Vivishek Sudhir

Measurement-based control of a mechanical oscillator at its thermal decoherence rate

In real-time quantum feedback protocols, the record of a continuous measurement is used to stabilize a desired quantum state. Recent years have seen successful applications of these protocols in a variety of well-isolated micro-systems, including microwave photons and superconducting qubits. However, stabilizing the quantum state of a tangibly massive object, such as a mechanical oscillator, remains very challenging: the main obstacle is environmental decoherence, which places stringent requirements on the timescale in which the state must be measured. Here we describe a position sensor that is capable of resolving the zero-point motion of a solid-state, 4.3-megahertz nanomechanical oscillator in the timescale of its thermal decoherence, a basic requirement for real-time (Markovian) quantum feedback control tasks, such as ground-state preparation. The sensor is based on evanescent optomechanical coupling to a high-Q microcavity, and achieves an imprecision four orders of magnitude below that at the standard quantum limit for a weak continuous position measurement—a 100-fold improvement over previous reports—while maintaining an imprecision–back-action product that is within a factor of five of the Heisenberg uncertainty limit. As a demonstration of its utility, we use the measurement as an error signal with which to feedback cool the oscillator. Using radiation pressure as an actuator, the oscillator is cold damped with high efficiency: from a cryogenic-bath temperature of 4.4 kelvin to an effective value of 1.1 ± 0.1 millikelvin, corresponding to a mean phonon number of 5.3 ± 0.6 (that is, a ground-state probability of 16 per cent). Our results set a new benchmark for the performance of a linear position sensor, and signal the emergence of mechanical oscillators as practical subjects for measurement-based quantum control.

Nonlinear Quantum Optomechanics via Individual Intrinsic Two-Level Defects

We propose to use the intrinsic two-level system (TLS) defect states found naturally in integrated optomechanical devices for exploring cavity QED-like phenomena with localized phonons. The Jaynes-Cummings-type interaction between TLS and mechanics can reach the strong coupling regime for existing nano-optomechanical systems, observable via clear signatures in the optomechanical output spectrum. These signatures persist even at finite temperature, and we derive an explicit expression for the temperature at which they vanish. Further, the ability to drive the defect with a microwave field allows for realization of phonon blockade, and the available controls are sufficient to deterministically prepare non-classical states of the mechanical resonator.

Quantum-Limited Amplification and Parametric Instability in the Reversed Dissipation Regime of Cavity Optomechanics

Cavity optomechanical phenomena, such as cooling, amplification, or optomechanically induced transparency, emerge due to a strong imbalance in the dissipation rates of the parametrically coupled electromagnetic and mechanical resonators. Here we analyze the reversed dissipation regime where the mechanical energy relaxation rate exceeds the energy decay rate of the electromagnetic cavity. We demonstrate that this regime allows for mechanically induced amplification (or cooling) of the electromagnetic mode. Gain, bandwidth, and added noise of this electromagnetic amplifier are derived and compared to amplification in the normal dissipation regime. In addition, we analyze the parametric instability, i.e., optomechanical Brillouin lasing, and contrast it to conventional optomechanical phonon lasing. Finally, we propose an experimental scheme that realizes the reversed dissipation regime using parametric coupling and optomechanical cooling with a second electromagnetic mode enabling quantum-limited amplification. Recent advances in high-Q superconducting microwave resonators make the reversed dissipation regime experimentally realizable.

Appearance and Disappearance of Quantum Correlations in Measurement-Based Feedback Control of a Mechanical Oscillator

Quantum correlations between imprecision and backaction are a hallmark of continuous linear measurements. Here, we study how measurement-based feedback can be used to improve the visibility of quantum correlations due to the interaction of a laser field with a nanomechanical oscillator. Backaction imparted by the meter laser, due to radiation-pressure quantum fluctuations, gives rise to correlations between its phase and amplitude quadratures. These quantum correlations are observed in the experiment both as squeezing of the meter field fluctuations below the vacuum level in a homodyne measurement and as sideband asymmetry in a heterodyne measurement, demonstrating the common origin of both phenomena. We show that quantum feedback, i.e., feedback that suppresses measurement backaction, can be used to increase the visibility of the sideband asymmetry ratio. In contrast, by operating the feedback loop in the regime of noise squashing, where the in-loop photocurrent variance is reduced below the vacuum level, the visibility of the sideband asymmetry is reduced. This is due to backaction arising from vacuum noise in the homodyne detector. These experiments demonstrate the possibility, as well as the fundamental limits, of measurement-based feedback as a tool to manipulate quantum correlations.

Quantum Correlations of Light from a Room-Temperature Mechanical Oscillator

We observe quantum correlations imprinted on an optical beam interacting with a room temperature nanomechanical oscillator, and show how this leads to an enhancement in the relative signal-to-noise ratio for the estimation of an arbitrary force.

A strongly coupled Λ-type micromechanical system

We study a classical Λ-type three-level system based on three high-Q micromechanical beam resonators embedded in a gradient electric field. By modulating the strength of the field at the difference frequency between adjacent beam modes, we realize strong dynamic two-mode coupling, via the dielectric force. Driving adjacent pairs simultaneously, we observe the formation of a purely mechanical “dark” state and an all-phononic analog of coherent population trapping—signatures of strong three-mode coupling. The Λ-type micromechanical system is a natural extension of previously demonstrated “two-level” micromechanical systems and adds to the toolbox for engineering of all-phononic micromechanical circuits and arrays.

Evidence for structural damping in a high-stress silicon nitride nanobeam and its implications for quantum optomechanics

Abstract We resolve the thermal motion of a high-stress silicon nitride nanobeam at frequencies far below its fundamental flexural resonance (3.4 MHz) using cavity-enhanced optical interferometry. Over two decades, the displacement spectrum is well-modeled by that of a damped harmonic oscillator driven by a 1 / f thermal force, suggesting that the loss angle of the beam material is frequency-independent. The inferred loss angle at 3.4 MHz, ϕ = 4.5 ⋅ 10 − 6 , agrees well with the quality factor (Q) of the fundamental beam mode ( ϕ = Q − 1 ). In conjunction with Q measurements made on higher order flexural modes, and accounting for the mode dependence of stress-induced loss dilution, we find that the intrinsic (undiluted) loss angle of the beam changes by less than a factor of 2 between 50 kHz and 50 MHz. We discuss the impact of such “structural damping” on experiments in quantum optomechanics, in which the thermal force acting on a mechanical oscillator coupled to an optical cavity is overwhelmed by radiation pressure shot noise. As an illustration, we show that structural damping reduces the bandwidth of ponderomotive squeezing.

Quantum limits on measurement and control of a mechanical oscillator

The precision measurement of position has a long-standing tradition in physics. Cavendishs verification of the universal law of gravitation using a torsion pendulum, Perrins confirmation of the atomic hypothesis via the precise measurement of the Brownian motion, and, the verification of the mechanical effect of electromagnetic radiation, all belong to this classical heritage. Quantum mechanics posits that the measurement of position results in an uncertain momentum; an idea developed to full maturity within the context of interferometric searches for gravity waves. Over the past decade, standing at the confluence of quantum optics and nanomechanics, cavity optomechanics has emerged as a powerful platform to study the quantum limits of position measurements. The subject of this thesis is the precision measurement of the position of a nano-mechanical oscillator, the fundamental limits of such measurements, and its relevance to measurement-based feedback control. The nano-mechanical oscillator is coupled to light confined in an optical micro-cavity via radiation pressure. The fluctuations in the position of the oscillator are transduced onto the phase of the light, while quantum fluctuations in the amplitude of the light leads to a disturbance in the momentum of the oscillator. We perform an interferometric position measurement with a sensitivity that is 10^5 times below what is required to resolve the zero-point motion of the oscillator, constituting the most precise measurement of thermal motion yet. The resulting disturbance -- measurement back-action -- is observed to be commensurate with the uncertainty principle, leading to a 10% contribution to the total motion of the oscillator. The continuous record of the measurement (performed in a 4 K cryogenic environment) furnishes the ability to resolve the zero-point motion of the oscillator within its decoherence rate - the necessary condition for measurement-based feedback control of the state of the oscillator. Using the measurement record as error signal, the oscillator is cooled towards its ground state, resulting in a factor 10^4 suppression of its total (thermal and back-action) motion, to a final occupation of 5 phonons on average. Measurements generally proceed by establishing correlations between the system being measured and the measuring device. For the class of quantum measurements employed here - continuous linear measurements - these correlations arise due to measurement back-action. These back-action-induced correlations appear as correlations between the degrees of freedom of the measuring device. For interferometric position measurements, quantum correlations are established between the phase and amplitude of the light. In a homodyne measurement, they lead to optical squeezing, while in a heterodyne measurement, they appear as an asymmetry in the sidebands carrying information about the oscillator position. Feedback is used to enhance sideband asymmetry, a first proof-of-principle demonstration of the ability to control quantum correlations using feedback. In the regime where amplified vacuum noise dominates the feedback signal, the disappearance of sideband asymmetry visualises a fundamental limit of linear feedback control. Using a homodyne detector, we also characterise these quantum correlations manifested as optical squeezing at the 1% level.

Phase properties of operator valued measures in phase space

Abstract The Wigner phase operator (WPO) is identified as an operator valued measure (OVM) and its eigenstates are obtained. An operator satisfying the canonical commutation relation with the Wigner phase operator is also constructed and this establishes a Wigner distribution based operator formalism for the Wigner phase distribution. It is then argued that the WPO cannot represent a projective measurement of the phase; but is in fact to be interpreted as an operator valued measure for the phase. The non-positivity of the latter can be overcome by defining a positive operator valued measure (POVM) via a proper filter function, based on the view that phase measurements are coarse-grained in phase space, leading to the well known Q-distribution. The identification of the Q phase operator as a POVM is in good agreement with the earlier observation regarding the relation between operational phase measurement schemes and the Q-distribution. The Q phase POVM can be dilated in the sense of Gelfand–Naimark, to an operational setting of interference at a beam-splitter with another coherent state – this results in a von Neumann projector with well-defined phase in the expanded Hilbert space of the two modes.