Attosecond physics

Quantum mechanics foundations

Attosecond physics typically deals with non-relativistic bounded particles and employs electromagnetic fields with a moderately high intensity (${\displaystyle {10}^{11}-{10}^{14}}$ 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, ${\displaystyle |\psi (t)⟩}$ is described by the Schrödinger equation (in atomic units):

${\displaystyle \hat{H}|\psi (t)⟩=i\hslash {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}}|\psi (t)⟩(1.0)}$

where the light–matter interaction Hamiltonian, ${\displaystyle \hat{H}}$, can be expressed in the length gauge, within the dipole approximation, as:

${\displaystyle \hat{H}=\frac{1}{2}{\widehat{\mathbf{\text{p}}}}^2+V_C+\widehat{\mathbf{\text{r}}}\cdot \mathbf{\text{E}}(t)}$

where ${\displaystyle V_C}$ is the Coulomb potential of the atomic species considered; ${\displaystyle \widehat{\mathbf{\text{p}}},\widehat{\mathbf{\text{r}}}}$ are the momentum and position operator, respectively; and ${\displaystyle \mathbf{\text{E}}(t)}$ 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:

${\displaystyle |\psi (t)⟩=e^{-i{\int }_{t_0}^t\hat{H}d{t}^{\prime }}|\psi (t_0)⟩(1.1)}$

where ${\displaystyle |\psi (t_0)⟩}$, 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 that the previous solution can also be written as:

${\displaystyle |\psi (t)⟩=-i{\int }_{t_0}^{t}d{t}^{\prime }[e^{-i{\int }_{t^{\prime }}^t\hat{H}(t^{″})d{t}^{″}}{\hat{H}}_I(t^{\prime })e^{-i{\int }_{t_0}^{t^{\prime }}{\hat{H}}_0(t^{″})d{t}^{″}}|\psi (t_0)⟩]+e^{-i{\int }_{t_0}^t{\hat{H}}_0(t^{″})d{t}^{″}}|\psi (t_0)⟩(1.2)}$

where,

${\displaystyle {\hat{H}}_0=\frac{1}{2}{\widehat{\mathbf{\text{p}}}}^2+V_C}$

is the bounded Hamiltonian and

${\displaystyle {\hat{H}}_I=\widehat{\mathbf{\text{r}}}\cdot \mathbf{\text{E}}(t)}$

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 ${\displaystyle t^{\prime }}$ 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, ${\displaystyle {\hat{H}}_I(t^{\prime })}$ that "excite" the electron. This event occurs at an arbitrary time that uni-vocally characterizes the quantum path ${\displaystyle t^{\prime }}$.
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.

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.

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) ${\displaystyle |\mathbf{\text{p}}⟩}$, of momentum ${\displaystyle \mathbf{\text{p}}}$, so that:

${\displaystyle c_{\mathbf{\text{p}}}(t)=⟨\mathbf{\text{p}}|\psi (t)⟩=-i{\int }_{t_0}^{t}d{t}^{\prime }⟨\mathbf{\text{p}}|e^{-i{\int }_{t^{\prime }}^t\hat{H}(t^{″})d{t}^{″}}{\hat{H}}_I(t^{\prime })e^{-i{\int }_{t_0}^{t^{\prime }}{\hat{H}}_0(t^{″})d{t}^{″}}|\psi (t_0)⟩+⟨\mathbf{\text{p}}|e^{-i{\int }_{t_0}^t{\hat{H}}_0(t^{″})d{t}^{″}}|\psi (t_0)⟩(1.3)}$

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

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, hence:

${\displaystyle a_{\mathbf{\text{p}}}(t)=-i{\int }_{t_0}^{t}d{t}^{\prime }⟨\mathbf{\text{p}}|e^{-i{\int }_{t^{\prime }}^t\hat{H}(t^{″})d{t}^{″}}{\hat{H}}_I(t^{\prime })e^{-i{\int }_{t_0}^{t^{\prime }}{\hat{H}}_0(t^{″})d{t}^{″}}|\psi (t_0)⟩(1.4)}$

Equation ${\displaystyle (1.4)}$ is also known as time reversed S-matrix amplitude 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, 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 ${\displaystyle (1.4)}$ into:

${\displaystyle a_{\mathbf{\text{p}}}^{SFA}(t)=-i{\int }_{t_0}^{t}d{t}^{\prime }⟨\mathbf{\text{p}}|e^{-i{\int }_{t^{\prime }}^t{\hat{H}}_V(t^{″})d{t}^{″}}{\hat{H}}_I(t^{\prime })e^{-i{\int }_{t_0}^{t^{\prime }}{\hat{H}}_0(t^{″})d{t}^{″}}|\psi (t_0)⟩(1.4)}$

where, ${\displaystyle {\hat{H}}_V=\frac{1}{2}(\widehat{\mathbf{\text{p}}}+\mathbf{\text{A}}(t){)}^2}$ is the Volkov Hamiltonian, here expressed for simplicity in the velocity gauge, with ${\displaystyle \mathbf{\text{A}}(t)}$, ${\displaystyle \mathbf{\text{E}}(t)=-\frac{\mathrm{\partial }\mathbf{\text{A}}(t)}{\mathrm{\partial }t}}$, the electromagnetic vector potential.

At this point, to keep the discussion at its basic level, lets consider an atom with a single energy level ${\displaystyle |0⟩}$, 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 ${\displaystyle t_0=-\mathrm{\infty }}$, and we can assume that initially the electron is in the atomic ground state ${\displaystyle |0⟩}$.

So that,

${\displaystyle {\hat{H}}_0|0⟩=-I_P|0⟩}$ and ${\displaystyle |\psi (t)⟩=e^{-i{\int }_{-\mathrm{\infty }}^{t^{\prime }}{\hat{H}}_{0}dt}|0⟩=e^{I_P{t}^{\prime }}|0⟩}$

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

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:

${\displaystyle ⟨\mathbf{\text{p}}|e^{-i{\int }_{t^{\prime }}^t{\hat{H}}_V(t^{″})d{t}^{″}}=⟨\mathbf{\text{p}}+\mathbf{\text{A}}(t)|e^{-i{\int }_{t^{\prime }}^t(\mathbf{\text{p}}+\mathbf{\text{A}}(t^{″}){)}^{2}d{t}^{″}}}$

here for consistency with Eq. ${\displaystyle (1.4)}$ 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 ${\displaystyle I_P}$, is expressed as:

${\displaystyle a_{\mathbf{\text{p}}}(t{)}^{SFA}=-i{\int }_{-\mathrm{\infty }}^t\mathbf{\text{E}}(t^{\prime })\cdot \mathbf{\text{d}}[\mathbf{\text{p}}+\mathbf{\text{A}}(t^{\prime })]e^{+i(I_P{t}^{\prime }-S(t,t^{\prime }))}d{t}^{\prime }(1.5)}$

where,

${\displaystyle \mathbf{\text{d}}[\mathbf{\text{p}}+\mathbf{\text{A}}(t^{\prime })]=⟨\mathbf{\text{p}}+\mathbf{\text{A}}(t^{\prime })|\widehat{\mathbf{\text{r}}}|0⟩}$

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

${\displaystyle S(t,t^{\prime })={\int }_{t^{\prime }}^t\frac{1}{2}(\mathbf{\text{p}}+\mathbf{\text{A}}(t^{″}){)}^{2}d{t}^{″}}$

is the semiclassical action.

The result of Eq. ${\displaystyle (1.5)}$ 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.

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 (${\displaystyle ({\mathbf{\text{E}}}_{IR}(t),{\mathbf{\text{A}}}_{IR}(t))}$,

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

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

${\displaystyle a_{\mathbf{\text{p}}}(t)=-i{\int }_{-\mathrm{\infty }}^t({\mathbf{\text{E}}}_{XUV}(t^{\prime })+{\mathbf{\text{E}}}_{IR}(t^{\prime }))\cdot \mathbf{\text{d}}[\mathbf{\text{p}}+{\mathbf{\text{A}}}_{XUV}(t^{\prime })+{\mathbf{\text{A}}}_{IR}(t^{\prime })]e^{+i(I_P{t}^{\prime }-S(t,t^{\prime }))}d{t}^{\prime }(1.6)}$

with

${\displaystyle S(t,t^{\prime })={\int }_{t^{\prime }}^t\frac{1}{2}(\mathbf{\text{p}}+{\mathbf{\text{A}}}_{IR}(t^{″})+{\mathbf{\text{A}}}_{XUV}(t^{″}){)}^{2}d{t}^{″}}$.

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.

Then, since the attosecond pulse is weaker than the infrared one, it holds ${\displaystyle {\mathbf{\text{A}}}_{IR}(t)>>{\mathbf{\text{A}}}_{XUV}(t)}$. Thus, ${\displaystyle {\mathbf{\text{A}}}_{XUV}(t)}$ 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, ${\displaystyle [{\mathbf{\text{A}}}_{IR}(t),{\mathbf{\text{E}}}_{XUV}(t-\tau )]}$.

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

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

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

with

${\displaystyle S(t)=\frac{1}{2}|\mathbf{\text{p}}{|}^{2}t+{\int }_t^{\mathrm{\infty }}(\mathbf{\text{p}}\cdot {\mathbf{\text{A}}}_{IR}(t^{\prime })+\frac{1}{2}|{\mathbf{\text{A}}}_{IR}(t^{\prime }){|}^2)d{t}^{\prime }}$

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.

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

Techniques

Here are listed and discussed some of the most common techniques and approaches pursued in attosecond research centers.

Metrology with photo-electron spectroscopy (FROG-CRAB)

A daily challenge in attosecond science is to characterize the temporal proprieties of the attosecond pulses used in any pump-probe experiments with atoms, molecules or solids.

The most used technique is based on the frequency-resolved optical gating for a complete reconstruction of attosecond bursts (FROG-CRAB).

The main advantage of this technique is that it allows to exploit the corroborated frequency-resolved optical gating (FROG) technique, developed in 1991 for picosecond-femtosecond pulse characterization, to the attosecond field.

Complete reconstruction of attosecond bursts (CRAB) is an extension of FROG and it is based on the same idea for the field reconstruction.

In other words, FROG-CRAB is based on the conversion of an attosecond pulse into an electron wave-packet that is freed in the continuum by atomic photoionization, as already described with Eq.${\displaystyle (1.7)}$.

The role of the low-frequency driving laser pulse( e.g. infra-red pulse) is to behave as gate for the temporal measurement.

Then, by exploring different delays between the low-frequency and the attosecond pulse a streaking trace (or streaking spectrogram) can be obtained.

This 2D-spectrogram is later analyzed by a reconstruction algorithm with the goal of retrieving both the attosecond pulse and the IR pulse, with no need of a prior knowledge on any of them.

However, as Eq. ${\displaystyle (1.7)}$ pinpoints, the intrinsic limits of this technique is the knowledge on atomic dipole proprieties, in particular on the atomic dipole quantum phase.

The reconstruction of both the low-frequency field and the attosecond pulse from a streaking trace is typically achieved through iterative algorithms, such as:

• Principal component generalized projections algorithm (PCGPA).
• Volkov transform generalized projection algorithm (VTGPA).
• extended ptychographic iterative engine (ePIE).