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Classical telegraph noise describes a discrete stochastic process denoted byQ(t),t∈[0,∞], where

the random variable Q can only have two possible values, Q(t'_{) =0/1, at some certain time} t'_{. Jumps between the two possible states of Q occur at a switching rate γ} cl _{such that in a} time-interval [0,t] the average number of flips is equal to γ clt. Figure. 6.2 (left panel) shows

a realization of the time evolution of Q(t) for increasing switching rate γ cl _{(from top to bot-} tom). Surprisingly, the regime where the fluctuator rarely flips in a given time-interval is most interesting and shows many unexpected features as compared to Gaussian noise.

Telegraph noise has the Markoff-property, i.e. in a given time-interval the probability for a flip to occur is independent of the history of preceding jumps. Furthermore, telegraph noise is an example of non-Gaussian noise. In contrast to the case of Gaussian-distributed noise where the two-point correlation function!Q(t)Q(0)"of the random variableQ(t)determines all higher

38 DECOHERENCE BY CLASSICAL TELEGRAPH NOISE 6.2 Frequencyω δQ(t) t δQ(t) t (a) (b) (c)

!δQδQ"

Figure 6.1:Two types of stochastic processes with the same noise-power: Shown are the time evolution for two types of stochastic processesδQ(t)which have the same noise-spectrum !δQδQ"ω. (a)

The upper panel displays Gaussian Lorentzian noise and in comparison with (b) non-Gaussian telegraph noise (lower panel). (c) Lorentzian noise-power !δQδQ"ω for the stochastic pro-

cesse displayed in (a), (b).

correlation functions, for non-Gaussian noise the knowledge of all higher correlation functions is necessary1_.

It is instructive to calculate the probability for the fluctuatorQ(t)to switchntimes between the two possible states during the time-interval[0,t]. Let us divide the interval[0,t]intoNpieces

of equal lengthδt such thatt =Nδt. The probability for the fluctuator to flipntimes inN trials is given by a binomial distribution

Pn(N) = _n!(NN! −n)!p

n_(1−p)N−n_{, (6.2)}

where p is the probability for the fluctuator to flip during the time-interval δt. The expected number of flips is ν =N p, taking the limit N →∞ while keeping ν constant we obtain the expected Poisson distribution for the fluctuator to switchntimes after the elapsed timet

Pn(t) = (γ cl_t)n

n! e−γ

cl_t

, (6.3)

whereγ cl_{=p/δt is theswitching rateof the fluctuator. The average number of flip-flops is then}

µn(t) =γ clt and the varianceσn2(t) =γ clt. Correlations between jumps fall off exponentially in time and the two-point correlation function is equal to

!δQ(t1)δQ(t2)"= ₄1e−2γcl|t1−t2|_{, (6.4)}

1_{This means for a Gaussian processX(t)with zero mean!X(t)"}

Gauss=0 and the two-point correlation function C(t1,t2) =_!X(t1)X(t2)_"Gauss:

6.2 CLASSICAL TELEGRAPH NOISE 39 whereδQ(t) =Q(t)_{− !}Q_"and!Q"=1/2. The time-scale on which these correlations decay is

set by the inverse of the switching rateτc=1/γ cl. The power-spectrum defined as the Fourier transform of the correlation function,!δQδQ"ω =0dteiωt!δQ(t)δQ(0)"of the charge fluctua- tions has a Lorentzian shape

!δQδQ"ω = γ

cl

ω2_{+ (2γ} cl₎2. (6.5)

Figure 6.1 displays possible realizations of the stochastic process δQ(t) for Gaussian (a) and non-Gaussian noise (b) which both have the same noise-spectrum!δQδQ"ω 2.

Let us start with a semi-classical analysis of the quantum model of a two-level system coupled to the charge fluctuations of a localized impurity level tunnel-coupled to a fermionic reservoir [see Eq. (5.1)]. In contrast to the full quantum mechanical model we now assume the charge on the impurity ˆQ to be a classical stochastic process of the “telegraph noise” type, which flips randomly between 0/1 at a rateγ cl. Models of this type have been studied in [42, 43] discussing

spectral difussion in glasses and in [34, 35] considering dephasing of qubits.

The switching rate γ cl _{can be derived from the full quantum mechanical model by second} order perturbation theory in the tunneling amplitudeTk. In the full quantum mechanical model the switching rate is connected to the classical rate by the relationγ cl_=γ(1−f

T(ε0)). In order to compare the classical results with the full quantum mechanical model [see Chap 7], we perform all calculations in this chapter at the rateγ cl_{=γ/2 forε}

0=0.

The two-level system coupled longitudinally to the random variable Q(t) experiences ran- domly fluctuating energy-levelsε_±(t) =_±(∆+vQ(t))/2. Averaging over these temporal fluctu-

ations results in the decay of an initially prepared coherent superposition of quantum states. For simplicity we assume the heat-bath to act like a classical potential, thereby neglecting the back- action of the qubit on the environment. The Hamiltonian in the semi-classical approximation then reads ˆ H =∆ 2σˆz+ v 2Q(t)σˆz, (6.7)

whereQ(t) =0/1, ∆ is the splitting between the qubit’s eigenenergies,vis the coupling of the

qubit to the fluctuator and ˆσz is the Pauli matrix acting on the qubit’s eigenstates. For a given realization of the random potentialQ(t)the solution of the time-dependent Schrödinger-equation is a superposition of the qubit’s eigenstates{|↑",_|↓"},

|ψ(t)"= √1 2

#

e−i∆t/2_eiϕ(t)/2_|↑"+ei∆t/2_e−iϕ(t)/2_|↓"$_{, (6.8)}

2_{The Gaussian stochastic process corresponds to anOrnstein-Uhlenbeckprocess{X}

t:t≥0}which is the solution

of the stochastic differential equation

dXt=−ρ(Xt−µ)dt+σdWt, (6.6)

where{Wt:t≥0}is a Wiener-process of unit variance andρ,µ,σ are constants. The mean!Xt"=µand variance

!δXtδXs"= (σ2/(2ρ)e−ρ|s−t|, whereδXt=Xt− !Xt". Figure 6.1(a) has been calculated using the Euler-Murayama

40 DECOHERENCE BY CLASSICAL TELEGRAPH NOISE 6.2 0 1 2 3 4 5 6 7 8 9 10 -5 -2.5 0 2.5 5 0 1 2 3 4 5 6 7 8 9 10 -1 0 1 0 1 2 3 4 5 6 7 8 9 10 -5 -2.5 0 2.5 5 0 1 2 3 4 5 6 7 8 9 10 -5 -2.5 0 2.5 5 0 1 2 3 4 5 6 7 8 9 10 -1 0 1 0 1 2 3 4 5 6 7 8 9 10 -1 0 1 Timet Phase ϕ ( t ) Q ( t ) Timet Timet Timet Timet Timet Phase ϕ ( t ) Phase ϕ ( t ) Q ( t ) Q ( t ) 0 0 0

Figure 6.2:Time evolution of possible realizations of random trajectories of the fluctuating charge Q(t)

on the impurity level and the random phase ϕ(t) =−v0₀tdt'_Q(t'_{)picked up during the time-}

interval [0,t] (right panel). The switching rate γ increases from top to bottom. For large

switching rateγ the phaseϕ is a sum of many random contributions and performs almost a random walk with a drift. The phase ϕ(t) is always negative and is drifting with a velocity

−v/2.

which acquire an additional phase-factor e∓iϕ(t)/2 _{to their oscillatory time evolution with the} random phase

ϕ(t) =₋v- t 0 dt

'_Q(t'_{). (6.9)}

Equation (6.8) is the solution of the Schrödinger-equation for a given specific realization of the random potentialQ(t). However, in an experiment one is not interested in a single measure- ment for a specific realization of the random phase ϕ(t)but in an average over many repeated runs on the same qubit. Suppose that we wish to measure an observable ˆA. Then the expectation value averaged over many realizations of the random phase is equal to

!!A""ˆ ϕ =tr

-

6.3 GAUSSIAN APPROXIMATION 41