CdSe nanoparticle dimers; Lie algebra; coherent quantum dynamics; computing by observables; Chemical Engineering (all); Materials Science (all)
Abstract :
[en] Dynamical symmetries, time-dependent operators that almost commute with the Hamiltonian, extend the role of ordinary symmetries. Motivated by progress in quantum technologies, we illustrate a practical algebraic approach to computing such time-dependent operators. Explicitly we expand them as a linear combination of time-independent operators with time-dependent coefficients. There are possible applications to the dynamics of systems of coupled coherent two-state systems, such as qubits, pumped by optical excitation and other addressing inputs. Thereby, the interaction of the system with the excitation is bilinear in the coherence between the two states and in the strength of the time-dependent excitation. The total Hamiltonian is a sum of such bilinear terms and of terms linear in the populations. The terms in the Hamiltonian form a basis for Lie algebra, which can be represented as coupled individual two-state systems, each using the population and the coherence between two states. Using the factorization approach of Wei and Norman, we construct a unitary quantum mechanical evolution operator that is a factored contribution of individual two-state systems. By that one can accurately propagate both the wave function and the density matrix with special relevance to quantum computing based on qubit architecture. Explicit examples are derived for the electronic dynamics in coupled semi-conducting nanoparticles that can be used as hardware for quantum technologies.
Disciplines :
Chemistry
Author, co-author :
Hamilton, James ; Université de Liège - ULiège > Département de chimie (sciences) > Laboratoire de chimie physique théorique ; Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
Levine, Raphael D; Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel ; Department of Molecular and Medical Pharmacology, David Geffen School of Medicine, University of California Los Angeles, Los Angeles, CA 90095, USA ; Department of Chemistry and Biochemistry, University of California Los Angeles, Los Angeles, CA 90095, USA
Remacle, Françoise ; Université de Liège - ULiège > Département de chimie (sciences) > Laboratoire de chimie physique théorique ; Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
Language :
English
Title :
Constructing Dynamical Symmetries for Quantum Computing: Applications to Coherent Dynamics in Coupled Quantum Dots.
Publication date :
23 December 2024
Journal title :
Nanomaterials
eISSN :
2079-4991
Publisher :
Multidisciplinary Digital Publishing Institute (MDPI), Switzerland
Cotton A.F. Chemical Applications of GroupTheory 3rd ed. Wiley & Sons Chichester, UK 1990
Lewis H.R. Jr. Riesenfeld W.B. An Exact Quantum Theory of the Time-Dependent Harmonic Oscillator and of a Charged Particle in a Time-Dependent Electromagnetic Field J. Math. Phys. 1969 10 1458 1473 10.1063/1.1664991
Magnus W. On the exponential solution of differential equations for a linear operator Commun. Pure Appl. Math. 1954 7 649 673 10.1002/cpa.3160070404
Pechukas P. Light J.C. On the Exponential Form of Time-Displacement Operators in Quantum Mechanics J. Chem. Phys. 1966 44 3897 3912 10.1063/1.1726550
Wilcox R.M. Exponential Operators and Parameter Differentiation in Quantum Physics J. Math. Phys. 1967 8 962 982 10.1063/1.1705306
Hioe F.T. Eberly J.H. N-Level Coherence Vector and Higher Conservation Laws in Quantum Optics and Quantum Mechanics Phys. Rev. Lett. 1981 47 838 841 10.1103/PhysRevLett.47.838
Rasetti M. Generalized definition of coherent states and dynamical groups Int. J. Theor. Phys. 1975 13 425 430 10.1007/BF01808325
Dattoli G. Di Lazzaro P. Torre A. SU(1,1), SU(2), and SU(3) coherence-preserving Hamiltonians and time-ordering techniques Phys. Rev. A 1987 35 1582 1589 10.1103/PhysRevA.35.1582 9898316
Dattoli G. Richetta M. Schettini G. Torre A. Lie algebraic methods and solutions of linear partial differential equations J. Math. Phys. 1990 31 2856 2863 10.1063/1.528937
Dattoli G. Torre A. Cayley–Klein parameters and evolution of two- and three-level systems and squeezed states J. Math. Phys. 1990 31 236 240 10.1063/1.529020
Dattoli G. Torre A. Matrix representation of the evolution operator for theSU(3) dynamics Il Nuovo Cimento B (1971–1996) 1991 106 1247 1256 10.1007/BF02728659
Altafini C. Explicit Wei-Norman formulae for matrix Lie groups Proceedings of the Proc 41st IEEE Conference on Decision and Control Las Vegas, NE, USA 10–13 December 2002 2714 2719
Altafini C. Parameter differentiation and quantum state decomposition for time varying Schrödiner equations Rep. Math. Phys. 2003 52 381 400 10.1016/S0034-4877(03)80037-X
Wei J. Norman E. Lie Algebraic Solution of Linear Differential Equations J. Math. Phys. 1963 4 575 581 10.1063/1.1703993
Wei J. Norman E. On Global Representations of the Solutions of Linear Differential Equations as a Product of Exponentials Proc. Am. Math. So. 1964 15 327 334 10.1090/S0002-9939-1964-0160009-0
Fano U. Description of States in Quantum Mechanics by Density Matrix and Operator Techniques Rev. Mod. Phys. 1957 29 74 93 10.1103/RevModPhys.29.74
Miller W. Lie Theory and Special Functions Academic Press New York, NY, USA 1968
Malkin I.A. Man’ko V.I. Trifonov D.A. Dynamical symmetry of nonstationary systems Il Nuovo Cimento A 1971 4 773 792 10.1007/BF02731519
Katzin G.H. Levine J. Dynamical symmetries and constants of the motion for classical particle systems J. Math. Phys. 1974 15 1460 1470 10.1063/1.1666832
Dragt A.J. Forest E. Computation of nonlinear behavior of Hamiltonian systems using Lie algebraic methods J. Math. Phys. 1983 24 2734 2744 10.1063/1.525671
Engelhardt G. Cao J. Dynamical Symmetries and Symmetry-Protected Selection Rules in Periodically Driven Quantum Systems Phys. Rev. Lett. 2021 126 090601 10.1103/PhysRevLett.126.090601
Kikoin K. Kiselev M. Avishai Y. Dynamical Symmetries for Nanostructures Springer Vienna, Austria 2012
Moehlis J. Knobloch E. Equivalent dynamical systems Scholarpedia 2007 2 2510 10.4249/scholarpedia.2510
Barut A. Bohm A. Ne’eman Y. Dynamical Groups and Spectrum Generating Algebras World Scientific Publishing Company Singapore 1988 1168
Alhassid Y. Levine R.D. Collision experiments with partial resolution of final states: Maximum entropy procedure and surprisal analysis Phys. Rev. C 1979 20 1775 1788 10.1103/PhysRevC.20.1775
Alhassid Y. Levine R.D. Connection between the maximal entropy and the scattering theoretic analyses of collision processes Phys. Rev. A 1978 18 89 116 10.1103/PhysRevA.18.89
Aronson E.B. Malkin I.A. Manko V.I. Dynamical Symmetries in quamtum Mechanics Sov. J. Part. Nucl. 1974 5 47 67
Pfeifer P. Levine R.D. A stationary formulation of time-dependent problems in quantum mechanics J. Chem. Phys. 1983 79 5512 5519 10.1063/1.445669
Vilemkin N.J. Special Functions and the Theory of Group Representations American Mathematical Society Providence, RI, USA 1968
Klauder J.R. Skagerstant B.-S. Coherent States World Scientific Singapore 1985
Sternberg S. Group Theory and Physics Cambridge University Press Cambridge, UK 1994
Perelomov A.M. Coherent states for arbitrary Lie group Commun. Math. Phys. 1972 26 222 236 10.1007/BF01645091
Gilmore R. On the properties of coherent states Rev. Mex. Física 1974 23 143 187
Dodonov V.V. Malkin I.A. Man’ko V.I. Integrals of the motion, green functions, and coherent states of dynamical systems Int. J. Theor. Phys. 1975 14 37 54 10.1007/BF01807990
Penson K.A. Solomon A.I. New generalized coherent states J. Math. Phys. 1999 40 2354 2363 10.1063/1.532869
Solomon A.I. Group Theory of Superfluidity J. Math. Phys. 1971 12 390 394 10.1063/1.1665601
Mukamel S. Principles of Nonlinear Optical Spectroscopy Oxford University Press Melbourne, VIC, Australia 1995
Hamilton J.R. Amarotti E. Dibenedetto C.N. Striccoli M. Levine R.D. Collini E. Remacle F. Harvesting a Wide Spectral Range of Electronic Coherences with Disordered Quasi-Homo Dimeric Assemblies at Room Temperature Adv. Quantum Technol. 2022 5 2200060 10.1002/qute.202200060
Collini E. 2D Electronic Spectroscopic Techniques for Quantum Technology Applications J. Phys. Chem. C 2021 125 13096 13108 10.1021/acs.jpcc.1c02693 34276867
Dibenedetto C.N. Fanizza E. De Caro L. Brescia R. Panniello A. Tommasi R. Ingrosso C. Giannini C. Agostiano A. Curri M.L. et al. Coupling in quantum dot molecular hetero-assemblies Mater. Res. Bull. 2022 146 111578 10.1016/j.materresbull.2021.111578
Kagan C.R. Bassett L.C. Murray C.B. Thompson S.M. Colloidal Quantum Dots as Platforms for Quantum Information Science Chem. Rev. 2021 121 3186 3233 10.1021/acs.chemrev.0c00831
Koley S. Cui J. Panfil Y.E. Banin U. Coupled Colloidal Quantum Dot Molecules Acc. Chem. Res. 2021 54 1178 1188 10.1021/acs.accounts.0c00691 33459013
Rebentrost P. Stopa M. Aspuru-Guzik A. Förster Coupling in Nanoparticle Excitonic Circuits Nano Lett. 2010 10 2849 2856 10.1021/nl1008647 20698598
Spittel D. Poppe J. Meerbach C. Ziegler C. Hickey S.G. Eychmüller A. Absolute Energy Level Positions in CdSe Nanostructures from Potential-Modulated Absorption Spectroscopy (EMAS) ACS Nano 2017 11 12174 12184 10.1021/acsnano.7b05300 29178801
Loss D. DiVincenzo D.P. Quantum computation with quantum dots Phys. Rev. A 1998 57 120 126 10.1103/PhysRevA.57.120
Kane B.E. A silicon-based nuclear spin quantum computer Nature 1998 393 133 137 10.1038/30156
Imamoglu A. Are quantum dots useful for quantum computation? Phys. E Low-Dimens. Syst. Nanostructures 2003 16 47 50 10.1016/S1386-9477(02)00581-7
Hanson R. Kouwenhoven L.P. Petta J.R. Tarucha S. Vandersypen L.M.K. Spins in few-electron quantum dots Rev. Mod. Phys. 2007 79 1217 1265 10.1103/RevModPhys.79.1217
Collini E. Gattuso H. Bolzonello L. Casotto A. Volpato A. Dibenedetto C.N. Fanizza E. Striccoli M. Remacle F. Quantum Phenomena in Nanomaterials: Coherent Superpositions of Fine Structure States in CdSe Nanocrystals at Room Temperature J. Phys. Chem. C 2019 123 31286 31293 10.1021/acs.jpcc.9b11153
Collini E. Gattuso H. Kolodny Y. Bolzonello L. Volpato A. Fridman H.T. Yochelis S. Mor M. Dehnel J. Lifshitz E. et al. Room-Temperature Inter-Dot Coherent Dynamics in Multilayer Quantum Dot Materials J. Phys. Chem. C 2020 124 1622 16231 10.1021/acs.jpcc.0c05572
Gattuso H. Levine R.D. Remacle F. Massively parallel classical logic via coherent dynamics of an ensemble of quantum systems with dispersion in size Proc. Natl. Acad. Sci. USA 2020 117 21022 10.1073/pnas.2008170117
Komarova K. Gattuso H. Levine R.D. Remacle F. Quantum Device Emulates the Dynamics of Two Coupled Oscillators J. Phys. Chem. Lett. 2020 11 6990 6995 10.1021/acs.jpclett.0c01880
Tishby N.Z. Levine R.D. Time evolution via a self-consistent maximal-entropy propagation: The reversible case Phys. Rev. A 1984 30 1477 1490 10.1103/PhysRevA.30.1477
Claveau Y. Arnaud B. Matteo S.D. Mean-field solution of the Hubbard model: The magnetic phase diagram Eur. J. Phys. 2014 35 035023 10.1088/0143-0807/35/3/035023
Montorsi A. Rasetti M. Solomon A.I. Dynamical superalgebra and supersymmetry for a many-fermion system Phys. Rev. Lett. 1987 59 2243 2246 10.1103/PhysRevLett.59.2243 10035493
Dirac P.A.M. The quantum theory of tghe emission and the absorption of radiation Proc. R. Soc. Lond. A 1927 114 243 265
Remacle F. Levine R.D. A quantum information processing machine for computing by observables Proc. Natl. Acad. Sci. USA 2023 120 e2220069120 10.1073/pnas.2220069120
Hioe F.T. Eberly J.H. Nonlinear constants of motion for three-level quantum systems Phys. Rev. A 1982 25 2168 2171 10.1103/PhysRevA.25.2168
Brus L. Electronic wave functions in semiconductor clusters: Experiment and theory J. Phys. Chem. 1986 90 2555 2560 10.1021/j100403a003
Efros A.L. Rosen M. Kuno M. Nirmal M. Norris D.J. Bawendi M. Band-edge exciton in quantum dots of semiconductors with a degenerate valence band: Dark and bright exciton states Phys. Rev. B 1996 54 4843 4856 10.1103/PhysRevB.54.4843
Norris D.J. Bawendi M.G. Measurement and assignment of the size-dependent optical spectrum in CdSe quantum dots Phys. Rev. B 1996 53 16338 16346 10.1103/PhysRevB.53.16338 9983472
Prezhdo O.V. Photoinduced Dynamics in Semiconductor Quantum Dots: Insights from Time-Domain ab Initio Studies Acc. Chem. Res. 2009 42 2005 2016 10.1021/ar900157s
Hyeon-Deuk K. Prezhdo O.V. Multiple Exciton Generation and Recombination Dynamics in Small Si and CdSe Quantum Dots: An Ab Initio Time-Domain Study ACS Nano 2012 6 1239 1250 10.1021/nn2038884
Seiler H. Palato S. Sonnichsen C. Baker H. Kambhampati P. Seeing Multiexcitons through Sample Inhomogeneity: Band-Edge Biexciton Structure in CdSe Nanocrystals Revealed by Two-Dimensional Electronic Spectroscopy Nano Lett. 2018 18 2999 3006 10.1021/acs.nanolett.8b00470 29589448
Trivedi D.J. Wang L. Prezhdo O.V. Auger-Mediated Electron Relaxation Is Robust to Deep Hole Traps: Time-Domain Ab Initio Study of CdSe Quantum Dots Nano Lett. 2015 15 2086 2091 10.1021/nl504982k 25639836
Palato S. Seiler H. Nijjar P. Prezhdo O. Kambhampati P. Atomic fluctuations in electronic materials revealed by dephasing Proc. Natl. Acad. Sci. USA 2020 117 11940 11946 10.1073/pnas.1916792117
Cassette E. Pensack R.D. Mahler B. Scholes G.D. Room-temperature exciton coherence and dephasing in two-dimensional nanostructures Nat. Commun. 2015 6 6086 10.1038/ncomms7086
Turner D.B. Hassan Y. Scholes G.D. Exciton Superposition States in CdSe Nanocrystals Measured Using Broadband Two-Dimensional Electronic Spectroscopy Nano Lett. 2012 12 880 886 10.1021/nl2039502
Caram J.R. Zheng H. Dahlberg P.D. Rolczynski B.S. Griffin G.B. Dolzhnikov D.S. Talapin D.V. Engel G.S. Exploring size and state dynamics in CdSe quantum dots using two-dimensional electronic spectroscopy J. Chem. Phys. 2014 140 084701 10.1063/1.4865832
Caram J.R. Zheng H. Dahlberg P.D. Rolczynski B.S. Griffin G.B. Fidler A.F. Dolzhnikov D.S. Talapin D.V. Engel G.S. Persistent Interexcitonic Quantum Coherence in CdSe Quantum Dots J. Phys. Chem. Lett. 2014 5 196 204 10.1021/jz402336t
Kim J. Wong C.Y. Scholes G.D. Exciton Fine Structure and Spin Relaxation in Semiconductor Colloidal Quantum Dots Acc. Chem. Res. 2009 42 1037 1046 10.1021/ar8002046
Wong C.Y. Scholes G.D. Biexcitonic Fine Structure of CdSe Nanocrystals Probed by Polarization-Dependent Two-Dimensional Photon Echo Spectroscopy J. Phys. Chem. A 2011 115 3797 3806 10.1021/jp1079197
Gattuso H. Fresch B. Levine R.D. Remacle F. Coherent Exciton Dynamics in Ensembles of Size-Dispersed CdSe Quantum Dot Dimers Probed via Ultrafast Spectroscopy: A Quantum Computational Study Appl. Sci. 2020 10 1328 10.3390/app10041328
Collini E. Gattuso H. Levine R.D. Remacle F. Ultrafast fs coherent excitonic dynamics in CdSe quantum dots assemblies addressed and probed by 2D electronic spectroscopy J. Chem. Phys. 2021 154 014301 10.1063/5.0031420
Hamilton J.R. Amarotti E. Dibenedetto C.N. Striccoli M. Levine R.D. Collini E. Remacle F. Time-Frequency Signatures of Electronic Coherence of Colloidal CdSe Quantum Dot Dimer Assemblies Probed at Room Temperature by Two-Dimensional Electronic Spectroscopy Nanomaterials 2023 13 2096 10.3390/nano13142096 37513107
Chae E. Choi J. Kim J. An elementary review on basic principles and developments of qubits for quantum computing Nano Converg. 2024 11 11 10.1186/s40580-024-00418-5
Lucas A. Ising formulations of many NP problems Front. Phys. 2014 2 5 10.3389/fphy.2014.00005
Xia R. Bian T. Kais S. Electronic Structure Calculations and the Ising Hamiltonian J. Phys. Chem. B 2018 122 3384 3395 10.1021/acs.jpcb.7b10371 29099600
