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Theoretical study of atoms dynamics in optical dipole trap
Denis N. Yanyshev, Boris A. Grishanin, and Victor N. Zadkov Faculty of Physics and International Laser Center, M. V. Lomonosov Moscow State University Moscow 119899, Russia

ABSTRACT
and computer simulation resultsfor stochastic dynamics oftwo atomstrapped in an optical dipole under action of a probe resonant radiationare presented. The radiationforce correlations resulting from our trap model lead, in addition to cold collisions, to a tendency for atoms escape in pairs from the trap.
Theoreticalstudy

1. INTRODUCTION Cold collisions between atoms in an atomic trap (magneto-optical trap or optical dipole trap) are thought to be one of the key mechanisms for atomic losses from the trap.' As it has been recentlyshown experimentally, this mechanism could result to a tendency for atoms escape in pairs (Fig. 1).23 At thesame time, anotherfundamental mechanism, long-range resonant dipole-dipole interactions (long-distanceRDDI), also resultsin correlationsbetween interactingatoms andtherefore to the sametendency for atoms escape in pairs.4 In thispaper, we discussthis second
mechanism.

U)

E 0 0
C,)

time(mm)

N=3

-- N=2
N=1 N=O

C

8

-

. Excerpt from a typical fluorescence signal from a magneto-optical trap observed with an avalanche photodiode. Five isolated cold collisions (two-atoms losses) are indicated by arrows. (Pictureis taken from Ref. 1.)

Figure 1

In our model, we consider dynamics of atoms in a red-detuned optical dipole trap.57 To clarify a role of longdistance RDDI correlations we shine the atoms additionally by a probe (weak) resonant laser filed. This additional field enhances significantly interactions between atoms via jointly emitted photons and reveals in increasing correlations of stochastic radiation forces acting on atoms. For simulation of atomic dynamics we use quasiclassical approximation for the emission radiationforce. In computer simulation, we analyze the atomic dynamics in the trap and the corresponding atomic losses by varying the frequency detuning of the probe laser field and its intensity. Send correspondenceto D.N.Y.: yanyshev©comsiml.phys.msu.su

104

ICONO 2001: Quantum and Atomic Optics, High-Precision Measurements in Optics, and Optical Information Processing, S. N. Bagayev, et al., Eds., SPIE Vol. 4750 (2002) © 2002 SPIE · 0277-786X/02/$15.00


(b)

(a)
Laser beam

Atoms
-1
ia25

U

J

-1.5 ia25

-2
--i

icr25

Figure 2. Scheme of a standing-wave optical dipole trap (a) and potentialenergy of the trap in radial and axial directions (b). The potentialsurface shows a regular pattern of micropotential holes.

2. MODEL OF OPTICAL DIPOLE TRAP An optical dipole trap for neutral atoms can be made of a tightly focused powerful laser beam frequencyofwhich is far-detuned from the working transition oftrapped atoms. One can then easily show that interaction of the induced dipole moments of atoms in the trap with the inhomogeneouselectric field along the beamprofilelead to the restoring force, which "traps" the atoms inside the beam.5 Surely, the described trapping mechanism works only for atoms with low energies (at temperaturesof about mK) because the potential of the dipole trap is shallow. Therefore, atoms should be cooled down, for example in a magneto-optical trap, before loading intothe optical dipole trap.3'6'7 In an experiment, for controlling positions of atoms in the trap it could be advantageous to keep single atoms in micropotential holes. Such a regular pattern of micropotential holes in the optical dipole trap potentialcan be formed with the use of counter-propagating laser beams, which form necessary structure due to the interferience. A scheme of a standing-wave optical dipole trap formed with the counter-propagating laser beams is shown in Fig. 2a. It uses only one laserbeam, which interferes with the beam reflected from the mirror that preserves the wave front and the polarization.8 The potentialof the trap is shown in Fig. 2b. Parametersofthe optical dipole trap we used in our model for numerical simulations have been taken similar to the parameters of the experimental setup of Refs. 6,7 Namely, we consider the optical dipole trap formed by focusing 2.5 W Nd:YAG laser beam (1.063 1am) with linear polarization along x-axis into an area with diameter of about 5 m. Cesium atomstrapped in the optical dipole trap we assume two-level atoms with the lifetime of the excited state l/'-y = 3.07 x 108 and the dipole moment ofthe working transition d = 8.01 x 1O_18 CGSE. Frequenciesof atomic oscillations in radial and axial directions are equalto Wr 60 kllz and w 1.5 MHz, respectively. In the following, we will consider the red-detuned optical dipole trap configuration. Such a trap is formed by a laserbeam tuned far below the atomic resonance frequency. We will also assume that the laserbeam with power P
forming the trap has the Gaussian intensity profile:

I(r,z) = w2(z)

exp

(_2))
+

cos2(kz),

(1)

where is the radial coordinate andthe half-waist beam diameterw(z) depends on the axial coordinate z as

r

w(z)

=

wo1

()2

(2)

Here w0 is the beam waist diameterand ZR = irw/A is the Rayleigh length. As we mentioned earlier, the thermal energy kBT of the atomic ensemble should be much smaller than the potential depth of the trap and, therefore, movement of the trapped atoms in radial direction is reasonably small

Proc. SPIE Vol. 4750

105


compared to the beam waist diameter and in the axial direction movement of atoms is smaller than the Rayleigh length. In this case, the optical potentialof the trap can be approximated as:5

Ud(r,z) =

--Uocos2(kz)

--

[i

2

(r)2 (z)2]
2

.

(3)

The oscillation frequencies of the trapped atoms are equal to Wr (4Uo/mw)'I2 in radial direction and k(2U0/m)'/2 in axial direction, so that the potentialenergy reads then as

w

=

Ud(r, z) =
where (ni
--

(n1 -- n2)'irc2y

I

2w

i--) I(r,z),

1 '\

(4)

z1 =

fl2) is the difference between population of the low and upper atom levels, Z2

w(P112

2

=

S112)

are the transition frequencies.

w(P312

2

S112)

and

3. MODEL FOR THE LONG-RANGERESONANTDIPOLE-DIPOLE INTERACTIONS We assume that interaction of atoms trapped in the trap is the long-range RDDI, which is governed by stochastic process that leads, in itsturn, to astochastic radiationforce between atoms. Then, we suppose that the laser detuning z << w0 and atoms are located at a distance of the order of laser wavelength, R12 A. With these assumptions, we can calculate the spectrum of atomic radiationforce, keeping in mind that the dipole radiationof atoms for this purpose can be approximated with a white noise. SpectraN11 and N22 of the interaction force fluctuations between two atoms are equal and spectrum of single atom fluctuations takes the form:
N11

=

N22

= tiwd2
2irc5

(LcT1

_ _

)Iii,

(5) (6)

N12
1"2 where d = of dipole moments parallel

N21

= hwd2

2irc5 (Zcr1

2

)112,

to each other and

the transition dipole moment and &± are the atomic transitionoperators. For the case orthogonal to the vector of displacement we have

I127I
where
-- '1--

(Ii 0

p
12

\o

0

, 13)
O\ 0
-- --

Iii=rrl
--

, oJ
(jg 0
0 0
o

(7)

-

4(9 -- Pi2) cos (p12 4(3 --

4(9 4(3

12-- 13--

2)
4

P12

4

42) 5
P12
5

sinP12

,
,

(8) (9) (10)

cos col2

--

2p2) sin
--

P12 --4(12
--

3Pi2) cos P12
4

+ 4(9

12co2

+

Pi2) sinOl2

5 (P12

(zoj

Lo)

is the correlation function of the transition operatorthat reads

(aLcT)=

[(1

+

g)2

+

4(

--4gL4(1+482)

+

52)

+

4(1 + g)252

+ 4(g2 +

.

(11)

252)212

Here = gL/F is thedimensionlessRabi frequency, gj. = Ed/h is the Rabi frequency,S laser detuning, and 3 p12 cos(P12 -- sin(P12 + (P12 sin Pl2
(P12

g

= z/F is the

dimensionless

3

106

Proc. SPIE Vol. 4750


The correlation function for fluctuations of a single atom in the trap,

(ajza),

can be written as
2

=
[(1

2g4 [1 + 2(g2 + 52)2 + g2(1 + + g)2 + 4(g2 + 52) + 4(1 + g)262 +

4(

452)]

+

(13)

262)2]

To clarify the roleofthe long-rangeRDDI we suggest to shine atoms in the optical dipole trap with an additional weak laser field tuned into the resonance with the atomic transition. This will increase the RDDI between atoms and, therefore, lead to the increase in fluctuation amplitude of each atom in the trap. Varying the laser detuning 6 of the weak laser field and its intensity we can control the radiationforce between atoms.

4. MODELINGOF ATOMS DYNAMICS IN THE TRAP Motion of atoms in the trap is governed by the followingequation:
..(k) -- dip

Dr

-'-F -'-F ' ' RDDI'
cool

k-- 1' 2 '

14

where index k numbers the atoms

and

FDI 5

in the trap, Udj 5 the trappingpotential, is the coolingforce (linear friction), the fluctuation force due to the RDDI. The coolingforce can be written as9
Fcoo1

F1

--moi,

(15)

where
-- a--

2hw2

I

26/'yo

mc2 Io[1+(2Æ/o)2]2'

I

is the intensity of the weak laserfield usedto enhance the RDDI interactions in the trap, and is the saturation intensity of the atomic transition. The fluctuation force FRDDI can be modeled with a whitenoise because the correlation time, determined by the radiationdecay rates, is small in comparison with all other time scales of atom's dynamics in the trap. The joint correlation matrix for the fluctuating force FRDDI has the form:

I

N--i'
L

N11
1V12
TtT

\\

--N12'\ TtT

11 J

I
Lt

where N11 and N12 are determined by Eqs. (5), (6). In accordance with Eq. (7), the only correlated projections in this matrix are because of the N12 matrix element. The corresponding correlated pair of randomvalues x1, x2 can be determined then as ____________ ____________ /6(Nii + N12) /6(N11 -- N12)

zt zt

X2V

/6(N11

+

N12)

1+\

/6(N11

--

N12)

where Lt is the time step in the course of computer simulation. Here are the independent randomvariables uniformlydistributedin the range (--1/2, 1/2), 50 that we can simply replacex/t with the integral x(t)dt, which for the white noise. gives us correct dispersion

i, 2

Nt

f

Proc. SPIE Vol. 4750

107


cto
010 coo
(ia)

I

I

I

cto
oio
.

(q)

0_z=7

coo

JE--

p

000
coo01.0-

P

000
coo-

010cLo___________________________________________ 0
0z.o-

Ãj-

0

I7

oq
ssoJuo!suouxp

qij

JWPT

oioj

oq

ADuonboij 76

smo SflSIOA uxoiou ousp u) pu ISIJ uunop à (q) ()
uoo

OAA (stp2inj ioj

uoiojjp

SOfl?A

jo oq

IaLfld1ATOD NOIJ1VIf1JATIS SJ'If1SEfli
uj iondmo 'uouxiodxo OM OZi1U OT OJOi JO UO1OU JO SWO UitJiAt Oi?I SOUSip uflJoiuoootJ s:!trnrnIcp jo smo u oq 'der tpqix spuodop uo oqj Asuoiiq prn uunop jo oq oqoid ios uroq ouonjjuj jo iojjnq s1 oq iojjnq swo u oq thi oveq ou uooq UO)[ 04W UflO3 UI 1110 pu po suosjjoz ujd xou OM poju pjoz suoso smo poddii u oq Tiuoodo1z!ux mupu jo oq o op

oj u is 01di js sj smo ujzso
oq uoooq
oie

qt u

si

uossnasp jo iondmo

uoenuissjnsoi

pooo u oq jinuoodoioiwununu suoiisod si jnbo o oioz SITJJ uoooq podthu smo s Auo onp o oq smo suonnjj (suoJi!zso) punoie swo
Jiuoodoi3pn
'osjy

poddi u oq Juoodo1!m uxuui jo oq piido ios iuoq uuuoj oq oodp dii put) oq

Azuopuodop uo oip

ousp

is

jdo

oj qp (°i
uooioq
osn jo

ioj

swo

SZYUfl?UAp

smo s uoqs u
oq poiod
sV

U

OTfl

dii

Jo 'g/y OiOA V s!

sJxoIToJ TuoiJ

oq

oApoJJo

jjp.j oioj

opnJdwi

SOSOiOp tJTAt UtSODUi 01P

'P' iq !mo1oU
OTfl

·!d oq IGEfl:I oioJ ioJ suom °-I IGG}I uopiu! mnpqJnbo suo!!sod u oq
·o3us!p

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ido
u!p?

qAt OtI

oq smo u

IGUIT OiOJ

oq dei

ujopoj,s oq Liopofu jo smo u oq uxopui SUOAO inoij oq mniqnbo suosod ponqisp uipioco °'-i TT°'JAI uonqqsp jnbo o oq puom1odx flJA jo o jr1 oos) joij

di

irnuui ioj oinb

j s os Anssoou oio sq ju omso oq Tiuoodoi3!m uxupn S1?i oqoid iosT 'moq
o&

uoJ .oU1

(

u

°u

oq uxuux jo oq

oq

y sq

io

oiniodmo

soupioo jo oq smo s (j) ot os sjuoodoiui pu 1IOq [i4iUi SOtT3OJA uo oq inioduio JO 'SfflO TpqM os spuodop

b

OAt

pozijo U

OT

IuOodoi!m

miso

oqoid

iosj uroq uuqs oq smo u oq
oq 'iouo
tLL

qq

s

poiiojsui

di ou

wo Aq i

pno

)UflJ UIOTJ

flO JO uooqd woJJ otI
(9T)

X g_OT ·M

ou pooxo oq qdop jo juood ooq jo oq dq u oq iouoc jo oq uroq 'pun 'oioJoiotI oq oqoid iosj moq iu' ou jupj oq smo no moij oq ituoodoicnui -ITh oTI %OOT oq suooqd moij ioye ojus uooqd uossuio sqj sojquo sn o osn u mo suonez uosnjpp qzoidth ioj oq AJqqoid

sqj

ioqmnu soop

ssoooid jo

mo-ouo iodS Jo suonnjj jo suxo u sTuoodoiD!m jo oq I?Dido OZWflJ OI UAOtJS '! T mnipods s Apuosso poippuo ioj oq os jo uiioui smo qi podsoi o oq OSZ JO oIU!s 'mow oq uo!Ioiio uoxoq OA Ia?:L!I sozioj y) = 'i ( U! jjpodso u oq oj Azuonboij uooi sqj 'sireow ooi boioq s qzq s ot Ajuo msutpouisunup jo A1A mo s upudopu moiju smo pui upuodsonoz (jT) ' uidnoo oip 'smo sjd u ·sDp1I1;rncp jj ou uoonoc 6) = (o ' oq qo ioqo

smo uo!pi odzso

o tj

J

oTLL

108

Proc. SPIE Vol. 4750


I
Frequency,MHz Frequency,MHz

Figure 4. Spectraof fluctuations of a single atom (a) andone of the two atoms in the trap (b) along z-axis.

0

N"

.
E
0.0 0.1 0.2 0.3 0.4 0.5

E

.

-5

-10
-15
0.1

0.2

0.3

0.4

0.5

t,

ms

t, ms

Figure 5. a) Temporal dependence of the axial coordinate of a single atom (solid line) and two atoms in the trap (dotted and dashed lines for each atom, respectively) with the same initial conditions. b) Temporal dependence of the axial projection R12 of the distance between two atoms in the optical dipole trap. spectrumis shownin Fig. 4a. Having fluctuation forces depending on both atomscoordinates (see Eqs. (6), (11)), we get an essential modificationsof their dynamics. Interaction via common photons emittedby the two atoms provides both correlation oftheir dynamics and its modification. Actually, the effect of a correlated escape of atoms in pairs from the trap shown in Fig. 5 can be associated with the nonzero fluctuation spectrum intensity at zero frequency, keeping in mind that the latter can correspond to the presence of zero center-of-mass velocity contribution in the total spectrum of an atom. Fluctuations of a single atom also lead to the escaping of atom from the trap. That is why for the studying of fluctuation power of interactionbetween atoms it is necessary to compare the dynamics of two atoms with the dynamics of single atom with the same initial conditions. As a result, we can determine how one atom affects the dynamics of another atom. For this purpose, we model the dynamics of a single atom and two atoms in the trap with the same initial conditions and with the same fluctuation statisticsof a single atom (Fig. 5). Analyzingatoms dynamics in the trap, we can see that both atoms leave the trap simultaneously (Fig. 5a). Simulating the dynamics of two atoms in the trap we can quantitatively and qualitatively compare the trajectories ofa singleatom and one ofthe two atomsin the trap at any moment oftime. This enables us to analyze the interaction between atoms long before the atoms escape from the trap. Analyzing, for example, temporal dependency of total energy of a single atom and one of the two atoms in the trap (Fig. 6) one can conclude that the increase of energy of one of the two interacting via RDDI atoms in the trap by contrastwith more or less a monotonous dependency of a single atom in the trap is a sign of escaping atoms from the trap. Surely, decreasing time along which we observe
Proc. SPIE Vol. 4750 109


-0.3

-0.4
-0.5

-0.6 -0.7 -0.8
0.0

0.5

1.0

1.5

2.0

1,

ms

Figure 6. Temporal dependency of total energy of a single atom (dotted line) and one of the two atoms (solid line) in the trap. Initial conditions for the atoms are the same. the atoms dynamics in a computer experiment, we decrease the accuracy of of our resulting conclusions. Therefore, special efforts have been made in our computer simulations towards selection of optimal parameters, namely the number of iterations in time, the time interval, an others.

6. CONCLUSIONS
In conclusion,theoretical study and computer simulation results for stochastic dynamics of two atoms trapped in an optical dipole trap underthe actionof a proberesonant radiationare presented. It is shown that interaction of atoms via common emitted photons reveals in an essential modification of the atoms dynamics. This modification leads to the enriched spectrumof atoms fluctuation dynamics and pair-correlated radiativeescape. The model discussed
here can be successfullyused for analyzing atoms dynamics in optical dipole traps.

ACKNOWLEDGMENTS
This work was partiallysupportedby the State Science-Technical Programs ofthe Russian Federation "Fundamental Metrology" and "Nano-technology"and INTAS grant no. 00--479.

REFERENCES
1. A. Gallagher and D. E. Pritchard, Phys. Rev. Lett. 63, 957 (1989). 2. P. A. Willems et al., Phys. Rev. Lett. 78, 1660 (1997). 3. B. Ueberholz, S. Kuhr, D. Frese, D. Meschede,and V. Gomer, quant-ph/9910120 (1999). 4. B. A. Grishanin, "Stochastic Dynamics of TrappedAtoms under the Radiation Force", 1997 (unpublished). 5. R. Grimm, M. Weidemuller,Y. B. Ovchinnikov,Adv. Atom. Mol. and Opt. Phys. 42, 95 (2000). 6. D. Schrader, S. Kuhr, W. Alt, M. Muller, V. Gomer, D. Meschede,quant-ph/0107029 (2001). 7. 5. Kuhr, W. Alt, D. Schrader, M. Muller, V. Gomer, D. Meschede, Science (2001), Published online June 14 2001; 10.1126/science.1062725. M. Ben Dahan, E. Peik, J. Reichel, Y. Castin,and C. Salomon, Phys. Rev. Lett. 76, 4508 (1996). 9. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge Univ. Press, p. 800 (1995). 10. Note that cold collisionswere suggested to be used for loading the optical lattice in the trap in a configuration when only one atom is located in every micropotential minima.11 11. M. T. DePue, C. McCormic, S. L. Winoto, Phys. Rev. Lett. 82, 2262 (1999).
8.

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