Stochastic Representation of Non-Markovian Fermionic Quantum Dissipation

Quantum Brownian motion plays a fundamental role in many areas of modern physics. In the path-integral formulation, environmental fluctuations can be characterized by auxiliary stochastic fields. Intriguingly, for fermionic environments the stochastic fields must be Grassmann valued so as to memorize the order of the random forces exerted on the system. Such nonclassical fields cannot be represented by conventional means. We propose a strategy to map the Grassmann-number fields to conventional $c$-number noises and a set of quantized pseudolevels. The resulting stochastic equation of motion (SEOM) enables direct stochastic simulation of the fermionic dissipative dynamics. The SEOM gives exact physical observables of noninteracting systems, and yields accurate approximate results for interacting systems. The practicality and accuracy of the proposed strategy and the SEOM are exemplified by numerical studies conducted on a single-impurity Anderson model.

Figure 1

Illustration of time evolution of a system-bath composite driven by stochastic fluctuations in the bath. For simplicity, the composite Hamiltonian is set to zero. The upper branch involves two dissipation events: a particle transfers from $Q_1$ (in the bath) to $A$ (in the system) at time $t_1$ driven by $v_1$, and a second particle transfers from $Q_2$ to $B$ at time $t_2$ driven by $v_2$, where $v_1$ and $v_2$ are two arbitrary dimensionless noises. The final state is $|{\mathrm{\Psi }}_t⟩=|{\mathrm{\Phi }}_t⟩-v_2{v}_1{\widehat{d}}_B^{†}{\widehat{d}}_{Q_2}{\widehat{d}}_A^{†}{\widehat{d}}_{Q_1}|{\mathrm{\Psi }}_0⟩$, where $|{\mathrm{\Psi }}_0⟩$ is the initial state and $|{\mathrm{\Phi }}_t⟩$ contains the rest of $|{\mathrm{\Psi }}_t⟩$. In the lower branch, the order of the two bath fluctuations is interchanged, and the final state is $|{\mathrm{\Psi }}_t^{\prime }⟩=|{\mathrm{\Phi }}_t^{\prime }⟩-v_1{v}_2{\widehat{d}}_B^{†}{\widehat{d}}_{Q_1}{\widehat{d}}_{\mathrm{A}}^{†}{\widehat{d}}_{Q_2}|{\mathrm{\Psi }}_0⟩$. Here, ${\widehat{d}}_{Q_1}$ and ${\widehat{d}}_{Q_2}$ commute (anticommute) for a boson (fermion) bath. To ensure $|{\mathrm{\Psi }}_t⟩$ and $|{\mathrm{\Psi }}_t^{\prime }⟩$ have the same statistical average over $\{v_1,v_2,Q_1,Q_2\}$, we must have $v_2{v}_1=v_1{v}_2$ for the boson bath, and $v_2{v}_1=-v_1{v}_2$ for the fermion bath, respectively. More details are given in the Supplemental Material [37]. $|{\mathrm{\Psi }}_s⟩$ in Eq. (1) is obtained by further tracing out the bath degrees of freedom in $|{\mathrm{\Psi }}_t⟩$.

Figure 2

Schematic illustration of our strategy to enable direction simulation of fermionic dissipative systems. Through the mapping (i), the system-bath coupling is replaced by stochastic $g$-number fields applied on both the system and the fermion bath. The dotted arrow indicates the influence of non-Markovian bath memory on the statistical average of the system state. Through the mapping (ii), the non-Markovian bath memory is captured by the memory-convoluted $g$-number fields applied on the system. Through the mapping (iii), all the $g$-number fields are represented by stochastic $c$-number fields and a set of pseudolevels. The mappings (i)–(iii) finally result in a numerically feasible SEOM.

Figure 3

(a) Time evolution of $n_s(t)$ calculated with the SEOM and HEOM methods. The parameters adopted are (in arbitrary unit): ${\epsilon }_{\uparrow }=0.5$, ${\epsilon }_{\downarrow }=-0.5$, $U=5$, $\mathrm{\Gamma }=0.5$, $\mathrm{\Omega }=0$, and $W=5$. The Euler-Maruyama algorithm [52] is employed to solve the SEOM with a time step of $dt=0.001$. The number of trajectories is $N_{\mathrm{traj}}=5\times {10}^6$ for all lines. (b) $n(t)=n_{\uparrow }(t)+n_{\downarrow }(t)$ calculated with the SEOM and HEOM methods. The impurity level energy is shifted by $\mathrm{\Delta }\epsilon =-5.0$ during the time interval $0.1<t<0.2$. Other parameters adopted are (in arbitrary unit): ${\epsilon }_{\uparrow }={\epsilon }_{\downarrow }=-2.5$, $U=10$, $\mathrm{\Gamma }=0.5$, $\mathrm{\Omega }=0$, $W=5$, and $T=0.01$. We choose $dt=0.001$ and $N_{\mathrm{traj}}=1\times {10}^6$ for SEOM calculations. To reveal the stochastic nature of the calculated $n(t)$, we display its time derivative $(dn/dt)$ in the inset of (b), so that the stochastic variance becomes recognizable.