Smoluchowski equation approach for quantum Brownian motion in a tilted periodic potential

Quantum corrections to the noninertial Brownian motion of a particle in a one-dimensional tilted cosine periodic 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 distribution function in configuration space. The theoretical predictions from two different forms of the quantum Smoluchowski equation already proposed—viz., J. Ankerhold et al. [Phys. Rev. Lett. 87, 086802 (2001)] and W. T. Coffey et al. [J. Phys. A 40, F91 (2007)]—are compared in detail in a particular application to the dynamics of a point Josephson junction. Various characteristics (stationary distribution, current-voltage characteristics, mean first passage time, linear ac response) are evaluated via continued fractions and finite integral representations in the manner customarily used for the classical Smoluchowski equation. The deviations from the classical behavior, discernible in the dc current-voltage characteristics as enhanced current for a given voltage and in the resonant peak in the impedance curve as an enhancement of the $Q$ factor, are, respectively, a manifestation of relatively high-temperature nondissipative tunneling (reducing the barrier height) and dissipative tunneling (reducing the damping of the Josephson oscillations) near the top of a barrier.

Figure 1

(Color online) Overdamped quantum Brownian motion of a particle in a tilted periodic potential: the particle may escape from a potential well both over (due to thermal agitation) and below (due to tunneling through) the higher and lower potential barriers.

Figure 2

The quantum correction factors $\Xi -1=\Gamma ∕{\Gamma }_{\mathit{cl}}-1=2b\Lambda$ (solid circles), ${\tau }_{\mathit{MFPT}}^{\mathit{cl}}∕{\tau }_{\mathit{MFPT}}-1$ (solid line), and ${\tau }_{\mathit{MFPT}}^{\mathit{cl}}∕{\tau }_{\mathit{MFPT}}^A-1$ (dashed line) vs $b$ (Josephson coupling energy) with $\Lambda =0.001$.

Figure 3

(a) Quantum effects on the current-voltage [Eq. (33)] and (b) differential resistance-current [Eq. (34)] characteristics of a Josephson junction in the presence of noise for $\Lambda =0.2$ and various values of the Josephson coupling energy $b=1$, 2, and 4. Solid and dashed lines are the predictions of Eq. (2) and of the Ankerhold equation (10), respectively. Dots: classical limit $\Lambda =0$.

Figure 4

(a) Quantum effects on the current-voltage, Eq. (33), and (b) differential resistance-current, Eq. (34), characteristics of a Josephson junction in the presence of noise for $b=3$ and various values of the quantum parameter $\Lambda =0$ (dots, classical limit), 0.1 (curves 1), and 0.2 (curves 2), 0.3 (curves 3), and 0.4 (curves 4). Dashed and solid lines: first- and second-order corrections, respectively.

Figure 5

Quantum effects on the real $[Z^{\prime }=\mathrm{Re}\left(Z\right)]$ and imaginary $[Z^{″}=\mathrm{Im}\left(Z\right)]$ parts of the normalized $(R=1)$ impedance vs normalized angular frequency $\omega$ for various values of $b$ (Josephson coupling energy). Comparison of the quantum [solid lines, $\Lambda =0.1$, Eqs. (39, 42)], classical [dashed lines, $\Lambda =0$, Eqs. (39, 42)], and approximate [stars, Eq. (46), and solid circles, Eq. (47)] solutions.

Figure 6

Quantum effects on the real and imaginary parts of the normalized $(R=1)$ impedance vs $\omega$ for various values of the tilt parameter $y=0.1$, 0.7, 1.0, and 1.5 and $b=5$. Key as in Fig. 5.

Figure 7

Quantum effects on the real and imaginary parts of the effective eigenvalue ${\lambda }_{\mathit{ef}}^{+}={\lambda }^{\prime }+i{\lambda }^{″}$ vs $b$ for various values of $y=0.1$, 0.5, 1.0, and 1.5. Solid and dashed lines: Eq. (30) for $\Lambda =0.1$ and $\Lambda =0$ (classical limit), respectively.

Figure 8

Quantum effects on the real and imaginary parts of the effective eigenvalue vs $y$ for various values of $b=1$, 5, 10, and 15. Key as in Fig. 7.