(p.279) Appendix C Bose–Einstein condensation
(p.279) Appendix C Bose–Einstein condensation
In this appendix we review the basic theory of Bose–Einstein condensation in a trapped ultracold gas. More advanced information can be found in Dalfovo et al. (1999) and Pitaevskii and Stringari (2004), on which this appendix is largely based.
In Bose–Einstein condensation (BEC) experiments, ultracold gases of neutral atoms are confined in harmonic trapping potentials provided by either magnetic (see Sec. 3.1.1 ) or optical traps (see Sec. 5.2.2 ). Here we consider the theory of Bose–Einstein condensation for a gas of N noninteracting identical bosons in a three‐dimensional (3D) harmonic confining potential
where m is the atomic mass and ω i are the (angular) trap frequencies. The eigenvalues of the single‐particle Hamiltonian problem are
where ni , are the quantum numbers identifying the 3D harmonic oscillator state. For a system in thermal equilibrium, the occupancy of the levels is described by the Bose–Einstein statistics, according to which the mean occupation number is given in the grand‐canonical ensemble by (Huang, 1987 )
where β = (kB T) −1 is the inverse reduced temperature and μ is the chemical potential, that accounts for the conservation of the total number of particles
In the classical limit (large T or small N) the mean occupation of the levels is much less than unity, f(nx, ny, nz) « 1 ∀ni , the chemical potential is much smaller than the ground‐state energy, μ « ∈0,0,0, and the Bose statistics reduces to the classical Maxwell‐Boltzmann statistics. As the number of particles N is increased, μ increases as well in order to satisfy the normalization condition in eqn (C.4). However, there is an upper (p.280) limit on the growth of μ, which is imposed by the condition that the occupation number in eqn (C.3) must be positive for all the quantum states: μ 〈 ∈nx,ny,nz ∀ni . Therefore, at most the chemical potential can approach the energy of the lowest‐energy state: μ ⋍ ∈ 0,0,0. When this happens, the occupation of the excited states becomes saturated and, increasing N, the new particles “condense” in the ground state, the occupation of which becomes macroscopic according to eqn (C.3). Substituting μ = ∈0,0,0 in eqn (C.4) we obtain
where we have isolated the population of the ground state N 0, that produces a divergence in the sum. If the level spacing ħω i is much smaller than kB T, we can replace the sum with an integral (semiclassical approximation): 1
By carrying out the integration, one finds that the number of atoms in the condensed fraction N0 as a function of T is
where the BEC critical temperature TC is defined as
in which ζ(n) is the Riemann function and ωho = (ωxωyωz) 1/3 is the geometric average of the trapping frequencies. This derivation is strictly valid in the thermodynamic limit, that for a harmonically trapped gas corresponds to the limit N → ∞, with constant. In Fig. C.1 we plot the condensate fraction as a function of temperature as obtained from eqn (C.7).
At T = 0 all the atoms occupy the harmonic oscillator ground state and the condensate density is given by
corresponding to the squared modulus of the Gaussian wavefunction of the harmonic oscillator ground state normalized with ∫ d r nho (r) = N. We note that, for a harmonically trapped gas, Bose–Einstein condensation occurs with a sudden narrowing of the density distribution both in momentum and coordinate space. This is different from the textbook description of a bosonic gas confined in a box, in which condensation occurs only in momentum space, while in coordinate space the condensed gas remains delocalized and cannot be spatially distinguished from the non‐condensed component. (p.281)
In the previous section we have considered the problem of Bose–Einstein condensation for a noninteracting gas, the thermodynamic behaviour of which is completely governed by quantum statistics. However, real systems are always characterized by interactions among their constituent particles. In strongly interacting systems the presence of interactions can blur the quantum effects and lead to a significant quantum depletion of the condensate phase even at zero temperature. This is the case of superfluid 4He, in which only a small fraction of the liquid (typically around 10%) is Bose‐condensed. In the case of weakly interacting systems, as for BEC of dilute gases, quantum depletion can be generally neglected. However, the presence of interactions strongly modifies the properties of the system, leading to superfluidity and to a nonlinear behaviour of the de Broglie matter‐wave that enriches the multitude of observable phenomena.
In second quantization, the many‐body Hamiltonian operator describing a system of N bosons in an external potential Vext is given by
where ̂Ψ(r) (̂Ψ†(r)) is the boson field annihilation (creation) operator and Vint describes low‐energy binary s‐wave collisions, that in an ultracold dilute gas are the only relevant interaction processes (see Sec. 3.1.4 ). The field operator can be written as
(p.282) where Ψj (r) is the wavefunction of the single‐particle state j and ̂bj is the annihilation operator of a boson in state j. From a very general point of view, the field operator ̂Ψ(r) can be written in Heisenberg representation as
where Ψ(r, t) = (̂Ψ(r, t)〉 is the expectation value of the field operator and the fluctuations (δ̂Ψ(r, t) describe the quantum and thermal excitations.
Bose–Einstein condensation occurs when the single‐particle ground state becomes macroscopically occupied. Using a standard approach in quantum field theory, when the number of particles in one state is macroscopic (N 0 » 1), the creation and annihilation operators can be substituted with c‐numbers, thus recovering the limit of a classical field. With this assumption the expectation value Ψ(r, t) naturally results from the macroscopic occupancy of the ground state, while δ̂Ψ(r, t) describes the excitations. Following the Bogoliubov approximation (Dalfovo et al., 1999 ) we neglect the contribution of δ̂Ψ;(r, t), thus the Heisenberg equation of motion for ̂Ψ(r, t) becomes an equation for the classical field Ψ(r, t):
where the complex function Ψ(r, t) stands for the condensate wavefunction. If the mean interparticle distance n−1/3 is much larger than the range of the two‐body potential Vint (diluteness condition), the latter can be substituted with the effective pseudopotential gδ(r − r′), that is independent of the details of the two‐body interaction. The scattering between two atoms can thus be entirely described by the scalar parameter
where a is the so‐called scattering length, that is positive for repulsive interactions and negative for attractive interactions (see Sec. 3.1.4 ). With this assumption for the interaction potential, which is typically well satisfied, eqn (C.13) becomes
This equation is known as the Gross‐Pitaevskii equation, after the two scientists that independently derived it in the early 1960s (Gross, 1961 ; Gross, 1963 ; Pitaevskii, 1961 ). In this equation, well verified in a great number of experiments, the effect of interactions is described by a nonlinear term for the condensate order parameter Ψ. This term, describing the condensate self‐interaction, results from a mean‐field approximation, in which the condensate wavefunction Ψ locally probes the average collisional potential g |Ψ |2 produced by itself.
The presence of the interaction term in eqn (C.15) is important since it has an essential role in determining the superfluid properties of atomic Bose–Einstein condensates discussed in Secs. 3.2.2 and 3.2.3 : among these, their collective excitations, (p.283) the propagation of sound, the existence of a critical superflow velocity. The interaction term in eqn (C.15) is also responsible for a rich multitude of nonlinear phenomena. In the context of coherent matter‐wave optics it plays the same role as the third‐order susceptibility χ(3) in nonlinear optics (see Appendix B.3 ), producing effects such as four‐wave mixing and solitonic propagation (Rolston and Phillips, 2002 ). However, it may also constitute a limitation, in particular for experiments of atom interferometry requiring long‐lived single‐particle coherence (see Secs. 3.2.6 and 6.4.2 ).
The stationary solutions of eqn (C.15) can be calculated using the ansatz Ψ(r, t) = e −iμt/ħΨ (r), where μ is the condensate chemical potential. Substituting this expression in eqn (C.15) we obtain the time‐independent Gross‐Pitaevskii equation
Repulsive interactions. We first consider positive values of the scattering length, a 〉 0, corresponding to repulsive interactions between the particles. The ground‐state condensate wavefunction can be analytically determined by solving eqn (C.16) in the so‐called Thomas‐Fermi approximation, holding for Na/aho » 1 (where is the average harmonic oscillator length) (Dalfovo et al., 1999 ). In this regime of a large number of atoms the kinetic energy (−ħ2∀ 2/2m) is much smaller than the interaction energy ⌫g|ψ|2 〉 and therefore it can be neglected. With this assumption, well satisfied in most of the experiments, the differential equation (C.16) becomes an algebraic equation from which we obtain the condensate density
In the case of an axially symmetric harmonic trap with ωx = ωy = ω⊥ (this is the geometry adopted in most of the experiments), the explicit shape of the condensate density distribution is an inverted parabola
with height and widths connected to the chemical potential by
We note that the repulsive interaction term in eqn (C.16) has the effect of broadening the density distribution of the single‐particle ground state. In Fig. C.2 we plot the condensate density for N = 3 × 105, m = 1.44 × 10−25 kg (87Rb mass), ω z = 2π × 9 Hz, and ω⊥ = 2π × 90 Hz in the noninteracting case (dotted line) and for a repulsive interaction (solid line) with strength a = 5.7 nm (value for 87Rb). 2 The ratio between the peak of the Thomas‐Fermi density in eqn (C.19) and the peak of the noninteracting density in eqn (C.9) is
corresponding to 1% for the parameters of Fig. C.2 , where Na/aho ⋍ 103.
Attractive interactions. BEC experiments are usually performed with atoms with repulsive interactions, because in this case condensates are stable for an arbitrarily large number of atoms. 3 If interactions are attractive (negative scattering length, a 〈 0) (p.285) a different scenario emerges: the gas tends to occupy a smaller volume, increasing its density with respect to the noninteracting case, in such a way as to have a larger negative interaction energy.
For a very small number of atoms the zero‐point kinetic energy term can still stabilize the system and a trapped condensate can be produced in a metastable state (Bradley et al., 1997 a). Above a critical value NC , however, the attractive interactions between atoms cause a collapse of the condensate, that is eventually destroyed by collisional processes out of the mean‐field description, including inelastic collisional processes (Gerton et al., 2000 ; Roberts et al., 2001 ; Donley et al., 2001 ).
( 1 ) This assumption corresponds to the requirement of a large number of levels to be thermally occupied. From the result of the calculation, given in eqn (C.8), at the critical temperature ħω i/kB Tc ⋍ N −1/3 = 0.01 for N = 106, so the assumption is reasonably satisfied.
( 2 ) These numbers, holding for one of the experiments running at LENS (Florence), are representative of typical values for a 87Rb BEC.
( 3 ) This is true provided that the density is not too large to determine excessive three‐body losses.