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Entanglement Dynamics and Noise Control in Test Masses for Quantum Gravitational Measurements

This study explores the intricate entanglement relationships between test masses in gravitational wave experiments, focusing on the mathematical frameworks for assessing entanglement via logarithmic negativity metrics. It highlights the role of radiation pressure and classical forces in maintaining entanglement, even under the influence of noise from laser systems. The use of optimal filtering techniques and homodyne detection strategies is discussed to enhance the fidelity of measurements. It aims to verify states through continuous measurement while ensuring robust performance against classical noise, paving the way for advancements in quantum sensing technologies.

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Entanglement Dynamics and Noise Control in Test Masses for Quantum Gravitational Measurements

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  1. Entanglement between test masses Helge Müller-Ebhardt, Henning Rehbein, Kentaro Somiya, Stefan Danilishin, Roman Schnabel, Karsten Danzmann, Yanbei Chen and the AEI-Caltech-MIT MQM discussion group Max-Planck-Institut für Gravitationsphysik (AEI) Institut für Gravitationsphysik, Leibniz Universität Hannover TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

  2. Compare with optics: Entangled output Input squeezing Entanglement between end mirrors Total system: Logarithmic negativity [Vidal, Werner] a quantitative measure of entanglement:

  3. Radiation pressure & classical forces. Conditional second-order moments not changed by any linear feedback control! Shot & classical sensing noise. Conditioning on continuous measurement Each (common and differential) mode: Prepare state by collecting data and finding optimal filter function. Need verification stage? → Stefan’s talk.

  4. known unknown Wiener Filtering Conditional moments of Gaussian state: Steady state!

  5. Differently squeezed for different . Homodyne detection angle Equality for:  = 0, Conditional test-mass state x / F > 2 → non-zero frequency band in which classical noise is completely below the SQL!

  6. c = 11.5 F, d = 0.87 F 5 dB c = 7.0 F, d = 0.8 F 10 dB c = 10.6 F, d = 0.3 F Optimal: Detect both modes at » -  / 2 → then independent from laser noise! x / F x / F- threshold larger than 2 for entanglement. x / F Constrains on conditional mirror entanglement [arXiv:quant-ph/0702258] Laser noise level. Common mode detection at » -  /2. Detection close to amplitude quadrature (» -  /2) → modes more squeezed – but at same time even more anti-squeezed!

  7. No laser noise! Conditional mirror entanglement x / F = 10 x / F = 8 Log [ d / F ] Log [ d / F ] Log [ c / F ] Log [ c / F ] x / F = 6 x / F = 4 Log [ d / F ] Log [ d / F ] Log [ c / F ] Log [ c / F ]

  8. No laser noise! F Lifetime c = 11.5 F, d = 0.87 F x / F c = 7.0 F, d = 0.8 F x / F = 5.6 x Survival of conditional mirror entanglement Second-order moments at t =  conditioned on measurement at t < 0: In “rotating frame”. Erase, if laser turned off at t = 0. x / F = 10

  9. Probe Detector Filtered output x / F = 10 decreasing  BAC Controlled test-mass state With optimal controller in terms of conditional second-order moments: • Controlled state always more mixed than conditional state. • Test masses fixed → can do simultaneous verification. • Weak measurement optimal for low noise - but phase transition. • Less mixed at quadrature close to amplitude → back-action-compensating (BAC).

  10. c = 18.3 F, d = 2.05 F c = 22.3 F, d = 2.07 F x / F Constrains on controlled mirror entanglement • Much higher demand on classical noise SQL-beating, i.e. higher x/F. • Optimal detection at » / 3. • More power needed: here even d > F.

  11. SQL Careful with low frequency noise! x / F» 10 Sub-SQL classical noise Noise budget advanced LIGO: Noise budget CLIO: [Report LIGO-060056-07-M] Herex»F» 30 Hz → ratio far too small!!! [Miyoki et al., Class. Quantum Grav. 21, S1173 (2004)]

  12. Hannover prototype • 10 m prototype with 100 g to 1 kg end mirrors. • Low mechanical loss (silica) in end mirrors and thin coating/ or just gratings. • Cool mirrors down to 20 K? • Inject squeezed input! • High power and frequency stabilized laser system… • Implement double readout or entangle differential motion of a pair of coherently operated interferometers. • → Need rigorous study of noise model.

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