Upload
christiaan-roca-catala
View
210
Download
2
Embed Size (px)
DESCRIPTION
My work in tunneling resolution for one of my seminars at LMU. Basically based on existen bibliography and recent publications.
Citation preview
Time-resolution of tunneling
Christian Roca CatalaSupervised by: Prof. Dr. Armin Scrinzi
Ludwing-Maximilians Universitat Munchen
Jan 16, 2014
1 Main ideas of tunneling
2 Three different approachesWave-packet treatmentDynamic paths treatmentPhysical clocks treatment
3 Time modulated barrier clockThe ideaButtiker & Landauer treatmentLow/High frequencyConclusion
4 Dudovich & Co ExperimentThe ideaThe setupThe GatesDisplacement GateVelocity GateReconstruction of ti , tr
5 ConclusionsChristian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 2 / 29
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 Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 3 / 29
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 CONSENSUSON THE EXISTENCE OF AN UNIQUE AND SIMPLE EXPRESSIONFOR IT
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 4 / 29
Contents
1 Main ideas of tunneling
2 Three different approachesWave-packet treatmentDynamic paths treatmentPhysical clocks treatment
3 Time modulated barrier clockThe ideaButtiker & Landauer treatmentLow/High frequencyConclusion
4 Dudovich & Co ExperimentThe ideaThe setupThe GatesDisplacement GateVelocity GateReconstruction of ti , tr
5 Conclusions
Three different approaches Wave-packet treatment
Wave-packet treatmentThe 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 havehigher velocity than incoming w.p.
Observing the w.p is an QM invasive procedure
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 5 / 29
Contents
1 Main ideas of tunneling
2 Three different approachesWave-packet treatmentDynamic paths treatmentPhysical clocks treatment
3 Time modulated barrier clockThe ideaButtiker & Landauer treatmentLow/High frequencyConclusion
4 Dudovich & Co ExperimentThe ideaThe setupThe GatesDisplacement GateVelocity GateReconstruction of ti , tr
5 Conclusions
Three different approaches Dynamic paths treatment
Dynamic paths treatmentDetermine a set of dynamic paths x(t) for the incoming particle and ask how long eachpath 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 Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 6 / 29
Contents
1 Main ideas of tunneling
2 Three different approachesWave-packet treatmentDynamic paths treatmentPhysical clocks treatment
3 Time modulated barrier clockThe ideaButtiker & Landauer treatmentLow/High frequencyConclusion
4 Dudovich & Co ExperimentThe ideaThe setupThe GatesDisplacement GateVelocity GateReconstruction of ti , tr
5 Conclusions
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 itwhenever 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 tothe tunneling process.
NOTE: A clock can be chosen to be minimally invasive
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 7 / 29
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 (Buttiker &Landauer 1982) - we will look at it!
Larmor clock (Buttiker 1983)
Oscillating spin clock (Buttiker &Landauer 1985)
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 8 / 29
Contents
1 Main ideas of tunneling
2 Three different approachesWave-packet treatmentDynamic paths treatmentPhysical clocks treatment
3 Time modulated barrier clockThe ideaButtiker & Landauer treatmentLow/High frequencyConclusion
4 Dudovich & Co ExperimentThe ideaThe setupThe GatesDisplacement GateVelocity GateReconstruction of ti , tr
5 Conclusions
Time modulated barrier clock The idea
The scheme of this approach:
A particle approaching from the left:
j =−i~2m
(ψ∗∇ψ − ψ∇ψ∗) =~km
A time dependent barrier V (t) = V0 + V1 cosωt between x ≤ d/2, otherwiseV (t) = 0.
The perturbation frequency ω is variable.
The time scale for the particle crossing is given by the traverse time τ .
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 9 / 29
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 thetraversal 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 ofbehaviour 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 =16k2κ2
k40
e−kd
It goes to 0 as d increases. Where:
k =√
2mE/~
k0 =√
2mV0/~
κ =√
k20 − k2
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 10 / 29
Contents
1 Main ideas of tunneling
2 Three different approachesWave-packet treatmentDynamic paths treatmentPhysical clocks treatment
3 Time modulated barrier clockThe ideaButtiker & Landauer treatmentLow/High frequencyConclusion
4 Dudovich & Co ExperimentThe ideaThe setupThe GatesDisplacement GateVelocity GateReconstruction of ti , tr
5 Conclusions
Time modulated barrier clock Buttiker & Landauer treatment
Using time- dependent perturbation theory, the solution within the barrier:
ψ±(x , t,E) =V1
~ωe±κxe−iEt/~e−iV1
~ωsinωt
Can be expressed as (*):
ψ±(x , t,E) =V1
~ωe±κxe−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 theupper/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 Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 11 / 29
Time modulated barrier clock Buttiker & 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 firstorder): E ,E ± ~ω.
ψ±(x , t,E) =V1
~ωe±κxe−iEt/~
(e−iωt
+eiωt)
For both sidebands of E, the intensities are given by the static potential problemtransition rate T:
T± =
(V1
2~ω
)2 (e±ωτ − 1
)2T
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 12 / 29
Time modulated barrier clock Buttiker & Landauer treatment
Summary
Oscillating perturbative potential leads a superposition of solutions within thebarrier.
At first order appear two extra solutions: absorption/emission of modulationquanta ~ω.
These new energies have their own transmission intensity given by theexpression above.
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 13 / 29
Contents
1 Main ideas of tunneling
2 Three different approachesWave-packet treatmentDynamic paths treatmentPhysical clocks treatment
3 Time modulated barrier clockThe ideaButtiker & Landauer treatmentLow/High frequencyConclusion
4 Dudovich & Co ExperimentThe ideaThe setupThe GatesDisplacement GateVelocity GateReconstruction of ti , tr
5 Conclusions
Time modulated barrier clock Low/High frequency
Question: What happens atlow frequencies ωτ � 1?
Answer: The particle sees aneffectively static barrier during itstraversal. No oscillating potential.
Therefore, the sidebands’intensities
T± =
(V1τ
2~
)2
T
Are the same. Actually this is thesame problem as for the staticbarrier.
Question: What happens athigh frequencies ωτ � 1?Answer: The particle sees manycycles of the oscillation. High
energy solutions have more chancesto be transmitted.
Therefore, the sidebands intensities
T+ =
(V1
2~ω
)2
e2ωτT
T− =
(V1
2~ω
)2
T
Are completely different. In factT+ � T−
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 14 / 29
Contents
1 Main ideas of tunneling
2 Three different approachesWave-packet treatmentDynamic paths treatmentPhysical clocks treatment
3 Time modulated barrier clockThe ideaButtiker & Landauer treatmentLow/High frequencyConclusion
4 Dudovich & Co ExperimentThe ideaThe setupThe GatesDisplacement GateVelocity GateReconstruction of ti , tr
5 Conclusions
Time modulated barrier clock Conclusion
Conclusions
An oscillating perturbative potential barrier is set up
The time modulation of the potential gives rise to ”sidebands” describingparticles which have absorbed or emitted modulation quanta ~ω.
At low frequencies ωτ � 1 the intensity of transmitted waves is equal for bothsidebands.
At high frequencies ωτ � 1 the intensity of transmitted upper sideband is higherthan the lower sideband.
Varying the frequency during an experiment and measuring the intensities forboth sidebands we can sketch a range for τ
A simpler way to observe de crossover is to compute the intensities in the followingway:
T+ − T−
T+ + T−= tanhωτ
Thus τ specifies the crossover.
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 15 / 29
Time modulated barrier clock Conclusion
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ωτ
tanh(ω
τ)
CrossoverRegion
ωτ ≈ 1
Unfortunately, there still are no successful experiments using the Buttiker approach.Although there are other experiments nowadays which are very interesting... Let’s see
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 16 / 29
Contents
1 Main ideas of tunneling
2 Three different approachesWave-packet treatmentDynamic paths treatmentPhysical clocks treatment
3 Time modulated barrier clockThe ideaButtiker & Landauer treatmentLow/High frequencyConclusion
4 Dudovich & Co ExperimentThe ideaThe setupThe GatesDisplacement GateVelocity GateReconstruction of ti , tr
5 Conclusions
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 inatoms and molecules.
Design a valid setup for further measurements.
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 17 / 29
Contents
1 Main ideas of tunneling
2 Three different approachesWave-packet treatmentDynamic paths treatmentPhysical clocks treatment
3 Time modulated barrier clockThe ideaButtiker & Landauer treatmentLow/High frequencyConclusion
4 Dudovich & Co ExperimentThe ideaThe setupThe GatesDisplacement GateVelocity GateReconstruction of ti , tr
5 Conclusions
Dudovich & Co Experiment The setup
Step 1
Use He atoms.
We use a strong laser field toinduce the tunneling:
~Fω = Fω cosωtex
The electrons exit the barrierat the time ti
Longitudinal displacement ofthe electron.
Step 2
We apply a SH weak field:
~F2ω = F2ω cos (2ωt + φ)ey
F2ω � Fω: perturbative!
Transversal displacement ofthe 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. Thiswill be a crucial parameter in the experiment.
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 18 / 29
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~ωWe measure the HHG intensity and polarization in terms of the delay φ.
IMPORTANT: The HHG carries the information about the ”gates” chosen, andtherefore of ti and tr
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 19 / 29
Contents
1 Main ideas of tunneling
2 Three different approachesWave-packet treatmentDynamic paths treatmentPhysical clocks treatment
3 Time modulated barrier clockThe ideaButtiker & Landauer treatmentLow/High frequencyConclusion
4 Dudovich & Co ExperimentThe ideaThe setupThe GatesDisplacement GateVelocity GateReconstruction of ti , tr
5 Conclusions
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 fixedrange. 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 arelooking for. It’s an indirect and very accurate way of measuring the times.
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 20 / 29
Dudovich & Co Experiment The Gates
Question: Where do weimplement the gates?
Answer: On the trajectory of theelectron after tunneling.
Displacement Gate Gy :controls the lateraldisplacement and, hence, theintensity at recombination(HHG intensity).
Velocity Gate Gv : controls thelateral velocity and, hence,the angle at recombination(HHG polarization).
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 21 / 29
Contents
1 Main ideas of tunneling
2 Three different approachesWave-packet treatmentDynamic paths treatmentPhysical clocks treatment
3 Time modulated barrier clockThe ideaButtiker & Landauer treatmentLow/High frequencyConclusion
4 Dudovich & Co ExperimentThe ideaThe setupThe GatesDisplacement GateVelocity GateReconstruction of ti , tr
5 Conclusions
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 frequenciesω, 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 Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 22 / 29
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−v20y2τT
Which corresponds to a gaussian distribution on the y-axis modulated by the tunnelingtime τT . This means, the gate is mapped onto the HHG intensity.
NOTE: the reconstruction procedure relies solely on the delay φ, and therefore isindependent of the value of τT . (*) Krausz, F. & Ivanov, M. Yu. Attosecond physics.Review of Modern Physics 81, 163 (2009)
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 23 / 29
Contents
1 Main ideas of tunneling
2 Three different approachesWave-packet treatmentDynamic paths treatmentPhysical clocks treatment
3 Time modulated barrier clockThe ideaButtiker & Landauer treatmentLow/High frequencyConclusion
4 Dudovich & Co ExperimentThe ideaThe setupThe GatesDisplacement GateVelocity GateReconstruction of ti , tr
5 Conclusions
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 longitudinalcomponents (recollision angle),
Velocity Gate:
Gv (ti , tr , φ) =vyvx
=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 lightand is mapped into the HHG polarization state.
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 24 / 29
Dudovich & Co Experiment Velocity Gate
In summary:
The two field configuration induces two independent gates which depend on thelateral displacement and lateral velocity.
Experimentally, we can decouple their contribution: we can measure Gy via HHGintensity and Gv via HHG polarization.
They impose a set of two equations for every N, from where we can extract tiand tr .
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 25 / 29
Contents
1 Main ideas of tunneling
2 Three different approachesWave-packet treatmentDynamic paths treatmentPhysical clocks treatment
3 Time modulated barrier clockThe ideaButtiker & Landauer treatmentLow/High frequencyConclusion
4 Dudovich & Co ExperimentThe ideaThe setupThe GatesDisplacement GateVelocity GateReconstruction of ti , tr
5 Conclusions
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 Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 26 / 29
Dudovich & Co Experiment Reconstruction of ti , tr
And finally
We find out the shift φ which maximizes the intensity (φymax) and the polarization
(φvmax) of the HHG:
∂Gy
∂φ|φy
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 Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 27 / 29
Conclusions
CONCLUSIONS
In general:
Although we know the main physics behind tunneling... We do not know thetime spent on it!
The tunneling time problem has as many approaches as researchers investigatingit!
The physical clock treatment is the most spread idea for attacking the problemof 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 thetunneling time t0 = ti + iτ
Provided the new ideas included, it gives fresh air to move the investigations onthe traversal time one step forward.
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 28 / 29
Conclusions
THANKS FOR WATCHING!
“This is not even wrong!” Wolfgang Ernst Pauli
Christian Roca Catala (LMU) Seminar: Dynamics in strong fields Jan 16, 2014 29 / 29