Attosecond physics

 ( Quantum Mechanics ) 

in krypton.

This technology is one of the most used techniques to generate attosecond bursts of light.]] Attosecond physics, also known as attophysics, or more generally attosecond science, is a branch of physics that deals with light-matter interaction phenomena wherein attosecond (10⁻¹⁸ s) photon pulses are used to investigate dynamical processes in matter with unprecedented temporal resolution.

The main research topics in this field are:

1. Atomic physics: investigation of electron correlation effects, photo-emission delay and ionization tunneling.
2. Molecular physics and molecular chemistry: role of electronic motion in molecular (e.g. charge-transfer processes), light-induced Ionization, and light-induced electron transfer processes.
3. Solid-state physics: investigation of exciton dynamics in advanced 2D materials, petahertz charge carrier motion in Solid, spin dynamics in Ferromagnetism.

One of the primary goals of attosecond science is to provide advanced insights into the quantum dynamics of electrons in , and with the long-term challenge of achieving real-time control of the electron motion in matter.

The advent of broadband solid-state (1986), chirped pulse amplification (CPA)

(1988), spectral broadening of high-energy pulses (e.g. gas-filled hollow-core fiber via self-phase modulation) (1996), mirror-dispersion-controlled technology () (1994), and carrier envelop offset stabilization (2000) had enabled the creation of isolated-attosecond light pulses (generated by the non-linear process of high harmonic generation in a noble gas) (2004, 2006), which have given birth to the field of attosecond science.

The current world record for the shortest light-pulse generated by human technology is 43 as.

In 2022, Anne L'Huillier, Paul Corkum, Ferenc Krausz were awarded with the Wolf Prize in physics for their pioneering contributions to ultrafast laser science and attosecond physics. This was followed by the 2023 Nobel Prize in Physics, where L'Huillier, Krausz and Pierre Agostini were rewarded “for experimental methods that generate attosecond pulses of light for the study of electron dynamics in matter.”

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Introduction .

The period of this states superposition (1s-2p) is around 400 as.]]

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Motivation The natural time scale of electron motion in atoms, molecules, and solids is the attosecond (1 as= 10⁻¹⁸ s).

For simplicity, consider a quantum particle in superposition between Ground state, of energy $\backslash epsilon\_0$, and the first Excited state, of energy $\backslash epsilon\_1$:

$|\backslash Psi\backslash rangle=c\_g|\backslash psi\_g\backslash rangle+c\_e|\backslash psi\_e\backslash rangle$

with $c\_e$ and $c\_g$ chosen as the square roots of the Born rule of observing the particle in the corresponding state.

$|\backslash psi\_g(t)\backslash rangle=\; |0\backslash rangle\; e^\{-\backslash frac\{i\backslash epsilon\_0\}\{\backslash hbar\}\; t\}\; \backslash qquad\; |\backslash psi\_e(t)\backslash rangle\; =|1\backslash rangle\; e^\{-\backslash frac\{i\backslash epsilon\_1\}\{\backslash hbar\}t\}$

are the time-dependent ground $|0\backslash rangle$ and excited state $|1\backslash rangle$ respectively, with $\backslash hbar$ the Planck constant

The expectation value of a generic hermitian and symmetric operator,

$\backslash hat\{P\}$, can be written as $P(t)=\backslash langle\backslash Psi|\backslash hat\{P\}|\backslash Psi\backslash rangle$, as a consequence the time evolution of this observable is:

$P(t)=|c\_g|^2\backslash langle0|\backslash hat\{P\}|0\backslash rangle+|c\_e|^2\backslash langle1|\backslash hat\{P\}|1\backslash rangle+2c\_ec\_g\backslash langle0|\backslash hat\{P\}|1\backslash rangle\backslash cos\backslash left(\backslash frac\{\backslash epsilon\_1-\backslash epsilon\_0\}\{\backslash hbar\}t\; \backslash right)$

While the first two terms do not depend on time, the third, instead, does. This creates a dynamic for the observable $P(t)$ with a characteristic time, $T\_c$, given by $T\_c=\backslash frac\{2\backslash pi\; \backslash hbar\}\{\backslash epsilon\_1-\backslash epsilon\_0\}$. As a consequence, for energy levels in the range of $\backslash epsilon\_1-\backslash epsilon\_0\; \backslash approx$ Electronvolt, which is the typical electronic energy range in matter, the characteristic time of the dynamics of any associated physical observable is approximately 400 as.

To measure the time evolution of $P(t)$, one needs to use a controlled tool, or a process, with an even shorter time-duration that can interact with that dynamic.

This is the reason why attosecond light pulses are used to disclose the physics of ultra-fast phenomena in the few-femtosecond and attosecond time-domain.

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Generation of attosecond pulses To generate a traveling pulse with an ultrashort time duration, two key elements are needed: bandwidth and central wavelength of the electromagnetic wave.

(2025). 9781420089387, CRC Press. ISBN 9781420089387

From Fourier analysis, the more the available spectral bandwidth of a light pulse, the shorter, potentially, is its time duration.

There is, however, a lower-limit in the minimum duration exploitable for a given pulse central wavelength. This limit is the optical cycle.

Indeed, for a pulse centered in the low-frequency region, e.g. infrared (IR) $\backslash lambda=$800 nm, its minimum time duration is around $t\_\{pulse\}=\backslash frac\{\backslash lambda\}\{c\}=$2.67 fs, where $c$ is the speed of light; whereas, for a light field with central wavelength in the extreme ultraviolet (XUV) at $\backslash lambda=$30 nm the minimum duration is around $t\_\{\backslash rm\; pulse\}=$100 as.

Thus, a smaller time duration requires the use of shorter, and more energetic wavelength, even down to the X-ray region.

For this reason, standard techniques to create attosecond light pulses are based on radiation sources with broad spectral bandwidths and central wavelength located in the XUV-SXR range.

The most common sources that fit these requirements are free-electron lasers (FEL) and high harmonic generation (HHG) setups.

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Physical observables and experiments Once an attosecond light source is available, one has to drive the pulse towards the sample of interest and, then, measure its dynamics.

The most suitable experimental observables to analyze the electron dynamics in matter are:

• Angular asymmetry in the velocity distribution of molecular photo-fragment.
• Quantum yield of molecular photo-fragments.
• XUV-SXR spectrum transient absorption.
• XUV-SXR spectrum transient reflectivity.
• Photo-electron kinetic energy distribution.
• Attosecond electron microscopy

The general strategy is to use a Femtochemistry scheme to "image" through one of the aforementioned observables the ultra-fast dynamics occurring in the material under investigation.

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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 ($10^\{11\}-10^\{14\}$ 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.

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.

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.

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Quantum mechanics foundations 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.

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Atoms

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Resolution of time dependent Schrödinger equation in an electromagnetic field The time evolution of a single electronic wave function in an atom, $|\backslash psi(t)\backslash rangle$ is described by the Schrödinger equation (in atomic units):

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

where the light-matter interaction Hamiltonian, $\backslash hat\{H\}$, can be expressed in the Gauge theory, within the dipole approximation, as:

$\backslash hat\{H\}=\backslash frac\{1\}\{2\}\backslash hat\{\backslash textbf\{p\}\}^2+V\_\{C\}+\; \backslash hat\{\backslash textbf\{r\}\}\backslash cdot\backslash textbf\{E\}(t)$

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

$|\backslash psi(t)\backslash rangle=e^\{-i\backslash int\_\{t\_0\}^\{t\}\backslash hat\{H\}dt\text{'}\}|\backslash psi\; (t\_0)\backslash rangle\; \backslash qquad(1.1)$

where $|\backslash psi\; (t\_0)\backslash rangle$, 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 Self-energy

that the previous solution can also be written as:

$|\backslash psi(t)\backslash rangle=-i\backslash int\_\{t\_0\}^\{t\}dt\text{'}\backslash Big+e^\{-i\backslash int\_\{t\_0\}^\{t\}\backslash hat\{H\}\_0(t$)dt}|\psi(t_0)\rangle \qquad(1.2)

where,

$\backslash hat\{H\}\_0=\backslash frac\{1\}\{2\}\backslash hat\{\backslash textbf\{p\}\}^2+V\_\{C\}$

is the bounded Hamiltonian and

$\backslash hat\{H\}\_I=\backslash hat\{\backslash textbf\{r\}\}\backslash cdot\backslash textbf\{E\}(t)$

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 $t\text{'}$ 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 $\backslash hat\{H\}\_0$ term in the integral.
2. Then, a "kick" from the electromagnetic field, $\backslash hat\{H\}\_I(t\text{'})$ that "excite" the electron. This event occurs at an arbitrary time that uni-vocally characterizes the quantum path $t\text{'}$.
3. A final evolution driven by both the field and the Coulomb potential, given by $\backslash hat\{H\}$.

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.

Equation $(1.2)$ 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. $(1.2)$ in a certain continuum state ( unbounded state or free state) $|\backslash textbf\{p\}\backslash rangle$, of momentum $\backslash textbf\{p\}$, so that:

$c\_\{\backslash textbf\{p\}\}(t)=\backslash langle\backslash textbf\{p\}|\backslash psi(t)\backslash rangle=-i\backslash int\_\{t\_0\}^\{t\}dt\text{'}\backslash langle\; \backslash textbf\{p\}|e^\{-i\backslash int\_\{t\text{'}\}^\{t\}\backslash hat\{H\}(t$)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 $|c\_\{\backslash textbf\{p\}\}(t)|^2$ is the probability amplitude to find at a certain time $t$, the electron in the continuum states $|\backslash textbf\{p\}\backslash rangle$.

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

For the majority of application, the second term in $(1.3)$ is not considered, and only the first one is used in discussions, hence:

$a\_\{\backslash textbf\{p\}\}(t)=-i\backslash int\_\{t\_0\}^\{t\}dt\text{'}\backslash langle\; \backslash textbf\{p\}|e^\{-i\backslash int\_\{t\text{'}\}^\{t\}\backslash hat\{H\}(t$)dt}\hat{H}_I(t')e^{-i\int_{t_0}^{t'}\hat{H}_0(t )dt}|{\psi(t_0)}\rangle \quad (1.4)

Equation $(1.4)$ is also known as time reversed S-matrix amplitude and it gives the probability of photoionization by a generic time-varying electric field.

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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 $(1.4)$ into:

$a\_\{\backslash textbf\{p\}\}^\{SFA\}(t)=-i\backslash int\_\{t\_0\}^\{t\}dt\text{'}\backslash langle\; \backslash textbf\{p\}|e^\{-i\backslash int\_\{t\text{'}\}^\{t\}\backslash hat\{H\}\_\{V\}(t$)dt}\hat{H}_I(t')e^{-i\int_{t_0}^{t'}\hat{H}_0(t )dt}|{\psi(t_0)}\rangle \quad (1.4)

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

At this point, to keep the discussion at its basic level, lets consider an atom with a single energy level $|0\backslash rangle$, 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 $t\_0=-\backslash infty$, and we can assume that initially the electron is in the atomic ground state $|0\backslash rangle$.

So that,

$\backslash hat\{H\}\_0|0\backslash rangle=-I\_P|0\backslash rangle$ and $|\backslash psi(t)\backslash rangle=e^\{-i\backslash int\_\{-\backslash infty\}^\{t\text{'}\}\backslash hat\{H\}\_0dt\}|0\backslash rangle=e^\{I\_Pt\text{'}\}|0\backslash rangle$

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

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:

$\backslash langle\backslash textbf\{p\}|e^\{-i\backslash int\_\{t\text{'}\}^\{t\}\backslash hat\{H\}\_\{V\}(t$)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:

$a\_\{\backslash textbf\{p\}\}(t)^\{SFA\}=-i\backslash int\_\{-\backslash infty\}^\{t\}\; \backslash textbf\{E\}(t\text{'})\backslash cdot\; \backslash textbf\{d\}\backslash textbf\{p\}+\backslash textbf\{A\}(t\text{'})\; e^\{+i(I\_Pt\text{'}-S(t,t\text{'}))\}dt\text{'}\; \backslash quad\; (1.5)$

where,

$\backslash textbf\{d\}\backslash textbf\{p\}+\backslash textbf\{A\}(t\text{'})=\backslash langle\backslash textbf\{p\}+\backslash textbf\{A\}(t\text{'})|\backslash hat\{\backslash textbf\{r\}\}|0\; \backslash rangle$

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

$S(t,t\text{'})=\backslash int\_\{t\text{'}\}^\{t\}\backslash frac\{1\}\{2\}(\backslash textbf\{p\}+\backslash textbf\{A\}(t$))^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.
  (2015). 9781557529688, Optical Society of America. . ISBN 9781557529688

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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. in a energy level. The XUV can Ionization 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. $(1.5)$, 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 ($(\backslash textbf\{E\}\_\{IR\}(t),\backslash textbf\{A\}\_\{IR\}(t))$,

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

Then, by substituting these fields to $(1.5)$ it results

$a\_\{\backslash textbf\{p\}\}(t)=-i\backslash int\_\{-\backslash infty\}^\{t\}\; (\backslash textbf\{E\}\_\{XUV\}(t\text{'})+\backslash textbf\{E\}\_\{IR\}(t\text{'}))\backslash cdot\; \backslash textbf\{d\}\backslash textbf\{p\}+\backslash textbf\{A\}\_\{XUV\}(t\text{'})+\backslash textbf\{A\}\_\{IR\}(t\text{'})\; e^\{+i(I\_Pt\text{'}-S(t,t\text{'}))\}dt\text{'}\; \backslash quad\; (1.6)$

with

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

At this point, we can divide Eq. $(1.6)$ in two contributions: 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 $\backslash textbf\{A\}\_\{IR\}(t)>>\backslash textbf\{A\}\_\{XUV\}(t)$. Thus, $\backslash textbf\{A\}\_\{XUV\}(t)$ is typically neglected in Eq. $(1.6)$.

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

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

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

$a\_\{\backslash textbf\{p\}\}(\backslash tau)=-i\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; \backslash textbf\{E\}\_\{XUV\}(t-\backslash tau)\backslash cdot\; \backslash textbf\{d\}\backslash textbf\{p\}+\backslash textbf\{A\}\_\{IR\}(t)\; e^\{+i(I\_Pt-S(t))\}dt\; \backslash quad\; (1.7)$

with

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

Equation $(1.7)$ 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.

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Techniques

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

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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,

(2025). 9781461354321, Springer US. ISBN 9781461354321

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.$(1.7)$.

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. $(1.7)$ 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).

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See also

• Femtochemistry
• Femtotechnology
• Ultrashort pulse
• Chirped pulse amplification
• Free-electron laser
• Attosecond chronoscopy

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Further reading

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