Guerra, Marta L.

Novotny, M.A.

Raedt, Hans De

University of Groningen

   University of GroningenReal-Time Quantum Dynamics of a Single Spin 1/2 Coupled to a Spin BathGuerra, Marta L.; Raedt, Hans De; Novotny, M.A.Published in:Proceedings of the 24th Workshop on Computer Simulation Studies in Condensed Matter Physics (CSP2011)IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2011Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Guerra, M. L., Raedt, H. D., & Novotny, M. A. (2011). Real-Time Quantum Dynamics of a Single Spin 1/2Coupled to a Spin Bath. In Proceedings of the 24th Workshop on Computer Simulation Studies inCondensed Matter Physics (CSP2011) (Vol. 15, pp. 33-36). University of Groningen, The Zernike Institutefor Advanced Materials.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019Available online at www.sciencedirect.comPhysics Procedia 15 (2011) 33–361875-3892 © 2011 Published by Elsevier Ltd.doi:10.1016/j.phpro.2011.05.056 Real-Time Quantum Dynamics of a Single Spin 1/2 Coupled to a Spin BathMarta L. Guerraa, Hans De Raedtb, M.A. NovotnyaaDepartment of Physics and Astronomy and HPC2 Center for Computational Sciences; Mississippi State University, P.O. Box 5167, MississippiState, MS, 39762-5167, USAbDepartment of Applied Physics, Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, TheNetherlandsAbstractWe performed real-time dynamics of a spin 12 quantum system that is coupled to a bath of N quantum spin12particles. The quantities we study are the time dependence of the magnetization and of the quantum purity, P(t). Ifthe spins that constitute the bath have no interactions with each other, the result is compared to a recent theoreticalresult. We also present simulations to study the eﬀect of interactions between the bath spins. In particular, we analyzehow P(t) approaches its asymptotic value for large t and N.Keywords: Quantum Dynamics, Spin Bath, Quantum Purity, Decoherence1. IntroductionAdvances in quantum information processing and quantum computing have been steady for over a decade [1,2]. However, additional theoretical understanding and improved physical implementations are necessary before aquantum computer becomes a practical device. At the simplest level a quantum computer can be viewed as a timedevelopment problem for a quantum spin system coupled to a bath that is its environment. At the computationallevel, the time development of the quantum system is in principle simple, since the underlying equation is just thetime-dependent Schro¨dinger equation. However in practice there are two complications that make calculations ofsuch quantum systems diﬃcult. First, the precise nature of the environment must be taken into account quantummechanically, and hence the system to be simulated is much larger than just the quantum system of the qubits of thequantum computer. Second, the development of real-time algorithms for the time-dependent Schro¨dinger equation arevery diﬃcult due to the imaginary phase in the propagator of the quantum system. In quantum statistical mechanicssimulations this diﬃculty is referred to as the ‘minus sign’ problem. There has however been significant progress insimulation of quantum spin systems in real time and the physical understanding of these systems [3, 4, 5, 6].Suppression of decoherence of spin systems is of fundamental importance for building quantum computers [2, 4].A recent paper [7] calculated the quantum frustration of dissipation of a single spin 12 coupled to one or two spinbaths, with no interactions between the spins of the bath. They showed that the quantum purity, P(t), decayed asan exponential for coupling to one bath but as a power law for coupling to two baths. In this paper we describe oursimulation to check their expressions for a spin 12 coupled to one spin bath. Simulations are restricted by the largestnumber of spins that can be simulated on modern supercomputers, so we study the approach to the asymptotic resultfor a finite number of spins. Furthermore in our calculations we are not restricted to zero interactions between thebath spins, and therefore we address the question of whether a spin bath with random interactions will allow a fasterapproach to the asymptotic region for P(t).34  Marta L. Guerra et al. / Physics Procedia 15 (2011) 33–362. Problem and TheoryWe consider a quantum spin system coupled to a quantum spin bath. The Hamiltonian for the system+bath isH = HS +HSB +HB (1)whereHS = ω0S z is the Hamiltonian of the system composed of a single spin 12 interacting with an external magneticfield, HSB = gS x ∑Nk=1 Ixk is the interactions of the system with the bath spins, HB is the Hamiltonian for interactionsamong the bath spins. The spin operators are S and I, for example S z = 12σz with σz the Pauli spin matrix.For the case whereHB = 0 the form of the interactions with the bath ensure that the total x component of the spinsof I are a conserved quantity. We consider the case where the system spin is initially in the down state. Therefore thetime dependence of the z component of the single spin that makes up the system is given by [7]〈S (t)〉= Tr(ρ(0)S (t))=N2∑m=− N2λmTr(S m(t)ρm(0))(2)where ρ(0) is the initial quantum density matrix andλm =( NN2 − m)=N!(N2 − m)!(N2 + m)!. (3)In addition 〈S x(t)〉 = 〈S y(t)〉 = 0. Define Ωm =√(mg)2 + ω20. Then one obtains [7]〈S z(t)〉 = − 12N+1N2∑m=− N2λmΩ2m[m2g2 cos (Ωmt) + ω20]. (4)Using the Laplace-de Moivre theorem12N∑mλm ≈√2πN∫ ∞−∞dm exp(−2m2N)(5)gives the asymptotic result for ω0 = 0 and large N, valid for all times t,〈S z(t)〉 = −12exp(−Ng2t28). (6)The quantum purity in our case is defined asP(t) = Tr(ρ2)=12+ 2 〈S z〉2 = 12+12exp(−Ng2t24). (7)Note the factor of 14 in the exponential, compared to Eq. (27) of Ref. [7]. A rederivation of their result [8] gives thecorrect factor in Eq. 7. For a pure quantum state P = 1, and hence the rate of decrease of the purity is a measure ofthe decoherence of the spin of the system.3. ResultsIn Fig. (1a) we show results for 2P(t) − 1 as a function of N t2. The quantum purity was calculated from Eq. (7)and Eq. (4), with g=1. To compare with the asymptotic form we have plotted 2P(t) − 1 versus N t2 rather than as afunction of either the number of spins N in the bath or the time t. Clearly to go to larger values of Nt2 the number ofspins in the bath, N, must increase. There is a limit to the number of bath spins which can be simulated, currently onMarta L. Guerra et al. / Physics Procedia 15 (2011) 33–36 35the largest supercomputers one is restricted to about N ≈ 40. Our calculation results performed for N = 8 and N = 16were indistinguishable from the results of Eq. (4), until numerical rounding error occurred for very small values of2P(t) − 1. In Fig. (1b) we show results for the fractional error due to finite N in 2P(t) − 1 as a function of N. Thefractional error is defined asfractional error =[2P(t) − 1]asymptotic − [2P(t) − 1]calculated[2P(t) − 1]asymptotic . (8)As long as N is large enough the fractional error for fixed Nt2 approaches the exact result as a power law in N. Thecomputer calculations are limited in the number of bath spins N, and hence can reach the power law approach to theasymptotic value only for small values of Nt2.Figure 1: (a) The decay of the quantum purity as a function of the variable N t2. For N = 8 and N = 16 the results from the simulation are the sameas the results shown here from Eq. (7). (b) The fractional error in 2P(t) − 1 as a function of N for three values of Nt2.The computer calculation is limited to values of N of less than about forty. However, unlike the exact calculationthe computer calculations are not limited to the case HB = 0. Therefore it is natural to ask whether or not havinginteractions within the bath can allow a closer approach to the asymptotic limit. This is shown in Fig. (2) for N = 10,20, and 30. The exact calculation is from Eq. (7) and Eq. (4). The label ‘No BI’ is from the program forHB = 0, and isindistinguishable on this scale from the exact results. For the case of interactions within the bath the bath HamiltonianisHB =N∑k=1N∑j=1, jk(gk j,xIxk Ixj + gk j,yIyk Iyj + gk j,zIzkIzj)(9)for the components of the spin 12 operators Ik. Each component of the values of gk j were chosen randomly from adistribution uniformly distributed between chosen bounds [−gmax, gmax]. Fig. (2) shows that there exists a value ofgmax for a given value of Nt2 that brings the value of the purity closer to the asymptotic value. However for a fixedvalue of Nt2 making the random bath interactions too strong hinders the approach with N to the asymptotic limit.4. Conclusions and DiscussionWe presented results for a single quantum spin 12 coupled to a bath of N quantum spin12 particles with a couplingg between the x-components of the spin operators. The comparison of the simulation results with the exact resultsfor no interactions between the bath spins is excellent. Furthermore, for large enough N for a fixed value of Nt2the approach to the exact result is a power law in N. Adding random interactions among the bath spins changes theapproach to the large N limit for a fixed value of Nt2.36  Marta L. Guerra et al. / Physics Procedia 15 (2011) 33–36Figure 2: 2P(t) − 1 is shown as a function of Nt2. The asymptotic result is shown, as is the theoretical and program result for no bath interaction(BI). Also shown are a single realization of the random bath interactions with diﬀerent strengths gmax. The graphs are for N = 10, 20, and 30 fromtop to bottom.A number of questions remain to be addressed. First, is there a mechanism to systematically pick the strengthof the random bath interactions, gmax, to approach the asymptotic limit more quickly for fixed N? Second, canadding interactions among the bath spins also allow a faster approach to the asymptotic limit when there is more thanone bath? Third, how can this approach improve on current simulations of quantum computers? Answers to thesequestions will hopefully be found in the near future.AcknowledgementsThis research was supported in part by the U.S. National Science Foundation through TeraGrid resources providedby TACC Lonestar under grant number DMR100059.[1] M. Le Bellac, A Short Introduction to Quantum Information and Quantum Computing, Cambridge University Press, Cambridge U.K., 2005.[2] M. Nakahara, T. Ohmi, Quantum Computing, From Linear Algebra to Physical Realizations, CRC Press, Boca Raton, FL, 2008.[3] V. V. Dobrovitski, H. De Raedt, An eﬃcient scheme for numerical simulation of the spin-bath decoherence, Phys. Rev. E 67 (056702) (2003)[7 pages].[4] V. V. Dobrovitski, H. De Raedt, Numerical simulations of decoherence suppression in open quantum spin systems, HAIT J. of Science andEngineering 1 (2004) 586–597.[5] K. De Raedt, K. Michielsen, H. De Raedt, B. Trieu, G. Arnold, M. Richter, T. Lippert, H. Watanabe, N. Ito, Massive parallel quantum computersimulator, Comp. Phys. Comm. 176 (2007) 642–651.[6] S. Yuan, M. I. Katsnelson, H. De Raedt, Decoherence by a spin thermal bath: Role of spin-spin interactions and initial state of the bath, Phys.Rev. B 77 (2008) 184301 [8 pages].[7] D. D. Rao, H. Kohler, F. Sols, Quantum frustration of dissipation by a spin bath, New Journal of Physics 10 (115017) (2008) [17 pages].[8] M. L. Guerra, Dynamics of classical particles and quantum spin systems, Ph.D. thesis, Mississippi State University (2011).

Real-Time Quantum Dynamics of a Single Spin 1/2 Coupled to a Spin Bath

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Real-Time Quantum Dynamics of a Single Spin 1/2 Coupled to a Spin Bath

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