The Dynamics of a Central Spin in Quantum Heisenberg XY Model

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 methods – 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 coupled spins in a quantum spin environment in the thermodynamic 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 operators 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 dynamics 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 matrix 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 quantum 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. As our 0 5 10 15 g t model corresponds to an extremely non-Markovian case, 0 the dynamics of Sz shows a two-step behavior. That z | 0 Fig. 1. Time evolution of S0 for an initial system state ψ = is, after an initially quicker decay of the amplitude, the | 1 for different temperatures. g = g0, T = 100g (dot dashed dynamics still shows a behavior of quantum oscillations curve), T =10g (solid curve), and T =10g, N = 280 (dashed at long times. The general trend for temperature depen- curve). dent dynamics is that the larger the temperature is, the quicker the initial decay is and the smaller the amplitude of the long-time oscillations is. This can be explained as z In Figure 1, we plot S0 as a function of time for follows. At high temperature, the environment appears to different values of temperature T . Except the dot dash be in very disorderly states. The central spin quickly loses curve which is for T =10g and N = 280, all the results its message to the environment and only a little message below are in the thermodynamic limit. It shows that ini- can return to it. At infinite high temperature, no message z tially S0 exhibits a coherent oscillatory decay. With the will return to the central spin. Then the revival behavior increase of temperature, the decay rate increases. After a will disappear. This is the case of reference [27], where turning point g0t ≈ 6, which is independent on the tem- the spin bath is assumed to be in an unpolarized infinite z perature, the oscillatory amplitude increases. Then S0 temperature state. In practice, only finite number of spins oscillates in certain low value regime at high temperature. in the bath interact with the central spin. Therefore it is After the initial fast decay, the central system exhibits necessary to investigate the working condition of the ther- long-time oscillation. This is quite an unexpected result. modynamic limit in our model. The good fitness between It is a common sense that interaction of the system with solid curve and dashed curve in Figure 1 shows that it its environment leads to a decay of the system’s initial is safe to use the thermodynamic limit if T<10g and pure state into a mixture of several states, i.e., non diag- N>280. onal elements of the density matrix vanish, and diagonal The von Neumann entropy of the central spin is de- elements achieve their equilibrium values. This causes an fined as damping of quantum oscillations and increase of the sys- P = −trB(ρS ln ρS). (19) tem’s entropy with time [5]. However our results are not unique. Recently, the authors of references [5,6] studied After diagonalising the reduced density matrix ρS,weob- numerically the dynamic behavior of a two spin system tain the von Neumann entropy decoherenced by a spin bath. Without considering the in- 2 fluence of the environmental temperature, they used a gen- P = − p ln p , (20) eral and realistic model and found that the system shows k k =1 a special dynamic behavior, i.e., an initial fast decay of k z z S1 or S2 is followed by revival oscillations. This new where pk are the eigenvalues of the reduced density matrix phenomenon was explained in detail, where the crucial fac- ρS and can be expressed as tor is whether the central system comprises even or odd 2 number of spin entities [5,6]. Here we consider a single ρ11 + ρ22 ± (ρ11 − ρ22) +4ρ12ρ21 central spin coupling with a spin bath. All the interac- p1,2 = . (21) tions belong to the XY type. At finite temperature, an 2 exact solution is given. We do not think the mechanism In Figure 2, we plot the von Neumann entropy P (t)ofthe of the revival oscillation in our paper is same as that in reduced system for different values of T . The initial state references [5,6]. Due to the symmetry of our model, the of the system is Hamiltonian after the Holstein-Primakoff transformation √ √ and in the thermodynamic limit is equivalent to that of a 2 2 qubit interacting with a single-mode thermal bosonic field. |ψ = |1 + |0. (22) 2 2 378 The European Physical Journal D

0.7 X.Z.Y. acknowledges support from the National Natural Sci- ence Foundation of China under Grant No. 10647137. H.S.G. 0.6 would like to acknowledge support from the National Science Council, Taiwan, under Grant No. 96-2112-M-002-007. 0.5

0.4 Appendix: Calculation of time evolution

0.3 of the operator |11| P(t)

+ − 0.2 The total Hamiltonian H contains operators b, b , S0 , + − + and S0 .HereS0 and S0 change the system state from 0.1 |1 to |0, and vice versa. Since

2 0 (iHt) e−iHt =1− iHt + + ..., (A.1) 2! −0.1 0 5 10 15 g t it is obvious that we can write 0 −iHt Fig. 2. Time√ evolution√ of entropy P (t) for an initial system e |1|x = γ|1|x + δ|0|x , (A.2) | 2 | 2 | state ψ = 2 1 + 2 0 for diﬀerent temperatures. g = g0, T =20g (solid curve), T =2g (dashed curve), and T =0.2g where |x is an arbitrary reference state on the bath and γ (dot dashed curve). and δ are two time dependent numbers. Deﬁning A|x = γ|x and B|x = δ|x,wehave

−iHt| | | | | At high temperature, our results agree with reference [27] e 1 x =(A 1 + B 0 ) x , (A.3) quite well, especially for the initial stage of evolution, + where the Von Neumann entropy increases quickly. How- where A and B are functions of operators b, b ,andtime ever, some kinds of oscillations appear at long times. This t.UsingtheSchrödinger equation identity shows the Non-Markovian behavior of the central spin sys- d tem. That is the system exchanges entropy (quantum in- i e−iHt|1|x = H e−iHt|1|x , (A.4) formation) with the spin bath for a long time at finite dt temperature, so that the entropy will not come close to and equation (A.3), we obtain saturation. With the decrease of temperature, the increas- + + ing rate of P (t) becomes smaller and P (t) oscillates in a d b b + b b A = −i g0 1 − bB +2gb b 1 − A , lower value regime. For T =2g, we see a large amplitude dt 2N N vibration which means that non-equilibrium properties of (A.5) the system become more obvious. At very low tempera- + + ture, the system still exchange entropy with the spin bath d + b b + b b B = −i g0b 1 − A +2gb b 1 − B , periodically. In such case, the bath is in a state with all dt 2N N spin down. The initial state of the central spin contains (A.6) spin up component. The interaction is XY type, then the central spin and the spin bath can still exchange entropy with initial conditions from equation (A.3) being A(0) = 1 at very low temperature. If the central spin is also in a and B(0) = 0. As A and B are functions of b+ and b, spin down state, no exchange will happen. they are operators and do not commute with each other. Equations (A.5) and (A.6) are thus coupled differential equations of non-commuting operator variables, which can 3 Conclusions not be solved by using conventional methods for ordinary number variables. We study the exact dynamics of a central spin in the quan- The crucial observation to solve the problem is that tum Heisenberg XY model in the thermodynamic limit. the Hamiltonian, equations (7) and (8), is of an effective The reduced dynamics of the central spin coupling with Jaynes-Cumming type and it can be block-diagonalized in its environment at finite temperature is obtained. From the dressed state subspace of |i; n,withi + n = constant. the reduced dynamics matrix, we calculate the expecta- Here |i represent the qubit states and |n are the bosonic z tion value of the central spin S0 . The results show that field number states. As a result, we may rewrite equa- it decreases quickly and oscillates in a low value regime tions (A.5) and (A.6) in such a subspace. By introducing at high temperature. We also calculate the von Neumann the following transformation entropy of the system. At√ very low√ temperature, with the 2 2 A = A1, (A.7) initial pure state |ψ = |1 + |0, the central spin 2 2 + evolves far from the completely mixed state Pmax =ln2. B = b B1, (A.8) X.-Z. Yuan et al.: The dynamics of a central spin in quantum Heisenberg XY model 379 equations (A.5) and (A.6) then become for n N. Therefore we demand M N for the integer M in equation (A.16). Also equation (A.16) must converge d nˆ nˆ quickly for M N, which is the case at low temperature A1 = −i g0(ˆn+1) 1− B1 +2gnˆ 1− A1 , dt 2N N (T g). In the thermodynamic limit (N →∞), we can (A.9) set M →∞and discard the limitation of the low temperature. In such case, the partition function d nˆ nˆ +1 B1 = −i g0 1− A1 +2g(ˆn+1) 1− B1 , dt 2N N 1 Z = . (A.18) (A.10) 1 − e−2g/T

+ wheren ˆ = b b. At this stage, we know that A1, B1 are Let −iHt functions ofn ˆ and t, and commute with each other. We can e |0|x =(C|1 + D|0)|x. (A.19) then treat equations (A.9) and (A.10) as coupled complex- Inthesameway,wehave number diﬀerential equations and solve them in a usual i2gt way. Considering initial condition C = bC1e , (A.20) i2gt D = D1e , (A.21) A1(0) = 1, (A.11) B1(0) = 0, (A.12) where − nˆ−1 we obtain g0 1 2N C1 = − 2 2 nˆ+1 2 2ˆn−1 2 nˆ−1 − 1 2 g 1 − + g0nˆ 1 − 2g(ˆn +1) 1 N + λ iλ1t N 2N A1 = e 2 2 2 2ˆn+1 2 nˆ iλ1t iλ2t 2 g 1 − + g0(ˆn +1) 1 − × e − e , (A.22) N 2N ˆ+1 2 − n (ˆn−1) 2g(ˆn +1) 1 + λ2 − − N iλ2t 2gnˆ 2g N + λ1 iλ t e , 1 1 2ˆ+1 2 ˆ 2 D = e 2 n 2 n 2ˆ−1 2 2 ˆ−1 2 2 g 1 − + g0(ˆn +1) 1 − 2 2 − n − n N N 2 g 1 N + g0nˆ 1 2N (A.13) 2 (ˆn−1) ˆ 2gnˆ − 2g + λ2 − n − N iλ2t g0 1 2N e , B1 = − 2ˆ−1 2 2 ˆ−1 2 2 2 2 − n − n 2 2ˆn+1 2 nˆ 2 g 1 N + g0nˆ 1 2N 2 g 1 − + g0(ˆn +1) 1 − N 2N (A.23) × eiλ1t − eiλ2t , (A.14) where where 2 − − 2ˆn 2ˆn +1 2 λ1,2 = g(2ˆn +1)+g 2ˆn +2ˆn +1 N λ1,2 = −g(2ˆn +1)+g N − 2 − 2 2 2ˆn 1 2 nˆ 1 2 2 ± g 1 − + g nˆ 1 − . (A.24) 2ˆn +1 nˆ N 0 2N ± g2 1 − + g2(ˆn +1) 1 − . N 0 2N (A.15) References Then ﬁnally we have 1. D. Loss, D.P. Divincenzo, Phys. Rev. A 57, 120 (1998) M 2. G. Burkard, D. Loss, D.P. Divincenzo, Phys. Rev. B 59, 1 ∗ −2gn 1− n /T E(t)=|11| A1A e ( N ) 2070 (1999) Z 1 n=0 3. A. Imamoglu, D.D. Awschalom, G. Burkard, D.P. M DiVincenzo,D.Loss,M.Sherwin,A.Small,Phys.Rev. 1 −2 1− n ∗ gn( N )/T Lett. 83, 4204 (1999) + |00| B1B1 (n +1)e , Z 4. R. Raussendorf, H.J. Briegel, Phys. Rev. Lett. 86, 5188 n=0 (A.16) (2001) 5. V.V. Dobrovitski, H.A. De Raedt, M.I. Katsnelson, B.N. where Harmon, Phys. Rev. Lett. 90, 210401 (2003) M 6. V.V. Dobrovitski, H.A. De Raedt, Phys. Rev. E 67, 056702 −2 1− n Z = e gn( N )/T . (A.17) (2003) n=0 7. J. Lages, V.V. Dobrovitski, M.I. Katsnelson, H.A. De Raedt, B.N. Harmon, Phys. Rev. E 72, 026225 (2005) In equation (A.16), the trace over the environmental de- 8. W.X. Zhang, V.V. Dobrovitski, K.A. Al-Hassanieh, E. grees of freedom has been performed and the operatorn ˆ Dagotto, B.N. Harmon, Phys. Rev. B 74, 205313 (2006) has been replaced by its eigenvalue n.Wehavemadea 9.F.M.Cucchietti,J.P.Paz,W.H.Zurek,Phys.Rev.A72, 1 N -expansion in equations (7) and (8), which is valid only 052113 (2005) 380 The European Physical Journal D

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