Experimental Control of Quantum-Mechanical Entanglement in an Attosecond Pump-Probe Experiment

Entanglement is one of the most intriguing aspects of quantum mechanics and lies at the heart of the ongoing second quantum revolution, where it is a resource that is used in quantum key distribution, quantum computing, and quantum teleportation. We report experiments demonstrating the crucial role that entanglement plays in pump-probe experiments involving ionization, which are a hallmark of the novel research field of attosecond science. We demonstrate that the degree of entanglement in a bipartite ion + photoelectron system, and, as a consequence, the degree of vibrational coherence in the ion, can be controlled by tailoring the spectral properties of the attosecond extreme ultraviolet laser pulses that are used to create them.

Figure 1

Setup for measuring the influence of quantum mechanical entanglement on vibrational wave packet dynamics in ${\mathrm{H}}_2^{+}$. (a) Neutral ${\mathrm{H}}_2$ molecules injected into the apparatus through a nozzle incorporated in the repeller (REP) of the velocity map imaging (VMI) spectrometer, are dissociatively ionized using a two-color $\mathrm{XUV}+\mathrm{NIR}$ laser field, where the XUV field consists of a two-pulse pair with a relative delay ${\tau }_{\mathrm{XUV}-\mathrm{XUV}}$, which is followed by the NIR pulse after a delay ${\tau }_{\mathrm{XUV}-\mathrm{NIR}}$. The XUV pulse pair is generated by passing ultrashort NIR pulses through a passively and actively stabilized Mach-Zehnder interferometer prior to their use within a high-harmonic generation gas cell (GC). The XUV and NIR pulses are collinearly combined using a drilled mirror (DM) and focused using a toroidal mirror (TM). The two-pulse XUV spectrum is monitored using an XUV spectrometer. Abbreviations: beam splitters (BS1-BS3); proportional-integral-derivative (PID) feedback for stabilization of ${\tau }_{\mathrm{XUV}-\mathrm{XUV}}$; cameras (Cam1-3); aluminum filter (AF); Extractor electrode (EX); flight tube (FT); microchannel plate (MCP); variable line-space grating (VLG); (b) and (c) typical example of a raw ${\mathrm{H}}^{+}$ VMI image and a slice through the 3D momentum distribution obtained via Abel inversion; (d) XUV spectrum recorded for two delays ${\tau }_{\mathrm{XUV}-\mathrm{XUV}}$, revealing fringes in the XUV spectrum resulting from the interference between the two XUV pulses.

Figure 2

Experimentally observed wave packet dynamics following ionization of ${\mathrm{H}}_2$ by a pair of XUV pulses separated by ${\tau }_{\mathrm{XU}V-\mathrm{XUV}}=29$ and 45 fs: (a)–(f) ${\beta }_0(v_{{\mathrm{H}}^{+}},{\tau }_{\mathrm{XUV}-\mathrm{XUV}},{\tau }_{\mathrm{XUV}-\mathrm{NIR}})$, ${\beta }_2(v_{{\mathrm{H}}^{+}},{\tau }_{\mathrm{XUV}-\mathrm{XUV}},{\tau }_{\mathrm{XUV}-\mathrm{NIR}})$, and ${\beta }_4(v_{{\mathrm{H}}^{+}},{\tau }_{\mathrm{XUV}-\mathrm{XUV}},{\tau }_{\mathrm{XUV}-\mathrm{NIR}})$ obtained by Abel inversion of the experimental VMI images. The vertical axis is given in pixel units of the CCD camera used in the VMI spectrometer, and represents the ${\mathrm{H}}^{+}$ velocity in arbitrary units; (g)–(l) Fourier transform power spectra $\mathrm{FTPS}(v_{{\mathrm{H}}^{+}},{\tau }_{\mathrm{XUV}-\mathrm{XUV}},\omega )$ of ${\beta }_0(v_{{\mathrm{H}}^{+}},{\tau }_{\mathrm{XUV}-\mathrm{XUV}},{\tau }_{\mathrm{XUV}-\mathrm{NIR}})$, ${\beta }_2(v_{{\mathrm{H}}^{+}},{\tau }_{\mathrm{XUV}-\mathrm{XUV}},{\tau }_{\mathrm{XUV}-\mathrm{NIR}})$, and ${\beta }_4(v_{{\mathrm{H}}^{+}},{\tau }_{\mathrm{XUV}-\mathrm{XUV}},{\tau }_{\mathrm{XUV}-\mathrm{NIR}})$, obtained by a Fast fourier transform. The vertical axis coincides with the vertical axis in (a)–(f), whereas the horizontal axis gives the Fourier frequency in ${\mathrm{cm}}^{-1}$, permitting a straightforward assignment to two-level quantum beats and a comparison with available literature values (see Table 1). Importantly, the intensity of the nearest-neighbor quantum beats in the FTPS are significantly different for the two different values of ${\tau }_{\mathrm{XUV}-\mathrm{XUV}}$.

Figure 3

Intensities of peaks in the Fourier transform power spectrum $\mathrm{FTPS}(v_{{\mathrm{H}}^{+}},{\tau }_{\mathrm{XUV}-\mathrm{XUV}},\omega )$ as a function of two-pulse delay ${\tau }_{\mathrm{XUV}-\mathrm{XUV}}$, for 4 selected values of the Fourier frequency: (a)–(c) ${\mathrm{FTPS}}_{{\beta }_0}(v_{{\mathrm{H}}^{+}},{\tau }_{\mathrm{XUV}-\mathrm{XUV}},\omega )$, ${\mathrm{FTPS}}_{{\beta }_2}(v_{{\mathrm{H}}^{+}},{\tau }_{\mathrm{XUV}-\mathrm{XUV}},\omega ),$ and ${\mathrm{FTPS}}_{{\beta }_4}(v_{{\mathrm{H}}^{+}},{\tau }_{\mathrm{XUV}-\mathrm{XUV}},\omega )$ with $\omega$ corresponding to the energy difference between $v^{+}=8$ and 9; (d)–(f) the same, with $\omega$ corresponding to the energy difference between $v^{+}=7$ and 8; (g)–(i), the same, with $\omega$ corresponding to the energy difference between $v^{+}=8$ and 10; (j)–(l) the same, with $\omega$ corresponding to the energy difference between $v^{+}=7$ and 9. The oscillations in the FTPS (experimental data points given by symbols) are fitted to sinusoidal curves (dashed lines, with shaded area according to the standard deviation of the fit parameters), leading to a determination of the oscillation periods shown at the top of each figure. The oscillation periods obtained for ${\mathrm{FTPS}}_{{\beta }_0}(v_{{\mathrm{H}}^{+}},{\tau }_{\mathrm{XUV}-\mathrm{XUV}},\omega )$, ${\mathrm{FTPS}}_{{\beta }_2}(v_{{\mathrm{H}}^{+}},{\tau }_{\mathrm{XUV}-\mathrm{XUV}},\omega ),$ and ${\mathrm{FTPS}}_{{\beta }_4}(v_{{\mathrm{H}}^{+}},{\tau }_{\mathrm{XUV}-\mathrm{XUV}},\omega )$ are combined to obtain values that are compared to available spectroscopic data in column 3 of Table 1.