Scheme for Entangling Micromeccanical Resonators by Entanglement Swapping Paolo Tombesi Stefano Mancini David Vitali Stefano Pirandola Microworld is quantum, macroworld is classical. Is there a boundary, or classical physics naturally emerges from quantum physics ? ow far can we go in the search and demonstration of macroscopic quantum phenomena ?

Recent spectacular achievements: sition of two magnetic flux states in a rf-SQUID (Stony Brook ement of internal spin states of two atomic ensembles (Aarhus nterference of macromolecules with hundred atoms (Vienna 2003 otons-microwave cavity field in a superposition of macroscopi distinct phases (Paris 2003) optical photons in a superposition with distinct phases (Ro un(r) normal modes mn =dd3r |un(r)|2 QuickTime e un decompressore TIFF (LZW) sono necessari per visualizzare quest'immagine.

Displacement x is generally given by the superposition of many acoustic modes. A single mode description is valid when the detection is limited to a frequency bandwidth including a single mechanical resonance. Lucent Techn. Lab. Very light mirrors

25 m A matter wave grating such as that created by cold atoms in an optical lattice acts as a dielectric mirror 1.5 1.4 5.3 m normalized counts 1.3

R. Scheunemann, F. S. Cataliotti, T. W. Hnsch, and M. Weitz Physical Review A (Rapid Communication) 62, 051801(R) (2000) 1.2 1.1 1.0 0.9 0.8 0.7 0.6 100

110 120 130 Pixel 140 150 160

QuickTime e un decompressore TIFF (LZW) sono necessari per visualizzare quest'immagine. Huang et al. Nature 2003 sed light beams are able to excite Gaussian tic modes in which only a small portion of the or vibrates (r,t)a e-it + a+eit)exp[-r2/w2] undamental Gaussian mode where is its waist. requency

and mass M = dd2r P(r,t) x(r,t) QuickTime e un decompressore TIFF (LZW) sono necessari per visualizzare quest'immagine. -

[Phys. Rev. A 68, 062317 (2003)] Tripartite ENTANGLEMT H i ac a c r 1

i a c' a c' QuickTime e un decompressore TIFF (LZW) sono necessari per visualizzare quest'immagine.

- 2 2 The mechanical oscillator mode is in a thermal state and the side modes

in vacuum inA = 0 c 0 a 0 i = |0>i< 0 | c n

1 N th a n n , 1 N th n 1 N th 1 N th / K B T e 1

Represented by the Gaussian characteristic fun QuickTime e un decompressore TIFF (LZW) sono necessari per visualizzare quest'immagine. (,) = e-th ) is the evolution of (,) (,) = e- V T which is still Gauss The 6x6 correlation matrix Where A,B,C,D,E, F

on r, Nth, , t , A 1/2I CZ FZ depend I CZ B 1/2 I DI

is the identity 2x2 matr FZ DI E 1/2and I Z =diag [1,-1] V = Vcac = At

1 1 cos2tN r 1 1 4r 1 cost 2 2 2r 2 1 th 2 Charlie performs a heterodyne meas.

on the anti-Stokes modes c and the two tripartite states become bipartite with Gaussian correlation matrices Vac , Vbc Charlie QuickTime e un decompressore TIFF (LZW) sono necessari per visualizzare quest'immagine. QuickTime e un decompressore TIFF (LZW) sono necessari per visualizzare quest'immagine.

Pirandola et al. PRA 2003 sic idea of entanglement tite entanglement within arlie-Bob to the distant able local operation and performed by Charlie. swapping is to transfer th the near pairs Alice-Charl pair Alice-Bob, by means o classical communica- Charlie performs a CV Bell state

measurement mixing the two Stokes modes on a 50%-50% beam splitter and measures the output quadratures (XcA - XcB) (pcA + pcB) The final output state ab is still Gaussian with the output 4x4 correlation matrix Charlie Vout For the entanglement we consider the logarithmic negativity

EN = max [ 0, ln2out] Vidal & Werner, PRA 65, (2002) Adesso et al. PRA 70, (2004) QuickTime e un decompressore TIFF (LZW) sono necessari per visualizzare quest'immagine. Optimal value t ~ 1s ENout ~ 1.1 Living time of entanglement depends on -1 with the vibrations damping constant The real living time is

2N 1 2 1 1 th out ln 1 1 2N th e 2e out

1 1 for T0 (2N th ) 1 as 2K B T 1

K B T Experimental detection Requires the measurement of the relative distance Xrel= xa-xb and the total momentum Ptot= pa+pb In this case 2 X rel 2

X rel and LO S QuickTime e un decompressore TIFF (LZW) sono necessari per visualizzare quest'immagine. Jacobs et al. PRA (1994)

Cohadon et al. _PRL (1999) Ptot 2out 2 QuickTime e un decompressore TIFF (LZW) sono necessari per visualizzare quest'immagine. Pinard et al. in Europhys. Lett. 05

Conclusion A scheme for entangling two micromechanical oscillators by entanglement swapping, exploiting the radiation pressure force and changing the nature of entanglement, from optomechanical to pure mechanical The mechanical entanglement could last for quite long time 1 1 ms @ T = 1 K, M 0.1g, 10 8 s 1 with -1 1s It could be a sensitive test to discriminate different theories of quantum gravity