Hierarchical Equations for Open System Dynamics in Fermionic and Bosonic Environments

Abstract

We present two approaches to the dynamics of an open quantum system coupled linearly to a non-Markovian fermionic or bosonic environment. In the first approach, we obtain a hierarchy of stochastic evolution equations of the diffusion type. For the bosonic case such a hierarchy has been derived and proven suitable for efficient numerical simulations recently (Suess et al. in Phys. Rev. Lett. 113, 150403, 2014). The stochastic fermionic hierarchy derived here contains Grassmannian noise, which makes it difficult to simulate numerically due to its anti-commutative multiplication. Therefore, in our second approach we eliminate the noise by deriving a related hierarchy for density matrices. A similar reformulation of the bosonic hierarchy of pure states to a master equation hierarchy and its relation to the hierarchical equations of motion of Tanimura and Kubo is also presented.

Appendices

Appendix 1: Derivation of the Fermionic Hierarchy of Pure States

The derivation for the fermionic hierarchy of pure states is very similar to that of the bosonic one discussed in [17]. To have a compact notation, we ignore the condition $k_j \in \{0,1\}$ for our derivation. In the end, it will turn out that these conditions are trivially incorporated due to the structure of the hierarchy.

We start by taking the time derivative of $\psi _t^{(\mathbf {k})}$. Using its definition (13) we find

$$\begin{aligned} \partial _t \psi _t^{(\mathbf {k})}= & {} ( \partial _t \mathbf {D}_{t}^{\mathbf {k}} ) \psi _t + \mathbf {D}_{t}^{\mathbf {k}} (\partial _t \psi _t ) \end{aligned}$$

(37)

For the first term on the right hand side we use⁴ $(\partial _t D_{j,t} ) \psi _t = - w_j D_{j,t} \psi _t$ due to the exponential BCF (16). For the second term on the right hand side we use that all system operators commute with all $D_{j,t}$ and obtain

$$\begin{aligned} \partial _t \psi _t^{(\mathbf {k})}= & {} - \mathbf {k}\cdot \mathbf {w}\, \psi _t^{(\mathbf {k})} \nonumber \\&- \mathrm {i}H \psi _t^{(\mathbf {k})} + \underbrace{\sum _j L_j \mathbf {D}_{t}^{\mathbf {k}} {Z}^*_j(t) \psi _t }_{(*)} - \underbrace{\sum _j {L}^\dagger _j \mathbf {D}_{t}^{\mathbf {k}} D_{j,t} \psi _t }_{(**)} \end{aligned}$$

(38)

To obtain a closed equation for the auxiliary states, we want the $D_{j,t}$ ordered as in the definition (13). In $(**)$ we have to move $D_{j,t}$ to the correct position (note the ordering in (13)):

$$\begin{aligned} \mathbf {D}_{t}^{\mathbf {k}} D_{j,t}= & {} (-1)^{k_N} D_{1,t} ^{k_1} \ldots D_{j,t} D_{N,t} ^{k_N} \nonumber \\= & {} (-1)^{k_{j+1} + \cdots + k_N} D_{1,t} ^{k_1} \ldots D_{j,t} ^{k_j + 1} \cdots D_{N,t} ^{k_N}. \end{aligned}$$

(39)

In $(*)$ we have to bring ${Z}^*_j(t)$ in front of $\mathbf {D}_{t}^{\mathbf {k}}$. This can be achieved by noting that $\{ D_{j,t} , {Z}^*_{j'}(s) \} = \delta _{jj'} \, \alpha _{j}(t_{j}-s)$. We then find

$$\begin{aligned} \mathbf {D}_{t}^{\mathbf {k}} {Z}^*_j(t)= & {} (-1)^{k_N} D_{1,t} ^{k_1} \ldots {Z}^*_j(t) D_{N,t} ^{k_N} \nonumber \\= & {} (-1)^{k_{j+1} + \cdots + k_N} D_{1,t} \ldots D_{j,t} ^{k_j} {Z}^*_j(t) \ldots \nonumber \\= & {} (-1)^{{\left| \mathbf {k} \right| }_j} D_{1,t} \ldots \left( - D_{j,t} ^{k_j-1} {Z}^*_j(t) D_{j,t} + D_{j,t} ^{k_j-1} p_j \right) \ldots \nonumber \\= & {} \ldots \left( D_{j,t} ^{k_j-2} {Z}^*_j(t) D_{j,t} ^2 - D_{j,t} ^{k_j-1} p_j + D_{j,t} ^{k_j-1} p_j \right) \ldots \nonumber \\= & {} \ldots \left( (-1)^{k_j} {Z}^*_j(t) D_{j,t} ^{k_j} + (k_j \,\mathrm{mod\, 2}) p_j D_{j,t} ^{k_j-1} \right) \ldots \nonumber \\= & {} (-1)^{\left| \mathbf {k} \right| } {Z}^*_j(t) \mathbf {D}_{t}^{\mathbf {k}} + (-1)^{{\left| \mathbf {k} \right| }_j} (k_j \,\mathrm{mod\, 2}) p_j \mathbf {D}_{t}^{\mathbf {k}- \mathbf {e}_j}, \end{aligned}$$

(40)

where ${\left| \mathbf {k} \right| }$ and ${\left| \mathbf {k} \right| }_j$ have been defined below Eq. (17).

Combining (39) and (40) leads to the (apparently) infinite hierarchy of pure states for fermionic environment

$$\begin{aligned} \partial _t \psi _t^{(\mathbf {k})}= & {} \left( -\mathrm {i}H - \mathbf {k}\cdot \mathbf {w}+ (-1)^{\left| \mathbf {k} \right| } \sum _j {Z}^*_j(t) L_j \right) \psi _t^{(\mathbf {k})} \nonumber \\&+ \sum _j (-1)^{{\left| \mathbf {k} \right| }_j} (k_j\, \mathrm{mod\, 2}) p_j L_j \psi _t^{(\mathbf {k}- \mathbf {e}_j)} \nonumber \\&- \sum _j (-1)^{{\left| \mathbf {k} \right| }_j} (-1)^{{\left| \mathbf {k} \right| }_j} {L}^\dagger _j \psi _t^{(\mathbf {k}+ \mathbf {e}_j)} . \end{aligned}$$

(41)

Note that all states with some $k_j \notin \{0,1\}$—which should be zero actually—only couple to other states also satisfying this condition: Due to the modulo function in the term coupling to states “below” in the hierarchy, states with some $k_j \notin \{0,1\}$ that are initially zero always remain zero. Therefore, the closed and finite hierarchy with all $k_j \in \{0,1\}$ and equation (41) can be written as (17).

Appendix 2: Derivation of Master Equation Hierarchy

In this appendix we provide the Novikov theorem, which is essential to get from Eqs. (21) to  (22). The Novikov theorem allows us to get rid of the explicit dependence of the Grassmann processes in the second and third line of (21) by a “partial integration”. For the fermionic case the Novikov theorem has been discussed in [15] (see Eqs. (22) and (23) therein). We need two variants of the Novikov theorem:

$$\begin{aligned} \mathbb {E}\left( | \psi _t^{(\mathbf {m})} \rangle \langle \tilde{\psi }_t^{(\mathbf {n})} | Z_j(t) \right)= & {} - \mathbb {E} \Bigg ( \int \mathrm {d}s \, \alpha _j(t-s) \frac{\overrightarrow{\delta }}{\delta {Z}^*_j(s)} | \psi _t^{(\mathbf {m})} \rangle \langle \tilde{\psi }_t^{(\mathbf {n})} | \Bigg ) \nonumber \\= & {} - \rho _t^{(\mathbf {m}+\mathbf {e}_j,\mathbf {n})} \end{aligned}$$

(42)

and

$$\begin{aligned} \mathbb {E}\left( {Z}^*_j(t) | \psi _t^{(\mathbf {m})} \rangle \langle \tilde{\psi }_t^{(\mathbf {n})} | \right)= & {} - \mathbb {E} \Bigg ( \int \mathrm {d}s \, {\alpha _j(t-s)}^* \left| \psi _t^{(\mathbf {m})} \rangle \langle \tilde{\psi }_t^{(\mathbf {n})} \right| \frac{\overleftarrow{\delta }}{\delta Z_j(s)} \Bigg ) \nonumber \\= & {} \rho _t^{(\mathbf {m},\mathbf {n}+\mathbf {e}_j)}, \end{aligned}$$

(43)

where in the second line of each equation we have used the definition of the auxiliary matrices (20) and the definitions (12) and (13). In the second equation the right-functional derivative appears and we have used

$$\begin{aligned} \frac{\overrightarrow{\delta } \tilde{\psi }_t ({Z}^*)}{\delta {Z}^*_j(s)} = \frac{\overrightarrow{\delta } \psi _t (-{Z}^*)}{\delta {Z}^*_j(s)} = - \left. \frac{\overrightarrow{\delta } \psi _t ({Z'}^*)}{\delta {Z_j'}^*(s)} \right| _{{Z'}^* = -{Z}^*}. \end{aligned}$$

(44)

These two equations (42) and (43) show that it is possible to express the averages in the second and third line of (21) containing the noise process explicitly by the auxiliary density operators.