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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
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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
R. Scheunemann, F. S. Cataliotti, T. W. Hänsch,
and M. Weitz
Physical Review A (Rapid Communication) 62,
051801(R) (2000)
normalized counts
1.3
1.2
1.1
1.0
0.9
0.8
0.7
0.6
100
110
120
130
Pixel
140
150
160
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Huang et al. Nature 2003
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.
Frequency W and mass M
W
Quic kT ime™ e un
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H = –∫d2r P(r,t) x(r,t)
W
-W
[Phys. Rev. A 68, 062317 (2003)]
Tripartite ENTANGLEMT
H  i ac  a c  i a c'a c'
 
  W

r   
 1

  W

W

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W


  
2
2
The mechanical oscillator mode is
in a thermal state and the side modes
in vacuum

rinA = r0 c  r0 c’  ra
-W

r0 i = |0>i< 0 |
n


1
N th
rˆ a 

 n n ,

1 N th n 1 N th 
N th 
1
e
W / KBT
1
Represented by the Gaussian characteristic function
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F(m,n,z,0) = e- |n|2Nth
F(m,n,z,t) is the evolution of F(m,n,z,0) which is still Gaussian
F(m,n,z,t) = e-  V  T
T m1,m2,n1,n2,z1,z2 )
The 6x6 correlation matrix
V = Vcac’ =
At  
1
2r 2 1

2
A 1/2I

 CZ
 FZ

CZ
B 1/2I
DI
FZ 

DI 
E 1/2I
Where A,B,C,D,E, F
depend on r, Nth, , t ,
I is the identity 2x2 matrix
and Z =diag [1,-1]
1 cos2tN r 11 4r 1 cost
2
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
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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 communication performed by 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 rab is still Gaussian
with the output 4x4 correlation matrix
Charlie
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)
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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
 2N  1 2

 th
out 

 1  1 ln
1
1

2N th  e  2e out 

 1  1 for
T 0

W
 
 (2N th)1 as
2K B T

1

K B T  W
Experimental detection
Requires the measurement of the relative distance Xrel= xa-xb
and the total momentum Ptot= pa+pb
In this case
X rel 
2
X rel   2out
2
and

LO

S
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Jacobs et al. PRA (1994)
Cohadon et al. _PRL (1999)
 Ptot 
2
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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  0.1mg, W  108 s1 with  -1 1s
It could be a sensitive test to discriminate different theories
of quantum gravity