Quantum Measurement Theory in Gravitational-Wave Detectors

Exceptional-Point Sensors Offer No Fundamental Signal-to-Noise Ratio Enhancement

Exceptional-point (EP) sensors exhibit a square-root resonant frequency bifurcation in response to external perturbations, making them appear attractive for sensing applications. However, there is an open debate as to whether or not this sensitivity advantage is negated by additional noise in the system. We settle this debate by showing that increased fundamental noises of quantum and thermal origin in EP sensors, and in particular self-excited (or PT-symmetric) EP sensors, negate the sensitivity benefit. Accordingly, EP sensing schemes are only beneficial either with further quantum enhancement or if compared to sensors limited by technical noise. As many modern sensors are limited by technical noise, EP sensors may still find practical uses despite their lack of a fundamental advantage. Alternatively, we propose a quantum-enhanced EP sensor that achieves a sensing advantage even when limited by quantum or thermal fluctuations.

Achieving the Fundamental Quantum Limit of Linear Waveform Estimation

Sensing a classical signal using a linear quantum device is a pervasive application of quantum-enhanced measurement. The fundamental precision limits of linear waveform estimation, however, are not fully understood. In certain cases, there is an unexplained gap between the known waveform-estimation quantum Cramér-Rao bound and the optimal sensitivity from quadrature measurement of the outgoing mode from the device. We resolve this gap by establishing the fundamental precision limit, the waveform-estimation Holevo Cramér-Rao bound, and how to achieve it using a nonstationary measurement. We apply our results to detuned gravitational-wave interferometry to accelerate the search for postmerger remnants from binary neutron-star mergers. If we have an unequal weighting between estimating the signal's power and phase, then we propose how to further improve the signal-to-noise ratio by a factor of sqrt[2] using this nonstationary measurement.

Quantum Precision Limits of Displacement Noise-Free Interferometers

Current laser-interferometric gravitational wave detectors suffer from a fundamental limit to their precision due to the displacement noise of optical elements contributed by various sources. Several schemes for displacement noise-free interferometers (DFI) have been proposed to mitigate their effects. The idea behind these schemes is similar to decoherence-free subspaces in quantum sensing; i.e., certain modes contain information about the gravitational waves but are insensitive to the mirror motion (displacement noise). We derive quantum precision limits for general DFI schemes, including optimal measurement basis and optimal squeezing schemes. We introduce a triangular cavity DFI scheme and apply our general bounds to it. Precision analysis of this scheme with different noise models shows that the DFI property leads to interesting sensitivity profiles and improved precision due to noise mitigation and larger gain from squeezing.

Acoustic frequency atomic spin oscillator in the quantum regime

Quantum noise reduction and entanglement-enhanced sensing in the acoustic frequency range is an outstanding challenge relevant for a number of applications including magnetometry and broadband noise reduction in gravitational wave detectors. Here we experimentally demonstrate quantum behavior of a macroscopic atomic spin oscillator in the acoustic frequency range. Quantum back-action of the spin measurement, ponderomotive squeezing of light, and virtual spring softening are observed at oscillation frequencies down to the sub-kHz range. Quantum noise sources characteristic of spin oscillators operating in the near-DC frequency range are identified and means for their mitigation are presented.

Mitigating Quantum Decoherence in Force Sensors by Internal Squeezing

The most efficient approach to laser interferometric force sensing to date uses monochromatic carrier light with its signal sideband spectrum in a squeezed vacuum state. Quantum decoherence, i.e., mixing with an ordinary vacuum state due to optical losses, is the main sensitivity limit. In this Letter, we present both theoretical and experimental evidence that quantum decoherence in high-precision laser interferometric force sensors enhanced with optical cavities and squeezed light injection can be mitigated by a quantum squeeze operation inside the sensor's cavity. Our experiment shows an enhanced measurement sensitivity that is independent of the optical readout loss in a wide range. Our results pave the way for quantum improvements in scenarios where high decoherence previously precluded the use of squeezed light. Our results hold significant potential for advancing the field of quantum sensors and enabling new experimental approaches in high-precision measurement technology.

Complete Physical Characterization of Quantum Nondemolition Measurements via Tomography

We introduce a self-consistent tomography for arbitrary quantum nondemolition (QND) detectors. Based on this, we build a complete physical characterization of the detector, including the measurement processes and a quantification of the fidelity, ideality, and backaction of the measurement. This framework is a diagnostic tool for the dynamics of QND detectors, allowing us to identify errors, and to improve their calibration and design. We illustrate this on a realistic Jaynes-Cummings simulation of a superconducting qubit readout. We characterize nondispersive errors, quantify the backaction introduced by the readout cavity, and calibrate the optimal measurement point.

Two-Carrier Scheme: Evading the 3 dB Quantum Penalty of Heterodyne Readout in Gravitational-Wave Detectors

Precision measurements using a traditional heterodyne readout suffer a 3 dB quantum noise penalty compared with a homodyne readout. The extra noise is caused by the quantum fluctuations in the image vacuum. We propose a two-carrier gravitational-wave detector design that evades the 3 dB quantum penalty of the heterodyne readout. We further propose a new way of realizing frequency-dependent squeezing utilizing two-mode squeezing in our scheme. It naturally achieves more precise audio frequency signal measurements with radio frequency squeezing. In addition, the detector is compatible with other quantum nondemolition techniques.

Fundamental limits of laser power stabilization via a radiation pressure transfer scheme

Traditional active laser power stabilization schemes are fundamentally limited by quantum shot noise on the in-loop photodetector. One way to overcome this limitation is to implement a nondemolition sensing scheme where laser power fluctuations are transferred to motion of a micro-oscillator, which can be sensed with a high signal-to-noise ratio. In this Letter, we analyze the power stability achievable in a nondemolition scheme limited by quantum and thermal noise. Under the assumption of realistic experimental parameters, we show that generation of a strong bright squeezed quantum state of light should be possible.

Demonstration of interferometer enhancement through EPR entanglement

The recent series of gravitational-wave (GW) detections by the Advanced LIGO and Advanced Virgo observatories launched the new field of GW astronomy. As the sensitivity of GW detectors is limited by quantum noise of light, concepts from quantum metrology have been adapted to increase the observational range. Since 2010, squeezed light with reduced quantum noise has been used for improved sensitivity at signal frequencies above 100 Hz. However, 100 m long optical filter resonators would be required to also improve the sensitivity at lower frequencies, adding significant cost and complexity. Here we report on a proof-of-principle setup of an alternative concept that achieves the broadband noise reduction by using Einstein-Podolsky-Rosen (EPR) entangled states instead. We show that the desired sensitivity improvement can then be obtained with the signal-recycling resonator that is already part of current observatories, providing the viable alternative to high-cost filter cavities.

Quantum expander for gravitational-wave observatories

The quantum uncertainty of laser light limits the sensitivity of gravitational-wave observatories. Over the past 30 years, techniques for squeezing the quantum uncertainty, as well as for enhancing gravitational-wave signals with optical resonators have been invented. Resonators, however, have finite linewidths, and the high signal frequencies that are produced during the highly scientifically interesting ring-down of astrophysical compact-binary mergers still cannot be resolved. Here, we propose a purely optical approach for expanding the detection bandwidth. It uses quantum uncertainty squeezing inside one of the optical resonators, compensating for the finite resonators' linewidths while keeping the low-frequency sensitivity unchanged. This quantum expander is intended to enhance the sensitivity of future gravitational-wave detectors, and we suggest the use of this new tool in other cavity-enhanced metrological experiments.

Squeezed vacuum states of light for gravitational wave detectors

A century after Einstein's formulation of general relativity, the detectors of the Laser Interferometer Gravitational-wave Observatory (LIGO) made the first direct detection of gravitational waves. This historic achievement was the culmination of a world-wide effort and decades of instrument research. While sufficient for this monumental discovery, the current generation of gravitational-wave detectors represent the least sensitive devices necessary for the task; improved detectors will be required to fully exploit this new window on the Universe. In this paper, we review the application of squeezed vacuum states of light to gravitational-wave detectors as a way to reduce quantum noise, which currently limits their performance in much of the detection band.

Fundamental Quantum Limits of Multicarrier Optomechanical Sensors

Optomechanical sensors involving multiple optical carriers can experience mechanically mediated interactions causing multimode correlations across the optical fields. One instance is laser-interferometric gravitational wave detectors which introduce multiple carrier frequencies for classical sensing and control purposes. An outstanding question is whether such multicarrier optomechanical sensors outperform their single-carrier counterpart in terms of quantum-limited sensitivity. We show that the best precision is achieved by a single-carrier instance of the sensor. For the current LIGO detection system this precision is already reachable.

Overcoming the Standard Quantum Limit in Gravitational Wave Detectors Using Spin Systems with a Negative Effective Mass

Quantum backaction (QBA) of a measurement limits the precision of observation of the motion of a free mass. This profound effect, dubbed the "Heisenberg microscope" in the early days of quantum mechanics, leads to the standard quantum limit (SQL) stemming from the balance between the measurement sensitivity and the QBA. We consider the measurement of motion of a free mass performed in a quantum reference frame with an effective negative mass which is not limited by QBA. As a result, the disturbance on the motion of a free mass can be measured beyond the SQL. QBA-limited detection of motion for a free mass is extremely challenging, but there are devices where this effect is expected to play an essential role, namely, gravitational wave detectors (GWDs) such as LIGO and Virgo. Recent reports on the observations of gravitational waves have opened new horizons in cosmology and astrophysics. We present a general idea and a detailed numerical analysis for QBA-evading measurement of the gravitational wave effect on the GWD mirrors, which can be considered free masses under relevant conditions. The measurement is performed by two entangled beams of light, probing the GWD and an auxiliary atomic spin ensemble, respectively. The latter plays the role of a free negative mass. We show that under realistic conditions the sensitivity of the GWD in m/sqrt[Hz] can be increased by 6 dB over the entire frequency band of interest.

A new quantum speed-meter interferometer: measuring speed to search for intermediate mass black holes

The recent discovery of gravitational waves (GW) by Advanced LIGO (Laser Interferometric Gravitational-wave Observatory) has impressively launched the novel field of gravitational astronomy and allowed us to glimpse exciting objects about which we could previously only speculate. Further sensitivity improvements at the low-frequency end of the detection band of future GW observatories must rely on quantum non-demolition (QND) methods to suppress fundamental quantum fluctuations of the light fields used to readout the GW signal. Here we present a novel concept of how to turn a conventional Michelson interferometer into a QND speed-meter interferometer with coherently suppressed quantum back-action noise. We use two orthogonal polarizations of light and an optical circulator to couple them. We carry out a detailed analysis of how imperfections and optical loss influence the achievable sensitivity. We find that the proposed configuration significantly enhances the low-frequency sensitivity and increases the observable event rate of binary black-hole coalescences in the range of 1 0 2 - 1 0 3 M ⊙ by a factor of up to ~300.

Gravitational wave detection using laser interferometry beyond the standard quantum limit

Interferometric gravitational wave detectors (such as advanced LIGO) employ high-power solid-state lasers to maximize their detection sensitivity and hence their reach into the universe. These sophisticated light sources are ultra-stabilized with regard to output power, emission frequency and beam geometry; this is crucial to obtain low detector noise. However, even when all laser noise is reduced as far as technically possible, unavoidable quantum noise of the laser still remains. This is a consequence of the Heisenberg Uncertainty Principle, the basis of quantum mechanics: in this case, it is fundamentally impossible to simultaneously reduce both the phase noise and the amplitude noise of a laser to arbitrarily low levels. This fact manifests in the detector noise budget as two distinct noise sources-photon shot noise and quantum radiation pressure noise-which together form a lower boundary for current-day gravitational wave detector sensitivities, the standard quantum limit of interferometry. To overcome this limit, various techniques are being proposed, among them different uses of non-classical light and alternative interferometer topologies. This article explains how quantum noise enters and manifests in an interferometric gravitational wave detector, and gives an overview of some of the schemes proposed to overcome this seemingly fundamental limitation, all aimed at the goal of higher gravitational wave event detection rates.This article is part of a discussion meeting issue 'The promises of gravitational-wave astronomy'.

Observation of an optical spring with a beam splitter

We present the experimental observation of an optical spring without the use of an optical cavity. The optical spring is produced by interference at a beam splitter and, in principle, does not have the damping force associated with optical springs created in detuned cavities. The experiment consists of a Michelson-Sagnac interferometer (with no recycling cavities) with a partially reflective GaAs microresonator as the beam splitter that produces the optical spring. Our experimental measurements at input powers of up to 360 mW show the shift of the optical spring frequency as a function of power and are in excellent agreement with theoretical predictions. In addition, we show that the optical spring is able to keep the interferometer stable and locked without the use of external feedback.

Beating the Standard Sensitivity-Bandwidth Limit of Cavity-Enhanced Interferometers with Internal Squeezed-Light Generation

The shot-noise limited peak sensitivity of cavity-enhanced interferometric measurement devices, such as gravitational-wave detectors, can be improved by increasing the cavity finesse, even when comparing fixed intracavity light powers. For a fixed light power inside the detector, this comes at the price of a proportional reduction in the detection bandwidth. High sensitivity over a large span of signal frequencies, however, is essential for astronomical observations. It is possible to overcome this standard sensitivity-bandwidth limit using nonclassical correlations in the light field. Here, we investigate the internal squeezing approach, where the parametric amplification process creates a nonclassical correlation directly inside the interferometer cavity. We theoretically analyze the limits of the approach and measure 36% increase in the sensitivity-bandwidth product compared to the classical case. To our knowledge, this is the first experimental demonstration of an improvement in the sensitivity-bandwidth product using internal squeezing, opening the way for a new class of optomechanical force sensing devices.

Optimal adaptive control for quantum metrology with time-dependent Hamiltonians

Quantum metrology has been studied for a wide range of systems with time-independent Hamiltonians. For systems with time-dependent Hamiltonians, however, due to the complexity of dynamics, little has been known about quantum metrology. Here we investigate quantum metrology with time-dependent Hamiltonians to bridge this gap. We obtain the optimal quantum Fisher information for parameters in time-dependent Hamiltonians, and show proper Hamiltonian control is generally necessary to optimize the Fisher information. We derive the optimal Hamiltonian control, which is generally adaptive, and the measurement scheme to attain the optimal Fisher information. In a minimal example of a qubit in a rotating magnetic field, we find a surprising result that the fundamental limit of T² time scaling of quantum Fisher information can be broken with time-dependent Hamiltonians, which reaches T⁴ in estimating the rotation frequency of the field. We conclude by considering level crossings in the derivatives of the Hamiltonians, and point out additional control is necessary for that case.

Quantum metrology investigates the improvement given to precision measurements by exploiting quantum mechanics, but it has been mostly limited to systems with static Hamiltonians. Here the authors study it in the general case of time-varying Hamiltonians, showing that optimizing the quantum Fisher information via quantum control provides an advantage.

Interferometer techniques for gravitational-wave detection

Several km-scale gravitational-wave detectors have been constructed worldwide. These instruments combine a number of advanced technologies to push the limits of precision length measurement. The core devices are laser interferometers of a new kind; developed from the classical Michelson topology these interferometers integrate additional optical elements, which significantly change the properties of the optical system. Much of the design and analysis of these laser interferometers can be performed using well-known classical optical techniques; however, the complex optical layouts provide a new challenge. In this review, we give a textbook-style introduction to the optical science required for the understanding of modern gravitational wave detectors, as well as other high-precision laser interferometers. In addition, we provide a number of examples for a freely available interferometer simulation software and encourage the reader to use these examples to gain hands-on experience with the discussed optical methods.

Simple delay-limited sideband locking with heterodyne readout

We present a robust sideband laser locking technique ideally suited for applications requiring low probe power and heterodyne readout. By feeding back to a high-bandwidth voltage-controlled oscillator, we lock a first-order phase-modulation sideband to a high-finesse Fabry-Perot cavity in ambient conditions, achieving a closed-loop bandwidth of 3.5 MHz (with a single integrator) limited fundamentally by the signal delay. The measured transfer function of the closed loop agrees with a simple model based on ideal system components, and from this we suggest a modified design that should achieve a bandwidth exceeding 6 MHz with a near-causally limited feedback gain as high as 4 × 10⁷ at 1 kHz. The off-resonance optical carrier enables alignment-free heterodyne readout, alleviating the need for additional lasers or optical modulators.

Overwhelming Thermomechanical Motion with Microwave Radiation Pressure Shot Noise

We measure the fundamental noise processes associated with a continuous linear position measurement of a micromechanical membrane incorporated in a microwave cavity optomechanical circuit. We observe the trade-off between the two fundamental sources of noise that enforce the standard quantum limit: the measurement imprecision and radiation pressure backaction from photon shot noise. We demonstrate that the quantum backaction of the measurement can overwhelm the intrinsic thermal motion by 24 dB, entering a new regime for cavity optomechanical systems.

Testing spontaneous wave-function collapse models on classical mechanical oscillators

We show that the heating effect of spontaneous wave-function collapse models implies an experimentally significant increment ΔT(sp) of equilibrium temperature in a mechanical oscillator. The obtained new form ΔT(sp) is linear in the oscillator's relaxation time τ and independent of the mass. The oscillator can be in a classical thermal state, also the effect ΔT(sp) is classical for a wide range of frequencies and quality factors. We note that the test of ΔT(sp) does not necessitate quantum state monitoring just tomography. In both the gravity-related and the continuous spontaneous localization models the strong-effect edge of their parameter range can be challenged in existing experiments on classical oscillators. For the continuous spontaneous localization theory, the conjectured highest collapse rate parameter values become immediately constrained by evidences from current experiments on extreme slow-ring-down oscillators.

Optomechanical sensing of spontaneous wave-function collapse

Quantum experiments with nanomechanical oscillators are regarded as a test bed for hypothetical modifications of the Schrödinger equation, which predict a breakdown of the superposition principle and induce classical behavior at the macroscale. It is generally believed that the sensitivity to these unconventional effects grows with the mass of the mechanical quantum system. Here we show that the opposite is the case for optomechanical systems in the presence of generic noise sources, such as thermal and measurement noise. We determine conditions for distinguishing these decoherence processes from possible collapse-induced decoherence in continuous optomechanical force measurements.

Macroscopic quantum mechanics in a classical spacetime

We apply the many-particle Schrödinger-Newton equation, which describes the coevolution of a many-particle quantum wave function and a classical space-time geometry, to macroscopic mechanical objects. By averaging over motions of the objects' internal degrees of freedom, we obtain an effective Schrödinger-Newton equation for their centers of mass, which can be monitored and manipulated at quantum levels by state-of-the-art optomechanics experiments. For a single macroscopic object moving quantum mechanically within a harmonic potential well, its quantum uncertainty is found to evolve at a frequency different from its classical eigenfrequency-with a difference that depends on the internal structure of the object-and can be observable using current technology. For several objects, the Schrödinger-Newton equation predicts semiclassical motions just like Newtonian physics, yet quantum uncertainty cannot be transferred from one object to another.