Cavity Quantum Optomechanical Nonlinearities and Position Measurement beyond the Breakdown of the Linearized Approximation.

Several optomechanics experiments are now entering the highly sought nonlinear regime where optomechanical interactions are large even for low light levels. Within this regime, new quantum phenomena and improved performance may be achieved; however, a corresponding theoretical formalism of cavity quantum optomechanics that captures the nonlinearities of both the radiation-pressure interaction and the cavity response is needed to unlock these capabilities. Here, we develop such a nonlinear cavity quantum optomechanical framework, which we then utilize to propose how position measurement can be performed beyond the breakdown of the linearized approximation. Our proposal utilizes optical general-dyne detection, ranging from single to dual homodyne, to obtain mechanical position information imprinted onto both the optical amplitude and phase quadratures and enables both pulsed and continuous modes of operation. These cavity optomechanical nonlinearities are now being confronted in a growing number of experiments, and our framework will allow a range of advances to be made in, e.g., quantum metrology, explorations of the standard quantum limit, and quantum measurement and control.

A Kerr polarization controller.

Kerr-effect-induced changes of the polarization state of light are well known in pulsed laser systems. An example is nonlinear polarization rotation, which is critical to the operation of many types of mode-locked lasers. Here, we demonstrate that the Kerr effect in a high-finesse Fabry-Pérot resonator can be utilized to control the polarization of a continuous wave laser. It is shown that a linearly-polarized input field is converted into a left- or right-circularly-polarized field, controlled via the optical power. The observations are explained by Kerr-nonlinearity induced symmetry breaking, which splits the resonance frequencies of degenerate modes with opposite polarization handedness in an otherwise symmetric resonator. The all-optical polarization control is demonstrated at threshold powers down to 7 mW. The physical principle of such Kerr effect-based polarization controllers is generic to high-Q Kerr-nonlinear resonators and could also be implemented in photonic integrated circuits. Beyond polarization control, the spontaneous symmetry breaking of polarization states could be used for polarization filters or highly sensitive polarization sensors when operating close to the symmetry-breaking point.

Non-Gaussian Mechanical Motion via Single and Multiphonon Subtraction from a Thermal State.

Quantum optical measurement techniques offer a rich avenue for quantum control of mechanical oscillators via cavity optomechanics. In particular, a powerful yet little explored combination utilizes optical measurements to perform heralded non-Gaussian mechanical state preparation followed by tomography to determine the mechanical phase-space distribution. Here, we experimentally perform heralded single-phonon and multiphonon subtraction via photon counting to a laser-cooled mechanical thermal state with a Brillouin optomechanical system at room temperature and use optical heterodyne detection to measure the s-parametrized Wigner distribution of the non-Gaussian mechanical states generated. The techniques developed here advance the state of the art for optics-based tomography of mechanical states and will be useful for a broad range of applied and fundamental studies that utilize mechanical quantum-state engineering and tomography.

Single-Phonon Addition and Subtraction to a Mechanical Thermal State.

Adding or subtracting a single quantum of excitation to a thermal state of a bosonic system has the counter-intuitive effect of approximately doubling its mean occupation. We perform the first experimental demonstration of this effect outside optics by implementing single-phonon addition and subtraction to a thermal state of a mechanical oscillator via Brillouin optomechanics in an optical whispering-gallery microresonator. Using a detection scheme that combines single-photon counting and optical heterodyne detection, we observe this doubling of the mechanical thermal fluctuations to a high precision. The capabilities of this joint click-dyne detection scheme adds a significant new dimension for optomechanical quantum science and applications.

Quantum Hypercube States.

We introduce quantum hypercube states, a class of continuous-variable quantum states that are generated as orthographic projections of hypercubes onto the quadrature phase space of a bosonic mode. In addition to their interesting geometry, hypercube states display phase-space features much smaller than Planck's constant, and a large volume of Wigner negativity. We theoretically show that these features make hypercube states sensitive to displacements at extremely small scales in a way that is surprisingly robust to initial thermal occupation and to small separation of the superposed state components. In a high-temperature proof-of-principle optomechanics experiment we observe, and match to theory, the signature outer-edge vertex structure of hypercube states.

Nonlinear optomechanical measurement of mechanical motion.

Precision measurement of nonlinear observables is an important goal in all facets of quantum optics. This allows measurement-based non-classical state preparation, which has been applied to great success in various physical systems, and provides a route for quantum information processing with otherwise linear interactions. In cavity optomechanics much progress has been made using linear interactions and measurement, but observation of nonlinear mechanical degrees-of-freedom remains outstanding. Here we report the observation of displacement-squared thermal motion of a micro-mechanical resonator by exploiting the intrinsic nonlinearity of the radiation-pressure interaction. Using this measurement we generate bimodal mechanical states of motion with separations and feature sizes well below 100 pm. Future improvements to this approach will allow the preparation of quantum superposition states, which can be used to experimentally explore collapse models of the wavefunction and the potential for mechanical-resonator-based quantum information and metrology applications.

Cooling-by-measurement and mechanical state tomography via pulsed optomechanics.

Observing a physical quantity without disturbing it is a key capability for the control of individual quantum systems. Such back-action-evading or quantum non-demolition measurements were first introduced in the 1970s for gravitational wave detection, and now such techniques are an indispensable tool throughout quantum science. Here we perform measurements of the position of a mechanical oscillator using pulses of light with a duration much shorter than a period of mechanical motion. Utilizing this back-action-evading interaction, we demonstrate state preparation and full state tomography of the mechanical motional state. We have reconstructed states with a position uncertainty reduced to 19 pm, limited by the quantum fluctuations of the optical pulse, and we have performed 'cooling-by-measurement' to reduce the mechanical mode temperature from an initial 1,100 to 16 K. Future improvements to this technique will allow for quantum squeezing of mechanical motion, even from room temperature, and reconstruction of non-classical states exhibiting negative phase-space quasi-probability.

Quantum state orthogonalization and a toolset for quantum optomechanical phonon control.

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.

Pulsed quantum optomechanics.

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.