attosecond physics

File:HHBreath.webm.

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

The natural time scale of electron motion in atoms, molecules, and solids is the attosecond (1 as= 10⁻¹⁸ s). This fact is a direct consequence of quantum mechanics.

For simplicity, consider a quantum particle in superposition between ground-level, of energy $\backslash epsilon\_0$, and the first excited level, 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 quantum probability 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 reduced Planck constant.

The expectation value of a generic hermitian and symmetric operator,{{Cite book| vauthors = Sakurai JJ |url=https://www.worldcat.org/oclc/1105708539|title=Modern quantum mechanics|date=2017|others=Jim Napolitano|isbn=978-1-108-49999-6|edition=2|location=Cambridge|oclc=1105708539}} $\backslash hat\{P\}$, 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 $T\_c=\backslash frac\{2\backslash pi\; \backslash hbar\}\{\backslash epsilon\_1-\backslash epsilon\_0\}$.

File:AtomicBreath2.png in hydrogen atoms. The color bar indicates the angular density (orientation of the wavepacket) as a function of the polar angle from 0 to π (x-axis), at which one can find the particle, and time (y-axis).]]

As a consequence, for energy levels in the range of $\backslash epsilon\_1-\backslash epsilon\_0\; \backslash approx$ 10 eV, 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.{{Cite journal| vauthors = Corkum PB, Krausz F |date=2007|title=Attosecond science|url=https://www.nature.com/articles/nphys620|journal=Nature Physics|language=en|volume=3|issue=6|pages=381–387|doi=10.1038/nphys620|bibcode=2007NatPh...3..381C|issn=1745-2481|url-access=subscription}}

To generate a traveling pulse with an ultrashort time duration, two key elements are needed: bandwidth and central wavelength of the electromagnetic wave.{{Cite book| vauthors = Chang Z |url= https://www.worldcat.org/oclc/713562984|title=Fundamentals of attosecond optics|date=2011|publisher=CRC Press|isbn=978-1-4200-8938-7|location=Boca Raton, Fla.|oclc=713562984}}

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.{{Cite book| vauthors = Zavelani-Rossi M, Vismarra F |title=High-intensity lasers for nuclear and physical applications. |publisher=ESCULAPIO|year=2020|isbn=978-88-9385-188-6|location=|oclc=1142519514}}

Thus, a smaller time duration requires the use of shorter, and more energetic wavelength, even down to the soft-X-ray (SXR) 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.{{cite journal | vauthors = Johnson AS, Avni T, Larsen EW, Austin DR, Marangos JP | title = Attosecond soft X-ray high harmonic generation | journal = Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences | volume = 377 | issue = 2145 | pages = 20170468 | date = May 2019 | pmid = 30929634 | pmc = 6452054 | doi = 10.1098/rsta.2017.0468 | bibcode = 2019RSPTA.37770468J }}

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

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.{{cite journal | vauthors = Sansone G, Kelkensberg F, Pérez-Torres JF, Morales F, Kling MF, Siu W, Ghafur O, Johnsson P, Swoboda M, Benedetti E, Ferrari F, Lépine F, Sanz-Vicario JL, Zherebtsov S, Znakovskaya I, L'huillier A, Ivanov MY, Nisoli M, Martín F, Vrakking MJ | display-authors = 6 | title = Electron localization following attosecond molecular photoionization | journal = Nature | volume = 465 | issue = 7299 | pages = 763–6 | date = June 2010 | pmid = 20535207 | doi = 10.1038/nature09084 | bibcode = 2010Natur.465..763S | s2cid = 205220785 | url = https://portal.research.lu.se/ws/files/1648850/2063932.pdf }}
• Quantum yield of molecular photo-fragments.{{cite journal | vauthors = Calegari F, Ayuso D, Trabattoni A, Belshaw L, De Camillis S, Anumula S, Frassetto F, Poletto L, Palacios A, Decleva P, Greenwood JB, Martín F, Nisoli M | display-authors = 6 | title = Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses | journal = Science | volume = 346 | issue = 6207 | pages = 336–9 | date = October 2014 | pmid = 25324385 | doi = 10.1126/science.1254061 | bibcode = 2014Sci...346..336C | hdl = 10486/679967 | s2cid = 5371103 | hdl-access = free }}
• XUV-SXR spectrum transient absorption.{{cite journal | vauthors = Kobayashi Y, Chang KF, Zeng T, Neumark DM, Leone SR | title = Direct mapping of curve-crossing dynamics in IBr by attosecond transient absorption spectroscopy | journal = Science | volume = 365 | issue = 6448 | pages = 79–83 | date = July 2019 | pmid = 31273121 | doi = 10.1126/science.aax0076 | bibcode = 2019Sci...365...79K | s2cid = 195804243 | url = https://yorkspace.library.yorku.ca/xmlui/handle/10315/37193 | doi-access = free }}
• XUV-SXR spectrum transient reflectivity.{{cite journal | vauthors = Lucchini M, Sato SA, Lucarelli GD, Moio B, Inzani G, Borrego-Varillas R, Frassetto F, Poletto L, Hübener H, De Giovannini U, Rubio A, Nisoli M | display-authors = 6 | title = Unravelling the intertwined atomic and bulk nature of localised excitons by attosecond spectroscopy | journal = Nature Communications | volume = 12 | issue = 1 | pages = 1021 | date = February 2021 | pmid = 33589638 | doi = 10.1038/s41467-021-21345-7 | pmc = 7884782 | arxiv = 2006.16008 | bibcode = 2021NatCo..12.1021L | hdl = 10810/50745 | hdl-access = free }}
• Photo-electron kinetic energy distribution.
• Attosecond electron microscopy{{Cite journal |last1=Hui |first1=Dandan |last2=Alqattan |first2=Husain |last3=Sennary |first3=Mohamed |last4=Golubev |first4=Nikolay V. |last5=Hassan |first5=Mohammed Th. |date=2024-08-23 |title=Attosecond electron microscopy and diffraction |journal=Science Advances |language=en |volume=10 |issue=34 |doi=10.1126/sciadv.adp5805 |issn=2375-2548 |pmc=11338230 |pmid=39167650|arxiv=2305.03014 |bibcode=2024SciA...10P5805H }}

File:Pump-probe techniques in physics.ogv are used to image ultra-fast processes occurring in matter.|325x325px]]

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

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.{{cite journal | vauthors = Lucarelli GD, Moio B, Inzani G, Fabris N, Moscardi L, Frassetto F, Poletto L, Nisoli M, Lucchini M | display-authors = 6 | title = Novel beamline for attosecond transient reflection spectroscopy in a sequential two-foci geometry | journal = The Review of Scientific Instruments | volume = 91 | issue = 5 | pages = 053002 | date = May 2020 | pmid = 32486725 | doi = 10.1063/5.0005932 | arxiv = 2002.10869 | bibcode = 2020RScI...91e3002L | s2cid = 211296620 }}

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.{{Cite journal| vauthors = Palacios A, Martín F |date=2020|title=The quantum chemistry of attosecond molecular science |journal=WIREs Computational Molecular Science|language=en|volume=10|issue=1|pages=e1430|doi=10.1002/wcms.1430|s2cid=199653256|issn=1759-0884|doi-access=free}}{{Cite journal| vauthors = Sato SA |date=2021|title=First-principles calculations for attosecond electron dynamics in solids |journal=Computational Materials Science|volume=194|pages=110274|doi=10.1016/j.commatsci.2020.110274|issn=0927-0256|arxiv=2011.01677|s2cid=226237040}}

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.

Attosecond physics typically deals with non-relativistic bounded particles and employs electromagnetic fields with a moderately high intensity ($10^\{11\}-10^\{14\}$ W/cm²).{{Cite web| vauthors = Mourou G |title=ICAN: The Next Laser Powerhouse|url=https://www.osa-opn.org/home/articles/volume_24/may_2013/features/ican_the_next_laser_powerhouse/|url-status=live|archive-url=https://web.archive.org/web/20210624201850/https://www.osa-opn.org/home/articles/volume_24/may_2013/features/ican_the_next_laser_powerhouse/ |archive-date=2021-06-24 }}

This fact allows to set up a discussion in a non-relativistic and semi-classical quantum mechanics environment for light-matter interaction.

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 length gauge, within the dipole approximation, as:{{cite book | vauthors = Reiss HR | chapter = Foundations of the Strong-Field Approximation|date=2008 | title = Progress in Ultrafast Intense Laser Science III|pages=1–31| veditors = Yamanouchi K, Chin SL, Agostini P, Ferrante G |series=Springer Series in Chemical Physics| volume = 89|place=Berlin, Heidelberg|publisher=Springer|language=en|doi=10.1007/978-3-540-73794-0_1|isbn=978-3-540-73794-0 }}{{Cite journal| vauthors = Maurer J, Keller U |date=2021-05-05|title=Ionization in intense laser fields beyond the electric dipole approximation: concepts, methods, achievements and future directions |journal=Journal of Physics B: Atomic, Molecular and Optical Physics|volume=54|issue=9|pages=094001|doi=10.1088/1361-6455/abf731|issn=0953-4075|hdl=20.500.11850/489253|s2cid=235281853|hdl-access=free}}

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

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 Dyson's equations{{Cite journal| vauthors = Ivanov MY, Spanner M, Smirnova O |date=2005-01-20|title=Anatomy of strong field ionization |journal=Journal of Modern Optics|volume=52|issue=2–3|pages=165–184|doi=10.1080/0950034042000275360|bibcode=2005JMOp...52..165I|s2cid=121919221|issn=0950-0340}}{{Cite book| vauthors = Mulser P, Bauer D |url=https://www.springer.com/gp/book/9783540506690|title=High Power Laser-Matter Interaction|date=2010|publisher=Springer-Verlag|isbn=978-3-540-50669-0|series=Springer Tracts in Modern Physics|volume=238|location=Berlin Heidelberg|doi=10.1007/978-3-540-46065-7|bibcode=2010hpli.book.....M|language=en}} that the previous solution can also be written as:

:$|\backslash psi(t)\backslash rangle=-i\backslash int\_\{t\_0\}^\{t\}dt\text{'}\backslash Big[\; e^\{-i\backslash int\_\{t\text{'}\}^\{t\}\backslash hat\{H\}(t$)dt}\hat{H}_I(t')e^{-i\int_{t_0}^{t'}\hat{H}_0(t)dt}|\psi(t_0)\rangle \Big]+e^{-i\int_{t_0}^{t}\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.{{Cite journal| vauthors = Faisal FH |date=2007-03-15|title=Gauge-invariant intense-field approximations to all orders|url=http://dx.doi.org/10.1088/0953-4075/40/7/f02|journal=Journal of Physics B: Atomic, Molecular and Optical Physics|volume=40|issue=7|pages=F145–F155|doi=10.1088/0953-4075/40/7/f02|s2cid=117984887 |issn=0953-4075|url-access=subscription}}

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 photoionized.

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.

Strong field approximation (SFA), or Keldysh-Faisal-Reiss theory is a physical model, started in 1964 by the Russian physicist Keldysh,{{Cite journal|last=V Popruzhenko|first=S|date=2014-10-08|title=Keldysh theory of strong field ionization: history, applications, difficulties and perspectives|url=https://iopscience.iop.org/article/10.1088/0953-4075/47/20/204001|journal=Journal of Physics B: Atomic, Molecular and Optical Physics|volume=47|issue=20|pages=204001|doi=10.1088/0953-4075/47/20/204001|bibcode=2014JPhB...47t4001P|s2cid=250736364|issn=0953-4075|url-access=subscription}} 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.{{cite journal | vauthors = Amini K, Biegert J, Calegari F, Chacón A, Ciappina MF, Dauphin A, Efimov DK, Figueira de Morisson Faria C, Giergiel K, Gniewek P, Landsman AS, Lesiuk M, Mandrysz M, Maxwell AS, Moszyński R, Ortmann L, Antonio Pérez-Hernández J, Picón A, Pisanty E, Prauzner-Bechcicki J, Sacha K, Suárez N, Zaïr A, Zakrzewski J, Lewenstein M | display-authors = 6 | title = Symphony on strong field approximation | journal = Reports on Progress in Physics | volume = 82 | issue = 11 | pages = 116001 | date = November 2019 | pmid = 31226696 | doi = 10.1088/1361-6633/ab2bb1 | arxiv = 1812.11447 | bibcode = 2019RPPh...82k6001A | s2cid = 118953514 }}

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,{{Cite web|last=University of Kassel|title=Physical phenomena in laser-matter interaction|url=https://www.pks.mpg.de/~lmi07/talk1_lein.pdf|url-status=live|archive-url=https://web.archive.org/web/20110101023252/http://www.pks.mpg.de/~lmi07/talk1_lein.pdf |archive-date=2011-01-01 }} with $\backslash textbf\{A\}(t)$, $\backslash textbf\{E\}(t)=-\backslash frac\{\backslash partial\; \backslash textbf\{A\}(t)\}\{\backslash partial\; t\}$, the electromagnetic vector potential.{{Cite book| vauthors = Jackson JD |url=https://www.worldcat.org/oclc/38073290|title=Classical electrodynamics|date=1999|publisher=Wiley|isbn=0-471-30932-X|edition=3|location=New York|oclc=38073290}}

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.{{Cite journal| vauthors = Milošević DB, Becker W |date=2019-04-10|title=Atom-Volkov strong-field approximation for above-threshold ionization|url=http://dx.doi.org/10.1103/physreva.99.043411|journal=Physical Review A|volume=99|issue=4|page=043411|doi=10.1103/physreva.99.043411|bibcode=2019PhRvA..99d3411M|s2cid=146011403|issn=2469-9926|url-access=subscription}}

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.{{cite arXiv| vauthors = Bechler A, Ślȩczka M |date=2009-12-25|title=Gauge invariance of the strong field approximation|class=physics.atom-ph|eprint=0912.4966}}

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,{{Cite journal| vauthors = Brabec T, Krausz F |date=2000-04-01|title=Intense few-cycle laser fields: Frontiers of nonlinear optics |journal=Reviews of Modern Physics |volume=72|issue=2|pages=545–591|doi=10.1103/RevModPhys.72.545|bibcode=2000RvMP...72..545B|issn=0034-6861}} 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.{{cite journal | vauthors = Yakovlev VS, Gagnon J, Karpowicz N, Krausz F | title = Attosecond streaking enables the measurement of quantum phase | journal = Physical Review Letters | volume = 105 | issue = 7 | pages = 073001 | date = August 2010 | pmid = 20868037 | doi = 10.1103/PhysRevLett.105.073001 | arxiv = 1006.1827 | bibcode = 2010PhRvL.105g3001Y | s2cid = 12746350 }}
• The debate on tunneling time.{{Cite journal| vauthors = Keller U |date=2015-05-10|title=Attosecond Ionization Dynamics and Time Delays|url=https://www.osapublishing.org/abstract.cfm?uri=CLEO_QELS-2015-FTh3C.1|journal=CLEO: 2015 (2015), Paper FTh3C.1|language=EN|publisher=Optical Society of America|pages=FTh3C.1|doi=10.1364/CLEO_QELS.2015.FTh3C.1|isbn=978-1-55752-968-8|s2cid=39531431|url-access=subscription}}{{Cite journal| vauthors = Kheifets AS |date=2020-03-06|title=The attoclock and the tunneling time debate |journal=Journal of Physics B: Atomic, Molecular and Optical Physics|volume=53|issue=7|pages=072001|doi=10.1088/1361-6455/ab6b3b|issn=0953-4075|arxiv=1910.08891|bibcode=2020JPhB...53g2001K|s2cid=204800609}}

Attosecond pump-probe experiments with simple atoms is a fundamental tool to measure the time duration of an attosecond pulse{{Cite journal| vauthors = Mairesse Y, Quéré F |date=2005-01-27|title=Frequency-resolved optical gating for complete reconstruction of attosecond bursts |journal=Physical Review A|volume=71|issue=1|pages=011401|doi=10.1103/PhysRevA.71.011401|bibcode=2005PhRvA..71a1401M}} and to explore several quantum proprieties of matter. File:QuantumInteraction3.jpg 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. $(1.5)$, 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: 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 $\backslash textbf\{A\}\_\{IR\}(t)>>\backslash textbf\{A\}\_\{XUV\}(t)$. 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.{{cite journal | vauthors = Itatani J, Quéré F, Yudin GL, Ivanov MY, Krausz F, Corkum PB | title = Attosecond streak camera | journal = Physical Review Letters | volume = 88 | issue = 17 | pages = 173903 | date = April 2002 | pmid = 12005756 | doi = 10.1103/PhysRevLett.88.173903 | bibcode = 2002PhRvL..88q3903I | s2cid = 40245650 | url = https://nrc-publications.canada.ca/eng/view/accepted/?id=bc8b5b86-6d18-4ce3-bc22-a04ef044bb3d }}

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

File:StreakingCamera.jpg

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,{{cite book | vauthors = Trebino R |chapter =FROG|date=2003| doi = 10.1007/978-1-4615-1181-6_5| title = Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses|pages=101–115|place=Boston, MA|publisher=Springer US|isbn=978-1-4613-5432-1 }} 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.{{Cite journal| vauthors = Zhao X, Wei H, Wei C, Lin CD |date=2017-10-23|title=A new method for accurate retrieval of atomic dipole phase or photoionization group delay in attosecond photoelectron streaking experiments|url=http://dx.doi.org/10.1088/2040-8986/aa8fb6|journal=Journal of Optics|volume=19|issue=11|pages=114009|doi=10.1088/2040-8986/aa8fb6|bibcode=2017JOpt...19k4009Z|s2cid=125209544 |issn=2040-8978|url-access=subscription}}

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).{{Cite journal| vauthors = Kane DJ |date=2008-06-01|title=Principal components generalized projections: a review [Invited]|url=https://www.osapublishing.org/josab/abstract.cfm?uri=josab-25-6-A120|journal=JOSA B|language=EN|volume=25|issue=6|pages=A120–A132|doi=10.1364/JOSAB.25.00A120|bibcode=2008JOSAB..25A.120K|issn=1520-8540|url-access=subscription}}
• Volkov transform generalized projection algorithm (VTGPA).{{Cite journal| vauthors = Keathley PD, Bhardwaj S, Moses J, Laurent G, Kaertner FX |date=2016-07-06|title=Volkov transform generalized projection algorithm for attosecond pulse characterization|url=http://dx.doi.org/10.1088/1367-2630/18/7/073009|journal=New Journal of Physics|volume=18|issue=7|pages=073009|doi=10.1088/1367-2630/18/7/073009|bibcode=2016NJPh...18g3009K|issn=1367-2630|hdl=1721.1/105139|s2cid=53077495 |hdl-access=free}}
• extended ptychographic iterative engine (ePIE).{{cite journal | vauthors = Lucchini M, Brügmann MH, Ludwig A, Gallmann L, Keller U, Feurer T | title = Ptychographic reconstruction of attosecond pulses | language = EN | journal = Optics Express | volume = 23 | issue = 23 | pages = 29502–13 | date = November 2015 | pmid = 26698434 | doi = 10.1364/OE.23.029502 | arxiv = 1508.07714 | bibcode = 2015OExpr..2329502L | s2cid = 33845261 }}

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

{{Reflist}}

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• {{cite journal | vauthors = Bucksbaum PH | title = Attophysics: Ultrafast control | journal = Nature | volume = 421 | issue = 6923 | pages = 593–4 | date = February 2003 | pmid = 12571581 | doi = 10.1038/421593a | hdl-access = free | s2cid = 12268311 | bibcode = 2003Natur.421..593B | hdl = 2027.42/62570 }}
• {{cite journal | vauthors = Cerullo G, Nisoli M | title = Ultrafast lasers: from femtoseconds to attoseconds. | journal = Europhysics News | date = March 2019 | volume = 50 | issue = 2 | pages = 11–4 | doi = 10.1051/epn/2019201 | bibcode = 2019ENews..50b..11C | s2cid = 132721942 | doi-access = free }}
• {{cite web | vauthors = Kennedy S, Burdick A | date = June 2003 | url = https://www.discovermagazine.com/technology/stopping-time | title = Stopping Time: What can you do in a billionth of a billionth of a second? }}
• {{cite journal | vauthors = Nisoli M | title = The Birth of Attochemistry. | journal = Optics and Photonics News | date = July 2019 | volume = 30 | issue = 7 | pages = 32–9 | doi = 10.1364/OPN.30.7.000032 | bibcode = 2019OptPN..30...32N | s2cid = 198445481 }}

{{refend}}

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