1/f Noise on the nano-scale Aeging Wiener-Khinchin approach. Pohang (2016)

Transcription

1 1/f Noise on the nano-scale Aeging Wiener-Khinchin approach Eli Barkai Bar-Ilan University Niemann, Kantz, EB PRL (213) MMNP (216) Sadegh, EB, Krapf NJP (214) Leibovich, EB PRL (215) Leibovich, Dechant, Lutz, EB PRE (216) Pohang (216)

2 Outline 1/f β noise and the low frequency cutoff paradox (Mandelbrot). Blinking quantum dots. Noise on the nano-scale: the conditional spectrum. Aging Wiener-Khinchin theorem.

3 1/f β noise and the infrared catastrophe Low frequency 1/f β power spectrum is widely observed with < β < 2 ( ). If β 1 the apparent total energy is infinite S(f)df = if β 1. The total energy cannot be infinite if the underlying process is bounded.

4 Power spectral density Sample spectrum I t (ω) = t I(t ) exp( iωt )dt S t (ω) = I t(ω) 2 If the process is stationary use Wiener-Khinchin theorem. t S(ω) = eiωτ I(t + τ)i(t) dτ The power spectrum is then normalizable.

5 Low frequency paradox (physics) If I(t) = x cos(ω t), i.e., one mode. S(ω) x 2 δ(ω ω ). x 2 is proportional to the energy of the mode. Sum over all the modes cannot be infinite. 1/f noise implies non countable normal modes.

6 Low frequency paradox (math) S t (ω)dω = 1 t dω t dt 1 exp( iωt 1 )I(t 1 ) t dt 2 exp(iωt 2 )I(t 2 ) = t 2π I 2 (t 1 )dt 1 } t {{} Variance < 2π(I max ) 2 The total energy of bounded stationary or non-stationary process, is finite. So how do we observe non-integrable 1/f noise?

7 Possible solution Think! Total energy must be finite z is called the aging exponent S t (ω) ω β t z 1/t S t (ω)dω t z t 1+β = const. Hence we find a relation between the exponents, based on the finite value of the total power z = β 1.

8 Stationary or Non-stationary that is the question A blind extrapolation of f β (β 1) to f = incorrectly suggests that the total energy is infinite (infrared catastrophe)... If β 1 one needs a non-wienerian spectral theory to account for f β noise. Mandelbrot IEEE Non-stationary: Manneville 198 Intermittency, and Bouchaud, Cugliandolo, Kurchan, Mezard 1997 Spin Glasses. As is customary in statistical physics, we shall henceforth assume that the noise process is stationary, for simplicity, and in the absence of overwhelming evidence to the contrary. Dutta and Horn RMP The statistical properties of 1/f noise in physical sources are fully consistent with the assumption of stationarity. Stoisiek and Wolf 1975

9 Quantum Dots Stefani, Hoogenboom, Barkai Physics Today 62, 34 (29).

10 Blinking Nano Crystals (coated CdSe) Intensity time (sec)

11 Power Law Distribution of on and Off times τ on 1 1 τ off.39τ PDF τ Power law waiting time ψ(τ) τ (1+α). Averaged time in States On and Off is infinite τ =. S(ω, t) Cω 2+α /t 1 α

12 Physical explanation for blinking Stefani, Hoogenboom, Barkai Physics Today 62, 34 (29).

13 Measured Power Spectrum Ages PSD (Hz -1 ) s f (Hz) Sadegh, EB, Krapf NJP (214)

14 Power Spectrum t z PSD x t.12 (Hz ) f (Hz) Sadegh, EB, Krapf NJP (214) Experiment

15 Hold your horses Measurement of most cond-mat 1/f noise sources do not exhibit aging. No signature of non-stationary noise. Fundamental difference between measurements of macroscopic source of noise and single molecule measurements. S i N S is time independent. When the number N of fluctuating subunits, on the time scale of the the measurement time is increasing with measurement time.

16 Distributed Kinetics: Popular Old Model S(ω) is time independent. S(ω) = p(τ) 1+ω 2 τ 2dτ τ Here exponential decay of the correlation function, and distributed kinetics yields 1/f noise. In experiment no Lorentzian spectrum for single QDs. Take p(τ) = 1/τ. But 1/τ is non-normalized. So maybe add a cutoff. What is the upper limit of the integral? Should we put the age of universe (as suggested)? Then S(ω) depends on age of universe? but measurement time is much less then that.

17 Answer: upper limit of integral is measurement time. Then S(ω) depends on measurement time. And exactly in the way we claim β 1 = z. How many fluctuating objects? N t z So N S is time independent and also independent of the non-relevant age of the universe. As observed in many macroscopic experiments.

18 Distributed Kinetics Model x x x x x Number of fluctuators increases with measurement time

19 Aging PSD in Single Molecule Measurements S(ω) sp t m =1 3 t m =1 4 t m =1 5 t m = ω Distributed kinetics model.

20 No Aging in Macroscopic Measurements 1 2 t m =1 3 t m =1 4 1 t m =1 5 ~ ω 1.5 S(ω) ens ω Distributed kinetics model.

21 Mechanisms of 1/f noise 2 SFD 2 1 LOG 1 2 BQD 2 time Long range correlations induced by many body effects, power law intermittency, burstines in amplitude, yield 1/f noise.

22 Aging Wiener Khinchin Theorem t m S tm (ω) = tm dt 1 tm dt 2 e iω(t 2 t 1 ) I(t 1 )I(t 2 ). (1) S tm (ω) = 2 t m tm dτ(t m τ) C T A (t m, τ) cos (ωτ). (2) C T A (t m, τ) = 1 t m τ tm τ dt 1 I(t 1 )I(t 1 + τ). (3) C T A (t m, τ) = (t m ) γ ϕ T A (τ/t m ), (4)

23 1 S tm (ω) = 2(t m ) 1+γ d τ (1 τ) ϕ T A ( τ) cos (ωt m τ). (5) I(t + τ)i(t) = t γ φ EN (τ/t). (6) ϕ T A (x) = x γ y(x) y(x) φ EA (z) dz (7) z2+γ 1 ( ) x ωx sin( ωx) + cos( ωx) 1 S tm (ω) = 2t m φ EA 1 x ( ωx) 2 dx (8)

24 S tm (ω) = 2(t m ) γ γ 1 ( ) ( x (1 x) γ φ EN 1F γ 1 x 2 ; 1 2, 2 + γ ( ) ) 2 ωx 2 ; dx. 2

25 Blinking dot power spectrum on natural frequencies 1 (tm) 1 Stm (ω) ωtm=2πn ωt m

26 Blinking dot power spectrum beyond the natural 1 Stm(ω) /tm ωt m

27 Single File Diffusion ta g g e d p a rtic le x(t + τ)x(t) 1 ρ ( D π t1/2 1 + τ ) τ t + 1. t x 2 t 1/2 Hard core point particles, free diffusion between collision events, infinite system, fixed density ρ. Harris (65)... Leibovich-Barkai (213).

28 PSD single file diffusion Stm (ω) ω

29 Model γ ν Unilayer Parisi s Tree < α < 1 α 1 Blinking Quantum Dot < α < 1 2α 2 α 1 1 α Laser-Cooled Atoms 1 < α < 3 2 α 2 α Single-File Diffusion 1/2 1/2 Generalized Elastic Model < α < 1 α α Coupled Oscillators [.58,.4].14 ±.3 1D Growing Interfaces.33 RC transmission Line 1 < α < 2 α 1 α 1 Table 1: The aging behavior of several models, where the correlation function is given in terms of I(t)I(t + τ) t γ φ EA (τ/t) and φ EA (x) A EA B EA x ν when x 1. S tm (ω) 2Γ (1 + ν) sin ( ) πν 2 BEA (γ ν + 1) (t m ) ν γ ω 1+ν.

30 Summary Critical exponents describe the spectrum of blinking dots and models of 1/f noise. Power spectrum ages, the longer the measurement time the noise is reduced. This solves the low frequency paradox, the total power is a constant. Fundamental difference between measurement of noise on the single molecule level, if compared with macroscopical measurements. Non-stationarity implies non-ergodicity, which was quantified. Aging Wiener Khinchin theorem: a tool for the calculation of power spectrum. A wide variety of mechanism responsible for 1/f noise: intermittency, single file diffusion and large amplitude burstiness. The unifying theme are scale invariant correlation functions.

31 Ref. and Thanks M. Niemann, H. Kantz, E. Barkai Fluctuations of 1/f noise and the low frequency cutoff paradox Phys. Rev. Lett. 11, 1463 (213). S. Sadegh, E. Barkai, and D. Krapf 1/f noise for intermittent quantum dots exhibits non-stationarity and critical exponents New. J. of Physics 16 (214) N. Leibovich, and E. Barkai Aging Wiener-Khinchin Theorem Phys. Rev. Lett. 115, 862 (215). N. Leibovich, A. Dechant, E. Lutz, and E. Barkai Aging Wiener-Khinchin theorem and critical exponents of 1/f β noise Phys. Rev. E. 94, 5213 (216). M. Niemann, E. Barkai, and H. Kantz Renewal theory for a system with internal states Mathematical Modelling of Natural Phenomena 11 3 (216)

32 The meaning of the PS I = Const as for the WK spectrum. I(t)I(t + τ) = t γ φ EN (τ/t):- scale invariant correlation function. This gives the nearly standard relation: S tm (ω)dω = 2π I2 1 + γ And since S tm (ω) it is resonably justified to call it a PS.

33 Single File Diffusion (x Τ ) 2 ~ D t N A s la n g u l E P L N ~ ρ L (t) ~ ρ (D t) 1 /2 (x Τ ) 2 ~ (D t) 1 /2 ρ H a rris 6 5, L e v itt 7 3

34 Models β z S ω β S t m single file diffusion 3/2 blinking quantum dot - finite mean on time < α < 1 α 1 α blinking quantum dot - infinite mean < α < 1 2 α 1 α logarithmic potential 1 < α < 2 3 α 2 < α < 3 Table 2: Summary of the critical exponents for the three systems discussed in sections VI-VIII.

35 Fluctuations of power spectrum PSD (1/Hz).1 1E-3 1E-4 1E f (Hz) 1 Amplitude of 1/f noise is random, varies from one measurement to the other. This is related to non-stationarity and non-ergodicity.

36 Fluctuations of the power spectrum Power spectrum ergodic processes, we expect P DF [S t (ω)] δ[s t (ω) S t (ω) ] This is correct, after binning/smoothing. For non ergodic process, this will not happen Where C is random S t (ω) S t (ω) C. C = Y α ξ Y α (ξ) is a ML (exp) RV. Y α n/ n

37 If I know the fluctuations of number of transitions, I have the fluctuations of the sample spectrum. Even better find PDF of (S(ω 1 ),...S(ω n )).

38 Transition frequency PSD (1/Hz).1 1E-3 1E-5 ~ f -.72 ~ f E f (Hz) Power spectrum exhibits a transition at f c =.1Hz. This is related to cutoff on the on times.

39 Things to do Do we believe in conservation of energy (total power cannot be infinite), a mathematical theorem (bound on total power), on the one hand, and the experimental fact that noise of many different sources exhibits non integrable 1/f noise. Better, believe in the basic laws of math and physics and the experiments, but throw away the notions of stationary process and ergodicity. Better explain why N increases with t and why it exactly cancels the aging. At start list of systems where this phenomena is found. Add some detail on the derivation, multipoint correlation functions. Show that you actually worked hard on the theory.

40 Wiki

41

42 2 SFD 2 1 LOG 1 2 BQD 2 time

43 Define the two state model better. Infinite energy or infinite Power?

44 History Wikipedia: In applied mathematics, the Wiener?ÄìKhinchin theorem, also known as the Wiener Khintchine theorem and sometimes as the Wiener?ÄìKhinchin?ÄìEinstein theorem or the Khinchin?ÄìKolmogorov theorem, states that the autocorrelation function of a wide-sensestationary random process has a spectral decomposition given by the power spectrum of that process From the web: This is the Einstein-Wiener-Khinchin theorem (proved by Wiener, and independently by Khinchin, in the early 193?Äôs, but?äî as only recently recognized?äî stated by Einstein in 1914). MIT OpenCourseWare Power Spectral Density Chapter 1.

45 Group Material Nu. Radii T α on α off Dahan CdSe-ZnS nm 3 K.58(.17).48(.15) Orrit CdS EXP.65(.2) Bawendi CdTe (.1).5(.1) Kuno CdSe-ZnS Cichos Si Ha CdSe(coat) 3 Exp? 1

46 Uncapped NC on (short) off (long) Capped NC on (short) on (long) off (long) Efros, Orrit, Onsager, Hong-Noolandi r Ons = e2 k b T ɛ 7nm

47 Critical exponents of nano-crystals Theory Experiment Simulation β > S f β β > = 2 α β < β < = α.7.6 z S t z z = 1 α.12.3 γ f c 1/t γ γ = ω S t () t ω ω = Aging exponent is lower than expected. Scaling relations between exponents hold γ(β > 1) z, ω = γ within errors.