Eur. Phys. J. D 46, 375–380 (2008) DOI: 10.1140/epjd/e2007-00300-9 THE EUROPEAN PHYSICAL JOURNAL D The dynamics of a central spin in quantum Heisenberg XY model X.-Z. Yuan1,a,K.-D.Zhu1,andH.-S.Goan2 1 Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, P.R. China 2 Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan Received 19 March 2007 / Received in final form 30 August 2007 Published online 24 October 2007 – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2007 Abstract. In the thermodynamic limit, we present an exact calculation of the time dynamics of a central spin coupling with its environment at finite temperatures. The interactions belong to the Heisenberg XY type. The case of an environment with finite number of spins is also discussed. To get the reduced density matrix, we use a novel operator technique which is mathematically simple and physically clear, and allows us to treat systems and environments that could all be strongly coupled mutually and internally. The expectation value of the central spin and the von Neumann entropy are obtained. PACS. 75.10.Jm Quantized spin models – 03.65.Yz Decoherence; open systems; quantum statistical meth- ods – 03.67.-a Quantum information z 1 Introduction like S0 and extend the operator technique to deal with the case of finite number environmental spins. Recently, using One of the promising candidates for quantum computa- numerical tools, the authors of references [5–8] studied the tion is spin systems [1–4] due to their long decoherence dynamic behavior of a central spin system decoherenced and relaxation time. Combined with nanostructure tech- by a spin bath. They considered a general and realistic nology, they have the advantage of being scalable. Just as model. However the influence of environmental tempera- other quantum systems, the spin systems are inevitably ture was not taken into consideration. Here our model just influenced by their environment, especially the spin envi- involves the Heisenberg XY coupling which has extensive ronment. As a result, decoherence will cause the transition applications for various quantum information processing of a system from pure quantum states to mixture classi- proposals [22–26]. Another benefit of this model is that we cal ones. This is a big obstacle in the development of the can obtain exactly the time evolution of the expectation z quantum computer. Therefore the dynamic behavior of a value of central spin S0 and the von Neumann entropy at single spin or several spins interacting with a spin bath has finite temperature. Also the special operator technique we attracted much attention in recent years [5–13]. A quan- used is mathematically simple and physically clear, and al- tum system exposed to environmental modes is described lows us to treat systems and environments that could be by the reduced density matrix. However, in most cases it is strongly coupled mutually. Furthermore, the internal dy- impossible to obtain an exact solution to the evolution of namics of the spin bath are taken into consideration. This the reduced density matrix with the environmental modes is the main difference between our model and that in refer- traced over. Then different approximate methods are used. ence [27] where the system-bath coupling is also XY type, Markovian-type approximations [14–17] are valid only for but the bath is assumed to be in an unpolarized infinite relatively long times and fail to represent behavior of the temperature states. system for short time regime [18]. On the other hand, some The paper is organized as follows. Section 2 introduces authors [18–20] develop short-time approximation to deal the model Hamiltonian. From the reduced density matrix, z with the problem. If we can construct a physically rel- the expectation value of S0 and the von Neumann entropy evant model which can be solved exactly, this will be a of the central spin are calculated. Conclusions are given quite important work. Using a novel operator technique, in Section 3. we have studied the entanglement evolution of two cou- pled spins in a quantum spin environment in the thermo- dynamic limit at finite temperature [21]. In the present 2 Model and calculations paper, we consider a similar model. However, to provide a complete picture of the dynamics, we study an observable We consider a single central spin (system) coupling with a e-mail: [email protected] the bath spins via a Heisenberg XY interaction. The bath 376 The European Physical Journal D correlations are also XY type. Both the central spin and to one. At absolute zero temperature, no excitation will 1 the bath consist of spin- 2 atoms. HB is the Hamiltonian exist. The bath is in a thoroughly polarized state with all of the bath, and HSB represents the interaction between spin down. With the increase of temperature, the number the central spin and the bath [27] of spin up atoms increases. ρS(t) is the reduced density matrix of the central spin system, it can be written as H = HSB + HB , (1) −iHt iHt ρS (t)=trB e [|ψψ|⊗ρB] e , (11) where where tr denotes the partial trace taken over the Hilbert N N B √g0 + − − + space of the spin bath. Using a novel operator technique, HSB = S0 Si + S0 Si , (2) we can exactly determine the matrix ρS(t)whichisa2×2 N =1 =1 i i matrix in the standard basis |0, |1.Asanexample,we g N just calculate time evolution of the operator |11|, i.e., H = S+S− + S−S+ , (3) B i j i j N = i j −iHt −HB /T iHt E(t)=trB e |11|⊗e e . (12) + − here S0 and S0 are the spin-flip operators of the central + − It is demonstrated in Appendix. spin, respectively. Si and Si are the corresponding oper- ators of the ith atom in the bath. N is the number of the From equations (9), (11), and results in Appendix, the bath atoms which have direct interaction with the central reduced density matrix can be written as spin. g0 is the coupling constant between the central spin ρ (t)=tr e−iHt [|ψψ|⊗ρ ] eiHt and the bath, whereas g√is that in the bath. Both con- S B B stants are rescaled as g0/ N and g/N [27–30]. Using the ρ11 ρ12 N ± = , (13) ρ21 ρ22 collective angular momentum operators J± = i=1 Si , we rewrite the Hamiltonians as where g0 + − HSB = √ S0 J− + S0 J+ , (4) N M 1 −2 1− n 2 ∗ gn( N )/T g ρ11 = |α| A1A1e HB = (J+J− + J−J+) − g. (5) Z N n=0 M 1 −2 1− n After the Holstein-Primakoff transformation: 2 ∗ gn( N )/T + |β| C1C1 ne , (14) Z √ + 1/2 √ + 1/2 n=1 + b b b b J+ = Nb 1 − ,J− = N 1 − b, M 1 −2 1− n N N ∗ −i2gt ∗ gn( N )/T ρ12 = αβ e A1D1e , (15) (6) Z n=0 + M 1 −2 1− n b and b being creation and annihilation bosonic operators ∗ i2gt ∗ gn( N )/T + ρ21 = α βe D1A e , (16) such that [b, b ] = 1, the Hamiltonians are written as Z 1 n=0 + + M + b b − + b b 1 −2 1− n 2 ∗ gn( N )/T HSB = g0S0 1 − b + g0S0 b 1 − , (7) ρ22 = |α| B1B (n +1)e 2N 2N Z 1 n=0 + + b b M HB =2gb b 1 − . (8) 1 −2 1− n 2 ∗ gn( N )/T N + |β| D1D1e . (17) Z n=0 1 We have made a N -expansion which is valid if the number of the atoms in excited states is much smaller than N [28]. As mentioned in Appendix, the operatorn ˆ has been re- In the following, we can obtain exactly the reduced density placed by its eigenvalue n. matrix for the central spin by tracing over the bosonic Assuming an initial condition α =1andβ =0,we z bath at finite temperature. obtain the expectation value of S0 We assume the initial density matrix of the composed z z system to be factorized, i.e., ρ(0) = |ψψ|⊗ρB. The initial S0 =trS(S0 ρS) z z state of the system is = 1|S0 ρS|1 + 0|S0 ρS|0 M | | | 1 −2 1− n ψ = α 1 + β 0 , (9) ∗ ∗ gn( N )/T = [A1A1 − B1B1 (n +1)]e , 2 2 2Z |α| + |β| =1. (10) n=0 (18) The density matrix of the bath satisfies the Boltzmann −HB /T distribution, that is ρB = e /Z ,whereZ is the par- where trS denotes the partial trace taken over the Hilbert tition function and the Boltzmann constant has been set space of the central spin. X.-Z. Yuan et al.: The dynamics of a central spin in quantum Heisenberg XY model 377 0.5 The effect of this single-mode environment on the dynam- ics of the central qubit is extremely non-Markovian. The 0.4 quantum information flowing into the environment may partially return to the central spin. This reflects onto, for 0.3 example, the revival behavior of the reduced density ma- trix of the central spin. This is different from the usual 0.2 environment model which consists of many bosonic modes 〉 z 0 and often causes the reduced dynamics of the system of S 〈 interest displaying an exponential decay in time behavior. 0.1 So the Markovian approximation usually used in quan- tum optics master equation will not work in our model. 0 One may do perturbation theory for weak-coupling case, but the single-mode environment in our model will not −0.1 remain in thermal equilibrium state as is usually assumed for an environment with very large degrees of freedom −0.2 in the weak-coupling master equation approach.
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