## Abstract

We investigate the long-term evolution of D_{2}
^{+} nuclear wave-packet after interacting with a 5fs, 800nm laser pulse at an intensity of 0.5•10^{14}W/cm^{2}. The nuclear wave-packet evolves in field-free condition for very long time after the laser pulse. The collapse and revival of nuclear wave-packet is studied. The scale of the classical time (~25fs), the revival time (~580fs) and the super-revival time (~12ps) are determined from the simulation as well as the calculation. The constituents of long-standing nuclear wave-packet are also analyzed.

©2009 Optical Society of America

## 1. Introduction

The interaction of the molecules with strong femtosecond laser pulses is of particular interest, since the laser pulse duration is comparable to the vibrational period [1]. The H_{2}
^{+} (D_{2}
^{+}) molecular ion is of fundamental interest in atomic and molecular physics. It represents the simplest three-body molecular system in the presence of strong laser field. As a consequence, a number of experiments and simulations have been performed on an H_{2}
^{+} target, as compared to the complexity in the case of H_{2} target. Irrespective of the target however, a number of interesting phenomena have been observed in the molecular hydrogen ion system, including bond-softening [2], bond-hardening [3], quantum encoding [4] and Coulomb explosion from a critical internuclear separation [5,6].

Femtosecond time-resolved Coulomb explosion by now is a frequently used technique to study dynamical processes in molecules. Recent the techniques of femtosecond laser [7] and Cold-target reaction microscopes (COLTRIMS) [8] have made it possible to trace electronic and nuclear motion in real time and also to achieve control over the nuclear wave-packet motion using femtosecond laser pulses [9,10]. Numerous theoretical studies of chirped pulse excitation have been performed, discussing effects as molecular dissociation in strong fields [11], population transfer in molecules [12], nonadiabatic dynamics [13], control of wave-packet evolution [14,15] or cold-molecule formation [16]. Experiments were performed on small molecules addressing phenomena as strong field dissociation [17], tailoring of wave packets [18], or high harmonic control [19]. Those works provide comprehensive understanding of coherent control of neutral molecular and molecular ions [20].

The evolution of quantum interference effects of nuclear wave packet created by an ultrashort laser pulse in a molecular system is an interesting topic (see e.g., the overview [21] and references therein). In the system quantum interference at an intersection of two potential energy curves could cause the nuclear vibration probability to oscillate. Later on, such interference was invoked to explain unexpectedly long-lived rotational and vibrational states in dissociative systems. The analysis on a molecular system makes clear that quantum wave-packet interference playes a decisive role in the molecular dissociation process. In this paper, we take attention on the long-term evolution of D_{2}
^{+} after interacting with an intense femtosecond laser pulse. Through solving time-dependent Shrödinger equation, we propagate the nuclear wave-packet as long as few nonaseconds and trace the evolution of vibration states. The phenomenon of nuclear wave-packet revivals will be investigated in the time domain and frequency domain. The constituents of nuclear wave-packet are also analyzed. Those findings provide a potential pathway of coherent control for a molecular system. The method using time-resolved Coulomb explosion to trace the evolution of nuclear wave-packet is suggested.

## 2. Method

We solve the time-dependent Schrödinger equation of hydrogen molecular ion (D_{2}
^{+}) system numerically on a grid using the Crank-Nicholson split-operator method [22]. The molecular ion D_{2}
^{+} is described in a reduced dimensionality collinear model, which is described in terms of the electronic coordinate *z* relative to the center of mass of the nuclei and the inter-nuclear distance *R*. Atomic units (a. u.) (ħ = me = e = 1) are used unless stated otherwise throughout this paper. The total Hamiltonian of D_{2}
^{+} system consists of the time-dependent field-free Hamiltonian${H}_{0}(R,t)$ and the interaction term $V(z,t)$:

*Tel*and

*Ti*are the electron and nuclear kinetic energy operator respectively, ${V}_{ei}$ is the soft-core Coulomb potential, and

*E*(

*t*) is the electric field of laser pulse. The interaction of the electron with the laser field is given in dipole approximation in the length gauge. The linear polarized laser field is used, therefore only the coordinates in the laser polarization plane play a decisive role. In order to simplify the calculation, the rotation of molecular ion is not taken into consideration. The form of the soft-core Coulomb potential is given by ${V}_{ei}=-1/\sqrt{{z}_{-}{}^{2}+1}-1/\sqrt{{z}_{+}{}^{2}+1}$, with ${z}_{\pm}=z\pm R/2$. The electric field waveform of laser pulse is described with$E(t)={\epsilon}_{0}{e}^{-{t}^{2}/{\tau}^{2}}\mathrm{cos}(\omega t+{\phi}_{0})$, where${\epsilon}_{0}$is the laser field strength,

*ω*is the carrier frequency and ${\phi}_{0}$is the carrier-envelope phase (CEP). The model includes the non-Born–Oppenheimer coupling between the electronic and nuclear motion. In the calculation, we imply the time step Δ

*t*= 1 and space step Δ

*r*= 0.05 in the range of

*R*= 0.05 to

*R*= 20. Due to the computation limitation, we can propagate the nuclear wave-packet up to few nanoseconds after femtosecond laser pulses. The probability density of nuclear wave-packet is calculated through:

*ρ*is time-dependent density matrix:

_{km}Thus the probability density includes two parts. One is the diagonal terms, which is time independent. Another part is coherent time-dependent off diagonal terms. The components of wave-packet are analyzed through Fourier transform of the temporal evolution of the expectation inter-nuclear distance.

## 3. Results and discussions

Figure 1
illustrates the coherent nuclear wave-packet evolution of D_{2}
^{+} up to 4ps, which is following the vertical transition from D_{2}
^{+} from ground vibration state ν = 0. In the simulation we use a laser pulse with an intensity of 0.5•10^{14}W/cm^{2} in a duration of 5 fs (Full With at Half Maximum) at the wavelength of 800nm. The CEP ${\phi}_{0}$ is zero in the simulations. The laser duration is chosen less than the classical time in order to resonantly excite the vibration states. The laser intensity is chosen to avoid the strong double ionization channel. The population density history represents the time evolution of a coherent superposition of states.

From Fig. 1, one can find that in the region *t*<100fs the dissociation channel is observed. In the time region of 450 ≤ *t* ≤ 650 fs, the probability density exhibits a 25 fs revival modulation with maximum contrast around 550 fs, which is the result of vibrational revival and is consistent with recent experiments findings [9]. Between 200 ≤ *t* ≤ 350 fs a half fractional revival is also observed and the period is about 12 fs. One can find the detailed features in subplots (a) and (b). After a few oscillations, the anharmonicity of the potential curve, i.e., the nonequal spacing of the vibrational levels, leads to the collapse of the wave packet. For 720 fs < *t <* 920 fs, there is another fractional revival wave-packet. Here the wave-packet consists essentially of two copies of the initial one, each shifted against the other by half of the vibrational period. Dynamical interference between these two counter-propagating waves produce a frequency doubled periodic pattern compared to the fundamental vibrational. The vibrational wave-packet revives again in the time region of 920 fs < *t <* 1300 fs. In general, collapse and revival phenomena are due to a de-phrasing and re-phrasing of a quantum system.

The revival patterns substantially overlap, resulting in constructive and destructive interferences, as indicated in Fig. 1. Then the revival patterns no longer represent individual full revivals, but their superposition gives rise to new structures. The anharmonicity of the vibrational potential provides an auxiliary and independent mechanism for revivals in the molecular vibration. We also illustrate the expectation inter-nuclear value $\u3008R\u3009$ and the auto-correlation function $\left|\u3008\phi (R,t=0)|\phi (R,t)\u3009\right|$ from the starting time to about 4ps in Fig. 2 . The time history of the expectation inter-nuclear value $\u3008R\u3009$can provide more direct information of the wave-packet evolution.

In order to have a look on the long-term evolution of nuclear wave-packet, the time history of the expectation inter-nuclear value $\u3008R\u3009$ up to 500ps is shown in Fig. 3 . Surprisely, one can find that the vibrational states can still revive. The inset (a) in Fig. 3 indicates that the interference stripes reappear, indicating a restored fractional periodicity of the wave-packet motion after about 12 ps. Already after a few oscillations, i.e., the non-equal spacing of the vibrational levels populated by the laser pulse, leads to the de-phasing, or collapse of the wave-packet, reflected in the apparent degeneration of the vibrational structures. However, after about every 12ps, the stripes reappear, indicating a restored periodicity of the wave-packet motion. The inset (b) in Fig. 3 shows the vibration period of the revival state is still about 25fs. This phenomenon is known as a quantum wave-packet super-revival [21]. Therefore, a longtime coherence is preserved in this interaction process.

A localized wave packet is excited with an energy spectrum which is tightly spread around a large central value of the vibrational number *v*
_{0}, i.e., *v*
_{0}>>Δ*v*>>1. In this case, we can expand the individual energy eigenvalues of *E*(*v*) using Talor formula,

Thus, the time-dependence of each individual vibrational eigenstate is through the factors,

_{2}

^{+}system. Here

*T*is corresponding to the vibration period,

_{cl}*T*gives the revival period and

_{rev}*T*represents the high-order super-revival time.

_{Super}For systems with purely quadratic energy dependence on a single quantum number, there are no independent time scales longer than the revival time, and the pattern of fractional and full revivals will repeat itself indefinitely with the *Trev* time scale. However, for a realistic system, with higher order terms in the expansion in Eq. (5), the super-revival time *Tsuper* becomes very important. In this case, the qualitatively new patterns of revival behavior, with periodicities in the motion of the packet characterized by periods which are fractions of *Trev*, giving a self-similar structure to the auto-correlation function plots for *t* >*Trev*. The wave-packet behavior on the *Tsuper* time scale is similar to that of the fractional revival structures seen on the *Trev* scale, with integral multiples of *Tsuper* appearing, explicitly due to the presence of the third-derivative term. According our simulation, we can roughly give about *T _{cl} ≈* 25 fs,

*T*580 fs and

_{rev}≈*T*12ps for the interaction system. The simulated results are consist with the calculated values.

_{Super}≈The Fourier transform spectrum of the evolution possibility of the nuclear wave-packet is performed, which is indicated in Fig. 4
. The frequency components present in the vibrational wave-packet are beats between vibrational states *v* and *v* ± 1, *v* ± 2, *v* ± 3. The Fourier amplitudes thus return a message of vibrational population. One can see that the largest population is through the selection rule of △*v* = 1.

Especially for the Franck–Condon distribution, known to overestimate the population of the higher excited vibrational states, the series of different Δν are seen to overlap each other. In particular, contributions from higher Δν>2 series are seen to reach into the range of frequencies for Δν = 2, even reach into the range of frequencies for Δν = 1. Clearly, without the presence of an external perturbation after the laser pulse, the population possibility is explicitly time–independent. The nuclear wave function will evolve on the adiabatic binding D_{2}
^{+} 1sσg potential curve. The anharmonicity of the potential results in a quick de-phasing of the wave-packet and only after a long propagation time the vibrational states overlap in–phase, leading to wave function revivals and fractional revivals.

Even without laser fields, nonadiabatic transitions take place frequently in chemistry of electronically excited states. Here, nonadiabatic effect is directly taken into account to nuclear wave-packet dynamics in the simulations and indeed nonadiabatic nuclear wave-packet dynamics is crucial in the molecular science.

## 4. Conclusions and applications

In a real experiment only the kinetic energy release after a probe pulse is measurable, which can be mapped using Coulomb’s law to reconstruct the evolution of the probability density$\rho (R,t)=\left|\varphi {(R,t)}^{2}\right|$. In a general application, the laser pulse duration shorter than the vibrational motion establishes a time-dependent variation of the refractive index in a Raman-active molecular medium. By employing the vibrational mode of a hydrogen molecular ion, it may be possible to spectrally broaden and self-compress, potentially to the laser sub-cycle level. The long-term evolution of nuclear wave-packet can also be retrieved using Coulomb explosion imaging pump-probe scheme method. The full vibration states involved in the evolution can be measured. Besides, intense laser chemistry is now under strong focus, where laser-induced nonadiabatic couplings between electronic and nuclear wave-packets are expected to frequently take place again in a very short time scale. The coupling between the electronic and nuclear wave-packets will be great interesting for both theory and experiment in the future.

In summary, we have investigated the long-term evolution of D_{2}
^{+} nuclear wave-packet after interacting with strong femtosecond laser field. The vibrational states revive a very long time after the interaction. The classical time, the revival time and the super-revival time for this system are simulated and calculated respectively. We find that the classical time, the revival time and super-revival time of the nuclear wave-packet is about 25fs, 580fs and 12ps respectively. Those time scales can be measured through precisely Coulomb explosion pump-probe imaging techniques. The components of the vibrational wave-packet are analyzed. We find that different Δν are seen to overlap each other. The goal of ultrafast coherent control of smallest hydrogen molecular ion is close at hand with the current advanced technique of ultrashort pulse lasers.

## Acknowledgement:

Yunquan Liu thanks the support through “Bairen” program of Peking University.

## References and links

**1. **A. H. Zewail, “Femtochemistry: Atomic-Scale Dynamics of the Chemical Bond,” J. Phys. Chem. **104**, 5660–5694 (2000). [CrossRef]

**2. **A. Zavriyev, P. H. Bucksbaum, J. Squier, and F. Saline, “Light-induced vibrational states in H_{2}^{+} and D_{2}^{+} in intense laser fields,” Phys. Rev. Lett. **70**(8), 1077–1080 (1993). [CrossRef] [PubMed]

**3. **L. J. Frasinski, J. H. Posthumus, J. Plumridge, and K. Codling, “Manipulation of Bond Hardening in H_{2}^{+} by Chirping of Intense Femtosecond Laser Pulses,” Phys. Rev. Lett. **83**(18), 3625–3628 (1999). [CrossRef]

**4. **D. S. Murphy, J. McKenna, C. R. Calvert, I. D. Williams, and J. F. McCann, “Ultrafast coherent control and quantum encoding of molecular vibrations in D_{+}^{2} using intense laser pulses,” N. J. Phys. **9**(8), 260 (2007). [CrossRef]

**5. **Ch. Ellert, H. Stapelfeldt, E. Constant, H. Sakai, J. Wright, D. M. Rayner, and P. B. Corkum, “Observing Molecular Dynamics with Timed Coulomb Explosion Imaging,” Phil. Trans. R. Soc. A **356**, 329 (1998). [CrossRef]

**6. **J. H. Posthumus, J. Plumridge, P. F. Taday, J. H. Sanderson, A. J. Langley, K. Codling, and W. A. Bryan, “Sub-pulselength time resolution of bond softening and Coulomb explosion using polarization control of laser-induced alignment,” J. Phys. B **32**(5), 029 (1999). [CrossRef]

**7. **A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature **421**(6923), 611–615 (2003). [CrossRef] [PubMed]

**8. **J. Ullrich, R. Moshammer, A. Dorn, R. Dörner, L. P. H. Schmidt, and H. Schmidt-Böcking, “Recoil-ion and electron momentum spectroscopy: reaction-microscopes,” Rep. Prog. Phys. **66**(9), 1463–1545 (2003). [CrossRef]

**9. **Th. Ergler, A. Rudenko, B. Feuerstein, K. Zrost, C. D. Schröter, R. Moshammer, and J. Ullrich, “Spatiotemporal Imaging of Ultrafast Molecular Motion: Collapse and Revival of the D_{2}^{+} Nuclear Wave Packet,” Phys. Rev. Lett. **97**(19), 193001 (2006). [CrossRef] [PubMed]

**10. **B. Feuerstein, T. Ergler, A. Rudenko, K. Zrost, C. D. Schröter, R. Moshammer, J. Ullrich, T. Niederhausen, and U. Thumm, “Complete characterization of molecular dynamics in ultrashort laser fields,” Phys. Rev. Lett. **99**(15), 153002 (2007). [CrossRef] [PubMed]

**11. **S. Chelkowski, A. D. Bandrauk, and P. B. Corkum, “Efficient molecular dissociation by a chirped ultrashort infrared laser pulse,” Phys. Rev. Lett. **65**(19), 2355–2358 (1990). [CrossRef] [PubMed]

**12. **J. Manz, H. Naundorf, K. Yamashita, and Y. Zhao, “Quantum Model Simulation of Complete S_{0} → S_{1} Population Transfer by Means of Intense Laser Pulses with Opposite Chirp,” J. Chem. Phys. **113**(20), 8969–8980 (2000). [CrossRef]

**13. **P. Král, I. Thanopulos, and M. Shapiro, “Colloquium: Coherently controlled adiabatic passage,” Rev. Mod. Phys. **79**(1), 53–77 (2007). [CrossRef]

**14. **B. Feuerstein and U. Thumm, “Mapping of coherent and decohering nuclear wave-packet dynamics in D_{2}^{+} with ultrashort laser pulses,” Phys. Rev. A **67**(6), 063408 (2003). [CrossRef]

**15. **Th. Niederhausen and U. Thumm, “Controlled vibrational quenching of nuclear wave packets in D_{2}^{+},” Phys. Rev. A **77**(1), 013407 (2008). [CrossRef]

**16. **J. Vala, O. Dulieu, F. Masnou–Seeuws, P. Pillet, and R. Kosloff, “Coherent control of cold-molecule formation through photoassociation using a chirped-pulsed-laser field,” Phys. Rev. A **63**(1), 013412 (2000). [CrossRef]

**17. **C. Trump, H. Rottke, and W. Sandner, “Strong-field photoionization of vibrational ground-state H_{2}^{+} and D_{2}^{+} molecules,” Phys. Rev. A **59**, 2858–2863 (1999). [CrossRef]

**18. **J. M. Papanikolas, R. M. Williams, and S. R. Leone, “Manipulation of rovibrational wave packet composition in the Li_{2}*E*(^{1}Σ_{g}^{+}) shelf state using intermediate state selection and shaped femtosecond laser pulses,” J. Chem. Phys. **107**(11), 4172–4178 (1997). [CrossRef]

**19. **S. Baker, J. S. Robinson, C. A. Haworth, H. Teng, R. A. Smith, C. C. Chirilǎ, M. Lein, J. W. G. Tisch, and J. P. Marangos, “Probing Proton Dynamics in Molecules on an attosecond time scale,” Science **312**(5772), 424–427 (2006). [CrossRef] [PubMed]

**20. **H. Niikura, D. M. Villeneuve, and P. B. Corkum, “Stopping a vibrational wave packet with laser-induced dipole forces,” Phys. Rev. Lett. **92**(13), 133002 (2004). [CrossRef] [PubMed]

**21. **R. W. Robinett, “Quantum wave packet revivals,” Phys. Rep. **392**(1-2), 1–119 (2004). [CrossRef]

**22. **J. Crank, P. Nicolson, and D. R. Hartree, “A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type,” Proc. Camb. Philos. Soc. **43**(01), 50–67 (1947). [CrossRef]

**23. **G. Hunter, A. W. Yau, and H. O. Pritchard, “Rotation-vibration level energies of the hydrogen and deuterium molecule-ions,” At. Data Nucl. Data Tables **14**, 11–20 (1974). [CrossRef]