Above-threshold dissociation of the molecular ion ${\mathrm{HD}}^{+}$ in a moderate-intensity femtosecond laser field from the calculation of time-of-flight spectra

The dissociation of the molecular ion ${\mathrm{HD}}^{+}$ in moderate-intensity laser pulses is studied using the time-dependent quantum wave-packet method. Simulations of the time-of-flight (TOF) neutral atom spectra produced from dissociation of the $v=10$ vibrational state at a laser intensity of ${10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ and from $v=0$ at the intensity of $5\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ are calculated. Good agreement is found with previous experimental results [P. A. Orr et al., Phys. Rev. Lett. 98, 163001 (2007) and J. D. Alexander et al., J. Phys. B 42, 154027 (2009)]. Furthermore, the kinetic energy distribution of the dissociated fragments of molecular ion ${\mathrm{HD}}^{+}$ is studied within the moderate laser-intensity region from ${10}^{12}$ to $5\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ by considering initial vibrational states from $v=0$ to 15. It is found that the above-threshold dissociation could be more obviously observed from the TOF spectra when the initial vibrational state is set as $v=6$ for a moderate-intensity femtosecond laser field.

Figure 1

(a) The adiabatic potential energies of the ${}^2{\mathrm{\Sigma }}_g^{+}$ ($1s{\sigma }_g$) and ${}^2{\mathrm{\Sigma }}_u^{+}$ ($2p{\sigma }_u$) states. (b) The nonadiabatic coupling $P_{12}\left(R\right)$, $Q_{11}\left(R\right)$, and $Q_{22}\left(R\right)$ versus internuclear distance. (c) $d_{11}$ and $d_{22}$ denote the permanent dipole moment of ${}^2{\mathrm{\Sigma }}_g^{+}$ and ${}^2{\mathrm{\Sigma }}_u^{+}$ states. $d_{12}$ displays the transition dipole moment between two electronic states. The relevant molecular data are taken from Ref. [24].

Figure 2

The theoretical TOF spectrum obtained from an initial vibrational state $v=10$ at a 1-keV energy $\mathrm{H}{\mathrm{D}}^{+}$ ion beam at a laser intensity ${10}^{12}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 3

Comparison of the TOF spectrum in the laboratory frame between experimental results (blue solid curve) [22] and our theoretical calculations (red dashed curve). In theoretical calculations, the initial central kinetic energy of the ion beam was $E=1\mathrm{keV}$ with an energy spread $\delta E=30\mathrm{eV}$, a laser intensity $I_0={10}^{12}\mathrm{W}/{\mathrm{cm}}^2$, and an initial vibrational state $v=10$. The corresponding theoretical emission probabilities are shown in the red font on the right.

Figure 4

The red solid line indicates the TOF spectrum in center-of-mass frame arising from H and D fragments in the theoretical simulation. The data points indicate the TOF spectrum in the center-of-mass frame from experimental measurements [21], where the vertical error bars indicate statistical uncertainties at the one sigma level. Vertical lines mark the expected positions of H and D fragments due to absorption of an integer number of photons by the molecular ion $\mathrm{H}{\mathrm{D}}^{+}$. The green dashed curve peaked around roughly 25 ns is the component of the D fragment, and the blue dashed curve peaked around roughly 50 ns is the component of the H fragment. In the theoretical calculations, the initial central kinetic energy of the ion beam was $E=2\mathrm{keV}$ with an energy spread $\delta E=60\mathrm{eV}$, a laser intensity of $I_0=5\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$, and for an initial vibrational state $v=0$.

Figure 5

The kinetic energy distribution of dissociation fragments from initial vibrational state $v=0$ with laser intensity $I_0=5\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$. The solid and dashed curves show the kinetic energy distributions from channel $1s{\sigma }_g$ and channel $2p{\sigma }_u$, respectively. The tick labels on the horizontal axis indicate the energy difference between the initial vibrational level ($v=0$) and the kinetic energy of the dissociated fragments.

Figure 6

The relative kinetic energy distributions of dissociated fragments from different initial vibrational states at a variety of laser intensities. Panels (a–e) denote kinetic energy distributions from channel $1s{\sigma }_g$ with the laser intensity from ${10}^{12}$ to $5\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$. Panels (f–j) denote the corresponding dissociation from channel $2p{\sigma }_u$.

Figure 7

The TOF spectrum starting from the vibrational state $v=6$ at a laser intensity of $I_0=5\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$. (a) The TOF spectrum of neutral fragments in the laboratory frame. (b) The green dashed (peaked around roughly 40 ns) and blue dashed (peaked around roughly 85 ns) curves indicate the TOF spectra of center-of-mass frame of D and H fragments, respectively. The red solid curve indicates the sum of the H and D components.

Figure 8

Diabatic dressed potential energy curves for $\mathrm{H}{\mathrm{D}}^{+}$ in the laser field of wavelength 800 nm. The horizontal solid lines denote the vibrational energy levels $v=0$, 6, and 10 of the ground electronic state ${}^2{\mathrm{\Sigma }}_g^{+}$ ($1s{\sigma }_g$). The blue squares labeled as $a$, $b$, $c$, and $d$ denote the diabatic crossing points of the field-free ground electronic state ${}^2{\mathrm{\Sigma }}_g^{+}$ ($1s{\sigma }_g$) with the first excited repulsive state ${}^2{\mathrm{\Sigma }}_u^{+}$ ($2p{\sigma }_u$) dressed by one, two, three, and four photons, respectively.

Figure 9

Same as Fig. 6 except that the nonadiabatic coupling terms, $P_{ij}$ and $Q_{ij}$ in Eq. (1), are set to be zero.

Figure 10

The kinetic energy distribution of dissociation fragments for initial vibrational state $v=6$ at the laser intensity $I_0=5\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$. (a) The nonadiabatic couplings are neglected; (b) the nonadiabatic couplings are included.

Figure 11

The TOF spectra in center-of-mass frame for initial vibrational state $v=6$ at the laser intensity $I_0=5\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$. The black solid curve denotes the calculation with the nonadiabatic couplings turned off (labeled as BO). For comparison, the red dashed curve, which is the same as the one in Fig. 7, denotes the calculation with consideration of the nonadiabatic couplings (labeled as non-BO).