Structured environments in solid state systems: crossover from Gaussian to non-Gaussian behavior
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Structured environments in solid state systems: crossover from Gaussian tonon-Gaussian behavior
E. Paladino (1) , A. G. Maugeri (1) , M. Sassetti (2) , G. Falci (1) and U. Weiss (3) (1)
MATIS CNR-INFM, Catania & Dipartimento di MetodologieFisiche e Chimiche, Universit´a di Catania, 95125 Catania, Italy. (2)
Dipartimento di Fisica, Universit`a di Genova & LAMIA CNR-INFM, 16146 Genova, Italy. (3)
II. Institut f¨ur Theoretische Physik, Universit¨at Stuttgart, D-70550 Stuttgart, Germany. (Dated: December 1, 2018)The variety of noise sources typical of the solid state represents the main limitation toward the re-alization of controllable and reliable quantum nanocircuits, as those allowing quantum computation.Such “structured environments” are characterized by a non-monotonous noise spectrum sometimesshowing resonances at selected frequencies. Here we focus on a prototype structured environmentmodel: a two-state impurity linearly coupled to a dissipative harmonic bath. We identify the timescale separating Gaussian and non-Gaussian dynamical regimes of the Spin-Boson impurity. Byusing a path-integral approach we show that a qubit interacting with such a structured bath mayprobe the variety of environmental dynamical regimes.
PACS numbers: 03.65.Yz, 03.67.Lx, 05.40.-aKeywords: Decoherence, quantum statistical methods, quantum computation
INTRODUCTION
Controlled coherent dynamics of solid state deviceshas been demonstrated in recent years [1, 2]. Comparedto other implementations, solid state qubits suffer fromstronger broadband noise originating from sources withdifferent character. The main limitation toward the re-alization of controllable and reliable quantum circuits al-lowing quantum computation is decoherence due to ma-terial (and device) dependent noise sources. The result-ing noise spectrum is non-monotonous and sometimescharacterized by resonances. Often these features maybe attributed to interaction with a nonlinear and non-Markovian environment [3].In superconducting nanocircuits a particularly detri-mental role is played by fluctuating impurities locatedin the insulating material surrounding the qubit, whichare responsible for charge noise and flux noise [4, 5].Background charges are known to be responsible forlow-frequency 1 /f noise [6], moreover experiments withJosepshon devices suggested that spurious two-level sys-tems may also affect high-frequency noise [7]. Connec-tions between low and high-frequency noise features havebeen suggested in the recent experiment Ref. [8]. Differ-ent microscopic mechanisms [9] and effective models [10]have been recently proposed to explain the observed spec-tral features.Predicting decoherence originating from such a struc-tured environment often responsible for non-Gaussiannoise is a non trivial task, which has attracted a lot of at-tention in the past years [3, 11, 12, 13, 14]. A well estab-lished scheme consists in studying the reduced dynamicsof an extended system composed of the qubit and of theenvironmental degrees of freedom responsible for non-Gaussian behavior. This strategy, combined with an ap- propriate classification of the noise sources (i.e. adiabaticor quantum noise), each treated via appropriate approx-imate tools, provides a general scheme to deal with thevariety of noise sources typical of the solid state [14, 15].In this paper we focus on a prototype impurity model, atwo-state impurity linearly coupled to a dissipative har-monic bath. Such Spin-Boson models have been thor-oughly investigated with a variety of methods since the’80s [16, 17]. Thus the impurity dynamics is well knownin a wide region of parameters space. This allows theidentification of the impurity characteristic time scalesand, therefore, of the conditions where deviations fromGaussian and/or weak coupling regimes are expected.Having this information at our disposal we will applystandard techniques developed for quantum dissipativesystems to find the qubit dynamics in the presence ofthis structured bath. The analysis will provide evidencefor the appearance of non-Gaussian effects. In partic-ular, their onset will be shown to be related to clearlyidentifiable effects in the qubit behavior.In Section 2 we introduce the qubit-impurity modeland identify the relevant impurity dynamical quantities.In Section 3 we review the equilibrium correlation func-tion for a Spin-Boson impurity and identify its correla-tion time. In Section 4 we study the qubit dynamics inthe presence of the Spin-Boson impurity within a path-integral approach and discuss the main features of thecrossover from weak to strong coupling. MODEL AND RELEVANT DYNAMICALQUANTITIES
To be specific we shall refer to superconducting qubitsbased on the Cooper-pair box [1, 18]. Under properconditions the device behaves as a two-state system de-scribed in terms of Pauli matrices by ( ~ = 1) H qubit = − E C σ z − E J σ x . (1)The charging energy E C gives the additional cost ofadding an extra Cooper pair to the superconducting is-land and the possibility of coherent transfer of pairsthrough the junction is given by the Josephson term E J σ x /
2. The charge on the superconducting island mayfluctuate because of interaction with uncontrolled impu-rities. Here we model a single impurity with a Spin-Bosonmodel, the overall Hamiltonian being given by H = H qubit + H SB − v σ z τ z (2) H SB = − ε τ z − ∆2 τ x − X τ z + H E . (3)The two-level-system impurity ( ~τ ) is coupled to a har-monic bath, described by H E = P α ω α a † α a α , via thecollective coordinate X . Its effect on the impurity de-pends only on the spectral density G ( ω ) or equivalentlyon the power spectrum S ( ω ) = Z ∞−∞ dt h X ( t ) X (0) + X (0) X ( t ) i e iωt == π G ( | ω | ) coth β | ω | , (4)where h ... i denotes the thermal average with respect to H E , and β = 1 /k B T . We consider the standard casewhen the coupling operator is a collective displacement X = P α λ α ( a α + a † α ) with ohmic spectral density G ( ω ) = X α λ α δ ( ω − ω α ) = 2 K | ω | e −| ω | /ω c , (5)where ω c represents the high frequency cut-off of the har-monic modes.A first step in understanding the effects of damping isto view the impurity ~τ and the ohmic bath as an envi-ronment for the qubit ~σ . This environment is in generalnon-Gaussian and non-Markovian. A Gaussian approxi-mation of this structured bath amounts to replace it withan effective harmonic model directly coupled to σ z andwith power spectrum S τ ( ω ) S τ ( ω ) = 12 Z ∞−∞ dt (cid:16) h τ z ( t ) τ z (0) + τ z (0) τ z ( t ) i − h τ z i ∞ (cid:17) e iωt , (6)the Fourier transform of equilibrium symmetrized auto-correlation function of the impurity observable which di-rectly couples to the qubit. Here the thermal average isperformed with respect to H SB and h τ z i ∞ is the thermalequilibrium value for τ z . Under this approximation andusing a master equation approach, the relaxation anddephasing rates for the qubit ~σ in lowest order in the T/ ∆ r Ω / ∆ r T/ ∆ r Ω / ∆ r ~ ~ FIG. 1: ˜Ω( T ) of Eqs.(11) and (13) as a function of tempera-ture. Inset: non-monotonous behavior for T ≪ T ∗ = 31∆ r .Parameters are K = 0 . ω c / ∆ r = 31, and we fixed k B = 1. coupling v read [17, 20],1 T = (cid:16) E J E (cid:17) v S τ ( E )2 (7)1 T = 12 T + 1 T ∗ == (cid:16) E J E (cid:17) v S τ ( E )4 + (cid:16) E C E (cid:17) v S τ (0)2 (8)where E = p E C + E J is the qubit splitting. The va-lidity of this standard approach is limited to couplings v ≪ /τ c [20], where τ c is the range of the correlationfunction in Eq.(6) (to be defined in the next Section).Clearly if the impurity τ has a slow dynamics v τ c ≫ vτ c > τ c increases [3].¿From a different perspective the failure of the Gaus-sian approximation can be understood by viewing thequbit ~σ as a measuring device [21] for the mesoscopicsystem described by the Spin-Boson model involving ~τ .A rather rough measurement protocol (short times, av-eraging of results) makes the dynamics of ~σ essentiallysensitive only to S τ ( ω ), whereas if the Spin-Boson has aslow dynamics the spin ~σ is able to detect also details ofthe dynamics of ~τ which go beyond S τ ( ω ), and have tobe described with more careful methods.In the following we will treat the impurity ~τ on thesame footing as the qubit ~σ , we will apply standard meth-ods to trace out the bosonic degrees of freedom withoutany approximation on the qubit-impurity coupling. T/ ∆ r Ω / γ ~ FIG. 2: Quality factor of the damped oscillations Q ( T ) =˜Ω( T ) /γ ( T ) from Eqs.(11), (12) and (13), (14). Parametersare fixed as in Fig.1. IMPURITY DYNAMICS AND CORRELATIONTIME
In this Section we will identify the characteristic timescale of the dynamics of the equilibrium fluctuations ofthe Spin-Boson impurity described by Eq.(6). The un-coupled (i.e. for v = 0) impurity dynamics strictly de-pends on the damping strenght K and on the tempera-ture [17]. We consider the small damping K ≪ equilibrium auto-correlation function of the ob-servable τ z which directly couples to the qubit. We re-mark that the thermal initial state of the Spin-Boson sys-tem, which is implied by the equilibrium correlation func-tion, may originate peculiar time-dependencies. Quali-tatively different behaviors may in fact be displayed bythe non-equilibrium correlation function, which is evalu-ated for a factorized initial state of the Spin-Boson sys-tem [17, 19]. Evaluation of equilibrium correlation func-tions for the Spin-Boson model is a non trivial task. How-ever it has been shown [17] that in the unbiased case ε = 0, and for small damping K ≪
1, the τ z auto-correlation function does not depend on the initial cor-related or factorized state. Here we focus on this caseand recall the characteristics of S τ ( ω ) more relevant forour analysis, the interested reader may find details of the T/ ∆ r γ / ∆ r FIG. 3: Dephasing rate γ ( T ) from Eqs.(12) and (14) (fullline), for temperatures T < T ∗ ( K ) and relaxation rates γ / ( T ) from Eq.(17) for T ≥ T ∗ ( K ). The rate γ (dotted)increases with temperature, γ (dashed) shows the Kondo be-havior. The value of T ∗ ( K ) / ∆ r ≈ . derivation in Refs. [17, 19].The crossover from under-damped to over-damped be-havior with increasing temperature takes place at T ∗ ( K ) ≈ ∆ r πKk B , (9)where ∆ r = ∆(∆ /ω c ) K/ (1 − K ) is the renormalized tun-neling amplitude in the Spin-Boson model.The explicit form for S τ ( ω ) depends ontemperature[22]For T < T ∗ ( K ) it reads S τ ( ω ) = γ + ( ˜Ω − ω ) tan φ ( ω − ˜Ω) + γ + γ + ( ˜Ω + ω ) tan φ ( ω + ˜Ω) + γ , (10)where tan φ = γ/ ˜Ω. The effective frequency ˜Ω and therelaxation rate are˜Ω( T )∆ r = 1 + K h Re ψ (cid:16) i ∆ r πk B T (cid:17) − ln (cid:16) ∆ r πk B T (cid:17) − ψ (1) i (11) γ ( T ) = S (∆ r )4 , (12)for temperatures k B T < ∆ r and˜Ω( T )∆ T = r − (cid:16) TT ∗ (cid:17) − K (13) γ ( T ) = πKk B T = Γ2 (14)for larger temperatures ∆ r ≤ k B T ≤ k B T ∗ ( K ).Here 2Γ is the white noise level S ( ω ) ≈
2Γ for fre-quencies ω ≪ π/β in (4). Oscillators with frequencies2 π/β ≤ ω ≪ ω c renormalize the tunneling amplitude to∆ T = ∆ r (2 πk B T / ∆ r ) K . (15)In Fig.1 we show the overall behavior of ˜Ω( T ). At verysmall temperatures k B T ≪ ∆ r (see inset in Fig.1) ˜Ω( T )increases ∝ T . The two forms (11) and (13) smoothlymatch on each other at k B T = ∆ r . A maximum isreached at T = K / − K ) T ∗ > ∆ r /k B , above this tem-perature ˜Ω( T ) decreases monotonously and approacheszero at T ∗ . Coherent oscillations are dephased on ascale 1 /γ ( T ) which decreases monotonously starting from1 /γ (0) = 2 / ( πK ∆ r ). The resulting quality factor ofthe damped oscillations Q ( T ) = ˜Ω( T ) /γ ( T ) is shown inFig.2.For temperatures higher than T ∗ the dynamics is in-coherent and S τ ( ω ) has a different form S τ ( ω ) = 2 γ γ − γ γ ω + γ + 2 γ γ − γ γ ω + γ (16) γ / = Γ2 ± r(cid:16) Γ2 (cid:17) − ∆ T . (17)Note that for T ≫ T ∗ ( K ) one of the two rates increaseswith temperature, γ → Γ, whereas the other shows thecharacteristic Kondo behavior, decreasing with tempera-ture γ → ∆ T Γ = ∆ r K ( ∆ r πk B T ) − K . (18)These features are illustrated in Fig.3.The typical scale of the equilibrium fluctuactions ofthe Spin-Boson environment described by S τ ( ω ) definesthe correlation time τ c . Usually in the literature oneis faced with environment models characterized by dy-namic fluctuations which tend rapidly to zero with time.In these cases τ c represents the order of magnitude ofthe width of the environment fluctuactions [20]. In thepresent case however looking at the Spin-Boson impurityas an environment characterized by S τ ( ω ) the identifica-tion of τ c is not immediate due to the different forms ofthe equilibrium fluctuations given by Eqs.(10) and (16).For high temperatures S τ ( ω ) is approximately a singleLorentzian centered at ω = 0 and width γ , leading tothe identification τ c ≈ /γ ( T ), see Fig.4. In the oppositelimit of very low temperatures S τ ( ω ) has a double peakstructure representing a bath responsible for oscillatingfluctuations very weakly damped. In this unusual envi-ronment regime, the typical scale of the impurity fluc-tuations is represented by 1 / ˜Ω( T ) which plays the roleof τ c . For intermediate temperatures in general two al-most superimposed Lorentzians contribute to S τ ( ω ) and τ c may be approximately identified from the condition S τ (1 /τ c ) = S maxτ /
2. The resulting τ c ( T ), illustratedin Fig.5, interpolates between the asymptotic behaviorsat low and high temperatures. The slight reduction of τ c ( T ) at intermediate temperatures is a consequence ofthe crossover from under-damped to incoherent dynam-ics, as shown in Fig.4.Once the impurity correlation time is identified, thecondition vτ c ( T ) = 1 separates the weak-coupling regimeof the qubit dynamics, from the strong coupling regime -2 -1 0 1 2 ω/∆ r ∆ r S τ ( ω ) -2 -1 0 1 2 ω/∆ r ∆ r S τ ( ω ) FIG. 4: Equilibrium correlation function S τ ( ω ) for increas-ing values of the temperature. Top: from under-damped toover-damped regime and T / ∆ r = 3 (light gray), T / ∆ r = 15(black). Dashed lines indicate the width 1 /τ c at half height.Bottom: from over-damped to incoherent regime T / ∆ r = 20(light gray), T / ∆ r = 30 (dashed), T / ∆ r = 40 (black). Pa-rameters are fixed as in Fig.1. where non-Gaussian behavior shows up. In the first case,when vτ c ( T ) ≪
1, the standard master equation predictsexponential decay with the Golden Rule rate T given inEq.(8). ¿From the above analysis we expect the masterequation result to be valid in the following regimes: Fortemperatures T < T ∗ if v/ ˜Ω( T ) ≈ v/ ∆ r ≪
1, for largertemperatures
T > T ∗ if v/γ ( T ) ≈ v Γ / ∆ T ≪
1, thiscondition can be cast in the following form v ∆ r ≪ (cid:16) T ∗ T (cid:17) − K . (19)Therefore, for small values of v/ ∆ r a crossover fromweak to strong coupling is expected with increasing tem-perature. For the ensuing discussion here we reportthe expected value of the pure dephasing rate 1 /T ∗ =( E C /E ) v S τ (0) / T ∗ = (cid:16) E C E (cid:17) v γ ( T ) γ ( T ) + ˜Ω( T ) T < T ∗ (20)1 T ∗ = (cid:16) E C E (cid:17) (cid:16) v ∆ T (cid:17) Γ T > T ∗ (21)The two forms match on each other at T ∗ , and it is easyto show that Eq.(21) approximates 1 /T ∗ also for T ≪ T ∗ . T/ ∆ r τ c ∆ r FIG. 5: Correlation time τ c ( T ): for sufficiently small temper-atures τ c ≈ / ∆ r . For T ≫ T ∗ ( K ) the asymptotic behavior ≈ /γ ( T ) is indicated (dashed). The interpolating form forintermediate temperatures has been obtained from the condi-tion S τ (1 /τ c ) = S maxτ /
2. Parameters are fixed as in Fig.1.
QUBIT DYNAMICS: PATH-INTEGRALAPPROACH
The discussion of the previous Section has evidenced theexistence of a large parameter regime where the Gaussianapproximation of the Ohmic Spin-Boson model does notapply. In this Section we study the qubit dynamics viaa path-integral approach which includes as a special casethe regime of Gaussian behavior of the impurity dynam-ics. We find exact expressions which we discuss for finitetemperatures, specifically for k B T ≥ p ∆ r + v .We focus on the so called pure-dephasing regime, E J =0, which represents the point of maximum noise sensitiv-ity of the qubit. Thus the more interesting in the per-spective of using the qubit as a “noise” analyzer. In thepure dephasing regime the charge on the qubit island isa constant of motion since [ H , σ z ] = 0, dephasing be-ing described by the decay of h σ x/y i or equivalently ofthe coherences h σ ± i . A simple analysis shows that thecoherences are related to correlation functions involvingthe Spin-Boson variables, specifically we found [3, 23] h σ − ( t ) ih σ − (0) i = e iE C t Tr SB (cid:8) e − i H SB − t ρ τ (0) ⊗ w β e i H SB + t (cid:9) ≡ e iE C t C − + ( t ) (22)where we have chosen a factorized initial density ma-trix for the qubit-impurity, ρ (0) = ρ σ (0) ⊗ ρ τ (0), withthe impurity τ initialized in the mixed state ρ τ (0) = ˆ I + δp (0) τ z and the bosonic bath in its thermalequilibrium state w β . The two conditional impurityHamiltonians H SB ± depend on the qubit state and read H SB ± = H SB ± v τ z .In Ref. [23] it has been shown that the Laplace trans- T/ ∆ r R e [ λ i / ∆ r ] I m [ λ i / ∆ r ] T/ ∆ r FIG. 6: Top: real parts of the exact solutions λ i of thepole equation D ( λ ) = 0 as a function of temperature for k B T ≥ √ v + ∆ r ≈ . r . Bottom: corresponding imagi-nary parts. Two λ i are complex conjugate below T − ≈ . r and above T + ≈ . r . All λ i are real at intermediate tem-peratures T − < T < T + . The dominant pole is real (blackdashed) until T + where the character of the dominant solu-tion changes. For T > T + the dominant poles are complexconjugate. Parameters are v/ ∆ r = 0 . ω c / ∆ r = 30 and K = 0 . form of the correlator C − + ( t ) reads b C − + ( λ ) = 1 D ( λ ) [ λ + K ( λ ) − ivδp (0) ] (23) D ( λ ) = λ + v + λK ( λ ) + ivK ( λ ) . (24)An exact formal series expression in ∆ for the ker-nels K ( λ ), K ( λ ) has been derived in Ref.[23]. In theMarkovian regime for the harmonic bath, i.e. for K ≪ p ∆ r + v ≤ k B T ≪ ω c , all contri-butions to K ( λ ) and K ( λ ) of order higher than ∆ cancel out exactly. The lowest order contributions [25]do not depend on the coupling v and coincide with thekernels entering the dynamics of ~τ in the uncoupled case( v = 0) [17] which read K ( λ ) = ∆ T λ + Γ ǫ + ( λ + Γ) (25) K ( λ ) = − πK ∆ T ǫǫ + ( λ + Γ) . (26)Inserting Eqs.(25) - (26) in (23) and (24) b C − + ( λ ) is read- t ∆ r C - + ( t ) -0.5 0 0.5 ω/∆ r ∆ r C - + ( ω ) ~ FIG. 7: Top: C − + ( t ) for v/ ∆ r = 0 .
25 and k B T / ∆ r = 4(black), k B T / ∆ r = 15 (gray). Bottom: correspondingFourier transform. Here K = 0 . ω c / ∆ r = 30, δp (0) = 0. ily found as b C − + ( λ ) = [( λ +Γ) + ǫ ] [ λ − ivδp (0)]+∆ T ( λ +Γ) D ( λ ) D ( λ ) = ( λ + v ) [( λ + Γ) + ǫ ] + ∆ T λ ( λ + Γ) − iπKvǫ ∆ T . (27)The scales entering the time evolution of C − + ( t ) arefound from the solution of the pole equation D ( λ ) = 0,which have been reported in Ref.[15]. Here we specifyto the unbiased case ǫ = 0, the goal being to elucidatethe correspondence with the expected Gaussian/non-Gaussian dynamical regimes as deduced from the equi-librium correlation function S τ ( ω ) discussed in Section . Pure dephasing due to a unbiased impurity
When ǫ = 0 the pole condition D ( λ ) = 0 with Eq.(27)reduces to a cubic equation which has either one real andtwo complex conjugate solutions, or three real solutions.We denote the three roots as λ = − Λ ∈ R e and λ / = − Λ ± iδE , where δE is either real or purely imaginary.The expression of C − + ( t ) in terms of the λ i is obtainedby inverting the Laplace transform (27) and reads T/ ∆ r R e [ λ i / ∆ r ] T/ ∆ r I m [ λ i / ∆ r ] FIG. 8: Top: real parts of the exact solutions λ i of thepole equation D ( λ ) = 0 as a function of temperature for k B T ≥ √ v + ∆ r ≈ . r . Bottom: corresponding imagi-nary parts. Two λ i are complex conjugate and one is realfor any temperature. The dominant poles are always com-plex conjugate. In this regime no crossing takes place amongthe R e [ λ i ] and the dominant root is non-monotonic with T .Parameters are v/ ∆ r = 0 . K = 0 . ω c / ∆ r = 30. C − + ( t ) = A e − Λ t + (1 − A ) cos ( δEt ) e − Λ t ++ B sin ( δEt ) e − Λ t (28) A = − Λ + ∆ T − i δp (0) v Λ(Λ − Λ ) + δE (29) B = (cid:2) Λ(1 − A ) + Λ A − iδp (0) v (cid:3) δE . (30)It is possible to show that the character of the roots de-pends on v/ ∆ T and on the temperature. In particularfor v < ∆ T / √ k B T ± ≈ πK h − (cid:16) v (cid:17) + 5∆ v ± √ ∆ − v ∆ + 192 v ∆ − v ∆ v i / (31)such that for T < T − and T > T + one solution isreal and two are complex conjugate, whereas for inter-mediate temperatures the three solutions are real. For v > ∆ T / √ t ∆ r -0.500.51 C - + ( t ) -2 -1 0 1 2 ω / ∆ r ∆ r C - + ( ω ) ~ FIG. 9: Top: C − + ( t ) for v/ ∆ r = 0 . k B T / ∆ r = 4(black), k B T / ∆ r = 15 (gray). Bottom: correspondingFourier transform. Here K = 0 . ω c / ∆ r = 30, δp (0) = 0. C − + ( t ). To this end we focus on the dominant λ i , i.e. thesmallest in absolute value. The analysis is convenientlyperformed distinguishing regimes where v/ ∆ T < v/ ∆ T > Case v/ ∆ T ≪ k B T > p v + ∆ r ). Fortemperatures low enough to fulfill the condition k B T ∆ T ≪ πKv ≈ k B T + ∆ r (32)the dominant scale is real and reads Λ ≈ ( v ∆ T ) Γ. In(30) A ≈ − (Γ / T ) and B ≈ − iδp (0) v/ ∆ T therefore C + − ( t ) ≈ exp [ − Λ t ] . (33)With increasing temperature, above T + the dominantscales are complex conjugate, λ / = − Λ ± iδE ≈− ∆ T / ± iv . Figure 6 illustrates the crossover betweenthe two regimes for v/ ∆ r = 0 .
25. Since A ≪ v/ Γ and B ≈ − iδp (0), we get C + − ( t ) ≈ cos ( vt ) e − Λ t − iδp (0) sin ( vt ) e − Λ t . (34)As expected, for small enough temperature the Gaussianapproximation for the structured bath applies and asingle scale dominates the qubit dynamics. It is easilyseen that the condition (32) corresponds to (19) whichwas derived from the weak coupling criterion vτ c ≪ = 1 /T ∗ = v S τ (0) / ≈ v in C + − ( t ) as shown in Fig.7,and beatings at frequencies E C ± v in the coherencesEq.(22). Note that the coupling strength v only entersthe induced frequency shift, whereas the decay rateshows the Kondo behavior Λ ∝ T K − , cfr Eq.(18). Case v/ ∆ T ≫ v/ ∆ T the systemstays in the regime where the dominant scales are com-plex conjugate and show a nontrivial temperature depen-dence. At temperatures T ≥ p v + ∆ r , the two polesread λ / ≈ − (∆ T /v ) Γ / ± iv , with increasing T in-stead λ / ≈ − ∆ T / (2Γ) ± iv . The qubit dynamics followsEq.(34). The dominant rate Λ is non-monotonous first in-creasing with temperature and than decreases ∝ T K − .For large enough temperatures the decay of the coher-ences does not depend on v , as expected.This qualitative behavior is already present for inter-mediate values of v/ ∆, as shown in Fig.8. The poles λ i never cross, therefore there is no change in the characterof the qubit dynamics. In the specific case consideredat small/intermediate temperatures the real parts of thethree poles are of the same order. This is reflected in theFourier transform of C + − ( t ) where three Lorentzians canbe identified, one centered at ω = 0, the others at ≈ ± v ,see Fig.9. DISCUSSION
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