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Physical Review A

Interferometer readout noise below the standard quantum limit of a membrane

T. Westphal, D. Friedrich, H. Kaufer, Kazuhiro Yamamoto, S. Goßler, H. Müller‐Ebhardt, S. L. Danilishin, Farid Ya Khalili, K. Danzmann, Roman Schnabel

Published Jun 6, 2012• 28 citations
Publisher

Abstract

Here we report on the realization of a Michelson-Sagnac interferometer whose purpose is the precise characterization of the motion of, i.e., membranes showing significant light transmission. Our interferometer has a readout noise spectral density (imprecision) of $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}\phantom{\rule{0.16em}{0ex}}\mathrm{m}/\sqrt{\mathrm{Hz}}$ at frequencies around the fundamental resonance of a SiN${}_{\mathrm{x}}$ membrane at about 100 kHz, without using optical cavities. The readout noise demonstrated is more than 16 dB below the peak value of the membrane's standard quantum limit (SQL). This margin is significantly bigger than those of previous works with nanowires [Teufel et al., Nature Nano. 4, 820 (2009); Anetsberger et al., Nature Phys. 5, 909 (2009)]. We discuss the meaning of the SQL for force measurements and its relation to the readout performance and emphasize that neither our nor previous experiments achieved a total noise spectral density as low as the SQL due to thermal noise.

DOI
10.1103/physreva.85.063806

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The quantum mechanical description of electromagnetic fields predicts fluctuations in their amplitude and phase quadrature, that underlay Heisenberg's uncertainty relation. When reflected by an object, the fields induce a radiation pressure force. The amplitude uncertainty causes fluctuations of this force, resulting in quantum radiation pressure noise. The standard quantum limit sets a lower bound for the sum of radiation pressure noise and shot noise, that will dominate the sensitivity of future gravitational-wave detectors. In optomechanics the radiation pressure noise is investigated. One fundamental goal is to undercut the standard quantum limit by applying squeezed states, that have a reduced variance in one quadrature and an amplified variance in the orthogonal. A shot noise reduction of 2.2 dB in an optomechanical table top experiment can be found in the literature. In this work squeezed states were produced in a nonlinear crystal via degenerate parametric down-conversion. 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In einem ersten Experiment mit dem Aufbau wurde demonstriert, wie intern produzierte gequetschte Zustande die Sensitivitat einer interferometrischen Messung verbessern konnen. Ein Anstieg des Sensitivitat-Bandbreite-Produkts von 36% wurde fur ein kunstlich generiertes Signal experimentell beobachtet. Es wird weiterhin ein kryogenes optomechanisches Experiment mit einer Siliziumnitrid-Membran als mechanischer Oszillator prasentiert. Die Topologie des Michelson-Sagnac-Interferometers wurde gewahlt um trotz ihrer geringen Reflektivitat ein Michelson-Interferometer mit der Membran als Endspiegel zu realisieren. Fur Temperaturen bis minimal 100K konnte ein Kontrast der Moden im Ausgang von 99.7% erreicht werden. Das Signal im dunklen Ausgang ermoglichte eine Messung mit einer vom Schrotrauschen limitierten Sensitivitat fur eine Eingangsleistung von 50 muW. 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•Oct 23, 2019•0 citations
Here we report on the realization of a Michelson-Sagnac interferometer whose purpose is the precise characterization of the motion of, i.e., membranes showing significant light transmission. Our interferometer has a readout noise spectral density (imprecision) of 3\ifmmode×\else\texttimes\fi10\ensuremath−16m/Hz3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}\phantom{\rule{0.16em}{0ex}}\mathrm{m}/\sqrt{\mathrm{Hz}}3\ifmmode×\else\texttimes\fi10\ensuremath−16m/Hz​ at frequencies around the fundamental resonance of a SiNx{}_{\mathrm{x}}x​ membrane at about 100 kHz, without using optical cavities. The readout noise demonstrated is more than 16 dB below the peak value of the membrane's standard quantum limit (SQL). This margin is significantly bigger than those of previous works with nanowires [Teufel et al., Nature Nano. 4, 820 (2009); Anetsberger et al., Nature Phys. 5, 909 (2009)]. We discuss the meaning of the SQL for force measurements and its relation to the readout performance and emphasize that neither our nor previous experiments achieved a total noise spectral density as low as the SQL due to thermal noise.
The quantum state of light changes its nature when being reflected off a mechanical oscillator due to the latter's susceptibility to radiation pressure. As a result, a coherent state can transform into a squeezed state and can get entangled with the motion of the oscillator. Full information of the state of light can only be gathered by a tomographic measurement. Here we demonstrate a tomographic interferometer readout by measuring arbitrary quadratures of the light field exiting a Michelson–Sagnac interferometer that contains a thermally excited high-quality silicon nitride membrane. A readout noise of 1.9 × 10 −16 m Hz −1/2 around the membrane's fundamental oscillation mode at 133 kHz has been achieved, going below the peak value of the standard quantum limit by a factor of 8.2 (9 dB). The readout noise was entirely dominated by shot noise in a rather broad frequency range around the mechanical resonance.
Interferometers enable ultrasensitive measurement in a wide array of applications from gravitational wave searches to force microscopes. The role of quantum mechanics in the metrological limits of interferometers has a rich history, and a large number of techniques to surpass conventional limits have been proposed. In a typical measurement configuration, the trade-off between the probe’s shot noise (imprecision) and its quantum backaction results in what is known as the standard quantum limit (SQL). In this work, we investigate how quantum correlations accessed by modifying the readout of the interferometer can access physics beyond the SQL and improve displacement sensitivity. Specifically, we use an optical cavity to probe the motion of a silicon nitride membrane off mechanical resonance, as one would do in a broadband displacement or force measurement, and observe sensitivity better than the SQL dictates for our quantum efficiency. Our measurement illustrates the core idea behind a technique known as variational readout, in which the optical readout quadrature is changed as a function of frequency to improve broadband displacement detection. And, more generally, our result is a salient example of how correlations can aid sensing in the presence of backaction.
When one seeks knowledge of the full dynamics of the displacement of a harmonic oscillator, non-commutation of the two mechanical quadratures requires a minimum added noise equal to the mechanical resonator's zero point motion. This fundamental quantum limit (QL) is a distinct bound from the standard quantum limit (SQL) that is often considered in interferometric displacement measurement. The SQL is a consequence of the non-commutation of the probe's quadratures in a specific measurement configuration and is characterized by a tradeoff between shot noise (SN) imprecision and quantum backaction that are uncorrelated. The QL and SQL reach the same limit when probing at the peak mechanical response, but the QL can be a significantly lower bound off resonance. In studies of micro-mechanical motion, there has been great interest in observing quantum backaction and approaching the SQL on mechanical resonance. However, much of what historically motivates SQL research is displacement monitoring over a wide frequency band, such as gravitational wave searches. One long-standing concept for surpassing the SQL is to introduce correlations by changing the readout quadrature as a function of frequency in a technique known as variational readout. Here we harness quantum correlations in the measurement of a macroscopic resonator by modifying the readout quadrature of an optical interferometer. By using an optomechanical device thermalized to a dilution refrigerator, we measure quantum noise far off resonance where we improve upon the off-resonance SQL for our quantum efficiency.
Thermal noise in optics constrains the precision of optical experiments, including Michelson interferometers, optical clocks, and optomechanical sensors. While high-power optical experiments such as gravitational-wave detectors are currently limited mainly by quantum noise, new experiments incorporating recent insights from quantum metrology are being developed that evade the quantum shot noise background. In particular, Michelson interferometers that use photon-counting readout will look for weak, high-frequency signals. Since shot noise is no longer the dominant noise source with these readout schemes, it is important to accurately model thermal noise to characterize signals and design more sensitive experiments. However, previous modeling uses approximations that are no longer valid in these frequency regimes. In the MHz band, the quasistatic approximation does not apply. We therefore develop more general models of substrate and coating mechanical (Brownian) noise, substrate and coating thermoelastic noise, and coating thermorefractive noise. We validate the models with comparisons to previous low-frequency modeling and high-frequency spectra from an experiment that has already taken data, the Holometer. We then apply the new models to GQuEST, an experiment under construction.
Quantum shot noise (QSN) and quantum back action (QBA) set the Standard Quantum Limit (SQL) for interferometric displacement measurements. We target broadband sub-SQL sensitivity using a cryogenic nanogram-scale SiN membrane in a fiber-based membrane-at-the-edge cavity[1]. Our approach employs bright squeezed light with a frequency-dependent angle to simultaneously suppress QSN off-resonance and QBA at resonance. We have implemented a filter cavity providing the required phase rotation and demonstrate 2 dB of detected bright squeezing. In parallel, we developed a fully fiber-integrated high-finesse cavity with coated fiber facets, compatible with cryogenic operation and homodyne readout. These results mark key progress toward broadband quantum-enhanced sensing beyond the SQL.
The quantum mechanical description of electromagnetic fields predicts fluctuations in their amplitude and phase quadrature, that underlay Heisenberg's uncertainty relation. When reflected by an object, the fields induce a radiation pressure force. The amplitude uncertainty causes fluctuations of this force, resulting in quantum radiation pressure noise. The standard quantum limit sets a lower bound for the sum of radiation pressure noise and shot noise, that will dominate the sensitivity of future gravitational-wave detectors. In optomechanics the radiation pressure noise is investigated. One fundamental goal is to undercut the standard quantum limit by applying squeezed states, that have a reduced variance in one quadrature and an amplified variance in the orthogonal. A shot noise reduction of 2.2 dB in an optomechanical table top experiment can be found in the literature. In this work squeezed states were produced in a nonlinear crystal via degenerate parametric down-conversion. At a sideband frequency of 400 kHz a squeeze factor of (8.7+/-0.2) dB was measured. In a first experiment the setup was used to demonstrate how internal squeezing can improve the sensitivity of an interferometric measurement. An increase of the sensitivity-bandwidth product of 36% was experimentally observed for an artificially generated signal. This thesis further presents a cryogenic optomechanical experiment with a silicon nitride membrane as mechanical oscillator. The topology of a Michelson-Sagnac interferometer was chosen to realize a Michelson interferometer with the membrane as end mirror despite its low reflectivity. A contrast of the outcoming modes of 99.7% was reached for temperatures down to 100 K. The signal at the dark fringe allowed a measurement with a shot-noise limited sensitivity for an input power of 50 muW. After calibrating the measured spectra with the theoretical shot noise, the quality factors of the membrane Q_293=2.7*10^5 and Q_100=1,5*10^6 at 293K and 100K could be determined from the resulting displacement spectral density. By injecting externally produced squeezed states, the quantum shot noise in the presented experiment was reduced by (3.1+-0.2) dB, which corresponds to a factor of sqrt(2) in the linear displacement spectral density. For a radiation-pressure-noise limited sensitivity, the demonstrated application of squeezed states could allow to overcome the standard quantum limit in the future. Aus der quantenmechanischen Beschreibung elektromagnetischer Felder folgen Fluktuationen in deren Amplituden- und Phasenquadratur, die der Heisenbergsche Unscharferelation unterliegen. Wenn die Felder von einem Objekt reflektiert werden, erzeugen sie eine Strahlungsdruckkraft. Die Fluktuationen dieser Kraft aufgrund der Amplitudenunscharfe verursachen das Quanten-Strahlungsdruckrauschen. Das Standardquantenlimit ist der Minimalwert fur die Summe aus Strahlungsdruckrauschen und Schrotrauschen, der in Zukunft die Sensitivitat von Gravitationswellendetektoren limitieren wird. In der Optomechanik wird das Strahlungsdruckrauschen erforscht. Ein fundamentales Ziel ist das Uberwinden des Standardquantenlimits durch die Anwendung gequetschter Zustande, deren Varianz in einer Quadratur vermindert aber in der orthogonalen erhoht ist. In der Literatur findet man eine Schrotrauschreduktion von 2.2 dB in einem optomechanischem Tischaufbau. In dieser Arbeit wurden gequetschte Zustande mithilfe der parametrischen Abkonversion in einem nichtlinearen Kristall generiert. Bei einer Seitenbandfrequenz von 400 kHz wurde ein Quetschfaktor von (8.7+-0.2) dB gemessen. In einem ersten Experiment mit dem Aufbau wurde demonstriert, wie intern produzierte gequetschte Zustande die Sensitivitat einer interferometrischen Messung verbessern konnen. Ein Anstieg des Sensitivitat-Bandbreite-Produkts von 36% wurde fur ein kunstlich generiertes Signal experimentell beobachtet. Es wird weiterhin ein kryogenes optomechanisches Experiment mit einer Siliziumnitrid-Membran als mechanischer Oszillator prasentiert. Die Topologie des Michelson-Sagnac-Interferometers wurde gewahlt um trotz ihrer geringen Reflektivitat ein Michelson-Interferometer mit der Membran als Endspiegel zu realisieren. Fur Temperaturen bis minimal 100K konnte ein Kontrast der Moden im Ausgang von 99.7% erreicht werden. Das Signal im dunklen Ausgang ermoglichte eine Messung mit einer vom Schrotrauschen limitierten Sensitivitat fur eine Eingangsleistung von 50 muW. Nach der Kalibrierung der gemessenen Spektren mit dem theoretischen Schrotrauschen konnten die Gutefaktoren der Membran Q_293=2.7*10^5 und Q_100=1.5*10^6 bei 293K und 100K aus der resultierenden spektrale Dichte der Ortsanderung bestimmt werden. Durch die Injektion extern produzierter gequetschter Zustande wurde das Quantenschrotrauschen im prasentierten Experiment um (3.1 +/-0.2) dB gesenkt, was einem Faktor Wurzel(2) in der linearen spektralen Rauschdichte entspricht. Fur eine vom Strahlungsdruckrauschen limitierte Sensitivitat konnte die demonstrierte Anwendungvon gequetschten Zustanden in Zukunft eine Unterschreitung des Standardquantenlimits ermoglichen.