Optimally driving multi-photon transitions in the perturbative single-mode regime

TL;DR

This work addresses how to optimally drive $m$-photon transitions in a weak, single-mode light–matter setting with a short-lived final state. It shows that the transition rate is governed by the $m$th order coherence function $G^{(m)}=\langle (\hat{a}^\dagger)^m \hat{a}^m \rangle$, and, under a fixed average photon number $n_{\mathrm{av}}$ within a truncated Hilbert space, the maximal $G^{(m)}$ is achieved by coin states whose enhancement scales as $\big(N_{\mathrm{max}}/n_{\mathrm{av}}\big)^{m-1}$. Remarkably, coin states can be effectively realized as a classical mixture of vacuum and a high-number coherent state, i.e., a purely classical strategy suffices to maximize multi-photon absorption in this perturbative regime. The findings imply that, for single-mode driving, classical statistics can outperform quantum states with the same mean photon number, clarifying the role of coherence in light-matter control and setting bounds for optimal single-mode strategies in quantum optics applications.

Abstract

The rate of $m$-photon transitions in matter, induced by an incident light field, depends on the field's $m$th order coherence function. Consequently, the coherence properties of the light field may be shaped to increase the rate of multi-photon transitions. Here, we determine the optimal state of a weak fixed-intensity, narrow-band incident light field, with a restricted maximal photon number, that optimally drives $m$-photon transitions in the case of a short-lived atomic multilevel system. We show that, in this case, no quantum properties of the light field need to be exploited, but that classical mixtures of coherent states are optimal.

Figures (1)

• Figure 1: Two-photon absorption rates. Probability $p_F$ of a two-photon transition, as a function of the average number $n_\mathrm{av}$ of photons initially in the field. We consider different initial states of the light field: the coin state $\ket{\mathrm{coin}}$, Eq. \ref{['eqP3Cont:coinState']}; the coin state approximated by a mixture of coherent states $\hat{\rho}_\mathrm{coin,coh}$, Eq. \ref{['eqP3Cont:coinDensCoh']}; a squeezed vacuum state $\ket{\xi} = \mathrm{e}^{(\xi^\ast \hat{a}^2 + \xi (\hat{a}^\dagger)^2)/2} \ket{0}$ determined by the squeezing parameter $\xi$; a thermal state $\hat{\rho}_\mathrm{th} = \mathrm{e}^{-\hbar \omega a^\dagger a/k_B T } / \mathrm{tr}\{\mathrm{e}^{-\hbar \omega a^\dagger a/k_B T } \}$ with Boltzmann constant $k_B$ and temperature $T$, for which $G^{(2)}/G^{(2)}_\mathrm{coh} = 2, \forall T$; a coherent state $\ket{\alpha_{n_\mathrm{av}}}$ with $|\alpha_{n_\mathrm{av}}|^2 = n_\mathrm{av}$. For $\ket{\xi}$, we obtained $G^{(2)}$ numerically for $\xi \in [0,2.65]$, which leads to the values of $n_\mathrm{av} = \mathrm{sinh}^2(|\xi|)$ shown in the Figure. If $\xi \gg 1$, we find $G^{(m)}/G^{(m)}_\mathrm{coh} \approx (2m-1)!!$janszky_many-photon_1987, i.e., $G^{(2)}/G^{(2)}_\mathrm{coh} \approx 3$. We restrict $n_\mathrm{av}$ to a regime in which all considered field states still lie within the restricted Hilbert space $\mathcal{H}_{N_\mathrm{max}}$, with $N_\mathrm{max} = 500$ (they have therefore negligible overlap with Fock states $\ket{n}$ with $n>N_\mathrm{max}$).