Femto group - LCAR - Toulouse

2nd Sino-German workshop on Quantum Engineering, Reisensburg (22-26/9/2007) 1

Femto group - LCAR - Toulouse

Antoine Monmayrant Jean-Christophe Delagnes

Jérôme Degert Sébastien Zamith

Sabine Stock Céline Nicole Emma Sokell Peter Zahariev

Giorgio Turri Arnaud Arbouet

Kevin Rafael Damien Bigourd Jean-Benoît Hamard

Elsa Baynard Valérie Blanchet

Aziz Bouchene Béatrice Chatel LCAR-IRSAMC

CNRS, Univ. P. Sabatier, Région Midi-Pyrénées Fondation Del Duca, Ministère de la Recherche Minist Affaires Etrangères, JSPS, IMS, ANR, IUF

Collaborations:

Ludger Wöste, E. Schreiber (Berlin) Marc Vrakking (Amsterdam)

Noureddine Melikechi (Delaware) Anne L’Huillier (Lund)

Dolorès Gauyacq (Orsay) Gustav Gerber (Würzburg) Thomas Baumert (Kassel)

Matthias Wollenhaupt (Kassel) Marcus Motzkus (MPQ Garching) Kenji Ohmori (IMS, Okazaki) Chris Meier (LCAR, Toulouse) Benjamin Whitaker (Leeds) Wolfgang Schleich (Ulm) Gerhard Paulus ( Texas) Ilya Averbukh (Rehovot) Bertrand Girard

Coherent control in Atoms with

ultrashort laser pulses

Toulouse : Romains’ wall

Femto group - LCAR - Toulouse

Place du Capitole

Le Pont Neuf

Basilique Saint-Sernin

(1180 – 1478)

Place du Capitole

Femto group - LCAR - Toulouse

- Molecules : Gerber, Science 98 - Clusters : Wöste, CP 2001

- Biological systems :Herek, Nature 2002 - Strong field : Levis, Science 2001

- ….

Control using shaped pulses

- 2 ph. transitions : Silberberg, Nature 98 - Small molecules : Leone, JCP 98

- Quantum info: Bucksbaum, Nature 99 - 1 photon transition : Girard PRL 2002 - ….

Pulse shaper Theory

System

Closed-loop Open-loop

System Pulse shaper

Optimization

Algorithm GOAL

- Silberberg, Nature 2002

- Wöste , Science 2003. PRL 2004 - Vrakking PRA 2004

- Dantus Optics Express 2004 - Bucksbaum PRL 2004 …..

In between…

Learning something on the interaction

Coherent control : opening the loop …

Open loop coherent control of simple systems

Meshulach, Silberberg Nature, 396 1998, PRA 99, PRL 01

Show a new effect ?

Simple System

Shaper Theory

Measure Shape

Many results:

2 photons transitions Silberberg, Nature 1998, PRA 99, PRL 01, Dantus JCP 03 …

Raman transitions Silberberg, Nature 02, Faucher 04, Motzkus 06…

Small molecules Leone et al., JCP 1998

1 photon transitions Girard - Motzkus, PRL 2002

Quantum information Bucksbaum-Weinacht, Nature 99, PRL 98, Science 00 ..

…

Femto group - LCAR - Toulouse

Quantum Interferences …

Many different ways to observe, study, control, use Quantum Interferences …

- External degree(s) of freedom : Matter waves (atoms, molecules, clusters, BEC …)

- Internal degree(s) of freedom : Ramsey fringes

In the Ultrashort world:

- Interferences of wave packets created by different laser pulses

- Interferences within the excitation process

Two level system laser pulses

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.000 0.002 0.004 0.006 0.008 0.010

-400 -200 200 400 600 time

0

0.000 0.002 0.004 0.006 0.008 0.010

-400 -200 200 400 600 time

0

Rabi Oscillations Adiabatic transfer

Coherent transients

0 1

τ

Ω >

0 1

τ

Ω <<

( )

( ) t μ E t Ω =

=

" 0

φ = φ " ≠ 0

Back to basics

FT limited Chirped

Strong field

Weak field

In molecular potentials, the effect of chirp is opposite between weak and strong field

Femto group - LCAR - Toulouse

Outline

I. Wave packet Interferences in atoms II. Quantum state holography

III. Interferences within the excitation process IV. Optics as an analog computer: Number

factorization

Interferences between identical time-delayed quantum paths : Ramsey fringes - Theory

τ L

2 π / ω _L

τ L

2 π / ω _L

t ₀ t ₀+ τ t

E ( t )

( ) ^{t i a e}

^{e a e}

^{( )}

^e

Ψ = + +

)weak field regime :

)interference phase oscillates rapidly with τ ª φ _cc ⁼ ω τ

φ _cc ⁼ 0 [2π] constructive interferences φ _cc ⁼ π [2π] destructive interferences

Time delay τ Excited state

population

Period = 2π/ω

( ) ^{t i a e}

^e

Ψ = +

|i 〉

|e 〉

ω

=

t

t

+ τ

0

1 a <<

Φ

Femto group - LCAR - Toulouse

Interferences between identical time-delayed quantum paths : Ramsey fringes - Experiment

V. Blanchet et al, Phys. Rev. Lett. 78, 2716 (1997).

M. A. Bouchene et al, Eur. Phys. J. D 2, 131 (1998).

-3 0 3 6 9 12 15

delay / ps 0

2 4 6 8 10

Cs+ / a.u

1.6 ps

T

=2.56 fs T

=1.28 fs

35040 35060

-10 0 10

Similar studies by

G. Fleming et al, JCP 93, 856 (1990); 95, 1487 (’91); 96, 4180 (92) R. Jones, P. Bucksbaum

T. Hänsch

K. Ohmori et al, PRL 91, 243003 (2003), 96, 093002 (2006) S.S. Vianna

PRA 74, 055402 (2006)

Pulse sequence produced by a Michelson interferometer

Beats due to spin-orbit precession

τ

7d

D

6s

S

τ=1.6 ps Cs⁺

(1)

(2)

769 nm 769 nm

τ

Interferences of free electron wavepackets

F. Krausz et al, Attosecond Double-Slit Experiment, PRL 95, 040401 (2005)

M. J. J. Vrakking, A. L'Huillier, et al, "Attosecond electron wave packet interferometry" Nat. Phys. 2, 323 (2006)

Femto group - LCAR - Toulouse

Wp interferences with a pulse shaper

1 ( ) ⁱ 2 ( )

E t + e ^θ E t

† High Res. Pulse Shaper:

„ Phase - Amplitude control with 640 pixels.

„ shaping window of 35 ps.

„ high complexity.

„ high amplitude dynamic (30 dB).

„ 75 % power transmission.

A. Monmayrant, B. Chatel. "A new phase and amplitude High Resolution Pulse Shaper."

Rev. Sci. Inst. 75, 2668 (2004)

|a

(t)|

2

(u. arb.)

Test of interferometric stability

0π 1π 2π 3π 4π

0 2 4

|a e2(t)|2 (u. arb.)

Phase θ (rad)

† A sequence of 2 FT limited pulses with a relative phase θ …

{ ⁽¹⁾ }

( ) 1 e x p ( _p ) / 2

H ω = + ^⎡ ⎣ i θ + i φ ω − ω ^⎤ ⎦

-1 0 1 2 3 4 5

0 1 2 3 4

Delay (ps)

θ=π

Intensit y (u. arb.) θ=0

θ=0

θ=π

Ramsey fringes

1+1 = 0 to 4

Femto group - LCAR - Toulouse

Outline

I. Wave packet Interferences in atoms II. Quantum state holography

Measuring the time evolution of the quantum state and not only the population

III. Interferences within the excitation process IV. Optics as an analog computer: Number

factorization

2 2

2ln 2

( )

2

t t

t i

a t e e ^τ e ^φ dt

⎛ ′ ⎞ ′

− ⎜ ⎟ −

⎝ ⎠ ′′

−∞ ′

∝ ∫

20 ps τ c ≈

t ₁ t ₂

2 2 " φ

5 2

at resonance 8 10 fs φ ′′ = −

-2 0 2 4 6 8 10

Delay (ps)

6p-5s Fluorescen ce

Application of wave packet interferences to Quantum State Holography

6d, 8s

5p²P_1/2

τ

5s

6p

Fluorescence (420 nm)

Rb

Excited state evolution through chirped pulse excitation

Zamith, et al, PRL 87, 033001 (2001) Degert et al, PRL 89, 203003(2002)

e

Im(a (t))

Re(a (t))

-0.1 0.0 0.1 0.2

-0.20 -0.15 -0.10 -0.05 0.00 0.05

Cornu spiral

Femto group - LCAR - Toulouse

Quantum state holography I

6d, 8s

5p²P_1/2

τ

5s

6p

Fluorescence (420 nm)

† Two level system: 5s- 5p²P1/2 in Rb

„ Red pump: resonant and chirped

„ Yellow probe: shorter than pump and CT dynamic

„ Fluorescence is monitored as function of the pump- probe delay

Rb † Principle:

„ Use a first pulse to create a local oscillator in the atom

„ The population in 5p (probed in real

time) reflects the beating between this

local oscillator and the wave function

excited by the driving field

Quantum state holography II

( ) ( )

(2)

(1) 2

( )

( ) 1 exp ( ) ( ) / 2

2

O I

p p

E H E

H i i i

ω ω ω

ω θ φ ω ω φ ω ω

=

⎡ ⎡ ⎤ ⎤

= + ⎢ ⎢ + − + − ⎥ ⎥

⎢ ⎣ ⎦ ⎥

⎣ ⎦

6d, 8s

5p²P_1/2

τ

5s

6p

Fluorescence (420 nm)

† Two level system: 5s- 5p²P1/2 in Rb

„ Red pump: resonant and chirped

„ Yellow probe: shorter than pump and CT dynamic

„ Fluorescence is monitored as function of the pump- probe delay

Rb

NOPA Ti:Sa

Oscillator

CPA

795 nm 1kHz 1 mJ 130 fs

607 nm 1kHz 5 µJ 20 fs

795 nm 1kHz 5 µJ

Delay line Rb

PM

Rb τ E

( ) ω

( )

E

ω

Femto group - LCAR - Toulouse

Quantum state holography III

From CT to …

2

1 2

( ) ( ) ( )

A e e

CT t ∝ a t + a t

1 ( ) 2 ( )

E t + E t

A)

2

1 2

( ) ( ) . ( )

B e e

CT t ∝ a t + i a t

1 ( ) 2 ( )

E t + i E t

B)

-1 0 1 2 3 4 5 6 7 8 9 10 11 0.0

0.1 0.2 0.3 0.4 0.5 0.6

Fl uorescence ( a rb. uni ts)

τ (ps) CT

CT

By combination of the two CT…

Quantum state holography IV

…the evolution of the atomic quantum state

By

combination of the two CT…

A. Monmayrant, B. Chatel, B. Girard, PRL, 96 103002 (2006) ; Opt. Lett.

31, 410 (2006) ;, Opt. Comm. 264, 256 (2006)

Femto group - LCAR - Toulouse

Quantum state holography

† Principle (Leichtle, Schleich, Averbukh, Shapiro PRL 80 (1998)):

Interference between a reference and an object quantum state.

Several measurements at different delays provide reconstruction of the object state

† Two experiments :

- Rydberg state wave packet measurement

(Weinacht, Ahn, Bucksbaum, PRL 80, 5508 (1998) Nature 397, 233 (1999))

- Vibrational wave packet measurement (Walmsley et al, PRL 74, 884 (1995);

J. Phys. B 31 (1998))

Our scheme is a direct measurement of the w.p. during the interaction

† See also Ohmori et al, PRL 96 , 093002 (2006)

Outline

I. Wave packet Interferences in atoms II. Quantum state holography

III. Interferences within the excitation process IV. Optics as an analog computer: Number

factorization

Femto group - LCAR - Toulouse

Interferences within the excitation process

† Takes advantage of the broad spectrum of ultrashort pulses:

„ Interferences between several quantum paths associated to different frequency components in multiphoton transitions

„ With or without intermediate states

„ Control of interferences by changing the pulse shape (spectral phase)

† Many examples

„ Chirped pulses

† Noordam et al, Chatel-Girard et al,

„ Shaped pulses

† Silberberg et al, Dantus et al, Faucher et al

eg

/ 2 ω

g i

e

1 pulse – Two-photon transition

-15000 -10000 -5000 0 5000 10000 0,0

0,5 1,0 1,5 2,0 2,5 3,0

Ex citation probability

φ '' (fs

)

( ( ) )

direc sequential

2

1 t

Fo 1

( ) ( ) sgn

( ) ( )

2

r : 1

2 ^{ei ig}

e

e g

i E E d E

a E ω ω φ δ

δ ω

ω ω ω

φ δ

ω δ

π ω

+∞

−∞ −

⎡ ⎤

′′ >> ⎢ ⎥

∝ ⎢ + ⎥

⎢ − ′′ + ⎥

= − ∫

i

e ^{φ δ ′′}

1 τ _c

≈ δ > 0

2 2

d s

e a

a = + a

Two-photon transitions with chirped pulses

2

a d 2

a s

g i e

ω ei

ω ig

Femto group - LCAR - Toulouse

φ ′′ < 0

h ν

, 1 g +

, 0 i

, 1 e −

2 δ δ

-15000 -10000 -5000 0 5000 10000

0,0 0,5 1,0 1,5 2,0 2,5 3,0

Excitation probability

i 2

e ^{φ δ ′′}

1 τ _c

≈

δ > 0 2

a e

Dressed states

Two-photon transitions

B. Noordam et al PRL 92, PRA 94 Rb

5s Æ 5p Æ 5d

Ladder climbing in Rb

D. J. Maas, C. W. Rella, P. Antoine, E. S. Toma, and L. D. Noordam, Phys. Rev. A 59, 1374 (1999).

Rb atom

Femto group - LCAR - Toulouse

Na cell

Na ladder climbing with chirped pulses

4P

Fluorescence 330 nm

602 nm

3S 3P

3P

5S 3 , s h ν

1 / 2

3P

3 / 2

3P

5 , S − h ν

φ ′′ < 0

h ν Dressed states

800 nm 1 kHz 1mJ 130 fs

NcOPA

Stretcher

PMT

CPA

605 nm, 30 fs width 25nm

10 µJ

SF58 rod

Variable chirp Fixed chirp

Fluorescence observed

as a function of chirp rate

Na (3s Æ 3p Æ 5s) at 605 nm

J L

S

L

S B. Chatel, et al,

PRA 68, 041402R (2003) PRA, 70, 053414 (2004).

FT limit

Femto group - LCAR - Toulouse

Outline

I. Wave packet Interferences in atoms II. Quantum state holography

III. Interferences within the excitation process IV. Optics as an analog computer: Number

factorization

† From Coherent transients and the Fresnel integral …

… to the Gauss sum and the factorization of numbers!

(with Wolfgang P. Schleich (Ulm)) 20 ps

τ c ≈

t ₁ t ₂

2 2 " φ

5 2

at resonance 8 10 fs φ ′′ = −

-2 0 2 4 6 8 10

Delay (ps)

6p-5s Fluorescen ce

2 2

2ln 2

( )

2

t t

t i

a t e e ^τ e ^φ dt

⎛ ′ ⎞ ′

− ⎜ ⎟ −

⎝ ⎠ ′′

−∞ ′

∝ ∫

Femto group - LCAR - Toulouse

From 2 to many levels : numbers factorization

Goal : Implement the Gauss sum in a physics experiment to factorize numbers

Number to factorize

If l is not a factor of N then the phase oscillates rapidly and the sum is small If l is a factor of N then the phase is a multiple of 2π and the sum is equal to 1

W. Merkel, W. P. Schleich, I. Averbukh, B. Girard, et al, Int. J. Mod. Phys. B 20, 1893 (2006).

W. Merkel, I. S. Averbukh, B. Girard, G. G. Paulus and W. P. Schleich, Fortschr Phys. 54, 856 (2006).

Trial number

Expected interferences with 23 levels

Multipath interferences in atoms

W. Merkel, I. S. Averbukh, B. Girard, G. G. Paulus and W. P. Schleich, Fortsch. Phys 54, 856 (2006)

W. Merkel, O. Crasser, F. Haug, E. Lutz, H. Mack, M. Freyberger, W. P. Schleich, I. S. Averbukh,

M. Bienert, B. Girard, H. Maier and G. G. Paulus, Int. J. Mod. Phys. B 20, 1893 (2006)

Femto group - LCAR - Toulouse

First successful attempts

- M. Mehring, K. Mueller, I. S. Averbukh, W. Merkel and W. P. Schleich,

"NMR experiment factors numbers with Gauss sums", PRL 98, 120502 (2007).

- T. S. Mahesh, N. Rajendran, X. Peng and D. Suter, "Factorizing numbers with the Gauss sum technique: NMR implementations", Phys. Rev. A 75, 062303 (2007).

- M. Gilowski, T. Wendrich, T. Muller, C. Jentsch, W. Ertmer, E. M. Rasel and W. P.

Schleich, "Gauss sum factorization with cold atoms", submitted.

- D. Bigourd, B. Chatel, B. Girard and W. P. Schleich, "Factorization of Numbers with the temporal Talbot effect: Optical implementation by a sequence of shaped ultrashort pulses", submitted

Each term of the Gauss

sum is « calculated »

separately !

Our experiment with shaped pulses

0 2

( ) exp ( )

2

m

H i im p

m N l

ω θ τ ω ω

θ π

⎡ ⎤

= ⎣ + − ⎦

=

∑

Laser fs Spectrometer

Sequence of shaped pulses with phases equal to the argument of the Gauss sum

( ) 2

( _p ) _N ^M ( ) I ω = A l

Hence we have

Femto group - LCAR - Toulouse

First results

N=19043= 137*139 with M=9 pulses

X Theory z Exp.

N=105= 3*5*7 with M=4 pulses

0 2 4 6 8 10 12 14 16

0,0 0,2 0,4 0,6 0,8

1,0 N=105

|A N

(M) (l)|2

l

130 132 134 136 138 140 142 144 146 0,0

0,2 0,4 0,6 0,8

1,0 N=19043

9 pulses

|A N

(M) (l)|2

l

Efficiency as a function of the number of pulses

2 4 6 8 10 12 14 16

0,0 0,2 0,4 0,6 0,8 1,0

130 135 140 145

0.0 0.2 0.4 0.6 0.8 1.0

|AN(M) (l)|2

l

|A N

(M) (138)|2

Number of pulses (M+1)

N=19043=137*139

Prediction (Schleich et al 2007) :

To obtain all truncated Gauss sums of Ghost factors below 0.5, one should have

0.7 4 8

M ≥ N ≈

Femto group - LCAR - Toulouse

Conclusion: Interferences, interferences !!!

† Observation of the atomic quantum state during the light interaction

„ Measurement with a reference, but no spectral condition for the reference (unlike conventional interferometric methods)

„ Extension to multiple states (molecular nuclear wave packets)

† Pulse shape measurements (from da

/dt):

„ Feasibility of reconstructing the phase and the amplitude using coherent transients

„ No intrinsic limitation to measure complex shapes

The only temporal limit is the linewidth of the atomic transition!

„ Requires a resonant bound state as a local oscillator!!

† Effect of chirp: hard to draw systematics !

† Factorization: Looking for direct schemes to generate the Gauss sum

… More interferences to come !

2 4 6 8 10 12 14 16

0,0 0,2 0,4 0,6 0,8 1,0

130 135 140 145

0.0 0.2 0.4 0.6 0.8 1.0

|AN(M)(l)|2

l

|AN (M) (138)|2

Number of pulses (M+1)

† Two counter propagating vibrational wave packets in I

B state

† High resolution (1 pm – 50 fs) required to observe this transient standing wave (5 pm – 300 fs periodicity)

† Partial mapping of the wave packet pattern

† Possible « Applications » ?

„ Create more complex structure

„ Probe decoherence (environment, induced perturbation) ?

„ Extension to polyatomic molecules (probe vibrational energy transfer)

H. Katsuki et al, Science 311, 1489 (2006)

Interferences of colliding vibrational wave packets

Femto group - LCAR - Toulouse H. Katsuki, H. Chiba, B. Girard, C. Meier and K. Ohmori, Science 311, 1489 (2006)