Direct observations of thermal fluctuations at sub-shot noise levels

Direct observations of thermal fluctuations at sub-shot noise levels 2012 T. Mitsui, K.A., Phys. Rev. E 80, 020602(R)-1 4 (2009) Phys. Rev. E 86, 011602 (2012) arxiv:1206.3828

Contents 1. Thermal fluctuations 2. Liquid surface fluctuations and their properties 3. Shot noise 4. Noise reduction 5. Surface height fluctuation measurements 6. Surface inclination fluctuation measurements 7. Discussion 1

Thermal fluctuations Thermal fluctuation phenomena: surfaces, Brownian motion, liquid surfaces, solid Thermal fluctuations are ubiquitous! Directly observe thermal fluctuations of common materials in an everyday environment. Fluctuations are small! noise! Measure at hitherto unseen low noise levels Test theory to previously unachieved precision. Observe new phenomena. 2

Liquid surface fluctuations Ripplons Typical thermal fluctuation phenomena which is relatively well understood. Liquid surfaces fluctuate thermally. (von Schmoluchowski, 1908; Mandelstam, 1913) Experimentally observed effectively as gratings. (Katyl, Ingard, 1967) Will not work for viscous liquids Size of fluctuations atomic scale z 2 k BT 4πσ log σ (0.4 nm)2 (σ gρl2 : surface tension, l: molecular scale) 3

Properties of thermal surface fluctuations Surface fluctuation properties determined from hydrodynamics. Interesting fundamental physics. Thermal fluctuation spectral properties known. Important test! Spectral function: (Bouchiat, Meunier, 1971) Analytic, but behavior not simple. (ρ: density, σ: surface tension, η: viscosity) P (k, ω) = k BT πω kτ 2 0 ρ Im [ (1 + s) 2 + y 1 + 2s ] 1 s iωτ 0, τ 0 ρ σρ 2ηk2, y 4η 2 k 4

P(k,2πf) [m 4 /Hz] 10-30 10-35 10-40 10-45 10-50 10 3 10 4 10 5 10 6 10 7 10 8 k [m -1 ] Low dissipation η 3 ω/(ρσ 2 ) < 1/(8 2) Compare with simple analytic behavior (black, dotted) 2 regions. Peak behavior disappears for higher ω Viscous f = 10 3, 10 5, 10 7 [s 1 ] P (k, ω) k BT π ) 1/3 ( 4ηk 3 ρω 2 k < 2 1/6 k ρ 2 ω 4 R (ω), k R (ω) σ 2η k > 2 1/6 k σ 2 k 3 R (ω) 5

P(k,2πf) [m 4 /Hz] 10-32 10-34 10-36 10-38 10-40 10-42 10-44 10-46 10-48 10-50 10 3 10 4 10 5 10 6 10 7 10 8 k [m -1 ] Highly dissipative case η 3 ω/(ρσ 2 ) > 1/(8 2) No peaks viscous Leading order behavior 3 regions, simple. f = 10 3, 10 5, 10 7 [s 1 ] P 0 (k, ω) = k BT π 4ηk 3 ρω k < 2 1/4 ρ 2 ω 4 2η 1 ρω 2 1/4 2ηkω 2 2η < k < 2ηω σ 6 2η σ 2 k 3 k > 2ηω σ

Measure surface fluctuations directly. Characterize the fluctuations spectra. Fluctuations too small to examine precisely. Noise, esp. shot noise 7

Shot noise Essentially quantum noise. Randomness of photon arrival times. Photoconversion noise in current ( I) 2 = 2eI f White noise (f independent). Unavoidable in photoconversion (cf. gravitational wave detection). quantum non- Reduce shot noise Squeezed light ( 1/2 so far), demolition(?). 8

Noise reduction A measurement: D 1 = S + N 1, S: signal, N 1 : noise S has no known periodicity ( eg. thermal fluctuations). Power spectrum: ( D 1 Fourier transform of D 1 ) D 1 2 = S 2 + Ñ1 2 Impossible to separate S, N 1 even in theory 9

Noise reduction A measurement: D 1 = S + N 1, S: signal, N 1 : noise S has no known periodicity ( eg. thermal fluctuations). Power spectrum: ( D 1 Fourier transform of D 1 ) D 1 2 = S 2 + Ñ1 2 Impossible to separate S, N 1 even in theory Make multiple measurements: D 1,2 = S + N 1,2 D 1 D2 S 2, N relative error 1/ N Uncorrelated noise statistically reduced. 9

Comments on the noise reduction method Simple! at least in theory Principle not limited to optical or surface measurements. Noise needs to be uncorrelated. crucial Any uncorrelated noise eliminated: Shot noise, AM noise, amplification noise, FM noise, directional fluctuations,. In practice, 10 3 reduction achieved. 10

Practical limitations of the noise reduction method Always applicable, but efficacy depends on the properties of the signal. Noise elimination statistical. Reltative reduction: 1/ N. Given f (frequency resolution), N = ft. (T : total measurement time). f = 1 MHz, noise reduction factor 10 3 T = 1 s. Roughly speaking, need many periods of the wave component for the method to be effective. 11

Surface height fluctuation measurements Basic setup Michelson interferometer, except for photodetection and data processing. applicable to any Michelson measurement Laser: 532 nm wavelength, 0.5 mw at sample, reference. Sample size: 0.96 µ m. 2.2 mm, beam diameter: Sample surface reflectivity few %. (water: 2 %, oil: 4 %) 12

Observed spectrum (summed over k, b: beam diameter): S(f) = 2 ã 0 (ω) 2 = 2 0 dk ke b2 k 2 /8 P (k, 2πf), a 0 : average height Uniquely determined, given the liquid properties (ρ, σ, η) and b. Sample size: long distance cutoff. Beam size: short distance cutoff (averages within the beam spot). 13

10-22 10-23 water surface spectrum S h [m 2 /Hz] (water) 10-23 10-24 10-25 10-26 10-27 10-28 10-29 10-30 ethanol water shot noise level (theory) 10 3 10 4 10 5 10 6 10 7 f [Hz] 10-24 10-25 10-26 10-27 10-28 10-29 10-30 10-31 S h [m 2 /Hz] (ethanol) ethanol surface spectrum Single signal shot noise Theoretical prediction! Theoretical shot noise level Good agreement, some lens aberration effects. 10 3 below shot noise level! Perfect agreement for water. Almost perfect for ethanol, very slight deviations at high frequencies. 14

S h [m 2 /Hz] 10-23 10-24 10-25 10-26 10-27 10-28 10-29 10-30 10 3 10 4 10 5 10 6 10 7 f [Hz] δα 0.105 0.1 0.095 0.09 0.085 0.08 0.075 0.07 0.065 0.06 0.055 290 300 310 320 330 340 350 Oil surface fluctuations: T = 21, 30, 59, 74 C. Good agreement, slight deviation at high frequencies. Mismatch larger for smaller T. Deviation: S h (f) f α, δα = C exp(u/k B T ), U = 7 kj/mol latent heat. T [K] 15

Quantum Optics Single light source: Classically, difficult to imagine uncorrelated noise from a single source. Photon arrival times are random in the coherent state. Photons partitioned at the mirror shot noise in the 2 detectors uncorrelated. Will not work with squeezed light (shot noise in two measurements correlated). 16

Surface inclination fluctuation measurements Basic concept: Surface = optical lever. Measure the fluctuation spectrum of the inclination. Dual PD for an inclination measurement. 2 Dual PD for correlation measurement Noise reduction Observed spectrum: S(f) = 0 dk k 3 e b2 k 2 /8 P (k, 2πf) 17

P(2πf) [rad 2 /Hz] 10-11 10-12 10-13 10-14 10-15 10-16 10-17 10-18 10-19 f -2 f -1 α 1000 10000 100000 f [Hz] 2 1.8 1.6 1.4 1.2 1 1 10 100 1000 t [min] Time dependent spectra Epoxy adhesive example. Gradually hardens Spectra S(f) f α, α: time dep. α : 2 1 f 2 : Viscous liquid surface f 1 : Solid surface Why? α(t)? 18

Discussion Directly measured surface thermal fluctuation spectra of ordinary materials at ordinary temperatures (simple liquids, biological materials, complex fluids, polymers, ) Modification to standard measurements minimal broad area of application Noise reduction: not limited to optical, surfaces, thermal fluctuations. Achieved 10 3 reduction or more. Small power 0.5 mw, small sample size few µm, short measurement time few sec. wide applicability 19

To do: Understand surface fluctuation properties of various materials. Applications in other areas, other than optical, surfaces measurements. More to come 20