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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