Bringing weak transitions to light

Introduction

Transitions between quantum states underlie modern spectroscopy and are integral to a variety of disciplines. Weak transitions are of particular relevance to optical clocks^{1, 2, 3, 4, 5–6} and fundamental-physics experiments⁷. Even dipole-allowed transitions can be very weak: In the first theoretical interpretation of the seminal observation of doubly excited states in helium using synchrotron radiation⁸, the unobserved transition from the ground state to the sp_2,n− series was described as quasi-forbidden⁹. The even weaker dipole-allowed transition to the 2pnd series was first experimentally resolved about 30 years later¹⁰. According to Fermi’s golden rule¹¹, the absorption cross section scales with the absolute square of the transition matrix element in traditional atomic and molecular spectroscopy performed with low-intensity laser light. Breaking this scaling law could boost the direct experimental detection of weak transitions.

The well-known optical theorem¹²

1

$\sigma \propto \mathrm{Im}\left(A\right)$relates the total cross section σ to the forward scattering amplitude A. For absorption near a resonance, the response function in the linear regime leads to A ∝ ∣T∣², with T being the complex-valued transition matrix element. In the presence of intense light at different frequencies, the response function modifies to

2

$\tilde{A}\propto T^{*}\left(T+T^{\prime }\right),$where $T^{\prime }$ accounts for the contribution of additional pathways beyond the direct single-photon excitation. For weak direct transitions, $T^{\prime }$ can be much larger than T, enabling a significant enhancement of the spectral visibility of weak transitions.

The key idea of our method is illustrated in Fig. 1, based on a few-level scheme. The interaction of the considered system with the weak broadband laser 1 results in a coherent excitation from the ground state $∣g⟩$ to two excited states $∣1⟩$ and $∣2⟩$. We consider the state $∣1⟩$ to be very weakly coupled to the ground state compared to state $∣2⟩$ owing to the small transition matrix element ∣T_g1∣ ≪ ∣T_g2∣. Its faint spectral signal [Fig. 1b] is easily lost or buried in noise. If state $∣1⟩$ can be strongly coupled to state $∣2⟩$ by an additional...