Stability in Bose-Einstein Condensates Author: Raquel Esteban Puyuelo. Facultat de Física, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain.∗ (Dated: June 16th 2014) Abstract: In this work the validity of an analytic approximation expression [1] for the critical number of atoms that a Bose-Einstein Condensate can hold before collapsing when the interaction between them is attractive is checked. In order to do this the system is treated in a more precise way by numerically solving the Gross-Pitaevskii equation. One finds that the variational approach underestimates the number of atoms that can fit in a Bose-Einstein Condensate before it collapses. I. INTRODUCTION can not be neglected. On the other side, there are other atoms such as alkali that can be in this state of matter. Bose-Einstein Condensation is among the most fas- cinating phenomena in nature as its surprising proper- ties are direct consequences of quantum mechanics. In A. The stability problem essence, it is a macroscopic quantum phenomena. Unlike the classical ideal gas or the Fermi-Dirac gas, the Bose- All BEC are essentially metastable gases that lack of Einstein ideal gas has a thermodynamic phase transition cohesion and therefore it is compulsory to use a trap to which is driven by the particle statistics and not by their confine them. Most of the times, the confining traps interaction. A Bose-Einstein Condensate (BEC from now are well approximated by harmonic potentials of frequen- on) is a state of matter of a dilute gas of cold atoms cies !x, !y, !z. Playing with these trap frequencies dif- obeying Bose-Einstein statistics below a certain critical ferent shapes of the condensate can be achieved. For temperature (T ), which is often called condensation tem- c !x = !y = !z the cloud will be spherical, while axially perature in analogy with the liquid-gas transition. This symmetric geometries such as the so called cigar shape transition occurs when the de Broglie wavelength of the and disk/pancake shape can also be created both numer- characteristic thermal motions becomes comparable to ically and experimentally. A characteristic length scale the mean interparticle separation. When this condition 1=2 for the systems can be defined: aho = [~=(m!ho)] , is attained, a macroscopic fraction of the bosons occupy 1=3 where !ho = (!x!y!z) is the geometric mean of the the lowest single particle quantum state, even if the tem- harmonic frequencies, m is the mass of the atoms in the perature is high enough to populate other states. gas and ~ is the reduced Plank constant. This omnipresent phenomenon plays remarkable roles Interactions between atoms can be either attractive or in atomic, nuclear, elementary particle and condensed repulsive. While a trapped repulsive interacting BEC matter physics, as well as in astrophysics [2]. The study will always be stable, it is different if the interactions are of BEC in weakly interacting systems has at its heart the attractive because that can imply instability to collapse possibility of revealing new macroscopic quantum phe- if the number of atoms in the condensate is large enough. nomena that can be understood from first principles, and That shows that attractive interacting condensates can may also help advance our comprehension of superfluid- exist, but only up to a critical number of atoms Ncr. ity and superconductivity. Thus, there are stability conditions for a small number BEC was predicted by Bose (1924) and Einstein (1924, of atoms such as lithium-7 [5] in trapped condensates. In 1925) but eluded direct and unquestioning observation these cases, the zero-point kinetic energy provided by the until 1995 as the experimental techniques for trapping trap can balance the increase of the center density due and cooling atoms in magnetic and laser traps had been to the attraction of the atoms for each other. It should improved. The time-of-flight measures carried on these be noted that cigar and pancake geometries are obviously experiments on gases of rubidium [3] and sodium [4] less stable than the spherical one if atoms interact attrac- showed a clear signature of the BEC: when the confining tively because less atoms can fit in one of the directions trap was switched off and the atoms were left to expand, of the trap and the critical number Ncr is thus reached a sharp peak in the velocity distribution below a certain sooner. Stability conditions in this last two geometries critical temperature appeared. are analytically more difficult to consider, only the spher- Not all bosons can form condensates. For example, He- ical geometry will be studied here because the aim of this lium, which is among the most famous bosons in Physics work is to check the approximate expression given in [1], can not form BEC when it is in its ground state. This is which treats only spherical symmetries. because it is liquid at T =0 K and therefore correlations There are different experimental ways in which a BEC can be produced, but they all share two common features: cooling down to the temperatures (between µK and nK) at which the condensates take place and trapping [6]. ∗Electronic address: [email protected] These techniques are laser cooling and magneto-optical Stability in Bose-Einstein Condensates Raquel Esteban Puyuelo trapping, which combines Doppler cooling and magnetic equation: trapping. This laser based-methods were developed in ~ 2 the 1980s and were the key to the first observations of − r '(~r) + Vext'(~r) + (N − 1) × this phenomenon. 2m Z dr~0V (j~r − r~0j)j'(~r)j2 '(~r) = µϕ(~r): (4) II. THEORY A. Gross-Pitaevskii equation deduction As condensate atoms interact by means of binary colli- sions, at low temperatures only s-wave collisions are im- Although in the standard literature the Gross- portant. Assuming that the gas is dilute and weakly in- Pitaevskii equation is usually obtained in the framework teracting, the interaction between particles can be writ- of the second quantisation formalism, it is possible to ten in term of the scattering length using a zero-range motivate the derivation of the GPE in relationship to interaction potential such that concepts from statistical physics. 4π 2 As it was discussed above, a BEC is obtained from 0 0 ~ 0 V (j~r − r~ j) = gδ(~r − r~ ) = asδ(~r − r~ ); (5) a collection of N bosons in their ground state at very m low temperatures. The general N-body Hamiltonian for where g is the coupling constant and as is the s-wave scat- identical particles of mass m can be expressed as follows tering length, which measures the interactions between the bosons and can be attractive (as < 0) or repulsive N 2 N N X 1 X X (as > 0). It is worth noting that this quantity is nowa- H^ = − ~ r2 + V (~r ) + V (j~r − ~r j); 2m i ext i 2 i j days tuneable by means of Feshbach ressonances [8]. i=1 i=1 i6=j For a large enough number of atoms, one can use the (1) approximation that N−1≈ N, which leads to the Gross- where the first term in the Laplacian is the kinetic en- Pitaevskii equation (GPE): ergy, Vext represents the external trapping potential and V represents the pair interaction between the N parti- 2 − ~ r2'(~r) + V (~r)'(~r) + gj'(~r)j2'(~r) = µϕ, (6) cles. 2m ext The ground state of the system corresponds to the min- where all the atoms are assumed to be in the condensate imum energy, which can be found approximately by min- at T =0 K. This equation considers that the system is imizing the expectation value of (1) using a totally sym- at zero temperature, which works fine for BEC as the metric trial wave function that is the product of single temperatures at which they take place are very low and particle states: close to the absolute zero. Furthermore, when the gas is very dilute and weakly interacting (na3 <<1), where n jΨ(1; 2; :::; N)i = j'(1)i ⊗ j'(2)i ⊗ ::: ⊗ j'(N)i : (2) s is the average density, considering that all the atoms are in the condensate fraction is very accurate. For example, This ansatz can be made as in the dilute situation one more than 99% of the alkali atoms are in the condensate can expect that correlations are negligible. Furthermore, at T = 0 K. the bosons interact weakly, so that a mean field approx- Solving this nonlinear equation for the wave function imation can be used, which means that the action felt must be done numerically but nonetheless it shows a form by a given particle due to the rest is substituted by the of BEC: the population of the lowest state in energy be- average effect of the interactions of this particle and the comes macroscopic under the right conditions. (N − 1) other particles. Following a similar deduction to the Hartree-Fock equations for fermions that can be found in standard B. Stability deduction (collapse for attractive textbooks [7], the GPE can be obtained. The total en- forces) ergy hΨj H jΨi will be minimized together with the con- 2 straint of normalization hΨ jΨi = N, so jΨj represents If the trapped gas has attractive interactions (as < the number of particles: 0) its central density will increase when new particles are added to the cloud. This happens in order to lower δ[hΨj H jΨi − λ hΨ jΨi] = 0: (3) the particle interaction density, which can be stabilized by the zero-point kinetic energy. However, if the cen- It can be proved that the Lagrange multiplier λ used in tral density increases too much, this kinetic energy is no the minimization can be identified with the chemical po- longer able to avoid the collapse of the gas.