Few-femtosecond IR-XUV/SXR attosecond pulse pump-probe experiments

As an example, in a typical pump-probe experimental apparatus, an attosecond (XUV-SXR) pulse and an intense (${\displaystyle 10^{11}-10^{14}}$ W/cm²) low-frequency infrared pulse with a time duration of few to tens femtoseconds are collinearly focused on the studied sample.

At this point, by varying the delay of the attosecond pulse, which could be pump/probe depending on the experiment, with respect to the IR pulse (probe/pump), the desired physical observable is recorded.^[24]

The subsequent challenge is to interpret the collected data and retrieve fundamental information on the hidden dynamics and quantum processes occurring in the sample. This can be achieved with advanced theoretical tools and numerical calculations.^[25][26]

By exploiting this experimental scheme, several kinds of dynamics can be explored in atoms, molecules and solids; typically light-induced dynamics and out-of-equilibrium excited states within attosecond time-resolution.^[20][21][23]

Resolution of time dependent Schrödinger equation in an electromagnetic field

The time evolution of a single electronic wave function in an atom, ${\displaystyle |\psi (t)\rangle }$ is described by the Schrödinger equation (in atomic units):

${\displaystyle {\hat {H}}|\psi (t)\rangle =i{\dfrac {\partial }{\partial t}}|\psi (t)\rangle \quad (1.0)}$

where the light-matter interaction Hamiltonian, ${\displaystyle {\hat {H}}}$, can be expressed in the length gauge, within the dipole approximation, as:^[28][29]

${\displaystyle {\hat {H}}={\frac {1}{2}}{\hat {\textbf {p}}}^{2}+V_{C}+{\hat {\textbf {r}}}\cdot {\textbf {E}}(t)}$

where ${\displaystyle V_{C}}$ is the Coulomb potential of the atomic species considered; ${\displaystyle {\hat {\textbf {p}}},{\hat {\textbf {r}}}}$ are the momentum and position operator, respectively; and ${\displaystyle {\textbf {E}}(t)}$ 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:

${\displaystyle |\psi (t)\rangle =e^{-i\int _{t_{0}}^{t}{\hat {H}}dt'}|\psi (t_{0})\rangle \qquad (1.1)}$

where ${\displaystyle |\psi (t_{0})\rangle }$, is the electron wave function at time ${\displaystyle t=t_{0}}$.

This exact solution cannot be used for almost any practical purpose.

However, it can be proved, using Dyson's equations^[30][31] that the previous solution can also be written as:

${\displaystyle |\psi (t)\rangle =-i\int _{t_{0}}^{t}dt'{\Big [}e^{-i\int _{t'}^{t}{\hat {H}}(t'')dt''}{\hat {H}}_{I}(t')e^{-i\int _{t_{0}}^{t'}{\hat {H}}_{0}(t'')dt''}|\psi (t_{0})\rangle {\Big ]}+e^{-i\int _{t_{0}}^{t}{\hat {H}}_{0}(t'')dt''}|\psi (t_{0})\rangle \qquad (1.2)}$

where,

${\displaystyle {\hat {H}}_{0}={\frac {1}{2}}{\hat {\textbf {p}}}^{2}+V_{C}}$

is the bounded Hamiltonian and

${\displaystyle {\hat {H}}_{I}={\hat {\textbf {r}}}\cdot {\textbf {E}}(t)}$

is the interaction Hamiltonian.

The formal solution of Eq. ${\displaystyle (1.0)}$, which previously was simply written as Eq. ${\displaystyle (1.1)}$, can now be regarded in Eq. ${\displaystyle (1.2)}$ as a superposition of different quantum paths (or quantum trajectory), each one of them with a peculiar interaction time ${\displaystyle t'}$ with the electric field.

In other words, each quantum path is characterized by three steps:

1. An initial evolution without the electromagnetic field. This is described by the left-hand side ${\displaystyle {\hat {H}}_{0}}$ term in the integral.
2. Then, a "kick" from the electromagnetic field, ${\displaystyle {\hat {H}}_{I}(t')}$ that "excite" the electron. This event occurs at an arbitrary time that uni-vocally characterizes the quantum path ${\displaystyle t'}$.
3. A final evolution driven by both the field and the Coulomb potential, given by ${\displaystyle {\hat {H}}}$.

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. ${\displaystyle (1.2)}$.

This process is entirely time-reversible, i.e. can also occur in the opposite order.^[30]

Equation ${\displaystyle (1.2)}$ is not straightforward to handle. However, physicists use it as the starting point for numerical calculation, more advanced discussion or several approximations.^[31][32]

For strong-field interaction problems, where ionization may occur, one can imagine to project Eq. ${\displaystyle (1.2)}$ in a certain continuum state (unbounded state or free state) ${\displaystyle |{\textbf {p}}\rangle }$, of momentum ${\displaystyle {\textbf {p}}}$, so that:

${\displaystyle c_{\textbf {p}}(t)=\langle {\textbf {p}}|\psi (t)\rangle =-i\int _{t_{0}}^{t}dt'\langle {\textbf {p}}|e^{-i\int _{t'}^{t}{\hat {H}}(t'')dt''}{\hat {H}}_{I}(t')e^{-i\int _{t_{0}}^{t'}{\hat {H}}_{0}(t'')dt''}|{\psi (t_{0})}\rangle +\langle {\textbf {p}}|e^{-i\int _{t_{0}}^{t}{\hat {H}}_{0}(t'')dt''}|\psi (t_{0})\rangle \quad (1.3)}$

where ${\displaystyle |c_{\textbf {p}}(t)|^{2}}$is the probability amplitude to find at a certain time ${\displaystyle t}$, the electron in the continuum states ${\displaystyle |{\textbf {p}}\rangle }$.

If this probability amplitude is greater than zero, the electron is photoionized.

For the majority of application, the second term in ${\displaystyle (1.3)}$ is not considered, and only the first one is used in discussions,^[31] hence:

${\displaystyle a_{\textbf {p}}(t)=-i\int _{t_{0}}^{t}dt'\langle {\textbf {p}}|e^{-i\int _{t'}^{t}{\hat {H}}(t'')dt''}{\hat {H}}_{I}(t')e^{-i\int _{t_{0}}^{t'}{\hat {H}}_{0}(t'')dt''}|{\psi (t_{0})}\rangle \quad (1.4)}$

Equation ${\displaystyle (1.4)}$ is also known as time reversed S-matrix amplitude^[31] and it gives the probability of photoionization by a generic time-varying electric field.

Strong field approximation (SFA)

Strong field approximation (SFA), or Keldysh-Faisal-Reiss theory is a physical model, started in 1964 by the Russian physicist Keldysh,^[33] 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.^[34]

This fact re-shapes equation ${\displaystyle (1.4)}$ into:

${\displaystyle a_{\textbf {p}}^{SFA}(t)=-i\int _{t_{0}}^{t}dt'\langle {\textbf {p}}|e^{-i\int _{t'}^{t}{\hat {H}}_{V}(t'')dt''}{\hat {H}}_{I}(t')e^{-i\int _{t_{0}}^{t'}{\hat {H}}_{0}(t'')dt''}|{\psi (t_{0})}\rangle \quad (1.4)}$

where, ${\displaystyle {\hat {H}}_{V}={\frac {1}{2}}({\hat {\textbf {p}}}+{\textbf {A}}(t))^{2}}$ is the Volkov Hamiltonian, here expressed for simplicity in the velocity gauge,^[35] with ${\displaystyle {\textbf {A}}(t)}$, ${\displaystyle {\textbf {E}}(t)=-{\frac {\partial {\textbf {A}}(t)}{\partial t}}}$, the electromagnetic vector potential.^[36]

At this point, to keep the discussion at its basic level, lets consider an atom with a single energy level ${\displaystyle |0\rangle }$, ionization energy ${\displaystyle I_{P}}$ and populated by a single electron (single active electron approximation).

We can consider the initial time of the wave function dynamics as ${\displaystyle t_{0}=-\infty }$, and we can assume that initially the electron is in the atomic ground state ${\displaystyle |0\rangle }$.

So that,

${\displaystyle {\hat {H}}_{0}|0\rangle =-I_{P}|0\rangle }$ and ${\displaystyle |\psi (t)\rangle =e^{-i\int _{-\infty }^{t'}{\hat {H}}_{0}dt}|0\rangle =e^{I_{P}t'}|0\rangle }$

Moreover, we can regard the continuum states as plane-wave functions state, ${\displaystyle \langle {\textbf {r}}|{\textbf {p}}\rangle =(2\pi )^{-{\frac {3}{2}}}e^{i{\textbf {p}}\cdot {\textbf {r}}}}$.

This is a rather simplified assumption, a more reasonable choice would have been to use as continuum state the exact atom scattering states.^[37]

The time evolution of simple plane-wave states with the Volkov Hamiltonian is given by:

${\displaystyle \langle {\textbf {p}}|e^{-i\int _{t'}^{t}{\hat {H}}_{V}(t'')dt''}=\langle {\textbf {p}}+{\textbf {A}}(t)|e^{-i\int _{t'}^{t}({\textbf {p}}+{\textbf {A}}(t''))^{2}dt''}}$

here for consistency with Eq. ${\displaystyle (1.4)}$ the evolution has already been properly converted into the length gauge.^[38]

As a consequence, the final momentum distribution of a single electron in a single-level atom, with ionization potential ${\displaystyle I_{P}}$, is expressed as:

${\displaystyle a_{\textbf {p}}(t)^{SFA}=-i\int _{-\infty }^{t}{\textbf {E}}(t')\cdot {\textbf {d}}[{\textbf {p}}+{\textbf {A}}(t')]e^{+i(I_{P}t'-S(t,t'))}dt'\quad (1.5)}$

where,

${\displaystyle {\textbf {d}}[{\textbf {p}}+{\textbf {A}}(t')]=\langle {\textbf {p}}+{\textbf {A}}(t')|{\hat {\textbf {r}}}|0\rangle }$

is the dipole expectation value (or transition dipole moment), and

${\displaystyle S(t,t')=\int _{t'}^{t}{\frac {1}{2}}({\textbf {p}}+{\textbf {A}}(t''))^{2}dt''}$

is the semiclassical action.

The result of Eq. ${\displaystyle (1.5)}$ is the basic tool to understand phenomena like:

• The high harmonic generation process,^[39] 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.^[40]
• The debate on tunneling time.^[41][42]

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^[43] and to explore several quantum proprieties of matter.^[40]

Scheme of a strong IR field and a delayed attosecond XUV pulse interacting with a single electron in a single-level atom. The XUV can ionize the electron, which "jumps" in the continuum by direct ionization (blue path in the figure). The IR pulse, later, "streaks" up and down in energy the photo-electron. After the interaction, the electron has final energy which can be subsequently detected and measured (e.g. time-of-flight apparatus). The multi-photon ionization process (red path in the figure) is also possible, but, since it is relevant in different energetic region, it can be disregarded.

This kind of experiments can be easily described within strong field approximation by exploiting the results of Eq. ${\displaystyle (1.5)}$, as discussed below.

As a simple model, consider the interaction between a single active electron in a single-level atom and two fields: an intense femtosecond infrared (IR) pulse (${\displaystyle ({\textbf {E}}_{IR}(t),{\textbf {A}}_{IR}(t))}$,

and a weak attosecond pulse (centered in the extreme ultraviolet (XUV) region) ${\displaystyle ({\textbf {E}}_{XUV}(t),{\textbf {A}}_{XUV}(t))}$.

Then, by substituting these fields to ${\displaystyle (1.5)}$ it results

${\displaystyle a_{\textbf {p}}(t)=-i\int _{-\infty }^{t}({\textbf {E}}_{XUV}(t')+{\textbf {E}}_{IR}(t'))\cdot {\textbf {d}}[{\textbf {p}}+{\textbf {A}}_{XUV}(t')+{\textbf {A}}_{IR}(t')]e^{+i(I_{P}t'-S(t,t'))}dt'\quad (1.6)}$

with

${\displaystyle S(t,t')=\int _{t'}^{t}{\frac {1}{2}}({\textbf {p}}+{\textbf {A}}_{IR}(t'')+{\textbf {A}}_{XUV}(t''))^{2}dt''}$.

At this point, we can divide Eq. ${\displaystyle (1.6)}$ in two contributions: direct ionization and strong field ionization (multiphoton regime), respectively.

Typically, these two terms are relevant in different energetic regions of the continuum.

Consequently, for typical experimental condition, the latter process is disregarded, and only direct ionization from the attosecond pulse is considered.^[31]

Then, since the attosecond pulse is weaker than the infrared one, it holds ${\displaystyle {\textbf {A}}_{IR}(t)>>{\textbf {A}}_{XUV}(t)}$. Thus, ${\displaystyle {\textbf {A}}_{XUV}(t)}$ is typically neglected in Eq. ${\displaystyle (1.6)}$.

In addition to that, we can re-write the attosecond pulse as a delayed function with respect to the IR field, ${\displaystyle [{\textbf {A}}_{IR}(t),{\textbf {E}}_{XUV}(t-\tau )]}$.

Therefore, the probability distribution, ${\displaystyle |a_{\textbf {p}}(\tau )|^{2}}$, of finding an electron ionized in the continuum with momentum ${\displaystyle {\textbf {p}}}$, after the interaction has occurred (at ${\displaystyle t=\infty }$), in a pump-probe experiments,

with an intense IR pulse and a delayed-attosecond XUV pulse, is given by:

${\displaystyle a_{\textbf {p}}(\tau )=-i\int _{-\infty }^{\infty }{\textbf {E}}_{XUV}(t-\tau )\cdot {\textbf {d}}[{\textbf {p}}+{\textbf {A}}_{IR}(t)]e^{+i(I_{P}t-S(t))}dt\quad (1.7)}$

with

${\displaystyle S(t)={\frac {1}{2}}|{\textbf {p}}|^{2}t+\int _{t}^{\infty }({\textbf {p}}\cdot {\textbf {A}}_{IR}(t')+{\frac {1}{2}}|{\textbf {A}}_{IR}(t')|^{2})dt'}$

Equation ${\displaystyle (1.7)}$ describes the photoionization phenomenon of two-color interaction (XUV-IR) with a single-level atom and single active electron.

This peculiar result can be regarded as a quantum interference process between all the possible ionization paths, started by a delayed XUV attosecond pulse, with a following motion in the continuum states driven by a strong IR field.^[31]

The resulting 2D photo-electron (momentum, or equivalently energy, vs delay) distribution is called streaking trace.^[44]