Erwin Schrödinger and his cat

Size: px

Start display at page:

Download "Erwin Schrödinger and his cat"

Geraldine Holland
5 years ago
Views:

Erwin Schrödinger (887-96) Acoustics: set of frequencies is associated with periodic solutions of linear differential equations m Ψ + (

1 Erwin Schrödinger and his cat How to relate discrete energy levels with Hamiltonian described in terms of continгous coordinate x and momentum p? Erwin Schrödinger (887-96) Acoustics: set of frequencies is associated with periodic solutions of linear differential equations m Ψ + ( E V ) Ψ = 0 Does the Ψ(x) yield the complete description of the physical reality? Does solution Ψ(x) describe a motion of particle? M. Born, L. Mandelstamm: Ψ(x) is probability density. Schrödinger equation determines the time evolution of statistical ensemble

2 Erwin Schrödinger and his cat Experimental setup Schrödinger Cat (930 -Present) Ψ( cat, atom) = + dead ( live,, ) Does the Ψ(x) yield the complete description of the physical reality?

3 Erwin Schrödinger and his cat Experimental setup Schrödinger Cat (930 -Present) 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. Ψ( cat, atom) = a live, + b dead, L. Mandelstamm: we can discuss statistics only when the ensemble is defined. For example we repeat the experiment under the same conditions

4 Einstein-Podolsky-Rosen paradox (935) Can Quantum-Mechanical Description of Physical Reality be Considered Complete?" Phys. Rev. 47, (935). x ( t) = x p = p ( t) x p <<? 0 x

5 One of topics: EPR paradox in quantum optics Detection of the coordinate Pump Detection of the momentum Let s try to understand what kind of media produces such correlated beams

6 ..0 Mind boggling experiments with entangled photons EPR paradox and Completeness of QM Nonlinear optics and parametric downconversion HOM experiment Mind boggling experiments with entangled photons

7 Non-linear optics: anharmonic oscillators. Lorentz model of atomic oscillators x + γ x + ω x ee p ( t) / m 0 = E p (t) medium consists of atomic oscillators Non-linear terms as perturbations: x( t) = x () + x P( t) = Nex( t) () + x (3) Non-linear oscillators x + γ x + ω x + ax ee p ( t) / m 3. Real pump field 0 = ( i ω t e + e i t ) E E t = ω p( ) 4. Non-linear term treated as small perturbation. First order (linear solution): x x () () ( ω ) = ee ~ () = x e p e iω t iω t / m + c. c. ( ω ω iω γ ) 0 + c. c

8 Non-linear optics: anharmonic oscillators 5. Second order approximation () () () x = x 0) + x (ω ) + c. c. ( 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

9 Non-linear optics: anharmonic oscillators 8. Consider non-resonance case (resonance case is UV- excitation) P( t) = Nex( t) P () / P () ~ eae p / mω Estimate the non-linear coefficient a. Consider large displacements of an oscillator nonlinear and linear terms are an order of atomic field. ee ee P atom atom () / P ~ ~ () mω x ( m / a) ~ 0 E / ~ ω 4 0 E ma x atom 0. Estimation for standard laser and standard atom. E E P atom laser () ~ 0 / P () 8 ~ 00 V / cm ~ 0 V / cm 6

10 Non-linear optics hardware: crystals Most of these crystals are known to be a piezo crystals

11 Non-linear optics hardware: lasers Ar ion laser. Gas discharge create inversion population It emits 8 lines Wavelengths nm Power up to 00 W

12 Second-harmonic generation and parametric down-conversion Second harmonic generation Parametric down-conversion P P P () () (3) = ε = ε E E E... ( n) ( n) P = ε χ E 0 = ε χ χ χ () () (3) Non-linear polarization of medium (gas, crystal) 3 n P () = ε ε 0χ = () 0 χ () E E E cosω t E [ cos( ω + ω ) t + cos( ω ω ) t] cosω t =

13 Parametric downconversion: phase matching k s,ω s k,ω Phase matching conditions: ω = ω s + ω i k i,ω i k = k s + k i n(ω ) = n (ω s )

14 Generation of entangled photon pairs Type : Signal and Idler waves of identical polarization (LiNbO 3, LiIO 3 ) Type : Signal and Idler waves of different polarization (KTP, KDP, BBO) ( ) ( ),,,, λ λ λ λ ϕ ϕ i PDC i PDC e e + = Ψ + = Ψ Count rate: ~0 4 photons/sec

15 Single photon interferes with another one Interference of entangled photons Look at HOM, Phys. Rev. Lett 59, 044 (987) PMT PMT Start Stop?

16 C.Hong, Z.Ou, and L. Mandel, Phys. Rev. Lett 59, 044 (987) Single photon interference: HOM experiment Crystal KDP is pumped with Ar-laser line at 35 nm Two photon interference Coherence time is about 00 fs corresponding to coherence length of 30 μm Measuring of coincidence rate, or g () -function

17 st and nd order interference effects V V pump amplitude (common pumps for and ) Are s and s waves are mutually coherent? State bares phase information ψ j M k 0 sk 0 ik + ηv k sk ik, k =, Ψ = ψ ψ Eˆ ( + ) A ( aˆ s + iaˆ s ), Eˆ ( + ) B ( aˆ i + iaˆ i ) g () Ψ Eˆ ( ) A Eˆ ( ) B Eˆ ( + ) B Eˆ ( + ) A Ψ η () [ θ + arg g ] ( ) () M M g cos,, Classical degree of coherence: pump mode-overlapping X. Zou et al., Phys. Rev A 4, 566 (990)

18 Inability to retrieve which-way information results in interference 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 BS 0 X. Zou et al., Phys. Rev A 4, 566 (990)

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?

19 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

20 Mind boggling experiment with two crystals Type parametric crystals, λ i = 633 nm, λ s = 789 nm Parametric pair originates wether in NL or in NL. The paths for i and s, and i and s are indistinguishable To destroy wich way information, ND-filter is introduced on i X. Zou et al., Phys. Rev. Lett. 67, 38 (99)

21 Mind boggling experiment: which way information Without ND filter: clear interference fringes by measuring count rate ND filter diminishes the visibility Obtaining which way information is sufficient to destroy interference X. Zou et al., Phys. Rev. Lett. 67, 38 (99)

22 Mind boggling experiment: which way information L. Mandel et al., Rev. Mod. Phys. 7, S74 (999)

23 Single photon interference: HOM experiment Generation of indistinguishable photons in type parametric down-conversion Entangled state of the field Ψ PDC = iϕ ( λ, λ + e λ λ ), What we do next? Trying to defeat quantum mechanics by doing EPR-like experiments Will we succeed?

24 EPR paradox with entangled photons Detection of the coordinate Pump p ~ / a0 λ x ~ λ α λ p x ~ << a 0 Detection of the momentum What would happen if we detect the coordinate for the photon and the momentum for the photon? Will we break uncertainty relations? Let s consider an experiment with entangled photon pairs

25 Mind boggling experiment to test EPR Type parametric light Choose the detection mode in a. Image plane detection: <x> b. Focal plane detection: <p> Case b): A momentum eigenstate can not carry position information: Interference pattern for photon is detected conditioned on registration of photon. Case a): Fringes vanishes when photon is projected on image plane. Neither of entangled photon possesses its own wave-function before the click of one of photodetectors A. Zeilinger, Rev. Mod. Phys. 7, S88 (000) A. Zeilinger et al., Nature 433, 30 (005)

Conclusion of the part: Copenhagen interpretation N. Bohr W. Heisenberg V.

Mandelstam In general case we can not assign neither momentum nor

Variables ( p,x ) has classical origin and can not characterize the field

Consider a length in relativity theory, one has to define at least the

26 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 (988)

27 Entangled State and EPR paradox Einstein: We measure the system without affecting the system. Why is that? Where the system has acquired its momentum? Apparently during its interaction with system. Therefore, by taking into account only specific momentum values of the system, we restrict the statistical ensemble. Wrong use of the probability theory. Systems or do not possess their own wave-function before the measurement Entangled state: 0 x Ψ(,) = ( ψ ( x) ψ ( x) + ψ ( x) ψ ( )) x 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

28 Mind boggling experiment with entangled pairs Principle of the experimental setup ~ 0 km D Analyzer Analyzer D EPR source Coincidence 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 0 km W. Tittel et al., Phys. Rev. A 57, 39 (998)

29 Mind boggling experiment with entangled pairs D δ EPR source δ D mode a mode b Ψ( a, b) Ψ( a, b) = 0,0 + e Coincidence ( + ) ( + ) ( + ) ( + ) ( E ( a) + E ( a) )( E ( b) + E ( b) ) i ( δ + δ ) short long short long, Interference of the paths: both long or both short The coincidence rate acquires total path phase shift δ +δ P c = Ψ * ( a, b) Ψ( a, b) 4 [ + cos( δ + δ )] Look also in book of M. Scully

Observation of quantum correlations over 0 km distance 8 mw pump diode laser at 655.

30 Observation of quantum correlations over 0 km distance 8 mw pump diode laser at nm Type I non-linear crystal KNbO 3 Parametric waves at 30 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, 39 (998)

31 Optical loss in fibers

32 Observation of quantum correlations over 0 km distance ~ 0 km D Interferometer Interferometer D δ δ EPR source P c + V 4 Coincidence λ( δ δ ) exp cos πlc ( δ + δ ) = Beam from EPR source is divided 50/50 δ - δ 0, δ i >> kl c. Single photon interference is excluded Entangled state: total phase shift affect the field propagation Term δ + δ due to two-photon interference of entangled pairs Detectors are widely separated and photon s trajectories do not mix

33 Observation of quantum correlations over 0 km distance Coherence length of photon is about 0 um. Much shorter then separation between them Observation of interference fringes (8% ): entanglement is not broken by large separation! W. Tittel et al., Phys. Rev. A 57, 39 (998)

Timeline: Bohm (1951) EPR (1935) CHSH (1969) Bell (1964) Theory. Freedman Clauser (1972) Aspect (1982) Weihs (1998) Weinland (2001) Zeilinger (2010)

Timeline: Bohm (1951) EPR (1935) CHSH (1969) Bell (1964) Theory. Freedman Clauser (1972) Aspect (1982) Weihs (1998) Weinland (2001) Zeilinger (2010) 1.EPR paradox 2.Bohm s version of EPR with spin ½ particles 3.Entangled states and production 4.Derivation of CHSH inequality - S parameter for mixed and entangled state 5. Loopholes 6.Experiments confirming