UNIVERSITÀ DEGLI STUDI DI BARI ALDO MORO DIPARTIMENTO INTERATENEO DI FISICA M. MERLIN Dottorato di ricerca in FISICA Ciclo XXV Settore Scientifico Disciplinare FIS/02 Bose-Einstein condensation: static and dynamical aspects Dottorando: Dott. Francesco Vincenzo Pepe Supervisore: Ch.mo Prof. Saverio Pascazio Coordinatore: Ch.mo Prof. Salvatore Nuzzo ESAME FINALE 2013
Contents Introduction 3 1 Bose-Einstein condensation: definition and general aspects 7 1.1 Condensation in an ideal Bose gas................ 7 1.2 General definition of BEC.................... 11 1.3 The weakly interacting Bose gas................. 15 1.4 Low energy excitations and healing length........... 21 2 Gross-Pitaevskii equation 25 2.1 Field equations for the condensate................ 25 2.2 The stationary Gross-Pitaevskii solutions............ 28 2.3 Thomas-Fermi limit and local density approximation..... 31 2.4 Quasi one-dimensional condensates............... 34 3 Two-component Bose-Einstein condensates 39 3.1 Boson mixtures: physical relevance and experiments...... 39 3.2 Mean field analysis of two-component condensates....... 41 3.3 Ground state of uniform mixtures................ 43 3.4 Elementary excitations in the mixed configuration....... 45 4 Binary mixtures in confining potentials 49 4.1 Thomas-Fermi solutions in generic potentials.......... 49 4.2 Mixed vs separated configurations................ 53 4.3 Phase separation at fixed numbers of particles......... 61 4.4 Maximal and non-maximal stable configurations........ 71 4.5 Phase separation at fixed chemical potentials.......... 74 4.6 Summary............................. 77 4.7 Computational relevance of the results............. 78 5 Binary mixtures beyond local density approximation 81 5.1 Thomas-Fermi singularities and regularization......... 81 1
5.2 Variational regularization of domain walls........... 83 5.3 Regularization of profile edges. Orders of magnitude...... 91 5.4 Domain wall suppression..................... 95 5.5 Outlook.............................. 102 6 Wave-function renormalization effects in Resonantly Enhanced Tunneling of a condensate 103 6.1 Cold atoms experiments and wave-function renormalization.. 103 6.2 Landau-Zener tunneling..................... 105 6.3 Interband tunneling in a lattice................. 108 6.4 Transient and asymptotic behavior............... 115 6.5 Simulation of Quantum Zeno and Inverse Zeno effects..... 119 6.6 Outlook.............................. 121 Conclusions and outlook 123 2
Introduction The phenomenon of Bose-Einstein condensation (BEC) is closely related to quantum statistics. The seminal idea dates back to 1925, when Einstein, on the basis of Bose s work on the statistical description of photons [1], predicted the occurrence of a phase transition in a gas of noninteracting atoms [2]. This phase transition is related to the macroscopic occupation of the lowest energy state and is a consequence of the symmetry of the N-body wave functions with respect to exchanges of any couple of particles, which leads to Bose statistics. This prediction had no practical impact until London in 1938 had the intuition that superfluidity in liquid Helium could be a manifestation of BEC [3]. Theoretical efforts and experimental studies provided a deep understanding of the relation between superfluidity and BEC. A microscopic theory of weakly interacting dilute Bose gases was developed by Bogoliubov in 1947 [4]. However, the first experimental studies on these systems started only in the 1970s, profiting from new techniques in atomic optical and magnetic trapping and advanced cooling mechanisms [5, 6]. Neutral alkali atoms are well suited for laser-based techniques, developed in the 1980s, such as laser cooling and magneto-optical trapping. Their internal energy level structure allows for cooling to extremely low temperatures, of the order of fractions of µk [7, 8]. The study of ultracold alkali atoms, together with trapping and cooling techniques, provided the first direct observations of BEC in a series of experiments in 1995, performed on vapors of 87 Rb by the team of Cornell and Wieman at Boulder [9] and on 23 Na by the Ketterle group at MIT [10]. The evidence for condensation emerged from time-of-flight measurements. The atoms were left to expand by switching off the confining trap and then imaged with optical methods. A sharp peak in the velocity distribution was then observed below a certain critical temperature, providing a clear signature for BEC. Signatures of condensation have been observed also in 7 Li [11], spin-polarized Hydrogen [12], metastable 4 He [13] and 41 K [14]. It is worth noticing that, in the low-temperature conditions attained in the aforementioned experiments, the equilibrium configuration of a system of alkali atoms 3
would be the solid phase. Thus, in order to observe BEC, one has to preserve the system in a metastable gas phase for a sufficiently long time. This is possible because three-body collisions are rare events in dilute and cold gases, causing the lifetime of the gaseous phase to be long enough to carry out the experiments. Cold alkali atoms are an interesting physical systems, since parameters leading to different physical regimes can be experimentally controlled. As a relevant example, the interaction between atoms can be tuned by placing the sample in a magnetic field close to a Feshbach resonance [15]. This leads to the possibility of implementing complicated configurations in experiments and trying to uncover new features and possible applications of Bose-Einstein condensation, both from an experimental and a theoretical point of view. The research work on which this Ph.D Thesis is based is devoted to two frontier aspect of such new features: the first one is the study of samples of multicomponent interacting condensates in trapping potentials, the second one is the possibility to use Bose-Einstein condensed cold gases in very dilute regimes to simulate single-particle systems for studying fundamental quantum mechanics. The first part of this Thesis is essentially a review of the fundamental properties of Bose-Einstein condensation and of all the physical features which will be relevant for the second part, devoted to the research made during my Ph.D course. In Chapter 1, a general introduction on BEC is made. We will start from the simple case of the noninteracting gas both for historical reasons and in order to highlight the key role of quantum statistics in the phenomenon of condensation. Then, a general definition of BEC will be given, which is valid for all physical systems of particles, regardless if they are interacting or not. We will analyze in detail Bogoliubov computations for the weakly interacting Bose gas, in which it emerges that if the gas is sufficiently dilute, almost all the particles are in the same quantum state. This will pave the way to the Gross-Pitaevskii equation, which will be the topic of Chapter 2. We will study the main properties of the Gross-Pitaevskii equation, and focus on the Thomas-Fermi limit, which leads to a semiclassical approximation in which dynamics and statics aspects are described by classical fluid-dynamics equations. We will consider the strategies to obtain quasi one-dimensional condensates, which will be the topic of the research part. Chapter 3 is devoted to the exposition of well-consolidated results in the field of binary mixtures of condensates. These results are mainly related to uniform systems, and will be extended to the general case in the research part. Chapter 4 contains a series of results for the static aspects of binary mixtures of condensates trapped in generic confining potentials. The results are 4
obtained by applying the aforementioned Thomas-Fermi approximation and are related to the search for the ground state and, more generally for states which are at least locally stable in energy. We will focus in particular on physical configurations which exhibit phase separation, namely, when condensates of different species are sharply divided by domain walls. In Chapter 5 we will go beyond the Thomas-Fermi approximation, and introduce analytical instruments to approximate the total energy of the mixture. In the Thomas-Fermi approximation, the estimate of the kinetic energy is not consistent, due to singularities in the density profiles. Our technique is aimed at healing the singularities and obtain proper estimate of both the kinetic and potential energies in the ground state of the system. In Chapter 6, we will shift back to a single-component condensate and consider a physical regime in which interactions are negligible. Under such a condition, the Gross-Pitaevskii equation for the condensate coincides with the Schrödinger equation for a single particle. The condensate can thus be used as an experimental tool to investigate fundamental quantum mechanics. It will be used, in particular, to analyze effects of wave-function renormalization in resonantly enhanced tunneling, a phenomenon which takes place in accelerated optical lattices. 5
6
Chapter 1 Bose-Einstein condensation: definition and general aspects In this chapter we briefly review the properties of an ideal uniform Bose- Einstein condensate. We introduce mathematical tools and describe those physical features which are useful for a quantitative description of condensation. Finally, we discuss how interactions affect BEC and expose the zero temperature Bogoliubov theory for weakly interacting Bose gases. 1.1 Condensation in an ideal Bose gas BEC in an ideal gas is conveniently described in the grand canonical formalism [16]. Let us consider a Bose system in thermal equilibrium at temperature T. The probability of realizing a configuration with N particles in state k with energy E k reads P N (E k ) = eβ(µn E k ) Z(β, µ), (1.1) with β = 1/k B T the inverse temperature and µ the chemical potential of the reservoir with which the system is in thermal equilibrium. The computation of the grand canonical partition function Z(β, µ) = e β(µn E k ), (1.2) N =0 with k representing the sum over a complete set of eigensates of the system Hamiltonian with energy E k, leads to a straightforward derivation of thermodynamical quantities by identifying the grand canonical potential k 7
Ω = E T S µn as a function of Z(β, µ): Ω = 1 ln Z. (1.3) β For an independent-particle Hamiltonian H = j H(1) j, with j the particle index and H (1) j denoting a single-particle operator, the many-body eigenstates are specified by the set {n i } of the occupation numbers of single-particle eigenstates { i } with eigenvalues {ɛ i }. This feature enables one to explicitly compute the partition function, since N = i n i (1.4) and E = i n i ɛ i. (1.5) In the case of Bose statistics, there is no bound on the occupation numbers, which can range from zero to infinity. Thus, the partition function reads Z = i n i =0 e β(µ ɛi)ni = i 1 1 e β(µ ɛ i). (1.6) The total average number of particles is obtained by performing a derivative of the grand canonical potential (1.3) with respect to µ. The result N = i 1 e β(ɛ i µ) 1 = i n i (1.7) is expressed as a sum of the average occupation numbers n i of single-particle states. Requiring the positivity of all n i s provides the physical constraint µ < ɛ 0 on the chemical potential, where ɛ 0 is assumed to be the ground state energy of the single-particle Hamiltonian. If µ ɛ 0, the occupation number of the ground state becomes increasingly large, leading to BEC. We can rewrite the number of particles separating the two contributions coming respectively from the ground state N = N 0 + N T, (1.8) N 0 = n 0 (β, µ) = 1 e β(ɛ 0 µ) 1 (1.9) 8
and from the excited states N T (β, µ) = i 0 n i (β, µ). (1.10) For a fixed temperature, the function N T increases as a function of µ and reaches its physical maximum at µ = ɛ 0, where N 0 diverges. If N c = N T (T, µ = ɛ 0 ) is larger than N, Eq. (1.8) is satisfied for µ < ɛ 0 and N 0 is negligible with respect to N T, which is proportional to the density of states. Since N c (T ) is an increasing function of temperature, this scenario takes place for T larger than a critical temperature T c [7], defined by N T (T c, µ = ɛ 0 ) = N. (1.11) If instead N c (T ) < N, or equivalently T < T c, the contribution of the condensate becomes crucial to satisfy the normalization condition (1.8), and the chemical potential approaches ɛ 0 in the thermodynamic limit, leading to a phase transition. Thus, the temperature T c defines the critical temperature below which the phenomenon of Bose-Einstein condensation, namely the macroscopic occupation of a single-particle state, takes place. Let us now review the properties of the simplest Bose system, namely an ideal gas in a box with periodic boundary conditions. In this case, the single-particle Hamiltonians are purely kinetic H (1) j = 2 2m 2 j. (1.12) Let us assume the particles to be confined in a cubic box of volume V = L 3. The eigenstates of the single-particle Hamiltonians are plane waves whose wave vectors k = 2π L (n x, n y, n z ) with n x,y,z Z. (1.13) are proportional to the eigenvalues of momentum operator p = k. In this case we have ɛ 0 = 0, implying µ < 0. In the thermodynamic limit, the discrete spectrum of the momentum operators becomes a continuum. The sum over states i in N T can be replaced by an integral in the phase space, which eventually leads to the result [16] N T = V g λ 3 3/2 (e βµ ), (1.14) T where λ T = 2π 2 /mk B T is the thermal wavelength and g p (z) = 1 Γ(p) 0 9 x p 1 dx z 1 e x 1, (1.15)
with Γ(p) being the Euler Gamma function [17]. The definition of critical temperature in Eq. (1.11) yields k B T c = 2π 2 m ( ) 2/3 n, (1.16) g 3/2 (1) where n = N/V is the density of the gas, which is kept constant in approaching the thermodynamic limit, and g 3/2 (1) 2.612. The critical temperature of a Bose gas is fully determined by the density and the mass of the particles. For T > T c, the chemical potential is obtained by solving the normalization condition (1.8) for N = N T. For T < T c, the same equation should be solved for µ = 0. This leads to the relation ( ) 3/2 T N T = N, (1.17) which yields the temperature dependence of the condensate fraction T c N 0 (T ) N ( ) 3/2 T = 1. (1.18) T c Thus, at temperatures close to zero, all the particles are in the single-particle ground state and the whole system is Bose condensed. The power of 3/2 appearing in (1.18) is typical of uniform systems. Analogous computations can be performed, for example, on an ideal Bose gas in a harmonic potential: in that case, the condensate fraction depends on (T/T c ) 3 and the critical temperature scales like N 1/3 [7]. Once the chemical potential is determined, the thermodynamics can be easily evaluated. It is possible, in particular, to obtain the isothermal curves in the (v, P ) plane, with v being the specific volume and P the pressure. For a fixed value of T, one defines a critical volume v c = λ3 T (1.19) g 3/2 (1) below which condensation takes place. It is at this point that a pathological feature of BEC in the ideal gas emerges. In the condensed phase (v < v c ) the pressure is indeed independent of the specific volume P = k BT λ 3 T g 5/2 (1), (1.20) with g 5/2 (z) defined as in Eq. (1.15). This implies that in the condensed phase the gas isothermal compressibility is infinite, and the volume per particle 10
can be eventually reduced to zero at all temperatures without increasing the pressure. This nonphysical feature is the result of neglecting interactions between particles. Real interatomic potentials usually have a repulsive core which would prevent the system to be infinitely compressible. Including two-body interactions in the model removes such a pathological aspect. Another remark should be done to emphasize the importance of interactions, concerning the macroscopic sensitivity of the ideal Bose-Einstein condensate to changes in boundary conditions [7]. The aforementioned results have been obtained under the hypothesis of cyclic boundary conditions, leading to a uniform ground state wave function φ (C) 0 = 1 V. (1.21) This means that the condensate at T = 0, with N 0 = N, would be uniform as well. If instead one imposes Dirichlet boundary conditions, forcing the wave function to vanish at the boundaries of the box, the ground state wave function would read φ (D) 0 = 8 ( πy ) V sin sin L ( πx ) sin L ( πz ) L (1.22) Thus, the density of the condensate, which is proportional to the square of the ground state wave function, would be nonuniform (oscillating) over macroscopic lengths of the order of the size of the system, no matter how large it is. This sensitivity is closely related to the problem of infinite compressibility [7]. Again, this feature is removed by including interactions in the model. If one switches from cyclic to Dirichlet boundary conditions in a weakly interacting condensate, the uniformity of the density at T = 0 is preserved up to microscopic distances from the boundaries, which are related to the healing length, to be defined and discussed in the following sections. 1.2 General definition of BEC In the previous section we exposed the main features of Bose-Einstein condensation in the case of an ideal gas in a box, following in some sense Einstein s original path. The simplicity and feasibility of this case lies in the fact that the many-body Hamiltonian is no more than the sum of single-particle operators, and thus the global eigenstates are tensor products of the single-particle ones. Condensation is characterized by a macroscopic number of particles occupying the single-particle ground state: this is clearly no longer possible if the particles interact. Thus, we are interested in extending the concept of 11
BEC to a wider class of Bose systems, by introducing a general quantitative definition. We will focus for simplicity on spinless bosons, having no internal degree of freedom. Let us consider a system composed of a large number N of identical bosons, otherwise arbitrary. At time t, the system can be either in a pure N-particle state Ψ s (t) whose wave function in the coordinate representation r 1,..., r N Ψ s (t) = Ψ s (r 1,..., r N ; t), (1.23) is symmetric under exchange of any couple of particle positions, or in a mixed state, which can be always written as a statistical superposition of normalized and mutually orthogonal pure states Ψ s with weight p s, represented by a Hermitian density matrix ρ N (r 1,..., r N ; r 1,..., r N; t) = p s Ψ s(r 1,..., r N ; t)ψ s (r 1,..., r N; t), s p s = 1. (1.24) s For further convenience, let us choose to normalize the pure N-particle states not to one but to the number of particles itself: dr 1... dr N Ψ s (r 1,..., r N ; t) 2 = N (1.25) This also implies that the trace of the density matrices will always be equal to N. The state of a single particle can be obtained by tracing the N-particle density matrix over the positions of the other N 1 particles: ρ 1 (r, r ; t) = dr 2... dr N ρ N (r, r 2,..., r N ; r, r 2,..., r N ; t). (1.26) The choice of integrating over (r 2,..., r N ) is arbitrary, due to the indistinguishability of the particles and the symmetry of the N-body density matrix. Being a Hermitian operator, the single particle density matrix can be properly diagonalized into a sum of single-particle, normalized and orthonormal projectors ρ 1 (r, r ; t) = n i (t)φ i (r, t)φ i (r, t). (1.27) i The eigenvalues {n i }, which sum to N, are called the occupation numbers of single particle states {φ i }. Notice that the eigenfuntcions {φ i } of ρ 1 need not be eigenfuction of the system Hamiltonian or of any other simple operator. Starting from the form (1.27), we can state the following definitions [18]: 12
if (at time t) all the occupation numbers n i are O(1) with respect to N, the system is normal; if one of the occupation numbers, say N 0 n 0, is O(N), while n i 0 = O(1), the system exhibits simple Bose-Einstein condensation; if more than one of the occupation numbers are O(N), the system exhibits fragmented Bose-Einstein condensation. The third definition is stated only for the sake of completeness, since in this Thesis, we will be interested only in simple condensation, to which we will refer simply as BEC or condensation. Let us assume that condensation occurs in the single-particle state φ 0. In this case, one can write the spatial density of the system, corresponding to the density matrix evaluated for r = r, with the condensate contribution isolated ρ(r, t) = N 0 (t) φ 0 (r, t) 2 + i 0 n i (t) φ i (r, t) 2. (1.28) The density of the Bose-Einstein condensed part of the system is given by the square modulus of the function Ψ 0 (r, t) N 0 (t)φ 0 (r, t). (1.29) Since this relation is analogous to the one between a single-particle wave function and the related probability density, apart from normalization, the quantity in Eq. (1.29) is called the condensate wave function. It can be identified as the order parameter of the BEC phase transition, which vanishes for T T c [7, 18]. It can be clearly observed how, by virtue of the previous definitions, the description of condensation has been extended to nonuniform systems, regardless of the interactions and more generally of the dynamics of the particles. A second quantization formalism is useful in the analysis of many-particles Bose systems [19]. We can introduce an annihilation operator ˆΨ(r, t) which destroys a particle in r at time t and the adjoint operator ˆΨ (r, t) which creates a boson at the same point. The bosonic field operators have to satisfy canonical commutation relations at all times [ [ ˆΨ(r, t), ˆΨ(r, t)] = 0, ˆΨ(r, t), ˆΨ (r, t)] = δ(r r ). (1.30) The annihilation operator ˆΨ(r, t) can be expanded as the sum over a complete set of single-particle states labelled with index i of operators â i which annihilate a particle in the state i, weighted with the correspondent wave 13
functions. One can conveniently choose as the complete set the basis {φ i } defined after Eq. (1.27): ˆΨ(r, t) = i φ i (r, t)â i (t). (1.31) It is easy to obtain an explicit expression of the operators â i by making use of the orthonormality relation for the wave functions {φ i }: â i (t) = drφ i (r, t) ˆΨ(r, t). (1.32) The canonical commutation relations at equal times for these new field operators easily follow from Eqs. (1.30)-(1.31), and read [ ] [â i (t), â j (t)] = drdr φ i (r, t)φ j(r, t) ˆΨ(r, t), ˆΨ(r, t) (1.33) and [â i (t), â j (t)] = = [ ] drdr φ i (r, t)φ j (r, t) ˆΨ(r, t), ˆΨ (r, t) drφ i (r, t)φ j (r, t) = δ ij, (1.34) respectively. In second quantization formalism, the single particle density matrix (1.27) corresponds to the expectation value of the operator product ˆΨ (r, t) ˆΨ(r, t) on the N-particle state. By comparing the results, we obtain â iâj = n i δ ij. (1.35) If the physical states we are considering are characterized by condensation in the i = 0 state, then â 0â 0 = N 0 = O(N). Since â 0 â 0 = â 0â 0 + 1 = N 0 + 1 N 0, (1.36) the commutator between operators with i = 0 can be neglected for all practical purposes. Thus, â 0 and â 0 can be regarded as c-numbers, with the replacement (Bogoliubov approximation) [19] â 0 N 0, â 0 N 0. (1.37) The field operator after the Bogoliubov approximation can thus be written as the sum of a c-number part, corresponding to the condensate wave function 14
defined in Eq. (1.29) and an operator part, describing particles out of the condensate ˆΨ(r, t) = Ψ 0 (r, t) + δ ˆΨ(r, t). (1.38) We stress that even for the Bogoliubov approximation no hypothesis on the interactions has been made, since it is based only on the assumption of large N 0. Different limits on its validity should follow a posteriori. The onset of condensation can be regarded as a spontaneous symmetry breaking for a U(1) invariant Hamiltonian, since the existence of a condensate wave function implies ˆΨ = Ψ 0 0 [7]. The phase S(r, t) of the condensate wave function Ψ 0 (r, t) = Ψ 0 (r, t) e is(r,t) (1.39) in the broken-symmetry phase is closely related, as we will clarify later, to the velocity field of the condensate, in a fluid-dynamics sense. Let us end this brief review of the general properties of BEC by mentioning a feature that can be observed in uniform systems, when the thermodynamic limit N, V can be taken, namely the existence of off-diagonal long range order in the single-particle density matrix [20, 7] ρ 1 (r, r ) C > 0 for r r. (1.40) If one considers the form (1.27) and assumes to have condensation for i = 0, unless pathological behaviors occur, the destructive interference of states in the sum for i 0, which in the thermodynamic limit is replaced by an integral, leads to the vanishing of all the contributions coming from noncondensed particles. Thus, the behavior of the density matrix (1.27) for r r is determined by the single-particle wave function φ 0. In an ideal gas, for example, we have φ 0(r, t)φ 0 (r, t) = 1/V > 0. This feature is obviously absent in trapped samples, where the external confinement forces the density of the system to vanish at long distances. 1.3 The weakly interacting Bose gas The original argument of Einstein, presented in Section 1.1, is related to the occurrence of BEC in a noninteracting system in thermal equilibrium. However, in all the real systems in which condensation is believed to occur the particles interact, and many of the interesting phenomena related to condensation and superfluidity occur well away from thermal equilibrium [7]. In the previous section we introduced general tools to characterize BEC in any physical system. We are now going to analyze how interactions affect condensation. Up to now, there is no general theorem on the occurrence of 15
BEC in an interacting Bose system, even at T = 0. Nevertheless, there are strong qualitative arguments which suggest that it is likely to occur when the interparticle interactions are repulsive and not too strong [18]. In this section we will expose the Bogoliubov theory for a weakly interacting Bose gas. It is based on a peculiar perturbation technique, in which the small parameter is the number of particles out of the condensate, and provides the basis of the modern approach to BEC in dilute gases. In a dilute gas the range r 0 of the interatomic forces is much smaller than the average distance d = n 1/3, fixed by the density n = N/V of the sample. This enables one to consider only two-body scattering, while configurations with three or more particles interacting simultaneously can be neglected [7]. Moreover, the distance between two particles is large enough to justify the use of asymptotic wave functions for their relative motion, which are determined by the scattering amplitude f(k, k). We shall consider Bose gases at temperatures well below the critical one for BEC: this implies that the relevant values p of the relative momentum will satisfy p r 0 1. (1.41) At such small momenta, the scattering amplitude becomes independent of the scattering angle and the energy and can be replaced by its low-energy approximation, which is determined by the s-wave scattering length a [19, 21] f(p, p) a for p, p 0, (1.42) where p and p are respectively the wave vectors of the incident and outgoing particle. In such a regime, one expects the scattering length to be the only parameter that characterizes all the effects of interactions on the physical properties of the gas. Let us consider a Hamiltonian describing N Bose particles with mass m interacting through a potential U(r) Ĥ = N i=1 2 2m 2 i + 1 2 N U(r i r j ), (1.43) i,j=1 where terms with i = j are excluded from the second summation, r i is the position of particle i and 2 i is the Laplace operator with respect to the coordinates of particle i. Shifting to a second quantization formalism, we can introduce a field operator ˆΨ(r, t), which enables us to write the system Hamiltonian as Ĥ = dr 2 2m ˆΨ (r) ˆΨ(r) + 1 drdr 2 ˆΨ (r) ˆΨ (r )U(r r ) ˆΨ(r ) ˆΨ(r), (1.44) 16
where only ˆΨ(r) ˆΨ(r, t = 0) appears, since the Hamiltonian is invariant under time evolution. The procedure to obtain the field Hamiltonian in Eq. (1.44) starting from the N-body operator in Eq. (1.43) is presented e.g. in [19]. The double summation of the interparticle interaction terms over a large number of particles disappears in Eq. (1.44) giving rise to a quartic term, representing the field s self interaction. Such a term gives rise to nonlinearity in the field evolution equations. For a uniform gas in a volume V, with cyclic boundary conditions, it is convenient to expand ˆΨ(r, t) in terms of operators a p (a p), annihilating (creating) particles with momentum p ˆΨ(r) = p e ip r/ V a p. (1.45) Since we are interested in the low energy physics, we perform an approximation by replacing the real interparticle potential U(r) with a point-like pseudopotential Ũ(r) = gδ(r), (1.46) where the constant g must be chosen in order to correctly reproduce the two-body scattering properties of the actual potential, namely the scattering amplitude at low energy. This request yields a relation between g and the scattering amplitude, which is a physically observable quantity [19]: 4π 2 a m = g mg2 dp 1 (2π ) 3 p +... (1.47) 2 The first-order result g = 4π 2 a/m is well defined, and we will make large use of it in the following chapters. In the second order result, however, the integral diverges for large momenta, and a cutoff should be introduced in order to regularize it. This seemingly annoying singular behavior will turn out to be useful to cancel another divergence arising from the diagonalization of the Hamiltonian. The origin of this singularities lies in the fact that the pseudopotential (1.46) is not suitable to describe the high-energy features of the system. The resulting approximated Hamiltonian reads Ĥ = p ɛ 0 pâ pâ p + g 2V p 1,p 2,q â p 1 +qâ p 2 qâ p2 â p1, (1.48) with ɛ 0 p p2 2m (1.49) 17
being the free-particle energy. Let us assume condensation to occur in the zero momentum state just as in the noninteracting case, and perform Bogoliubov approximation by replacing â 0 and â 0 with the square root of the number of particles in the condensate N 0 (see Eq. (1.37)). Interaction terms in the Hamiltonian (1.48) can be classified according to the power of N 0 appearing in them. By retaining only terms containing N0 2 or N 0, one can obtain the approximated Bogoliubov Hamiltonian Ĥ B = p ɛ 0 pâ pâ p + g 2V [ N0 2 + 2N 0 (â pâ ) p + â pâ p +N 0 p 0 p 0 ( â pâ p + â pâ p ) ] (1.50) It follows from Eq. (1.50) that the condensate acts as a source for non condensed particles, which are thus not conserved. It would be possible to introduce a chemical potential but it is more useful to work with a fixed total number of particles N and explicitly eliminate N 0 by using the normalization relation N = N 0 + 1 (â pâ ) p + â 2 pâ p (1.51) p 0 and by keeping in the Hamiltonian (1.50) only terms of order N 2 and N, based on the assumption that N N 0 N [19]. The final model Hamiltonian reads Ĥ B = gn2 V 2 + 1 2 p 0 [ ] (ɛ 0 p + gn)(â pâ p + â pâ p ) + gn(â pâ p + â pâ p ), (1.52) where n = N/V is the particle density. The approximated Hamiltonian (1.52), in which all the coefficients are known, can be solved exactly by applying a canonical transformation [4, 22], introducing a new set of field operators {b p } in the following way â p = u pˆbp + v pˆb p, (1.53) with the coefficient u p and v p real and spherically symmetric. We request that the ˆb p s follow the same bosonic algebra as the â p s, namely This is accomplished by imposing [ˆb p, ˆb p ] = 0, [ˆb p, ˆb p ] = δ pp. (1.54) u 2 p v 2 p = 1 p. (1.55) 18
In order to fulfill diagonalization, the coefficients are chosen in a way that terms depending on ˆb pˆb p and ˆb pˆb p cancel in the Hamiltonian (1.52). The remaining terms will be proportional to the number operators ˆb pˆb p. Thus, condition 2u p v p = gn (1.56) u 2 p + vp 2 ɛ 0 p + gn should be imposed. Solving Eq. (1.56) together with Eq. (1.55) yields the solution vp 2 = u 2 p 1 = 1 ( ɛ 0 ) p + gn 1 p, (1.57) 2 ɛ p with ɛ p (ɛ 0 p) 2 + 2ɛ 0 pgn. (1.58) After these manipulations, the Bogoliubov Hamiltonian finally reads Ĥ B = E + p ɛ pˆb pˆbp. (1.59) Thus, ˆb pˆb p is the number operator which counts the elementary excitations, which are quasiparticles characterized by momentum p and energy ɛ p as in Eq. (1.58). In the next section we shall analyze the relevant features of these excitation and their role in thermodynamics. The quantity E in Eq. (1.59) is the energy of the ground state GS of the interacting system, which corresponds to the vacuum of the quasiparticles, satisfying ˆb p GS = 0 for all p. Taking the thermodynamic limit V p dp (2π ) 3, (1.60) one can replace sums in E with integrals in momentum space and consider ɛ 0 (p) and ɛ(p) as functions of the continuous variable p: E = gn2 V + V dp [ ( ɛ(p) ɛ 0 (p) + gn )]. (1.61) 2 2 (2π ) 3 If g is approximated at the first order in a, it can be verified that the integral in momentum space in Eq. (1.61) diverges linearly for high momenta, and it is necessary to introduce a cutoff. However, this divergence can be healed by observing the analogy with the singularity in the second order term of a as a function of g [23]. It is indeed possible to invert Eq. (1.47) and obtain g = 4π 2 a m ( 1 + 4π 2 a 19 dp 1 (2π ) 3 p 2 ). (1.62)
If this result is inserted in (1.61), the integral appearing in the ground state energy remains finite as the cutoff momentum is sent to infinity. We can thus perform the integration and obtain the result [19] E N = 2π 2 an m [ 1 + 128 15π 1/2 (na3 ) 1/2 ], (1.63) which is cutoff-independent and expressed in terms of physically observable quantities. In the thermodynamic limit, it is also possible to compute the number of particles out of the condensate (depletion), which interestingly reads N N 0 N = 8 3π 1/2 (na3 ) 1/2. (1.64) It is evident from Eqs. (1.63)-(1.64) that the quantity na 3, which is called the gas parameter, plays the role of a control parameter of the Bogoliubov approximation, since if na 3 1, the results are a small perturbation of the ones obtained by assuming that all the particles are in the condensate. If the gas parameter is sufficiently small, the role of particles out of the condensate can be neglected. These results shows how in the Bogoliubov theory it is in fact a proper combination of weak interactions and low density that ensures the BEC to exist in the non ideal case. Using Eq. (1.63) at the lowest order, one can get the zero-temperature values of the pressure P = E V = gn2 2, (1.65) the compressibility n P = 1 gn, (1.66) which is finite as expected and goes to infinity when g 0, and the chemical potential µ = E = gn. (1.67) N It is remarkable that the uniform condensate is stable only if g is positive, since when g < 0 the compressibility becomes negative and the system tends to collapse. The speed of sound c can be obtained using the hydrodynamic relation n/ P 1/mc 2, yielding gn c = m. (1.68) This quantity will emerge again in the analysis of the elementary excitations. 20
1.4 Low energy excitations and healing length In this section we discuss the main features and implications of the peculiar form (1.58) of the excitation spectrum in a weakly interacting Bose gas. This analysis will eventually lead to the identification of the healing length, which determines the length scale below which collective phenomena are physically more relevant than free-particle-like ones. Since the weakly interacting gas of Bose particles has been mapped onto a gas of noninteracting quasiparticles, we are able to analyze the thermodynamics of the system and find how temperature affect the depletion of the condensate. The dispersion law (1.58) of Bogoliubov excitations appears as the square root of the sum of two terms ɛ(p) = ( p 2 2m ) 2 + gn m p2, (1.69) one of which corresponds to the dispersion law of a free particle squared. For sufficiently small momenta, this quartic term is negligible, and the dispersion relation is approximately linear: gn ɛ(p) p = cp. (1.70) m In the second equality, we used the hydrodynamic sound velocity c defined in (1.68). For small momenta, the quantized Bogoliubov excitations behave like sound waves (phonons) propagating through the condensate with velocity c. These excitation can be regarded as the Goldstone modes associated with the U(1) symmetry breaking after condensation [18]. The existence of a gapless phononic spectrum in a Bose gas with repulsive interactions is the result of a general theorem obtained by Hugenholtz and Pines [24]. In the phonon regime, the coefficients (u p, v p ) mixing the free particle operators tend to be equal in magnitude. In the opposite limit of large momenta, the dispersion law approaches that of a free particle ɛ(p) p2 p2 + gn = + µ, (1.71) 2m 2m with the result (1.67) being used in the second equality. The law (1.71) appears as the spectrum of a free particle, with an additional energy due to the motion of the particle in the condensate background. The free-particle features of high momentum excitations are proved also by the fact that the 21
coefficients in Eq. (1.53) satisfy v p u p 1. The transition between the phonon and the particle regimes takes place for values of momentum corresponding to the length scale p 2mgn, (1.72) ξ = 2mgn, (1.73) which is called the healing length [7]. On length scales smaller than ξ, namely for momenta larger than /ξ, the independent-particle features dominate the physics of the system. On the other hand, on length scales greater than ξ collective properties related to the interactions emerge. This is evident in the phonon character of low-momentum Bogoliubov excitations, but the role of the healing length is indeed general, as we shall see in dealing with the mean field treatment of BEC in the next chapter. The thermodynamics of a weakly interacting Bose gas is easily obtained, since the system has been reduced through (1.59) to a gas of independent excitations. It is sufficient to set the chemical potential to zero and use the dispersion relation (1.69) to compute the average occupation numbers of quasiparticle levels at temperature k B T = 1/β [7] N p ˆb pˆb p = 1 exp(βɛ(p)) 1. (1.74) The free energy of the system is obtained by adding the quasiparticle contribution to the ground state energy dp A(T, V ) = E + k B T V (2π ) ln [ 1 e βɛ(p)], (1.75) 3 which, at low temperatures, reads A(T, V ) = E π2 90 3 c 3 V (k BT ) 4 (1.76) and reproduces the typical T 3 dependence of the phonon specific heat [16]. It is important to analyze how the depletion of the condensate is affected by interactions. This can be done by observing that the particle occupation numbers can be obtained from the quasiparticle ones through ( ) n p â pâ p = u 2 p ˆb pˆb p + vp 2 1 + ˆb pˆb p, (1.77) 22
that can be used to compute N 0 = N V [ dp u 2 v 2 p + v 2 ] p (2π ) 3 p +. (1.78) exp(βɛ(p)) 1 Computation of (1.78) for T 0 yields the result (1.64) previously discussed, while a low-temperature expansion for k B T mc 2 enables one to get the temperature dependence of the condensate fraction n 0 = N 0 /N, namely n 0 (T ) n 0 (T = 0) n 0 (T = 0) = m 12 3 cn (k BT ) 2. (1.79) Thus, a purely thermal depletion scales quadratically in the temperature. This result enable us to discuss the range of validity of Bogoliubov approximation, which relies on the assumption N N 0 that almost all the particles are in the condensate, implying that the quantity (1.79) must be small. This is certainly true for temperatures up to the order of the chemical potential for which δn 0 /n 0 (na 3 ) 1/2 1. At higher temperatures, the elementary excitations contributing to the thermal depletion are mainly free-particlelike. This implies that the temperature dependence of n 0 approaches that of an ideal gas. On the other hand, in a dilute gas the condition (na 3 ) 1 is equivalent to mc 2 k B T c. Thus, provided the condition T T c is satisfied, both the phonon regime mc 2 k B T and the free-particle regime mc 2 k B T are compatible with the use of Bogoliubov approach [7]. 23
24
Chapter 2 Gross-Pitaevskii equation In this second chapter we will introduce a fundamental tool in investigating the zero temperature properties of a generally nonuniform condensate. In the weak interaction regime, the relevant aspects concerning statics and dynamics of a condensate can be deduced by a mean-field equation, called the Gross-Pitaevskii equation (GP), which is a kind of nonlinear Schrödinger equation for the condensate wave function. We will first obtain GP under the hypothesis that all the particles are in the condensate. Then we will use the equation to characterize the physical significance and the dynamics of the modulus and phase of the condensate wave function. The stationary form of GP will be investigated and the local density approximation (Thomas- Fermi limit) will be discussed, with a focus on the ground state of a trapped condensate. We will generally deal with three-dimensional systems, but in the end we will analyze how is it possible to obtain a quasi one-dimensional condensate by tight transverse confinement. 2.1 Field equations for the condensate In order to study nonuniform condensates at zero temperature, it is necessary to extend the Bogoliubov formalism, introduced in the previous chapter, to include the effects of an external potential. For the T = 0 study of the system, we are going to use Bogoliubov approximation (1.38). Moreover, it will be assumed that the gas is sufficiently dilute (na 3 1), in order to neglect all the effects due to the particles out of the condensate and approximate the field operator ˆΨ with its mean value Ψ 0. This replacement is analogous to the transition from quantum to classical electrodynamics [7]. In the previous chapter we introduced the Hamiltonian in Eq. (1.44) to describe an interacting condensate in a uniform background. Here we are 25
interested in a more general form, including the effect of an external potential V (r), which reads ) Ĥ = dr ˆΨ (r) ( 2 2m 2 + V (r) ˆΨ(r) + 1 drdr 2 ˆΨ (r) ˆΨ (r )U(r r ) ˆΨ(r ) ˆΨ (r). (2.1) The dynamical field equation for ˆΨ(r, t) is obtained by the Heisenberg prescription [21] i t ˆΨ(r, [ t) = ˆΨ(r, t), Ĥ]. (2.2) The commutator in the right hand side is easily evaluated, yielding the nonlinear equation i ] t ˆΨ(r, t) = [ 2 2m 2 + V (r) + dr ˆΨ (r )U(r r ) ˆΨ(r ) ˆΨ(r, t). (2.3) By assuming that almost all particles are in the condensate, and that the number of particles is so large that fluctuations can be neglected, it is possible to obtain a differential equation for the function Ψ 0 (r, t). Moreover, since we are interested in the T = 0 physics and we expect the particles to scatter at low energies, we are allowed to replace the potential U(r) with a pseudopotential Ũ(r) = gδ(r) = 4π 2 a δ(r), (2.4) m with a the s-wave scattering length. All these manipulations yield the Gross- Pitaevskii equation [25, 26] i ) t Ψ 0(r, t) = ( 2 2m 2 + V (r) + g Ψ 0 (r, t) 2 Ψ 0 (r, t). (2.5) The GP appears to be very similar to the Schrödinger equation for a single quantum particle. The difference lies in the presence of an additional potential g Ψ 0 (r, t) 2, generated by the mean field of the condensate. The presence of such nonlinear term is related to the fact that Ψ 0 (r, t) is not a single-particle wave function with a probabilistic interpretation, but rather a collective wave function, representing at the same time a large quantity of particles which are mutually interacting. The nonlinear term in Eq. (2.5) is analogous to the cubic term appearing in the mean-field analysis of the Ising model [27]. It should be remarked that Eq. (2.5) is valid only on length scales which are much larger than the scattering length. The assumption leading to 26
GP, especially replacing U with a point-like potential, are no longer justified on a microscopic scale. Since, as a consequence of the gas diluteness, one can ignore as a first approximation correlations between particles, a symmetric N-body wave function can be built starting from a solution Ψ 0 (r, t) of Eq. (2.5) [7]: Φ(r 1,..., r N ; t) = N i=1 1 N Ψ 0 (r i, t). (2.6) This wave function is normalized to one, since applying Bogoliubov prescription and neglecting particles out of the condensate yield dr Ψ 0 (r, t) 2 = N, (2.7) with N being the total number of particles in the system. Due to this normalization condition, the square modulus of the condensate wave function can be regarded as the local density of particles ρ(r, t) Ψ 0 (r, t) 2. (2.8) Starting from GP, it is possible to obtain a dynamic equation for the local density and give physical meaning to the other independent parameter of the complex condensate wave function, namely its phase: Ψ 0 (r, t) = ρ(r, t)e is(r,t). (2.9) By multiplying Eq. (2.5) by Ψ 0 and subtracting the complex conjugate equation, one observes that the density obeys the continuity equation ρ t + j = 0, (2.10) where the current density j is related to the phase of the wave function by j(r, t) = i 2m (Ψ 0 Ψ 0 Ψ 0 Ψ 0) = ρ(r, t) S(r, t). (2.11) m The continuity equation (2.10), together with Eq. (2.11), shows that GP describes the motion of a fluid with density ρ and current density j ρv, whose velocity field is given by v(r, t) = S(r, t). (2.12) m 27
Thus, the phase itself has no physical meaning, but its spatial derivatives yield the velocity field of the condensate. The quantity (2.12) is called the superfluid velocity of the condensate [7, 18], since if the phase is not singular, the velocity field is irrotational v = 0, (2.13) which is a common feature of superfluidity. It must be remarked that relations (2.8) and (2.11), despite being formally equivalent to those obtained for a single quantum particle following the Schrödinger equation, are very different in their interpretation, since the quantities ρ and j are no longer a probability density and a probability density current but the actual density of particles and density current of particles of the condensate. Some relevant conservation laws are associate with the GP [7]. The most immediate of them is conservation of the number of particles dn/dt = 0, which follows from the continuity equation (2.10) if the contour integral of the current vanishes, which is the case for confined systems or cyclic boundaries. Another important law is conservation of the energy functional [ E[Ψ 0, Ψ 2 0] = dr 2m Ψ 0 Ψ 0 + V Ψ 0 2 + g ] 2 Ψ 0 4, (2.14) which can be also obtained by applying Bogoliubov prescription to the initial system Hamiltonian. Finally, one can get a continuity equation for the momentum density with Π ik 2 4m 2 m j i t + Π ik x k = ρ V x i, (2.15) [ ] Ψ0 Ψ 0 2 Ψ 0 Ψ 0 + c.c. + gρ2 x i x k x i x k 2 δ ik, (2.16) which, in the absence of an external potential and under suitable boundary conditions, leads to the conservation of the total momentum of the condensate P = drmj(r, t). 2.2 The stationary Gross-Pitaevskii solutions The Gross-Pitaevskii equation (2.5) is a nonlinear Schrödinger equation with a cubic term. Like its linear counterpart, it preserves normalization and total energy of the condensate. The Schrödinger equation admits stationary solutions of the form ψ(r, t) = ψ(r) exp( iεt/ ), where ψ(r) ψ(r, t = 0) is an eigenstate of the single-particle Hamiltonian operator and ε is the 28
corresponding eigenvalue. Let us look by analogy for stationary GP solutions in the form [7] ( Ψ 0 (r, t) = Ψ 0 (r) exp iµt ), (2.17) with the physical meaning of the energy µ still to be clarified. This kind of solutions is characterized by the feature that the density ρ(r, t) = Ψ 0 (r) 2 is time-independent, as well as the velocity field, which depends only on the phase of Ψ(r). By inserting the form (2.17) in Eq. (2.5), we obtain a time-independent equation for Ψ(r): ) ( 2 2m 2 + V (r) µ + g Ψ 0 (r) 2 Ψ 0 (r) = 0. (2.18) It is easy to verify that the solutions of (2.18) are stationary points for the functional E[Ψ 0, Ψ 0] = E[Ψ 0, Ψ 0] µn [Ψ 0, Ψ 0], (2.19) where E is the energy functional, defined in (2.14), and N [Ψ 0, Ψ 0] = dr Ψ 0 (r) 2 (2.20) counts the number of particles corresponding to a given wave function Ψ 0. Finding stationary points of the energy under the constraint that the number of particles be fixed to some N is equivalent to finding stationary points of the functional (2.19) which are function of µ, and then adjusting µ as a Lagrange multiplier in order to get the required value of N. This means that µ physically corresponds to the chemical potential µ = E N, (2.21) while E is the grand canonical energy functional. Since the stationary GP (2.18) is nonlinear, solutions Ψ a and Ψ b with the same N but corresponding to different chemical potential are not necessarily orthogonal. However, if one builds the corresponding time-independent N-body wave functions as in Eq. (2.6), they turn out to be orthogonal in the large N limit dr 1... r N Φ aφ b = ( 1 N drψ aψ b ) N, (2.22) since drψ aψ b is smaller than N, unless Ψ a = Ψ b. This result confirms that the consistency of the mean field picture is ensured only in the large N limit 29
[7]. A rigorous proof that (2.18) correctly describes the ground state of a dilute Bose gas with repulsive interactions has been given by Lieb et al. in Ref. [28]. In order to observe how interactions affect GP solutions when interactions are present, let us consider stationary solutions in a cubic box of side L, with cyclic boundary conditions and V 0. In this case, the simplest solution for a given chemical potential µ is µ Ψ 0 (r) = g, (2.23) with constant density n = N/V and µ = gn (compare with (1.67)). Let us now shift to Dirichlet boundary condition in the x direction, Ψ 0 (0, y, z) = Ψ 0 (L, y, z) = 0. (2.24) Close to the boundary at x = 0, the stationary GP admits the solution ( ) µ x Ψ 0 (x, y, z) g tanh, (2.25) 2ξ with ξ being the healing length ξ = 2mgn, defined in Eq. (1.73). The density corresponding to the solution (2.25) saturates to the value n = µ/g at a distance of order ξ from x = 0. A specular solution is obtained in a neighborhood of x = L. Thus, if L ξ, the condensate appears almost uniform on a macroscopic scale, and deviates from uniformity only at distances of order ξ from the rigid boundaries. This does not happen in ideal gases (ξ ), where the shift in boundary conditions affects the macroscopic shape of the condensate. Once again, the healing length fixes the length scale above which collective physics dominates over single-particle physics. The healing length is recurrent in all the physical cases where a microscopic impurity is present in a macroscopically uniform condensate. This happens in the case of vortex-line solution [29, 30, 25, 26] Ψ 0 (r, φ, z) = ρ(r)e isφ, s = 1, 2, 3,... (2.26) where φ is the azimuthal angle and r = x 2 + y 2. The condensate is repelled from the vortex line r = 0, where the phase of (2.26) is singular, and the density becomes uniform at a distance of order ξ from the line itself. Another 30