I)}(t) = \mathcal{U}_0^\dagger(t)\, \tilde{\mathcal{U}}^k\!\left(\frac{t}{k}\right)\,\hat{w}(0),\]

where $\tilde{\mathcal{U}}(t)\hat{O} = \hat{O} - \frac{i}{\hbar}[\hat{H}, \hat{O}]\,t$ is the linearised evolution superoperator. With more corrections (larger $k$), the L2 ansatz approaches the exact evolution, but divergence is still possible for $k$ too small.

In the package: ansatz=:linear2_sch.

U1 ansatz (block-diagonal unitary evolution)

Instead of a polynomial approximation, we evolve only the diagonal (population) blocks of the RBP with the corresponding block Hamiltonian:

\[\hat{w}^{(I)}(t) = \mathcal{U}_0^\dagger(t) \left[\sum_a \hat{U}_a(t)\,\hat{w}_{eq}\,\hat{U}_a^\dagger(t)\,|a\rangle\langle a| + \sum_{a \neq b} \hat{w}_{eq}\,|a\rangle\langle b|\right],\]

where $\hat{U}_a(t) = \exp\bigl[-\frac{i}{\hbar}\langle a|\hat{H}|a\rangle\,t\bigr]$ is the evolution operator for the $a$-th population block of the Hamiltonian. Off-diagonal blocks of the RBP are left at equilibrium.

This ansatz correctly captures the separate evolution of each population's bath state without the divergence issues of polynomial approximations.

In the package: ansatz=:upart1_sch or :upart1_int.

U2 ansatz (full-block unitary evolution)

The U2 ansatz extends U1 by evolving all blocks of the RBP, including coherences:

\[\hat{w}^{(I)}(t) = \mathcal{U}_0^\dagger(t) \sum_{ab} \hat{U}_{ab}(t)\,\hat{w}_{eq}\,\hat{U}_{ab}^\dagger(t),\]

where $\hat{U}_{ab}(t) = \exp\bigl[-\frac{i}{\hbar}\langle a|\hat{H}|b\rangle\,t\bigr]$ uses the full $(a,b)$ block of the Hamiltonian. Numerically, U1 and U2 give nearly identical results for the systems tested, suggesting that the off-diagonal bath evolution is less important than the diagonal part.

In the package: ansatz=:upart2_sch or :upart2_int.

Comparison of ansatze

The constant ansatz performs similarly to Redfield equations. The L1, U1, and U2 ansatze do not systematically outperform Redfield for finite systems, though L2 with many corrections can. For best results in the weak-coupling regime, use the iterative ansatz (see Iterative Quantum Master Equation) which systematically improves the bath state.