Enhanced decoherence in the vicinity of a phase transition
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Enhanced decoherence in the vicinity of a phase transition
S. Camalet and R. Chitra Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, UMR 7600,Universit´e Pierre et Marie Curie, Jussieu, Paris-75005, France (Dated: today)We study the decoherence of a spin-1/2 induced by an environment which is on the verge of acontinuous phase transition. We consider spin environments described by the ferromagnetic andantiferromagnetic Heisenberg models on a square lattice. As is well known, these two dimensionalsystems undergo a continuous phase transition at zero temperature, where, the spins order sponta-neously. For weak coupling of the central spin to these baths, we find that as one approaches thetransition temperature, critical fluctuations make the central spin decohere faster. Furthermore,the decoherence is maximal at zero temperature as signalled by the divergence of the Markoviandecoherence rate.
PACS numbers: 03.65.Yz,05.70.Jk
The dynamics of a real system is determined not onlyby its internal Hamiltonian but also by its environment.Due to the coupling to the numerous environmental de-grees of freedom, an initial pure quantum state of thesystem evolves into an incoherent mixture. This pro-cess, decoherence , is a major hurdle in the constructionof quantum computers where sufficiently long coherencetimes of the qubits (the basic units of quantum infor-mation) are fundamental requisites. The recent spate ofexperimental work on qubits has generated a great dealof interest in the question of decoherence induced by dif-ferent environments. The most studied environments arebosonic baths modeled as baths of harmonic oscillators .More recently, due to the numerous experimental realiza-tions of qubits which involve real spin-1/2 objects , therehas been an increasing focus on spin baths as a primarysource of decoherence.The rich and varied physics of spin systems make spinbaths fundamentally interesting. Spin baths comprisingnon-interacting spins have been among the first studiedmodels to understand the decoherence process . How-ever, since the physical origin of the decoherence of asystem lies in the dynamical fluctuations of the bath de-grees of freedom to which it couples, we expect the re-sulting decoherence to reflect the non-trivial nature ofthe fluctuations induced by the interactions in the bath.From this perspective, the vicinity of a continuous phasetransition is especially interesting given the existence ofcritical fluctuations. Clearly, these divergent fluctuationsare expected to have dramatic consequences for the de-coherence. Though some authors have studied spin bathmodels which exhibit continuous phase transitions, theeffect of critical fluctuations has been occulted either be-cause of the purely mean field nature of the models ordue to the temperature regimes considered .In this Letter, we address the issue of the decoherenceof a central spin weakly coupled to a spin bath which is onthe verge of a phase transition. We model the spin bathas a two-dimensional system of Heisenberg spins withnearest neighbour interactions on a square lattice. Weconsider both ferromagnetic and antiferromagnetic inter- actions. In both cases, the spin bath undergoes sponta-neous symmetry breaking at T c = 0 to a magnetic phase.These spin bath models are interesting for our purposeas their critical fluctuations are well understood . Stud-ies on fluctuation free mean field models indicate thatthe decoherence time scale increases as temperature islowered and the system undergoes a transition . More-over, this result is in accord with the conventional wis-dom that decoherence is minimal at T = 0 and increaseswith increasing thermal fluctuations. In what follows, wewill show that contrary to the above scenario, as one ap-proaches the critical point at zero temperature, criticalfluctuations result in a faster decoherence of the centralspin with the decoherence being maximal at zero tem-perature.We consider a central spin-1/2 coupled isotropically viaa hyperfine like contact interaction to a spin- S bath withuniform nearest neighbor interactions. For simplicity, weassume that the central spin has no internal dynamics.The total Hamiltonian describing the central spin and itsenvironment is given by H = H I + H B ≡ σ · X i λ i S i − J X ( ij ) S i · S j (1)where σ are the Pauli matrices, S i the spin operatorsof the bath spin at site i and ( . . . ) denotes summationover nearest neighbor spin pairs of a square lattice withperiodic boundary conditions. For the case of an elec-tronic spin interacting with nuclear spins through thehyperfine contact interaction, the coupling constant λ i is simply related to the electron envelope wave functionat the site i . Antiferromagnetic interactions ( J <
J >
0) lead to Neel or-der and ferromagnetic order respectively, at T = 0. Tostudy the decoherence for weak coupling to the spin bathand a finite temperature T , we use the resolvent operatorapproach in conjunction with the Schwinger bosontechnique. To simplify the calculation, we assume a fac-torizable initial density matrix for the composite system,Ω = ρ (0) ⊗ ρ B where ρ (0) and ρ B ∝ exp( − H B /T ) denote,respectively, the central spin state and the thermal equi-librium of the bath. We use units k B = ~ = 1 throughoutthis paper.Since the total Hamiltonian (1) is rotationally invariantand the finite temperature phase of the bath is isotropic,the central spin Bloch vector at time t is related to itsinitial value as given by h σ i ( t ) = r ( t ) h σ i (0) (2)where r is a scalar function of the time t and h . . . i denotesthe average over the density matrix of the composite sys-tem. Due to the coupling to the bath, the function r vanishes in the long time limit. To determine r ( t ), itis useful to write the reduced density matrix ρ ( t ) of thecentral spin as ρ ( t ) = i π Z R + iη dze − izt Tr B h ( z − L ) − ρ (0) ρ B i (3)where η is a real positive number, Tr B denotes the par-tial trace over the bath degrees of freedom and L is theLiouvillian corresponding to the total Hamiltonian H ,i.e., L A = [ H, A ] for any operator A . Using the decom-position of the density matrix ρ ( t ) in the basis of 2 × { I, σ α } and the projection operator techniqueexplained in Refs. 9 and 10, we obtain from (3) the ex-pression (2) where r ( t ) = i π Z R + iη dz e − izt z − Σ( z ) . (4)The self-energy Σ is given byΣ( z ) = Tr (cid:2) σ α Tr B (cid:2) L I ( z − Q L Q ) − L I ρ B σ α (cid:3)(cid:3) (5)where σ α is any Pauli matrix and L I is the Liouvilliancorresponding to the interaction Hamiltonian H I . Theprojection operator Q is defined by its action on any op-erator A as Q A = A − Tr B ( A ) ρ B . We note that forany arbitrary Hamiltonian, the decoherence of the cen-tral spin cannot be described by a single self-energy, thefull time evolution of the state ρ ( t ) is given by a 4 × z . From this4 × ρ ( t ) is in general characterised by two differenttimes, a decoherence time and a relaxation time whichdetermines the time for thermal equilibration of the cen-tral spin . For the Hamiltonian (1), the time-scales forrelaxation and decoherence are the same. However, forthe generic case of the central spin having its own inter-nal dynamics, these two times are in principle different.This generic case is studied in the last part of the paper.The decoherence in the weak coupling regime is deter-mined by the lowest order contribution of the interactionHamiltonian H I to the self-energy Σ. The first term ofthis expansion is given by (5) with L replaced by the bathLiouvillian L B . Using the properties of the Pauli matri-ces, this second-order contribution to the self-energy can be written asΣ ( z ) = − i X i , j λ i λ j Z ∞ dt e izt Re h S i ( t ) · S j i B (6)where S i ( t ) = exp( iH B t ) S i exp( − iH B t ) and h A i B =Tr( ρ B A ) denotes the thermal average of any bath op-erator A . Neglecting higher order contributions to Σ in(4) is equivalent to the Born approximation . It can beshown that in the weak coupling limit, the expression(4) can be simplified to obtainln r ( t ) ≃ − π Z dE sin( tE/ E Γ ( E ) (7)where Γ ( E ) = − ImΣ ( E + i + ). Eq.(7) describes the de-coherence at all time scales. It yields the usual quadraticdecrease r ( t ) ≃ − t R dE Γ ( E ) / π for very short timesand Markovian decay r ( t ) ≃ exp[ − Γ (0) t ] for asymptotictimes. The full time evolution of the central spin is de-termined by the dynamic spin correlation function of thebath via the relations (7) and (6). Typically, dynamicalcorrelations are rather difficult to calculate for spin sys-tems. In the following, we use the successful Schwingerboson mean field theory to evaluate the dynamical spincorrelation of the bath described by H B . We will showthat the critical fluctuations in the bath lead to a diver-gence of Γ (0) at the transition, implying a faster thanexponential asymptotic decay of r ( t ). This divergence ismerely a reflection of the divergence of the underlyingcorrelation time in the bath at the phase transition.In the Schwinger representation , at every site i , thespin operators S i are replaced by two bosonic opera-tors a i and b i with the constraint a † i a i + b † i b i = 2 S .In the ferromagnetic case, the correspondence relationsread 2 S z i = a † i a i − b † i b i and S + i = a † i b i . In the antifer-romagnetic case, due to the possibility of Neel orderingof spins, two sub-lattices must be distinguished. On oneof the sub-lattices, the spin and boson operators are re-lated as above, while on the other sub-lattice the cor-respondence relations take the form 2 S z i = b † i b i − a † i a i and S + i = − b † i a i . In both ferromagnetic and antiferro-magnetic cases, the Hamiltonian (1) can be interpretedas describing a spin-1/2 coupled to a boson bath. Theresulting problem is nonetheless very different from thestandard spin-boson model for the following reasons: i)the central spin couples to two species of bosons ii) thebath bosons interact with each other and are subject toconstraints which conserve the number of bosons. In theSchwinger boson mean field theory, the local constraintson the bosons are replaced by a global constraint via auniform chemical potential and the boson Hamiltonianderived from H B is studied using a Hartree-Fock meanfield scheme . We now present analytical results forthe decoherence obtained within this theory in the lowtemperature regime T ≪ | J | S .For the ferromagnetic Heisenberg model ( J > H B yields, up to a constantenergy, H fmMF = P k ω k ( a † k a k + b † k b k ) where a k and b k are the Fourier transforms of a i and b i , respectively. Themagnon dispersion is given by ω k = 2 JQ (2 − cos k x − cos k y ) − µ (8)where k x and k y are the components of the wave-vector k .The mean field parameter Q and the chemical potential µ are determined by the self consistent equations S = 1 N X k n k Q = 1 N X k n k cos k x (9)where n k = [exp( ω k /T ) − − is the Bose occupationfactor. Note that here the chemical potential µ is nec-essarily negative. At low temperatures, since the abovesums are dominated by the modes ω k ≪ T , Q ≃ S and µ → T → µ ≃ − T exp( − πJS /T ). The transition to the orderedstate is explained by a pseudo Bose condensation of thebosons at the critical temperature .Using the above results, we find the decoherence in theweak coupling regime is given by (7) withΓ ( E ) = 4 π N X k , q | Λ( k − q ) | δ ( E + ω k − ω q ) × ( n k + n q + 2 n k n q ) (10)where Λ( k ) = P i λ i exp( i k · i ). In accordance with Ref.12,a multiplicative factor 2 / n k vanishes for ω k ≫ T , the last term of (10) is a Diracfunction at E = 0 whereas the first two terms converge toa continuous function of E with the characteristic energy JS and a finite value, Λ( ) / J , at E = 0. For times t ≪ ( JS ) − , the sine function in (7) can be expandedleading to the usual short-time quadratic decoherence.For ( JS ) − ≪ t ≪ T − , the last term of (10) is essen-tially a Dirac function in (7) as it practically vanishesfor | E | & T . Moreover, the contribution of the first twoterms of (10) to ln r ( t ), − t Λ( ) / J , is negligible andthen ln r ( t ) ≃ −
43 (Λ fm St ) (11)where Λ fm ≡ Λ( ) = P i λ i . To obtain the decoherencefor longer times, we evaluate Γ ( E ) for energies | E | ≪ T using the approximations ω k ≃ JS ( k x + k y ) and Λ( k − q ) ≃ Λ( ). We findΓ ( E ) = Λ fm T πJ S | µ | (cid:12)(cid:12)(cid:12) µE (cid:12)(cid:12)(cid:12) ln (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) Eµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . (12)For times T − ≪ t ≪ | µ | − , (12) results in a smallpositive correction to the Gaussian decay (11), ln r ( t ) +4(Λ fm St ) / ≃ T (Λ fm t ) ln( T t ) / πJ. For longer times, t ≫ | µ | − , the decoherence is Markovian with the rateΓ (0). The non-Markovian corrections arising from thelow energy behaviour of (12) are logarithmic, ln r ( t ) +Γ (0) t ≃ (Λ fm T /πJSµ ) ln( | µ | t ) /
6. As anticipated, therate Γ (0) diverges in the zero temperature limit, im-plying that the asymptotic decoherence is Gaussian at T = 0. Note that the cross-over from the Gaussian de-coherence given by (11) to the Markovian decoherence isextremely long at low temperatures.For the antiferromagnetic bath, following the stepsoutlined in Refs. 12 and 14, we obtain the followingmean field Hamiltonian for the bath (up to a constant) H afmMF = P k ω k ( α † k α k + β † k β k ) where α k and β k are lin-ear combinations of the Fourier transforms of the originaloperators a i and b i , respectively. The magnon dispersionis now given by ω k = q µ − A k (13)where A k ≡ JQ (cos k x + cos k y ) and the correspondingself consistency conditions are S + 12 = 1 N X k | µ | ω k (cid:18) n k + 12 (cid:19) JQ = 1 N X k A k ω k (cid:18) n k + 12 (cid:19) . (14)In the paramagnetic phase, T >
0, there exists a gap∆ = ( µ − J Q ) / in the magnon dispersion for k = and the Neel ordering wavevector k = ( π, π ). An anal-ysis of (14) shows that as T → Q → Q ≃ S + 0 . ≃ T exp( − π | J | ¯ ρ s /T ) where | J | ¯ ρ s is the spin stiffness of the bath . The dimensionlessparameter ¯ ρ s depends only on S , ¯ ρ s ≃ .
176 for S = 1 / ρ s ≃ S for S ≫ Γ ( E ) = π N X k , q | Λ( k − q ) | X ǫ = ± ǫ (cid:20) µ − A k A q ω k ω q + ǫ (cid:21) (cid:20) n k n q + n k + n q + 1 − ǫ (cid:21) δ ( | E | + ǫω k − ω q ) (15)As in the ferromagnetic case, in the zero temperaturelimit, the sum of the terms ∝ n k n q is a Dirac func-tion at E = 0 whereas the other terms converge to acontinuous function of E with finite characteristic en-ergy, | J | Q , and value at E = 0, 2Λ( π, π ) ¯ ρ s / | J | Q .Here also, for temperatures T ≪ | J | S , this continuousfunction contributes to the decoherence only in the veryshort time regime t ≪ ( | J | S ) − . To evaluate the decoher-ence for longer times, we remark that, for T ≪ | J | S and | E | ≪ | J | S , the sums over k and q in (15) are dominatedby the vicinities of ( k , q ) = ( , π ) and ( k , q ) = ( π , )where π = ( π, π ) permitting us to expand ω k and ω q toquadratic order in k and π − q or vice-versa in conjunctionwith the consistent approximations | Λ( k − q ) | ≃ Λ( π ) and µ − A k A q ≃ | J | Q . Using these, we find the decoher-ence is Markovian for times t ≫ ∆ − with the rateΓ (0) = Λ afm T πJ Q ∆ (16)where Λ afm ≡ Λ( π ). For times ( | J | S ) − ≪ t ≪ T − ,taking into account the terms with ǫ = − r ( t ) ≃ − (cid:18) Λ afm ¯ ρ s Q t (cid:19) . (17)As in the ferromagnetic case, the rate Γ (0) diverges inthe zero temperature limit and the asymptotic decoher-ence is Gaussian at T = 0. We note that the decoherenceis faster as S → ∞ i.e., when the spins become classical .In this case, since the associated scales µ, ∆ →
0, theMarkovian regime is not accessible and the decoherenceis Gaussian with a characteristic time ∝ S − .We observe that the decoherence is qualitatively sim-ilar for ferromagnetic and antiferromagnetic interactionsin the bath but with the important difference that ln r ( t )is proportional to Λ fm = Λ( ) or to Λ afm = Λ( π ) ,respectively. This can be understood as follows : an enhanced decoherence is observed in the vicinity of thetransition temperature only if the central spin couplesto the critical mode of the bath. We now show that thiscritical enhancement of the decoherence is not contingenton the absence of internal dynamics for the central spin.Due to the rotational invariance of H , any intrinsic dy-namic for the central spin can be described by the totalHamiltonian H ′ = H + ǫσ z /
2. In this case, as mentionedearlier, the Markovian asymptotic behavior of the centralspin state is characterized by two times , a relaxationtime T = Γ ( ǫ ) − and a decoherence time T given by T − = T − / (0) /
2. At the critical point, for anyvalue of ǫ , T vanishes and the resulting decoherence isfaster than exponential. On the other hand, the behav-ior of T depends on the value of ǫ . For | ǫ | ≪ | J | S , theabove study of the function Γ shows that T − reaches amaximum at T ∼ | ǫ | . This maximum grows as ǫ → ǫ = 0, i.e., T = T →
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