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.

Phononic resonators play important roles in settings that range from gravitational wave detectors to cellular telephones. They serve as high-performance transducers, sensors, and filters by offering low dissipation, tunable coupling to diverse physical systems, and compatibility with a wide range of frequencies, materials, and fabrication processes. Systems of phononic resonators typically obey reciprocity, which ensures that the phonon transmission coefficient between any two resonators is independent of the direction of transmission 1,2 . Reciprocity must be broken to realize devices (such as isolators and circulators) that provide one-way propagation of acoustic energy between resonators. Such devices are crucial for protecting active elements, mitigating noise, and operating full-duplex transceivers. To date, nonreciprocal phononic devices 3,4,5,6,7,8,9,10,11 have not combined the features necessary for robust operation: strong nonreciprocity, in situ tunability, compact integration, and continuous operation. Furthermore, they have been applied only to coherent signals (rather than fluctuations or noise), and have been realized exclusively in travellingwave systems (rather than resonators). Here we describe a cavity optomechanical scheme that produces robust nonreciprocal coupling between phononic resonators. This scheme provides ~ 30 dB of isolation and can be tuned in situ simply via the phases of the drive tones applied to the cavity. In addition, by directly monitoring the resonators' dynamics we show that this nonreciprocity can be used to control thermal fluctuations, and that this control represents a new resource for cooling phononic resonators.Reciprocity is a generic feature of linear, time-invariant oscillator systems. It may be broken in various ways; for example by introducing bias, nonlinearity, or parametric time dependence 1,2 .In phononic systems, nonreciprocal bias can be introduced by imposing rotational motion 9 or a

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...

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