Conditional $1/f^{\alpha }$ noise: From single molecules to macroscopic measurement

We demonstrate that the measurement of $1/f^{\alpha }$ noise at the single molecule or nano-object limit is remarkably distinct from the macroscopic measurement over a large sample. The single-particle measurements yield a conditional time-dependent spectrum. However, the number of units fluctuating on the time scale of the experiment is increasing in such a way that the macroscopic measurements appear perfectly stationary. The single-particle power spectrum is a conditional spectrum, in the sense that we must make a distinction between idler and nonidler units on the time scale of the experiment. We demonstrate our results based on stochastic and deterministic models, in particular the well-known approach of superimposed Lorentzians, the blinking quantum dot model, and deterministic dynamics generated by a nonlinear mapping. Our results show that the $1/f^{\alpha }$ spectrum is inherently nonstationary even if the macroscopic measurement completely obscures the underlying time dependence of the phenomena.

Figure 1

An illustration for eight dichotomous Poisson processes with a characteristic time scale ${\tau }_j$ varying from one realization to another. The realization with the longest $\tau$ is given at the top (dark red), and then $\tau$ changes gradually where the shortest $\tau$ realization is given at the bottom (dark blue). It is clear that while changing the measurement time, the size of the set of moving realizations changes as well.

Figure 2

Simulation results for the macroscopic spectrum [panel (a)] and the single-particle conditional spectrum [panel (b)]. We use $N={10}^5$ particles all following the two-state telegraph process with $I_0=1$ and relaxation times $\left\{{\tau }_j\right\}$ drawn from the fat-tailed PDF with $\beta =1/2$, ${\tau }_{\min }=1$, ${\tau }_{max}={10}^8$, and $K=0$. The spectrum was measured at measurement times; $t={10}^3$ (yellow squares), $t={10}^4$ (cyan circles), $t={10}^5$ (green triangles), $t={10}^6$ (blue stars), and $t={10}^7$ (pink crosses). The lines represent Eqs. (13) and (15). The macroscopic approach appears stationary while the conditional measurements reveal aged spectra.

Figure 3

Single-particle conditional spectrum, using the same parameters as in Fig. 2, however now altering the condition on the number of transitions, $K=10$ [panel (a)] and $K=100$ [panel (b)]. The aging effect is recovered, however now we have a cutoff frequency ${\omega }_c\sim K/t$ below which we observe a flattening effect of the spectrum. Solid lines represent Eq. (19).

Figure 4

Simulation results for the Ornstein-Uhlenbeck process with $N={10}^4$, $k_{B}T=1$, and $m{\omega }_0^2=1$. The relaxation times are fat-tailed distributed with $\beta =1/2$, ${\tau }_{\min }=1$, and ${\tau }_{max}={10}^6$. The measurement time of the spectrum was $t={10}^3$ (cyan circles), $t=3162$ (green crosses), and $t={10}^4$ (blue stars). The analytic predictions Eqs. (23) are presented as solid lines. The conditional measurements of spectra reveal an aging effect, while the macroscopic approach obscures the nonstationarity.

Figure 5

Simulation results for the blinking-quantum-dot model with $\beta =1/2$, $N={10}^4$, $I_0=1$, and $t_w={10}^6$ at measurement time $t={10}^3$ (red squares), $t={10}^4$ (pink crosses), and $t={10}^5$ (purple dots). The lines represent Eqs. (24) and (26). The conditional spectrum ages, unlike the macroscopic measurements that appear stationary.

Figure 6

The deterministic signal $I_t$ [panel (a), blue] generated from the Pomeau-Manneville map (28) with $a=1$ and $\beta =1/2$. A realization is considered as a mover where $I_t$ crosses the threshold $I^{*}=\frac{1}{2}$ (represented in a dashed line) at least once in the measurement-time period. The signal $I_t$ is modeled with a two-state stochastic process $\tilde{I}\left(t\right)$ [panel (b), pink].

Figure 7

A comparison between macroscopic (a) and single-particle measured spectrum (b) in the two-unstable-fixed points deterministic map (28). The waiting time is $t_w={10}^6$, $\beta =0.5$, and $N={10}^3$ for three measurement times: $t={10}^3$ (green), $t={10}^4$ (pink), and $t={10}^5$ (blue). The black lines represent Eqs. (25) and (26).

Figure 8

The simulation results for the Ornstein-Uhlenbeck process. In panel (a) we present the macroscopic measured spectra at three measurement times: $t={10}^3$ (cyan squares), $t=3162$ (green crosses), and $t={10}^4$ (blue dots). The black line represents the analytic prediction Eq. (23). In panels (b) and (c) we present the microscopic measured spectra, where we use the spectrum-based method for classification. The main result, the aging effect, is clearly visible in both classification methods.

Figure 9

(a),(c) Parallel measured power spectrum (green line) vs the total spectrum corresponds to the macroscopic signal (blue line) in two processes; the two-state random telegraph noise with $\beta =1/2$, ${\tau }_{\min }=1$, ${\tau }_{\max }={10}^5$, $N={10}^4$, and $t={10}^4$ [panels (a) and (b)] and the Ornstein-Uhlenbeck process with the same parameters [panels (c) and (d)]. (b),(d) The smoothed total spectrum (red stars) vs the paralleled measured spectrum (green line) shows that the parallel measurements provide a reasonable approximated measuring method.

Figure 10

The corresponding spectra of five blinking quantum dots. The two particles' spectra types are clearly observed; trapped particles exhibit white noise (green, blue, cyan) vs the nontrapped particles with $1/f^{\beta }$ noise (red, pink). Notice that differentiation between the two populations is manifest in sufficiently low frequencies, where in higher frequencies all the spectra are observed in a similar order of magnitude. For that reason, detecting the nonfrozen particles requires a long measurement time (corresponding to low frequencies).

Figure 11

The spectrum for the deterministic intermittent map with $\alpha =0.5$, waiting time $t_w={10}^6$, and measurement time $t={10}^5$. In panel (a) we present the single-particle measurements (red line) and its $1/f^{\beta }$ noise prediction Eq. (24) (black curve). The blue line represents the averaged spectrum of the trapped particles (i.e., the particles that do not cross the threshold). In panel (b) five realizations of the spectrum are presented. The trapped particles (four bottom realizations) exhibit significantly low power related to the nontrapped particle (upper blue curve).