True and apparent motion of optomechanical resonators, with applications to feedback cooling of gravitational wave detector test masses

Abstract

Modern optomechanical systems employ increasingly sophisticated quantum-mechanical states of light to probe and manipulate mechanical motion. Squeezed states are now used routinely to enhance the sensitivity of gravitational-wave interferometers to small external forces, and they are also used in feedback-based trapping and damping experiments on the same interferometers to enhance the achievable cooling of fluctuations in the differential test mass mode (arXiv:2102.12665). In this latter context, an accurate accounting of the true test mass motion, incorporating all sources of loss, the effect of feedback control, and the influence of classical force and sensing noises, is paramount. We work within the two-photon formalism to provide such an accounting, which extends a previously described decomposition of the quantum-mechanical noise of the light field (arxiv:2105.12052). This decomposition provides insight, rooted in physically motivated parameters, into the optimal squeezed state and feedback control configuration that should be employed to achieve the lowest fluctuations. We apply this formalism to feedback damping experiments in current and possible future gravitational-wave interferometers -- LIGO A+, LIGO Voyager, Cosmic Explorer (CE), and CE Voyager -- and discuss how these multi-degree-of-freedom systems might be compared to a single degree-of-freedom oscillator. We find that, for the oscillator definition used most commonly in the literature so far, occupation numbers below 1 are possible in these interferometers over a frequency range comparable to the bandwidth of the trapped and cooled oscillator. We also discuss several technical issues in cooling experiments with gravitational-wave detectors

Figures (8)

• Figure 1: A Fabry--Perot Michelson interferometer with power recycling and signal extraction (left) underlying a simplified three-mirror coupled cavity model (right) for our study of true and apparent motion under measurement-based feedback control. In a Fabry--Perot Michelson interferometer, the two arm cavities are pumped with a coherent optical carrier field (solid lines) from a laser on the bright port side of the beamsplitter. Our effective three-mirror description does not include such a port, so we simply assume the presence of an optical carrier field in the arm that does not propagate into the signal extraction cavity or out the dark port. Optical sidebands are extracted from the arms, through the partially transmissive signal extraction cavity, and to the photodetector readout at the interferometer's dark port. (The readout uses balanced homodyne detection; the coherent local oscillator field for this is not drawn.) The modeling of this system and the relevant optical fields, including loss fields not shown in this diagram, is described in the text and shown in the signal flow graph in \ref{['fig:signal_flow_full']}. A squeezed vacuum source and a two-mirror filter cavity are used to injected squeezed vacuum sidebands (dashed lines) into the coupled cavity via an optical circulator. The photodetector output is used as an error signal to apply a feedback control force to the end mirror of the arm cavity; in gravitational-wave detectors, this feedback control force is typically applied by electrostatic or magnetic actuation.
• Figure 2: Signal flow graph of a dual-recycled Fabry--Perot Michelson inteferometer with squeezed light injection. The system is modeled as a three mirror coupled cavity, with an end test mass (ETM) and input test mass (ITM) comprising a Fabry--Perot cavity, and with a signal extraction mirror (SEM) that broadens the cavity bandwidth. (A diagram is shown in \ref{['fig:sec_diagram']}.) The measurement $y$ is read out in reflection of the interferometer, while the true motion $x$ can only be inferred and cannot be directly measured in an experiment. (Freerunning, or apparent, motion $x_\text{free}$ produces the same excitations as $\chi_0F_\text{ext}$.) The squeezed vacuum state $\vec{\mathbb{s}}$ is injected into the interferometer after being reflected off of a Fabry--Perot filter cavity. For simplicity, the filter cavity has already been algebraically reduced to a reflection matrix $\mathbb{H}_\text{fc}$ and noise transmission matrix $\mathbb{T}_\text{fc}$. The measurement $y$ is passed through a control filter $C$ and fed back to the end test mass as a feedback force, which is summed into the same point as $F_\text{ext}$. Note that quantum radiation pressure and shot noise are sourced by the squeezed field $\vec{\mathbb{s}}$ and cannot be treated as entering with $F_\text{ext}$ or $x_\text{sens}$. The various sources of unsqueezed vacuum $\{\vec{\mathbb{a}}_\mu\}$, responsible for the optical losses, are also shown.
• Figure 3: Reduced signal flow graph for the optomechanical system in \ref{['fig:signal_flow_full']}. A further reduced representation can be made by eliminating the feedback control path labeled "$C$" and then replacing all quantities suffixed with "om" by their counterparts suffixed by "eff" (\ref{['eq:controled_tfs']}). The additional loss fields $\{\vec{\mathbb{a}}_\mu\}$ have been omitted from this graph.
• Figure 4: An example budget of the physical test mass motion for an oscillator in LIGO A+ trapped and cooled with measurement-based feedback control, with frequency-independent squeezed light (no filter cavity). The frequency-independent squeeze angle $\phi$ is adjusted to counteract the squeezed-state rotation $\theta_x$ near the resonance frequency $\Omega_\text{eff}$, which results in the null in the budgeted antisqueezing. The injected squeeze amplitude $r$ is tuned so that the dephasing, which increases with $r$, is roughly equal to the injected squeezing, which decreases with $r$. The two largest contributions to the true motion $x$ near the resonance are the quantum noise entering the detector's dark port and the classical sensing noise from the Brownian motion of the test mass coatings. Referring back to \ref{['fig:signal_flow_reduced']}, the quantum noise, labeled by $\vec{\mathbb{s}}$, contributes to the true motion via the transfer function $\vec{\mathbb{D}}^\dag_\text{eff}$, while the coating noise, part of the total sensing noise $x_\text{sens}$, contributes to the true motion via the transfer function $X_\text{eff}$. These are distinct from the transfer functions ($\vec{\mathbb{v}}^\dag \mathbb{H}_\text{eff}$ and $\vec{\mathbb{v}}^\dag \vec{\mathbb{Y}}_\text{eff}$, respectively) that propagate these noises into the measurement record $y$, which is in turn used to characterize the interferometer sensitivity when used for detecting gravitational waves; therefore, the relative magnitudes of the noises that form the budget of the true motion $x$ differ in general from the relative magnitudes of the noises in a budget of the measurement $y$.
• Figure 5: An example budget of the physical test mass motion for a trapped and cooled oscillator in a cryogenic silicon Cosmic Explorer, with frequency-independent squeezing.