Measuring the internal temperature of a levitated nanoparticle in high vacuum

The interaction of an object with its surrounding bath can lead to a coupling between the object's internal degrees of freedom and its center-of-mass motion. This coupling is especially important for nanomechanical oscillators, which are among the most promising systems for preparing macroscopic objects in quantum mechanical states. Here we exploit this coupling to derive the internal temperature of a levitated nanoparticle from measurements of its center-of-mass dynamics. For a laser-trapped silica particle in high vacuum, we find an internal temperature of $1000\left(60\right)\mathrm{K}$. The measurement and control of the internal temperature of nanomechanical oscillators is of fundamental importance because black-body emission sets limits to the coherence of macroscopic quantum states.

Figure 1

Experimental setup. In the focus of a $1064-\mathrm{nm}$ laser (red), we trap a ${\mathrm{SiO}}_2$ nanoparticle. The trap is placed in a vacuum chamber to adjust the gas pressure. The light scattered by the particle is collected and detected with a balanced detector. A switchable parametric feedback allows us to cool the particle motion by modulating the laser beam intensity with an electro-optic modulator (EOM). To increase the internal temperature of the particle, we irradiate it from the side with a ${\mathrm{CO}}_2$ laser (blue) whose intensity we adjust with an acousto-optic modulator (AOM).

Figure 2

Relaxation experiments. (a) Using parametric feedback (FB), we first cool the particle's COM motion. We switch the feedback off at time $t=0$, such that the particle's average COM energy equilibrates toward the effective bath temperature. After $900\mathrm{ms}$, the feedback is turned on again to return to the initial conditions. (b) At a pressure of $1\times {10}^{-3}\mathrm{mbar}$, we observe a relaxation of the COM temperature $T_{\mathrm{com}}$ to the bath temperature $T_{\mathrm{bath}}$ along all three oscillation axes. (c) When reducing the pressure to $1\times {10}^{-5}\mathrm{mbar}$, we only observe the initial slope of the relaxation curve that rises linearly with the heating rate ${\mathrm{\Gamma }}^{\prime }$. (d) For measuring the particle dynamics at elevated internal temperatures, we activate the ${\mathrm{CO}}_2$ laser $900\mathrm{ms}$ before starting the relaxation experiment. (e) At ${10}^{-3}\mathrm{mbar}$, in the presence of the ${\mathrm{CO}}_2$ heating laser, the particle's COM temperature equilibrates to a bath temperature higher than in the absence of the heating laser. (f) At ${10}^{-5}\mathrm{mbar}$, the slope of the heating curve ${\mathrm{\Gamma }}^{\prime }$ is larger in the presence of the heating laser as compared to panel (c), where the heating laser is off. Quantities extracted from these measurements are summarized in Table 2.

Figure 3

(a) Heating rates as a function of pressure extracted from relaxation measurements without additional heating of the internal temperature (blue) and with maximal heating (${\mathrm{CO}}_2$ laser intensity of $I_{{\mathrm{CO}}_2}=0.47\mu \mathrm{W}/\mu {\mathrm{m}}^2$, orange). The dashed lines are linear fits to the data points. Error bars are smaller than the marker size. (b) Heating rates at different intensities of the ${\mathrm{CO}}_2$ laser measured at a pressure of $1\times {10}^{-5}\mathrm{mbar}$. The error bars indicate the standard deviation of the measurements.

Figure 4

(a) Calculated temperature dependence of oscillation frequency. When the particle is heated with the ${\mathrm{CO}}_2$ laser, its oscillation frequency increases due to changes in the particle's material properties. This leads to a nearly linear relation between relative frequency change and increase of the internal particle temperature with $\mathrm{\Delta }{\mathrm{\Omega }}_0/{\mathrm{\Omega }}_0=(1.43\times {10}^{-5}{\mathrm{K}}^{-1})\mathrm{\Delta }{T}_{\mathrm{int}}$, where ${\mathrm{\Omega }}_0$ denotes the oscillation frequency at an internal particle temperature of $300\mathrm{K}$. (b) Measured relative frequency shift due to heating of particle with ${\mathrm{CO}}_2$ laser for different heating laser powers at a pressure of $1\times {10}^{-3}\mathrm{mbar}$. Here, ${\mathrm{\Omega }}_0$ is the particle oscillation frequency without ${\mathrm{CO}}_2$ laser irradiation. The time axis corresponds to the one shown in the relaxation cycle in Fig. 2. According to the functional dependence extracted from panel (a), the maximally observed frequency shift corresponds to a temperature rise of $\mathrm{\Delta }{T}_{\mathrm{int}}=601\left(16\right)\mathrm{K}$.

Figure 5

(a) Internal particle temperatures derived using different methods: (i) using the absolute heating rate of the internally nonheated particle (blue), (ii) using the ratio of heating rates for an internally heated and internally nonheated particle (orange). The combination of methods (i) and (ii) is shown in black [see panel (d)]. Marked are room temperature ($300\mathrm{K}$), the melting temperature of silica ($\sim 2000\mathrm{K}$), and the average of the internal temperatures $T_{\mathrm{int}}=1000\left(60\right)\mathrm{K}$ at pressures below ${10}^{-3}\mathrm{mbar}$. (b) Corresponding to the combined method in panel (a), we plot the deduced energy accommodation coefficients and mark their mean value ${\alpha }_{\mathrm{G}}=0.65\left(3\right)$. (c) To illustrate method (ii), we plot the ratio of the heating rates with and without the ${\mathrm{CO}}_2$ laser illuminating the particle as a function of $\mathrm{\Delta }{T}_{\mathrm{int}}$ for two pressures. To each dataset, we fit Eq. (5) and infer an internal particle temperature $T_{\mathrm{int}}$ as a fit parameter. (d) Combining methods (i) and (ii), we plot the derived internal particle temperature depending on the chosen accommodation coefficient for methods (i) and (ii) at a pressure of $1\times {10}^{-6}\mathrm{mbar}$. From the intersection point, we infer the best-fit energy accommodation coefficient and the internal particle temperature [see panels (a) and (b), black data points]. The error bars and shaded areas in all plots indicate the $1\sigma$ confidence intervals.

Figure 6

Internal particle temperature calculated as a function of pressure from the balance of emission and absorption rates (blue solid line). At pressures above ${10}^{-1}\mathrm{mbar}$, convective gas cooling dominates the dissipation of internal energy and the internal particle temperature equilibrates to the environment temperature of $300\mathrm{K}$ (black dotted line). Below ${10}^{-3}\mathrm{mbar}$, convection is not efficient anymore and the internal temperature approaches $1020\mathrm{K}$. Additionally, we calculate the resulting COM equilibrium temperature $T_{\mathrm{bath}}$ from the two-bath model (Sec. 4) using the parameters determined in the experiments (orange dashed line).