We report the first numerical calculations in which converged Matsubara dynamics is compared directly with exact quantum dynamics with no artificial damping of the time-correlation functions (TCFs). The system treated is a Morse oscillator coupled to a harmonic bath. We show that, when the system-bath coupling is sufficiently strong, the Matsubara calculations can be converged by explicitly including up to M = 200 Matsubara modes, with the remaining modes included as a harmonic "tail" correction. The resulting Matsubara TCFs are in near-perfect agreement with the exact quantum TCFs, for non-linear as well as linear operators, at a temperature at which the TCFs are dominated by quantum thermal fluctuations. These results provide compelling evidence that incoherent classical dynamics can arise in the condensed phase at temperatures at which the statistics are dominated by quantum (Boltzmann) effects, as a result of smoothing of imaginary-time Feynman paths. The techniques developed here may also lead to efficient methods for benchmarking system-bath dynamics in the overdamped regime.
We report the first numerical calculations in which converged Matsubara dynamics is compared directly with exact quantum dynamics with no artificial damping of the time-correlation functions (TCFs). The system treated is a Morse oscillator coupled to a harmonic bath. We show that, when the system-bath coupling is sufficiently strong, the Matsubara calculations can be converged by explicitly including up to M = 200 Matsubara modes, with the remaining modes included as a harmonic "tail" correction. The resulting Matsubara TCFs are in near-perfect agreement with the exact quantum TCFs, for non-linear as well as linear operators, at a temperature at which the TCFs are dominated by quantum thermal fluctuations. These results provide compelling evidence that incoherent classical dynamics can arise in the condensed phase at temperatures at which the statistics are dominated by quantum (Boltzmann) effects, as a result of smoothing of imaginary-time Feynman paths. The techniques developed here may also lead to efficient methods for benchmarking system-bath dynamics in the overdamped regime.
We develop an exact quantum thermostat that generates the quantum Boltzmann distribution in path integral phase space and so correctly captures the imaginary position–momentum correlations. This extends beyond the capability of existing quantum thermostats such as the PILE thermostat, which can only achieve the marginal of this distribution in position, with the momentum acting purely as a sampling tool and satisfying classical Maxwell–Boltzmann statistics. The dynamics correspond to the treatment of the Caldeira–Leggett system–bath model with Matsubara dynamics, a theory for approximating nuclear quantum dynamics by propagating smooth imaginary-time Feynman paths in real time. A Matsubara generalised Langevin equation that obeys certain quantum fluctuation–dissipation relations is mapped onto a Markovian stochastic process by introducing auxiliary variables; this allows the construction of a corresponding Fokker–Planck equation from which the stationary distribution in the extended variable space is determined. We show that this corresponds to the joint quantum Boltzmann distribution in position and momentum, a finding that holds for any bath spectral density and for anharmonic systems. The sole approximation in Matsubara dynamics is the neglect of real-time quantum coherence; consequently, provided there is sufficient quantum decoherence, the dynamics of the exact quantum thermostat also well approximates the exact quantum dynamics of the system–bath. In the limit of infinite system–bath coupling, the level of real-time coherence becomes negligible and so the Matsubara dynamics of the thermostat tends to the exact dynamics. The quantum thermostat can therefore be used to study this limit exactly and, in so doing, we demonstrate the existence of a quantum mechanical overdamped regime analogous to its classical counterpart that is characterised by a Matsubara Smoluchowski equation. The drawback of the quantum thermostat is that one of its associated quantum fluctuation–dissipation relations is imaginary. This results in complex noise and dynamical instabilities that often preclude the numerical implementation of the thermostat. Instead, it primarily serves as a theoretical framework that can be used as a starting point for introducing approximations that yield more practical methods. One such approximation that results in real noise is considered here and leads to a practical approach for calculating linear time-correlation functions of bound open quantum systems. These exhibit good agreement with exact quantum mechanical calculations, particularly at strong system–bath coupling, where the real-time quantum coherence is increasingly suppressed.
Beyond the weak-coupling limit, open quantum systems equilibrate to a highly entangled thermal state. For continuous-variable systems, this state can be written explicitly as an imaginary-time phase-space path integral, in which the positions are directly entangled with the bath, and the momenta are correlated with the positions through a phase term. Here, we ask to what extent this state can be reached by propagating stochastic classical trajectories in path-integral phase space. Surprisingly, we find that the trajectories equilibrate to the exact quantum equilibrium state, recovering the purely imaginary momentum-position correlation in the phase term. The trajectories are generated using a recently derived Matsubara generalized Langevin equation, which produces the imaginary correlations by evolving the stochastic variables into the complex plane. This makes the dynamics numerically unstable, but we are nonetheless able to demonstrate the equilibration of a quartic oscillator coupled to a white-noise bath. These unexpected findings could lead to new approximate methodologies for simulating continuous-variable open quantum systems.
Matsubara dynamics has recently emerged as the most general form of a quantum-Boltzmann-conserving classical dynamics theory for the calculation of single-time correlation functions. Here, we present a generalization of Matsubara dynamics for the evaluation of multitime correlation functions. We first show that the Matsubara approximation can also be used to approximate the two-time symmetrized double Kubo transformed correlation function. By a straightforward extension of these ideas to the multitime realm, a multitime Matsubara dynamics approximation can be obtained for the multitime fully symmetrized Kubo transformed correlation function. Although not a practical method, due to the presence of a phase-term, this multitime formulation of Matsubara dynamics represents a benchmark theory for future development of Boltzmann preserving semiclassical approximations to general higher order multitime correlation functions. It also reveals a connection between imaginary time-ordering in the path integral and the classical dynamics of multitime correlation functions.
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.
The dissipative quantum dynamics of a two-level system interacting with a structured reservoir consisting of a damped harmonic mode is investigated by means of the numerically exact path integral Monte Carlo method. This approach provides benchmark results in a broad range of parameter space, in particular, in those domains which are not accessible by approximate methods and alternative numerical schemes, that is, strong coupling between system and harmonic mode and from low to high temperatures. For low temperatures the numerical data are quantitatively in agreement with the noninteracting blip approximation only in the regimes of very weak and very strong coupling. It turns out that the entangled dynamics of the two-level system and the harmonic mode is relatively robust so that its signatures are observable up to relatively strong friction and high temperatures. Nonequilibrium initial preparations of the reservoir with respect to the initial state of the system give rise, for strong interactions, to a stepwise decay of the population, thus displaying coherent dynamics of the bath. The impact of an additional Ohmic bath coupled directly to the two-level system is studied as well, including the case where both reservoirs carry different temperatures.
In this paper, we present benchmark results for dissipative dynamics of a harmonic oscillator coupled to an anharmonic bath of Morse oscillators. The microscopic Hamiltonian has been chosen so that the anharmonicity can be adjusted as a free parameter, and its effect can be isolated. This leads to a temperature dependent spectral density of the bath, which is studied for ohmic and lorentzian cases. Also, we compare numerically exact multiconfiguration time-dependent Hartree results with approximate solutions using continuous configuration time-dependent self-consistent field and local coherent state approximation.