time resolution of quantum tunneling - dynamics in strong fields seminar lmu

Time-resolution of tunneling

Christian Roca CatalaSupervised by: Prof. Dr. Armin Scrinzi

Ludwing-Maximilians Universitat Munchen

[email protected]

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


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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


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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


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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


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)


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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 .


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


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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)



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



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.


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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−


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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.


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


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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.


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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.


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


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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.


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).


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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


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)


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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.


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 .


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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


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.


Conclusions

THANKS FOR WATCHING!

“This is not even wrong!” Wolfgang Ernst Pauli


time resolution of quantum tunneling - dynamics in strong fields seminar lmu

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