GoKawiilExperimental details
Both coherent and quantum light sources are pumped with the same femtosecond laser pulse (790 nm, 28 fs, 10 kHz) generated by a Ti:sapphire multipass amplifier laser system (Femtolaser). The coherent light source centred at 1,580 nm and with a pulse duration of 70 fs, is produced by a commercial optical parametric amplifier (Light Conversion, TOPAS-Prime). For the quantum BSV light source, the pump beam collimated to a 4-mm diameter is propagated through two cascaded 3-mm BBO crystals. Both BBO crystals are cut for type-I collinear frequency-degenerate phase matching to generate high-gain parametric down-conversion. Here, the optical axes are oriented oppositely in the horizontal plane to minimize the spatial walk-off. The distance between two crystals is set at 80 cm so that only the spatial mode with the lowest diffraction undergoes the phase-sensitive amplification. The pulse duration of the BSV light is measured to be approximately 150 fs using the technique of cross-correlation frequency-resolved optical gating based on sum-frequency generation of the to-be-calibrated BSV pulse and a reference infrared pulse. The effect of different pulse durations for coherent and BSV lights is discussed in the section ‘Effect of pulse duration’.
The second-order correlation function g(2) of the generated BSV is measured using the standard Hanbury Brown–Twiss technique. The BSV source is split at a nonpolarizing 50/50 beam splitter into two channels, the photon number distributions of which are measured with two fast InGaAs photodiodes. The signals from the photodiodes are recorded by a multichannel oscilloscope and then integrated over time to present the total photon numbers. Finally, the second-order correlation function is calculated by \({g}^{(2)}=\langle {n}_{1}{n}_{2}\rangle /(\langle {n}_{1}\rangle \langle {n}_{2}\rangle )\), where n 1 and n 2 are photon numbers measured for two channels. Considering the typical measured g(2) value of 1.5, there are multiple frequency modes amplified in the BSV generation process, which has been further confirmed by a spectral filtering of the BSV light using a standard 4-f monochromator (see section ‘Mode analysis of photon number statistics’ for details). It is worth noting that to preserve the peak intensity required for tunnelling ionization, no spectral filtering is applied in our photoionization experiments. The details of controlling the correlation function g(2) and validating the multi-mode BSV light are demonstrated in the section ‘Mode analysis of photon number statistics’.
The beam diameters for both the coherent light and the BSV light are approximately 3 mm and are tightly focused by a concave silver mirror (focal length f = 75 mm) inside an ultrahigh-vacuum chamber onto the sodium vapour jet. The sodium vapour is produced in a resistively heated crucible at 170 °C and collimated by a 2-mm skimmer. The peak intensity of the coherent light in the interaction region is estimated to be 1 × 1013 W cm−2. The corresponding Keldysh parameter is calculated to be 1.48, which places the ionization process in the non-adiabatic tunnelling regime. The photoelectron energy spectrum (Fig. 2b) featuring a smooth structure and the absence of resonant peaks61 also indicates that ionization proceeds through a tunnelling pathway, without significant population of intermediate states such as Na(4s). This distinguishes our results from regimes in which atomic saturation and resonances suppress quantum-optical enhancement effects62 and allows for an observation of the BSV-induced enhancement.
The strong-field tunnelling ionization of a single atom generates both photoelectron and photoion, whose momenta are coincidently measured using the cold-target recoil ion momentum spectroscopy reaction microscope52,53. These charged particles are accelerated by a homogeneous electric field (7 V cm−1) with the assistance of a weak magnetic field (11 G) and finally strike the detectors at the opposite ends of the spectrometer. The momenta of the emitted electron p(ele) and the corresponding parent ion p(ion) for each laser–atom interaction event are reconstructed from the measured time of flight and position of impact during the offline analysis. The principle of momentum conservation in the centre-of-mass frame dictates that the momentum sum of the electron and ion from the same interacting atom should vanish, that is, p(ele) + p(ion) = 0. In practice, the coincidence condition of \(| {p}_{z}^{({\rm{e}}{\rm{l}}{\rm{e}})}+{p}_{z}^{({\rm{i}}{\rm{o}}{\rm{n}})}| < 0.2\,{\rm{a.u}}.\) along the time-of-flight axis of the spectrometer, which has the highest momentum resolution, is used to select the right coincidence events from other background signals or false coincidences.
Both coherent and quantum light sources are converted from linear to elliptical polarization by a broadband quarter-wave plate. By rotating the fast axis of the plate relative to the initial linear polarization, the relative intensity between the two orthogonal components is controlled while introducing a fixed π/2 phase shift, thereby producing the desired ellipticity of 0.7, which is appropriate for performing angular streaking analysis using the accessible BSV light (see section ‘Choice of elliptical polarization’ for details).
QADK theory
The semiclassical ADK theory57,58,59, commonly used in strong-field ionization, treats the electron quantum mechanically while describing the light as a classical field. It represents the adiabatic limit of the strong-field approximation12,63,64 and the Perelomov–Popov–Terent’ev theory65,66,67,68. In this framework, the doubly differential ionization rate is given by
$${\varGamma }_{{\rm{A}}{\rm{D}}{\rm{K}}}({p}_{{\rm{e}}},t)=F(t)\exp \left\{-\frac{2{[2{I}_{{\rm{p}}}+{({p}_{{\rm{e}}}-A(t))}^{2}]}^{3/2}}{3F(t)}\right\},$$ (2)
where Coulomb-related prefactors are not included here for consistency. For circular polarization, the time dependence factorizes out, allowing us to directly replace A(t) with A 0 and F(t) with F 0 . For elliptical polarization, we replace A(t) with εA 0 and F(t) with F 0 for ionization along the major axis or replace A(t) with A 0 and F(t) with εF 0 for that along the minor axis. For simplicity of the expression, we adopt the substitution A(t) → A 0 and F(t) → F 0 , and the singly differential rate is expressed as
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