Decoherence from ensembles of two-level fluctuators

1/f noise, the major source of dephasing in Josephson qubits, may be produced by an ensemble of two-level systems. Depending on the statistical properties of their distribution, the noise distribution can be Gaussian or non-Gaussian. The latter situation is realized, for instance, when the distribution of coupling strengths has a slowly decaying power-law tail. In this regime questions of self-averaging and sample-to-sample fluctuations become crucial. We study the dephasing process for a class of distribution functions and analyze the self-averaging properties of the results.


Introduction
In Josephson qubits dephasing is dominated by low-frequency noise, often with a 1/f power spectrum, due to fluctuations of background charges, magnetic fluxes, or critical currents [1,2,3]. While irrelevant for the relaxation process with time scale T 1 , lowfrequency noise dominates the dephasing time T * 2 . Standard NMR echo techniques allow one to reduce dephasing by rendering the low frequency spectrum ineffective [1]. Operation at optimal bias points, chosen such that the linear longitudinal coupling of the qubit to the 1/f noise source vanishes, proved to be very successful in increasing the dephasing time [2]. Further progress in this direction may require an improved understanding of the mechanisms causing 1/f noise and of its statistical properties. It was realized recently that qubits themselves can be used to study the noise properties of their environment [4,5], and an interesting relation between the low-frequency 1/f and the high-frequency charge noise was observed [6]. An extensive study of dephasing due to both charge and flux noise was undertaken in Ref. [7].
Still, many questions remain open. If the number of fluctuators contributing to the 1/f noise is large, one could expect Gaussian statistics [1,8]. In Ref. [9] and following work the role of individual, strongly coupled fluctuators was emphasized, and it was suggested [10] that even ensembles of many fluctuators may produce strong non-Gaussian effects, emerging as a result of rare configurations in which dephasing is dominated by a small number of very strongly coupled fluctuators. As far as we can judge, the decay laws observed in Ref. [7] cannot be fully explained by either of these theories.
As the experiments are performed on individual systems with a particular configuration of the fluctuators, it is important to understand whether the predicted decay laws are self-averaging or have strong sample-to-sample fluctuations. Here we will analyze a class of distribution functions for the coupling strengths of the fluctuators. We determine the ensemble-averaged decay laws (extending the results of Ref. [10]) and analyze which of them are self-averaging. We study both dephasing due to linear longitudinal coupling and dephasing at the optimal point where the coupling is quadratic.
2. 1/f noise from two-level fluctuators (TLF) 1/f noise is often attributed to a collection of bistable systems, switching randomly between two states [11]. On one hand, such a model provides a natural explanation of 1/f noise. On the other hand, in many samples distinct two-level fluctuators (TLF's) were detected. In metals this switching causes conductance fluctuations [12,13] and, consequently, 1/f noise of the transport current. In Josephson junctions it causes the critical current to fluctuate [14,15]. More generally, spin bath environments were analyzed in Ref. [16]. In charge qubits the TLF's contribute to the fluctuations of the gate charge controlling the qubit. The TLF's are characterized by their coupling strengths to the qubit, v n , which may vary depending on the location of the respective TLF. The fluctuating quantity that couples to the qubit, X(t), contains contributions from all TLF's: Each fluctuator switches randomly between two positions, denoted by σ n,z = ±1, with rate γ n (for simplicity, we assume equal rates in both directions for the relevant TLF's) and thus contributes to the noise power S X (ω) = n S n (ω) with A set of TLF's produces 1/f noise when the switching rate γ depends exponentially on a physical quantity, l, with a smooth distribution. For instance, γ ∝ e −l/l 0 , with l distributed uniformly over a range much wider than l 0 , translates in a loguniform distribution of the switching rates, with probability density P (γ) ∝ 1/γ in the corresponding exponentially wide range γ min ≪ γ ≪ γ max . In this range the total noise power thus scales as An example is a particle trapped in a double-well potential, whose tunneling rate through the potential barrier depends exponentially on both the height and the width of the barrier, leading to 1/f noise. Another example is thermally activated tunneling with rate γ 0 e −E/k B T , where E denotes an activation energy. In this way the 1/f power spectrum observed in metals can be attributed to a broad (much wider than k B T ) distribution of activation energies [17].

Distribution of coupling strengths, self-averaging
The analysis of decoherence in the presence of many fluctuators requires the study of probability distributions of coupling strengths and switching rates. In each particular sample one deals with specific fluctuators, i.e., with a realization of the set of parameters v and γ, drawn from this distribution. One should distinguish between quantities averaged over a statistical ensemble of samples and the results for a specific sample. This difference is essential, if the quantity under consideration is not self-averaging, i.e., if it has considerable sample-to-sample fluctuations. Such a situation arises if a quantity is dominated by contributions from a small number of TLF's.
In Ref. [10], a continuous distribution of the parameters v n and γ n was considered, with a long tail of the distribution of coupling strengths v n , such that rare configurations with very large v n dominate certain ensemble properties. It arises, e.g., from a uniform spatial distribution of fluctuators on a d-dimensional surface and a powerlaw TLF-qubit coupling [10], v(r) ∝ 1/r b . This results in a distribution of coupling strengths P (v) ∝ 1/v 1+d/b . The joint distribution P (v, γ), defined in the domain [v min , ∞] × [γ min , γ max ] and normalized to describe N fluctuators is thus Here µ = d/b > 0, c = 1/ ln(γ max /γ min ), and η = v min N 1/µ . One can also allow for fluctuations of N, but this does not change the results significantly. We consider a d-dimensional volume of typical size r max around the qubit containing a uniform distribution of TLF's. The typical distance between the strongest (closest) fluctuator and the qubit thus scales as r min ∼ (V /N) 1/d ∼ r max /N 1/d . On the other hand, since the coupling strength was assumed to decay as v(r) ∝ 1/r b the relation between the strongest and weakest coupling strength is given by v max /v min = (r max /r min ) b . Combining both results we find that the typical maximal coupling strength scales as v typ max ∼ v min N 1/µ . This does not exclude the existence of fluctuators with v ≫ v typ max in certain realizations, as the long tail of the distribution function suggests.
As examples of averaging over the distribution of coupling strengths and switching rates we calculate the noise produced by the ensemble of fluctuators, distinguishing two cases: In one case, for µ < 2 the integral over v diverges at the upper limit. Hence the noise is dominated by the strongest fluctuator(s). Thus the result is sensitive to the properties of one or a few fluctuators and is therefore not self-averaging. Estimates below are based on cutting the integral at v = η = v typ max but one has to remember that for a comparison with experiments averaging (including the averaging over γ) makes little sense, since only a few TLF's contribute.
In contrast, for µ > 2 the integral is dominated by fluctuators with v < η. The weak fluctuators are most important, and due to their large number the noise is given by a sum of many comparable independent contributions. Consequently, the result is self-averaging, i.e., in different samples or runs of the experiment with µ > 2 one should observe the same noise amplitude.
We now summarize the typical/average results for the noise, retaining only the leading contributions For µ > 2 we defined the average coupling strength of the TLFs, v 2 = 1 N dvP (v)v 2 . Note that (6) is only valid for frequencies γ min ≪ |ω| ≪ γ max . At lower frequencies, |ω| < γ min , S X (ω) tends to a constant, whereas at higher frequencies, |ω| > γ max , S X (ω) crosses over to a faster power law decay ∝ 1/ω 2 .

Longitudinal and transverse noise coupling
We consider a qubit controlled (for simplicity) by a single parameter λ and Hamiltonian After an initial preparation in a coherent superposition of the qubit's eigenstates, the effective spin precesses under the influence of the static field H 0 , set by the control parameter λ 0 . Coupling to the environment disturbs this evolution, leading to decoherence. In many cases the effect of the environment can be modeled by classical and quantum fluctuations of λ(t) = λ 0 + X(t), where X(t) fluctuates. For instance, in a charge qubit electromagnetic fluctuations of the control circuit as well as the background charge noise influence the gate voltage which controls the qubit.
To proceed we expand the Hamiltonian H qb to second order in the perturbation X, Introducing the notations where ω 01 ≡ | H 0 (λ 0 )|, δω z ≡ D λ,z X + D λ2,z X 2 /2 + ..., and δω ⊥ ≡ D λ,⊥ X + .... Here σ ⊥ denotes the transverse spin components (i.e., σ x or σ y ). The coefficients D are related to the derivatives of ω 01 (λ): and Thus, in general, the coupling of noise to the qubit contains both transverse (δω z ) and longitudinal (δω ⊥ ) parts, and both may have linear as well as higher order (e.g., quadratic) contributions.

Bloch-Redfield theory
For weak, short-correlated noise the dynamics of the two-level systems (spins, qubits) can be summarized by Bloch's equations [18,19] in terms of two rates: the longitudinal relaxation (depolarization) rate Γ 1 = T −1 1 , and the transverse relaxation (dephasing) rate Γ 2 = T −1 2 . Evaluated perturbatively, using the golden rule, the rates are given by and where The dephasing process (13) is a combination of depolarization effects (Γ 1 ) and of the so called 'pure' dephasing, characterized by the rate Γ ϕ = T * 2 −1 . The pure dephasing is usually associated with the inhomogeneous level broadening in ensembles of spins, but occurs also for a single spin due to the longitudinal low-frequency noise.
6. Pure dephasing for Gaussian noise, µ > 2 If there are sufficiently many fluctuators in the environment, the central limit theorem (CLT) applies, and the noise is Gaussian. More specifically, since the central limit theorem applies to a large collection of equally distributed random quantities, one needs to have a large number of TLF's of each (relevant) "kind" (i.e., for each pair v, γ). This implies a regular distribution of coupling strengths, so that the relevant physical quantities are not dominated by a few TLF's at a boundary of the distribution. In particular, the distribution (4) gives rise to Gaussian noise if µ > 2. We will discuss now the pure dephasing derived from such Gaussian noise. The random phase accumulated at time t, is then also Gaussian-distributed. Hence the decay law, due to longitudinal noise (coupling to σ z ) in a free induction decay (Ramsey signal) is given by f R (t) = exp(i∆φ) = exp(−(1/2) ∆φ 2 ). Averaging here is over the different trajectories of X(t) in repeated runs of the dephasing experiment. We obtain where sinc x ≡ sin x/x. If most of the noise power is concentrated at frequencies ω ≪ 1/t (static noise), then one can approximate sinc ωt 2 ≈ 1 and obtain where σ 2 X = +∞ −∞ (dω/2π)S λ (ω) is the dispersion of X. In general, for static noise with (not necessarily Gaussian) distribution function P (X) the Ramsey decay is given by i.e., by the Fourier transform of P (X). Static noise corresponds to a situation when X is constant during each run of the experiment but fluctuates between different runs.
In an echo experiment, the phase acquired is the difference between the two free evolution periods: which after averaging over the trajectories of X(t) gives 1/f spectrum: Here and below we assume that the 1/f law extends over a wide range of frequencies, limited by infrared and ultraviolet cut-offs, The infra-red cutoff ω ir is usually determined by the measurement protocol, as discussed further below. The decay rates typically depend only logarithmically on ω ir , and details of the noise power below ω ir are irrelevant to logarithmic accuracy. For most of our analysis, the same applies to the ultra-violet cut-off ω c . However, for some specific questions considered below, frequency integrals may be dominated by ω ≈ ω c , and thus the detailed behavior near and above ω c (i.e. the "shape" of the cut-off) is relevant. We will refer to an abrupt suppression above ω c (S(ω) ∝ θ(ω c − |ω|)) as a 'sharp cut-off', and to a crossover at ω ∼ ω c to a faster decay 1/ω → 1/ω 2 (motivated by modeling of the noise via a set of bistable fluctuators, see below), as a 'soft cut-off'.
For 1/f noise, at times t ≪ 1/ω ir , the free induction (Ramsey) decay is dominated by the frequencies ω < 1/t, i.e., by the quasistatic contribution [20], and (15) reduces to Here the logarithmically large part of the exponent originates from a static contribution of frequencies ω < 1/t. Indeed it can be obtained from Eq. (16) with σ 2 X = 2 1/t ω ir (dω/2π)S X (ω) = A ln(1/ω ir t). This contribution dominates the decay of f R (t).
For the echo decay we obtain The echo method thus increases the decay time only by a logarithmic factor. This low efficiency of the echo has its origin in the high-frequency tail of the 1/f noise, which, as we note, influences the results strongly. For 1/f noise with a low cut-off ω c the integral in Eq. (19) over the interval ω ω c is dominated by the upper limit. For instance, in the case of a sharp cutoff, i.e., S = (A/|ω|)θ(ω c − ω), we obtain On the other hand, for a soft cut-off, which we expect when the noise is produced by a collection of bistable fluctuators with Lorentzian spectrum, the integral in Eq. (19) is dominated by frequencies ω c < ω < 1/t, and we find ln f E (t) ∝ D 2 λ,z A ω c t 3 . In either case, one finds that the decay is slower by a factor ∼ (ω c t) 2 or ω c t, respectively, than for 1/f noise with a high cutoff, ω c > D λ,z A 1/2 .

Individual fluctuators
We consider a single fluctuator coupled longitudinally to the qubit, whose contribution to the level splitting, v n (t) ≡ v n σ n,z (t), switches between ±v n . For this case the free induction (Ramsey) and echo decays have been evaluated in Refs. [9,10]. In the limit of high effective temperature, i.e., when the transition rates in both directions are equal, the decay functions, obtained by averaging over the switching history of σ n,z (t), are given by f R,n (t) = e −γnt cos µ n t + γ n µ n sin µ n t , and f E,n (t) = e −γnt 1 + γ n µ n sin µ n t + γ 2 where µ n ≡ (Dv n ) 2 − γ 2 n and D ≡ D λ,z . In order to derive these expressions we introduce the averaged phase factors χ ± (t) = exp(i t dt ′ D v n (t ′ )) , averaged over the switching histories ending at v n (t) = +v n or −v n , respectively. Their dynamics is governed by the rate equationṡ χ + = iDv n χ + − γ n χ + + γ n χ − , The solution for f R,n (t) = χ + (t) + χ − (t) is obtained by solving the coupled equations for the initial conditions χ ± = 1/2, which gives Eq. (24). Similarly, for more general protocols, we have to analyze phase factors exp(i t dt ′ D g(t ′ ) v n (t ′ )) with appropriate time dependence of g(t). In this case the first terms on the right hand side of Eqs. (26) are modified accordingly. For the echo experiment we obtain in this way Eq. (25). The decay produced by a number of fluctuators is the product of the individual contributions, i.e., f R (t) = Π n f R,n (t) and f E (t) = Π n f E,n (t). If the noise is dominated by a few fluctuators (this includes the case of many fluctuators in total, but a few of them with similar rates γ) the fluctuations of X(t) may be strongly non-Gaussian.

Non-Gaussian effects, µ < 2
Since we consider uncorrelated TLF's the total decay of coherence is the product of all single-TLF contributions, f (t) = n f n (t), where f n (t) is given by (24) and (25) for the free induction and echo experiment, respectively. In Ref. [10] an ensemble-averaged value of ln f (t), denoted as ln f (t) F , was calculated for µ = 1. Here · · · F denotes the average over the distribution of coupling strengths and switching rates (4). Both free induction decay and the "phase memory decay" (a protocol similar but not equivalent to the spin echo decay) were analyzed in the regimes t < γ −1 max and t > γ −1 max . Below we will generalize these results to the range 0 < µ < 2.
As discussed above, the quantity ln f (t) F is relevant for experiments with specific samples only if the sample-to-sample fluctuations of ln f (t) are weak, i.e., if ln f (t) is self-averaging. Then, the experimentally observable decay law f (t) would be well approximated by exp( ln f (t) F ). In Ref. [10] the self-averaging was numerically confirmed for the phase memory decay in the regime t < γ −1 max . Here we analyze the self-averaging in four regimes: for the free induction and the echo cases, both in the limits t < γ −1 max and t > γ −1 max . Specifically, we evaluate the ensemble-average ln f (t) F , given by an integral over the (v, γ)-space. In some cases this integral is dominated by a range in the 'bulk' of the distribution, which contains many fluctuators on average; this indicates that sample-to-sample fluctuations are weak. In other cases, the integral is dominated by the boundary of the distribution, indicating that the studied quantity is not self-averaging. Our analysis confirms the conclusion of Ref. [10], obtained in one regime: for the echo decay at short times t < γ −1 max . We show further that in all other three regimes investigated the dephasing law is not self-averaging. In the calculations we assume that v min , γ min are very low frequency scales, and 1/t always exceeds them.

Free induction decay
For short times, t < γ −1 max , we are effectively in the static regime, and the ensembleaveraged free induction decay is described by This result is dominated by the fluctuators with strength of order v ∼ v typ max ∼ η and thus is not self-averaging. For an experiment with a specific sample the results should be fitted by a contribution of one (24) or a few fluctuators, rather than by the ensemble-averaged behavior (27). We can also estimate the typical decay law for short times t < η −1 . In every realization there will be a few strongest fluctuators, typically with strength v max .
For distributions with µ < 2 the sum n v 2 n is dominated by the largest v n 's, and thus, ln |f R (t)| ∝ −D 2 λ,z t 2 v 2 max . Finally, we can calculate the distribution function for the strength of the strongest fluctuator v max and obtain Most of the weight of this distribution is around v max ∼ η. Thus, in a typical sample for t < η −1 the decay is given by ln |f R (t)| ∝ −(D λ,z ηt) 2 rather than by (27). To understand the difference we note that the average decay law (27) can also be obtained by averaging the realization-dependent −D 2 λ,z v 2 max t 2 (valid for t < v −1 max ) over the distribution (28). This average is dominated by rare samples with a fluctuator of strength v max ∼ 1/t rather than by typical samples.
For longer times, t > γ −1 max , the integration gives Both results are not self-averaging.

Echo signal decay
For short times, t < γ −1 max , we find For c 1/µ D λ,z η > γ max the echo decay is dominated by this quasi-static contribution; the decay takes place on the time scale shorter than the flip time of the fastest fluctuators, 1/γ max . In this regime (c 1/µ D λ,z η > γ max ) the result is self averaging since it is dominated by fluctuators with D λ,z v ∼ (cD µ λ,z η µ γ max ) 1/(1+µ) < c 1/µ D λ,z η < D λ,z v typ max . For longer times, t > γ −1 max the dephasing is due to multiple flips of the fluctuators. These times are relevant if c 1/µ D λ,z η < γ max . The decay law is given by • for µ = 1 • for µ < 1 All these results are not self-averaging.

Quadratic coupling
At the optimal working point, the first-order longitudinal coupling D λ,z vanishes. Thus, to first order, the decay of the coherent oscillations is determined by the relaxation processes and for regular power spectra at low frequencies one expects from Eq. (13) that Γ 2 = Γ 1 /2. On the other hand, for power spectra which are singular at low frequencies the second-order contribution of the longitudinal noise can be comparable or even dominate over Γ 1 /2. To evaluate this contribution, one has to calculate where for the analysis of the free induction decay (Ramsey signal) one sets g(t ′ ) = 1, while for decay of the echo-signal g(t ′ < t/2) = −1 and g(t ′ > t/2) = 1.
Quasi-static case: In this case, i.e., when the cutoff ω c is lower than 1/t for all relevant times, the Ramsey decay is simply given by the static contribution (36). At all relevant times the decay is algebraic and the crossover to the exponential law is not observed. More generally, in the static approximation with a distribution P (δλ), the dephasing law is given by the Fresnel-type integral transform, which reduces to Eq. (36) for a Gaussian P (X) ∝ exp (−X 2 /2σ 2 X ). In general, any distribution P (X), finite at X = 0, yields a long time decay of f st 2,R proportional to t −1/2 .
For µ < 2 the analysis is technically more complicated. In that case the distribution of initial conditions P (X 0 ), and equivalently the sum X 0 = n v n σ n,z (t = 0), are no longer Gaussian-distributed and, in particular, they cannot be characterized by a typical width σ, due to the divergence of the second moment v 2 of (4). The generalized central limit theorem tells us that x = X 0 /η is then distributed according to a Lévy distribution L µ,0 (x) and consequently, according to (38) the free induction decay in the quasi-static regime is given by For some values of µ explicit expressions are known. An example is the Cauchy distribution, L 1,0 (x) = 1/[π(1 + x 2 )]. Using (39) the free induction decay in the static regime, t < γ −1 max , is then given by Here we introduced the rate α = 1 2 and Φ(z) = 2π −1/2 z 0 dx e −x 2 denotes the error function. One can expand Φ(z) in (40) to find the asymptotic behavior of f st 2,R (t) for µ = 1: The initial decay for t ≪ α −1 is thus very fast, but at times t ≈ α −1 the decay crosses over to a much slower power law ∝ 1/ √ t. The dephasing time scales as α −1 , but with a relatively large prefactor due to the slow algebraic decay. For other values of µ < 2 the asymptotic behavior of f st 2,R has been obtained in Ref. [24]: where C(µ) and D(µ) are factors of order 1.
Let us now discuss the shape of the decay of f st R qualitatively and comment on their validity. The initial decay of f st R for µ < 2 is singular and thus very fast compared to the Gaussian case µ > 2. This initial decay is dominated by strongly coupled fluctuators, i.e., by the tail of the distribution (4). It is, thus, not self-averaging.
On the other hand, for longer times, t ≫ α −1 , the decay goes over to a much slower power law. The exponent −1/2 is independent of µ and coincides even with the prediction of the Gaussian model (µ > 2). Hence, the 1/ √ t decay law appears to be universal in the presence of quasi-static noise, independent of the considered statistics. For low enough γ max such that α ≫ γ max the free induction signal decays already in the quasi-static regime, t < γ −1 max , and is thus given by (43). Otherwise, further analysis characterizing the contribution of fast fluctuators, γ > t −1 , is needed to describe decoherence.

Conclusions
We have shown that non-Gaussian 1/f noise of ensembles of two-level fluctuators frequently leads to non-self-averaging dephasing laws. Non-Gaussian noise arises, for instance, when the distribution of coupling strengths between the two-level fluctuators and a qubit has a long algebraic tail. In this case, since experiments are performed on specific samples, one should study the typical rather than ensemble averaged behavior. Interestingly, in certain regimes, e.g., for short-time echo decay, the decay law is selfaveraging.