1. School of something FACULTY OF OTHER School of Physics and Astronomy FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES Nonlocality of a single particle Jacob Dunningham Vlatko Vedral Paraty, 16 August 2007

2. Nonlocality is usually confirmed in an EPR-type experiment Alice and Bob look for “better than perfect correlations” in their measurements Two or more particles in the system

3. “…. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives [the quantum states] have become entangled." Schrödinger (Cambridge Philosophical Society, 1935) It would be strange if the characteristic trait of quantum mechanics did not apply to a single particle, but was an emergent property of two or more particles.

4. Single particle nonlocality Tan, Walls and Collett (1991) first suggested that a single particle could exhibit nonlocality

5. Single particle nonlocality Tan, Walls and Collett (1991) first suggested that a single particle could exhibit nonlocality Lucien Hardy (1994) tightens up some assumptions in TWC

6. Single particle nonlocality Tan, Walls and Collett (1991) first suggested that a single particle could exhibit nonlocality Lucien Hardy (1994) tightens up some assumptions in TWC Greenberger, Horne, and Zeilinger (GHZ) raise objections: not a real experiment/ multiparticle effect in disguise (1995)

7. Single particle nonlocality Tan, Walls and Collett (1991) first suggested that a single particle could exhibit nonlocality Lucien Hardy (1994) tightens up some assumptions in TWC Greenberger, Horne, and Zeilinger (GHZ) raise objections: not a real experiment/ multiparticle effect in disguise (1995) Other schemes…lots of debate…no clear consensus (1995 - 2007)

8. Single particle nonlocality Tan, Walls and Collett (1991) first suggested that a single particle could exhibit nonlocality Lucien Hardy (1994) tightens up some assumptions in TWC Greenberger, Horne, and Zeilinger (GHZ) raise objections: not a real experiment/ multiparticle effect in disguise (1995) Other schemes…lots of debate…no clear consensus (1995 - 2007) What is needed is a feasible experiment to resolve the issue….

9. “The term ‘particle’ survives in modern physics but very little of its classical meaning remains. A particle can now best be defined as the conceptual carrier of a set of variates... It is also conceived as the occupant of a state defined by the same set of variates... It might seem desirable to distinguish the ‘mathematical fictions’ from ‘actual particles’; but it is difficult to find any logical basis for such a distinction. ‘Discovering’ a particle means observing certain effects which are accepted as proof of its existence.” A. S. Eddington, Fundamental Theory, (Cambridge University Press., Cambridge, 1942) pp. 30-31.

10. What is a beam splitter? For atoms, this is equivalent to Josephson coupling Any physical process that transforms states in the same way as a beam splitter.

11. Hardy Scheme 2 1 Reference: L. Hardy, Phys. Rev. Lett. 73, 2279 (1994)

12. Hardy Scheme 2 1 Reference: L. Hardy, Phys. Rev. Lett. 73, 2279 (1994)

13. Hardy Scheme Alice 2 1 Experiment 1: Alice and Bob both measure the number of photons on their path Bob They never both detect one

14. Hardy Scheme Alice 2 1 Experiment 2: Alice makes a homodyne detection and Bob detects the number in path 2 If Bob detects no particles the state at Alice’s detectors is So, if Alice detects one, it must be at c 1 Conversely, if Alice detects one particle at d 1 and none at c 1 then Bob cannot detect none, i.e. he must detect one! Bob

15. Hardy Scheme Alice 2 1 Experiment 3: The roles of Alice and Bob are reversed. Alice measures the number of particles on path 1 and Bob makes a homodyne detection. If Bob detect one particle at d 2 and nothing at c 2 then Alice must detect one particle. Bob

16. Hardy Scheme Alice 2 1 Experiment 4: Alice and Bob both make homodyne detections Bob There is a finite probability that: Alice detects one particle at d 1 and none at c 1 AND Bob detects one particle at d 2 and none at c 2

17. Recap Experiment 1: Alice and Bob cannot both detect a particle in their path. Experiment 2: If Alice detects one particle at d 1 and nothing at c 1 it follows that Bob must detect a particle on path 2. Experiment 3: If Bob detects one particle at d 2 and nothing at c 2 it follows that Alice must detect a particle on path 1. Experiment 4: One possible outcome is that Alice detects one particle at d 1 and nothing at c 1 AND Bob detects one particle at d 2 and nothing at c 2.

18. Recap Experiment 1: Alice and Bob cannot both detect a particle in their path. Experiment 2: If Alice detects one particle at d 1 and nothing at c 1 it follows that Bob must detect a particle on path 2. Experiment 3: If Bob detects one particle at d 2 and nothing at c 2 it follows that Alice must detect a particle on path 1. Experiment 4: One possible outcome is that Alice detects one particle at d 1 and nothing at c 1 AND Bob detects one particle at d 2 and nothing at c 2. CONTRADICTION NONLOCALITY

19. Objections Greenberger, Horne, and Zeilinger (1995) “Partly-cle” states are unobservable - violate superselection rules Does not correspond to a real experiment

20. Objections Greenberger, Horne, and Zeilinger (1995) “Partly-cle” states are unobservable - violate superselection rules Does not correspond to a real experiment They proposed an alternative scheme that required additional particles. These particles also introduced additional nonlocality They concluded that ‘apparent’ single particle nonlocality is really a multiparticle effect However, they did not disprove single particle nonlocality more generally.

21. Mixed States Convenient to use coherent state inputs and average over the phases at the end where Classical mixture of number states

22. State truncation Reference: D. T. Pegg, L. S. Phillips, and S. M. Barnett, Phys. Rev. Lett. 81 1604 (1998)

23. State truncation Reference: D. T. Pegg, L. S. Phillips, and S. M. Barnett, Phys. Rev. Lett. 81 1604 (1998) Input to top beam splitter: Output from lower beam splitter

24. State truncation Reference: D. T. Pegg, L. S. Phillips, and S. M. Barnett, Phys. Rev. Lett. 81 1604 (1998) Input to top beam splitter: We need: Total output:

25. State creation

26. x

27. x

28. x

29. Not Entangled Will become a mixed state when we average over phases at the end

36. Results This gives all the same results as Hardy’s scheme Overall state just before Alice and Bob’s beam splitters is: Each only makes a local operation Therefore nonlocality is due to the single particle state

37. Results This gives all the same results as Hardy’s scheme Overall state just before Alice and Bob’s beam splitters is: Each only makes a local operation Therefore nonlocality is due to the single particle state Results are independent of  - so we can average over all phases and get the same result, I.e. a mixed state input also works This takes care of the GHZ objections

38. Massive particles Both Hardy and GHZ said that nonlocality with single massive particles could not be observed Hardy: Superselection rules GHZ: Not really a single particle effect This scheme works equally well for atoms as photons The only components required are variable beam splitters and efficient detectors

39. Conclusions Feasible experimental scheme for demonstrating the nonlocality of a single particle Should work equally well for massive particles as for photons All we need are variable beam splitters and efficient detectors