Cold Damping of an Optically Levitated Nanoparticle to micro-Kelvin Temperatures

TL;DR

This paper demonstrates a feedback cooling technique that reduces the temperature of an optically levitated nanoparticle to 100 micro-Kelvin, approaching ground-state conditions, with results matching theoretical models.

Contribution

The study achieves record-low temperatures in nanoparticle cooling using cold damping, providing a detailed model and a pathway toward ground-state cooling.

Findings

Achieved nanoparticle cooling to 100 micro-Kelvin.

Measured and confirmed the temperature dependence on feedback gain.

Provided a roadmap for reaching quantum ground state.

Abstract

We implement a cold damping scheme to cool one mode of the center-of-mass motion of an optically levitated nanoparticle in ultrahigh vacuum from room temperature to a record-low temperature of 100 micro-Kelvin. The measured temperature dependence on feedback gain and thermal decoherence rate is in excellent agreement with a parameter-free model. We determine the imprecision-backaction product for our system and provide a roadmap towards ground-state cooling of optically levitated nanoparticles.

Equations

$$\hat{y}\left[{\Omega }_y^2-{\Omega }^2+\text{i}\gamma \Omega -H(\Omega )\right]=\frac{{\hat{f}}_{\text{fluct}}}{m}+H(\Omega ){\hat{y}}_n,$$

$$S_{yy}^{\text{ol}}(\Omega )=\frac{S_{ff}/m^2+{\gamma }_{\text{FB}}^2{\Omega }^2{S}_{nn}}{{\left({\Omega }_y^2-{\Omega }^2\right)}^2+{\left(\gamma +{\gamma }_{\text{FB}}\right)}^2{\Omega }^2}+S_{\nu \nu },$$

$$⟨y^2⟩=\frac{\pi S_{ff}}{m^2{\gamma }_{FB}{\Omega }_y^2}+\pi {\gamma }_{FB}{S}_{nn},$$

Full text

Introduction.

The interaction of light and matter is at the heart of a host of precision measurements, ranging from the detection of gravitational waves to the definition of the international unit system Abbott and et al. (2016); Mohr et al. (2016). What makes electromagnetic fields our probe of choice is the availability of detectors and laser light sources that operate at the noise limits dictated by the laws of quantum mechanics. Shortly after the invention of the laser, the scientific community started to explore the possibilities of mechanical manipulation of matter using the forces of light in optical traps Ashkin (1980, 2006). These forces can be interpreted as the inevitable consequence of the measurement process resulting from light-matter interaction Braginsky and Khalili (1992). Thus, optical forces and measurement precision are linked according to the Heisenberg uncertainty principle. The investigation of these measurement backaction effects has generated the field of optomechanics, which has developed experimental platforms that allow both measurement and control of mechanical motion at the quantum limit using light fields Teufel et al. (2009); Anetsberger et al. (2010); Verlot et al. (2010); Purdy et al. (2013); Aspelmeyer et al. (2014).

Dielectric particles levitated in optical traps are particularly versatile optomechanical systems Chang et al. (2010); Romero-Isart et al. (2011a); Li et al. (2011); Gieseler et al. (2012); Yin et al. (2013). Next to applications for precision measurements Geraci et al. (2010); Arvanitaki and Geraci (2013); Moore et al. (2014); Rider et al. (2016), one exciting prospect is the investigation and control of quantum states of massive objects Romero-Isart et al. (2011b). The starting point for any experiment in this direction is cooling an optically levitated nanoparticle to its quantum ground state of motion, a feat already achieved for cryogenically precooled mechanically clamped systems using autonomous cavity cooling Chan et al. (2011); Teufel et al. (2011) and active feedback control Rossi et al. (2018). While cavity-based feedback methods have made remarkable progress in recent years Kiesel et al. (2013); Millen et al. (2015); Fonseca et al. (2016); Windey et al. (2018), the most successful method to cool the center-of-mass motion of a levitated particle to date has been parametric feedback cooling in a single-beam optical dipole trap Gieseler et al. (2012); Vovrosh et al. (2017), where cooling from room temperature to occupation numbers below a hundred phonons has been achieved Jain et al. (2016).

Recently, the finite net charge carried by levitated nanoparticles has moved to the center of attention in the context of force sensing  Frimmer et al. (2017); Ranjit et al. (2015). Importantly, the Coulomb force that can be applied to a charged optically levitated particle provides the possibility to implement a cooling method termed cold damping Steixner et al. (2005); Bushev et al. (2006); Iwasaki et al. (2018). This measurement-based feedback technique applies a direct force to the oscillator in proportion to its speed, effectively leading to an increased damping rate Mancini et al. (1998). Cold damping has successfully been used in optomechanics to cool clamped mechanical oscillators Cohadon et al. (1999); Poggio et al. (2007); Wilson et al. (2015); Rossi et al. (2018) and optically levitated micron-sized particles Ashkin and Dziedzic (1977); Li et al. (2011), using the radiation pressure force. Surprisingly, the potential of cold damping for ground-state cooling the motion of an optically levitated nanoparticle has remained unexplored to date.

In this Letter, we cool the center-of-mass motion of an optically levitated nanoparticle to a temperature of $100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{K}$ using cold damping. To this end, we exploit the Coulomb force acting on the net electric charge carried by the particle. We investigate the cooling performance as a function of gas pressure and feedback gain to explore the limitations of the method. Our system operates a factor of one thousand from the Heisenberg limit of the imprecision-backaction product and provides a platform for studying ground-state cooling of optically levitated oscillators.

Experimental.

Our experimental setup is shown in Fig. 1. We optically trap a silica nanoparticle (diameter 136 nm) in a linearly polarized laser beam (wavelength 1064 nm, focal power 130 mW), focused by a microscope objective (0.85NA) resulting in oscillation frequencies of the particle’s center-of-mass $\Omega_{z}=2\pi\times 45\leavevmode\nobreak\ {\rm kHz}$, $\Omega_{x}=2\pi\times 125\leavevmode\nobreak\ {\rm kHz}$, and $\Omega_{y}=2\pi\times 146\leavevmode\nobreak\ {\rm kHz}$, where $z$ denotes the direction along the optical axis, while $x$ ($y$) are the coordinates in the focal plane along (orthogonal to) the axis of polarization. We collect the forward scattered light with a lens and guide it to a standard homodyne detection system for the particle’s motion along all three axes, which we call the in-loop detector (only shown for the $y$ axis in Fig. 1) Gieseler et al. (2012). Throughout our work, the particle’s motion along the $x$ and $z$ directions is cooled using parametric feedback to temperatures below $1\text{\,}\mathrm{K}$, rendering non-linearities of the trapping potential irrelevant Gieseler et al. (2014); Jain et al. (2016). From here on, we solely focus on the motion of the particle along the $y$ axis. We exert a Coulomb force on the net charge carried by the optically trapped nanoparticle by applying a voltage to a pair of electrodes enclosing the trap Frimmer et al. (2017). To cool the particle’s motion, this voltage is a feedback signal derived from the measurement signal $y_{\text{il}}$ acquired from the forward scattered light. Our linear feedback filter with transfer function $H(\Omega)$ consists of a series of digital, second-order biquad filters, which essentially mimics a derivative filter, such that the feedback signal is proportional to the particle’s velocity. More specifically, we use a band-pass filter whose center-frequency is set to above the particle’s oscillation frequency $\Omega_{y}$, such that the transfer function at $\Omega_{y}$ increases linearly with frequency while preserving a flat phase response Rossi et al. (2018). Finally, we measure the out-of-loop signal $y_{\text{ol}}$ with a heterodyne detection system for the backscattered light, using a local oscillator which is frequency shifted by $1\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ from the trapping light. We calibrate our detectors in the mildly underdamped regime at a pressure of $10\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$ using the equipartition theorem in the absence of feedback cooling Hebestreit et al. (2018).

Cooling performance.

We now investigate the performance of our cold damping scheme at a pressure of $1.4\text{\times}{10}^{-8}\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$. In Fig. 2(a) we show the single-sided power spectral density (PSD) $\tilde{S}_{yy}^{\text{ol}}$ 111We define our PSDs according to $\Braket{y^{2}}=\int_{0}^{\infty}{\rm d}f\leavevmode\nobreak\ \tilde{S}_{yy}(f)=\int_{-\infty}^{\infty}{\rm d}\Omega\leavevmode\nobreak\ S_{yy}(\Omega)$ of the out-of-loop signal for different feedback gains, which we express as damping rates $\gamma_{\text{FB}}$. We extract the damping rate $\gamma_{\text{FB}}$ from ring-down measurements as detailed further below. The measured signal $\tilde{S}_{yy}^{\text{ol}}$ corresponds to a Lorentzian function added to a spectrally flat noise floor due to the photon shot noise on our detector. The spectral width of the Lorentzian is a measure for the total damping rate arising from feedback cooling and residual gas damping. The latter is largely negligible under feedback at the low gas pressures of our experiments. The area under the Lorentzian, on the other hand, is a measure for the energy (i.e., temperature) of the particle’s oscillation mode. As expected, as we increase the feedback gain, the Lorentzian broadens in width and simultaneously shrinks in area. Thus, from the PSD of the out-of-loop signal $\tilde{S}_{yy}^{\text{ol}}$, we extract the energy $k_{B}T_{y}$ in the $y$ mode of the levitated particle.

In Fig. 2(b), we plot the measured mode temperature $T_{y}$ as a function of feedback damping rate $\gamma_{\text{FB}}$ at a pressure of $1.4\text{\times}{10}^{-8}\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$ as black circles. At small feedback gains, we observe a decrease in oscillator temperature with increasing feedback gain. However, there exists an optimal feedback gain of about $1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$. For gain values larger than the optimum, the oscillator temperature increases with increasing feedback gain. For comparison, we show the PSD of the measured in-loop signal $\tilde{S}_{yy}^{\text{il}}$ in the inset of Fig. 2(b) at the same gain values as in Fig. 2(a). For large feedback gain, we observe that $\tilde{S}_{yy}^{\text{il}}$ drops below the shot noise level. This effect, termed noise squashing, arises from correlations between the particle’s position and the measurement noise that is fed back by the control loop Poggio et al. (2007); Wilson et al. (2015). In Fig. 2(b), we additionally show measurements performed at a higher pressure of $1.2\text{\times}{10}^{-7}\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$ (grey triangles), where the increased gas damping rate leads to a larger mode temperature as compared to the low-pressure data.

Analysis.

To understand our results, let us analyze our system from a theoretical perspective. The Fourier transform $\hat{y}(\Omega)$ of the time-dependent particle position $y(t)$ follows the equation of motion

[TABLE]

where $\Omega_{y}$ is the $y$ mode’s eigenfrequency, $m$ the particle’s mass, and $\hat{y}_{n}$ is the measurement shot noise on the in-loop detector, which measures $\hat{y}_{\text{il}}=\hat{y}+\hat{y}_{n}$. The damping rate $\gamma$ arises from the interaction with residual gas molecules. The term $\hat{f}_{\text{fluct}}$ describes the fluctuating force generated by the interaction with the gas and from radiation pressure shot noise. Via the fluctuation dissipation theorem, $\hat{f}_{\text{fluct}}$ is inextricably linked to $\gamma$ Clerk et al. (2010). Within the bandwidth of interest, the transfer function of our feedback circuit is well described by $H(\Omega)=-\text{\rm i}\gamma_{\text{FB}}\Omega$. The feedback damping rate $\gamma_{\text{FB}}$ can be set by adjusting the feedback gain and incorporates the exact geometry of the capacitor electrodes and the number of charges carried by the levitated particle. Importantly, the feedback transfer function $H(\Omega)$ appears twice in Eq. (1), which results from the fact that the input to the feedback circuit $\hat{y}_{\text{il}}$ is the sum of the true position $\hat{y}$ and the measurement shot noise $\hat{y}_{n}$. From Eq. (1), we obtain the two-sided PSD on the out-of-loop detector

[TABLE]

where $S_{ff}$ denotes the PSD of the fluctuating force $\hat{f}_{\text{fluct}}$, and $S_{nn}$ ($S_{\nu\nu}$) are the PSDs of the in-loop (out-of-loop) detector noise. Integrating the first term of Eq. (2), which corresponds to the PSD of the true position $y$, in the limit $\gamma_{\text{FB}}\gg\gamma$ yields the variance

[TABLE]

which is a direct measure for the temperature $T_{y}=m\Omega_{y}^{2}\Braket{y^{2}}/k_{B}$ of the oscillator mode. The first term contributing to the expression in Eq. (3) scales with the inverse of the feedback cooling rate $\gamma_{\text{FB}}$. This term resembles the desired action of the feedback, which is to reduce the impact of the heating term given by the fluctuating force $S_{ff}$. Importantly, the second term is proportional to the feedback damping rate, which multiplies with the measurement noise $S_{nn}$. This term resembles the undesired but inevitable effect of the control loop heating the particle by feeding back measurement noise. Accordingly, our model predicts the existence of an optimum feedback cooling rate, where the mode temperature reaches its minimum value $T_{\text{min}}=2\pi\Omega_{y}\sqrt{S_{ff}S_{nn}}/k_{B}$, a behavior that we observe in our measurements in Fig. 2(b).

For a quantitative comparison of measurement and theory, we have to determine all parameters entering Eq. (3). We extract the in-loop measurement noise $S_{nn}$ from the PSD shown in the inset of Fig. 2(b). To obtain the feedback damping rate $\gamma_{\text{FB}}$, we perform ring-down measurements. To this end, we toggle the feedback gain back and forth between $\gamma_{\text{FB}}$ for $30\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ and a much lower feedback gain $\gamma_{\text{FB}}^{\text{low}}=\gamma_{\text{FB}}/300$ for $50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$. As shown in Fig. 3, we measure the mode temperature as a function of time after the gain was switched from $\gamma_{\text{FB}}^{\text{low}}$ to $\gamma_{\text{FB}}$ at time $t=0$. The blue triangles in Fig. 3(a) are the ensemble average over 100 such decay curves. We observe an exponential decay of the temperature and extract its time constant, which equals $\gamma_{\text{FB}}$. When the feedback gain is switched from $\gamma_{\text{FB}}$ to $\gamma_{\text{FB}}^{\text{low}}$ at time $t=0$, we observe the mode temperature increasing linearly in time [red circles in Fig 3(a), averaged over 100 reheating experiments]. Since the observed time is much shorter than the inverse damping rate $\gamma$, we expect the temperature to increase as $T(t)=\gamma T_{\text{bath}}\leavevmode\nobreak\ t$. Together with the fluctuation dissipation theorem $S_{ff}=m\gamma k_{B}T_{\text{bath}}/\pi$, the measured slope of the reheating curve therefore provides us with a direct measurement of the first term in Eq. (3) Clerk et al. (2010). Equipped with the experimentally determined values for $\gamma_{\text{FB}}$, $S_{nn}$, and $S_{ff}$, we calculate the mode temperature as a function of feedback gain according to Eq. (3) and display it as the solid black line in Fig. 2(b). The dashed lines show the two separate contributions from the bath (blue) and measurement noise (red) to Eq. (3). Our model describes our experimental findings very well. We stress that there is no free parameter or fit involved.

Finally, we investigate the reheating speed and the ring-down rate $\gamma_{\text{FB}}$ as a function of pressure. The results are displayed in Fig. 3(b). We find that the ring-down rates (blue triangles) do not depend on pressure. This observation confirms that the damping rate under feedback is indeed fully dominated by and therefore equivalent to the cold-damping rate $\gamma_{\text{FB}}$. The red circles in Fig. 3(b) show the measured reheating speeds ${\rm d}T_{y}/{\rm d}t$ as a function of pressure, which follows the expected linear behavior (dash-dotted line).

Discussion.

Let us discuss the current limitations and future prospects of our cold-damping approach for levitated optomechanics. To this end, we return to Eq. (3), whose two contributions are fundamentally related by the imprecision-backaction product $S_{ff}S_{nn}=\left(\frac{\hbar}{4\pi}\right)^{2}\frac{1}{\eta}$, with the measurement efficiency $\eta\leq 1$ Clerk et al. (2010). At the optimal feedback gain, we find an effective phonon occupation number $n=k_{B}T/(\hbar\Omega_{y})-1/2$ that solely depends on $\eta$ as $n_{\text{min}}=\frac{1}{2}(\frac{1}{\sqrt{\eta}}-1)$. At the Heisenberg limit of unit efficiency $\eta=1$, when the fluctuating force $S_{ff}$ driving the system under investigation is purely due to measurement backaction, and the imprecision noise $S_{nn}$ is minimized by optimally detecting all photons scattered by the levitated particle, the particle’s motion could, in principle, be brought to its quantum ground state $n_{\text{min}}=0$. In our case, at the lowest investigated pressure of $1.4\text{\times}{10}^{-8}\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$, we extract a total efficiency of $\eta=9\times 10^{-4}$ and hence an occupation number of about 16. Our measurements in Fig. 3(b) suggest that we can further reduce $S_{ff}$ by moving to even lower pressures, before entering the regime where reheating is fully dominated by photon recoil Jain et al. (2016). The factor $S_{nn}$ in our case is limited by the finite collection and detection efficiency. The latter is restricted by the non-ideal mode-overlap between the scattered dipole field and the Gaussian trapping beam on the detector. Exploiting the Purcell-enhanced collection and detection efficiency of a cavity, a suppression of $S_{nn}$ by more than one order of magnitude seems realistic Kiesel et al. (2013); Fonseca et al. (2016); Windey et al. (2018). Accordingly, occupation numbers approaching unity appear within reach.

Conclusion.

In conclusion, we have demonstrated cold damping of the center-of-mass motion of an optically levitated nanoparticle from room temperature to $100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{K}$, corresponding to less than 20 phonons. We have determined the optimal feedback-damping rate for our system, in excellent agreement with a parameter-free model. Together with photonic techniques under development Windey et al. (2018); Kuhn et al. (2017), our results put ground-state cooling of optically levitated nanoparticles firmly within reach. Besides setting a new temperature benchmark, we believe that our feedback control scheme will serve as a model system for the levitated optomechanics community. Putting our work into context, our approach is complementary to parametric feedback cooling, the method of choice to control charge-neutral optically levitated particles. In contrast, our system relies on the levitated object carrying finite net charge. Importantly, our work provides the direct connection to established optomechanical technologies Poggio et al. (2007); Wilson et al. (2015); Rossi et al. (2018). This fact generates the opportunity to leverage the insights gained with mechanically clamped systems to drive levitated optomechanics forward.

Acknowledgements.

This research was supported by ERC-QMES (Grant No. 338763) and the NCCR-QSIT program (Grant No. 51NF40-160591). We thank R. Diehl, E. Hebestreit, F. van der Laan, R. Reimann, and D. Windey for valuable input and discussions.