Capturing Transient Polaron Formation Dynamics: A Time-Dependent Polaron Transformation Approach

Static polaron transformations impose an equilibrium environmental displacement at the initial time and can therefore misrepresent transient polaron formation. We introduce a time-dependent (TD) polaron transformation that describes the dynamical formation of polarons. We first present the TD polaron transformation for the spin–boson model via an infinite sequence of interaction-picture transformations. Each transformation in the sequence cancels the residual term generated by its predecessor; the ordered transformation sequence combines exactly into a displacement operator whose amplitude is zero at the initial time. We then generalize this recursive elimination to any suitably partitioned Hamiltonian. When the generated operator algebra closes, the ordered sequence of transformations reduces to an exact closed form; otherwise, a truncated sequence provides a systematic short-time approximation. For the transformed spin–boson Hamiltonian, we derive a second-order Nakajima–Zwanzig quantum master equation with two-time bath correlation functions. We benchmark this quantum master equation against numerically exact matrix-product-state dynamics and hierarchical equations of motion using Landau–Zener sweeps coupled to a thermal bath, whose tunable crossing time acts as a clock for polaron formation. We find that the TD-polaron quantum master equation outperforms the static formulation in regimes where transient polaron formation competes with the system evolution. The TD-polaron transformation is therefore applicable to nonequilibrium processes such as sudden charge injection, while the underlying sequential elimination scheme extends to other suitably partitioned Hamiltonians.