A cumulant expansion of the influence functional entering the path integral representation of the reduced density matrix is presented. Truncation of the expansion at the quadratic terms allows mapping of a many-body environment onto an effective harmonic model with the same force autocorrelation function, recovering the conventional linear response limit. It is shown that the latter becomes exact in the thermodynamic limit if the bath consists of independent modes coupled to the system via coefficients that scale appropriately with the size of the bath. Low-order correction terms to the linear response approximation are found to be given by multitime correlation functions of the force exerted by the bath on the system. Within a classical approximation, such correction terms can be obtained from molecular dynamics calculations and incorporated in the path integral, which can be evaluated by means of iterative procedures. Numerical calculations on an intrinsically nonlinear bath of two-level systems confirm the above behaviors and indicate that the deviations from the linear response approximation are more significant at low temperatures where quantum mechanical effects play an important role to the bath dynamics.
A cumulant expansion of the influence functional entering the path integral representation of the reduced density matrix is presented. Truncation of the expansion at the quadratic terms allows mapping of a many-body environment onto an effective harmonic model with the same force autocorrelation function, recovering the conventional linear response limit. It is shown that the latter becomes exact in the thermodynamic limit if the bath consists of independent modes coupled to the system via coefficients that scale appropriately with the size of the bath. Low-order correction terms to the linear response approximation are found to be given by multitime correlation functions of the force exerted by the bath on the system. Within a classical approximation, such correction terms can be obtained from molecular dynamics calculations and incorporated in the path integral, which can be evaluated by means of iterative procedures. Numerical calculations on an intrinsically nonlinear bath of two-level systems confirm the above behaviors and indicate that the deviations from the linear response approximation are more significant at low temperatures where quantum mechanical effects play an important role to the bath dynamics.
The influence functional (IF) encodes all of the information required for calculating the dynamical properties of a system in contact with its environment. A direct and simple procedure is introduced for extracting from a few numerical evaluations of the IF, without computing time correlation functions or evaluating integrals, the parameters required for path integral calculations, either within or beyond the harmonic mapping, and for assessing the accuracy of the harmonic bath approximation. In addition, the small matrix decomposition of the path integral (SMatPI) is extended to anharmonic environments and the required matrices are constructed directly from the IF.
The recently developed (energy) sudden representation of the Schrödinger propagator is examined in conjunction with the path integral method. The perturbation expansion of the propagator in this representation (with the sudden approximation as the zeroth order term) is easily understood using Feynman diagrams. The zeroth order term is a form involving only sudden paths (i.e., paths resulting from the sudden approximation). Each higher order term is a series of sudden paths which are connected to one another due to the nonsudden effects of the dynamics. The sudden representation, and in particular its perturbation expansion, is also used to study the bath effects in many body systems which are expressed in the form of influence functionals in the path integral formalism. In our treatment, the bath coordinates are assumed to be the slow coordinates. The zeroth order (sudden approximation) influence functionals are expressed in a very simple form; they involve only an ordinary integral over the bath coordinates. Finally, to illustrate the approach, a harmonic oscillator coupled linearly to an unspecified primary system is used as a simple model bath. The reduction of the exact influence functional to the sudden approximation form is also considered.
We review an iterative path integral method that allows the computation of exact, long-time dynamics of small systems interacting with a dissipative bath. The method takes advantage of the finite memory of large environments at finite temperature to allow an iterative evaluation of the dynamics, thereby replacing an exponential scaling in simulation time with a linear scaling. This method is applied to calculate the dynamics of two model systems. The first consists of two-qubits interacting with a common bath. In this case we observe a variety of entanglement effects. If the qubits are initially separable, through an indirect coupling, the bath can create steady state entanglement between the qubits. This effect is due to the existence of a decoherence-free subspace as a result of the form of the qubit Hamiltonian and system-bath coupling. Entanglement created by the bath is shown to decrease with increasing temperature and system-bath coupling strength. However large system-bath coupling causes a faster increase in the entanglement. Initially entangled qubits lose their entanglement as a result of interactions with the bath, an effect that is heightened at higher temperatures. Direct coupling between the qubits is shown to slow the decay of entanglement and preserve some entanglement at long times; however at high temperatures this steady state entanglement becomes negligible. The second system we consider is vibrational degree of freedom coupled to a bath of harmonic oscillators or two-level systems. We compute four-time correlation functions which are used to calculate response functions relevant to third order infrared or seventh order Raman experiments for harmonic, Morse, and quadratic-quartic potentials. Our calculations reveal the role of potential features (anharmonicity and eigenvalue spectrum), both on short and long time scales, on the response function. Further, thermal excitation causes dramatic changes in the appearance of the response function, introducing symmetry with respect to the main diagonal. Finally, coupling to harmonic dissipative baths leads to decay of the response function (primarily along the t3 direction) and a broadening of the peaks in its Fourier transform. At high temperatures two-level-system baths are less efficient in destroying coherence than harmonic baths of similar parameters.
For common condensed phase problems described by a low-dimensional system coupled to a harmonic bath, Feynman’s path integral formulation of time-dependent quantum mechanics leads to expressions for the reduced density matrix of the system where the effects of the harmonic environment enter through an influence functional that is nonlocal in time. In a recent Letter [Chem. Phys. Lett. 221, 482 (1994)], we demonstrated that the range of the nonlocal interactions is finite even at zero temperature, such that the nonlocal kernel extends over only a few time steps if the path integral is expressed in terms of accurate quasiadiabatic propagators. This feature arises from disruption of phase coherence in macroscopic environments and leads to Markovian dynamics for an augmented reduced density tensor, permitting iterative time evolution schemes. In the present paper we analyze the structure and properties of the relevant tensor propagator. Specifically, we show that the tensor multiplication scheme rigorously conserves the trace of the reduced density matrix, and that in cases of short-range nonlocality it leads to Redfield-type equations which are correct to all orders in perturbation theory and which take into account memory effects. We also argue that a simple eigenvector analysis reveals (without actual iteration) the nature of the dynamics and of the equilibrium state, and directly yields quantum reaction or relaxation rates.
We present a novel approximation scheme to describe the influence of a harmonic bath on the dynamics of a two-level particle over almost the whole regime of temperatures and coupling to the environment, for a wide class of bath spectral densities. Starting from the exact path integral solution for the two-level system density matrix, effective intra-blip correlations are fully included, while inter-blip and blip-sojourn interactions are considered up to first order. In the proper regimes, an excellent agreement with conventional perturbative approaches and ab initio path-integral results is found.
We use the “generalized hierarchical equation of motion” proposed in Paper I [C.-Y. Hsieh and J. Cao, J. Chem. Phys. 148, 014103 (2018)] to study decoherence in a system coupled to a spin bath. The present methodology allows a systematic incorporation of higher-order anharmonic effects of the bath in dynamical calculations. We investigate the leading order corrections to the linear response approximations for spin bath models. Two kinds of spin-based environments are considered: (1) a bath of spins discretized from a continuous spectral density and (2) a bath of localized nuclear or electron spins. The main difference resides with how the bath frequency and the system-bath coupling parameters are distributed in an environment. When discretized from a continuous spectral density, the system-bath coupling typically scales as ∼1/NB where NB is the number of bath spins. This scaling suppresses the non-Gaussian characteristics of the spin bath and justifies the linear response approximations in the thermodynamic limit. For the nuclear/electron spin bath models, system-bath couplings are directly deduced from spin-spin interactions and do not necessarily obey the 1/NB scaling. It is not always possible to justify the linear response approximations in this case. Furthermore, if the spin-spin Hamiltonian is highly symmetrical, there exist additional constraints that generate highly non-Markovian and persistent dynamics that is beyond the linear response treatments.