In addition, I was deeply impressed by his natural and humorous attitude towards life and it is a great honor to allow him to be one of my best men at my wedding. We give examples of different couplings, discuss in detail one particular case of coherent coupling and show its advantages in optomechanical experiments.
INTRODUCTION
Preface
Linear quantum measurement device for gravitational wave detection In the early days of GW detection, excitations of mechanical oscillators (resonantIn the early days of GW detection, excitations of mechanical oscillators (resonant
Optomechanics and the interaction classification
Novel schemes to bypass bandwidth-sensitivity trade-off
Matter-wave interferometer
Conventionally, the coupling between the optical and mechanical degrees of freedom is classified based on an intuitive physical picture of the setup. However, EQL limits the useful bandwidth of the detector according to the gain-bandwidth trade-off.
Modeling of BBH ringdown gravitational waves
Gwtc-1: A Gravitational Wave Transient Catalog of Compact Binary Mergers Observed by Ligo and Virgo During the First and Second Observing Runs. Testing general relativity with binary black holes from the second LIGO-Virgo Gravitational-Wave Transient Catalog, 10 2020.
CLASSIFICATION OF OPTOMECHANICAL INTERACTION AND THE DISCOVERY OF COHERENT COUPLING
Introduction
However, this Hamiltonian would have two different forms based on the parameters of the system. Such a Hamiltonian is dispersive: the resonant frequencies of the modulations are modulated by the mechanical oscillations.
Figure 2.3: On-chip optomechanical coupling between the curved input waveguide and the optical racetrack cavity, adapted from FIG
Purely coherent coupling in a ring cavity system
Cavity modes and the Hamiltonian
Application: Enhanced cooling
ˆ cAAAB73icbVBNS8NAEJ34WetX1aOXxSJ4KkkR9Fj04rGC/YA2lMl20y7dbOLuKLuRi8BW4B000F800008686666666868686666666668686686868886668686868686868686868686868 LEsG1cd1vZ219Y3Nru7RT3t3bPzisHB23dZwqylo0FrHqBqiZ4JK1DDeCdRPFMAoE6wST29zvPDGleSwfzDRhfoQjyUNO0Vip2x+jIXRQH1SqVSimL65GH1SwSimL60V6V1SqVSiL6VQH1SqSimL6V6GN1SqVSiL6V6V6 J8TNUhlPBZuV+qlmCdIIj1rNUYsS0n83vnZFzqwxJGCtb0u7P1d8TGUZaT6PAdkZoxnrZy8X/vF5qwms/4zJJDZN0sShMBTExyZ8NFJWq25Ej lbTrNc/y+8tq46aIowSncAYX4MEVNOAOmtACCgKe4RXenEfnxXl3Phata04xcwJ/4Hz+AEtuj3c=AAAB73icbVBNS8NAEJ34WetX1aOXxSJ4KkkR9Fj04rGC/YA2lMl20y7dbOLuRiihf8KLB0W8+ne8+W/ctDlo64OBx3szzMwLEsG1cd1vZ219Y3Nru7RT3t3bPzisHB23dZwqylo0FrHqBqiZ4JK1DDeCdRPFMAoE6wST29zvPDGleSwfzDRhfoQjyUNO0Vip2x+jIXRQH1Sqbs2dg6wSryBVKNAcVL76w5imEZOGCtS657mJ8TNUhlPBZuV+qlmCdIIj1rNUYsS0n83vnZFzqwxJGCtb0u7P1d8TGUZaT6PAdkZoxnrZy8X/vF5qwms/4zJJDZN0sShMBTExyZ8nQ64YNWJqCVLF7a2EjlEhNTaisg3BW355lbTrNc/y+8tq46aIowSncAYX4MEVNOAOmtACCgKe4RXenEfnxXl3Phata04xcwJ/4Hz+AEtuj3c=AAAB73icbVBNS8NAEJ34WetX1aOXxSJ4KkkR9Fj04rGC/YA2lMl20y7dbOLuRiihf8KLB0W8+ne8 +W/ctDlo64OBx3szzMwLEsG1cd1vZ219Y3Nru7RT3t3bPzisHB23dZwqylo0FrHqBqiZ4JK1DDeCdRPFMAoE6wST29zvPDGleSwfzDRhfoQjyUNO00VSQVQV20JUNO00VBQVBQJYUNO00VBQVB200VBQJYUNO0VBQV2 76w5imEZOGCtS657mJ8TNUhlPBZuV+qlmCdIIj1rNUYsS0n83vnZFzqwxJGCtb0u7P1d8TGUZaT6PAdkZoxnrZy8X/vF5qwms/VF5qwms/4zW8F5qwms/4zV8F5qwms/4zV8F5Qwms 2EjlEhNTaisg3BW355lbTrNc/y+8tq46aIowSncAYX4MEVNOAOmtACCgKe4RXenEfnxXl3Phata04xcwJ/4Hz+AEtuj3c= 2. MAAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeqx6MWLUNF+QBvKZjtpl24LNH74X WHd2bYmTdIBNfGdb+dwtr6xuZWcbu0s7u3f1A+PGrpOFUMmywWseoEVKPgEpuGG4GdRCGNAoHtYHwzq7efUGkey0czSdCP6FDykDNqrPVw13fciqw75Y+13f6fDykDNqrPVw13fcwqjXR+ buemxg/o8pwJnBa6qUaE8rGdIhdi5JGqP1svuqUnFlnQMJY2ScNmbu/JzIaaT2JAtsZUTPSy7WZ+V+tm5rwys+4TFKDki0+ClNBTExmd5MBTHWgSDKV2FQVQVQVQVQVQVQVQVQVQVQNBTExmd5MBTHQVgdKV2 z/ L9ZaV+ncdRhBM4hXPwoAZ1uIUGNIHBEJ7hFd4c4bw4787HorXg5DPH8EfO5w/InY10Figure 2.6: Ring cavity configuration and field labeling. Here 𝑀 0 is the front mir- mir-ror with amplitude reflectivity 𝑟 0 and transmittance 𝑡 0 , 𝑀 1 , 2 are two fixed totally reflective end mirror, 𝑀 is the movable membrane with amplitude reflectivity
Discussion
Appendix: Dispersive Hamiltonian
Appendix: Optoacoustic Hamiltonian
Input-output relation
Resonance structure
Electric field standing wave distribution
Conservative cavity Hamiltonian
The origin of this 𝑧-coordinate. is the front mirror 𝑀0 and it increases clockwise along the optical axis of the ring cavity. It becomes 𝑧𝑥 =𝐿/2+𝑥at the instantaneous position of the membrane and finally becomes 𝑧= 𝐿when the front mirror is reached again.
Appendix: Optomechanical cooling limit in the ring cavity system .1 Coupled optical and mechanical equations of motion.1Coupled optical and mechanical equations of motion
Sideband feature and optical damping
Quantum limit of mechanical occupation number
To obtain the mechanical occupation number, we need to calculate the second-order correlation function of mechanical operators. Based on all derivations above, under conditionΩ𝑚 𝛾, the mechanical occupation number defined in Eq.
PT-SYMMETRIC AMPLIFIER: BROADBAND SENSITIVITY IMPROVEMENT VIA COHERENT QUANTUM FEEDBACK
Introduction
SYMMETRICAL AMPLIFIER: BROADBAND SENSITIVITY ENHANCEMENT VIA COHERENT QUANTUM FEEDBACK . interferometer type [6]) use optical resonators to increase the interaction between the space-time voltage and the laser light field. 22] (Fig. 3.1), an additional coherent quantum feedback controller [34, 35] (activated by an unstable optomechanical oscillator) is attached to a laser interferometer to provide an "anomalous dispersion" whose negative group time delay cancels the positive group time delay in the interferometer, which achieves broadband signal amplification without increasing noise.
Figure 3.1: Coherent quantum feedback for laser interferometer GW detectors. The feedback controller (parametric amplifier) is achieved by the filter cavity pumped at 𝜔 0 + 𝜔 𝑚 and the movable mirror with oscillation frequency 𝜔 𝑚
PT-symmetric amplifier .1 Theoretical model.1Theoretical model
Connection to EP, PT-symmetry, and multi-mode generalization
Figure 3.2: Mode interaction and stability analysis. a, A weak signal sensor [mode ( 𝑎,ˆ 𝑎ˆ † ), plus ( 𝑥 , 𝑝 ) for the test mass in case of GW detectors] plus coherent quantum feedback with modes ( 𝑏,ˆ 𝑏ˆ † ) , ( 𝑐,ˆ 𝑐 ˆ † )
Comparison with the unstable amplifier
The stability of the system is jointly determined by the stability of the open-loop gain 𝐺𝑜and the condition when𝐺𝑜 =11. The feedback gain at uWLC is inherently unstable, and the instability will be inherited by the entire system; while for sWLC, the stability depends on the ratio of the feedback gain to the open-loop transfer function, i.e. the condition to reach 𝐺𝑜 = 1, which is determined by the value of 𝜒/𝜅.
Figure 3.4: The comparison between sWLC and uWLC. a, Mode interaction structure (same as Fig
Application to laser interferometer GW detectors .1 PT-symmetric amplifier for GW detection.1PT-symmetric amplifier for GW detection
BAE with effective negative mass
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Figure 3.6: Example of GW noise spectra for sWLC and uWLC (both with 𝜅 /( 2 𝜋 ) = 5 kHz , 𝜒 /( 2 𝜋 ) = 4
Application to microwave axion detectors .1 Introduction of microwave axion detector.1Introduction of microwave axion detector
Figure 3.7: Illustration of the axion search. The entire bandwidth Γ is divided into bins with width Γ 𝐴 , and to be covered by a (large) number of detectors (three shown in the figure, in terms of their 1 / 𝑆 Ψ (Ω) )
Discussion
Converting a signal recycling cavity into an astable optomechanical filter to improve the detection bandwidth of gravitational wave detectors. Phys. Overcoming the standard quantum limit in gravitational wave detectors using spin systems with negative effective mass.
Figure 3.9: Sensitivity improvement for axion detection. a, The enhancement of effective scan rate R 𝑎 achievable by sWLC (black solid lines) and uWLC (red dashed lines) axion detectors over the single-cavity one, as a function of the amplifier gain 𝜒 and
OPTOMECHANICAL REALIZATION OF PT-SYMMETRIC INTERFEROMETER
Introduction
In the original proposal [30], in addition to the filter cavity, there are several auxiliary optics, either for impedance matching with the input mirror of the arm cavity or for directing the field to the filter, leading to a rather complex arrangement. We recently realized that when the optomechanical interaction strength is less than or equal to the coupling frequency between the arm cavity and the filter cavity, the system self-stabilizes [1].
Figure 4.2: Idealized mode interaction structure of the optomechanical system illustrated in Fig
Frequency-domain analysis
Formalism
Noise spectral density
Stability analysis
4.7, where a greater amount of signal will leak into the idler channel than is output into the signal channel. At low frequencies, the signal information contained in the idler channel is even more than that contained in the signal channel.
Figure 4.5: Coupling between the optical modes and the mechanical modes rep- rep-resented in frequency domain
Time-domain analysis
Connecting the system
Step-response stability analysis
Numerical noise spectral density
The steady-state response is the most essential part of the sensitivity analysis of the system. One such test we can perform is to determine the stability of the system using the step response.
Figure 4.11: Logical flowchart of the time-domain simulation.
Optimal readout scheme
In the simulation discussed in Sect. 4.4, the signal can be extracted both from the signal channel around 𝜔0 and from the inactive channel near 𝜔0+2𝜔𝑚. 4.37), the entire noise spectrum can be exploited by any channel with better behavior at one frequency. 4.15, the free channel contributes more below 25 Hz, while the signal channel dominates at higher frequencies above 25 Hz, and the optimal mixing curve follows better at all frequencies.
Figure 4.14: Optical gain transfer functions for the signal and idler channels in the undamped, optical-spring-compensated system.
Discussion
First focused search for gravitational wave bursts from collapsing supernovae in the data of first-generation laser interferometer detectors. Conversion of conventional gravitational wave interferometers to quantum non-demolition interferometers by changing their input and/or output optics.
ATOM INTERFEROMETRY: LINEAR QUANTUM MEASUREMENT THEORY AND THE PRECISION LIMIT
Introduction
Early resonant rod GW detectors and current laser interferometer GW detectors have been extensively studied and understood using this framework of quantum measurement theory [38–40] . In this work, we set up a framework of quantum measurement theory to analyze the physics of an atomic interferometer based on the interaction between an atomic cloud and two optical fields (a passive and a control laser).
Effective Hamiltonian of an atom interferometer
Effective Hamiltonian and dynamics of an interaction kernel
Back-action noise
The first terms in brackets of r.h.s. 5.10) is much smaller than the other terms (the ratio ∼ p. Inserting this relation into the Langevin force equation 5.12) for the second interaction nucleus, and retains only those terms which due to the atomic fluctuations brought from the first interaction nucleus, we obtain the "reaction force ", acting on the atomic fields of the second interaction nucleus as:
Figure 5.2: A four-boson Raman interaction kernel and its corresponding WKB trajectory
Interferometry solution
Input-output relation
Standard quantum limit for a single atom interferometer
Let us denote the 𝜋/2 processes for the 𝐴 and 𝐵 channels to be stage-2a and stage-2b, respectively. On the optical Mach-Zender interferometer, part of the quantum noise injected at 2a and 2b stages is reflected away and left unmeasured.
Figure 5.3: Comparison between atom interferometer and optical Mach-Zender interferometer
Back-action in atom interferometer pair
It is important to note that this standard quantum limit can only be understood in terms of extrapolation. The distance between two interferometers in a real device is much larger than the length scale of the atomic interferometers themselves.
Figure 5.4: Atom interferometer gravitational wave detector configuration: this is the configuration proposed by Dimopoulos et.al
Dynamics of the effective operators: A more exact treatment
Perturbative solution to the optical fields: Effective operator for atoms Typically, in an interferometric process, the light-atom interaction time is very
The physical interpretation of the effective operators is that it describes the entire wave packet of the atomic field. Now we want to rewrite the equations of motion for the atomic fields in a more concise way using these creation and annihilation operators.
Figure 5.5: Atom-light interaction kernel. The atoms are undergoing the state swapping interaction, and during the very short interaction time (typically ∼ 𝜇 s), the A and B atoms do not have enough time to fly apart
Conceptual comparison with the laser interferometer GW detector Now we can make some comparison between Laser Interferometer GW detectorNow we can make some comparison between Laser Interferometer GW detector
It is also worth commenting on the similarity between the atom interferometer GW detector and the LISA detector. For a detailed study for the comparison between atom interferometer GW detector with LISA detector, see Refs.
Discussion
Zero point fluctuations in the test masses on LISA can be eliminated as it is also a displacement sensor. The diffraction loss will erase most of the information from the atomic cloud on the first interferometer carried by the light field.
Appendix: Field formalism for atom interferometer
Free fields
Effective interaction
Definition of field operators
Quantum states of fields
Equations of motion: Structures
It is probably instructive to study the method to mitigate these issues in the future. Since the typical atomic width is much smaller than the light pulse width, it can be approximated as almost a plane wave in the atom-light interaction region by lim𝑎→∞𝑎sinc(𝜋 𝑎𝑥) =𝛿(𝑥).
Appendix: Mean field solutions
In Gerard Auger and Eric Plagnol, editors, An Overview of Gravitational Waves: Theory, Sources and Detection, pages 285–313. Noise in gravitational wave detectors and other classical force measurements is unaffected by test mass quantization.
HIGH-PRECISION MODELING FOR GRAVITATIONAL WAVES OF BBH REMNANTS
Introduction
However, the sensitivity of proposed next-generation detectors, including the Einstein Telescope [ 53 , 54 ], Cosmic Explorer [ 55 , 56 ] and NEMO [ 57 ], will be significantly improved [ 58 ], especially at the high frequencies that open up. more options in post-merger BBH studies [25]. In this work, we study annular gravity waves and show that spin-weighted spheroidal harmonics are essential for an accurate representation of the downward spatio-temporal emission pattern [59], further confirming that the Teukol-.
Ringdown models
Coordinate frames
QNM decomposition models
Temporal and spatial dependences of the modes
Excitations of prograde and retrograde modes
QNM conventions
Beyond linear ringdowns
The most natural coordinate system for describing non-processing binaries has its ˆ𝑧 axis aligned with the direction of the orbital angular momentum. These are consistent with the fact that the direction of rotation of the black hole is clockwise as seen from the North Pole.
Figure 6.1: Convention of coordinate continuation from positive to negative values of the final spin 𝜒 𝑓
Fit memory-free NR waveforms with QNM expansions
Target waveforms and templates
The memory waveform can be calculated from those waveforms obtained by perturbation theory that do not take into account the memory effect [84, 85]. In this study, we use memoryless waveforms and show that they can be decomposed into QNM in terms of temporal and spatial distribution.
Fitting the benchmark GW150921-like binary
Strategy for choosing angular modes
Fittings results for the benchmark binary G0
Since the 𝑡trans values generally agree between 𝜒2 and the accuracy of (𝑀𝑓 ,est, 𝜒𝑓 ,est), we simply refer to the optimal distance when comparing the fitting results in the following discussions. Thus, apart from the limitation of the model itself, the fitting errors mostly come from numerical noise in the NR waveforms.
Table 6.3: SXS BBH waveforms used in Secs. 6.4–6.5.
Nonspinning binaries with different mass ratios
Once these (𝑙 , 𝑚) modes are included, the𝑆model is consistently better than the𝑌model in both𝜒2. Now we only consider the 𝑆 model since we have demonstrated that the 𝑆 model is a better representation.
Figure 6.7: The minimum distance 𝜒 2
Spinning binaries and retrograde excitation
Adding harmonics brings 𝑡trans forward, while adding (𝑙, 𝑚) modes postpones 𝑡trans — as the model becomes more accurate, the achievable 𝜒2. In this section we discussed the discriminability of the 𝑆/𝑌 model and the contribution of modes and higher order harmonics (𝑙, 𝑚).
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![Figure 3.2: Mode interaction and stability analysis. a, A weak signal sensor [mode ( 𝑎,ˆ 𝑎ˆ † ), plus ( 𝑥 , 𝑝 ) for the test mass in case of GW detectors] plus coherent quantum feedback with modes ( 𝑏,ˆ 𝑏ˆ † ) , ( 𝑐,ˆ 𝑐 ˆ † )](https://thumb-ap.123doks.com/thumbv2/123dok/10412367.0/80.918.240.679.158.701/figure-interaction-stability-analysis-detectors-coherent-quantum-feedback.webp)
