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

▶
quantum noise in mirror field systems beating the
SMART_READER_LITE
LIVE PREVIEW

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

Dec 13, 2023 168 likes •406 views

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

slide-1
SLIDE 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, 2016

Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 1 / 23

slide-2
SLIDE 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 1980,1981).

Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 2 / 23

slide-3
SLIDE 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 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

slide-4
SLIDE 4

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

  • ne 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

slide-5
SLIDE 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) = 0 , where q(t) is the displacement along the z-direction from its original position at z = L.

Figure: Schematic diagram of the field-mirror system.

Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 5 / 23

slide-6
SLIDE 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)) =

  • dk

(2π)2 ∞ dkz (2π) 1 k ak e−ikt ×eik·x(eikzz − e−ikz(z−2L−2q(t−(L−z)))) for L2 ≫ A.

Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 6 / 23

slide-7
SLIDE 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) =

  • A

dx Tzz(x, z = L, t) , where Tzz = 1 2

  • (∂tφ)2 + (∂zφ)2 − (∂xφ)2 − (∂yφ)2

. Thus, the equation of motion for the position operator is then ( Wu & Ford 2001) q(t) = 1 m t dτ τ dt′

  • A

dx′

Tzz(x′) |z′=L .

Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 7 / 23

slide-8
SLIDE 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(E1 → E2(E1 < E2)) =| E2 | monopole operator | E1 |2 ×Πφ(E2 − E1) . The response function is defined as Πφ(E) = δ(E − ω0)

  • dtφ−(x)φ+(x)α ,

where we have assumed that the incident field is in a single-mode coherent state, α, with frequency ω0. Thus, the quantity of interest is obtained by further integration over the area located at an arbitrary z = z plane as IT(z0, t; q + L) = t dt′

  • dx I(x, z0, t′; q + L) ,

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

slide-9
SLIDE 9

The idea

The measurement of IT(z0, t; q + L) is to measure the effective distance between the mirror and detector. Thus, the variation of I(z0, t; q + L) under small q approximation can be approximated by ∆IT(z0, t; q + L) = ∆IT(z0, t; L) + ∆q(t − (L − z0))∂LIT(z0, t; L)α +∆∂LIT(z0, t; L)q(t − (L − z0))α (1) The overall uncertainty of the effective distance can be defined as, ∆z = ∆I(z0, t; q + L) | ∂LI(z0, t; L) α| (2) The normalization factor ∂LI(z0, t; L) α is to measure the change of I(z0, 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

slide-10
SLIDE 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, ∆z2sn = ∆I 2(z0, t; L) α ∂LI(z0, t; L) 2

α

, (3) ∆z2rp = ∆q2(t)α , (4) ∆z2mf = q(t) 2

α

∆(∂LI)2(z0, t; L) α ∂LI(z0, t; L) 2

α

. (5) In addition and more importantly, there exist the cross terms owing to correlation between different sources of uncertainty. ∆z2cor = 1 ∂LI(t) α

  • ∆I(t) , ∆q(t)
  • α + q(t) α

∂LI(t) 2

α

  • ∆I(t) , ∆∂LI(t)
  • + q(t) α

∂LI(t) α

  • ∆∂LI(t) , ∆q(t)
  • α + q(t) α ∂2

LI(t) α

∂LI(t) 2

α

  • ∆I(t) , ∆q(t)
  • + 1

2 q(t)2

α

∂LI(t) 2

α

  • ∆∂2

LI(t) , ∆I(t)

  • α .

(6)

Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 10 / 23

slide-11
SLIDE 11

The square of the position uncertainty

To compute the square of the position uncertainty, we may use the follow identity: φ1φ2φ3φ4 = : φ1φ2φ3φ4 : + : φ1φ2 : φ3φ40+ : φ1φ3 : φ2φ40 + : φ1φ4 : φ2φ30+ : φ2φ3 : φ1φ40+ : φ2φ4 : φ1φ30 + : φ3φ4 : φ1φ20 + φ1φ20φ3φ40 + φ1φ30φ2φ40 + φ1φ40φ2φ30 , 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, φ1φ2φ3φ4α − φ1φ2αφ3φ4α = : φ1φ3 :αφ2φ40 + cross terms + φ1φ20φ3φ40 + 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

slide-12
SLIDE 12

The square of the position uncertainty

Shot noise: intrinsic quantum fluctuations of the incident field: (∆IT)2α = I 2

Tα − IT2 α

= t dt1

  • dx1

t dt2

  • dx2 φ−(x1, z, t1; L)φ+(x2, z, t2; L)α

×φ+(x1, z, t1; L)φ−(x2, z, t2; L)0 = 16| α |2 Ω A 2ω0 sin2[ω0(L − z)] t dt1 t dt2 ∞ dk 2π 1 2k sin2[k(L − z)] ×e−i(k−ω0)t1ei(k−ω0)t2 ≈ 8| α |2 Ω A 2ω0 sin4[ω0(L − z)] × t (for t ≫ 1/ω0) The normalization term: ∂LITα = 2| α |2 Ω A sin[2ω0(L − z)] × t Thus, ∆z2sn = (∆IT)2α ∂LIT2

α

≈ 1 Pω0t 1 4tan2[ω0(L − z)] ≥ 0

Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 12 / 23

slide-13
SLIDE 13

The square of the position uncertainty

Noise due to radiation pressure fluctuations: ∆z2rp = (∆q)2α = q2α − q2

α

= t dτ1 τ1

  • dx1

t dτ2 τ2

  • dx2

(: Tzz(x1) : : Tzz(x2) :α − : Tzz(x1) :α: Tzz(x2) :α) ≈ 1 m2 | α |2 Ω A ω2

0 × t3

= Pω0t3 m2 ≥ 0 (for t ≫ 1/ω0; (L − z)) Noise from modified field fluctuations due to motion of the mirror under radiation pressure: ∆z2mf = z2

α

∂LIT2

α

∆∂LI2

α ≈ Pω0t3

m2 ≥ 0 (for t ≫ 1/ω0; (L − z))

Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 13 / 23

slide-14
SLIDE 14

The square of the position uncertainty

Correlation between shot noise and noise from radiation pressure fluctuations ( and noise from modified field fluctuations): 1 ∂LITα (∆IT ∆zα + ∆z ∆ITα) ≈ zα ∂LIT2

α

(∆IT ∆∂LITα + ∆∂LIT ∆ITα) ≈ t m2tan[ω0(L − z)] ≤ or ≥ 0 (for t ≫ 1/ω0; (L − z)) Correlation between noises of the q2 terms: zα ∂LITα (∆∂LIT ∆qα + ∆z ∆∂LITα) +qα∂2

LITα

∂LIT2

α

(∆IT ∆zα + ∆z ∆ITα) +1 2 z2

α

∂LIT2

α

(∆∂LIT ∆ITα + ∆IT ∆∂LITα) ≈ Pω0t3 m2 9 2 − 3 2tan2[ω0(L − z)]

  • (for t ≫ 1/ω0; (L − z))

Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 14 / 23

slide-15
SLIDE 15

The square of the position uncertainty

Correlations between various sources can be established as a consequence

  • f 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.

Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 15 / 23

slide-16
SLIDE 16

The square of the position uncertainty

Beating the standard quantum limit (SQL): Putting together all terms gives:

  • 1

1 2 3

  • 5
  • 4
  • 3
  • 2
  • 1

1 1.0 1.5 2.0 2.5 3.0

  • 3
  • 2
  • 1

1 2

Figure: a) Log-log plot of ∆z2 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 ω0

m = 10−2 and P m2 = 10−4 are chosen. b) Log-log plot

  • f ∆z2 α versus

√ Pt for various values of ω0

m . The straight line is the result of

min ∆z2 α.

P m2 = 10−4 is chosen.

Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 16 / 23

slide-17
SLIDE 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 = ξα .

(7) The mean radiation pressure is given by F(t)α = |α|2 4π3 ω0A

  • 1 − cos
  • 2ω0(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 2π3 ω0A sin

  • ω0(t − L) − ϕ

ω0 cos

  • ω0(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))

Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 17 / 23

slide-18
SLIDE 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 AT(x) =

  • d3k

(2π)3/2 1 √ 2ω

  • λ=1,2

ˆ ǫλ k aλ k eik·x−iωt + h.c. , (10) with ω = |k| , and the polarization unit vectors ˆ ǫλ k. The squeezed vacuum states can be constructed from the normal vacuum state through the squeeze operator, |0ζk = S(ζk) |0 . S(ζk) = exp ζ∗

k

2 a2

λ k − ζk

2 a† 2

λ k

  • , ,

The squeeze parameter ζk = rk eiθk is an arbitrary complex number with rk ≥ 0 and θk ∈ R where µk = coshrk, νk = sinhrkeiθk, and ηk = |νk|.

Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 18 / 23

slide-19
SLIDE 19

Squeezing

The influence from the quantum field is expected to give an effect to the velocity uncertainty of the particle (Lee et. al.(2012)). The velocity dispersion of a nonrelativistic particle due to electromagnetic field fluctuations under the dipole approximation is expressed as ∆v2

i (t)E = e2

m2 t

ti

du t

ti

du′ Ei(0, u)Ei(0, u′)ζ . (11) Let Ξ and ∆ be the mean frequency and width of the band, respectively, and suppose the wave vectors are distributed over small solid angle dΩs about a certain direction. Thus δ∆v2

i (t)ξ = e2

m2 A(dΩs) Ξ+∆/2

Ξ−∆/2

dω 4ω

  • η2 + µη cos(ωt − θ)
  • sin2 ωt

2 . Ξ+∆/2

Ξ−∆/2

dω ω η

  • η + µ cos(ωt − θ)
  • sin2 ωt

2 = ∆Ξ 4 η

  • 2η − µ cos θ
  • + the terms ∝ 1/t2 .

Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 19 / 23

slide-20
SLIDE 20

Squeezing

The saturated value is as δ∆v2

i (t)E = e2

m2 A(dΩs) (2Ξ∆)

  • η2 − 1

2 µη cos θ

  • .

(12) R(r, θ) = 2η2 − µη cos θ , (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.

0.2 0.4 0.6 2 0.5 0.0 0.5 1.0

Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 20 / 23

slide-21
SLIDE 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!!

Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 21 / 23

slide-22
SLIDE 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)

Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 22 / 23

slide-23
SLIDE 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. 647.

◮ C. M. Caves, Rev. Lett. 45, 75 (1980); Phys. Rev. D 23, 1693

(1981).

◮ A. Buonanno and Y. Chen, Phys. Rev. D 64, 042006 (2001); ibid,

65, 042001 (2002).

◮ V. B. Braginsky, M. L. Gorodetsky, and F. Ya. Khalili, Phys. Lett. A

232, 340 (1997); V. B. Braginsky and F. Ya. Khalili, ibid. 257, 241 (1999).

◮ C.-H. Wu and L.H. Ford, Phys. Rev. D 64, 045010 (2001). ◮ C.-H. Wu and D.-S. Lee, Phys. Rev. D71, 125005 (2005). ◮ J.-T. Hsiang, T-H. Wu, and D.-S. Lee, Phys. Rev. D 77, 105021

(2008).

◮ T-H. Wu, J-T. Hsiang, and D.-S. Lee, Annals Phys. 327, 522 (2012). ◮ J.-T. Hsiang, T.-H. Wu, D.-S. Lee, S.-K. King, and C.-H. Wu, Annals

  • Phys. 329, 28-50 (2013).

Da-Shin Lee (NDHU) Quantum Noise in Mirror-Field Systems: Beating the Standard Quantum Limit IARD2010 23 / 23