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The Two-Body Universal Energy Spectrum of Interacting Ultracold Atoms Near Feshbach Resonances Han, Alex C. 2009-05-05

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The Two-Body Universal EnergySpectrum of InteractingUltracold AtomsNear Feshbach ResonancesbyAlex C. HanA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFBACHELOR OF SCIENCEinFaculty of Science(Honours Physics and Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)May 2009c Alex C. Han 2009AbstractIn this undergraduate thesis, we discuss the two-body quantum mechanical problem of two ultra-cold bosonic atoms at low energy limit, interacting through a square potential with and withoutan external con nement. The two-body s-wave scattering length is introduced in terms of theradial wavefunction. In free space with no con nement, the interaction-scattering length rela-tion is calculated and compared in 3- and 1-dimensions; we show the recurring resonance regionswhere the scattering length diverges and the existence of dimer bound state with positively largescattering length. Lastly, in the case of a 3-dimensional isotropic harmonic con nement, againunder the ultracold limit, we obtain the universal (s-state) energy spectrum of the two atoms interms of the scattering length.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 The Two-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Reduction to One-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 The S-wave Scattering Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Scattering in Free Space: 3D and 1D Cases . . . . . . . . . . . . . . . . . . . . . . 92.4 Resonance and The Bound State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Energy Spectrum for Harmonic Con nement . . . . . . . . . . . . . . . . . . . . 173.1 The Schrodinger’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 Harmonic Well in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30iiiList of Figures2.1 Di erent Orders of Magnitude for Scattering Length . . . . . . . . . . . . . . . . . 82.2 Scattering Length a vs Interaction Strength V0 in 3D . . . . . . . . . . . . . . . . . 112.3 Square Potentials in 3D and 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Scattering Length a vs Interaction Strength V0 in 1D . . . . . . . . . . . . . . . . . 133.1 S-state Energy Levels in the 3D Harmonic Well . . . . . . . . . . . . . . . . . . . . 223.2 Bound State Scaling in 3D Harmonic Well . . . . . . . . . . . . . . . . . . . . . . . 233.3 Derivative at Each Energy Level in 3D Harmonic Well . . . . . . . . . . . . . . . . 243.4 Energy Levels in the 1D Harmonic Well . . . . . . . . . . . . . . . . . . . . . . . . 263.5 S-state Energy Levels in the 3D Harmonic Well (in a) . . . . . . . . . . . . . . . . 27ivAcknowledgmentsThe author thanks greatly Dr. Fei Zhou of the Department of Physics and Astronomy, TheUniversity of British Columbia, for the supervision of this undergraduate thesis; also many thanksto graduate students Junliang Song and Zhen Zhu for valuable discussions.1Chapter 1Introduction and MotivationUltracold atom physics has been a  eld of intense research and attracted great deal of inter-ests from both theoretical and experimental areas since the experimental achievements such aslaser trapping and cooling of atoms and the  rst Bose-Einstein Condensation of Rubidium atoms[1]. As the temperature cools down very close to absolute zero, the thermal motion of atomsis greatly suppressed and the quantum nature of matter becomes very apparent, which providesresearchers a clean playground for exploring atomic and many-body quantum e ects in quantumgases. Quantum gases of ultracold atoms are of special attention due to their ultracold tem-peratures where collisions of the atoms happen at very low energies, and the quantum (wave)nature of the matter dominates their behaviours. Novel quantum phases have also been stud-ied for ultracold atomic gases, such as the exhibition of super uidity and Mott-insulator phase [2].The wheel of ultracold atom research spins forward through advances in experimental tech-niques such as optical lattices and Feshbach resonance scattering methods, where optical latticesare constructed by counter-propagating laser beams and form periodic trapping potential withexperimentally controlled trap size; while Feshbach resonance scattering methods are used to tunethe interaction between ultracold atoms [3]. With these experimental abilities of control over thethe system parameters, more and more theoretical research has been in place in order to fullyunderstand the underlying principles, and to give support and predictions for future experimentalworks.In view of these recent development to study quantum many-body systems, new doors havebeen opened for few-body physics at ultracold temperatures. The few-body problems focus moreon the details of the quantum mechanical scattering of a small number (usually two or three) ofultracold atoms. In addition, the con nement for these atoms, namely optical traps, are usuallyeasy to deal with theoretically. The optical con nement is in essence in the shape of a harmonic2Chapter 1. Introduction and Motivationoscillator potential, whose quantum mechanical properties are well known in textbooks on mod-ern quantum physics. At appropriate conditions, the interaction potential between atoms canalso be very well represented by the quantity called the two-body s-wave scattering length. Oneamong many of its interesting properties of the scattering length, is that its scale of value canvary from as small as typical cold atom interaction range (angstroms) to as large as comparableto the harmonic oscillator length of the laser trap [4], where the physics of the situation is greatlysimpli ed in terms of this scattering length alone and the system has, namely, universal proper-ties: the details of the interaction are unimportant and same physics apply for given scatteringlength.The universal properties apply for both fermionic and bosonic ultracold atoms. In this re-search project, we focus on the bosonic atoms due to the ease of theoretical calculations on thesymmetric requirement of the two-body wavefunction. In addition, the s-partial-wave approxi-mation is valid for bosonic atoms whereas the behaviours of fermionic atoms are dominated byp-waves due to their spins.In summary, in this undergraduate research project we focus on the study of two interactingbosonic atoms at very low energies and their scattering properties. In the following sectionswe  rst reduce the two-body problem in general to a one-body problem and then study it interms of the s-wave scattering length. Then we will look at a speci c problem of a 3-dimensionalisotropic harmonic trapping potential and solve the energy spectrum of the s-states in terms ofthe scattering length alone (universal spectrum).3Chapter 2The Two-Body Problem2.1 Reduction to One-Body ProblemThe two-body low energy scattering formalism has been associated with nuclear interactions inthe past, but it is borrowed and utilized here in the context of ultracold atoms. Physically, weconcern ourselves with two interacting bosonic atoms may or may not be in a con nement. We rst of all pose the two-body problem by writing down the Schrodinger’s equation as the energyeigenvalue problem for the stationary states:H (r1;r2) = E12 (r1;r2)H =  h22m1r21  h22m2r22 +V12(r1 r2) +Vtrap(r1) +Vtrap(r2) (2.1)where the Hamiltonian involves the kinetic energies of atoms denoted 1 and 2, the interactionpotential between them V12 and there might also be a con nement providing a trapping potentialVtrap. The eigen-energies are continuous if there is no trap and is quantized in traps.Similar to the method a two-body problem has been solved in the classical mechanics, weintroduce the centre of mass frame of reference and the relative motion of the two particles, by aseparation of spatial wavefunction  (r1;r2) =   (r) M(R) with new variables and massesR = m1r1 +m2r2m1 +m2; r = r1 r2; M = m1 +m2;  = m1m2m1 +m2(2.2)Now, assuming the interaction is isotropic, as most practical cases are, then V12(r1  r2) =V12(jr1 r2j) = V12(r). And change of variables makes 1m1r21 + 1m2r22 = 1Mr2R + 1 r2r, then theHamiltonian becomes42.1. Reduction to One-Body ProblemH =  h22Mr2R  h22 r2r +V12(r) +Vtrap(R) +Vtrap(r) (2.3)Note here we rely on the trapping potential to be "nice" such that Vtrap(r1) + Vtrap(r2) =Vtrap(R) + Vtrap(r) (and to be seen in Chapter 3 that a 3-dimensional isotropic harmonic willsatis es this condition as well). The original Schrodinger’s equation then decouples into twoseparate Schrodinger’s equations for the centre of mass motion and the relative motion (withE12 = EM +E )   h22Mr2R +Vtrap(R)  M(R) = EM M(R) (2.4)   h22 r2r +Vtrap(r) +V12(r)   (r) = E   (r) (2.5)where we note that the equation for the motion of the centre of mass is exactly solvable if thegiven trapping potential is exactly solvable, and the problem of  nding the eigen-energies is re-duced to solving the problem posed by equation (2.5).At low energy limit, the momenta (k) of the atoms are extremely low and thus the de Brogliewavelengths (2 =k) of the atoms are extremely large, resulting in the poor resolution of theinteraction potential the atom scatters. E ectively then, we can write the s-wave approximationin the partial wave expansion of the relative motion wavefunction, assuming azimuthal symmetry,  (r) =   (r; ) = Pl ul(r)r Pl(cos ) and let l = 0 (retain only s-wave term), then  (r) =   (r) = u0(r)r (2.6)and if the trapping potential is also isotropic, then equation (2.5) becomes   h22 r2r +Vtrap(r) +V12(r)   (r) = E   (r) (2.7)writing out the Laplacian operator in polar form and insert the ansatz in equation (2.6), we have52.1. Reduction to One-Body Problem   h22 @2@r2 +Vtrap(r) +V12(r) (r  (r)) = E (r  (r)) (2.8)which is in form a 1-dimensional scattering problem, with boundary conditions speci ed by theinteraction term and the trapping potential term.In order to proceed, we now look at the interaction potential and justify that a good e ectivemodel is an isotropic square potential for V12, namelyV12(r) =8<:V0 for 0 <r<r00 for r>r0(2.9)where the range of the potential is r0, and the strength V0 can be either positive, zero or nega-tive. The features of this square potential include: (i) isotropy - justi ed in most ultracold atominteractions; in addition, the low energy limit also implies a low angular resolution of the objectthe atoms scatters; (ii)  nite range - for typical ultracold atoms, the interaction is dominatedby van der Waals interaction potential, which can be thought of essentially zero after a certain nite distance, combined with the large de Broglie wavelength, the interaction can be considerednegligible after a  nite range r0; (iii) uniform potential value inside - again justi ed by the longde Broglie wavelength (comparing to r0), and the scattering atom can’t "feel" the details of theinteraction inside r0.As a summary of the preconditions that we used to reduce problem of equation (2.1) to thatof equation (2.8), we require An isotropic,  nitely ranged, (square) interaction, A "nice" trapping potential (if any): isotropic and is felt by the two atoms separately(namely the requirement that Vtrap(r1) +Vtrap(r2) = Vtrap(R) +Vtrap(r)), and Ultra-low energy levels at which the s-wave approximation is valid.62.2. The S-wave Scattering Length2.2 The S-wave Scattering LengthIn this section we introduce the concept of two-body s-wave scattering length. The de nition isrooted in the low energy limit of the Schrodinger’s equation. Let’s consider the case where thereis no external con nement, and outside the interaction range r0 we take the limit of energy goingto zero in equation (2.8)  h22 @2@r2u(0)(r) = 0 (2.10)where we denoted u(r) = r  (r). The solution to above di erential equation is linear in r andcan be written in the form [5]u(0)(r) = const(r a) (2.11)therefore, the solution to equation (2.8) with no con nement and outside the interaction ranger0 must go to u(0)(r) under low energy limit. i.e.limk!0u(r) = const(r a) (2.12)which is to saylimk!0u0(r)u(r) =1r a (2.13)The parameter a here is de ned to be the s-wave scattering length. Graphically, we can setr = r0 and a is just the intercept of the tangent line to u(r) at r0, shown by rearranging theterms in above equation, at low k limit,u0(r0)(a r0) +u(r0) = 0 (2.14)Note that we can set r = 0 in equation (2.13). Even though the radial wavefunction is NOT u(r)there, it does not violate the equality of equation (2.13). Then we have a more compact de nitionfor the scattering lengthlimk!0u0(0)u(0) = 1a (2.15)In free space, outside the interaction u(r) has the asymptotic function form u(r) sin(kr  (k))with s-wave phase shift  (k), thus equation (2.15) has a more often seen form [6],limk!0kcot( (k)) = 1a (2.16)72.2. The S-wave Scattering LengthFigure 2.1: Di erent situations in 3D where the scattering length varies greatly in magnitude:(a) small and positive due to a very repulsive potential; (b) very large and positive (intercept notseen in graph) due to a relatively large attractive potential and (3) large and negative due to asmall attractive potential.This is an important equation by which we will calculate the relation between the strength of thesquare potential (V0) and the scattering length in the next section.Physically, the value of scattering length a can vary greatly. For a repulsive interaction,scattering length is on the same order of magnitude as the range of the interaction potential r0;and for a attractive interaction it can be as large as comparable to the characteristic size of thecon nement, if any, which is orders of magnitude larger than r0. Figure 2.1 shows the cases ofdi erent scales of scattering length. Notice that by the graphical interpretation of a, it is possiblefor the tangent line to the radial wavefunction u(r) at r0 to be  at, i.e. jaj=1. When scatteringlength diverges without bound, we say the system is at resonance. And we will see speci callyhow this resonance situation is achieved in the next section.82.3. Scattering in Free Space: 3D and 1D Cases2.3 Scattering in Free Space: 3D and 1D CasesWe now utilize equation (2.16) and by calculating the phase shift (k), we can obtain a relationshipbetween scattering length a and interaction strength V0. Let’s  rst consider the 3-dimensionalcase of two bosonic atoms interact through the square interaction de ned in equation (2.9), andthe problem is posed by equation (2.8) with Vtrap = 0 everywhere. First let’s say V0  E > 0(low energy scattering states), then the di erential equation from (2.8) is changed to@2@r2u(r) =8<: 2 E h2 u(r) r>r02 (V0 E) h2 u(r) 0 <r<r0(2.17)and the radial wavefunction u(r) = r (r) that satis es above di erential equation has formu(r) 8<:sin(kr  ) r>r0sinh( r) 0 <r<r0(2.18)with =p2 (V0 E) h =p2 V0 h ; k =p2 E h (2.19)It is a hyperbolic trigonometric function inside  rst because the exponential decay of the wave-function, and secondly, in 3D, the radial wavefunction has to vanish at r = 0 (thus it is sinhinstead of cosh). Now we impose the boundary condition that the radial wavefunction has to becontinuous and continuously di erentiable at r0, then it is necessary to requiretan(kr0  ) = k tanh( r0) (2.20)but note that low energy limit corresponds to kr0  1 thus left hand side is tan(kr0  ) = tan ,then we have phase shift = arctan k tanh( r0) (2.21)using equation (2.16), we havear0 = 1 tanh( r0) r0 (2.22)This is the relation of a vs. V0 ( =  (V0)) for a positive V0. Now let’s assume V0 < 0 andjV0j E > 0, then the di erential equation we want to solve becomes92.3. Scattering in Free Space: 3D and 1D Cases@2@r2u(r) =8<: 2 E h2 u(r) r>r0 2 (E V0) h2 u(r) 0 <r<r0(2.23)and the solution isu(r) 8<:sin(kr  ) r>r0sin( r) 0 <r<r0(2.24)with again =p2 (E V0) h =p2 jV0j h ; k =p2 E h (2.25)so the di erence from a positive V0 is that now inside the interaction potential the radial wave-function is a standing wave in nature, rather than an exponential decay in the repulsive case.With the same boundary condition requirement at r0, we have phase shift = arctan k tan( r0) (2.26)Putting into equation (2.16) again, we have the relation between a and V0 for negative V0ar0 = 1 tan( r0) r0 (2.27)Equations (2.22) and (2.27) together give us the whole picture of a vs. V0, or rather, in anormalized way: a=r0 vs. 2 V0r20= h2. The graph is presented in Figure 2.2. Let’s  rst note thatas V0 !1, tanh( r0)= r0 ! 0, and a=r0 ! 1 in equation (2.22). This is the so-called HardSphere limit where the wavefunction is completely pushed out of the interaction potential andthe scattering length is just the range of the interaction potential a = r0. Near zero, i.e. jV0j issmall, the relationship of a and V0 is approximately linear, i.e.ar0 = 2 r20V03 h (2.28)which is also given by the Born Approximation at low energy [7]a = f( ; ) =  2  h2ZVdr =  2  h2 43 r30V0 = 2 r30V03 h (2.29)Another very important feature is that jaj diverges due to the tan( r0) function and thisdivergence (resonance) region is repeated (although with di erent periods) as V0 deepens. Theresonance values of V0 is thus 2 jV0jr20= h2 = ( =2 +n )2 =  2(n+ 1=2)2, n = 0;1;2;3:::102.3. Scattering in Free Space: 3D and 1D CasesFigure 2.2: Scattering Length a vs. interaction strength V0 in 3-dimensions.Now we can repeat the above procedure to two interacting bosonic atoms in 1 dimension.The di erence in the set-up of the problem includes,  rst of all, that there is no "s-wave" in1D: the relative coordinate wavefunction   has no "radial function x  (x)"; it is simply itself.Another di erence is that the interaction potential covers the spatial region from  x0 to x0 andthe wavefunction does not have to vanish at x = 0. The Schrodinger’s equation in 1D is   h22 d2dx2 +V(x)   (x) = E  (x) (2.30)withV(x) =8<:V0 jxj<x00 jxj>x0(2.31)considering the symmetry of bosonic wavefunction, we can simply solve the wavefunction insideand outside the interaction potential; let’s  rst solve for the case where V0  E > 0,  (x) 8<:cosh( x) jxj<x0sin(kx  ) jxj>x0(2.32)again with =p2 (V0 E) h =p2 V0 h ; k =p2 E h (2.33)112.3. Scattering in Free Space: 3D and 1D CasesFigure 2.3: The square potentials in 3- and 1-dimensions. Solid line shows the square wallwith height V0 and dashed line shows the square well with depth V0. The 1-dimensional squarepotential is symmetric about the origin and therefore extends to negative values.but the boundary conditions of continuity and continuous derivative at x0 now yieldstan(kx0  ) = k 1tanh( x0)(2.34)which gives us the scattering length vs. interaction strength according to equation (2.16) 1ax0 = 1 1 x0 tanh( x0) (2.35)And solve for jV0j E > 0 but V0 < 0 case,  (x) 8<:cos( x) jxj<x0sin(kx  ) jxj>x0(2.36)the di erence is again, inside the interaction potential, the wavefunction is a standing wave inessence instead of exponential decay. The boundary condition yields nowtan(kx0  ) = k 1tan( x0)(2.37)which gives the relation of a vs V0 for negative V0ax0 = 1 +1 x0 tan( x0) (2.38)1Although equation (2.16) is obtained as a 3D example, but it is yet valid anywhere as long as phase shift exists;a more general derivation is in [6].122.3. Scattering in Free Space: 3D and 1D CasesFigure 2.4: Scattering Length a vs. interaction strength V0 in 1-dimension.Equations (2.35) and (2.38) together gives us the whole picture of a vs V0 in 1 dimension. Wecan observe there are several similarities between the a vs V0 relations in 3D and 1D: they bothhave a hard sphere limit where V0 !1; scattering length a in each case diverges at certain  nitevalue of V0: for 3D, 2 jV0jr20= h2 =  2(n+ 1=2)2, n = 0;1;2;3::: and for 1D, 2 V0x20= h2 = (n )2,n = 0;1;2;3:::; and the resonance region appears repeatedly (again with di erent periods) asV0 deepens. However, there is a signi cant di erence between the two cases, that is, the BornApproximation limit. In 3D, scattering length is always positive for a repulsive interaction, butin 1D, very near zero, the scattering length not only diverges, but also changes sign and does notfollow the Born Approximation.The nature of this di erence lies in how the system responds to perturbation in these two cases.Imagine two non-interacting atoms in a box and we turn on a square interacting potential as asmall perturbation. Then perturbation theory is valid only if the ratio of the  rst order correctionto the energy to the di erence of energies between ground state and  rst excited state is verysmall, namely E(1)0 = E01  1. In a 3D box with size L, the energy associated with each quantumstate scales like 1=L2 (since momentum k scales like 1=L), and the  rst order perturbation to the132.4. Resonance and The Bound Stateenergy scales like 1=L3 (the unit of the amplitude squared of the wavefunction in 3D is 1/volume,thus E(1)0  h jV0j i = V0h j i), thus E(1)0 = E01  1=L, and as L!1, perturbation theorybecomes extremely accurate. However, in 1D the amplitude squared of the wavefunction scaleslike 1=L, resulting that E(1)0 = E01  L, i.e. perturbation theory becomes invalid for L!1.Therefore, since the energy shift is not perturbative, any small change in interaction between twoatoms can result in very strong interaction between them, manifested in the divergence of thescattering length near zero.2.4 Resonance and The Bound StateAn interesting observation we have from the previous section is that V0 has these resonancevalues at which jaj!1: in 3D it is 2 V0r20= h2 =  n2 2=4 for n = 1;2;3::: and in 1D it is2 V0x20= h2 =  (n 1)2 2 for n = 1;2;3:::. Across these values of V0 the scattering length  rstdiverges to negative in nity, then comes back from positive in nity and then slowly comes to zeroand repeat the whole process again. This is re ected in the graphical interpretation of the conceptof scattering length, which is the intercept of the tangent line to the radial wavefunction at ranger0 (back in Figure 2.1, this process is partly shown): as most part of the radial wavefunction,u(r), is outside the range r0, the slope of u(r) at r0 is positive and small, then as the body ofu(r) moves towards the interaction potential, this slope becomes more and more  at; when it iseventually  at, the intercept of the tangent line to the spatial axis can be thought of as being atpositive and negative in nity, but as the tangent line keeps rotating, its slope becomes negativeand intercepts the spatial axis at positive side, signally that most part of u(r) is now inside therange r0. And as this whole resonance process happens when V0 < 0, one may wonder whathappens to the bound states, if any.Now let’s continue from the free space scattering situation in previous section and try to solvethe problem of negative eigen-energy and look for possible bound states for V0 < 0. Let’s considerthe 3D situation  rst and let E < 0 (and jV0j jEj), in order to satisfy radial equation, theradial wavefunction has the formu(r) 8<:sin( r) 0 <r<r0e kr r>r0(2.39)142.4. Resonance and The Bound Statewith =p2 (E V0) h =p2 jV0j h ; k =p2 jEj h (2.40)contrasting with equation (2.18) that the outside wavefunction is exponential decay instead of freesinusoidal wave, i.e. the very essence of a bound state. Posing again the continuity and continuousderivative boundary condition at range r0, we have the following quantization conditiontan( r0) =  k (2.41)and if we express  in terms of V0 using equation (2.40) above, we havetans2 jV0jr20 h2 =q2 jV0jr20 h2 kr0 or kr0 = q2 jV0jr20= h2tanq2 jV0jr20= h2(2.42)which gives us the value of the bound state energy for a given value of V0: notice that for2 jV0jr20= h2 in the range from zero to  2=4, the value of k is negative, i.e. there is no boundstate; when 2 jV0jr20= h2 =  2=4, k is zero and a bound state is "born" (notice that aboveequation reproduces the resonance values of V0 where scattering length diverges, as predicted byequation (2.27), bottom of page 10); lastly for 2 jV0jr20= h2 slightly larger than  2=4, k is positive.So let’s expand the above expression around 2 jV0jr20= h2 =  2=4, say 2 jV0jr20= h2 =  2=4 + tanr 24 + =q 24 +  kr0 ) tan  2 +    =  2 +    kr0)  cot    = 2 +   kr0) kr0  =  2 (2.43)Now on the other hand, recall equation (2.27) gives the a vs. V0 relation, we can now expand aaround 2 jV0jr20= h2 =  2=4 + , thenar0 = 1 tanq 24 + q 24 +  = 1 tan( 2 +   ) 2 +   = 1 + cot    =2 = 1 + 2  = 2 (2.44)and combining (2.43) and (2.44), we havek = 1a (2.45)152.4. Resonance and The Bound Statein other words, the bound state energy is e ectivelyE =  h22 1a2 (2.46)Physically, the signi cance of k  = 1=a is that the radial wavefunction outside becomesu(r)  e r=a, which describes a diatomic molecule, or a dimer: two atoms separated over longdistance in a weakly bounded state, and its momentum (thus energy) is dominantly described bythe inverse of the scattering length alone.We can apply the above procedure to the 1-dimensional case as well. ForE < 0 andjV0j jEj,the wavefunction is (x) 8>>><>>>:cos( x)  x0 <x<x0e kx x>x0ekx x< x0(2.47)with =p2 (E V0) h =p2 jV0j h ; k =p2 jEj h (2.48)a patching boundary condition gives a similar quantization to (2.41)k =  tan( x0) (2.49)and again we expand around the resonance value 2 jV0jx20= h2 = 0 and let 2 jV0jx20= h2 =  , thenabove equation becomeskx0 =  tan  =  2 (2.50)and use (2.38) to get a in this expansionax0 = 1 +1 tan  = 1 + 1 2 = 1 2 (2.51)combining (2.50) and (2.51)k = 1a (2.52)thus again we recovered this relation similarly found in (2.45). So the 1-dimensional dimermolecule near the resonance also has this binding energy large dependent on scattering lengthaloneE =  h22 1a2 (2.53)16Chapter 3Energy Spectrum for HarmonicCon nement3.1 The Schrodinger’s EquationNow we consider a 3-dimensional isotropic harmonic well as a con nement of the two interact-ing atoms. The motivation for this set-up is that it is a very good model to the modern lasertraps. Thus this chapter is devoted to an e ective model to the realistic experimental situation oftrapped interacting ultracold bosonic atoms in optical lattices, and through Feshbach resonancescattering, their interaction is tunable. We consider the trapped atomic gas is so dilute that thereare no more than two atoms per cell, hence our set-up of two interacting ultracold atoms in aisotropic 3-dimensional harmonic well.Since all the assumptions summarized in Section 2.1 are met, our main task is then to solveequation (2.8). The relative coordinate trapping potential is found  rst by looking atVtrap = 12m1!2r21 + 12m2!2r22 (3.1)then after centre-of-mass transformation,Vtrap = 12M!2R2 + 12 !2r2 (3.2)therefore, the Vtrap(r) term in equation (2.8) isVtrap(r) = 12 !2r2 (3.3)rewriting (2.8) as follows   h22 @2@r2 +12 !2r2 +V12(r) u(r) = E u(r) (3.4)173.2. Boundary Conditionswhere recall u(r) = r (r) is the radial wavefunction, and V12 is again the square interactionde ned as equation (2.9). Employing the convention of harmonic oscillator units, namely lettingnew spatial variable x = r=(p h= !) and energy  = 2E = h!, then equation (3.4) readily changesto   d2dx2 +x2 +V12(x) u(x) =  u(x) (3.5)with square interactionV12(x) =8<:2V0= h! for 0 <x<x00 for x>x0(3.6)where x0 = r0=(p h= !).3.2 Boundary ConditionsTo tackle the problem posed by equation (3.5), we  rst see that for a general solution satisfyingdi erential equation (3.5), boundary conditions are needed in order to obtain exact quantizationof energy. So aside from the usual normalization boundary condition that wavefunction has tovanish at in nite distances, there is also some boundary condition set by the interaction potentialV12 near zero, and the whole existence of this V12 term is to impose that condition.An immediate idea that follows is that, since we devoted Chapter 2 on the relation betweenthe s-wave scattering length and the interaction potential, we can somehow relate the interactionpotential here with the scattering length. And that is, the very de nition of scattering lengthfrom equation (2.15) (in new variable x)limk!0u0(0)u(0) = 1a (3.7)which is precisely a (mixed) boundary condition for solution u(x) set by interaction potential V12(since a sets V12).Another way of looking at how we can obtain this boundary condition is the  -pseudo-potentialmethod where we replace V12 by an e ective potential involving a regularized Dirac- function[8]. Physically to justify the e ectiveness of the  potential (also known as a zero-range or contact183.2. Boundary Conditionspotential), we  rstly have the low energy limit where large de Broglie wavelengths result in poorresolution, therefore the object atoms scatter o are essentially point-like. Furthermore, we cansee the interaction range r0 is much smaller than the harmonic oscillator lengthp h= ! (or equiv-alently x0  1) for our experimental setting: the cold atom interaction is dominated by van derWaals potentials with range on the order of angstroms, whereas the smallest optical traps havefrequencies on the order of 100 kHz, corresponding to a harmonic oscillator length on the order of100 nm. And the "strength" of the  potential is represented by a factor in front of the  -function.So mathematically, we recall the original Schrodinger’s equation in relative coordinate (equa-tion (2.5))    h22 r2r +12 !2r2 +V12(r)  (r) = E  (r) (3.8)and we replace the interaction potential by the e ective potentialV12(x)! 2 a h2  3(r) @@rr (3.9)where  3(r) is the 3-dimensional Dirac- function and the @@rr term is there to make the Hamil-tonian self-adjoint, thus the term "regularized" [8]. Then what we can do to the new di erentialequation    h22 r2r +12 !2r2 + 2 a h2  3(r) @@rr  (r) = E  (r) (3.10)is to integrate both sides over a small sphere in space with radius going to the limit of zero, uponwhich the right hand side of the equation and the 12 !2r2 term on the left go to zero, and we areleft with is Zr r (r)d3r = 4 aZ 3(r) @@r(r (r))d3r (3.11)by divergence law from calculus we haveIr (r) dA = 4 a @@r(r (r))r=0 = 4 au0(0) (3.12)and as the radius goes to zero, the closed integral over the surface of the sphere becomes193.3. Energy SpectrumIr (r) dA =I @@r (r)dA =I   u(r)r2 + u0(r)r!dA = 4 r2  u(0)r2!= 4 u(0) (3.13)equating to the right hand side of (3.12) gives usu0(0)u(0) = 1a (3.14)Therefore, the  -pseudo-potential is a good e ective potential to our square interaction as long asthe range of the interaction is much smaller than the harmonic oscillator length and the scatteringis kept at low energy, then the e ective potential reproduces the same boundary condition (3.7)as the de nition of the scattering length.3.3 Energy SpectrumSo continuing from previous section, we have had two boundary conditions for the solution to thedi erential equation (3.5), namely u(x)!0 as x!1and the mixed one in equation (3.7). Nowlet’s look at what general solution we have for (3.5) outside the interaction range x0. Outside x0,we want to solve for u(x) in   d2dx2 +x2 u(x) =  u(x) (3.15)Motivated by the classical textbook problem on solving harmonic oscillator eigenstates, we factorout a Gaussian term from u(x) and let u(x) = e x2=2w(x), then we have a new equation for w(x)  d2dx2 + 2xddx + 1 w(x) =  w(x) (3.16)and changing variable y = x2, thus letting d=dx = (dy=dx)(d=dy) changes above equation to theso-called con uent hypergeometric equation (or sometimes called Kummer’s equation) [9]yd2wdy2 + 12  y dwdy  1  4 w = 0 (3.17)with parameters c = 12 and b = 1  4 . The solutions of the con uent hypergeometric equation aregiven by two linearly independent con uent hypergeometric functions [9],203.3. Energy SpectrumF(b;c;y) =1Xn=0(b)nyn(c)nn! and G(b;c;y) = y1 cF(b c+ 1;2 c;y) (3.18)where (b)n denotes the forward factorial (b)n := b(b+ 1)(b+ 2):::(b+n 1) with (b)0 = 18b. AndG is a solution only when parameter c is not an integer [9](which is true our case, c = 1=2). Sowriting out the solution for w(y(x)) in linear combinations of F and Gw(x) = C1F 1  4 ;12;x2 +C2(x2)1 12F 1  4  12 + 1;2 12;x2 = C1F 1  4 ;12;x2 +C2xF 3  4 ;32;x2 (3.19)sou(x) = e x22"C1F 1  4 ;12;x2 +C2xF 3  4 ;32;x2 #(3.20)Applying the boundary condition in (3.7) immediately gives us a relation between constants C1and C2 1a = u0(x = 0)u(x = 0) =C2F 3  4 ;32; 0 dxdrC1 =C2C11p h= ! (3.21)where F(b;c;0) = 18b;c by the de nition of F in (3.18). Whereas the condition to require u!0as x!1 is bit complicated: we look at the asymptotic behaviour of F(b;c;y) for large yF(b;c;y)  (c) (b)yb cey (3.22)with the usual  -function. By this, we see at large x,u(x)  ex2 x22"C1  (12) (1  4 )(x2)1  4  12 +C2x  (32) (3  4 )(x2)3  4  32#= ex22 x 1  2"C1  (12) (1  4 ) +C2 (32) (3  4 )#!0(3.23)only ifC1  (12) (1  4 ) +C2 (32) (3  4 ) = 0 (3.24)orC1C2 =  (3  4 ) (12) (1  4 ) (32) =  (3  4 )p  (1  4 )p =2 = 2 ((3  )=4) ((1  )=4) (3.25)213.3. Energy SpectrumFigure 3.1: The s-state energies of two bosonic atoms interacting through a square potential withscattering length a, inside a 3-dimensional isotropic harmonic well with frequency !. The darkerlines are the energy levels in the harmonic oscillator energy unit vs. the inverse of the scatteringlength in the unit of the inverse of the harmonic oscillator length; shadowed lines are asymptoticvalues E = (12 +n) h!2 , n = 1;3;5;7:::Combining (3.25) above with (3.21), we havep h= !a = 2 ((3  )=4) ((1  )=4) (3.26)or more clearly, p h= !a = 2 ((3 2E h!)=4) ((1 2E h!)=4) (3.27)which is the quantization of energy (and is also theoretically obtained by some previous papers[8], [10], [11]). In Figure 3.1, the s-state energy levels are plotted according to equation (3.27) inthe harmonic oscillator energy unit ( h!=2) against the inverse of the scattering length in the unitof the inverse of the harmonic oscillator length (1=p h= !). This is the most important graphfor this Chapter, and it has several interesting features:223.3. Energy SpectrumFigure 3.2: The scaling property of the bound state energy (negative in sign), comparing to thefunction of quadratic dependence in the graph units (darker curve). Universality: the energy spectrum is dependent on the inverse of the scattering length alone.The details of the interaction potential is not concerned. Equally spaced energy levels at resonance (1=a = 0): a special feature of energy spectrum tothe harmonic oscillator. The energies are calculated by the zeros of equation (3.27), whichis equivalent to solving  ((1 2E h!)=4) = 0. Since the zeros of the  -function are negativeintegers, say  m, m = 1;2;3:::, then we have energies E = (4n + 1) h!2 , n = 0;1;2;3:::.This equally spaced spectrum implies the system is still harmonic at resonance even thoughthere is a strong interaction between the two atoms. Forming of a bound state and the binding energy: reproduced as shown in Section 2.4. Asthe scattering length a is positive and very large, corresponding to 1=a is small and positive,there is bound state with energy scaling like 1=a2, shown in Figure 3.2, where the parabolicfunction is E =  h22 1a2 . This also recon rms what we saw in Section 2.4, that as we increasethe well depth V0 across each resonance value, 1=a goes across the origin from negative topositive values, and a bound state forms.233.3. Energy SpectrumFigure 3.3: The numerical values of the derivatives of each energy level in harmonic oscillatorunit with respect to 1=a at resonance 1=a = 0; graph shows  rst 15 levels. Energy shifts from a = 0 to a = 1: in both cases the energy levels are equally spaced.At a = 0, there is no interaction, the energy spectrum is merely the one for the har-monic oscillator but with only half of the spatial axis (r can’t be negative), which corre-sponds to the well known harmonic oscillator spectrum with only odd quantum numbers,namely, E = (12 + n) h!2 , n = 1;3;5;7:::. When a = 1, the energies are E = (4n + 1) h!2 ,n = 0;1;2;3:::, thus giving the uniform energy shifts for each state, with value  h!. Anharmonicity: the interaction introduces anharmonicities slightly away from resonanceshown by the  rst derivative of the energy with respect to 1=a at 1=a = 0 (Figure 3.3). Acomputer interpolation shows the derivatives approach zero at a rate of (2E= h!) 0:4065 as(2E= h!) takes discrete values 1;5;9:: in Figure 3.1. The derivatives of each energy levelwith respect to 1=a at resonance (1=a = 0) are not equal, thus when 1=a is slightly awayfrom zero the energy spectrum is no longer harmonic (equally spaced). The anharmonicitiesimply an analog to the classical damping to the quantum harmonic motion: for a coherent243.4. Harmonic Well in 1Dquantum state, at resonance, oscillating in time between these equally spaced energy levels,when we slightly shift the scattering length away from resonance, the energy levels are nolonger equally spaced and thus causes the coherent quantum state to dephase, resulting inthe coherent state  nally collapsing to one energy state (decoherence) after certain dephas-ing time. The decay rate of the derivatives in Figure 3.3 may be of future use to determinethe change in this dephasing time for higher energy levels, since their change in energy withrespect to inverse of the scattering length slightly away from zero is small (and asymptoti-cally approaching zero).Finally, we look back to the assumption we made to solve this energy spectrum in order toverify the validity range to which the spectrum is e ectively correct. The short range (comparingto harmonic oscillator length) of the interaction has already been justi ed by order-of-magnitudecomparison with actual experimental facts (Section 3.2). As of the low energy limit, where werequired the de Broglie wavelength (2 )1=k>>r0, we can estimate an appropriate bound for k:say r0 =1 nm (since r0 is on the order of angstroms), then our result is valid for 2E= h! 104,which is much larger than the top of our graph in Figure 3.1, where 2E= h! is only around 30.3.4 Harmonic Well in 1DHere we brie y look at the case of the same scenario as previous sections but in a harmoniccon nement in 1-dimension. The Schrodinger’s equation in 1D is   h22 d2dx2 +12 !2x2 +V12(x)   (x) = E   (x) (3.28)which is exactly in the same form as equation (3.4), but with the relative coordinate wavefunctionhere instead of the radial wavefunction, and with variable x instead of r (here x is the 1D spatialvariable, not the same as the r dressed in harmonic oscillator unit in previous sections). Thesquare interaction V12(x) is now extended to 1D as Figure 2.3 showed.253.4. Harmonic Well in 1DFigure 3.4: The energies of two bosonic atoms interacting through a square potential with scat-tering length a, inside a 1-dimensional harmonic well with frequency !.Now notice that we still have the boundary condition relating to the scattering length speci- ed by equation (3.7). The general solution to the Schrodinger’s equation (in harmonic oscillatorunits again) again involves the linear combination of con uent hypergeometric function (equation(3.20) but for   (x) instead of u(x)). Now to require normalization condition for   (x), we needit to go to zero at both1and 1. However, since we are looking at bosonic atoms, the relativecoordinate wavefunction is of even parity (also re ected in the solution involving the con uenthypergeometric functions of x2). Thus, the negative side normalization is naturally taken careof by the requirement on the positive side. Apply the same procedure in Section 3.3 to theSchrodinger’s equation in this 1D case, and we then have exactly the same quantization of energycondition as equation (3.27) and same plot (Figure 3.4).However, even with the same energy spectrum, the 3D and 1D systems have very di erentbehaviours near the non-interacting point (i.e. the point when we  st turn on a non-zero inter-action between the two atoms), because with no interaction (V0 = 0), the 1D case is already atresonance (i.e. 1=a = 0) but the 3D case is not. The scaling for the correction to the energy263.4. Harmonic Well in 1DFigure 3.5: The s-state energies of two bosonic atoms interacting through a square potentialwith scattering length a, inside a 3-dimensional isotropic harmonic well with frequency !. Theenergies are plotted against 1D is like h njV0j ni1D  V0   1=a where V0   1=a as Figure 2.4 showed. So near thenon-interacting point where V0 = 0, i.e. 1=a = 0, the derivative of the energy with respect to1=a is negative, which is consistent with Figure 3.4. In 3D, the energy correction scaling be-haviour near non-interacting point is di erent: the correction is proportional to a, speci cally,h njV0j ni3D V0  a, where V0  a as in Born approximation; thus when V0 = 0, a = 0, so thescaling of the correction energy is an expansion around a = 0. This is directly seen if we re-plotFigure 3.1 with E vs. a (Figure 3.5): the derivative of the energy with respect to scatteringlength a at a = 0 is positive.27Chapter 4ConclusionIn conclusion, let’s summarize the main points discussed in this research project. We started outby looking at the 3-dimensional two-body quantum mechanical problem of interacting atoms inthe context of ultracold limit, where we pose the Schrodinger’s equation for their eigen-energies.We use the low energy scattering limit and approximate the interaction of the ultracold atoms byan isotropic square potential with  nite range and a variable strength which can be either repul-sive or attractive. By the classic change of reference frame transformation to the centre of masscoordinate and relative coordinate, the two-body problem is e ectively reduced to a one-bodyquantum scattering problem with the interaction potential as the scattering centre.We then turn to the scattering of the two atoms in free space without any con nement, whereof course the reduced one-body scattering problem applies. In this context the concept of s-wavescattering length is introduced and several properties, such as its sign and scale, are discussed.Then by solving the Schrodinger’s equation and apply patching boundary conditions at the rangeof interaction, we obtain the relation between the scattering length and the interaction strength;the results in 3-dimensions and 1-dimension are compared and contrasted. An interesting fea-ture of a bound state, also known as the di-atomic molecular state, that comes into existencenear resonance, is presented, and its binding energy in terms of the scattering length is calculated.Then we looked at a speci c con nement for the two interacting atoms, namely a 3-dimensionalisotropic harmonic well. We discussed in detail the boundary conditions for the problem andsolved its s-state energy spectrum, which is e ectively the complete spectrum in the ultracoldlimit, in terms of the scattering length as a parameter alone. The universal energy spectrum hasseveral interesting properties including the harmonicity at resonance, the anharmonicity away butnear resonance, and the forming of the bound state and its scaling property in terms of scatteringlength near resonance as shown in previous sections.28Chapter 4. ConclusionContinuing from the results of this project, the future work may include several aspects of thedynamics of the system near resonance, which may be explored by considering the anharmonicitywhen the system is slightly away from the resonance, causing the classical analogue of damping.When some parameter varies with time, interesting dynamical properties can also be derived;for example the harmonic oscillator frequency ! may vary with time, with applications to thesituation where the harmonic trap is turned o slowly in time such that !(t)!0 as time goes on[12]. Similar discussions may also apply to fermionic atoms where the universal properties shouldalso apply; technical di erences include the partial wave approximation and the requirement onthe symmetric of two-body wavefunctions, where the complete energy spectrum may apply tobosonic atoms occupying the energy states with even-parity wavefunctions, and fermionic atomsoccupying the states with odd-parity wavefunctions.29Bibliography[1] C. Wieman, E. Cornell et al, Nature 412, 295-299 (19 July 2001).[2] M. Greiner et al, Nature 415,39 (2002)[3] W. Ketterle et al, Nature 392, 151 (1998).[4] F. Dalfovo, S Giorgini, L.P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999)[5] J.J. Sakurai, Modern Quantum Mechanics Revised ed. (Adisson-Wesley,1994)[6] H.P.Noyes, "The nucleon-nucleon e ective range expansion paramters," Ann. Rev. Nucl. Sci.465-484 (1972).[7] D.J. Gri ths, Introduction to Quantum Mechanics, (Prentice Hall, NJ, 1995).[8] T. Busch et al, Foundations of Physics, Vol 28, No 4, 1998.[9] A.S. Davydov, Quantum Mechanics, (NEO Press, 1966).[10] S. Jonsell, Few-body Systems 31, 255-260 (2002).[11] M. Block and M. Holthaus, Phys. Rev. Lett. A 65 052102.[12] Y. Castin, arXiv: cond-mat/0406020.30


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