Molecules in Strong Laser Fields

Abstract

Molecules, compared to atoms, have additional degrees of freedom. Not only do the electrons move around the nuclei, but also the nuclei move relative to each other. This allows for a multitude of new phenomena, already without the action of external fields.

Appendices

5.A Relative and Center of Mass Coordinates for H$_2^+$

In order to derive the Hamiltonian for H$_2^+$ in a laser field, we follow Hiskes’s general treatment of diatomic molecules [104]. Specializing to the single electron case, the Hamiltonian in length gauge and atomic units is first expressed by using the coordinates of the nuclei $\varvec{R}_{a}$, $\varvec{R}_{b}$ and of the electron, $\varvec{r}_\mathrm{e}$, according to

$$\begin{aligned} \hat{H}_{\mathrm{mol}}=-\frac{1}{2}\left\{ \Delta _{a}/M_\mathrm{p} +\Delta _{b}/M_\mathrm{p}+\Delta _\mathrm{e}\right\} +V_1 -\mathcal{E}(t)[Z_{a}+Z_{b}-z_\mathrm{e}], \end{aligned}$$

(5.183)

with the proton mass $M_\mathrm{p}$, where $V_1=V_\mathrm{CC}+1/R$, with $V_\mathrm{CC}$ from (5.25), contains all Coulomb interaction terms and the laser is polarized in z-direction.

Now center of mass and relative coordinates

$$\begin{aligned} \varvec{R}_\mathrm{S}= & {} \frac{M_\mathrm{p}\varvec{R}_{a} +M_\mathrm{p}\varvec{R}_{b}+\varvec{r}_\mathrm{e}}{M_\mathrm{S}}, \end{aligned}$$

(5.184)

$$\begin{aligned} \varvec{R}= & {} \varvec{R}_{a}-\varvec{R}_{b}, \end{aligned}$$

(5.185)

$$\begin{aligned} \varvec{r}_i= & {} \varvec{r}_\mathrm{e}-\frac{\varvec{R}_{a}+\varvec{R}_{b}}{2} \end{aligned}$$

(5.186)

are introduced. $M_\mathrm{S}=2M_\mathrm{p}+1$ is the total mass of the system (in a.u.) and the coordinate of the electron is measured relative to the center of mass of the nuclei.

In matrix form the old and the new coordinates are related by

$$\begin{aligned} \left( \begin{array}{c} \varvec{R}_\mathrm{S} \\ \varvec{R} \\ \varvec{r}_i \end{array} \right) = \left( \begin{array}{ccc} \frac{M_\mathrm{p}}{M_\mathrm{S}}&{}\frac{M_\mathrm{p}}{M_\mathrm{S}}&{}\frac{1}{M_\mathrm{S}} \\ 1&{}-1&{}0 \\ -1/2&{}-1/2&{}1 \end{array} \right) \left( \begin{array}{c} \varvec{R}_{a} \\ \varvec{R}_{b} \\ \varvec{r}_\mathrm{e} \end{array} \right) . \end{aligned}$$

(5.187)

With the help of the inverse matrix, the back transformation can be derived, which amounts to

$$\begin{aligned} \left( \begin{array}{c} \varvec{R}_{a} \\ \varvec{R}_{b} \\ \varvec{r}_\mathrm{e} \end{array} \right) = \left( \begin{array}{ccc} 1&{}1/2&{}-\frac{1}{M_\mathrm{S}} \\ 1&{}-1/2&{}-\frac{1}{M_\mathrm{S}} \\ 1&{}0&{}\frac{2M_\mathrm{p}}{M_\mathrm{S}} \end{array} \right) \left( \begin{array}{c} \varvec{R}_\mathrm{S} \\ \varvec{R} \\ \varvec{r}_i \end{array} \right) . \end{aligned}$$

(5.188)

Finally, not only the old coordinates, but also their time derivatives in the classical form of the Hamiltonian are expressed in terms of the new coordinates. It turns out that the center of mass with charge e moves “freely” in the electrical field [104]. The relative motion, however, is governed by the Hamiltonian

$$\begin{aligned} \hat{H}_\mathrm{{rel}}=-\frac{1}{2}\left\{ \Delta _R/M_\mathrm{r}+\Delta _i/m_i\right\} +V_1+\mathcal{E}(t)\left[ 1+1/M_\mathrm{S}\right] z_i, \end{aligned}$$

(5.189)

where the reduced masses $M_\mathrm{r}=M_\mathrm{p}/2$ and $m_i=\frac{2M_\mathrm{p}}{M_\mathrm{S}}$ have been introduced and the Coulomb interaction is expressed in terms of the relative coordinates.

In the case of H$_2^+$, no kinetic couplings between the different degrees of freedom are present, nor does the field couple directly to the relative coordinate of the nuclei, which is not true in the general case [104].

5.B Perturbation Theory for Two Coupled Surfaces

In the case of a laser driven two level system with a $2\,\times \,2$ (matrix-) Hamilton operator

$$\begin{aligned} \hat{\mathbf{H}}=\hat{\mathbf{H}}_0+\hat{\mathbf{W}}(t), \end{aligned}$$

(5.190)

perturbation theory is best performed in the interaction picture of Sect. 2.2.4. In first order and after back transformation to the Schrödinger picture, analogous to (2.104),

$$\begin{aligned} |\varvec{\chi }(t)\rangle =\mathrm{e}^{-\mathrm{i}\hat{\mathbf{H}}_0t}|\varvec{\chi }(0)\rangle +\frac{1}{\mathrm{i}}\int _0^t\mathrm{d}t'\mathrm{e}^{-\mathrm{i}\hat{\mathbf{H}}_0(t-t')} \hat{\mathbf{W}}(t')\mathrm{e}^{-\mathrm{i}\hat{\mathbf{H}}_0t'}|\varvec{\chi }(0)\rangle \end{aligned}$$

(5.191)

can be written for the vector valued wavefunction in atomic units.

Invoking the Born-Oppenheimer approximation , we assume that the unperturbed Hamiltonian $\hat{\mathbf{H}}_0$ has only diagonal elements $\hat{H}_\mathrm{g},\hat{H}_\mathrm{e}$. Furthermore, the initial wavefunction shall be restricted to the electronic ground state

$$\begin{aligned} |\varvec{\chi }(0)\rangle =\left( \begin{array}{c} |\chi _\mathrm{g}(0)\rangle \\ 0 \end{array} \right) \;. \end{aligned}$$

(5.192)

Under the influence of the perturbation (having only off diagonal elements), the component of the wavefunction

$$\begin{aligned} |\varvec{\chi }(t)\rangle =\left( \begin{array}{c} |\chi _\mathrm{g}(t)\rangle \\ |\chi _\mathrm{e}(t)\rangle \end{array} \right) \end{aligned}$$

(5.193)

in the excited electronic state as a function of time is the desired quantity. Under the assumptions mentioned above, for this quantity the golden rule expression in position representation

$$\begin{aligned} \chi _\mathrm{e}(\varvec{R},t)=\frac{1}{\mathrm{i}}\int _0^t\mathrm{d}t'\mathrm{e}^{-\mathrm{i}\hat{H}_\mathrm{e}(t-t')} \hat{W}_{\mathrm{e.g.}}(\varvec{R},t')\mathrm{e}^{-\mathrm{i}\hat{H}_\mathrm{g}t'}\chi _\mathrm{g}(\varvec{R},0) \end{aligned}$$

(5.194)

is found. Here the definitions

$$\begin{aligned} \hat{W}_{\mathrm{e.g.}}(\varvec{R},t)= \langle \mathrm{e}(\varvec{R}) |\hat{W}(t)| \mathrm{g}(\varvec{R})\rangle \qquad \hat{H}_{j}= \langle j(\varvec{R}) |\hat{H}_0| j(\varvec{R})\rangle \end{aligned}$$

(5.195)

of the matrix elements of the (electronic plus nuclear) Hamiltonian¹⁶ have been introduced, and j can either be e(xited) or g(round).

The physical interpretation of this final result is straightforward. The wavefunction propagates for a time $t'$ on the lower surface is then multiplied by the perturbation and propagates on the upper surface until the final time t. All the possibilities to split the time interval [0, t] have to be integrated over.

5.C Reflection Principle of Photodissociation

The dynamical reflection principle plays a major role for the interpretation of the photoelectron spectrum in a pump-probe experiment. It has an analog in the field of photodissociation. In [23] it is shown that the absorption spectrum of photodissociation is given by the Fourier transform of the auto-correlation function of the ground state wavepacket,

$$\begin{aligned} \chi _\mathrm{g}(R)=\left( \frac{2\alpha _R}{\pi }\right) ^{1/4}\mathrm{e}^{-\alpha _R(R-R_\mathrm{e})^2}\;, \end{aligned}$$

(5.196)

approximated by a Gaussian centered around $R_\mathrm{e}$ and with inverse width parameter $\alpha _R$, that is instantaneously lifted to the excited state, where it is evolving in time.

In order to perform analytic calculations, this antibinding surface is approximated by a straight line

$$\begin{aligned} V(R)\approx V_\mathrm{e}-V_R(R-R_\mathrm{e}). \end{aligned}$$

(5.197)

Using the short-time approximation (i.e., neglecting the kinetic energy) the wavefunction on the antibinding surface is given by

$$\begin{aligned} \chi _\mathrm{e}(R,t)\sim \mathrm{e}^{-\mathrm{i}[V_\mathrm{e}-V_R(R-R_\mathrm{e})]t} \mathrm{e}^{-\alpha _R(R-R_\mathrm{e})^2}\;. \end{aligned}$$

(5.198)

The Fourier transformation of the auto-correlation $c(t)=\langle \chi _\mathrm{e}(0)|\chi _\mathrm{e}(t)\rangle$ can be done analytically, yielding the absorption spectrum

$$\begin{aligned} \sigma (E)\sim \frac{\mathrm{e}^{-2\beta (E-V_\mathrm{e})^2}}{V_R} \end{aligned}$$

(5.199)

with $\beta =(V_R^2/\alpha _R)^{-1}$. The same result can also be obtained by a purely classical calculation [23]. The maximum of the spectrum is at $E=V_\mathrm{e}$ and its FWHM $\Delta E= V_R\Delta R$ is proportional to the negative slope of the antibinding surface and the FWHM of the squared initial wavepacket. In Fig. 5.50 it is shown that the reflection of the squared initial wavepacket at the antibinding surface yields the spectrum.

Fig. 5.50

Reflection principle of photodissociation [23]. $\Delta E$ is the FWHM of the absorption spectrum, whereas $\Delta R$ is the FWHM of the absolute square of the initial wavepacket

5.D The Undriven Double-Well Problem

Figure 5.51 shows the unperturbed double-well potential,

$$\begin{aligned} V_\mathrm{DW}(x)\equiv -\frac{1}{4}x^{2}+\frac{1}{64 D}x^{4}, \end{aligned}$$

(5.200)

in the units introduced in Sect. 5.5.1 for $D=2$, including the five energy eigenvalues which lay below the barrier.

The coherent tunneling of a particle in the double-well potential emerges by considering an initial state that is a superposition of the two lowest (real-valued) eigenfunctions, $\chi _{1}(x), \chi _{2}(x)$, depicted in Fig. 5.52, with the energies $E_{1},E_{2}$.¹⁷ At time $t=0$ this leads to a state that is localized in the left well

$$\begin{aligned} \chi _{l}(x,0)=\frac{1}{\sqrt{2}} \left[ \chi _{1}(x)-\chi _{2}(x)\right] . \end{aligned}$$

(5.201)

In the case $D\rightarrow \infty$ it is identical to the ground state of the harmonic approximation to the left well. Its absolute value has the time evolution

$$\begin{aligned} |\chi _{l}(x,t)|^{2}=\frac{1}{2} \left\{ |\chi _{1}(x)|^{2}+|\chi _{2}(x)|^{2} -2\chi _{1}(x)\chi _{2}(x) \cos [(E_{2}-E_{1})t] \right\} . \end{aligned}$$

(5.202)

Defining the tunneling splitting as

$$\begin{aligned} \varDelta =E_{2}-E_{1}, \end{aligned}$$

(5.203)

the corresponding tunneling time

$$\begin{aligned} T_\mathrm{tu}=\frac{2\pi }{\varDelta } \end{aligned}$$

(5.204)

follows. The eigenvalues of the time-independent Schrödinger equation with a quartic potential and thus also the tunneling splitting are not available exactly analytically. Using a semiclassical approximation, $\varDelta$ can be determined, however. The result of such a calculation is [105]

$$\begin{aligned} \varDelta _\mathrm{s}=8\sqrt{\frac{2D}{\pi }}\exp \left( -\frac{16D}{3}\right) , \end{aligned}$$

(5.205)

depending exponentially on the dimensionsless barrier height. In Table 5.4 some values of $\varDelta$ for different barrier heights can be found.

Table 5.4 Numerical and semiclassical (index s) results for the tunneling splitting as a function of the barrier height [53]

Fig. 5.51

Symmetric double-well potential with five energy eigenvalues for $D=2$

Fig. 5.52

Eigenfunctions of the lowest two eigenvalues for $D=2$. Solid line: $\chi _{1}(x)$, dashed line: $\chi _{2}(x)$

5.E The Quantum Mechanical Adiabatic Theorem

To derive the adiabatic theorem in quantum theory, let us consider a system with discrete levels, whose state vector is given by

$$\begin{aligned} |\varvec{\varPsi }(t)\rangle =\left( \begin{array}{c} |\psi _1(t)\rangle \\ |\psi _2(t)\rangle \\ \vdots \end{array}\right) \;. \end{aligned}$$

(5.206)

In case of a time-dependent perturbation, the Hamilton matrix is given by

$$\begin{aligned} \mathbf{H}(t)=\left( \begin{array}{ccc} E_1&{}V_{12}(t)&{}\cdots \\ V_{21}(t)&{}E_2&{}\cdots \\ \vdots &{}\vdots &{}\ddots \end{array} \right) \;. \end{aligned}$$

(5.207)

Clearly, an eigenstate stays an eigenstate without the external perturbation. Even in the presence of a slowly changing perturbation an analogous statement holds, however.

To show this, one first defines a unitary transformation , diagonalizing the instantaneous Hamiltonian

$$\begin{aligned} \mathbf{U}^{-1}(t)\mathbf{H}(t)\mathbf{U}(t)=\mathbf{D}(t)\;. \end{aligned}$$

(5.208)

The transformed state vector is given by

$$\begin{aligned} |\varvec{\varPsi }'(t)\rangle =\mathbf{U}^{-1}|\varvec{\varPsi }(t)\rangle \;. \end{aligned}$$

(5.209)

It fulfills the time-dependent Schrödinger equation

$$\begin{aligned} \mathrm{i}|\dot{\varvec{\varPsi }}'(t)\rangle =\mathbf{D}|\varvec{\varPsi }'(t)\rangle -\mathrm{i}\mathbf{U}^{-1}\dot{\mathbf{U}}|\varvec{\varPsi }'(t)\rangle \;. \end{aligned}$$

(5.210)

If the Hamilton matrix $\mathbf{H}$ is slowly time-dependent, then also $\mathbf{U}$ depends only weakly on time and the second term on the right hand side of the equation above can be neglected. An eigenfunction of the original Hamiltonian thus stays an eigenfunction of the instantaneous Hamiltonian. In [106] it has been shown that for periodically driven systems the adiabatic theorem has to be modified.

Finally, it is worthwhile to mention that by choosing the perturbation in such a way that the Hamiltonian switches from a simple to a complex one, the eigenstates of the complex Hamiltonian can be gained numerically [107]. In addition, an application to two-level systems has been given in [108].