Models for Time-Dependent Phenomena I. Phenomena in laser-matter interaction: atoms II. Phenomena in laser-matter interaction: molecules III. Model systems and TDDFT Manfred Lein p.
Outline Phenomena in laser-matter interaction: atoms Classical and quantum description Floquet theory, Volkov states Multiphoton processes, tunneling Recollision, harmonics, ATI, double ionization Strong-field approximation p.2
Laser-matter interaction Weak light field (normal light, synchrotron) ω ω Single-photon absorption P r E 2 Strong light field (laser pulses) ω ω ω ω ω ω ω Multiphoton absorption perturbative or nonperturbative p.3
Ultrashort laser pulses State-of-the-art few-cycle pulse:.2. I = 5x 4 W/cm 2 FWHM = 6 fs Field (a.u.)...2 2 2 Time (fs) allows ultrafast time-resolved measurements (pump-probe) carrier-envelope phase becomes important p.4
Classical preliminaries Classical electron in a monochromatic laser field Equation of motion: r(t) = E sin(ωt) (using dipole approximation; E sin(ωt) = electric field, linearly polarized) Velocity: ṙ(t) = v drift + E ω cos(ωt) Position: r(t) = r + v drift t + E ω 2 sin(ωt) x drift + oscillation t Oscillation amplitude: α = E /ω 2 p.5
Classical preliminaries Kinetic energy: T(t) = v2 drift 2 +v drift E ω cos(ωt) + E2 2ω 2 cos 2 (ωt) Average kinetic energy: T = v 2 drift 2 + E2 4ω 2 Define ponderomotive potential: U p = E2 4ω 2 If field amplitude is position dependent, there will be a ponderomotive force F p = U p (r). (But in an ultrashort laser pulse, the electron has not enough time to follow this force). p.6
Quantum mechanical description Time evolution is described by the time-dependent Schrödinger equation (TDSE): i t Ψ(t) = H(t)Ψ(t). Hamiltonian in dipole approximation (λ >> system size): H(t) = H + E(t) j r j with E(t) = electric field. This is called length gauge. Alternatively: H (t) = H + A(t) j [p j + A(t)/2] with A(t) = t E(t )dt. This is called velocity gauge. p.7
Quantum mechanical description The velocity-gauge wave function Ψ (t) is related to the length-gauge wave function Ψ(t) by Ψ (t) = e ia(t) Pj r j Ψ(t) Are there problems with TDDFT and velocity gauge (momentum-dependent interaction)? No, because gauge transformation does not change density. TDKS equations may be solved in either gauge. But: orbitals change under gauge transformation. p.8
Floquet theory Consider monochromatic field E(t) = E sin(ωt) periodic Hamiltonian H(t + T) = H(t) Floquet theorem (cf. Bloch theorem in solid-state physics): TDSE has solutions of the form Ψ(t) = e iet Φ(t) with time-periodic wave functions Φ(t), Φ(t + T) = Φ(t). p.9
Floquet theory The Floquet states Φ(t) are eigenstates of the Floquet operator H(t) = H(t) i t, where E is the quasienergy. H(t)Φ(t) = EΦ(t), If E and Φ(t) are solutions, then also E = E + nω and Φ (t) = Φ(t)e inωt are solutions. Φ(t) are called dressed states (analog to stationary eigenstates for time-independent Hamiltonian). p.
Volkov states Free electron in the presence of a time-dependent electric field is described by the Hamiltonian (length gauge): H(t) = 2 2 + E(t) r Possible solutions of the TDSE are Volkov states: Ψ V p(r,t) = e is(p,t,t ) e i[p+a(t)] r with the action integral S(p,t,t ) = 2 arbitrary, fixed t. t t [p + A(t )] 2 dt and These are plane waves with momenta depending on time as in classical mechanics. p.
Volkov states For monochromatic field, the Volkov states can be written as Ψ V p(r,t) = e i(p2 /2+U p )t Φ p (r,t) with a time periodic function Φ p, i.e. this is a Floquet state with quasienergy E p = p 2 /2 + U p. The ponderomotive potential is the ac Stark shift of plane waves! p.2
Tunneling Static electric field E potential barrier, allows tunneling. z V(z) Tunneling rate for the hydrogen atom (see Landau & Lifshitz): w = 4 E e 2/(3E) (derived from quasiclassical theory) p.3
Over-barrier ionization For sufficiently large field E > critical field E BS ground-state energy above barrier maximum z V(z) Classical escape of the electron. E BS = barrier suppression field strength H atom: E BS =.3 a.u. (corresponds to laser intensity I BS = 4.5 4 W/cm 2 ) p.4
Ionization regimes multiphoton ionization tunnel ionization over-barrier ionization z z z V(z) γ = ω/ω t > V(z) γ = ω/ω t < V(z) E > E BS γ = tunneling time laser period (Keldysh parameter) H atom: γ = ω/e in general: γ = I p /(2U p ), I p = ionization potential, U p = ponderomotive potential p.5
Simple man s model of ionization At each instant t, the ionization rate is given by a simple estimate (e.g. tunneling formula) using the instantaneous field strength. Electron appears with zero velocity at position zero. Subsequent dynamics is described classically. p.6
Simple man s model of ionization At each instant t, the ionization rate is given by a simple estimate (e.g. tunneling formula) using the instantaneous field strength. Electron appears with zero velocity at position zero. Subsequent dynamics is described classically. For monochromatic field: ṙ(t) = E ω [cos(ωt) cos(ωt )] i.e. drift velocity v drift = E ω cos(ωt ) Estimate of maximum photoelectron energy: cos(ωt ) = E max = E2 2ω 2 = 2U p p.6
Above-threshold ionization Absorption of more photons than needed to overcome the ionization threshold Peaks separated by the photon energy in the electron spectrum Example: experiment with Xe atoms, Agostini et al. PRA 36, R4 (987). p.7
Above-threshold ionization Ponderomotive shift of the ATI peaks - continuum dressed states have energies E d k = k2 /2 + U p - shift of ground-state energy is small: E d g E g, so absorption of n photons yields electrons with energy E g + nω. Photoelectron kinetic energies k 2 /2 = E g + nω U p p.8
Recollision mechanism 3-step process:. ionization 2. acceleration by the field 3. return to the core p.9
Recollision mechanism 3-step process:. ionization 2. acceleration by the field 3. return to the core Possible consequences: recombination (high harmonic generation coherent light) elastic scattering fast photoelectrons inelastic scattering e.g. double ionization p.9
Mechanism of high-harmonic generation Photon picture ω ω ω ω ω Ω = Νω ω N photons of frequency ω photon of frequency Nω. p.2
Mechanism of high-harmonic generation Recollision picture Ionization Free acceleration Recombination V(x) V(x) V(x) x x x p.2
Mechanism of high-harmonic generation Recollision picture Ionization Free acceleration Recombination V(x) Maximum return energy: E max = 3.7U p Cut-off at ω = 3.7U p + I p V(x) V(x) [Corkum, PRL 7, 994 (993)] x x x plateau cut off Harmonic order p.2
Calculation of spectra Calculation of the time-dependent dipole acceleration and Fourier transform a(t) = ψ(t) V + E(t) ψ(t) a(ω) = a(t)e iωt gives emission spectrum S(Ω) a(ω) 2 In practice: time integration over pulse duration T, a(ω) = T a(t)f(t)eiωt with some window function f(t). Alternatively: a(t) = D(t) from time-dependent dipole moment D(t) or: a(t) = v(t) from time-dependent dipole velocity v(t) p.22
Application of high harmonics source of coherent extreme UV radiation and attosecond pulses p.23
Application of high harmonics source of coherent extreme UV radiation and attosecond pulses investigation of molecular properties: molecular tomography: J. Itatani et al., Nature 432, 867 (24) measured N 2 orbital ab-initio N 2 orbital p.23
Rescattered photoelectrons Scattered electrons (high-order above-threshold ionization) direct electrons scattered electrons G.G. Paulus et al. PRL 72, 285 (994) 2U p Electron energy U p p.24
Double ionization Walker et al., PRL 73, 227 (994) Double ionization is enhanced due to electron correlations by orders of magnitude. Identification of the recollision mechanism: A. Becker, F.H.M. Faisal, J. Phys. B 29, L97 (996) R. Moshammer et al., PRL 84, 447 (2) T. Weber et al., Nature 45, 658 (2) M.L., E.K.U. Gross, V. Engel, PRL 85, 477 (2) p.25
Quantum mechanical methods for ultrashort pulses Numerical solution of the TDSE (or TDKS) equations - accurate, - but time consuming and hard to interpret, - approximations for TDDFT xc potential have deficiencies Strong-field approximation ( Keldysh-Faisal-Reiss theory, intense field S-matrix formalism ) - less reliable (e.g. strong dependence on gauge), - but fast and amenable to interpretation. p.26
Strong-field approximation for ionization Time evolution operator U(t,t ) obeys Schrödinger equation: i t U(t,t ) = [H + H int (t)]u(t,t ), where H int is the system-field interaction. The solution can be written in integral form: t U(t,t ) = U (t,t ) i U(t,t )H int (t )U (t,t )dt. t Goal: calculation of transition amplitudes from ground state to continuum states with momentum p at final time t f : M p (t f,t i ) = Ψ p (t f ) U(t f,t i ) Ψ (t i ) p.27
Strong-field approximation for ionization Assumption : time evolution after ionization is governed by the laser field only, not by the binding potential, i.e. U(t,t ) U V (t,t ) (Volkov-Propagator). Then t U(t,t ) = U (t,t ) i U V (t,t )H int (t )U (t,t )dt. t Assumption 2: final state with momentum p is approximated as a Volkov state. Ionization amplitude in strong-field approximation (SFA): Mp SFA (t i,t f ) = i t f t i Ψ V p(t) H int (t) Ψ (t) dt p.28
Strong-field approximation for ionization SFA is not gauge invariant: results for ionization of N 2 T.K. Kjeldsen and L.B. Madsen, J. Phys. B 37, 233 (24) Length gauge (r E(t)) appears favourable (except for large molecules) p.29
Conclusions Strong laser fields require nonperturbative description and large grids. In general, theoretical description is not quantitative and relies heavily on models. TDDFT is the only tractable first-principles approach, already for atoms. Many-body wave-function methods are restricted to 2 particles. Next part: Phenomena in laser-matter interaction: molecules p.3