Ultrafast electron dynamics and decoherence in surface bands Branko Gumhalter Institute of Physics Zagreb

MOTIVATION FOR STUDYING THE PROPERTIES OF SURFACE ELECTRONIC BANDS: Surface bands provide local electronic environment for catalytic processes and formation of nanostructures. The (de)coherence of electronic states in surface bands and nanostructures is an important phenomenon that sets limits to implementation of quantum computing.

Examples of electronic surface bands: Surface State (n=0) and Image Potential (n=1) bands Surface bands arise on low index planes of fcc and bcc metals for effective V(z) of the form:

Formation of the surface bands due to a combined effect of the surface projected bulk band gap and image potential

Investigations of surface bands by electron spectroscopies PES of occupied states IPES of unoccupied states

Time resolved two-photon-photoemission (TR2PPE) from surface bands Intermediate IP-state Initial SS-state pump photon probe photon Time delay between pump and probe pulses in TR2PPE in ~fs range !

Final 2PPE yield strongly depends on ultrafast decoherence and decay of the intermediate states Evolution of the intermediate states in the interval (0,t) assessed from the response function describing pump photon ( ) absorption: electron in |K,IP> state 2 R(K,t) renormalized through IP-electron and SS-hole interactions with heatbath R(Q, t hole in |K,SS> state

Interaction of quasiparticles with the heatbath (intraband transitions only) = bosonized excitation of the heatbath IP-electron-boson interaction IP-SS electron-hole pair boson interaction (entangled) … disentangled from SS-hole-boson interaction

Disentangled decoherence of quasiparticles Amplitude of fundamental IP-electron decoherence process exp( )= t HF-correlation correction Exchange correlation correction N.B. Completely analogous expressions for SS-holes !!

Estimate of the correlation corrections on the example of 1D electron coupled to 1D tomonagon heatbath Tomonagon dispersion: 1D bath Q 1D band K V(Z) 1D-electron energy dispersion:

Off-the-energy-shell relaxation and decoherence (pure dephasing)

On-the-energy-shell relaxation and decoherence

Decoherence of optically excited IP-SS electron-hole pair on Cu(111) for K=0 E(K) IP band SS band K

Ultrafast dynamics of IP-electron & SS-hole pairs on Cu(111) Heatbath: bosonized substrate e-h pairs Transition amplitudes involving n-boson excitation: Disentangled decoherence: 1-boson emission by IP-electron or SS-hole: Entangled decoherence: 1-boson exchange between IP-SS e-h pair = Σ| |² = |Σ |²

Disentangled decoherence by emission of one and two bosons

Asymptotic behaviour of disentangled decoherence for K=0 DWF FGR

Entangled decoherence by exchange of one and two bosons

Total asymptotic decoherence of initial IP-SS electron-hole pair for K=0 decay

Conclusions Intermediate electronic states in 2PPE from surface bands are affected by relaxation and decoherence. These effects are caused by (i) separate interactions of electrons and holes with the substrate heatbath in disentangled processes, and (ii) mutual interactions mediated by the excitations of the heatbath in entangled processes. Ultrafast relaxation of intermediate IP-SS e-h pairs on Cu(111) surface is non-Markovian in the first ~1-5 fs. Thereafter the initial coherence of e-h pairs decays as ~exp(-Γt), with total Γ close to the sum of FGR values.

A glimpse of formalism Example: Hamiltonian of IP-electron coupled to boson field

Single IP-electron propagator from cumulant expansion: Second order cumulant (only even terms survive):

Fourth order cumulants (direct & exchange)