Weakly bound states of polar molecules in bilayers

0.08 MB·English

by  D. V. Fedorov

Noname manuscript No.
(will be inserted by theeditor)
Weakly bound states of polar molecules in bilayers⋆
D.V. Fedorov · J.R. Armstrong ·
N.T. Zinner · A.S. Jensen
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1
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2 Received: date/Accepted: date

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a
Abstract We investigate a system of two polarized molecules in a layered trap. The
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moleculesresideinadjacentlayersandinteractpurelyviathedipole-dipoleinteraction.
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Wedeterminethepropertiesofthegroundstateofthesystemasafunctionofthedipole
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moment and polarization angle. A boundstate is always present in thesystem and in

] the weak binding limit the bound state extends to a very large distance and shows
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universal behavior.
u
cl Keywords dipole-dipoleinteraction·2Dgeometry·weaklyboundstates·universality
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m
t 1 Introduction
a
.
s Cold gases of polar molecules represent an interesting opportunity to study the in-
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terplay of long- and short-range interactions in the controllable environment of the
i
s magneto-opticaltrap[1].Oneofthepossibleconfigurationsisanopticallatticewitha
y
multi-layered stack of planar traps.
h
p Weconsiderasystemoftwopolarmoleculesinadjacentlayers.Theuniqueproperty
[ of the system is that the interaction between molecules is pure dipole-dipole. This

interaction has the integrated strength equal to zero – the attractive and repulsive
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partsofthepotentialareofequalintegratedstrength.In2D-geometrysuchinteraction
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always supports a bound state [3]. We investigate the energy and root-mean-square
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6 radius of theground state as a function of the dipole moment and polarization angle.
5 Inparticularweshowthatintheweakbindinglimitthestateshowsuniversalbehavior.
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. ⋆ Presentedatthe21stEuropeanConferenceonFew-BodyProblemsinPhysics,Salamanca,
1 Spain,30August-3September 2010
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1 D.V.Fedorov
1 AarhusUniversity,8000AarhusC,Denmark
: J.R.Armstrong
v
AarhusUniversity,8000AarhusC,Denmark
i
X
N.T.Zinner
r AarhusUniversity,8000AarhusC,Denmark
a
A.S.Jensen
AarhusUniversity,8000AarhusC,Denmark
2
0.6
D2>0
0.4 D2<0
2 0.2
b
/
2
D 0
,
W
r -0.2
-0.4
-0.6
0 1 2 3 4 5
r,b
Fig. 1 The dipole-dipole potential (2) in the rotationally symmetric case, θ = π/2, as a
functionofr= x2+y2.Thepotential ismultipliedbyr forbettervisibility.
p
2 The Schr¨odinger equation for two dipoles in bilayer geometry
ThesystemoftwopolarmoleculesinabilayergeometryisdescribedbytheSchr¨odinger
equation,
2 2 2
¯h ∂ ∂
− + +W(x,y) ψ(x,y)=Eψ(x,y), (1)
(cid:20) 2µ(cid:18)∂x2 ∂y2(cid:19) (cid:21)
whereµisthereducedmassofthetwomolecules,xandy aretherelativecoordinates
in thelayer plane, and W is thedipole-dipole potential,
2 2 2 2
x +y +b −3(xcosθ+bsinθ)
W(x,y)=D2 (x2+y2+b2)5/2 , (2)
where b is the distance between the layers, θ is the polarization angle measured from
the layer, and D2 is the product of the dipole moments of the molecules taken with
thepositiveornegativesignifthedipolesarecorrespondinglyparalleloranti-parallel.
The dipole-dipole potential has the integrated strength identically equal to zero –
ithasattractiveandrepulsivepartswithequalintegratedstrength,seefigure1.Sucha
potential is expectedto always haveabound state in atwo-dimensional geometry [3].
3 Ground state energy
Wehavecalculatednumericallythebindingenergyofthegroundstateofthesystemof
two polarized molecules for different dipole moments as a function of the polarization
angle for parallel and anti-parallel orientation of thedipoles. Theresults areshown in
figure 2.
Forparallel dipolesthestrongest bindingisachievedwhenthedipoles areperpen-
dicular to the layer, θ = π/2. The binding decreases monotonically with increasing
polarization angle. Approaching θ=0, thebinding decreases significantly but still re-
mainsfinite.Foranti-paralleldipolesthesituationisopposite–thebindingisstrongest
forθ=0anddecreasestowardsθ=π/2.Inbothcasesastrongerdipolemomentleads
to stronger binding.
3
0
-5
)
2
b
µ
2
( -10
/
2
¯h
E, D2=−7
-15 D2=−9
D2=+7
D2=+9
-20
0 π/4 π/2
θ
Fig.2 GroundstateenergyofthesystemoftwopolarmoleculesdescribedbytheSchro¨dinger
equation (1) as a function of the polarization angle θ for parallel, D2 >0, and anti-parallel,
D2<0,orientationofthedipoles.D2 isinunitsofh¯2b/(2µ).
4 Root-mean-square radius and universality in the weak binding limit
Forshort-rangepotentialsintwodimensionstherelationbetweentheroot-mean-square
radius,Rrms,andthebindingenergy,|E|,isuniversalinthelimitofweak binding[4],
2
2 E→0 2 ¯h
Rrms −→ . (3)
32µ|E|
−3
The dipole-dipole potential asymptotically decreases as r , that is faster than
−2
the centrifugal barrier, r . It is thus of a short-range type and must obey the uni-
versal relation. Indeed the numerical calculations show that the universal regime is
approachedat|E|∼0.001 h¯2 ,seefigure3.Fortheparallel(anti-parallel)orientation
2µb2
1.2
D2>0
D2<0
1 2/3
2¯hµE||
/2 0.8
ms
2Rr
0.6
0.4
0.0001 0.001 0.01 0.1 1
|E|, 2hµ¯2b2
Fig. 3 Root-mean-squareradiusRrms ofthesystemoftwopolarmoleculesdescribedbythe
Schro¨dinger equation (1) as a function of the binding energy |E| for parallel, D2 > 0, and
anti-parallel, D2 <0, orientation of the dipoles. The binding energy was varied by changing
the dipole-dipole strength D2 while the polarization angle was fixed at θ = π/2. The model
independent (universal)asymptoticvalueforvanishingbindingis2/3.
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ofthedipolestheuniversallimitisreachedfrombelow(above)reflectingthegeometry
ofthedipole-dipole potentialforthetwoconfigurations–eitherabarrieroracore,as
illustrated in figure 1.
5 Conclusion
We have investigated the ground state of a system of two polar molecules in a bi-
layer geometry. The ground state is always bound with the binding energy strongly
dependentonthepolarization angle.Intheweakbindinglimitthesystemexhibitsthe
universal relation between the root-mean-squareradius and thebindingenergy.
References
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Tailoringinteractionswithexternalfieldsfornovelquantumphases,Phys.Rev.A76043604
(2007)
2. J.R.Armstrong,N.T.Zinner,D.V.Fedorov,andA.S.Jensen,BoundStatesandUniversality
inLayersofColdPolarMolecules,EurophysicsLetters9116001(2010)
3. B. Simon, The Bound State of Weakly Coupled Schro¨dinger Operators in One and Two
Dimensions,AnnalsofPhysics97279-288(1976)
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