Quantum simulation of many-body effects in steady-state nonequilibrium: Electron-phonon coupling in quantum dots

We develope a method of mapping quantum nonequilibrium steady-state to an effective equilibrium system and present an algorithm to calculate electron-transport using an equilibrium technique. A systematic implementation of boundary conditions in steady-state nonequilibrium is made in the statistical operator $\widehat{Y}$ constructed from scattering state operators. We explicitly demonstrate the equivalence of this method to nonequilibrium Green function techniques for a noninteracting quantum dot model. In electron-phonon coupled quantum dot systems, we formulate an algorithm to construct the statistical bias operator $\widehat{Y}$ and perform a full many-body calculation with the quantum Monte Carlo technique. The results coherently demonstrate various transport behaviors such as phonon dephasing, $I-V$ staircase, and phonon-assisted tunneling phenomena. This formulation makes the existing computational quantum many-body techniques applicable to quantum steady-state nonequilibrium problems, which will complement the theories based on the diagrammatic approach.

Figure 1

(a) Schematic diagram of a quantum transport device. The source and drain reservoirs are biased by a chemical potential difference $\Phi$. The quantum dot (QD) level is set in the middle for simplicity. (b) Each reservoir is mapped to a single site and a new reservoir. The three discrete sites for the QD, $L$, and $R$ sites are treated in a fully quantum mechanical manner.

Figure 2

Simulated $I\text{−}V$ characteristics in the broadband limit, $D⪢{\omega }_{ph}$. Parameters are $t_L=t_R=0.1$, ${\omega }_{ph}=0.5$, $D=8.0$, and $T=1∕32=0.03125$. The current decreased with increasing electron-phonon coupling constant $g$. Noninteracting phonon level aligns with the reservoir Fermi energy at the unit bias $\Phi =2{\omega }_{ph}=1$. The phonon dephasing effect is so strong that the phonon satellite peak is invisible at all coupling constant while the phonon satellites at zero bias (inset) are clearly separated from the main peak.

Figure 3

(Color online) $I\text{−}V$ characteristics for narrow bandwidth limit. $t_L=t_R=0.1$ [except for (c)], ${\omega }_{ph}=1$, $g=0.2$, and $T=1∕12=0.0833$. (a) The saturated current in the large bandwidth limit gradually turns into negative differential resistance regime as $\Phi$ becomes comparable to $D$. (b). A three-peak structure emerges as $D$ gets further reduced. The peaks correspond to ballistic coherent transport, phonon-assisted tunneling (B, $\Phi ={\omega }_{ph}$) and sequential off-resonant transport (A, $\Phi =2{\omega }_{ph}$). (c) Reduction of tunneling parameter $t_L,t_R$ dramatically suppresses the phonon-assisted tunneling compared to the sequential current.

Figure 4

(a) Initial and (b) final states in the phonon assisted tunneling process. The particle-hole and phonon excitations are resonant at the bias $\Phi ={\omega }_{ph}$.