Anisotropic velocity distributions in 3D dissipative optical lattices

7 Pages·0.26 MB·English

by  Johan Jersblad

EPJ manuscript No.
(will be inserted by the editor)
Anisotropic velocity distributions in 3D dissipative optical lattices
J.Jersblad1, H.Ellmann1, L.Sanchez-Palencia2, and A.Kastberg1,3
3 1 Department of Physics, Stockholm University,S-106 91 Stockholm, Sweden
0 2 Laboratoire Kastler-Brossel, D´epartementdePhysiquedel’Ecole NormaleSup´erieure,24rueLhomond,F-75231 Paris cedex
0 05, France
2 3 Department of Physics, Ume˚aUniversity,S-901 87, Ume˚a, Sweden

n
a Received: date/ Revised version: date
J

9 Abstract. We present a direct measurement of velocity distributions in two dimensions by using an ab-
sorptionimagingtechniqueina3Dnearresonantopticallattice.Theresultsshowacleardifferenceinthe

1 velocitydistributionsforthedifferentdirections.Theexperimentalresultsarecompared withanumerical
v 3D semi-classical Monte-Carlo simulation. The numerical simulations are in good qualitative agreement
4 with theexperimental results.
3
0
PACS. 32.80.Pj Optical cooling of atoms; trapping
1
0
3
1 Introduction where v2 is the mean square velocity of the released
0
h i
/ atoms,kB istheBoltzmannconstantandM istheatomic
h
mass.Arobustresultinallstudiesisthatthetemperature
p
Anopticallatticeis aperiodicopticallightshiftpotential scales linearly with the irradiance divided by the detun-
-
t createdby the interference of laserbeams in which atoms ing, that is linearly with the light shift at the bottom of
n
canbetrapped.Usuallyonedistinguishbetweentwotypes the optical potential (U). This is in excellent qualitative
a
u oflattices,near-resonanceopticallattices(NROL)[1]and agreement with 1D-theoretical predictions [7].
far-off resonance lattices (FOROL). In the later type, an
q Nevertheless,inseveralworks(for example[3],[5]and
: atom can only be trapped, whereas the former (the one
v thiswork)afourlaserbeamconfigurationresultsinaface
considered in this paper) also exhibits an inherent cool-
i centeredtetragonallattice that cannotsimply be reduced
X ing mechanism (Sisyphus cooling). The Sisyphus cooling
to three 1D cases. Indeed, all spatial directions are not
mechanisminanNROLhasbeen the subjectofextensive
r equivalent(seesection2.1)andtheparticulargeometryof
a research due to its high cooling efficiency, but also since
the lattice hasto be takeninto account.Inarecentpaper
an optical lattice is a very pure quantum system suitable
by the Grynberg group [8], the dependence of tempera-
for fundamental studies of atom-light interaction.
ture and spatial diffusion on geometric parameters con-
Theoretical studies of the atomic motion in NROLs
trolling the lattice spatial periods (lattice constants) in
have been done in 1D and 2D, both analytically and nu-
different directions was studied. For different laser beam
merically. The extension to 3D configurations is however
configurationsproducingtheNROL,thetemperatureand
cumbersome. Analytical solutions become unwieldy and
spatial diffusion coefficient were measured for tetragonal
numerical simulations require long computation time, es-
lattices(seesection2.1)withdifferentaspectratios,i.e.as
peciallyforhighangularmomentumtransitions.Thusvery
afunctionoflattice constants.Itwasshownthatthe spa-
few detailed studies have been made in 3D. An exception
tialdiffusioncoefficientstronglydependsonthedirection.
is the workby CastinandMølmer [2] who studied spatial
The temperature, which was measured in one direction,
andmomentumlocalizationviafullquantumMonteCarlo
was found to be independent of the lattice spacing. The
wavefunction simulations in the case of optical molasses.
difference between spatial directions lies not only in the
Measurements of temperature have been made on 3D latticeconstants,butalsointhemodulationsofthelaser-
NROLsbyourgroup[3,4],andbygroupsatNIST[5]and atom interaction parameters (optical potentials and opti-
in Paris [6]. In all these experiments, and in this work, calpumping)insuchawaythatdifferentbehaviorsofthe
the kinetic temperature is derived from measured veloc- temperaturealongdifferentaxesispossible.In[9]theSisy-
ity distributions along one axis and is defined as a direct phus cooling effect in a 3D tetragonal NROL was studied
measure of the kinetic energy through theoretically. With a simplified choice of atomic angular
momentum,itwasshownbyasemi-classicalMonte-Carlo
M v2 calculationthatthetemperaturealongagivencoordinate
T = h i, (1)
axisisindependentofthe latticeconstant,butindeeddif-
kB
2 J.Jersblad et al.: Anisotropic velocity distributions in 3D dissipative optical lattices
ferentalongdifferentdirections.Forthesamegeometryas z
considered here, the linear scaling parameter of the tem-
perature differs by a factor of 1.4. Moreover, a compari-
sonbetween[3]and[5]suggestssuchananisotropyofthe ex
2q
velocity distribution. In both experiments, the direction ex
of measurement coincided with the direction of gravity, y
but this direction did not correspond to the same lattice
axis.Itturnsoutthattheseworksyieldaquantitativedis-
crepancy.Thederivedtemperaturewasfoundtobelinear ey
with U with proportionality constants of 12 nK/ER and 2q x
24 nK/ER (in [3] and [5] respectively), where ER is the
recoil energy 1. The difference in scaling factor called out ey
for a more thorough investigation, which would rule out
any systematic error.
This work aims at a direct comparison between the Fig. 1. Beam configuration of the3D lin⊥ lin opticallattice.
kinetic temperatures along different directions in a 3D
Twobeam pairsinthexz-andyz-planesrespectively,orthog-
NROL. Measurements of velocity distributions along dif-
onallypolarized alongthey-andx-axesrespectively,makean
ferentdirectionsweremadefordifferentlatticeparameters angle θ=45◦ with thez-axis.
(potential depth and detuning) by absorption imaging of
an expanding atomic cloud. The experimental results are
compared with a 3D semi-classical Monte-Carlo simula-
tion performedfor the actualatomic angularmomentum. the kinetic temperature. This short falltime of the AOM
Thepaperisorganizedasfollows.Insection2.1wede- avoidsadiabaticreleaseoftheatomsintheopticallattice.
scribe the experimental set-up. The experimental data is The optical lattice is a 3D generalization of the 1D
presentedwithderivedkinetictemperaturesinsection2.2. lin lin configuration created by two orthogonally polar-
⊥
In section 3, we describe the numerical calculation and ized pairs of laser beams that propagate in the yz- and
present the result for the kinetic temperatures. In sec- xz-planesrespectively[1].Theanglebetweenthebeamsof
tion 4 we discuss the results from the experiment and the eachpairis90◦,andeachbeamformsanangleofθ =45◦
simulations. Finally, in section 5 we draw conclusions on with the (vertical) quantization (z-) axis (see figure 1).
our work. This results in a tetragonal structure with alternating
sites of pure σ+- and σ−-light, where potential minima
areformed.Fromfigure1,itisclearthatdirectionsx and
y are equivalent but that direction z is different. It fol-
2 Experiment
lows that the optical pumping rates and the light shift
modulations are different along z compared to x or y. In
2.1 Experimental setup
figure2weplotthe projectionalongx andz ofthe lowest
adiabaticpotential,which is where the atomsspend most
Initially, a magneto-optical trap (MOT) is loaded with
of their time [1]. Two main anisotropic properties arise.
N 2 106 cesium atoms (133Cs) from a chirped deceler-
ate≈d at·omic beam in 4 s. This gives a peak number den- First,thelatticeconstantsaz =λ/(2√2)andax,y =λ/√2
aredifferent.Second,the shapes ofthe potentials arealso
sity of n0 ≈ 5 · 1010 cm−3. The MOT operates at the clearly different. In particular, they show different poten-
(Fg = 4 Fe = 5) transition at 852 nm (the D2 line),
→ tial barriers to escape adiabatically from a potential well
where F is the total angular momentum quantum num-
(lower along the z-direction than what it is along the x-
ber.Due to off-resonantexcitationto Fe =4,arepumper
and y-directions by a factor of 1.65) and show different
beam resonant with the (Fg =3 Fe =4) transition is
alsoused.Afterturningofftheloa→ding,theatomsarefur- reduced oscillating frequencies (ωiai/λ) at the bottom of
the potential wells.
ther cooled in an optical molasses for about 20 ms. From
The velocity distributions along the z- and x-axes are
the opticalmolasses,an atomic cloudat a temperature of
measured using a well known absorption imaging tech-
T =3µK is loadedinto the opticallattice witha transfer
nique [10]. After release from the lattice, a short (50 µs)
efficiency of about 50%. The filling factor of the lattice is
resonant probe pulse (Fg =4 Fe =5) hits the atomic
around 0.2 %. The optical lattice beams are red detuned →
cloud. The irradiance of the probe pulse is I I0, where
fromthe (Fg =4→Fe =5)resonance,typicallybetween I0 =1.1mW/cm2isthesaturationirradiance.≪Theshadow
∆5 =−10Γ and ∆5 =−40Γ, where Γ/2π=5.2 MHz is intheprobebeamisimagedontoaCCDcamera.Bycap-
the natural linewidth. The atoms equilibrate in the lat-
turingimagesatdifferenttimedelaysafterturningoffthe
tice for 10 ms and are then released by turning off the
opticallatticebeams,weextractthedifferentspatialden-
optical lattice beams, with an acousto-optical modulator
sitydistributionsfromwhichvelocitydistributionscanbe
(AOM) in less than 1µs, followed by a measurement of
derived. The velocity distribution in the z-direction (di-
1 The recoil energy ER=(¯hk)2/2M, wherek=2π/λ isthe rection of gravity) was compared to the results obtained
wave vector, λ is the wavelength of the light, and M is the with a ”time-of-flight” (TOF) method [11], showing good
atomic mass. agreement.
J.Jersblad et al.: Anisotropic velocity distributions in 3D dissipative optical lattices 3
Fig. 3. Typical2D densityprofilesacquired at twodifferenttimes τ after releasing theatomsfrom thelattice. Theleft image
shows an atomic cloud after τ = 12.8 ms expansion together with density profiles in the z- and x-directions. The right image
shows an atomic cloud after τ =36.8 ms.
Fig. 2. Lowest adiabatic optical lightshift potential projected
in thexz-planein units of theoptical wavelength, λ.
2.2 Measured Kinetic Temperatures Fig. 4. Tx (filled) and Tz (open) as a function of modula-
tion depth, U0/ER, for three different detunings (∆5 =−10Γ
(squares) , −20Γ (triangles), −30Γ (circles)). The solid and
The 2D projection (in the xz-plane) of the expanding
dashed line are linear fitsto thedata.
cloud is recorded at two different time delays, τ1,2, af-
ter extinction of the optical lattice beams. Typical values
are τ1 = 12 ms and τ2 = 35 ms. Examples of 2D den-
well. For sufficiently high irradiances and temperatures,
sity profiles are shown in figure 3 together with Gaus-
sian fits to the spatial density profile along x and z. Ex- it is obvious that the universal scaling with U0 prevails
cellent agreement with Gaussian distributions is found. foreachdirection.However,this scalingwithU0 isclearly
From the fits, we extract the rms radius, σ , (i = x,z), differentfordifferentdirections.ForlargeU0,thetemper-
i ature alongz is found to be significantly smaller thanthe
of the clouds which increases with time, t, according to
σ2(t)=σ2(0)+v2t2 [12]. temperature along x. Linear fits to the data yield
i i i
The kinetic temperature in different directions is de-
Tx = 0.55+0.022(U0/ER) µK (4)
fined as
T = M σi2(τ2)−σi2(τ1). (2) Tz =(cid:0)0.62+0.012(U0/ER)(cid:1) µK. (5)
i kB τ22−τ12 That is, the ratio(cid:0)between the scaling p(cid:1)arameters along x
In figure 4 we plot derived kinetic temperatures along x andz isdeterminedtobe 1.8(0.3).However,atlowmod-
and z for three different detunings, as a function of U0, ulation depths and low temperatures, the temperatures
which is the modulation depth of the diabatic optical po- are found to be approximately the same along z and x.
tential. Here, U0 is defined as
¯h∆5 44 Ω2 3 Numerical Simulations
U0 = |2 |ln 1+ 45 2∆2 , (3)
(cid:20) (cid:18) (cid:19) 5(cid:21)
3.1 Theoretical framework
where Ω2 = (Γ2/2)/(I/I0) is the square of Rabi fre-
quency and the irradiance is I = 8Ibeam (Ibeam is the Wehaveperformedsemi-classicalMonte-Carlosimulations
irradiance of a single beam), at the center of a potential in3Dfortheactual(Fg =4 Fe =5)transitionof133Cs.
→
4 J.Jersblad et al.: Anisotropic velocity distributions in 3D dissipative optical lattices
Themainfeaturesofthemethodhavebeendiscussedelse- are related to the position-dependent coefficients appear-
where [9,13] so here we just recall the main elements and ing within the FPE. In a (µ-indexed) space base where
peculiarities for our multidimensional configuration. themomentumdiffusionmatrix D (seeappendixB)
n,m
{ }
The optical Bloch equations (OBE), which describe is diagonal, the first two moments of fm and δpn,m read
the evolution of a sample of two-level atoms (with Zee-
man degeneracy)coupledto both laser fields and vacuum 2Dµ,µ (r)
fµ =0 and (fµ)2 = m,m
modes, are the starting point of the analysis. Because of h mi h m i dt
thecoolingeffectsandthedecoherenceduetophotonscat- 2Dµ,µ (r)
tering, the atomic cloud dynamics can be reduced to a δpµ =0 and (δpµ )2 = n,m . (8)
h n,mi h n,m i γ
semi-classical picture for a large range of lattice parame- n,m
ters [14]. The OBE are therefore converted into a set of
coupledsemi-classicalFokker-Planckequations(FPE)via
Wignertransforms.ProjectingtheFPEontotheposition-
3.2 Numerical results
dependentadiabaticstatesbase Φ (r) (seeappendixA)
m
| i
and neglecting the coherence terms which are unimpor-
The numerical simulations are performed for a typical
tant in a semi-classical description, one gets a new set of
sample of 300 independent atoms. For the lattice param-
FPE only involvingthe localpopulations of the adiabatic
states2. eters considered in this work, the kinetic energy reaches
′
steady-state in a time of approximately 4000/Γ , where
By physical interpretation of the FPE, it follows that
′
the atomic cloud dynamics can be reduced to internal Γ =Γs0/2 is the totalscatteringrate ands0 is the satu-
state transitions via optical pumping at a rate γ from rationparameter(seeappendixA).Theaveragesoftheki-
n,m
Φ to Φ ,andtheevolutionofeachatominagivenin- neticenergiesinsteadystateinthex-,y-andz-directions
n m
|ternial Φ| -istateduetodeterministicforces.Theseforces provide the kinetic temperatures in the corresponding di-
m
are firs|t ofiall due to the optical potential modulation (- rections,
∇Um) and secondly, due to the radiation pressure force T = Mhvi2i. (9)
(F). Moreover, the atomic cloud undergoes momentum i kB
diffusion due to photon scattering.
The simulations were made for three different detun-
It is then straightforward to show that the FPE solu-
ings (∆5 = 10Γ, 20Γ, 30Γ). For each detuning we
tion is formally equivalent to the integration of a set of − − −
acquired velocity distributions, in each direction, at six
Langevin equations interrupted by internal states quan-
different modulation depths. Note that the chosen mod-
tumjumps,eachoneaccountingfortherandomtrajectory
ulation depths are much higher than in the experiment
of a single atom. The quantum jumps are taken into ac-
since the semi-classical model breaks down when the mo-
countbygeneratingarandomnumberr ateachtimestep
mentumdistributionbecomestoonarrow.Thisisbecause
which is compared to the transition probability γ dt
m,n deep modulation depths are required to avoid non-adia-
from Φ to Φ (with n=m) during the time step dt.
m n baticmotionalcouplingsbetweenadiabaticsublevelsthat
| i | i 6
Inthefollowing,wedefiner as1ifaquantumjumpoc-
n,m arenotincludedinourtreatment[13].Moreover,thetime
curs from n to m and 0 otherwise. Between two quantum
toreachsteadystateincreasesforlowmodulationdepths.
jumps, the elementary evolution of the atom is
However,thelinearscalingshouldstillhold.Theresultsof
thenumericalsimulationsareshowninfigure5.Here,the
P(t)
dR(t)= dt (6) kinetictemperatureisplottedasafunctionofmodulation
M depth for the detunings mentioned above. The tempera-
dP(t)= ∇U dt+ r δp +F dt ture scales linearly with the light shift independently of
− m n,m n,m n,m the detuning according to
n6=m
X (cid:0) (cid:1)
+ 1− rn,m (fm+Fm,m)dt, (7) Tx ∝0.035(U0/ER)µK (10)
(cid:0) nX6=m (cid:1) Ty 0.035(U0/ER)µK (11)
∝
where R and P are the atomic position and momentum Tz 0.013(U0/ER)µK (12)
∝
respectively. The Hamiltonian force, (-∇U ), is derived
m Asintheexperiments,theresultsofthesimulationsshow
from the adiabatic potential in state Φ , and F is
| mi n,m a clear difference in scaling of the kinetic temperature
the averageradiationpressurein case of a quantum jump
along the z-axis compared to the x- and y-axes, here, by
from n to m (if m=n, no jump occurs). The momentum
a factor of 2.7.
diffusion is determinedby randomvalues:the momentum
kick undergone by the atom in case of a quantum jump
from n to m, δp and the recoil mean force in the ab-
n,m 4 Discussion
sence of a quantum jump, f . Note that δp and f
m n,m m
2 Note that the adiabatic approximation is justified by the Theresultsfromtheexperimentalworkandthenumerical
fact that the adiabatic state splittings are generally greater simulationsarecompiledintable1.Acomparisonshowsa
than themotional couplings in the regime of deep potentials. quantitativeexcellentagreementbetweenourexperiments
J.Jersblad et al.: Anisotropic velocity distributions in 3D dissipative optical lattices 5
tice wells in agreement with former experimental inves-
tigations, for instance [5], and full quantum Monte-Carlo
simulations[2].Thedifferenceinthescalingfactorsispro-
portionalto the difference inthe modulationdepth ofthe
lowestadiabaticopticalpotentialinthe correspondingdi-
rections. Therefore we conclude that it is this difference
which induce anisotropic kinetic temperatures in the op-
tical lattice. This conclusion is not incompatible with the
resultsof[8]inwhichthesteady-statekinetictemperature
was measured for different lattice constants showing that
the steady-state kinetic temperature was independent of
the lattice spacing, because the geometricalanisotropyin
thelatticedonotreducetoasimplescalingfactorbetween
directions x,y and z.
At low modulation depths, the lattice reaches a min-
imum temperature followed by a sharp increase in tem-
perature,usuallycalledd´ecrochage.Whenlasercoolingis
Fig. 5. Kinetic temperature, along the x- (circles), y-
still effective there exists a region where the temperature
(squares)andz-direction(filledtriangles)asafunctionofmod-
is isotropic.However,this regionis difficult to analyzefor
ulation depth, U0/ER, for three different detunings (∆5 = several reasons. For instance, at low modulation depths
−10Γ,−20Γ,−30Γ). The solid and dashed lines are linear fits
the atomic localization in a trapping site is less strong,
to thedata.
andthus the anharmonicityof the potential wellbecomes
more important.This could lead to anincreasedcoupling
betweenthedifferentmotionaldirectionsandalsoabroad-
Table 1. The scaling parameter ξx,y,z, (in units of nK/ER),
eningofthevibrationallevels,i.e.increasingthetunneling
intheequationTx,y,z =T0+ξx,y,zU0fordifferentstudies.The
rate inthe lattice.Another effect that must be takeninto
experimental errors of the slope for this work is the quadratic
accountatlowmodulationdepths is increasedspatialdif-
sum of the statistical error and an estimated maximum sys-
tematical error. The errors in the simulation is the statistical fusion [8,16]. This means that the loss rate of the atoms
error from thefit. in the lattice becomes larger,and thus the signal-to-noise
in the absorption images decreases. Furthermore, if the
ref [3] ref [5] thiswork this work thermalexpansionoftheatomiccloudintherecordedab-
(experimental) (simulations)
sorptionimagesissmallcomparedtothesizeofthecloud,
ξx - 24(2.4) 22(3.5) 35(1.2) due to spatial diffusion, there will be large uncertainties
ξy - - - 35(1.2) in the extracted temperatures.
ξz 12(1.2) - 12(2.5) 13(1.0)
5 Conclusions
and former studies in which the kinetic temperature was
measuredalongx [5]oralongz [3].Thenumericalsimula-
We have measured the velocity distributions in a 3D op-
tionsalsoreproducethedifferenceinscalingparameterfor
ticallattice of cesium alongtwo non-equivalentdirections
differentdirectionswasmeasuredinthe experiments,and as a function of lightshift (U0). In agreement with previ-
confirms the appearanceof a discrepancy between kinetic
ousworks,thekinetictemperaturescaleslinearlywithU0.
temperatures along x-y and z.
Asanoriginalresult,wehavefoundthatthedistributions
The inherent cooling process in an optical lattice for areclearlyanisotropic(withT >T ).Theexperimental
x,y z
atoms with kinetic energy EK > U0 is Sisyphus cool- resultsareingoodagreementwitha3DnumericalMonte-
ing. This process was explained by Dalibard and Cohen- Carlosimulationandweconcludethatitisthemodulation
Tannoudji in [7] in the case of a theoretical transition depth of the adiabatic optical potential that determines
(Jg =1/2 Je =3/2).TheSisyphuscoolingcycleoccurs the steady-state kinetic temperatures. The anisotropy in
→
untilthe atomickineticenergyislowerthanthe potential kinetic temperature is not paradoxical. In fact the ”ki-
barrierinaparticulardirectionandthusdoesnotdepend netic temperature” here is defined as a simple measure
onanyotheranisotropy(thelatticespacingsforexample). of the atomic kinetic energy (see Eq. (1)) and not as a
However, for higher angular momentum transitions, thermodynamical temperature. This is because thermal-
thecoolingprocessdoesnotstopbecauseotherrelaxation ization in Sisyphus cooling do not result from energy ex-
processesthanstandardSisyphuscoolingcouldstilloccur change beetween particles via collisions, but from atom-
[13,15]. For example, atoms in bound states within a lat- photon interactions. Our result show that no thermody-
tice well can be excited to unbound states, followed by namical temperature can be defined for Sisyphus cooled
decay to lower lying vibrational states. We find that the atomic samples because of the violation of the equipar-
atomic kinetic energy is EK U0/10 and thus that the tition theorem [17]. Our results can give important clues
∼
atoms are very well localized at the bottom of the lat- for a full understanding of the cooling mechanism in an
6 J.Jersblad et al.: Anisotropic velocity distributions in 3D dissipative optical lattices
optical lattice. Furthermore, knowledge about the veloc- plus a relaxation part. In the semi-classical limit, the
ity distributions in all directions is importantin precision position-dependentadiabaticstatesaredefinedastheeigen-
experiments utilizing optical lattices. states of the light-shift operator h¯∆s0A:
2
LSPthankstheswedishgroupforwarmhospitalityduringthe ¯h∆s0A(r) Φ (r) =U (r)bΦ (r) . (19)
m m m
2 | i | i
period when a part of this work was achieved. He also ac-
knowledgesfinancialsupportfromtheSwedishFoundationfor Note that in gebneral Φ (r) and U (r) cannot be cal-
m m
International Cooperation in Research and Higher Education | i
culated analytically.
(STINT). We would like to thank Dr. Peter Olsson at Ume˚a
University for letting us use the LINUX cluster and also for
support during the simulations at the theoretical physics de-
B Dynamics coefficients for the Langevin
partmentatUme˚aUniversity.Thisworkwassupportedbythe
Swedish Natural Sciences Research Council (NFR), the Carl equation
TryggerFoundation,theMagnusBergwallFoundationandthe
Knut& Alice Wallenberg Foundation. In this appendix, we give the general expressions for the
dynamics coefficients involved in the FPE and Langevin
equations for Sisyphus cooling in the low saturation and
A Optical Bloch equations and adiabatic semi-classical regime. The transition rate from state Φn
| i
to state Φ (for m=n) is
states | mi 6
γ =Γ′ Φ B Φ 2. (20)
n,m 0 n q m
Thisappendixaimsatintroducingtheadiabaticstatesfor |h | | i|
q=±,0
a general J J +1 transition atomic sample. Consider X
→ b
an atom of dipole operator D = (d+ +d−) , with d± The average radiation pressure term in the direction i
D
being the raising and lowering components of D, and (i=x,y,z) is
D
the reduced dipole moment. Tbhis atobm intebracts with tbhe
laser field b Fi = ¯hΓ′ Im Φ ∂ B† Φ Φ B Φ
n,m − 0 h m| i q| nih n| q| mi!
EL(r,t)= E20ǫ(r)e−iωLt+c.c. (13) q=X±,0 b b (21)
and the momentum diffusion matrix is
,whereE0 is the amplitude ofthe electric field,ωL is the ¯h2Γ′
laser frequency and ǫ(r) is a vector describing the spatial Di,j = 0 Φ ∂2 AΦ δ
n,m 8 h n| i,j | mi n,m
varying profile of the laser polarization. The operators A
¯h2k2Γ′
and Bq represent the hermitian conjugates of the opti- + 0δ b Φ B† Φ Φ B Φ
cal pumping cycles (absorption of laser photons followedb 4 i,j h m| u| nih n| u| mi
u∈x,y,z
by embission ofstimulated or spontaneous photons respec- X
u6=i,j b b
tively), and are defined as
¯h2Γ′
A = d− ǫ∗(r) d+ ǫ(r) − 8 0 hΦm|∂i2,jBq†|ΦnihΦn|Bq|Φmi
· · · q=X±,0(cid:16)
h i h i
Bbq = db−·ǫ∗(r) · db+·eq (14) −hΦm|∂biBq†|ΦnihΦnb|∂jBq|Φmi
with qh=0, oriq =h x,y,zi (15) +c.c. (22)
b b ± b b b
(cid:17)
where
where δ is the Kroneckersymbol(1whenα=β and0
e ie α,β
e± = ∓ x− y and e0 =ez (16) else) and i,j denotes the spatial directions (x,y,z). Note
√2
that for the sake of simplicity, the spontaneous emission
are the circular basis vectors. After elimination of the pattern is simplified in a way that the photons are re-
excited state in the low saturation regime, stricted to be emitted only along the x-, y- and z-axes.
Thisapproximationis justifiedbecausethekinetic energy
Ω2/2 is expected to be greater than the recoil energy [18].
s0 = ∆2+Γ2/4 ≪1, (17)
the atomic sample dynamics is governed by the OBE References
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H = +h¯ A(r) (18) (2001)
2M 2
b
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