Figure 1

Experimental setup. An acousto-optical deflector (AOD) (ISOMET LS55 NIR) steers a $1060\mathrm{nm}$ optical beam from a laser coupled into a single-mode fiber (ManLight, ML10-CW-P-OEM/TKS-OTS, maximal power 3 W). The AOD modulation voltage is obtained from an arbitrary waveform generator (Tabor Electronics WW1071) controlled by a labview program. The beam deflected by the AOD is expanded and inserted through an oil-immersed objective O1 (Nikon, CFI PL FL 100X NA 1.30) into a custom-made fluid chamber. An additional $532\mathrm{nm}$ optical beam from a laser coupled to a single-mode fiber (OZOptics) is collimated by a $(10\times ,\text{NA}=0.10)$ microscope objective and passes through the trapping objective. The forward scattered detection beam is collected by a $(10\times ,\text{NA}=0.10)$ microscope objective O2, and its back focal-plane field distribution is analyzed by a quadrant position detector (QPD) (New Focus 2911) at an acquisition rate of $20\mathrm{kHz}$. A $532\mathrm{nm}$ bandpass filter in front of the QPD blocks beams with wavelengths different from the detection beam wavelength. The AOD permits control of the position of the beam focus. We also used it for the calibration procedures and for mapping the force distribution in the optical trap, as described in the text. A fluid chamber with microspheres was placed on a piezoelectric-controlled calibrated stage (Piezosystem Jena, Tritor 102) allowing three-dimensional translation. The inset shows an optically trapped sphere between the electrodes fed by noisy signals with controllable PSD.Reuse & Permissions

Figure 2

(a) The output signal of the QPD has a nonlinear character when the sphere displacements exceed $\pm 1000$ nm. The minor deviations from the linear dependence in the linear part of the QPD response are due to the nonuniformity in the detection beam intensity as well as the Brownian fluctuations of the sphere. (b) The force exerted on the 1 $\mu$m optically trapped sphere versus the radial distance from the trapping beam focus. The incident intensity of the trapping beam at the input pupil of the trapping objective is 5 mW. We confirmed that the force map is not affected by a further increase in the strong trap stiffness while maintaining the same intensity of the weak trap. (c) The amplitude of the sphere displacement in an ac electric voltage 30 V at 1 Hz over 3000 s.Reuse & Permissions

Figure 3

PSD of the position of the sphere without (blue curve, 1) and with (green curve, 2) the additional stochastic force. Solid black lines correspond to the Lorentzian fits. The power spectral density of the input noisy signal measured at the electrodes of the fluid cell is also shown (black line, 3). Both axes are in ${\text{log}}_{10}$ scale.Reuse & Permissions

Figure 4

Histograms of the sphere position without (1) and with the additional stochastic force (2) corresponding to the PSD curves (1) and (2) shown in Fig. 3.Reuse & Permissions

Figure 5

The sphere position (blue solid curve, left axis) and amplitude of electric signal (red dot curve, right axis) as a function of time if the electric field changes abruptly.Reuse & Permissions

Figure 6

(a) The position traces (green lines, left axis) of the sphere in the double-well potential. The black lines (right axis) show the voltage on the electrodes. The additional noisy signal was switched on at 35.5 s. As can be seen, the frequency of the Kramers transitions increases. (b) The trapping potential obtained at room temperature. (c) Probability of the residence time of the Kramers transitions (in counts, ${\text{log}}_{10}$ scale) at room temperature (green squares) and at 3000 K (blue circles).Reuse & Permissions

Figure 7

The position of the center of the trap (black line) and the positions of the sphere as functions of time without external electric field (blue curve) and with noisy electric field (green curve).Reuse & Permissions

Figure 8

The probability density function of the work obtained in around 7000 realizations of the forward process [$\rho \left(W\right)$], and the probability density function obtained from the same number of realizations of the reverse process [${\rho }^{*}(-W)$]. The vertical black lines represent the analytical value of the average work in the forward process calculated using (13) (solid line) and the average of $-W$ in the reverse process (dashed line) which is calculated in a way analogous to (13). The squares correspond to the forward process [$\rho \left(W\right)$] and the circles to the reverse [${\rho }^{*}(-W)$]. The solid and dashed curves are Gaussian fits.Reuse & Permissions

Figure 9

$\ln \left[\rho \left(W\right)/{\rho }^{*}(-W)\right]$ as a function of $W/k_{B}T$ without (blue squares) and with additional noise (green circles) corresponding respectively to the PSD curves (1) and (2) shown in Fig. 3. The solid lines are linear fits to the experimental data.Reuse & Permissions

Figure 10

Effective kinetic temperatures as functions of $T_e={\sigma }^2/\left(2k_B\gamma \right)$. The filled symbols correspond to experimental values of the effective temperature measured from the fluctuations and the response of the bead along the $x$ axis (direction of the additional noise): $T_{\mathrm{PSD}}$ (blue filled squares), $T_{\mathrm{hist}}$ (red filled circles), and $T_C$ (green filled triangles). Statistical errors are plotted in the error bars. Lines represent the analytical values obtained for nonwhite external noise $\zeta \left(t\right)$ with correlation function given by (12) for $T_{\mathrm{PSD}}$ (7) (blue solid line), $T_{\mathrm{hist}}$ (8) (red dashed line), and $T_C$ (green dotted line), which was obtained analytically using (14). We also show, in open symbols, the experimental values of the effective temperature measured from the fluctuations along the $y$ axis: $T_{\mathrm{PSD}}$ (blue open squares) and $T_{\mathrm{hist}}$ (red open circles). The statistical errors are smaller than the size of the symbols. The black dashed line is set to 300 K.Reuse & Permissions