Search for entanglement in the decay dynamics and anisotropic emission of a pair of $\mathrm{H}\left(2p\right)$ atoms produced by XUV photodissociation of ${\text{H}}_2$

The production of pairs of correlated Lyman-$\alpha$ photons upon XUV excitation of molecular hydrogen is studied using coincidence measurements to evaluate at which level the entanglement is transferred from atoms to photons. By referencing the timing of fluorescence photon detection to the synchrotron light pulse, we were able to determine branching ratios for the $2p+2p$ and $2p+3\ell$ (with $\ell =s,d$) channels separately. Time-dependent analysis of the spectral signature recorded around 33.6 eV and close examination of the doubly excited states lying in the Franck-Condon window confirm prior assignments of the $2p+2p$ being the main dissociation channel of the $Q_2{}^1\mathrm{\Pi }_u$ (1) molecular state. The angular dependence of the two-photon detection probability measured with respect to the polarization axis is found to be in agreement with earlier measurements [Y. Torizuka et al., Phys. Rev. A 99, 063426 (2019)]. A simple model, assuming a transition from Hund's case (a) to Hund's case (c), reproduces the measured angular distributions satisfactorily.

Figure 1

(a) Experimental setup. The interaction region between the synchrotron radiation and the ${\mathrm{H}}_2$ gas is highlighted in purple. In this configuration, the angle between the synchrotron polarization direction $\widehat{\varepsilon }$ and the interdetector axis (aligned along $\widehat{x}$) is given by ${\theta }_{\text{D}}$. (b) Schematic representation of the experimental setup. CFD, constant fraction discriminator; TDC, time-to-digital converter; MCP, multichannel plate; and FC, Faraday cup.

Figure 2

Decay time histograms of $\mathrm{\Delta }{t}_1$ (shaded area, experiment with $1\sigma$ uncertainty interval) and $\mathrm{\Delta }{t}_2$ (black dots with error bars, experiment with $1\sigma$ uncertainty interval) at a ${\mathrm{H}}_2$ pressure of (a) 0.02 Pa and (b) 0.3 Pa. Recording time was adapted to reach the same statistics.

Figure 3

Decay time histograms of $t_f$ (shaded area, experiment with $1\sigma$ uncertainty interval) and $t_s$ (black dots with error bars, experiment with $1\sigma$ uncertainty interval), the respective time of arrival of the first and second photon, at a ${\text{H}}_2$ pressure of (a) 0.02 Pa and (b) 0.3 Pa.

Figure 4

Radiative cascade following photodissociation of ${\mathrm{H}}_2$ into a pair of $\mathrm{H}\left(2p\right)$ atoms. Green arrows indicate the first Lyman-$\alpha$ (${\mathrm{Ly}}_{\alpha }$) photon, red arrows the second Lyman-$\alpha$ photon.

Figure 5

Comparison between theory and experiment for the normalized emission rate of the (a) first and (b) second photon at 0.02 Pa, $h{\nu }_{\text{XUV}}=33.6$ eV, ${\theta }_{\text{D}}={90}^{\circ }$ (see Fig. 1). Shaded area, experiment with $1\sigma$ uncertainty interval; dotted curves, superradiance; dashed curves, $(2p,2p)$ spontaneous emission; solid curves, combined $(2p,2p)$ and $(2p,3d)$ spontaneous emission. Theoretical curves are convoluted by a Gaussian of 120 ps FWHM (see text).

Figure 6

Radiative cascade including $(2p,3\ell )$ channels, with $\ell =s,d$. Green arrows indicate the first Lyman-$\alpha$ (${\mathrm{Ly}}_{\alpha }$) photon, red arrows the second Lyman-$\alpha$ photon, and gray arrows the Balmer-$\alpha$ (${\mathrm{H}}_{\alpha }$) photons. Only the Lyman-$\alpha$ photons are detected in the present experiment.

Figure 7

Comparison between theory and experiment for the normalized photon emission probability of the (a) first and (b) second photon at 0.02 Pa, $h{\nu }_{\text{XUV}}=33.6$ eV, ${\theta }_{\text{D}}={90}^{\circ }$ (see Fig. 1). Shaded area, experiment with $1\sigma$ uncertainty interval; solid blue curves, $2p\text{−}2p$ spontaneous emission component; red dashed curves, $2p\text{−}3\ell$ spontaneous emission component; black dotted curves, total emission signal.

Figure 8

Pressure dependence of the total (black triangles), partial $(2p,2p)$ (blue dots), and $(2p,3d)$ (red squares) coincidence rates measured with vertically polarized (along the $z$ axis) incident photons at 33.6 eV. The partial coincidence rates are obtained using the fit shown in Fig. 7. Quadratic fit (thick lines) and linear fit (thin lines) are shown for the total (dotted lines), the $(2p,2p)$ (blue solid lines), and the $(2p,3d)$ (red dashed lines) components.

Figure 9

Decay dynamics at various photon energies (33.6–37 eV). (a) First photon emission and (b) second photon emission. The increasing contribution of the $(2p,3\ell )$ is clearly visible.

Figure 10

Doubly differential cross section for the emission of two Lyman-$\alpha$ photons as a function of incident photon energy. Polarization is set perpendicular to the detection axis (${\theta }_{\text{D}}={90}^{\circ }$). $(2p,2p)$ partial cross section: solid blue circles, present experiment; open gray triangles, Odagiri et al. [22]; open black circles, Hosaka et al. [28]. $(2p,3\ell )$ partial cross section: Solid red squares, present experiment. $Q_2{}^1\mathrm{\Pi }_u\left(1\right)$: solid blue line, linear dipole matrix element; long dashed blue line, constant dipole matrix element; dot-dashed blue line, matrix element from Borges and Bielschowsky [29]; short dashed blue line, model calculation from Glass-Maujean and Schmoranzer [30]. $Q_2{}^1\mathrm{\Sigma }_u^{+}\left(2\right)$: solid red line, linear dipole matrix element; long dashed red line, constant dipole matrix element; dot-dashed red line, matrix element from Borges and Bielschowsky [29]; short dashed red line, model calculation from Glass-Maujean and Schmoranzer [30].

Figure 11

Singly differential cross section for the emission of Lyman-$\alpha$ (blue circles) or Balmer-$\alpha$ (red squares) photons as a function of the XUV photon energy. Solid circles and squares, this work (${\theta }_{\text{D}}={90}^{\circ }$ and Balmer-$\alpha$ values are deduced from delayed $2p$ emission); open circles and squares, Glass-Maujean [27].

Figure 12

Potential energy curves of the lowest $Q_2{}^1\mathrm{\Pi }_u$ (solid black lines) and ${}^1\mathrm{\Sigma }_u^{+}$ (dashed red lines) autoionizing states of ${\mathrm{H}}_2$. Also shown are the $Q_2{}^1\mathrm{\Pi }_u$ potential curves of Glass-Maujean and Schmoranzer [30] (short dashed black lines), and the potential energy curves of the $2p{\pi }_u{}^2\mathrm{\Pi }_u$ and $2s{\sigma }_g{}^2\mathrm{\Sigma }_g^{+}$ states of ${\mathrm{H}}_2^{+}$ (solid gray lines). The energy difference between the ${}^1\mathrm{\Pi }_u$ and ${}^2\mathrm{\Pi }_u$ states is shown in the inset. The gray area marks the Franck-Condon region. Arrows point to the avoided crossings (see text).

Figure 13

Coincidence yield for different detection geometries and relative orientations of the detectors with respect to the polarization axis given by the angles ${\theta }_{\text{D1}}$ and ${\theta }_{\text{D2}}$. Top: Configuration where the two detectors are facing each other. This configuration corresponds to the measurements performed in this study. All data are averaged over a 0.64 sr solid angle and an interaction length equal to the detector's diameter [17], and scaled for their average to be equal to one. Bottom: Other detector orientations detailed below the graph. Data are scaled using the same factor as for the top panel. Models from this work and from Torizuka et al. [26] are presented in both graphs. Experiment: solid blue circles, this work; open red circles, Ref. [26]. Theory: dot-dashed red curve, Ref. [26]; solid blue line, this work.

Figure 14

Coincident integrated decay rate for various distributions. First photon $P_f$, red; second photon $P_s$, blue. Solid lines, $2p\text{−}3\ell$, reduced coincidence window $t_{max}=\mathrm{\Delta }t$; dash-dotted lines, $2p\text{−}3\ell$, infinite coincidence window $t_{max}=\infty$; dashed lines, $2p\text{−}BG$, reduced coincident window. (a) $2p\text{−}3d, \mathrm{\Delta }t=10\mathrm{ns}$; (b) $2p\text{−}3d, \mathrm{\Delta }t=30\mathrm{ns}$; (c) $2p\text{−}3d, \mathrm{\Delta }t=60\mathrm{ns}$; (d) $2p\text{−}3s, \mathrm{\Delta }t=60\mathrm{ns}$.