We propose a cavity QED approach to describe light-matter interaction between an individual anharmonic molecular vibration and an infrared cavity field. Starting from a generic Morse oscillator with quantized nuclear motion, we derive a multi-level quantum Rabi model to study vibrational polaritons beyond the rotating-wave approximation. We analyze the spectrum of vibrational polaritons in detail and compare with available experiments. For high excitation energies, the spectrum exhibits a dense manifold of true and avoided level crossings as the light-matter coupling strength and cavity frequency are tuned. These crossings are governed by a pseudo parity selection rule imposed by the cavity field. We also analyze polariton eigenstates in nuclear coordinate space. We show that the bond length of a vibrational polariton at a given energy is never greater than the bond length of a bare Morse oscillator with the same energy. This type of bond hardening of vibrational polaritons occurs at the expense of the creation of virtual infrared cavity photons, and may have implications in chemical reactivity.Cavity quantum electrodynamics (QED) has been intensely studied for the development of quantum technology over the last decade [1,2]. Precision experiments under carefully controlled conditions have been implemented to reach the regime where quantum optical effects become relevant for applications [3][4][5]. Chemical systems and molecular materials at ambient conditions for long have been considered to be unnecessarily complex and uncontrollable to enable useful quantum optical effects. In recent years, the demonstration of reversible modifications of chemical properties in molecular materials via strong coupling to confined light has stimulated the study of cavity QED as an emerging research direction in chemical physics [6]. Light-matter interaction in the strong coupling (SC) and ultrastrong coupling (USC) regimes opens the possibility of creating novel hybrid photon-molecule states whose unique properties may enable novel applications in chemistry and material science.In the infrared regime, the coupling of an intramolecular vibration to the quantized electromagnetic vacuum of a Fabry-Pérot cavity can lead to the formation of vibrational polaritons [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. These hybrid light-matter states exhibit fundamentally novel properties in comparison with free-space vibrations. For instance, vibrational polaritons may enable the selective control of chemical reactions [21-23], a long-standing goal in physical chemistry [24]. Strong light-matter coupling provides a reversible way of modifying reactive processes without changing the chemical composition of materials, and also modify the radiative and non-radiative dynamics of molecular vibrations [25][26][27][28][29][30][31]. Several recent studies on vibrational strong coupling (VSC) within the ground electronic state have shown that chemical reactions can proceed through novel pathways in comparison with free space. Under VSC, r...