Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD2010 Limit 1 / 23

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1 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, 216 Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 1 / 23

2 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 198,1981). Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 2 / 23

3 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 198,1981) GEO6 was upgraded with a source of squeezed light in mid-21 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 GEO6 and Advanced LIGO.(See Buonanno and Chen (22) for details). and others Modifying test mass dynamics to suppress displacement noise, for example Optical Bar (an effective oscillator)(braginsky et. al (1997, 1999)). Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 3 / 23

4 Quantum Noise in Mirror In our work (Lee et al (213)), 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!! Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 4 / 23

5 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) =, where q(t) is the displacement along the z-direction from its original position at z = L. Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 5 / 23

6 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) φ + (x, z, t; L + q(t)) = for L 2 A. dk (2π) 2 dk z 1 (2π) k a k e ikt e ik x (e ikz z e ikz (z 2L 2q(t (L z))) ) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 6 / 23

7 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) = dx T zz (x, z = L, t), where T zz = 1 2 A [ ( t φ) 2 + ( z φ) 2 ( x φ) 2 ( y φ) 2]. Thus, the equation of motion for the position operator is then ( Wu & Ford 21) q(t) = 1 m t dτ τ dt A dx T zz(x ) z =L. Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 7 / 23

8 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 Π φ (E) = δ(e ω ) dt φ (x)φ + (x) α, where we have assumed that the incident field is in a single-mode coherent state, α, with frequency ω. Thus, the quantity of interest is obtained by further integration over the area located at an arbitrary z = z plane as t I T (z, t; q + L) = dt dx I (x, z, t ; q + L), where I (x) = φ (x)φ + (x). Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 8 / 23

9 The idea The measurement of I T (z, t; q + L) is to measure the effective distance between the mirror and detector. Thus, the variation of I (z, t; q + L) under small q approximation can be approximated by I T (z, t; q + L) = I T (z, t; L) + q(t (L z )) L I T (z, t; L) α + L I T (z, t; L) q(t (L z )) α (1) The overall uncertainty of the effective distance can be defined as, z = I (z, t; q + L) L I (z, t; L) α (2) The normalization factor L I (z, t; L) α is to measure the change of I (z, t) due to variation of the mirror s position. Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 9 / 23

10 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, t; L) α L I (z, t; L) 2, α (3) z 2 rp = q 2 (t) α, (4) z 2 mf = q(t) 2 ( L I ) 2 (z, t; L) α α L I (z, t; L) 2. (5) α In addition and more importantly, there exist the cross terms owing to correlation between different sources of uncertainty. z 2 1 cor = { I (t), q(t) } α + q(t) α L I (t) α L I (t) 2 { I (t), L I (t) } α + q(t) α { L I (t), q(t) } α + q(t) α L 2I (t) α L I (t) α L I (t) 2 { I (t), q(t) } α + 1 q(t) 2 α 2 L I (t) 2 { L 2 I (t), I (t)} α. (6) α Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 1 / 23

11 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 + : φ 1 φ 3 : φ 2 φ 4 + : φ 1 φ 4 : φ 2 φ 3 + : φ 2 φ 3 : φ 1 φ 4 + : φ 2 φ 4 : φ 1 φ 3 + : φ 3 φ 4 : φ 1 φ 2 + φ 1 φ 2 φ 3 φ 4 + φ 1 φ 3 φ 2 φ 4 + φ 1 φ 4 φ 2 φ 3, 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 + cross terms + φ 1 φ 2 φ 3 φ 4 + 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. Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 11 / 23

12 The square of the position uncertainty Shot noise: intrinsic quantum fluctuations of the incident field: = ( I T ) 2 α = IT 2 α I T 2 α t t dt 1 dx 1 dt 2 dx 2 φ (x 1, z, t 1 ; L)φ + (x 2, z, t 2 ; L) α = 16 α 2 Ω 8 α 2 Ω The normalization term: Thus, A 2ω sin 2 [ω (L z)] φ + (x 1, z, t 1 ; L)φ (x 2, z, t 2 ; L) t t dk dt 1 dt 2 2π e i(k ω )t 1 e i(k ω )t 2 A sin 4 [ω (L z)] t (for t 1/ω ) 2ω L I T α = 2 α 2 Ω A sin[2ω (L z)] t z 2 sn = ( I T ) 2 α L I T 2 α 1 1 Pω t 4 tan2 [ω (L z)] 1 2k sin2 [k(l z)] Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 12 / 23

13 The square of the position uncertainty Noise due to radiation pressure fluctuations: = z 2 rp = ( q) 2 α = q 2 α q 2 α t τ1 t τ2 dτ 1 dx 1 dτ 2 dx 2 ( : T zz (x 1 ) : : T zz (x 2 ) : α : T zz (x 1 ) : α : T zz (x 2 ) : α ) 1 m 2 α 2 Ω A ω2 t 3 = Pω t 3 m 2 (for t 1/ω ; (L z)) Noise from modified field fluctuations due to motion of the mirror under radiation pressure: z 2 mf = z 2 α L I T 2 L I 2 α Pω t 3 α m 2 (for t 1/ω ; (L z)) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 13 / 23

14 The square of the position uncertainty Correlation between shot noise and noise from radiation pressure fluctuations ( and noise from modified field fluctuations): 1 ( I T z α + z I T α ) L I T α z α L I T 2 ( I T L I T α + L I T I T α ) α t m 2tan[ω (L z)] or Correlation between noises of the q 2 terms: z α L I T α ( L I T q α + z L I T α ) (for t 1/ω ; (L z)) + q α L 2I T α L I T 2 ( I T z α + z I T α ) α + 1 z 2 α 2 L I T 2 ( L I T I T α + I T L I T α ) α Pω t 3 ( 9 m ) 2 tan2 [ω (L z)] (for t 1/ω ; (L z)) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 14 / 23

15 The square of the position uncertainty Correlations between various sources can be established as a consequence of interference between the incident field and the reflected field out of the mirror in the read-out. Figure: Schematic diagram of the field-mirror system. Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 15 / 23

16 The square of the position uncertainty Beating the standard quantum limit (SQL): Putting together all terms gives: Figure: a) Log-log plot of z 2 α versus Pt for various values of ζ (the distance between the mirror and the detector). The straight line corresponds to the SQL. The parameters ω m = 1 2 and P m = 1 4 are chosen. b) Log-log plot 2 of z 2 α versus Pt for various values of ω m. The straight line is the result of min z 2 P α. m = 1 4 is chosen. 2 Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 16 / 23

17 The square of the position uncertainty The equation of motion for the mirror with backreaction effects from the incident field, described as m q(t) i t dt Θ(t t ) [ F (t), F (t ) ] α q(t ) F (t) α = ξ α. (7) The mean radiation pressure is given by F (t) α = α 2 { 4π 3 ω A 1 cos [ 2ω (t L) 2ϕ ]}, (8) The backreaction term is an expression of: i t dt Θ(t t ) [ F (t), F (t ) ] α q(t ) (9) = α 2 2π 3 ω A sin [ ω (t L) ϕ ] { ω cos [ ω (t L) ϕ ] q(t) + sin [ ω(t L) ϕ ] } q(t), The backreaction effect depends on the position/velocity of the mirror with time-dependent coefficients that might enhance the response to perturbations to improve its sensitivity. (Braginsky et. al (1997, 1999)) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 17 / 23

18 Squeezing All interferometer configurations can benefit from squeezing quantum states. Consider the effects from electromagnetic squeezed vacuum on a particle: The plane-wave expansion of the vector potential is of the form A T (x) = d 3 k 1 (2π) 3/2 2ω λ=1,2 ˆɛ λ k a λ k e ik x iωt + h.c., (1) with ω = k, and the polarization unit vectors ˆɛ λ k. The squeezed vacuum states can be constructed from the normal vacuum state through the squeeze operator, [ ζ ζk = S(ζ k ). S(ζ k ) = exp k 2 a2 λ k ζ ] k 2 a 2 λ k,, The squeeze parameter ζ k = r k e iθ k is an arbitrary complex number with r k and θ k R where µ k = coshr k, ν k = sinhr k e iθ k, and η k = ν k. Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 18 / 23

19 Squeezing The influence from the quantum field is expected to give an effect to the velocity uncertainty of the particle (Lee et. al.(212)). The velocity dispersion of a nonrelativistic particle due to electromagnetic field fluctuations under the dipole approximation is expressed as t t vi 2 (t) E = e2 m 2 du du E i (, u)e i (, u ) ζ. (11) t i t i Let Ξ and be the mean frequency and width of the band, respectively, and suppose the wave vectors are distributed over small solid angle dω s about a certain direction. Thus Ξ+ /2 δ vi 2 (t) ξ = e2 [ ] m 2 A(dΩ s) dω 4ω η 2 + µη cos(ωt θ) sin 2 ωt 2. Ξ+ /2 Ξ /2 Ξ /2 [ ] dω ω η η + µ cos(ωt θ) sin 2 ωt 2 = Ξ [ ] 4 η 2η µ cos θ + the terms 1/t 2. Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 19 / 23

20 Squeezing The saturated value is as δh vi2 (t)ie 1 e2 2 = 2 A(dΩs ) (2Ξ ) η µη cos θ. m 2 R(r, θ) = 2η 2 µη cos θ, (12) (13) can be negative by choosing the proper value of squeezing parameters with Rmin = (2 3)/2, namely, that the observed velocity dispersion is smaller in the electromagnetic squeezed vacuum background than in the normal vacuum background, leading to the subvacuum effect Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 2 / 23

21 Squeezing Perhaps we can design an experiment on small scales to examine the above results!! Then the design of new subsystems for noise reduction based upon what we have learned and the method we developed is anticipated!! Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 21 / 23

22 Collaborations Collaborators Jen-Tsung Hsiang (Fudan University) Sun-Kun King (Institute of Astronomy and Astrophysics, AS) Tai-Hung Wu (Former Ph.D student, National Dong Hwa University) Chun-Hsien Wu (Department of Physics, Soochou University) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 22 / 23

23 References References W. G. Unruh, in Quantum Optics, Experimental Gravitation, and Measurement Theory, edited by P. Meystre and M. O. Scully (Plenum, New York, 1982), p C. M. Caves, Rev. Lett. 45, 75 (198); Phys. Rev. D 23, 1693 (1981). A. Buonanno and Y. Chen, Phys. Rev. D 64, 426 (21); ibid, 65, 421 (22). V. B. Braginsky, M. L. Gorodetsky, and F. Ya. Khalili, Phys. Lett. A 232, 34 (1997); V. B. Braginsky and F. Ya. Khalili, ibid. 257, 241 (1999). C.-H. Wu and L.H. Ford, Phys. Rev. D 64, 451 (21). C.-H. Wu and D.-S. Lee, Phys. Rev. D71, 1255 (25). J.-T. Hsiang, T-H. Wu, and D.-S. Lee, Phys. Rev. D 77, 1521 (28). T-H. Wu, J-T. Hsiang, and D.-S. Lee, Annals Phys. 327, 522 (212). J.-T. Hsiang, T.-H. Wu, D.-S. Lee, S.-K. King, and C.-H. Wu, Annals Da-Shin Phys. Lee 329, (NDHU) 28-5 (213). Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum IARD21 Limit 23 / 23

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