1. Scheme for Entangling Micromeccanical Resonators
by Entanglement Swapping
Paolo Tombesi
Stefano Mancini
David Vitali
Stefano Pirandola

2. 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)

3. 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.

5. Lucent Techn. Lab.

6. 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, (R) (2000)

7. Huang et al. Nature 2003

8. 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, (2003)]
Tripartite ENTANGLEMT

10. 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 |

11. 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]

12. 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

13. 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.

14. 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)

15. Optimal value t ~ 1µs ENout ~ 1.1

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

17. 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

18. Pinard et al. in Europhys. Lett. ‘05

19. 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