• How to relate discrete energy levels with Hamiltonian described in terms of continгous coordinate x and momentum p?

• Acoustics: set of frequencies is associated with periodic solutions of linear differential equations

• Does solution Ψ(x) describe a motion of particle ?

• M. Born, L. Mandelstamm: |Ψ(x)|2 is probability density. Schrödinger equation determines the time evolution of statistical ensemble

Erwin Schrödinger and “his” cat

Erwin Schrödinger (1887 -1961)

0)(2

22

=Ψ−+Ψ∇ VEm Does the Ψ(x) yield the complete

description of the physical reality?

Erwin Schrödinger and “his” cat

Schrödinger Cat (1930 -Present)“Experimental setup”

( )1,2,2

1),( deadliveatomcat +=Ψ

Does the Ψ(x) yield the complete description of the physical reality?

Erwin Schrödinger and “his” cat

Schrödinger Cat (1930 -Present)“Experimental setup”

• A. Einstein: … Is the quantum state of a cat created upon a “measurement”? However, nobody doubts whether the cat state is something independent from measurement process.

• L. Mandelstamm: we can discuss statistics only when the ensemble is defined. For example we repeat the experiment under the same conditions

1,2,),( deadbliveaatomcat +=Ψ

?2

<<∆∆ px

Einstein-Podolsky-Rosen paradox (1935)

“Can Quantum-Mechanical Description of Physical Reality be Considered Complete?" Phys. Rev. 47 , 777–780 (1935).

0 x

21

21 )()(pp

txtx

−=−=

One of topics: EPR paradox in quantum optics

Detection of the momentum

Detection of the coordinate

Pump

Let’s try to understand what kind of media produces such correlated beams

Mind boggling experiments with entangled photons

2.12.2012

• EPR paradox and Completeness of QM

• Nonlinear optics and parametric downconversion

• HOM experiment

• Mind boggling experiments with entangled photons

1. Lorentz model of atomic oscillators

2. Non-linear oscillators

3. Real pump field

4. Non-linear term treated as small perturbation . First order (linear solution):

Non-linear optics: anharmonic oscillators

medium consists of atomic oscillators

mteExxx p /)(20 =++ ωγ

Ep(t) mteEaxxxx p /)(220 =+++ ωγ

( )titip eeEtE 11

2)( 1 ωω −+=

( )..~

./)(1

1

)1(21)1(

121

201

)1(

ccexx

ccimeeExti

tip

+=

+−−=−

−

ω

ω γωωωω)()(

...)( )3()2()1(

tNextPxxxtx

=+++=

Non-linear terms as perturbations:

Non-linear optics: anharmonic oscillators

5. Second order approximation

6. As we see non-linear polarization components on sum/difference and doubled frequencies has appeared

7. Let’s estimate the magnitude of non-linear polarization

..)2()0( 1)2()2()2( ccxxx ++= ω

8. Consider non-resonance case (resonance case is UV- excitation)

9. Estimate the non-linear coefficient “a”. Consider large displacements of an oscillator nonlinear and linear terms are an order of atomic field.

10. Estimation for “standard laser” and “standard atom”.

Non-linear optics: anharmonic oscillators

40

)1()2( /~/

)()(

ωmeaEPP

tNextP

p

=

( )

atom

atom

atom

EEPP

ameE

xmaxmeE

/~/

/~

~~

)1()2(

40

220

ω

ω ⋅

6)1()2(

8

10~/

/ 200~/ 102~

−

⋅

PP

cmVEcmVE

laser

atom

Non-linear optics hardware: crystals

http://www.eksmaoptics.com/en/c/nonlinear-crystals-42

Most of these crystals are known to be a piezo crystals

Non-linear optics hardware: lasers

• Ar ion laser. Gas discharge create inversion population

• It emits 8 lines

• Wavelengths 351 -514 nm

• Power up to 100 W

Second-harmonic generation and parametric down-conversion

Second harmonic generation Parametric down-conversion

nnn EP

EP

EP

EP

)(0

)(

3)3(0

)3(

2)2(0

)2(

)1(0

)1(

......................

χε

χε

χε

χε

=

=

=

=

( ) ( )[ ]ttEE

tEtEP

212121

)2(0

2211)2(

0)2(

coscos2

coscos

ωωωωχε

ωωχε

−++=

=××=

Non-linear polarization of medium (gas, crystal)

Parametric downconversion: phase matching

k1,ω1

ks,ωs

ki,ωi

Phase matching conditions:

ω1 = ωs + ωi

k1 = ks + ki

n(ω1) = n (ωs)

Generation of entangled photon pairs

• Type 1: Signal and Idler waves of identical polarization (LiNbO3, LiIO3)

• Type 2: Signal and Idler waves of different polarization (KTP, KDP, BBO)

( )

( )1221

2121

,,2

1

,,2

1

λλλλ ϕ

ϕ

iPDC

iPDC

e

e

+=Ψ

↑→+→↑=Ψ

Count rate: ~104 photons/sec

Single photon interferes with another one

PMT 1

Start

Stop

PMT 2

?

Interference of entangled photons

Look at HOM, Phys. Rev. Lett 59, 2044 (1987)

Single photon interference: HOM experiment

• Crystal KDP is pumped with Ar-laser line at 351 nm

• Two photon interference

• Coherence time is about 100 fs corresponding to coherence length of 30 μm

• Measuring of coincidence rate, or g(2)-function

C.Hong, Z.Ou, and L. Mandel, Phys. Rev. Lett 59, 2044 (1987)

1st and 2nd order interference effects

X. Zou et al., Phys. Rev A 41, 566 (1990)

V1 V2 pump amplitude (common pumps for 1 and 2)

Classical degree of coherence: pump mode-overlapping

Are s1 and s2 waves are mutually coherent?

[ ]( ))1(2,1

)1(2,121

2)()()()()2(

21)(

21)(

21

argcos1ˆˆˆˆ

)ˆˆ(ˆ , )ˆˆ(ˆ

2,1 , 1100

ggMMEEEEg

aiaEaiaE

kVM

ABBA

iiBssA

ikskkikskkj

+∆−∝ΨΨ∝

+∝+∝

⊗=Ψ

=+≈

++−−

++

θη

ψψ

ηψState bares phase information

Inability to retrieve “which-way” information results in interference

X. Zou et al., Phys. Rev A 41, 566 (1990)

• No single photon interference

• There is a two photon interference

• Interference of photon paths(can not distinguish where photon pair was created)

• Coincidence rate varies by scanning BS0

A. Einstein versus N. Bohr

• Einstein: Is it possible to observe interference pattern and at the same time know through which slit photon went through?

• Bohr: Yes, if we construct apparatus which determines the path of the photon, then interference vanishes

• Today: Interference pattern arises when it is impossible to obtain so called “which-way information”

Mind boggling experiment with two crystals

X. Zou et al., Phys. Rev. Lett. 67, 318 (1991)

• Type 1 parametric crystals, λi = 633 nm, λs = 789 nm

• Parametric pair originates wether in NL1 or in NL2.

• The paths for i1 and s1, and i2 and s2 are indistinguishable

• To destroy „wich way information“, ND-filter is introduced on i1

Mind boggling experiment: which way information

X. Zou et al., Phys. Rev. Lett. 67, 318 (1991)

• Without ND filter: clear interference fringes by measuring count rate

• ND filter diminishes the visibility

• Obtaining which way information is sufficient to destroy interference

Mind boggling experiment: which way information

L. Mandel et al., Rev. Mod. Phys. 71, S274 (1999)

Single photon interference: HOM experiment

• Generation of indistinguishable photons in type 2 parametric down-conversion

• Entangled state of the field

• What we do next? – Trying to defeat quantum mechanics by doing EPR-like experiments

• Will we succeed? …

( )1221 ,,2

1 λλλλ ϕiPDC e+=Ψ

EPR paradox with entangled photons

Detection of the momentum

Detection of the coordinate

• What would happen if we detect the coordinate for the photon 1 and the momentum for the photon 2?

• Will we break uncertainty relations?

• Let’s consider an experiment with entangled photon pairs

Pump

2~

~

/~

0

0

<<∆∆

≈∆

∆

axp

x

ap

λ

λαλ

A. Zeilinger, Rev. Mod. Phys. 71, S288 (2000)A. Zeilinger et al., Nature 433, 230 (2005)

• Type 1 parametric light

• Choose the detection mode in 1a. Image plane detection: <x>b. Focal plane detection: <p>

• Case b): A momentum eigenstate can not carry position information: Interference pattern for photon 2 is detected conditioned on registration of photon1.

• Case a): Fringes vanishes when photon 1 is projected on image plane.

• Neither of entangled photon possesses its own wave-function before the click of one of photo-detectors

Mind boggling experiment to test EPR

Conclusion of the part: Copenhagen interpretation

N. Bohr W. Heisenberg V. Fock L. Mandelstam

• In general case we can not assign neither momentum nor coordinate or polarization to the entangled photons before the measurement!

• Variables ( p,x ) has classical origin and can not characterize the field propagation, rather they relate measurement devices and quantum object. Consider a length in relativity theory, one has to define at least the reference frame.

• Comprehensive treatment is made with the full state vector. Even better to use density matrix.

Klyshko D. N., Sov. Phys. Usp. 31 74–85 (1988)

• Einstein: “We measure the system 1 without affecting the system 2”. Why is that?

• Where the system 2 has acquired its momentum? Apparently during its interaction with system 1. Therefore, by taking into account only specific momentum values of the system 2, we restrict the statistical ensemble. Wrong use of the probability theory.

• Systems 1 or 2 do not possess their own wave-function before the measurement

• Entangled state:

• Measurement on any system destroys the quantum correlations between them!

• Indeed, the wave-function of single system does not imply complete description of the reality

Entangled State and EPR paradox

( ))()()()(2

1)2,1( 2121 xxxx ψψψψ −+−=Ψ

0 x

• Principle of the experimental setup

• Type I process. Degenerate (the same wavelength) emission of photon pairs of the same polarizations.

• Arms are nearly equal and much larger then coherence length

• Analyzer: Michelson interferometer or Franson interferometer

• Observation of coincidence between different arms separated by 10 km

Mind boggling experiment with entangled pairs

W. Tittel et al. , Phys. Rev. A 57, 3229 (1998)

EPR source

Analyzer 2Analyzer 1D1 D2

Coincidence

~ 10 km

Look also in book of M. Scully

EPR source

δ1

D1 D2

Coincidence

mode a

( )( )( )

( )[ ]21*

)()()()(

cos141),(),(

1),(

1,1)()()()(0,0),(21

δδ

δδ

++∝ΨΨ=

+∝Ψ

++=Ψ+

++++

babaP

eba

bEbEaEaEba

c

i

longshortlongshort

δ2

mode b

• Interference of the paths: both long or both short

• The coincidence rate acquires total path phase shift δ1+δ2

Mind boggling experiment with entangled pairs

Observation of quantum correlations over 10 km distance

• 8 mW pump diode laser at 655.7 nm

• Type I non-linear crystal KNbO3

• Parametric waves at 1310 nm

• Michelson fiber interferometers

• Each single photon counter triggers lasers which send pulses to Geneva

• TPHC – time to pulse hight converter. Coincidence hystogramm.

W. Tittel et al. , Phys. Rev. A 57, 3229 (1998)

Optical loss in fibers

Observation of quantum correlations over 10 km distance

( )

+

−−+= 21

21 cos2

)(exp141 δδ

πδδλ

cc L

VP

EPR source

Interferometer 2Interferometer 1D1 D2

Coincidence

~ 10 km

• Beam from EPR source is divided 50/50

• δ1- δ2 ≈ 0, δi >> kLc . Single photon interference is excluded

• Entangled state: total phase shift affect the field propagation

• Term δ1+ δ2 due to two-photon interference of entangled pairs

• Detectors are widely separated and photon‘s trajectories do not mix

1δ 2δ

Observation of quantum correlations over 10 km distance

• Coherence length of photon is about 10 um. Much shorter then separation between them

• Observation of interference fringes (81% ): entanglement is not broken by large separation !W. Tittel et al. , Phys. Rev. A 57, 3229 (1998)