Molecules in Strong Laser Fields

Abstract

Before entering the wide field of molecule-laser interaction in this chapter, some basics of molecular theory will be repeated by discussing the solution of the time-independent Schrödinger equation of the hydrogen molecular ion. The concept of electronic potential energy surfaces, on which the nuclear dynamics occurs, will thereby be introduced and the analytical Morse potential will be reviewed in some detail. The solution of the time-dependent Schrödinger equation for the hydrogen molecular ion then serves as an example for the increasing numerical effort that has been taken in the course of time, starting with frozen nuclei and then taking the nuclear dynamics into account.

After the discussion of the fundamental Born-Oppenheimer approximation, dynamics on a single as well as on laser-coupled potential energy surfaces will be reviewed. From the field of pump-probe femtosecond spectroscopy two examples, photoelectron spectroscopy and flourescence spectroscopy will be dealt with in some detail. The remaining part of this chapter is then devoted to the concept of control of the molecular dynamics by suitably chosen laser fields, including a possible future application in quantum computing.

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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.

By coupling an external electric field to the molecular dynamics, such diverse topics as femtosecond spectroscopy, control of molecular dynamics, and the realization of quantum logic operations emerge. Before entering that wide field, some basics of molecular theory will be repeated by discussing the simplest molecule, the hydrogen molecular ion. The concept of electronic potential energy surfaces, on which the nuclear dynamics occurs, will thereby be introduced and the analytical Morse potential curve will be reviewed in some detail. The hydrogen molecular ion then serves as an example for the increasing numerical effort that has been taken in the course of time, starting with frozen nuclei and then taking the nuclear dynamics into account.

After the discussion of the Born-Oppenheimer approximation, dynamics on a single as well as on laser-coupled potential energy surfaces will be reviewed. From the field of pump-probe femtosecond spectroscopy two examples, photoelectron spectroscopy and flourescence spectroscopy will be dealt with in some detail. The remaining part of this chapter is then devoted to the concept of control of the molecular dynamics by suitably chosen laser fields.

1 The Molecular Ion $\mathrm{H}_{2}^{+}$

In order to understand laser driven molecular dynamics, we first review the basics of molecular theory. The time-independent Schrödinger equation of the hydrogen molecular ion is approximately as well as exactly [1] solvable. In order to keep the discussion simple, we first review the approximate treatment of $\mathrm{H}_{2}^{+}$ in the stationary case, leading to the notion of electronic potential energy surfaces, which will be central for the understanding of almost all the material presented in this chapter.

1.1 Electronic Potential Energy Surfaces

The hydrogen molecular ion consists of two protons, apart by a distance of R, and a single electron with the distances r _a and r _b to proton a, respectively proton b. A schematic representation of the molecule is given in Fig. 5.1.

Fig. 5.1

$\mathrm{H}_{2}^{+}$ molecular ion consisting of two protons labeled by a and b, with distance R, and a single electron with the distances r _a and r _b from the protons

In the following, we try to understand how the energy of the electron is changing as a function of the internuclear distance. It will turn out that there are several possible solutions to this problem, and the outcome has a very intuitive meaning. To make progress, the electron shall first be close to either nucleus a, or nucleus b, while keeping the two nuclei very far apart, i.e. R→∞. These two limiting cases are simple hydrogen atoms, undisturbed by the other proton, for which the exact solution of the respective time-independent Schrödinger equation

$$\begin{aligned} \hat{H}_{a}\psi_{a} =&E_{a} \psi_{a} \end{aligned}$$

(5.1)

$$\begin{aligned} \hat{H}_{b}\psi_{b} =&E_{b} \psi_{b} \end{aligned}$$

(5.2)

is given in Sect. 4.1. Furthermore, the two hydrogen atoms are completely equivalent, and therefore the energies are degenerate E _a =E _b .

The full electronic eigenvalue problem, i.e. the time-independent Schrödinger equation for $\mathrm{H}_{2}^{+}$ with a fixed (finite) distance R is given in atomic units by

$$ \biggl\{ -\frac{1}{2}\Delta -\frac{1}{r_{a}} - \frac{1}{r_{b}} \biggr\}\psi_{{\mathrm{e}}}(r_{a},r_{b},R)= E(R)\psi_{{\mathrm{e}}}(r_{a},r_{b},R) , $$

(5.3)

where the Laplacian can be expressed either by $\nabla_{r_{a}}^{2}$ or by $\nabla_{r_{b}}^{2}$. The internuclear repulsion leads to an R-dependent shift of the energy scale and shall be neglected for the time being. Due to the fact that we know the solution of the problem for R→∞, let us calculate the energy E(R) and the corresponding eigenfunction ψ _e(r _a ,r _b ,R), both depending parametrically on R, as a linear combination of atomic orbitals (LCAO) of the two hydrogen atoms. In the simplest case we can use just a single hydrogen 1s ground state function from Sect. 4.1

$$ \psi_{{a,b}}=\frac{1}{\sqrt{\pi}}\exp\{-r_{{a,b}}\} $$

(5.4)

per proton. The LCAO Ansatz is then given by

$$ \psi_{{\mathrm{e}}}=c_1\psi_{a}+c_2 \psi_{b} . $$

(5.5)

Inserting it into the time-independent Schrödinger equation yields

$$ \biggl(\hat{H}_{a}-\frac{1}{r_{b}} \biggr)c_1 \psi_{a} + \biggl(\hat{H}_{b}-\frac{1}{r_{a}} \biggr)c_2\psi_{b} =E(R) (c_1 \psi_{a}+c_2\psi_{b}) . $$

(5.6)

The application of the Hamiltonian to the ground state leads to a multiplication of the wavefunction by the ground state energy E _g , and therefore the equation above can be rewritten as

$$ \biggl(E_g-E(R)-\frac{1}{r_{b}} \biggr)c_1 \psi_{a} + \biggl(E_g-E(R)-\frac{1}{r_{a}} \biggr)c_2\psi_{b} =0 . $$

(5.7)

This equation can be transformed into a linear system of equations in the usual way by multiplying it from the left with the real eigenfunctions ψ _a,b and integration over electronic coordinates. The following definitions are appropriate:

• 1. overlap integral:

  $$ \int\mathrm{d}V \psi_a\psi_b=:S(R)= \bigl(1+R+R^2/3\bigr)\exp\{-R\} $$

  (5.8)

• 2. Coulomb integral:

  $$ -\int\mathrm{d}V \psi_a\frac{1}{r_b} \psi_a=:C(R)=-\bigl(1-(1+R)\exp\{-2R\}\bigr)/R $$

  (5.9)

• 3. exchange integral (having no classical analog):

  $$ -\int\mathrm{d}V \psi_a\frac{1}{r_a} \psi_b=:D(R)=-(1+R)\exp\{-R\} $$

  (5.10)

• 4. energy difference:

  $$ \Delta E(R):=E_g-E(R) $$

  (5.11)

5.1 :

Calculate the integrals needed for the LCAO solution procedure for the electronic eigenvalue problem of $\mathrm{H}_{2}^{+}$ with the help of the hydrogen 1s functions. Use prolate spheroidal coordinates (i.e. elliptic coordinates rotated around the focal line) μ=(r _a +r _b )/R, ν=(r _a −r _b )/R, φ, for which the volume element is given by $\mathrm{d}V=\frac{1}{8}R^{3}(\mu^{2}-\nu^{2})\mathrm{d}\mu \mathrm{d}\nu\mathrm{d}\varphi$ , and where 1≤μ≤∞, −1≤ν≤1, 0≤φ≤2π.

The coefficients then fulfill the linear system of equations

$$\begin{aligned} \bigl[\Delta E(R)+C(R) \bigr]c_1+ \bigl[\Delta E(R) S(R)+D(R) \bigr]c_2 =&0 \end{aligned}$$

(5.12)

$$\begin{aligned} \bigl[\Delta E(R) S(R)+D(R) \bigr]c_1+ \bigl[\Delta E(R)+C(R) \bigr]c_2 =&0 . \end{aligned}$$

(5.13)

The eigenvalue problem defined above is a generalized one, due to the fact that the eigenvalues E(R) are also appearing in the off-diagonals because of the non-vanishing overlap integral. The condition for solubility leads to the symmetric so-called 1σ _g solution, where

$$ c_1=c_2= \biggl(\frac{1}{2(1+S(R))} \biggr)^{1/2} , $$

(5.14)

and the antisymmetric 1σ _u solution, where

$$ c_1=-c_2= \biggl(\frac{1}{2(1-S(R))} \biggr)^{1/2} , $$

(5.15)

with the corresponding energies (E _g =0)

$$ E_{\pm}(R)=\frac{C(R)\pm D(R)}{1\pm S(R)} . $$

(5.16)

The eigenfunctions displayed in Fig. 5.2 have decisively distinct character. Whereas in the symmetric superposition a sizable part of the wavefunction is located between the two nuclei, the antisymmetric solution 1σ _u with the minus sign has a node between the nuclei!

Fig. 5.2

Symmetric (binding) (left panel) and antisymmetric (anti-binding) (right panel) LCAO solution for nuclei located at $R_{\mathrm{a,b}}=\pm 1$ a.u. as a function of cylindrical coordinates (ρ,z), with the z-axis along the nuclear axis

For the energies it is important to note that C(R) as well as D(R) are negative, and therefore the energy is reduced in the symmetric case as compared to two infinitely separated nuclei. To discuss the binding character of the solutions, we have to include the nuclear repulsion in our discussion, however, by considering the quantity

$$ E_{\pm}^{\mathrm{tot}}(R)=\frac{C(R)\pm D(R)}{1\pm S(R)}+ \frac{1}{R} . $$

(5.17)

For R→0 the nuclear repulsion dominates due to the fact that both C and D are finite in that limit, see Fig. 5.3. Furthermore, as it should be, for R→∞ both curves have the H+p case with energy E _g =0 as the limiting case. At an intermediate value of R _e, the symmetric solution displays a minimum in the energy curve, with a binding energy (dissociation threshold) of D _e, whereas the antisymmetric one is a continuously decreasing function of R as can be seen in Fig. 5.3. A comparison of experimental and theoretical values for the two parameters of the binding potential can be found in Table 5.1.

Fig. 5.3

Upper panel: LCAO binding (solid black line) and anti-binding (dashed red line) electronic potential energy curve of $\mathrm{H}_{2}^{+}$ (in atomic units) according to (5.17) as a function of internuclear distance in atomic units; note that at the minimum of the binding curve (around 2.5 a.u.), the binding effect is much smaller than the anti-binding effect. The zero of energy is indicated by a dotted line and corresponds to the H+p case. Lower panel: Coulomb (solid black line), exchange (dashed blue), and overlap integral (dotted red)

Table 5.1 Comparison of experimental and theoretical LCAO results for the equilibrium distance and the dissociation energy of $\mathrm{H}_{2}^{+}$

We have reviewed an “electronic structure calculation” for a molecule with only a single electron and furthermore, we have used the smallest possible set of basis functions for this system. Nevertheless, the results are quite reasonable as could be seen by comparing them to experimental results. In general, many electron systems have to be considered, however. A prototypical diatomic example is the hydrogen molecule with two electrons. The methods that are typically used to obtain the electronic wavefunction of that molecule are the molecular orbital method or the Heitler-London method. A discussion of both approaches can be found in [2]. Going beyond the treatment of simple, small systems is done in the field of quantum chemistry, which is dealing with the calculation of the electronic energy curves as a function of internuclear distances in the general case [3].

For our present case, a method to dramatically improve the results by using a slightly modified basis set shall be mentioned. Finkelstein and Horowitz [4] have shown that the variation of the 1s basis functions according to

$$ \psi_{{a,b}}\sim\exp\bigl\{-\alpha(R)r_{{a,b}}\bigr\} , $$

(5.18)

where α(R) is allowed to vary between the Helium value of 2 at R=0 and the hydrogen value of 1 at R=∞ does improve the results tremendously to R _e=106 pm and D _e=2.35 eV. This is a version of the so-called variational LCAO method. An alternative would be to use many more basis function in the standard LCAO method. The convergence of this approach to the experimental value is rather slow, however. In Fig. 5.4 the variationally improved total energy curve is shown together with the exact one [5].

5.2 :

Prove that the ground state energy of $\mathrm{H}_{2}^{+}$ calculated with the variational molecular orbital [(2π/α ³)(1+S)]^−1/2(exp{−αr _a }+exp{−αr _b }) is given by

$$E_+=\alpha^2F_1(w)+\alpha F_2(w), $$

where the abbreviations w=αR and

$$\begin{aligned} F_1(w) =&\frac{1+\mathrm {e}^{-w}(1+w-w^2/3)}{2+2\mathrm {e}^{-w}(1+w+w^2/3)} \\ F_2(w) =&-\frac{1+2\mathrm {e}^{-w}(1+w)-1/w-(1/w+1)\mathrm {e}^{-2w}}{1+\mathrm {e}^{-w}(1+w+w^2/3)} \end{aligned}$$

have been used.

Now minimize the energy with respect to α at constant R and plot α as a function of R.

Fig. 5.4

LCAO binding (solid black line) electronic potential energy curve of $\mathrm{H}_{2}^{+}$ according to (5.17) together with the variational LCAO (dashed blue line) and the exact result (dotted red line)

1.2 The Morse Potential

Another, arguably more crude, possibility to construct an analytic binding potential energy curve for a diatomic molecule (the two nuclei shall have the masses M _a and M _b ) is to use the prototypical Morse potential [6]

$$ V_{\mathrm{M}}(R)\equiv D_{\mathrm{e}}\bigl[1-\exp \bigl\{-\alpha(R-R_{\mathrm{e}})\bigr\}\bigr]^2 , $$

(5.19)

displayed in Fig. 5.5, and determine the free parameters from experimental values. To do this, we use that the kinetic energy of the relative motion is given by

$$ T_R=\frac{M_{\mathrm{r}}}{2}\dot{R}^2 , $$

(5.20)

with the reduced mass M _r=M _a M _b /(M _a +M _b ). Parameters that are available experimentally are, e.g.,

• D ₀: dissociation energy (from vibrational ground state)

• ω _e: angular frequency of harmonic oscillations around the minimum

From these, the Morse potential parameters can be extracted according to

• D _e≈D ₀+ω _e/2: dissociation threshold (from minimum of potential curve)

• $\alpha=\omega_{\mathrm{e}}\sqrt{M_{\mathrm{ r}}/(2D_{\mathrm{e}})}$: range parameter (not to be confused with α(R) in the previous section),

where the α parameter follows from the harmonic approximation to the Morse potential.

Fig. 5.5

Morse potential in atomic units (solid line) with D _e=0.103 a.u. and R _e=2.00 a.u. and α=0.72 a.u., corresponding to the experimental values of $\mathrm{H}_{2}^{+}$ [8], together with the bound eigenvalues and the (unnormalized) two lowest eigenfunctions

The solution of the time-independent Schrödinger equation for the Morse potential can be given analytically, although it should be noted that it is only approximate in character [7]. Defining the anharmonicity constant by x _e=ω _e/(4D _e) the eigenvalues can be calculated according to

$$ E_n=\omega_{\mathrm{e}}(n+1/2)-x_{\mathrm{e}} \omega_{\mathrm{e}}(n+1/2)^2,\quad n=0,1,\dots. $$

(5.21)

The above definition of the anharmonicity is assuming the exact Morse form of the potential. One could, however, also use the experimental value for x _e which is typically slightly different from the one that is obtained by inserting the experimental values for ω _e and D _e. In the case of $\mathrm{H}_{2}^{+}$, the frequency and the dissociation energy are given by ω _e≈0.0105 a.u. and D _e≈0.103 a.u. leading to an anharmonicity of x _e≈1/39.2 a.u., whereas the direct experimental value as of 1950 is x _e≈1/37.0 a.u. [8] (the value given in the revised 1979 version of [8] is slightly lower). Choosing x _e≈1/39.2 a.u. leads to a better agreement of the energies in (5.21) with the exact vibrational eigenvalues of $\mathrm{H}_{2}^{+}$ [9], gained without resorting to the Morse potential, as can be seen in Table 5.2. Finally, also the range parameter can be calculated from the anharmonicity according to $\alpha=\sqrt{2 M_{\mathrm{ r}}\omega_{\mathrm{e}}x_{\mathrm{e}}}$. Again α differs slightly if the direct experimental anharmonicity or the one derived from the Morse potential is used!

Table 5.2 Comparison of the exact vibrational eigenvalues of $\mathrm{H}_{2}^{+}$ [9] in the electronic ground state and different Morse oscillator eigenvalues in atomic units

The bound eigenvalues of the Morse oscillator are depicted in Fig. 5.5 together with the two lowest eigenfunctions according to [7]. Because of the anharmonicity, the distance between the levels decreases with increasing energy (for the parameters of $\mathrm{H}_{2}^{+}$ this is barely visible at low quantum numbers). In contrast to the case of the hydrogen atom, where infinitely many levels below the ionization threshold exist, however, only a finite number of levels lies below the threshold of dissociation. The maximal bound state index in the Morse potential can be determined from (5.21) by setting E _max≤D _e and is given by

$$ n_{\max}=\operatorname{Int}\bigl(1/(2x_{\mathrm{e}})-1/2\bigr) . $$

(5.22)

For the direct experimental anharmonicity parameter of $\mathrm{H}_{2}^{+}$ this number is 18, corresponding to 19 bound states (the last entry in the third column of Table 5.2 is already larger than D _e =ω _e /(4x _e )), whereas the exact numerical solution displays 20 bound states.

2 $\mathrm{H}_{2}^{+}$ in a Laser Field

The hydrogen molecular ion is the simplest molecule and therefore it has been the first molecule that has been studied in detail numerically under the influence of an external laser field. Restricting the electronic dynamics to two coupled potential surfaces is allowing the treatment of dissociation via the laser field [10]. Ionization cannot be studied in this framework. We want to focus on both ionization and dissociation phenomena, however. Historically, first the numerical solution of the electron’s time-dependent Schrödinger equation with fixed nuclei [11], allowing the investigation of ionization probabilities, has been given. Only after a further increase of computer power, the fully coupled molecular dynamics has been studied [12].

2.1 Frozen Nuclei

To study the dynamics of a small molecule under the influence of an external electric field, usually several well founded assumptions are made. First, one neglects the translational motion of the center of mass and rotations of the molecule. Then the z-axis, along which the nuclei are assumed to be aligned, shall also be the polarization direction of the incident radiation. The effect of molecular alignment [13] is at the heart of this assumption. Fortunately, the problem then has a cylindrical symmetry and the time-dependent Schrödinger equation with fixed nuclei in atomic units is given by

$$ \mathrm {i}\dot{\varPsi}(\rho,z,t)= \biggl[-\frac{1}{2} \frac{\partial^2 }{\partial z^2}+\hat{T}_\rho+V_{\mathrm{ CC}}(\rho,z) +z \mathcal{E}(t) \biggr]\varPsi(\rho,z,t) , $$

(5.23)

where ρ and z are cylindrical coordinates, and the Hamiltonian does not depend on the azimuthal angle φ. Furthermore,

$$ \hat{T}_\rho=-\frac{1}{2}\frac{\partial^2 }{\partial\rho^2}- \frac {1}{2\rho} \frac{\partial}{\partial\rho} $$

(5.24)

stands for the radial part of the kinetic energy and

$$ V_{\mathrm{CC}}(\rho,z)=-\bigl[\rho^2+(z-R/2)^2 \bigr]^{-1/2}-\bigl[\rho^2+(z+R/2)^2 \bigr]^{-1/2} $$

(5.25)

is the Coulomb potential of the electron bi-nuclear interaction.

The singularity of $\hat{T}_{\rho}$ at ρ=0 as well as of the Coulomb potential at the position of the nuclei can be treated very elegantly, because of the cylindrical symmetry, by expansion in a so-called Fourier-Bessel series [11], see also p. 126 in [14]. If L is the largest distance from the z-axis that is to be described (i.e. the wavefunction shall be zero for ρ≥L), then a complete orthonormal system of functions for the expansion of the ρ-dependence of the wavefunction is given by

$$ v_n(\rho)=\frac{\sqrt{2}}{LJ_1(x_n)}J_0(x_n \rho/L) , $$

(5.26)

where

$$ J_\nu(x)\equiv \biggl(\frac{x}{2} \biggr)^\nu\sum_{j=0}^{\infty} \frac{(-1)^j}{j!\varGamma(j+\nu+1)} \biggl(\frac{x}{2} \biggr)^{2j} $$

(5.27)

are Bessel functions of νth order and the x _n are the zeros of the Bessel function of 0th order. The basis functions are orthonormal according to

$$ \int_0^L \mathrm {d}\rho \rho v_n( \rho)v_m(\rho)=\delta_{nm} . $$

(5.28)

Application of the radial part of the operator of kinetic energy to the basis functions yields

$$ \hat{T}_\rho v_n(\rho)= \frac{1}{2}(x_n/L)^2v_n(\rho) , $$

(5.29)

i.e., the v _n are eigenfunctions of that operator. This fact can be proven explicitly by using the definition in (5.27).

5.3 :

Using the definition of the Bessel function of 0th order show that v _n is an eigenfunction of the radial part of the kinetic energy.

One now expands the wavefunction according to

$$ \varPsi(\rho,z,t)=\sum_{n=1}^Mv_n( \rho)\chi_n(z,t) . $$

(5.30)

After multiplication of the time-dependent Schrödinger equation from the left with v _k and integration over ρ, the system of partial differential equations

$$ \mathrm {i}\dot{\boldsymbol{\chi}}(z,t)= \biggl[-\frac{1}{2} \frac{\partial^2}{\partial z^2} +\mathbf{A}(z) +z\mathcal{E}(t) \biggr]{\boldsymbol{\chi}(z,t)} $$

(5.31)

is found for the vector of coefficients and the non-singular matrix A (ρV _c is finite at the position of the nuclei) with the elements

$$ A_{kn}=\frac{1}{2}(x_n/L)^2 \delta_{kn} +\int_0^L \mathrm {d}\rho \rho v_k(\rho)V_{\mathrm{CC}}(\rho,z)v_n(\rho) $$

(5.32)

has been defined.

The time-dependent Schrödinger equation can be solved by again using the split-operator FFT method from Sect. 2.3.2. This leads to the propagated wavefunction vector

$$\begin{aligned} \boldsymbol{\chi}(z,t+\Delta t) =&\exp(-\mathrm {i}\hat{T}_z\Delta t/2) \exp\bigl[-\mathrm {i}\mathcal{E}\bigl(t'\bigr)z\Delta t\bigr] \exp \bigl[-\mathrm {i}\mathbf{A}(z)\Delta t\bigr]\exp(-\mathrm {i}\hat{T}_z\Delta t/2) \\ &{}\times\boldsymbol{\chi}(z,t) , \end{aligned}$$

(5.33)

where $\hat{T}_{z}=-1/2\frac{\partial^{2}}{\partial z^{2}}$ and t′=t+Δt/2. The only difference to what we have already encountered is the fact that the wavefunction is not a scalar but a vector and correspondingly the non-diagonal matrix A appears in the exponent. In order to cope with this exponentiated matrix, it is diagonalized using the matrix U and one finally uses

$$ \exp\bigl[-\mathrm {i}\mathbf{A}(z)\Delta t\bigr]=\mathbf{U}\exp\bigl[-\mathrm {i}\mathbf{A}_D(z)\Delta t\bigr]\mathbf{U}^T . $$

(5.34)

The matrix is typically of a dimensionality, such that the solution of the eigenvalue problem cannot be done analytically but has to be performed numerically.

2.1.1 Numerical Details and Results

It is worthwhile to note some numerical details of the benchmark calculations of the Bandrauk group [11]. First of all, the numerical grid for the remaining z direction was restricted to |z|<128 a.u. Furthermore, it turned out that L=8 and M=16 for the number of Bessel functions was adequate for moderate field intensities around 10¹⁴ W cm⁻². The laser was assumed to be turned on over five cycles of the field with a frequency of ω=0.2 a.u., corresponding to a wavelength of 228 nm. Finally, the initial state was taken as the electronic ground state 1σ _g . It can be approximated to a good degree by the variational LCAO of the previous section. Alternatively, the LCAO-Ansatz may serve as an initial condition for propagation in imaginary time (see Sect. 2.1.3), distilling the true ground state.

In [11] a measure for ionization was defined with a certain degree of arbitrariness by introducing

$$ P_V(t)=2\pi\int_{-z_{\mathrm{I}}}^{z_{\mathrm{I}}}\mathrm {d}z\int _0^Ld\rho\rho\bigl|\varPsi (z,\rho,t)\bigr|^2 $$

(5.35)

with z _I=16 a.u. as the non-ionized part of the probability. Together with the probability to be in the initial state, i.e., the survival probability

$$ P_0(t)=\bigl|\bigl\langle\varPsi(0)\big|\varPsi(t)\bigr\rangle\bigr|^2 , $$

(5.36)

this quantity is plotted in Fig. 5.6 for two different internuclear distances. In the case R=3 a.u. Rabi oscillations appear for the parameters chosen, which correspond to a one photon transition between the 1σ _g and 1σ _u state, whereas for R=2 a.u., two photons would be needed for the same transition. Although we have plotted only the LCAO result for the surfaces in Fig. 5.3, the drastic decrease of the distance between the two surfaces as a function of R can be observed also there.

Fig. 5.6

Different probabilities (defined in the text) for different fixed inter-nuclear distances of $\mathrm{H}_{2}^{+}$ as a function of time: upper panel: R=2 a.u., lower panel: R=3 a.u. [11]

2.1.2 Charge Resonance Enhanced Ionization

From the quantities defined above, one can extract an ionization rate by using an exponential fit to P _V (t). These rates as a function of intensity can then be compared to the case of the hydrogen atom. In some early work, see, e.g., Fig. 5.7, it turned out that the molecular results tend toward the atomic case for increasing excitation, i.e., increasing R from 2 to 3 a.u. [11].

Fig. 5.7

Ionization rate for $\mathrm{H}_{2}^{+}$ compared to the atom case (asterisks) for a wavelength of 228 nm as a function of laser intensity. Nuclei fixed at the equilibrium distance (squares) and nuclei fixed at R=3 a.u. (circles) [11]

For even larger distances, however, dramatic rate enhancement far beyond the atom limit was found in [15] for a 1064 nm laser with a five cycle rise. For a fixed intensity this is shown in Fig. 5.8. The explanation of this effect is that a pair of charge resonant states (here the almost degenerate 1σ _g and 1σ _u states, which have a similar charge distribution at large R) are strongly coupled to the field at large R, when the dipole moment between them (see Sect. 5.3) diverges linearly. The effect was therefore termed charge resonance enhanced ionization (CREI).

5.4 :

Show that the matrix element of z between the 1σ _g and 1σ _u of $\mathrm{H}_{2}^{+}$ diverges linearly with the interatomic distance R.

Fig. 5.8

Ionization rate for $\mathrm{H}_{2}^{+}$ as a function of internuclear distance for fixed laser intensity of 10¹⁴ W cm⁻² and wavelength of 1,064 nm; atom result indicated by a filled square [15]

The success of the static tunneling picture in the atomic case of Sect. 4.2.1 led the authors of [15] to consider the tunneling out of the statically distorted double well potential that the electron experiences in a $\mathrm{H}_{2}^{+}$ molecule with a field induced potential of $\mathcal{E}_{0}z$. The results for the energies evolving out of the two lowest, unperturbed electronic eigenstates are displayed in Fig. 5.9. For the distance R=10 a.u., at which a maximum in the ionization rate can be observed, the width of the upper level has a maximum, because it lies just above the inner barrier and the left outer barrier is rather narrow as compared to the R=6 a.u. case. In addition, due to the rapid turn on of the field, the population of the upper level is almost equal to the one of the lower level after the amplitude is constant [15] and therefore the system ionizes to a substantial degree.

Fig. 5.9

Lowest two static field induced levels and their line widths for $\mathrm{H}_{2}^{+}$ for three different internuclear distances: a R=6 a.u., b R=10 a.u., c R=14 a.u. [15]

2.2 Nuclei in Motion

Nuclear dynamics in $\mathrm{H}_{2}^{+}$ can nowadays be treated on the same level as the electronic dynamics by the solution of the full time-dependent Schrödinger equation. The wavefunction then depends on the additional degree of freedom R, describing the relative motion of the nuclei. The coupling to the field shall again be given in the length gauge. In this case, the motion of the center of mass can be separated by introducing a center of mass and the relative coordinate between the two nuclei as well as an electron coordinate which is measured with respect to the center of mass of the nuclei. This is a lengthily calculation, however, which is reviewed in Appendix A.

As the final result it turns out that the Hamiltonian of (5.23) has to be augmented by the kinetic and the potential energy of the nuclei

$$ \hat{T}_R+V_R=-\frac{1}{2M_{\mathrm{r}}}\frac{\partial^2}{\partial R^2}+ \frac {1}{R} , $$

(5.37)

where M _r=M _p/2≈918 is the reduced nuclear mass in atomic units. Furthermore, as can be seen in Appendix A the electron mass (which is unity in atomic units) has to be modified slightly to read m _i =2M _p/(2M _p+1) and the term with the laser field is to be multiplied by the factor [1+1/(2M _p+1)]. Both modifications are marginal due to the large mass ratio, however.

The total wavefunction

$$\varPsi(R,\rho,z,t) $$

now also depends on R, and distance dependent quantities like

$$ f_1(R,t)=2\pi\int_0^L\mathrm {d}\rho\rho \int_{-z_{\mathrm{I}}}^{z_{\mathrm{I}}}\mathrm {d}z\bigl|\varPsi (R,\rho,z,t)\bigr|^2 $$

(5.38)

can be studied. This is the probability density to find the protons a distance R apart and the electron within a cylinder of height 2z _I, such that $\mathrm{H}_{2}^{+}$ is not fully ionized. z _I is later on chosen to be 32 a.u. From the R-dependent quantity just defined some integrated quantities can be calculated. These are the dissociation probability without ionization, i.e., the probability for the “reaction” $\mathrm{H}_{2}^{+} \to \mathrm{H}+\mathrm{H}^{+}$ ¹

$$ P_{\mathrm{D}}(t)=\int_{R_{\mathrm{D}}}^{R_{\max}} \mathrm {d}Rf_1(R,t) , $$

(5.39)

where again somewhat arbitrarily R _D=9.5 a.u. can be chosen as the onset of dissociation and R _max is the nuclear grid boundary. Furthermore, the probability of ionization

$$ P_{\mathrm{I}}(t)=1-\int_{0}^{R_{\max}} \mathrm {d}Rf_1(R,t) , $$

(5.40)

is given by the probability to find the electron outside of a cylinder with |z|≤z _I.

2.2.1 Molecular Stabilization

Numerical results for the case of a molecule initially in the electronic 1σ _g ground state and in an excited vibrational state with quantum number n=6, according to

$$ \varPsi(R,z,\rho,0)=\phi_{1\sigma_{g}}(z,\rho,R)\chi_6(R) $$

(5.41)

are depicted in Fig. 5.10. Apart from the probabilities defined above also the probability P ₆ to stay in the 6th vibrational state is displayed there. For relatively low intensities (I=3.5×10¹³ W/cm²) a stabilization of the initial state, i.e., after a short initial decay, an increase of P ₆ is observed. It can be understood due to stimulated emission (Rabi oscillation) from the dissociative 1σ _u state [12]. For higher intensities (I=10¹⁴ W cm⁻²) this effect vanishes, however, because the system is already ionized to a substantial degree. This is in contrast to the prediction of two-state calculations, which are also displayed in Fig. 5.10 and which show the stabilization effect for both intensities. Furthermore, it is worthwhile to note that the fine oscillations in P ₆, well visible for low intensities, occur at twice the laser frequency and are due to the counter-rotating term, which is neglected in the RWA (but not in the full numerical calculations reviewed here).

Fig. 5.10

Different probabilities for $\mathrm{H}_{2}^{+}$ in a laser field (λ=212 nm) as a function of time for two different intensities: a I=3.5×10¹³ W cm⁻², b I=10¹⁴ W cm⁻² [12]

Further light can be shed on the stabilization (or bond hardening) effect by looking at the nuclear wavefunction at a fixed time. For different field intensities these results are displayed in Fig. 5.11. Sharp peaks near R=3 and R=3.6 a.u. can be observed. In a two state picture the 1σ _g and 1σ _u surfaces form an avoided crossing² if they are dressed by the laser field [17, 18] and the peaks are at the turning points of the bound motion in the upper adiabatic potential well. For the lower intensity the shape of the peaks does not vary much as a function of time, whereas for the higher intensity the peaks decrease considerably due to ionization.

Fig. 5.11

Time evolved nuclear wavefunction of $\mathrm{H}_{2}^{+}$ in a laser field of different intensity: a I=3.5×10¹³ W cm⁻², b I=10¹⁴ W cm⁻² [12]. In these plots also two-surface calculations, which do not account for ionization are displayed

2.2.2 Coulomb Explosion Versus Dissociation

In the results that we have discussed so far, the initial vibrational state was fixed to be the n=6 state. What happens for different initial vibrational quantum numbers?

As already discussed, the fragmentation of the molecular ion under the external field can occur via different channels. One channel, in which the electron stays with one (or each) of the two nuclei, is the dissociation channel (without ionization)

$$\begin{aligned} \mathrm{H}_2^+\to{\mathrm{p}}+\mathrm{H} . \end{aligned}$$

The alternative is complete fragmentation, which is our ionization case, given by

$$\begin{aligned} \mathrm{H}_2^+\to{\mathrm{p}}+{\mathrm{p}}+{\mathrm{e}}^- . \end{aligned}$$

This last channel is also referred to as the Coulomb explosion channel.

The outcome of numerical calculations using a modified 1d soft-core potential, where the softening parameter depends on the internuclear distance [19], and the Crank-Nicolson method has been used for the propagation is displayed in Fig. 5.12. It can be observed that for vibrational levels n≥2 for the field parameters chosen, the Coulomb explosion channel dominates the dissociation channel. Recently, quantitatively different but qualitatively similar results with up to 60 % dissociation probability have been reported for the case of full 3d hard core Coulomb interaction [20].

Fig. 5.12

Comparison of the two fragmentation channels as a function of initial vibrational excitation for a 25 fs laser pulse with an intensity of 0.2 PW cm⁻² and 800 nm central wavelength. Black squares show the overlaps $|\langle\mathrm{H}_{2}^{+},n|\mathrm{H}_{2},n=0\rangle|^{2}$ [19]

3 Adiabatic and Nonadiabatic Nuclear Dynamics

The neutral hydrogen molecule H₂ has an additional electron compared to the hydrogen molecular ion of the previous section. Its dynamics has been treated fully quantum mechanically only recently [21]. In order to deal with even more complex systems numerically, approximations and/or restrictions of the dynamics to some relevant degrees of freedom have to be made. An important, if not the most important approximation of molecular theory is the Born-Oppenheimer approximation to be discussed in the following.

This approximation is closely related to what we have already done for the case of $\mathrm{H}_{2}^{+}$ in Sect. 5.1. There the binding potential energy surface 1σ _g was taken as a sum of the repulsive potential between the nuclei and the attraction due to the electron in-between the two nuclei. If no other (external) force is acting, then the nuclear motion would be attracted by the minimum of the potential. The motion has to be described quantum mechanically, however, and thus a probability distribution with its maximum at the minimum of the potential curve will result. In the following we will realize that the notion above is approximate in nature. Even in the case without an external field the dynamics can, in principle, not be restricted to a single electronic surface.

3.1 Born-Oppenheimer Approximation

In (5.3) we have neglected the kinetic as well as the potential energy of the nuclei. The total Hamiltonian for a general molecule with M nuclei and N electrons, however, is given by

$$ \hat{H}_{\mathrm{mol}}=\hat{T}_{\mathrm{N}}+\hat{H}_{\mathrm{e}}= \hat{T}_{\mathrm{N}}+\hat{T}_{\mathrm{e}}+V(\boldsymbol{x},\boldsymbol{X}) , $$

(5.42)

with the kinetic energies (switching back to SI units for this section)

$$\begin{aligned} \hat{T}_{\mathrm{N}} =&\sum_{i=1}^M- \frac{\hbar^2}{2M_i} \Delta _i^2 \end{aligned}$$

(5.43)

$$\begin{aligned} \hat{T}_{\mathrm{e}} =&\sum_{j=1}^N- \frac{\hbar^2}{2m_{\mathrm{e}}} \Delta _j^2 . \end{aligned}$$

(5.44)

The potential energy V=V _ee+V _eN+V _NN contains the electron-electron interaction

$$ V_{\mathrm{ee}}(\boldsymbol{x})=\frac{e^2}{4\pi\epsilon_0}\frac{1}{2} \sum _{i\neq j}^N\frac{1}{|\boldsymbol{r}_i-\boldsymbol{r}_j|} $$

(5.45)

as well as the electron-nucleus (nuclear charge Z _i e)

$$ V_{\mathrm{eN}}(\boldsymbol{x},\boldsymbol{X})=-\frac{e^2}{4\pi\epsilon_0}\frac{1}{2} \sum _{j=1}^N\sum_{i=1}^M \frac{Z_i}{|\boldsymbol{r}_j-\boldsymbol{R}_i|} $$

(5.46)

and the internuclear interaction

$$ V_{\mathrm{NN}}(\boldsymbol{X})=\frac{e^2}{4\pi\epsilon_0}\frac{1}{2} \sum _{i\neq j}^M\frac{Z_iZ_j}{|\boldsymbol{R}_i-\boldsymbol{R}_j|} . $$

(5.47)

All nuclear coordinates are contained in the symbol

$$\begin{aligned} \boldsymbol{X}=(\boldsymbol{R}_1,\dots,\boldsymbol{R}_M) \end{aligned}$$

whereas the electronic coordinates including spin are denoted by

$$\begin{aligned} \boldsymbol{x}=(\boldsymbol{r}_1,s_1;\dots;\boldsymbol{r}_N,s_N) . \end{aligned}$$

As usual, the nuclear coordinates are distinguished from the electronic ones by using capital letters for the former and lower case letters for the latter.

The electronic part of the Hamiltonian commutes with the nuclear coordinates, i.e. $[\hat{H}_{\mathrm{e}},\boldsymbol{X}]=0$. The nuclear coordinates therefore are “good quantum numbers” for the electronic operator and can be viewed as parameters. One now first solves the electronic eigenvalue problem (the time-independent Schrödinger equation of the electrons), which is the generalized analog of (5.3),

$$ \hat{H}_{\mathrm{e}}\phi_\nu= \biggl[ -\sum _j\frac{\hbar^2}{2m_{\mathrm{e}}}\Delta _j+V( \boldsymbol{x},\boldsymbol{X}) \biggr] \phi_{\nu}(\boldsymbol{x},\boldsymbol{X})=E_\nu( \boldsymbol{X})\phi_{\nu}(\boldsymbol{x},\boldsymbol{X}) , $$

(5.48)

where the electronic energy, as well as the corresponding wavefunction depend parametrically ³ on the nuclear coordinates and ν is a suitable set of quantum numbers of the electronic system. For the electronic functions orthonormality and completeness relations hold according to

$$\begin{aligned} \int \mathrm {d}^{4N} x\,\phi^{\ast}_\nu(\boldsymbol{x},\boldsymbol{X}) \phi_\mu(\boldsymbol{x},\boldsymbol{X}) =& \delta_{\mu\nu} \end{aligned}$$

(5.49)

$$\begin{aligned} \sum_{\nu}|\phi_\nu\rangle \langle \phi_\nu| =&\hat{1} . \end{aligned}$$

(5.50)

The total wavefunction can thus be expanded by using the electronic states as basis states according to the Born-Huang expansion

$$ \psi(\boldsymbol{x},\boldsymbol{X})=\sum_\nu \phi_{\nu}(\boldsymbol{x},\boldsymbol{X})\chi_\nu(\boldsymbol{X}) $$

(5.51)

with the nuclear functions χ _ν (X). Inserting this Ansatz into the time-independent Schrödinger equation

$$ \biggl[ -\sum_i \frac{\hbar^2}{2M_i}\Delta _i-\sum_j \frac{\hbar^2}{2m_{\mathrm{e}}}\Delta _j +V(\boldsymbol{x},\boldsymbol{X}) \biggr]\psi( \boldsymbol{x},\boldsymbol{X})= \epsilon\psi(\boldsymbol{x},\boldsymbol{X}) , $$

(5.52)

one can now use the electronic Schrödinger equation to replace the electronic Hamiltonian by its eigenvalue and arrives at (written suggestively)

$$\begin{aligned} &\sum_\nu\phi_{\nu}(\boldsymbol{x},\boldsymbol{X}) \biggl[-\sum_i\frac{\hbar^2}{2M_i} \Delta _i+E_{\nu}(\boldsymbol{X}) \biggr] \chi_{\nu}(\boldsymbol{X}) \\ &\quad=\sum_\nu \phi_{\nu}(\boldsymbol{x},\boldsymbol{X})\epsilon\chi_{\nu}(\boldsymbol{X}) \\ &\quad\quad{}+\sum_\nu\sum _i\frac{\hbar^2}{2M_i} \bigl[\chi_\nu(\boldsymbol{X}) \Delta _i\phi_{\nu}(\boldsymbol{x},\boldsymbol{X})+ 2\boldsymbol{ \nabla}_i\chi_\nu(\boldsymbol{X})\cdot\boldsymbol{\nabla}_i \phi_{\nu}(\boldsymbol{x},\boldsymbol{X})\bigr] . \end{aligned}$$

(5.53)

The third line in the equation above follows from the application of the product rule of differentiation and is due to the change of the electronic states induced by the nuclear motion.

If we neglect this third line and multiply the equation from the left by $\phi_{\mu}^{\ast}(\boldsymbol{x},\boldsymbol{X})$, after integration over the electronic coordinates x, we get

$$ \biggl[-\sum_i\frac{\hbar^2}{2M_i} \Delta _i+E_\mu(\boldsymbol{X}) \biggr] \chi_\mu(\boldsymbol{X}) \approx\epsilon\chi_\mu(\boldsymbol{X}) . $$

(5.54)

This is the time-independent Schrödinger equation for the nuclear degrees of freedom in a potential that is given by the μth eigensolution of the electrons. We have used this notion already in the last section. As we can appreciate now, an approximation has been made along the way, however. The equation above and also its time-dependent analog, which results in the dynamics on a single potential energy surface are based on the Born-Oppenheimer or adiabatic approximation that resulted due to the neglect of the second line in (5.53).⁴

The approximation that has been made still needs to be justified. That is we have to argue why the terms in the second line of (5.53) might by small compared to the terms in the first line of that equation. Let us just study the terms

$$\begin{aligned} \frac{\hbar^2}{2M_i}\Delta _i\phi_{\nu}(\boldsymbol{x}, \boldsymbol{X}) \end{aligned}$$

containing the second derivative. The electronic wavefunction depends on X in a similar fashion as it depends on x (depending mainly on the differences R _i −r _j ). Therefore the respective derivatives are of the same order of magnitude. The corresponding prefactors in (5.52), however, are smaller by almost 3 orders of magnitude due to the large mass ratio M _i /m _e. That is why the adiabatic approximation is so successful.

In general, however, one has to multiply also the second line in (5.53) from the left with $\phi_{\mu}^{\ast}(\boldsymbol{x}, \boldsymbol{X})$ and has to integrate over the electronic coordinates. The terms that emerge then lead to transitions between the different electronic surfaces, the so-called nonadiabatic or non Born-Oppenheimer transitions. As we have already stressed, the notion of nuclear motion on a single surface is only approximate and breaks down especially close to avoided crossings of potential surfaces. These transitions can be described by using the Born-Oppenheimer surfaces and allowing for a coupling via the derivative terms of the second line of (5.53). Alternatively they can also be described in another, so-called diabatic basis. A schematic plot of adiabtic and diabatic levels is given in Fig. 5.13. The diabatic levels can cross and their coupling is given by non-diagonal potential terms and not by derivative terms, which might be advantageous computationally. However, the construction of diabatic surfaces is not unique [23].

Fig. 5.13

Adiabatic (solid lines) and diabatic (dashed lines) levels in the vicinity of an avoided crossing

Before leaving this section let us come back to the mixed quantum classical methods that we have discussed in Chap. 2. In the framework of an adiabatic description the classical dynamics is restricted to a single surface, most frequently, the ground state. For frozen nuclear coordinates the electronic quantum problem is solved for the ground state and the forces which govern the nuclear motion are determined. After a short time step of the nuclear motion the same procedure is repeated. In this way the forces (or the potential surface) are calculated “on the fly”. We will not consider this approach and also its generalization to the non-adiabatic case any further but will assume that the potential is at our disposal as the outcome of a quantum chemical calculation.

3.1.1 Relative and Center of Mass Coordinates

Similar to the case of $\mathrm{H}_{2}^{+}$, the treatment of the nuclear problem is preferably done in center of mass and relative coordinates. All the relative coordinates shall be contained in the vector R. The center of mass is moving freely due to the fact that the potential does not depend on the corresponding coordinate. We therefore concentrate on the description of the relative motion.⁵ The relevant masses then are reduced masses. In the diatom case this has already been used in Sect. 5.2. For more atoms things become complicated rather quickly. Already in the case of collinear motion of three atoms cross terms do appear in the kinetic energy.

5.5 :

For three collinear masses M ₁ , M ₂ , M ₃ give relative and center of mass coordinates and write the kinetic energy with the help of the canonically conjugate relative momenta.

3.1.2 Coupling to a Laser Field

In order to couple a laser field to a molecular system, we will use the length gauge. Generalizing the discussion in Sect. 3.1.2 to the many particle case, this leads us to consider the dipole operator

$$ \boldsymbol{d}(\boldsymbol{x},\boldsymbol{X})=\sum_i Z_ie\boldsymbol{R}_i-\sum_je \boldsymbol{r}_j . $$

(5.55)

After having solved the electronic problem, the dipole matrix element (or dipole moment)

$$ \boldsymbol{\mu}_{ba}(\boldsymbol{X})=\int\mathrm{d}^{4N}x\, \phi_{b}^\ast(\boldsymbol{x},\boldsymbol{X})\boldsymbol{d}(\boldsymbol{x},\boldsymbol{X}) \phi_{a}(\boldsymbol{x},\boldsymbol{X}) $$

(5.56)

has to be calculated. This is a generalization of the dipole matrix element of Sect. 3.2.1, due to the fact that it still may depend on the internuclear distance.

If we consider different electronic levels, i.e., if b≠a, then due to the orthogonality of the corresponding states only the parts proportional to ∑ _j e r _j of the dipole operator survive. If a corresponding transition is not forbidden then even if one neglects the nonadiabatic terms in the spirit of the adiabatic approximation, still a coupled surface time-dependent Schrödinger equation has to be solved due to the presence of the laser.

3.2 Dissociation in a Morse Potential

Before treating the problem of coupled surfaces, the influence of a laser on the wavepacket dynamics in the electronic ground state shall be studied. Frequently used systems for theoretical calculations are “diatomics” like HF and CH- or OH-groups of larger molecules. Heteronuclear systems are chosen due to the fact that symmetric homonuclear molecules as, e.g., H₂ do not have a permanent electric dipole moment in the electronic ground state (see, e.g., Sect. 11.1 in [2]). If the parameters of a typical infrared laser are chosen appropriately, the diatomic can be driven into dissociation. In contrast to the studies of dissociation of the hydrogen molecular ion of Sect. 5.2, in the following only the nuclear part of the Schrödinger equation will be considered.

5.6 :

Show explicitly that, for the relative motion, the permanent electronic dipole moment of $\mathrm{H}_{2}^{+}$ in the electronic ground state (approximated by 1σ _g of Sect.  5.1 ) is zero.

After the separation of the center of mass motion, the Hamiltonian for the relative motion of the two nuclei in a Morse potential modeling the electronic ground state is given by

$$ \hat{H}=-\frac{1}{2M_{\mathrm{r}}}\frac{\partial^2}{\partial R^2}+V_{\mathrm{M}}(R) +\mu(R) \mathcal{E}_0f(t)\cos\bigl(\omega(t)t\bigr) . $$

(5.57)

The potential parameters for HF are D _e=0.225, R _e=1.7329, α=1.1741 in atomic units [24]. In general, one allows for a chirp, see also Sect. 1.3.3, in the laser frequency.

The R-dependent dipole matrix element (or dipole moment) in principle has to be determined by quantum chemical methods and we assume it to be given analytically in the form studied in detail by Mecke [25]

$$ \mu(R)=\mu_0R \mathrm {e}^{-R/R^\ast} . $$

(5.58)

Alternatively, other powers of R than the first may appear in the exponential function. For HF the power of 4 is frequently used [26]. As can be seen from Fig. 5.14, around the minimum R _e of the Morse potential the dipole function in (5.58) can (up to an irrelevant constant) be approximated by a linear function

$$ \mu(R)\approx-\mu'(R-R_{\mathrm{e}}). $$

(5.59)

The slope of this linear function is referred to as the dipole gradient or effective charge. For HF this quantity is given by μ′=0.297 in atomic units [27].

Fig. 5.14

Mecke form for the dipole function (solid line) of an X-H diatomic and its linear approximation (dashed line) around the minimum of the corresponding Morse potential; parameters used are the ones for the OH stretch in H₂O: R _e=1.821, R ^∗=1.134 in atomic units [28]

What is the reason for allowing a chirp in the laser frequency? As we have seen in Chap. 3 complete population transfer between two levels is only possible in the case of resonance. So if the oscillator is to be excited resonantly on the ladder of energy levels, in the case of the Morse energies (5.21), the frequency has to decrease as a function of time. Furthermore, the excitation “pulse” has to be a π-pulse. Following these arguments, for the HF molecule, the authors of [27] have constructed an analytic form of a pulse that leads to a large final dissociation probability. The envelope of that pulse and its time varying central frequency are depicted in Fig. 5.15 and we stress that the chirp appears in the simple form cos[ω(t)t] in the Hamiltonian (not integrated over time like in (1.25)). The frequency decreases from an initial value of ω ₀₁,⁶ equal to the energy difference of the lowest two levels. In the figure also the probabilities for dissociation and the occupation of different vibrational levels of the HF molecule are displayed. The results have been gained by numerically solving the TDSE with the vibrational ground state as the initial state. Apart from the slight generalization of the chirp, this study is completely analogous to the investigation of multi-photon ionization in the Gauss potential of Sect. 4.2.2.

Fig. 5.15

a Chirped pulse frequency ω(t) and envelope function f(t), b several probabilities in a driven Morse oscillator, all as a function of time in cycles of the external field; adapted from [27]

Analogous results have been found by applying a classical mechanics optimization procedure [29] and are reproduced in Fig. 5.16. It is not surprising that also the classical result displays a down chirp of the frequency. Also in classical mechanics, a softening of the bond occurs for higher energies.

Fig. 5.16

Optimal field (upper panel) and quantum mechanical expectation value of position (solid line in lower panel) and classical trajectory (dashed line in lower panel) for the dissociation of a Morse oscillator [29]

The material just presented is already a glimpse of what we will discuss in detail in Sect. 5.4 on the control of quantum systems. There we will, e.g., review the use of optimal control methods to steer a Morse oscillator into a desired excited vibrational state [30].

3.3 Coupled Potential Surfaces

Now we are ready to deal with the problem of nonadiabatic dynamics on coupled potential energy surfaces. The wavefunction that describes the system in this case is a vector and each component evolves on a specific surface. Due to the fact that the equations are coupled the problem is also referred to as a coupled channel problem. In Sect. 5.2 we have encountered such a situation at least formally already. However, the different channels there were the different basis functions in the Fourier-Bessel series expansion.

In this section we will show how the formalism for the solution of the time-dependent Schrödinger equation has to be augmented to cope with the new situation. First this will be done fully quantum mechanically and then we will deal in some detail with the semiclassical approximation to coupled surface quantum dynamics.

3.3.1 Quantum Mechanical Approach

As a simple example let us start with the case of a homonuclear diatomic molecule and consider two diagonal (diabatic) potential matrix elements V _nn (R), n=1,2. Their coupling shall be given by arbitrary non diagonal matrix elements V ₁₂(R,t)=V ₂₁(R,t), which may depend on time.

The corresponding two surface time-dependent Schrödinger equation is given by⁷

$$\begin{aligned} \mathrm {i}\dot{\chi}_1(R,t) =& \biggl[-\frac{1}{2M_{\mathrm{r}}} \frac{\partial^2}{ \partial R^2}+V_{11}(R) \biggr]\chi_1(R,t)+V_{12}(R,t) \chi_2(R,t) \end{aligned}$$

(5.60)

$$\begin{aligned} \mathrm {i}\dot{\chi}_2(R,t) =& \biggl[-\frac{1}{2M_{\mathrm{r}}} \frac{\partial^2}{ \partial R^2}+V_{22}(R) \biggr]\chi_2(R,t)+V_{12}(R,t) \chi_1(R,t) . \end{aligned}$$

(5.61)

Its solution can be gained by an extension of the split-operator FFT method of Sect. 2.3. A generalization of the known procedure is necessary due to the fact that the potential is a 2×2 matrix

$$ \mathbf{V}(R,t)= \begin{pmatrix} V_{11}(R)&V_{12}(R,t) \\ V_{12}(R,t)&V_{22}(R) \end{pmatrix} $$

(5.62)

now, in complete analogy to the matrix A of Sect. 5.2. In order to exponentiate it, one first has to diagonalize it, as in the previous section. In contrast to the previous section, in the case of two levels the diagonalization can be done exactly analytically, however, leading to [31]

$$ \exp\{-\mathrm {i}\Delta t\mathbf{V}\}=\exp \biggl\{-\mathrm {i}\Delta t \biggl( \frac{V_{11}+V_{22}}{2} \biggr) \biggr\} \begin{pmatrix} A&B\\ B&A^\ast \end{pmatrix} . $$

(5.63)

Here A and B are complex numbers given by

$$ A=\cos\phi-\mathrm {i}\Delta t \lambda\frac{\sin\phi}{\phi} , \qquad B=\mathrm {i}\Delta t V_{12}\frac{\sin\phi}{\phi} $$

(5.64)

with the phase

$$ \phi(R,t)=\Delta t\sqrt{V_{12}^2(R,t)+ \lambda^2(R)} $$

(5.65)

and half the energy difference

$$ \lambda(R)=\frac{V_{11}(R)-V_{22}(R)}{2} . $$

(5.66)

The presented approach is still exact if maximally two surfaces are taking part in the dynamics.

The formalism shall now be applied to the case of coupling between two adiabatic surfaces due to a laser pulse in length gauge using the RWA. The nondiagonal coupling is then given by

$$ V_{12}(R,t)=\mu(R)f(t)\mathcal{E}_0\cos(\omega t) $$

(5.67)

with the dipole moment μ. To apply the rotating wave approximation of Sect. 3.2.3.1, the second component of the wavefunction vector is transformed according to

$$ \tilde{\chi}_2=\exp\{\mathrm {i}\omega t\}\chi_2 . $$

(5.68)

Inserting the transformed wavefunction into the time-dependent Schrödinger equation and neglecting the counter-rotating term, proportional to exp{−2iωt}, the only time-dependence that remains is the one due to the envelope. Furthermore, because of the product rule to be used for the time-derivative, the second surface is shifted by −ω as displayed in Fig. 5.17. The transformed coupled surface equations in RWA thus contain the modified potential matrix elements

$$\begin{aligned} \tilde{V}_{22}(R) =&V_{22}(R)-\omega \end{aligned}$$

(5.69)

$$\begin{aligned} \tilde{V}_{12}(R,t) =&\mu(R) f(t) \mathcal{E}_0/2 . \end{aligned}$$

(5.70)

In addition the Condon approximation can be made. The dipole moment then does not depend on R. Before presenting an application of this formalism, we first discuss a semiclassical approach to the coupled surface problem.

Fig. 5.17

Upper panel: Excited electronic surface V ₂₂ (dashed blue). Lower panel: the coupling to a field in the resonance case and for non-resonance leads to differently modified excited states $\tilde{V}_{22}$ (dashed blue: resonance, dashed-dotted green: nonresonance). In both panels also the ground state wavefunction (dotted red) and the ground state potential surface V ₁₁ (solid black) are depicted

3.3.2 Semiclassical Approach: Mapping Hamiltonian

The classical and semiclassical description of the motion on coupled potential surfaces seems to be problematic. A way must be found to let trajectories switch from the motion on one to motion on another surface. A method that is ad hoc in nature, but is rather successful numerically, is the surface hopping technique developed by Tully [32].

Modern semiclassical methods can be derived from the underlying quantum mechanics, however, as we will see in the following. These methods have a historical precursor in the so-called “classical electron analog model” by Meier and Miller [33]. We will now review the method by Stock and Thoss, which is based on Schwinger’s “mapping formalism” [34]. The N discrete electronic levels that shall be taking part in the dynamics are mapped onto N continuous, harmonic degrees of freedom in this approach.

The analogy between uncoupled harmonic oscillators and a spin system shall be dealt with in the following for the simple case of N=2. In this case an oscillator of plus type and an oscillator of minus type are defined with the respective annihilation and creation operators

$$ \hat{a}_+,\hat{a}_+^\dagger,\qquad\hat{a}_-,\hat{a}_-^\dagger, $$

(5.71)

where operators of the same type are fulfilling the standard commutation relations (2.144) and any operators of different type are commuting with each other. Furthermore, occupation number operators

$$ \hat{N}_+=\hat{a}_+^\dagger\hat{a}_+,\qquad\hat{N}_-=\hat{a}_-^\dagger \hat{a}_- $$

(5.72)

can be defined. Simultaneous eigenkets of $\hat{N}_{+}$ and $\hat{N}_{-}$ fulfill the eigenvalue equations

$$ \hat{N}_+|n_+,n_-\rangle=n_+|n_+,n_-\rangle, \qquad \hat{N}_-|n_+,n_- \rangle=n_-|n_+,n_-\rangle $$

(5.73)

with the eigenvalues n _±, and an arbitrary state can be created from the vacuum state by the application of $\hat{a}_{+}^{\dagger}$ and $\hat{a}_{-}^{\dagger}$, using (2.145), according to

$$ |n_+,n_-\rangle=\frac{(\hat{a}_+^\dagger)^{n_+}(\hat{a}_-^\dagger)^{n_-}}{ \sqrt{n_+!n_-!}}|0,0\rangle. $$

(5.74)

One can now define the products

$$ \hat{J}_+\equiv\hat{a}_+^\dagger\hat{a}_- , \qquad \hat{J}_-\equiv \hat{a}_-^\dagger\hat{a}_+ , \qquad \hat{J}_z= \frac{1}{2}\bigl(\hat{a}_+^\dagger\hat{a}_+- \hat{a}_-^\dagger \hat{a}_-\bigr) , $$

(5.75)

where $\hat{J}_{z}=\frac{1}{2}(\hat{N}_{+}-\hat{N}_{-})$. It can be shown [35], that these operators fulfill the angular momentum commutation relations

$$ [\hat{J}_z,\hat{J}_{\pm}] = \pm\hat{J}_{\pm} $$

(5.76)

and

$$ [\hat{J}_{+},\hat{J}_{-}] = 2\hat{J}_z . $$

(5.77)

Furthermore,

$$ \hat{\boldsymbol{J}}^2=\hat{J}_z^2+ \frac{1}{2}(\hat{J}_+\hat{J}_-+\hat{J}_-\hat{J}_+) =\frac{\hat{N}}{2} \biggl(\frac{\hat{N}}{2}+1 \biggr) $$

(5.78)

with total occupation number operator $\hat{N}\equiv\hat{N}_{+}+\hat{N}_{-}$holds.

For our purposes the restriction to the subspace n ₊+n ₋=1 is appropriate. This is due to the fact that a spin 1/2 particle can be mapped onto two uncoupled oscillators. In case the eigenvalues of the plus and minus oscillator are n ₊=1 and n ₋=0, respectively, the particle is in the spin up state. Application of the operator $\hat{J}_{-}$ leads to n ₊=0 and n ₋=1 and the particle is in the spin-down state. The equivalent description of such a system in terms of harmonic oscillators and by using the occupation number representation is depicted in Fig. 5.18.

Fig. 5.18

Mapping analogy and occupation number representation

Now we return to the problem of a Hamilton operator for the coupled dynamics on N surfaces

$$ \hat{H}=\sum_{n,m}h_{nm}| \phi_n\rangle \langle\phi_m| . $$

(5.79)

Rewriting this Hamiltonian with the help of continuous bosonic variables, one introduces N harmonic degrees of freedom by using the mapping procedure

$$\begin{aligned} |\phi_n\rangle \langle\phi_m| \to& \hat{a}_n^\dagger\hat{a}_m \end{aligned}$$

(5.80)

$$\begin{aligned} |\phi_n\rangle \to& |0_1,\dots,1_n, \dots,0_N\rangle, \end{aligned}$$

(5.81)

discussed above. This leads to the Hamiltonian

$$ \hat{H}=\sum_{n,m}h_{nm} \hat{a}_n^\dagger\hat{a}_m . $$

(5.82)

A classical analog of the mapped quantum dynamics can be defined by replacing the position and momentum operators appearing in the creation and annihilation operator

$$\begin{aligned} \hat{a}^{\dagger}_n =&(\hat{y}_n-\partial/ \partial y_n)/\sqrt{2} \end{aligned}$$

(5.83)

$$\begin{aligned} \hat{a}_n =&(\hat{y}_n+\partial/\partial y_n)/\sqrt{2} \end{aligned}$$

(5.84)

by the respective classical variables y _n , p _n . The lower case variables (y,p)={y _n ,p _n } with n=1,…,N are thus the coordinates and momenta of the auxiliary harmonic oscillators and (R,P), are the phase space variables of the relative nuclear motion with the reduced mass M _r. The classical “mapping” Hamiltonian of an N level system is then given by

$$ H(\boldsymbol{y},\boldsymbol{p},\boldsymbol{R},\boldsymbol{P})= \frac{\boldsymbol{P}^2}{2M_{\mathrm{r}}}+H_{\mathrm{e}} , $$

(5.85)

with the “electronic” Hamiltonian

$$ H_{\mathrm{e}}=\sum_{n=1}^{N}V_{nn}( \boldsymbol{R})\frac{1}{2}\bigl(p_n^2+y_n^2-1 \bigr) +\sum_{n<m=1}^{N}V_{nm}( \boldsymbol{R}) (y_ny_m+p_np_m) . $$

(5.86)

A semiclassical implementation of the coupled time-dependent Schrödinger equation (5.60), (5.61) can now be done by using the Herman-Kluk propagator of Sect. 2.3.4. Each of the vectors (x,p _i ,q _i ) in the multi-dimensional formulation of the semiclassical propagator (2.197) contains the nuclear as well as the harmonic degrees of freedom. The initial state is a direct product |Ψ _α 〉 of, e.g., a Gaussian wavepacket in the nuclear coordinate times the ground state eigenfunction of the initially unoccupied harmonic mode and the first excited state in the occupied harmonic mode [34]. The overlap with the coherent state in (2.197) can again be determined analytically.

3.3.3 Application to a Model System

In the following, full quantum as well as semiclassical results of the solution of the coupled surface time-dependent Schrödinger equation (5.60), (5.61) will be reviewed for a model that has been used in order to study the breakdown of the Rosen-Zener “approximation” [36]. The dimensionless variables

$$ Q\equiv R/R_{\mathrm{c}} \quad \mbox{and} \quad \tau\equiv t/t_{\mathrm{c}} $$

(5.87)

are the same as used there, with

$$ R_{\mathrm{c}}=\sqrt{\hbar/(\sqrt{2}M_{\mathrm{r}}\omega_{\mathrm{e}})} \quad \mbox{and} \quad t_{\mathrm{c}}=\sqrt{2}/\omega_{\mathrm{e}} . $$

(5.88)

Here ω _e is the frequency of the harmonic ground electronic surface and the model potentials are given by

$$\begin{aligned} V_{11}(Q) =&Q^2/2 \end{aligned}$$

(5.89)

$$\begin{aligned} \tilde{V}_{22}(Q) =&-AQ+B , \end{aligned}$$

(5.90)

whereas the off-diagonal potential, proportional to the envelope of the external field with dimensionless pulse length parameter T _p, is given by

$$ \tilde{V}_{12}(\tau)=D\operatorname{sech} \biggl[\frac{\tau-\tau_0}{T_{\mathrm{p}}} \biggr] . $$

(5.91)

For an inverse hyperbolic cosine pulse as above, a driven two-level system can be treated analytically and its Rosen-Zener solution has been reviewed in Sect. 3.2.4. The only difference to the case we study here is the absence of the kinetic energy in the Rosen-Zener model. Therefore, although the problem without kinetic energy is solvable exactly analytically, now this solution is an approximation! With these remarks, it is clear that the Rosen-Zener (RZ) approximation is the exact analytical solution of the approximate coupled time-dependent Schrödinger equation

$$\begin{aligned} \mathrm {i}\dot{\chi}_1^{\mathrm{RZ}}(\tau) =&\lambda(Q) \chi_1^{\mathrm{RZ}}(\tau) +\tilde{V}_{12}(\tau)\tilde{ \chi}_2^{\mathrm{RZ}}(\tau) \end{aligned}$$

(5.92)

$$\begin{aligned} \mathrm {i}\dot{\tilde{\chi}}_2^{\mathrm{ RZ}}(\tau) =& \tilde{V}_{12}(\tau)\chi_1^{\mathrm{ RZ}}(\tau) -\lambda(Q) \tilde{\chi}_2^{\mathrm{ RZ}}(\tau) \end{aligned}$$

(5.93)

with

$$ \lambda(Q)=\frac{\tilde{V}_{22}(Q)-V_{11}(Q)}{2} . $$

(5.94)

In the spirit of the so-called Franck-Condon approximation [37, 38] electronic transitions take place at fixed nuclear positions Q. We can therefore use the solution of Rosen and Zener

$$ \bigl|\chi_1^{\mathrm{ RZ}}({\lambda} ,\tau)\bigr|^2 =\biggl \vert F \biggl[{D} {T_{\mathrm{p}}}, -{D} {T_{\mathrm{p}}}; \frac{1}{2}-\mathrm {i}{\lambda} {T_{\mathrm{p}}};{z(\tau)} \biggr]\biggr \vert ^2 $$

(5.95)

with the hypergeometric function F [39] and

$$ {z(\tau)}=\frac{1}{2}\bigl[\tanh(\tau/{T_{\mathrm{p}}})+1\bigr] . $$

(5.96)

$|\chi_{1}^{\mathrm{RZ}}|^{2}$ depends on Q via λ and provides the probability to be in the ground state at a given Q. The total probability to be in the ground state can be gained by multiplying the Rosen-Zener solution with the initial probability density and integrating over position. For the probability to be in the excited state

$$ P_2^{\mathrm{RZ}}(\tau)=1-\int \mathrm {d}Q \bigl|\chi_1(Q,- \infty)\bigr|^2\bigl|\chi_1^{\mathrm{RZ}}\bigl(2 \lambda = \Delta V(Q),\tau\bigr)\bigr|^2 $$

(5.97)

then follows.

In Table 5.3 the model parameters for the results to be presented are gathered. In both cases, the initial nuclear wavepacket is the ground state wavefunction of the harmonic surface. The Gaussian part of the 3d wavefunction in the semiclassical case is thus centered around the origin and has the width parameters (γ ₁₁,γ ₂₂,γ ₃₃)=(2^−1/2,1,1). Quantities of interest are the auto-correlation function

$$ c(\tau)=\bigl\langle\chi_1(\tau)\big|\chi_1(0)\bigr\rangle+ \bigl\langle\chi_2(\tau)\big|\chi_2(0)\bigr\rangle, $$

(5.98)

as well as the occupation probabilities

$$ P_{1,2}(\tau)=\bigl\langle\chi_{1,2}(\tau)\big| \chi_{1,2}(\tau) \bigr\rangle $$

(5.99)

of the different levels.

Table 5.3 Dimensionless model parameters for the problem of two coupled surfaces

For model I, a considerable number of Rabi oscillations occurs as can be seen in Fig. 5.19. The quality of the semiclassical results is good. This is so, although the parameters chosen for model I lead to a substantial nonlinearity of the classical equations of motion.⁸ Also the semiclassical wavefunctions at time τ=0.135 in the different levels in Fig. 5.20 give a good account of the quantum wavefunction. Especially the double humped structure of the wavefunction in the ground state is correctly reproduced.

Fig. 5.19

Real part of the auto-correlation function (a) and the population of level 1 (b) and level 2 (c) as a function of time for model I; solid line: semiclassical result, dotted line: full quantum result; adapted from [40]

Fig. 5.20

Absolute value of the wavefunction in level 1 (a) and level 2 (b) at time τ=0.135 for model I; solid line: semiclassical result, dotted line: full quantum result [40]

For model I, the Rosen-Zener approximation (which is not shown) would be very well founded due to the shortness of the pulse and there would be almost no difference compared to the full quantum results. Let us consider a case where the neglect of the kinetic energy, sometimes referred to as the short time approximation, breaks down, however. This is the case of model II, that uses the same wavepacket parameters as model I. In Fig. 5.21 a comparison of three different results is displayed. The semiclassical as well as the full quantum and the approximate Rosen-Zener quantum results for the probability to be in the second level are shown. As in case I above, the semiclassical results are representing the full quantum results quite well. In the Rosen-Zener approximation for longer times, a deviation from the quantum result can be observed, however, which is due to the neglect of the kinetic energy in (5.92), (5.93).

Fig. 5.21

Comparison of the semiclassical (solid), the quantum (long-dashed) and the Rosen-Zener result (short dashed) for the population of level 2 as a function of time for model II [40]

3.4 Femtosecond Spectroscopy

The study of ultra-fast molecular processes is a rapidly growing research field that has gained considerable attention due to the 1999 Nobel prize in Chemistry for Zewail. Before we concentrate on some theoretical aspects of that field let us get acquainted with orders of magnitude of time scales in molecules.

To this end we will convert times into energy (and vice versa) using the formula

$$ E/h=1/T . $$

(5.100)

In different units this reads

$$ E(\mathrm{eV})=\frac{4.134}{T(\mathrm{fs})},\qquad E\bigl(\mathrm{cm}^{-1}\bigr)= \frac{33\mbox{,}368}{T(\mathrm{fs})}. $$

(5.101)

From molecular spectra [41] the following ranges for times (periods) in which typical phenomena occur can be extracted⁹:

• Rotation: from 1–100 ps ($\bar{\nu}_{J=0\to J=1}$ from 20 cm⁻¹ to 0.25 cm⁻¹)

• Normal vibrations: from 10–300 fs ($\bar{\nu}_{n=0\to n=1}$ from 4,000 cm⁻¹ to 100 cm⁻¹)

• Vibrational relaxation: 100 fs–100 ps

• Direct photodissociation: up to 100 fs [23]

In order to investigate molecular phenomena on a femtosecond scale, time-resolved measurements, which are frequently referred to as “pump-probe” experiments, are performed. In such experiments a sample is excited by a first, so-called pump pulse. After a variable time delay T _d a second, so-called probe pulse is impinging on the excited system and a signal is measured. This may, e.g., be the absorption of the system, the fluorescence of the system or the probability to emit an electron with a certain energy. The time resolution of the experiment is given by the full width at half maximum of the pulses, which shall be used in the following to characterize their shortness.

3.4.1 Pump-Probe Photoelectron Spectroscopy of Na₂

As a first example of a pump-probe experiment, we consider the excitation of the $(2)^{1}\varSigma_{u}^{+}$-state of the sodium dimer by a 40 fs laser pulse of the fundamental wavelength 340 nm (pump-pulse) and subsequent ionization by a probe-pulse of wavelength 530 nm, arriving after a variable time delay T _d. The energy of the emitted electrons can then be measured as a function of T _d [42]. Theoretical investigations of the same system have been performed by Christoph Meier in his Ph.D. thesis [43], which we will follow closely.

Before the action of the pump pulse the molecule is described by the vibrational ground state wavepacket in the electronic ground state $\mathrm{X}^{1}\varSigma_{g}^{+}$, depicted in the left panel of Fig. 5.22. The perturbation of the system by the pump pulse

$$ V_{\mathrm{L}}=\mu_{10}f(t)\frac{\mathcal{E}_0}{2} \bigl(\mathrm {e}^{\mathrm {i}\omega_{\mathrm{ P}} t}+\mathrm {e}^{-\mathrm {i}\omega_{\mathrm{ P}} t}\bigr) $$

(5.102)

leads to excitation of the wavefunction onto an excited electronic surface. Using the Condon approximation, the dipole moment is assumed to be independent of position. Furthermore, for the following investigation, perturbation theory is adequate to describe the laser molecule interaction. In a form applicable to a vector-type time-dependent Schrödinger equation as in (5.60), (5.61) it is reviewed in Appendix B.

Fig. 5.22

Left panel: potential surfaces of the sodium dimer together with the action of the pump pulse onto the initial wavepacket in the electronic ground state; right panel: emergence of the wavepacket on the excited state surface due to the action of the pulse (dotted line); adapted from [43]

In first order in the perturbation and in rotating wave approximation (neglecting the counter-rotating term $\sim \mathrm {e}^{\mathrm {i}\omega_{\mathrm{ P}} t}$ of the perturbation)

$$ \chi_1(R,t)=\frac{1}{\mathrm {i}}\int_0^t \mathrm {d}t'\mathrm {e}^{-\mathrm {i}\hat{H}_1(t-t')} \mu_{10}f\bigl(t' \bigr)\frac{\mathcal{E}_0}{2}\mathrm {e}^{-\mathrm {i}\omega_{\mathrm{ P}} t'}\mathrm {e}^{-\mathrm {i}E_0t'} \chi_0(R,0) $$

(5.103)

for the wavefunction on the excited state surface is found.

For a numerical implementation the integral above has to be discretized according to

$$ \chi_1(R,t)=\frac{\Delta t}{\mathrm {i}}\sum _{j=0}^n \mathrm {e}^{-\mathrm{i}\hat{H}_1(n-j)\Delta t} \mu_{10}f(j\Delta t)\frac{\mathcal{E}_0}{2}\mathrm {e}^{-\mathrm {i}(E_0+\omega_{\mathrm{ P}})j\Delta t} \chi_0(R,0). $$

(5.104)

The vibrational ground state with the energy E=E ₀ is propagated on the shifted ground state surface until an intermediate time jΔt is reached. The resulting state then is propagated on the excited surface with Hamiltonian $\hat{H}_{1}$ until the final time nΔt is reached. It is not known at what time the photon is being absorbed, however, and therefore all the intermediate times have to be integrated over. This procedure can be also formulated iteratively according to [44]

$$\begin{aligned} \chi_1(R,t_n+\Delta t) =& \mathrm {e}^{-\mathrm {i}\hat{H}_1\Delta t}\chi_1(R,t_n) \\ &{}+\frac{\Delta t}{\mathrm {i}}\mu_{10}f(t_n+\Delta t) \frac{\mathcal{E}_0}{2} \mathrm {e}^{-\mathrm {i}(E_0+\omega_{\mathrm{ P}})(t_n+\Delta t)}\chi_0(R,0) . \end{aligned}$$

(5.105)

Here the first term can be calculated with the split-operator method, whereas the second term is given analytically. At each time step, a further part of the wavefunction is lifted on the excited electronic surface. The result of such a calculation is depicted in the right panel of Fig. 5.22, which shows a vibrationally excited wavepacket on the excited electronic surface, that moves almost dispersionless to larger internuclear distances.

5.7 :

Rewrite the sum in the discretized representation of first order perturbation theory for n=1,2,3 and verify the iterative prescription

$$\chi_1(R,t_n+\Delta t)=\hat{U}_1( \Delta t)\chi_1(R,t_n) +\frac{\Delta t}{\mathrm {i}}\mu_{10}\mathcal{E}(t_n+\Delta t) U_0(t_n+\Delta t)\chi_0(R,0) $$

with $\hat{U}_{1}(t)=\mathrm {e}^{-\mathrm {i}\hat{H}_{1}t}$ , $U_{0}(t)=\mathrm {e}^{-\mathrm {i}(E_{0}+ \omega_{\mathrm{ P}})t}$ and $\mathcal{E}(t)=f(t)\mathcal{E}_{0}/2$ for the propagation of the component of the wavefunction on the excited surface.

The probe pulse, delayed by a time T _d and centered around the frequency ω _T, now allows the detection of the nuclear wavepacket motion on the excited potential energy surface via the measurement of the energy of the emitted photo electrons after ionization, as can be seen in Fig. 5.23. The key to the understanding of this measurement is the reflection principle. The use of that principle in the theory of photodissociation is reviewed in Appendix C.

Fig. 5.23

Potential surfaces of the sodium dimer with the action of the probe pulse on the wavepacket in the excited state [43]

In order to invoke the reflection principle, the wavefunction in the ionization continuum has to be considered. The basis of bound states ϕ _e,j (r,R) is extended by the continuum states $\phi_{E,V_{0}^{\mathrm{I}}}$ (free electron with energy E, ionic core in the ground state with potential $V_{0}^{\mathrm{I}}$). In the adiabatic approximation the nuclei would then fulfill the uncoupled equations

$$\begin{aligned} \mathrm {i}\dot{\chi}_{j}(R,t) =&\bigl\{\hat{T}_R+V_j(R) \bigr\}\chi_{j}(R,t) \end{aligned}$$

(5.106)

$$\begin{aligned} \mathrm {i}\dot{\chi}_{E}(R,t) =&\bigl\{\hat{T}_R+V_0^{\mathrm{I}}(R)+E \bigr\}\chi_{E}(R,t) . \end{aligned}$$

(5.107)

Coupling to a laser field (that starts to interact with the system at time T _d) in length gauge and using again the RWA, we get in first order perturbation theory

$$ \chi_E(R,t)= \frac{1}{\mathrm {i}}\int _{T_{\mathrm{d}}}^t\mathrm {d}t' \mathrm {e}^{-\mathrm {i}(\hat{H}_0^{\mathrm{I}}+E-\omega_{\mathrm{ T}})(t-t')} \mu_{E1}f\bigl(t'-T_{\mathrm{d}}\bigr) \frac{\epsilon_0}{2}\chi_1\bigl(R,t'\bigr) $$

(5.108)

for the wavefunction in the ionized state. From this, the spectrum of the emitted electron can be extracted according to

$$ P^{\mathrm{I}}(E,T_{\mathrm{d}})=\lim_{t\to\infty}\int \mathrm {d}R\bigl| \chi_E(R,t)\bigr|^2 . $$

(5.109)

Further progress toward an understanding is made by using the short-time approximation, that we have already encountered in the previous subsection. In this approximation the kinetic energy of the nuclei is neglected, which means for (5.108):

• Replace $\hat{H}_{0}^{\mathrm{I}}$ by $V_{0}^{\mathrm{I}}$

• Replace χ ₁(R,t′) by $\mathrm {e}^{-\mathrm {i}V_{1}(t'-T_{\mathrm{d}})}\chi_{1}(R,T_{\mathrm{d}})$

Equation (5.108) thus becomes the Fourier transformation of the pulse envelope. Using the definition¹⁰

$$ F(x)=\int \mathrm {d}t \mathrm {e}^{\mathrm {i}xt}f(t) $$

(5.110)

and the short-time approximation, we get

$$ P^{\mathrm{I}}(E,T_{\mathrm{d}})\sim\int \mathrm {d}R\bigl|\mu_{E1} \chi_1(R,T_{\mathrm{d}})\bigr|^2 \bigl|F\bigl(D(R)+E- \omega_{\mathrm{ T}}\bigr)\bigr|^2 . $$

(5.111)

Here the definition of the difference potential

$$ D(R)\equiv V_0^{\mathrm{I}}(R)-V_1(R) $$

(5.112)

has been used. The largest contributions to the expression for the electron emission probability come from regions of vanishing argument of the Fourier transform. This is yet another application of the SPA from Sect. 2.2.1. The SPA condition leads to the definition of so-called transient Franck-Condon regions [45]

$$ D(R_{\mathrm{ tr}})\approx\omega_{\mathrm{ T}}-E . $$

(5.113)

In case of a monotonous function D(R) only a single stationary phase point exists and the remaining integral can be approximated by

$$ P^{\mathrm{I}}(E,T_{\mathrm{d}})\sim\bigl|\mu_{E1}\chi_1 \bigl(R_{\mathrm{ tr}}(E),T_{\mathrm{d}}\bigr)\bigr|^2 . $$

(5.114)

This is the mathematical formulation of the dynamical reflection principle, saying that the electron spectrum is proportional to the absolute square of the wavefunction at time T _d. The structure of its argument tells us that the squared wavefunction is reflected at the difference potential [43]. If it is steep then P ^I is broad in energy. If it has a small slope, however, a sharp peak of P ^I emerges. Both situations are depicted graphically in Fig. 5.24. In this figure two different probe pulse delays, T ₁=0.2 ps and T ₂=1 ps are compared with each other. Due to the motion of the wavepacket on surface V ₁, a dramatic change of the electron spectrum occurs. For the longer delay, first it is shifted to smaller energies, and second, it becomes much narrower. By knowing the potential surfaces, one can monitor the motion of the nuclear wavepacket by means of the measured photoelectron spectrum.

Fig. 5.24

Dynamical reflection principle for two different pulse delays [43]

The validity of the dynamical reflection principle hinges on the applicability of the short-time approximation. This can be judged by looking at the full time-evolution operators appearing in (5.108)

$$ \mathrm {e}^{\mathrm {i}\hat{H}_0^{\mathrm{I}}t'}\mathrm {e}^{-\mathrm {i}\hat{H}_1t'}=\mathrm {e}^{\hat{\varLambda}} . $$

(5.115)

Using the Baker-Campbell-Haussdorff formula from Sect. 2.3.2 in the form

$$ \mathrm {e}^{\hat{A}}\mathrm {e}^{\hat{B}}\approx \mathrm {e}^{\hat{A}+\hat{B}+1/2[\hat{A},\hat{B}]} $$

(5.116)

one finds that

$$ \hat{\varLambda}=\mathrm {i}D(R)t'-\bigl[\hat{T}_R,D(R) \bigr]t'^2 $$

(5.117)

holds. Retaining only the first term in this expression leads to the short-time approximation. The term proportional to t′² with the prefactor

$$ -\bigl[\hat{T}_R,D(R)\bigr]=\frac{1}{2M_{\mathrm{r}}}\bigl(D''(R)+ 2D'(R)\partial_R\bigr)\approx\frac{\mathrm {i}}{M_{\mathrm{r}}}D'p $$

(5.118)

should be small. If one interprets (p/M _r)t′ classically as the change of the internuclear distance during the pulse then the condition for the applicability of the short time approximation is

$$ D'\frac{p}{M_{\mathrm{r}}}t'\ll D(R), $$

(5.119)

i.e. the difference potential should not change much in the range that the wavepacket crosses during the pulse.

3.4.2 Fluorescence Spectroscopy of ICN

Instead of detecting the motion of the wavepacket on the excited surface via the measurement of the energy of emitted electrons as in the previous case, also the fluorescence after excitation into a second excited state can be used. An example that has been studied experimentally as well as theoretically is the ICN molecule. Theoretically it suffices to consider only the dynamics of the C-I stretch coordinate. The corresponding dynamics on the 3 coupled potential surfaces

• Electronic ground state

• I + CN(X ² Σ ⁺) excited (dissociative) state

• I + CN(B ² Σ ⁺) excited (dissociative) state

two of which can be seen in Fig. 5.25, has been investigated in [46].

Fig. 5.25

Dissociative potential surfaces and different probe frequencies for ICN; adapted from [46]

The coupled surface time-dependent Schrödinger equation for the laser driven system is given by

$$ \mathrm {i}\partial_t \begin{pmatrix} \chi_0 \\ \chi_1 \\ \chi_2 \end{pmatrix} = \begin{pmatrix} \hat{H}_0&\hat{H}_{01}&0 \\ \hat{H}_{10}&\hat{H}_1&\hat{H}_{12} \\ 0&\hat{H}_{21}&\hat{H}_{2} \end{pmatrix} \begin{pmatrix} \chi_0 \\ \chi_1 \\ \chi_2 \end{pmatrix} , $$

(5.120)

where in the rotating wave approximation the coupling by the pump, respectively the probe pulse is given by

$$\begin{aligned} \hat{H}_{01} =&\mu_{01}A_1(t)\mathrm {e}^{-\mathrm {i}\omega_{\mathrm{ P}}t} \end{aligned}$$

(5.121)

$$\begin{aligned} \hat{H}_{12} =&\mu_{12}A_2(t-T_{\mathrm{d}}) \mathrm {e}^{-\mathrm {i}\omega_{\mathrm{ T}}t} . \end{aligned}$$

(5.122)

The occupation of the second excited state is proportional to the measured fluorescence signal and is therefore to be calculated theoretically. In first order perturbation theory we get for the wavefunctions in the different electronic states (if E ₀=0)

$$\begin{aligned} \bigl|\chi_0(t)\bigr\rangle =&\mathrm {e}^{-\mathrm {i}E_0t}\bigl|\chi_0(0) \bigr\rangle=\bigl|\chi_0(0)\bigr\rangle \end{aligned}$$

(5.123)

$$\begin{aligned} \bigl|\chi_1(t)\bigr\rangle \sim&\int_{-\infty}^{t} \mathrm {d}t'\mu_{01}A_1\bigl(t'\bigr) \mathrm {e}^{-\mathrm {i}\hat{H}_1'(t-t')}\bigl|\chi_0(0)\bigr\rangle \end{aligned}$$

(5.124)

$$\begin{aligned} \bigl|\chi_2(t)\bigr\rangle \sim&\int_{-\infty}^{t} \mathrm {d}t'\mu_{12}A_2\bigl(t'-T_{\mathrm{d}} \bigr) \mathrm {e}^{-\mathrm {i}\hat{H}_2'(t-t')}\bigl|\chi_1(t)\bigr\rangle \end{aligned}$$

(5.125)

with $\hat{H}_{1}'=\hat{H}_{1}-\omega_{\mathrm{ P}}$ and $\hat{H}_{2}'=\hat{H}_{2}-\omega_{\mathrm{T}}$, due to the fact that the potentials are shifted in RWA by ω _P, respectively ω _T.¹¹ The transfer of probability density to an excited surface can only be large if the crossing with the shifted level is at the maximum of the wavepacket. This so-called resonance case was depicted together with the off-resonance case in Fig. 5.17. Only in the case of resonance a total population transfer is possible by a so-called π-pulse.

In the following the results of a simulation [46] of an experiment of the Zewail group [47] will be reviewed.

• Pump- and probe pulse have a FWHM of 125 fs.

• The pump wavelength is fixed at the off-resonant value of 306 nm.

• Four different (only three in the experiment) probe wavelengths have been applied

In Fig. 5.26, the time evolution of the wavepacket on the first excited state surface is displayed, with the pump pulse being centered around t=0. The wavepacket dissociates and spreads simultaneously. In Fig. 5.27, the population of the first excited state is depicted. That this population is of the order of a few times 10⁻⁴ reflects the fact that an off-resonant pump frequency is used in the experiment.

Fig. 5.26

Time evolution on the first excited potential energy surface [46]

Fig. 5.27

Occupation probability of the first excited surface [46]

The probe part of the experiment has been performed with three different probe frequencies, leading to resonance at different internuclear displacements as depicted in Fig. 5.28. The additional probe wavelength of 433 nm is not depicted in this plot. For the long wavelengths the resonance condition is more or less well localized in space and therefore as a function of the probe pulse delay, a peaked structure is to be expected. This is exactly what can be observed in Fig. 5.29! For the additional theoretical wavelength the signal is barely visible but peaked. The peak tends to become a plateau for the shorter wavelengths. In these cases the resonance condition is fulfilled for a long interval of internuclear distances as can be seen in Fig. 5.28. The plateau is perfectly developed in the case of 388.9 nm. The experimental results are very well reproduced by the calculations as can be seen by comparing the two panels in Fig. 5.29.

Fig. 5.28

First excited surface and second surface shifted by different amounts, corresponding to different probe frequencies [46]

Fig. 5.29

Comparison of calculated (left panel) and experimental (right panel) fluorescence spectra of ICN for 4 (theory), respectively 3 (experiment) values of the probe wavelength, from [46, 47]

4 Control of Molecular Dynamics

Up to now we have encountered a multitude of partly counter-intuitive phenomena, appearing in atomic or molecular systems exposed to a laser field. Quite naturally one might ask if a suitable field can be found that drives a system into a desired quantum state or steers a chemical reaction into a desired channel.

Let us start to find an answer to that question for a system that exhibits one of the most fundamental quantum phenomena: a symmetric double well potential allowing for coherent tunneling between its two minima. This system has been studied in detail under the influence of an external periodic laser field. In the following we will then refrain from the restriction to periodic fields and will consider pulsed fields with arbitrary pulse shapes, that, e.g., allow for the control of chemical reactions or for the selective excitation of vibrational modes.

4.1 Control of Tunneling

Tunneling in a stationary double well is a phenomenon, which is discussed in almost every textbook on quantum mechanics. In the heyday of quantum theory it has been used to explain the vibrational spectrum of pyramidal molecules, like NH₃, by F. Hund [48]. The influence of a periodic external force (mediated, e.g., by a cw-laser) on coherent tunneling has been investigated in my Ph.D. thesis and the results have been published in [49]. The basis for the understanding of those results is Floquet theory as will be seen in the following.

4.1.1 The Model System

A model potential for a particle of mass M _r moving in a symmetric quartic double well is given by

$$ V_{\mathrm{DW}}(R)\equiv-\frac{M_{\mathrm{r}}\omega_ {\mathrm{e}}^{2}}{4}R^{2} + \frac{M_{\mathrm{r}}^{2}\omega_{\mathrm{e}}^{4}}{64 E_{\mathrm{B}}}R^{4} . $$

(5.126)

The frequency of small oscillations around the minima

$$ R_{\mathrm{r},l}=\pm\sqrt{\frac{8 E_{\mathrm{B}}}{M_{\mathrm{r}}\omega_ {\mathrm{e}}^{2}}} $$

(5.127)

of this potential is ω _e, and E _B denotes the height of the barrier between the two wells. The dynamics of a wavepacket, that can be written as a superposition of the ground and the first excited state of that single surface, is the well-known coherent tunneling dynamics, reviewed in Appendix D.

A realization of the potential of (5.126) is given, e.g. by the pyramidal NH₃ molecule. The relevant coordinate R refers to the umbrella mode (see Fig. 1.4 of Chap. 1) and measures the distance between the nitrogen atom and the hydrogen plane. The reduced mass is M _r=M _N M _3H/(M _N+M _3H) (see footnote on p. 566 of [2]). In this system an external periodic force can be generated by a monochromatic laser field of amplitude $\mathcal{E}_{0}$ coupling to the dipole moment

$$ \mu(R)=\mu' R $$

(5.128)

of the molecule with the dipole gradient μ′. The amplitude of the external force is then given by

$$ F_0=\mu' \mathcal{E}_0 . $$

(5.129)

In this subsection, we measure energies in units of ħω _e, such that D=E _B/ħω _e. Time is measured in units of 1/ω _e, $x=\sqrt{(M_{\mathrm{r}}\omega_{\mathrm{e}})/\hbar}R$, and the dimensionless amplitude is given by $S=F_{0}/\sqrt{\hbar M_{\mathrm{r}}\omega_{\mathrm{e}}^{3}}$. The dimensionless Hamiltonian is then given by

$$ \hat{H}(x,t)=-\frac{1}{2}\partial_{x}^{2}- \frac{1}{4}x^{2} +\frac{1}{64 D}x^{4}+xS\sin(wt) , $$

(5.130)

where w=ω/ω _e is the dimensionless ratio of driving frequency and harmonic well frequency.

4.1.2 Coherent Destruction of Tunneling

One of the most counter-intuitive effects that external forcing can have is the suppression of the tunneling dynamics in a double well. For frequencies in the middle of the interval

$$ \frac{\varDelta }{2}\leq w\leq w_{\mathrm{res}} $$

(5.131)

with Δ=E ₂−E ₁ and w _res=E ₃−E ₁ (E _i being the unperturbed eigenvalues of the double well) this suppression is found along a one-dimensional curve in the (w,S) parameter space [50].

The reason for this behavior is an exact crossing of the quasi-energies emerging from the lowest two unperturbed energies as a function of the amplitude. The quasi-energies can be determined by diagonalizing the Floquet matrix of Sect. 2.2.7. For D=2 (which is close to the NH₃ value of 2.18) and an external frequency of w=0.01, near the geometric mean of the unperturbed tunneling frequency Δ=1.895×10⁻⁴ and the first resonance frequency w _res=0.876, the behavior shown in Fig. 5.30 is found. In panel (a) of this figure the quasi-energies cross at two different values of the external force. These crossings are exact (see panel (b)), due to the fact that the quasi-eigenfunctions have different symmetry under the generalized parity transformation defined in (3.69). The non-crossing rule does therefore not hold, and as a function of a parameter the quasi-energies may approach each other arbitrarily closely.

Fig. 5.30

Exact crossing of two quasi-energies of the driven double well (D=2, w=0.01): a 0<S<0.01; b zoomed region around S≈3×10⁻³ [50]

Now the time-evolution of a Gaussian wavepacket, initially centered in the left well, $\chi_{\mathrm{ l}}^{\mathrm{GW}}(x)$ ¹² for the parameters at the first exact crossing has been investigated. Quantities of interest are the probability to stay (survival probability) in the initial state, i.e. the absolute square of the auto-correlation function of that wavefunction

$$ P(t):=\bigl|\bigl\langle\chi_{\mathrm{l}}^{\mathrm{GW}}(t)\big|\chi_{\mathrm{l}}^{\mathrm{GW}}(0) \bigr\rangle\bigr|^2 $$

(5.132)

and the probability to be to the left of the barrier

$$ \rho_{\mathrm{l}}(t):=\int_{-\infty}^0 \mathrm {d}x\bigl| \chi_{\mathrm{l}}^{\mathrm{GW}}(x,t)\bigr|^2 . $$

(5.133)

These quantities are displayed in Fig. 5.31.

Fig. 5.31

Staying probability and probability to be to the left of the barrier (D=2) for the exact crossing parameters S=3.171×10⁻³, w=0.01: a–b Stroboscopic time evolution over 2¹⁰ periods $\frac{2\pi}{w}$; c–d time evolution inside the first period, adapted from [50]

The unperturbed tunneling period for the parameters chosen here (D=2, w=0.01) is at around 50 periods, T, of the external field. In Fig. 5.31, panels (a)–(b) we can see, however, that in the presence of driving the particle is almost completely localized at the initial position even after 2¹⁰ T. This is due to the fact that the initial wavefunction consists mainly of two Floquet functions whose quasi-energies cross exactly. Small deviations of the staying probability from unity are due to the finite overlap of the initial state with other Floquet functions, whose energies are not crossing exactly. The dynamics has been considered only stroboscopically, so far. Time-evolution during a period of the external field is shown in panels (c)–(d) of Fig. 5.31. It can be observed that the periodic time-dependence of the quasi-eigenfunctions does not destroy the tunneling suppression for the present parameters.

In order to illustrate the localization effect in position space, in Fig. 5.32, the absolute square of the initial state $\chi_{\mathrm{l}}^{\mathrm{GW}}(x,0)$ and the time-evolved state with the lowest overlap (occurring at t=458T) during the first 1,024 periods is depicted. Apart from a small shift of the center of the wavepacket to the right, there is almost no dynamics observable. This picture also explains why the probability to stay left to the barrier deviates less from unity than the staying probability. In calculating ρ _l, one has to integrate over the whole range −∞<x≤0 and motion of the wavefunction in the left well will not show up directly in the dynamics of ρ _l. The probability to stay left thus deviates only by maximally 2 % from unity, whereas P(n) maximally looses around 8 % of its initial value.

Fig. 5.32

Absolute square of wavefunctions at t=0 (solid line) and at t=458T (dashed line), and unperturbed potential (dotted line) with D=2, adapted from [50]

4.1.3 Crossing Manifold and Two-Level System

As already mentioned, the localization phenomenon occurs along a 1d manifold in the (w,S) parameter space, along which two relevant quasi-energies, having different parity, cross. This manifold has been determined in [50] and is shown in Fig. 5.33. The crossing of the two quasi-energies is a necessary but not a sufficient condition for the localization phenomenon, however. This is studied in great detail in [50], where it is shown how the old unperturbed tunneling behavior is recovered for small driving frequencies w≈Δ and what happens at resonance w≈w _res, where the third level comes into play.

Fig. 5.33

Double logarithmic plot of the one dimensional manifold in (w,S) parameter space along which the relevant quasi-energies cross for the first time. Driven double well (D=2): crosses; driven two-level system: solid line. The vertical line crosses the manifold at w=0.01, S=3.171×10⁻³ [50]

A deeper understanding of the linear part of the manifold along which real localization is observed can be gained by studying the two-level system describing just the lowest two levels of the double well problem [51]. The corresponding time-dependent Schrödinger equation is given by

$$\begin{aligned} \mathrm {i}\dot{c}_{1}(t) =&E_{1}c_{1}(t)+ \langle\chi_{1}|x|\chi_{2}\rangle S\sin(wt)c_{2}(t) \end{aligned}$$

(5.134)

$$\begin{aligned} \mathrm {i}\dot{c}_{2}(t) =&E_{2}c_{2}(t)+ \langle\chi_{1}|x|\chi_{2}\rangle S\sin(wt)c_{1}(t) \end{aligned}$$

(5.135)

with c _1,2(t)≡〈χ _1,2|χ(t)〉. The unperturbed Hamiltonian is now a 2×2 matrix and the quasi-energies can again be determined according to the scheme reviewed in Sect. 2.2.7. The location in parameter space of the exact crossing of the two quasi-energies, emerging out of the two lowest unperturbed states, is plotted in Fig. 5.33 as a solid line. For frequencies w≪w _res it is very close to the manifold of the full problem and it stays a perfect straight line also for frequencies w>w _res due to the non-existence of a third unperturbed level.

The slope of the manifold in the linear range can be determined analytically for a two-level system. In the case of w≫Δ (defining the linear region), it has been shown by Shirley that the first crossing is approximately given by the first zero of the Bessel function $J_{0}\bigl(\frac{2b}{w}\bigr)$ [52]. Here b denotes the field strength multiplied by the dipole matrix element

$$ b\equiv\langle\chi_{1}|x|\chi_{2}\rangle S \stackrel{D=2}{\approx}3.791 S . $$

(5.136)

Using the simple form of the argument of the Bessel function and its first zero [53] the straight line

$$ S=\frac{2.40482\dots}{2\langle\chi_{1}|x|\chi_{2}\rangle}w \stackrel{D=2}{\approx}0.3172 w $$

(5.137)

in (w,S) parameter space is found. Higher zeroes of the Bessel function give straight lines along which the quasi-energies cross each other exactly again (see Fig. 5.30(a)). Tuning the parameters to an exact crossing, localization is also found in the time-dependent two-level Schrödinger equation (5.134), (5.135) with the initial conditions $c_{1}(0)=-c_{2}(0)=1/\sqrt{2}$.

Furthermore, it is worthwhile to note that in the strong field limit, the suppression phenomenon can also be understood in the transfer matrix formalism [54].

4.1.4 The Asymmetric Double-Well Potential

What is the effect of a finite asymmetry on the localization phenomenon? In order to study this question, a potential of the form

$$ V_{\sigma}(x,t)=-\frac{1}{4}x^{2}+\frac{1}{64 D}x^{4}+ \sigma x +xS\sin(wt) $$

(5.138)

with a static asymmetry, σ>0, can be used. In this case no symmetry under the generalized parity transformation (3.69) does exist any more. This leads to the fact that all Floquet energies, due to the non-crossing rule, do not cross exactly (except maybe at singular points). Therefore no 1d manifold along which two quasi-energies cross does exist.

What happens to the allowed crossings that we have observed in the symmetric case? In order to answer this question a relatively small asymmetry with parameter σ can be applied. In Fig. 5.34 the same field and potential parameters have been used as in Fig. 5.30(b), except for the asymmetry. We can see that the allowed exact crossing becomes an avoided crossing in the presence of asymmetry. Localization therefore goes away gradually. The splitting of the levels is rather small and the wavepacket would still be localized for relatively long times. It would not be localized forever any more as in the symmetric case, however. For strong asymmetry two effects have to be considered. First the avoided crossing becomes broader but secondly a partial localization does result from the fact that the lowest eigenstate becomes similar to the coherent state in the lower well.

Fig. 5.34

Quasi-energies as a function of S for D=2 and w=0.01 in the asymmetric (σ=10⁻⁶) double well [50]

4.1.5 More Driven Double-Well Systems

The realization of driven double well systems is possible in many different branches of physics. Recently, driven double wells have, e.g., been realized in optical fiber systems. In such a system a light beam propagating through a periodically curved waveguide is coupled to a parallel fiber. In this setup the first experimental realization of the effect of coherent destruction of tunneling has been performed [55]!

Another physical system whose dynamics can be described with the help of a multistable potential is the rf-SQUID. There the macroscopic flux through the ring is the tunneling degree of freedom. An external perturbation maybe given by a magnetic field. This is an example from the realm of solid state physics, however, and shall not be dealt with here. Very recently, the direct observation of suppression of single particle tunneling of atoms in light shift double-well potentials has been reported [56].

Let us finally come back to molecular physics. Apart from the NH₃-molecule where the nitrogen atom experiences a double-well potential, also the electron in the $\mathrm{H}_{2}^{+}$- (resp. $\mathrm{D}_{2}^{+}$-) molecule sees a double well potential, due to the electron-nuclear interaction. Recently it has been shown both theoretically as well as experimentally that the dissociation of the electron can be steered by the carrier envelope phase such that the electron is localized preferentially at a specific proton (deuteron) [16, 57].

4.2 Control of Population Transfer

The transfer of population into a desired state is one of the central challenges of control theory. Before we discuss a direct approach to that field, using optimal control theory, a counter-intuitive method to control population transfer shall be reviewed.

In molecular systems, this is the stimulated Raman adiabatic passage or short STIRAP method. In this scheme a three level system, displayed in Fig. 5.35, is coupled via two different laser pulses. A direct coupling of level |1〉, which might by a rotational level in the vibrational ground state and the highly excited vibrational state |3〉 shall be dipole forbidden. The methodology is used experimentally to selectively excite vibrational states [58].

Fig. 5.35

A three level system (so-called Λ-system) coupled via pump and Stokes pulse with the respective detunings Δ _P and Δ _S [58]

The pump-pulse couples levels 1 and 2, while the Stokes pulse couples levels 2 and 3. The total Hamilton matrix H=H ₀ +W is given by

$$ \mathbf{H}= \begin{pmatrix} E_1&-\mu_{12}E_{\mathrm{P}}\cos(\omega_{\mathrm{P}}t)&0\\ -\mu_{21}E_{\mathrm{P}}\cos(\omega_{\mathrm{P}}t) &E_2&-\mu_{23}E_{\mathrm{S}}\cos(\omega_{\mathrm{S}}t)\\ 0&-\mu_{32}E_{\mathrm{S}}\cos(\omega_{\mathrm{S}}t)&E_3 \end{pmatrix} . $$

(5.139)

Transformation into the interaction picture with the help of the unperturbed Hamiltonian

$$ \mathbf{H}_{\mathbf{0}}= \begin{pmatrix} E_1&0&0\\ 0&E_2&0\\ 0&0&E_3 \end{pmatrix} , $$

(5.140)

and invoking the rotating wave approximation leads to

$$ \mathbf{W}_{\mathbf{I}}=\mathbf{U}_{\mathbf{0}}^\dagger \mathbf{W U}_{\mathbf{0}} = - \frac{1}{2} \begin{pmatrix} 0&\varOmega_{\mathrm{P}}\mathrm{e}^{-\mathrm{i}\varDelta _{\mathrm{P}}t}&0\\ \varOmega_{\mathrm{P}}\mathrm{e}^{\mathrm{i}\varDelta _{\mathrm{P}}t} &0&\varOmega_{\mathrm{S}}\mathrm{e}^{\mathrm{i}\varDelta _{\mathrm{S}}t}\\ 0&\varOmega_{\mathrm{S}}\mathrm{e}^{-\mathrm{i}\varDelta _{\mathrm{S}}t}&0 \end{pmatrix} $$

(5.141)

for the Hamiltonian in the interaction representation. Here the abbreviations Ω _P(t)=μ ₂₁ E _P(t), Ω _S(t)=μ ₃₂ E _S(t) for the Rabi frequencies without detuning and Δ _P=(ω ₂−ω ₁)−ω _P, Δ _S=(ω ₂−ω ₃)−ω _S for the detunings that are plotted in Fig. 5.35 have been defined.

In the case of vanishing detunings, the time-dependent eigenvalues and eigenstates (dressed states) are given by

$$\begin{aligned} E_{0,\pm} =&0,\pm\frac{(\varOmega_{\mathrm{P}}^2+\varOmega_{\mathrm{S}}^2)^{1/2}}{2} \end{aligned}$$

(5.142)

$$\begin{aligned} |g_0\rangle =&\cos\varTheta|1\rangle-\sin \varTheta|3\rangle \end{aligned}$$

(5.143)

$$\begin{aligned} |g_{\pm}\rangle =&\sin\varTheta|1\rangle-\cos \varTheta|3\rangle \mp\frac{1}{\sqrt{2}}|2\rangle, \end{aligned}$$

(5.144)

with $\varOmega=(\varOmega_{\mathrm{P}}^{2}+\varOmega_{\mathrm{S}}^{2})^{1/2}$ and the definition

$$ \varTheta\equiv\arctan \biggl(\frac{\varOmega_{\mathrm{P}}}{\varOmega_{\mathrm{S}}} \biggr) . $$

(5.145)

Cosines and sines of this angle can be resolved by using the relations given in footnote 3 of Chap. 3.

With the help of the dressed states and of the quantum mechanical adiabatic theorem of Appendix E, the pulse sequence can be understood. Starting from state |1〉 only the dressed state |g ₀〉 is occupied initially if Ω _S≫Ω _P. This amounts to the counter-intuitive pulse sequence depicted in panel (a) of Fig. 5.36, where the Stokes pulse comes before the pump pulse! If the field changes adiabatically¹³ then according to the adiabatic theorem the system stays in the dressed state |g ₀〉 of the instantaneous Hamiltonian. For large positive times Ω _P≫Ω _S holds, however, and thus the system finally is in state |3〉, without having occupied the “dark state” |2〉 in the meantime. This dynamics is depicted in Fig. 5.36, where also the mixing angle and the dressed eigenvalues are displayed.

Fig. 5.36

STIRAP dynamics: a Pump- and Stokes-pulse, b angle Θ, c dressed eigenvalues, d occupation probabilities [58]

The ordering of the pulses is counter-intuitive. They have to have a non-vanishing overlap, however, in order for population transfer to be achieved (see also Fig. 5.36). This can be seen by doing Exercise 15.8 in [22], the reference we are following closely throughout this subsection. Furthermore, an alternative perspective on STIRAP can be gained by demanding constant probability to be in the second (dark) state. For this the time derivative

$$ \frac{\mathrm {d}|a_2|^2}{\mathrm {d}t}=2\operatorname{Re}\bigl[a_2^\ast \dot{a}_2\bigr] =-\bigl[ \varOmega_{\mathrm{P}}(t)\operatorname{Im} \bigl(a_2^{\ast}a_1\bigr) + \varOmega_{\mathrm{S}}(t)\operatorname{Im} \bigl(a_2^{\ast}a_3\bigr)\bigr] $$

(5.146)

must vanish and again, we have assumed resonance Δ _S=Δ _P=0. This requirement leads to the conditions

$$\begin{aligned} \varOmega_{\mathrm{P}} =&-\varOmega_0(t)\operatorname{Im} \bigl[a_3^\ast(t)a_2(t)\bigr] \end{aligned}$$

(5.147)

$$\begin{aligned} \varOmega_{\mathrm{S}} =&\varOmega_0(t)\operatorname{Im} \bigl[a_1^\ast(t)a_2(t)\bigr] . \end{aligned}$$

(5.148)

The two terms on the RHS of (5.146) then cancel each other. The counter-intuitive ordering of the pulses follows from the fact that a ₁ is initially large and therefore also Ω _S is large compared to Ω _P. When the system is in state |3〉 the pump pulse takes over.

4.3 Optimal Control Theory

Optimal control theory deals with the search for external fields that steer a system into a desired state. This can be a certain vibrational excitation, which was also the goal of STIRAP, just discussed. One of the most demanding goals that can be reached with lasers is the control of a chemical reaction, however. One can, e.g., try to design laser pulses in such a way that chemically bound species dissociate in a predetermined way. In a triatomic system several different reaction channels exist. Two of them are

$$\begin{aligned} \mathrm{ABC}\to\mathrm{ A}+\mathrm{ BC}\quad\mbox{channel 1} \end{aligned}$$

and

$$\begin{aligned} \mathrm{ABC}\to\mathrm{AB}+\mathrm{C}\quad\mbox{channel 2} \end{aligned}$$

and a laser field that discriminates channel 1 in favor of channel 2 might for example be looked for.

In the following we will discuss two important scenarios in the field of chemical reactions, starting with the “precursor” of the optimal control schemes, the so-called “pump-dump”-scheme and then reviewing in detail the Krotov method, which gives a mathematical prescription to find the optimal field. Finally, we will come back to the question of steering a system into a desired quantum state.

4.3.1 Pump-Dump Control

The so-called pump-dump method is a very intuitive way to approach the field of optimal control [59]. One tries to steer the breaking of a specific bond by first lifting the system (e.g. a collinear ABC system) onto an electronically excited state and then using the motion of the nuclei in that state in such a way that the system is deexcited exactly at a time when the subsequent motion in the electronic ground state leads to dissociation in the desired channel.

In order to understand the physics behind the pump-dump method we first need to look at a typical potential landscape of a collinear ABC system, shown in Fig. 5.37. The potentials are drawn as functions of two degrees of freedom corresponding to the two interatomic distances. The lower surface has a local minimum and two channels which are separated from the minimum via saddle points. The upper surface is almost harmonic. The basic idea of how to steer the reaction into a desired channel becomes clear, if we consider the classical Lissajous motion on the electronically excited surface after excitation with the pump pulse. This motion is depicted in the leftmost panel in Fig. 5.38. Now the dump-pulse arrives with a specific time delay. The Husimi transform of a typical pump-dump pulse sequence has been displayed in Fig. 1.10 of Chap. 1 already. Choosing the time delay accordingly, the Lissajous motion can be intercepted at any desired point. If it is intercepted at t ₁, such that the motion on the electronic ground state continues in channel 1, then the dissociation has been steered to proceed in this channel, as depicted in the middle panel of Fig. 5.38.

Fig. 5.37

Electronic potential curves of a collinear ABC system with pictorial representation of the pump and the dump pulse [60]

Fig. 5.38

Classical mechanical understanding of the pump-dump scenario in the ABC system: a Lissajous motion on upper surface, b classical trajectory exiting in channel 1, c classical trajectory exiting in channel 2 [22]

Quantum mechanically, the dynamics of wavefunctions and not of a single classical trajectory has to be considered. It turns out, however, that due to the harmonic nature of the excited electronic state, the physical picture of the pump-dump method stays intact [61]. The wavepacket evolves almost dispersionless on the upper surface and the description in terms of classical trajectories is sufficient. After action of a dump pulse with a time delay of 810 a.u. the wavepacket exits in channel 2 on the electronic ground state, as depicted in Fig. 5.39. This case corresponds to the rightmost panel of Fig. 5.38.

Fig. 5.39

Quantum mechanical pump-dump scenario in the HHD system for a time delay of 810 a.u.: a wave function in the electronic ground state at the initial time t=0 a.u., b wave function in the electronic ground state at the time t=1,000 a.u., c wave function in the electronic ground state at the time t=1,200 a.u. [22]

4.3.2 Krotov Method

The pump-dump method that we have just discussed is the precursor of modern control methods, that try to achieve higher yields, i.e., to achieve the desired goal to a higher degree.

The goal can by formulated mathematically by using a projection operator $\hat{P}_{\alpha}$, projecting the wavefunction on the desired channel and trying to maximize

$$ J_P= \bigl\langle\chi(T_{\mathrm{t}})\bigl| \hat{P}_\alpha\bigr|\chi(T_{\mathrm{t}})\bigr\rangle. $$

(5.149)

Here T _t is the total time allowed for the control process.

In order that the energy content of the field does not grow indefinitely, the functional above is usually augmented by a term

$$ J_\mathcal{E}=\lambda\int_0^{T_{\mathrm{t}}} \mathrm {d}t\bigl|\mathcal{E}(t)\bigr|^2 , $$

(5.150)

proportional to a Lagrange multiplier λ. Furthermore, the time-dependent Schrödinger equation is introduced again via a Lagrange multiplier 〈ξ(t)| into the functional by the real term

$$ J_H=2\operatorname{Re} \int_0^{T_{\mathrm{t}}}\mathrm {d}t \bigl\langle\xi(t)\bigr| \biggl(-\partial_t+\frac{\hat{H}}{\mathrm {i}} \biggr)\bigl| \chi(t)\bigr\rangle, $$

(5.151)

which deconstrains $\mathcal{E}$ and χ to first order in the field [22]. The functional to be extremized is finally given by

$$\begin{aligned} \tilde{J} \equiv& J_P+J_H-J_\mathcal{E} . \end{aligned}$$

(5.152)

To be specific, the case of a two component wavefunction χ=(χ _g,χ _e) and a corresponding 2×2 Hamilton matrix (see also Appendix B)

$$ \hat{\mathbf{H}}= \begin{pmatrix} \hat{H}_{\mathrm{g}}&\mu\mathcal{E}^\ast(t) \\ \mu\mathcal{E}(t)&\hat{H}_{\mathrm{e}} \end{pmatrix} $$

(5.153)

is considered in the following. After integration by parts of the J _H term,

$$ \tilde{J}=J_P-2\operatorname{Re}\langle\boldsymbol{\xi}|\boldsymbol{\chi} \rangle |_0^{T_{\mathrm{t}}}+ 2\operatorname{Re}\int_0^{T_{\mathrm{t}}} \mathrm {d}t \biggl\{\bigl\langle\boldsymbol{\xi}(t)\bigr| \frac{\hat{\mathbf{H}}}{\mathrm {i}}\bigl| \boldsymbol{\chi}(t)\bigr \rangle +\langle\dot{\boldsymbol{\xi}}|\boldsymbol{\chi}\rangle \biggr\} -J_{\epsilon} $$

(5.154)

is found. The variation of this expression can now be performed according to the rules that are gathered in Appendix B to Chap. 2. Extremalizing with respect to χ, i.e., the condition

$$ \frac{\delta\tilde{J}}{\delta|\boldsymbol{\chi}(t)\rangle}=0 $$

(5.155)

leads to the equation

$$ -\mathrm {i}\langle\dot{\boldsymbol{\xi}}|=\langle\boldsymbol{\xi}|\hat{\mathbf{H}} , $$

(5.156)

which is a backward Schrödinger equation for the Lagrange parameter. Its final condition is found by doing the variation

$$ \frac{\delta\tilde{J}}{\delta|\boldsymbol{\chi}(T_{\mathrm{t}})\rangle}=0 , $$

(5.157)

leading to

$$ \bigl\langle\boldsymbol{\xi}(T_{\mathrm{t}})\bigr| = \bigl\langle \boldsymbol{ \chi}(T_{\mathrm{t}})\bigr|\hat {P}_\alpha. $$

(5.158)

In addition to this equation, also the initial value equation

$$\begin{aligned} \mathrm {i}|\dot{\boldsymbol{\chi}}\rangle =&\hat{\mathbf{H}}|\boldsymbol{\chi}\rangle \end{aligned}$$

(5.159)

$$\begin{aligned} \bigl|\boldsymbol{\chi}(0)\bigr\rangle =&|\boldsymbol{\chi}_0\rangle, \end{aligned}$$

(5.160)

has to hold.

Extremalizing with respect to $\mathcal{E}^{\ast}$, i.e., the condition

$$ \frac{\delta\tilde{J}}{\delta\mathcal{E}^\ast(t)}=0 $$

(5.161)

leads to

$$ \mathcal{E}(t)=\frac{-\mathrm {i}}{\lambda}\bigl[\langle \xi_{\mathrm{g}}|\mu|\chi_{\mathrm{e}}\rangle- \langle\chi_{\mathrm{g}}| \mu|\xi_{\mathrm{e}}\rangle\bigr] $$

(5.162)

for the field [60].

5.8 :

Show that setting the variation of $J_{H}-J_{\mathcal{E}}$ with respect to $\mathcal{E}^{\ast}$ equal to zero leads to the expression ( 5.162 ) for the field.

The five equations (5.156), (5.158), (5.159), (5.160), (5.162) contain a double sided boundary value problem. The easiest solution procedure is given by the following steps:

• 1. Propagate χ(t) from t=0 to t=T _t forward in time

• 2. Apply $\hat{P}_{\alpha}$ to χ(T _t) yielding ξ(T _t)

• 3. Propagate ξ from t=T _t to t=0 backward in time

The field has to be guessed, however, and does not necessarily fulfill the equation coming out of the variation procedure! Therefore the scheme above has to be augmented by an iterative procedure, due to Krotov [62]:

• 1. Choose an initial field $\mathcal{E}^{0}(t)$

• 2. Propagate χ(t) under $\mathcal{E}^{0}(t)$ forward in time

• 3. Projection of χ(T _t) gives ξ(T _t)

• 4. Propagate ξ backward in time

• 5. Commonly propagate ξ(t) (with the old field) and χ(t) with the new instantaneously calculated field

  $$ \mathcal{E}^1(t)=\frac{-\mathrm {i}}{\lambda} \bigl[\bigl\langle \xi_{\mathrm{g}}^0\bigr|\mu\bigl|\chi_{\mathrm{e}}^1\bigr \rangle- \bigl\langle\chi_{\mathrm{g}}^1\bigr|\mu\bigl|\xi_{\mathrm{e}}^0 \bigr\rangle\bigr] $$

  (5.163)

  forward in time

• 6. Project χ(T _t) and continue the procedure until convergence is achieved.

The propagation of ξ(t) forward in time seems to be superfluous, because the result is already known. Keeping the wavefunction in computer memory would be barely possible for most cases of interest, however, and therefore it is cheaper to calculate ξ(t) once more. In general the propagated wavefunction has amplitude in both the desired and the undesired channel, see, e.g., Fig. 2f in [60]. The above procedure iterates the field in such way that the undesired portion of the wavefunction is minimized. If this minimum is an absolute or a local one is a question that goes far beyond the scope of this book. The method just laid out goes back to Krotov. Other methods to solve for the optimal field have been devised, however, see [22] and [63] and the references therein. Whereas in the simple pump-dump scheme, low yields of 10⁻² and selectivity ratios of 3:2 have been reported, in the optimally controlled case, typical yields increase to more than 10 % and selectivity ratios can be as high as 13:3 [60].

As an example let us review results that were obtained for an ABC system. As the initial guess for the electric field in the iterative process a pump-dump pulse as depicted in Fig. 1.10 has been used. One can try to steer the reaction either into channel 1 or into channel 2. In the first case the resulting optimal field is still rather similar to the original pump-dump pulse [22], whereas in the second case the field displayed in Fig. 5.40 is resulting. Around 50 iterations are typically necessary to converge the results.

Fig. 5.40

Steering the breakup of the ABC system into the second channel: a optimal field, b Husimi transform of the field, c norm of the wavefunction in the ground and excited state, d final wavefunction, from [22]

4.3.3 Optimally Controlled Excitation of Quantum States

Optimal control schemes do not only work for the breakup reaction just considered. They have been shown to be applicable also for the case of vibrational excitation.

In [30] optimal control theory has been applied with the objective to steer a Morse oscillator, representing a CH-stretch, with Morse parameters D _e=0.199, R _e=1.5, and α=0.9386, all in atomic units, into a specific excited target state |n _T〉. The projection operator therefore is given by

$$ \hat{P}_{\mathrm{T}}=|n_{\mathrm{T}}\rangle \langle n_{\mathrm{T}}| . $$

(5.164)

The dipole moment was assumed to be of Mecke form with the parameters μ ₀=1.76 a.u. and R ^∗=1 a.u. The result of the optimization starting from the vibrational ground state and fixing the final time to be T _t=0.1 ps are shown in Fig. 5.41.

Fig. 5.41

Steering a Morse oscillator into selected excited states via optimal control theory, left panels show probabilities as a function of time and quantum number, right panels show the corresponding electric fields [30]

Optimal control theory has been applied in a lot of other physical systems. One out of many other examples is the control of cis-trans isomerization [64].

4.4 Genetic Algorithms

The theory of optimal control of the last section rests on the availability of analytically (or numerically) given potential energy surfaces and on the validity of the underlying Born-Oppenheimer approximation. Both requirements may be violated, however, and even the Hamiltonian might not be known. Therefore, alternative control schemes are sought for.

A recent development in the field of control therefore is the application of genetic algorithms. Their application is based on an experimental “analog computer”. The system to be controlled is exposed to a laser whose temporal shape can be varied. By a feedback mechanism a digital computer using a genetic (evolutionary) algorithm can vary the field iteratively in such a way that the desired goal is reached. The principal setup is displayed in Fig. 5.42.

Fig. 5.42

Principal setup of an “analog computer” for feedback control. The control pulse excites the system and the probe pulse measures the outcome (the measurement could, e.g., also be performed by a mass spectrometer) which is fed into the evolutionary algorithm; a good initial guess helps to achieve convergence quickly [65]

The first theoretical study that showed the feasibility of such an approach is due to Judson and Rabitz [66]. These authors have shown that it is based on the following three paradigms:

• “Survival of the fittest”

• Crossover

• Mutation

As an example the transition from the n=j=0 ground vibrational state to the rotationally excited vibrational ground state n=0, j′=3 of the KCl molecule was investigated. An “individuum” of the genetic algorithm is a specific laser pulse sequence. Its initial gene consists of N_gene=128 entries of random numbers, uniformly distributed between zero and one, and was then scaled to a maximum field strength of 5 kV cm⁻¹. The total number of individuals was chosen as N_pop=50. The first paradigm can be tested by introducing the “cost-function”

$$\begin{aligned} \sum_j(\delta_{jj'}- \rho_j)^2 , \end{aligned}$$

where ρ _j is the occupation probability of state j. It is zero if the desired state with label j′ is fully populated. Individuals can then be ranked according to their fitness. The highest ranked ones were taken over to the next generation without change, whereas the other ones had to undergo crossover and a small probability of mutation. In the upper panel of Fig. 5.43 the decay of the cost function as a function of generation is displayed for the average population, as well as for the best individuum. Furthermore, in the lower panel of that figure, the spectrum corresponding to the best gene is shown. It displays maxima at the resonant transitions between the rotational states j=0,1,2,3. It is important to stress that the spectral information was not input into the generation of the optimal field but was found by the learning loop.

Fig. 5.43

Upper panel: Cost function of the average population and the best gene; lower panel: Spectrum of the optimal pulse (arrows indicate positions of resonant transitions between rotational sublevels of the vibrational ground state of the KCl molecule), from [66]

An experimental realization of control based on evolutionary algorithms was performed by the Gerber group [67]. The goal of the experiment was to steer the photo-fragmentation of CpFe(CO)₂Cl into a desired channel. A pulse shaper that allows to split the laser light into 128 spectral components and vary them separately has been used. Selectivity ratios of about 5 have been achieved.

4.5 Toward Quantum Computing with Molecules

A recent new development in the field of laser-molecule interaction is the realization of quantum logic operations with the help of molecular vibrational states. We will not deal with that exciting new field in much detail but will discuss the realization of the basic ingredient of every setup used for computing: the flipping of a bit. As we will see, anharmonic vibrational modes have to be used to this end. Using the OH diatomic, this has been shown by Babikov [68] and by Cheng and Brown [69].

The flipping of a bit is based on the realization of the NOT-Operation. In a two-level system this corresponds to the complete transfer of population from level 0 to level 1 or vice versa

$$\begin{aligned} \mathrm{NOT}|0\rangle =&|1\rangle \end{aligned}$$

(5.165)

$$\begin{aligned} \mathrm{NOT}|1\rangle =&|0\rangle. \end{aligned}$$

(5.166)

Each deviation from the complete population transfer reduces the so-called fidelity, defined as the occupation probability of the initially unpopulated level. Unfortunately a harmonic oscillator cannot be controlled to switch completely to a desired state, because of the equidistance of its levels. One has to choose two levels of an anharmonic system, as e.g., the Morse potential of Sect. 5.1.2 in order to realize the NOT operation.

5.9 :

Derive the maximal probability to populate the n-th excited state of a harmonic oscillator by using an external field in length gauge and starting from the (vibrational) ground state χ ₀ .

• (a) Calculate the time-dependent wavefunction χ ₀(x,t) under the influence of the external field.

• (b) Determine the overlap

  $$\begin{aligned} a_{n0}(t)=\bigl\langle\chi_n\big|\chi_0(t)\bigr \rangle \end{aligned}$$

  by using ∫dxexp[−(x−y)²]H _n (x)=π ^1/2(2y)ⁿ with the Hermite polynomial H _n (x).

• (c) Show that the maximum of the absolute value |a _n0(t)|² is given by n ⁿe⁻ⁿ/n!

As a specific example the driven OH stretch with the Morse parameters D _e=0.1994, R _e=1.821, and α=1.189 in atomic units and the Mecke parameters μ ₀=1.634 a.u. and R ^∗=1.134 a.u. (see also caption of Fig. 5.14) has been used. Although the anharmonicity constant of that molecule is fixed in nature, it can be viewed as a parameter in theoretical considerations [69]. These authors have looked at the fidelity

$$ P_{10}(T_{\mathrm{t}})=\bigl|\bigl\langle1\big|0(T_{\mathrm{t}})\bigr \rangle\bigr|^2 $$

(5.167)

with T _t=750 fs as a function of anharmonicity, and found the results reproduced in Fig. 5.44. Two different results are shown there. First the system has been exposed to an optimal control pulse, in close analogy to the work of Shi and Rabitz [30], and secondly to a simple π-pulse, we know already from Chap. 3. For large anharmonicity it can be seen that the π-pulse is superior to the “optimal” pulse! Furthermore, the statement that the harmonic oscillator cannot be controlled to 100 % can be read off from the results at small anharmonicity.

Fig. 5.44

Fidelity of the NOT gate as a function of the anharmonicity ω _e x _e for an optimal pulse (squares) and for the π-pulse (circles); adapted from [69]

The anharmonic properties of suitable candidates for molecular quantum computing and the realization of additional gates are discussed in [70].

5 Notes and Further Reading

The authoritative reference on the hydrogen molecular ion is the classic book by Slater [1]. More general material on molecular spectra and molecular structure can be found in the textbook by Bransden and Joachain [2]. Spectroscopic constants that can be used for the generation of analytic Morse potentials for diatomic molecules are contained in [8]. Further information on diatomic molecules is available in the references given in [71]. A modern exposition of quantum chemistry which is the main theoretical tool for the calculation of electronic potential surfaces is given in the book by Szabo and Ostlund [3].

Reviews covering both theoretical as well as experimental facts on the dynamics of $\mathrm{H}_{2}^{+}$ in intense laser fields are given in [10, 72]. More information on molecules in laser fields can also be found in the book edited by Bandrauk [73]. The recent review by Posthumus [72] contains a very insightfull discussion of field dressed states (for strong fields these are Floquet states) of $\mathrm{H}_{2}^{+}$ and their use to explain phenomena like molecular stabilization (bond hardening) and bond softening.

The transformation from adiabatic to diabatic states and the non-uniqueness of that transformation are discussed in Chap. 15.2 of [23]. Additional material on this transformation can be found in Chap. 12.2 of [22]. Nonadiabatic molecular dynamics can be tackled in many different ways. Further information on that topic with additional references can be found for the case without an external laser in [74] and with an external laser in [75]. The semiclassical initial value method applied to the problem of coupled surfaces is reviewed in [76]. Reviews of experimental and theoretical approaches to femtosecond chemistry are collected in [77, 78].

Driven quantum tunneling is reviewed in depth in [79] and STIRAP is discussed in greater detail than here in the books by Rice and Zhao [63] and by Tannor [22]. Some experimental aspects of STIRAP can be found in the overview article by Bergmann et al. [58]. The formulation of most of the section on optimal control is based on Chap. 16 of [22]. Tannor’s book as well as [63] and [80] contain a wealth of additional material on the coherent control of quantum dynamics. Also in Tannor’s book (Chap. 15.6 of [22]) an intuitive local control scheme for the heating of an electronic ground state wavepacket without substantial excitation of electronically excited states is reviewed.

A natural extension of the scope of this chapter would be to study clusters in intense laser fields. A recent review of that field is given in [81]. Furthermore, the phenomenon of HHG in molecules has become a hot topic which is not covered here. A tutorial review of that field, including the concept of molecular orbital tomography is contained in [82].

Appendices

Appendix A: Relative and Center of Mass Coordinates for $\mathrm{H}_{2}^{+}$

In order to derive the Hamiltonian for $\mathrm{H}_{2}^{+}$ in a laser field we follow Hiskes’s general treatment of diatomic molecules [83]. Specializing to the single electron case, the Hamiltonian in length gauge and atomic units is first expressed by using the coordinates of the nuclei R _a , R _b and of the electron, r _e, according to

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

(5.168)

with the proton mass M _p, where V ₁ contains all Coulomb interaction terms, the laser is polarized in z-direction, and we have used atomic units.

Now center of mass and relative coordinates

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

are introduced. M _S=2M _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

$$ \begin{pmatrix} \boldsymbol{R}_{\mathrm{S}} \\ \boldsymbol{R} \\ \boldsymbol{r}_i \end{pmatrix} = \begin{pmatrix} \frac{M_{\mathrm{p}}}{M_{\mathrm{S}}}&\frac{M_{\mathrm{p}}}{M_{\mathrm{S}}}&\frac{1}{M_{\mathrm{S}}} \\ 1&-1&0 \\ -1/2&-1/2&1 \end{pmatrix} \begin{pmatrix} \boldsymbol{R}_{a} \\ \boldsymbol{R}_{b} \\ \boldsymbol{r}_{\mathrm{e}} \end{pmatrix} . $$

(5.169)

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

$$ \begin{pmatrix} \boldsymbol{R}_{a} \\ \boldsymbol{R}_{b} \\ \boldsymbol{r}_{\mathrm{e}} \end{pmatrix} = \begin{pmatrix} 1&1/2&-\frac{1}{M_{\mathrm{S}}} \\ 1&-1/2&-\frac{1}{M_{\mathrm{S}}} \\ 1&0&\frac{2M_{\mathrm{p}}}{M_{\mathrm{S}}} \end{pmatrix} \begin{pmatrix} \boldsymbol{R}_{\mathrm{S}} \\ \boldsymbol{R} \\ \boldsymbol{r}_i \end{pmatrix} . $$

(5.170)

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 [83]. The relative motion, however, is governed by the Hamiltonian

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

(5.171)

where the reduced masses M _r=M _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 $\mathrm{H}_{2}^{+}$ no kinetic couplings between the different degrees of freedom are present nor does the field couple to the relative coordinate of the nuclei, which is not true in the general case [83].

Appendix B: Perturbation Theory for Two Coupled Surfaces

In the case of a laser driven two level system with a 2×2 (matrix-) Hamiltonian

$$ \hat{\mathbf{H}}=\hat{\mathbf{ H}}_0+\hat{\mathbf{ W}}(t) $$

(5.172)

perturbation theory is best performed in the interaction picture of Sect. 2.2.3. In first order and after back transformation to the Schrödinger picture, analogous to (2.87),

$$ \bigl|\boldsymbol{\chi}(t)\bigr\rangle=\mathrm {e}^{-\mathrm {i}\hat{\mathbf{ H}}_0t}\bigl|\boldsymbol{\chi}(0)\bigr\rangle + \frac{1}{\mathrm {i}}\int_0^t\mathrm {d}t' \mathrm {e}^{-\mathrm {i}\hat{\mathbf{ H}}_0(t-t')} \hat{\mathbf{ W}}\bigl(t'\bigr)\mathrm {e}^{-\mathrm {i}\hat{\mathbf{ H}}_0t'}\bigl|\boldsymbol{ \chi}(0)\bigr\rangle $$

(5.173)

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

$$ \bigl|\boldsymbol{\chi}(0)\bigr\rangle= \begin{pmatrix} |\chi_{\mathrm{g}}(0)\rangle \\ 0 \end{pmatrix} . $$

(5.174)

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

$$ \bigl|\boldsymbol{\chi}(t)\bigr\rangle= \begin{pmatrix} |\chi_{\mathrm{g}}(t)\rangle \\ |\chi_{\mathrm{e}}(t)\rangle \end{pmatrix} $$

(5.175)

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

$$ \chi_{\mathrm{e}}(\boldsymbol{R},t)=\frac{1}{\mathrm {i}}\int_0^t \mathrm {d}t'\mathrm {e}^{-\mathrm {i}\hat{H}_{\mathrm{e}}(t-t')} \hat{W}_{\mathrm{eg}}\bigl( \boldsymbol{R},t'\bigr)\mathrm {e}^{-\mathrm {i}\hat{H}_{\mathrm{g}}t'}\chi_{\mathrm{g}}(\boldsymbol{R},0) $$

(5.176)

is found. Here the definitions

$$ \hat{W}_{\mathrm{eg}}(\boldsymbol{R},t)= \bigl\langle{\mathrm{e}}(\boldsymbol{R}) \bigr|\hat{W}(t)\bigl| { \mathrm{g}}(\boldsymbol{R})\bigr\rangle \qquad \hat{H}_{j}= \bigl\langle j( \boldsymbol{R}) \bigr|\hat{H}_0\bigl| j(\boldsymbol{R})\bigr\rangle $$

(5.177)

of the matrix elements of the (electronic plus nuclear) Hamiltonian¹⁴ have been introduced, and j can either be e(xited) or g(round). The physical interpretation of this final result is straightforward. The system propagates for a time t′ on the lower surface is then lifted to the upper surface and propagates there until the final time t. All the possibilities to split the time interval [0,t] have to be integrated over.

Appendix 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. For example, 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,

$$ \chi_{\mathrm{g}}(R )= \biggl(\frac{2\alpha_R}{\pi} \biggr)^{1/4} \mathrm {e}^{-\alpha _R(R-R_{\mathrm{e}})^2} , $$

(5.178)

approximated by a Gaussian centered around R _e and with inverse width parameter α _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

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

(5.179)

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

$$ \chi_{\mathrm{e}}(R,t)\sim \mathrm {e}^{-\mathrm {i}[V_{\mathrm{e}}-V_R(R-R_{\mathrm{e}})]t} \mathrm {e}^{-\alpha_R(R-R_{\mathrm{e}})^2} . $$

(5.180)

The Fourier transformation of the auto-correlation c(t)=〈χ _e(0)|χ _e(t)〉 can be done analytically, yielding the absorption spectrum

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

(5.181)

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 _e and its FWHM ΔE=V _R ΔR is proportional to the negative slope of the antibinding surface and the FWHM of the squared initial wavepacket. In Fig. 5.45 it is shown that the reflection of the squared initial wavepacket at the antibinding surface yields the spectrum.

Fig. 5.45

Reflection principle of photodissociation [23]. ΔE is the FWHM of the absorption spectrum, whereas ΔR is the FWHM of the absolute square of the initial wavepacket

Appendix D: The Undriven Double-Well Problem

Figure 5.46 shows the unperturbed double-well potential

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

(5.182)

in the units introduced in Sect. 5.4.1 for D=2, including the five energy eigenvalues which lay below the barrier.

Fig. 5.46

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

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, χ ₁(x), χ ₂(x), depicted in Fig. 5.47, with the energies E ₁, E ₂.¹⁵ At time t=0 this leads to a state that is localized in the left well

$$ \chi_{l}(x,0)=\frac{1}{\sqrt{2}} \bigl[ \chi_{1}(x)-\chi_{2}(x) \bigr] . $$

(5.183)

In the case D→∞ it is identical to the ground state of the harmonic approximation to the left well. Its absolute value has the time evolution

$$ \bigl|\chi_{l}(x,t)\bigr|^{2}= \frac{1}{2} \bigl\{\bigl|\chi_{1}(x)\bigr|^{2}+\bigl| \chi_{2}(x)\bigr|^{2} -2\chi_{1}(x) \chi_{2}(x) \cos\bigl[(E_{2}-E_{1})t\bigr] \bigr \}. $$

(5.184)

Defining the tunneling splitting as

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

(5.185)

the corresponding tunneling time

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

(5.186)

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, Δ can be determined, however. The result of such a calculation is [84]

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

(5.187)

depending exponentially on the dimensionless barrier height. In Table 5.4 some values of Δ for different barrier heights can be found.

Fig. 5.47

Eigenfunctions of the lowest two eigenvalues for D=2. Solid line: χ ₁(x), dashed line: χ ₂(x)

Table 5.4 Numerical and semiclassical (index s) results for the tunneling splitting as a function of the barrier height [50]

Appendix 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

$$\begin{aligned} \bigl|\boldsymbol{\varPsi}(t)\bigr\rangle= \begin{pmatrix} |\psi_1(t)\rangle \\ |\psi_2(t)\rangle \\ \vdots \end{pmatrix} . \end{aligned}$$

(5.188)

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

$$\begin{aligned} \mathbf{ H}(t)= \begin{pmatrix} E_1&V_{12}(t)&\cdots\\ V_{21}(t)&E_2&\cdots\\ \vdots&\vdots&\ddots \end{pmatrix} . \end{aligned}$$

(5.189)

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

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

(5.190)

The transformed state vector is given by

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

(5.191)

It fulfills the time-dependent Schrödinger equation

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

(5.192)

If the Hamilton matrix H is slowly time-dependent, then also 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 [85] 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 [86]. In addition, an application to two-level systems has been given in [87].