Levitodynamics is a branch of optomechanics that represents the study, control and manipulation of nano and microparticles levitated through optical or radiofrequency forces. The field of levitodynamics has seen a fast and steady growth in the past decade, enabled by its potential to study transitions from classical to quantum mechanics, to investigate nonequilibrium statistical physics and to sense extremely small forces and torques [1-10].

In our group we focus our efforts on trapping silica nanoparticles (diameter typically in the order of 100 nm) through optical forces. Recently, we started exploring hybrid traps as well, that make use of both an optical trap and a Paul trap.

Our research pursues different directions:

- Cooling of center of mass motion through active feedback.
- Cooling of center of mass motion through passive cavity cooling.
- Study and manipulation of the rotational degrees of freedom of the nanoparticle.
- Nonequilibrium statistical mechanics of a trapped nanoparticle.
- Development of an inertial sensing platform based on optically levitated nanoparticles

**Optical trapping**

Silica nanoparticles are dielectric materials. An external electric field can thus induce a dipole on the nanoparticle and consequently generate a force proportional the field gradient. Optical trapping of silica nanoparticles consists of a strongly focused laser beam, whose strong field gradient generates an optical force strong enough to stably trap the particle in its focus.

The figure on the side displays one of our optical traps. A strong infrared laser beam propagating towards the right of the image is focused with a high numerical aperture lens. A weaker, green laser beam is overlapped with the trapping beam to make the particle visible by eye. The optical trap is enclosed in a vacuum chamber, within which the air pressure can be freely tuned.

The motion of the particle in the optical trap can be described to first order approximation by a damped harmonic oscillator:

In the above equation, *x* is one of the three spatial degrees of freedom of the particle, *m* is the mass, Ω_{0} is the resonance frequency induced by the optical trap, F_{th }is a stochastic term representing thermal fluctuations and Γ_{0} is the damping coefficient. The strength of the thermal fluctuations is connected to the damping force through the fluctuation-dissipation theorem, so that the equipartition theorem holds. An analogous equation holds for the other two degrees of freedom.

Below we show typical power densities of the three translational modes together with the dependency of the damping coefficient on the vacuum chamber's pressure.

**Free space optomechanics**

The importance of trapping and cooling techniques is evidenced by the series of Nobel Prizes awarded to this theme (1989 ion trapping, 1997 atomic trapping, 2001 Bose-Einstein condensation, 2012 trapping of photons and ions, optical tweezers in 2018). Trapping and cooling of mesoscopic objects, such as the nanoparticles we use, is a new territory that lies at the boundary of classical and quantum physics. The objective of this project is to cool the centre of mass motion of the nanoparticle through a feedback force based on the measurement of its position.

When trapped by an optical beam, the nanoparticle scatters coherent radiation whose phase encodes its position [11]. By extracting the phase through an interferometric measurement, we can reconstruct the position of the nanoparticle and apply a real time feedback force to control its motion.

We make use of two feedback schemes in our experiments:

- Parametric feedback [1, 4]. This method relies on modulating the trapping beam intensity at a frequency equal to twice the particle’s eigenfrequency. When the phase of the modulation is suitably locked to the phase of the particle’s motion, the feedback effectively increases the particle’s damping. The advantage of this scheme is that the same laser beam used to trap the nanoparticle can be additionally used to cool its motion in all three dimensions.
- Cold damping, also called linear feedback [12]. This method applies a force on the particle proportional to its velocity, i.e. a damping force. Cold damping is much easier to model mathematically than parametric feedback, it requires however additional experimental overhead. In general, cold damping allows to reach much lower phonon occupation numbers.

Typically, we employ cold damping on the longitudinal axis (along the beam propagation) and parametric feedback for the other two axes. At low feedback strengths, the phonon occupation is determined by the thermal fluctuations induced by the surrounding air molecules. When the linear feedback is increased, the additional damping contributes to lowering the average phonon occupation number. For very high feedback strengths, measurement back-action is expected: the measurement noise is fed back to the particle, increasing the phonon occupation as a function of the feedback strength [2, 12]. Recently, we have used cold damping to resolve for the first time the motional sideband asymmetry of a free space oscillator (figure below).

**Cavity optomechanics**

A different paradigm for the cooling of the center of mass motion of the nanoparticle is to combine the optical trap with an optical cavity [3, 13], as shown on the right (schematic) and below (actual setup in lab).

The light scattered by the particle is captured by the cavity and populates the mode closest to the trapping beam’s frequency. Such experimental scheme is called *coherent scattering*. As the particle moves away from its equilibrium position, the resonance condition of the cavity changes. In turn, the field of the cavity’s mode exerts an optical force on the nanoparticle through dipole interaction. The resulting coupling between the mechanical mode of the particle and the optical mode of the cavity is a rich source of interesting phenomena, including sideband cooling and parametric heating among others.

In our experiments, we blue detune the cavity’s resonance from the trapping beam by an amount equal to the mechanical resonance frequency. Under this condition, the cavity enhances the anti-Stokes photon scattering (which extract mechanical energy from the particle) and hampers the Stokes photon scattering (which in turn increase the particle’s energy). A typical position measurement of the particle is shown below, where the cavity-induced asymmetry in the two sidebands visible.

When carefully selecting the position of the particle within the cavity, we have shown that all three dimensions of the particle’s motion can be cooled at the same time [3]. Currently, we are concentrating our efforts towards reaching the motional quantum ground state and towards integrating multiple optical traps within the same cavity.

**Hybrid optical-Paul traps**

As mentioned above, when a nanoparticle is trapped by an optical beam, its scattered light encodes information about the position of the nanoparticle. Whether an experimentalist records the phase of the particle or not, the position of the nanoparticle is being continuously measured by the trapping beam. Such a continuous measurement is a source of quantum decoherence, since the environment is continuously monitoring the particle’s state, thus contaminating its purity.

Despite their deleterious effect on the state’s purity, optical traps are essential for state preparation, as they allow to efficiently measure and control its degrees of freedom. Since the charge of the nanoparticle can in general be controlled, a particularly promising solution to the decoherence problem is to combine the optical trap with a Paul trap. A Paul trap makes use of a strong oscillating voltage that generates a three-dimensional trap through the ponderomotive force induced on the nanoparticle. The figure on the side shows our first Paul trap.

**Rotational optomechanics**

When trapped in a circularly polarised laser beam, the dumbbell starts to rotate very rapidly (see plot below) [15]. For pressures below 10^-5 mbar the rotation frequency reaches the GHz regime. This makes optically rotating dumbbells the fasted rotating man-made objects in the world. Due to the high rotation speeds, the particle’s material undergoes enormous strain, which makes rotating particles an interesting platform for investigating material properties.

When trapped in a linearly polarised laser beam, the dumbbell experiences an alignment torque which tends to align the dumbbell to the polarisation direction. The resulting libration around the equilibrium orientation shares many similarities with the translational dynamics of trapped nanoparticles. However, the provided rotational symmetry provides interesting quantum protocols which cannot be performed using translational degrees of freedom. Therefore we aim to bring the libration mode into the quantum regime via feedback cooling.

**Nonequilibrium statistical mechanics**

**Sensing with trapped nanoparticles**

The plot above is the theoretical probability distribution of of the nanoparticle before the fall (a), after the free fall when only gravity is present (b) and after the fall when both gravity and a 20 aN force drive the particle. Free falls can be applied to particles trapped at subwavelength distances from dielectric surfaces to detect Casimir forces.

Nanodumbbells trapped in a nearly circularly polarised beam are an example of optical rotors that hold great promise as pressure, temperature, acceleration and torque sensors. Unlike in the case of a linearly polarised trap, optical rotors are not subject to an alignment torque and are free to rotate. A change in the external parameters (pressure, temperature, torque, etc.) translate into a change in the angular velocity of the rotor, which can in turn be detected. We have recently implemented an optical rotor whose sensitivity is thermal limited, i.e. the smallest detectable torque is only limited by the strength of the thermal fluctuations [9].

The plot above displays the detected angular velocity of the optical rotor as a function of the pressure of the vacuum chamber. Lower pressures correspond to a lower coupling to the thermal bath and a consequent higher average angular velocity. In contrast, the standard deviation of the angular velocity depends only on the temperature of the coupled thermal bath and remains thus independent from pressure.

At the moment, we are pushing towards the implementation of a trapped nanoparticle based inertial system. The goal is to develop a small and flexible platform for translational and rotational velocity and for acceleration detection. We focus on building a levitated particle based gyroscope. Compared to commercial MEMS gyroscopes, the gyroscope based on a levitated particle is less affected by the environment noise thanks to levitation. At the same time, the optical tweezer provides a stable and flexible potential to manipulate the movement of the particle, for instance by cooling the centre of mass motion of the particle. These factors considerably enhance the performance of the gyroscopes.

In future, techniques based on quantum detection with noise cancellation can allow us to achieve ground-breaking inertial sensing performance in the low frequency range. Such a product would fit some of the current needs of industry.

Related publications

[1] J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, ”Sub-Kelvin parametric feedback cooling of a laser-trapped nanoparticle,” *Phys. Rev. Lett.* **109**, *103603* (2012).

[2] F. Tebbenjohanns et al., "Motional Sideband Asymmetry of a Nanoparticle Optically Levitated in Free Space", *Phys. Rev. Lett.* **124**, *013603* (2020).

[3] D. Windey et al., "Cavity-Based 3D Cooling of a Levitated Nanoparticle via Coherent Scattering", *Phys. Rev. Lett.* **122**, *123601* (2019).

[4] V. Jain et al., "Direct Measurement of Photon Recoil from a Levitated Nanoparticle", *Phys. Rev. Lett.* **116**, *243601* (2016)

[5] J. Gieseler et al., "Nonlinear mode coupling and synchronisation of vacuum-trapped nanoparticles", *Phys. Rev. Lett.* **112**, *103603* (2014)

[6] L. Rondin et al., "Direct measurement of the Kramers turnover", *Nat. Nanotech.* **12**, *1130 *(2017)

[7] E. Hebestreit et al., "Sensing Static Forces with Free-Falling Nanoparticles", *Phys. Rev. Lett.* **121**, *063602* (2016)

[8] G. Ranjit et al., "Zeptonewton force sensing with nano spheres in an optical lattice", *Phys. Rev. *A (2016)

[9] F. van der Laan et al., "Optically levitated rotor at its thermal limit of frequency stability", *Phys. Rev. *A. **102**, *013505* (2020)

[10] R. Diehl, "Optical levitation and feedback cooling of a nanoparticle at sub wavelength distances from a membrane", *Phys.Rev.A*, **98**, (2018)

[11] F. Tebbenjohanns et al., "Optimal position detection of a dipolar scatterer in a focused field", *Phys. Rev. *A. **100**, *043821* (2019)

[12] F. Tebbenjohanns et al., "Cold Damping of an Optically Levitated Nanoparticle to Microkelvin Temperatures", *Phys. Rev. Lett.* **122**, *223601* (2019)

[13] M. Aspelmeyer, T. J. Kippenberg, F. Marquardt, "Cavity Optomechanics", Springer (2014)

[14] The illustration has been provided by S. Papadopoulos.

[15] R. Reimann et al., "GHz Rotation of an Optically Trapped Nanoparticle in Vacuum", *Phys. Rev. Lett.* **121**, *033602* (2018)

[16] J. Gieseler et al., "Thermal nonlinearities in a nano mechanical oscillator", *Nat. Phys. 9, 806 *(2013)

[17] J. Gieseler et al., "Dynamic relaxation of a levitated nanoparticle from a non-equilibrium steady state", *Nat. Nanotech. 9, 358* (2014)