Quantum Bocce: Magnon–magnon collisions between propagating and bound states in 1D spin chains
Introduction
Spin waves are a fundamental concept in solid-state physics which dates back to the early days of quantum mechanics [1], [2], [3], [4]. For many decades they have been studied as part of the ongoing efforts to understand the properties of magnetic materials. More recently the study of spin networks or spin chains [5] has taken on a new and exciting context. Progress in controllable quantum systems and quantum information processing coupled with advances in nano-fabrication are leading towards spin networks which can be fabricated and controlled at will. In much the same way as fabrication of nanostructures and low-dimensional electronic devices led to ‘quantum electronics’ [6], [7], [8], understanding and controlling quantized spin excitations will lead to ‘quantum magnonics’ [9].
The Heisenberg model is one of the corner stones in the theoretical description of classical and quantum magnetism [10] and has become indispensable in the field of condensed matter physics in general. Besides the applicability in the context of ferromagnetism in conventional bulk materials, the enormous success of the Heisenberg model can (at least partly) be attributed to the wide range of analogous physical realizations whose properties resemble those of a Heisenberg model even though “true” magnetic interactions are absent. One such example is ultra-cold atoms trapped in optical lattices that can be designed to emulate the dynamics of quantum spin systems [11], providing the advantage of single-site addressability and optical readout. Moreover, atom–cavity arrays are believed to be another route towards a large-scale realization of so-called quantum emulators that mimic the dynamics of a spin chain in certain limits [12], [13], [14], [15], as well as excitons in semiconductor self-assembled quantum dots [16] or chains of superconducting qubits [17]. Furthermore, spin waves share many properties with light pulses in optical waveguides [18], [19] and display soliton solutions obtained in the continuum limit of the classical Heisenberg model [20], [21]. Although throughout this Letter we use the language of magnons and spin physics, many of our results apply directly to these ‘Heisenberg model analogy’ systems.
As demonstrated in [18], [22], a time-dependent external magnetic field applied to a one-dimensional Heisenberg spin chain allows one to deterministically transfer a single magnon by storing 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 potential. Our results are obtained by means of a wave-function-based numerical framework originally developed in the context of few-photon quantum optics [23], [24].
This Letter is organized as follows. We begin by reviewing the theoretical foundations relevant to this Letter in Section 2. This includes the formulation of a Heisenberg Hamiltonian and its transformation to a form suitable for subsequent consideration. In Section 3, we introduce the exact two-body eigenstates and energies of a one-dimensional -Heisenberg spin chain in the absence of an external magnetic field. We identify two classes of states — two-body scattering states and propagating pairs of bound magnons. Then, in Section 4, we present the central setup of this Letter. We explain the initial state consisting of a spin wave impinging on a magnon which is stored in the ground state of the external potential. The spin waveʼs initial momentum is related to the bound state energy of the potential and adjusted in such a way that the transfer of the initial excitation to propagating magnon–magnon pairs is efficient. This extraction of excitation from the potential is then investigated numerically in Section 5, where we also determine the single-particle transport properties. We find that the underlying physical mechanism allows for an interaction-induced interconversion of bound and propagating magnons.
Section snippets
Fundamentals
In this section, we formulate the Hamiltonian of a one-dimensional Heisenberg spin chain under the influence of an external magnetic field. We then transform the Hamiltonian to a system of interacting bosons by means of a Holstein–Primakoff transformation. The resulting Hamiltonian is exact in the single- and two-excitation subspace.
Exact two-body eigenstates in the absence of an external potential
In the absence of an external potential (), the two-excitation eigenstates for the Hamiltonian (10) can be calculated analytically (see [28], [29] and Appendix A). The wave function for an eigenstate can be decomposed into a centre-of-mass wave function (coordinate , momentum K) and a wave function in the relative coordinate (), i.e., .
To form a complete basis of the two-excitation subspace, two classes of solutions need to be
External potential induced magnon–magnon state population
In the absence of an external potential, the Hamiltonian (11) already exhibits an interesting interaction-induced effect — the existence of bound magnon–magnon pairs. However, there is no mechanism of transferring excitation from the scattering states to the bound states or vice versa. We therefore introduce localized single-particle bound states by virtue of an external potential, e.g., an external magnetic field. We exploit the discrete levels introduced by the potential to connect the
Dynamics
In the following, we study the wave packet dynamics with the help of our numerical framework developed in the context of few-photon transport [23], [37], [24]. Time is measured in units of J−1 and the time-dependent Schrödinger equation is dimensionless, i.e., . All simulations are performed for a system with lattice sites and the initial condition (19) with parameters as described in Sections 4.2 Initial states, 4.3 Parameters for maximized overlap. We choose as the
Conclusion and outlook
We numerically analyzed the interaction-induced conversion of single bound and propagating magnons into a propagating two-magnon state. In essence, we probed the localized bound state with an impinging spin wave whose parameters were adjusted in such a way that the initial excitation can be transferred efficiently to propagating pairs of bound magnons. Specifically, we related the spin waveʼs initial momentum to the bound state energy. The width of the potential was chosen such that it supports
Acknowledgements
We thank Melissa Makin for helpful discussions. P.L. acknowledges financial support by the Karlsruhe House of Young Scientists (KHYS) for his stay abroad at the RMIT University, Melbourne. The Ph.D. education of P.L. is embedded in the Karlsruhe School of Optics and Photonics (KSOP). P.L. and K.B. acknowledge financial support by the DFG within the priority programme SPP 1391 “Ultrafast Nanooptics” (grant BU 1107/7-1). A.D.G. acknowledges the Australian Research Council for financial support
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Present address: Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany.