Topological operations can achieve certain goals without requiring accurate control over local operational details; for example, they have been used to control geometric phases and have been proposed as a way of controlling the state of certain systems within their degenerate subspaces. More recently, it was predicted that topological operations can be used to transfer energy between normal modes, provided that the system possesses a specific type of degeneracy known as an exceptional point. Here we demonstrate the transfer of energy between two vibrational modes of a cryogenic optomechanical device using topological operations. We show that this transfer arises from the presence of an exceptional point in the spectrum of the device. We also show that this transfer is non-reciprocal. These results open up new directions in system control; they also open up the possibility of exploring other dynamical effects related to exceptional points, including the behaviour of thermal and quantum fluctuations in their vicinity.

Controlling a quantum system based on the observation of its dynamics is inevitably complicated by the backaction of the measurement process. Efficient measurements, however, maximize the amount of information gained per disturbance incurred. Real-time feedback then enables both canceling the measurement's backaction and controlling the evolution of the quantum state. While such measurement-based quantum control has been demonstrated in the clean settings of cavity and circuit quantum electrodynamics, its application to motional degrees of freedom has remained elusive. Here we show measurement-based quantum control of the motion of a millimetre-sized membrane resonator. An optomechanical transducer resolves the zero-point motion of the soft-clamped resonator in a fraction of its millisecond coherence time, with an overall measurement efficiency close to unity. We use this position record to feedback-cool a resonator mode to its quantum ground state (residual thermal occupationn = 0.29 ± 0.03), 9 dB below the quantum backaction limit of sideband cooling, and six orders of magnitude below the equilibrium occupation of its thermal environment. This realizes a long-standing goal in the field, and adds position and momentum to the degrees of freedom amenable to measurement-based quantum control, with potential applications in quantum information processing and gravitational wave detectors. 1 arXiv:1805.05087v2 [quant-ph] 10 Sep 2018Controlling the state of a quantum system is a delicate task, since observation of the system will inevitably perturb it. 1, 2 Coherent quantum control avoids this issue, by coupling the system to another "controller" quantum system in such a way that the joint system converges to the target state without the need for measurement-at the expense of quantum resources in the controller. Measurement-based quantum control 3-5 is based on a different paradigm. It exerts control by measuring the quantum state, and applying feedback that depends on the measurement outcome, much alike classical control systems. In the quantum regime, however, the effect of the measurement's backaction must be taken into account, and effectively canceled. This requires an overall measurement efficiency η-in essence the amount of information gained per decoherence induced-close to unity, a challenging demand yet met only with the impeccable systems of cavity and circuit QED 6, 7 (e.g. η = 40 % in ref. 7 ).To prepare high-purity motional quantum states, researchers have traditionally relied on sideband cooling, a form of coherent quantum control. An engineered quantum optical bath acts as controller, to which the motional degree of freedom couples through optical forces. The motion thermalizes to this bath, at a temperature determined by the forces' quantum fluctuations. This temperature sets a fundamental limit to sideband cooling. In optomechanics, this limit requires that the cavity linewidth resolves the motional sidebands to enable ground state cooling with coherent light. 8 Systems operating in this regime have b...

Quantum mechanics dictates that the precision of physical measurements must be subject to certain constraints. In the case of inteferometric displacement measurements, these restrictions impose a 'standard quantum limit' (SQL), which optimally balances the precision of a measurement with its unwanted backaction 1 . To go beyond this limit, one must devise more sophisticated measurement techniques, which either 'evade' the backaction of the measurement 2 , or achieve clever cancellation of the unwanted noise at the detector 3, 4 . In the half-century since the SQL was established, systems ranging from LIGO 5 to ultracold atoms 6 and nanomechanical devices 7, 8 have pushed displacement measurements towards this limit, and a variety of sub-SQL techniques have been tested in proof-of-principle experiments 9-13 . However, to-date, no experimental system has successfully demonstrated an interferometric displacement measurement with sensitivity (including all relevant noise sources: thermal, backaction, and imprecision) below the SQL. Here, we exploit strong quantum correlations in an ultracoherent optomechanical system to demonstrate off-resonant force and displacement sensitivity reaching 1.5dB below the SQL. This achieves an outstanding goal in mechanical quantum sensing, and further enhances the prospects of using such devices for state-of-the-art force sensing applications.

Evaluating the performance of optical flow algorithms has been difficult because of the lack of ground-truth data sets for complex scenes. We describe a simple modification to a ray tracer that allows us to generate ground-truth motion fields for scenes of arbitrary complexity. The resulting flow maps are used to assist in the comparison of eight optical flow algorithms using three complex, synthetic scenes. Our study found that a modified version of Lucas and Kanade's algorithm has superior performance but produces sparse flow maps. Proesmans et al.'s algorithm performs slightly worse, on average, but produces a very dense depth map.

Continuous weak measurement allows localizing open quantum systems in state space, and tracing out their quantum trajectory as they evolve in time. Efficient quantum measurement schemes have previously enabled recording quantum trajectories of microwave photon and qubit states. We apply these concepts to a macroscopic mechanical resonator, and follow the quantum trajectory of its motional state conditioned on a continuous optical measurement record. Starting with a thermal mixture, we eventually obtain coherent states of 78% purity-comparable to a displaced thermal state of occupation 0.14. We introduce a retrodictive measurement protocol to directly verify state purity along the trajectory, and furthermore observe state collapse and decoherence. This opens the door to measurement-based creation of advanced quantum states, and potential tests of gravitational decoherence models.

We describe measurements of the motional sidebands produced by a mechanical oscillator (with effective mass 43 ng and resonant frequency 705 kHz) that is placed in an optical cavity and cooled close to its quantum ground state. The red and blue sidebands (corresponding to Stokes and anti-Stokes scattering) from a single laser beam are recorded simultaneously via a heterodyne measurement. The oscillator's mean phonon numbern is inferred from the ratio of the sidebands, and reaches a minimum value of 0.84 ± 0.22 (corresponding to a mode temperature T = 28 ± 7 μK). We also infern from the calibrated area of each of the two sidebands, and from the oscillator's total damping. The values ofn inferred from these four methods are in close agreement. The behavior of the sidebands as a function of the oscillator's temperature agrees well with theory that includes the quantum fluctuations of both the cavity field and the mechanical oscillator. Cavity optomechanical systems operating in the quantum regime are expected to play an important role in advancing the control of electromagnetic fields and mechanical oscillators, interfacing disparate quantum systems, detecting gravitational waves, constraining modifications to orthodox quantum mechanics, and testing hypotheses about quantum gravity [1][2][3][4][5][6][7][8][9][10][11]. The utility of optomechanical systems in these areas reflects their particular combination of long relaxation times, unitary coupling to electromagnetic fields in the microwave and near-infrared domains, and access to the quantum behavior of massive objects.Optomechanical experiments have been based primarily on systems in which the mechanical oscillator and the cavity field are prepared in Gaussian states, couple weakly to each other at the quantum level (i.e., the bare optomechanical coupling rate g 0 is much less than the oscillator frequency ω m and the cavity damping rate κ), and are probed via linear measurements of the fields leaving the cavity. (Some optomechanics experiments have demonstrated nonlinear measurements of the cavity fields [12,13], although without resolving non-Gaussian behavior.) Within this paradigm of Gaussian states, weak coupling, and linear measurements, quantum effects can manifest themselves as apparent fluctuations of quantities which, according to classical mechanics, could be noiseless [14]. Depending on the specific type of measurement, these quantum fluctuations may be ascribed to the cavity field, the mechanical oscillator, or both [15,16].One such experiment is a heterodyne measurement of the light leaving an optomechanical cavity that is driven on resonance by a single laser. Classically, the thermal motion of the mechanical oscillator inside the cavity adds modulation sidebands to the laser beam. In the spectrum of the heterodyne signal, the area of these sidebands will be equal, and will be proportional to the oscillator's temperature.In the quantum treatment described in Refs. [15,16] of the same measurement, the heterodyne spectrum arises from four distinct c...

Cavity optomechanics offers powerful methods for controlling optical fields and mechanical motion. A number of proposals have predicted that this control can be extended considerably in devices where multiple cavity modes couple to each other via the motion of a single mechanical oscillator. Here we study the dynamic properties of such a multimode optomechanical device, in which the coupling between cavity modes results from mechanically induced avoided crossings in the cavity's spectrum. Near the avoided crossings we find that the optical spring shows distinct features that arise from the interaction between cavity modes. Precisely at an avoided crossing, we show that the particular form of the optical spring provides a classical analogue of a quantum non-demolition measurement of the intracavity photon number. The mechanical oscillator's Brownian motion, an important source of noise in these measurements, is minimized by operating the device at cryogenic temperature (500 mK).

Many applications of quantum information processing (QIP) require distribution of quantum states in networks, both within and between distant nodes [1]. Optical quantum states are uniquely suited for this purpose, as they propagate with ultralow attenuation and are resilient to ubiquitous thermal noise. Mechanical systems are then envisioned as versatile interfaces between photons and a variety of solid-state QIP platforms [2, 3]. Here, we demonstrate a key step towards this vision, and generate entanglement between two propagating optical modes, by coupling them to the same, cryogenic mechanical system. The entanglement persists at room temperature, where we verify the inseparability of the bipartite state and fully characterize its logarithmic negativity by homodyne tomography. We detect, without any corrections, correlations corresponding to a logarithmic negativity of E N = 0.35. Combined with quantum interfaces between mechanical systems and solid-state qubit processors already available [4, 5, 6, 7] or under development [8,9], this paves the way for mechanical systems enabling long-distance quantum information networking over optical fiber networks.Entanglement is a crucial resource for QIP [10]. As such, the ability to entangle fields of arbitrary wavelength will be important for linking nodes in heterogeneous QIP networks. Mechanical oscillators are uniquely poised in their ability to create such links, thanks to the frequency-independence of the radiation pressure interaction. The ability to entangle two radiation fields via a common mechanical interaction was outlined 20 years ago [11,12], and the intervening decades have seen the development of optomechanical devices [13] which are robustly quantum mechanical and routinely integrated into hybrid systems.Recently, mechanically-mediated entanglement has been reported between propagating microwave fields [14] as well as two superconducting qubits [15]. In both cases, the entanglement remained confined to the dilution refrigerator in which it was created. Here, we utilize an extremely coherent mechanical platform 1 arXiv:1911.05729v2 [quant-ph]