The Langevin Equation

Femtonewton Force Sensing with Optically Trapped Nanotubes

“…Finally, to explain the large Brownian motion asymmetry of Figure 2a, we plot in Figure 4d a parameter which quantifies the trap aspect ratio: (k x + k y )/2k z . Due to the equipartition theorem, 28 the variances of translational displacements (giving the widths of the histograms in Figure 2, panels c, d, and e) are linked to the force constants 〈x i 2 (t)〉 ) k B T/k i . From our data, this parameter is ∼45, while for the latex spherical particle of Figure 2b it is ∼5.…”

Nonlinear noninertial response of a quantum Brownian particle in a tilted periodic potential to a strong ac force as applied to a point Josephson junction

“…As a particular example, we shall estimate quantum effects on the nonlinear dynamics of a point Josephson junction in the zero-capacitance limit ͑so-called RSJ model͒ [19][20][21] initiated by Ambegaokar and Halperin 28 and by Ivanchenko and Zil'berman. 29 Despite its limitations, [19][20][21] the RSJ model yields for the linear response of the junction to an ac driving current a relatively simple treatment of both the dc current-voltage characteristic and the impedance both in the classical [19][20][21]30 and quantum 31 cases. In the quantum case, the dc current-voltage characteristic and the differential resistance may be expressed 31 as modified Bessel functions of the first kind just as in the classical limit.…”

“…In the quantum case, the dc current-voltage characteristic and the differential resistance may be expressed 31 as modified Bessel functions of the first kind just as in the classical limit. 30 Moreover, the linear impedance characteristic resembles that of a simple resonant circuit, where the real part exhibits a pronounced minimum at the resonant frequency and the imaginary part a pronounced maximum. In particular, the quantum effects are discernible in the linear response as an enhanced current for a given voltage in the dc current-voltage characteristic and an enhanced slope in the differential-resistance, which, besides the impedance, are the other quantities of physical interest.…”

“…Hence, the results of classical theory ͑in particular, those arising from the classical Smoluchowski equation͒ require modification when quantum effects become important, e.g., at very low temperatures T Ͻ 0.1 K. 21 Noting that the mass m and the friction coefficient of the mechanical Brownian particle are replaced in Eq. ͑3͒ by the corresponding electrical parameters R and C via m = C͑ប / 2e͒ 2 and = ͑ប / 2e͒ 2 / R, [19][20][21]30,31 the QSE for the reduced Wigner function in configuration space P͑ , t͒ is given by Eq. ͑3͒, viz.,…”

“…In order to calculate the nonlinear impedance of the Josephson junction from the QSE ͑6͒, we note that the spatially periodic function P͑ , t͒ can be expanded in a Fourier series in , 18,30 viz.…”