Semiclassical treatment of a Brownian ratchet using the quantum Smoluchowski equation

Quantum effects in the noninertial Brownian motion of a particle in a one-dimensional ratchet potential are treated in the high temperature and weak bath-particle coupling limit by solving a quantum Smoluchowski equation for the time evolution of the Wigner function in configuration space. In particular, an analytical expression for the stationary average drift velocity for constant driving forces is presented including quantum corrections to any order in Planck’s constant. The corresponding frequency response is determined using continued fractions in both the linear approximation holding for small ac driving amplitude and in the nonlinear regime for arbitrary driving amplitude exhibiting pronounced ac induced frequency dependence of the dc component of the average drift velocity. Moreover, Shapiro steps are apparent in the dc characteristics for strong ac driving just as in the dc current-voltage characteristics of a point Josephson junction.

Figure 1

The average drift velocity in the slow-switching limit ${⟨\dot{x}⟩}_{ssl}$ vs two-state driving force $a$ for the ratchet potential $V_0=-2$, $b_1=0.25$, and $b_2=0$ (see right inset). The quantum case $\left(\Lambda =0.02\right)$ for the quantum Smoluchowski Eq. (13) and that of Ref. [12] is shown along with the classical case $\left(\Lambda =0\right)$. In all cases there is zero probability current (no Maxwell-demon behavior) at zero tilt (see left inset).

Figure 2

(Color online) The average drift velocity in the low switching limit ${⟨\dot{x}⟩}_{ssl}$ vs two-state driving force $a$ in the ratchet potential $V_0=1$, $b_1=0.5$, and $b_2=0.5$ (see inset) for various values of the quantum parameter, $\Lambda =0$ (curve 1, classical limit), $\Lambda =0.02$ (curves 2) and $\Lambda =0.04$ (curves 3). Dashed and solid lines are the predictions of the quantum Smoluchowski equation to first- and second-order quantum corrections, respectively. Clearly as $\Lambda$ increases the higher order terms in the diffusion coefficient must be included.

Figure 3

(Color online) The real and imaginary parts of the linear dynamic mobility $\mu ={\mu }^{\prime }-i{\mu }^{″}$ vs angular frequency $\omega$ in the ratchet potential $b_1=0.5$ and $b_2=0.5$ (see inset of Fig. 2) for various values of the barrier height $V_0=2$ (curves 1), $V_0=3$ (curves 2), $V_0=5$(curves 3) and $V_0=7$ (curves 4), and the constant tilt $a=15$. The classical (dashed lines, $\Lambda =0$) and quantum (solid lines, $\Lambda =0.02$) cases are shown.

Figure 4

(Color online) The real and imaginary parts of the linear dynamic mobility $\mu ={\mu }^{\prime }-i{\mu }^{″}$ vs $\omega$ in the ratchet potential $V_0=5$, $b_1=0.5$ and $b_2=0.5$ for various values of the tilt $a=1$ (curves 1), $a=10$ (curves 2), $a=15$ (curves 3), $a=20$ (curves 4) and $a=30$ (curves 5). The classical (dashed lines, $\Lambda =0$) and quantum (solid lines, $\Lambda =0.02$) cases are shown.

Figure 5

(Color online) The frequency-dependent dc term of the average drift velocity ${⟨\dot{x}⟩}_{dc}$ vs the tilt $a$ in the ratchet potential $V_0=5$, $b_1=0.5$, and $b_2=0.5$ for various values of the ac driving amplitude $a^{\prime }=10$ (curves 1), $a^{\prime }=15$ (curves 2), $a^{\prime }=20$ (curves 3) and $a^{\prime }=30$ (curves 4) for fixed frequency $\omega =10$. The Shapiro steps (curves 1–4) appear as a result of phase modulation due to nonlinear effects, in contrast to the frequency-independent average drift velocity ${⟨\dot{x}⟩}_0$ (curve 0) plotted using Eq. (26). The classical (dashed lines, $\Lambda =0$) and quantum (solid lines, $\Lambda =0.02$) cases are shown.

Figure 6

(Color online) The frequency-dependent dc term of the average drift velocity ${⟨\dot{x}⟩}_{dc}$ vs angular frequency $\omega$ in the ratchet potential $V_0=5$, $b_1=0.5$, and $b_2=0.5$ for various values of the ac driving amplitude $a^{\prime }=1$ (curves 1), $a^{\prime }=5$ (curves 2), $a^{\prime }=10$ (curves 3), $a^{\prime }=15$ (curves 4), and $a^{\prime }=25$ (curves 5), and the constant tilt $a=15$. The classical (dashed lines, $\Lambda =0$) and quantum (solid lines, $\Lambda =0.01$) cases are shown.

Figure 7

(Color online) The real and imaginary parts of the nonlinear dynamic mobility $\mu ={\mu }^{\prime }-i{\mu }^{″}$ vs angular frequency $\omega$ in the ratchet potential, $V_0=5$ $b_1=0.5$ and $b_2=0.5$ for various values of the stimulus amplitude $a^{\prime }=0.01$ (curves 1), $a^{\prime }=5$ (curves 2), $a^{\prime }=10$ (curves 3) and $a^{\prime }=15$ (curves 4), and the constant tilt $a=15$. The classical (dashed lines, $\Lambda =0$) and quantum (solid lines, $\Lambda =0.02$) cases are shown. In the limit of $a^{\prime }⪡1$ the linear response is reproduced, i.e., curves 1 correspond exactly to curves 3 of Fig. 3.