Ultrafast Optimal Sideband Cooling under Non-Markovian Evolution

A sideband cooling strategy that incorporates (i) the dynamics induced by structured (non-Markovian) environments in the target and auxiliary systems and (ii) the optimally time-modulated interaction between them is developed. For the context of cavity optomechanics, when non-Markovian dynamics are considered in the target system, ground state cooling is reached at much faster rates and at a much lower phonon occupation number than previously reported. In contrast to similar current strategies, ground state cooling is reached here for coupling-strength rates that are experimentally accessible for the state-of-the-art implementations. After the ultrafast optimal-ground-state-cooling protocol is accomplished, an additional optimal control strategy is considered to maintain the phonon number as close as possible to the one obtained in the cooling procedure. Contrary to the conventional expectation, when non-Markovian dynamics are considered in the auxiliary system, the efficiency of the cooling protocol is undermined.

Figure 1

(a) Time evolution of the phonon number $\widehat{n}(t)$ during the optimal modulation of the interaction with $n_{\mathrm{c}}=0$, $\gamma =10^{-6}{\omega }_{\text{m}\mathrm{m}}$ and $\kappa =2.15\times 10^{-4}{\omega }_{\mathrm{m}\text{m}}$. In particular, for $n_{\mathit{T}}=10^2$ ($n_{\mathit{T}}=10^3$) and final times $t_{\text{cool}}=0.55{\tau }_{\mathrm{m}\text{m}}$ and $t_{\text{cool}}=1.25{\tau }_{\mathrm{m}\text{m}}$, the dashed white (gray) curve depicts the dynamics of the phonon number under Markovian dynamics (${\omega }_{\mathrm{D}}={\mathrm{\Omega }}_{\mathrm{D}}=100{\omega }_{\mathrm{m}\text{m}}$) while the continuos black curves depicts the dynamics under non-Markovian dynamics (${\omega }_{\mathrm{D}}={\mathrm{\Omega }}_{\mathrm{D}}={\omega }_{\text{m}\mathrm{m}}$). The $y$-axis is in logarithmic scale. (b) Optimal coupling function with $t_{\text{cool}}=0.55{\tau }_{\mathrm{m}\text{m}}$ under non-Markovian (continuous lines) and Markovian dynamics (dashed lines). (c) Optimal coupling function with $t_{\text{cool}}=1.25{\tau }_{\mathrm{m}\text{m}}$ under non-Markovian (continuous lines) and Markovian dynamics (dashed lines). For (b) and (c) the continuos black curve corresponds to $n_{\mathit{T}}=10^2$ and the continuos lightgray curve correponds to $n_{\mathit{T}}=10^3$.

Figure 2

(a) Phonon number as a function of time during the optimal control protocol aimed at maintaining the minimum phonon number for different values of the dissipation rate. The initial parameters are $n_{\mathit{T}}=⟨n(t_{\text{cool}})⟩$, $n_{\mathit{c}}=0$, $\kappa =2.15\times {10}^{-4}{\omega }_{\text{om}}$. (b) The optimal optomechanical coupling function $c(t)$ for $\gamma =10^{-6}{\omega }_{\text{mm}}$. (c) The optimal optomechanical coupling function $g_{\mathit{m}}(t)$.