In this paper, multidimensional dissipative quantum dynamics is studied within a system–bath approach in the Markovian regime using a model Lindblad operator. We report on the implementation of a Monte Carlo wave packet algorithm in the Heidelberg version of the Multi-Configuration Time-Dependent Hartree (MCTDH) program package, which is henceforth extended to treat stochastic dissipative dynamics. The Lindblad operator is represented as a sum of products of one-dimensional operators. The new form of the operator is not restricted to the MCTDH formalism and could be used with other multidimensional quantum dynamical methods. As a benchmark system, a two-dimensional coupled oscillators model representing the internal stretch and the surface–molecule distance in the O2/Pt(111) system coupled to a Markovian bath of electron–hole-pairs is used. The simulations reveal the interplay between coherent intramolecular coupling due to anharmonic terms in the potential and incoherent relaxation due to coupling to an environment. It is found that thermalization of the system can be approximately achieved when the intramolecular coupling is weak.
In this paper, multidimensional dissipative quantum dynamics is studied within a system–bath approach in the Markovian regime using a model Lindblad operator. We report on the implementation of a Monte Carlo wave packet algorithm in the Heidelberg version of the Multi-Configuration Time-Dependent Hartree (MCTDH) program package, which is henceforth extended to treat stochastic dissipative dynamics. The Lindblad operator is represented as a sum of products of one-dimensional operators. The new form of the operator is not restricted to the MCTDH formalism and could be used with other multidimensional quantum dynamical methods. As a benchmark system, a two-dimensional coupled oscillators model representing the internal stretch and the surface–molecule distance in the O2/Pt(111) system coupled to a Markovian bath of electron–hole-pairs is used. The simulations reveal the interplay between coherent intramolecular coupling due to anharmonic terms in the potential and incoherent relaxation due to coupling to an environment. It is found that thermalization of the system can be approximately achieved when the intramolecular coupling is weak.
In this article, we explore the dissipation dynamics of a strongly coupled multidimensional system in contact with a Markovian bath, following a system-bath approach. We use in this endeavor the recently developed stochastic multi-configuration time-dependent Hartree approach within the Monte Carlo wave packet formalism [S. Mandal et al., J. Chem. Phys. 156, 094109 (2022)]. The method proved to yield thermalized ensembles of wave packets when intramolecular coupling is weak. To treat strongly coupled systems, new Lindblad dissipative operators are constructed as linear combinations of the system coordinates and associated momenta. These are obtained by a unitary transformation to a normal mode representation, which reduces intermode coupling up to second order. Additionally, we use combinations of generalized raising/lowering operators to enforce the Boltzmann distribution in the dissipation operators, which yield perfect thermalization in the harmonic limit. The two ansatz are tested using a model two-dimensional Hamiltonian, parameterized to disentangle the effects of intramolecular potential coupling, of strong mode mixing observed in Fermi resonances, and of anharmonicity.
We investigate the effect of inter-mode coupling on the vibrational relaxation dynamics of molecules in weak dissipative environments. The simulations are performed within the reduced density matrix formalism in the Markovian regime, assuming a Lindblad form for the system-bath interaction. The prototypical two-dimensional model system representing two CO molecules approaching a Cu(100) surface is adapted from an ab initio potential, while the diatom-diatom vibrational coupling strength is systematically varied. In the weak system-bath coupling limit and at low temperatures, only first order non-adiabatic uni-modal coupling terms contribute to surface-mediated vibrational relaxation. Since dissipative dynamics is non-unitary, the choice of representation will affect the evolution of the reduced density matrix. Two alternative representations for computing the relaxation rates and the associated operators are thus compared: the fully coupled spectral basis, and a factorizable ansatz. The former is well-established and serves as a benchmark for the solution of Liouville-von Neumann equation. In the latter, a contracted grid basis of potential-optimized discrete variable representation is tailored to incorporate most of the inter-mode coupling, while the Lindblad operators are represented as tensor products of one-dimensional operators, for consistency. This procedure results in a marked reduction of the grid size and in a much more advantageous scaling of the computational cost with respect to the increase of the dimensionality of the system. The factorizable method is found to provide an accurate description of the dissipative quantum dynamics of the model system, specifically of the time evolution of the state populations and of the probability density distribution of the molecular wave packet. The influence of intra-molecular vibrational energy redistribution appears to be properly taken into account by the new model on the whole range of coupling strengths. It demontrates that most of the mode mixing during relaxation is due to the potential part of the Hamiltonian and not to the coupling among relaxation operators.
Modeling the dynamics of a quantum system coupled to a dissipative environment becomes particularly challenging when the system's dimensionality is too high to permit the computation of its eigenstates. This problem is addressed by introducing an eigenstate-free formalism, where the open quantum system is represented as a mixture of high-dimensional, time-dependent wave packets governed by coupled Schrödinger equations, while the environment is described by a multi-component quantum master equation. An efficient computational implementation of this formalism is presented, employing a variational mixed Gaussian/multiconfigurational time-dependent Hartree (G-MCTDH) ansatz for the wave packets and propagating the environment dynamics via hierarchical equations, truncated at the first or second level of the hierarchy. The effectiveness of the proposed methodology is demonstrated on a 61-dimensional model of phonon-driven vibrational relaxation of an adsorbate. G-MCTDH calculations on 4- and 10-dimensional reduced models, combined with truncated hierarchical equations for the mean fields, nearly quantitatively replicate the full-dimensional quantum dynamical results on vibrational relaxation while significantly reducing the computational time. This approach thus offers a promising quantum dynamical method for modeling complex system-bath interactions, where a large number of degrees of freedom must be explicitly considered.
Modeling the non-equilibrium dissipative dynamics of strongly interacting quantized degrees of freedom is a fundamental problem in several branches of physics and chemistry. We implement a quantum state trajectory scheme for solving Lindblad quantum master equations that describe coherent and dissipative processes for a set of strongly coupled quantized oscillators. The scheme involves a sequence of stochastic quantum jumps with transition probabilities determined by the system state and the system-reservoir dynamics. Between consecutive jumps, the wave function is propagated in a coordinate space using the multi-configuration time-dependent Hartree method. We compare this hybrid propagation methodology with exact Liouville space solutions for physical systems of interest in cavity quantum electrodynamics, demonstrating accurate results for experimentally relevant observables using a tractable number of quantum trajectories. We show the potential for solving the dissipative dynamics of finite size arrays of strongly interacting quantized oscillators with high excitation densities, a scenario that is challenging for conventional density matrix propagators due to the large dimensionality of the underlying Hilbert space.
We extend a standard stochastic theory to study open quantum systems coupled to a generic quantum environment. We exemplify the general framework by studying a two-level quantum system coupled bilinearly to the three fundamental classes of non-interacting particles: bosons, fermions, and spins. In this unified stochastic approach, the generalized stochastic Liouville equation (SLE) formally captures the exact quantum dissipations when noise variables with appropriate statistics for different bath models are applied. Anharmonic effects of a non-Gaussian bath are precisely encoded in the bath multi-time correlation functions that noise variables have to satisfy. Starting from the SLE, we devise a family of generalized hierarchical equations by averaging out the noise variables and expand bath multi-time correlation functions in a complete basis of orthonormal functions. The general hierarchical equations constitute systems of linear equations that provide numerically exact simulations of quantum dynamics. For bosonic bath models, our general hierarchical equation of motion reduces exactly to an extended version of hierarchical equation of motion which allows efficient simulation for arbitrary spectral densities and temperature regimes. Similar efficiency and flexibility can be achieved for the fermionic bath models within our formalism. The spin bath models can be simulated with two complementary approaches in the present formalism. (I) They can be viewed as an example of non-Gaussian bath models and be directly handled with the general hierarchical equation approach given their multi-time correlation functions. (II) Alternatively, each bath spin can be first mapped onto a pair of fermions and be treated as fermionic environments within the present formalism.
A variational approach for the quantum dynamics of statistical mixtures is developed, which is based upon the representation of the natural states of the mixture in terms of hybrid Gaussian-based Multiconfiguration Time-Dependent Hartree (G-MCTDH) wavefunctions. The method, termed ρG-MCTDH, is combined with a treatment of dissipation and decoherence based on the nonstochastic open-system Schrödinger equations. The performance and the convergence properties of the approach are illustrated for a two-dimensional tunneling system, where the primary tunneling coordinate, represented by flexible single-particle functions, is resonantly coupled to a second harmonic mode, represented by Gaussian wave packets. The harmonic coordinate is coupled to the environment and two different processes are studied: (i) vibrational relaxation at zero temperature described by a master equation in the Lindblad form and (ii) thermalization induced by the Caldeira-Leggett master equation. In the second case, the evolution from a quantum tunneling regime to a quasistationary classical-limit distribution, driven by the heat bath, is visualized using a flux analysis.