Quantum decoherence of a charge qubit in a spin-fermion model
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Quantum decoherence of a charge qubit in a spin-fermion model
Roman M. Lutchyn,
1, 2
Lukasz Cywi´nski, Cody P. Nave, and S. Das Sarma
1, 2 Condensed Matter Theory Center, Department of Physics,University of Maryland, College Park, MD 20742-4111, USA Joint Quantum Institute, Department of Physics,University of Maryland, College Park, MD 20742-4111, USA (Dated: October 27, 2018)We consider quantum decoherence in solid-state systems by studying the transverse dynamics ofa single qubit interacting with a fermionic bath and driven by external pulses. Our interest is ininvestigating the extent to which the lost coherence can be restored by the application of externalpulses to the qubit. We show that the qubit evolution under various pulse sequences can be mappedonto Keldysh path integrals. This approach allows a simple diagrammatic treatment of differentbath excitation processes contributing to qubit decoherence. We apply this theory to the evolutionof the qubit coupled to the Andreev fluctuator bath in the context of widely studied superconductingqubits. We show that charge fluctuations within the Andreev-fluctuator model lead to a 1 /f noisespectrum with a characteristic temperature depedence. We discuss the strategy for suppression ofdecoherence by the application of higher-order (beyond spin echo) pulse sequences. I. INTRODUCTION
The loss of coherence of a quantum two-level system(quantum bit) is caused by its unavoidable coupling tothe surrounding environment. For solid-state qubits, thedecoherence process can be quite fast due to couplingto a large number of internal degrees of freedom. Ourunderstanding of quantum decoherence and methods forits suppression in a realistic solid-state environment ismainly confined to the cases of a qubit interacting withbosonic and nuclear spin baths, the so-called (andextensively studied) spin-boson and spin-bath models, re-spectively. A less well understood, but very relevant casefor solid-state quantum architectures is that of a qubitcoupled to a fermionic bath, which dramat-ically differs from the previous examples. In this paperwe study quantum decoherence in the context of a su-perconducting charge qubit interactingwith the non-trivial bath of Andreev fluctuators.
This problem is a paradigmatic spin-fermion decoher-ence problem and applies to many situations involvingthe quantum coupling of a qubit (“spin”) to a generalfermionic environment. Using a many-body Keldyshpath integral approach, we obtain a quantum-mechanical description of the qubit evolution under pulsesequences aimed at prolonging the coherence of the sys-tem. The simplest case of decoherence under pulsesis the spin echo dephasing experiment, which has beenshown to extend the coherence time of solid state (su-perconducting) qubits, by essentially eliminatingthe quasi-static shifts of qubit energy splitting (inho-mogeneous broadening) due to the slow environmentalfluctuations. However, sequences involving more pulses,for example, CPMG and Uhrig’s sequences, are ex-pected to lead to a further increase of the coherencetime. In this paper, we consider an experimentally relevantexample - a superconducting qubit coupled to fluctuat-ing background charges, e.g. electrons residing on Anderson-impurity sites. Due to a large on-site Coulombrepulsion forbidding double occupancy, this example rep-resents a non-trivial interacting bath. The dynamics ofthe charge fluctuations on the impurity sites is deter-mined by the hybridization of impurity levels with thequasiparticle band of the superconductor. To the low-est order in tunneling at the superconductor/insulatorinterface, the hybridization of the impurity levels canbe described by a correlated tunneling events of twoelectrons with opposite spin to/from the superconduc-tor. We show that in the small background-charge den-sity limit, these fluctuations lead to a 1 /f spectral den-sity of noise. Using these results, we finally obtain thequantum-mechanical description of the qubit evolutiondriven by external pulses, and discuss optimal strategyfor the suppression of the decoherence with designed com-posite pulse sequences.The paper is organized as follows: In Sec. II we providea general derivation of the qubit evolution with pulsesand map the calculation of decoherence function ontoKeldysh path integral formalism. In Secs. III and IVwe introduce Andreev fluctuator bath and derive spec-tral density of noise for this model. Finally, in Sec. Vwe discuss the influence of pulse sequences on the qubitdecoherence. II. GENERAL THEORY FOR QUBITEVOLUTION
The transverse dynamics of a qubit interacting with itsenvironment is determined by the following Hamiltonianˆ H = E σ z + ˆ σ z ˆ V + ˆ H B . (1)Here the environment is represented by a fermionic bathˆ H B , and the qubit is coupled to the environment throughthe density fluctuation operator:ˆ V = X lσ v l ( c † lσ c lσ − h n lσ i ) . (2)This model corresponds precisely to the coupling of asuperconducting charge qubit to the density fluctuationson the impurities in the substrate. Here c lσ and c † lσ arethe fermionic annihilation and creation operators at l -thsite with spin σ , and v l and h n lσ i are, respectively, thestrength of the coupling and average occupation of l -thimpurity, i.e. h n lσ i = h c † lσ c lσ i . Equations (1) and (2)define our spin-fermion model.We study the evolution of the qubit in contact witha fermionic bath assuming the qubit energy relaxationtime T to be much longer than the quantum dephasingtime T (thus only ˆ σ z coupling is present in the Hamil-tonian). Qubit decoherence under the influence of theenvironment is given by the off-diagonal matrix elementsof the qubit’s reduced density matrix, and for the freeevolution of the qubit we get ( ~ = 1) ρ + − ( t ) = h + | Tr B { ˆ ρ ( t ) }|−i = ρ + − (0) e − iEt W ( t ) . (3)In the above ˆ ρ ( t ) is the density matrix of the whole sys-tem (qubit+bath), which is assumed factorizable at t = 0,Tr B { ... } is the trace with respect to the bath degrees offreedom, and W ( t ) is the decoherence function definedas W ( t ) = D e i ( ˆ H B − ˆ V ) t e − i ( ˆ H B + ˆ V ) t E (4)with the brackets representing the thermal average withrespect to the bath Hamiltonian ˆ H B , i.e. h ... i =Tr B { ˆ ρ B ... } . The time t always refers here to the totaltime of the evolution.In addition to the free evolution of the qubit(free induction decay), one is often interested inthe dynamics of the system subject to external π -pulses which could, in principle,prolong or restore quantum coherence. The π -pulses con-sidered here correspond to rotations of the qubit’s Blochvector by angle π about, e.g., the ˆ x axis, and are shortenough for the bath dynamics during the pulse durationto be negligible. Then, the evolution operator for qubitand bath is given byˆ U ( n ) ( t ) = ( − i ) n e − i ˆ Hτ n +1 ˆ σ x e − i ˆ Hτ n ... ˆ σ x e − i ˆ Hτ (5)with n and τ i being the number of applied pulses andtime delays between the pulses,respectively, and the to-tal evolution time t = P n +1 i =1 τ i . One can see that thewell-known Hahn spin echo (SE) sequence, for example,corresponds to a single pulse with τ = τ = t/ |±i are the eigenstatesof the Hamiltonian (1), we can write the decoherencefunction under the influence of pulses as W n ( t ) = (cid:28)(cid:16) ˆ U ( n ) − ( t ) (cid:17) † ˆ U ( n )+ ( t ) (cid:29) (6) with the evolution operators ˆ U ( n ) ± ( t ) given byˆ U ( n )+ ( t )= e − i ( ˆ H B + ˆ V ) τ n +1 e − i ( ˆ H B − ˆ V ) τ n ...e − i ( ˆ H B + p ˆ V ) τ , (7)ˆ U ( n ) − ( t )= e − i ( ˆ H B − ˆ V ) τ n +1 e − i ( ˆ H B + ˆ V ) τ n ...e − i ( ˆ H B − p ˆ V ) τ , where p = ( − n is the parity of the sequence. Then,the off-diagonal elements of the qubit density matrix aregiven by ρ + − ( t ) = ρ p, − p (0) e − ipE ( τ − τ + ... + pτ n +1 ) W n ( t ) . (8)Here the phase factor is zero for all balanced sequences(for which the total times of evolution due to ˆ H + ˆ V andˆ H − ˆ V are the same in Eq. (7)). The evolution of thequbit under SE sequence, for example, acquires a simpleform ρ SE + − ( t ) = ρ − + (0) D e i ˆ H + t e i ˆ H − t e − i ˆ H + t e − i ˆ H − t E (9)with ˆ H ± = ˆ H B ± ˆ V .Decoherence under pulses has been analyzed withmethods specific to the spin-boson model and the spinbath model, or using operator algebra. The lat-ter approach, although very general, does not allow fortransparent understanding of physics of the bath. How-ever, the evaluation of W n ( t ) defined in Eq. (6) can bemapped onto the evolution on the Keldysh contour, putting the calculation of decoherence into the frame-work of many-body theory. Similar formalism has beenused to study full counting statistics of a general quan-tum mechanical variable and has proved to be quiteconvenient. The evolution operators ˆ U ( n ) ± can bewritten asˆ U ( n ) ± ( t ) = T exp (cid:20) − i Z t ( ˆ H B ± f n ( t ′ ) ˆ V ) dt ′ (cid:21) , (10)where T is the time ordering operator. The function f n ( t ′ ) encodes a particular sequence, and is defined as f n ( t ′ ) = p n X k =0 ( − k Θ( t k +1 − t ′ )Θ( t ′ − t k ) , (11)where Θ( t ′ ) is the Heaviside step function, t k with k =1 ..n are the times at which the pulses are applied, t = 0,and t n +1 = t . Thus, the product of operators inside theaverage in Eq. (6) corresponds to (reading from left toright) the time-ordered evolution from 0 to t (with + ˆ V coupling), followed by the time anti-ordered evolutionfrom t to 0 (with − ˆ V coupling). We can then introducethe Keldysh contour C (see Fig. 2a) together with thenotion of contour-ordering of operators. The qubit-bath coupling takes then two opposite signs on the up-per/lower branch of the contour: ˆ V C = ± ˆ V . While f n ( t ′ )is non-zero only for t ′ ∈ [0 , t ], we can extend the limitsof time integration on both branches to [ −∞ , ∞ ]. Theevolution from t ′ = −∞ allows one to include the adia-batically turned-on interactions in ˆ H B (see, for example,Ref. [26]), paving the way to the treatment of decoher- ence in an interacting fermionic bath. The final resultis most compactly written as a functional integral withthe Grassmann fields ¯ ψ l and ψ l defined on the Keldyshcontour: W n ( t )= (cid:28) T C exp (cid:18) − i Z C dt ′ h ˆ H B + ˆ V C f n ( t ′ ) i(cid:19)(cid:29) = 1 Z B Z D ¯ ψ l D ψ l exp iS B (cid:2) ¯ ψ, ψ (cid:3) − i Z C dt ′ X lσ v l ( t ′ ) f n ( t ′ ) (cid:2) ¯ ψ lσ ( t ′ ) ψ lσ ( t ′ ) −h n lσ i (cid:3) , (12) FIG. 1: (color online). Correlated tunneling of two electronswith opposite spins from the impurity sites in the insulatorinto the superconductor. An electron from the i -th impuritywith energy below the gap ∆ tunnels into superconductor,propagates over distances of the order of coherence length ξ and recombines with another electron with opposite spinfrom j -th site into a Cooper-pair. The amplitude for suchAndreev process decays exponentially with distance betweenthe impurity sites A lj ∝ exp( −| r l − r j | /πξ ), see Eq. (14). where the integration is performed on the contour C shown in Fig. 2a, v l ( t ′ ) = ± v l on the upper/lower branchof the contour, and the normalization constant is definedas the functional integral with ˆ V = 0. The bath action S B = S + S int , and the functional integration with non-interacting S corresponds to averaging over an equilib-rium noninteracting density matrix at t ′ = −∞ . Thisformulation of the decoherence problem enables one touse techniques and approximations developed in many-body theory. It also allows for a transparent treatment ofthe physics of the bath while simply encoding the drivingof the qubit in a single function of time f n ( t ′ ). III. ANDREEV FLUCTUATOR BATH
In order to evaluate the functional integral (12),one needs to specify the bath Hamiltonian. Here,as an example, we consider a non-trivial bath of An-dreev fluctuators, which describes the fluctua-tions of the occupation of impurities close to insula-tor/superconductor interface due to Andreev processes.This model takes into account coherent processes of cre-ation (destruction) of the Cooper pair in the supercon- ductor by correlated tunneling of two electrons from(to) different impurity sites in the insulator, see alsoFig. 1. In the limit when the superconducting gap en-ergy ∆ is the largest relevant energy scale in the prob-lem (
T, E, ε j , ≪ ∆), the effective Hamiltonian for theAndreev fluctuator bath, after integrating out supercon-ducting degrees of freedom, is given byˆ H B = X lσ ε l c † lσ c lσ + U X l ˆ n l ↑ ˆ n l ↓ + X l = j h A ∗ lj c † l ↑ c † j ↓ +H . c . i . (13)Here, ε l and U are the energy of a localized electronon l -th impurity (measured with respect to the Fermienergy ε F of the conduction electrons) and repulsive on-site interaction (assumed to be large enough to preventdouble occupation of the sites), respectively. The matrixelements A lj , in the limit of low transparency barrierbetween the insulator and superconductor, are given by A lj ≈ A sin( p F | r l − r j | ) p F | r l − r j | e −| r l − r j | /πξ . (14)Here p F is the Fermi momentum, ξ is the coherencelength in a clean superconductor. The amplitude A =2 π d aN (0) T is determined by the tunneling matrix el-ement between the insulator and superconductor T , thenormal density of states in the metal N (0) = mp F /π ,the localization length under the barrier d and the sizeof the impurity wavefunction a . Given the Hamiltonian (13), the action for the bath onthe Keldysh contour can be written as S B (cid:2) ¯ ψ, ψ (cid:3) = X lj Z C dt ′ X σ δ lj ¯ ψ lσ ( t ′ )( i∂ t ′ − ε l − U h n l, − σ i ) ψ lσ ( t ′ )+ A ∗ lj ¯ ψ l ↑ ( t ′ ) ¯ ψ j ↓ ( t ′ ) + A lj ψ j ↓ ( t ′ ) ψ l ↑ ( t ′ ) . (15)Here we used the mean-field approximation for the An-derson impurity model assuming that the Kondo tem-perature T K is smaller than the superconducting gap ∆,which is reasonable in the situation at hand when theimpurities are located in the substrate and the tunnel-ing matrix element T coupling them to the states in thesuperconductor is small. The occupation probabilities h n lσ i are obtained self-consistently using h n lσ i = Z dω π n F ( ω )[ G Allσ ( ω ) − G Rllσ ( ω )] , (16)see Ref. [40] for more details. Performing a Keldyshrotation, one can calculate the full Green’s function ˆG llσ ( t, t ′ ) for the bath (see Fig. 3) ˆG − llσ ( t, t ′ ) = ˆ G − llσ ( t, t ′ ) − ˆΣ llσ ( t, t ′ ) . (17)Here ˆ G − llσ ( t, t ′ ) is the bare Green’s function, see Eq. (15),and the self energy ˆΣ llσ ( t, t ′ ) is calculated to second orderin A ij giving the components of the self-energy matrixΣ A/Rllσ ( t, t ′ ) = X j = l | A lj | G R/Ajj, − σ ( t ′ , t ) , (18)Σ Kllσ ( t, t ′ ) = X j = l | A lj | G Kjj, − σ ( t ′ , t ) . (19)In Eqs. (18) and (19) we have neglected the off-diagonalterms in the impurity indices, i.e. Σ ljσ ≈ δ lj Σ llσ . Since the amplitude A lj oscillates on the length scale of p − F ,the contribution of these off-diagonal terms to the self-energy is small.Using the above results, the action for the bath can bewritten in terms of the full Green’s function ˆG llσ ( t, t ′ ).Then, the decoherence function becomes FIG. 2: a) Dependence of ˆ V C ( t ) on time along the Keldyshcontour. b) The plot of the function f n ( t ′ ) for the Spin Echosequence ( n = 1). W n ( t ) ≡ exp [ − χ n ( t )] = 1 Z B Z D ¯ ψ D ψ (20) × exp i X lσ ∞ Z −∞ dt ∞ Z −∞ dt X a,b =1 ¯ ψ ( a ) lσ ( t ) h ˆG − llσ ( t , t ) i ab ψ ( b ) lσ ( t ) − δ ( t − t ) v l f ( n ) ( t ) [ ρ lσ ( t ) −h n lσ i ] . Here, ρ lσ ( t ) corresponds to the fermion density opera-tor ρ lσ ( t ) = h ¯ ψ (1) lσ ( t ) ψ (2) lσ ( t )+ ¯ ψ (2) lσ ( t ) ψ (1) lσ ( t ) i ; the fields ψ (1) ( t ) and ψ (2) ( t ) are given by the appropriate super-position of the fermionic fields on the upper and lowerparts of the Keldysh contour, see Ref. [26]. After per-forming the functional integral over the fermionic fieldsand expanding to second order in v l , one finds χ n ( t ) = X lσ v l Z t Z t dt dt f n ( t ) f n ( t ) (cid:2) G Allσ ( t ,t ) G Rllσ ( t ,t )+ G Rllσ ( t , t ) G Allσ ( t , t )+ G Kllσ ( t , t ) G Kllσ ( t , t ) (cid:3) . (21)Equation (21) holds whenever the short-time expansionis valid. The long-time asymptote can be obtained byresumming the whole series. By introducing the Fourier transform of the Green’sfunctions, Eq. (21) can be formally recast as χ n ( t ) = Z ∞−∞ dω π F n ( ωt ) ω S Q ( ω ) . (22) Here F n ( ωt ) = ω | f n ( ω ) | / FIG. 3: (color online). Dyson’s equation for the retardedGreen’s function G R in the Born approximation. Here wehave adopted the convention of Ref. [38]. The advanced andKeldysh Green’s functions are obtained analogously resultingin Eqs. (18)-(19). second order in v i we obtain χ n ( t ) having the same struc-ture as in the case of a qubit coupled to the spin-bosonbath or classical noise, i.e. χ n ( t ) is the integralof the product of the environment-specific spectral den-sity of noise S Q ( ω ) and sequence-specific filter function F n ( ωt ). The spectral density of quantum noise S Q ( ω ) inthe spin-fermion problem is given by S Q ( ω )= X lσ v l Z ∞−∞ d Ω2 π h G Allσ (cid:16) Ω+ ω (cid:17) G Rllσ (cid:16) Ω − ω (cid:17) + G Rllσ (cid:16) Ω+ ω (cid:17) G Allσ (cid:16) Ω − ω (cid:17) + G Kllσ (cid:16) Ω+ ω (cid:17) G Kllσ (cid:16) Ω − ω (cid:17)i . (23)In the frequency domain, the full Green’s functions are G A/Rllσ ( ω ) = 1 ω − ε l − U h n l, − σ i − Σ A/Rllσ ( ω ) , G Kllσ ( ω ) = tanh (cid:16) ω T (cid:17) (cid:2) G Rllσ ( ω ) − G Allσ ( ω ) (cid:3) , (24)where the self energy Σ A/Rllσ ( ω ) is defined asΣ A/Rllσ ( ω ) = X j = l | A lj | ω + ε j + U h n jσ i ∓ iδ . (25)Equation (23), defining the noise spectral density in thequantum-mechanical many-body language enables a di-rect calculation of decoherence in various situations, aswe consider next. IV. SPECTRAL DENSITY OF NOISE DUE TOANDREEV FLUCTUATORS
In general, the solution for S Q ( ω ) with many Andreevfluctuators, can be obtained numerically by randomlygenerating the energies ε l , and positions r l of the im-purities at the insulator/superconductor interface. Thenumerically obtained spectral density of noise S Q ( ω ) for50 fluctuators is shown in Fig. 4. At low frequencies thenoise power spectrum has 1 /f dependence.For ω and A much smaller than the typical impuritylevel spacing δε and temperature T , the analytical solu-tion for the spectral density of noise (23) is given by S Q ( ω ) ≈ X lσ v l (cid:20) − tanh (cid:18) ˜ ε lσ T (cid:19)(cid:21) γ lσ (˜ ε lσ ) ω + 4 γ lσ (˜ ε lσ ) , (26)where ˜ ε lσ = ε l + U h n l, − σ i , and γ lσ (˜ ε lσ ) = Im Σ Allσ (˜ ε lσ ) isthe broadening of the impurity energy levels due to An-dreev processes. This broadening corresponds to the fluc-tuations of the impurity occupations changing the elec-trostatic environment of the qubit, and thus causing de-phasing. From Eq. (26), one can see that S Q ( ω ) is givenby a sum of Lorentzians with different widths, which un-der proper distribution of γ l gives rise to a 1 /f noise spec-trum (see below). Given that the charge density fluctua-tions via Andreev processes involve two impurities withenergies of the order of T , the probability to find twosuch impurities is proportional to ( T /D ) with D beingthe impurity energy bandwidth, and thus, S Q ( ω ) ∝ T at low frequencies as seen experimentally. For 1 /f spectrum to arise from Eq. (26), the distribu-tion of γ l has to be log-normal. In order to have such dis-tribution, the density of the charge traps has to be small, FIG. 4: (color online). Log-log plot of the noise spectral den-sity S Q ( ω ) for Andreev fluctuator model. The plot is obtainedby randomly generating the positions r l ∈ [ − , ξ and ener-gies ε l ∈ [ − , v i = v , on-site re-pulsion U → ∞ , and the sites are occupied with equal numberof electrons with spins up and down. We used p − F = 10 − ξ , A = 0 .
1K and T = 0 . /f and1 /f . noise spectra. so that the dominant contribution to the self energy inEq. (25) comes from few pairs of impurity sites, whichare selected from the sum because of the energy conser-vation and distance constraint. Then, the switching rate γ l ∝ exp( − | r l − r j | /πξ ) for a certain j (see Eq. (14)).Since the distances between the charge traps are uni-formly distributed, the probability of finding a switchingrate γ is P ( γ ) ∝ /γ , leading to 1 /f noise. In the op-posite limit of large density of charge traps, many sites j contribute to the sum in Eq. (25), and the switchingrates γ l self average and become approximately the samefor all sites. Note that unlike in the theory of 1 /f chargenoise produced by fluctuating two level systems (TLS) inthe substrate with log-uniform distribution in the tunnelsplitting, the emergence of the 1 /f noise within An-dreev fluctuator model has a qualitatively different geo-metrical origin due to the exponential dependence of therate γ l on the distance between different impurity sites.This finding of the geometric origin of the 1 /f noise inthe Andreev fluctuator model is an important result ofour work.We note that the model of charge traps with no on-site repulsion U = 0 [21] does not lead to 1 /f noise be-cause in this case the self-energy is dominated by the two-electron tunneling from the same site. The contributionsto the self energy from Andreev processes involving othersites are exponentially smaller than the dominant term,and the distribution of the rates in Eq. (26) is not log-normal. Therefore, we emphasize that the realistic modelfor 1 /f noise due to Andreev processes should includeboth spinful fermions (to correctly describe the dynam-ics of charge fluctuations), and large on-site repulsion (toprevent double-electron occupation).At high frequencies ω ≫ δε, T , the spectral density S Q ( ω ) has resonances corresponding to the virtual pro-cesses of correlated two-electron tunneling from (to) theimpurity sites in the insulator. These resonances, de-scribing manifestly quantum-mechanical processes, canbe seen in Fig. 4 at high frequencies. Their contribu-tion to the decoherence of the qubit is suppressed by afactor F n ( ωt ) /ω , see Eq. (22). However, going beyondthe pure dephasing model, T ≫ T , considered here, onecan show that correlated two-electron tunneling processescontribute to the energy relaxation of the qubit. V. THE INFLUENCE OF PULSES ONDECOHERENCE
The time dependence of the decoherence function W n ( t ) under a pulse sequence is given by Eqs. (20)-(22), showing that the noise contribution is modulatedby a filter function F n ( ωt ). For the free induction de-cay F ( ωt ) = 2 sin [ ωt/ F ( ωt ) = 8 sin [ ωt/
4] suppressing the low-frequency( ω ≪ /t ) part of S Q ( ω ). In general, higher-order pulsesequences act as more efficient high-pass filters of envi-ronmental noise, i.e. for n pulses applied in time t onlyfrequencies ω > n/t contribute to χ n ( t ). Due to the for-mal analogy between Eq. (22) and the solution for thedecoherence under classical Gaussian noise, the analysisgiven for the latter case in Ref. [37] also applies here aslong as the time expansion is valid. The results rele-vant for the noise spectral density derived here can besummarized as follows. For S Q ( ω ) ∝ /ω α , we obtain χ n ( t ) ∝ t α /n α for all sequences applicable to the puredephasing case, i.e. the CPMG sequence, periodic dy-namical decoupling, concatenations of spin echo, and Uhrig’s sequence. Thus, sequences beyond spin echoshould lead to a further increase in coherence time for 1 /f spectral density of noise. For noise spectrum with- out sharp ultra-violet cutoff, which is the case consideredhere, the CPMG sequence marginally outperforms othersequences.
Furthermore, taking into account the sim-plicity of CPMG sequence (defined by τ = τ n +1 = t/ n and all the other τ i = t/n ), we believe that it is a pre-ferred approach of noise suppression for the problem athand. We therefore propose that detailed experimentalinvestigation of superconducting charge qubit dephasingbehavior be carried out in order to test our specific pre-dictions.
VI. CONCLUSION
We consider the spin-fermion model for quantum de-coherence in solid-state qubits in the pure dephasing(i.e. T ≫ T ) situation. We map the evolution ofthe qubit interacting with the fermionic environment,possibly subject to various π -pulse sequences, onto theKeldysh path integral. This approach is very general andallows one to apply well-developed many-body techniquesto the problem of the evolution of the qubit coupled tothe environment and driven by pulses. In the short-timelimit, we derive the expression for the qubit decoherencewhich involve the product of the noise spectral densitydue to quantum fluctuations of the bath and the filterfunction representing a particular pulse sequence. For anon-trivial interacting model of the bath, the Andreevfluctuator model, we show that the spectral density has1 /f dependence at low frequencies. Finally, we discussthe optimal strategy for the suppression of 1 /f chargenoise by the application of higher-order (beyond spinecho) pulse sequences for the problem at hand. One ofour concrete conclusions of experimental significance isthat the well-established CPMG pulse sequence shouldbe an optimal method for fighting T dephasing whenthe noise spectrum has no sharp ultra-violet cutoff. Acknowledgments
We thank A. Kamenev, J. Koch, Y. Nakamura,E. Rossi, B. Shklovskii, F. Wellstood and N. Zimmermanfor stimulating discussions. This work was supported bythe LPS-NSA-CMTC grant and by the Joint QuantumInstitute (RL). A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A.Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. ,1 (1987). G. S. Uhrig, Phys. Rev. Lett. , 100504 (2007). Y. Makhlin and A. Shnirman, Phys. Rev. Lett. , 178301(2004). A. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. B ,195329 (2003). W. M. Witzel and S. Das Sarma, Phys. Rev. B , 035322 (2006). S. K. Saikin, W. Yao, and L. J. Sham, Phys. Rev. B ,125314 (2007). L.-D. Chang and S. Chakravarty, Phys. Rev. B , 154(1985). E. Paladino, L. Faoro, G. Falci, and R. Fazio, Phys. Rev.Lett. , 228304 (2002). A. Grishin, I. V. Yurkevich, and I. V. Lerner, Phys. Rev.B , 060509(R) (2005). A. Grishin, I. V. Yurkevich, and I. V. Lerner, cond-mat/0608445 (2006). R. de Sousa, K. B. Whaley, F. K. Wilhelm, and J. vonDelft, Phys. Rev. Lett. , 247006 (2005). D. Segal, D. R. Reichman, and A. J. Millis, Phys. Rev. B , 195316 (2007). B. Abel and F. Marquardt, arXiv:0805.0962 (2008). Y. Makhlin, G. Sch¨on, and A. Shnirman, Rev. Mod. Phys. , 357 (2001). Y. Nakamura, Y. A. Pashkin, T. Yamamoto, and J. S.Tsai, Phys. Rev. Lett. , 047901 (2002). D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier,C. Urbina, D. Esteve, and M. H. Devoret, Science ,886 (2002). G. Ithier, E. Collin, P. Joyez, P. J. Meeson, D. Vion, D. Es-teve, F. Chiarello, A. Shnirman, Y. Makhlin, J. Schriefl,et al., Phys. Rev. B , 134519 (2005). O. Astafiev, Y. A. Pashkin, Y. Nakamura, T. Yamamoto,and J. S. Tsai, Phys. Rev. Lett. , 267007 (2004). O. Astafiev, Y. A. Pashkin, Y. Nakamura, T. Yamamoto,and J. S. Tsai, Phys. Rev. Lett. , 137001 (2006). J. A. Schreier, A. A. Houck, J. Koch, D. I. Schuster, B. R.Johnson, J. M. Chow, J. M. Gambetta, J. Majer, L. Frun-zio, M. H. Devoret, et al., Phys. Rev. B , 180502(R)(2008). L. Faoro, J. Bergli, B. L. Altshuler, and Y. M. Galperin,Phys. Rev. Lett. , 046805 (2005). V. I. Kozub, A. A. Zyuzin, Y. M. Galperin, and V. Vi-nokur, Phys. Rev. Lett. , 107004 (2006). L. Faoro and L. B. Ioffe, Phys. Rev. Lett. , 047001(2006). L. S. Levitov, in
Quantum Noise in Mesoscopic Sys-tems , edited by Y. Nazarov (Kluwer, Boston, 2003),(cond-mat/0210284). Y. V. Nazarov and M. Kindermann, Eur. Phys. J. B ,413 (2003). A. Kamenev, in
Nanophysics: Coherence and Transport ,edited by H. Bouchiat (Elsevier, 2005), pp. 177–246,(cond-mat/0210284). P. J. Leek, J. M. Fink, A. Blais, R. Bianchetti, M. Goppl,J. M. Gambetta, D. I. Schuster, L. Frunzio, R. J.Schoelkopf, and A. Wallraff, Science , 1889 (2007). U. Haeberlen,
High Resolution NMR in Solids, Advancesin Magnetic Resonance Series, Supplement 1 (Academic,New York, 1976). G. S. Uhrig, arXiv:0803.1427 (2008). L. Viola and S. Lloyd, Phys. Rev. A , 2733 (1998). L. Viola, J. Mod. Opt. , 2357 (2004). G. Falci, A. D’Arrigo, A. Mastellone, and E. Paladino,Phys. Rev. A , 040101(R) (2004). K. Khodjasteh and D. A. Lidar, Phys. Rev. Lett. ,180501 (2005). K. Khodjasteh and D. A. Lidar, Phys. Rev. A , 062310(2007). L. Faoro and L. Viola, Phys. Rev. Lett. , 117905 (2004). B. Lee, W. M. Witzel, and S. Das Sarma, Phys. Rev. Lett. , 160505 (2008). L. Cywi´nski, R. M. Lutchyn, C. P. Nave, and S. Das Sarma,Phys. Rev. B , 174509 (2008). J. Rammer,
Quantum Field Theory of Non-equilibriumStates (Cambridge University Press, New York, 2007). A. Larkin and A. Varlamov,
Theory of Fluctuations in Su-perconductors (Oxford University Press, New York, 2005). H. Bruus and K. Flensberg,
Many-Body Quantum FieldTheory in Condensed Matter Physics (Oxford UniversityPress, Oxford, 2004). J. M. Martinis, S. Nam, J. Aumentado, K. M. Lang, andC. Urbina, Phys. Rev. B , 094510 (2003). A. Shnirman, G. Sch¨on, I. Martin, and Y. Makhlin, Phys.Rev. Lett.94