Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum - PowerPoint PPT Presentation

Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit Da-Shin Lee Department of Physics National Dong Hwa University Hualien,Taiwan Presentation to Workshop on Gravitational Wave activities in Taiwan (GWTW) Institute of Physics, Academia Sinica January 15, 2016 Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 1 / 23

Quantum Noise in Interferometer SQL of Quantum Noise in Interferometer ◮ Quantum noise in a laser interferometer detector arises from the quantum nature of the light directly via the photon number √ fluctuations (shot noise ∝ 1 / N (N: number of photons)) or indirectly via random motion of the mirror under a fluctuating force ( √ radiation pressure fluctuations ∝ N ). ◮ To minimize the uncertainty from the sum of two uncorrelated nose effects may give the SQL when an input power is appropriately chosen.( Caves 1980,1981). Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 2 / 23

Quantum Noise in Interferometer The ideas to reduce quantum noise for potential upgrades in GW interferometer detector include: ◮ Manipulating ubiquitous quantum field vacuum: Squeezed vacuum is injected into the dark port of the beam splitter to improve the sensitivity. (Caves 1980,1981) GEO600 was upgraded with a source of squeezed light in mid-2010 and has since been testing it under operating conditions. ◮ Modifying input-output fields to enhance the signals and also establish the correlation between Shot Noise and Radiation Pressure fluctuations for noise reduction such as signal recycle employed in GEO600 and Advanced LIGO.(See Buonanno and Chen (2002) for details). and others ◮ Modifying test mass dynamics to suppress displacement noise, for example Optical Bar (an effective oscillator)(Braginsky et. al (1997, 1999)). Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 3 / 23

Quantum Noise in Mirror In our work (Lee et al (2013)), ◮ We try to find out a fundamental mechanism to establish the correlation between Shot Noise (intrinsic quantum fluctuations of light sources) and Radiation Pressure fluctuations (induced noise due to the movement of a mirror) for reducing the net noise effect that might need to mix the incident field with the reflected field from the mirror. ◮ We provide a consistent approach (quantum field theory) that on the one hand naturally incorporates all sources of noise on the mirror from the quantum field and (manipulated) quantum field vacuum fluctuations as well, and on the other hand allows us to derive a dynamical equation to account for backreaction effects from the (incident) quantum field to the mirror in a consistent manner. We will give an example!! Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 4 / 23

Quantum Noise in Mirror We consider that a single mirror with perfect reflection is illuminated by a massless quantum scalar field propagating along the z direction that gives motion of the mirror. The mirror of mass m and area A is originally placed at the z = L plane. Thus, the boundary condition on the field evaluated at the mirror surface can be expressed in the specific form: φ ( x � , z = L + q ( t ) , t ) = 0 , where q ( t ) is the displacement along the z -direction from its original position at z = L . Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 5 / 23 Figure: Schematic diagram of the field-mirror system.

The idea The model The approximate solution to the field equation subject to the boundary condition if we assume slow motion ( ˙ q ≪ 1) is that φ = φ + + φ − , where the positive (negative) energy solution φ + ( φ − ) is respectively given by (Unruh 1982) � ∞ d k � 1 � dk z φ + ( x � , z , t ; L + q ( t )) k a k e − ikt = (2 π ) 2 (2 π ) 0 × e i k � · x � ( e ik z z − e − ik z ( z − 2 L − 2 q ( t − ( L − z ))) ) for L 2 ≫ A . Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 6 / 23

The idea The force acting on the mirror is given by the area integral of the z − z component of the stress tensor in terms of the field operators: � F ( t ) = d x � T zz ( x � , z = L , t ) , A where T zz = 1 ( ∂ t φ ) 2 + ( ∂ z φ ) 2 − ( ∂ x φ ) 2 − ( ∂ y φ ) 2 � � . 2 Thus, the equation of motion for the position operator is then ( Wu & Ford 2001) � t � τ q ( t ) = 1 � dt ′ d x ′ � T zz ( x ′ ) | z ′ = L . d τ m 0 0 A Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 7 / 23

The idea Here we assume that the scalar particle detection is based upon the processes of stimulated absorption by the detector due to the coupling between the scalar field and the monopole moment of the detector. The transition rate between states of the detector can be given by P ( E 1 → E 2 ( E 1 < E 2 )) = | � E 2 | monopole operator | E 1 � | 2 × Π φ ( E 2 − E 1 ) . The response function is defined as � dt � φ − ( x ) φ + ( x ) � α , Π φ ( E ) = δ ( E − ω 0 ) where we have assumed that the incident field is in a single-mode coherent state, α , with frequency ω 0 . Thus, the quantity of interest is obtained by further integration over the area located at an arbitrary z = z plane as � t � dt ′ d x � I ( x � , z 0 , t ′ ; q + L ) , I T ( z 0 , t ; q + L ) = 0 where I ( x ) = φ − ( x ) φ + ( x ) . Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 8 / 23

The idea The measurement of I T ( z 0 , t ; q + L ) is to measure the effective distance between the mirror and detector. Thus, the variation of I ( z 0 , t ; q + L ) under small q approximation can be approximated by ∆ I T ( z 0 , t ; q + L ) = ∆ I T ( z 0 , t ; L ) + ∆ q ( t − ( L − z 0 )) � ∂ L I T ( z 0 , t ; L ) � α +∆ ∂ L I T ( z 0 , t ; L ) � q ( t − ( L − z 0 )) � α (1) The overall uncertainty of the effective distance can be defined as, ∆ z = ∆ I ( z 0 , t ; q + L ) (2) |� ∂ L I ( z 0 , t ; L ) � α | The normalization factor � ∂ L I ( z 0 , t ; L ) � α is to measure the change of I ( z 0 , t ) due to variation of the mirror’s position. Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 9 / 23

The idea They respectively come from the shot noise (sn) contribution associated with intrinsic fluctuations of the incident coherent fields, and the contributions of radiation pressure fluctuations (rp) and modified field fluctuations (mf), both of which are induced by the mirror’s motion, � ∆ z 2 � sn = � ∆ I 2 ( z 0 , t ; L ) � α , (3) � ∂ L I ( z 0 , t ; L ) � 2 α � ∆ z 2 � rp = � ∆ q 2 ( t ) � α , (4) � ∆( ∂ L I ) 2 ( z 0 , t ; L ) � α � ∆ z 2 � mf = � q ( t ) � 2 . (5) α � ∂ L I ( z 0 , t ; L ) � 2 α In addition and more importantly, there exist the cross terms owing to correlation between different sources of uncertainty. 1 � α + � q ( t ) � α � ∆ z 2 � cor = � � � � � ∆ I ( t ) , ∆ q ( t ) � ∆ I ( t ) , ∆ ∂ L I ( t ) � ∂ L I ( t ) � 2 � ∂ L I ( t ) � α α � α + � q ( t ) � α � ∂ 2 + � q ( t ) � α L I ( t ) � α � � � � � ∆ ∂ L I ( t ) , ∆ q ( t ) � ∆ I ( t ) , ∆ q ( t ) � � ∂ L I ( t ) � 2 � ∂ L I ( t ) � α α � q ( t ) � 2 + 1 α ∆ ∂ 2 � � � L I ( t ) , ∆ I ( t ) � α . (6) � ∂ L I ( t ) � 2 2 Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 10 / 23 α

The square of the position uncertainty To compute the square of the position uncertainty, we may use the follow identity: φ 1 φ 2 φ 3 φ 4 = : φ 1 φ 2 φ 3 φ 4 : + : φ 1 φ 2 : � φ 3 φ 4 � 0 + : φ 1 φ 3 : � φ 2 φ 4 � 0 + : φ 1 φ 4 : � φ 2 φ 3 � 0 + : φ 2 φ 3 : � φ 1 φ 4 � 0 + : φ 2 φ 4 : � φ 1 φ 3 � 0 + : φ 3 φ 4 : � φ 1 φ 2 � 0 + � φ 1 φ 2 � 0 � φ 3 φ 4 � 0 + � φ 1 φ 3 � 0 � φ 2 φ 4 � 0 + � φ 1 φ 4 � 0 � φ 2 φ 3 � 0 , The first term is fully normal ordered term, the next six terms are cross terms and the final three terms are pure vacuum terms. For a coherent state, � φ 1 φ 2 φ 3 φ 4 � α − � φ 1 φ 2 � α � φ 3 φ 4 � α = � : φ 1 φ 3 : � α � φ 2 φ 4 � 0 + cross terms + � φ 1 φ 2 � 0 � φ 3 φ 4 � 0 + pure vacuum terms The fully normal terms cancel. However, cross terms and pure vacuum terms involve quantum field vacuum fluctuations, and may give singularity as the fields in the end will be evaluated in the same point, and the finite results can be obtained by finding its principle values. Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 11 / 23