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arXiv:1102.0385 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Pass-through Mach-Zehnder topologies for macroscopic quantum measurements

Farid Ya Khalili

Published Jun 1, 2011• 1 citation
arXivHTMLPDF

Abstract

Several relatively small-scale experimental setups, aimed on prototyping of future laser gravitational-wave detectors and testing of new methods of quantum measurements with macroscopic mechanical objects, are under development now. In these devices, not devoted directly to the gravitational-wave detection, Mach-Zehnder interferometer with pass-through Fabry-Perot cavities in the arms can be used instead of the standard Michelson/Fabry-Perot one. The advantage of this topology is that it does not contain high-reflectivity end mirrors with multilayer coatings, which Brownian noise could constitute the major part of the noise budget of the Michelson/Fabry-Perot interferometers. We consider here two variants of this topology: the ``ordinary'' position meter scheme, and a new variant of the quantum speed meter.

DOI
10.1103/physrevd.83.122001

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Several relatively small-scale experimental setups, aimed on prototyping of future laser gravitational-wave detectors and testing of new methods of quantum measurements with macroscopic mechanical objects, are under development now. In these devices, not devoted directly to the gravitational-wave detection, Mach-Zehnder interferometer with pass-through Fabry-Perot cavities in the arms can be used instead of the standard Michelson/Fabry-Perot one. The advantage of this topology is that it does not contain high-reflectivity end mirrors with multilayer coatings, which Brownian noise could constitute the major part of the noise budget of the Michelson/Fabry-Perot interferometers. We consider here two variants of this topology: the ``ordinary'' position meter scheme, and a new variant of the quantum speed meter.
In the quest to develop viable designs for third-generation optical interferometric gravitational-wave detectors (e.g., LIGO-III and EURO), one strategy is to monitor the relative momentum or speed of the test-mass mirrors, rather than monitoring their relative position. A previous paper analyzed a straightforward but impractical design for a speed-meter interferometer that accomplishes this. This paper describes some practical variants of speed-meter interferometers. Like the original interferometric speed meter, these designs in principle can beat the gravitational-wave standard quantum limit (SQL) by an arbitrarily large amount, over an arbitrarily wide range of frequencies. These variants essentially consist of a Michelson interferometer plus an extra ``sloshing'' cavity that sends the signal back into the interferometer with opposite phase shift, thereby cancelling the position information and leaving a net phase shift proportional to the relative velocity. In practice, the sensitivity of these variants will be limited by the maximum light power Wcirc{W}_{\mathrm{circ}}Wcirc​ circulating in the arm cavities that the mirrors can support and by the leakage of vacuum into the optical train at dissipation points. In the absence of dissipation and with squeezed vacuum (power squeeze factor ${e}^{\ensuremath{-}2R}\ensuremath{\simeq}0.1)$ inserted into the output port so as to keep the circulating power down, the SQL can be beat by ${h/h}_{\mathrm{SQL}}\ensuremath{\sim}\sqrt{{W}_{\mathrm{circ}}^{\mathrm{SQL}}{e}^{\ensuremath{-}2R}{/W}_{\mathrm{circ}}}$ at all frequencies below some chosen ${f}_{\mathrm{opt}}\ensuremath{\simeq}100\mathrm{Hz}.$ Here ${W}_{\mathrm{circ}}^{\mathrm{SQL}}\ensuremath{\simeq}800\mathrm{kW}{(f}_{\mathrm{opt}}/100\mathrm{}\mathrm{Hz}{)}^{3}$ is the power required to reach the SQL in the absence of squeezing. (However, as the power increases in this expression, the speed meter becomes more narrow band; additional power and reoptimization of some parameters are required to maintain the wide band. See Sec. III B.) Estimates are given of the amount by which vacuum leakage at dissipation points will debilitate this sensitivity (see Fig. 12); these losses are 10% or less over most of the frequency range of interest $(f\ensuremath{\gtrsim}10\mathrm{Hz}).$ The sensitivity can be improved, particularly at high freqencies, by using frequency-dependent homodyne detection, which unfortunately requires two 4-km-long filter cavities (see Fig. 4).
The intracavity topologies of laser gravitational-wave detectors proposed several years ago are the promising way to obtain sensitivity of these devices significantly better than the Standard Quantum Limit (SQL). In essence, the intracavity detector is a two-stage device where the end mirrors displacement created by the gravitational wave is transferred to the displacement of an additional local mirror by means of the optical rigidity. The local mirror positions have to be monitored by an additional local meter. It is evident that the local meter precision defines the sensitivity of the detector. To overcome the SQL, the quantum variational measurement can be used in the local meter. In this method a frequency-dependent correlation between the meter backaction noise and measurement noise is introduced, which allows us to eliminate the backaction noise component from the meter output signal. This correlation is created by means of an additional filter cavity. In this article the sensitivity limitations of this scheme imposed by the optical losses both in the local meter itself and in the filter cavity are estimated. It is shown that the main sensitivity limitation stems from the filter cavity losses. In order to overcome it, it is necessary to increase the filter cavity length. In a preliminary prototype experiment, an approximate 10 m long filter cavity can be used to obtain sensitivity approximately 2--3 times better than the SQL. For future Quantum Non-Demolition (QND) gravitational-wave detectors with sensitivity about 10 times better than the SQL, the filter cavity length should be within kilometer range.
Abstract 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 102−103 M⊙10^2 - 10^3\,M_ \odot102−103M⊙​ 1 0 2 - 1 0 3 M ⊙ by a factor of up to ~300.
The double-pass interferometer scheme was proposed in Ref.\,[Light Sci. Appl. {\bf 7}, 11 (2018)] as the method of implementation of the quantum speed meter concept in future laser gravitational-wave (GW) detectors. Later it was shown in Ref.\,[Phys. Rev. D {\bf 110}, 062006 (2024)] that it allows to implement the new type of the optical spring that does not require detuning of the interferometer. Here we show that both these regimes can coexist, combining the speed meter type broadband sensitivity gain with the additional lows-frequency minimum in the quantum noise originated from the optical spring. We show that the location of this minimum can be varied without affecting the core optics of the interferometer, allowing to tune the quantum noise shape in real time to follow the ``chirp'' GW signals.
Long-baseline laser-interferometer gravitational-wave (GW) detectors are operating at a factor of $\ensuremath{\sim}10$ (in amplitude) above the standard quantum limit (SQL) within a broad frequency band (in the sense that $\ensuremath{\Delta}f\ensuremath{\sim}f$). Such a low-noise budget has already allowed the creation of a controlled 2.7 kg macroscopic oscillator with an effective eigenfrequency of 150 Hz and an occupation number of $\ensuremath{\sim}200$. This result, along with the prospect for further improvements, heralds the possibility of experimentally probing macroscopic quantum mechanics (MQM)---quantum mechanical behavior of objects in the realm of everyday experience---using GW detectors. In this paper, we provide the mathematical foundation for the first step of a MQM experiment: the preparation of a macroscopic test mass into a nearly minimum-Heisenberg-limited Gaussian quantum state, which is possible if the interferometer's classical noise beats the SQL in a broad frequency band. Our formalism, based on Wiener filtering, allows a straightforward conversion from the noise budget of a laser interferometer, in terms of noise spectra, into the strategy for quantum-state preparation and the quality of the prepared state. Using this formalism, we consider how Gaussian entanglement can be built among two macroscopic test masses and the performance of the planned Advanced LIGO interferometers in quantum-state preparation.
In the quest to develop viable designs for third-generation optical interferometric gravitational-wave detectors (e.g. LIGO-III and EURO), one strategy is to monitor the relative momentum or speed of the test-mass mirrors rather than monitoring their relative position. This paper describes and analyzes the most straightforward design for a speed meter interferometer that accomplishes this---a design (due to Braginsky, Gorodetsky, Khalili and Thorne) that is analogous to a microwave-cavity speed meter conceived by Braginsky and Khalili. A mathematical mapping between the microwave speed meter and the optical interferometric speed meter is developed and is used to show [in accord with the speed being a quantum nondemolition observable] that in principle the interferometric speed meter can beat the gravitational-wave standard quantum limit (SQL) by an arbitrarily large amount, over an arbitrarily wide range of frequencies, and can do so without the use of squeezed vacuum or any auxiliary filter cavities at the interferometer's input or output. However, in practice, to reach or beat the SQL, this specific speed meter requires exorbitantly high input light power. The physical reason for this is explored, along with other issues such as constraints on performance due to optical dissipation. This analysis forms a foundation for ongoing attempts to develop a more practical variant of an interferometric speed meter and to combine the speed meter concept with other ideas to yield a promising LIGO-III/EURO interferometer design that entails low laser power.