在一個和二個組成的玻色-愛因斯坦凝結系統的研究主題

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(1)TOPICS ON ONE AND TWO-COMPONENT BOSE-EINSTEIN CONDENSED SYSTEMS. by Chao-Chun Huang. A dissertation submitted to National Taiwan Normal University in partial fulfillment of the requirements for the degree of. Doctor of Philosophy. Department of Physics National Taiwan Normal University January 2010.

(2) c 2010 Chao-Chun Huang Copyright ° All Rights Reserved.

(3) ABSTRACT. This thesis concerns the ground-state property and dynamics of one- and twocomponent trapped Bose-Einstein condensates (BEC) in a variety of states and regimes. Variational method and finite-difference numerical method are applied throughout this thesis. Starting from the conditions of energetic stability, coupled time independent and dependent Gross-Pitaveskii (GP) equations are re-derived for a two-component system. With the phenomenon of phase separation built in, we introduce a trial wavefunction, called “modified Gaussian (MG) function”. MG function is shown to be more suitable for a two-component as well as one-component system, providing that the (nonlinear) interaction effect is not too strong. Using MG trial wavefunction, the equilibrium and dynamical properties of a two-component system are studied in details. With the MG trial wavefunction in hand, we then study a BEC system of strong dipolar interaction. Since dipolar interaction is long-range and can be tuned to be resonant, a more realistic treatment for scattering should go beyond the first Born approximation (FBA). It is shown that the effect going beyond FBA is significantly enhanced when the system is close the phase boundary of collapse. To simulate the environment of a real crystalline solid, we also consider a one-dimensional optical lattice with a basis, i.e., a superlattice. Analytical results of acoustic and optical phonons are reported. Measurements of these modes can give unambiguous evidence to see whether the system is in the superfluid or Mott insulting regimes. Finally, we consider.

(4) the effect of anharmonic trap on vortex arrays of a one-dimensional rapid rotating BEC. It is shown that due to the anharmonic quartic trap, the system remains stable at high rotating velocity (Ω) and vortex lattices form even in the absence of the repulsive s-wave interaction (g). When g is present, the interplay between g and the quartic trap potential can lead to rich vortex lattice transition states as a function of Ω, to which vortex lattices vanish eventually at some higher Ω..

(5) ACKNOWLEDGMENTS. Firstly it is my pleasure to acknowledge my supervisor, Prof. Wen-Chin Wu, for his excellent support and guidance over the past six years. His enthusiasm and patience constantly encourage me for continuing my research works. More importantly, he cares about students not only on the studies, but also on the daily life. I am grateful to the long-time support of my family, including my wife, daughter, parents, and sister. Without their encouragement and support, I can not finish my PhD degree. I also want to acknowledge the fellow graduate students in my office. I enjoyed the discussion with them. Having them as my officemates, the life becomes much more enjoyable and delightful. I thank Prof. Cheng-Shi Liu of Yanshan University (former postdoc and visitor of our group) who helped me a lot on the numerical methods. I also thank Prof. Daw-Wei Wang of NTHU who introduces and teaches me the exciting field of dipolar systems. Finally I thank all professors of NTNU who ever taught me. I have learned a lot from you..

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(7) Contents Table of Contents. vii. List of Figures. viii. 1 Introduction. 1. 2 Theory of one and two-component BEC systems 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . 2.2 Coupled GP energy functionals . . . . . . . . . . 2.3 Energetic stability and ground-state wavefunction 2.4 Properties of two-component BEC system . . . . 2.4.1 Motivation . . . . . . . . . . . . . . . . . . 2.4.2 Remarks on Gaussian function . . . . . . . 2.4.3 Modified Gaussian function . . . . . . . . 2.4.4 Equilibrium and dynamical properties . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . 3 BEC in optical superlattice 3.1 Introduction . . . . . . . . . . . . . . . . . . . 3.1.1 Optical lattice . . . . . . . . . . . . . . 3.1.2 Relevant energy scales . . . . . . . . . 3.2 Tight-binding model . . . . . . . . . . . . . . 3.3 Oscillations in a 1D optical superlattice . . . . 3.3.1 Motivation . . . . . . . . . . . . . . . . 3.3.2 An optical superlattice . . . . . . . . . 3.3.3 Phonon modes in optical superlattices 3.4 Summary . . . . . . . . . . . . . . . . . . . . 4 BEC with dipolar interaction 4.1 Introduction . . . . . . . . . . . . . . . . . . 4.1.1 Pseudopotential of dipole interaction 4.2 Generalized scattering amplitude . . . . . . 4.3 Beyond the first Born approximation . . . . 4.3.1 Motivation . . . . . . . . . . . . . . . vii. . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. 5 5 6 8 11 11 11 12 16 23. . . . . . . . . .. 25 25 26 27 28 30 30 31 32 39. . . . . .. 41 41 42 44 46 46.

(8) viii. 4.4. CONTENTS 4.3.2 Effective Hamiltonian . . 4.3.3 Ground-state properties 4.3.4 Elementary excitations . Summary . . . . . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 5 Vortex state of fast rotating BEC 5.1 Introduction . . . . . . . . . . . . . . . . . . . . 5.1.1 Quantized Vortices . . . . . . . . . . . . 5.1.2 Slow and fast rotation . . . . . . . . . . 5.1.3 Lowest Landau levels (LLL) . . . . . . . 5.2 More topics of rapidly rotating system . . . . . 5.3 Vortices under extreme elongation in a harmonic 5.3.1 Motivation . . . . . . . . . . . . . . . . . 5.3.2 Energy functionals . . . . . . . . . . . . 5.3.3 Noninteracting system . . . . . . . . . . 5.3.4 Effect of interaction . . . . . . . . . . . . 5.4 Summary . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 47 50 53 57. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . plus quartic trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. 59 59 60 60 61 63 64 64 65 68 72 74. A Conditions of energetic stability. 77. B Variational method. 81. Bibliography. 85.

(9) List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7. Density profile by the modified Gaussian . . . . . . . . One component ground-state energy . . . . . . . . . . One component density profile . . . . . . . . . . . . . . Two-component ground-state energy . . . . . . . . . . Two-component density profile . . . . . . . . . . . . . The parameter r1 . . . . . . . . . . . . . . . . . . . . . Breathing mode dispersions for a two-component boson. . . . . . . .. 13 14 15 17 19 20 22. 3.1 3.2 3.3. Optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supperlattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dispersions relation of collective mode . . . . . . . . . . . . . . . . .. 27 32 37. 4.1 4.2 4.3 4.4 4.5. Dipolar figure . . . . . . . . . . . . . . . . . . . . . . . Aspect ratio as function of ζd or ζ0,2 . . . . . . . . . . . Schematic plot of breathing modes . . . . . . . . . . . Lower breathing mode frequencies as a function of λ . . Lower breathing mode and γ ratio as a function of ζ2,2. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 43 51 53 55 57. 5.1 5.2 5.3 5.4 5.5. Lowest Landau level . . . . . . . . . . . Lowest energy in noninteracting system . Density profile in noninteracting system Value of Nc in interaction system . . . . Density profile in interaction system . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 63 69 71 72 75. ix. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . system.. . . . . . . .. . . . . . . ..

(10) x. LIST OF FIGURES.

(11) Chapter 1 Introduction All real particles can be classified into two types, bosons and fermions. Their behaviors are quite distinct from quantum statistical description, especially when temperature is low. For bosons, this marks the onset of Bose-Einstein condensation (BEC) – a phenomenon first predicted by Satyendra Nath Bose and Albert Einstein in 1925. A gas of noninteracting bosons would undergo a phase transition at a temperature TBEC , below which a macroscopic number of atoms occupy the lowest-energy single-particle state. Within a few months only in 1995, three groups independently reported the successful generation of BEC in ultracold rubidium. 87. Rb [1], sodium. 23. Na [2] and. lithium 7 Li [3] gases. Six years later, Carl Wieman, Eric Cornell and Wolfgang Ketterle were awarded the 2001 Nobel prize in physics for their pioneering works on the physics of this new state of matter. Since then, the field of ultracold bosonic atoms has attracted a great attention both experimentally and theoretically. In fact, BEC has been successfully generated in many different species, including vapours of H [4] in 1998,. 85. Rb [5] in 2000,. 41. K [6] and 4 He [7, 8] in 2001,. 131. Cs [9] and. 174. Yb [10] in. 2003, 52 Cr [11] (with large magnetic moment and hence strong dipolar interaction) in 2005,. 39. K [12] and. 170. Yb [13] in 2007,. 40. Ca [14] (first alkaline earth element) in 2009, 1.

(12) 2. Chapter 1 Introduction. and. 84. Sr [15, 16] in 2009.. For fermions, in contrast, degenerate Fermi sea as a consequence of Pauli exclusion principle is obtained without passing through a phase transition. The first achievement of quantum degeneracy in trapped Fermi gases was in the group at JILA in 1999 [17]. Degenerate. 40. 40. K obtained by. K fermions can also been obtained in. a boson-fermion mixture by sympathetic cooling with. 87. Rb atoms in 2002 [18]. In a. similar way, fermionic 6 Li atoms were cooled to degeneracy by sympathetic cooling with bosonic 7 Li [19, 20] in 2001, with. 23. Na [21] in 2002, and with. 87. Rb [22] in 2002.. The ultracold fermion-fermion mixtures have also been tested and accomplished, such as the combinations of two different spin states in 40 K atoms [23] in 2003 and in 6. Li atoms [24, 25] in 2003. The latter has led to the formation of bosonic molecules. using Feshbach resonance and has advanced the observation of BEC-BCS crossover [26–28]. More recently, imbalanced spin population in fermion-fermion mixture has also been created and boosted great attentions on these kinds of systems [29, 30]. Theoretically in an ultracold Bose condensed system, macroscopic component of the field operator can be treated as a classical field. One can write the filed operator as ˆ t) = Ψ0 (r, t) + δ Ψ(r, ˆ t), where Ψ0 (r, t) corresponds to a single wave function (the Ψ(r, ˆ t) corresponds to thermal and quantum fluctuations. order parameter), while δ Ψ(r, When the system is dilute, quantum fluctuations is negligible; while when temperature is low (T ¿ TBEC ), thermal fluctuations is negligible. Therefore the field operator can ˆ t) ' Ψ0 (r, t). Nevertheless complex Ψ0 (r, t) can be be treated as a classical field, Ψ(r, written as |Ψ0 (r, t)| eiS(r) . The square of its magnitude gives the superfluid density, while the phase S(r) determines the superfluid velocity via vs = (¯h/M )∇S(r). In fact, the treatment not only can be applied to the bosonic condensed systems, but also to the system of weakly bound fermion pairs, which are the building blocks of the BCS picture of superfluidity in trapped Fermi systems..

(13) 3 Ultracold dilute bosonic and fermionic gases do provide a concrete realization of several models (e.g., Bose Hubbard model, Fermi Hubbard model, etc.) of many-body physics, and many of their characteristic properties have been verified quantitatively. Along these lines, for Bose condensed gases, good reviews are given by Dalfovo et al. (1999) [31], Leggett (2001) [32], Pethick and Smith (2002) [33], Pitaevskii and Stringari (2003) [34], Morsch and Oberthaler (2006) [35], and Bloch et al. (2008) [36]. For ultracold atomic Fermi gases, good reviews are given by Giorgini et al. (2008) [37] and Bloch et al. (2008) [36]. This thesis will focus on the studies of one and two-component Bose-Einstein condensed systems. We will discuss four topics on ultracold bosonic trapped atoms and molecules, namely (1) energetic stability, ground-state wavefunction, and dynamics of one and two-component BEC system, (2) BEC in optical superlattices, (3) dipolar BEC system of stronger interaction, and (4) vortex states in a rapid rotating BEC system. Before turning to next few chapters of my completed works, it is useful to mention some of my on-going works and prospects of my future works. I am developing finiteelement method (FEM) in one and two-dimensional systems. FEM can handle more complicated boundary conditions than the finite-difference method (FDM) and the quality of a FEM approximation is often better than that of the FDM approach. Besides, FEM does not need high computing facility and it can be applied equally well for one and two-dimensional systems. With this tool, many properties of the one and two-dimensional systems can be more accurately calculated, such as the rotating BEC in two dimensions which is related to the quantum Hall physics and the Tonks-Girardeau gas in one dimension. I would also like to develop the variational quantum Monte Carlo method which is very powerful when the variational parameters are many and finding the minimum energy becomes tedious. Although I focused on.

(14) 4. Chapter 1 Introduction. bosons in my thesis work, I am also very interested in ultracold fermionic systems. I am particularly interested in the system of fermion-boson mixtures. In my opinion, this is a good approach to study both the boson and fermion physics in ultracold atom systems..

(15) Chapter 2 Theory of one and two-component BEC systems 2.1. Introduction. As mentioned in Chapter 1, there are rich experiments on one-component BEC systems. For two component BEC system by the sympathetic cooling technique various combinations of two-component bosons also have been mixed and cooled. The first was done on the mixture of the same kind of bosonic atoms with different hyperfine spin states. For instance, the combinations of and of. 87. Rb|2, 1i and. 87. 87. Rb|2, 2i and. 87. Rb|1, −1i states [38],. Rb|1, −1i states [39]. Soon after, the experiment has been. extended to cool the mixture of different kinds of bosons. This includes the mixture of. 87. Rb and. 41. K atoms [6] and. 133. Cs and. 87. Rb atoms [40]. In the theoretical side,. there are also rich theoretical works on various two-component ultracold atomic systems, including boson-boson, boson-fermion, and fermion-fermion mixtures. These involve studies of the systems in various optical lattice environments [41, 42], in various vortex states [43, 44], in various scattering regimes, and the related dynamics. 5.

(16) 6. Chapter 2 Theory of one and two-component BEC systems. In general, finding an exact ground-state wavefunction of the system is always an priority of any theoretical work. For the nonuniform ultracold atomic systems under consideration, it is always a challenge to find the exact ground-state wavefunction for various cases. For example in Refs. [45, 46], the ground states of a boson-boson mixture are studied in details by Thomas-Fermi approximation (TFA) and by numerical calculation. For a boson-fermion mixture, the ground states have also been studied in Ref. [47, 48]. From the density profile point of view, the ground-state wavefunction of any two-component system can be classified into two distinct states, mixing v.s. demixing. In addition to the mixing-demixing issue, another important issue of studies is to verify whether the ground state is energetic stable or not [49]. In this chapter, I will introduce the coupled two-component GP equations in Sec. 2.2. Sec. 2.3 devotes to the discussion of the condition of energetic stability. In Sec. 2.4, a new trial wavefunction, called “Modified Gaussian”, is introduced to study properties of a two-component BEC system.. 2.2. Coupled GP energy functionals. This section focuses on the Gross-Pitaveskii (GP) theory of a two-component BEC system. The theory can be easily reduced to the one for a one-component BEC system. For a trapped two-component boson-boson mixture, the Hamiltonian can be written as H = H1 + H2 + H12 ,. (2.1). where Z. Hi =. ". ¯ 2 ∇2 1 ˆ †i (r) − h + Vi,ext (r) + drΨ 2mi 2. Z. #. ˆ †i (r0 )Vii (r − r0 )Ψ ˆ i (r0 ) Ψ ˆ i (r) dr0Ψ (2.2).

(17) 2.2 Coupled GP energy functionals. 7. and Z Z. H12 =. ˆ †1 (r)Ψ ˆ †2 (r0 )V12 (r − r0 )Ψ ˆ 2 (r0 )Ψ ˆ 1 (r). drdr0 Ψ. (2.3). ˆ †i (r) and Ψ ˆ i (r) are creation and Here i = 1, 2 denote the species i of the mixture; Ψ annihilation operators on the particle of species i at r. Vi,ext (r) is the external trap respectively for each component i and Vij (r − r0 ) is the interaction potential between species i and j. For a low-energy and dilute BEC system, the interaction potential Vij is dominated by the s-wave scattering process, consequently it can be written in a unified way 4π¯h2 aij δ(r − r0 ), Vij (r − r ) = gij δ(r − r ) = mij 0. 0. (2.4). where gij is the coupling constant and aij is the scattering length for a particle in species i scattered with a particle in species j. The “mass” mij ≡ 2mi mj /(mi + mj ) with mi the bare mass of the particle in species i. For simplicity, we shall denote g11 ≡ g1 , g22 ≡ g2 and a11 ≡ a1 , a22 ≡ a2 afterwards. At sufficiently low temperatures, considering that both components are fully condensed, the thermal average of Hi in (2.2) becomes the famous GP energy functional: ". Z. Ei = hHi i =. drΨ∗i (r). ¸. h ¯ 2 ∇2 gi + Vi,ext (r) + |Ψi (r)|2 Ψi (r), − 2mi 2. (2.5). ˆ i i, while the thermal average of H12 in (2.3) becomes where Ψi = hΨ. Z. V12 = hH12 i = g12. dr|Ψ1 (r)|2 |Ψ2 (r)|2 .. (2.6). Putting (2.5) and (2.6) together, the coupled GP energy functional for a two-component BEC system is then Z. E = hHi =. dr. X i=1,2. ". Ψ∗i. #. h ¯ 2 ∇2 gi + Vi (r) + |Ψ|2 Ψi + g12 − 2m 2. Z. dr |Ψ1 |2 |Ψ2 |2 . (2.7).

(18) 8. Chapter 2 Theory of one and two-component BEC systems. The GP energy functional for a single-component BEC system is just Eq. (2.5) with all ”i” omitted for brevity.. 2.3. Energetic stability and ground-state wavefunction. Starting from the coupled GP energy functional (2.7), ground-state wave function of the two-component BEC system can be obtained by minimizing it subject to the conservation of total particle number. Equivalently, one is to minimize G = E − µ1 N1 − µ2 N2 with Ni =. R. dr |Ψi (r)|2 and µi the chemical potential for species i. To. minimize G, one can write Ψi ≡ Ψi,0 + δΨi with Ψi,0 the equilibrium condensate wavefunction and δΨi the fluctuation around Ψi,0 and then expand G to second order in δΨi . It is found that G = G0 [Ψi,0 ] + δG1 + δG2 , where δG1 is linear in δΨi while δG2 is quadratic in δΨi . Moreover, δG1 vanishes so long as Ψi,0 satisfy the following coupled time-independent GP equations " ". #. h ¯ 2 ∇2 + V1 (r) + g11 |Ψ1,0 |2 + g12 |Ψ2,0 |2 Ψ1,0 = µ1 Ψ1,0 − 2m1 #. h ¯ 2 ∇2 + V2 (r) + g22 |Ψ2,0 |2 + g12 |Ψ1,0 |2 Ψ2,0 = µ2 Ψ2,0 . − 2m2. (2.8). As for a extremum of G, linear term δG1 needs to vanish for any small δΨi . This results that the ground-state wavefunctions must satisfy the above coupled timeindependent GP equation (2.8). Furthermore, to fully justify the condition of energetic stability, second-order term δG2 needs be determined. It is found that 4 1 X δG2 = 2 i,j=1. Z. d3 xBi Mij Bj† ,. (2.9).

(19) 2.3 Energetic stability and ground-state wavefunction. 9. where the 4-vector Bi = (B1 , B2 , B3 , B4 ) ≡ (δΨ1 , δΨ2 , δΨ†1 , δΨ†2 ) and the 4 × 4 matrix M is a Hermitian. The elements of M are M11 = M33 = h1 + g12 Ψ22,0 + 2g1 Ψ21,0 M22 = M44 = h2 + g12 Ψ21,0 + 2g2 Ψ22,0 M12 = g12 Ψ1,0 Ψ∗2,0 ,. M34 = g12 Ψ∗1,0 Ψ2,0. M23 = M14 = g12 Ψ1,0 Ψ2,0 M13 = g1 Ψ21,0 , 2. M24 = g2 Ψ22,0 ,. (2.10). 2. where hi ≡ − ¯h2m∇i + Vi (r) − µi . Detailed derivations of δG1 and δG2 can be found in Appendix A. The two-species system is energetically stable if all eigenvalues of M are positive. This criterion is justified by the fact that for a minimum G, arbitrary small nonzero fluctuation δΨi should not decrease the value of G. Of course the criterion can be generalized to include the fluctuations of particle numbers (δNi ), in addition to the fluctuations of the wavefunction (δΨi ) considered above. In Ref. [49], the authors considered the energetic stability of a two-component BEC system, taking into account both number and wavefunction fluctuations. It was found that the system can be unstable even when a12 is positive and this instability is mainly caused by the number fluctuations [49]. This means that when the system is energetically unstable, one can still find a “ground-state” wavefunction that satisfies δG1 = 0. However, this wavefunction may just be a metastable ground state. To solve Eq. (2.8) exactly, it generally requires some numerical methods. Nevertheless, it is instructive to obtain some analytical results by some approximate ways. For this purpose, the most common approaches are variational methods and ThomasFermi approximation. Thomas-Fermi approximation has been applied to study onecomponent as well as two-component GP energy functionals [45]. The details can be found in the book [33]. For the variational method, simple Gaussian function (SG) is.

(20) 10. Chapter 2 Theory of one and two-component BEC systems. usually used as the trial wave function in one-component GP energy functional [33]. For a two-component BEC system, due to the possibility of phase separation, a more suitable trial wave function is needed. In next section, we will introduce a new trial wave function (called Modified Gaussian) to study the two-component BEC system. In addition to the time-independent GP equation (2.8), if one wants to study the dynamics of the system, the time-dependent GP equations need to be solved. For a two-component BEC system, they are " ". #. h ¯ 2 ∇2 ∂Ψ1 (r, t) + V1 (r) + g11 |Ψ1 (r, t)|2 + g12 |Ψ2 (r, t)|2 Ψ1 (r, t) = i¯h − 2m1 ∂t #. h ¯ 2 ∇2 ∂Ψ2 (r, t) + V2 (r) + g22 |Ψ2 (r, t)|2 + g12 |Ψ1 (r, t)|2 Ψ2 (r, t) = i¯h − . 2m2 ∂t (2.11). The above time-dependent GP equation can be derived from the action principle R. δ Ldt = 0, where the Lagrangian L=. X Z. dr. i=1,2. ∂Ψ∗ i¯ h ∗ ∂Ψi (Ψi − Ψi i ) − E. 2 ∂t ∂t. (2.12). Here E is the two-component energy functional, Eq. (2.7). With the time-dependent GP equation one can study the dynamics of the system numerically. Alternatively, if one adds some suitable (dynamical) parameters into a suitable trial wave function and makes use of the Lagrangian (2.12), one can also study the dynamics of the system based on the variational method. In Appendix B, we show in details how the variational method together with the Lagrangian equation can be used to study the dynamics of the system. Some related numerical methods are also mentioned in Appendix B..

(21) 2.4 Properties of two-component BEC system. 2.4 2.4.1. 11. Properties of two-component BEC system Motivation. In this section, variational method is applied to study the properties of a twocomponent BEC system. As mentioned before, simple Gaussian (SG) is not generally suitable for a two-component system, hence one of the keys here is to find a more suitable trial wavefunction. In this regard, we will introduce a new trial wavefunction, called “modified Gaussian (MG)” function. Before the MG trial wavefunction is introduced, it is instructive to first examine the validity of the SG function.. 2.4.2. Remarks on Gaussian function. When temperature approaches zero, isotropic single-component trapped Bose condensate is governed by the GP energy functional given in Eq. (2.5). When the coupling g is infinitesimally small, the ground-state wavefunction of the system is exactly the Gaussian distribution: ³. ´. Φ(r) = C exp −Ar2 . Here we shall define the wavefunction Ψ(r) =. (2.13). √ N Φ(r) in Eq. (2.5) with normalized Φ. . Gaussian function is “non-perturbative” and has the advantage of easily integrating out over the space. When g starts to increase, Gaussian function may no longer be a good one to describe the system. In the limit of large g, Gaussian distribution fails completely and the most common approximation to the ground state of the system is the one called “Thomas-Fermi approximation” (TFA), which ignores the kinetic term in (2.5). The validity of the Gaussian distribution upon the increase of g can be sorted out as follows. First, let’s rescale the GP energy functional: E/(¯hω) → E, tω → t, r/` → r with ` ≡. q. h ¯ /mω being the length scale and the external potential.

(22) 12. Chapter 2 Theory of one and two-component BEC systems. Vext = mω2 /2 throughout this section. Next, by dividing the energy functional by N (total number of particles), Eq. (2.5) becomes Z. E/N =. ·. ¸. 1 m 1 drΦ∗ − ∇2 + r2 + N g|Φ|2 Φ. 2 2 2`. (2.14). By writing this way, it is seen clearly that in addition to g, both the magnetic trap potential (through `) and N are also coupled to the nonlinear term. As a consequence, whether a Gaussian is a good approximation will depend on the magnitude of the √ front factor for the nonlinear term, mN g/2` ∝ N gm3/2 ω. Provided that g is fixed, √ the smaller the value of N ω, the better the Gaussian distribution is. In current experiments performed on the single-component systems, nonlinear effect is not small typically. Application of the Gaussian distribution to a real system thus poses a serious question mark. Moreover, in a two-component system, “phase separation” can exist when the interspecies coupling g12 is strong and in this case, Gaussian distribution for individual component is not suitable from the beginning.. 2.4.3. Modified Gaussian function. To better describe the system and also capture the advantage of easy integration, we propose the following trial wavefunction [50], (. ". #. ". A(r − r0 )2 A(r + r0 )2 + exp − Φ(r) = C exp − 2 2 ! Ã 2 Ar cosh(Ar0 r), = C 0 exp − 2. #). (2.15). which is the superposition of two Gaussian functions, with centers deviated from the origin by r0 and −r0 respectively. As written alternatively in the second line of (2.15), the resulting function is a Gaussian function modified (multiplied) by a hypercosine function. Hence, function (2.15) is called “modified Gaussian”. In (2.15), C and C 0 = 2C exp(−Ar02 /2) are normalization constants. A and r0 are parameters.

(23) 2.4 Properties of two-component BEC system. 13. Figure 2.1 The normalized density profile, |Φ(r)|2 , with Φ given by the modified Gaussian function in (2.15). Here A = 0.8 is fixed for all cases. corresponding to the amplitude and overall density profile of the atom cloud. For a given C, the values of A and r0 are determined upon minimizing the groundstate energy E of the system. In Fig. 2.1, r0 dependence of the normalized density profile |Φ(r)|2 r2 is shown with a fixed A. As can be seen clearly, the maximum of |Φ(r)|2 r2 has a tendency to move outwards when r0 is increased. When r0 is large, the maximum could move significantly away from the origin. Having this feature, which is important when phase separation occurs, the modified Gaussian function is considered to be more effective for a two-component mixture. Compared to the simple Gaussian distribution [Eq. (2.13)], the modified Gaussian consists one more parameter (r0 ), that allows for a better description of the groundstate equilibrium properties of the system. As an illustration, in Fig. 2.2 we plot the ground-state energy E [Eq. (2.14)] as a function of the scattering length a for the modified Gaussian distribution. The parameters are taken to be: ω = 2π ×160Hz and N = 104 . Results obtained from using the simple Gaussian trial wavefunction, the TFA, and by exact numerical method are also shown for comparison. In the small-a.

(24) 14. Chapter 2 Theory of one and two-component BEC systems. 7 6. N. 4. =10. E/(N. ). 5 G. 4. MG. 3. TFA Numerical. 2 1 0 0. 1. 2. 3. 4. 5. 6. 7. a(nm). Figure 2.2 Ground-state energy E divided by h ¯ ω (ω = trap frequency) versus the scattering length a for single-component condensed bosons. The curves are for Gaussian (G), modified Gaussian (MG), TFA, and exact (numerical) calculation. The parameters used are ω = 2π × 160Hz and N = 104 . regime (small nonlinear effect), the curve of modified Gaussian merges to the one of simple Gaussian and numerical result, as it must be. Gaussian distribution is exact in the a → 0 limit. The ground-state energy E is seen to be much underestimated for TFA at small a. In the intermediate a (= 2 ∼ 5) regime, nonlinear effect becomes more and more important, the curve of modified Gaussian starts to deviate (becomes lower) from the one of a simple Gaussian. More importantly, the curve of MG is very close to the exact numerical one, indicating in the intermediate a regime that MG is a good trial wavefunction from the ground-state energy point of view. It should be emphasized again that most of current experiments lie in the intermediate a regime. In the large-a regime, the results of TFA should be more close to the exact one (not shown in Fig. 2.2). To further check the validity of MG, spatial distribution (density profile) of the.

(25) 2.4 Properties of two-component BEC system. N. 20 15. 15. =200. G MG. 10. TFA. 5. Num.. 0 0.0. 0.5. 1.0. 60. N. 45. 1.5. 2.0. 2.5. 3.0. 3. =10. 30 15 0 0.0. 0.5. 160. 1.0. 1.5. N. 120. 2.0. 2.5. 3.0. 3.5. 4. =10. 80 40 0 0. 1. 2. 3. 4. 5. r/l. Figure 2.3 The spatial dependence of the density profile |Φ(r)|2 for a onecomponent system. Four kinds of distribution (Gaussian, modified Gaussian, TFA, and exact numerical calculation) are compared. Frame (a)-(c) are for atom number N = 200, 103 , and 104 respectively. four cases (Gaussian, modified Gaussian, TFA, and numerical calculation) are compared for different total numbers of particle N in Fig. 2.3. The case of larger N corresponds to the case of larger nonlinear effect. In frame (a) of smaller N , the curve of modified Gaussian is closer to the numerical calculation – with the maximum at the origin. When N is larger [frame (b) & (c)], the curves of modified Gaussian behave closer to the ones of TFA and also close to numerical calculation , although the maxima may deviate from the origin. As far as a real distribution is concerned, this feature is somewhat defective. It should be emphasized, however, that modified Gaussian has an overall distribution which is closer to the one of TFA and is more accurate for large nonlinear effect. Moreover, modified Gaussian inherits the.

(26) 16. Chapter 2 Theory of one and two-component BEC systems. advantage of easy integration of simple Gaussian, which makes variational method feasible for the problems. In next section, equilibrium property and dynamics of a two-component system are studied. Breathing modes of such system will be shown to depend crucially on whether the system is miscible or phase-separate.. 2.4.4. Equilibrium and dynamical properties. The analyses in previous section are limited to single-component systems. In the case of a two-component mixture, the system could undergo a phase transition from a miscible state to a phase-separate state when the inter-species coupling is strong. Considering an isotropic system, when phase separation occurs one component of the atoms will have maximum density deviated from the origin, while the density of the other component remains maximum at the origin. As pointed out before, a simple Gaussian is ineffective to describe such phenomenon. Here we show how the phase separation occurs and their related physics when the modified Gaussian distributions are in use. At the same time, the dynamics (breathing modes) of the two-component system will also be studied starting from the same modified Gaussian distribution. Equilibrium Property The ground-state properties of the two-component bosonic atoms are first studied. Considering that both components are described by the modified Gaussian distribution µ. ¶. 1 Φi (r) = Ci exp − Ai r2 cosh(Ai ri r), 2. (2.16). where i = 1, 2 and the normalization factor Ci is dependent of the total number Ni of individual component i. The ground-state energy E of the system can be calculated.

(27) 2.4 Properties of two-component BEC system. 17. 20. N. =10. 3. N. =10. 18. ). 9. 4. E/ a. 12. E/(N. MG 8. Num.. 16. 7. 14. 0.5. 0.16. 0.4. 0.12. 0.3. 0.08. 0.2. 0.04. 0.1 0. 2. 4. a (nm) 12. 6. 8. 0.00 0. 2. 4. 6. 8. a (nm) 12. Figure 2.4 The dependence of ground-state energy E, divided by N h ¯ ω1 , on the inter-component scattering length a12 of a two-component BEC system. Modified Gaussian distribution and numerical calculation are undertaken. The slopes of ∂E/∂a12 vs. a12 are also shown [E 0 ≡ E/(N h ¯ ω1 )]. Left column 3 corresponds to N1 = N2 ≡ N = 10 , while right column corresponds to N = 104 . See text for the values of other parameters. by substitution of (2.16) into Eq. (2.7). Given three coupling strength a1 , a2 , and a12 , and the particle number N1 and N2 , final result of E can be obtained by minimizing E against the variation of A1 and A2 , and r1 and r2 . In Fig. 2.4, the results of E are shown as a function of the inter-component scattering length a12 . Two cases of equal population N1 = N2 ≡ N = 103 and N = 104 are presented. The slopes, ∂E/∂a12 vs. a12 are also shown. For easy comparison, throughout this section the parameters are taken to be the same as those in Ref. [46]. Based on the experiment of Na-Rb mixture, the scattering lengths of Na and Rb atoms are chosen to be 3 nm and 6 nm respectively. Regarding the external magnetic trap, the corresponding trap frequencies are ω1 = 2π × 310Hz for Na and ω2 = 2π × 160Hz for Rb respectively. In both plots in Fig. 2.4 (especially ∂E/∂a12 vs. a12 ), one sees clear evidence that the system undergoes a transition from one phase to another at a12 ∼ 2 − 2.5 when N = 104 and at a12 ∼ 4 − 5 when N = 103 . Numerical results are also shown.

(28) 18. Chapter 2 Theory of one and two-component BEC systems. for comparison. It is found that MG results are very close to those of numerical calculations for the (intermediate) N = 103 case. Fig. 2.5 shows how this transition behaves in the density profiles of the twocomponent system and we have presented both MG and numerical results for comparison. The figures are plotted in a variety of interspecies scattering lengths (a12 ) with fixed a1 = 3 nm and a2 = 6 nm and by varying the atom number N . In the case of N = 104 , it is found that both |Φ1 (r)|2 and |Φ2 (r)|2 have maximum away from the center for the MG cases when a12 = 0. It has to be emphasized again, that modified Gaussian distribution has a feature that the maximum density could deviate from the origin due to large nonlinear effect. Even though a12 = 0, the individual nonlinear effect is still large, which lead to such “unphysical” result. As mentioned before, it is the fact that modified Gaussian has an overall distribution more close to TFA, that one has to look into. When a12 becomes larger (a12 ∼ 2 − 2.5), the maximum of |Φ1 (r)|2 is pushed significantly away from the center, while the maximum of |Φ2 (r)|2 is right at the center – the phenomenon of phase separation. To see the phase-separation phenomenon in an alternative way, in Fig. 2.6 we plot the value of parameter r1 (for the modified Gaussian) versus a12 . It is reminded that r1 is referred to the Na component in Fig. 2.5. In the case of N = 104 , there is a sudden change for r1 at a12 ∼ 2 − 2.5. This is in great consistence with the ground-state energy shown in Fig. 2.4 where a sharp change also occurs at this range of a12 . In view of the original definition of r1 in the first line of Eq. (2.15) (i.e., r0 there), a sudden change of r1 means a sudden change of the coordinate of density maximum (see also Fig. 2.1). This sudden change indeed corresponds to the transition of “phase separation”. In the case of N = 103 , we also see a consistent picture among the results in Figs. 2.4–2.6. A phase separation occurs at a12 ∼ 4−5. When N = 200, the phenomenon is less obvious. However, one can still see a phase separation that.

(29) 2.4 Properties of two-component BEC system. 19. Rb-Num.. Rb-MG. Na-Num.. 120. density profile. 100. a. 12. 120. 100. = 0. a. 80. 80. 60. 60. 60. 40. 40. 40. 20. 20. 0 0. 1. 2. 3. 4. 5. 6. 120. 0 0. 2. 3. 120 a. 100. 12. = 3 nm. 4. a. 100. 12. 5. 6. 0 0. = 4 nm. 80. 60. 60. 60. 40. 40. 40. 20. 20 3. 4. 5. 6. 0 0. 50. a. 12. = 0. 2. 3. 4. 5. 6. 0 0. = 2 nm. 30. 20. 20. 20. 10. 10. 10. 0. 0. 2. 3. 4. 5. 0. 1. 2. 3. 4. 5. 50 = 6 nm. a. 40. 12. 00. a. 40. 12. = 8 nm. 30. 30. 20. 20. 10. 10. 10. 0. 0. 2. 3. 4. 5. 0. 1. 2. 3 r/l. 20. N = 200 a. 16. 12. = 0. 20. 5. = 3 nm. a. 16. 4. 0. 0. 12. 8. 4. 4. 20. a. 16. = 9 nm. 12. 0 0 20. 2 a. 16. 3. 4. = 12 nm. 12. 0 0. 12. 8. 8. 4. 4 3. 4. 1. 0 0. 1. 2. 6. = 4 nm. 3. 4. 5. = 10 nm. 2. 3. a. 4. 5. = 6 nm. 12. 1. 2 a. 16. 12. 2. 12. 20. 8. 1. 5. 4 1. 12. 0 0. 4. a. 16 12. 4. 3. 20. 8. 3. 2. 2. 12. 2. = 5 nm. 12. 8. 1. 1. a. 12. 0 0. 6. 12. 40. 20. 1. 5. 50. 30. 0. 4. a. 40. 30. 1. 3. 50 12. 30. 0. 2. 2. a. 40. 50. = 2 nm. 20 1. 50. 3. N = 10. 40. 1. 100. 80. 2. 12. 120. 80. 1. a. 20 1. r/l. density profile. 100. = 1 nm. 12. 80. 0 0. density profile. Na-MG. 120. 4. N = 10. 12. 3. 4. = 15 nm. 4 1. 2. 3. 4. 0 0. 1. 2. 3. 4. r / l. 2. Figure 2.5 The density profile of individual component is plotted in a variety of the interspecies scattering length a12 for a two-component system. Both MG and numerical results are presented for comparison. From the top to the bottom are for N1 = N2 ≡qN = 104 , 103 and 200 respectively. Note that in ¯ /m2 ω2 . x axis, r is scaled to `2 = h.

(30) 20. Chapter 2 Theory of one and two-component BEC systems. 4.0 3.5. 4. N = 10 3 N = 10 N = 200. r1 / l2. 3.0 2.5 2.0 1.5 1.0 0. 2. 4. 6. a12 (nm). 8. 10. 12. Figure 2.6 The value of the parameter r1 divided by `2 (r1 is referred to Na component in Fig. 2.5) for the modified Gaussian plotted as a function of the interspecies scattering length a12 . appears at a12 ∼ 9 − 10. When MG results are compared to those of the numerical calculation, there is a good matching for them for the small N = 200 and intermediate N = 103 cases. For the large N = 104 case, apart from the defect that density profile could show a unphysical maximum away from the origin (see Fig. 2.5 for example), MG results still match well with the numerical ones. Thus it is convincing that MG is a good trial wave function for one- and two-component BEC systems when the nonlinear effect is not too large. Breathing Modes Here we investigate how the phase separation affects the dynamics of a two-component BEC system. It was shown in Ref. [46] that for a two-component mixture, the collective mode dispersion will have a drastic change near the occurrence of phase separation. Since the property of collective mode is intimately connected to the equilibrium property of the system, the sharp transition in the ground-state energy of the system.

(31) 2.4 Properties of two-component BEC system. 21. could imply a drastic collective mode dispersion change in the same regime. For the dynamics, we shall focus on the breathing modes of the system. We first extend the modified Gaussian distribution for each component [Eq. (2.16)] to include the dynamical variables [51, 52]: ½. 1 Φi (r, t) = Ci exp − Ai [1 + εi (t) + iε0i (t)]r2 2. ¾. × cosh(Ai ri [1 + εi (t)]1/2 r).. (2.17). Here εi and ε0i correspond to the fluctuations of local amplitude and local phase of the atom cloud along the radial r direction. Ai and ri are determined earlier through minimizing the ground-state energy of the system (see previous subsection). Since we are interested in the breathing modes of the lowest energy, the fluctuation of local amplitude, εi , is coupled to the r and r2 terms in (2.17). When the value of ε1 /ε2 is obtained to be positive (negative), the corresponding breathing mode is called in-phase (out-of-phase) mode. Substitution of the above dynamical wave function into corresponding Lagrangian equation and Eq. (2.7), one can derive the coupled equations of motion [through the Euler-Lagrange equations] for εi and ε0i , and then solve a corresponding dynamical matrix to obtain the dispersions of relations for the breathing modes. In Fig. 2.7, we present the breathing mode dispersion as a function of the inter-component scattering length a12 . Again, all parameters are the same as before. The lowest-energy mode results given by numerical calculations are also shown for comparison [46]. Much information is revealed in Fig. 2.7. Taking the N = 104 case as an example, when interspecies scattering length a12 is negative, the system is miscible. In-phase breathing mode has lower energy and is more easily excited. When a12 is positive, the results are divided into three regimes. When 0 ≤ a12 ≤ 2.5, the system is miscible and thus out-of-phase breathing mode has lower energy and more easily gets excited..

(32) 22. Chapter 2 Theory of one and two-component BEC systems. 9. 6 N=10. Mode Frequency. 8. 4. out-of-phase (v). 3. N=200. 5. in-phase (v). 7. 5 N=10. 4. out-of-phase (n) in-phase (n). 6. 4. 5. 3. 3. 4 3. 2. 2. +. +. +. 2 1. 1 -2. 0. 2. 4. 6. 8. 1. -2 0. 2. 4. 6. a. (nm). 8 10. -3. 0. 3. 6. 9 12 15. 12. Figure 2.7 Based on the variational method (V), in-phase and out-ofphase breathing mode dispersions are plotted as a function of a12 for a two-component boson system. In each frame, the turning point in lower curves corresponds to where the system starts to be phase separated. The lowest-energy mode result given by numerical calculation (N) are shown and compared. The crosse signs at a12 = 0 correspond to the breathing modes of individual component 1 and 2 (decoupled regime). When the system starts to phase separate but most parts of the two components are still overlapping (2.5 ≤ a12 ≤ 4), the energy of the out-of-phase breathing mode remains lower than that of in-phase mode . When the two components are well separated (not much overlap) at a12 ≥ 4 (see also Fig. 2.5), in-phase breathing mode turns out to have lower energy again. The switch of the in-phase and out-of-phase breathing modes in the above latter two cases can be understood alternatively as follows. When phase separation occurs but the two components are still very much overlapped, out-of-phase mode is more easily excited because it corresponds to the decrease of the overlap (and the energy). However, when the two components are well separated, in-phase mode is in turn more easily excited because in this case out-of-phase mode simply increases the overlap (and the energy)..

(33) 2.5 Summary. 23. As seen in Fig. 2.7 and N = 104 case, there is one turning point in the dispersion curves. The turning point is the signature of the system being phase separated. In comparison with the numerical results (where only the lowest-energy mode is given), our result of lowest-energy mode matches theirs quantitatively before phase separation arises. After the phase separation occurs, our result is qualitatively similar to numerical result. It is worth noting that the two points of a12 = 0 correspond actually to the breathing modes of two decoupled components (lower-frequency one is for Rb, while higher-frequency one is for Na). From the goodness of the match of our results to numerical result, it strongly suggests that modify Gaussian is also a good function to describe the dynamics of a one-component system when the nonlinear effect is important. Importantly the in-phase and out-of phase breathing modes in numerical result are verified by a nonlinear response analysis [46]. So there is no the exchange of the in-phase and out-of-phase breathing modes by numerical calculation in the regime where the two component are well separated. It is because the definition is different in variation method and numerical calculation.. 2.5. Summary. This chapter mainly applies the variational method to study the properties of one and two-component BEC systems. Our approach can be applied to other two-component systems such as boson-fermion and fermion-fermion mixtures. In Sec. 2.2, the coupled GP energy functional for a two-component BEC system is introduced. By using the perturbation expansion around the equilibrium ground-state wavefunction, one is able to derive the time-independent GP equation for studying the dynamics. The formalism is easily reduced to a single-component BEC system. In Sec. 2.3, I consider the condition of energetic stability by using the second-derivative test. Together with.

(34) 24. Chapter 2 Theory of one and two-component BEC systems. the Lagrangian of the system, one obtains the same time-dependent GP equations. In Sec. 2.4, the density profile, the ground-state energy, and the breathing modes of a two-component Bose condensed system are studied in details using the variational method. We propose a more suitable trial wavefunction, called “modified Gaussian (MG) distribution function”, which is shown to give a better description for the twocomponent as well as one-component BEC systems when the nonlinear effect is not too large. The MG trial wavefunction can be used to study the vortices of a twocomponent system, the dipolar system (see Chapter 4), and possibly the system of imbalanced spin population..

(35) Chapter 3 BEC in optical superlattice 3.1. Introduction. In past few years, Bose-Einstein condensates (BECs) in optical lattices have been the subject of extremely intense and rewarding research, both theoretically and experimentally. It is an active research topic in the cross-disciplined field of atomic molecular and optics physics (AMO) and condensed-matter physics (CMP). From the condensed matter point of view, because (i) laser power can be tuned to vary the ratio of on-site repulsion to intersite coupling of cold atoms, (ii) different geometry of laser beams can be taken to manipulate different dimensions and configurations of optical lattices, and (iii) different types of atoms (fermions, bosons, ions, or their mixtures) can be loaded into the optical lattice, consequently the simulation of condensed matter environment is considerably easy and flexible. Recent investigations on the cold-atom system with an optical lattice have been made in various aspects, including the superfluid -Mott insulator quantum phase transition [53–57], the band-structure phenomena [58, 59], 2D and 1D systems [55, 60–63], fermionic systems [64–69], and the quantum informatics [70–73]. 25.

(36) 26. Chapter 3 BEC in optical superlattice. 3.1.1. Optical lattice. An optical lattice is a periodic potential which is generated by overlapping two counterpropagating laser beams. Due to the interference between the two laser beams, an optical standing wave with a period of half wavelength λ is formed, in which atoms can be trapped. The physical origin of the confinement of cold atoms with laser light is the dipole force, 1 F = α(ωL )∇[|E(r)|2 ], 2. (3.1). due to a spatially varying ac Stark shift to which atoms experience in an off-resonant light field. Since the time scale for the center-of-mass motion of atoms is much slower than the inverse laser frequency ωL , only the time-averaged intensity |E(r)|2 enters. The direction of the force depends on the sign of the polarizability α(ωL ). In the vicinity of an atomic resonance from the ground state |gi to an excited state |ei at ¯. ¯2. frequency ω0 , the polarizability has the form α(ωL ) ≈ ¯¯he| dÊ |gi¯¯ /¯h(ω0 −ωL ), with dÊ the dipole operator in the direction of the field. Atoms are thus attracted to the nodes or to the antinodes of the laser intensity for blue-detuned (ω0 > ωL ) or red-detuned (ω0 < ωL ) laser light, respectively. For example in a 1D optical lattice the potential seen by the atoms is given by V (x) = V0 cos2 (πx/c) ,. (3.2). where the lattice spacing c = λ/2 and V0 is the lattice depth. Similarly periodic potentials in two or three dimensions can be formed by overlapping two or three optical standing waves along different (usually orthogonal) directions, like those in Fig 3.1. In addition, by controlling the polarizations of the lattice beams, it is also possible to create state-dependent potentials that can be shifted relative to each other. This kind potential is called spin-dependent optical lattice potentials..

(37) 3.1 Introduction. 27. Figure 3.1 (a) Two- and (b) three- dimensional optical lattice potentials formed by superimposing two or three orthogonal standing waves. The figure is borrowed from Ref. [36].. 3.1.2. Relevant energy scales. A basic theory of optical lattice systems can be classified to different regimes depending on the relative importance of the nonlinear interaction and the lattice parameters. In general, there are four important energy scales (takeing 1D case as an example): (i) Bandwidth Ew : the energy difference between q = π/c and q = 0 of the lowest band. (ii) On site interaction U : gives the on-site interaction energy per atom on a single lattice site. (iii) Energy gap Eg : represents the energy difference between the bands at q = π/c. In a deep optical lattice, this corresponds to the energy difference between the lowest and first vibrational state in a single potential well of the lattice. (iv) Tunneling rate J : the nearest-neighbor Josephson coupling responsible for con-.

(38) 28. Chapter 3 BEC in optical superlattice densate tunneling. In the deep periodic potential limit, there is also a direct connection between. Ew and J; that is, Ew = 4J. In this regime, the nonlinearity will be the smallest energy scale if U is smaller than the bandwidth Ew and the band gap Eg . When Ew < U < Eg , the nonlinear energy scale is in the intermediate range. The nonlinear energy scale is dominant when U is larger than Ew and Eg . We shall focus on the nonlinear energy scale being the smallest one and the optical lattice is in the deep limit, called the “tight-binding limit”. For other regimes, a detailed account can be found in the reviews [35, 36]. In the tight-binding limit mentioned above, the condensate wave function can be described by localized Wannier function associated with the lowest band. The condensate energy functional in this regime is derived in next section.. 3.2. Tight-binding model. This section studies BEC condensates with an optical lattice in the tight-binding limit [52]. The time-dependent Gross-Pitaveskii (GP) energy functional of the system is given by Z ∗. E [Ψ , Ψ] =. (. ¸. ¾. h ¯2 ∗ 2 g Ψ ∇ Ψ + [Vext (r) + |Ψ|2 |Ψ|2 , dr − 2m 2 3. (3.3). where Ψ = Ψ(r, t) is the time-dependent wave function of the condensate, µ is the chemical potential, and g = 4π¯ h2 a/m with a the s-wave scattering length of the twobody interaction. Vext is the external potential. In the tight-binding approximation the condensate wavefunction can be written as Ψ(r, t) =. X n. ψn (t)φ(r − rn ),. (3.4).

(39) 3.2 Tight-binding model. 29. where φ(r − rn ) is the condensate wave function localized in site n with the norR. malization condition. dr|φn |2 = 1. For tight-binding limit,. q. R. drφ∗n φn+1 ' 0. ψn ≡. Nn (t)eiνn (t) corresponds to the amplitude at site n, where Nn and νn are associated. with particle number and the phase, respectively. Substituting the approximation (3.4) into (3.3), the GP energy functional reduces to a discrete nonlinear Schrödinger equation (DNLSE) E=. X n. {²n |ψn |2 − J(ψn∗ ψn+1 + c.c.) +. U |ψn |4 }, 2. (3.5). where ". Z. h ¯ 2 ³ ~ ´2 ∇φn + Vext |φn |2 dr 2m. ²n =. Z. J ' − Z. #. (3.6). ". h ¯2 ∗ 2 φ ∇ φn+1 + φ∗n Vext φn+1 dr 2m n ¯ ¯. #. (3.7). ¯ ¯. dr¯φ4n ¯.. U = g. (3.8). The quantization of the DNLSE in (3.5) is straightforward. The corresponding Hamiltonian is called “Bose-Hubbard model”, [53] ˆ = H. X n. U {²n ψˆn† ψˆn − J(ψˆn† ψˆn+1 + H.c.) + ψˆn† ψˆn† ψˆn ψˆn }, 2. (3.9). with ψˆn† , ψˆn the bosonic creation and annihilation operators. The Bose-Hubbard model is usually used to discuss the phenomenon of Superfluid-Mott-insulator transition. In the following, let us consider the case of a one-dimensional (1D) optical lattice, within the tight-binding approximation. ) and let Ψ(r, t) = y 2 ) + V0 cos2 ( πz c. P n. Consider that Vext ≡. ψn (t)φ(r − rn ) =. P n. 1 mωρ2 (x2 2. +. wn (z)Φn (x, y, t). Here. we assume that the local wave function φn (r) is divided to wn (z)ϕn (x, y, t) and Φn (x, y, t) = ψn (t)ϕn (x, y, t). Consequently the energy functional of the system becomes ∗. E(Φ , Φ) =. XZ n. (. ". #. 1 h ¯2 ∗ 2 U¯ Φn ∇ Φn + mωρ2 (x2 + y 2 ) + |Φn |2 |Φn |2 dxdy − 2m 2 2.

(40) 30. Chapter 3 BEC in optical superlattice − J¯. X Z <n,m>.  . dxdyΦ∗m ∇2 Φn ,. (3.10). . where J¯ ≡ − U¯ = g. Z. Z. ". h ¯2 ∗ πz wn (z)∇2 wn+1 (z) + V0 cos2 ( )wn∗ (z)wn+1 (z) dz 2m c. #. dz|wn (z)|4 .. (3.11) (3.12). Comparing J¯ and U¯ to J and U in (3.7) and (3.8) and assuming that the system is in the ground state [(Φn (x, y, t) = Φn (x, y) = ψn ϕn (x, y))], one finds that J = J¯. Z. [dxdyϕ∗n (x, y)ϕn+1 (x, y)]. Z. + ' J¯. Z. dzwn∗ (z)wn+1 (z) Z. ". 1 h ¯2 ∗ 2 ϕn ∇ ϕn+1 + mωρ2 (x2 + y 2 )ϕ∗n ϕn+1 dxdy 2m 2. ¯ [dxdyϕ∗n (x, y)ϕn+1 (x, y)] ' J,. where we have used the conditions R. R. #. (3.13). dxdyϕ∗n (x, y)ϕn+1 (x, y) =. R. dxdyϕ∗n ϕn = 1 and. dzwn∗ (z)wn+1 (z) ' 0. Moreover, U = U¯. Z. dxdy|ϕn (x, y)|4 '. U¯ , 2π`ρ. (3.14). where ϕn (x, y) = √. 3.3 3.3.1. 1 −(x2 +y2 )/2`2ρ 2 h ¯ . e , `ρ ≡ π`ρ mωρ. (3.15). Oscillations in a 1D optical superlattice Motivation. Since the achievement of atomic BEC in 1995, dynamics of single Bose condensate had been under detailed investigation both theoretically and experimentally [31]. When condensates are distributed in an optical lattice, in addition to collective modes,.

(41) 3.3 Oscillations in a 1D optical superlattice. 31. phonons can also propagate among the lattice. How the collective and phonon modes are coupled in such a condensed system thus has attracted a large number of studies [51, 58, 74–77]. In fact, condensate excitations in a 3D optical lattice have been measured recently in the superfluid (SF) regime [78]. Among different theoretical approaches, Martikainen and Stoof [51] have recently applied the variational approach to study the transverse breathing mode and longitudinal acoustic phonon in a simple 1D Bose-condensed optical lattice. The advantage of this approach is that it enables naturally the coupling between the breathing and phonon modes, and at the same time, allows for analytical results. In next section, we shall generalize their approach to study how the breathing and phonon modes of Bose condensates are coupled and behave in a 1D optical superlattice.. 3.3.2. An optical superlattice. We shall consider a physical environment very common in crystalline solids [52]. It is often in a solid that the system is better described by a superlattice (i.e., lattice with a basis). This occurs when there are more than one type of atoms (or sites) within a unit cell. Correspondingly in the cold-atom system, a 1D optical lattice with n-point basis can be set up as follows. One can shine two laser beams coherently towards the same direction (say, z-axis), with one having n times of frequency (and thus n times of wave vector) of the other. Along with their reflected beams, the resulting electric field is E(z, t) = A1 cos(nkz)einωt + A2 cos(kz)eiωt + c.c. and the dipole energy, which is proportional to the time average of electric field square, is thus U (z) ∝ E(z, t)2 ∼ A21 cos2 (nkz) + A22 cos2 (kz).. (3.16). The cross term vanishes due to time average. Consequently Bose condensate, which is pre-confined in the x-y plane by the magnetic trap, can be optically trapped and form.

(42) 32. Chapter 3 BEC in optical superlattice (a). c. nω. nω. potential energy (arb. unit). n=2. ω. J1 1. 2. 1. 1 1. 2. J2. 2. 1. 2. d c. n=3. (c). J1 1. ω. (b). 2. 3. 1. 2. 3. J3. J2 1. 3. 2. 1. 3 2. lattice s pacing. Figure 3.2 (a) Illustration of how a 1D optical lattice with an n-point basis is formed by two laser beams and their reflected beams. Frame (b) and (c) show the resulting potential energies (thick solid lines) as a function of lattice spacing of the n = 2 and n = 3 case. c denotes the lattice constant and J’s denote the Josephson tunnelling couplings. In frame (b), d denotes the spacing between any potential maximum and its nearby minimum. a z-direction 1D optical lattice with an n-point basis. Since the ratio of amplitude A1 /A2 can be adjusted and the factor n can be chosen, 1D optical superlattice can be created for a great flexibility. Figure 3.2 illustrates how the 1D optical superlattice with a 2 and 3-point basis is formed and their resulting potential energies (A1 /A2 = √ 5 is taken). Josephson coupling J’s that differ due to condensates tunnelling across different potential barriers are also indicated.. 3.3.3. Phonon modes in optical superlattices. At sufficiently low temperatures, dynamics of Bose condensates is governed by the time-dependent Gross-Pitaveskii (GP) energy functional (. Z ∗. E [Ψ , Ψ] =. 3. dr −. ¸. ¾. h ¯2 ∗ 2 g Ψ ∇ Ψ + [Vtrap (r) + |Ψ|2 − µ |Ψ|2 , 2m 2. (3.17).

(43) 3.3 Oscillations in a 1D optical superlattice. 33. where Ψ = Ψ(r, t) is the time-dependent wave function of the condensate, µ is the chemical potential, and g = 4π¯ h2 a/m with a the s-wave scattering length of the twobody interaction. In the present case of interest, the external trap potential has two contributions, Vtrap (r) = Vm (r) + Vo (r), where Vm (r) = 21 m[ωρ2 ρ2 + ωz2 z 2 ] with ρ2 ≡ x2 + y 2 is due to the magnetic trap, while Vo (r) = Vo (z) = V1 cos2. ³. 2πz c. ´. + V2 cos2. ³. πz c. ´. is due to the optical trap. When ωρ À ωz for the magnetic trap, optical potential then results a 1D optical superlattice (along z axis) with a two-point basis. Here c corresponds to the lattice constant which is related to the wavelength (λ) or the wavevector (k) of the laser beam with frequency ω by c = λ/2 = π/k (see Fig. 3.2). It is convenient to work with dimensionless quantities and rescale the energy, time, and length as E/(¯ hωρ ) → E, tωρ → t, and r/`ρ → r (`ρ ≡. q. h ¯ /mωρ ). Eq. (3.17) thus. becomes Z ∗. E [Ψ , Ψ] =. ½. ·. ¸. ¾. 1 g¯ 1 ¯ |Ψ|2 , (3.18) d r − Ψ∗ ∇2 Ψ + (x2 + y 2 ) + V¯o (z) + |Ψ|2 − µ 2 2 2 3. where the dimensionless coupling V¯o (z) ≡ Vo (z)/(¯hωρ ), g¯ ≡ 4πa/`ρ , and µ ¯ ≡ µ/(¯hωρ ). The frequency ωz is completely dropped in (3.18). For longitudinal phonon modes to be considered under the harmonic approximation, tight-binding (TB) limit is applied. This means that anharmonic effect is negligibly small. This also means that optical potential barriers need to be large, so condensate wave functions are strongly localized in z-direction around the potential minimums. In this limit, Ψ(r, t) can be taken to be the form Ψ(r, t) =. X. {w1 [z − `c − d]Φ1 (x, y, `; t) + w2 [z − (` + 1)c + d]Φ2 (x, y, `; t)} ,(3.19). `. where ` sums over all unit cells and d is the spacing between any potential maximum and its nearby minimum (see Fig. 3.2). Pertaining to site 1 and site 2, w1 and w2 are the strongly localized Wannier functions depending on z only, while Φ1 and Φ2 are the ones associated with x, y coordinates which in turn contain the periodic factor.

(44) 34. Chapter 3 BEC in optical superlattice. ∼ exp(i`kc) for any propagating wave (along the lattice direction) of wave vector k. It is noted that w1 (z) = w2 (−z) for the reflection symmetry. Moreover, for simplicity, fluctuations (time dependence) are considered to be through Φi (x, y, `; t) only. Substitution of (3.19) reduces (3.18) into E=. X X Z ` i=1,2. +. XZ. 2. d. #. ". ½. U¯ 1 1 ¯ |Φi (`)|2 d ρ − Φ∗i (`)∇2 Φi (`) + ρ2 + |Φi (`)|2 − µ 2 2 2. ). 2. ρ [J1 Φ∗1 (`)Φ2 (`. − 1) + c.c.] +. `. XZ. d2 ρ [J2 Φ∗1 (`)Φ2 (`) + c.c.] , (3.20). `. where Φi (`) ≡ Φi (x, y, `; t) and ". Z. J1 J2. #. 1 ∂2 ≡ − dz − d) − + V¯o (z) w2 (z + d) 2 ∂z 2 " # Z 1 ∂2 ∗ ¯ ≡ − dz w1 (z − d) − + Vo (z) w2 (z − c + d) 2 ∂z 2 w1∗ (z. U¯ ≡ g¯. Z. Z. 4. dz |w1 (z)| = g¯. dz |w2 (z)|4 .. (3.21). In (3.20) and (3.21), only nearest-neighbor intersite couplings are considered. J1 and J2 are thus the two different nearest-neighbor Josephson couplings responsible for condensate tunneling. In principle, w1 , w2 can be solved numerically, which in turn solve J1 , J2 , and U¯ . Typical value of J1 or J2 is less than 0.1 in the TB limit. In the following, as mentioned before, we will focus on the breathing and phonon modes. Similar to Martikainen and Stoof [51], a Gaussian ansatz s. N [1 + δ`,i (t)]B0 [1 + ²0`,i (t)] π # " B0 [1 + ²0`,i (t) + i²00`,i (t)](x2 + y 2 ) + iν`,i (t) × exp − 2. Φi (x, y, `; t) =. (3.22). is assumed for the wave functions, where N represents the average number of atoms per site and B0 represents the condensate size in the equilibrium state. In fact, when N is fixed, the value of B0 can be calculated through minimizing the GP energy in √ (3.20). This was done in Ref. [51] where B0 = 1/ 1 + 2U 0 with U 0 ≡ 4N U¯ /π is.

(45) 3.3 Oscillations in a 1D optical superlattice. 35. given. No cross (xy) term is considered in (3.22) because breathing mode is isotropic in the x-y plane. For each site (`, i), dimensionless dynamical variable ²0`,i (t), ²00`,i (t), δ`,i (t), and ν`,i (t) corresponds respectively to fluctuations of the local amplitude, the local phase, the number of atoms, and the global phase of the condensate. One crucial aspect on (3.22) is however that it enables naturally the coupling between the transverse breathing and longitudinal phonon modes [51]. In a onecomponent Bose-condensed system with repulsive interaction, the fluctuation of condensed atom number is coupled to the fluctuation of condensate size. When condensates are distributed in an optical lattice, the fluctuating degrees of freedom are complicated by the intersite coupling of Josephson tunnelling. For the present optical superlattice, the case is further complicated by the possible out-of-phase motion in addition to usual in-phase one. How important are the various couplings and how the condensates are fluctuating in such a system are thus the subject of the following studies. By variational approach, one starts from the Lagrangian of the system, L = T −E. Here T =. R. dr. i¯ h (Ψ∗ ∂Ψ/∂t 2. − Ψ∂Ψ∗ /∂t) and E is given by (3.18) [and hence (3.20)].. Applying (3.22) in T and E and expanding to second order in dynamical variables, one obtains (. " ´ X X ²˙00 ³ T = (1 + δ`,i ) −ν˙ `,i + `,i 1 − ²0`,i N 2 ` i=1,2. #). , (3.23). and E N. =. X X ½. ·. (1 + δ`,i ). ` i=1,2. + EJ ,. ¸¾ ³ ´ ´ B0 1 ³ 2 + (²0`,i )2 + (²00`,i )2 + U 0 B0 1 + ²0`,i δ`,i − µ ¯ 2B0 2. (3.24).

(46) 36. Chapter 3 BEC in optical superlattice. where the Josephson-tunneling term EJ =. J1 X 0 [(²`,1 − ²0`−1,2 )2 + 2(²00`,1 − ²00`−1,2 )2 + 4(ν`,1 − ν`−1,2 )2 + (δ`,1 − δ`−1,2 )2 4 `. − 4(ν`,1 − ν`−1,2 )(²00`,1 − ²00`−1,2 ) − 4(δ`,1 + δ`−1,2 )] +. J2 X 0 [(²`,1 − ²0`,2 )2 + 2(²00`,1 − ²00`,2 )2 + 4(ν`,1 − ν`,2 )2 + (δ`,1 − δ`,2 )2 4 `. − 4(ν`,1 − ν`,2 )(²00`,1 − ²00`,2 ) − 4(δ`,1 + δ`,2 )].. (3.25). It is noted in (3.24) and (3.25) that when chemical potential µ ¯ is chosen to be µ ¯= 3/2B0 − B0 /2 − J1 − J2 (as given by the stability condition of ∂E/∂δ`,i |var.=0 = 0), linear in δì terms vanish. In this case, only second order terms are left with EJ in (3.25) and it justifies the smallness of EJ in accordance with TB. With (3.23)-(3.25), one can then derive the Euler-Lagrange equations of motion for all eight dynamical variables. They can be linearized and written as ²˙0`,1 = 2B0 ²00`,1 + J1 (²00`,1 − ²00`−1,2 ) + J2 (²00`,1 − ²00`,2 ), ²˙0`,2 = −2B0 ²00`,2 + J1 (²00`,2 − ²00`+1,1 ) + J2 (²00`,2 − ²00`,1 ), 2 0 ²`,1 − 2U 0 B0 δ`,1 − J1 (²0`,1 − ²0`−1,2 ) − J2 (²0`,1 − ²0`,2 ), B0 2 = − ²0`,2 − 2U 0 B0 δ`,2 − J1 (²0`,2 − ²0`+1,1 ) − J2 (²0`,2 − ²0`,1 ), B0. ²˙00`,1 = − ²˙00`,2. δ˙`,1 = J1 [2(ν`,1 − ν`−1,2 ) − (²00`,1 − ²00`−1,2 )] + J2 [2(ν`,1 − ν`,2 ) − (²00`,1 − ²00`,2 )], δ˙`,2 = J1 [2(ν`,2 − ν`+1,1 ) − (²00`,2 − ²00`+1,1 )] + J2 [2(ν`,2 − ν`,1 ) − (²00`,2 − ²00`,1 )], ν˙ `,1 = −(1 + 3U 0 )B0 ²0`,1 − 3B0 U 0 δ`,1 − −. J2 [(δ`,1 − δ`,2 ) + (²0`,1 − ²0`,2 )], 2. ν˙ `,2 = −(1 + 3U 0 )B0 ²0`,2 − 3B0 U 0 δ`,2 − −. J2 [(δ`,2 − δ`,1 ) + (²0`,2 − ²0`,1 )]. 2. J1 [(δ`,1 − δ`−1,2 ) + (²0`,1 − ²0`−1,2 )] 2 J1 [(δ`,2 − δ`+1,1 ) + (²0`,2 − ²0`+1,1 )] 2 (3.26). Note that the above eight linear first-order differential equations can be cast into four second-order ones in terms of any two sets of variables ²0`,i (t), ²00`,i (t), δ`,i (t), and.

(47) 3.3 Oscillations in a 1D optical superlattice. 37. 4. 5. (a). (b). (c). OB. 4. 0. 3. 5. OB. ω/ωρ. 3. 0. OP. OB. 2. 5. IB. IB 2. 0. IB. 1. 5. OP 1. 0. OP. 0. 5. AP AP. AP. 0. 0. 0.0. 0.5. kc/π. 0.0. 0.5. 0.0. 0.5. 1.0. kc /π. kc/π. Figure 3.3 Dispersions of four-branch modes resulting from ( 3.26). The parameters J1 = 0.09 and J2 = 0.1 are the same for all three frames, while U 0 = 1, 10, and 100 respectively for frame (a), (b), and (c). Here IB, OB, AP, and OP denote for in-phase breathing, out-of-phase breathing, acoustic phonon, and optical phonon modes. See text for more description. ν`,i (t) (i = 1, 2) only. (That is in the lowest-order limit, the other two sets will share the same excitation spectra.) Considering the coupled equations between ²0`,i (t) and δ`,i (t), for example, and searching for solutions of the type: ²0`,i (t) ≡ ²0i ei(`kc−ωt) and δ`,i (t) ≡ δi ei(`kc−ωt) , a 4 × 4 dynamical matrix (not shown) will be attained, which is then diagonalized to obtain four branches of modes. In Fig. 3.3, dispersions of the four branches of modes are shown for three cases: with same J1 = 0.09 and J2 = 0.1, but different U 0 = 1, 10, and 100. In the study of Bose-Hubbard model on the SF-Mott insulator quantum phase transition in 1D, it has been established that (U/J)c = 2.2N is a critical point for the case N À 1. Below (above) it the system is in the SF ¯. √. 0. √U [see Eq. (3.14)] (insulating) phase [36, 79]. In our case, U = U 2πB0 = 8N 4 1+πU 0 /4N √ with the scale `ρ = 1/ B0 . Besides, we are interested in collective excitation of condensates with average atom number per site being N ∼ 10−100. Thus the system.

(48) 38. Chapter 3 BEC in optical superlattice. is justified to be in the SF phase when U 0 /J < √ 4. U0 /J 1+πU 0 /4N. < (17.2N 2 ) ≈ 2 × 103 .. Consequently the cases U 0 = 1, 10, and 100 studied in Fig. 3.3 are considered to be in the SF phase. The four branches of modes are labelled by in-phase breathing (IB), out-of-phase breathing (OB), acoustic phonon (AP), and optical phonon (OP) modes respectively (see Fig. 3.3). Strictly speaking, these descriptions are exact only when phonon modes are completely decoupled from breathing modes and at the same time, the propagating wave comprised of ²0`,1 (t), for instance, has a phase angle π shift (out-ofphase) by the the propagating wave comprised of ²0`,2 (t). The coupling effect, which is proportional to the magnitude of tunnelling strength J, is indeed small in the TB limit. One should be noted that the coupling effect is also accompanied by the factor of B0 U 0 (recall that B0 corresponds to the size of condensate), to which it becomes more and more important in the large U 0 limit. At the long-wavelength limit (kc ¿ 1), dispersions of the four branches of modes can be analytically solved and uniformly written as 2 ωi2 = ω0i + u2i k 2 ,. (3.27). where, at the TB limit (keeping to J linear order), one obtains. 2 ω0i =.  ! Ã  B02 U 0 2  0   i = OB 4 + 8B0 (J1 + J2 ) 1 + U +   2        4 i = IB ! Ã 0 2    8B U 0 (J + J ) 1 − B0 U  i = OP 0 1 2   2       . 0. (3.28). i = AP,. and u2IB = −u2OB = 2B0 (1 + U 0 )c2 u2AP = −u2OP = 2B0 U 0 c2. J1 J2 J1 + J2. J1 J2 . J1 + J2. (3.29).

(49) 3.4 Summary. 39. As shown in (3.27)–(3.29), the slopes (ui ) and the k = 0 intercept (ω0i ) of the four branches of modes depend crucially on the values of J1 , J2 , and U 0 . Therefore observation of these modes will be a direct way to access them. Some features worth noting are as follows. For IB mode, the k = 0 frequency (ω = 2ωρ ) is robust regardless of the values of J1 and J2 . It is a clear result to the uniform (single-condensate) limit. For AP mode, on the other hand, the phonon velocity at small k behaves like √ u ∼ B0 U 0 Jc, an expected result for the current system of a bulk modulus ∼ B0 U 0 J. In the large U 0 case [see Fig. 3.3(c)], IB and OP modes start to hybridize, to which a small gap (not visible in the scale potted) develops where the two branches cross. This feature may be a signature when the system undergoes a quantum phase transition from SF into insulating phase.. 3.4. Summary. This chapter studies the BEC condensates in optical lattices. In Sec. 3.1, I consider the relevant energy scales in an optical lattice system. In Sec. 3.2, I discuss the tight-binding model in an optical lattice. Detailed derivations are given for a 1D system. In Sec. 3.3, oscillations of Bose condensates in a 1D optical lattice with a two-point basis is investigated. Focuses are made to the transverse breathing modes and longitudinal phonon modes. Pertaining to condensates in site 1 and site 2, inphase and out-of-phase modes are obtained. The mode dispersion relations depend crucially on the sizes of the two intersite Josephson tunnelling strengths J1 and J2 and the on-site repulsion U . For the phonons, there are optical as well as acoustic modes, in a close resemblance to those in a 1D crystalline solid. While direct observation of these in-phase and out-of-phase oscillations of condensates may be resolution limited by current instruments, dispersion relations of these modes should be probable by.

(50) 40 inelastic scattering measurements.. Chapter 3 BEC in optical superlattice.

(51) Chapter 4 BEC with dipolar interaction 4.1. Introduction. In recent years dipolar gases become a fast growing field of theoretical and experimental interests in the studies of ultracold atoms and molecules. Among several dipolar systems, chromium (52 Cr) atoms have been successfully realized and studied to great extents [11, 80–84]. The dipolar interaction effects for as. 39. 87. Rb atoms as well. K atoms are also observed in different groups [85, 86]. Several polar molecule. systems such as CO [87], ND3 [88], RbCs [89], LiCs [90], and CsCl have also been trapped, cooled, and studied [91,92]. More recently, a high phase-space density gas of polar 40 K87 Rb molecules have also been produced [93]. These stimulate great interest in the studies of dipolar systems at low temperatures. Within the first Born approximation (FBA), Yi and You [94] proposed a pseudopotential to study the long-range dipolar interaction. Based on this approximation, which is justified only in the weak dipole moment limit, various theoretical studies of the excitations, collapses, instability, etc. of the dipolar BEC system have been carried out in these years. However these results do not become justified when applied to systems of polar molecules, 41.