Yaron Silberbergwww.weizmann.ac.il/~feyaron

Physics of Complex SystemsWeizmann Institute of Science

Rehovot, Israel

Optical Solitons

Coherent Quantum Control

NonlinearMicroscopy

NonclassicalLight

Ultrafast Optics group

On the Shape of the Photon:On the Shape of the Photon:

Quantum Coherent Control with Single Photons

www.weizmann.ac.il/~feyaron

Cargese August 2008

Narrow Transitions, Broad LightNarrow Transitions, Broad Light

Atomic transitions ~ 1 GHz

10 fs pulse ~ 100,000 GHz

( ) dttita fgf ∫∝∞ )exp()( 2 ωε

( ) ( ) ωωωω dEEa fgf ∫ −∝∞)(g

f

Nonresonant TwoNonresonant Two--Photon AbsorptionPhoton Absorption

Transition is induced by interference ofmany trajectories:

Perturbation analysis yields:

( ) ( )

( ) ( ) ( ) ( )[ ]δωωδωωδωωδωωδω

δωωδωωδω−Φ++Φ⋅−+=

=−+∝∞

∫∫

0000

00)(ieEEd

EEda f

Transform limited pulses are most efficient, but:

Antisymmetric phase has no effect on transition probability

g

f

ω0= ωfg/2

Nonresonant TPANonresonant TPA

Nonresonant TPANonresonant TPAscan of a periodic phase maskscan of a periodic phase mask

πI

2π

00

1

Φ

ω

Sinusoidal (antisymmetric) phase

Cosinusoidal(symmetric) phase –dark pulse

Long, weak AM modulated pulses induce TPA just like transform limited pulses with the same energy

How long is long?

20 fs pulse modulated by a shaper could becomes ~10 ps

g

f

ω0= ωfg/2

( )

AntiAnti--Symmetric Phase ModulationSymmetric Phase Modulation

( )*00 Δ+=Δ− ωω EE

)cos()( 0ttA ω

Broadband downBroadband down--converted light (squeezed vacuum)converted light (squeezed vacuum)

Each beam is a broadband incoherent noise, ( ) ( )*21

21 Δ+=Δ− PSPI EE ωωBut :

Pω21

SIGNALIDLER

E(ω)

Is (t)

Ii (t)

( ) ( )*tAtA Si =

χ(2)

SIGNAL

IDLER

PUMP

Together they form a single quadrature AM modulated field )cos()( 21 ttA pω

S. E. Harris, M. K. Oshman, and R. L. Byer, "Observation of Tunable Optical Parametric Fluorescence," Phys. Rev. Lett. 18, 732-734 (May 1967).

Broadband down-converted light beams can induce TPA just like an ultrashort pulse with the same bandwidths

( ) ( )*21

21 Δ+∝Δ− PSPI EE ωω

when the pump frequency ωp is tuned to the two-photon overall frequency ωfg :

( )22

∫∝ ωω dEp Sf

A complete constructive interference, just like with a transform-limited pulse

TPA with SPDC LightTPA with SPDC Light

Broadband downBroadband down--converted light beams induce TPA just like a converted light beams induce TPA just like a pair of ultrashort pulse with the same bandwidthspair of ultrashort pulse with the same bandwidths

χ(2)

SIGNAL

IDLER

3 ns1 MW

20 fs150 GW =

PUMP100 nm

0.04 nm= χ(2)

SIGNAL

IDLER

100 nm

But :

Temporal resolution of ultrashort pulses(though the light can be continuous)

Spectral resolution of a narrowband laser(though the light is as broadband as a pulse)

Rb cell

PMT

Digitaloscilloscope

Computer

SignalSLM

Delay line

Pulse-Shaper

516.65 nm

4D5/2,1/2

5S1/2

Signal~ 870

nm

Idler~ 1270

nm

TPA with downTPA with down--converted lightconverted light

Pump (516.65nm, 3ns)

Signal (870+-50nm)

Idler(1270+-50nm)

Experimental ResultsExperimental ResultsTemporal resolution as of 23 fs pulses,

5 orders of magnitudeBelow the duration of the light (3 ns).

Spectral resolution as of the pump (0.04nm)3 orders of magnitude

Below bandwidth of light (~100nm / beam)

Calculated

Experimental

B. Dayan, A. Pe’er, A.A. Friesem, Y. Silberberg, Phys. Rev. Lett, 93, 023005 (2004)

Rb cell

PMT

Digitaloscilloscope

Computer

Idler

SignalSLM

Delay line

Pulse-Shaper

516.65 nm

4D5/2,1/2

5S1/2

Signal~ 870

nm

Idler~ 1270

nm

Controlling TPAControlling TPA

Pump

Controlling TPAControlling TPA

B. Dayan, A. Pe’er, A.A. Friesem, Y. Silberberg, Phys. Rev. Lett, 93, 023005 (2004)

Spontaneous Parametric DownSpontaneous Parametric Down--Conversion Conversion --the bithe bi--photon sourcephoton source

a pump photon is spontaneously converted into two lower frequency photons

pωsω

iωenergy conservation

momentum conservation(phase matching)

pkr

skr

ikr

non linear crystal

pumpsignal

idler

QCC with NonQCC with Non--classical Lightclassical Light

Can we shape a single photon?Can we control with single photons?

Spontaneous DownSpontaneous Down--Conversion:Conversion:TimeTime--Energy Entangled PhotonsEnergy Entangled Photons

5p MHzδ ≈

THznm 1030 ≈≈Δ

PUMP (cw)

χ(2)

SIGNAL (cw)

IDLER (cw)

( ) ( ) ( ), 0s i s s i it t E t E t ϕ+ +Ψ =

The two-photon wavefunction

1 0.2p sδ μ≈

( )is tt +21

is tt −1 100 fsΔ ≈

∫ +−+−= ωωωωωωεεϕ2/2/

1,1)(0)1(pp

gd

Gate

• The time DIFFERENCE between the photons behaves as a fs pulse

TimeTime--Energy Entangled PhotonsEnergy Entangled Photons

non linear crystal

pump (cw)

signal (cw)

idler (cw)

… so lets shape the two-photon correlation function !

• But electronics limits temporal resolution to ~ns

Shaper

Δc~

“Measurement of Subpicosecond Time Intervals between Two Photons by Interference”C.K. Hong, Z.Y. Ou and L. Mandel, PRL 59 (1987)

IDLER

SIGNAL

PUMP χ(2)

d

TwoTwo--Photon Coincidence Interference :Photon Coincidence Interference :HongHong--OuOu--Mandel DipMandel Dip

1

Indistinguishable Paths

21

21⋅

T R

RT

Indistinguishable paths which lead to the same event interfere

22ii

⋅ 0=+

1

22

destructive interference no coincidence

1

2BS

delay

delay distinguishable paths no interference

Δc~

“Measurement of Subpicosecond Time Intervals between Two Photons by Interference”C.K. Hong, Z.Y. Ou and L. Mandel, PRL 59 (1987)

IDLER

SIGNAL

PUMP χ(2)

d

TwoTwo--Photon Coincidence Interference :Photon Coincidence Interference :HongHong--OuOu--Mandel DipMandel Dip

Shaper

HOM in polarizationHOM in polarization

Pump364 nm

Computer

SLM

FourierPlane

PBS

V

1

2H

)(ωΦ

V

φ

2 type-I crystals generatepolarization entanglementand broad spectrum

( )

( )isis

isis

XYYX

VVHH

+=

−

21

21

H

XV

Y( )

isi

isVVeHH ϕ+

21

A. V. Burlakov et. al. , PRA 64, (2001)

Experimental SetupExperimental Setup

Pump364 nm

Computer

crystalsSLM

FourierPlane

PBS

V

1

2H

)(ωΦ

Phase-and-polarization SLMControls independently the ±45° axes (X,Y)

Experimental ResultsExperimental Results

B. Dayan, Y. Bromberg, I. Afek and Y. Silberberg, in preparation.

Experimental ResultsExperimental Results

We can shapethe two-photoncorrelationfunction

Polarization Bell States Polarization Bell States

=Φ − )(

=Φ + )(

=Ψ − )(

=Ψ + )(

phase with LCC ππ step with SLM

Entanglement of signal (ω>ωp/2) and idler (ω<ωp/2) photons

Nonlinear Optics with Single Photons ?Nonlinear Optics with Single Photons ?

χ(2)

HOM correlations are nice, but HOM correlations are nice, but wouldnwouldn’’t it be nicer to havet it be nicer to havedirect detection of photons ?direct detection of photons ?

TPA ? SFG ?

Nonlinear Optics with Single PhotonsNonlinear Optics with Single Photons

SumSum--Frequency GenerationFrequency Generation

( ) ( ) ωωω dEEE ∫ −Ω∝Ω)(

Coincidence detection through Coincidence detection through SumSum--Frequency Generation (SFG)Frequency Generation (SFG)

CW PUMP χ(2)

SIGNAL (CW)

IDLER (CW)

χ(2)

910~ −1610~ −s 1310 −−≈ s×typical flux SFG efficiency SFG signal

Delay

Delay

13 -1max 10 s 2 !!WμΦ ≈ Δ ≈ ≈

ωh ωh

1 Δ

ωh ωh

1 Δ

ωh ωh

1 Δ

A photon-pair per time-bin

How many How many ‘‘single photonssingle photons’’ can arrive in one second ?can arrive in one second ?

(How high can (How high can ‘‘low light levelslow light levels’’ be ?)be ?)

n=1 photon per mode

The photon-pair arrives within 1/Δ

Down-convertingcrystal

SFGcrystal

pump 532nm5W

IR detector

Beam dump

SPCM

Dispersioncompensation

Computer

SFG with Entangled PhotonsSFG with Entangled Photons

PP-KTPPP-KTP

SFG 532nm~40,000 s-1

Quantum mechanical analysis of SFG

02

0 nnRSFG αα +∝

2I∝ I

0n - photons per mode

∝Entangled photonsClassical

1995: Kimble’s groupmeasures a slope of 1.3at low photon numbers

0

Intensity Dependence of SFG with Entangled Photons

0nα( )02

0 nn +α

"Nonlinear Interactions with an Ultrahigh Flux of Broadband Entangled Photons", B. Dayan, A. Pe’er, A.A. Friesem and Y. Silberberg, Phys. Rev. Lett. 94, 043602 (2005)

Intensity Dependence of SFG with Entangled Photons

"Nonlinear Interactions with an Ultrahigh Flux of Broadband Entangled Photons", B. Dayan, A. Pe’er, A.A. Friesem and Y. Silberberg, Phys. Rev. Lett. 94, 043602 (2005)

T

SFG

0

T

Intensity dependence of SFG with entangled photons

nα

"Nonlinear Interactions with an Ultrahigh Flux of Broadband Entangled Photons", B. Dayan, A. Pe’er, A.A. Friesem and Y. Silberberg, Phys. Rev. Lett. 94, 043602 (2005)

( )nn +2α

22Tnα

Down-convertingcrystal

up-convertingcrystal

Pump532nm

IR detector

Beam dump

SPCM

Computer

Shaping of Shaping of EntangeledEntangeled PhotonsPhotons

Fourierplane

SLM )(ωΦ

"Temporal Shaping of Entangled Photons",A. Pe’er, B. Dayan, A.A. Friesem and Y. Silberberg, Phys. Rev. Lett. 94, 073601 (2005)

Temporal shaping of the twoTemporal shaping of the two--photon wavefunctionphoton wavefunction

Down-convertingcrystal

SFGcrystal

Pump 532nm

IR detector

SPCM

Beam dump

Fourierplane

SLMComputer

Mach-Zehnderinterferometer

TwoTwo--photon interferencephoton interference

Two-photon interference

Two-photon interference

Mach-Zehnderinterferometer

ωh

B

Two-photon interference

ωh

B

: 1for B>>Δτ

%50

%50

Two-photon interference

ωh

: 1for B>>Δτ

ωhSFG

χ(2) ωhωhωh

Two-photon interference

ωh

: 1for B>>Δτ

ωh

B/1RT RT

Electronic detection is not fast enough,

χ(2)

…But SFG is !

Two-photon interference

: 1for B>>Δτ

ωhωh

B/1RR RR

Two-photon interference

: 1for B>>Δτ

ωhωh

TT TT

ωhωh

Two-photon interference

ωh

: 1for B>>Δτ

ωh

pumpδ

Background-free two-photon interference oscillations

Background-free two-photon interference oscillations

HOM Interference in a couplerHOM Interference in a coupler

In coupled waveguides there is a π/2 phase between light in adjacent waveguides

Discrete DiffractionDiscrete Diffraction

Shaped laser Beam

Slab waveguide

Linear propagation

2D core

Discrete diffraction

The 1d waveguide latticeThe 1d waveguide lattice

[ ] nnnnnnnnn UUUUCU

zUi 2

111, γβ +++=∂∂

−+±

• The discrete nonlinear Schrödinger equation (DNLSE)

Slab waveguide

2D corex

y

z

4 μm 8 μm

[ ]111, −+± ++=∂∂

− nnnnnnn TE

ti ψψψψ

• The Tight Binding Model (Discrete Schrödinger Equation)

Key Results:

Periodic Lattices:

Discrete Spatial Optical Solitons (1998)Diffraction Management, (2000).Self-Focusing and Defocusing, dark solitons (2001).Modulational Instability (2002)Vector Solitons (2003)Band Structure and Floquet-Bloch Solitons (2003). Gap solitons (2004)Surface states (2006)Spatio-temporal effects (x-waves) (2007)Quantum & Classical Correlations (2008)

Non-uniform arrays

Bloch Oscillations (1999).Defect States (1999).Binary Arrays (2004) Anderson Localization (2007)

Quantum Correlations in ArraysQuantum Correlations in Arrays

TwoTwo--Photon CorrelationsPhoton Correlations

Experiments with entangled photons in waveguide arrays are tough

But there is a simple classical version…

19561956

HB&T claim that they have measured the angular size of Sirius by intensity correlations

The HBT experimentThe HBT experiment

2)(

)()()2( )(xI

xxIxIxg δδ +=

D

How could photons generated at two different sides of a star interfere?

δx Corr.

I(x)

W

δx

g(2)

WLD // λδθ ≈=

HBT and QOHBT and QO

HBT was a key point in the development of quantum optics. It was later explained in terms of particle interference.

HBT has been demonstrated since with electrons, pions and matter waves. It reflects quantum statistics, leading to bunching (bosons) or anti-bunching (fermions).

Discrete HBTDiscrete HBT

Discrete HBT CorrelationsDiscrete HBT Correlations

101

11

Discrete HBT CorrelationsDiscrete HBT Correlations

101

11

Nonlinear Discrete HBT !Nonlinear Discrete HBT !

We have seenWe have seen……

• Photon Correlations behave much like short pulses

• Shaping of photon correlations

• SFG for photon correlation measurements

• Quantum correlations in periodic structures

UltrafastUltrafast Optics groupOptics group

www.weizmann.ac.il/~feyaron

Coherent Control:

Doron MeshulachNirit DudovichDan OronThomas PolackEvgeny FrumkerAdi NatanHaim SuchowskiBarry BrunerV Prabhudesai

Nonclassical Light:

Barak DayanAvi Pe’erItay Afek

Microscopy:

Yaniv BaradDvir YelinEran TalOri Katz

Optical Lattices:

Hagai EisenbergRoberto MorandottiDaniel MandelikAsaf AvidanYoav LahiniYaron BrombergGilad Tauber