Quantum field theory (QFT) is considered to be most correct for the description of quantum systems. For the reason, we study local physical quantities defined in QFT, such as local polarizability, local conductivity, local spin torque [1–4]. In these works, for quantum states, we used wave packets derived by first principles calculation based on quantum mechanics. To treat states and physical quantities consistently, quantum systems should be treated in QFT. Hence, we are developing a first principles calculation program package based on quantum electrodynamics (QED). We name this program package QEDynamics [5]. The purpose of this program package is to study time evolution of quantum systems consisting of molecules or condensed matter. This challenge has two points of difficulties. One is the non-perturbativity, and the other is the time evolution of quantum systems. The formulation of QFT has been developed in a perturbative method. Hence perturbative calculations are widely used. For non-perturbative systems i.e. bound states, many methods are explored, lattice quantum chromodynamics, NRQED [6], Bethe-Salpeter method, and so on. All methods have various advantages and disadvantages, and hence we should be careful with the treatment of non-perturbative systems. Moreover, in most perturbative calculations, the existence of asymptotic states is assumed and the time integration from −∞ to∞ is performed for calculations, such as cross section. Thus the time evolution i.e. the study of nonequilibrium systems is restricted in specific cases. In literature, nonequilibrium systems are studied by Boltzmann equation, closed time path formalism [7], thermo field dynamics [8] and so on. We solve directly Heisenberg equations of field operators in QED in the present version of QEDynamics for the study of time evolution. In computational simulations, operators are difficult to handle. In our program, the evolution of the operators is calculated as the transition between modes prepared by the method of first principles computations of quantum structures. This mode expansion is based on the method used in Furry picture. This choice of the expansion functions of field operator has an advantage representing bound states. In this article, we show the present status of our program. In particular, the computation method of the evolution of photon fields are explained. Hence most other detail information cannot be mentioned in this article. For the information, please see Ref. [5].
An approximate numerical method is proposed and discussed for solving the evolution of the spin density operator when the quantum system has an interaction with an external electromagnetic field.In this method by separating the relaxation and field interaction processes at small steps, instead of solving the conventional Liouville-von Neumann or Bloch differential equations, the time evolution of the density operator is efficiently obtained by a two-stage numerical algorithm.Here we have compared the results of this approach with Bloch equation results for a twolevel quantum system.The proposed approach has potential applications in calculation of the time evolution for different atomic system including nuclear or electron spin resonance systems.
In this paper we propose an ab initio method to solve quantum many-body problems of molecular dynamics where both electronic and nuclear degrees are represented by ensembles of trajectories and guiding waves in physical space. Both electrons and nuclei can be treated quantum mechanically where the guiding waves obey a set of coupled Schrodinger equations (quantum-quantum description) or, alternatively, coupled Schrodinger-Newtonian equations are solved for the quantum-classical approximation. The method takes into account local and nonlocal quantum correlation effects in a self-consistent manner. The general formalism is applied to one- and two-dimensional hydrogen molecules subjected to a strong ultrashort optical pulse. Comparison is made with the results from the "exact" Ehrenfest molecular dynamics for the molecular ionization and for the evolution of the internuclear distance as the molecule dissociates.
The time evolution of quantum many body systems in the relativistic quantum field theory is studied. We use Shibata-Hashitsume’s projection operator method which gives a systematic expansion of the Heisenberg equation of motion in terms of the interaction Hamiltonian. We calculate the time evolution of expectation values of creation and annihilation operators in a renormalizable model with unstable particles in a vacuum and in a medium. It is shown how to renormalize the ultraviolet divergences appearing in the equation of motion and calculate the medium effects on the time evolution. The mass and the wavefunction renormalizations are carried out in the lowest order. We find that the results coincide with the on-mass-shell values of the self-energy and its derivative which are calculated by the usual Feynman rule. We also discuss the initial divergence which remains at the initial time after mass renormalization. It is suggested that the projection operator method can be applied to the relativistic quantum field theory.
ConspectusSimulating molecular dynamics (MD) within a comprehensive quantum framework has been a long-standing challenge in computational chemistry. An exponential scaling of computational cost renders solving the time dependent Schrödinger equation (TDSE) of a molecular Hamiltonian, including both electronic and nuclear degrees of freedom (DOFs), as well as their couplings, infeasible for more than a few DOFs. In the Born-Oppenheimer (BO), or adiabatic, picture, electronic and nuclear parts of the wave function are decoupled and treated separately. Within this framework, the nuclear wave function evolves along potential energy surfaces (PESs) computed as solutions to the electronic Schrödinger equation parametrized in the nuclear DOFs. This approximation, together with increasingly elaborate numerical approaches to solve the nuclear time dependent Schrödinger equation (TDSE), enabled the treatment of up to a few dozens of degrees of freedom (DOFs). However, for particular applications, such as photochemistry, the BO approximation breaks down. In this regime of non-adiabatic dynamics, solving the full molecular problem including electron-nuclear couplings becomes essential, further increasing the complexity of the numerical solution. Although valuable methods such as multiconfigurational time-dependent Hartree (MCTDH) have been proposed for the solution of the coupled electron-nuclear dynamics, they remain hampered by an exponential scaling in the number of nuclear DOFs and by the difficulty of finding universal variational forms.In this Account, we present a perspective on novel quantum computational algorithms, aiming to alleviate the exponential scaling inherent to the simulation of many-body quantum dynamics. In particular, we focus on the derivation and application of quantum algorithms for adiabatic and non-adiabatic quantum dynamics, which include efficient approaches for the calculation of the BO potential energy surfaces (PESs). Thereafter, we study the time-evolution of a model system consisting of two coupled PESs in first and second quantization. In a first application, we discuss a recently introduced quantum algorithm for the evolution of a wavepacket in first quantization and exploit the potential quantum advantage of mapping its spatial grid representation to logarithmically many qubits. For the second demonstration, we move to the second quantization framework and review the scaling properties of two alternative time-evolution algorithms, namely, a variational quantum algorithm (VQA) (based on the McLachlan variational principle) and conventional Trotter-type evolution (based on a Lie-Trotter-Suzuki formula). Both methods clearly demonstrate the potential of quantum algorithms and their favorable scaling compared to the available classical approaches. However, a clear demonstration of quantum advantage in the context of molecular quantum dynamics may require the implementation of these algorithms in fault-tolerant quantum computers, while their application in near-term, noisy quantum devices is still unclear and deserves further investigation.