0 10 20 0 2 4 6 $_{σ}$reg  /  0 0.5 1 1.5 2 1 2 3 0 0.05 0.1 0 0.01 0.02 0 1 2 3 4  /  p 0 2 10 -4 4 10 -4 S reg S reg  S reg $_{0}$=0.01,  =0.25 $_{0}$=0.01,  =0.5 $_{0}$=0.1,  =0.4 $_{0}$=10, =40 $_{0}$=1, =4 $_{0}$=0.5,  =2 0.5 1 0.5 1 0.5 1 0 0.5 1 1.5 2 0.5 1 -E kin /2, S reg / , D 0.5 1 0.5 1 0.5 1 0 2 4 6 8 g$^{2}$= /2 0 0 0.5 1 $_{0}$=0.01 $_{0}$=0.05 $_{0}$=0.1 $_{0}$=0.5 $_{0}$=0.5 $_{0}$=1 $_{0}$=5 $_{0}$=10 S reg E$_{kin}$ D 6 FIG. 9: (Color online) Optical conductivity σ reg ( ω ) in 1D. Left column: For small ω 0 . The red dashed curves in the upper two panels give the conductivity in the static limit ω 0 = 0, with same λ . Right column: For large ω 0 . The red dashed curves show the approximate Gaussian expression for the conductivity obtained in small-polaron theory $^{46,47}$. The insets compare the integrated conductivity S reg ( ω ) = ∫ ω 0 σ reg ( ω $^{′}$) dω ′ (in arbitrary units). FIG. 8: (Color online) Depiction of the f-sum rule, with kinetic energy E$_{kin}$ (solid), integrated regular conductivity S reg = ∫ ∞ 0 σ reg ( ω $^{′}$) dω ′ (dashed), and Drude weight D (dash- dotted) in 1D. The long-dashed red curve in the left column gives the kinetic energy in the static limit for comparison. For ω 0 = 0 . 01, it lies on top of the E$_{kin}$ -curve. are correct, but it should be clear that tiny spikes on top of the curves are a kind of numerical noise. We show the evaluation of the f-sum rule in Fig. 8. Note that, working on an infinite lattice, the f-sum rule is satisfied in all parameter regimes. In particular, the Drude weight obtained either from the f-sum rule via σ$_{reg}$ ( ω ) or from the polaron mass differs by less than 10 − $^{4}$. mass increase is not so much a result of dressing the elec- tron with phonons, but rather of the immobility of the extended lattice displacement as a whole. The electron motion relative to the lattice displacement is less sup- pressed, which results in the large radius. Again, we see this behavior in the f-sum rule in Fig. 8 (left top panel). Now, even if the Drude weight close to zero indicates the immobility of the polaron, the kinetic energy and opti- cal conductivity remain large. The opposite behavior of these quantities can be attributed to the different con- stituents of the polaron. The Drude weight is small since the phonon part suppresses motion of the whole polaron, but the kinetic energy is large due to the electron motion relative to the phonons. Transitions between different electronic states in the almost rigid phonon background give a large contribution to the optical conductivity. If electron and phonon motion decouple in this way an adi- abatic polaron can be understood as a quasiparticle only with certain restrictions. The polaron is rather a compos- ite electron-phonon object, where electron and phonons influence each other but remain separable, instead of that joint electron-phonon entity found in the antiadiabatic regime where the electron and the phonons lose their in- dependence in favor of a new quasiparticle. As the f-sum rule shows this difference persists through the entire cou- pling range. As we have discussed above, the polaron in the an- tiadiabatic strong coupling regime with R < 1 attains a large mass renormalization only through local phonon excitations tightly bound to the electron. Such a polaron is always a very small particle, with strong correlations between electron and phonon dynamics. We therefore expect that, with increasing coupling, the kinetic en- ergy and Drude weight simultaneously become small as the polaron gets less mobile, and that at the same time the optical conductivity decreases since a very small ob- ject is less susceptible to optical fields. As this expecta- tion is confirmed in Fig. 8 (right bottom panel), we can say that in this type of polaron the electron is dressed with phonons, resulting in a new polaronic quasiparticle emerging as a joint electron-phonon entity. The physics is essentially different in the adiabatic regime of heavy but large polarons. For such adiabatic polarons electron and phonon dynamics partially decou- ple, anticipating the static limit where the electron moves in the rigid phonon configuration of the lattice displace- ment. That there translational symmetry is broken and the polaron has infinite mass is equivalent to the fact that states of a static oscillator with different position, and vanishing width, have zero overlap. At small but finite ω$_{0}$ the translated phonon configurations have small over- lap, and the polaron acquires a finite but eventually large mass even for large radius. That the radius remains large means that in contrast to the antiadiabatic situation the Differences between the adiabatic and antiadiabatic regime appear also in the optical conductivity (see Fig. 9). At large phonon frequencies and strong coupling the optical conductivity is dominated by phonon-assisted transitions to excited polaron states, which change the average phonon number by an integer. The single peaks are thus separated by ω$_{0}$ , and resolve the Poisson dis- tribution of phonons in the polaron groundstate. The