We develop a fully quantum mechanical methodology to describe the static properties and the dynamics of a single anharmonic vibrational mode interacting with a quantized infrared cavity field in the strong and ultrastrong coupling regimes. By comparing multiconfiguration time-dependent Hartree (MCTDH) simulations for a Morse oscillator in a cavity, with an equivalent formulation of the problem in Hilbert space, we describe for the first time the essential role of permanent dipole moments in the femtosecond dynamics of vibrational polariton wavepackets. We show that depending on the shape of the electric dipole function $d_e(q)$ along the vibrational mode coordinate $q$, molecules can be classified into three general families. For molecules that are polar and have a positive slope of the dipole function at equilibrium, we show that an initial diabatic light-matter product state without vibrational or cavity excitations can evolve into a polariton wavepacket with a large number of intracavity photons, for interaction strengths at the onset of ultrastrong coupling. This build up of intracavity photon amplitude is accompanied by an effective $lengthening$ of the vibrational mode of nearly $10%$, comparable with a laser-induced vibrational excitation in free space. In contrast, molecules that are also polar at equilibrium but have a negative slope of the dipole function, experience an effective mode $shortening$ under equivalent coupling conditions. Our model predictions are numerically validated using realistic $ab$-$initio$ potentials and dipole functions for HF and CO$_2$ molecules in their ground electronic states. We finally propose a non-adiabatic state preparation scheme to generate vibrational polaritons using nanoscale infrared antennas and UV-vis photochemistry or electron tunneling, to enable the far-field detection of spontaneously generated infrared quantum light.
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