1. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 Scheme for Entangling Micromeccanical Resonators by Entanglement Swapping Paolo Tombesi Stefano Mancini David Vitali Stefano Pirandola

2. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 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. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 m n =  ∫d 3 r |u n (r)| 2 u n (r) normal modes 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.

4. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006

5. Lucent Techn. Lab.

6. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 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)

7. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 Optical lattices with large spacing @ 20  m (horizontal) LENS

8. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 From Roukes’ Group Caltech webpage

9. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 Huang et al. Nature 2003

10. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 M.L.Roukes - Nano Electromechanical systems - Technical Digest of the 2000 Solid-state Sensor and Actuator Workshop

11. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 Frequency  and mass M x(r,t)  b e -i  t + b + e i  t )exp[-r 2 /w 2 ] fundamental Gaussian mode where w is its waist. H = –∫d 2 r P(r,t) x(r,t) Focused light beams are able to excite Gaussian acustic modes in which only a small portion of the mirror vibrates [Phys. Rev. A 68, 062317 (2003)] Tripartite ENTANGLEMT   --

12. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006

13. entanglement: polarization of two photons (  H  1  H  2 ±  V  1  V  2 )/√2(  H  1  V  2 ±  H  2  V  1 )/√2 or In general, for a bipartite system, it is separable   =  i w i  i1  i2 w i ≥ 0  i w i =1 Simple criterion for inseparability or entanglement was derived by Peres (PRL 77, 1413 (1996) These are the so-called Bell states  (  ) and  (  )

14. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 Given an orthonormal basis in H 12 = H 1 H 2 the arbitrary state of the bipartite state 1+2 is described by the density matrix (  12 ) m ,n (Latin indices for the first system and Greek indices for the second one). To have the transpose operation it means to invert row indices with column indices (  12 ) n,m  The partial transpose operation (PT) is given by the the inversion of Latin indices (Greek) PT : (  12 ) m ,n  (  12 ) n ,m  (  T 1 12 ) m ,n We ask if the operator  T 1 12 is yet a density operator i.e. Tr (  T 1 12 ) = 1 and  T 1 12 ≥ 0 It easy to prove this because the transposition does not change the diagonal elements, Thus the Trace remains invariant, and the positivity is connected with the positivity of the eigenvalues of the matrix, which do not change under transposition. Then the violation of the positivity of the partial transpose is a sufficient criterion for entanglement It easy to prove that the positivity of partial transpose of the state is a necessary condition for separability. i.e.  12 separable   T 1 12 ≥ 0  T 1 12 < 0   12 entangled In 2x2 and 2x3 dimension for the Hilbert space  12 separable   T 1 12 ≥ 0 Horodecki 3 Phys Lett A 223, 8 (1996) O x

15. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 Non-linear crystal Pump laser

16. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006

17. The tripartite state  1 b 2 is fully entangled (class 1) at any n th By tracing out one mode of the three we study the entanglement of a bipartite subsystem We find that mode a 2 and b are never entangled Modes a 1 and a 2 are entangled (extremely robust with respect to the mirror temperature n th ) Modes a 1 and b are entangled even though the region of entanglement is small and depends on n th a1a1 a2a2 b

18. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 The mechanical oscillator mode is in a thermal state and the side modes in vacuum  in A =  0 c   0 c’   a  0 i = |0> i < 0 |   --

19. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 Represented by the Gaussian characteristic function  ( ,  ) = e -    th  ( ,  ) is the evolution of  ( ,  ) which is still Gaussian  ( ,  ) = e -  V  T               ) The 6x6 correlation matrix V = V cac’ = Where A,B,C,D,E, F depend on r, N th, , t, I is the identity 2x2 matrix and Z =diag [1,-1]

20. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 Charlie Charlie performs a heterodyne meas. on the anti-Stokes modes c’ and the two tripartite states become bipartite with Gaussian correlation matrices V ac, V bc Pirandola et al. PRA 2003

21. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 Charlie Charlie performs a CV Bell state measurement mixing the two Stokes modes on a 50%-50% beam splitter and measures the output quadratures (X c A - X c B )(p c A + p c B ) Obtaining the output 4x4 correlation matrix V out For the entanglement we consider the logarithmic negativity E N = max [ 0, –ln2  out ]

22. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 Optimal value t ~ 1µs E N out ~ 1.1

23. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 Living time of entanglement depends on  -1 with  the vibration’s damping constant The real living time is for as

24. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 Experimental detection In this case and Requires the measurement of the relative distance X rel = x a -x b and the total momentum P tot = p a +p b

25. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 Pinard et al. Europhys. Lett.

26. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 f In a frame rotating at  Can we use this tripartite entangled state for measuring very weak forces? 00 0-0- 0+0+ a1a1 a2a2 b

27. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006  Heterodyne measurement  Im{Z   Y 1 +Y 2 = S f   (Y 1 +Y 2 ) 2   (Y 1 +Y 2 ) 2 = N  signal noise SNR = S √N f  1 S=(  cos(  t)  cos(  t  /  2  2   =√(  2 -  2 )

28. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006 Mirror in a thermal state Sidebands in entangled state Solid line SQL Dashed line sidebands initially with x = 0 N th = 300 Dotted line sidebands initially entangled and squuezed x = 0

29. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006

30. CONCLUSIONS With entangled beams one can beat the SQL when detecting a constant force acting on a MOMS. By means of radiation pressure force we can entangle two MOMS

31. IEN-Galileo Ferraris - Torino - 16 Febbraio 2006