Dipole-dipole interaction in
quantum logic gates and
quantum reflection
Angela M. Guzmán
Departamento de Física, Universidad Nacional de Colombia, Bogotá,
Colombia, and visiting Professor,
School of Physics, The Georgia Institute of Technology, Atlanta, GA
30332, USA.
angela.guzman@guzgon.com.
Outline
1. Quantum dipole-dipole interaction
2. Controlled collisions between neutral atoms.
s-scattering .vs. dipole-dipole interaction in
a phase gate.
Marco Dueñas, Universidad Nacional de Colombia
3. van der Waals interaction in an external field:
Quantum reflection in evanescent-wave mirrors:
static .vs. dynamic van der Waals (dipole-dipole)
potential.
Brian Kennedy, Georgia Institute of
Technology.
DIPOLE-DIPOLE INTERACTION
d1
H dip  Vdd  i dd
s0

2
( R )  
where,
q , q '  ,0

0


1



ˆ
ˆ
[ D  e ( z )]q [( R)]q , q ' [ D2  e ( z )] q '  H .c.
ck
dkk
4 2
2
d2
ˆˆ)  d ][eik  R I  e  ik  R I ]
ˆ  kk
d

[
d

(
I
2


 k 1
I   i (k   L ) 
P
k   L
DIPOLE-DIPOLE INTERACTION
Controlled collisions
between
adjacent
atoms in an optical
lattice
Atom-wall interaction
in quantum reflection
Cold atoms
n1 m2 | H dip | n1' m'2 
Wannier functions
d2 
s  0
d x1 xˆ  d y1 yˆ  d z1 zˆ
s  0


Two-qubit: Phase-gate
s-scattering
(Fermi Potential)
4at  2
V (r1  r2 ) 
 (r1  r2 )
m
| 0 | 0  | 0 | 0
| 0 |1  | 0 |1
| 1 | 0  |1 | 0
|1 |1   |1 | 1
D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller. Phys. Rev.
Lett. 82,1975 (1999).
A 1D moving optical lattice
(with polarization gradient)
x

Θ
E2
E1
4
 
Θ
z
y
U0
U  ( z,  ) 
 2 cos 2(kL z  )  sin 2kL z sin 2  
2
U 0  s0
2
s0  2
  2 / 4
A 1D moving optical lattice
Optical potential U+,UU-
2U0
}Θ=0.1
U+
U-
U0
}Θ=0.25
U+
U0
U+
kLz
1 
sin 4 
  4 

z  z  r 
2
2k L 
1  sin 2  

osc  2 U 0 ER 1  sin 2 2 
}Θ=0.5
CONTROLLED
COLLISION
Controlled Collisions
Sinusoidal variation
of the angle:
 (t )  c sin b
with  
Adiabaticity
tcol
1
osc
,
U0

2 ER
b
ER
t

Operation
time
tcol
ER
2 ER
U0
DIPOLE-DIPOLE INTERACTION
Atom 1
Atom 2
K2
K1
k
k
VACUUM PHOTONS
Induced
dipole moment.
r(t )  [ / 2  2(t )]/ k L
Selection rules
V00 nn '  0 ifV n  n ' is odd
0000
V0011
1 ER

V0000
2 U0
V0002 
1
V0011
2
Forbidden
Transition
probabilities
2

2
Elastic collisions
Two-qubit: Phase-gate
| Q1 , Q2   C0,0 (t ) | 0,0  C0,1 (t ) | 0,1  C1,0 (t ) |1,0  C1,1 (t ) |1,1
|0,0  ei00 |0,0
|01  ei01 |01
i10
|10  e |10
i11
|11  e |11
Two-qubit: Phase-gate
Elastic collisions, dipole-dipole interaction
i

Ci , j (t )   [Vij,ij (t )  iij ,ij (t )]Ci , j (t )

Cij ,ij (t )  e
 ij , ij ( c , b ) 
1

tcol
0
iij ,ij (t )  ij ,ij (t )
Vij ,ij (t )dt
Interaction energy
 ij ,ij (c, b) 
1

tcol
0
ij ,ij (t )dt
Spatially modulated losses.
MATRIX ELEMENTS
1
i
1
V00,00 (  ,  )  i00,00 (  ,  )  V0 e  3  2   T (  ,  )  V00
 

i
  kL r
3
V0 
U0
16
T (  ,  )  e (  ) [cos   sin 2 ]
ER
 ( ) 
U 0 (1  sin 2 2 )
1
 3 3i
2
i
Vll ',ll ' (  ,  )  V00  V0 (l  l ')  4  3  2   sin  ei , l , l '  0,1
  

Interaction energy
  0,   10 , U 0  100 ER
Im[Dipole-Dipole interaction potential]
ij ,ij  

4
U0
  0,   10 , U 0  100 ER
ORDERS OF MAGNITUDE
•
Relative phase difference with respect to

•
00,00
  U0
16 b
ER
The probability losses (probability of having the
atoms in the original two-qubit state)
| Cij ,ij |  e
2
Adiabatic
criterion

2 ij ,ij
2U 0
16 ER
|  ij ,ij | 
U0

ER
Using a commutation
frequency b=3
For c=0.4:
Phase Logic Gate
| 0 | 0 | 0 | 0
| 0 | 1   | 0 | 1
| 1 | 0   | 1 | 0
| 1 | 1 | 1 | 1
Probability losses
of 84%
Remarks
1. Long range potentials rather than s-scattering
determine the table of truth of logic gates based on
atomic collisions.
2. Logic operations based in the dipole-dipole
interaction can not be performed in a time scale
shorter than that of the spatially modulated losses.
3. Dissipation diminishes fidelity and does not allow
for successive quantum operations.
4. Same limitations apply to schemes with enhanced
dipole-dipole interaction [G. K. Brennen, C. M. Caves, P.
S. Jessen, and I. H. Deutsch, Phys. Rev. Lett. 82, 1060 (1999)],
unless special bichromatic engineering is used to
balance losses.
Atom-wall interaction in atomic reflection &
the dipole-dipole interaction
Perfect conductor
J.E. Lennard-Jones, Trans. Faraday Soc. 28,33 (1932).
U int
r
r
Image
dipole
K L J
e2  R 2 
  3 , where K L-J 
r
12
r
dipole
• H.B. Casimir and D. Polder, Phys. Rev. 73,
360 (1948). Radiative corrections
& retardation effects
Perfect conductor
Perfect conductor
|{0k },|{1m , 0k m }
EM Vacuum
U int
r 3
r 0
EM Vacuum
U int
r 4
r 
ALKALI ATOMS & GOLD SURFACE
 R2 
12
Exp. [1]
Theor. [2]
Cs
1.087
4.143
0.59
Rb
0.938
3.362
0.65
K
0.791
2.877
0.73
U int
r
r  0
4
U int
C3val / C3
C4

(r  0 / 2 ) r 3
[3]
[1] A. Shih, V.A. Parsegian , Phys. Rev. A 12, 835 (1975)
[2] A. Derevianko, W. R. Johnson, M. S. Safranova, J. F. Babb
Phys. Rev. Lett. 82, 3589 (1999).
[3] F. Shimizu, Phys. Rev. Lett. 86, 987 (2001) (Neon)
QUANTUM REFLECTION
Na BEC
U int
C4

(r  3 / 2 2 )r 3
T. A. Pasquini, Y-I Shin, C. Sanner, M. Saba, A. Schirotzek,
D.E. Pritchard, and W. Ketterle, arXiv.org/cond-mat/0405530.
EVANESCENT-WAVE ATOMIC MIRRORS
M. Kasevich, K. Moler, E. Riis, E. Sunderman, D. Weiss,
and S. Chu, Atomic Physics 12, AIP Conf. Proc. 233, 47
(1991).
A means of measuring atom-surface forces
U  U opt ( x)  U vdW ( x)
U opt ( x)   s ( x)
2 2 x
0
s ( x)   e
/(   )
2
2
 1 
1  1 1 e  R 
U vdW ( x)  
 0.11  

3
1  1 4 0 12 x
 kL x 
2
2
3
DIPOLE-DIPOLE INTERACTION
H dip = Vdd + i dd


MM ' M '' M '''
 D1 ˆ  ( x, z )   ( R)  i  ( R) 
 D2 ˆ  ( x, z )   H .c.
 M ' M '' 

 Mb 
 bM '''
D1,2  d1,2 / | d1,2 |
 (R)=
k 3L
d1 V(k L R) d 2 ,
 (R)=
k 3L
d1 (k L R) d 2
ˆ ˆ cos k L R
ˆ ˆ  sin k L R cos k L R 
V(k L R)=(I -RR)
- (I -3RR) 

2
3 
kLR
(k
R
)
(k
R
)
L
 L

ˆ ˆ sin k L R
ˆ ˆ  cos k L R sin k L R 
(k L R)=(I -RR)
+ (I-3RR) 

2
3 
kLR
(k
R
)
(
k
R
)
L
 L

Atomic levels

M=-1


M=0
M=1
J=0
Dynamic van der Waals potential
between a ground
2
s 

state atom and
a dielectric
surface in the
0
U dyn ( x)  i dyn ( x) 
U dyn ( x) 


presence of
evanescent
wave
and
the
EM
12an



s
0


vacuum.
Dynamic Potential
Dissipation
 cos  sin  cos  
U dyn (  )   
+ 2 
s( )
3 

 
 
  2k L x
 3
 sin  cos  sin  
 dyn (  )   
- 2  3  s(  )

 
 
Dissipation due to the interaction through the vacuum
Dynamic van der Waals potential
2
 s ( )   s   0   cos  sin  cos  
R e[U dyn ( x)]  
+ 2 



12   s   0   

 3 
Static van der Waals potential
 s  0 
U vdW ( x)   3 
2    s   0 
Effective potential
2



  s   0  cos  sin  cos   

U eff ( x, z )  s( )    
+ 2 
 
3 
12   s   0   

 


U opt (normalized)
Optical potential
Dynamic
 100
van der Waals potential
Effect of van der Waals potential
Quantum reflection
• Evanescent
waves.
A. Landragin,
J.-Y.T.
Courtois,
G. Labeyrie,
From
a solid surface
at normal
incidence.
A. Pasquini,
Y-I
N. Vansteenkiste,
C.Saba,
I. Westbrook,
and A. Aspect,
Phys. Rev.and
Shin,
C. Sanner, M.
A. Schirotzek,
D.E. Pritchard,
Lett. 77, 1464 (1996).
W. Ketterle, arXiv.org/cond-mat/0405530.
U U r 3 r 3
2.0
E
(U dyn )
U vdW

0.0
1.5
2.5
3.5
(U dyn )
-2.0
V0  100ER , =10
4.5
Quantality of the potentials
q(r ) 
2
( p(r ))1/ 2
q
d2
1/ 2
(
p
(
r
))
, p(r )  2m( E  U (r ))
2
dr
25
20
15
10
5
0
U dyn
Quantum
region
U vdW
0
2
4
q 1
WKB
Remarks
1. Atom-wall and atom-atom van
der
Waals potential in
external fields relate to the dynamic rather than to the static
polarizability.
2. The shape of the reflecting potential is not controlled by S0
alone. Variations in field intensity scale the potential but
variations in detuning shift the maximum.
3. Quantum reflection from solid surfaces occurs only for atomic
velocities close to zero (heating has been observed). Quantum
reflection from evanescent-wave atomic mirrors occurs at
finite energies, but the reflectivity will be less than one
because of dissipative effects.
4. Applications in atomic funnels,
quantum reflection
engineering, optical traps for quantum gases, Rydberg atoms
in optical lattices (a power dependent line width of the
fluorescence spectrum has already been observed, FiO 2004).