Non-Stationary Dephasing by a Classical Intermittent Noise

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1 Non-Stationary Dephasing by a Classical Intermittent Noise (CNRS & ENS Lyon) David Carpentier (CNRS & ENS Lyon) Maxime Clusel (ENS Lyon) Josef Schriefl (ENS Lyon & University of Karlsruhe) May 2005 Europhysics Letters 69 (2005) 156 Preprint cond-mat/ (to appear in Phys. Rev. B) Typeset by FoilTEX

2 Dephasing of Josephson qubits Introduction Josephson qubits: circuits. E J g Φ x C J Q=2ne C g V controlled two level quantum systems built from superconducting Sources of dephasing: Resistances in the circuit : Johnson-Nyquist noise for the voltage Measurement device for write/read operations Background charge fluctuations : 1/f noise Yu. Makhlin et al, Rev. Mod. Phys (2001) Today, 1/f noise is currently the dominant limiting factor. Mostly analyzed using phenomenological models based on independent bistable fluctuators. E. Paladino et al, Phys. Rev. Lett (2002) Y.M. Galperin et al, proc. of NATO/Euresco Conf., cond-mat/ Typeset by FoilTEX 1

3 e SET ance of demonapping f power rf-set ence of details dentify re that tion of g sites rphous low the [17]. the reral difn, with e. The T outcharge ception legraph events trongly les ob- 10MHz ken on r bandcharge signals, etected itching al elecidth of search g. Our Firstly, cal poof the f 200ns Buehler et al, J. Appl. Phys. 96 (2004) 6827 Probing the origin of 1/f noise Introduction FIG. 2: Mapping the dependence of telegraph noise on gate bias. accumulated At each gate biasbetween configuration, the data preparation is taken for 2.0ms. time t p and t p + t. a) Intensity plot showing the variance of each trace as a function of the gate bias. Note the zone moving diagonally across the plotdependance that primarily corresponds of the to dephasing the switchingfactor of a two on the preparation time? level fluctuator. b) Data taken at the point indicated by the star in Decay a). Dashed oflines theindicate dephasing the dynamic factor range atof fixed the t p? SET. c) Shows signal histograms corresponding to the two states of a two-level trap. d) Shows the trap capture and emission time histograms. For the case shown, equal time is spent Typeset in bothby thefoiltex up (triangles) and down (circles) states with 2 a characteristic decay time of 3.4µs. ing diagonally across the plot). In addition, the variation in sensitivity of the rf-set can be seen as light regions Open questions: Non Gaussian effects? Dynamics and localization of the fluctuators? Non stationarity and intermittency properties of the 1/f noise? Our phenomenological approach: compute the dephasing scenario resulting from an intermittent noise (random spike field). The dephasing factor: D t p(t) = exp (iφ(t p, t)) where Φ(t p, t) is the relative phase

4 Model for an intermittent noise A simple model for intermittent noise Intermittent noise: succession of independent random phase pulses X(τ) = X i x i δ(t t i ) Described by: Can be viewed as the intermittent limit of a telegraphic noise. The probability distribution of phase pulses. x i. The distribution of waiting times between two pulses t i+1 t i. The phase Φ(t p, t) performs a continouus time random walk. See Tunaley, J. Stat. Phys. 15 (1976), 149 and Monthus and Bouchaud, J. Phys. A 29 (1996), 3847 and also Barkai and Cheng, J. Chem. Phys. 118 (2003), Typeset by FoilTEX 3

5 Model for an intermittent noise A simple model for intermittent noise Probability distribution of phase pulses Symmetric case: p( x) = p(x). Asymmetric case: p( x) p(x) (relevant for the case of transverse noise cf Makhlin). Waiting time distribution: Poissonian case ψ(τ) = γe γτ (γ = τ 1 ). Boad waiting time distribution with an algebraic tail parametrized by µ ψ(τ) = µ τ 0 τ0 τ 0 + τ «1+µ. Typeset by FoilTEX 4

6 Model for an intermittent noise A simple model for intermittent noise Three different cases: 2 µ: first and second momentas of the waiting time distribution are finite. 1 < µ < 2: the average waiting time is finite but there are broad fluctuations µ 1: no average waiting time. No time scale Average amplitude decays as times goes on: where S(t) is the average density of events. X(t) = x S(t). µ < 1 1 < µ < 2 2 < µ τ `τ 0t µ 1 τ 1 sin (πµ) πτ 0 `τ0t µ For µ 1, events become more and more rare. Typeset by FoilTEX 5

7 Model for an intermittent noise A simple model for intermittent noise Noise power spectrum: extracted from the quasi-stationary regime t t p : S X (t p, ω) = 2 Z + 0 X(t p ) X(t p + τ) c cos (iωτ) dτ For ωt p 1:: S X (t p, ω) 2h 2 S(t p ) R(L[S]( iω)). Frequency dependence: µ < 1 µ = 1 1 < µ < 2 (ωτ 0 ) µ (ωτ 0 ) 1 log (ωτ 0 ) 2 (ω τ ) µ 2 Algebraic power spectrum, slowly decaying in time for µ 1. Typeset by FoilTEX 6

8 The first waiting time distribution Origin of non stationarity All the t p dependance is hidden in the first waiting time distribution ψ t p(τ 1 ) 3. Origin of the non-stationarity in CTRW : the first waiting time of the first e waiting time distribution. (23) does not depend on Model for an intermittent tionarity noise of the CTRW fo Within our model, the waiting times between successive pulses are chosen independently according to the distribution ψ. Consequently, all the t p dependence of Φ(t p,t) will come from the choice of τ 1 defined as the waiting time between t p and the first spike that follows t p (see figure 4). Indeed, at time t p + τ 1, the CTRW x(t) τ 1 t 0 t 1 t p τ 1 t t p + τ exp FIG. 4: Intermittent noise between t p and t p + t Renewal equation: Godrèche and Luck, J. Stat. Phys. 104 (2001), 489 ψ t p(τ 1 ) = ψ(t p + τ 1 ) + Z t p starts anew: τ 2 is chosen without any correlation to the history of the CTRW. Hence given the probability distribution of τ 1, we can forget about the history of the CTRW of the phase and describe its behavior starting dτ at t p. ψ(τ This 1 remark + τ)s(t is at the core p of τ) the use. of renewal theory. In the following, the probability distribution of τ 1 will be denoted by ψ tp and apriori, it may depend on t p. In fact, as we shall see later, its behavior can be quite counter-intuitive. First of all, note that a given τ 1 can be obtained from many different noise configurations that differ from the time of the last event occurring before t p. Separating noise configurations (starting at t = 0) that have their first spike at time t p + τ 1 from the others leads to an integral equation that determines ψ tp in terms of ψ (S is determined from ψ throughanintegralequation(a1)): If ψ(τ) = γ e γτ, then ψ t p = ψ. For µ > 2: τ 1 = τ 2 + τ2 τ 2 2 τ For 1 < µ < 2: non stationarity induced by large fluctuations of ψ: τ 1 t 2 µ 0 tp Typeset by FoilTEX ψ tp (τ 1 )=ψ(t p + τ 1 )+ dτ ψ(τ 1 + τ)s(t p τ). (22) 0 7 The integral in the r.h.s. comes from noise configurations that have a spike between 0 and t p. Equation (22) is the starting point for deriving analytic results about ψ tp in appendix B and C using Laplace transform techniques. Before computing exactly the dephasing factor, p regime at small t p. On t (23) is expected to break divergence signals that in can become of the order the average properties of t this age of the noise. Inde τ 1 ψtp scales with the age sion regime, many phase p the qubit. Therefore, we e aging effect on ψ tp (τ 1 ) on the dephasing scenario as act computations. Howeve that the t p dependence of spectacular consequences the 1 <µ<2. We will now show that, of ψ tp, an explicit express of the average dephasing f B. Exact dephas 1. Dep Among all noise config into account, some of the have any event between t p is given by Π 0 (t p,t)which Π 0 (t p,t)= All other histories have at t p + t. Let us assume th τ lies between 0 and t. T starts anew. The jump i dephasing factor and the contributes by D 0 (t τ) probability that the first t p + τ is nothing but ψ tp to D tp (t) from all possibl form of the following rene D tp (t) =Π 0 (t p,t)+e ix

9 Results for the dephasing factor Analytic solution for the dephasing factor Renewal equation for the dephasing factor: Z t D t p(t) = Π 0 (t p, t) + e ix ψ t p(τ)d 0 (t τ) dτ. where Π 0 (t p, t) = 0 Z + t ψ t p(τ) dτ. is the probability that no event occurs between t p and t p + t. Laplace transform solution: introduce f = 1 e ix and S t p(t) = S(t p + t) L[D t p](s) = 1 s L[D t p](s) = 1 s «1 L[ψ tp ] 1 (1 f) L[ψ] «1 + f L[S tp ] 1+f L[S] Series expansion in the number of phase pulses Random phase partition noise e iφ(t p,t) = Q τ (1 + (eix(τ) 1)n(τ)) Typeset by FoilTEX 8

10 Renewal equation for the dephasing factor: Physical discussion Results for the dephasing factor D t p(t) = Π 0 (t p, t) + (1 f) Z t 0 ψ t p(τ)d 0 (t τ) dτ. Widely spread phase pulses: f 1. Narrow distribution: f = 1 e i x. Very strong coupling (f = 1): after one event, dephasing is total. D t p(t) = Π 0 (t p, t) Aging behavior for µ < 1: dephasing factor depends on t/t p. Very weak coupling (f 0): phase diffusion is so slow that many events are necessary to dephase the qubit. D t p(t) Π 0 (t p, t) + (1 Π 0 (t p, t))d 0 (t) D 0 (t) Typeset by FoilTEX 9

11 Renewal equation for the dephasing factor: Physical discussion Results for the dephasing factor D t p(t) = Π 0 (t p, t) + (1 f) Z t 0 ψ t p(τ)d 0 (t τ) dτ. Competition between time scales: Π 0 (t p, t) and D 0 (t). Weak coupling: Π 0 (t p, t) decays much faster than D 0 (t). Strong coupling: D 0 (t) decays much faster than Π 0 (t p, t). Case µ > 1: there is a finite average waiting time τ. Therefore Π 0 (t p, t) decays on a scale comparable to τ independently of t p. Dephasing time at t p = 0: diverges when the phase distribution becomes narrow. Thus the weak coupling region extends to some non zero value g > 0. Case µ 1: the decaying time of Π 0 (t p, t) scales as t p and therefore the weak coupling region extends to some g c (t p ) which goes to zero as t p increases! Typeset by FoilTEX 10

12 τ φ Couplage critique g c Results: symmetric model and µ = 0.8. Results for the dephasing factor p g c = K µ t µ/2 Coupla Pe D Coupla D Dephasing factor function of time. Dephasing times function of g = p x 2. τ φ Fo vi Maxime Clusel Aging in the strong coupling phase: the dephasing factor only depends on t/t p. Anomalous dephasing scenario at weak coupling: stretched exponential decay below τ φ and algebraic decay above τ φ. History dependent cross over coupling: any qubit will end up in the strong coupling regime! Non ergodicity: sample averages differ from time averages. Déphasage pa Typeset by FoilTEX 11

13 Results for the dephasing factor Results: symmetric model and µ < 1. Very small coupling: f x 2 /2 1. Introduce γ f such that γ f τ 0 = (f/γ(1 µ)) 1/µ, then: " # D 0 (t) L 1 s 1 = E 1 + f(sτ 0 ) µ µ [ (γ f t) µ ]. Behavior for µ 1: crosses over from e (γ f t)µ to an algebraic decay (γ f t) µ. Very large coupling and t p τ 0 Π 0 (t p, t) = sin (πµ) πµ «µ tp t 2F 1 (1, µ; 1 + µ; t p t ). Cross over coupling: in the limit f 0, we have: f c (t p ) C(µ) «µ τ0 t p Typeset by FoilTEX 12

14 D tp ( t/ τ φ ) Results: symmetric model and µ = 1.1. µ = 1.1, t p =10 5 τ 0 g=1 g=10-1 g=10-2 g=5x10-3 exp(-t / τ φ ) t/τ φ Dephasing factor function of time. Couplage critique g c 1 Results for the dephasing factor Dephasing times function of the coupling. Coupla Pe µ Un co so µ Coupla Sa De se Maxime Clusel Dephasing scenario at weak coupling: exponential decay below τ φ but algebraic decay above τ φ. Weak history dependance of the cross over coupling: decays with increasing t p but remains above a non zero value. Weaker effects when µ 2: much weaker history dependance of the dephasing time for µ = 1.7 than for µ = 1.1. Déphasage pa Typeset by FoilTEX 13

15 Results: symmetric model and 1 < µ < 2. Very small coupling: introduce γ f such that γ f τ = f, then: D 0 (t) e γ f t f c(µ) t τ0 2 µ. Standard exponential decay corrected by a stretched exponential. Very large coupling and t p τ 0 Π 0 (t p, t) τ0 τ 0 + t «µ 1 τ 0 τ 0 + t + t p Results for the dephasing factor «µ 1. Cross over coupling: in the limit f 0, we have: f c (t p ) 1 µ 1 " e 1 + τ0 t p «µ 1 # 1/(µ 1). Typeset by FoilTEX 14

16 Results for the dephasing factor Results: symmetric model and µ = 1. Very small coupling: for t τ 0 log (f) /f: D 0 (t) exp f Crosses over to D 0 (t) 1/t at large times. «t/τ 0. log (t/τ 0 ) Very large coupling and t p τ 0 Π 0 (t p, t) 1 ln(1 + t p /τ 0 ) ln 1 + t «p τ 0 + t. Cross over coupling: in the limit f 0, we have: f c (t p ) τ 0 t p log (t p /τ 0 ). Typeset by FoilTEX 15

17 Results: asymmetric model and µ = 0.8. Huge asymmetry limit: h = x much larger than g = p x 2. Results for the dephasing factor Dephasing factor function of time. Dephasing times function of h = h. Aging in the strong coupling phase Anomalous dephasing scenario at weak coupling: stretched exponential decay below τ φ and algebraic decay above τ φ. Oscillations between these two limiting behaviors. History dependent cross over coupling Typeset by FoilTEX 16

18 On the origin of oscillations Results for the dephasing factor Non gaussian shape of the phase distribution: for µ < 1, introducing τ φ (h) = τ 0 (Γ(1 µ)/h) 1/µ : «t P 0,t (φ) = µ τ φ (h) φ L t/τφ (h) 1+1/µ µ. φ 1/µ 0 < µ 1/2: P 0,t decreases monotonically. No oscillations. 1/2 < µ < 1: P 0,t has a maximum at some non zero value. Oscillations are present. Works at t p = 0 but not easy to extend to t p > 0. Interference effects between rare configurations: the dephasing factor is dominated by rare configurations that do not move after some given time. Analogous to the Griffiths effect in disordered systems. Typeset by FoilTEX 17

19 Fig. 6 Module du taux de cohérence, dans leon cas constant, thepour origin µ = 1.1, t p = of 10 oscillations 5, et g = Contributions de la partie réelle et de la partie imaginaire. La ligne horizontale pointillée signale la valeur e 1. Results for the dephasing factor Dephasing factor in the complex plane for µ = 0.8 and t p = 10 5 τ 0. Dephasing factor in the complex plane for µ = 1.1 and t p = 10 5 τ 0. Fig. 7 Taux de cohérence dans le plan complexe, dans le cas constant, Fig. pour 8 Taux µ = 0.8, de cohérence t p = 10 5, dans et le plan complexe, dans le cas constant, pour µ = 1.1, t p = 10 différents g. différents g. 5 Typeset by FoilTEX 18

20 ing atare s. als. um zereof el- Ds eresce to gh reies on ayon, ose of to on nd ow ite, ce of vy lds Relevance of these results in other contexts In this Letter, we show that single QD measurements can be used to explicitly compare ensemble- and timeaveraged properties and explore some of the unusual phenomena induced by Lévy statistics, such as statistical aging and ergodicity breaking. Using an epifluorescence microscopy setup and a low-noise CCD camera, we simultaneously recorded at room temperature the fluorescence intensity of 215 individual QDs for the duration of 10 min with a time resolution of 100 ms [23]. The blinking of the fluorescence intensity was observed for each QD detected in the field of the camera (Fig. 1). Because of the binary behavior of the blinking process, each intensity time trace was considered simply as a sequence of n on and off times f 1 on ; 1 off ; 2 on ; 2 off ;...; n on ; n off g from which the distributions P on and P off were derived. In our measurements, the on and off periods both followed power-law distributions [25]. After adjustment of the cumulative distributions of the on and off periods for Relevance in other contexts Statistics of events All pulses having equal phase h: the dephasing factor is the generating function for the number of pulses in a given time interval. D t p(t) = e ih N[t p,tp+t] Blinking of CdSe nanocrystals: Brokman et al, Phys. Rev. Lett. 90 (2003), each of the 215 QDs, the exponents on and off were Intensity (a.u.) Time (s) A single nanocrystal alternates on and off periods of fluorescence. Long waiting time distributions were observed with µ on/off 0.5. Questions: can we probe the on/off model by studying the aging properties of the faction of time spent in each of the states? FIG. 1. Fluorescence intermittency of a single CdSe nanocrystal measured over 10 min with 100 ms time bins. Because of the broad distribution of the on and off states, the signal is dominated Typeset by aby few FoilTEX long events The American Physical Society

21 Extension of the model Relevance in other contexts Generalized telegraphic noise: bivalued noise. X = 0 plateaux: ψ (τ): exponent µ + X = 1 plateaux: ψ + (τ): exponent µ Telegraphic noise: ψ ± (τ) = γ ± e γ ± τ. Dephasing factor: D t p(t) = e iλ R tp+t tp X(τ) dτ. Results: Exact solution using renewal theory for the Laplace transform of the dephasing factor. Assume that ψ is much broader than ψ +, then in the long time limit, the generalized telegraphic model is equivalent to the intermittent noise model where ψ = ψ and 1 f = Z + 0 e iλτ ψ(τ) dτ. Work in progress: numerical study, analysis of shorter time behaviours and crossovers. Typeset by FoilTEX 20

22 Conclusion Conclusion To summarize Detailed study of the dephasing induced by an intermittent low frequency noise. Intermittency and 1/f µ (µ 1) spectrum imply non stationarity of the noise. Aging behavior of the dephasing factor in the long time limit or in the strong coupling regime are clear signatures of non stationarity. Questions and perspectives More realistic description of non stationary noise experienced by qubits? Clarification of Griffiths effects in this model. Application of this model to other contexts (fluorescence of nanocrystals, Internet traffic, etc). Use of these results as characterization tools for intermittent signals? Typeset by FoilTEX 21