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:
Atomic physics: investigation of electron correlation effects, photo-emission delay and ionization tunneling. Molecular physics and molecular chemistry: role of electronic motion in molecular excited states (e.g. charge-transfer processes), light-induced photo-fragmentation, and light-induced electron transfer processes. Solid-state physics: investigation of exciton dynamics in advanced 2D materials, petahertz charge carrier motion in solids, spin dynamics in ferromagnetic materials. One of the primary goals of attosecond science is to provide advanced insights into the quantum dynamics of electrons in atoms, molecules and solids with the long-term challenge of achieving real-time control of the electron motion in matter. The advent of broadband solid-state titanium-doped sapphire based (Ti:Sa) lasers (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 (chirped mirrors) (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."
Introduction
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-level, of energy ϵ 0 {\displaystyle \epsilon _{0}} , and the first excited level, of energy ϵ 1 {\displaystyle \epsilon _{1}} :
| Ψ ⟩ = c g | ψ g ⟩ + c e | ψ e ⟩ {\displaystyle |\Psi \rangle =c_{g}|\psi _{g}\rangle +c_{e}|\psi _{e}\rangle }
with c e {\displaystyle c_{e}} and c g {\displaystyle c_{g}} chosen as the square roots of the quantum probability of observing the particle in the corresponding state.
| ψ g ( t ) ⟩ = | 0 ⟩ e − i ϵ 0 ℏ t | ψ e ( t ) ⟩ = | 1 ⟩ e − i ϵ 1 ℏ t {\displaystyle |\psi _{g}(t)\rangle =|0\rangle e^{-{\frac {i\epsilon _{0}}{\hbar }}t}\qquad |\psi _{e}(t)\rangle =|1\rangle e^{-{\frac {i\epsilon _{1}}{\hbar }}t}}
are the time-dependent ground | 0 ⟩ {\displaystyle |0\rangle } and excited state | 1 ⟩ {\displaystyle |1\rangle } respectively, with ℏ {\displaystyle \hbar } the reduced Planck constant. The expectation value of a generic hermitian and symmetric operator, P ^ {\displaystyle {\hat {P}}} , can be written as P ( t ) = ⟨ Ψ | P ^ | Ψ ⟩ {\displaystyle P(t)=\langle \Psi |{\hat {P}}|\Psi \rangle } , as a consequence the time evolution of this observable is:
P ( t ) = | c g | 2 ⟨ 0 | P ^ | 0 ⟩ + | c e | 2 ⟨ 1 | P ^ | 1 ⟩ + 2 c e c g ⟨ 0 | P ^ | 1 ⟩ cos ( ϵ 1 − ϵ 0 ℏ t ) {\displaystyle P(t)=|c_{g}|^{2}\langle 0|{\hat {P}}|0\rangle +|c_{e}|^{2}\langle 1|{\hat {P}}|1\rangle +2c_{e}c_{g}\langle 0|{\hat {P}}|1\rangle \cos \left({\frac {\epsilon _{1}-\epsilon _{0}}{\hbar }}t\right)}
While the first two terms do not depend on time, the third, instead, does. This creates a dynamic for the observable P ( t ) {\displaystyle P(t)} with a characteristic time, T c {\displaystyle T_{c}} , given by T c = 2 π ℏ ϵ 1 − ϵ 0 {\displaystyle T_{c}={\frac {2\pi \hbar }{\epsilon _{1}-\epsilon _{0}}}} .
As a consequence, for energy levels in the range of ϵ 1 − ϵ 0 ≈ {\displaystyle \epsilon _{1}-\epsilon _{0}\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 ) {\displaystyle 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.
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. 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) λ = {\displaystyle \lambda =} 800 nm, its minimum time duration is around t p u l s e = λ c = {\displaystyle t_{pulse}={\frac {\lambda }{c}}=} 2.67 fs, where c {\displaystyle c} is the speed of light; whereas, for a light field with central wavelength in the extreme ultraviolet (XUV) at λ = {\displaystyle \lambda =} 30 nm the minimum duration is around t p u l s e = {\displaystyle t_{\rm {pulse}}=} 100 as. 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. The most common sources that fit these requirements are free-electron lasers (FEL) and high harmonic generation (HHG) setups.
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 pump-probe scheme to "image" through one of the aforementioned observables the ultra-fast dynamics occurring in the material under investigation.
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 {\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. 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.
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 {\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, | ψ ( t ) ⟩ {\displaystyle |\psi (t)\rangle } is described by the Schrödinger equation (in atomic units):
H ^ | ψ ( t ) ⟩ = i ℏ ∂ ∂ t | ψ ( t ) ⟩ ( 1.0 ) {\displaystyle {\hat {H}}|\psi (t)\rangle =i\hbar {\dfrac {\partial }{\partial t}}|\psi (t)\rangle \quad (1.0)}
where the light–matter interaction Hamiltonian, H ^ {\displaystyle {\hat {H}}} , can be expressed in the length gauge, within the dipole approximation, as:
H ^ = 1 2 p ^ 2 + V C + r ^ ⋅ E ( t ) {\displaystyle {\hat {H}}={\frac {1}{2}}{\hat {\textbf {p}}}^{2}+V_{C}+{\hat {\textbf {r}}}\cdot {\textbf {E}}(t)}
where V C {\displaystyle V_{C}} is the Coulomb potential of the atomic species considered; p ^ , r ^ {\displaystyle {\hat {\textbf {p}}},{\hat {\textbf {r}}}} are the momentum and position operator, respectively; and E ( t ) {\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:
| ψ ( t ) ⟩ = e − i ∫ t 0 t H ^ d t ′ | ψ ( t 0 ) ⟩ ( 1.1 ) {\displaystyle |\psi (t)\rangle =e^{-i\int _{t_{0}}^{t}{\hat {H}}dt'}|\psi (t_{0})\rangle \qquad (1.1)}
where | ψ ( t 0 ) ⟩ {\displaystyle |\psi (t_{0})\rangle } , is the electron wave function at time t = t 0 {\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:
| ψ ( t ) ⟩ = − i ∫ t 0 t d t ′ [ e − i ∫ t ′ t H ^ ( t ″ ) d t ″ H ^ I ( t ′ ) e − i ∫ t 0 t ′ H ^ 0 ( t ″ ) d t ″ | ψ ( t 0 ) ⟩ ] + e − i ∫ t 0 t H ^ 0 ( t ″ ) d t ″ | ψ ( t 0 ) ⟩ ( 1.2 ) {\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,
H ^ 0 = 1 2 p ^ 2 + V C {\displaystyle {\hat {H}}_{0}={\frac {1}{2}}{\hat {\textbf {p}}}^{2}+V_{C}}
is the bounded Hamiltonian and
H ^ I = r ^ ⋅ E ( t ) {\displaystyle {\hat {H}}_{I}={\hat {\textbf {r}}}\cdot {\textbf {E}}(t)}
is the interaction Hamiltonian. The formal solution of Eq. ( 1.0 ) {\displaystyle (1.0)} , which previously was simply written as Eq. ( 1.1 ) {\displaystyle (1.1)} , can now be regarded in Eq. ( 1.2 ) {\displaystyle (1.2)} as a superposition of different quantum paths (or quantum trajectory), each one of them with a peculiar interaction time t ′ {\displaystyle 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 H ^ 0 {\displaystyle {\hat {H}}_{0}} term in the integral. Then, a "kick" from the electromagnetic field, H ^ I ( t ′ ) {\displaystyle {\hat {H}}_{I}(t')} that "excite" the electron. This event occurs at an arbitrary time that uni-vocally characterizes the quantum path t ′ {\displaystyle t'} . A final evolution driven by both the field and the Coulomb potential, given by H ^ {\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. ( 1.2 ) {\displaystyle (1.2)} . This process is entirely time-reversible, i.e. can also occur in the opposite order. Equation ( 1.2 ) {\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. ( 1.2 ) {\displaystyle (1.2)} in a certain continuum state (unbounded state or free state) | p ⟩ {\displaystyle |{\textbf {p}}\rangle } , of momentum p {\displaystyle {\textbf {p}}} , so that:
c p ( t ) = ⟨ p | ψ ( t ) ⟩ = − i ∫ t 0 t d t ′ ⟨ p | e − i ∫ t ′ t H ^ ( t ″ ) d t ″ H ^ I ( t ′ ) e − i ∫ t 0 t ′ H ^ 0 ( t ″ ) d t ″ | ψ ( t 0 ) ⟩ + ⟨ p | e − i ∫ t 0 t H ^ 0 ( t ″ ) d t ″ | ψ ( t 0 ) ⟩ ( 1.3 ) {\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 | c p ( t ) | 2 {\displaystyle |c_{\textbf {p}}(t)|^{2}} is the probability amplitude to find at a certain time t {\displaystyle t} , the electron in the continuum states | p ⟩ {\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 ( 1.3 ) {\displaystyle (1.3)} is not considered, and only the first one is used in discussions, hence:
a p ( t ) = − i ∫ t 0 t d t ′ ⟨ p | e − i ∫ t ′ t H ^ ( t ″ ) d t ″ H ^ I ( t ′ ) e − i ∫ t 0 t ′ H ^ 0 ( t ″ ) d t ″ | ψ ( t 0 ) ⟩ ( 1.4 ) {\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 ( 1.4 ) {\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 ( 1.4 ) {\displaystyle (1.4)} into:
a p S F A ( t ) = − i ∫ t 0 t d t ′ ⟨ p | e − i ∫ t ′ t H ^ V ( t ″ ) d t ″ H ^ I ( t ′ ) e − i ∫ t 0 t ′ H ^ 0 ( t ″ ) d t ″ | ψ ( t 0 ) ⟩ ( 1.4 ) {\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, H ^ V = 1 2 ( p ^ + A ( t ) ) 2 {\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, with A ( t ) {\displaystyle {\textbf {A}}(t)} , E ( t ) = − ∂ A ( t ) ∂ t {\displaystyle {\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 ⟩ {\displaystyle |0\rangle } , ionization energy I P {\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 t 0 = − ∞ {\displaystyle t_{0}=-\infty } , and we can assume that initially the electron is in the atomic ground state | 0 ⟩ {\displaystyle |0\rangle } . So that,
H ^ 0 | 0 ⟩ = − I P | 0 ⟩ {\displaystyle {\hat {H}}_{0}|0\rangle =-I_{P}|0\rangle } and | ψ ( t ) ⟩ = e − i ∫ − ∞ t ′ H