# Levitated Nanoparticles for Microscopic Thermodynamics—A Review

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## Abstract

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## 1. Introduction

## 2. Forces and Potentials

#### 2.1. Deterministic Forces

#### 2.1.1. Optical Potential

#### 2.1.2. Rotation

#### 2.2. Stochastic Forces

#### 2.2.1. Gas Damping

#### 2.2.2. Radiation Damping

#### 2.2.3. Artificial Damping and Heating

## 3. Brownian Motion

#### 3.1. Harmonic Brownian Motion

#### 3.2. Power Spectral Density and Calibration

#### 3.3. Quantum Brownian Motion

## 4. Trap Stability and Kramers Turnover

## 5. Effective Potentials in the Steady State

#### 5.1. Effective Potential for the Energy

#### 5.2. Effective Temperature

## 6. Relaxation

## 7. Fluctuation Theorems

## 8. Heat Engines

## 9. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic of optical levitation setup. A nanoparticle is trapped by a tightly focused laser beam. The translational degrees of freedom of the nanoparticle are measured with photodetectors and the center-of-mass motion is cooled by parametric feedback. In addition to feedback, external modulation allows excitation of the particle to drive it far from equilibrium. The top inset highlights the dominant forces in a typical optical levitation experiment, which are the optical gradient and scattering forces and gravity. The bottom inset shows the temperatures involved in a collision with a heated sphere: the sphere’s centre-of-mass temperature (${T}_{\mathrm{CM}}$) and surface temperature (${T}_{\mathrm{int}}$), and the temperatures of the impinging gas particles (${T}_{\mathrm{gas}}$) and emerging gas particles (${T}_{\mathrm{em}}$) with ${T}_{\mathrm{gas}}\le {T}_{\mathrm{CM}}\le {T}_{\mathrm{em}}\le {T}_{\mathrm{int}}$. The collision with the air molecules leads to damping ${\Gamma}_{\mathrm{gas}}$, which depends on the pressure. Main figure taken from [23] with permission from Physical Review Letters. Inset adapted from [24] with permission from Nature Nanotechnology.

**Figure 2.**First experimental observation of the instantaneous velocity of a Brownian particle. (

**a**) The mean-square displacement for short times is proportional to ${t}^{2}$, a signature of ballistic motion. (

**b**) The normalized velocity autocorrelation functions for different pressures in perfect agreement with Equation (11b). Figures taken from [9] with permission from Science.

**Figure 3.**Feedback cooling of a levitated nanoparticle. (

**a**) Power spectral density under phase locked feedback cooling at three different pressures and constant feedback gain. The area under the power spectral densities is a measure for the effective center-of-mass temperature. (

**b**) The effective temperature expressed in terms of the phonon occupation as a function of gas pressure. Figures reproduced from Physical Review Letters [18].

**Figure 4.**Measurement of the Kramers turnover with a levitated nanoparticle. Data illustrating the first experimental observation of Kramers turnover, taken from [66]. The full theory from [64] (solid line) is shown as a solid blue line together with the limiting cases as predicted by Kramers [63] (dot-dashed lines). The red dashed line highlights the expected turnover point as predicted from the measured shape of the double well potential and is in excellent agreement with the experimental observations.

**Figure 5.**Relaxation from a non-equilibrium steady state. (

**a**) Individual trajectories of the energy as the system relaxes toward equilibrium (

**b**) Position distribution during the relaxation process. The energy distribution is given by Equation (23). (

**c**) Energy distribution in the steady state ($t\le 0$) in agreement with Equation (18). The deviation from a thermal state due to the nonlinear feedback is clearly visible. (

**d**) Experimental verification of the detailed fluctuation theorem (c.f. Equation (26)). All figures taken from [47] with permission from Nature Nanotechnology.

**Figure 6.**Single particle engines. (

**a**) Realization of a Stirling engine by Blickle & Bechinger [91] using laser absorption to change the temperature of the environment. (

**b**) Martinez et al., [107] realized a microscopic Carnot engine. The adiabatic steps of the Carnot engine requires to change the temperature and the trap stiffness synchronously. Figures reproduced from Nature Physics.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Gieseler, J.; Millen, J.
Levitated Nanoparticles for Microscopic Thermodynamics—A Review. *Entropy* **2018**, *20*, 326.
https://doi.org/10.3390/e20050326

**AMA Style**

Gieseler J, Millen J.
Levitated Nanoparticles for Microscopic Thermodynamics—A Review. *Entropy*. 2018; 20(5):326.
https://doi.org/10.3390/e20050326

**Chicago/Turabian Style**

Gieseler, Jan, and James Millen.
2018. "Levitated Nanoparticles for Microscopic Thermodynamics—A Review" *Entropy* 20, no. 5: 326.
https://doi.org/10.3390/e20050326