3.1. Photoionization dynamics#
The core physics of photoionization has been covered extensively in the literature, and only a very brief overview is provided here with sufficient detail to introduce the metrology/reconstruction/retrieval problem; the reader is referred to Quantum Metrology Vol. 1 [4] (and references therein) for further details and general discussion.
Photoionization can be described by the coupling of an initial state of the system to a particular final state (photoion(s) plus free photoelectron(s)), coupled by an electric field/photon. Very generically, this can be written as a matrix element \(\langle\Psi_f|\hat{\Gamma}(\boldsymbol{\mathbf{E}})|\Psi_i\rangle\), where \(\hat{\Gamma}(\boldsymbol{\mathbf{E}})\) defines the light-matter coupling operator (depending on the electric field \(\boldsymbol{\mathbf{E}}\)), and \(\Psi_i\), \(\Psi_f\) the total wavefunctions of the initial and final states respectively.
There are many flavours of this fundamental light-matter interaction, depending on system and coupling. For metrology, the focus is currently on the simplest case of single-photon absorption, in the weak field (or perturbative), dipolar regime, resulting in a single photoelectron. (For more discussion of various approximations in photoionzation, see Refs. [80, 81].) In this case the core physics is well defined, and tractable (albeit non-trivial), via the separation of matrix elements into radial (energy) and angular-momentum (geometric) terms pertaining to couplings between various elements of the problem; the retrieval of such matrix elements is then a well-defined problem in general, achieved by making use of the analytic geometric terms in combination with fitting methodologies for the unknown radial matrix elements, and this is the approach explored herein. Again, more extensive background and discussion can be found in Quantum Metrology Vol. 1 [4], and references therein. Note, however, that whilst the general approach taken here is sound, many outstanding questions remain regarding the information content required for such an approach to be successful, and if other limitations prevent a unique set of solutions to be found in a given case (e.g. symmetry restrictions, phase ambiguities) - such questions are explored further herein, particularly in the case studies presented Part II.
The basic case also provides a strong foundation for extension into more complex light-matter interactions, in particular cases with shaped laser-fields (i.e. a time-dependent coupling \(\hat{\Gamma}(\boldsymbol{\mathbf{E,t}})\)) and multi-photon processes (which require multiple matrix elements, and/or different approximations). Note, however, that non-perturbative (strong field) light-matter interactions are, typically, not amenable to description in a separable picture in this manner. In such cases the laser field, molecular and continuum properties are strongly coupled, and are typically treated numerically in a fully time-dependent manner (although some separation of terms may work in some cases, depending on the system and interaction(s) at hand).
Underlying the photoelecton observables is the photoelectron continuum state \(\left|\mathbf{k}\right>\), prepared via photoionization. The photoelectron momentum vector is denoted generally by \(\boldsymbol{\mathbf{k}}=k\mathbf{\hat{k}}\), in the molecular frame (MF). The ionization matrix elements associated with this transition provide the set of quantum amplitudes completely defining the final continuum scattering state,
where the sum is over states of the molecular ion \(\left|\Psi_{+}\right>\). The number of ionic states accessed depends on the nature of the ionizing pulse and interaction. For the dipolar case,
Hence,
Where the notation implies a perturbative photoionization event from an initial state \(i\) to a particular ion plus electron state following absorption of a photon \(h\nu\), \(|\Psi_{i}\rangle+h\nu{\rightarrow}|\Psi_{+};\boldsymbol{\mathbf{k}}\rangle\), and \(\hat{\mu}.\boldsymbol{\mathbf{E}}\) is the usual dipole interaction term [82], which includes a sum over all electrons \(s\) defined in position space as \(\mathbf{r_{s}}\):
The position space photoelectron wavefunction is typically expressed as a partial-wave expansion, expanded as (asymptotic) continuum eignstates of orbital angular momentum, with angular momentum components \((l,m)\),
where \(\boldsymbol{r}\) are MF electronic coordinates and \(Y_{lm}(\mathbf{\hat{k}})\) are the spherical harmonics. Note the lower-case notation for the partial wave angular-momentum components, distinct from upper-case for the similar terms \((L,M)\) in the observables (Sect. 3.6).
Similarly, the ionization dipole matrix elements can be separated generally into radial (energy-dependent or ‘dynamical’ terms) and geometric (angular momentum) parts (this separation is essentially the Wigner-Eckart Theorem, see Ref. [83] for general discussion), and written generally as (using notation similar to [84]):
Provided that the geometric coupling parameters (the geometric part of the matrix elements) \(\gamma_{l,m}\) - which includes the geometric rotations into the LF arising from the dot product in Eq. (3.6) and other angular-momentum coupling terms - are known, knowledge of the so-called radial matrix elements elements, at a given \(k\), thus equates to a full description of the system dynamics (and, hence, the observables). Determination of these radial matrix elements - which are complex quantities with magnitudes and phases - is the aim of the reconstruction methodologies discussed herein (see Chpt. 4).
For the simplest treatment, the radial matrix element can be approximated as a 1-electron integral involving the initial electronic state (orbital), and final continuum photoelectron wavefunction:
As noted above, the geometric terms \(\gamma_{l,m}\) are analytical functions which can be computed for a given case - minimally requiring knowledge of the molecular symmetry and polarization geometry, although other factors may also play a role (see Sect. 3.3.2 for details).
The photoelectron angular distribution (PADs) at a given \((\epsilon,t)\) can then be determined by the squared projection of \(\left|\Psi_f\right>\) onto a specific state \(\left|\Psi_{+};\bf{k}\right>\); very generally this can be written in terms of the energy and angle-resolved observable, which arises as the coherent square:
Expansion in terms of the components of the matrix elements as detailed above then yields a separation into radial and angular components (see Quantum Metrology Vol. 1 [4], Sect. 2.1 for a full derivation), which can be written (at a single energy) as (following Eq. 2.45 of Quantum Metrology Vol. 1 [4]):
In this form \(\alpha\) denotes all other quantum numbers required to define the initial state, and \(\alpha_{+}\) the final state of the molecular ion. The radial matrix elements \(\boldsymbol{r}_{kl\lambda}\), denote an integral over the radial part of the wavefunctions, in this case labelled by the MF quantum numbers, and the associated scattering phase is given by \(\eta_{l\lambda}(k)\) (i.e. the matrix elements are written in magnitude-phase form, rather than complex form). The \(\gamma\) term denotes a general set of geometric coupling parameters arising from the coherent square. A tensor form is also given herein, see Sect. 3.3.2, and includes a full breakdown of these terms and details of numerical implementations.
Comparison of Eq. (3.9) with Eq. (3.37) indicates that the amplitudes in Eq. (3.6) also determine the observable anisotropy paramters \(\beta_{L,M}(\epsilon,t)\) (Eq. (3.37)), which basically collect all the terms in Eq. (3.9) and the product over spherical harmonics, into a resultant set of \((L,M)\) parameters which describe the observable PADs. (Note that the photoelectron energy \(\epsilon\) and momentum \(k\) are used somewhat interchangeably herein, with the former usually preferred in reference to observables.) Further discussion of the observables, including computation and form of these observables, can be found in Sect. 3.6.
The radial matrix elements - hence observables - are a sensitive function of molecular geometry and electronic configuration in general, as they depend on the overlap of the initial and final state wavefunctions, which includes the ionization continuum (scattering wavefunction) of the photoelectron. Hence, they may be considered to be responsive to molecular dynamics as well as photoionization dynamics, although they are formally time-independent in a Born-Oppenheimer basis. For further general discussion and examples see Ref. [85] and Quantum Metrology Vol. 1 [4]; discussions of more complex cases with electronic and nuclear dynamics can be found in Refs. [81, 86, 87, 88].
Note, also, that in the treatment above there is no time-dependence incorporated in the notation; however, a time-dependent treatment readily follows, and may be incorporated either as explicit time-dependent modulations in the expansion of the wavefunctions for a given case, or implicitly in the radial matrix elements. Examples of the former include, e.g. a rotational or vibrational wavepacket, or a time-dependent laser field. The rotational wavepacket case is discussed herein (see Sect. 3.3.2).
Typically, for reconstruction experiments, a given measurement will be selected to simplify this as much as possible by, e.g., populating only a single ionic state (or states for which the corresponding observables are experimentally energetically-resolvable), and with a bandwidth \(d\bf{k}\) which is small enough such that the radial matrix elements can be assumed constant over the observation window. Importantly, the angle-resolved observables are sensitive to the magnitudes and (relative) phases of these radial matrix elements - as emphasised in the magnitude-phase form of Eq. (3.9) - and can be considered as angular interferograms. It is the interferometric nature of the PADs which enables a phase-sensitive reconstruction protocol to be pursued.