1. Observing the quantum nonlocality in the state of a massive particle Koji Maruyama RIKEN (Institute of Physical and Chemical Research) with Sahel Ashhab (RIKEN) Franco Nori (RIKEN / Michigan) 22/3/07 IMS, IC Phys. Rev. A 75, 022108 (2007)

2. Is this state entangled? Trailer Even if it’s in number basis? Can this single massive particle violate the Bell inequalities? YES! A single photon? Or a single massive particle? with some extra resources/manipulations…

3. Backgrounds ? beam splitter a photon vacuum A single photon going through a beam splitter just a superposition In (one-photon) mode basis

4. Backgrounds ? beam splitter a photon vacuum A single photon going through a beam splitter In the number basis looks entangled Is this (mode) entanglement just formal or physical?

5. Backgrounds (Question) A single photon going through a beam splitter Does this single photon state show quantum nonlocality?

6. Backgrounds (Question) A single photon going through a beam splitter Does this single photon state violate the Bell inequalities? In other words: Can this single photon state be converted to another state (possibly a different physical system) locally so that we can perform the Bell test on it? N.B. Measurement in the basis is physically impossible!

7. History of arguments Tan et al., 1991; Hardy, 1994: Observing the nonlocality of a single photon with a homodyne detection on each arm. Vaidman, 1995; GHZ, 1995: No! Hardy’s proposal involves multi-particle states. Jacobs and Knight, 1996: Eight-port homodyne detection and quantum correlations in quadratures. Banaszek and Wodkiewicz, 1999: Proposal of Bell-type inequalities with Q- and Wigner functions. Babichev et al., 2004: Experimental Bell test in the phase space. Violation observed. van Enk, 2005: Some arguments supporting the existence of ‘real/useful’ entanglement in. Drezet, 2006: Entanglement in is just formal (exists only on paper). van Enk, 2006: Drezet is wrong! Useful entanglement can be withdrawn from.

8. Mode entanglement of a photon A simple conversion of mode entanglement into a Bell-testable form local interactions only N.B. Measurement in the basis is possible.

9. Mode entanglement of a massive particle The same scheme doesn’t work for massive particles. BS for particles a particle No entanglement

10. The conversion scheme (1) We shouldn’t throw the flying atom away to keep the entanglement. Assume this is given as a ‘resource’. auxiliary atom In order to erase the which-path information, we use another mode-entangled single particle. the same species as the flying atom being tested

11. Then, shoot a particle to the BS. flying atomflying aux.target atom target Combine two atoms. The conversion scheme (1)

12. flyingtargetaux. # of atoms in modes c and d Measure the number of atoms in each mode. With probability 1/2, we get. The conversion scheme (1) (The concurrence of is 1/2.)

13. However, counting the atoms is a bit tricky. In the previous slide, we did the following: The conversion scheme (1)

14. But, in principle, and are orthogonal, thus distinguishable. (They aren’t bound at the same location.) The conversion scheme (1)

15. flyingaux. Assume that the auxiliary and flying atoms are each bound in a potential well. merge flyingaux. aha! distinguishable! Superposition of the lowest levels with amp. & phases depending on the initial state and the details of the merging procedure. The conversion scheme (1)

16. Need to make and indistinguishable. How? The same for the mode d. But, in principle, and are orthogonal, thus distinguishable. (They aren’t bound at the same location.) The conversion scheme (1)

17. Recall that there is a perfect correlation between the flying and target atoms. flyingtargetaux. The conversion scheme (1)

18. flyingaux. Recall that there is a perfect correlation between the flying and target atoms. flying aux. Voila, indistinguishable! target Let lower the potential well for the aux atom. equivalent to a CNOT operation. The conversion scheme (1)

19. The conversion scheme (2): more efficient scheme Yet, this conversion succeeds only with probability 1/2. Can it be more efficient? - Mode entanglement of an auxiliary atom - Indistinguishability between the flying and aux atoms The keys were: More auxiliary atoms may be useful in hiding (erasing) the which-path information of the flying atom.

20. The conversion scheme (2) Spatially split BEC has large (quantum) fluctuations in the number of atoms in each side, thus could be a good eraser of the which-path information. BEC of N non-interacting atoms BS where, # of atoms in c # of atoms in d

21. The conversion scheme (2) In the single aux atom case, the whole state before counting the atoms was flyingtargetaux. Replacing the auxiliary atom with the BEC, we have

22. The conversion scheme (2) Merging the flying atom and the BEC gives

23. The conversion scheme (2) Tracing out the BEC gives the density matrix of the target atom Concurrence An (accidental) coincidence with the one-auxiliary-particle case (N=1).

24. The conversion scheme (2) The state of BEC after detaching the target atoms is where and. Whichever we have, the (amplitude) distribution is the same as that of BEC before the 1 st run. The density matrix depends only on, not the number of atoms in each mode.

25. The conversion scheme (2) Another shot of a flying atom results in the same amount of entanglement in the (new pair of) target atoms. The resulting BEC will be in a mixed state --- a classical mixture of states, each of which has again the same amplitude distribution (for different number states). Shooting flying atoms of the same species as the BEC will generate equally entangled pairs of the target atoms indefinitely.

26. The conversion scheme (2) mode entanglement catalyst (BEC) ‘useful’ entanglement massive particle Repeatable many times!

27. Conclusions Mode entanglement of massive particles carries indeed real entanglement. It can be converted into Bell testable entanglement with an auxiliary mode-entangled atom pair with probability 1/2. with an auxiliary mode-entangled BEC with probability. The BEC can be used as many times as possible (like a catalyst) to generate highly entangled ‘useful’ atom pairs from incident mode-entangled atoms. Possibly useful for some quantum information processing. Similar scheme for fermions?

28. References Beam splitter for massive particles e.g., see Zhang et al., PRL (2006).