A minimal observable length is a common feature of theories that aim to merge quantum physics and gravity. Quantum mechanically, this concept is associated with a nonzero minimal uncertainty in position measurements, which is encoded in deformed commutation relations. In spite of increasing theoretical interest, the subject suffers from the complete lack of dedicated experiments and bounds to the deformation parameters have just been extrapolated from indirect measurements. As recently proposed, low-energy mechanical oscillators could allow to reveal the effect of a modified commutator. Here we analyze the free evolution of high-quality factor micro- and nano-oscillators, spanning a wide range of masses around the Planck mass mP (≈22 μg). The direct check against a model of deformed dynamics substantially lowers the previous limits on the parameters quantifying the commutator deformation.

We realise a phase-sensitive closed-loop control scheme to engineer the fluctuations of the pump field which drives an optomechanical system, and show that the corresponding cooling dynamics can be significantly improved. In particular, operating in the counter-intuitive "anti-squashing" regime of positive feedback and increased field fluctuations, sideband cooling of a nanomechanical membrane within an optical cavity can be improved by 7.5 dB with respect to the case without feedback. Close to the quantum regime of reduced thermal noise, such feedback-controlled light would allow going well below the quantum backaction cooling limit.Feedback loops based on real-time continuous measurements [1] are commonly used for stabilisation purposes, and they have also been successfully applied to the stabilisation of quantum systems [2][3][4]. Typically a system is continuously monitored and the acquired signal drives the actuator which in turn drives the system to the desired target. Here we demonstrate a novel approach to closed-loop control in which the feedback acts on an additional control field which is used to drive the system of interest. In particular, the actuator acts on the control field in order to engineer its phase and amplitude fluctuations. The resulting feedback-controlled inloop field is then exploited to manipulate the system and improve its performance. In-loop optical fields have been studied for decades both theoretically [5][6][7][8] and experimentally [9, 10]. A lot of effort has been made to reduce (squash) the noise exhibited by the field fluctuations inside the loop. However, in-loop sub-shot-noise fluctuations cannot be recognised as squeezed below the vacuum noise level, for two different reasons: firstly, the free field commutation relations are no longer valid for time events separated by more than the loop delay-time, since in-loop fields are not free fields [6]; secondly, the corresponding out-of-loop fields exhibit supershot-noise fluctuations [7]. Nevertheless, useful applications of these fields have been proposed and realised, e.g. suppression of the radiation pressure noise [9], removal of classical intensity noise [10], and atomic line narrowing [8]. The common basis of these works is the negative feedback regime. Negative feedback has also been successfully employed in mechanical [11][12][13], and cavity optomechanical systems [4], where an electromagnetic field is used to probe a mechanical resonator, and in turn to control the feedback actuator, which acts directly on the mechanical oscillator. Engineered light fluctuations in the form of squeezed light have also been used in optomechanical systems to improve both the detection sensitivity [14][15][16][17] and the cooling efficiency [18][19][20]. In the present work we show that it is possible to manipulate, with a feedback system [see Figure 1 (a)], the fluctuations of the laser field that drives an optomechanical system to enhance optomechanical sideband cooling [21][22][23][24]. Our analysis demonstrates the effectiveness of t...

Normal-mode splitting is the most evident signature of strong coupling between two interacting subsystems. It occurs when two subsystems exchange energy between themselves faster than they dissipate it to the environment. Here we experimentally show that a weakly coupled optomechanical system at room temperature can manifest normal-mode splitting when the pump field fluctuations are anti-squashed by a phase-sensitive feedback loop operating close to its instability threshold. Under these conditions the optical cavity exhibits an effectively reduced decay rate, so that the system is effectively promoted to the strong coupling regime.Keywords: cavity optomechanics, active feedback, squashed states, strong coupling regime, normal mode spltting Normal-mode splitting is the hallmark of strongly coupled systems. In this regime two interacting systems exchange excitations faster than they are dissipated, and form collective normal modes the hybridized excitations of which are superpositions of the constituent systems' excitations [1,2]. This regime is necessary for the observation of coherent quantum dynamics of the interacting systems and is a central achievement in research aimed at the control and manipulation of quantum systems [3]. In cavity opto/electro-mechanics, where electromagnetic fields and mechanical resonators interact via radiation pressure, normal-mode splitting and strong coupling have already been obtained, using sufficiently strong power of the input driving electromagnetic field [4], or working at cryogenic temperatures with relatively large singlephoton coupling [5,6].In this letter we report on the oxymoron of observing normal-mode splitting in a weakly coupled system. Specifically, we have designed and implemented a feedback system [7,8] which permits the formation of hybridized normal modes also at room temperature and in a relatively modest device, in terms of single-photon optomechanical interaction strength (as compared to the devices used in Refs. [4][5][6]). Our system is basically weakly coupled at the driving power that we can use (limited by the onset of optomechanical bistability at stronger power), and the emergence of hybridized optomechanical modes is observed when the light amplitude at the cavity output is detected and used to modulate the amplitude of the input field driving the cavity itself. The feedback works in the anti-squashing regime, close to the feedback instability, where light fluctuations are enhanced over a narrow frequency range around the cavity resonance. In this regime the system behaves effectively as an equivalent optomechanical system with reduced cavity linewidth. This allows coherent energy oscillations between light and vibrational degrees of freedom when, for example, a coherent light pulse is injected into the cavity mode, similar to what has been discussed in Ref. [6].Light (anti-) squashing [9][10][11] refers to an in-loop (enhancement) reduction of light fluctuations within a (positive) negative feedback loop. Even if the sub-shot noise features of in-l...