quantum nonlinear optics maria chekhova max … · strong amplification along the pump poynting...

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



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

6

PARAMETRIC DOWN-CONVERSION

7

PARAMETRIC DOWN-CONVERSION

A rope!

A snake!

A tree!

A leaf!

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


15

FREQUENCY ENTANGLEMENT

λ

)(λS

signal idler

sλ

iδλ

iλΔ

1>>Δδλλ entanglement

16

EINSTEIN-PODOLSKY-ROSEN ‘GEDANKENEXPERIMENT’

Entangled state

S

A B

00

=+=−

BA

BA

ppxx!!"!

17

EPR ‘GEDANKENEXPERIMENT’

S A

B pp /⊥=θ

θ

momentums coordinates

BA

BA

ppxx!!!!

,,

x

18

FIRST OBSERVATION OF TRANSVERSE ENTANGLEMENT

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).

26

NONLINEAR AMPLIFICATION AND ANISOTROPY

Transverse walk-off

Optic axis

ordinary

extraordinary

27

LOW-GAIN PDC

pump

Only slight asymmetry observed in the angular spectrum

28

HIGH-GAIN PDC

pump

Preferable emission

Strong amplification along the pump Poynting vector!

29

TUNING THE PHASE MATCHING

Amplification of green

red

yellow

Visualization of the Poynting vector

30

SPONTANEOUS FOUR-WAVE MIXING

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


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

38

KERR SQUEEZING IN FIBRES

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


J.Heersink, V.Josse, G. Leuchs, and U.Andersen, Optics Lett. 30, 1192 (2005).

22jki SSS Δ<<Δ

43


The sidebands are within the pump bandwidth-> no local oscillator is needed to see squeezing.

Further, polarization entanglement can be demonstrated.

44

GENERATION OF PHOTON TRIPLETS

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]

48

QUESTIONS?

49

OUTLINE



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


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

68

CONCLUSION:

THERE IS MUCH INTERESTING AT THE BOUNDARY BETWEEN

QUANTUM AND NONLINEAR OPTICS!

69

THANK YOU!

70

WHY NOISE BELOW SNL IS NONCLASSICAL?

)2(2)ˆˆ(Var )2()2()2(2siiissis gggNNNN −++=−

02 )2()2()2( ≥−+ siiiss ggg Cauchy-Schwarz inequality

1≥NRF for classical sources 0=NRF for SV

Noise reduction factor is

is

NNNNNRF ˆˆ)ˆˆ(Var

+−≡

NNN is == ˆˆLet s

i

quantum nonlinear optics maria chekhova max … · strong amplification along the pump poynting...

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