- Properties of a Bose–Einstein Condensate. - By cooling down a collection of integer spin atoms to a temperature of less than one micro-Kelvin, one can observe the phenomenon of Bose–Einstein condensation. - This results in a situation where a large fraction of the atoms are in the same quantum state. - We study here the ground state of such an N particle system, hereafter called a condensate. - We will show that the nature of the system depends crucially on whether the two-body interactions between the atoms are attractive or repulsive.. - 19.1 Particle in a Harmonic Trap. - The Hamiltonian of the system is. - where ˆ r = (ˆ x, y, ˆ z) and ˆ ˆ p = (ˆ p x , p ˆ y , p ˆ z ) are respectively the position and momentum operators of the particle. - Recall the energy levels of this system, and its ground state wave function φ 0 ( r. - We wish to obtain an upper bound on this ground state energy by the variational method. - (19.1) The values of a relevant set of useful integrals are given below.. - By varying σ, find an upper bound on the ground state energy. - 194 19 Properties of a Bose–Einstein Condensate Formulas. - 19.2 Interactions Between Two Confined Particles. - We now consider two particles of equal masses m, both placed in the same harmonic potential. - We denote the position and momentum operators of the two particles by ˆ r 1 , ˆ r 2 and ˆ p 1 , ˆ p 2. - In the absence of interactions between the particles, the Hamiltonian of the system is. - (b) What is the ground state wave function Φ 0 ( r 1 , r 2. - (19.2) The quantity a, which is called the scattering length, is a characteristic of the atomic species under consideration. - (a) Using perturbation theory, calculate to first order in a the shift of the ground state energy of ˆ H caused by the interaction between the two atoms. - 19.3 Energy of a Bose–Einstein Condensate. - We now consider N particles confined in the same harmonic trap of angular frequency ω. - In order to find an (upper) estimate of the ground state energy of the system, we use the variational method with factorized trial wave functions of the type:. - Calculate the expectation values of the kinetic energy, of the potential energy and of the interaction energy, if the N particle system is in the state. - Cast the result in the form E(σ. - For a = 0, recall the ground state energy of ˆ H. - 19.4 Condensates with Repulsive Interactions. - 196 19 Properties of a Bose–Einstein Condensate. - Draw qualitatively the value of ˜ E as a function of ˜ σ. - Discuss the variation with η of the position of its minimum ˜ E min. - Show that the contribution of the kinetic energy to ˜ E is negligible. - In that approximation, calculate an approximate value of ˜ E min. - In this variational calculation, how does the energy of the conden- sate vary with the number of atoms N? Compare the prediction with the experimental result shown in Fig. - Figure 19.1 has been obtained with a sodium condensate (mass m kg) in a harmonic trap of frequency ω/(2π. - (b) Above which value of N does the approximation η 1 hold?. - (c) Within the previous model, calculate the value of the sodium atom scat- tering length that can be inferred from the data of Fig. - Is it possible a priori to improve the accuracy of the variational method?. - Energy per atom E/N in a sodium condensate, as a function of the number of atoms N in the condensate. - 19.5 Condensates with Attractive Interactions. - Comment on the approximation (19.2) in the region σ → 0.. - Show that there exists a critical value η c of | η | above which ˜ E no longer has a local minimum for a value ˜ σ = 0. - In an experiment performed with lithium atoms (m kg), it has been noticed that the number of atoms in the condensate never exceeds 1200 for a trap of frequency ω/(2π. - 19.6 Solutions. - Section 19.1: Particle in a Harmonic Trap. - We therefore use a basis of eigenfunctions of ˆ H of the form φ(x, y, z. - The ground state wave function, of energy (3/2)¯ hω, corresponds to n x = n y = n z = 0, i.e.. - In order to obtain an upper bound for the ground-state energy of ˆ H , we must calculate E(σ. - Using the formulas given in the text, one obtains. - This is due to the fact that the set of trial wave functions contains the ground state wave function of ˆ H.. - 198 19 Properties of a Bose–Einstein Condensate. - Section 19.2: Interactions between Two Confined Particles. - A basis of eigenfunctions of ˆ H is formed by considering products of eigenfunctions of ˆ H 1 (functions of the variable r 1 ) and eigenfunctions of ˆ H 2 (functions of the variable r 2. - (b) The ground state of ˆ H is:. - (a) Since the ground state of ˆ H is non-degenerate, its shift to first order in a can be written as. - 0), there is an increase in the energy of the system. - Conversely, in the case of an attractive interaction (a <. - 0), the ground state energy is lowered.. - the scattering length must be small compared to the spreading of the ground state wave function.. - Section 19.3: Energy of a Bose–Einstein Condensate 19.3.1. - Using the formulas provided in the text, one obtains:. - With the change of variables introduced in the text, one finds E(σ. - The ground state of the system is the product of the N functions φ 0 ( r i ) and the ground state energy is E = (3/2)N¯ hω.. - Section 19.4: Condensates with Repulsive Interactions. - Figure 19.2 gives the variation of ˜ E(˜ σ) as a function of ˜ σ for increas- ing values of η. - The value of the function for a given value of ˜ σ increases as η increases. - Since the interactions are repulsive, the size of the system is larger than in the absence of interactions, and the corresponding energy is also increased.. - In fact it is always smaller than one of the two other contributions to ˜ E:. - 200 19 Properties of a Bose–Einstein Condensate. - For a number of atoms N 1, the energy of the system as calculated by the variational method is. - 19.3 we have plotted a fit of the data with this law. - Fit of the experimental data with an N 2/5 law. - (b) Consider the value a = 3.4 nm given in the text. - This is due to the fact that the result (19.3), E/(N¯ hω) 1.142 (N a/a 0 ) 2/5 , obtained in a variational calculation using simple Gaussian trial functions, does not yield a sufficiently accurate value of the ground state energy. - With more appropriate trial wave functions, one can obtain, in the mean field ap- proximation and in the limit η 1: E gs /(N ¯ hω) 1.055 (N a/a 0 ) 2/5 . - The fit to the data is then in agreement with the experimental value of the scattering length.. - Section 19.5: Condensates with Attractive Interactions. - For such small sizes, approximation (19.2) for a “short range”. - This happens for a critical value of η determined by the two conditions:. - This leads to the system. - 202 19 Properties of a Bose–Einstein Condensate. - one can hope to obtain a metastable condensate, whose size will be of the order of the minimum found in this variational approach. - On the other hand, if one starts with a value of | η | which is too large, for instance by trying to gather too many atoms, the condensate will collapse, and molecules will form.. - 19.7 Comments. - The first Bose–Einstein condensate of a dilute atomic gas was observed in Boulder (USA) in 1995 (M.H. - The measurement of the energy E/N is done by suddenly switching off the confining potential and by measuring the resulting ballistic expansion.. - The motion of the atoms in this expansion essentially originates from the conversion of the potential energy of the atoms in the trap into kinetic energy.. - Wieman for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates.
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