Backaction-evading Measurement Research Articles

Entanglement-enhanced optomechanical sensing

Optomechanical systems have been exploited in ultrasensitive measurements of force, acceleration and magnetic fields. The fundamental limits for optomechanical sensing have been extensively studied and now well understood—the intrinsic uncertainties of the bosonic optical and mechanical modes, together with backaction noise arising from interactions between the two, dictate the standard quantum limit. Advanced techniques based on non-classical probes, in situ ponderomotive squeezed light and backaction-evading measurements have been developed to overcome the standard quantum limit for individual optomechanical sensors. An alternative, conceptually simpler approach to enhance optomechanical sensing rests on joint measurements taken by multiple sensors. In this configuration, a pathway to overcome the fundamental limits in joint measurements has not been explored. Here we demonstrate that joint force measurements taken with entangled probes on multiple optomechanical sensors can improve the bandwidth in the thermal-noise-dominant regime or the sensitivity in the shot-noise-dominant regime. Moreover, we quantify the overall performance of entangled probes with the sensitivity–bandwidth product and observe a 25% increase compared with that of classical probes. The demonstrated entanglement-enhanced optomechanical sensors would enable new capabilities for inertial navigation, acoustic imaging and searches for new physics. Joint force measurements with entangled optical probes on two optomechanical sensors are demonstrated. The force sensitivity is improved by 40% in the shot-noise-dominant regime. The sensing bandwidth is improved by 20% in the thermal noise limit.

Quantum backaction evading measurements of a silicon nitride membrane resonator

Quantum backaction disturbs the measurement of the position of a mechanical oscillator by introducing additional fluctuations. In a quantum backaction measurement technique, the backaction can be evaded, although at the cost of losing part of the information. In this work, we carry out such a quantum backaction measurement using a large 0.5 mm diameter silicon nitride membrane oscillator with 707 kHz frequency, via a microwave cavity readout. The measurement shows that quantum backaction noise can be evaded in the quadrature measurement of the motion of a large object.

Broadband quantum back action evading measurements of a resonant force

Quantum back action imposes fundamental sensitivity limits to the majority of quantum measurements. The effect results from the unavoidable contamination of the measured parameter with the quantum noise of a meter. Back action evading measurements take advantage of the quantum correlations introduced by the system under study to the meter and allow overcoming the fundamental limitations. The measurements are frequently restricted in their bandwidth due to a finite response time of the system components. Here we show that probing a mechanical oscillator with a dichromatic field with frequencies separated by the oscillator frequency enables independent detection in two outputs and complete subtraction of the measurement noise associated with the quantum back action.

Unconditional mechanical squeezing via backaction-evading measurements and nonoptimal feedback control

Backaction-evading (BAE) measurements of a mechanical resonator, by continuously monitoring a single quadrature of motion, can achieve precision below the zero-point uncertainty. When this happens, the measurement leaves the resonator in a quantum squeezed state. The squeezed state so generated is however conditional on the measurement outcomes, while for most applications it is desirable to have a deterministic, i.e., unconditional, squeezed state with the desired properties. In this work we apply feedback control to achieve deterministic manipulation of mechanical squeezing in an optomechanical system subject to a continuous BAE measurement. We study in details two strategies, direct (Markovian) and state-based (Bayesian) feedback. We show that both are capable to achieve optimal performances, i.e., a vanishing noise added by the feedback loop. Moreover, even when the feedback is restricted to be a time-varying mechanical force (experimentally friendly scenario) and an imperfect BAE regime is considered, the ensuing nonoptimal feedback may still obtain significant amount of squeezing. In particular, we show that Bayesian feedback control is nearly optimal for a wide range of sideband resolution. Our analysis is of direct relevance for ultrasensitive measurements and quantum state engineering in state-of-the-art optomechanical devices.

Combining Floquet and Lyapunov techniques for time-dependent problems in optomechanics and electromechanics

Cavity optomechanics and electromechanics form an established field of research investigating the interactions between electromagnetic fields and the motion of quantum mechanical resonators. In many applications, linearised form of the interaction is used, which allows for the system dynamics to be fully described using a Lyapunov equation for the covariance matrix of the Wigner function. This approach, however, is problematic in situations where the Hamiltonian becomes time dependent as is the case for systems driven at multiple frequencies simultaneously. This scenario is highly relevant as it leads to dissipative preparation of mechanical states or backaction-evading measurements of mechanical motion. The time-dependent dynamics can be solved with Floquet techniques whose application is, nevertheless, not straightforward. Here, we describe a general method for combining the Lyapunov approach with Floquet techniques that enables us to transform the initial time-dependent problem into a time-independent one, at the acceptable cost of enlarging the drift and diffusion matrix. We show how the lengthy process of applying the Floquet formalism to the original equations of motion and deriving a Lyapunov equation from their time-independent form can be simplified with the use of properly defined Fourier components of the drift matrix of the original time-dependent system. We then use our formalism to comprehensively analyse dissipative generation of mechanical squeezing beyond the rotating wave approximation. Our method is applicable to various problems with multitone driving schemes in cavity optomechanics, electromechanics, and related disciplines.

Stroboscopic quantum optomechanics

We consider an optomechanical cavity that is driven stroboscopically by a train of short pulses. By suitably choosing the inter-pulse spacing we show that ground-state cooling and mechanical squeezing can be achieved, even in the presence of mechanical dissipation and for moderate radiation-pressure interaction. We provide a full quantum-mechanical treatment of stroboscopic backaction-evading measurements, for which we give a simple analytic insight, and discuss preparation and verification of squeezed mechanical states. We further consider stroboscopic driving of a pair of non-interacting mechanical resonators coupled to a common cavity field, and show that they can be simultaneously cooled and entangled. Stroboscopic quantum optomechanics extends measurement-based quantum control of mechanical systems beyond the good-cavity limit.

Laser Cooling of a Nanomechanical Oscillator to Its Zero-Point Energy.

Optomechanical systems in the well-resolved-sideband regime are ideal for studying a myriad of quantum phenomena with mechanical systems, including backaction-evading measurements, mechanical squeezing, and nonclassical states generation. For these experiments, the mechanical oscillator should be prepared in its ground state, i.e., exhibit negligible residual excess motion compared to its zero-point motion. This can be achieved using the radiation pressure of laser light in the cavity by selectively driving the lower motional sideband, leading to sideband cooling. To date, the preparation of sideband-resolved optical systems to their zero-point energy has eluded laser cooling because of strong optical absorption heating. The alternative method of passive cooling suffers from the same problem, as the requisite milliKelvin environment is incompatible with the strong optical driving needed by many quantum protocols. Here, we employ a highly sideband-resolved silicon optomechanical crystal in a ^{3}He buffer-gas environment at ∼2 K to demonstrate laser sideband cooling to a mean thermal phonon occupancy of 0.09_{-0.01}^{+0.02} quantum (self-calibrated using motional sideband asymmetry), which is -7.4 dB of the oscillator's zero-point energy and corresponds to 92% ground state probability. Achieving such low occupancy by laser cooling opens the door to a wide range of quantum-optomechanical experiments in the optical domain.

Floquet dynamics in the quantum measurement of mechanical motion

The radiation-pressure interaction between one or more laser fields and a mechanical oscillator gives rise to a wide range of phenomena: from sideband cooling and backaction-evading measurements to pondermotive and mechanical squeezing to entanglement and motional sideband asymmetry. In many protocols, such as dissipative mechanical squeezing, multiple lasers are utilized, giving rise to periodically driven optomechanical systems. Here we show that in this case, Floquet dynamics can arise due to presence of Kerr-type nonlinearities, which are ubiqitious in optomechanical systems. Specifically, employing multiple probe tones, we perform sideband asymmetry measurements, a macroscopic quantum effect, on a silicon optomechanical crystal sideband-cooled to 40% ground-state occupation. We show that the Floquet dynamics, resulting from the presence of multiple pump tones, gives rise to an artificially modified motional sideband asymmetry by redistributing thermal and quantum fluctuations among the initially independently scattered thermomechanical sidebands. For pump tones exhibiting large frequency separation, the dynamics is suppressed and accurate quantum noise thermometry demonstrated. We develop a theoretical model based on Floquet theory that accurately describes our observations. The resulting dynamics can be understood as resulting from a synthetic gauge field among the Fourier modes, which is created by the phase lag of the Kerr-type response. This novel phenomenon has wide-ranging implications for schemes utilizing several pumping tones, as commonly employed in backaction-evading measurements, dissipative optical squeezing, dissipative mechanical squeezing and quantum noise thermometry. Our observation may equally well be used for optomechanical Floquet engineering, e.g. generation of topological phases of sound by periodic time-modulation.

Two-Tone Optomechanical Instability and Its Fundamental Implications for Backaction-Evading Measurements

Experiments reveal a novel optomechanical instability in devices that attempt to surpass quantum limits in ultraprecise displacement measurements, an effect that must be considered in future quantum sensing experiments.

Structural Characterization of Linear Quantum Systems With Application to Back-Action Evading Measurement

The purpose of this paper is to study the structure of quantum linear systems in terms of their Kalman canonical form, which was proposed in a recent paper \cite{ZGPG18}. The spectral structure of quantum linear systems is explored, which indicates that a quantum linear system is both controllable and observable provided that it is Hurwitz stable. A new parameterization method for quantum linear systems is proposed. This parameterization is designed for the Kalman canonical form directly. Consequently, the parameters involved are in a blockwise form in correspondence with the blockwise structure of the Kalman canonical form. This parameter structure can be used to simplify various quantum control design problems. For example, necessary and sufficient conditions for the realization of quantum back-action evading (BAE) measurements are given in terms of these new parameters. Due to their blockwise nature, a small number of parameters are required for realizing quantum BAE measurements.

Conditional Dynamics of Optomechanical Two-Tone Backaction-Evading Measurements.

Backaction-evading measurements of mechanical motion can achieve precision below the zero-point uncertainty and quantum squeezing, which makes them a resource for quantum metrology and quantum information processing. We provide an exact expression for the conditional state of an optomechanical system in a two-tone backaction-evading measurement beyond the standard adiabatic approximation and perform extensive numerical simulations to go beyond the usual rotating-wave approximation. We predict the simultaneous presence of conditional mechanical squeezing, intracavity squeezing, and optomechanical entanglement. We further apply an analogous analysis to the multimode optomechanical system of two mechanical and one cavity mode and find conditional mechanical Einstein-Podolski-Rosen entanglement and genuinely tripartite optomechanical entanglement. Our analysis is of direct relevance for ultrasensitive measurements and measurement-based control in high-cooperativity optomechanical sensors operating beyond the adiabatic limit.

Optical backaction-evading measurement of a mechanical oscillator

Quantum mechanics imposes a limit on the precision of a continuous position measurement of a harmonic oscillator, due to backaction arising from quantum fluctuations in the measurement field. This standard quantum limit can be surpassed by monitoring only one of the two non-commuting quadratures of the motion, known as backaction-evading measurement. This technique has not been implemented using optical interferometers to date. Here we demonstrate, in a cavity optomechanical system operating in the optical domain, a continuous two-tone backaction-evading measurement of a localized gigahertz-frequency mechanical mode of a photonic-crystal nanobeam cryogenically and optomechanically cooled close to the ground state. Employing quantum-limited optical heterodyne detection, we explicitly show the transition from conventional to backaction-evading measurement. We observe up to 0.67 dB (14%) reduction of total measurement noise, thereby demonstrating the viability of backaction-evading measurements in nanomechanical resonators for optical ultrasensitive measurements of motion and force.

Entanglement-Enhanced Radio-Frequency Field Detection and Waveform Sensing.

We demonstrate a new technique for detecting the amplitude of arbitrarily chosen components of radio-frequency waveforms based on stroboscopic backaction evading measurements. We combine quantum nondemolition measurements and stroboscopic probing to detect waveform components with magnetic sensitivity beyond the standard quantum limit. Using an ensemble of 1.5×10^{6} cold rubidium atoms, we demonstrate entanglement-enhanced sensing of sinusoidal and linearly chirped waveforms, with 1.0(2) and 0.8(3)dB metrologically relevant noise reduction, respectively. We achieve volume-adjusted sensitivity of δBsqrt[V]≈3.96 fTsqrt[cm^{3}/Hz], comparable to the best rf magnetometers.

Quantum Backaction Evading Measurement of Collective Mechanical Modes.

The standard quantum limit constrains the precision of an oscillator position measurement. It arises from a balance between the imprecision and the quantum backaction of the measurement. However, a measurement of only a single quadrature of the oscillator can evade the backaction and be made with arbitrary precision. Here we demonstrate quantum backaction evading measurements of a collective quadrature of two mechanical oscillators, both coupled to a common microwave cavity. The work allows for quantum state tomography of two mechanical oscillators, and provides a foundation for macroscopic mechanical entanglement and force sensing beyond conventional quantum limits.

Quantum nondemolition measurement by pulsed oscillation

Paramagnetic Faraday rotation is a quantum nondemolition measurement method that can generate spin squeezing and improve the measurement precision of a collective spin component beyond the standard quantum limit. In practice, a constant bias magnetic field is used to drive the spin precessing at sufficiently high frequency in order to lift the signal out of low-frequency technical noises. However, continuous measurement of precessing spins introduces back-action noise (BAN) due to the light-shift effect. Two types of back-action-evading (BAE) measurement of collective spin components have been demonstrated recently: continuous measurement of a two-ensemble system and stroboscopic measurement of a single ensemble. Here we propose another single ensemble BAE measurement by periodically modulating the bias field with $\ensuremath{\pi}$ pulses. Our theoretical calculation shows that under experimental settings where pulse-field modulation does not introduce significant decoherences, the proposed method can suppress the BAN and generate spin squeezing faster than the stroboscopic one at the same probe light power. Moreover, if it is combined with synchronous stroboscopic probing, light-shift BAN can be completely eliminated.

QUANTUM MECHANICS. Quantum squeezing of motion in a mechanical resonator.

According to quantum mechanics, a harmonic oscillator can never be completely at rest. Even in the ground state, its position will always have fluctuations, called the zero-point motion. Although the zero-point fluctuations are unavoidable, they can be manipulated. Using microwave frequency radiation pressure, we have manipulated the thermal fluctuations of a micrometer-scale mechanical resonator to produce a stationary quadrature-squeezed state with a minimum variance of 0.80 times that of the ground state. We also performed phase-sensitive, back-action evading measurements of a thermal state squeezed to 1.09 times the zero-point level. Our results are relevant to the quantum engineering of states of matter at large length scales, the study of decoherence of large quantum systems, and for the realization of ultrasensitive sensing of force and motion.

Quantum mechanics. Mechanically detecting and avoiding the quantum fluctuations of a microwave field.

Quantum fluctuations of the light field used for continuous position detection produce stochastic back-action forces and ultimately limit the sensitivity. To overcome this limit, the back-action forces can be avoided by giving up complete knowledge of the motion, and these types of measurements are called "back-action evading" or "quantum nondemolition" detection. We present continuous two-tone back-action evading measurements with a superconducting electromechanical device, realizing three long-standing goals: detection of back-action forces due to the quantum noise of a microwave field, reduction of this quantum back-action noise by 8.5 ± 0.4 decibels (dB), and measurement imprecision of a single quadrature of motion 2.4 ± 0.7 dB below the mechanical zero-point fluctuations. Measurements of this type will find utility in ultrasensitive measurements of weak forces and nonclassical states of motion.

Optomechanical backaction-evading measurement without parametric instability

We review a scheme for performing a backaction-evading measurement of one mechanical quadrature in an optomechanical setup. The experimental application of this scheme has been limited by parametric instabilities caused in general by a slight dependence of the mechanical frequency on the electromagnetic energy in the cavity. We find that a simple modification to the optical drive can effectively eliminate the parametric instability even at high intracavity power, allowing realistic devices to achieve sub-zero-point uncertainties in the measured quadrature.

Cavity optomechanics with ultrahigh-Qcrystalline microresonators

We present the first observation of optomechanical coupling in ultra-high Q crystalline whispering-gallery-mode (WGM) resonators. The high purity of the crystalline material enables optical quality factors in excess of 10^{10} and finesse exceeding 10^{6}. Simultaneously, mechanical quality factors greater than 10^{5} are obtained, still limited by clamping losses. Compared to previously demonstrated cylindrical resonators, the effective mass of the mechanical modes can be dramatically reduced by the fabrication of CaF2 microdisc resonators. Optical displacement monitoring at the 10^{-18} m/sqrt{Hz}-level reveals mechanical radial modes at frequencies up to 20 MHz, corresponding to unprecedented sideband factors (>100). Together with the weak intrinsic mechanical damping in crystalline materials, such high sindeband factors render crystalline WGM micro-resonators promising for backaction evading measurements, resolved sideband cooling or optomechanical normal mode splitting. Moreover, these resonators can operate in a regime where optomechanical Brillouin lasing can become accessible.

Back-action-evading measurements of nanomechanical motion

When carrying out ultrasensitive continuous measurements of position, one must ultimately confront the fundamental effects of detection back-action. Back-action forces set a lower bound on the uncertainty in the measured position, the ‘standard quantum limit’ (SQL). Recent measurements of nano- and micromechanical resonators are rapidly approaching this limit. Making measurements with sensitivities surpassing the SQL will require a new kind of approach: back-action-evading (BAE), quantum non-demolition measurement techniques. Here we realize a BAE measurement based on the parametric coupling between a nanomechanical and a microwave resonator. We demonstrate for the first time BAE detection of a single quadrature of motion with sensitivity four times the quantum zero-point motion of the mechanical resonator. We identify a limiting parametric instability inherent in BAE measurement, and describe how to improve the technique to surpass the SQL and permit the formation of squeezed states of motion.