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We demonstrate a strong influence of the phonon environment on the coherentdynamics of the quantum dot (QD)-cavity system in the quantum strong couplingregime. This regime is implemented in the nonlinear QD-cavity QED and can bereliably measured by heterodyne spectral interferometry. We present asemi-analytic asymptotically exact path integral-based approach to thenonlinear optical response of this system, which includes two key ingredients:Trotter's decomposition and linked-cluster expansion. Applied to thefour-wave-mixing optical polarization, this approach provides access todifferent excitation and measurement channels, as well as to higher-orderoptical nonlinearities and quantum correlators. Furthermore, it allows us toextract useful analytic approximations and analyze the nonlinear opticalresponse in terms of quantum transitions between phonon-dressed states of theanharmonic Jaynes-Cummings (JC) ladder. Being well described by theseapproximations at low temperatures and small exciton-cavity coupling, the exactsolution deviates from them for stronger couplings and higher temperatures,demonstrating remarkable non-Markovian effects, spectral asymmetry, and strongphonon renormalization of the JC ladder.

Impact of the phonon environment on the nonlinear quantum-dot-cavity QED. I. Path-integral approach