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SIMONS FOUNDATION 160 Fifth Avenue New York, NY 10010 Tensor train algorithms for quantum dynamics simulations of excited state nonadiabatic processes and quantum control Victor S. Batista Yale University, Department of Chemistry & Energy Sciences Institute NSF DOE NIH AFOSR 2/5/2019 Yale University - Chemistry Department

The TT-SOFT Method Multidimensional Nonadiabatic Quantum Dynamics Samuel Greene and Victor S. Batista Department of Chemistry, Yale University Samuel Greene 2/5/2019 Yale University - Chemistry Department

Yale University - Chemistry Department SOFT Propagation 2/5/2019 Yale University - Chemistry Department

Yale University - Chemistry Department 2/5/2019 Yale University - Chemistry Department

Yale University - Chemistry Department Generic Tensor Wavepacket Kinetic Propagator 2/5/2019 Yale University - Chemistry Department

Yale University - Chemistry Department 2/5/2019 Yale University - Chemistry Department

Yale University - Chemistry Department 25-Dimensional S1/S2 Wavepacket Components 2/5/2019 Yale University - Chemistry Department

Samuel M. Greene and Victor S. Batista Tensor-Train Split-Operator Fourier Transform (TT-SOFT) Method: Multidimensional Nonadiabatic Quantum Dynamics Samuel M. Greene and Victor S. Batista J. Chem. Theory Comput. 2017, 13, 4034−4042 2/5/2019 Yale University - Chemistry Department

Samuel M. Greene and Victor S. Batista Tensor-Train Split-Operator Fourier Transform (TT-SOFT) Method: Multidimensional Nonadiabatic Quantum Dynamics Samuel M. Greene and Victor S. Batista J. Chem. Theory Comput. 2017, 13, 4034−4042 2/5/2019 Yale University - Chemistry Department

Samuel M. Greene and Victor S. Batista Tensor-Train Split-Operator Fourier Transform (TT-SOFT) Method: Multidimensional Nonadiabatic Quantum Dynamics Samuel M. Greene and Victor S. Batista J. Chem. Theory Comput. 2017, 13, 4034−4042 Comparison TT-SOFT versus Experimental Spectra of Pyrazine 2/5/2019 Yale University - Chemistry Department

Tensor-Train Split-Operator Fourier Transform (TT-SOFT) Method E. Rufeil Fiori, H.M. Pastawski Chem. Phys. Lett. 420: 35-41 2/5/2019 Yale University - Chemistry Department

Yale University - Chemistry Department 2/5/2019 Yale University - Chemistry Department

Yale University - Chemistry Department 2/5/2019 Yale University - Chemistry Department

=1 FGWT Qian, J.; Ying, L. Fast Gaussian Wavepacket Transforms and Gaussian Beams for the Schrödinger Equation. J. Comp. Phys. (2010) 229: 7848–7873. Kong, X.; Markmann, A.; Batista, V.S., A Time-Sliced Thawed Gaussian Propagation Method for Simulations of Quantum Dynamics. J. Phys. Chem. A 120: 3260-3269(2016).

Time Sliced Thawed Gaussian Propagation Method Kong, X.; Markmann, A.; Batista, V.S., A Time-Sliced Thawed Gaussian Propagation Method for Simulations of Quantum Dynamics. J. Phys. Chem. A 120: 3260-3269(2016). FGWT HT

Yale University - Chemistry Department Time Sliced Thawed Gaussian Propagation Method Kong, X.; Markmann, A.; Batista, V.S., A Time-Sliced Thawed Gaussian Propagation Method for Simulations of Quantum Dynamics. J. Phys. Chem. A 120: 3260-3269(2016). 2/5/2019 Yale University - Chemistry Department

Heller, E. J. Time-Dependent Approach to Semiclassical Dynamics. J Heller, E. J. Time-Dependent Approach to Semiclassical Dynamics. J. Chem. Phys. (1975) 62: 1544–1555.

Gaussian Beam Thawed Gaussian Propagation Hermite Gaussian Basis Hagedorn, G. A.: Raising and lowering operators for semiclassical wave packets. Ann. Phys. 269: 77-104 (1998)] and [Faou, E.,Gradinaru, V.; Lubich, C: Computing Semiclassical Quantum Dynamics with Hagedorn Wavepackets. SIAM J. Sci. Comput. 31: 3027-3041 (2009)]:

Time Sliced Thawed Gaussian Propagation Method Kong, X.; Markmann, A.; Batista, V.S., A Time-Sliced Thawed Gaussian Propagation Method for Simulations of Quantum Dynamics. J. Phys. Chem. A 120: 3260-3269(2016). Lagrangian propagation HT Eulerian propagation

Yale University - Chemistry Department Time Sliced Thawed Gaussian Propagation Method n-dimensional implementation Kong, X.; Markmann, A.; Batista, V.S., A Time-Sliced Thawed Gaussian Propagation Method for Simulations of Quantum Dynamics. J. Phys. Chem. A 120: 3260-3269(2016). 2/5/2019 Yale University - Chemistry Department

Yale University - Chemistry Department Time Sliced Thawed Gaussian Propagation Method n-dimensional implementation Kong, X.; Markmann, A.; Batista, V.S., A Time-Sliced Thawed Gaussian Propagation Method for Simulations of Quantum Dynamics. J. Phys. Chem. A 120: 3260-3269(2016). 2/5/2019 Yale University - Chemistry Department

Yale University - Chemistry Department Time Sliced Thawed Gaussian Propagation Method n-dimensional implementation Kong, X.; Markmann, A.; Batista, V.S., A Time-Sliced Thawed Gaussian Propagation Method for Simulations of Quantum Dynamics. J. Phys. Chem. A 120: 3260-3269(2016). 2/5/2019 Yale University - Chemistry Department

Yale University - Chemistry Department Time Sliced Thawed Gaussian Propagation Method n-dimensional propagation Kong, X.; Markmann, A.; Batista, V.S., A Time-Sliced Thawed Gaussian Propagation Method for Simulations of Quantum Dynamics. J. Phys. Chem. A 120: 3260-3269(2016). 2/5/2019 Yale University - Chemistry Department

Yale University - Chemistry Department 2/5/2019 Yale University - Chemistry Department

Yale University - Chemistry Department 2/5/2019 Yale University - Chemistry Department

A New Bosonization Technique The Single Boson Representation Kenneth Jung, Xin Chen and Victor S. Batista Department of Chemistry, Yale University Kenneth Jung Xin Chen 2/5/2019 Yale University - Chemistry Department

Holstein-Primakoff Representation Spin-Boson Hamiltonian Angular momentum operators are transformed by harmonic oscillator operators The commutation relations of the harmonic oscillator operators ensures that the computation relations of the angular momentum operators are satisfied. The mapping is based on a single boson but, unfortunately, it introduces momenta under a square root very much like the relativistic energy E = sqrt(m^2 c^2 (c^2 + v^2), although the square root would remain even when trying to formulate a Schrodinger-like equation, like the Klein-Gordon equation. [T. Holstein and H. Primakoff Phys. Rev. 58, 1098-1113 (1940)] 2/5/2019 Yale University - Chemistry Department

Schwinger-Bosonization Schwinger-Jordan Transform An alternative bosonization technique to transform angular momentum operators by harmonic oscillator operators was introduced by Jordan and Schwinger. The main difference when compared to the Holstein-Primakoff transformation is that each angular momentum operator is represented by products of operators of pairs of harmonic oscillators [P. Jordan, Z. Phys. 94, 531 (1935)] [J. Schwinger, in Quantum Theory of Angular Momentum, edited by L. C. Biedenharn and H. V. Dam Academic, New York (1965)] 2/5/2019 Yale University - Chemistry Department

Schwinger-Jordan Representation Jordan-Schwinger Transform Constraint: As in the HP transformation, the commutation relations relations of the harmonic oscillator operators ensures that the computation relations of the angular momentum operators are fulfilled. However, the two bosons are constrained by the eigenvalues of S^2. [P. Jordan, Z. Phys. 94, 531 (1935)] [J. Schwinger, in Quantum Theory of Angular Momentum, edited by L. C. Biedenharn and H. V. Dam Academic, New York (1965)] 2/5/2019 Yale University - Chemistry Department

Schwinger-Jordan Representation Spin-Boson Hamiltonian When implemented for representing the Spin-Boson Hamiltonian (s=1/2), the Schwinger representation gives what in our family we call the Meyer-Miller Hamiltonian, which is essentially the Hamiltonian of two coupled Harmonic oscillators. The zero-point energy of the oscillators is introduced directly due to the commutation relation between coordinates and momenta. So, modifying that value is equivalent to modifying the commutation relation between x and p. The constraint, given by the eigenvalue of S^2, imposes an additional constraint on the operators of the harmonic oscillators. [P. Jordan, Z. Phys. 94, 531 (1935); J. Schwinger, in Quantum Theory of Angular Momentum, edited by L. C. Biedenharn and H. V. Dam Academic, New York (1965)] 2/5/2019 Yale University - Chemistry Department

Schwinger-Jordan Representation N-Level Hamiltonian The Schwinger representation of the N-level Hamiltonian for non-equally spaced states requires N-harmonic oscillators. While the Hamiltonian is exact, as we discussed several times throughout this meeting, the outstanding question is how to implement it efficiently to obtain accurate results. For example, SC-IVR implementations of this Hamiltonian are usually very demanding since they involve high dimensional integrals. [H.-D. Meyer and W. H. Miller J. Chem. Phys. (1979) 70, 3214; M. Thoss and G. Stock Phys. Rev. Lett. 78, 578 (1997); V.S. Batista and W. H. Miller J. Chem. Phys. 108, 498 (1998)] 2/5/2019 Yale University - Chemistry Department

Yale University - Chemistry Department Harmonic Basis Set The basis of 2/5/2019 Yale University - Chemistry Department

Matrix Representation of Boson Operators 2/5/2019 Yale University - Chemistry Department

Single Boson Representation of Pjk= |j><k| Operators 6-level System: Single Boson Representation 2/5/2019 Yale University - Chemistry Department

Single Boson Representation of Pjk= |j><k| Operators 6-level System: Single Boson Representation 2/5/2019 Yale University - Chemistry Department

Single Boson Representation of Pjk= |j><k| Operators 6-level System: Single Boson Representation Recurrence Relation: 2/5/2019 Yale University - Chemistry Department

N-level System: Single Boson Representation N-level System: Jordan-Schwinger Representation 2/5/2019 Yale University - Chemistry Department

Two-level Systems: Single Boson Representation 2/5/2019 Yale University - Chemistry Department

Two-level Systems: Single Boson Representation 2/5/2019 Yale University - Chemistry Department

Single Boson DVR Hamiltonian H(x, p=0) x 2/5/2019 Yale University - Chemistry Department

Single Boson DVR Hamiltonian 2/5/2019 Yale University - Chemistry Department

Single Boson DVR Hamiltonian |Ψ(x)|, Re[Ψ(x)] P(t) t x 2/5/2019 Yale University - Chemistry Department

Single Boson Representation of Pauli Matrices 2/5/2019 Yale University - Chemistry Department

Single Boson Representation of Pauli Matrices 2/5/2019 Yale University - Chemistry Department

Single Boson Representation of Pauli Matrices 2/5/2019 Yale University - Chemistry Department

Single Boson Representation of Pauli Matrices 2/5/2019 Yale University - Chemistry Department