Introduction

Transitions between quantum states underlie modern spectroscopy and are integral to a variety of disciplines. Weak transitions are of particular relevance to optical clocks1,2,3,4,5,6 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 sp2,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

$$\sigma \propto {{{{\rm{Im }}}}}(A)$$
(1)

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

$$\tilde{A}\propto {T}^{*}(T+T^{\prime} ),$$
(2)

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 \(\left\vert g\right\rangle\) to two excited states \(\left\vert 1\right\rangle\) and \(\left\vert 2\right\rangle\). We consider the state \(\left\vert 1\right\rangle\) to be very weakly coupled to the ground state compared to state \(\left\vert 2\right\rangle\) owing to the small transition matrix element ∣Tg1∣ ≪ ∣Tg2∣. Its faint spectral signal [Fig. 1b] is easily lost or buried in noise. If state \(\left\vert 1\right\rangle\) can be strongly coupled to state \(\left\vert 2\right\rangle\) by an additional (intense) laser, as illustrated in Fig. 1a, the resulting transfer of quantum-state amplitude from \(\left\vert 2\right\rangle\) to \(\left\vert 1\right\rangle\) will lead to an enhanced and more detectable spectral signal as shown in Fig. 1b.

Fig. 1: Schematic illustration of the concept of enhancing spectroscopically weak transitions.
figure 1

a Laser 1 creates a coherent superposition of ground and excited states, with state \(\left\vert 1\right\rangle\) being very weakly excited compared to state \(\left\vert 2\right\rangle\) as a result of a vanishingly small transition matrix element ∣Tg1∣ ≪ ∣Tg2∣. Laser 2 opens an additional pathway by coupling state \(\left\vert 1\right\rangle\) to the more strongly populated excited state \(\left\vert 2\right\rangle\). The resulting spectral signal on the originally weak transition \(\left\vert g\right\rangle \to \left\vert 1\right\rangle\) gets enhanced as shown in panel (b). Parameters for the calculation results shown in panel (b) are as follows: state \(\left\vert 1\right\rangle\) (Eg1 = 62 eV, 400-fs decay time); state \(\left\vert 2\right\rangle\) (Eg2 = 60 eV, 17-fs decay time); transition matrix elements for Tg1, Tg2, and T21 are 0.0008, 0.04, and 2 a.u., respectively; laser 1 [0.2-fs FWHM (full width at half maximum) duration, 9.1-eV spectral bandwidth, 1 × 1011-W/cm2 peak intensity, 60 eV central frequency] overlaps temporally with laser 2 (4-fs FWHM duration, 2 × 1012-W/cm2 peak intensity, 700 nm central wavelength).

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Results

Experimental demonstration

To experimentally demonstrate this concept, we turn to the helium atom and its 2p3d and sp2,4− (1Po) doubly excited states, whose transition probabilities from the ground state 1s2 (1Se) are a few orders of magnitude lower than that of the 2s2p (1Po) state (see Supplementary Table 1). Throughout this work, we followed the early notation of the 1Po doubly excited states below the N = 2 threshold of He+ first introduced by Cooper, Fano, and Prats9: sp2,n± (the strong mixing of 2snp and 2pns) and 2pnd, in which the electron configurations are from the shell-model picture (e.g., 2snp implies that one electron is in the 2s state and the other in the np state). For more classification schemes of these states, see13,14. However, both the 2p3d and the sp2,4− states can be strongly coupled to the 2s2p state with two visible (VIS) photons, via the intermediate 2p2 (1Se) state as shown in the relevant energy-level scheme in Fig. 2b. This additional coupling pathway opens the possibility of enhancing the spectral visibility of these two weak resonances. A common beam-path transient absorption beamline15 is utilized in the experiment as depicted in Fig. 2a, in which the few-cycle VIS pulses propagate collinearly with the extreme-ultraviolet (XUV) pulses produced by high-harmonic generation in neon. The ultrashort, few-cycle VIS pulse allows the transfer of quantum-state amplitude to happen within the short lifetime of the 2s2p state. A variable time delay between the VIS and XUV pulses is introduced by a piezo-driven split mirror. Both pulses are focused into a helium-filled target gas cell with a backing pressure of  ~ 200 mbar. The transmitted XUV radiation is dispersed by a grating and detected by a CCD camera. The absorption spectrum is characterized by the optical density \({{{{\rm{OD}}}}}(\omega,\tau )=-{\log }_{10}[I(\omega,\tau )/{I}_{0}(\omega )]\), where I(ω, τ) and I0(ω) are the transmitted and incident XUV spectra, respectively. ω is the frequency, and τ denotes the time delay between the two pulses. The delay convention is such that a positive time delay means that the target helium is first excited by the weak XUV pulse and then exposed to the stronger VIS pulse. The spectral resolution of the spectrometer is 40 meV near 63.7 eV, determined by fitting the spectral profile of sp2,3+ with a natural linewidth of 8.3 meV16. The sub-20 meV energy spacing between the 2p3d and sp2,4− states makes them indistinguishable in our measured spectroscopic data.

Fig. 2: Outline of the attosecond transient absorption measurement.
figure 2

a Illustration of the experimental setup. CCD, charge-coupled device. b Relevant energy-level diagram in helium. c Absorption profile at 1.5-fs time delay (reddish-orange line) along with the result in the absence of the VIS pulse (blue line).

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When only the weak XUV pulse is present, the absorption spectrum shown in Fig. 2c agrees with the reported measurements performed with synchrotron radiation10,16,17,18, with the spectral features of the sp2,n+ series being predominant and the sp2,n− series barely visible. When the additional VIS pulse is employed, the spectral profiles of the sp2,n+ series are modified as shown in Fig. 2c. The variation of the strongly coupled states of the sp2,n+ series was studied earlier and enabled the extraction of amplitude and phase modifications19 and the reconstruction of a correlated two-electron wave packet20 from modified Fano absorption line shapes21.

We now focus on the spectral feature around the weakly coupled 2p3d and sp2,4− resonances at  ~ 64.1 eV, the visibility of which is much enhanced in the presence of the VIS laser pulse. While hardly visible in the XUV-only case, the additional VIS laser turns their relative spectral amplitude comparable to the neighboring lines. We note that these neighboring lines belonging to the sp2,n+ series have much larger couplings to the ground state compared to the 2p3d and sp2,4− states (see Supplementary Table 1). Figure 3a (upper panel) shows the transient absorption spectrum in the vicinity of these two resonances. The spectral profile is barely visible for negative time delays, then starts to become very pronounced in the pulse-overlap region, and gradually fades away with increasing time delay. Furthermore, it exhibits a significant spectral dip below the background absorption by the action of the VIS pulse arriving later in time [lower panel of Fig. 3a], which persists for positive time delays and shows oscillations in amplitude. The observed spectral profile thus resembles window-type resonances, while neither the sp2,4− nor the 2p3d states naturally exhibit such line shapes.

Fig. 3: Experimental and theoretical demonstration.
figure 3

a Upper panel: measured transient absorption spectrum in the vicinity of the 2p3d and sp2,4− states; lower panel: absorption profiles at two representative time delays [blue (reddish-orange) line: negative (positive) time delay] marked as white dashed lines in the upper panel. b Simulated spectrum and absorption profiles after implementing the spectral broadening and the spectrometer resolution.

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Theoretical modeling

To model our experimental observations, we performed a large-scale numerical simulation by solving the two-electron time-dependent Schrödinger equation (TDSE) using the hyperspherical close-coupling method22. We use an XUV pulse with a full width at half maximum (FWHM) duration of 0.2 fs and a peak intensity of 1 × 1011 W/cm2, centered at 60 eV. The VIS pulse has a central wavelength of 700 nm, a FWHM duration of 4 fs, and a peak intensity of 2 × 1012 W/cm2. The generalized photoabsorption cross section23,24 is calculated, which can be both positive and negative and reflects the photon-energy dependent loss or gain for a helium atom interacting with the two-color field. Compared to the case of large negative time delays which corresponds to effectively XUV-only interaction, the generalized absorption cross section for both 2p3d and sp2,4− states gets enhanced by an order of magnitude (see Supplementary Fig. 1). In order to compare with the measured results shown in Fig. 3a, one needs to include the spectral broadening and the finite spectrometer resolution (see ’Data processing’ in Methods). The simulated transient absorption spectrum and its lineouts after this procedure are shown in Fig. 3b, which agree well with the experimental profiles and the dynamical evolution with time delay. The lower contrast of the oscillation observed in the experiment is explained by experimental time-delay jitter and spatial beam inhomogeneity. The 2s2p → 2p2 → 2p3d/sp2,4− two-VIS-photon coupling is identified to be the dominant pathway for the enhanced spectral visibility and its temporal behavior, indicated by the 1.1-fs beating period, which corresponds to the energy difference between 2s2p and 2p3d/sp2,4− states of  ~ 3.9 eV. Its resonant enhancement is further confirmed by the calculation through artificial removal of the 2p2 state (see Supplementary Fig. 2). The finite time-delay window for the observation of the spectral enhancement is linked to the  ~ 17-fs lifetime of the 2s2p state25,26, which serves as the initial state of the two-VIS-photon coupling pathway. We note that due to our experimental grazing-incidence geometry15, the walk-off of the focus of the XUV beam leads to less spatial overlap with the VIS beam in the interaction region at large time delays. This effect gives rise to a faster decay of the experimental absorption signal than the theory results. The intersecting yellowish features shown in the transient absorption spectra in Fig. 3 are hyperbolic sidebands of the sp2,3+ state, which converge to its energy position with increasing time delay and result from its perturbed free-induction decay24,27,28.

Discussion

In summary, we have presented and experimentally verified a general concept to enhance the spectroscopic signal of weak transitions by exploiting a stronger laser-coupled pathway. As an experimental demonstration, we report the first time-resolved observation of the weakly XUV-populated 2p3d and sp2,4− (1Po) doubly excited states in helium by attosecond transient absorption spectroscopy. With tunable coupling lasers, more weakly coupled autoionizing states are expected to be observed and studied by ultrafast time-resolved spectroscopy in helium and other atomic and molecular species, enabling the tracking and coherent control of correlated multielectron dynamics in the time domain. Similar to other well-established methods like cavity-enhanced spectroscopy29 and surface-enhanced infrared spectroscopy30, enhancing single-photon-suppressed transitions in matter can find technological applications in analytical spectroscopy and spectral fingerprinting with enhanced sensitivity in life, chemical, and material sciences. Moreover, with the aid of intermediate states31 or bridge processes32, this concept holds substantial potential for fundamental physics: We envision it to enable identification and efficient driving of extremely weakly coupled states (including nuclear states) of relevance to future atomic clocks and precision spectroscopy for fundamental-physics tests in searches for physics beyond the standard model1,2,3,4,5,6,7,33.

Methods

Transition matrix elements

To compute the transition matrix elements for the doubly excited states, we developed a program to solve the two-electron Schrödinger equation for helium. A similar strategy can be found in the work by Feist34. In short, the angular degree of freedom of the two-electron wave function Ψ(r1, r2) is expanded in a coupled angular basis:

$$\Psi ({{{{{\bf{r}}}}}}_{{{{{\rm{1}}}}}},{{{{{\bf{r}}}}}}_{{{{{\rm{2}}}}}})=\sum\limits_{L,M,{l}_{1},{l}_{2}}{f}_{{l}_{1},{l}_{2}}^{L,M}({r}_{1},{r}_{2})\left\vert {l}_{1}{l}_{2}LM\right\rangle,$$
(3)

with total angular momentum L and magnetic number M. The radial coordinates are discretized with a finite-element discrete-variable representation35. According to R-matrix theory36, the continuum wave function \(\left\vert 1sEl\right\rangle\) is the solution to a linear equation:

$$(H-E-{E}_{1s})\left\vert 1sEl\right\rangle={\left\vert 1s\right\rangle }_{1}\delta ({r}_{2}-{r}_{\max })+\delta ({r}_{1}-{r}_{\max }){\left\vert 1s\right\rangle }_{2},$$
(4)

where H is the Hamiltonian of the He atom. A minimal residual method37 is applied to solve this linear equation. The doubly excited states are approximately represented by the continuum states with the same real part of energy, normalized in a finite box. With the wave function, the transition dipole matrix elements could be computed directly as listed in Supplementary Table 1.

Experimental apparatus

The commercial laser system (FEMTOPOWERTM HE/HR CEP4) delivers 20 fs, 3 mJ pulses with a central wavelength of 780 nm at 3 kHz repetition rate. In order to broaden the spectrum, the laser pulses are guided through a 1.5 m-long double differentially pumped hollow-core fiber, which is filled with helium with a gas pressure of  ~ 2 bar. The pulses are temporally compressed by 7 pairs of chirped mirrors (Ultrafast Innovations PC70) and are characterized by a dispersion-scan (D-scan) setup. The D-scan measurement yields a FWHM pulse duration of 4.6 fs. Afterwards, the few-cycle VIS laser pulses (725 nm central wavelength,  ~ 1 mJ pulse energy) are sent into the vacuum beamline15 to perform the transient absorption measurements. The XUV pulses are produced by high-harmonic generation (HHG) in a cell containing neon gas with a backing pressure of 100 mbar. To account for the plasma-induced blue shift introduced in the HHG process, a VIS pulse with a 700 nm central wavelength and 4-fs FWHM duration is employed in the numerical simulation.

Data processing

In order to compare with the experimental results, the spectral broadening and the finite spectrometer resolution have to be taken into account in the simulation results, especially for the narrow resonances as considered here. To this end, we first convolute the calculated generalized absorption cross section σ(ω) as shown in Supplementary Fig. 1 with a normalized Gaussian function G(ω, Δω1) of FWHM width Δω1 to account for the Doppler broadening (the pressure broadening is much smaller and is hence ignored)

$${\sigma }_{{{{{\rm{D}}}}}}(\omega )=\sigma (\omega ) * G(\omega,\Delta {\omega }_{1}).$$
(5)

Regarding the consideration of spectrometer resolution, we would like to point out that the commonly used approach by directly convolving the cross section with a Gaussian function is inappropriate, since what is recorded by the spectrometer in the experiment is the transmitted laser spectrum. After obtaining the Doppler-broadened cross section σD(ω) from Eq. (5), one should apply the convolution to the transmitted laser spectrum before computing the optical density OD, which takes the following form after applying the linear Beer-Lambert law

$${{{{\rm{OD}}}}}(\omega )=-{\log }_{10}\frac{[{I}_{0}(\omega ){e}^{-{\sigma }_{{{{{\rm{D}}}}}}(\omega )\rho l} * G(\omega,\Delta {\omega }_{2})]}{{I}_{0}(\omega )}.$$
(6)

Here, one assumes the target medium to be homogeneous and the cross section σD space-independent. However, it has been shown that the temporal reshaping of the driving pulse resulting from resonant pulse propagation effects inside the medium leads to reduced absorption compared to what is predicted by the Beer-Lambert law38. To account for this effect and reach a comparable enhancement of resonant OD relative to the background as in the experiment, we used a pathlength-density product ρl = 3.6 × 1017 cm−2 rather than 7.7 × 1017 cm−2 retrieved from the experiment. This explains the smaller background OD shown in Fig. 3b as compared to that in Fig. 3a. For the simulation results shown in the main text, the Doppler broadening Δω1 = 400 μeV and the spectrometer resolution Δω2 = 40 meV are employed.

According to the Beer-Lambert law, the transmitted laser spectrum will be exponentially damped or amplified depending on the sign of the cross section σD. Due to the convolution with the laser spectrum in Eq. (6), absorption features in the transmitted spectrum are highly suppressed as compared to the emission, which underlies the persistent window-like resonant feature in the absorption spectrum despite the comparable amplitudes of positive and negative absorption cross sections.