This document discusses several approaches to determining tunneling time: - Wave-packet treatment compares incoming and outgoing wave packet peaks. - Dynamic paths treatment averages time spent in barrier for different particle paths. - Physical clocks treatment uses measurable degrees of freedom coupled to tunneling. It then focuses on the time-modulated barrier clock approach, which uses sideband intensities at different modulation frequencies to estimate traversal time. Finally, it describes an experiment that uses two laser fields to independently resolve ionization and recombination times during tunneling.

1. Time-resolution of tunneling
Christian Roca Catal´a
Supervised by: Prof. Dr. Armin Scrinzi
Ludwing-Maximilians Universit¨at M¨unchen
Christian.Roca@campus.lmu.de
Jan 16, 2014

2. 1 Main ideas of tunneling
2 Three different approaches
Wave-packet treatment
Dynamic paths treatment
Physical clocks treatment
3 Time modulated barrier clock
The idea
B¨uttiker & Landauer treatment
Low/High frequency
Conclusion
4 Dudovich & Co Experiment
The idea
The setup
The Gates
Displacement Gate
Velocity Gate
Reconstruction of ti , tr
5 Conclusions
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 2 / 29

3. Main ideas of tunneling
Question: What is the tunnel effect?
Answer: A particle crossing a energetic region classically forbidden,
usually called potential barrier.
Question: What do we know?
Answer: In all basic QM books the tunneling problem is solved
Transition/reflection probability
Lifetimes
What about tunneling/traverse time?
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 3 / 29

4. Main ideas of tunneling
The traverse time is commonly accepted either as
Interaction time between the penetrating particle and the potential.
Crossing time spent by the particle throughout the barrier.
AFTER MORE THAN 60 YEARS THERE STILL IS NOT CONSENSUS
ON THE EXISTENCE OF AN UNIQUE AND SIMPLE EXPRESSION
FOR IT
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 4 / 29

5. Contents
1 Main ideas of tunneling
2 Three different approaches
Wave-packet treatment
Dynamic paths treatment
Physical clocks treatment
3 Time modulated barrier clock
The idea
B¨uttiker & Landauer treatment
Low/High frequency
Conclusion
4 Dudovich & Co Experiment
The idea
The setup
The Gates
Displacement Gate
Velocity Gate
Reconstruction of ti , tr
5 Conclusions

6. Three different approaches Wave-packet treatment
Wave-packet treatment
The peak of the incident w.p is identified with the peak of the transmitted w.p. Then,
the delay between the peaks is a measure of the traverse time.
The main critics:
Incoming peak does not turn into an outgoing peak necessarily
Transmission effectiveness depends on the w.p form: outgoing w.p will have
higher velocity than incoming w.p.
Observing the w.p is an QM invasive procedure
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 5 / 29

7. Contents
1 Main ideas of tunneling
2 Three different approaches
Wave-packet treatment
Dynamic paths treatment
Physical clocks treatment
3 Time modulated barrier clock
The idea
B¨uttiker & Landauer treatment
Low/High frequency
Conclusion
4 Dudovich & Co Experiment
The idea
The setup
The Gates
Displacement Gate
Velocity Gate
Reconstruction of ti , tr
5 Conclusions

8. Three different approaches Dynamic paths treatment
Dynamic paths treatment
Determine a set of dynamic paths x(t) for the incoming particle and ask how long each
path spend in the barrier. Then average them to calculate the main time of tunneling.
Feynman path-integral formalism
Bohm approach
Wigner distribution
The traversal time is now viewed as a distribution better than a concrete value
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 6 / 29

9. Contents
1 Main ideas of tunneling
2 Three different approaches
Wave-packet treatment
Dynamic paths treatment
Physical clocks treatment
3 Time modulated barrier clock
The idea
B¨uttiker & Landauer treatment
Low/High frequency
Conclusion
4 Dudovich & Co Experiment
The idea
The setup
The Gates
Displacement Gate
Velocity Gate
Reconstruction of ti , tr
5 Conclusions

10. Three different approaches Physical clocks treatment
Question:What do we mean by clocks?
Answer: Literally we give a small pocket clock to the electron. We can watch at it
whenever we want and compare the times given before and after the barrier crossing.
Question:How many clocks do we have?
Answer: A clock can be any measurable degree of freedom of the system coupled to
the tunneling process.
NOTE: A clock can be chosen to be minimally invasive
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 7 / 29

11. Three different approaches Physical clocks treatment
Question: Different clocks give the same results?
Answer: Generally they don’t. But there is a wide range of overlap.
Some examples:
Time-modulated barrier (B¨uttiker &
Landauer 1982) - we will look at it!
Larmor clock (B¨uttiker 1983)
Oscillating spin clock (B¨uttiker &
Landauer 1985)
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 8 / 29

12. Contents
1 Main ideas of tunneling
2 Three different approaches
Wave-packet treatment
Dynamic paths treatment
Physical clocks treatment
3 Time modulated barrier clock
The idea
B¨uttiker & Landauer treatment
Low/High frequency
Conclusion
4 Dudovich & Co Experiment
The idea
The setup
The Gates
Displacement Gate
Velocity Gate
Reconstruction of ti , tr
5 Conclusions

13. Time modulated barrier clock The idea
The scheme of this approach:
A particle approaching from the left:
j =
−i
2m
(ψ∗
ψ − ψ ψ∗
) =
k
m
A time dependent barrier V (t) = V0 + V1 cos ωt between x ≤ d/2, otherwise
V (t) = 0.
The perturbation frequency ω is variable.
The time scale for the particle crossing is given by the traverse time τ.
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 9 / 29

14. Time modulated barrier clock The idea
Question: What are we going to study?
Answer: We will use the physical clock treatment to find a way to measure the
traversal time τ. For this purpose is of key importance to control the perturbation
frequency ω, which leads TWO different behaviours of the system at high/low values.
Question: Then, where is the clock?
Answer: As the modulation frequency is varied the crossover between the two types of
behaviour occurs when ωτ ≈ 1. From the transition between these two behaviours we
can sketch a range for τ .
The usual treatment for time-independent potential:
For an opaque (kd 1) barrier, the transmission rate is:
T =
16k2
κ2
k4
0
e−kd
It goes to 0 as d increases. Where:
k =
√
2mE/
k0 =
√
2mV0/
κ = k2
0 − k2
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 10 / 29

15. Contents
1 Main ideas of tunneling
2 Three different approaches
Wave-packet treatment
Dynamic paths treatment
Physical clocks treatment
3 Time modulated barrier clock
The idea
B¨uttiker & Landauer treatment
Low/High frequency
Conclusion
4 Dudovich & Co Experiment
The idea
The setup
The Gates
Displacement Gate
Velocity Gate
Reconstruction of ti , tr
5 Conclusions

16. Time modulated barrier clock B¨uttiker & Landauer treatment
Using time- dependent perturbation theory, the solution within the barrier:
ψ±(x, t, E) =
V1
ω
e±κx
e−iEt/
e−iV1
ω
sin ωt
Can be expressed as (*):
ψ±(x, t, E) =
V1
ω
e±κx
e−iEt/
n
Jn
V1
ω
e−inωt
Question: What does this mean?
Answer: Inside the barrier the energies E ± n ω are also solutions. There appear the
upper/lower sidebands, corresponding to the w.f absorbing/emitting modulation
quanta.
(*) P.K.Tien and J.P.Gordon, Phys. Rev. 129, 647 (1963)
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 11 / 29

17. Time modulated barrier clock B¨uttiker & Landauer treatment
And now, what?
Restrict to first order corrections n = 1
V1
ω
n
≈ Jn
V1
ω
for small argument.
Calculate the transmission intensities for the three possible energies (at first
order): E, E ± ω.
ψ±(x, t, E) =
V1
ω
e±κx
e−iEt/
e−iωt
+ eiωt
For both sidebands of E, the intensities are given by the static potential problem
transition rate T:
T± =
V1
2 ω
2
e±ωτ
− 1
2
T
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 12 / 29

18. Time modulated barrier clock B¨uttiker & Landauer treatment
Summary
Oscillating perturbative potential leads a superposition of solutions within the
barrier.
At first order appear two extra solutions: absorption/emission of modulation
quanta ω.
These new energies have their own transmission intensity given by the
expression above.
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 13 / 29

19. Contents
1 Main ideas of tunneling
2 Three different approaches
Wave-packet treatment
Dynamic paths treatment
Physical clocks treatment
3 Time modulated barrier clock
The idea
B¨uttiker & Landauer treatment
Low/High frequency
Conclusion
4 Dudovich & Co Experiment
The idea
The setup
The Gates
Displacement Gate
Velocity Gate
Reconstruction of ti , tr
5 Conclusions

20. Time modulated barrier clock Low/High frequency
Question: What happens at
low frequencies ωτ 1?
Answer: The particle sees an
effectively static barrier during its
traversal. No oscillating potential.
Therefore, the sidebands’
intensities
T± =
V1τ
2
2
T
Are the same. Actually this is the
same problem as for the static
barrier.
Question: What happens at
high frequencies ωτ 1?
Answer: The particle sees many
cycles of the oscillation. High
energy solutions have more chances
to be transmitted.
Therefore, the sidebands intensities
T+ =
V1
2 ω
2
e2ωτ
T
T− =
V1
2 ω
2
T
Are completely different. In fact
T+ T−
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 14 / 29

21. Contents
1 Main ideas of tunneling
2 Three different approaches
Wave-packet treatment
Dynamic paths treatment
Physical clocks treatment
3 Time modulated barrier clock
The idea
B¨uttiker & Landauer treatment
Low/High frequency
Conclusion
4 Dudovich & Co Experiment
The idea
The setup
The Gates
Displacement Gate
Velocity Gate
Reconstruction of ti , tr
5 Conclusions

22. Time modulated barrier clock Conclusion
Conclusions
An oscillating perturbative potential barrier is set up
The time modulation of the potential gives rise to ”sidebands” describing
particles which have absorbed or emitted modulation quanta ω.
At low frequencies ωτ 1 the intensity of transmitted waves is equal for both
sidebands.
At high frequencies ωτ 1 the intensity of transmitted upper sideband is higher
than the lower sideband.
Varying the frequency during an experiment and measuring the intensities for
both sidebands we can sketch a range for τ
A simpler way to observe de crossover is to compute the intensities in the following
way:
T+ − T−
T+ + T−
= tanh ωτ
Thus τ specifies the crossover.
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 15 / 29

23. Time modulated barrier clock Conclusion
0 0.5 1 1.5 2 2.5 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ωτ
tanh(ωτ)
Crossover
Region
ωτ ≈ 1
Unfortunately, there still are no successful experiments using the B¨uttiker approach.
Although there are other experiments nowadays which are very interesting... Let’s see
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 16 / 29

24. Contents
1 Main ideas of tunneling
2 Three different approaches
Wave-packet treatment
Dynamic paths treatment
Physical clocks treatment
3 Time modulated barrier clock
The idea
B¨uttiker & Landauer treatment
Low/High frequency
Conclusion
4 Dudovich & Co Experiment
The idea
The setup
The Gates
Displacement Gate
Velocity Gate
Reconstruction of ti , tr
5 Conclusions

25. Dudovich & Co Experiment The idea
Resolving the time when an electron exits a tunnel barrier
Objectives:
NOT TO MEASURE THE TUNNELING TIME ITSELF
To calibrate the internal attosecond clock on which the experiments are based.
High resolution measurements for ionization time (ti ) and recombination time
(tr )
To provide a general tool for time-resolving multi-electron rearrangements in
atoms and molecules.
Design a valid setup for further measurements.
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 17 / 29

26. Contents
1 Main ideas of tunneling
2 Three different approaches
Wave-packet treatment
Dynamic paths treatment
Physical clocks treatment
3 Time modulated barrier clock
The idea
B¨uttiker & Landauer treatment
Low/High frequency
Conclusion
4 Dudovich & Co Experiment
The idea
The setup
The Gates
Displacement Gate
Velocity Gate
Reconstruction of ti , tr
5 Conclusions

27. Dudovich & Co Experiment The setup
Step 1
Use He atoms.
We use a strong laser field to
induce the tunneling:
Fω = Fω cos ωt ˆex
The electrons exit the barrier
at the time ti
Longitudinal displacement of
the electron.
Step 2
We apply a SH weak field:
F2ω = F2ω cos (2ωt + φ) ˆey
F2ω Fω: perturbative!
Transversal displacement of
the electron.
Semiclassical approximation!
Electron with v0x = 0!
Question: What is φ?
Answer: This is the delay or phase shift between both fields and can be controlled. This
will be a crucial parameter in the experiment.
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 18 / 29

28. Dudovich & Co Experiment The setup
Question: Why do we need the SH field?
Answer: The independent characterization of ionization and recombination times
(using ”gates”) requires another field that is both perturbative and fast enough to
monitor these electron trajectories on the system timescale.
Step 3
If the trajectory is closed, then the recombination happens at tr
If not, there is no recombination: those trajectories are rejected (”gates”).
When the electron recombines, there is a HHG (High Harmonic Generation) N ω
We measure the HHG intensity and polarization in terms of the delay φ.
IMPORTANT: The HHG carries the information about the ”gates” chosen, and
therefore of ti and tr
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 19 / 29

29. Contents
1 Main ideas of tunneling
2 Three different approaches
Wave-packet treatment
Dynamic paths treatment
Physical clocks treatment
3 Time modulated barrier clock
The idea
B¨uttiker & Landauer treatment
Low/High frequency
Conclusion
4 Dudovich & Co Experiment
The idea
The setup
The Gates
Displacement Gate
Velocity Gate
Reconstruction of ti , tr
5 Conclusions

30. Dudovich & Co Experiment The Gates
Question: What is a gate?
Answer: A gate controls a given variable in the sense that restricts its value to a fixed
range. Mathematically our gates are functions of the ionization time and the
recombination time, as well as of the delay φ.
Question: Why do we need gate?
Answer: The gates provide us with measurable functions of the times ti ,tr that we are
looking for. It’s an indirect and very accurate way of measuring the times.
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 20 / 29

31. Dudovich & Co Experiment The Gates
Question: Where do we
implement the gates?
Answer: On the trajectory of the
electron after tunneling.
Displacement Gate Gy :
controls the lateral
displacement and, hence, the
intensity at recombination
(HHG intensity).
Velocity Gate Gv : controls the
lateral velocity and, hence,
the angle at recombination
(HHG polarization).
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 21 / 29

32. Contents
1 Main ideas of tunneling
2 Three different approaches
Wave-packet treatment
Dynamic paths treatment
Physical clocks treatment
3 Time modulated barrier clock
The idea
B¨uttiker & Landauer treatment
Low/High frequency
Conclusion
4 Dudovich & Co Experiment
The idea
The setup
The Gates
Displacement Gate
Velocity Gate
Reconstruction of ti , tr
5 Conclusions

33. Dudovich & Co Experiment Displacement Gate
Question: What happens after the ionization?
Answer:
1 Semiclassical approx. v0x = 0, and residual v0y from the tunneling
2 The total field (strong plus weak) acts over the electron and give him motion.
3 The motion (classical) is given by the relative delay φ and both frequencies
ω, 2ω.
4 Recombination does happen if the condition yi = yr is fulfilled.
Question: What is the condition of recombination?
Answer: Zero transversal (y) displacement. That is:
tr
ti
v0y − A2ω(ti ) − A2ω(t)dt = 0
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 22 / 29

34. Dudovich & Co Experiment Displacement Gate
The initial velocity must compensate the action of the field:
v0y (ti , tr , φ) =
F2ω
2ω
sin (2ωti + φ) +
cos (2ωtr + φ) − cos (2ωti + φ)
2ω(tr − ti )
With this condition over the velocity we can define the displacement gate:
Displacement Gate (*)
Gy (ti , tr , φ) = e−
v2
0y
2
τT
Which corresponds to a gaussian distribution on the y-axis modulated by the tunneling
time τT . This means, the gate is mapped onto the HHG intensity.
NOTE: the reconstruction procedure relies solely on the delay φ, and therefore is
independent of the value of τT . (*) Krausz, F. & Ivanov, M. Yu. Attosecond physics.
Review of Modern Physics 81, 163 (2009)
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 23 / 29

35. Contents
1 Main ideas of tunneling
2 Three different approaches
Wave-packet treatment
Dynamic paths treatment
Physical clocks treatment
3 Time modulated barrier clock
The idea
B¨uttiker & Landauer treatment
Low/High frequency
Conclusion
4 Dudovich & Co Experiment
The idea
The setup
The Gates
Displacement Gate
Velocity Gate
Reconstruction of ti , tr
5 Conclusions

36. Dudovich & Co Experiment Velocity Gate
Question: What do we look at now?
Answer: At the lateral velocity at the recombination time. It’s given by:
vy (ti , tr , φ) = v0y (ti , tr , φ) − A2ω(ti ) + A2ω(tr )
We define then the velocity gate as the ratio between transversal and longitudinal
components (recollision angle),
Velocity Gate:
Gv (ti , tr , φ) =
vy
vx
=
F2ω/2ω
2(Nω − Ip)
sin (2ωtr + φ) +
cos (2ωtr + φ) − cos (2ωti + φ)
2ω(tr − ti )
Where vx = 2(Nω − Ip), Ip is the ionization potential and Nω is the energy of the N
harmonic generated. This gate dictates the vectorial properties of the emitted light
and is mapped into the HHG polarization state.
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 24 / 29

37. Dudovich & Co Experiment Velocity Gate
In summary:
The two field configuration induces two independent gates which depend on the
lateral displacement and lateral velocity.
Experimentally, we can decouple their contribution: we can measure Gy via HHG
intensity and Gv via HHG polarization.
They impose a set of two equations for every N, from where we can extract ti
and tr .
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 25 / 29

38. Contents
1 Main ideas of tunneling
2 Three different approaches
Wave-packet treatment
Dynamic paths treatment
Physical clocks treatment
3 Time modulated barrier clock
The idea
B¨uttiker & Landauer treatment
Low/High frequency
Conclusion
4 Dudovich & Co Experiment
The idea
The setup
The Gates
Displacement Gate
Velocity Gate
Reconstruction of ti , tr
5 Conclusions

39. Dudovich & Co Experiment Reconstruction of ti , tr
Question: What do we have?
Answer:
We can measure both gates Gy , Gv
We can vary the delay between fields φ
Both gates depend on ti and tr : two equations
φ (delay)
a) Displacement gate chart / b) Velocity gate chart
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 26 / 29

40. Dudovich & Co Experiment Reconstruction of ti , tr
And finally
We find out the shift φ which maximizes the intensity (φy
max ) and the polarization
(φv
max ) of the HHG:
∂Gy
∂φ
|φ
y
max
= 0,
∂Gv
∂φ
|φv
max
= 0
And substitute in the gate equations to obtain the desired times
Reconstructed ionization and recollision times(red dots). The pink shaded areas represent the
uncertainty in the reconstruction procedure. The extracted times are compared to the calculated
times according to the semiclassical model (grey curves) and the quantum stationary solution
(black curves).
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 27 / 29

41. Conclusions
CONCLUSIONS
In general:
Although we know the main physics behind tunneling... We do not know the
time spent on it!
The tunneling time problem has as many approaches as researchers investigating
it!
The physical clock treatment is the most spread idea for attacking the problem
of tunneling.
About Dudovich & Co Experiment:
Is a very clean and ingenious experiment in the attosecond physics sector.
Gives high resolution measurements of time at the tunneling time scale.
Measures the ionization time ti which is half of the way of measuring the
tunneling time t0 = ti + iτ
Provided the new ideas included, it gives fresh air to move the investigations on
the traversal time one step forward.
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 28 / 29

42. Conclusions
THANKS FOR WATCHING!
“This is not even wrong!” Wolfgang Ernst Pauli
Christian Roca Catal´a (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 29 / 29