# Simulation of Nuclear Quantum Effects in Condensed Matter Systems via Quantum Baths
## 2 The Langevin Equation as a Thermal Bath
### Quasiclassical Limit of Harmonic Bath  
The Hamiltonian (9) or closely related harmonic bath models are common starting points to derive GLE also when looking at quasiclassical approximations for quantum dynamical properties. In general, these developments start from a fully quantum dynamical description of a generic system coupled to the harmonic bath and then approximate the dynamics in some flavor of the classical limit. The specific form of the results depends both on the procedure enforced to impose the classical limit (e.g., high temperature, formal limit for \(\hbar\to 0\), stationary phase approximation of the propagator...), and on the adopted representation of quantum mechanics. For example, starting from the Wigner representation of quantum averages, nuclear quantum effects can be approximated using generalized Langevin equations where the friction and random forces are kept Markovian but become position-dependent [29; 30]. Furthermore, one can take the classical limit for all degrees of freedom (semiclassical methods) or exploit knowledge of the exact solution of the harmonic part of the problem to derive a mixed quantum-classical scheme in which the bath is described quantum mechanically, while the dynamics of the system, including the coupling to the harmonic bath, is treated classically (quasiclassical or mixed quantum-classical limit). Mixed quantum-classical limits, however, are delicate. For instance, [PERSON] and [PERSON] [31] considered the--apparently simple--case of a harmonic system linearly coupled to a harmonic bath to investigate the problem of vibrational energy relaxation. They recovered the well-known result that, for this kind of system, quantum and classical dynamics lead to the same time-evolution for averages and showed that the quantum relaxation time can be evaluated from a purely classical calculation. However, they also proved that--when a mixed quantum-classical dynamics is imposed on the system--the result for the relaxation time is incorrect and different depending on whether the bath is treated quantum mechanically and the system classically or vice-versa. A careful analysis of the mixed quantum-classical limit that nicely sets the stage for the methods presented in the next chapter was performed by [PERSON] [32]. [PERSON] focused on the time evolution of the density matrix of a generic quantum system linearly coupled to a bath of quantum harmonic oscillators (as in (9)) as written in the path integral formulation of quantum mechanics (see also Section 6). In the coordinate representation, the density matrix determines the probability to find the system at a given position at time \(t=0\) and at another assigned position at time \(t\). Within the path integral formalism, the density matrix can be computed as a functional integral over the set of all possible paths connecting the initial and final positions of the system and bath in the assigned time \(t\). The harmonic nature of the bath enables an explicit expression for the marginal density matrix of the system to be obtained by integrating over the paths of the bath. This still leaves an integral over the system's paths to perform. When the coupling with the bath is sufficiently strong, quantum coherence effects are strongly suppressed. The integral can then be approximated to argue that the relevant paths for the system become stochastic trajectories of the form:  
\[m\ddot{x}=-\frac{dV}{dx}-m\gamma\dot{x}+R(t) \tag{13}\]  
where \(R(t)\) is a Gaussian stochastic process such that, in Fourier space,  
\[C_{RR}(\omega)=2m\gamma\theta(\omega,T) \tag{14}\]  
The result above is obtained by introducing a non-trivial (and somewhat heuristic) probabilistic interpretation of the path integral representation of the density matrix and enforcing a quasiclassical approximation based on a truncation of an expansion of the potential that is not always easy to justify.