Dynamical properties of few bosons trapped in an harmonic potential
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MASTER THESIS Dynamical properties of few bosons trapped in an harmonic potential Author: David LEDESMA MORAGA Supervisors: Dr. Artur POLLS MARTÍ Dr. Bruno JULIÁ-DÍAZ Dep. Física Quàntica i Astrofísica Facultat de Física Universitat de Barcelona September 8, 2018
iii Acknowledgements I would like to sincerely thank my advisors, Dr. Artur Polls and Dr. Bruno Juliá- Díaz, specially for their constant help and support during the project, and particularly their patience. I would like also to thank them for providing me all their knowledge and giving me the possibility of assisting to very special meetings. I also would like to thank my family and friends for their companion and enlivenment during this project and my life.
v Contents Acknowledgements iii 1 Introduction 1 2 Homogeneous system 3 2.1 Non-interacting limit and mean-field . . . . . . . . . . . . . . . . . . . 3 2.2 Tonks-Girardeau limit and Fermi-Bose mapping . . . . . . . . . . . . 4 3 1D system trapped in an H.O. potential 9 3.1 Hamiltonian of the system . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 VirialTheorem................................ 9 3.3 Non-interacting and Tonks-Girardeau limits . . . . . . . . . . . . . . 11 3.4 Gross Pitaevskii and Thomas Fermi approaches . . . . . . . . . . . . 12 4 Two particles trapped in a 1D H.O. potential 17 4.1 Separation of the center of mass . . . . . . . . . . . . . . . . . . . . . . 17 4.2 First order perturbation theory . . . . . . . . . . . . . . . . . . . . . . 18 4.3 Solution of the relative Hamiltonian: exact diagonalization . . . . . . 18 4.4 Convergence of the method . . . . . . . . . . . . . . . . . . . . . . . . 19 4.5 Energyspectrum .............................. 20 4.6 Contributions to the total energy . . . . . . . . . . . . . . . . . . . . . 22 5 Breathing mode 25 5.1 Dynamic Structure Function (DSF): Excitation operator . . . . . . . . 25 5.2 CalculationsforN=2 ............................ 26 5.3 Results for two particles . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.4 ExcitationoftheC.M............................. 30 6 Sum Rules 33 6.1 Theenergysumrules............................ 33 6.2 Derivation of the sum rules . . . . . . . . . . . . . . . . . . . . . . . . 34 6.2.1 M−1sumrule ............................ 34 6.2.2 M1sumrule............................. 35 6.2.3 M3sumrule............................. 37 6.3 Non-interacting and Tonks-Girardeau limits . . . . . . . . . . . . . . 38 7 Sum Rules applied to a 2 bosons system 41 7.1 Behaviour of the sum rules and contribution of the C.M. peak . . . . 41 7.2 Excitation energy and reentrant behaviour . . . . . . . . . . . . . . . 43
vi 8 Summary and Conclusions 45
1 Chapter 1 Introduction Quantum few-body problems appear in many branches of physics, ranging from nuclear and high energy physics, to ultra-cold atomic gases and condensed matter physics. Notorious examples are for instance the study of small nuclei [1], like the triton, the study of hadrons as compounds of constituents quarks [2], the trapped few-atom problem which can be prepared in ultra-cold atomic labs [3]-[6] or fewelectron quantum dots in condensed matter [7]. In all cases, there are some common features and techniques which can be applied to many of the systems despite the very different energy and size scales involved in the problem. All these few-body systems can exhibit different kind of phases, depending on the interaction among the constituents, e.g. long or short ranged, spin dependent, etc., which give rise to different quantum mechanical properties. Among the latter, the onset of quantum mechanical correlations among the constituents, requiring beyond mean-field approaches in the description, are particularly interesting nowadays. In this work we will concentrate on systems of ultra-cold atoms trapped in external potentials. Whenever possible, e.g. non-interacting, we will explore manybody properties of the system to then concentrate on a more detailed study of the few-body case. Recent experiments have shown a great improvement in control of the experimental parameters of the Bose-Einstein condensates (BEC), such as their geometry, the number of particles condensed and the interactions induced in the system. Nowadays, it is possible to generate a BEC confined to one dimension just by increasing substantially the transversal frequencies of the optical lattices which sustain the condensate [8]. It has also been reported the possibility of precisely controlling the interaction among particles by means of a magnetically induced Feshbach resonances [9]. Thus open us a new way to understand how many-body quantum mechanical correlations build in few-body systems [3]-[6] & [10]-[13], as in these systems, the clash between a strong interaction and the confinement to a low-dimensional geometry ends in a magnification of the quantum effects, inducing important quantum correlations in the system. These new experimental procedures provide us a theoretical many-body laboratory that permits to either corroborate or discard some mean-field theories developed in the last century, which are thought to be good approaches for weakinteracting systems. Another interesting example to study this interplay between confinement and interaction, is the Tonks-Girardeau gas, which basically consists
2Chapter 1. Introduction on having atoms with a very strong repulsive interaction confined to a one dimensional geometry. Taking advantage of these new experimental methods, it is possible to study the evolution of the static and dynamical properties of this kind of systems from the non-interacting to the strongly interacting regimes. The minimal example of the Tonks-Girardeau system is that of two particles trapped in a harmonic oscillator potential [14]. Such system offers an appropriate playground to study this interplay between quantum many-body correlations and statistics. For the case of a contact interaction potential, two noteworthy limits are well known. This kind of contact interaction potentials are realistic in the case of ultra-cold atomic systems [15], and even have also been used in effective interaction theories in nuclear physics [16]. In the particular case of two particles, in absence of interactions the two atoms populate the lowest energy single particle state, producing the minimal version of a Bose-Einstein condensed state, whereas in the infinite interaction limit the two bosons resemble in many ways two spin-less non-interacting fermions, producing the so called Tonks-Girardeau (TG) limit [17]. This thesis is organized in the following way. First, in chapter 2 we introduce the homogeneous system, which is most simple system of this kind only containing aδ-contact interaction. Next, in chapter 3 we describe the Nparticles one dimensional Tonks-Girardeau system with a confinement given by an harmonic potential, whereas in chapter 4 we do analyze more extensively the same system but only for 2 particles. In the following chapter, the breathing mode (excitation) of this system is explained; and in chapter 6 the general energy sum rules of the dynamic structure function associated to a given excitation operator are derived. After that, in chapter 7 , these sum rules are applied to the two particle system. Finally, in chapter 8 we summarize the whole project and draw the conclusions.
3 Chapter 2 Homogeneous system We start by considering the homogeneous system, characterized by a linear density n=N/L where Nis the number of bosons enclosed in a linear box of length L. To simulate the homogeneous system, we implement periodic boundary conditions and take the thermodynamic limit with N→ ∞ and L→ ∞ keeping the ratio n constant. We concentrate our efforts on the Lieb-Liniger model ([18]), which takes a δ-contact interaction to describe the inter-atomic interactions which has been used frequently in the literature. The Hamiltonian for this system has two terms : the kinetic energy and the interaction energy. H=−~2 2m N X i d2 dx2 i +g N X i<j δ(xi−xj),(2.1) where gis a parameter which characterizes the strength of the δ-contact interaction. In this work we consider only the repulsive case with g≥0. 2.1 Non-interacting limit and mean-field In the non-interacting limit, the system would simply consists on having Nfree bosons of mass menclosed in a linear box of length Lwith periodic boundary conditions. The system could be described as a product state of Nfree-particles properly symmetrized. The use of periodic boundary conditions allows us to consider the plane wave single-particle basis. ψ0(x1, x2, ..., xN) = ˆ S( N Y i=1 φk(xi)),(2.2) where the operator ˆ Sis the responsible for the symmetrization of the wave function. The single particle wave functions are defined as : φk(xi) = 1 √Leipkxi, pk=2πnk L,(2.3)
10 Chapter 3. 1D system trapped in an H.O. potential H=T+VHO +Vint with T= N X i=1 −1 2 d2 dx2 i ,VHO = N X i=1 x2 i 2, Vint = N X i<j gδ(xi−xj).(3.3) When the system is composed by bosons, one can compute analytically the energy in two limiting cases: 1) the non-interacting case and, 2) the extremely repulsive case. But in this section we want to concentrate on the virial theorem which allows to establish a relation between the different energy contributions. The virial theorem is based on the scaling transformation of the many-body wave-function and how the expectation values of the different pieces of the Hamiltonian transform under these scaling transformations. To this end, we first define a new wave function through the scaling transformation, Ψλ(x1, x2, ..., xn) = λN/2Ψ(λx1, λx2, ..., λxN).(3.4) The new wave-function ψλhas exactly the same normalization than ψ, since hψλ|ψλi=Zdx1dx2...dxNψλ(x1, x2, ..., xn)∗ψλ(x1, x2, ..., xn) =λNZdx1dx2...dxNψ(λx1, λx2, ..., λxn)∗ψ(λx1, λx2, ..., λxn) =λNZ1 λNdy1dy2...dyNψ(y1, y2, ..., yn)∗ψ(y1, y2, ..., yn) = hψ|ψi, (3.5) where we have implemented the change of variable yi=λxi. The harmonic oscillator energy is transformed as VHO(λ) = hψλ| N X i=1 x2 i 2|ψλi =N 2λNZdx1dx2...dxNψ(λx1, λx2, ..., λxn)∗x2 1ψ(λx1, λx2, ..., λxn) =N 2λNZ1 λNdy1...dyNψ(y1, ..., yN)∗y2 1 λ2ψ(y1, ..., yN) =1 λ2hψ|VHO |ψi=1 λ2VHO(λ= 1). (3.6) The transformation of the interaction energy reads: Vint(λ) = hψλ|X i<j gδ(xi−xj)|ψi =N(N−1) 2gλNZdx1...dxNψ(λx1, λx2, ..., λxN)δ(x1−x2)ψ(λx1, λx2, ..., λxN) =N(N−1) 2gλNZ1 λNdy1...dyNψ(y1, y2, ..., yN)λδ(y1−y2)ψ(y1, y2, ..., yN) =λhψ|Vint |ψi=λVint(λ= 1), (3.7)
3.3. Non-interacting and Tonks-Girardeau limits 11 and for the kinetic energy we have T(λ) = hψλ| N X i=1 −1 2 d2 dx2 i|ψλi=−N 2λNZdx1...dxNψ(λx1, ..., λxN)∗d dx2 1 ψ(λx1, ..., λxN) =−N 2λNZ1 λNdy1...dyNψ(y1, ..., yN)∗λ2d2 dy2 1 ψ(y1, ..., yN) =λ2T(λ= 1). (3.8) After all these transformation, the final effect of the scaling can be written as Eλ=hψλ|H|ψλi=λ2T(λ= 1) + 1 λ2VHO(λ= 1) + λVint(λ= 1).(3.9) Now we know that at λ= 1, the energy function Eλhas a stationary point, i.e. dEλ dλ = 0λ=1 ,(3.10) since ψ(x1, ..., xN)) is taken to be an eigenfunction, which in particular could be the ground state. Therefore, dEλ dλ = 2λT(λ= 1) −2 λ3VHO(λ= 1) + Vint(λ= 1),(3.11) and imposing the stationary condition, we reach dEλ dλ λ=1 = 0 = 2T(λ= 1) −2VHO(λ= 1) + Vint(λ= 1),(3.12) which is valid for any eigenstate of the system. Also notice that the virial relation is valid for both type of particles; for bosons and for fermions. 3.3 Non-interacting and Tonks-Girardeau limits It is easy to figure out how the non-interacting limit behaves: as we have no interaction, we just deal with a system composed by particles confined by an harmonic oscillator. Therefore, the Hamiltonian of the system reduces to H=−1 2X i d2 dx2 i +1 2X i x2 i,(3.13) and the total energy of the system will be the sum of the single-particle states of one dimensional harmonic oscillator. Knowing that the eigenvalues of a one dimensional H.O. are EHO n=n+1/2(in H.O. units), the ground state energy of the system is E0= N X i EHO 0=N 2,(3.14)
12 Chapter 3. 1D system trapped in an H.O. potential having all the particles in the single-particle state n= 0 of the H.O. Therefore, the wave-function for the ground state of the system is given by ψ0(x1, x2, ..., xN) = N Y i=1 φ0(xi),(3.15) where φ0(xi) = 1 π1/4e−x2 i/2is the ground state of the single-particle harmonic oscillator. In the Tonks-Girardeau limit, g→ ∞ we rely again on the Fermi-Bose mapping to calculate the energy of the ground-state of the system. . As has been explained in previous sections, in this case, the ground state is the absolute value of the Slater determinant built as if the particles were fermions. In the case of Nparticles they occupy the first Nlowest single-particle energy levels of the harmonic oscillator potential, therefore E0=1 2+3 2+... +2N−1 2=N2 2.(3.16) Notice that the absolute value of the Slater determinant guarantees the symmetric character of the wave function under the exchange of particles, which is a necessary condition to describe identical bosons. Is also worth to mention the different behavior of the total energy on the number of particles in these two limits: in the noninteracting case is just linear in the number of particles while becomes quadratic a function in the strongly interacting case. 3.4 Gross Pitaevskii and Thomas Fermi approaches In the mean field approach we assume a trial wave function for our system such as ψ(x1, x2, ..., xN) = N Y i=1 φ(xi),(3.17) with all the particles in the same single-particle state. The expectation value of the Hamiltonian in this wave function is given by hψ|H|ψi=Nhφ|−1 2 d2 dx2+1 2x2|φi+gN(N−1) 2hφ(x1)φ(x2)|δ(x1−x2)|φ(x1)φ(x2)i, (3.18) where the two-body matrix element hφ(x1)φ(x2)|gδ(x1−x2)|φ(x1)φ(x2)i=gRdx |φ(x)|4. To find the optimal (less energy) wave function of that type, we perform a functional variation of the previous energy functional, which is complemented with the introduction of of a Lagrange multiplier µto impose that the variations in the wave function respect the normalization condition. The Lagrange multiplier has the meaning of the chemical potential of the system δ δφ∗[hψ|H|ψi−µhψ|ψi] = 0,(3.19)
3.4. Gross Pitaevskii and Thomas Fermi approaches 13 from which we get a non-linear differential Hartree-Bose equation for φ, usually known as the Gross-Pitaevskii equation −1 2 d2 dx2φ(x) + 1 2x2φ(x)+(N−1)g|φ(x)|2φ(x) = µφ(x),(3.20) where φ(x)is normalized to unity. It is important to remark that the virial relation is also valid in the mean-field approximation 2t−2vHO +vint = 0,(3.21) where t=hφ|− 1 2 d2 dx2|φi, vHO =hφ|1 2x2|φi,vint =gN−1 2hφ(x1)φ(x2)|δ(x1−x2)|φ(x1)φ(x2)i (3.22) being φthe solution to the Gross-Pitaevskii equation. A useful approximation to the Gross-Pitaevskii equation which is known as the Thomas-Fermi approach, which is valid when the kinetic energy contribution can be neglected in front of both potentials energies; the interaction and the harmonic oscillator ones. Then the Gross-Pitaevskii equation reduces to 1 2x2+ (N−1)g|φ(x)|2=µ. (3.23) From this equation we are able to isolate |φ(x)|2, and impose that the density should be always positive. From that assumption we obtain φ(x) = 1 √g(N−1) qµ−1 2x2,− √2µ < x <√2µ 0,|x|>√2µ . (3.24) To determine the value of µ, we impose the normalization of the wave function φ(x), so that 1 g(N−1) Z√2µ − √2µµ−1 2x2dx = 1.(3.25) Performing the integral we obtain one equation in terms of the chemical potential of the system Z√2µ − √2µµ−1 2x2dx = 2 Z√2µ 0µ−1 2x2dx =25/2µ3/2 3.(3.26) Then, imposing the normalization condition explained above, we obtain the chemical potential
14 Chapter 3. 1D system trapped in an H.O. potential 1 g(N−1) 25/2µ3/2 3= 1 ⇒µ=(3g(N−1))2 251/3 .(3.27) Once the chemical potential is known the wave function is fully determined and the total energy can be calculated assuming always that the kinetic energy is zero. We can start by the interaction energy per particle vint =gN−1 2hφ(x1)φ(x2)|δ(x1−x2)|φ(x1)φ(x2)i=gN−1 2Z√2µ − √2µ|φ(x)|4dx =(N−1) g2(N−1)2gZ√2µ 0µ2−µx2+1 4x4dx =1 g(N−1) µ2x−µx3 3+1 5x5√2µ 0 =1 g(N−1) 21/2−23/2 3+25/2 5µ5/2=1 53g(N−1) 22/3 . (3.28) and also compute the harmonic oscillator potential energy per particle vHO =hφT F |1 2x2|φTF i=1 2Z√2µ − √2µ φ(x)2x2dx =1 g(N−1) Z√2µ 0µ−1 2x2x2dx =1 g(N−1) µx3 3−1 2 x5 5√2µ 0 =1 g(N−1) 25/2 15 µ5/2 =1 10 3g(N−1) 22/3 (3.29) By adding both vHO and vint, the total energy per particle reads e=1 10 3g(N−1) 22/3 +1 53g(N−1) 22/3 =3 10 3g(N−1) 22/3 .(3.30) This expression for the energy coincides with the one obtaining by integrating respect to N, the chemical potential as a function of the number of particles: µ∼(3gN)2/3 25/3,(3.31) and E=ZN 0 µ(N)dN =ZN 0 (3g)2/3 25/3N2/3dN =N3 10 3gN 22/3 ,(3.32) which coincide with the previous expression when N≈(N−1) (i.e. for N1). Notice also that the virial theorem is still fulfilled in the TF approach. Since t≡0, 2t−2vHO +vint = 0 ⇒vint(N) = 2vHO(N),(3.33)
3.4. Gross Pitaevskii and Thomas Fermi approaches 15 expression which we can clearly see that is fulfilled since vTF int =1 53g(N−1) 22/3 = 2vTF HO.(3.34) In general, mean-field theories derived above describe properly the behaviour of our system when the interaction is weak and no correlations are present in the system. We have been able to describe the two extreme limits, interacting and strongly interacting, for any number of particles. It is very interesting, to see how the transition between the two regimes takes place, and the emergence of the quantum correlations that made useless the mean-field approach. Unfortunately, to solve the problem for intermediate gand any number of particles is numerically difficult. However, in the next chapter we will present the first steps towards a general solution by considering the simple case with N= 2.
17 Chapter 4 Two particles trapped in a 1D H.O. potential Recent experiments with few ultra-cold atoms are able to probe the statistical properties of bosons and fermions in the smallest possible settings. Impressively, there are nowadays experiments with as few as just two fermions [3]-[6] or two bosons [13]. In the first case, the authors where able to produce the smallest version of the Fermi sea, populating it atom by atom. In the second, they managed to observe effects stemming from bosonic statistics with just to bosons. Thus, in this chapter we will consider the minimal setting of just two bosons trapped in a harmonic oscillator potential. We will be able to study the transition from the non-interacting to the strongly interacting systems, thus going from the two-particle analog of a condensed system to the corresponding to the TG limit. 4.1 Separation of the center of mass The Hamiltonian can be splitted into two parts: the first one corresponding to the center of mass of the system (C.M.), and the second one describing the relative motion. To this end, we define µ≡m/2 M≡2m xr≡x1−x2 X≡x1+x2 2, (4.1) where µis the reduced mass, Mis the total mass of the system, Xis the center of mass coordinate, and xris the relative coordinate. With these new variables, the Hamiltonian is splitted in two pieces H=HCM +Hrwith HCM =−~2 2M d2 dX2+1 2Mω2X2(4.2) Hr=−~2 2µ d2 dx2 r +1 2µω2x2 r+gδ (xr).(4.3)
18 Chapter 4. Two particles trapped in a 1D H.O. potential The frequencies associated to both Hamiltonians are the same. However, the oscillator lengths are different. Now, it is convenient to express the Hamiltonians in the oscillator units of the original one: HCM =−1 4 d2 dX2+X2(4.4) Hr=−d2 dx2 r +1 4x2 r+gδ (xr).(4.5) HCM is a harmonic oscillator Hamiltonian which is easy to solve. The problem reduces to the solution of Hr. 4.2 First order perturbation theory In order to get a first idea of the dependence of the energy e system with the strength of the interaction, we can perform a first order perturbation calculation, which should provide a reasonable result for small interacting strengths. We assume that the relative Hamiltonian is given by Hr=−d2 dx2 r +1 4x2 r | {z } unperturbed +gδ (xr) |{z} perturbation ≡H0+gVpert.(4.6) Then, using the analytical solution of the relative H.O., we obtain a spectrum at first order perturbation which is slightly shifted Er=E(0) r,n +ghψ(0) r,n|δ(xr)|ψ(0) r,ni=n+1 2+gψ(0) r,n(0)2,(4.7) Notice that only the states with non zero value of the wave function are affected by the interaction. For the specific case of the ground state we have: Eint =ghψ(0) r,0|δ(xr)|ψ(0) r,0i=g1 2π1/2 ,(4.8) and consequently, it provides a perturbed energy for the ground state of relative motion E0 r= 1/2 + g1 2π1/2. 4.3 Solution of the relative Hamiltonian: exact diagonalization As HCM and Hrcommute, the wave functions associated to the C.M. and relative motion factorize, so we can solve each one separated Ψ(X, xr) = φ(X)ψ(xr).(4.9)
4.4. Convergence of the method 19 The HCM is an harmonic oscillator Hamiltonian with the spectrum ECM k= (k+ 1/2) where kare the number of quantums associated to the center of mass motion. The wave-functions associated to the C.M. are given by Ψk=a π1/41 √2kk!Hk( √aX)e−aX2/2,(4.10) where Hkare the Hermite’s polynomials and a≡Mω/~. Notice that acm = 2, whereas ar=1 2in the H.O. units. To diagonalize the relative part we will use the following procedure. We choose a finite number of non-interacting energy eigenstates of the harmonic oscillator, {ψ0(xr), ψ2(xr), ..., ψ2M(xr)}corresponding to H(0) r(here M+1 will be the total number of modes we want to use). Notice that in order to fulfill the requirements of the bosonic symmetry, we consider only the modes associated to even functions. The matrix elements of Hrare hΨm|Hr|Ψni=n+1 2δm,n +gZdxrΨn(xr)Ψm(xr)δ(xr) =n+1 2δm,n +gΨn(0)Ψm(0), (4.11) where Ψi(0) are the H.O. wave functions at the origin. Diagonalizing the truncated Hamiltonian we get approximate solutions, whose eigenvalues are upper bounds of the corresponding exact solutions. Hr¯ Ψl=¯ E(r) l¯ Ψl.(4.12) 4.4 Convergence of the method In practice we use a huge number of modes, M up to 1500, to guarantee the convergence of the calculations. In fact, in Fig. 4.1 we report the dependence of the ground state energy on the number of modes used to diagonalize Hrfor different values of the strength interaction g. Obviously, the energy decreases as the number of modes increases reaching in all cases a limit saturating value with the number of modes which defines an asymptotic value. The horizontal line at Egs = 3/2corresponds to the ground-state energy of Hrin the limit g→ ∞. For any finite value of g one can always find a number of modes (mcrit) such that for a larger number of modes the energy will be always below 3/2. The mcrit becomes larger as gincreases. Therefore, one can conclude that when g→ ∞ the ground-state energy of Hr, as a function of the number of modes, approaches 3/2from below. For the case of two bosons trapped in a harmonic oscillator potential, we could have used the exact solutions of Ref. [14]. Instead we have decided to present a procedure which, although more tedious for this precise case, is more general and can readily be applied to other
26 Chapter 5. Breathing mode 5.2 Calculations for N=2 The monopole excitation operator b F=PN i=1 x2 ican be separated in two pieces, one that corresponds to the center of mass and the other to the intrinsic motion. In the particular case of N= 2 we have, b F=x2 1+x2 2= 2X2+x2 r 2,(5.3) so that we compute the matrix elements ha|b F|0itaking advantage of the factorization of the state, |ai=|λCM i|νri=|Φλ CM i|Φν ri, where λ&νare the quantum numbers associated to the C.M. and the relative part of the state |aiof our system ha|b F|0i=hΦλ CM |hΦν r|2X2+x2 r 2|Φ0 CM i|Φ0 ri =hΦλ CM |2X2|Φ0 CM ihΦν r|Φ0 ri+hΦν r|x2 r 2|Φ0 rihΦλ CM |Φ0 CM i. (5.4) The product hΦλ CM |2X2|Φ0 CM ican be obtained using the orthogonality properties of the Hermite’s polynomials in the following way: First of all we write the product explicitly hΦλ CM |2X2|Φ0 CM i=Z∞ −∞ Ψλ∗(X)2X2Ψ0(X)dX =2 π1/21 √2λλ!Z∞ −∞ Hλ( √2X)e−X2/22X2H0( √2X)e−X2/2dX =2 π1/21 √2λλ!Z∞ −∞ Hλ( √2X)e−X22X2dX. (5.5) After that, we can perform a variable change y≡√2Xobtaining hΦλ CM |2X2|Φ0 CM i= 2 2 π1/21 √2λλ!2−3/2Z∞ −∞ Hλ(y)e−y2y2dy (5.6) and taking advantage of the fact that y2can be expressed in therms of the Hermite’s Polynomials, we can use H2(y)=4y2−2 = 4y2−2H0(y)⇒y2=H2(y) 4+H0(y) 2,(5.7) to rewrite the eq. (5.6) hΦλ CM |2X2|Φ0 CM i= 2 2 π1/21 √2λλ!2−3/2Z∞ −∞ Hλ(y)e−y2H2(y) 4+H0(y) 2dy. (5.8) We have used that the integral value will be the same for y1and y2, as we integrate over all the space, in order to do the summation. Then, using the orthogonality properties, we do know
5.2. Calculations for N=2 27 Z∞ −∞ e−x2Hα(x)Hβ(x)dx =√π2αα!δα,β.(5.9) Finally, we observe that the only possible values of λfor the C.M. contribution to the breathing mode are 0 and 2 hΦλ CM |2X2|Φ0 CM i= 2 2 π1/21 √2λλ!2−3/2√π222! 4δ2,λ +√π 2δ0,λ =4δ2,λ +δ0,λ 2·√2λλ!. (5.10) As we have diagonalized the relative Hamiltonian the eigenstates obtained are orthogonal, and orthonormal when normalized, and consequently we can obtain the matrix elements hΦν r|Φ0 rias hΦν r|Φ0 ri=δν,0.(5.11) With all this, we observe that the contribution of the C.M. operator to the breathing mode is very limited, since only accepts two possible states, corresponding to λ= 0,2as explained above. On the other hand, the Hrwill contribute to the breathing mode very differently. As we are using many H.O. states to diagonalize and obtain our eigenstates, every excited state will connect many H.O. states by pairs, as we are going to demonstrate. First of all, we obtain hΦλ CM |Φ0 CM i=δλ,0simply because of the normalization of the wave-function in the H.O. that describes the C.M. motion hΦν r|x2 r 2|Φ0 rihΦλ CM |Φ0 CM i=hΦν r|x2 r 2|Φ0 riδλ,0.(5.12) Then, as the states |Φriare linear combinations of all the eigenstates of a H.O. system, the matrix elements hΦν r|x2 r 2|Φ0 rimust be expressed in terms of H.O. matrix elements in the following way |Φν ri=X α Cν α|ΦH.O. αi hΦν r|x2 r 2|Φ0 ri=X α,β Cν αC0 β 2hΦH.O. α|x2 r|ΦH.O. βi, (5.13) where the matrix elements hΦH.O. α|x2|ΦH.O. βican be expressed as a solvable integral which solution is hΦH.O. α|x2 r|ΦH.O. βi= √β(β−1) 2arα=β−2 2β+1 2arα=β √(β+1)(β+2) 2arα=β+ 2 0Otherwise. (5.14)
28 Chapter 5. Breathing mode Using these integral solutions, we can rewrite the breathing mode operator’s mean value as ha|b F|0i=hΦλ CM |2X2|Φ0 CM ihΦν r|Φ0 ri+hΦν r|x2 r 2|Φ0 rihΦλ CM |Φ0 CM i =4δλ,2+δλ,0 2·√2λλ!δν,0+X α,β Cν αC0 β 2hΦH.O. α|x2 r|ΦH.O. βiδλ,0. (5.15) Notice that actually, we have clearly two contributions to the DSF; the first one corresponds to the C.M., and can only connect the ground state with the first excitation of the C.M. However, in the second contribution, which belongs to the relative Hamiltonian excitations, we are able to connect many H.O. states since the relative motion states are not pure H.O. states. Therefore, if we want to depict the exact value, we just need to use this formula in the eigenstates diagonalized above. 5.3 Results for two particles For g= 0 and g→ ∞ we can also compute SF(E)analytically. In the case of g= 0 both contributions of the excitation operator, i.e. C.M. and intrinsic, excite only two different states, both at the same energy E= 2, being both strengths the center of mass excitation and the one of the relative motion 1/4. The C.M. peak will remain at the same energy and with the same strength independently of the interaction strengths. In the g→ ∞case the two excitations are again peaked at E= 2 but they have different strengths, 1/4for the C.M. and 3/4for the internal excitation. This implies that even though both excitations have exactly the same energy it should be more likely to excite the breathing mode in the TG limit than in the BEC one. In Fig. 5.1 we depict the strengths of the dynamic structure function for N= 2 for four different values of the interaction g. Notice that the strength contains the factor 1/N included in the definition of SF(E). For g= 0, we have only one peak, corresponding to the sum of the C.M. and relative motion. As mentioned before, the C.M. peak will remain constant within for all the interaction strengths. For nonzero values of the interaction, the strength is distributed over several excited states, but in all cases the center of mass peak remains visible and located at E= 2. The strength of the higher intrinsic peaks is always significantly smaller, two orders of magnitude, than those of the center of mass and the breathing mode. The strength of the breathing mode and the next three excited states of the relative motion is shown in Fig. 5.2 as a function of g. In all cases, at g= 0, the strength of these secondary peaks is zero, and increases with gup to a maximum located around g≈2, and decreases to zero for g→ ∞. The maximum decreases when the excitation energy increases. Apparently, this dependence is very much correlated to the dependence of the interaction energy with g(Fig. 4.3). In Fig. 5.2 we report the dependence of the strength of the breathing mode as a function of g, which turns out to be an increasing function of g. This behavior will be relevant to understand the evolution of the energy weighted sum rules of SF(E).
5.3. Results for two particles 29 SF(E) ExcitationEnergy(H.O.units) g=1.75 CM 1 2 4 6 10 g=30 CM 10-6 10-5 10-4 10-3 10-2 10-1 100 1 2 4 6 10 g=10 CM 10-6 10-5 10-4 10-3 10-2 10-1 100 g=0 FIGURE 5.1: Dynamic structure function computed for a system composed by 2 bosons, with a delta contact potential. The value is computed for different interaction’s g values [0,30], and with a number of modes m=200 (101 modes). The breathing mode energy decreases ( it lies below the C.M. excitation) for increasing gup to g.2, above that, it increases again towards E= 2 when g→ ∞. This is the so called reentrant behavior of the mono-polar excitation reported in Ref. [23]. In an experiment, separating the contribution from the center of mass from that of the breathing mode seems involved. The response of the system will thus easily mix both contributions. The reentrant behavior affects not only to the ground state but to all excited states of the relative motion. The reentrant energy, defined as Er=Ex(g)−Ex(g= 0) is zero, by definition, at g= 0, it decreases reaching a negative minimum and then grows to approach asymptotically zero when g→ ∞. The reentrant behaviour is more pronounced for the higher excited states. The maximum reentrance slightly shifts to higher values of gfor the higher excited states. Again, there seems to exist a correlation between the interaction energy and the reentrant energy as a function of g. Computing the structure function explicitly becomes difficult for more than 2 particles. In particular a naive second quantization scheme using the one particle modes as single particle states runs into difficulties as it mixes the center of mass with the relative motion in a nontrivial way [24].
30 Chapter 5. Breathing mode 0.2 0.3 0.4 0.5 0.6 0.7 0.8 SF(E) g 1ststateDSF 0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0 5 10 15 20 25 30 2ndstateDSF 3rdstateDSF 4thstateDSF FIGURE 5.2: Strength of excitations of Hras a function of g. The breathing mode can be observed in red, whereas the other three excited states are shown in green, black and blue, respectively. The breathing mode is at least two orders of magnitude more intense than the other excitations in the whole range of interaction strength [0,30]. FIGURE 5.3: Reentrant energy, Ex(g)−Ex(g= 0) as a function of the strength of the interaction for several excited states of the relative motion. 5.4 Excitation of the C.M. For the monopole operator, the contribution of the C.M. piece of the excitation operator can be exactly calculated for any number of particles. In fact, starting from the
5.4. Excitation of the C.M. 31 definition of the DSF SF(E) = 1 NX j|hj| N X i=1 x2 i|0i|2δ(E−(Ej−E0)),(5.16) where |jiis a general excited state of the system. However, the excited states can be factorized |ji=|jCM i|jreliand the excitation operator can be also splitted into two pieces, in such a way that the matrix elements responsible of the DSF can be written as: hj|X i x2 i|0i=hjCM , jr|NX2+1 NX i<j (xi−xj)2|0CM ,0ri(5.17) which can be expressed as hj|Xx2 i|0i=hjCM |NX2|0CM ihjr|0ri+hjCM |0CM ihjr|1 NX i<j (xi−xj)2|0ri.(5.18) Taking into account the orthogonality of the states, it can be rewritten: hj|Xx2 i|0i=hjCM |NX2|0CM iδjr,0r+δjCM ,0CM hjr|1 NX i<j (xi−xj)2|0ri.(5.19) Therefore, we have separated the excitations of the C.M. from the intrinsic excitations for the general case of Nparticles. Now, we should take into account that the spectrum of the C.M. is identical to the harmonic oscillator: 1/2,3/2,5/2... and that the ground state will correspond always to the jCM = 0 state with energy 1/2. With this excitation operator, the only excitation of the C.M. will correspond always to excite the state jCM = 2 with energy ECM = 5/2, with an excitation energy ECM exc = 2. The matrix element, for jCM 6= 0 fulfills that: hjCM |NX2|0CM i=δjCM ,2Nh2CM |X2|0CM i.(5.20) Actually, one can evaluate this matrix element so that h2CM |X2|0CM i=1 √2N,(5.21) and hjCM |NX2|0CM i2=N1 √2N 2 =1 2.(5.22) The C.M. excitation appears always at an excitation energy E= 2 with strength 1 2N, independently of the interactions between the particles. The factor 1 Nis due to our definition of SF(E)in 5.16. Also notice that this is an alternative and more generalist way to obtain 5.10.
33 Chapter 6 Sum Rules In this chapter we introduce the energy sum rules of the dynamic structure function associated to a given excitation operator. The sum rules encode the response of the system to an excitation, in particular we focus our attention on the mono-polar excitation. We derive explicitly and with complete generality for the sum rules M−1, M1and M3for Nparticle systems. The most important outcome from the these sum rules is that they allow us to compute an estimate of the main excitation energy of the system, i.e., the breathing mode energy in our case, by only knowing some properties of the ground state of the system. 6.1 The energy sum rules The energy moments (Mn) of the dynamic structure function associated to a given excitation operator b F Mn(E) = Z∞ 0 EnSF(E)dE =1 NX a (Ea−E0)nha|b F|0i 2. (6.1) provide a full characterization of the response of the system. If the dynamic structure function is concentrated around a single excitation energy, SF≃Smδ(E−Eex), then a few energy moments are enough to reconstruct the DSF. In this extreme case, we have Eex ≃pMn/Mn−2. For instance, the simplest one would be Eex ≃pM1/M−1. Usually, the DSF associated to a given operator b Fis difficult to calculate. It turns out, however, that the energy moments of the DSF, can be calculated without the explicit knowledge of the DSF. As it is well known, there is a great number of theorems that allow one to compute several of the Mnwithout any explicit knowledge of all the eigenvalues and eigenvectors of the many-body systems [25]. This is done through so-called Sum Rules. These sum rules can be computed by the expectation values on the ground state of certain suitable operators. This strategy is specially appropriate when we have a good knowledge of the ground-state of the system, and is the way that Monte-Carlo calculations take advantage of its capability to calculate the ground
34 Chapter 6. Sum Rules state properties for large number of particles to provide information on the response of the system [23]. In this chapter we derive the M−1,M−1and M−3energy sum rules associated to the single-particle operator b F. Some of the derivations are valid for any excitation operator but finally we give the explicit results for b F=Pix2 i. The derivation is valid for any number of particles and in some aspects applies to any inter-particle interaction. However, at the end we report the results for N= 2 and for the contact interaction. The contribution of the C.M. motion to the considered sum rules is also discussed. 6.2 Derivation of the sum rules In this section we will derive the sum rules M−1,M1and M3with complete generality. These sum rules can be applied to any system of Nparticles, either they are fermions or bosons, with a Hamiltonian of the kind from 3.2. 6.2.1 M−1sum rule The sum rule M−1can be obtained through perturbation theory, by defining a new Hamiltonian as b H0=b H+λb F, where b His the unperturbed Hamiltonian with an unperturbed ground state energy E0, and λb Fis the perturbation , which in our case is b F=Pix2 i. The expansion of the energy up to the second order, can be written as E0(λ) = E0+λh0|b F|0i+λ2X q |hq|b F|0i|2 E0−Eq ,(6.2) and therefore the M−1sum rule can be obtained as M−1=−1 2 1 N ∂2E0(λ) ∂λ2λ=0 ,(6.3) which can be calculated numerically. Alternatively, related with this procedure, we have another way to calculate this sum rule, which can be used as a test M−1=−1 2 1 N d dλ h˜ 0|F|˜ 0iλ=0 ,(6.4) where |˜ 0iis the ground state of H+λF. In fact, the expansion of the ground state of H+λF around λ= 0 can be written as |˜ 0i=|0i+λX k6=0 Fk0 E0−Ek|ki+... , (6.5)
6.2. Derivation of the sum rules 35 and h˜ 0|F|˜ 0i=h0|F|0i+ 2λX k6=0 |hk|F|0i|2 E0−Ek +... (6.6) Therefore, dh˜ 0|F|˜ 0i dλ λ=0 = 2 X k6=0 |hk|F|0i|2 E0−Ek ,(6.7) and recovering expression 6.4 M−1=−1 2 1 N d dλ h˜ 0|F|˜ 0iλ=0 .(6.8) 6.2.2 M1sum rule Using the definition of M1, one can express the sum rule as the expectation value of a certain commutator of the excitation operator b Fand the Hamiltonian in the ground-state of the system. M1=ZESF(E)dE =1 NZdEE X n|hn|F|0i|2δ(E−(En−E0)) =1 NX n (En−E0)h0|F†|nihn|F|0i =1 NX(h0|F†|nihn|EnF|0i−h0|F†|nihn|FE0|0i) =1 NX(h0|F†|nihn|HF |0i−h0|F†|nihn|FH |0i) =1 Nh0|F†[H, F]|0i=1 2 1 Nh0|F†,[H, F]|0i. (6.9) With this new expression we are able to compute M1using only the ground state. To do so, we must start computing the commutators for the specific b F=Pix2 i, [H, F]=[Ekin +VH.O. +Vint, F] = [Ekin, F]+[VH.O., F]+[Vint, F].(6.10) Since both potentials VH.O. and Vint depend only on the coordinates, we have that [VH.O., F]=[Vint, F]=0. Therefore, we only need to calculate the kinetic energy commutator [Ekin, F] = −"N X i=1 1 2 d2 dx2 i , N X j=1 x2 j#=−1 2X i,j d2 dx2 i , x2 jδi,j =−1 2X i=1 d2 dx2 i , x2 i.(6.11)
42 Chapter 7. Sum Rules applied to a 2 bosons system 4 5 6 7 8 SumRules M3DSF M3GS 1 1.2 1.4 1.6 1.8 2 M1DSF M1GS 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0 5 10 15 20 25 30 g M-1DSF M-1GS FIGURE 7.1: Comparison of the sum rules M−1,M1and M3computed through the DSF or using the ground-state properties. In the DSF method we have truncated the spectra by taking the first 10 energy levels and both have been computed with m=200 (101 modes). All the sum rules are increasing functions of the strength of the interaction and approach an asymptotic value for g→ ∞. The increasing character of the sum rules with gis mainly due to the increment of the strength of the breathing mode with g. As expected the convergence to the asymptotic values is faster for M−1and slower for M3. However, in all cases for g= 30, the largest value of gconsidered in the figure, the convergence has been almost reached. For N= 2, the expressions derived in the previous chapter in the non-interacting limit reduce to M1= 1 and M3= 4, while for the Tonks- Girardeau regime we have M1= 2 and M3= 8. Also relevant is the contribution of the C.M. excitation to the total sum rules, as mentioned before, the contribution is independent of gand for N= 2 we have: M−1,CM = 1/8, M1,CM = 1/2and M3,CM = 2. While the strength and energy of the c.m. excitation remains constant as a function of g, the energy and strength of the peak due to the intrinsic motion depends on g. In particular the strength is an increasing function of ggoing from 1/4at g= 0 to 3/4when g→ ∞. This increment of the strength of the intrinsic peak associated to the mono-pole vibration is the main reason for the increasing behaviour of the different sum rules as a function of g. For all values of g, the peaks associated to the C.M. and to the breathing mode exhaust more than the 99 % of the sum rules and the higher peaks have a small but not negligible influence. It is also illustrative to compare the g→ ∞ value with the Thomas-Fermi one, in this case the kinetic energy is neglected and the estimate of the excitation energy from the sum rules is, Eexc =√3. Thus, as emphasized in [23], the reentrant behavior is a clear consequence of the "fermionization" taking place as the g→ ∞. In 1D gases with contact interaction increasing the interaction strength, the approximation which is sensible is to neglect the interaction energy keeping the kinetic one, unlike
7.2. Excitation energy and reentrant behaviour 43 in the Thomas-Fermi limit where the kinetic energy is neglected in front of the interaction. 7.2 Excitation energy and reentrant behaviour In the calculation of SF(E)in a previous chapter, we have observed that the two important peaks (C.M. and breathing mode ) of SF(E)are rather close to each other. Therefore, , one expects that the estimation of the breathing excitation energy through the sum rules should be rather accurate. In Fig. 7.2 we compare the mono-polar excitation energies estimated by means of the sum rules described above with the values of the excitation energy of the breathing mode. 1.84 1.86 1.88 1.9 1.92 1.94 1.96 1.98 2 2.02 0 5 10 15 20 25 30 ExcitationEnergy(H.O.units) g (M1/M-1)(1/2) (M3/M1)(1/2) Intrinsic FIGURE 7.2: Mono-polar excitation energy estimated using Eex = pM3/M1and Eex =pM1/M−1compared to the one from the breathing mode itself. In the non-interacting limit and in the Tonks-Girardeau limit they reach the value Eex = 2. For g≈2we can depict a minimum of Eex leaving us with the reentrant behaviour. The sum rules have been computed using m=200 (101 modes). Notice that in all cases, the estimatepM3/M1is larger thanpM1/M−1for any value of gand both are larger or equal than the minimum excitation energy defined by the breathing mode. First we note that the two quantities obtained frompM3/M1 andpM1/M−1do not produce the same estimate for the intrinsic mono-polar excitation energy. This reflects the fact that the structure function contains two relevant excited states not located at the same energy, one from the center of mass and the second from the breathing mode. The difference between the two estimates provided by the sum rules increases with gand is larger for the value of gthat maximizes the interaction energy. For this value of gthe difference in energy between the C.M. excitation and the energy of the breathing mode is also maximal. Then, when gincreases further and the breathing mode gets closer again to the C.M. excitation, the
44 Chapter 7. Sum Rules applied to a 2 bosons system estimates get closer and both reach asymptotically the Eex = 2 when g→ ∞. The reentrant behaviour of the breathing mode previously discussed in cleanly reproduced by the estimations of the excitation energies provided by the sum rule analysis. If we subtract the contribution of the C.M. to the sum rules, is equivalent to study only the intrinsic excitations, see Fig. 7.3. In this case there is only one dominating peak: the breathing mode and consequently, the estimations of the excitation energy from the sum rules are much closer. The differences in this case should be assigned to the higher excitations beyond the breathing mode. 1.84 1.86 1.88 1.9 1.92 1.94 1.96 1.98 2 2.02 0 5 10 15 20 25 30 ExcitationEnergy(H.O.units) g (M1/M-1)(1/2) (M3/M1)(1/2) Intrinsic FIGURE 7.3: Intrinsic excitation energy of the breathing mode compared to the mono-polar excitation estimations using Eex =pM3/M1 and Eex =pM1/M−1, where the C.M. contribution has been extracted. In both limits they all reach the value Eex = 2, and there is a minimum that signals the reentrant behaviour around g≈2. These new estimations are a good approach to the breathing mode excitation energy. The sum rules have been computed using m=200 (101 modes).
45 Chapter 8 Summary and Conclusions In this thesis we have considered a 1D gas of ultra-cold bosonic atoms, where the atom-atom interaction is assumed to be well represented by a delta contact interaction. This system has already been realized experimentally in two ultra-cold atomic laboratories and is expected to be studied in more detail in future experiments. Our main aim has been to explore the properties of the system, i.e. wave functions, excitations, correlations, etc, as the strength of the atom-atom interaction is varied. The latter is possible in ultra-cold atom experiments by means of Feshbach resonances. In the first chapter, we considered the homogeneous system. In this case, we introduced already the two noteworthy limiting cases, the zero interaction case, in which the system fully condenses and the infinitely strong interaction case, which gives rise to the Tonks-Girardeau gas. We have shown some of the properties of the TG gas, like the impossibility of finding two particles at the same position, as revealed by pair correlations. Then we have concentrated in the case of N bosons trapped in a harmonic oscillator potential. In this case, we have first derived a virial relation which holds for both bosons and fermions and which is very useful to cross check numerical calculations. Afterwards we have discussed the two limiting cases, discussing the properties of both the condensate and TG gas. Then we have derived the Gross-Pitaevskii equation, which is obtained in the mean-field approximation. As expected, this description fails to capture the properties of the N-body system when the interactions are not small or for very small number of particles, where more involved quantum correlations appear in the system. After the general case, we have concentrated in the two-boson case. We have explained in detail the spectral properties of the system as a function of the interaction parameter, discussing also the different contributions of the energy. Notably, the interaction energy provides a good witness of the presence of correlations: for small interaction strength the interaction energy increases with the strength. At sufficiently large interaction strength, the system starts to build quantum correlations to avoid the contact, which in turn result in a decrease of the interaction energy as we further increase the interaction strength. The study of the spectral properties has been the main objective of the last two chapters. This has been done by computing the dynamic spectral function for a
46 Chapter 8. Summary and Conclusions mono-polar excitation. In particular, we have thus considered the excitation of the breathing mode, i.e. the first excitation of the relative motion of the atoms. This breathing mode had been previously shown to exhibit a reentrant behavior as the interaction strength was varied from 0 to large values. We have demonstrated that this is indeed the case for the two-particle case, and also that other relative excitation exhibit a similar reentrant behavior. Finally, we have explained how information about the mono-polar excitation can be obtained from ground state properties by means of energy sum rules, which were originally introduced in nuclear physics. We have shown how the squared ratio of two momenta of the distribution can be used to obtained estimates for the breathing mode energy. This is the technique used in quantum Monte-Carlo calculations in which only ground state properties can be computed. Our method allows us to explore the quality of this approach for different squared ratios of the sum rules in the whole range of interaction strengths considered.
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