Motivated by recent experiments with Josephson qubits we propose a new
phenomenological model for 1/f noise due to collective excitations of
interacting defects in the qubit's environment. At very low temperatures the
effective dynamics of these collective modes are very slow leading to
pronounced non-Gaussian features and nonstationarity of the noise. We analyze
the influence of this noise on the dynamics of a qubit in various regimes and
at different operation points. Remarkable predictions are absolute time
dependences of a critical coupling and of dephasing in the strong coupling
regime.Comment: 4 pages, 2 figures, to be published in the proceedings of the Vth
  Rencontres de Moriond in Mesoscopic Physic

Carpentier, D.

Clusel, M.

Degiovanni, P.

Makhlin, Yu.

Schriefl, J.

English

arXiv

KITopen

arXiv:cond-mat/0404641v1  [cond-mat.mes-hall]  27 Apr 2004DEPHASING DUE TO NONSTATIONARY 1/f NOISEJ. SCHRIEFL1,2, M. CLUSEL1, D. CARPENTIER1, P. DEGIOVANNI1, Y. MAKHLIN2,31CNRS-Laboratoire de Physique de l’Ecole Normale Supe´rieure de Lyon, 69007 Lyon, France2Institut fu¨r Theoretische Festko¨rperphysik, Universita¨t Karlsruhe, 76128 Karlsruhe, Germany3Landau Institute for Theoretical Physics, Kosygin st. 2, 117940 Moscow, RussiaMotivated by recent experiments with Josephson qubits we propose a new phenomenologicalmodel for 1/fµ noise due to collective excitations of interacting defects in the qubit’s environ-ment. At very low temperatures the effective dynamics of these collective modes are very slowleading to pronounced non-Gaussian features and nonstationarity of the noise. We analyze theinfluence of this noise on the dynamics of a qubit in various regimes and at different operationpoints. Remarkable predictions are absolute time dependences of a critical coupling and ofdephasing in the strong coupling regime.Despite a large number of investigations, the detailed explanation of the origin of 1/f noisestays an open problem in condensed matter physics. The diversity of systems that exhibit 1/ffluctuations suggests that its physical origin is not universal.1 Still, because of its ubiquity, thestudy of underlying mechanisms, relevant at least in some cases, is interesting.In the context of quantum-information processing2 the field has attracted new attention. Inparticular in solid state devices, e.g. Josephson qubits, low frequency noise, typically with a 1/fpower spectrum, is considered as the most important limitation for the preservation of the phasecoherence. The microscopic origin of this noise depends thereby on the considered system. Inthe case of charge noise, for instance, it is believed to be due to the activity of random traps forcharges in the dielectric substrate and in the oxide barrier of the Josephson junction.3 Similarly,neutral defects in the junction barriers can cause critical current fluctuations and low-frequencyflux noise may take its origin from trapped vortices and magnetic impurities. In practice, 1/fnoise appears difficult to suppress and, since dephasing is dominated by low-frequency noise, itis particularly destructive.Decoherence of a qubit due to 1/f noise has already been studied in literature using variousmodels for the noise: One possibility is to assume Gaussian-distributed fluctuations, e.g. ofa large collection of microscopic modes, which in some cases may be modeled by a harmonicoscillator bath.5 Alternatively, one can consider a microscopic model with independent bistablefluctuators, each one characterized by a switching rate γi and an effective bias voltage vi inducedat the qubit.6,7 If the switching rates are distributed according to p(γ) ∼ 1/γ the resulting powerspectrum exhibits a 1/f divergence at low frequencies. If only a few fluctuators (microscopicmodes) dominate, the effect of the noise differs.In this work, we present a new model of 1/f noise and discuss its effects on the phasecoherence of a qubit. In contrast to the mentioned proposals we consider situations wherethe mutual interaction of the localized defects in the background cannot be neglected, e.g. atvery low temperatures. Since neutral and charged defects behave as elastic and electric dipolesrespectively 7 the strength of the interaction is long-range in both cases decaying as 1/r3 asa function of the distance r. Although weak, this interaction may play a crucial role at verylow temperatures, leading to formation of highly cooperative clusters of interacting defects.9The resulting slow dynamics can be described as a perpetual search of the absolute free-energyminima: the system spends most of the time in local free energy minima corresponding tometastable configurations, occasionally flipping between these states. The exponential relationbetween the typical trapping time τ and the energy barrier to overcome E, τ(E) ≃ τ0eE/T ,implies that the distribution of the trapping times, P (τ) is generally broad, decaying veryslowly for large times: P (τ) ≃ τ−1−µ. Here µ is a positive dimensionless number that can bewritten as the ratio of temperature T and a typical height E0 of the energy barriers, µ ∼ T/E0.The fluctuating extra bias induced at the qubit is described by an asymmetric telegraphnoise, where the times between two successive plateaux are distributed according to an algebraicdistributionP (τ) =µτ0(τ0τ0 + τ)1+µ. (1)The heights and lengths of the plateaux are distributed according to narrow distributions P(v),which is assumed to have zero mean, and ρ(τ↑) respectively. The coupling strength to the qubitis characterized by the dimensionless parameter g2 = 〈(vτ↑)2〉. For µ > 2, when τ =∫∞0 dτ τP (τ)and τ2 are finite our model is equivalent to usual Poissonian telegraph noise as proposed in6,7.However, at ultra-low temperatures, i.e. µ < 2, important differences emerge.a We thereby haveto distinguish two additional classes of noise: (i) for 1 < µ < 2, τ is still finite but τ2 diverges.Consequently, the rate of occurring plateaux fluctuates strongly around the average τ−1. Inparticular, due to the diverging τ2 the statistics of the noise are only approximately Gaussian,even in the weak coupling regime. (ii) for 0 < µ < 1, when both τ and τ2 diverge, it is evenimpossible to define an average rate of occurring plateaux, i.e. the noise is scale invariant. Inthis case, the two point noise correlator decays asX(t+ t′)X(t) −X(t+ t′) X(t) ≃ X(t)2 (t/t′)1−µ , (2)as opposed to the exponential decay in the case µ > 2. For 0 < µ < 1, the power spectrum thusexhibits a 1/fµ divergence at low frequencies. Furthermore, the t/t′-scaling in (2) implies anintrinsic nonstationarity of this noise.In this work we focus on the influence of such a noise on the phase coherence of a qubit.The full Hamiltonian of a dissipative quantum two-level system can be written in the form:H = −12ǫ σz − 12(sin η σx + cos η σz)X +Henv(X) , (3)where ǫ and η are control parameters which we assume to be time-independent and X is thefluctuating extra bias induced by the environment described by Henv. For definiteness we con-sider in the following two special working points for the qubit: (i) the optimal point (η = π/2),where the linear longitudinal coupling to the noise is tuned to zero and consequently decoher-ence is reduced considerably, 4 and (ii) the case of longitudinal coupling (η = 0). Below we willshow that transverse linear coupling is - regardless of the considered noise model - equivalentto quadratic longitudinal coupling and a remaining transverse noise component with suppressedlow-frequency tail.Optimal point (η = π/2): In general, transverse noise leads to relaxation, but also contributesto pure dephasing in higher orders. The effect of low-frequency transverse noise (ω ≪ ǫ) canbe treated in the adiabatic approximation: the state of the qubit follows the effective magneticfield and (3) can be diagonalized to −σz√ǫ2 +X2/2 ≈ −σz[ǫ+X2/(2ǫ)]/2. Therefore, low-frequency transverse noise contributes to pure dephasing as quadratic longitudinal noise. Onthe other hand, higher frequencies, ω ∼ ǫ, mainly contribute to relaxation and the adiabatic ap-proximation breaks down. Situations where both low and high frequencies are present thereforeneed further analysis.The unitary transformation U(t) = exp{iΘ(t)σy/2} with Θ(t) = arctan(X(t)/ǫ) transformsthe Hamiltonian (3) to a spin frame that follows the effective field ǫzˆ +X(t)xˆ:H˜ = U(t)HU †(t) + i U(t) ddtU †(t) = −12√ǫ2 +X2(t) τz +12Θ˙(t) τy , (4)aOur predictions also differ from those for an ensemble of independent Poissonian fluctuators.6,7 A detailedcomparison is not straightforward and will be addressed elsewhere.where the Pauli matrices τi remind that the spin is now measured in the time-dependent basis,defined by τz(t) = U(t)σzU†(t). Note that for small amplitude of the noise |X(t)| ≪ ǫ the newbasis is almost identical to the original basis.As we can see from the expression of the transformed Hamiltonian (4) linear transversenoise is indeed equivalent to quadratic longitudinal noise - as predicted in the adiabatic ap-proximation. But due to the explicit time dependence of the transformation U(t) an addi-tional transverse component proportional to the time derivative of the original noise X appears,Θ˙(t) = (X˙(t)/ǫ)/(1 + (X(t)/ǫ)2) ≈ X˙(t)/ǫ. One might think that the transformed Hamiltonianis therefore even more complicated than the original one since we are now dealing with bothlongitudinal and transverse fluctuations. But a closer look at the expression reveals that thelow-frequency transverse noise is suppressed, SΘ˙(ω) = 2〈Θ˙(t)Θ˙(t′)〉ω ≈(ωǫ)2SX(ω). Hence, inthe experimentally relevant situation where the spectrum SX is less singular than 1/ω2 at lowfrequencies, SΘ˙ is a regular function at low frequencies with SΘ˙(ω = 0) = 0. As a consequencethe transformed transverse noise does not contribute considerably to dephasing but only to re-laxation. Dephasing is dominated by the longitudinal part of the noise proportional to X2 thatstill contains all low-frequency contributions. We can therefore add up the two contributions in(4) independently and extract simple expressions for dephasing and relaxation rates:T−11 =14SΘ˙(ω = ǫ) ≃14SX(ω = ǫ) (5)T−12 =12T−11 + T−1ϕ (6)where T1 and the first contribution to T2 are due to the second term in (4) and can be calculatedsimply using the golden rule. The pure dephasing time Tϕ is defined as the characteristic decaytime due to quadratic longitudinal noise X2(t)/(2ǫ) in (4). In the case of a non-singular noisespectrum it is given by T−1ϕ = SX2(ω = 0)/(4ǫ2). For Gaussian 1/f noise it has been calculatedin Ref.8 In this case the dephasing law differs from a simple exponential and one should add twodecay laws rather than the rates in (6). For the model of nonstationary 1/f noise considered inthis work the evaluation of Tϕ can be reduced straightforwardly to the case of linear longitudinalcoupling: X(t) andX2(t)/(2ǫ) both describe an asymmetric telegraph noise, the major differencebeing a change in the amplitude and thus in the coupling strength.b The coupling strength atthe optimal point is thereby considerably reduced in comparison to linear longitudinal coupling.Longitudinal coupling (η = 0): We now focus on the case of pure dephasing, η = 0 in (3).Therefore, we follow the time evolution of the off-diagonal entries of the qubit’s density matrix,〈σ+(t)〉 = D(tp, t)eiǫt〈σ+(0)〉, where the dephasing factor is defined by D(tp, t) = eiΦ(tp,t) withΦ(tp, t) =∫ tp+ttp X(t′)dt′. Because of the intrinsic nonstationarity of the noise (2) we expectto find an explicit dependence of dephasing on the preparation time tp. In the following wewill therefore investigate this explicit dependence of D(tp, t) on tp. This can be achieved usingCTRW techniques and taking special care of initial conditions at tp.12As already discussed previously we have to distinguish three different cases depending onthe value of µ: When µ > 2 the noise is equivalent to Poissonian telegraph noise6 and thereforestationary (for tp > τ0). Consequently, dephasing is insensitive to the absolute time tp.For 1 < µ < 2 it turns out that nonstationarity only plays a role in the strong couplingregime g > 1. For weak coupling g < 1 the noise is qualitatively equivalent to the Poissoniancase µ > 2, i.e. only subdominant corrections appear.12 For g > 1 the dephasing law differsfrom a simple exponential and the dephasing time, defined as D(tp, τφ) = e−1, scales as τφ ≃τ0[1/e + (τ0/(τ0 + tp))µ−1]−1/(µ−1). The dependence of τφ on tp thus only matters for values ofµ close to 1 and disappears as µ increases to higher values (see Figure 1).bIn addition X2(t)/(2ǫ) has a finite mean. A thorough discussion of this case will be presented elsewhere.0.001 0.01 0.1 1 10g110001e+061e+091e+12τ φ/τ0tp=108τ0tp=106τ0tp=104τ0tp=102τ0µ=0.8 µ=0.40.01 0.1 1 10g1100100001e+06τ φ/τ0µ = 1.1µ = 1.7tp=108τ0tp=106τ0tp=104τ0tp=107τ0tp=102τ0Figure 1: Dephasing time τφ as a function of the coupling strength g for µ < 1 (left) and µ > 1 (right).As expected nonstationarity turns out to be most prominent in the case 0 < µ < 1. Again,we have to distinguish two regimes of weak and strong coupling separated by a critical couplingconstant gc. But here gc is not of order one, but depends explicitly on the age of the noise:gc(tp) ≃ λ(µ)t−µ/2p ≪ 1. Therefore, the range of the strong coupling regime increases with theage of the system implying that any qubit subject to a noise with 0 < µ < 1 eventually ends upin the strong coupling regime.For weak coupling g < gc(tp) dephasing is essentially insensitive to the age of the noise. Awayfrom a (short) initial aging regime D(tp, t) decreases as exp(−(t/τφ)µ), where the dephasingtime scales as τφ ∼ τ0g−2/µ (in the marginal case µ = 1 we get logarithmic corrections τφ ∼τ0g−2| ln g|). Indeed, for t < τφ, D(tp, t) is accurately given within a Gaussian approximationof the phase: D(tp, t) ≃ exp(−12Φ(tp, t)2). Only for t > τφ the Gaussian approximation breaksdown and the exponential decay goes over to a much slower algebraic one, D(tp, t) ∝ 1/tµ.In the strong coupling regime g > gc(tp) the dephasing depends strongly on the age of thenoise. The Gaussian approximation of the phase breaks down even at short times and the decayis not exponential any more. The dephasing time becomes proportional to tp, τφ ≃ C(µ)tp, andsaturates as a function of g. We emphasize that for µ→ 1, C(µ) takes smaller and smaller valuesimplying that the dephasing time can be orders of magnitude shorter than the preparation time.In summary, we have proposed a new model of 1/fµ noise describing phenomenologicallya slow collective environment. The present model shows a cross-over from nonstationary 1/fµnoise to a memoryless Poissonian telegraph noise as a function of temperature. We analyzedthe decay of coherence of a qubit subject to this noise at different operation points. For lowtemperatures and strong coupling the dephasing time was found to depend explicitly on the ageof the noise.References1. M. B. Weissman, Rev. Mod. Phys. 60, 537 (1998).2. M. Nielsen and I. Chuang Quantum Computation and Quantum Communication, Cam-bridge University Press, (2000).3. Y. Nakamura et al., Phys. Rev. Lett. 88, 047901 (2002).4. D. Vion et al., Science 296, 886 (2002).5. A. Shnirman, Y. Makhlin, and G. Scho¨n, Physica Scripta T102 (2002) 147.6. E. Paladino et al., Phys. Rev. Lett 88 (2002), 228304.7. Y. M. Galperin, B. L. Altshuler, and D. V. Shantsev, cond-mat/0312490.8. Yu. Makhlin and A. Shnirman, cond-mat/0308297, Phys. Rev. Lett. (2004).9. A. L. Burin, J. Low Temp. Phys. 100, 309 (1995).10. Le´vy Statistics and Laser Cooling, F. Bardou et al., Cambridge University Press (2002).11. J.-P. Bouchaud, J. Physique I 2 (1992), 1705.12. J. Schriefl, M. Clusel, D. Carpentier, P. Degiovanni, cond-mat/0403750 .

Dephasing due to nonstationary 1/f noise

4 pages, 2 figures, to be published in the proceedings of the Vth Rencontres de Moriond in Mesoscopic PhysicsInternational audienceMotivated by recent experiments with Josephson qubits we propose a new phenomenological model for 1/f noise due to collective excitations of interacting defects in the qubit\'s environment. At very low temperatures the effective dynamics of these collective modes are very slow leading to pronounced non-Gaussian features and nonstationarity of the noise. We analyze the influence of this noise on the dynamics of a qubit in various regimes and at different operation points. Remarkable predictions are absolute time dependences of a critical coupling and of dephasing in the strong coupling regime

Dephasing due to nonstationary 1/f noise

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