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Unitary quantum lattice simulations for Maxwell equations in vacuum and in dielectric media

Published online by Cambridge University Press:  28 October 2020

George Vahala Open the ORCID record for George Vahala [Opens in a new window] ,
Linda Vahala ,
Min Soe  and
Abhay K. Ram
George Vahala*
Affiliation:
Department of Physics, William & Mary, Williamsburg, VA23185, USA
Linda Vahala
Affiliation:
Department of Electrical & Computer Engineering, Old Dominion University, Norfolk, VA23529, USA
Min Soe
Affiliation:
Department of Mathematics and Physical Sciences, Rogers State University, Claremore, OK74017, USA
Abhay K. Ram
Affiliation:
Plasma Science and Fusion Center, MIT, Cambridge, MA02139, USA
*
†Email address for correspondence: gvahala@gmail.com
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Abstract

Utilizing the similarity between the spinor representation of the Dirac and the Maxwell equations that has been recognized since the early days of relativistic quantum mechanics, a quantum lattice algorithm (QLA) representation of unitary collision-stream operators of Maxwell's equations is derived for both homogeneous and inhomogeneous media. A second-order accurate 4-spinor scheme is developed and tested successfully for two-dimensional (2-D) propagation of a Gaussian pulse in a uniform medium whereas for normal (1-D) incidence of an electromagnetic Gaussian wave packet onto a dielectric interface requires 8-component spinors because of the coupling between the two electromagnetic polarizations. In particular, the well-known phase change, field amplitudes and profile widths are recovered by the QLA asymptotic profiles without the imposition of electromagnetic boundary conditions at the interface. The QLA simulations yield the time-dependent electromagnetic fields as the wave packet enters and straddles the dielectric boundary. QLA involves unitary interleaved non-commuting collision and streaming operators that can be coded onto a quantum computer: the non-commutation being the very reason why one perturbatively recovers the Maxwell equations.

Keywords

plasma wavesplasma simulation
Type
Research Article
Information
Journal of Plasma Physics , Volume 86 , Issue 5 , October 2020 , 905860518
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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References

Bialynicki-Birula, I. 1996 Photon wave function. In Progress in Optics (ed. E. Wolf), vol. 34, pp. 248–294. Elsevier.Google Scholar
Childs, A. M. & Wiebe, N. 2012 Hamiltonian simulation using linear combinations of unitary operations. Quantum Inf. Comput. 12, 901–924.Google Scholar
Coffey, M. W. 2008 Quantum lattice gas approach for the Maxwell equations. Quantum Inf. Process. 7, 275–281.Google Scholar
Dirac, P. A. M. 1928 The quantum theory of the electron. Proc. R. Soc. A 117, 610–624.Google Scholar
Jackson, J. D. 1998 Classical Electrodynamics, 3rd edn. Wiley.Google Scholar
Jestadt, R., Appel, H. & Rubio, A. 2014 Real time evolution of Maxwell systems in spinor representation. In Conference on Proceedings.Google Scholar
Jestadt, R., Ruggenthaler, M., Oliveira, J. T., Rubio, A. & Appel, H. 2018 Real-time solutions of coupled Ehrenfest-Maxwell-Pauli-Kohn-Sham equations: fundamentals, implementation and nano-optical applications. arXiv:1812.05049.Google Scholar
Khan, S. A. 2005 Maxwell optics: I. An exact matrix representation of the Maxwell equations in a medium. Phys. Scr. 71, 440–442. also arXiv:0205083v1 (2002).Google Scholar
Kulyabov, D. S., Kolokova, A. V. & Sevatianov, L. A. 2017 Spinor representation of Maxwell's equations. J.Phys.: Conf. Ser. 788, 012025.Google Scholar
Laporte, O. & Uhlenbeck, G. E. 1931 Application of spinor analysis to the Maxwell and Dirac equations. Phys. Rev. 37, 1380–1397.CrossRefGoogle Scholar
Moses, E. 1959 Solutions of Maxwell's equations in terms of a spinor notation: the direct and inverse problems. Phys. Rev. 113, 1670–1679.CrossRefGoogle Scholar
Oganesov, A., Flint, C., Vahala, G., Vahala, L., Yepez, J. & Soe, M. 2016 b Imaginary time integration method using a quantum lattice gas approach. Radiat. Eff. Defect Solids 171, 96–102.CrossRefGoogle Scholar
Oganesov, A., Vahala, G., Vahala, L. & Soe, M. 2018 Effects of Fourier transform on the streaming in quantum lattice gas algorithms. Radiat. Eff. Defect Solids 173, 169–174.Google Scholar
Oganesov, A., Vahala, G., Vahala, L., Yepez, J. & Soe, M. 2016 a Benchmarking the Dirac-generated unitary lattice qubit collision-stream algorithm for 1D vector Manakov soliton collisions. Comput. Maths Applics 72, 386.CrossRefGoogle Scholar
Oppenheimer, J. R. 1931 Note on light quanta and the electromagnetic field. Phys. Rev. 38, 725–746.CrossRefGoogle Scholar
Vahala, G., Soe, M. & Vahala, L. 2020 a Qubit unitary lattice algorithm for spin-2 Bose Einstein condensates: II – Vortex reconnection simulations and non-Abelian vortices. Radiat. Eff. Defect Solids 175, 113–119.CrossRefGoogle Scholar
Vahala, L., Soe, M., Vahala, G. & Yepez, J. 2019 a Unitary qubit lattice algorithms for spin-1 Bose-Einstein condensates. Radiat. Eff. Defect Solids 174, 46–55.Google Scholar
Vahala, G., Vahala, L. & Soe, M. 2020 b Qubit unitary lattice algorithm for spin-2 Bose Einstein condensates: I – Theory and pade initial conditions. Radiat. Eff. Defect Solids 175, 102–112.CrossRefGoogle Scholar
Vahala, G., Vahala, L., Soe, M. & Ram, A. K. 2020 c Unitary quantum lattice simulations for Maxwell equations in vacuum and in dielectric media. arXiv:2002.08450.Google Scholar
Vahala, L., Vahala, G., Soe, M., Ram, A. & Yepez, J. 2019 b Unitary qubit lattice algorithm for three-dimensional vortex solitons in hyperbolic self-defocusing media. Commun. Nonlinear Sci. Numer. Simul. 75, 152–159.Google Scholar
Vahala, G., Vahala, L. & Yepez, J. 2003 a Quantum lattice gas representation of some classical solitons. Phys. Lett A 310, 187–196.CrossRefGoogle Scholar
Vahala, L., Vahala, G. & Yepez, J. 2003 b Lattice Boltzmann and quantum lattice gas representations of one-dimensional magnetohydrodynamic turbulence. Phys. Lett A 306, 227–234.Google Scholar
Vahala, G., Vahala, L. & Yepez, J. 2004 Inelastic vector soliton collisions: a lattice-based quantum representation. Phil. Trans. R. Soc. A 362, 1677–1690.CrossRefGoogle ScholarPubMed
Vahala, G., Vahala, L. & Yepez, J. 2005 Quantum lattice representations for vector solitons in external potentials. Physica A 362, 215–221.CrossRefGoogle Scholar
Vahala, G., Yepez, J., Vahala, L. & Soe, M. 2012 a Unitary qubit lattice simulations of complex vortex structures. Comput. Sci. Disc. 5, 014013.CrossRefGoogle Scholar
Vahala, G., Yepez, J., Vahala, L., Soe, M., Zhang, B. & Ziegeler, S. 2011 Poincaré recurrence and spectral cascades in three-dimensional quantum turbulence. Phys. Rev. E 84, 046713.CrossRefGoogle ScholarPubMed
Vahala, G., Zhang, B., Yepez, J., Vahala, L. & Soe, M. 2012 b Unitary qubit lattice gas representation of 2D and 3D quantum turbulence. In Advanced Fluid Dynamics (ed. H. W. Oh), chap. 11, pp. 239–272. InTech Publishers.CrossRefGoogle Scholar
Yepez, J. 2002 An efficient and accurate quantum algorithm for the Dirac equation. arXiv:0210093.Google Scholar
Yepez, J. 2005 Relativistic path integral as a lattice-based quantum algorithm. Quantum Inf. Process 4, 471–509.Google Scholar
Yepez, J. 2016 Quantum lattice gas algorithmic representation of gauge field theory. SPIE paper 9996-22.Google Scholar
Yepez, J., Vahala, G. & Vahala, L. 2009 a Vortex-antivortex pair in a Bose-Einstein condensate, quantum lattice gas model of ${\phi ^4}$ theory in the mean-field approximation. Eur. Phys. J. Spec. Top. 171, 9–14.Google Scholar
Yepez, J., Vahala, G., Vahala, L. & Soe, M. 2009 b Superfluid turbulence from quantum Kelvin wave to classical Kolmogorov cascades. Phys. Rev. Lett. 103, 084501.CrossRefGoogle ScholarPubMed