Attosecond physics typically deals with
non-relativistic bounded particles and employs electromagnetic fields with a moderately high intensity (10^{11}-10^{14} W/cm2). This fact allows to set up a discussion in a
non-relativistic and
semi-classical quantum mechanics environment for light–matter interaction.
Atoms Resolution of time dependent Schrödinger equation in an electromagnetic field The time evolution of a single electronic
wave function in an atom, |\psi(t)\rangle is described by the
Schrödinger equation (in
atomic units): :\hat{H}|\psi(t)\rangle=i\dfrac{\partial}{\partial t}|\psi(t)\rangle \quad (1.0) where the light–matter interaction
Hamiltonian, \hat{H} , can be expressed in the
length gauge, within the dipole approximation, as: :\hat{H}=\frac{1}{2}\hat{\textbf{p}}^2+V_{C}+ \hat{\textbf{r}}\cdot\textbf{E}(t) where V_C is the
Coulomb potential of the atomic species considered; \hat{\textbf{p}}, \hat{\textbf{r}} are the momentum and position operator, respectively; and \textbf{E}(t) is the total
electric field evaluated in the neighbor of the atom. The formal solution of the Schrödinger equation is given by the
propagator formalism: :|\psi(t)\rangle=e^{-i\int_{t_0}^{t}\hat{H}dt'}|\psi (t_0)\rangle \qquad(1.1) where |\psi (t_0)\rangle, is the electron
wave function at time t=t_0. This exact solution cannot be used for almost any practical purpose. However, it can be proved, using
Dyson's equations that the previous solution can also be written as: :|\psi(t)\rangle=-i\int_{t_0}^{t}dt'\Big[ e^{-i\int_{t'}^{t}\hat{H}(t
)dt}\hat{H}_I(t')e^{-i\int_{t_0}^{t'}\hat{H}_0(t
)dt}|\psi(t_0)\rangle \Big]+e^{-i\int_{t_0}^{t}\hat{H}_0(t
)dt}|\psi(t_0)\rangle \qquad(1.2) where, :\hat{H}_0=\frac{1}{2}\hat{\textbf{p}}^2+V_{C} is the bounded Hamiltonian and :\hat{H}_I=\hat{\textbf{r}}\cdot\textbf{E}(t) is the interaction Hamiltonian. The formal solution of Eq. (1.0), which previously was simply written as Eq. (1.1), can now be regarded in Eq. (1.2) as a superposition of different
quantum paths (or quantum trajectory), each one of them with a peculiar interaction time t' with the electric field. In other words, each quantum path is characterized by three steps: • An initial evolution without the electromagnetic field. This is described by the left-hand side \hat{H}_0 term in the integral. • Then, a "kick" from the electromagnetic field, \hat{H}_I(t') that "excite" the electron. This event occurs at an arbitrary time that uni-vocally characterizes the quantum path t' . • A final evolution driven by both the field and the
Coulomb potential, given by \hat{H} . In parallel, you also have a quantum path that do not perceive the field at all, this trajectory is indicated by the right-hand side term in Eq. (1.2). This process is entirely
time-reversible, i.e. can also occur in the opposite order. For strong-field interaction problems, where
ionization may occur, one can imagine to project Eq. (1.2) in a certain continuum state (
unbounded state or free state) |\textbf{p}\rangle, of
momentum \textbf{p} , so that: :c_{\textbf{p}}(t)=\langle\textbf{p}|\psi(t)\rangle=-i\int_{t_0}^{t}dt'\langle \textbf{p}|e^{-i\int_{t'}^{t}\hat{H}(t
)dt}\hat{H}_I(t')e^{-i\int_{t_0}^{t'}\hat{H}_0(t
)dt}|{\psi(t_0)}\rangle +\langle \textbf{p} |e^{-i\int_{t_0}^{t}\hat{H}_0(t
)dt}|\psi(t_0)\rangle \quad (1.3) where |c_{\textbf{p}}(t)|^2 is the
probability amplitude to find at a certain time t, the electron in the continuum states |\textbf{p}\rangle. If this probability amplitude is greater than zero, the electron is
photoionized. For the majority of application, the second term in (1.3) is not considered, and only the first one is used in discussions, is currently used to describe the behavior of atoms (and molecules) in intense laser fields. SFA is the starting theory for discussing both high harmonic generation and attosecond pump-probe interaction with atoms. The main assumption made in SFA is that the free-electron dynamics is dominated by the laser field, while the Coulomb potential is regarded as a negligible perturbation. This fact re-shapes equation (1.4) into: :a_{\textbf{p}}^{SFA}(t)=-i\int_{t_0}^{t}dt'\langle \textbf{p}|e^{-i\int_{t'}^{t}\hat{H}_{V}(t
)dt}\hat{H}_I(t')e^{-i\int_{t_0}^{t'}\hat{H}_0(t
)dt}|{\psi(t_0)}\rangle \quad (1.4) where, \hat{H}_V=\frac{1}{2}(\hat{\textbf{p}}+\textbf{A}(t))^2 is the Volkov Hamiltonian, here expressed for simplicity in the velocity gauge, with \textbf{A}(t) , \textbf{E}(t)=-\frac{\partial \textbf{A}(t)}{\partial t} , the electromagnetic
vector potential. At this point, to keep the discussion at its basic level, lets consider an atom with a single energy level |0\rangle,
ionization energy I_P and populated by a single electron (single active electron approximation). We can consider the initial time of the wave function dynamics as t_0=-\infty, and we can assume that initially the electron is in the atomic ground state |0\rangle. So that, :\hat{H}_0|0\rangle=-I_P|0\rangle and |\psi(t)\rangle=e^{-i\int_{-\infty}^{t'}\hat{H}_0dt}|0\rangle=e^{I_Pt'}|0\rangle Moreover, we can regard the continuum states as
plane-wave functions state, \langle\textbf{r}|\textbf{p}\rangle=(2\pi)^{-\frac{3}{2}}e^{i\textbf{p}\cdot{\textbf{r}}} . This is a rather simplified assumption, a more reasonable choice would have been to use as continuum state the exact atom scattering states. The time evolution of simple plane-wave states with the Volkov Hamiltonian is given by: :\langle\textbf{p}|e^{-i\int_{t'}^{t}\hat{H}_{V}(t
)dt}=\langle\textbf{p}+\textbf{A}(t)|e^{-i\int_{t'}^{t}(\textbf{p}+\textbf{A}(t
))^2dt} here for consistency with Eq. (1.4) the evolution has already been properly converted into the length gauge. As a consequence, the final momentum distribution of a single electron in a single-level atom, with ionization potential I_P, is expressed as: a_{\textbf{p}}(t)^{SFA}=-i\int_{-\infty}^{t} \textbf{E}(t')\cdot \textbf{d}[\textbf{p}+\textbf{A}(t')] e^{+i(I_Pt'-S(t,t'))}dt' \quad (1.5) where, :\textbf{d}[\textbf{p}+\textbf{A}(t')]=\langle\textbf{p}+\textbf{A}(t')|\hat{\textbf{r}}|0 \rangle is the dipole expectation value (or
transition dipole moment), and :S(t,t')=\int_{t'}^{t}\frac{1}{2}(\textbf{p}+\textbf{A}(t
))^2dt is the semiclassical
action. The result of Eq. (1.5) is the basic tool to understand phenomena like
: • The high harmonic generation process, which is typically the result of strong field interaction of noble gases with an intense low-frequency pulse, • Attosecond pump-probe experiments with simple atoms. • The debate on
tunneling time.
Weak attosecond pulse-strong-IR-fields-atoms interactions Attosecond pump-probe experiments with simple atoms is a fundamental tool to measure the time duration of an attosecond pulse and to explore several quantum proprieties of matter. == Techniques ==