Eli Barkai. Wrocklaw (2015)

Transcription

1 1/f Noise and the Low-Frequency Cutoff Paradox Eli Barkai Bar-Ilan University Niemann, Kantz, EB PRL (213) Theory Sadegh, EB, Krapf NJP (214) Experiment Stefani, Hoogenboom, EB Physics Today (29) Wrocklaw (215)

2 Outline General remarks on 1/f noise. Fluctuations of 1/f and the low frequency cutoff paradox. Blinking quantum dots. Aging and weak ergodicity breaking.

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 power is infinite S(f)df = if β 1. The total power 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 t If the process is stationary use Wiener-Khintchine theorem. Correlation function is aging, e.g., I(t)I(t + t ) = C(t /t).

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 ) = 2π t t I 2 (t 1 )dt 1 < 2π(I max ) 2 The total power, of bounded stationary or non-stationary process, is finite. So how come we do we observe non-integrable 1/f noise?

7 This is a moment in our lifes where we have to ask ourselves if 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.

8 Possible solution Think! Total power 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.

9 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.

10 History Non-stationary source of 1/f noise Mandelbrot. Intermittency dynamical systems Manneville 198. Spin glasses Bouchaud, Cugliandolo, Kurchan, Mezard We focus on ergodicity and blinking quantum dots using stochastic approach.

11 Quantum Jumps: Atoms Stefani, Hoogenboom, Barkai Physics Today 62, 34 (29).

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

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

14 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 off ). Averaged time in States On and Off is infinite τ =.

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

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

17 Sample power spectrum is random different realizations of S t (f) e-5 1e-6 1e-7 a) ensemble average of S t (f) e-5 1e-6 1e-7 t=.1 t=1. t=1. t=1. Eq. (9) b) frequency

18 Non-stationarity means that underlying process is non-ergodic. Sample spectrum is not equal to its ensemble average. Niemann, Kantz, EB PRL (213) Theory Similarly, time averaged intensity correlation function is not equal to the ensemble average. Long tailed power law distribution of waiting times induce weak ergodicity breaking Bouchaud.

19 Fluctuations of the power spectrum The average decreases with measurement time S t (ω) C t 1 α 1 ω 2 α The value of S t (ω) is fluctuating ( ) St (ω 1 ) S t (ω 1 ),, S t (ω n ) S t (ω n ) Y α (ξ 1,, ξ n ) Y α has a normalized Mittag-Leffler distribution. The ξ i are independent exponential random variable with mean 1. The whole spectrum has a common random prefactor Y α.

20 Motivation of the result t dτ exp(iωτ)i(τ) n(t) j=1 d j (ω) d j (ω) iχ j exp(iωt j ) 1 exp(iωτ j) ω With this approximation S t (ω) 1 t n(t) k,l=1 d k (ω)d l ( ω) Assuming that n(t) is independent of waiting times

21 S t (ω) n(t) d l (ω)d l ( ω), t S t (ω) (I ) 2 n(t) t 2 ˆψ(iω) ˆψ( iω) ω 2 The spectrum is time dependent since n(t) t α. We get β = 2 α and z = 1 α so β = z + 1. Repeat this for S t (ω 1 ) S t (ω 2 ). The Mittag-Leffler distribution describes number of renewals n/ n(t). To isolate the Mittag Leffler fluctuations N S t (ω j ) with large N. M = 1 N i=1 S t (ω j )

22 Mittag Leffler distribution α=.2 probability density of M α=.5 α= value of M When α 1 we get an ergodic behavior. Niemann, Kantz, EB PRL (213) Theory

23 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

24 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

25 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.

26 Summary Critical exponents describe the spectrum of blinking dots. 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. Power law sojourn times are found in diverse systems. Non-stationarity implies non-ergodicity, which was quantified.

27 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) F. D. Stefani, J. P. Hoogenboom, and E. Barkai Beyond Quantum Jumps: Blinking Nano-scale Light Emitters Physics Today 62 nu. 2, p. 34 (February 29).

28 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.

29 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.

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

31 Aging PSD in Single Molecule Measurements t m =1 3 t m =1 4 2 t m =1 5 t m = Distributed kinetics model.

32 No Aging in Macroscopic Measurements 4 2 t m =1 3 2 t m =1 4 t m =1 5 4 t m =1 6 ~ω Distributed kinetics model.

33 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.

34 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

35 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 )).

36 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.

37 Things to do Ask the listener: This is a moment in your life where you have to ask your self if you 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. Find quatations from reviews on stationarity.

38 Quotation from mathematical book on 1/f noise.