Quantum Bocce: Magnon–Magnon Collisions Between Propagating and Bound States in 1D Spin Chains ∗ Paolo Longo A, ,1, Andrew D
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Physics Letters A 377 (2013) 1242–1249 Contents lists available at SciVerse ScienceDirect Physics Letters A www.elsevier.com/locate/pla Quantum Bocce: Magnon–magnon collisions between propagating and bound states in 1D spin chains ∗ Paolo Longo a, ,1, Andrew D. Greentree b, Kurt Busch c,d,JaredH.Colee a Institut für Theoretische Festkörperphysik, Karlsruher Institut für Technologie (KIT), 76131 Karlsruhe, Germany b Applied Physics, School of Applied Sciences, RMIT University, Melbourne 3001, Australia c Humboldt-Universität zu Berlin, Institut für Physik, AG Theoretische Optik & Photonik, Newtonstr. 15, 12489 Berlin, Germany d Max-Born-Institut, Max-Born-Str. 2A, 12489 Berlin, Germany e Chemical and Quantum Physics, School of Applied Sciences, RMIT University, Melbourne 3001, Australia article info abstract Article history: The dynamics of two magnons in a Heisenberg spin chain under the influence of a non-uniform magnetic Received 26 September 2012 field is investigated by means of a numerical wave-function-based approach using a Holstein–Primakoff Received in revised form 19 February 2013 transformation. The magnetic field is localized in space such that it supports exactly one single-particle Accepted 5 March 2013 bound state. We study the interaction of this bound mode with an incoming spin wave and the interplay Available online 15 March 2013 between transmittance, energy and momentum matching. We find analytic criteria for maximizing the Communicated by P.R. Holland interconversion between propagating single-magnon modes and true propagating two-magnon states. The Keywords: manipulation of bound and propagating magnons is an essential step towards quantum magnonics. Magnons © 2013 Elsevier B.V. All rights reserved. Spin-chains Quantum magnonics Wave packet propagation Magnon–magnon interactions 1. Introduction range of analogous physical realizations whose properties resemble those of a Heisenberg model even though “true” magnetic interac- Spin waves are a fundamental concept in solid-state physics tions are absent. One such example is ultra-cold atoms trapped in which dates back to the early days of quantum mechanics [1–4]. optical lattices that can be designed to emulate the dynamics of For many decades they have been studied as part of the ongoing quantum spin systems [11], providing the advantage of single-site efforts to understand the properties of magnetic materials. More addressability and optical readout. Moreover, atom–cavity arrays recently the study of spin networks or spin chains [5] has taken on are believed to be another route towards a large-scale realization a new and exciting context. Progress in controllable quantum sys- of so-called quantum emulators that mimic the dynamics of a spin tems and quantum information processing coupled with advances chainincertainlimits[12–15], as well as excitons in semiconduc- in nano-fabrication are leading towards spin networks which can tor self-assembled quantum dots [16] or chains of superconducting be fabricated and controlled at will. In much the same way as fab- qubits [17]. Furthermore, spin waves share many properties with rication of nanostructures and low-dimensional electronic devices light pulses in optical waveguides [18,19] and display soliton solu- led to ‘quantum electronics’ [6–8], understanding and controlling tions obtained in the continuum limit of the classical Heisenberg quantized spin excitations will lead to ‘quantum magnonics’ [9]. model [20,21]. Although throughout this Letter we use the lan- The Heisenberg model is one of the corner stones in the the- guage of magnons and spin physics, many of our results apply oretical description of classical and quantum magnetism [10] and directly to these ‘Heisenberg model analogy’ systems. has become indispensable in the field of condensed matter physics As demonstrated in [18,22], a time-dependent external mag- in general. Besides the applicability in the context of ferromag- netic field applied to a one-dimensional Heisenberg spin chain netism in conventional bulk materials, the enormous success of the allows one to deterministically transfer a single magnon by storing Heisenberg model can (at least partly) be attributed to the wide it in the ground state of the moving potential. Here, we investi- gate the properties of two-magnon systems to shed light on the dynamics of interacting magnons in the discrete one-dimensional Heisenberg model under the influence of a static external poten- * Corresponding author. E-mail address: [email protected] (P. Longo). tial. Our results are obtained by means of a wave-function-based 1 Present address: Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 numerical framework originally developed in the context of few- Heidelberg, Germany. photon quantum optics [23,24]. 0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.03.020 P. Longo et al. / Physics Letters A 377 (2013) 1242–1249 1243 This Letter is organized as follows. We begin by reviewing context of the time-evolution of two-magnon wave packets. In this the theoretical foundations relevant to this Letter in Section 2. Letter, we therefore focus on the Holstein–Primakoff approach. This includes the formulation of a Heisenberg Hamiltonian and its transformation to a form suitable for subsequent consideration. In 2.2. Holstein–Primakoff transformation Section 3, we introduce the exact two-body eigenstates and ener- gies of a one-dimensional S = 1/2-Heisenberg spin chain in the Following [10],weset absence of an external magnetic field. We identify two classes of S z = S − n , (5) states — two-body scattering states and propagating pairs of bound i √ i magnons. Then, in Section 4, we present the central setup of this + S = 2Sφ(ni)ai, (6) Letter. We explain the initial state consisting of a spin wave im- i √ − pinging on a magnon which is stored in the ground state of the = † Si 2Sai φ(ni), (7) external potential. The spin wave’s initial momentum is related to = † † the bound state energy of the potential and adjusted in such a way where ni ai ai is the usual number operator, ai (ai )arebosonic that the transfer of the initial excitation to propagating magnon– creation (annihilation) operators, and magnon pairs is efficient. This extraction of excitation from the po- tential is then investigated numerically in Section 5, where we also ni φ(ni) = 1 − (8) determine the single-particle transport properties. We find that the 2S underlying physical mechanism allows for an interaction-induced is a nonlinear operator function. The original spin algebra thus re- interconversion of bound and propagating magnons. sults in a system of interacting bosons (magnons). The nonlinearity in Eq. (8) usually renders an analytical and a numerical solution to 2. Fundamentals the general many-body problem intractable. A common strategy is to formally expand Eq. (8) in orders of the normal-ordered number In this section, we formulate the Hamiltonian of a one- operator [10], i.e., dimensional Heisenberg spin chain under the influence of an ex- ternal magnetic field. We then transform the Hamiltonian to a 1 2 φ(ni) = 1 − 1 − 1 − ni + O :n : , (9) system of interacting bosons by means of a Holstein–Primakoff 2S i transformation. The resulting Hamiltonian is exact in the single- :·: 2 = + † † and two-excitation subspace. where denotes normal ordering (note that ni ni ai ai aiai and : 2:= † † ni ai ai aiai ). Usually, the so-called low-temperature approxima- 2.1. The Hamiltonian tion ni /2S 1isappliedandonlyafewlowerordertermsof Eq. (9) are taken into account. The Heisenberg–Hamiltonian of a one-dimensional spin chain In the following, we restrict our investigations to the dynam- in the presence of an external magnetic field Bi reads ics of at most two magnons in the system. Hence, terminating the expansion in Eq. (9) after the first order results in an exact Hamil- H =− · − · J ijSi S j Bi Si, (1) tonian for the single- and two-excitation subspace. Furthermore, i= j i we focus on a chain of S = 1/2-spins. By neglecting all constant y energy terms and on-site terms we arrive at where S = (Sx, S , S z) is the total spin operator for a spin S on i i i i lattice site i and J ij denotes the exchange coupling between sites † † † H =− a a + + a a − + V ia a + n n + + n n − i and j. We assume throughout our discussion that there exists a i i 1 i i 1 i i i i 1 i i 1 i background field BDC which imparts a uniform Zeeman shift across − † − † − † − † the chain so that we may ignore thermal effects, i.e., |BDC|kT. ai+1niai ai−1niai ai+1ni+1ai ai−1ni−1ai . (10) ± y We introduce ladder operators according to S = Sx ± iS and as- i i i † By performing an index shift and exploiting the identities a n a sume nearest-neighbor hopping, i.e., J ij = J(δi, j+1 + δi, j−1). Then, i+1 i i the Hamiltonian (in units of J ) becomes = † − † † = † − † ai+1aini ai+1ai and ai ni+1ai+1 ai ai+1ni+1 ai ai+1 Hamilto- H nian (10) can be recast into + − − + z z z H = =− S S + + S S + + 2S S + + V i S , (2) J i i 1 i i 1 i i 1 i =− † + † − − i H ai+1ai ai ai+1 (2 ni ni+1) i where V i is the magnetic field (in units of J ) at lattice site i.For + + the remainder, lengths are measured in units of the lattice constant 2nini+1 V ini . (11) a ≡ 1. Because the spin operators obey the algebra (h¯ ≡ 1) This Hamiltonian describes bosons with nearest-neighbor interac- tion in the presence of an external potential. z ± ± S , S =±δijS , (3) i j i + − z 3.