Dissipative quantum transport in a nanowire

The coherence-to-decoherence transition is studied in a nanowire modeled as a one-dimensional tight-binding lattice in the presence of an external field and in linear interaction with a boson heat bath, characterized by Ohmic dissipation. The focus of attention is the probability propagator which quantifies the likelihood of a quantum particle, such as an electron to end up at an arbitrary site at a time $t$, given that it was at the origin initially—and from it—the particle-current and the mean-squared displacement. If the bath is absent, the probability operator exhibits quantum coherence as can be captured by say, Bloch oscillation and Wannier-Stark (dynamic) localization. The coupling with the bath—which can be weak or strong—leads to decoherence that will be quantified in the text. The Ohmic model of the spectral function of bath excitations contains a cutoff frequency, which if greater than the temperature (in energy units) defines the “low-temperature” regime, whereas the opposite limit implies “high temperatures.” Results will be presented in both these regimes separately.

Figure 1

Plot of current as a function of dimensionless time ${\omega }_{c}t$ for three different external bias fields $a=\frac{F_{0}d}{{\omega }_c}$ (electric field) for the weak dissipative regime at (a) low temperatures and (b) at high temperatures. For the plotting purpose, we use $d=\frac{\mathrm{\Delta }}{2F_{0}d}=1.0,\frac{\gamma }{{\omega }_c}=0.005$. Furthermore, we consider $\beta {\omega }_c=1.5$ for low temperatures and $\beta {\omega }_c=0.05$ for high temperatures. The stationary state current for $a=0.2,0.3,$ and 0.4 are 0.06, 0.09, and 0.11.

Figure 2

Plot of the mean-squared displacement (MSD) as a function of dimensionless time ${\omega }_{c}t$ for three different external bias fields (electric field) at (a) high temperatures and (b) low temperatures. We also use $\frac{\mathrm{\Delta }}{2F_{0}d}=1.0$ and $\frac{\gamma }{{\omega }_c}=0.005$. Furthermore, we consider $\beta {\omega }_c=1.5$ for low temperatures and $\beta {\omega }_c=0.05$ for high temperatures.

Figure 3

Plot of current as a function of dimensionless time $x={\omega }_{c}t$ at long times for three different noise strengths at zero temperature. For the plotting purpose we use $F_0=0.1,\mathrm{\Delta }=0.1,d=1.0,{\omega }_c=1,T=1.5$.

Figure 4

Plot of the current as a function of dimensionless time ${\omega }_{c}t$ for three different external bias fields $a=\frac{F_{0}d}{{\omega }_c}$ (electric field) for the strong dissipative regime at (a) high temperatures and (b) at low temperatures. For the plotting purpose we use $d=\frac{\mathrm{\Delta }}{2F_{0}d}=1.0,\frac{\gamma }{{\omega }_c}=0.5$. Furthermore, we consider $\beta {\omega }_c=1.5$ for low temperatures and $\beta {\omega }_c=0.05$ for high temperatures. At high temperatures, the stationary current magnitudes are 0.012, 0.018, and 0.025 for $a=0.1,0.15,\mathrm{and}0.2$, respectively.

Figure 5

Plot of MSD as a function of dimensionless time ${\omega }_{c}t$ for three different external bias fields (electric field) for the strong dissipative regime at (a) high temperatures and (b) at low temperatures. We also use $\frac{\mathrm{\Delta }}{2F_{0}d}=1.0$ and $\frac{\gamma }{{\omega }_c}=0.5$. Furthermore, we consider $\beta {\omega }_c=1.5$ for low temperatures and $\beta {\omega }_c=0.05$ for high temperatures.