2. Niels Bohr Institute Copenhagen University Eugene PolzikLECTURE 3

3. Einstein-Podolsky-Rosen (EPR) entanglement Experimental techniques for observing quantum noise and entanglement Light Photon statistics – dc measurements Entanglement in the spectral modes of light – ac measurements Atoms Addition of magnetic field – ac measurements with atoms Generation of Entangled state of two distant Atomic Objects

4. Einstein-Podolsky-Rosen paradox – entanglement; 1935 2 particles entangled in position/momentum L Simon (2000); Duan, Giedke, Cirac, Zoller (2000) Necessary and sufficient condition for entanglement

5. 2 particles entangled in position/momentum L What does it mean in practice? 1.Prepare many identical pairs of particles 2.Measure X 1 - X 2 on some of those pairs 3.Measure P 1 +P 2 on others 4.Plot statistical distributions of the results 5.Measure the width of these distributions L X 1 - X 2 0 P 1 + P 2  (X 1 - X 2 )  (P 1 + P 2 )

6. Why does it make sense? Two independent particles Minimal symmetric uncertainties 0 P 1 + P 2  (P 1 + P 2 )= L X 1 - X 2  (X 1 - X 2 )= < Entangled state

7. Position – momentum uncertainty The best optical interferometry measurement: Macroscopic object – a mirror Assume M=1mg

8. Instead of two EPR particles we use Two beams of light 1998 Two atomic ensembles 2001

9. Coherent state of light. Poissonian photon statistics. Delta-correlated noise. Probability of counting n photons Variance: Compare to For coherent state

10. (vacuum units) Strong field A(t) Polarizing cube Polarizing cube 45 0 /-45 0 0 X coherent state 1 -1

11. Phase squeezed Amplitude squeezed Quadrature operators. Squeezed light

12. Single photon state

13. x Polarization beamsplitter Spectral measurements of Quadrature Operators Homodyne detector RF spectrum analyzer

14. x Polarization beamsplitter Spectral measurements on vacuum state (coherent state) Homodyne detector RF spectrum analyzer vacuum Delta correlated noise Flat spectrum of noise

15. In real world Flat spectrum of noise Technical (classical) noise Quantum noise n > n > n > n

16. Spectral measurements of Quadrature Operators Spectrum analyzer Advantage: getting rid of technical noise Same for sine modes 1 MHz

17. Optical Spectrum of the strong field and the quantum field

18. 0,00,20,40,60,81,01,21,41,61,82,0 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 Atomic Quantum Noise Atomic noise power [arb. units] Atomic density [arb. units] y z Measuring rotating spin states

19. 300000310000320000330000340000 0,0000 0,0002 0,0004 0,0006 0,0008 Shot noise level Noise power [arb. units] Frequency (Hz) Density [arb. units] ---------------------------------- 1.00±0.02 0.56±0.01 0.21±0.01, Probe polarization noise spectrum Larmor frequency  320kHz Detecting quantum projections of the spin in rotating frame Atomic density (a.u.) 1.0 0.5 0.2 RF frequency z B probe y

21. Einstein-Podolsky-Rosen paradox – entanglement; 1935 2 particles entangled in position/momentum L Simon (2000); Duan, Giedke, Cirac, Zoller (2000) Necessary and sufficient condition for entanglement

22. 10 12 spins in each ensemble yz x yz x Spins which are “more parallel” than that are entangled Experimental long-lived entanglement of two macroscopic objects. B. Julsgaard, A. Kozhekin and EP, Nature, 413, 400 (2001)

23. X Z or Y 2nd 1st If the two macroscopic spins are collinear they must be entangled: Compare

24. Stern-Gerlach projection on any axis to x: 1 21+2 JJ J Along y,z: ideally no misbalance between heads and tails of the two ensembles, or, at least, less than random misbalance

25. 10 12 atoms in each ensemble 2 gas samples 6S 1/2 Cesium )3,...,3(  m 4 3

26. Total z and y components of two ensembles with equal and opposite macroscopic spins can be determined simultaneously with arbitrary accuracy xx y z z Therefore entangled state with Can be created by a measurement Top view: Parallel spins must be entangled

27. How to measure the total spin projections? Send off-resonant light through two atomic samples Measure polarization state of light Duan, Cirac, Zoller, EP 2000

28.   pump   pump Y Z Z Y Entangling beam Polarization detection Entangled state of 2 macroscopic objects J1J1 J2J2 B B

30. Establishing the entanglement bound Two independent ensembles Minimal symmetric uncertainties 0 Jy 1 + Jy 2 0 Jz 1 + Jz 2

31. S p e c t r a l v a r i a n c e o f t h e p r o b e p u l s e Collective spin of the atomic sample J x [10 ] 0246 0 10 20 30 40 12 CSS Entanglement criterion: =

32. 0 Jy 1 + Jy 2 0 Jz 1 + Jz 2 Entangled state < Proving the entanglement condition:

33. B-field PBS Time Verifying pulse Entangling pulse 0.5 ms m=4 700MHz 6S 6P Entangling and verifying beams S Entangling and verifying pulses F=3 F=4  = 325kHz m=4 1/2 3/2 y out x2 Optical pumping Pumping beams J  x1 + J  - 0,00,20,40,60,81,01,21,41,61,82,0 0,0 0,2 0,4 0,6 Atomic density [arb. units]

34. Entangled spin states Create entangled state and measure the state variance Julsgaard, Kozhekin, EP Nature 413, 400 (2001).

35. Material objects deterministically entangled at 0.5 m distance Niels Bohr Institute December 2003 0246810121416 0,70 0,75 0,80 0,85 0,90 0,95 1,00 10-12- 2003/noise.opj Atom/shot(comp) / PN Mean Faraday angle [deg] Quantum uncertainty

36. Decoherence issues only collective spin states are entangled particles are indistinguishable - high symmetry of the system – - robustness against losses. This is not a Schrodinger’s cat made of 10 12 atoms! no free lunch: limited capabilities compared to ideal maximal entanglement Sources of decoherence: stray magnetic fields decoherence time 3 milliseconds collisions decoherence time 1-2 milliseconds

37. Realism – two noncommuting spin components cannot be measured – therefore do not exist? But can be entangled Non-locality – two entangled macroscopic objects can be used to violate Bell-type inequalities via distillation All entangled Gaussian two-mode states are distillable (Giedke et al)