quantum nonlinear optics maria chekhova max … · strong amplification along the pump poynting...
TRANSCRIPT
1
QUANTUM NONLINEAR OPTICS
Maria Chekhova
Max-Planck Institute for the Science of Light, Erlangen, Germany
M.V.Lomonosov Moscow State University
Max-Planck Institute for the Science
of Light
Quantum Radiation group
2
OUTLINE
1. Nonlinear effects producing quantum light - Parametric down-conversion at low and high gain - Spontaneous four-wave mixing - Kerr-effect squeezing - Generation of photon triplets
2. Quantum light producing nonlinear effects - up-conversion of single photons - seeding PDC with single photons - two-photon effects with squeezed vacuum 3. Both: nonlinear interferometers 4. Conclusions
3
OUTLINE
1. Nonlinear effects producing quantum light - Parametric down-conversion at low and high gain - Spontaneous four-wave mixing - Kerr-effect squeezing - Generation of photon triplets
2. Quantum light producing nonlinear effects - up-conversion of single photons - seeding PDC with single photons - two-photon effects with squeezed vacuum 3. Both: nonlinear interferometers 4. Conclusions
4
NONLINEAR OPTICS
Expansion of the polarization in powers of the field:
ann E
EEEEEEP/~
...)()1(
)3()2()1(
χχχχχ
+
+++=!!!!!!!
)3()2( ,χχ
Second-harmonic generation
Parametric down-conversion
Third harmonic generation
Three-photon PDC
Kerr effect Four-wave
mixing
5
QUANTUM LIGHT
Nonclassical states of light Their theoretical description Their measurement in experiment
In the lecture by Jeff Lundeen
Here, only - single-photon Fock states - two-photon Fock states - three-photon Fock states - squeezed states and squeezed vacuum
8
FOR A THEORETICIAN: A HAMILTONIAN…
Two photon creation operators –> creation of photon pairs – biphotons.
...
,)(~ˆ
)3()2()1(
3
+++=
= ∫EEEEEEP
EPdEdH!!!!!!!
!!!!
χχχ
rr
EEEH!!!
~ˆ ..~ˆ chbaEH p +++
(2)χ
E!
d!
..)(~ˆ 2 chaEH p ++
9
PHASE MATCHING CONDITIONS
s ..~ˆ chaaEH isp +++ pump
i
}exp{~
}exp{~
}exp{~
tirkia
tirkia
tirkiE
iii
sss
ppp
ω
ω
ω
−
−
+−
+
+
!!!!
!!
...})()(exp{~ +−−+−−− tirkkkiH ispisp ωωω!!!!
Fast oscillation in space
Fast oscillation in time
ispisp kkk ωωω +=+= ,!!!
10
…OR NONCOMMUTATIVITY OF a AND a+
LEchaaiH pis)2(~.,.ˆˆˆ χΓ+Γ= ++!
++++
++++
+++=
=++=
002
0000002
0000
ˆˆˆˆˆˆˆˆ)ˆˆ()ˆˆ(ˆˆ
iiisisss
isisss
aaVaaUVaaUVaaUaVaUaVaUaa
)(sinhˆˆ,0ˆˆ 22`00`0 tVaaNaaN ssssss Γ==≡=≡ ++
)sinh(),cosh(,ˆˆˆ,ˆˆˆ
00
00
tVtUaVaUaaVaUa
sii
iss
Γ=Γ=+=
+=+
+ Output-input relations = Bogoliubov transformations
Nonzero output photon number with nothing at the input
++++
++++
+++=
=++=
002
0000002
0000
ˆˆˆˆˆˆˆˆ
)ˆˆ()ˆˆ(ˆˆ
iiisisss
isisss
aaVaaUVaaUVaaU
aVaUaVaUaa
]ˆ,ˆ[ˆ
],ˆ,ˆ[ˆ
Haadtdi
Haadtdi
ii
ss
=
=
!
!
11
FOR AN EXPERIMENTALIST: NOISE OF AN OPA
Optical parametric amplifier, a device known from the 1960-s
(2)χpump
signal signal idler
pump
signal idler
wi=wp-ws
Spontaneous parametric down-conversion
S.E.Harris, M.K.Oshman, and R.L.Byer, PRL 18, 732 (1967); D.Magde and H.Mahr, PRL 18, 905 (1967); S.A.Akhmanov, V.V.Fadeev, R.V.Khokhlov, and O.N.Chunaev, JETP Lett. 6, 575 (1967)
(Never look into the laser beam!)
12
LOW PARAMETRIC GAIN
(2)χpump
1)(sinhˆ 2
)2(
<<Γ==
∝Γ+ taaN
LE
sss
pχ
1,100)ˆˆ1()0()(
0)0(,ˆ
..ˆˆˆ
ˆ1
taatet
Hdtdi
chaaiH
is
dtHi
is
Γ+=Γ+≈Ψ=Ψ
=ΨΨ=Ψ
+Γ=
++∫
++
!
!
! Generation of photon pairs - biphotons
signal
idler
Single-photon Fock states
2200))ˆ(1()0()(
0)0(,ˆ
..)ˆ(ˆ
2ˆ1
2
tatet
Hdtdi
chaiH
s
dtHi
s
Γ+=Γ+≈Ψ=Ψ
=ΨΨ=Ψ
+Γ=
+∫
+
!
!
!
Two-photon Fock states
signal
13
ENTANGLEMENT
S
QWP QWP
PBS PBS
A B
Polarization-entangled photon pairs
BABABAHVVH ΨΨ≠−=Ψ − )(
21)(
Polarization of each photon is uncertain but correlated with the polarization of the other one.
14
TRANSVERSE (DIRECTIONAL) ENTANGLEMENT
c(2) Dq dq
Entanglement: Dq>>dq
Uncertainty for one subsystem plus correlations between the two subsystems
qi
qs
Spontaneous parametric down-conversion
19
BRIGHT SQUEEZED VACUUM
(2)χpump
collinear degenerate PDC: degenerate BSV
1)(sinhˆ 2
)2(
>>==
∝Γ≡+ GaaN
LEtG pχ
q
p (2)χpump s
i
collinear nondegenerate PDC: twin-beam BSV
Bright
(2)χpump s
i
noncollinear degenerate PDC: twin-beam BSV
(2)χpump si
collinear degenerate type-II PDC: twin-beam BSV
132 102)16(sinh ⋅≈
20
WHY ‘SQUEEZED VACUUM’?
+
+
Γ=
=
Γ+Γ=
adtad
Hadtadi
LEchaiH p
ˆ2ˆ
]ˆ,ˆ[ˆ
~.,.)ˆ(ˆ )2(2
!
! χ
pdtpdq
dtqd
piqa
ˆ2ˆ
,ˆ2ˆ
ˆˆˆ
Γ−=Γ=
+=
tt eptpeqtq Γ−Γ == 22 )0(ˆ)(ˆ,)0(ˆ)(ˆ
21
WHY ‘SQUEEZED VACUUM’?
t
t
eptpeqtq
Γ−
Γ
==
2
2
)0(ˆ)(ˆ,)0(ˆ)(ˆ
EapEaq
Im)ˆIm(ˆRe)ˆRe(ˆ
→=→=
EIm
ERe
Quantum vacuum
p
q
2/)0(ˆ)0(ˆ !=Δ=Δ pq
constpqtptq == )0(ˆ)0(ˆ)(ˆ)(ˆ
Squeezed vacuum
The ellipse rotates with time
Squeezed coherent state
22
BRIGHT SQUEEZED VACUUM
‘Macroscopic’ state
Interactions with matter (atoms, mechanical systems, ...), with itself (nonlinear optics)
Degenerate BSV: Quadrature squeezing, superbunching N
gN
Ng ˆ
13;ˆ
:ˆ:)2(
deg2
2)2( +=≡
Nondegenerate BSV: photon-number correlations = noise reduction
is
is
NNNNNRF ˆˆ)ˆˆ(Var
+−≡
23
NOISE REDUCTION
Twin-beam squeezed vacuum
(2)χpump s
i
Photons are always born in pairs:
is NN =
0)(Var =− is NNIdeally, This is very unusual!
2121 )(Var NNNN +=−
For a split coherent beam,
2N1N
Shot noise
24
SHOT NOISE
2121 )(Var NNNN +=−
For a split coherent beam,
2N1N
Shot noise
2N1N
Light does consist of photons!
Poissonian distribution: NN
NN
=Δ
=)(Var SNL: shot-noise level
25
POLARIZATION ENTANGLEMENT OF BSV
S
QWP QWP
PBS PBS
BA
m
m n
n
T.Sh.Iskhakov et al., PRL 109, 150602 (2012).
31
THE HAMILTONIAN
The same type of Hamiltonian as for PDC
...
,)(~ˆ
)3()2()1(
3
+++=
= ∫EEEEEEP
EPdEdH!!!!!!!
!!!!
χχχ
rr
EEEEH!!!!
~ˆ ..~ˆ 2 chaaEH isp +++
(3)χ
32
SPDC VERSUS SFWM
..2)3( chaLaEH isp +∝ ++χ
pk pk
ik sk
2, pis II ∝
ispisp kkk ωωω +=+= 2,2!!!
pk
ik sk
..)2( chaLaEH isp +∝ ++χ
pis II ∝,
ispisp kkk ωωω +=+= ,!!!
Spontaneous four-wave mixing
Spontaneous parametric down-conversion
33
SFWM IN FIBRES
Advantages: - single spatial mode;
- possibility to increase parametric gain via focusing;
- possibility to engineer dispersion dependence;
- integrability into fibre networks
Disadvantages: - Raman scattering;
- difficulties with eliminating the pump
34
TYPES OF PHASE MATCHING
.2,22 2
effisp A
nPkkkλπγγ ≡++=,2 isp ωωω +=
W. J. Wadsworth, N. Joly, J. C. Knight, T. A. Birks, F. Biancalana, P. St. J. Russell, Optics Exp. 12, 299 (2004).
Possible in any fibres
Requires strong higher-order dispersion: possible in PCF
35
USUAL SINGLE-MODE FIBRES
Signal (Stokes) and idler (anti-Stokes) wavelengths are very close to the pump one, which is at zero dispersion.
X.Li, J.Chen, P.Voss, J.Sharping, and P.Kumar, Optics Express 12, 3737 (2004).
0.5 THz detuning
36
PHOTONIC-CRYSTAL FIBRES (PCF)
Signal and idler frequencies can be much separated
J.Fan, A.Migdall, L.-J.Wang, Optics Lett. 30, 3368 (2005).
100 nm separation
37
PCF, NORMAL DISPERSION RANGE
J.G.Rarity, J.Fulconis, J. Duligall, W. J. Wadsworth, and P. St. J. Russell, Optics Exp. 13, 534 (2005).
570 nm separation
39
THE HAMILTONIAN: THE SAME AS FOR SFWM
...)3()2()1( +++= EEEEEEP!!!!!!!
χχχ ..)(~ˆ 22 chaEH p ++
p
q
Squeezing in a certain quadrature
)(Inn =Kerr effect
C. Silberhorn et al., PRL 86, 4267 (2001).
40
SIDEBANDS
The effect takes place in anomalous-dispersion range (modulation instability)
λ
)(λS
signal idler
pump
Signal (anti-Stokes) and idler (Stokes) sidebands are nearly within the pump bandwidth.
p
q
pump (carrier)
sidebands (squeezed vacuum)
41
POLARIZATION SQUEEZING IN FIBRES
p
q
Polarization-maintaining fibre
H H
V
V
Polarization-squeezed light
J.Heersink, V.Josse, G. Leuchs, and U.Andersen, Optics Lett. 30, 1192 (2005).
42
POLARIZATION SQUEEZING IN FIBRES
J.Heersink, V.Josse, G. Leuchs, and U.Andersen, Optics Lett. 30, 1192 (2005).
22jki SSS Δ<<Δ
43
POLARIZATION SQUEEZING IN FIBRES
The sidebands are within the pump bandwidth-> no local oscillator is needed to see squeezing.
Further, polarization entanglement can be demonstrated.
45
THE HAMILTONIAN
...
,)(~ˆ
)3()2()1(
3
+++=
= ∫EEEEEEP
EPdEdH!!!!!!!
!!!!
χχχ
rr..~ˆ chaaaEH risp ++++
1,1110 <<+=Ψ ccris
1,30 <<+=Ψ cc
Or, at degeneracy, three-photon Fock state
(3)χ
46
WHY IT IS INTERESTING
New nonlinear effect – not observed yet
Non-Gaussian squeezing
..)(~ˆ 3 chaH ++
GHZ state: Bell paradox without inequalities
[ ]risris
VVVHHH +=Ψ21
p
q
Heralded generation of photon pairs
(3)χ2
47
PHOTON TRIPLETS: ATTEMPTS
Cascaded PDC in two crystals: asymmetric states in 3 beams [1]
[1] H.Hübel et al., Nature 466, 601 (2010). [2] J. Douady and B. Boulanger, Opt. Lett. 29, 2794 (2004). [3] M. Corona, K. Garay-Palmett, and A. B. U’Ren, Opt. Lett. 36, 190 (2011).
Direct 3-photon PDC in a nonlinear crystal [2]
Direct 3-photon PDC in an optical fibre using inter-modal dispersion [3]
49
OUTLINE
1. Nonlinear effects producing quantum light - Parametric down-conversion at low and high gain - Spontaneous four-wave mixing - Kerr-effect squeezing - Generation of photon triplets
2. Quantum light producing nonlinear effects - up-conversion of single photons - seeding PDC with single photons - two-photon effects with squeezed vacuum 3. Both: nonlinear interferometers 4. Conclusions
50
NONLINEAR OPTICS WITH QUANTUM LIGHT
Obvious problem: quantum states are faint. Are they?
What is ‘faint’?
ω!cohN E≡mode
Coherence volume=mode volume
cohl 2cohρ
1mode <<N
faint
1mode >>N
bright
Bright light is efficient for nonlinear interactions
51
PHOTON NUMBER PER MODE FOR VARIOUS STATES
Heralded single photons 1mode =N
Bright squeezed vacuum 13mode 10...10=N
Kerr squeezed light 6mode 10~N
52
FREQUENCY UP-CONVERSION
lphlph kkk!!!
+=+= ,ωωω
LEGGNNGNN plllh)2(2
02
0 ~,cos,sin χ==
lk!
pk!
hk!
L
0
2lh NN
G
=
= π Every photon is up-converted G
hNlN
pω hωlωω
I
0
53
UP-CONVERSION OF SINGLE PHOTONS
M. A. Albota and F. N.C. Wong, Opt. Lett. 29,1449 (2004).
90% efficiency of single-photon up-conversion
PPLN: larger L, higher nonlinear tensor component
54
UP-CONVERSION OF TWIN BEAMS
J. Huang and P. Kumar, PRL 68,2153 (1992).
Noise reduction survives up-conversion!
Every photon is up-converted
KTP2
532 nm 1064 nm
1064 nm H
V
KTP1 532 nm
NRF<1
NRF<1
55
UP-CONVERSION OF SINGLE PHOTONS: IR DETECTORS
Telecom range (1320 and 1550 nm): InGaAs APD
High level of dark counts and afterpulsing -> gating needed; QE<30%
Si APD: response times as short as 40 ps; QE up to 70%; dark counts below 10 Hz
Idea: up-convert IR single photons into visible range and then detect
State of the art: ~10%QE, room temperature operation, tunability, photon-number resolution
56
PARAMETRIC AMPLIFICATION OF SINGLE PHOTONS
pωsωiωω
I
ispisp kkk!!!
+=+= ,ωωω
ik!
pk!
sk!
L
LEG p)2(~ χ
02
02
0 sinh)1(,sinh)1( iiiis NGNNGNN ++=+=
GNNNG sii2
0 sinh2:1,1 =≈=>>
Spontaneous Stimulated
A single photon seeding an OPA: two-fold increase compared to spontaneous case
57
LOW GAIN: PHOTON ADDITION
tG
aGaaGtchaaiH
iisisisis
is
Γ≡
Ψ+Ψ=Ψ+≈Ψ
+Γ=+++
++
,ˆ100)ˆˆ1()(
..ˆˆˆ !
Ψ 1<<G
Ψ+a
1
A. Zavatta, S. Viciani, M. Bellini, Science 306, 660 (2004)
A single photon at the OPA output heralds a photon added to the conjugate mode!
58
TWO-PHOTON EFFECTS WITH SQUEEZED VACUUM
Two-photon effects:
- two-photon absorption
- two-photon ionization
- second harmonic generation
2)2( ~ NN
2)2()2( ~ NgN
Ng 13)2( +=Single-mode
squeezed vacuum: (2)χpump SV
NN ~)2(Low gain:
2)2( ~ NNHigh gain: N
)2(N
1 photon/mode
NNN +2)2( 3~
59
SHG WITH TWO-PHOTON LIGHT
B. Dayan et al., PRL 94, 043602 (2005)
Attenuation of the two-photon beam: quadratic dependence
Attenuation of the pump: linear dependence
60
OUTLINE
1. Nonlinear effects producing quantum light - Parametric down-conversion at low and high gain - Spontaneous four-wave mixing - Kerr-effect squeezing - Generation of photon triplets
2. Quantum light producing nonlinear effects - up-conversion of single photons - seeding PDC with single photons - two-photon effects with two-photon light 3. Both: nonlinear interferometers 4. Conclusions
61
NONLINEAR INTERFEROMETERS
Nonlinear crystal 1
Nonlinear crystal 2
pump
B. Yurke, S.L. McCall, and J.R. Klauder, PRA 33, 4033 (1986)
Nonlinear Mach-Zehnder interferometer
Linear Mach-Zehnder interferometer
62
SHG INTERFEROMETRY
K.Kemnitz et al., Chemical Physics Letters 131, 285 (1986)
Measurement of surface c(2) including the phase Fringes: due to the interference between the reference sample (quartz) and the surface contribution
63
PDC INTERFEROMETRY
Nonlinear crystal 1
Nonlinear crystal 2
pump
idler
signal
Low-gain PDC High-gain PDC
64
PDC INTERFEROMETRY: INDUCED COHERENCE
Nonlinear crystal 1
Nonlinear crystal 2
pump
idler
signal
signal2
signal1
signal1 signal2
L.J. Wang, X.Y. Zou, and L. Mandel, PRA 44, 7 (1991)
interference
Measurement of absorption and dispersion; imaging
65
PDC INTERFEROMETRY: SUPERSENSITIVITY
Nonlinear crystal 1
Nonlinear crystal 2
pump idler
signal
A.M. Marino, N. V. Corzo Trejo, and P. D. Lett, PRA 86, 023844 (2012)
ϕ
N1~ϕΔ Heisenberg limit
To be compared with the standard quantum limit (in a linear interferometer) N
1~ϕΔ
66
PDC INTERFEROMETER: A SPATIALLY SINGLE-MODE SOURCE OF BSV
cLLa+
Δ ~θ
When the crystals are spatially separated, only a small solid angle of BSV from the first crystal is amplified in
the second one.
2
⎟⎠⎞⎜
⎝⎛ Δ≈δθθmThe number of modes
Angular width of correlations
67
SPATIALLY SINGLE-MODE SOURCE: RESULTS
a
0.01rad
0 2 4 6 8 10 12 14 16 18
0.0
0.2
0.4
0.6
0.8
1.0
inte
nsity
,arb
.uni
ts
distance, cm
b0 2 4 6 8 10 12 14 16 18
1.2
1.3
1.4
1.5
1.6
1.7
1.8
g(2)
distance, cm
a
b
0 2 4 6 8 10 12 14 16 18
0.0
0.1
0.2
0.3
0.4
0.5
0.6
inte
nsity
,arb
.uni
ts
distance, cm
0 2 4 6 8 10 12 14 16 18
1.2
1.3
1.4
1.5
1.6
1.7
1.8
g(2)
distance, cm
A. Perez et al., Optics Letters 39, 2403 (2014).
Frequency filtering (monochro-mator): 1.25 frequency modes.
Angular structure: 1.1 angular modes
Experiment Theory