Dephasing by a nonstationary classical intermittent noise

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Josef Schriefl, Maxime Clusel, David Carpentier, Pascal Degiovanni

To cite this version:

Josef Schriefl, Maxime Clusel, David Carpentier, Pascal Degiovanni. Dephasing by a nonstationary

classical intermittent noise. Physical Review B: Condensed Matter and Materials Physics (1998-2015),

American Physical Society, 2005, 72, pp.035328. �10.1103/PhysRevB.72.035328�. �hal-00641673�

Dephasing by a nonstationary classical intermittent noise

J. Schriefl,^1,2M. Clusel,¹D. Carpentier,¹ and P. Degiovanni¹

1CNRS-Laboratoire de Physique de l’Ecole Normale Supérieure de Lyon, 46, Allée d’Italie, 69007 Lyon, France

2Institut für Theoretische Festkörperphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany 共Received 12 January 2005; revised manuscript received 4 April 2005; published 14 July 2005兲

We consider a phenomenological model for a 1 /f^␮classical intermittent noise and study its effects on the dephasing of a two-level system. Within this model, the evolution of the relative phase between the兩±典states is described as a continuous time random walk共CTRW兲. Using renewal theory, we find exact expressions for the dephasing factor and identify the physically relevant various regimes in terms of the coupling to the noise.

In particular, we point out the consequences of the nonstationarity and pronounced non-Gaussian features of this noise, including some anomalous and aging dephasing scenarios.

DOI:10.1103/PhysRevB.72.035328 PACS number共s兲: 73.23.⫺b, 03.65.Yz, 73.50.Td, 05.40.Ca

I. INTRODUCTION

Recent experimental progress in the study of solid-state quantum bits共Josephson qubits兲¹has stressed the importance of low-frequency noise in the dephasing or decoherence of these two-level systems.^2–4It now appears that the coupling to low-frequency noise is the main limitation in obtaining long lived phase coherent states of qubits necessary for quan- tum computation. However, a complete understanding of the microscopic origin of 1 /f noise in solid state physics is not available yet⁵ and therefore, theoretical studies of the dephasing by such a noise are based on phenomenological models. In the spin-boson model, the environment of the qubit is modeled by a set of harmonic oscillators, with an adequate frequency spectrum.^6,7 Another commonly used model for a low-frequency noise consists in considering the contributions from many independent bistable fluctuators.^8,9 In the semiclassical limit, the noise from each fluctuator is approximated by a telegraph noise of characteristic switching rate␥. For a broad distribution⬃1 /␥of switching rates␥^{, a} 1 /f spectrum is recovered when summing contributions of all fluctuators. Such a model is based on observations of telegraph like fluctuations in nanoscale devices,^10,11 but a precise characterization and justification of the broad distri- bution of switching rates is still lacking although the local- ization of these fluctuators^12,13as well as their collective or individual nature¹⁶ have been investigated for a long time.^14,15

In this paper, we consider a phenomenological model for the classical low-frequency noise. This model can be viewed as the intermittent limit of the sum of telegraphic signals. In this limit, the duration of each plateau of the telegraphic signal is assumed to be much shorter than the waiting time between plateaus.¹⁰A 1 /f power spectrum for the intermit- tent noise is then recovered for a distribution of waiting times␶behaving as␶⁻²for large times. Because the average waiting time is infinite, no time scale characterizes the evo- lution of the noise which is nonstationary. The purpose of this paper is to study the effects of such a low-frequency intermittent noise on the dephasing of a two-level system in order to identify possible signatures of intermittence.

As we will show, in this model the relative phase⌽ be- tween the states of the qubit performs a continuous time

random walk^17,18 共CTRW兲 as time goes on. Such a CTRW was considered in the context of 1 /f current noise by Tunaley,¹⁹ extending the previous work of Montroll and Scher on electronic transport.²⁰However, in the present pa- per it is the integral of noise, and not the noise itself, which performs a CTRW. Moreover to our knowledge, the precise consequences of CTRWs nonstationarity on dephasing have not been studied. On the other hand aging CTRW were pre- viously considered in the context of trap models in glassy materials²¹ and in the study of fluorescence of single nanocrystals.^22,23 Technically, the dephasing factor that we will consider corresponds to the average Fourier transform of the positional correlation function of the random walk. Some of the asymptotic behaviors of this correlation function were already obtained in Ref. 21. However, in the present paper we will extend these results to all possible regimes and we will present all of these results in a unified framework. The use of renewal theory²⁴greatly enlightens the origin of non- stationarity and enables us to interpret some features of the dephasing scenario.

This paper is organized as follows: in Sec. II, we present our model for the noise and define the quantity of interest, i.e., the dephasing factor of a two-level system coupled to this noise. In Sec. III, the exact expression for the single Laplace transform of the dephasing factor will be derived and, from this result, the physically relevant weak and strong coupling regimes are identified. Moreover, we clarify the ori- gin of nonstationarity and show the relation of our problem to renewal theory. For completeness and pedagogy, the ef- fects of standard anomalous diffusion of the phase and of randomness of waiting times on dephasing are compared showing the importance of intermittence in the nonstationar- ity properties of the dephasing scenario. In Secs. IV and V, we present a complete study of the behavior of the dephasing factor, respectively, for a noise with a vanishing average am- plitude 共symmetric noise兲 and with a finite average one 共asymmetric noise兲. The general discussion of the results is postponed to Sec. VI.

II. MODEL

A. Pure dephasing by an intermittent noise

In this paper, we consider a quantum bit defined as a two-level system with controllable energy difference ប␻0

1098-0121/2005/72共3兲/035328共16兲/$23.00 035328-1 ©2005 The American Physical Society

and tunneling amplitude ⌬ between the two states 兩−典 and 兩+典 共eigenstates of␴z兲. The effect of the environment on this two-level system will be accounted for by a fluctuating shift បX of the energy differenceប␻0. Thus the Hamiltonian de- scribing this model is written as

H=ប

2共␻0␴z+⌬␴x−X␴z兲. 共1兲 In this paper, we will mainly focus on the case of pure dephasing共⌬= 0兲. However, as explained below in Sec. II C, our discussion will also apply to other operating points共⌬

⫽0兲, including the special points where a careful choice of control parameters considerably lowers the qubit sensitivity to low frequency noise.³

Here, we will focus on the effects of a low-frequency classical noise on the qubit. The noise is represented by a classical stochastic function corresponding to the fluctuations of the noise in a given sample. Within this statistical ap- proach, we focus on the statistical properties共e.g., the aver- age兲of physical quantities associated with the qubit such as the so-called 共average兲 dephasing factor. As we shall see now, its meaning can be understood by considering a typical Ramsey共interference兲experiment on the qubit.^25,26

In such an experiment, the qubit is prepared at initial time t_p in a superposition of the eigenstates of ␴z, e.g., 兩+典

=共兩↑典+兩↓典兲/

冑

2. Note that throughout this paper, t= 0 will correspond to the origin of time for the noise共e.g., the time at which the sample reached the experiment’s temperature兲.

At some later timetp+␶exp⬎tp, we consider the projection of the evolved qubit state on兩↑典. In the meantime, the state has evolved under Hamiltonian 共1兲 共⌬= 0兲 and both states 兩↑典 and兩↓典have accumulated a random relative phase⌽共t_p,␶exp兲 defined by

⌽共tp,␶exp兲=

冕

t_p t_p+␶exp

X共t兲dt. 共2兲

For a given accumulated phase ⌽=⌽共tp,␶exp兲, the quantum probabilityP_⌽,␶

exp共兩↑典兲 to find the qubit in state兩+典at time tp+␶expis given by

P_⌽,␶

exp共兩+典兲=1

2兵1 + cos关␻0␶exp−⌽共tp,␶exp兲兴其. 共3兲 Note that in a given sample,P_⌽,␶

exp共兩+典兲oscillates between 0 and 1 as a function of ␶exp 共although possibly nonperiodi- cally兲. The experimental determination of the probability for finding the qubit in the兩+典state at timetp+tusually requires many experimental runs of same duration ␶exp. The phase fluctuations between different runs induce an attenuation of the amplitudes of these oscillations共analogously to destruc- tive interference effects in optics兲. Using Bayes theorem, the corresponding statistical frequency to find the qubit in state 兩+典after a duration␶expis given by the probability

Pt_p,␶_exp共兩+典兲=

冕

^{d⌽P⌽,␶exp共兩+典兲P关⌽=⌽共tp,␶exp兲兴}

=1

2兵1 +R关Dt_p共␶exp兲e^{−i␻0␶exp}兴其. 共4兲 In this expression, the decay rate of these oscillations is en- coded in the dephasing factorDt_p共␶exp兲 defined as

Dt_p共␶exp兲= exp关i⌽共tp,␶exp兲兴. 共5兲 In this formula共and only here兲, the overline denotes an av- erage over all possible configurations of noiseX共t兲during the experiment.

Note that in deriving Eq.共4兲, statistical independence of the phases⌽between different runs has been assumed. This is not necessarily true for successive runs in a given sample as correlations of the noise might lead to a dependence of the distribution of the phase⌽共t_p,␶exp兲on the starting datet_pof the run. Hence throughout this paper, for self-consistency, we will keep track of this effect through a possible tp depen- dence of the dephasing factorDt_p共␶exp兲. Its possible implica- tions will be discussed together with our results in Sec. VI.

B. Model for classical intermittent noise

In several experimental situations, the low-frequency noise acting on the qubit is supposed to be due to contribu- tions from background charges in the substrate.^8,9When the dephasing is dominated by the low-frequency fluctuators, a semiclassical approach, in which the noise is modeled by a classical field, appears sufficient.^8,9,27In this case, the noise is described by a Dutta-Horn model.²⁸ In its simplest form, the potentialX共t兲is written as the sum of the contributions of many telegraphic signals, each with a characteristic switch- ing rate␥between the up and down states关see Fig. 1共a兲兴. For switching rates distributed according to an algebraic distri- butionp共␥兲⯝1 /␥, the power spectrum of the corresponding noise has a 1 /f low-frequency behavior.

In this paper we propose a different noise model, moti- vated by several noise signatures in various setups.^10–15In- FIG. 1. Representation of a low-frequency noise as a sum of contributions from telegraphic signals 共a兲. In this first case, the switching rates for the up共␥+兲and down共␥−兲states are comparable, and a 1 /fspectrum is recovered for a distribution of switching rates

⬃1 /␥. The intermittent limit共b兲corresponds to the limit where the noise stays in the down states most of the time共␥+Ⰷ␥−兲. In this paper, we will approximate this intermittent noise by a spike field.

For this intermittent noise, a 1 /fspectrum implies a nonstationarity whose consequences on dephasing are studied in this paper.

deed asymmetric telegraphic signals, with longer stays in the down state than in the up state, have been observed in MOS tunnel diodes共see Fig. 3 in Ref. 12兲and tunnel junctions共see Fig. 1 in Ref. 11兲. The extreme asymmetric case of a spike field was even detected in共SET兲electrometer共see Fig. 3 of Ref. 13兲. In all cases, the signal exhibits a 1 /f spectrum at low frequencies. This suggests a description of the 1 /f noise in terms of a single asymmetric telegraphic fluctuator. More- over, as we will show below, such a description is naturally associated with nonstationarity, which has already been re- ported and studied in other experiments on tunnel junctions by Covingtonet al.¹⁴ and Kautz et al.¹⁵ More precisely the amplitude of this 1 /f nonequilibrium noise was shown to decay over several weeks共Fig. 2 of Ref. 15兲. These experi- mental evidences call for a description of a signal which captures the 1 /f spectrum, the intermittence, and the nonsta- tionarity. The model we propose in this paper can be seen as the intermittent limit for the Dutta-Horn model mentioned above. We will focus on the consequence of this intermittent noise on the qubit dephasing. By intermittence, we mean for the case of the telegraphic noise that the switching rate␥+

from the up to the down states is much larger than the switching rate ␥− from the down to the up states 共or vice versa兲. In this limit, the total noise reduces to a collection of well defined events, separated by waiting times␶i关see Fig.

1共b兲兴. If considered on times much longer than the typical duration ␶0 of these events 共or at frequencies smaller than 1 /␶0兲, we can approximate this noise by a spike field con- sisting of a succession of delta functions of weightx_icorre- sponding to the integral over time of the corresponding events of the intermittent field关Fig. 1共b兲兴.

More precisely, denoting byt= 0 the origin of time, and by ␶i the successive waiting times between the spikes 共or events兲, we know that the nth spike occurs at time t_n

=兺i=1,n␶i. The value of the stochastic intermittent classical noiseX共t兲 共see Fig. 2兲is then

X共t兲=

兺

_i ^{xi␦共t−ti兲. 共6兲}

In the following, we will consider the dephasing produced by this spike field. We expect any short time details like the specific shape of the real pulses to be irrelevant in the limit of typical dephasing time long compared to ␶0. Within this approximation, a noise signal is fully characterized by the collection of waiting times ␶i and pulse amplitudes xi that occur as time goes on. We will assume these two quantities

to be independent from each other, and completely uncorre- lated in time. We will then characterize such a noise solely by two independent probability distributions ␺共␶兲 and p共x兲 for the␶iandxi, respectively.

The distribution of pulse amplitudesp共x兲will be assumed to have at least its two first moments finite, and denoted in the following by

h=¯x=

冕

_−⬁^{+⬁xp共x兲dx, 共7a兲}

g=

冑

x²=

冉 ^冕

−⬁

+⬁

x²p共x兲dx

冊

^{1/2. 共7b兲}

We will consider separately the case of zero average共sym- metric noise兲 and of nonzero average 共asymmetric noise兲 since the latter induces specific features of the dephasing factor, discussed in Sec. V.

We will consider algebraic distribution of waiting times

␺共␶兲, parametrized by a single parameter␮

␺共␶兲= ␮

␶0

冉

␶0^␶+⁰␶

冊

^{1+␮, ␶⬎0. 共8兲}

As we will show in Sec. II D, a 1 /f^␮power spectrum for this intermittent spike field follows naturally from such a choice for ␺共␶兲. As we shall see in this paper, the above algebraic distribution of waiting times allows us to correctly capture the essential features of the dephasing scenario generated by an intermittent noise. The first and second moments of␺are finite, respectively, for␮⬎1 and ␮⬎2 and are given by

具␶典= ␶0

␮− 1; 具␶²典= 2␶₀²

共␮− 1兲共␮− 2兲. 共9兲 Let us note that the above defined-pulse noise contains two independent potential sources of dephasing: The randomness of the pulse weights and the randomness of the waiting times. Their respective effects will be compared in Sec. III.

C. Decoherence at optimal points

Before turning to the detailed study of pure dephasing 关⌬= 0 in共1兲兴, let us mention that our discussion can be easily extended to the study of dephasing in the presence of a trans- verse coupling in Eq.共1兲, in particular at the so-called opti- mal points. They correspond to configurations where the fluctuations of the effective qubit level splitting are only qua- dratic in the noise amplitude. The qubit can be operated at these optimal points by a careful choice of the control pa- rameters ␻0 and ⌬ of the qubit and then, the influence of low-frequency noise can be reduced considerably.³ For the Hamiltonian considered in the present work共1兲such an op- timal point is reached for transverse coupling to the noise 共␻0= 0 and⌬⫽0兲. In this case and assuming the amplitude of the noise to be small compared to the control parameter⌬, the effective qubit level splitting is given by

冑

^{⌬2+X共t兲2}

⬇⌬+X共t兲²/共2⌬兲. Hence, the dephasing effect of a linear transverse noise can be accounted for using an effective qua- dratic longitudinal noise. The corresponding dephasing fac- FIG. 2. Representation of the random spike field used to model

the intermittent low-frequency noise in this work. This noise is described by the distributions of the phase pulses x_i and of the waiting time intervals␶ibetween the spikes.

tor is then given by Eqs. 共2兲 and 共5兲 with the replacement X→X²/共2⌬兲.

In addition, the transverse noise at an optimal point in- duces transitions between the eigenstates of the qubit, i.e., it leads to relaxation. The Fermi golden rule relaxation rate⌫r, which is used for estimating the relaxation contribution to dephasing, involves the power spectrum of the noise at the resonance frequency of the qubit.¹The total dephasing rate is obtained by summing the contribution of relaxation given by

⌫r/ 2 and the contribution of pure dephasing due to the above effective longitudinal noise.

In general the statistics ofXandX²are very different and a special treatment is needed to derive the dephasing factor at optimal points.⁶ However, for the noise considered in the present work, the effective quadratically coupled longitudi- nal noise can be viewed as en effective linearly coupled noise of the same type but with renormalized parameters.

The renormalized distribution of the spike intensities is now given by共x艌0兲

p

˜共x兲=

冑

^⌬8x^␶0兵p共

冑

2x⌬␶0兲+p共−

冑

2x⌬兲其, 共10兲 where␶0is a microscopic time scale needed to regularize the square of delta functions. Therefore, our results for the lon- gitudinal noise presented in Sec. II A can also be used to describe the effect of a transverse noise.

D. Spectral properties of the intermittent noise 1. One and two point correlation functions

To make contact with other descriptions of low-frequency noise, we will determine the behavior of the two-time corre- lation function of our noise, or equivalently of its spectral density. However, as we will see, this spectral density is far from enough to characterize the statistics of the noise for small␮, in particular due to its nonstationarity. We will con- sider for simplicity the case of a nonzero averageh=¯.x

a. Time dependent average. Let us consider the average of the noise amplitudeX共t兲. In our case, it reduces to two inde- pendent averages: over the amplitudes x_i of the spikes and over the waiting times ␶i between them. Noting that X共t兲 vanishes except if a spike occurs at timet, we can express its average in terms of the average densityS共t兲of pulses at time talso called the sprinkling time distribution in Ref. 29

具X共t兲典=h S共t兲. 共11兲 Using the expressions共A4兲and共A5兲forS共t兲 共see Appendix A兲, we obtain for the average of X

具X共t兲典= h 具␶典= h

␶0

共␮^{− 1}兲 for␮⬎2, 共12a兲

具X共t兲典= h

具␶典

冋

^{1 +}

^冉

^␶t0

^冊

^␮−1

册

^{for 1⬍␮⬍2, 共12b兲}

具X共t兲典=sin共␲␮兲

␲ h

␶0

冉

^␶t0

冊

^{1−␮ for␮⬍1. 共12c兲}

Hence the nonstationarity of the noise manifests itself al- ready in the time dependence of this single time average.

While it is only a subleading algebraic correction for 1⬍␮

⬍2, this time dependence becomes dominant for 0⬍␮⬍1.

b. Two time function. Following the same lines of reason- ing, we can derive the expression of the two time correlation functions共witht⬎0兲

具X共tp兲X共tp+t兲典=h²S共tp兲S共t兲. 共13兲 The first factor S共t_p兲 corresponds to the probability that a spike occurs at timet_p, while the second factorS共t兲reads the probability of having a pulse at timet_p+t, knowing that there was one at tp. This reflects the reinitialization of the noise once a spike has occurred at timetp. From Eqs.共11兲and共13兲, we obtain the connected two points functions

C共t_p,t兲=具X共t_p兲X共t_p+t兲典c=h²S共t_p兲关S共t兲−S共t_p+t兲兴. 共14兲

2.1 /f noise spectrum

We will define the spectral density of the noise as the Fourier transform of the connected two points functions re- stricted tot⬎0

SX共tp,␻兲= 2

冕

0

+⬁C共tp,t兲cos共␻t兲dt. 共15兲

The correlation functionC共tp,t兲generically depends on both times tp and tp+t thus showing that in general X is not a stationary process. However, to extract the low-frequency behavior of SX共tp,␻兲, it appears sufficient to consider the quasistationary regime 兩t兩Ⰶtp which corresponds to experi- ment durations much smaller than the age of the noise. In this regime, the connected correlation function共14兲reduces fort⬎0 to

C共t_p,t兲 ⯝h²S共t_p兲关S共t兲−S共t_p兲兴. 共16兲 The associated effective power spectrum is defined for fre- quencies␻large compared to 1 /t_p and reads

SX共tp,␻兲 ⯝2h²S共tp兲R„L关S兴共−i␻兲…, 共17兲 whereL关S兴denotes the Laplace transform ofS共t兲. Note that in this quasistationary regime, the nonstationarity of the noise manifests itself only through thetp dependent ampli- tudeS共tp兲. Using explicit expressions forL关S兴 共see Appendix A兲, we obtain the effective power spectra

S_X共t_p,␻兲 ⯝

冋

^{h2S共tp兲cos⌫共1 −共␲␮␮/2兲兲}

册

^{共␻␶0兲−␮ 共18兲}

for 0⬍␮⬍1, and

S_X共t_p,␻兲 ⯝

冋

^{h2S共tp兲sin}

^冉

^{␲共␮2− 1兲}

^冊

^{共␮− 1兲␮−1}

册

^{共␻具␶典兲␮−2}

共19兲 for the intermediate class 1⬍␮⬍2. The common 1 /␻ ^de- pendence of the spectral density is recovered in the limit␮

→1. More precisely, the Laplace transform of S for ␮= 1, obtained in Appendix A, gives a logarithmic correction to a 1 /␻effective power spectrum

SX共tp,␻兲 ⯝ ␲^h2S共tp兲

␻␶0关log共␻␶0兲兴². 共20兲 Let us stress finally that this effective power spectrum, al- though useful to compare our approach with other noise models, is not sufficient to characterize the statistical prop- erties of the spike field noise relevant for dephasing. As we shall see in this paper, it does not account precisely for the nonstationarity of the dephasing factor. Besides this, non- Gaussian properties of the noise have strong effects on the dephasing factor in many regimes.

III. DEPHASING, CONTINUOUS TIME RANDOM WALK OF THE PHASE AND RENEWAL THEORY A. Continuous time random walk of the phase Having defined our model for the intermittent noise field X共t兲, we will now study its effects on the dephasing of the two-level system, characterized by the dephasing factor共5兲.

Note that a priori this dephasing has several origins: the randomness of the pulses amplitudesx_i, and the randomness of the waiting times ␶i. These two dephasing sources are assumed to be independent from each other in this work. For clarity and pedagogical reasons, we first start by considering the effect of randomness of the pulse amplitudes before turn- ing to the effect of waiting time randomness.

1. Dephasing by random phase pulses: The random walk of the phase

Here, we assume that all waiting times are equal ␶i=ti

−t_i−1=˜. In this case, the phase␶ ⌽共tp,t兲accumulated between

tp andtp+t performs a usual random walk characterized by the distribution p共x兲 of phase pulses: At the dates t_n

=n^˜␶,⌽共tp,t兲 is increased by a random valuexi. In the limit 兩x兩Ⰶ1, the phase slowly diffuses and dephasing is achieved only after a large number of pulsesn=tn/˜. Then, the distri-␶ bution of the phase⌽共tp,tn兲 can be well approximated by a Gaussian distribution共apart from some irrelevant tails兲and the dephasing factor is easily computed. It decays exponen- tially with characteristic time␶␾= 2^˜␶^/共g²−h²兲.

Note that this diffusive regime can also be studied when the distribution p共x兲 lacks a finite second or even first mo- ment. An anomalous diffusion of the phase is expected.³¹Let us assume thatx_i=h+␬␰iwhere the probability distribution of␰ihas zero average and belongs to the attraction bassin of the stable lawL_␯,␤characterized by the exponent 0⬍␯⬍2 of the algebraic tails for large values of ␰iand the asymmetry parameter兩␤兩艋1. Then, according to the generalized central limit theorem, the accumulated phase after n pulses is ⌽n

=nh+␬n^1/␯␰^where␰is distributed according toL_␯,␤. Conse- quently, in the diffusive limit共nⰆ1兲, the dephasing factor is the product of the homogenous phasee^iht/␶˜by the character- istic function of L_␯,␤ evaluated for k⯝g共t/^˜␶兲^1/␯. For ␯⬍2 and␯⫽1, this leads to

Dt_p共t兲=e^{−C␬␯t/␶˜}e^{i关h−C␤tan共␲␯/2兲␬␯兴t/␶˜}, 共21兲 whereCis a numerical constant which can be absorbed in a rescaling of␬. Thus, in this case, the decay is still exponen- tial and stationary although non-Gaussian features of the noise lead to an anomalous dependance of the dephasing time on the coupling constant␬ that characterizes the scale of the fluctuations of phase pulses.³⁰

2. Dephasing by random waiting times and continuous time random walk

Let us now turn to the situation where the phase pulses happen at random times. In this case, the accumulated rela- tive phase⌽共tp,␶兲 关see Eq.共2兲兴does not perform a random walk as ␶ increases, but rather a continuous time random walk^17,18 共CTRW兲 on the unit circle. In other words, after FIG. 3. A configuration of the noiseX共t_p+␶兲 in our model 共as a function of ␶兲 between t_p

= 10⁵␶0 andt_p+␶= 1.1 10⁵␶0. In this figure, h= 0 and the waiting times are distributed with an al- gebraic distributionP共␶i兲⯝␶i

−2.1. The bottom part of the figure shows the corresponding continuous time random walk of the accumulated phase of the TLS betweent_pandt_p+␶.

some random waiting time␶i,⌽共t_p,␶兲 is incremented by a random valuex_i共see Fig. 3兲. Thus, on a technical level, the dephasing properties of the two level system are now related to some correlation function of the corresponding CTRW.

The corresponding dephasing factor can differ from results obtained in the previous section due to the additional source of dephasing given by the randomness of waiting times. To illustrate this point, let us consider the case where all phase pulses have the same intensity p共x兲=␦共x−h兲. The accumu- lated phase after N events is exactly Nh. Thus, dephasing comes only from the randomness of the numberN关tp,tp+t兴 of events occurring betweent_pandt_p+tand the phase diffu- sion is governed by the probability distribution for N关t_p,t_p +t兴. The time needed to obtainNevents is the sum over the Nfirst waiting times aftertp:tN=兺^N_j=1␶j. At tp= 0, all␶is are distributed according to the same probability distribution and therefore, in the limit of largeN, the generalized central limit theorem³¹can be used共the case tp⫽0 will be discussed be- low兲. It provides the limit law fortNat largeNwhich in turn determines the probability law for the number of events. Ac- cording to this theorem, three classes must be considered depending on whether the moments共9兲are finite:

1. The case␮⬎2 where both具␶典 and具␶²典 are finite. In this case, the probability distribution for the number of events is Gaussian with a vanishing relative uncertainty.

2. The case 1⬍␮⬍2 where the average of␶is finite but the second moment diverges.

3. The case␮⬍1 where all moments diverge.

The usual model for telegraphic noise assumes a Poissonian distribution for the number of events in a given time interval and corresponds to␺共␶兲=␥^e−␥␶. We will refer it as thePois- sonian caseand it belongs to the␮⬎2 class.

Note that nontrivial behavior is expected in the last two cases where␺ has infinite first or second moments. In par- ticular, for␮⬍1, as␺共␶兲does not have any average, no time scale characterizes the evolution of the noise and, as we will see, nonstationarity follows. This case deserves a special at- tention as we showed that a 1 /f spectrum is found precisely for␮→1. Before turning to the general formal expressions for the dephasing factor, we will focus on the origin of this nonstationarity.

3. Origin of the non-stationarity in CTRW: The first waiting time distribution

Within our model, the waiting times between successive pulses are chosen independently according to the distribution

␺. Consequently, all thetp dependence of⌽共tp,t兲will come from the choice of␶1defined as the waiting time betweentp

and the first spike that followstp共see Fig. 4兲. Indeed, at time t_p+␶1, the CTRW starts anew:␶2is chosen without any cor- relation to the history of the CTRW. Hence given the prob- ability distribution of␶1, we can forget about the history of the CTRW of the phase and describe its behavior starting at tp. This remark is at the core of the use of renewal theory. In the following, the probability distribution of ␶1 will be de- noted by␺t_panda priori, it may depend ontp. In fact, as we shall see later, its behavior can be quite counter-intuitive.

First of all, note that a given ␶1 can be obtained from many different noise configurations that differ from the time of the last event occurring beforetp. Separating noise con- figurations共starting att= 0兲that have their first spike at time t_p+␶1 from the others leads to an integral equation that de- termines␺t_p in terms of␺ 关S is determined from␺through an integral Eq.共A1兲兴

␺t_p共␶1兲=␺共t_p+␶1兲+

冕

0 t_p

d␶ ␺共␶1+␶兲S共t_p−␶兲. 共22兲

The integral in the right-hand side共r.h.s.兲comes from noise configurations that have a spike between 0 andtp. Equation 共22兲is the starting point for deriving analytic results about

␺t_p in Appendix B and C using Laplace transform tech- niques. Before computing exactly the dephasing factor, let us show some of the counter-intuitive properties of ␺t_p共␶1兲. In particular from Eq.共22兲, we can derive the following expres- sion for the average of␶1 valid for␮⬎2:

具␶1典_␺t

p

= 具␶²典 2具␶典=具␶典

2 +具␶²典−具␶典²

2具␶典 . 共23兲 This first term corresponds to the case of regularly spaced pulses averaged over the origin of times. The second term is the contribution of fluctuations. This result means that ir- regularities in the event spacings increase the average wait- ing time of the first event followingt_p. The r.h.s. of Eq.共23兲 does not depend ontp, as expected from the stationarity of the CTRW for␮⬎2 after a short transient regime at smalltp. On the other hand, for ␮⬍2, Eq. 共23兲 is expected to break down since 具␶²典 diverges. This divergence signals that in some noise configurations␶1 can become of the order of tp

and, as a consequence, the average properties of the CTRW aftert_p, depend on this age of the noise. Indeed, we can show from Eq. 共22兲 that 具␶1典_␺t

p

scales with the age as t^2−␮_p . Note that in the diffusion regime, many phase pulses are necessary to dephase the qubit. Therefore, we expect that, in this re- gime, the aging effect on ␺t_p共␶1兲 only brings weak correc- tions to the dephasing scenario as will be confirmed below by exact computations. However, we shall see in the follow- ing that the tp dependence of␺t_p for ␮艋1 has much more spectacular consequences on the dephasing factor than in the 1⬍␮⬍2.

We will now show that, knowing the Laplace transform of

␺t_p, an explicit expression for the Laplace transform of the average dephasing factor can be obtained.

FIG. 4. Intermittent noise betweent_pandt_p+t.

B. Exact dephasing via renewal theory 1. Dephasing factor

Among all noise configurations that are to be taken into account, some of them共possibly very few兲 do not have any event between tp and tp+t. Their total weight is given by

⌸0共t_p,t兲which is

⌸0共tp,t兲=

冕

t +⬁

␺t_p共␶兲d␶^. 共24兲 All other histories have at least one event between tp and tp+t. Let us assume that it happens at tp+␶ ^where ␶ ^lies between 0 andt. Then after this event, the noise starts anew.

The jump itself contributes bye^ixto the dephasing factor and the rest of the noise configuration contributes by D₀共t−␶兲 关i.e.,tp= 0 in Eq.共5兲兴. The probability that the first event after tp happens at time tp+␶ is nothing but ␺t_p共␶兲. Hence the contributions toDt_p共t兲from all possible noise configurations take the form of the following renewal equation:

Dt_p共t兲=⌸0共tp,t兲+e^ix

冕

0 t

␺t_p共␶兲D0共t−␶兲d␶^. 共25兲 Note that the Laplace transform of this expression is very simple

L关Dt_p兴=L关⌸0兴+共1 −f兲共L关␺t_p兴·L关D0兴兲, 共26兲 where f= 1 −e^ix. Specializing tp= 0, one gets an expression forD₀共t兲that contains⌸0共0 ,t兲. Since⌸0共0 ,t兲=兰t⬁␺共␶兲d␶an explicit expression for the Laplace transform ofD₀ can be found

L关D0兴=1 s

1 −L关␺兴

1 −共1 −f兲L关␺兴. 共27兲 Plugging expression共27兲into Eq.共26兲gives

L关Dt_p兴=1

s

冉

^{1 −1 −}共1 −^fL关␺f^tp兲L关^兴 ␺兴

冊

^{. 共28兲}

This exact expression will be extensively used to derive both analytical expressions and numerical plots by Laplace inver- sion. Before proceeding along, let us expressL关Dt_p兴in terms of the density of events. It is then convenient to define St_p共t兲=S共tp+t兲 共see Appendix A兲. From the expressions in this appendix ofL关St_p兴andL关S兴in terms ofL关␺兴andL关␺t_p兴, we obtain

L关Dt_p兴=1

s

冉

^{1 −1 +fL关S}fL关S兴^tp兴

冊

^{. 共29兲}

Equations 共28兲 and 共29兲 enable analytical estimates of the dephasing factor in various limiting regimes. Equation共29兲 is useful in the limit of small coupling共i.e., f→0兲 whereas Eq.共28兲will be useful in the opposite limit of a wide distri- bution of spikes’ heights共i.e., f→1兲.

2. Weak and strong coupling regimes

In the limit of very strong coupling 共f= 1兲, the phase spreading of a single spike is sufficient to dephase the qubit.

In this case, Eq.共25兲 immediately leads toDt_p共t兲=⌸0共tp,t兲.

This means that the whole average is dominated by the rare noise configurations that do not evolve during the experi- ment. As we shall see later, this leads to a strong t_p depen- dence ofDt_p共t兲for ␮⬍2.

At much lower couplings共兩f兩Ⰶ1兲, it is nota priori clear whether the renewal equation forD_t

p共t兲induces a strong de- pendence ofD_t

p共t兲ont_p. Pushing the above analysis forward amounts to comparing the time dependence of the two terms in the r.h.s. of the renewal Eq.共25兲. As we will see now, it will provide a better understanding of the physics underlying the dephasing scenario.

Let us first assume that⌸0共tp,t兲decays much faster than D₀共t兲. Since ␺t_p共␶兲= −⳵␶⌸0共tp,␶兲, it also decays much faster than D₀. Consequently, we can approximate ␺t_p共␶兲D0共t−␶兲 by␺t_p共␶兲D0共t兲 in Eq.共25兲. Then, after a short initial decay due to bothD₀共t兲and⌸0共tp,t兲,Dt_p共t兲decays asD₀共t兲. Con- sequently, the dephasing time does not depend on t_p. For instance, we expect this situation to occur at weak coupling when the average waiting time具␶典 is finite since the prob- ability that no event occurs between tp and tp+t vanishes quite fast whentincreases. Note that in the limit of vanish- ing coupling, the spreading of the phase can become much slower than the decay of⌸0共tp,t兲and thus our starting point hypothesis is valid.

The opposite case where the decay ofD₀共t兲is much faster than the decay of ⌸0共tp,t兲 can be discussed more conve- niently by integrating the renewal Eq.共25兲by parts. Consid- ering againgⰆ1 we get

Dt_p共t兲 ⯝D₀共t兲−

冕

0 t

⌸0共tp,t−␶兲D₀⬘共␶兲d␶^. 共30兲 Approximating⌸0共tp,t−␶兲⬇⌸0共tp,t兲in this equation yields Dt_p共t兲=D₀共t兲+⌸0共tp,t兲关1 −D0共t兲兴. As a consequence, once D₀共t兲 has decayed, the dephasing factor is almost equal to

⌸0共tp,t兲. This is the same behavior than in the very strong coupling regime f= 1, although here, we assumed 兩f兩Ⰶ1.

This regime is expected to occur when the decaying time scales ofD₀共t兲and of⌸0共tp,t兲are comparable. This is obvi- ously the case at very strong coupling but, surprisingly, as we shall see now it can also be obtained for兩f兩Ⰶ1.

First of all, the above discussion shows that such a regime can only happen if the average waiting time is infinite, i.e., for␮⬍1. In this case,⌸0共t_p,t兲can be evaluated analytically 关see Eq. 共D1兲兴: It is shown to be independent of ␶0 and to exhibit aging behavior共i.e., to depend only ont/tp兲. There- fore, the decaying time scale of⌸0共tp,t兲 scales astp. Com- paring this time scale with the dephasing time scale for兩f兩 Ⰶ1, leads to a t_p dependent crossover coupling constant f_c共t_p兲. In the case of aging⌸0, the crossover coupling decays to lower and lower values by increasingtp.

For兩f兩ⲏfc共tp兲, the dephasing factor behaves as ⌸0共tp,t兲 and the dephasing time saturates as a function of f. Such a saturation of the dephasing time as a function of the ampli- tude of the noise has already been discussed for a Poissonian fluctuator.^9,27 In this case ⌸0共t_p,t兲=⌸0共0 ,t兲=e^{−t/具␶典} decays very fast, on a time scale具␶典. The crossover between weak

and strong coupling regime happens precisely at the point where the dephasing time assuming weak coupling具␶典/g²is of the same order as this decay time, i.e., forg⬃1. Note that in this case, as expected from our discussion, the crossover scale is independent of the age of the noisetp. We expect this reasoning to break down in our model because of the broad distribution of waiting times共␮⬍2兲. Understanding the vari- ous dephasing scenario and computing thetp dependence of the crossover coupling in our case requires the computation of ⌸0共t_p,t兲 and of D₀共t兲. These quantitative results will be presented in forthcoming sections.

To summarize the above discussion, we have argued that the dephasing time is bounded by the typical decay time of both ⌸0共tp,t兲 and D₀共t兲. This suggests distinguishing be- tween two regimes: On one hand, aweak coupling regimefor whichD₀共t兲decays much slower than⌸0共tp,t兲and for which the dephasing time—in that case just the decay time of D₀共t兲—is not sensitive to tp. On the other hand, a strong coupling regimefor whichD₀共t兲decays faster than⌸0共tp,t兲.

In that case, the dephasing time is given by the decay time of

⌸0共tp,t兲 and thus possibly tp dependent. As the above dis- cussion shows, the crossover between these two regimes may happen for a possibly t_p dependent crossover coupling fc共tp兲Ⰶ1. Note that in the strong coupling regime, the dephasing time becomes independent of the amplitude of the noise whereas in the weak coupling regime, it is expected to depend on the amplitude and to diverge in the vanishing coupling limit.

IV. DEPHASING IN THE SYMMETRIC MODELS In this section we present our results for the situation of a symmetric distribution of the spikes,p共−x兲=p共x兲. Under this assumption the average random accumulated phase vanishes, i.e., f and consequently Dt_p共t兲 are real. ForgⰆ1 , f may be expanded in moments ofp共x兲,f⬇g²Ⰶ1 whereg measures the typical scale of the fluctuations of the spikes.

A. Dephasing att_p= 0

Before discussing the decoherence factor for arbitrary preparation timetp, it is useful to investigate the casetp= 0.

Rewriting the Laplace transform ofD₀共t兲 共27兲as L关D0兴共s兲=1

s

冉

^{1 +1 −fL关}L关^␺␺^兴兴

冊

^{−1 共31兲}

suggests to introduce a scale␥grelated to the strength of the coupling

冏

^{1 −fL}L关^关␺␺^兴兴

冏

_s=␥g^{= 1. 共32兲}

Note that␥gvanishes with the coupling g. Investigating the behavior of D₀ for tⰆ␶0 requires evaluating the Laplace transform fors␶0Ⰶ1. In this regime, the Laplace transform L关␺兴/共1 −L关␺兴兲can be approximated by 1 /共1 −L关␺兴兲. Within this approximation, we shall now derive explicit expressions for L关D0兴 which can be Laplace inverted explicitly. This

leads to expressions forD₀共t兲valid at tⰇ␶0 for the different classes of␮^.

In the case of finite average waiting time共␮⬎1兲, we can expand L关␺兴⬇1 −s具␶典 to find the leading contribution to D₀共t兲for 具␶典Ⰶ␥gt⬍1. This gives

D₀共t兲 ⯝L⁻¹

冋

^{s具␶具典␶典+f}

册

^{=e−t/␶␾, 共33兲}

with the dephasing time␶␾=␥g−1=具␶典/f 关see Eq.共32兲兴. Note that this expression is exact only for ␺共␶兲= 1 /具␶典e^{−␶/具␶典}. At finite noninteger ␮, taking into account the fluctuations of the waiting times requires keeping all terms in 1 −L关␺兴up to the first noninteger power共s␶b兲^␮. For 1⬍␮⬍2, we get alge- braic subleading corrections to Eq. 共33兲: log关D0共t兲兴⯝−␥gt

−fc共␮兲共t/␶0兲^2−␮, where c共␮兲=共␮− 1兲/共2 −␮兲. These correc- tions being weak for gⰆ1, the dephasing time ␶_␾ ^remains equal to␥g

−1=具␶典/g²in this regime.

For ␮⬍1, the first term in the expansion of 1 −L关␺兴 is proportional to 共␶0s兲^␮. Therefore, plugging in 共1

−L关␺兴兲/L关␺兴⯝⌫共1 −␮兲共s␶0兲^␮ and performing the inverse Laplace transform of共31兲gives

D₀共t兲 ⯝L⁻¹

冋

^{1 +fs共s−1␶0兲−␮}

册

^{=E␮关−共␥gt兲␮兴, 共34兲}

where E_␮共z兲=兺_n=0^⬁ zⁿ/⌫共1 +␮n兲 denotes the Mittag-Leffler function.³² For values of␮close to one and zⱗ1 this func- tion can be approximated by a simple exponential, whereas for ␮⬃0 it is similar to an algebraic function E_␮共z兲⬃共1 +z兲⁻¹. For large values of the argument 共兩z兩Ⰷ1兲 and 兩arg共−z兲兩⬍共1 −␮/ 2兲␲, we obtain³² E_␮共z兲⬇关−z⌫共1 −␮兲兴⁻¹. This change of behavior from an exponential to an algebraic behavior was interpreted in Ref. 33 as a Griffith effect.

Computing long time behavior共␥gtⰇ1兲of the dephasing factor can be done by expanding Eq.共31兲as follows:

L关D₀兴共s兲 ⯝1 s

1 −L关␺兴

fL关␺兴 . 共35兲

For␥gtⰇ1共sⰆ␥g兲we can safely replaceL关␺兴⬇1 in the de- nominator. The inverse Laplace transform can then be done easily and reads for␥gtⰇ1

D₀共t兲 ⯝1 f

冕

t

⬁␺共␶兲d␶=1

f

冉

␶0^␶+⁰ t

冊

^{␮. 共36兲}

For␮⬍1, Eq.共36兲is nothing but the asymptotic behavior of Eq.共34兲for ␥gtⰇ1. Note that forgⰇ1 ,␥g−1 is of the order of␶0 and the decay is algebraic at all timest⬎␶0, given by Eq. 共36兲. For 1⬍␮⬍2 and gⰆ1, only at large times 关␥gt

⬃ln共1 /f兲兴, when the qubit has almost completely dephased, the decay crosses over to the power law共36兲.

B. Influence of a finite preparation timet_p

We will now discuss in detail the tp dependence of the dephasing scenario and of the crossover coupling strengthg_c for the different classes of␮. Simple analytical expressions forDt_p共t兲 can be derived in the weak共gⰆgc兲and strong 共g Ⰷgc兲 coupling regimes.