ARTICLE IN PRESS

1386-9477/$ - se

doi:10.1016/j.ph

�CorrespondE-mail addr

Physica E 40 (2007) 198–205

www.elsevier.com/locate/physe

Structured environments in solid state systems: Crossover fromGaussian to non-Gaussian behavior

E. Paladinoa,�, A.G. Maugeria, M. Sassettib, G. Falcia, U. Weissc

aMATIS CNR-INFM, Catania & Dipartimento di Metodologie Fisiche e Chimiche (DMFCI), Universita di Catania. Viale A. Doria 6, 95125 Catania, ItalybDipartimento di Fisica, Universita di Genova, & LAMIA CNR-INFM, Via Dodecaneso 33, 16146 Genova, Italy

cII Institute fur Theoretische Physik, Universitat Stuttgart, Pfaffenwaldring 57, Germany

Available online 21 May 2007

Abstract

The variety of noise sources typical of the solid state represents the main limitation toward the realization of controllable and reliable

quantum nanocircuits, as those allowing quantum computation. Such ‘‘structured environments’’ are characterized by a non-

monotonous noise spectrum sometimes showing resonances at selected frequencies. Here we focus on a prototype structured

environment model: a two-state impurity linearly coupled to a dissipative harmonic bath. We identify the time scale separating Gaussian

and non-Gaussian dynamical regimes of the Spin-Boson impurity. By using a path-integral approach we show that a qubit interacting

with such a structured bath may probe the variety of environmental dynamical regimes.

r 2007 Elsevier B.V. All rights reserved.

PACS: 03.65.Yz; 03.67.Lx; 05.40.�a

Keywords: Decoherence; Quantum statistical methods; Quantum computation

1. Introduction

Controlled coherent dynamics of solid state devices hasbeen demonstrated in recent years [1,2]. Compared to otherimplementations, solid state qubits suffer from strongerbroadband noise originating from sources with differentcharacter. The main limitation toward the realization ofcontrollable and reliable quantum circuits allowing quan-tum computation is decoherence due to material (anddevice) dependent noise sources. The resulting noisespectrum is non-monotonous and sometimes characterizedby resonances. Often these features may be attributed tointeraction with a non-linear and non-Markovian environ-ment [3].

In superconducting nanocircuits a particularly detrimen-tal role is played by fluctuating impurities located in theinsulating material surrounding the qubit, which areresponsible for charge noise and flux noise [4,5]. Back-ground charges are known to be responsible for low-

e front matter r 2007 Elsevier B.V. All rights reserved.

yse.2007.05.002

ing author.

ess: epaladino@dmfci.unict.it (E. Paladino).

frequency 1=f noise [6], moreover experiments withJosephson devices suggested that spurious two-levelsystems may also affect high-frequency noise [7]. Connec-tions between low- and high-frequency noise features havebeen suggested in the recent experiment Ref. [8]. Differentmicroscopic mechanisms [9] and effective models [10] havebeen recently proposed to explain the observed spectralfeatures.Predicting decoherence originating from such a struc-

tured environment often responsible for non-Gaussiannoise is a non-trivial task, which has attracted a lot ofattention in the past years [3,11–14]. A well establishedscheme consists in studying the reduced dynamics of anextended system composed of the qubit and of theenvironmental degrees of freedom responsible for non-Gaussian behavior. This strategy, combined with anappropriate 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]. Inthis paper we focus on a prototype impurity model, a two-state impurity linearly coupled to a dissipative harmonic

ARTICLE IN PRESSE. Paladino et al. / Physica E 40 (2007) 198–205 199

bath. Such Spin-Boson models have been thoroughlyinvestigated with a variety of methods since the 1980s[16,17]. Thus the impurity dynamics is well known in awide region of parameters space. This allows the identifica-tion of the impurity characteristic time scales and, there-fore, of the conditions where deviations from Gaussianand/or weak coupling regimes are expected. Having thisinformation at our disposal we will apply standardtechniques developed for quantum dissipative systems tofind the qubit dynamics in the presence of this structuredbath. The analysis will provide evidence for the appearanceof non-Gaussian effects. In particular, their onset will beshown to be related to clearly identifiable effects in thequbit behavior.

In Section 2 we introduce the qubit-impurity model andidentify the relevant impurity dynamical quantities. InSection 3 we review the equilibrium correlation functionfor a Spin-Boson impurity and identify its correlation time.In Section 4 we study the qubit dynamics in the presence ofthe Spin-Boson impurity within a path-integral approachand discuss the main features of the crossover from weakto strong coupling.

2. Model and relevant dynamical quantities

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 systemdescribed in terms of Pauli matrices by (_ ¼ 1),

Hqubit ¼ �EC

2sz �

EJ

2sx. (1)

The charging energy EC gives the additional cost of addingan extra Cooper pair to the superconducting island andthe possibility of coherent transfer of pairs through thejunction is given by the Josephson term EJsx=2. The chargeon the superconducting island may fluctuate because ofinteraction with uncontrolled impurities. Here we model asingle impurity with a Spin-Boson model, the overallHamiltonian being given by

H ¼Hqubit þHSB �v

2sztz, (2)

HSB ¼ �e2tz �

D2tx �

1

2Xtz þHE . (3)

The two-level-system impurity (s) is coupled to a harmonicbath, described by HE ¼

Paoaa

yaaa, via the collective

coordinate X. Its effect on the impurity depends only onthe spectral density GðoÞ or equivalently on the powerspectrum

SðoÞ ¼Z 1�1

dt1

2hX ðtÞX ð0Þ þ X ð0ÞX ðtÞieiot

¼ pGðjojÞ cothbjoj2

, ð4Þ

where h. . .i denotes the thermal average with respect toHE , and b ¼ 1=kBT . We consider the standard case when

the coupling operator is a collective displacement X ¼Palaðaa þ ayaÞ with ohmic spectral density

GðoÞ ¼Xa

l2adðo� oaÞ ¼ 2K joje�joj=oc , (5)

where oc represents the high frequency cut-off of theharmonic modes.A first step in understanding the effects of damping is to

view the impurity s and the ohmic bath as an environmentfor the qubit r. This environment is in general non-Gaussian and non-Markovian. A Gaussian approximationof this structured bath amounts to replace it with aneffective harmonic model directly coupled to sz and withpower spectrum StðoÞ,

StðoÞ ¼1

2

Z 1�1

dtðhtzðtÞtzð0Þ þ tzð0ÞtzðtÞi � htzi21Þe

iot, (6)

the Fourier transform of equilibrium symmetrized auto-correlation function of the impurity observable whichdirectly couples to the qubit. Here the thermal average isperformed with respect to HSB and htzi1 is the thermalequilibrium value for tz. Under this approximation andusing a master equation approach, the relaxation anddephasing rates for the qubit r in lowest order in thecoupling v read [17,20],

1

T1¼

EJ

E

� �2v2StðEÞ

2, ð7Þ

1

T2¼

1

2T1þ

1

T�2

¼EJ

E

� �2v2StðEÞ

4þ

EC

E

� �2v2Stð0Þ

2, ð8Þ

where E ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2

C þ E2J

qis the qubit splitting. The validity of

this standard approach is limited to couplings v51=tc [20],where tc is the range of the correlation function in Eq. (6)(to be defined in the next section). Clearly if the impurity thas a slow dynamics vtcb1, this picture does not apply andwe have to resort to other methods. However previousstudies on a similar model have shown that the Gaussianapproximation may give good results even for vtc41 butfor shorter and shorter times, as long as tc increases [3].From a different perspective the failure of the Gaussian

approximation can be understood by viewing the qubit r asa measuring device [21] for the mesoscopic systemdescribed by the Spin-Boson model involving s. A ratherrough measurement protocol (short times, averaging ofresults) makes the dynamics of r essentially sensitive onlyto StðoÞ, whereas if the Spin-Boson has a slow dynamicsthe spin r is able to detect also details of the dynamics of s

which go beyond StðoÞ, and have to be described withmore careful methods.In the following we will treat the impurity s on the same

footing as the qubit r, we will apply standard methods totrace out the bosonic degrees of freedom without anyapproximation on the qubit-impurity coupling.

ARTICLE IN PRESS

0.5

1

1.5

2

0 6 8 10T/Δ r

0.98

1

1.02

1.04

Ω/Δ

r~

42

Ω/Δ

r~

E. Paladino et al. / Physica E 40 (2007) 198–205200

3. Impurity dynamics and correlation time

In this section we will identify the characteristic timescale of the dynamics of the equilibrium fluctuations of theSpin-Boson impurity described by Eq. (6). The uncoupled(i.e. for v ¼ 0) impurity dynamics strictly depends on thedamping strength K and on the temperature [17]. Weconsider the small damping K51 regime where series ofcrossovers from under-damped to over-damped oscilla-tions and to relaxation dynamics with increasing tempera-ture are observed. This analysis is also relevant tounderstand the effect of ensemble of impurities when awide distribution of parameters has to be taken intoaccount as in Ref. [10]. In fact at any fixed temperatureeach impurity may display a specific dynamical behavior.As a consequence, various sets of impurities may affect thequbit dynamics in qualitatively different ways.

Our goal is to establish to which extent the qubit mayprobe the variety of regimes of its environment dynamics.The important scale allowing the identification of Gaus-sian/non-Gaussian dynamical regimes is extracted from theequilibrium auto-correlation function of the observable tz

which directly couples to the qubit. We remark that thethermal initial state of the Spin-Boson system, which isimplied by the equilibrium correlation function, mayoriginate peculiar time-dependencies. Qualitatively differ-ent behaviors may in fact be displayed by the non-

equilibrium correlation function, which is evaluated for afactorized initial state of the Spin-Boson system [17,19].Evaluation of equilibrium correlation functions for theSpin-Boson model is a non-trivial task. However it hasbeen shown [17] that in the unbiased case e ¼ 0, and forsmall damping K51, the tz auto-correlation function doesnot depend on the initial correlated or factorized state.Here we focus on this case and recall the characteristics ofStðoÞ more relevant for our analysis, the interested readermay find details of the derivation in Refs. [17,19].

The crossover from under-damped to over-dampedbehavior with increasing temperature takes place at

T�ðKÞ �Dr

pKkB, (9)

where Dr ¼ DðD=ocÞK=ð1�KÞ is the renormalized tunneling

amplitude in the Spin-Boson model.The explicit form for StðoÞ depends on temperature.1

For ToT�ðKÞ it reads

StðoÞ ¼gþ ð ~O� oÞ tanf

ðo� ~OÞ2 þ g2þ

gþ ð ~Oþ oÞ tanf

ðoþ ~OÞ2 þ g2, (10)

1The above analytic forms have been derived in Ref. [17] within the so

called non-interacting-blip-approximation (NIBA) which is exact in the

considered regime (K51 and � ¼ 0).

where tanf ¼ g= ~O. The effective frequency ~O and therelaxation rate are

~OðTÞDr

¼ 1þ K Rec iDr

2pkBT

� �� ln

Dr

2pkBT

� �� cð1Þ

� �,

(11)

gðTÞ ¼SðDrÞ

4, (12)

for temperatures kBToDr and

~OðTÞDT

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

T

T�

� �2�2Ks

, (13)

gðTÞ ¼ pKkBT ¼G2

(14)

for larger temperatures DrpkBTpkBT�ðKÞ.Here 2G is the white noise level SðoÞ � 2G for

frequencies o52p=b in (4). Oscillators with frequencies2p=bpo5oc renormalize the tunneling amplitude to

DT ¼ Drð2pkBT=DrÞK . (15)

In Fig. 1 we show the overall behavior of ~OðTÞ. At verysmall temperatures kBT5Dr (see inset in Fig. 1) ~OðTÞincreases / T2. The two forms (11) and (13) smoothlymatch on each other at kBT ¼ Dr. A maximum is reachedat T ¼ K1=2ð1�KÞT�4Dr=kB, above this temperature ~OðTÞdecreases monotonously and approaches zero at T�.Coherent oscillations are dephased on a scale 1=gðTÞ whichdecreases monotonously starting from 1=gð0Þ ¼ 2=ðpKDrÞ.The resulting quality factor of the damped oscillationsQðTÞ ¼ ~OðTÞ=gðTÞ is shown in Fig. 2.For temperatures higher than T� the dynamics is

incoherent and StðoÞ has a different form

StðoÞ ¼ 2g2

g2 � g1

g1o2 þ g21

þ 2g1

g1 � g2

g2o2 þ g22

, (16)

0 10 20 30 40

T/Δ r

0

Fig. 1. ~OðTÞ of Eqs. (11) and (13) as a function of temperature. Inset: non-

monotonous behavior for T5T� ¼ 31Dr. Parameters are K ¼ 0:01,oc=Dr ¼ 31, and we fixed kB ¼ 1.

ARTICLE IN PRESS

0 10 20 30 40

T/Δ r

0

10

20

30

40

50

60

70

Ω/γ

~

Fig. 2. Quality factor of the damped oscillations QðTÞ ¼ ~OðTÞ=gðTÞ fromEqs. (11)–(14). Parameters are fixed as in Fig. 1.

0 20 40 600

1

2

3

4

T/Δr

γ/Δ r

Fig. 3. Dephasing rate gðTÞ from Eqs. (12) and (14) (full line), for

temperatures ToT�ðKÞ and relaxation rates g1=2ðTÞ from Eq. (17) for

TXT�ðKÞ. The rate g1 (dotted) increases with temperature, g2 (dashed)

shows the Kondo behavior. The value of T�ðKÞ=Dr � 31:8 is indicated by

the dashed gray line. Parameters are fixed as in Fig. 1.

-2 -1 0 1 2ω/Δr

0

5

10

Δ rSτ(

ω)

-2 -1 0 1 2

ω/Δr

0

1

2

3

4

5

6

Δ rSτ(

ω)

Fig. 4. Equilibrium correlation function StðoÞ for increasing values of the

temperature. Top: from under-damped to over-damped regime and

T=Dr ¼ 3 (light gray), T=Dr ¼ 15 (black). Dashed lines indicate the width

1=tc at half height. Bottom: from over-damped to incoherent regime

T=Dr ¼ 20 (light gray), T=Dr ¼ 30 (dashed), T=Dr ¼ 40 (black). Para-

meters are fixed as in Fig. 1.

E. Paladino et al. / Physica E 40 (2007) 198–205 201

g1=2 ¼G2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG2

� �2

� D2T

s. (17)

Note that for TbT�ðKÞ one of the two rates increases withtemperature, g1 ! G, whereas the other shows the char-acteristic Kondo behavior, decreasing with temperature

g2!D2

T

G¼

Dr

K

Dr

2pkBT

� �1�2K

. (18)

These features are illustrated in Fig. 3.The typical scale of the equilibrium fluctuations of the

Spin-Boson environment described by StðoÞ defines thecorrelation time tc. Usually in the literature one is facedwith environment models characterized by dynamic fluc-tuations which tend rapidly to zero with time. In thesecases tc represents the order of magnitude of the width of

the environment fluctuations [20]. In the present casehowever looking at the Spin-Boson impurity as anenvironment characterized by StðoÞ the identification oftc is not immediate due to the different forms of theequilibrium fluctuations given by Eqs. (10) and (16). Forhigh temperatures StðoÞ is approximately a single Lor-entzian centered at o ¼ 0 and width g2, leading to theidentification tc � 1=g2ðTÞ, see Fig. 4. In the opposite limitof very low temperatures StðoÞ has a double peak structurerepresenting a bath responsible for oscillating fluctuationsvery weakly damped. In this unusual environment regime,the typical scale of the impurity fluctuations is representedby 1= ~OðTÞ which plays the role of tc. For intermediatetemperatures in general two almost superimposed Lor-entzians contribute to StðoÞ and tc may be approximatelyidentified from the condition Stð1=tcÞ ¼ Smax

t =2. Theresulting tcðTÞ, illustrated in Fig. 5, interpolates betweenthe asymptotic behaviors at low and high temperatures.The slight reduction of tcðTÞ at intermediate temperaturesis a consequence of the crossover from under-damped toincoherent dynamics, as shown in Fig. 4.

ARTICLE IN PRESS

0 10 20 30 40

T/Δr

0.5

1

1.5

2

2.5

τ cΔr

Fig. 5. Correlation time tcðTÞ: for sufficiently small temperatures

tc � 1=Dr. For TbT�ðKÞ the asymptotic behavior � 1=g2ðTÞ is indicated(dashed). The interpolating form for intermediate temperatures has been

obtained from the condition Stð1=tcÞ ¼ Smaxt =2. Parameters are fixed as in

Fig.1.

2In the Spin-Boson literature the approximation of the kernels to order

D2 is referred to as non-interacting-blip-approximation (NIBA). Its

validity regimes include the considered Markovian case K51 and

kBTX

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2 þ D2

r

q[17].

E. Paladino et al. / Physica E 40 (2007) 198–205202

Once the impurity correlation time is identified, thecondition vtcðTÞ ¼ 1 separates the weak-coupling regimeof the qubit dynamics, from the strong coupling regimewhere non-Gaussian behavior shows up. In the first case,when vtcðTÞ51, the standard master equation predictsexponential decay with the Golden Rule rate 1=T2 given inEq. (8). From the above analysis we expect the masterequation result to be valid in the following regimes: fortemperatures ToT� if v= ~OðTÞ � v=Dr51, for largertemperatures T4T� if v=g2ðTÞ � vG=D2

T51. This lastcondition can be cast in the following form:

v

Dr

5T�

2T

� �1�2K

. (19)

Therefore, for small values of v=Dr a crossover from weakto strong coupling is expected with increasing temperature.For the ensuing discussion here we report the expectedvalue of the pure dephasing rate 1=T�2 ¼ ðEC=EÞ2v2Stð0Þ=2with Eqs. (10) and (16):

1

T�2¼

EC

E

� �24v2gðTÞ

gðTÞ2 þ ~OðTÞ2; ToT�, ð20Þ

1

T�2¼

EC

E

� �2v

DT

� �2

G; T4T�. ð21Þ

The two forms match each other at T�, and it is easy toshow that Eq. (21) approximates 1=T�2 also for T5T�.

4. Qubit dynamics: path-integral approach

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 via apath-integral approach which includes as a special case the

regime of Gaussian behavior of the impurity dynamics. Wefind exact expressions which we discuss for finite tempera-

tures, specifically for kBTX

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2

r þ v2q

.

We focus on the so called pure-dephasing regime,EJ ¼ 0, which represents the point of maximum noisesensitivity of the qubit. Thus the more interesting in theperspective of using the qubit as a ‘‘noise’’ analyzer. In thepure dephasing regime the charge on the qubit island is aconstant of motion since ½H;sz� ¼ 0, dephasing beingdescribed by the decay of hsx=yi or equivalently of thecoherences hs�i. A simple analysis shows that thecoherences are related to correlation functions involvingthe Spin-Boson variables, specifically we found [3,22]

hs�ðtÞihs�ð0Þi

¼ eiECt TrSBfe�iHSB�trtð0Þ � wbe

iHSBþtg

� eiECtC�þðtÞ, ð22Þ

where we have chosen a factorized initial density matrix forthe qubit-impurity, rð0Þ ¼ rsð0Þ � rtð0Þ, with the impurityt initialized in the mixed state rtð0Þ ¼

12I þ 1

2dpð0Þtz and the

bosonic bath in its thermal equilibrium state wb. The twoconditional impurity Hamiltonians HSB� depend on thequbit state and read HSB� ¼HSB � v=2tz.In Ref. [22] it has been shown that the Laplace transform

of the correlator C�þðtÞ reads

bC�þðlÞ ¼ 1

DðlÞ½lþ K1ðlÞ � ivdpð0Þ�, ð23Þ

DðlÞ ¼ l2 þ v2 þ lK1ðlÞ þ ivK2ðlÞ. ð24Þ

An exact formal series expression in D for the kernelsK1ðlÞ, K2ðlÞ has been derived in Ref. [22]. In theMarkovian regime for the harmonic bath, i.e. for K51

and temperaturesffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2

r þ v2q

pkBT5oc, all contributions

to K1ðlÞ and K2ðlÞ of order higher than D2 cancel outexactly. The lowest order contributions2 do not depend onthe coupling v and coincide with the kernels entering thedynamics of s in the uncoupled case (v ¼ 0) [17] which read

K1ðlÞ ¼ D2T

lþ G

�2 þ ðlþ GÞ2, ð25Þ

K2ðlÞ ¼ �pKD2T

�

�2 þ ðlþ GÞ2. ð26Þ

Inserting Eqs. (25), (26) in (23) and (24) bC�þðlÞ is readilyfound as

bC�þðlÞ ¼ ½ðlþ GÞ2 þ �2�½l� ivdpð0Þ� þ D2T ðlþ GÞ

DðlÞ,

DðlÞ ¼ ðl2 þ v2Þ½ðlþ GÞ2 þ �2� þ D2Tlðlþ GÞ � ipKv�D2

T .

ð27Þ

ARTICLE IN PRESS

0 4 8 12–2.0

–1.5

–1.0

–0.5

0.0

T/Δr

T/Δr

Re

[λ i

/Δr]

0 4 8 12

–1.0

0.0

1.0

Im [

λ i/Δ

r]

Fig. 6. Top: real parts of the exact solutions li of the pole equation

DðlÞ ¼ 0 as a function of temperature for kBTX

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2 þ D2

r

q� 0:87Dr.

Bottom: corresponding imaginary parts. Two li are complex conjugate

below T� � 6:1Dr and above Tþ � 7:2Dr. All li are real at intermediate

temperatures T�oToTþ. The dominant pole is real (black dashed) until

Tþ where the character of the dominant solution changes. For T4Tþ the

dominant poles are complex conjugate. Parameters are v=Dr ¼ 0:25,oc=Dr ¼ 30 and K ¼ 0:05.

E. Paladino et al. / Physica E 40 (2007) 198–205 203

The scales entering the time evolution of C�þðtÞ are foundfrom the solution of the pole equation DðlÞ ¼ 0, whichhave been reported in Ref. [15]. Here we specify to theunbiased case � ¼ 0, the goal being to elucidate thecorrespondence with the expected Gaussian/non-Gaussiandynamical regimes as deduced from the equilibriumcorrelation function StðoÞ discussed in Section 3.

4.1. Pure dephasing due to a unbiased impurity

When � ¼ 0 the pole condition DðlÞ ¼ 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 l0 ¼ �L0 2 Re andl1=2 ¼ �L� idE, where dE is either real or purelyimaginary. The expression of C�þðtÞ in terms of the li isobtained by inverting the Laplace transform (27) and reads

C�þðtÞ ¼ Ae�L0t þ ð1� AÞ cosðdEtÞe�Lt

þ B sinðdEtÞe�Lt, ð28Þ

A ¼�2L0Lþ D2

T � i2dpð0ÞvL

ðL� L0Þ2þ dE2

, ð29Þ

B ¼½Lð1� AÞ þ L0A� idpð0Þv�

dE. ð30Þ

It is possible to show that the character of the rootsdepends on v=DT and on the temperature. In particular forvoDT=2

ffiffiffi2p

we can identify two temperatures

kBT� �1

pK�

v

2

� �2þ

5D2

8þ

D4

32v2

�

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD8 � 24v2D6 þ 192v4D4 � 512v6D2

p32v2

#1=2ð31Þ

such that for ToT� and T4Tþ one solution is real andtwo are complex conjugate, whereas for intermediatetemperatures the three solutions are real. For v4DT=2

ffiffiffi2p

there is always one real and two complex conjugatesolutions. Analytic expressions for the roots are quitecumbersome, approximate forms have been reported inRef. [22]. Here we discuss the physically relevant regimeswhere crossover are expected in the behavior of C�þðtÞ. Tothis end we focus on the dominant li, i.e. the smallest inabsolute value. The analysis is conveniently performeddistinguishing regimes where v=DTo1 or v=DT41.

Case v=DT51: In this regime the characteristics of thedominant pole change qualitatively with increasing tem-

peratures (with lower bound kBT4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2 þ D2

r

q). For tem-

peratures low enough to fulfill the condition

kBT

DT

5Dr

4pKv�

kBTþ

Dr

(32)

the dominant scale is real and reads L0 � ðv=DT Þ2G. In

Eq. (30) A � 1� ðG=2DT Þ2 and B � �idpð0Þv=DT therefore

Cþ�ðtÞ � exp½�L0t�. (33)

With increasing temperature above Tþ the dominant scalesare complex conjugate, l1=2 ¼ �L� idE � �D2

T=2G� iv.Fig. 6 illustrates the crossover between the two regimes forv=Dr ¼ 0:25. Since A5v=G and B � �idpð0Þ, we get

Cþ�ðtÞ � cosðvtÞe�Lt � idpð0Þ sinðvtÞe�Lt. (34)

As expected for small enough temperature the Gaussianapproximation for the structured bath applies and a singlescale dominates the qubit dynamics. It is easily seen thatthe condition (32) corresponds to Eq. (19) which wasderived from the weak coupling criterion vtc51. MoreoverL0 ¼ 1=T�2 ¼ v2Stð0Þ=2 as given in Eq. (21). For highertemperature non-Gaussian effects show up. The qualitativechange is characterized by damped oscillations at fre-quency � v in Cþ�ðtÞ as shown in Fig. 7, and beatings atfrequencies EC � v in the coherences of Eq. (22). Note thatthe coupling strength v only enters the induced frequency

ARTICLE IN PRESS

0 20 40 60

t Δr

0

0.4

0.8

C- +(t

)

-0.5 0 0.5

ω/Δr

0

10

20

30

ΔrC

- +

(ω)

Fig. 7. Top: C�þðtÞ for v=Dr ¼ 0:25 and kBT=Dr ¼ 4 (black), kBT=Dr ¼

15 (gray). Bottom: corresponding Fourier transform. Here K ¼ 0:05,oc=Dr ¼ 30, dpð0Þ ¼ 0.

0 4 8 12–2.0

–1.5

–1.0

–0.5

0.0

T/Δr

0 4 8 12

T/Δr

Re [

λ i/Δ

r]

–1

0

1

Im [

λ i/Δ

r]

Fig. 8. Top: real parts of the exact solutions li of the pole equation

DðlÞ ¼ 0 as a function of temperature for kBTX

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2 þ D2

r

q� 1:2Dr.

Bottom: corresponding imaginary parts. Two li are complex conjugate

and one is real for any temperature. The dominant poles are always

complex conjugate. In this regime no crossing takes place among the

Re½li � and the dominant root is non-monotonic with T. Parameters are

v=Dr ¼ 0:8, K ¼ 0:05, oc=Dr ¼ 30.

E. Paladino et al. / Physica E 40 (2007) 198–205204

shift, whereas the decay rate shows the KondobehaviorL / T2K�1, cf. Eq. (18).

Case v=DTb1: For larger values of v=DT the systemstays in the regime where the dominant scales are complexconjugate and show a non-trivial temperature dependence.

At temperatures TX

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2 þ D2

r

q, the two poles read

l1=2 � �ðDT=vÞ2G=2� iv, with increasing T instead

l1=2 � �D2T=ð2GÞ � iv. The qubit dynamics follows

Eq. (34). The dominant rate L is non-monotonous firstincreasing with temperature and then decreases / T2K�1.For large enough temperatures the decay of the coherencesdoes not depend on v, as expected.

This qualitative behavior is already present for inter-mediate values of v=D, as shown in Fig. 8. The poles li

never cross, therefore there is no change in the character ofthe qubit dynamics. In the specific case considered at small/intermediate temperatures the real parts of the three polesare of the same order. This is reflected in the Fouriertransform of Cþ�ðtÞ where three Lorentzians can beidentified, one centered at o ¼ 0, the others at � �v, seeFig. 9.

5. Discussion

The presented analysis has shown that a dampedimpurity may behave as an effective short-time correlatedor as a non-Gaussian environment depending both on itscoupling with the qubit and temperature. The relevantscale separating the two regimes is given by tc. In Section 3the correlation time of the unbiased Spin-Boson model hasbeen found for small damping K51 at any temperature.We have shown that the qubit may act as a detector of non-Gaussian dynamical behavior, the most evident effect beingthe occurrence of beatings which are expected withincreasing temperature. We remark that the results wehave illustrated have been derived within the NIBA which

limits temperatures to values larger thanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2 þ D2

r

q. A

interesting issue is to analyse the small temperature regimewhere we expect that crossover effects may take place alsofor large values of v=DT . A detailed analysis of the low

ARTICLE IN PRESS

0 10 20 30

t Δr

-0.5

0

0.5

1

C- +

(t)

-2 -1 0 1 2

ω/Δr

0

4

8

ΔrC

- +(ω

)

Fig. 9. Top: C�þðtÞ for v=Dr ¼ 0:8 and kBT=Dr ¼ 4 (black), kBT=Dr ¼ 15

(gray). Bottom: corresponding Fourier transform. Here K ¼ 0:05,oc=Dr ¼ 30, dpð0Þ ¼ 0.

E. Paladino et al. / Physica E 40 (2007) 198–205 205

temperature regime can be performed within the systematicweak damping approximation [17] and will be reportedelsewhere.

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