Generation and dynamics of entangled fermion–photon–phonon states in nanocavities

Abstract

We develop the analytic theory describing the formation and evolution of entangled quantum states for a fermionic quantum emitter coupled simultaneously to a quantized electromagnetic field in a nanocavity and quantized phonon or mechanical vibrational modes. The theory is applicable to a broad range of cavity quantum optomechanics problems and emerging research on plasmonic nanocavities coupled to single molecules and other quantum emitters. The optimal conditions for a tripartite entanglement are realized near the parametric resonances in a coupled system. The model includes dissipation and decoherence effects due to coupling of the fermion, photon, and phonon subsystems to their dissipative reservoirs within the stochastic evolution approach, which is derived from the Heisenberg–Langevin formalism. Our theory provides analytic expressions for the time evolution of the quantum state and observables and the emission spectra. The limit of a classical acoustic pumping and the interplay between parametric and standard one-photon resonances are analyzed.

The universal model of strong coupling at the nonlinear parametric resonance in open cavity-QED systems

“…If the phonon mode is classical, the parametric process is simply the modulation of the electron-photon coupling by molecular vibrations which serve as an external driving force for the electron-photon quantum dynamics. If the phonon mode is quantized, the strong coupling between photon, phonon, and electron degrees of freedom near parametric resonance ω e = ω ± Ω leads inevitably to the formation of tripartite entangled states belonging to the family of Greenberger-Horne-Zeilinger (GHZ) states [17].…”

“…In [12] resonant Raman scattering of single molecules when the two-wave exciton-photon coupling frequency is comparable to the vibrational frequency was analyzed. The situation when the Rabi frequency becomes comparable to the vibrational frequency was also considered in [17].…”

“…Note that the system must be in the RWA regime for the nonlinear parametric resonance and all associated physics to exist and make sense. Otherwise the three-wave and two-way resonances overlap [17] and the GHZ-like entangled states cannot be created. Also, it becomes impossible to build a universal Hamiltonian.…”

“…This approach is well known [33], but it is usually applied for numerical Monte-Carlo simulations [34][35][36][37][38][39][40][41][42]. We recently developed a version of stochastic Schrödinger equation suitable for analytic solutions in open strongly-coupled cavity QED problems [17,43].…”

“…Section IV includes the effects of dissipation, decoherence, and fluctuations within the stochastic equation for the state vector which describes the evolution of an open system in contact with dissipative reservoirs. As compared to our recent work [17,43], we develop a model of fluctuations and dissipative processes which includes all effects of phonon dissipation on the dynamics of the parametric process and the emission spectra. In Section V we describe the formation of entangled electron-photon-phonon states for an open system.…”