Gauge transformation for pulse propagation and time ordered integrals

Abstract

We investigate a gauge transformation based on the successive elimination of time-dependent onsite potentials at individual sites in finite or infinite systems. Our analysis shows that this transformation renormalizes the inward hoppings by a phase factor $e^{i φ(t)}$ and the outward hoppings by $e^{-i φ(t)}$. We further demonstrate how this procedure facilitates the reduction and simulation of pulse propagation in scattering systems, while significantly simplifying the time-ordered integrals involved in the time evolution operator for time-dependent Schrodinger equation.

Figures (4)

• Figure 1: The gauge transformation effect on a graph. The hoppings from the central site (site 1) are renormalized by a phase factor $e^{+i\phi_1}$ when they are inward and by $e^{-i\phi_1}$ when they are outward. It is worth noting that $\gamma_{45}$ and $\gamma_{54}$ are not changed since they do not involve the site 1
• Figure 2: a) 1D scattering system (orange color) connected to two semi-infinite leads. The left lead is rised to a potential $V(t)$. b) The potential $V(t)$ can be eliminated by a gauge transformation leading to unchanged hoppings in the lead and only the hopping at te interface (in red color) is affected by a phase factor $e^{\pm i\phi(t)}$ (depending on the hopping left to right or the opposit.).
• Figure 3: Eliminating the uniform onsite potential (orange) in (a) results in a central region without any potential (white sites), while the hoppings within the central system remain unchanged. This occurs because neighboring sites contribute opposite phase factors that cancel out. Only the hoppings at the interface are modified, acquiring the corresponding phase factors (the thick red bonds in (a) become thick green in (b))
• Figure 4: A graph representing the Hamiltonian matrix. The self-loops (red and blue) represent the onsite potentials. To reduce the number of possible paths starting from a given node (a site) to itself, one can get rid of the self loops by a gauge transformation leaving only $C_1$ and $C_3$ with renormalized hoppings by a phase factor.