Decoherence from ensembles of two-level fluctuators

Josef Schriefl¹, Yuriy Makhlin², Alexander Shnirman¹ and Gerd Schön¹

¹ Institut für Theoretische Festkörperphysik, Universität Karlsruhe, D-76128 Karlsruhe, Germany
² Landau Institute for Theoretical Physics, Kosygin st. 2, 119 334 Moscow, Russia

Email: shnirman@tfp.uni-karlsruhe.de

Received 21 October 2005
Published 20 January 2006

Contents

• 1. Introduction
• 2. 1/f noise from two-level fluctuators (TLFs)
• 3. Distribution of coupling strengths, self-averaging
• 4. Longitudinal and transverse noise coupling
• 5. Bloch-Redfield theory
• 6. Pure dephasing for Gaussian noise, μ  >  2
  • 6.1.  1/f spectrum
• 7. Individual fluctuators
• 8. Non-Gaussian effects, μ  >  2
  • 8.1. Free induction decay
  • 8.2. Echo signal decay
• 9. Quadratic coupling
  • 9.1. 1/f noise
  • 9.2. Quasi-static case
• 10. Conclusions
• Acknowledgments
• References

1. 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]-[3]. While irrelevant for the relaxation process with a timescale T₁, low-frequency noise dominates the dephasing time T₂*. 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 [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 [9] and following work [10], the role of individual, strongly coupled fluctuators was emphasized, and it was suggested 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 [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 analyse a class of distribution functions for the coupling strengths of the fluctuators. We determine the ensemble-averaged decay laws (extending the results of [10]) and analyse 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 (TLFs)

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 TLFs were detected. In metals, this switching causes conductance fluctuations [12, 13] and, consequently, 1/f noise of transport current. In Josephson junctions, it makes the critical current to fluctuate [14, 15]. More generally, spin bath environments were analysed in [24]. In charge qubits, the TLFs contribute to the fluctuations of the gate charge controlling the qubit. The TLFs 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 TLFs:

Each fluctuator switches randomly between two positions, denoted by σ_n,z  =  ±1, with a rate γ_n (for simplicity, we assume equal rates in both directions for the relevant TLFs) and thus contributes to the noise power with

A set of TLFs produces 1/f noise when the switching rate γ depends exponentially on a physical quantity, l, with a smooth distribution. For instance, , with l distributed uniformly over a range much wider than l₀, translates in a log-uniform 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 tunnelling 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 tunnelling with rate , 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_BT) distribution of activation energies [16].

3. 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 TLFs.

In [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 power-law 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_minN^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 TLFs. 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_max^typ ~ v_minN^1/μ. This does not exclude the existence of fluctuators with v    v_max^typ 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_max^typ but one has to remember that for a comparison with experiment averaging (including the averaging over γ) makes little sense, since only a few TLFs 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, . 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/ω².

4. 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 , set by the control parameter λ₀. Coupling to the environment disturbs this evolution, leading to decoherence. In many cases the effect of the environment can be modelled by classical and quantum fluctuations of λ(t)  =  λ₀  +  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 and , we find in the eigenbasis of :

where , δω_z ≡ D_λ,zX  +  D_{λ2, z} X²/2 +  ... 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 ω₀₁(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.

5. Bloch-Redfield theory

For weak, short-correlated noise the dynamics of the two-level systems (spins, qubits) can be summarized by Bloch's equations [17, 18] in terms of two rates: the longitudinal relaxation (depolarization) rate Γ₁  =  T₁ ^- 1, and the transverse relaxation (dephasing) rate Γ₂  =  T₂ ^- 1. Evaluated perturbatively, using the golden rule, the rates are given by

and

where

The dephasing process (13) is a combination of depolarization effects (Γ₁) and of the so-called `pure' dephasing, characterized by the rate Γ  =  T₂^* - 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 CLT applies to a large collection of equally distributed random quantities, one needs to have a large number of TLFs 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 TLFs 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(iΔ)  =  exp( - (1/2)Δ²). 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 and obtain

where 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 tragectories of X(t) gives

6.1.  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 infrared cut-off 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 ultraviolet cut-off ω_c. However, for some specific questions considered below, frequency integrals may be dominated by ω ≈ ω_c, and thus the detailed behaviour near and above ω_c (i.e. the `shape' of the cut-off) is relevant. We will refer to an abrupt suppression above ω<sub>c</sub> (S(ω) θ(ω<sub>c</sub>  -  |ω|)) as a `sharp cut-off', and to a crossover at ω ~ ω_c to a faster decay 1/ω → 1/ω² (motivated by modelling 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 quasi-static contribution [19], 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 equation (16) with . 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 equation (19) over the interval ω    ω_c is dominated by the upper limit. For instance, in the case of a sharp cut-off, 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 equation (19) is dominated by frequencies ω_c  <  ω  <  1/t, and we find ln f_E(t) D_λ,z² A ω_c t³. In either case, one finds that the decay is slower by a factor ~(ω_ct)² or ω_ct, respectively, than for 1/f noise with a high cut-off, ω_c  >  D_λ,zA^1/2.

7. Individual fluctuators

We consider a single fluctuator coupled longitudinally to the qubit, whose contribution to the level splitting, v_n(t) ≡ v_nσ_n,z(t), switches between ±v_n. For this case, the free induction (Ramsey) and echo decays have been evaluated in [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

and

where and D ≡ D_λ,z. In order to derive these expressions, we introduce the averaged phase factors , averaged over the switching histories ending at v_n(t)  =   + v_n or  - v_n, respectively. Their dynamics are governed by the rate equations

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 equation (24). Similarly, for more general protocols, we have to analyse phase factors with appropriate time dependence of g(t). In this case the first terms on the right-hand side of equations (26) are modified accordingly. For the echo experiment we obtain in this way, equation (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.

8. Non-Gaussian effects, μ  >  2

Since we consider uncorrelated TLFs the total decay of coherence is the product of all single-TLF contributions, f(t)  =  Π_nf_n(t), where f_n(t) is given by (24) and (25) for the free induction and echo experiment, respectively. In [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 analysed in the regimes t  <  γ_max ^- 1 and t  >  γ_max ^- 1. 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, experimentally observable decay law f(t) would be well approximated by exp(ln f(t)_F). In [10] the self-averaging was numerically confirmed for the phase memory decay in the regime t  <  γ_max ^- 1. Here we analyze the self-averaging in four regimes: for the free induction and the echo cases, both in the limits t  <  γ_max ^- 1 and t  >  γ_max ^- 1. 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 given in [10], obtained in one regime: for the echo decay at short times t  <  γ_max ^- 1. 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 and γ_min are very low-frequency scales, and 1/t always exceeds them.

8.1. Free induction decay

For short times, t  <  γ_max ^- 1, we are effectively in the static regime, and the ensemble-averaged free induction decay is described by

This result is dominated by the fluctuators with strength of order v ~ v_max^typ ~ 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 behaviour (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 t    v_max ^- 1, we obtain . For distributions with μ  <  2, the sum is dominated by the largest v_n's, and thus, ln|f_R(t)|  - D_λ,z²t²v_max². 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,zηt)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_λ,z²v_max²t² (valid for t  <  v_max ^- 1) 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_max ^- 1, the integration gives

• for 1 ≤ μ  <  2

  

• for μ  <  1

  

Both results are not self-averaging.

8.2. Echo signal decay

For short times, t  <  γ_max ^- 1, we find

For c^1/μD_λ,zη  >  γ_max, the echo decay is dominated by this quasi-static contribution; the decay takes place on the timescale shorter than the flip time of the fastest fluctuators, 1/γ_max. In this regime (c^1/μD_λ,zη  >  γ_max), the result is self-averaging since it is dominated by fluctuators with D_λ,zv ~ (cD_λ,z^μη^μγ_max)^{1/(1  +  μ)}  <  c^1/μD_λ,zη  <  D_λ,zv_max^typ.

For longer times, t  >  γ_max ^- 1, the dephasing is due to multiple flips of the fluctuators. These times are relevant if c^1/μD_λ,zη  <  γ_max. The decay law is given by

• for 1  <  μ  <  2

  

• for μ  =  1

  

• for μ  <  1

  

All these results are not self-averaging.

9. 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 equation (13) that Γ₂  =  Γ₁/2. 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 Γ₁/2. 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.

9.1. 1/f noise

The free induction decay for the 1/f noise with a high cut-off ω_c (the highest energy scale in the problem) has been analysed in [20]. Depending on the time t, the decay is dominated by low- or high-frequency noise, and the decay law can be approximated by a product of low-frequency (ω  <  1/t, quasi-static) and high-frequency (ω  >  1/t) contributions, f_2,R(t)  =  f_2,R^lf(t)·f_2,R^hf(t). The contribution of low frequencies is given by [20]-[22]

For 1/f noise the variance of the low-frequency fluctuations is σ_X²  =  2A ln(1/ω_irt). This contribution dominates at short times t  <  [(∂²ω₀₁/∂λ²) A/2] ^- 1. At longer times, the high-frequency contribution

takes over. When t    [(∂²ω₀₁/∂λ²) A/2] ^- 1 (provided ω_c    π(∂²ω₀₁/∂λ²) A), we obtain asymptotically ln f_2,R^hf(t) ≈  - (π/2)(∂²ω₀₁/∂λ²) At. Otherwise, the quasi-static result (36) is valid at all relevant times. One can also evaluate the pre-exponential factor in the long-time decay. This pre-exponent decays very slowly (algebraically) but differs from 1 and thus further reduces f_2,R(t) [23].

9.2. Quasi-static case

In this case, i.e., when the cut-off 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 equation (36) for a Gaussian P(X) exp( - X²/2σ_X²). In general, any distribution P(X), finite at X  =  0, yields a long-time decay of f_2,R^st proportional to t ^- 1/2.

For μ  <  2 the analysis is technically more complicated. In that case the distribution of initial conditions P(X₀) and equivalently the sum 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 v² of (4). The generalized CLT tells us that x  =  X₀/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  +  x²)]. Using (39), the free induction decay in the static regime, t  <  γ_max ^- 1, is then given by

Here we introduced the rate

and denotes the error function. One can expand Φ(z) in (40) to find the asymptotic behaviour of f_2,R^st(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 . 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 behaviour of f_2,R^st has been obtained in [23]:

where C(μ) and D(μ) are factors of order 1.

Let us now discuss the shape of the decay of f_R^st qualitatively and comment on their validity. The initial decay of f_R^st 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 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_max ^- 1, and is thus given by (43). Otherwise, further analysis characterizing the contribution of fast fluctuators, γ  >  t ^- 1, is needed to describe decoherence.

10. Conclusions

We have shown that non-Gaussian 1/f noise of ensembles of TLFs frequently leads to non-self-averaging dephasing laws. Non-Gaussian noise arises, for instance, when the distribution of coupling strengths between the TLFs 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 behaviour. Interestingly, in certain regimes, e.g., for short-time echo decay, the decay law is self-averaging.

Acknowledgments

The work is part of the CFN of the DFG and of the EU IST Project SQUBIT. YM acknowledges support from the Dynasty Foundation and the grant MD-2177.2005.2.

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