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

2. 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.

3. 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:

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

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

6. 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 !

7. 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

8. 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

9. 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 !!

10. 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:

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

12. On-the-energy-shell relaxation and decoherence

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

14. 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

15. Disentangled decoherence by emission of one and two bosons

16. Asymptotic behaviour of disentangled decoherence for K=0
DWF
FGR

17. Entangled decoherence by exchange of one and two bosons

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

19. 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.

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

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

22. Fourth order cumulants (direct & exchange)