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 ? How far can we go in the search and demonstration of macroscopic quantum phenomena ? Recent spectacular achievements: Superposition of two magnetic flux states in a rf-SQUID (Stony Brook, 2000) Entanglement of internal spin states of two atomic ensembles (Aarhus, 2001) Interference of macromolecules with hundred atoms (Vienna 2003) 40 photons-microwave cavity field in a superposition of macroscopically distinct phases (Paris 2003) several optical photons in a superposition with distinct phases (Roma 2004)

un(r) normal modes mn =r∫d3r |un(r)|2 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 A matter wave grating such as that created by cold atoms in an optical lattice acts as a dielectric mirror R. Scheunemann, F. S. Cataliotti, T. W. Hänsch, and M. Weitz Physical Review A (Rapid Communication) 62, 051801(R) (2000)

Huang et al. Nature 2003

Tripartite ENTANGLEMT Focused light beams are able to excite Gaussian acustic modes in which only a small portion of the mirror vibrates x(r,t)a e-iWt + a+eiWt)exp[-r2/w2] fundamental Gaussian mode where w is its waist. W  - Frequency W and mass M H = –∫d2r P(r,t) x(r,t) [Phys. Rev. A 68, 062317 (2003)] Tripartite ENTANGLEMT

W  inA = 0 c  0 c’  a - 0 i = |0>i< 0 | The mechanical oscillator mode is in a thermal state and the side modes in vacuum inA = 0 c  0 c’  a 0 i = |0>i< 0 |

F(m,n,z,t) is the evolution of F(m,n,z,) which is still Gaussian Represented by the Gaussian characteristic function F(m,n,z,0) = e- | n |2Nth F(m,n,z,t) is the evolution of F(m,n,z,) which is still Gaussian F(m,n,z,t) = e-  V  T xT = (m1,m2,n1,n2,z1,z2 ) The 6x6 correlation matrix V = Vcac’ = Where A,B,C,D,E, F depend on r, Nth, , t , I is the identity 2x2 matrix and Z =diag [1,-1]

Vac , Vbc Charlie performs a heterodyne meas. on the anti-Stokes modes c’ and the two tripartite states become bipartite with Gaussian correlation matrices Vac , Vbc Pirandola et al. PRA 2003

The basic idea of entanglement swapping is to transfer the bi-partite entanglement within the near pairs Alice-Charlie and Charlie-Bob to the distant pair Alice-Bob, by means of a suitable local operation and classical communica- tion performed by Charlie.

with the output 4x4 correlation matrix Charlie 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 Vout For the entanglement we consider the logarithmic negativity EN = max [ 0, –ln2out] Vidal & Werner, PRA 65, (2002) Adesso et al. PRA 70, (2004)

Optimal value t ~ 1µs ENout ~ 1.1

Living time of entanglement depends on -1 with  the vibration’s damping constant The real living time is for as

Experimental detection Requires the measurement of the relative distance Xrel= xa-xb and the total momentum Ptot= pa+pb In this case and Jacobs et al. PRA (1994) Cohadon et al. _PRL (1999) LO S

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 It could be a sensitive test to discriminate different theories of quantum gravity