Quantum Bocce: Magnon–Magnon Collisions Between Propagating and Bound States in 1D Spin Chains ∗ Paolo Longo A, ,1, Andrew D

Physics Letters A 377 (2013) 1242–1249

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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 investigate 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].

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 energies 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 potential 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 external 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 interaction in the presence of an external potential. z ± ± S , S =±δijS , (3) i j i + − z 3. Exact two-body eigenstates in the absence of an external S , S = 2δijS , (4) i j i potential they are neither bosonic nor fermionic. However, the Hamilto- nian (2) can be transformed to a system of bosons or fermions by In the absence of an external potential (V i = 0), the two- means of a Holstein–Primakoff [10,25] or a Jordan–Wigner trans- excitation eigenstates for the Hamiltonian (10) can be calculated formation [26,27], respectively. The latter is only applicable for a analytically (see [28,29] and Appendix A). The wave function for † † system with S = 1/2, whereas the Holstein–Primakoff transforma- an eigenstate |Ψ = φ a a |0 can be decomposed into a x1x2 x1x2 x1 x2 tion is exact for arbitrary spin S [10,25]. centre-of-mass wave function (coordinate c = (x1 + x2)/2, momen- For S = 1/2, the Hamiltonian (2), when mapped onto a sys- tum K ) and a wave function in the relative coordinate (r = x1 −x2), = iKc tem of fermions, is a one-dimensional tight-binding chain with i.e., φx1x2 e Ψr . nearest-neighbor interactions. However, the underlying Jordan– To form a complete basis of the two-excitation subspace, two Wigner transformation is nonlocal, which is inconvenient in the classes of solutions need to be considered: scattering and bound 1244 P. Longo et al. / Physics Letters A 377 (2013) 1242–1249

of states. In the following, we shift the energy of the free dispersion relations such that the lowest energy (in the absence of a potential) is zero, i.e., we set ωk → ωk + 2, Kp → Kp + 4 and bound → bound + K K 4 in the previous expressions.

4.1. Bound states of the Pöschl–Teller potential

In principle, the effects we investigate in the following do not depend on the actual spatial profile of the external potential as long as the potential is such that it supports at least one bound state. We therefore choose a potential for which the analytical bound-state solution is available in the continuum limit [33].One Fig. 1. (Color online.) Eigenenergies of the two-particle scattering states (solid black smooth potential of this kind is the Pöschl–Teller potential [34–36] lines) and the propagating magnon–magnon bound states (dashed red line). The with black lines correspond to different equidistant values of the relative momentum p from the first Brillouin zone (see Eq. (14)). The energies of the scattering states and − 2 x x the propagating magnon–magnon bound states never cross. V (x) =−B0 sech , (17) w states. For both classes, Ψ = 0 holds, which indicates that the two- 0 where B0 is the depth, w the width, and x the centre of the po- particle excitations of Hamiltonian (11) are hard-core bosons. The tential. scattering states are described by According to [36] and adapted to our notation and parameters, | | − | | theboundstateenergiesare = = ip r + iδKp ip r ∀ = Ψr Ψ−r e e e r 0, (12) 1 BS =− − + + + 2 2 where δ is the scattering phase shift given by En (1 2n) 1 4B0 w . (18) Kp 4w2 ip = BS K + 2e The bound states are labeled by n 0, 1, 2,...,nmax.SinceE < 0, eiδKp =− (13) n −ip there is only a finite number of supported bound states. For sim- K + 2e plicity, we restrict ourselves to the situation where the potential =− and K 2 cos(K /2) is the dispersion relation of the centre-of- supports exactly one bound state. The condition EBS = 0with √ nmax mass. The relative momentum is denoted by p. The eigenenergy of nmax = 1yieldsw = 2/B0. this two-particle state reads (see Fig. 1) In the discrete system, the Pöschl–Teller potential is only defined at the lattice sites, i.e., V = V (x). For the remainder of this = = · x Kp K 2cos(p). (14) Letter,wechooseafinitesystemwithN lattice sites where the = The magnon–magnon bound states have a relative wave func- potential is located in the centre, i.e., x N/2(N is even). tion of the form 4.2. Initial states |r| K Ψr = Ψ−r = − ∀r = 0 (15) 2 In all discussions below, we choose the initial state of the system to be a two-magnon state which is a symmetrized product of with eigenenergy (Fig. 1) single-particle wave functions according to bound 1 2 | = † † | = =−2 − . (16) Ψ φx1x2 ax ax 0 , (19) K 2 K 1 2 x1x2 In the single-excitation subspace, the eigenstates are plane = √1 + waves with momentum k and an eigenenergy given by the disper- φx1x2 (ϕx1 χx2 ϕx2 χx1 ), (20) 2 sion relation of the tight-binding lattice, i.e., = ωk =−2 cos(k). − 2 (x x0) Note that ignoring the nearest-neighbor interaction terms in − ik x ϕ ∝ e 2s2 e 0 , (21) Hamiltonian (11) removes the magnon–magnon bound states from x the spectrum since a particle–particle interaction is required for where ϕx is a single-particle Gaussian wave function — a spin wave the formation of bound states. Conversely, ignoring those hopping — with initial centre x0,widths, and momentum k0. The ground- terms which depend on the occupation number (last two lines in state wave function of the Pöschl–Teller potential is denoted by χx Eq. (10)) results in modified wave functions for which Ψ0 = 0is and is obtained by means of an exact numerical diagonalization of allowed. Such a system does not describe hard-core bosons as re- the finite system in the single-excitation subspace. In the follow- quired for a Heisenberg chain, but still modified magnon–magnon ing, we utilize the numerical scheme as outlined and applied in bound states exist (see [30–32] for similar calculations). the context of few-photon scattering [23,24,37]. The incoming spin wave is launched from the left-hand side of the potential with a 4. External potential induced magnon–magnon state population carrier momentum k0 > 0. A schematic of the setup with two excitations is depicted in Fig. 2. In the absence of an external potential, the Hamiltonian (11) The goal in the following is to extract as much excitation already exhibits an interesting interaction-induced effect — the as possible from the “stored” ground state wave function and existence of bound magnon–magnon pairs. However, there is no to transfer it to propagating two-particle magnon–magnon bound mechanism of transferring excitation from the scattering states to states. To this end, the overlap of the initial state with the potential the bound states or vice versa. We therefore introduce localized target states in the Hilbert space needs to be maximized. Alterna- single-particle bound states by virtue of an external potential, e.g., tively, the situation where the stored excitation is transferred to an external magnetic field. We exploit the discrete levels intro- the two-particle scattering states could also be investigated. How- duced by the potential to connect the hitherto separated classes ever, this is issue is not addressed in this Letter. P. Longo et al. / Physics Letters A 377 (2013) 1242–1249 1245

Fig. 2. (Color online.) Schematic of the setup. The Pöschl–Teller potential (indicated by the dashed violet curve) is initialized with the single-particle ground state wave function (red curve). A spin wave in the form of a Gaussian wave packet (black curve) is launched towards the potential.

4.3. Parameters for maximized overlap Fig. 3. (Color online.) Ground state energy of the Pöschl–Teller potential√ as a function of the potential depth B0 (bottom axis). The potential width is w = 2/B0 (top In a ﬁrst step, we obtain a relation between the single-particle axis). The solid black line corresponds to values obtained by an exact numerical di- = input momentum k0 and the two-particle centre-of-mass output agonalization of the discrete system with N 400 lattice sites. The dashed blue line =− momentum K as a function of the bound-state energy b < 0of refers to the analytical result for the continuum case where b B0/2. Deviations occur when the potential is narrow, i.e., its width is comparable to or smaller than the potential. To this end, we consider global energy conservation. the lattice spacing (which is unity in our units). The green dashed line shows the For the transfer of the initial excitation to magnon–magnon bound lower bound for allowed bound state energies as imposed by constraint (27). states (see Eq. (16)), we require As mentioned above, we focus in this Letter on the transfer ω + b = bound, (22) k0 K of the initial excitation to the magnon–magnon bound states. We which results in therefore use Eq. (27) to initialize the momentum of the incoming spin wave. To ensure that the overlap with the scattering states b does not seriously affect the transfer to bound magnon–magnon K = 2 arccos ± cos(k0) − . (23) 2 pairs, we equate Eqs. (27) and (28), yielding

| − | Note that condition (23) is only defined when cos(k0) b/2 1 4 + b 1, which is already a weak constraint for possible combinations of cos(p) = √ √ . (29) 2 2 2 + b ground state energies and input momenta. Similarly, the transfer of excitation to the scattering states (see Here, a real-valued solution for the relative momentum p ex- Eq. (14))requires ists only for b = 0, i.e., in the absence of a potential (this can be verified by demanding that cos2(p) 1). Hence, the condi- + = ωk0 b Kp, (24) tions (27) and (28) are incompatible. After scattering, all excitation so that is therefore either transferred to magnon–magnon bound states or remains in the form of a scattered spin wave and a localized exci- 1 b tation in the potential. K = 2 arccos cos(k0) + 1 − . (25) 2cos(p) 2 Note that Eq. (25) depends on the relative momentum p of the 5. Dynamics two-body scattering states to which the excitation is transferred. To maximize the overlap of the initial state with the target In the following, we study the wave packet dynamics with states, we enforce the magnitudes of the centre-of-mass momenta the help of our numerical framework developed in the context of few-photon transport [23,37,24].Timeismeasuredinunitsof before and after scattering to be equal, i.e., − J 1 and the time-dependent Schrödinger equation is dimension- | = | |k0 + 0|=|K |. (26) less, i.e., i∂t Ψ H Ψ . All simulations are performed for a system with N = 400 lattice sites and the initial condition (19) with pa- Then,wearriveat = rameters as described in Sections 4.2 and 4.3. We choose x0 36 = b as the initial centre of the spin wave. A width of s 12 ensures k0 = 2arcsin − (27) that the wave packet’s width in momentum space only covers a 2 small spectral window when compared to the full bandwidth of for the transfer of the initial excitation to magnon–magnon bound the single-particle cosine dispersion relation (which is 4 in units states (see Eq. (23)), whereas we need to choose of J). 1 2 k0 = 2 arccos cos(p) ± b + cos (p) (28) 5.1. Single-particle transmittance 2 in the case of the scattering states (see Eq. (25)). In Eqs. (27) So far, we only discussed the effect of the external potential in and (28), we only accounted for positive initial momenta k0 so that terms of the existence of an additional discrete level — the bound the initial wave packet is a right-moving spin wave. These equa- state. However, the potential also affects the two-particle eigen- tions determine the initial momentum of the incoming spin wave states discussed in Section 3. In general, incoming plane waves as a function of the bound state energy of the potential. Note that are partly reflected and partly transmitted. In order to better −2 b < 0 holds according to condition (27).Asanestimate,the understand the two-particle dynamics, we first obtain the trans- BS =− bound state energy can be approximated by E0 B0/2, which mission characteristics for a single spin wave impinging on the is only exact in the continuum limit [36]. In the numerical simu- empty Pöschl–Teller potential. Interestingly, the Pöschl–Teller po- lations, we use the exact ground state energy b of the potential in tential belongs to a class of reflectionless potentials [38].However, the finite system (see Fig. 3). the property of unity transmittance only exists in the continuous 1246 P. Longo et al. / Physics Letters A 377 (2013) 1242–1249

† = (λ) † Then, by employing the change of basis aλ x ux ax, the quantity ∗ † (λ) (λ) † = = nλ aλaλ ux ux axax (30) xx is the occupation of the single-particle state |λ. Consequently, the amount of extracted excitation is given by

= = − = η nλ0 (t 0) nλ0 (t tend) = − = 1 nλ0 (t tend). (31) At the beginning of the simulation, the potential is initialized in | = ˆ = = its bound state λ0 b so that nλ0 (t 0) 1. The time at the end of the simulation is denoted by tend, which is long enough after the incoming spin wave scattered at the potential, but before the boundaries of the computational domain corrupt the results (cf. [23,24] for details). Fig. 4(b) displays the amount of extracted excitation from the potential as a function of the potential depth B0. The actual shape of the curve is a result of the interplay of the incoming spin wave’s reflected and transmitted parts (Fig. 4(a)) and the non-zero group- velocity dispersion which affects the shape of the wave packet. Those regions in parameter space where the potential induces ei- Fig. 4. (Color online.) (a) Single-particle transmittance through the Pöschl–Teller po- ther unity transmittance or reflectance belong to the nonlinear tential obtained from a time-dependent transport simulation for an initial Gaussian regime of the cosine dispersion. In other words, the group-velocity spin wave with parameters s = 12 and x0 = 36. The carrier momentum k0 and the potential depth B0 are varied. The dashed line corresponds to pairs of B0 and k0 dispersion causes the wave packet to spread out rapidly so that its for which constraint (27) holds. This constraint assures momentum matching of two maximum is significantly decreased when arriving at the poten- magnons (see Section 5.2). (b) Amount of excitation extracted from the potential tial. This causes an inefficient extraction of the bound state (see according to Eq. (31) as a function of the potential depth B0 with k0 varied ac- B0 < 0.82 and B0 > 3.14 in Figs. 4(a) and 4(b)). Conversely, under cording to the optimal solution given by the dashed line in panel (a). Under the the constraints imposed, those regions in parameters space where constraints imposed, B0 0.82 represents an optimal set of parameters for conversion to a magnon–magnon bound state propagating in the forward direction and the influence of the cosine dispersion’s nonlinearity is minimal similarly B0 3.14 is optimal for a reflected bound state (corresponding values in (around k0 = π/2) are subject to both significant reflection and (a) are indicated by crosses). transmission, resulting only in a moderate extraction efficiency. In between these two regimes, we can find the maxima of the extrac- system for waves with special constraints. Since the present system tion efficiency at B0 = 0.82 and B0 = 3.14. At these points, a sin- is discrete and we do not want to impose any special restriction gle spin wave is either almost completely transmitted or reflected on the incoming spin wave (besides being a Gaussian wave packet (Fig. 4(a)) whilst the influence of the nonlinear dispersion relation with a defined momentum), the potential causes non-zero reflec- is moderate. In all cases, the extraction efficiency is smaller than tion. unity, which is a consequence of the fact that the wave packets ex-

Fig. 4(a) displays the transmittance Tk0 (B0) through the poten- perience a finite dwell time, i.e., a finite interaction time with the tial. The quantity Tk0 (B0) is defined as the sum of all occupation “stored” bound state in the potential. numbers on the right-hand side of the system at a time after In Fig. 5, we display a series of “snapshots” of the wave packet = N the wave packet passed the potential, i.e., Tk0 (B0) i=N/2+1 ni . dynamics in real space for B0 = 0.82. In order to demonstrate the Here, the potential depth B0 and the input momentum k0 are importance of the interaction terms of Hamiltonian (11),thedy- varied independently.√ Note that the parameter for the potential’s namics of an interaction-free tight-binding chain is also plotted width is w = 2/B0. The dashed line in the color plot signifies for comparison. In the latter case, the two excitations pass each pairs of B0 and k0 for which constraint (27) holds. All following other without any effect besides wave interference. In the absence two-particle simulations correspond to such pairs of B0 and k0. of the interaction, the spin wave passes through the potential as Hence, B0 is the only free parameter left (according to Eq. (27), k0 if it was empty. The delay induced by the interaction terms, i.e., depends on the ground state energy b which is a function of B0). the presence of a magnon in the potential, can be interpreted as a signature for an occupied potential. In addition, Fig. 6 displays space–time plots of the transport process with and without 5.2. Interaction-induced extraction of the bound-state and transfer to interaction. The slopes of the “rays” leaving the potential region propagating magnon–magnon bound states after scattering clearly indicate that the excitation is transferred to states with different group velocities. In order to further visualize the extraction of the stored Now, we discuss the physical mechanism central to this Letter magnon, we depict the momentum distribution, i.e., the Fourier — the interaction-induced extraction of excitation from the poten- transform of the two-particle real-space wave function [24,23] in tial and its transfer to propagating magnon–magnon bound states. Fig. 7. In line with the analytical considerations in Section 4.3,the To quantify the efficiency of extraction, we find it advantageous incoming spin wave’s momentum (indicated by red lines) is trans- to transform to the number basis of the single-particle energy ferred to a centre-of-mass momentum according to Eqs. (23), (26), eigenstates (which are obtained by means of exact numerical di- (27) (indicated by black dashed lines). Furthermore, the extracted (λ) agonalization). Let ux be the (real space) wave function of the excitation is centred around zero relative momentum (indicated by single-particle energy eigenstate with eigenenergy λ in the pres- the blue line), which demonstrates that the two particles leave the | = | | | = (λ) ence of the external potential, i.e., H λ λ λ and 0 ax λ ux . potential as a magnon–magnon bound state. The transfer of the P. Longo et al. / Physics Letters A 377 (2013) 1242–1249 1247

Fig. 5. (Color online.) Expectation values of the occupation numbers in real space at different instances of time for B0 = 0.82, i.e., for the set of parameters representing the most eﬃcient extraction under the constraints imposed. The solid black line denotes the wave packet dynamics for the full Hamiltonian (11), whereas the dashed blue line refers to an interaction-free system of a tight-binding chain. The delay induced by the interaction terms, i.e., the presence of excitation in the potential, depends on whether or not a magnon was initially stored in the potential.

Fig. 6. (Color online.) Space–time plot of the occupation numbers in real space for B0 = 0.82. In the left panel, the dynamics according to Hamiltonian (11) is shown, whereas the right panel corresponds to the system of an interaction-free tight-binding chain. Note the different slopes of the “rays” after scattering, corresponding to different group velocities. The magnon–magnon interaction induces a delay for the transmitted spin wave when compared to the interaction-free case. In the latter, the dynamics of the spin wave is not affected by the existence of a bound state. stored excitation to scattering states can be excluded since scat- width of the potential was chosen such that it supports exactly one tering states with a relative momentum of zero would require a bound state. We furthermore determined and discussed the extrac- centre-of-mass momentum different from the values appearing in tion efficiency as a function of the potential depth. In addition, we Fig. 7 (this can be verified by setting p = 0inEq.(25)). This con- illustrated the wave packet dynamics in real space by monitoring firms the conclusion of Eq. (29), stating that, under the parameters the time evolution of the occupation numbers. chosen, an efficient transfer to both magnon–magnon bound states A variety of extensions and modifications to our work presented and scattering states is not realizable. in this Letter can be envisioned for future investigations. In partic- ular, as the efficiency of the creation of propagating two-magnon 6. Conclusion and outlook states with a single spin wave is less than unity, one can imag- ine a sequence of probe pulses to overcome this limitation. In We numerically analyzed the interaction-induced conversion of analogy to optics, by controlling the actual shape of the incom- single bound and propagating magnons into a propagating two- ing spin wave, one should gain further control over the extraction magnon state. In essence, we probed the localized bound state efficiency. In the spirit of [18], time-dependent external fields add with an impinging spin wave whose parameters were adjusted in a novel twist to the dynamics of interacting magnons and could such a way that the initial excitation can be transferred efficiently open up a variety of interesting physical mechanisms including a to propagating pairs of bound magnons. Specifically, we related form of magnonic molecular physics. Although precision engineer- the spin wave’s initial momentum to the bound state energy. The ing of inter-spin coupling poses considerable technical challenges, 1248 P. Longo et al. / Physics Letters A 377 (2013) 1242–1249

−1 Fig. 7. (Color online.) Momentum distribution of the two-magnon wave function before (t = 0) and after (t = 180 J ) scattering at the potential for B0 = 0.82. The red lines mark the value of the spin wave’s initial momentum, whereas the black dashed lines correspond to centre-of-mass momenta after scattering according toconstraint(26). Zero relative momenta are marked by the blue diagonal line. After scattering, the extracted excitation mainly propagates with zero relative momentum and a centre-of-mass momentum which is equal to the initial momentum of the spin wave (each magnon carries half the momentum of the initial spin wave). engineered hopping amplitudes would in principle allow for a per- A.1. Eigenproblem and reduction to an effective single-particle problem fect state transfer [5] and minimize the effect of group-velocity dispersion. In general, more complicated scenarios involve more Using Eqs. (A.1), the eigenproblem yields than two excitations for which higher-order terms in the expansion of the operator function (8) need to be taken into account. More- 0 =−(φx +1x + φx −1x + φx x +1 + φx x −1) over, a more realistic description of spin chain systems is needed 1 2 1 2 1 2 1 2 − to address the issue of decoherence, i.e., relaxation and dephasing φx1x2 of individual spins, which will be critical for practical implementa- − 2 − + − tions. (δx1x2 1 δx1 1x2 )φx1x2 + + 2(δx11x2 φx1−1x2 δx1+1x2 φx1+1x2 ) Acknowledgements + + + + δx1x2 (φx1−1x2 φx1x2−1 φx1x2+1 φx1+1x2 ) (A.2)

We thank Melissa Makin for helpful discussions. P.L. acknowl- for the wave function coefficients. The introduction of a centre- edges financial support by the Karlsruhe House of Young Sci- of-mass coordinate c ≡ (x1 + x2)/2 and a relative coordinate r ≡ entists (KHYS) for his stay abroad at the RMIT University, Mel- x1 − x2 transforms Eq. (A.2) to bourne. The Ph.D. education of P.L. is embedded in the Karlsruhe School of Optics and Photonics (KSOP). P.L. and K.B. acknowledge =− ˜ + ˜ + ˜ + ˜ 0 (φc+ 1 r+1 φc− 1 r−1 φc+ 1 r−1 φc− 1 r+1) financial support by the DFG within the priority programme SPP 2 2 2 2 1391 “Ultrafast Nanooptics” (grant BU 1107/7-1). A.D.G. acknowl- ˜ − φcr edges the Australian Research Council for financial support (project ˜ No. DP0880466). − 2(δr1 + δr−1)φcr + ˜ + ˜ 2(δr1φc− 1 0 δr−1φc+ 1 0) Appendix A. Two-body eigenstates 2 2 ˜ ˜ ˜ ˜ + δr0(φ − 1 − + φ − 1 + φ + 1 − + φ + 1 ), (A.3) c 2 1 c 2 1 c 2 1 c 2 1 In this appendix, we derive the exact two-body eigenstates of = ˜ Hamiltonian (10) in the absence of an external potential (V i 0). where φcr denote the wave function coefficients in the new coor- − | = | = In order to solve the eigenproblem (H ) Ψ 0with Ψ dinates. Next, we separate the centre-of-mass motion in order to † † φ a a |0, the action of the different terms in the Hamil- arrive at an effective single-particle problem. The ansatz x1x2 x1x2 x1 x2 tonian on the basis states needs to be calculated, i.e., ˜ = iKc · = φcr e Ψr ,Ψr Ψ−r (A.4) † † † † † † † a a a a |0= δx na a + δx na a |0, m n x1 x2 2 m x1 1 m x2 yields the single-particle problem † † † † † † a a a a a a |0=2 a a |0 i+1 i i i x1 x2 δx1iδx2i i+1 i , 0 =−K (Ψr+1 + Ψr−1) † † † † † † a a a a a a |0=2 a a |0 i−1 i i i x1 x2 δx1iδx2i i−1 i , + Ψr † † † † † † a a a a a a |0= + + + a a |0 i+1 i+1 i+1 i x1 x2 (δx1iδx2i 1 δx2iδx1i 1) i+1 i+1 , + 2Ψ1(δr1 + δr−1) † † † † † † K K a a a − a a a |0=(δ δ − + δ δ − )a a |0, −i i i−1 i−1 i 1 i x1 x2 x1i x2i 1 x2i x1i 1 i−1 i−1 − 2Ψ0 δr1e 2 + δr−1e 2 † † † † † † | = + + + | + amamananax1 ax2 0 (δx1iδx2i 1 δx1i 1δx2i)aman 0 , (A.1) 2Ψ1K δr0, (A.5) where n and m are arbitrary indices. where K =−2 cos(K /2). P. Longo et al. / Physics Letters A 377 (2013) 1242–1249 1249

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