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.

angela.guzman@guzgon.com.

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

Interaction energy

00, 10 , 100 RU E

Im[Dipole-Dipole interaction potential]

, 04ij ij U

00, 10 , 100 RU E

• 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).