Librational feedback cooling

Librational motion, whereby a rigid body undergoes angular oscillation around a preferred direction, can be observed in optically trapped, silica microspheres. We demonstrate the cooling of 1 librational degree of freedom for $\sim 5-\mu \text{m-diameter}$ spheres that have been induced to rotate with an external electric field coupled to their electric dipole moment. Cooling is accomplished by adding a phase modulation to the rotating field. The degree of cooling is quantified by applying a $\pi /2$ shift to the phase of the electric field and fitting the resulting exponential decay of the librational motion to obtain a damping time, as well as estimating a mode temperature from the observed libration in equilibrium. The result is an important step in the study of the dynamics of trapped microspheres, crucial to cooling the mechanical motion to its ground state, as well as providing insights regarding the charge mobility in the material at microscopic scales.

Figure 1

A schematic depiction of the central features of the apparatus including the optical trap, the surrounding electrode structure, and parts of the imaging system. The lower right inset demonstrates a typical electric field configuration when driving a trapped MS to rotate. The upper right inset is an idealized version of the expected signal from the cross-polarized light monitor as the MS is rotating with the angular velocity ${\omega }_0$. Note that the coordinate pair relevant for the inset is different from that in the main panel. A complete description of the apparatus can be found in Ref. [22]

Figure 2

Block diagram of the feedback architecture, where all of the elements within the dotted border are integrated with the FPGA, such that the entire module is mutually clocked by the same top-level oscillator. The power of the cross-polarized light incident on the photodiode is modulated at the angular frequency $2{\omega }_0$ for a MS rotating at ${\omega }_0$. The $2{\omega }_0$ carrier is demodulated by phase-locked sampling, yielding $\varphi$, the angular position of the dipole moment relative to the electric field. The derivative can then be computed and scaled by an arbitrary and user-controlled gain parameter $K_d$. The quantity ${\varphi }_{\mathrm{ext}}$ represents an arbitrary user-defined phase that can be added to the phase modulation ${\varphi }_m$ of the electric field.

Figure 3

The amplitude envelope of the libration in response to an applied step, for a variety of derivative gain values, including $k_d=0$, where distinct colors (and lightness values) distinguish different values of derivative gain. A dashed line indicates the result of exponential fitting, with the extracted damping time showing in the legend. Inset: An example of the underlying oscillation of the librational motion for the largest value of derivative gain. Some filtering artifacts are present immediately following the step and are excluded from the exponential fit.

Figure 4

Summary of the libration phase step measurements. The damping time extracted from exponential fits is plotted as a function of the applied derivative gain. Different colors (lightness values) indicate different electric field amplitudes, while different marker shapes indicate distinct MSs. (Left) Damping times in the absence of applied feedback, showing a clear dependence on both drive amplitude and MS, with the former being a monotonic relation. Data from distinct MSs have been offset horizontally to aid their visibility. (Right) Damping times with feedback on. Dashed lines indicate fits to the expected scaling relation $\widehat{\tau }=2/(\gamma +C{k}_d)$, where $C$ is an arbitrary scaling constant found to be necessary to match the observed relation between $k_d$ and $\widehat{\tau }$.

Figure 5

PSDs of the librational motion depicting a drifting central frequency. Each panel contains one 200-s dataset, with 5.4 h between the two. The power spectral density of each full dataset is shown in light gray, while the power spectral densities of individual 10-s integrations within the datasets are shown in curves of varying lightness and color. It appears that some of the observed spectral width is driven by slow fluctuations in the center frequency, distinct from additive dissipation and subsequent line broadening.

Figure 6

A few examples of the mean PSDs of the librational motion of one particular MS, where distinct colors (and lightness values) distinguish different values of derivative gain. For measurements with $k_d\ne 0$, each PSD shown is the mean of the PSDs of 20 individual and consecutive 10-s integrations, following the averaging procedure discussed in the text. The dashed lines indicate fits of the PSD to the expression in Eq. (6), with the extracted value of ${\widehat{\gamma }}_d=\gamma +k_d$ and the ratio of the effective mode temperature from Eq. (7) to the zero-feedback mode temperature both shown in the legend. For the data with $k_d=0$, there are 10 distinct PSDs from 10 measurement series all plotted together, where each PSD is again the mean of 20 consecutive integrations, with a delay of the order of 1 h between each measurement series to ensure thermalization of the system. The dashed line for $k_d=0$ represents the mean of the fits to each of the 10 measurement series and is used to derive an estimate of zero-feedback damping $\gamma$, whereas the zero-feedback mode temperature is derived from a direct integration of the spectra and the assumption of equipartition.

Figure 7

Summary of the libration thermalization measurements, where, as before, different colors (and lightness values) indicate different electric field amplitudes, while different marker shapes indicate distinct MSs. Top panel: Ratio of the effective mode temperature calculated via Eq. (7) to the zero-feedback mode temperature, with the values of ${\omega }_{\varphi }, \gamma , {\widehat{k}}_d, S_{\mathrm{th}}$, and $S_{\xi }$ extracted from fitting Eq. (6) to the observed PSDs. The reheating observed for large $k_d$ is consistent with broadband noise injection from the feedback loop as the gain is increased. Bottom panel: Ratio of the extracted value of ${\widehat{k}}_d$, relative to the expected value $k_d=K_d{\omega }_{\varphi }^2$.