Nonperturbative quantum dynamics

Transcription

1 Nonperturbative quantum dynamics I. Laser interactions with atoms/molecules II. Model systems and TDDFT Manfred Lein, TDDFT school Benasque 24 p.

2 Outline Laser interactions with atoms/molecules Classical models and quantum description Multiphoton processes, tunneling ionization Recollision, high-harmonic generation, double ionization Strong-field approximation Molecules, laser-induced alignment Examples of open problems in strong-field dynamics p. 2

3 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

4 Ultrashort laser pulses Few-femtosecond light pulses are available:.2. I = 5x 4 W/cm 2 FWHM = 6 fs Field (a.u.) Time (fs) cause ionization of atoms, fragmentation of molecules allow ultrafast time-resolved measurements (pump-probe) carrier-envelope phase becomes important p. 4

5 Classical preliminaries Free 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

6 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

7 Quantum mechanical description Time evolution is described by the time-dependent Schrödinger equation (TDSE): i Ψ(t) = H(t)Ψ(t). 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

8 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) j 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

9 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. 9

10 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.

11 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 = W/cm 2 ) p.

12 Ionization regimes multiphoton ionization tunnel ionization over-barrier ionization z z z V(z) γ = ω/ω t > γ = V(z) γ = ω/ω t < tunneling time laser period (Keldysh parameter) H atom: γ = ω/e V(z) E > E BS in general: γ = I p /(2U p ), I p = ionization potential, U p = ponderomotive potential p. 2

13 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. 3

14 Recollision mechanism 3-step process:. ionization 2. acceleration by the field 3. return to the core p. 4

15 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. 4

16 High-harmonic generation Photon picture ω ω ω ω ω Ω = Νω ω N photons of frequency ω photon of frequencynω. p. 5

17 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. 6

18 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. 7

19 Attosecond pulses Currently shortest achieved coherent light pulses: 67as Zhao et al., Opt. Lett. 37, 389 (22) p. 8

20 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. 9

21 Double ionization Double ionization is enhanced due to electron correlations by orders of magnitude. Identification of the recollision mechanism: Walker et al., PRL 73, 227 (994) 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. 2

22 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 physical interpretation. p. 2

23 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. 22

24 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. 23

25 Molecules Set of nuclear and electronic coordinates R j and r k. Hamiltonian: H = Pj 2 2M j + p 2 k 2 + w j,e (R j,r k )+ j k j,k w j,j 2 (R j,r j2 )+ w ee (r k,r k2 ) j j 2 k k 2 with w j,j 2 nucleus-nucleus interaction, w j,e nucleus-electron interaction, and w ee electron-electron interaction. Light-molecule interaction: H(t) = H D E(t) with dipole moment D = ( j Z jr j ) ( k r k) p. 24

26 Born-Oppenheimer approximation Separation of time scales for nuclear and electronic motion due to great mass difference Electrons adjust instantaneously to nuclear positions. Born-Oppenheimer (BO) Ansatz for wave function: Ψ(R,r,t) = m χ m(r,t)φ m (R,r), Φ m (R,r) = electronic eigenstates at fixed nuclear positions. Inserting into the field-free TDSE yields i χ t m(r,t) = [ T n +Vm BO (R) ] χ m BO approximation + m Φ m T n Φ m χ m nonadiabatic couplings (T n acting on both Φ m and χ m ) p. 25

27 Born-Oppenheimer approximation Including the laser-molecule interaction, the BO TDSE becomes: i t χ m(r,t) = [ T n +V BO m (R) ] χ m (R,t) E m Φ m D Φ m χ m Functions χ m coupled only by the dipole matrix elements. BO approximation breaks down for highly excited electrons: Rydberg molecules Electrons in the continuum p. 26

28 Controlling harmonic generation with molecular alignment CS 2, H 2, N 2 N 2 9th harmonic, adiabatic alignment Hay et al. PRL 87, 839 (2) impulsive alignment Itatani et al., PRL 94, 2392 (25) p. 27

29 Adiabatic alignment of molecules Linear molecule with dipole moment µ and polarizabilities α, α interacts with laser field E = E cos(ωt) as V µ (θ) = µecosθ, V α (θ) = 2 E2 (α cos 2 θ +α sin 2 θ) Sufficiently high frequencies: E, E 2 E 2 /2 V(θ) θ Aligning laser field is switched on slowly pendular states Friedrich and Herschbach, PRL 74, 4623 (995) experiment: Larsen et al., JCP, 7774 (999) p. 28

30 Impulsive alignment Kick with a femtosecond pulse shorter than the rotational period Rotational revival structure exhibits times of field-free alignment. Seideman, PRL 83, 497 (999), Rosca-Pruna/Vrakking, PRL 87, 5392 (2) Detection by Coulomb explosion of N 2 in circularly polarized pulse / ion detection: Dooley et al., PRL 68, 2346 (23) p. 29

31 Two examples of open problems in laser atom/molecule interactions Tunneling times from the attoclock Photoelectron circular dichroism p. 3

32 Tunneling times from the attoclock Attoclock principle: ionization by near-circular polarized pulses measurement of angular offset of the distribution peak Pfeiffer et al., Nat. Phys. 8, 76 (2) p. 3

33 Tunneling times from the attoclock Previous experiment: consistent with instantaneous tunneling Pfeiffer et al., Nat. Phys. 8, 76 (2) p. 32

34 Tunneling times from the attoclock Recent experiment: appears to indicate non-zero real tunneling time Landsman et al., arxiv:32766v2 (23) p. 33

35 Strong-field photoelectron circular dichroism Ionization of chiral molecules with circularly polarized laser pulses Observable: three-dimensional photoelectron distribution Search for forward-backward asymmetry θ z [Ritchie, PRA 3, 4 (976); Cherepkov, CPL 87, 344 (982); Powis, JCP 2, 3 (2); Böwering et al., PRL 86, 87 (2)] p. 34

36 Strong-field photoelectron circular dichroism Femtosecond pulse PECD [Lux et al., Angew. Chem. Int. Ed. 5, 5 (22)] p. 35

37 Camphor and fenchone HOMOs R-camphor R-fenchone p. 36

38 Calculated electron distributions (SFA+Born approximation) R-fenchone, single orientation, LCP pulse (total length 2 cycles) Three-, four- and five-photon rings are visible. Asymmetry due to single orientation. p. 37

39 Photoelectron distributions R-fenchone, random orientation (numerically averaged over Euler angles) Asymmetry is reduced by angle averaging. Camphor distribution looks practically the same as in fenchone. p. 38

40 Forward-backward asymmetries R camphor R fenchone Lux et al. Lux et al. Nodal structure in good agreement with experiment Asymmetries smaller and of different sign compared to experiment Fenchone gives stronger asymmetry (matches experiment) p. 39

41 Conclusions Strong laser fields require nonperturbative description; even one-electron problem is demanding. In general, theoretical description relies heavily on models; open problems in simple systems TDDFT is the only tractable first-principles approach, already for atoms. Next part: Model systems and TDDFT p. 4