1. Attosecond Larmor clock: how long does it take to create a hole?
Olga Smirnova
Max-Born Institute, Berlin

2. Work has been inspired by:
Alfred Maquet
Armin Scrinzi
Work has been done with:
Jivesh Kaushal
MBI, Berlin
Misha Ivanov,
MBI Berlin, Imperial College
Lisa Torlina,
PhD students

3. Attosecond spectroscopy: Goals & Challenges
Observe & control electron dynamics at its natural time-scale (1asec=10-3fsec)
One of key challenges:
Observe non-equilibrium many-electron dynamics
This dynamics can be created by photoionization
Electron removal by an ultrashort pulse creates coherent hole
X 2Pg ~
A 2Pu
B 2S+u
4.3eV
3.5 eV ħw
CO+2 ħW ħw ħw ħw
CO2
Ionization by XUV
Ionization by IR
Coherent population of several ionic states

4. Attosecond spectroscopy: Questions
How long does it take to remove an electron and create a hole?
– the time scale of electron rearrangement
Experiment & Theory: Eckle, P. et al Science 322, 1525–1529 (2008).
Goulielmakis, E. et al, Nature 466 (7307), (2010)
Schultze, M. et al. Science 328, 1658–1662 (2010).
Klunder, K. et al., Phys. Rev. Lett. 106, (2011)
Pfeiffer, A. N. et al. Nature Phys. 8, 76–80 (2012)
Nirit’s talk: Shafir, D. et al. Nature 485 (7398), (2012)
How does this time depend on the number of absorbed photons (strong IR vs weak XUV)?
How does electron-hole entanglement affect this process and its time-scale?
Can we find a clock to measure this time?

5. The Larmor clock for tunnelling
I. Baz’, 1966 S S H
distance
DE=gH/2
Beautiful but academic ? –
No! There is a built-in Larmor-like clock in atoms!
Based on Spin-Orbit Interaction
Good for any number of photons N

6. Spin-orbit interaction: the physical picture
Take e.g. L=Lz=1xħ
Lz => H
For e-, the core rotates around it
Rotating charge creates current
Current creates magnetic field
This field interacts with the spin
Results in DESO for nonzero Lz - S +
We have a clock! ...
But we need to calibrate it:
How rotation of the spin is mapped into time?
Consider one-photon ionization, where the ionization time is known
Wigner-Smith time
E. Wigner Phys. Rev. 98, (1955)
F. T. Smith Phys. Rev. 118, (1960)
Find angle of rotation of the spin in one-photon ionization

7. Gedanken experiment for Calibrating the clock
One-photon ionization of Cs by right circularly polarized pulse
Define angle of rotation of electron spin during ionization Cs 5s S S ħw +
No SO interaction in the ground state

8. SO Larmor clock as Interferometer
Initial state
Final state
Record the phase between the spin-up and spin-down pathways
Looks easy, but … -- the initial and final states are not eigenstates, thanks to the spin-orbit interaction

9. SO Larmor clock as Interferometer
U. Fano, 1969
Phys Rev 178,131
j=1/2
j=3/2
Radial photoionization matrix element
A crooked interferometer:
arm + double arm
j=3/2

10. SO Larmor clock as Interferometer
U. Fano, 1969
Phys Rev 178,131
j=1/2
j=3/2
Radial photoionization matrix element
A crooked interferometer:
arm + double arm
j=3/2
Wigner-Smith time hides here

11. The appearance of Wigner-Smith time
0.38 eV
J. Cond. Matter 24 (2012)
Wigner-Smith time
We have calibrated the clock

12. Strong Field Ionization in IR fields
Keldysh, 1965
Multiphoton Ionization: N>>1
xFLcoswt
Adiabatic (tunnelling) perspective (w/Ip << 1)
-xFLcoswt ħw
Find time it takes to create a hole in general case for arbitrary Keldysh parameter

13. Starting the clock: Ionization in circular field
N>>1 ionization preferentially removes p- (counter-rotating) electron
- Theoretical prediction: Barth, Smirnova, PRA, 2011
- Experimental verification: Herath et al, PRL, 2012
P electrons Kr
4s24p6 +
P +
Kr+
4s24p5 +
Nħw
P -
Closed shell, no Spin-Orbit interaction
Open shell, Spin-Orbit interaction is on
Ionization turns on the clock in Kr+
Clock operates on core states: P3/2 (4p5,J=3/2) and P1/2 (4p5,J=1/2)

14. SO Larmor clock operating on the core
electron
At the moment of separation
core
J=3/2
J=3/2
J=1/2
Ionization amplitude

15. The SFI Time One photon, weak field Many photons, strong field
- Looks like a direct analogue of tWSDESO
- Does f13 /DESO correspond to time?

16. The appearance of SFI time
Kr+
P3/2
Kr+
P1/2 e-
Part of f13 yields Strong Field Ionization time
What about Df13 ?

17. The phase that is not time
- Df13 does not depend on DESO
- Trace of electron – hole entanglement
‘Chirp’ of the hole wave-packet imparted by ionization: compression / stretching of the hole wave-packet
Proper time delay in hole formation
Time is phase, but not every phase is time!

18. Stopping the clock: filling the p- hole
Final s - state
P +
4s24p6
4s24p5
4s 4p6
Asec XUV,
Left polarized
J=3/2
J=1/2
Kr+
4s24p5 s s
Few fs IR,
Right polarized
Pump: Few fs IR creates p-hole and starts the clock
Probe: Asec XUV pulse fills the p-hole and stops the clock
Observe: Read the attosecond clock using transient absorption measurement

19. Strong-field ionization time & tunnelling time
Larmor tunneling time : V
Hauge,E. H. et al, Rev. Mod Phys, 61, 917 (1989)
-xFLcoswt Ip
SFI time:
We can calculate this phase analytically (Analytical R-Matrix: ARM method):
L. Torlina & O.Smirnova, PRA,2012, J. Kaushal & O. Smirnova, arXiv:

20. Delays : Results and physical picture
Exit point, Bohr
Kr atom:
Ip=14 eV
Kr+
DESO=0.67 eV
2.5x1014W/cm2
Approaches WS delay as N -> 1
WS-like delay
Ip-3/2
Number of photons
-xFLcoswt
N>10
N=4
N=2
Delay, as
Apparent ‘delay’
0.4F2/DESOIp5/2
Number of photons
Phase and delays are accumulated after exiting the barrier
Larger N – more adiabatic, exit further out
Phase accumulated under the barrier signifies current created during ionization

21. Conclusions Using SO Larmor clock we defined delays in hole formation:
Actual delay in formation of hole wave-packet
Larmor- and Wigner-Smith – like,
Applicable for any number of photons, any strong-field ionization regime
Apparent ‘delay’ – trace of electron-hole entanglement:
Clock-imparted ‘delay’ (encodes electron – hole interaction )
Analogous to spread of an optical pulse due to group velocity dispersion
does not depend on clock period
Absorbing many photons takes less time than absorbing few photons, but not zero
The SO Larmor clock allowed simple analytical treatment, but the result is general
Moving hole = coherent population of several states: This set of states is a clock
Reading the clock = finding initial phases between different states
Not all phases translate into time! This will be general for any attosecond measurements of electronic dynamics.