Dipole-dipole interaction in quantum logic gates and
quantum reflection
Angela M. GuzmánDepartamento de Física, Universidad Nacional de Colombia, Bogotá,
Colombia, and visiting Professor,
School of Physics, The Georgia Institute of Technology, Atlanta, GA 30332, USA.
OutlineOutline
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
Brian Kennedy, Georgia Institute of Technology.
3. van der Waals interaction in an external field: Quantum reflection in evanescent-wave mirrors: static .vs. dynamic van der Waals (dipole-dipole) potential.
21 220
ˆ ˆˆ( ) [ ( ) ][ ]4
ik R ik Rk
ckR dkk d d I kk d e I e I
�
where,Lk
LkiI
P
)(
DIPOLE-DIPOLE INTERACTION
01 , ' 2 '
, ' ,0
ˆ ˆ[ ( )] [ ( )] [ ( )] . .2
dip dd dd
q q q qq q
H V i
sD e z R D e z H c
�
1d
2d
Controlled collisions between adjacent atoms in an optical lattice
DIPOLE-DIPOLE INTERACTION
'1 1| | '
2 dip 2n m H n m
Wannier functions
Atom-wall interaction in quantum reflection
Cold atoms
1 10
20
1ˆ ˆ ˆ
xs
zs
yd x d y d zd
D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller. Phys. Rev.
Lett. 82,1975 (1999).
Two-qubit: Phase-gate
s-scattering (Fermi
Potential)
| 0 | 0
| 0 | 0
|
| 0 | 0
|1 |1
| 01 |1 | 0
|1|1 |1 |
1
)(
4)( 21
2
21 rrm
arrV t
A 1D moving optical lattice (with polarization gradient)
z
x
y
Θ
Θ
E1E2
0( , ) 2 cos 2( ) sin 2 sin 22 L L
UU z k z k z
4
0 0U s 2
0 2 2 / 4s
Optical potential U+,U-
0
U0
2U0
0 1.60.8
kLz
U-
U+
}Θ=0.1
U-
U+
}Θ=0.25
U-
U+
}Θ=0.5
U-
U+
}Θ=0.25
U-
U+
}Θ=0.1
A 1D moving optical lattice
2
1 sin 44
2 1 sin 2L
z z rk
202 1 sin 2osc RU E
Controlled Collisions
Sinusoidal variation of the angle:
( ) sinct b
with tER
Adiabaticity 1
0
,
2
col osc
R
t
bU
E
Operation time
0
2 Rcol
R
Et
E U
CONTROLLED COLLISION
DIPOLE-DIPOLE INTERACTION
K1K2
Atom 1 Atom 2
VACUUM PHOTONS
k k
r( ) [ / 2 (2 ]/) Lktt
Induced dipole -moment.
Selection rules
00 ' 0 ' nnV if n n is odd
Forbidden
V0000
1
2V V0002 0011
0
1
2RE
UVV 0011 0000
Transition probabilities
2
2
Elastic collisions
01
10
|01 |01
|10 |10
i
i
e
e
00 |0,0 |0, 0ie
11 |11 |11ie
Two-qubit: Phase-gate
1 2 0,0 0,1 1,0 1,1| , ( ) | 0,0 ( ) | 0,1 ( ) |1,0 ( ) |1,1Q Q C t C t C t C t
)()]()([)( ,,,, tCtitVi
tC jiijijijijji
,, , (( ) )
( )ij ijij i ijj iji t t
C t e
, ,0( ,
1) ( )
colt
ii j jj iji c b t dt , ,0( ,
1) ( )
colt
ii j jj iji c b V t dt
Two-qubit: Phase-gate
Elastic collisions, dipole-dipole interaction
Interaction energy Spatially modulated losses.
0000,00 00,00 0 3 2
1 1( , ) ( , ) ( , )i i
V V e T Vi
20
( ) 1(1 sin 2 )
RE
U
0 0
3
16V U
Lk r ( )( , ) [cos sin 2 ]T e
00', ' 0 4 3 2
3 3 2( , ) ( ') sin , , ' 0,1ll ll
ii iV V l l e lV l
MATRIX ELEMENTS
• Relative phase difference with respect to 00,00
0
16 R
U
b E
• The probability losses (probability of having the
atoms in the original two-qubit state)
ORDERS OF MAGNITUDE
0 0,
2 | |
16 ij ijR R
U U
E E
2,
,2| |ij ij
ij ijC e
Adiabaticcriterion
Probability losses of 84%
Using a commutation frequency b=3
1|1|1|1|
0|1|0|1|
1|0|1|0|
0|0|0|0|
Phase Logic Gate
For c=0.4:
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.
J.E. Lennard-Jones, Trans. Faraday Soc. 28,33 (1932).
2 2
int L-J3, where K
12L JK e R
Ur
dipoleImagedipole
r r
P
erfe
ct c
ondu
ctor
rrrrrr
P
erfe
ct c
ondu
ctor
r
P
erfe
ct c
ondu
ctor
r
P
erfe
ct c
ondu
ctor
r
Atom-wall interaction in atomic reflection & the dipole-dipole
interaction
Per
fect
con
duct
or EM Vacuum
• H.B. Casimir and D. Polder, Phys. Rev. 73,
360 (1948). Radiative corrections
Per
fect
con
duct
or EM Vacuum
|{0 } ,|{1 ,0 }k m k m & retardation effects
4int
U r
r
3
int
0
U r
r
ALKALI ATOMS & GOLD SURFACE
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
4int
0
U r
r
4
int 30( / 2 )
CU
r r
[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)
2
12
R 3 3/valC C
[3]
4int 2 3( 3 / 2 )
CU
r r
T. A. Pasquini, Y-I Shin, C. Sanner, M. Saba, A. Schirotzek, D.E. Pritchard, and W. Ketterle, arXiv.org/cond-mat/0405530.
QUANTUM REFLECTION
Na BEC
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
( ) ( )opt vdWU U Ux x
( ) ( )optU x s x
12 3
31 0
21 1 10.11
1 12(
4)
LvdW
e R
xx
kU
x
2 2 2 20( ) /( )xs x e
1 ' '' '''' ''2
'''
ˆ ˆ( ( ) ( )) , ) . ., (M M bMMM M M Mb
D R i R H cDx z x z
dip dd ddH =V +i
3 3L
1L
2 1 2d , k k
(R)= ( R) (R d)= ( R)d dL Lk k
V
1,2 1,2 1,2/ | |D d d
DIPOLE-DIPOLE INTERACTION
L L L2 3
L L L
L L L2 3
L L L
cos k sin k cos k= -
k (k ) (kˆ ˆ ˆ ˆ( -RR) ( -3RR)
ˆ ˆ ˆ ˆ(
)
si
( R
n k cos k sin k= +
k (k ) (-RR) ( -3
)
( R) Rk
)
R)
L
L
R R R
R R R
R
R
k
R R
R Rk
I I
I I
V
2
s 0
s 0
( )) ( )(12dyn dyn dynU U ix xx
Dynamic van der Waals potential between a ground state atom and a dielectric surface in the presence of an evanescent wave and the
EM vacuum.
Atomic levels
J=0
M=-1 M=0 M=1
Dissipation Dynamic Potential
32
coscos sin
( ) s( )+dynU
2 3
sin cos sin ( ) ( )-dyn s
3
2 Lk x
Dissipation due to the interaction through the vacuum
s 02
s 0
2
3
cos sin [ ( )] +
1
( co ) s
2dyne xs
U
Re
s
s3
0
0
( )2vdWU x
Dynamic van der Waals potential
Static van der Waals potential
03
2
20
cos sin +
12
co,
s ( () ) s
seffU x sz
100
(normalized) optU
Effect of van der Waals potential
Effective potential
Optical potential
Dynamic van der Waals potential
Quantum reflection
3U r
• Evanescent waves. A. Landragin, J.-Y. Courtois, G. Labeyrie, N. Vansteenkiste, C. I. Westbrook, and A. Aspect, Phys. Rev. Lett. 77, 1464 (1996).
-2.0
0.0
2.0
1.5 2.5 3.5 4.5
( )dynU
vdWU
( )dynU
3U r E
0 100 ,RV E =10
From a solid surface at normal incidence. T. A. Pasquini, Y-I Shin, C. Sanner, M. Saba, A. Schirotzek, D.E. Pritchard, and W. Ketterle, arXiv.org/cond-mat/0405530.
05
10152025
0 2 4
Quantality of the potentials2 2
1/ 21/ 2 2
( ) ( ( )) , ( ) 2 ( ( ))( ( ))
dq r p r p r m E U r
p r dr
dynU
q
1q
WKB
Quantumregion
vdWU
RemarksRemarks
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).