LEY GENERAL DE EDUCACIÓN: GARROTE Y ZANAHORIAS

Levitation, the chance to defy the pull of gravity and hover free from tangible con- straints, has held a place in mankind’s aspirations for as long as historical records can confirm. Even today, with countless aircraft weaving trails through the skies and astronauts regularly experiencing weightlessness in a permanently occupied space sta- tion, the levitation of ground-based objects can still induce awe and inspiration in the general public. As with any other phenomenon, however, the interest on levitation would be quite short-lived if it had to depend merely on its wonder factor. The shift from novelty devices used to demonstrate physical principles to practical engineering tools with functional applications is nowadays being completed by more and more levitated systems. Levitated trains are a notable example [186], using the action at a distance from permanent electromagnets or superconducting circuits to guarantee fast, friction-less transportation. Other examples include contact-free manipulation of small particles by acoustic standing waves [187,188], and “levitation” of graphene using oxygen intercalation to lift and decouple the structure from a metal substrate [189].

Optical levitation dates back to the early 1970s, when micron-sized dielectric par- ticles were trapped by radiation pressure force alone for the first time [190, 191]. The technique, now known as optical tweezers [192–194], exploits the high refractive index of the particle to deflect the beam and produce a back-action force that pushes the object towards the point of highest intensity, the focus. Since the first experiments the development of optical tweezers has been fuelled by continuous upgrades, which include single beam realization over diverse scales [195] and the broadening of the trap- ping range thanks to regenerative Bessel beams [47]. Advancements continue even to

(e) (d) Trap Cooling Trap + Cooling (c) Trap Cooling (b) (a) (a)

Figure 8.1: Only few of the many examples of optically levitated systems. (a)The principle of optical tweezers uses the intensity gradient to trap particles in the focus of the beam [199].

(b) Multiple optical tweezers along the three axes create an “optical molasses” that cools the centre-of-mass motion of the particle [152]. (c)Doubly resonant optical cavities can also create a trapping potential for tiny particles [200]. (d) In air, photophoretic forces allow manipulation of objects on a much bigger scale [198]. (e) A proposed system based on a cavity mirror attached to a silica disk which is suspended by two optical tweezers [97].

this date, for example with the establishment of robust techniques to deliver particles to high-vacuum environments [196]. Photophoresis can provide an alternative method of levitation for particles that are not required to be in vacuum conditions: the non- uniform heating of the air surrounding the particle causes the gas molecules to rebound o↵the surface with di↵erent velocities, producing a net force that can trap bodies sev- eral order of magnitude heavier compared to the particles lifted by radiation pressure force [197, 198]. A collection of representative systems based on optical levitation is shown in Fig. 8.1.

The success of optical levitation is largely attributable to the flexibility it provides in manipulating objects without direct contact [201, 202]. The technique is virtually suitable for anything ranging from single atoms to particles of a few micrometres in size, and has been successfully applied to manipulate DNA molecules [203] and col- loidal systems [204, 205]. By transferring the orbital angular momentum of light onto the particle, it is possible to rotate and manoeuvre micro-machines [206] and micro- gyroscopes [207]. One could even think of using optical tweezers to operate a micro- scopic steam engine [208], evidence of the fact that only inventiveness poses a limit to the versatility of optically levitated systems.

Trapping in an optical potential makes particle’s centre of mass behave like a har- monic oscillator. Thanks to the complete detachment from any mechanical support, which removes a direct coupling to a thermal reservoir, these suspended oscillators demonstrate ideal qualities for optomechanics experiments. In principle, in high vac- uum the mechanical damping rate _m could be reduced almost indefinitely and the mechanical quality factor Q_m = !_m/ _m of the centre-of-mass motion could become higher than 1012 [200] once it is decoupled from the internal degrees of freedom. A long coherence time and a high quality factor would make techniques like laser cooling, state transfer, and quantum superposition incredibly accessible [209].

Multiple approaches could be considered in connection to laser cooling. Doppler cooling along the optical axis can be achieved if the levitated particle enables resonance of whispering gallery modes around its perimeter [210]. A combination of three cool- ing beams encompassing all three spatial directions generates an “optical molasses”, an expression derived from the viscous nature of the forces experienced by the levi- tated object. Any generic cooling technique can be applied to the principle of optical molasses, including modulation of the intensity of the trapping beam to realize active feedback damping rather than passive cooling [152, 153]. Alternatively, the principle of optical tweezers can be applied to suspend the object within an optical resonator, extending the possibility of passive Doppler cooling to any type of particle. With a doubly resonant arrangement, two optical fields can cooperate to simultaneously trap and cool the target [88, 211]. Sympathetic cooling by coupling of the levitated sys- tem with ultra-cold atoms has also been suggested to reach the quantum regime for macroscopic resonators with less demanding cavity requirements [212].

The absence of environmental noise makes levitated systems particularly useful platforms for metrological measurements. Force sensitivities down to a few zN Hz 1/2 have been accomplished by optically suspended nanoparticles [199], although feedback control is required to cancel the e↵ect of thermal non-linearities that arise even at the lowest power in the absence of a thermal bath. Accurate measurements of tem- perature [213] and electric charge [214] are also possible, in the latter case with a resolution 10 5smaller than the fundamental charge of the electron. A high-frequency gravitational wave detector based on optically trapped particles has also been pro- posed [215]. Levitated systems in general are particularly functional as accelerometers and gravimeters, and superconductive levitation of microspheres [216] or of magne- tized macrospheres [217] has been suggested for long-term scanning of force gradients induced by surface gravity di↵erentials.

Precision and sensitivity are critical for tests of fundamental physics. For instance, the extremely high resolution of levitated particles may be applied to the detection of short-range non-Newtonian forces or the characterization of Casimir interaction [218], and thanks to the availability of long integration times possible violations of the inverse- law of gravity may be expressly ruled out [219]. Oscillators that are completely de- coupled from a thermal reservoir are more predisposed to anomalous dynamics which allow for example the study of non-equilibrium fluctuation theorems [220], important for chemical and biological processes based on irreversibility. The atypical suscepti- bility of levitated systems could in principle promote the achievement of strong cou- pling [221], also facilitating the observation of quantum dynamics. This leads to the inevitable appeal towards exotic operations such as matter-wave interferometry [222] and superposition of living organisms [223]. Other audacious proposals close the loop between fundamental physics and levitation by suggesting that repulsive quantum vac- uum forces may be used to levitate an ultra-thin mirror [224].

Unavoidably, all the systems mentioned so far are a↵ected by distinct limiting factors often tied to the levitation process itself. While certain noise processes may be monitored and counterbalanced by feedback control, such as classical laser amplitude noise in optical levitation, other e↵ects may irreparably impair the measurements and hold back the sensitivity. Superconductive levitation su↵ers decoherence because of the generation of eddy currents, whereas optical tweezers are subject to scattering losses that become especially pronounced in cavity-enhanced systems. If the cavity is a pre-requisite for more refined sensitivity, a solution could be found by separating the trapping process from the measuring component. For example, two optical tweezers can be applied to a silica disk to trap it in the horizontal plane while balance in the vertical direction is obtained by the radiation pressure force on a cavity mirror attached to the disk [97] (cf. Fig. 8.1d). To completely eradicate scattering losses, however, a more extreme approach is necessary. The system that will be discussed in the following chapters, based on the fully coherent optical levitation of a cavity mirror [14], is designed to accomplish absolute detachment from the environment while preserving all of the information about the system. Using the resonantly amplified fields from three optical cavities in a tripod configuration, the weight of the common end mirror on top of the tripod can be cancelled by radiation pressure force. The trapping potential is provided by the optical spring e↵ect, which induces restoring forces when the fields driving the cavities are blue-detuned with respect to resonance. Each “leg” of the tripod behaves like an extremely rigid spring, with the sti↵ness determined by

the finesse of the corresponding cavity. Like levitated particles, the quality factor of the motional eigenfrequencies of the levitated mirror may grow to exceptionally high values in vacuum. Now, however, the read-out from all the cavities provides a complete picture of the state of the oscillator, making the system more robust and suitable for state-of-the-art applications.