One of the main challenges in physics today is to merge quantum theory and the theory of general relativity into a unified framework. Researches are developing various approaches towards such a theory of quantum gravity, but a major hindrance is the lack of experimental evidence of quantum gravitational effects. Yet, the quantization of space-time itself can have experimental implications: the existence of a minimal length scale is widely expected to result in a modification of the Heisenberg uncertainty relation. Here we introduce a scheme to experimentally test this conjecture by probing directly the canonical commutation relation of the center-of-mass mode of a mechanical oscillator with a mass close to the Planck mass. Our protocol utilizes quantum optical control and readout of the mechanical system to probe possible deviations from the quantum commutation relation even at the Planck scale. We show that the scheme is within reach of current technology. It thus opens a feasible route for table-top experiments to explore possible quantum gravitational phenomena.It is currently an open question whether our underlying concepts of space-time are fully compatible with those of quantum mechanics. The ongoing search for a quantum theory of gravity is therefore one of the main challenges in modern physics. A major difficulty in the development of such theories is the lack of experimentally accessible phenomena that could shed light on the possible route for quantum gravity. Such phenomena are expected to become relevant near the Planck scale, i.e. at energies on the order of the Planck energy E P = 1.2 × 10 19 GeV or at length scales near the Planck length L P = 1.6 × 10 −35 m, where spacetime itself is assumed to be quantized. However, such a minimal length scale is not a feature of quantum theory. The Heisenberg uncertainty relation, one of the cornerstones of quantum mechanics (1), states that the position x and the momentum p of an object cannot be simultaneously known to arbitrary precision. Specifically, the indeterminacies of a joint measurement of these canon- * Electronic address: igor.pikovski@univie.ac.at † Electronic address: m.kim@imperial.ac.uk ical observables are always bound by ∆x∆p ≥ /2. Yet, the uncertainty principle still allows for an arbitrarily precise measurement of only one of the two observables, say position, on the cost of our knowledge about the other (momentum). In stark contrast, in many proposals for quantum gravity the Planck length constitutes a fundamental bound below which position cannot be defined. It has therefore been suggested that the uncertainty relation should be modified in order to take into account such quantum gravitational effects (2). In fact, the concept of a generalized uncertainty principle is found in many approaches to quantum gravity, for example in string theory (3-5), in the theory of doubly special relativity (6, 7), within the principle of relative locality (8) and in studies of black holes (9-11 rent approaches to test quantum gravitational effects mainly ...

Studying mechanical resonators via radiation pressure offers a rich avenue for the exploration of quantum mechanical behavior in a macroscopic regime. However, quantum state preparation and especially quantum state reconstruction of mechanical oscillators remains a significant challenge. Here we propose a scheme to realize quantum state tomography, squeezing, and state purification of a mechanical resonator using short optical pulses. The scheme presented allows observation of mechanical quantum features despite preparation from a thermal state and is shown to be experimentally feasible using optical microcavities. Our framework thus provides a promising means to explore the quantum nature of massive mechanical oscillators and can be applied to other systems such as trapped ions.optomechanics | quantum measurement | squeezed states C oherent quantum mechanical phenomena, such as entanglement and superposition, are not apparent in the macroscopic realm. It is currently held that on large scales quantum mechanical behavior is masked by decoherence (1) or that quantum mechanical laws may even require modification (2-5). Despite substantial experimental advances, see for example ref. 6, probing this regime remains extremely challenging. Recently however, it has been proposed to utilize the precision and control of quantum optical fields in order to investigate the quantum nature of massive mechanical resonators by means of the radiation-pressure interaction (7-13). Quantum state preparation and the ability to probe the dynamics of mechanical oscillators, the most rigorous method being quantum state tomography, are essential for such investigations. These important elements have been experimentally realized for various quantum systems, e.g., light (14, 15), trapped ions (16, 17), atomic ensemble spin (18,19), and intracavity microwave fields (20). By contrast, an experiment realizing both quantum state preparation and tomography of a mechanical resonator is yet to be achieved. Also, schemes that can probe quantum features without full tomography [e.g., (9, 10, 21)] are similarly challenging. In nanoelectromechanics, cooling of resonator motion and preparation of the ground state have been observed (22, 23) but quantum state reconstruction (24) remains outstanding. In cavity optomechanics significant experimental progress has been made towards quantum state control over mechanical resonators (11-13), which includes classical phase-space monitoring (25,26), laser cooling close to the ground state (27, 28), and low noise continuous measurement of mechanically induced phase fluctuations (29-31). Still, quantum state preparation is technically difficult primarily due to thermal bath coupling and weak radiation-pressure interaction strength, and quantum state reconstruction remains little explored. Thus far, a common theme in proposals for mechanical state reconstruction is state transfer to and then read-out of an auxillary quantum system (32)(33)(34)(35). This technique is a technically demanding approach and remains a c...

We introduce a method that can orthogonalize any pure continuous variable quantum state, i.e., generate a state |ψ (perpindicular)} from |ψ} where {ψ|ψ(perpindicular)}= 0, which does not require significant a priori knowledge of the input state. We illustrate how to achieve orthogonalization using the Jaynes-Cummings or beamsplitter interaction, which permits realization in a number of physical systems. Furthermore, we demonstrate how to orthogonalize the motional state of a mechanical oscillator in a cavity optomechanics context by developing a set of coherent phonon level operations. As the mechanical oscillator is a stationary system, such operations can be performed at multiple times providing considerable versatility for quantum state engineering applications. Utilizing this, we additionally introduce a method how to transform any known pure state into any desired target state.

Utilizing the tools of quantum optics to prepare and manipulate quantum states of motion of a mechanical resonator is currently one of the most promising routes to explore non-classicality at a macroscopic scale. An important quantum optomechanical tool yet to be experimentally demonstrated is the ability to perform complete quantum state reconstruction. Here, after providing a brief introduction to quantum states in phase space, the current proposals for state reconstruction of mechanical motional states are reviewed and contrasted and experimental progress is discussed. Furthermore, it is shown that mechanical quadrature tomography using back-action-evading interactions gives an s-parameterized Wigner function where the numerical parameter s is directly related to the optomechanical measurement strength. The effects of classical noise in the optical probe for both state reconstruction and state preparation by measurement are also discussed.

We present a table-top quantum estimation protocol to measure the gravitational acceleration g by using an optomechanical cavity. In particular, we exploit the nonlinear quantum light-matter interaction between an optical field and a massive mirror acting as mechanical oscillator. The gravitational field influences the system dynamics affecting the phase of the cavity field during the interaction. Reading out such a phase carried by the radiation leaking from the cavity, we provide an estimate of the gravitational acceleration through interference measurements. Contrary to previous studies, having adopted a fully quantum description, we are able to propose a quantum analysis proving the ultimate bound to the estimability of the gravitational acceleration and verifying optimality of homodyne detection. Noticeably, thanks to the light-matter decoupling at the measurement time, no initial cooling of the mechanical oscillator is demanded in principle