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Bose-Einstein Condensation of Ultracold Atoms a Contribution to the Quantum Mechanics Seminar 2019/20 Supervised by Prof

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Bose-Einstein Condensation of Ultracold Atoms a Contribution to the Quantum Mechanics Seminar 2019/20 Supervised by Prof Bose-Einstein condensation of ultracold atoms A contribution to the quantum mechanics seminar 2019/20 supervised by Prof. Wolschin Omar Al-Bazaz January 30, 2020 Abstract The Bose-Einstein condensation represents a macroscopic phenomenon of quantum mechanics which was already theoretically predicted by Albert Einstein in 1924 but could only be experimentally confirmed more than 70 years later. It is one of the prominent macroscopic quantum effects that occur at very low temperatures, similar to superconductivity and superfluidity. Below the critical temperature of Bose-Einstein condensation, the bosons of a bose gas occupy the energetic ground state. In a potential the condensate is distributed according to a single ground state wave function, normal- ized by the number of bosons. The wave function is ether derived from the Schr¨odinger equation for non-interacting or from the Gross-Pitaevskii equation for weak interacting bosons. In the present summarizing work, the history and experimental findings are predomi- nantly based on [1], [2], [6] and [7]. Formalisms and derivations are mainly taken from [10], although [9] and [12] were also used. 1 Introduction Essential for the discovery of the Bose-Einstein condensation was the work of the Indian physicist Satyendranath Bose. He was dissatisfied with the original derivation of Planck's radiation formula, because it contains a factor that at that time could only be explained by classical physics. In 1924 [2] he derived the radiation formula using a completely new approach. His treatment of photons as indistinguishable particles laid the foundation for the Bose-Einstein statistics. Only Albert Einstein, to whom Bose sent his work, recognized the importance of Bose's new approach. In the same year he used Bose's approach in his own work [6] to derive a new quantum theory of the ideal monoatomic gas. In this work, the condensation effect at very low temperatures is also mentioned for the first time. A Bose-Einstein condensate (BEC) is a collection of bosonic particles where a macroscopic number of them occupy a specific single-particle quantum state. It is a prominent ex- ample of a macroscopic quantum phenomenon. In general, a necessary condition for the occurrence of macroscopic quantum phenomena is that the thermal de Broglie wavelength p 2 λth = 2π~ =mkBT , where kB is the Boltzmann constant, ~ the Planck constant, m the particle mass and T the temperature, becomes comparable with the mean free path. Intro- 3 ducing the phase space density nλth, where n is the particle density, we have 2 3=2 3 2π~ nλth = n & 1 (1) mkBT 1 By isolating the temperature 2 2π~ 2=3 T . n (2) mkB we see that macroscopic quantum phenomena are inherently low-temperature phenomena. However, for light particles with high density, the temperature in (2) can be quite high. The atomic vapour quantum gases used in the laboratories since 1995 are very diluted, which means that the interatomic interactions are weak and that the sample is optically transpar- ent enough to allow the cloud to be imaged with laser light. The price of dilution is a much lower transition temperature, but this obstacle can be overcome because diluted clouds of some elements can be cooled with laser light. The first BEC of this kind contained about 2×104 87Rb atoms and was prepared in 1995 in the group of Carl Wieman and Eric Cornell [1] and shortly afterwards condensates with up to 5 × 105 23Na atoms were prepared in the group of Wolfgang Ketterle [5]. The BECs of diluted atomic gases consist almost exclusively of alkali metal isotopes 7Li[3];23 Na[5];39 K[11];41 K[8];85 Rb[4];87 Rb[1] and 233Cs[13], since well-established laser cooling and magnetic trapping techniques can be applied to the sta- ble isotopes. Additional to laser cooling evaporative cooling is used to reach even lower temperatures, like 170 nK in the example of [1]. 2 Bose-Einstein statistics Before the behaviour of bosons at ultra-low temperatures can be investigated, we first need to establish the foundations of Bose-Einstein statistics. This is done for the ideal bose gas, which consists of non interacting particles with integer spin. For derivation we classify the state of the bose gas with the occupation number representation: 1 X 1 n0 1 n1 jN; n0; n1;:::i = p Q P φ0 ⊗ · · · ⊗ jφ0 i ⊗ φ1 ⊗ · · · ⊗ jφ1 i ::: (3) N! i ni! P where ni is the number of particles in the i-th energy state of the single-particle Hamiltonian P operator and N the total number of bosons i ni in the gas. The jφii are the i-th eigenstates, which are tensored with each other and symmetrized by the permutations P to satisfy the symmetry property of bosons. The normalization factor results from the orthogonality of the states: 0 0 Y N ; n ;::: jN; n ;::: = δ 0 δ 0 (4) 0 0 N ;N ni;ni i Q For each of the N! summations in jN; n0;:::i there are exactly i ni! identical summations 0 0 in hN ; n0;:::j, whereby the interchangeability of identical energy states was used. In this representation, the particle number operator of the i-th energy eigenvaluen ^i is defined as follows: n^i jN; n0; : : : ; ni;::: i = ni jN; n0; : : : ; ni;::: i (5) The particle number operator N^ and the Hamilton operator H^ can then be expressed by the i-th particle number operators. X N^ = n^i (6) i X H^ = in^i (7) i To determine the distribution of the bosons on the energy levels, the grand canonical en- semble is used, since in this ensemble the summation over the total particle numbers is 2 P observed and thus the restriction N = i ni does not cause any problems when deter- mining the partition function Ξ. In the grand canonical ensemble the partition function is defined as the trace of the exponential operator exp(−β(H^ − µN^)), where µ is the chemical potential and β := 1=kBT . We take advantage of the fact that the states in the occupation number representation are eigenstates of this operator. Ξ(T; V; µ) = Tr(exp(−β(H^ − µN^))) 1 X X D E = N; n0; n1;::: j exp(−β(H^ − µN^))jN; n0; n1;::: N=0 ni 1 1 X X Y = ··· exp (−βni (i − µ)) (8) n0=0 n1=0 i 1 ! Y X = exp (−βni (i − µ)) i ni=0 Y 1 = 1 − exp (−β ( − µ)) i i The last step of this calculation is only possible if the chemical potential is really smaller than all energy eigenvalues, otherwise the geometric series would not converge. From this follows directly that the chemical potential is in particular smaller than the ground state. With this expression, the formalism of the grand canonical ensemble can be used to determine expected values for occupation numbers ni, N and the internal energy U. 1 @ 1 ni(T; V; µ) = − ln Ξ(T; V; µ) = (9) β @i exp (β (i − µ)) − 1 X X 1 N(T; V; µ) = n = (10) i exp (β ( − µ)) − 1 i i i @ X i U(T; V; µ) = − ln Ξ(T; V; µ) + µN = (11) @β exp (β ( − µ)) − 1 i i 3 Bose-Einstein condensation In the following we will look at the transition of the bose gas into the Bose-Einstein con- densation and determine a general term for the critical temperature at which condensation takes place. For this we consider a model of an ideal bose gas in a box with the volume V = L3, where L is the edge length of the box. In this case the single-particle Hamiltonian operator given by the free Hamiltonian operator H^ =p ^2=2m. It is important to mention that we do not consider here a box, which is limited by an infinitely high potential well, but rather we restrict the wave function with the help of a periodicity condition. '(x + L; y; z) = '(x; y; z) '(x; y + L; z) = '(x; y; z) (12) '(x; y; z + L) = '(x; y; z) This provides an even distribution of the bose gas within the box and a momentum quan- 3 tization according to p = 2π~n=L for n 2 Z . In this model, an expression for the particle number can be determined by evaluating the sum of the thermally excited bosons. For that reason we go from the explicit summation in (10) to a integration over p. To make 3 the transition from summation to integration, the particle number of the bosons in the i-th energy state is only allowed to differ a little of those in the (i+1)-th energy state. To ensure this, it is sufficient to require that the thermal energy of the system is much larger than the energy difference between the first excited state and the ground state. In the case of the box in which the energy of the ground state is zero, the following must therefore apply h2 k T − = (13) B 1 0 2mV 2=3 Since we will see later that Bose-Einstein condensation already occurs at temperatures far above this so-called microscopic criterion, this demand can be made without further ado. That being said we evaluate N by splitting it into the number of bosons n0 already in the ground state and the number of thermally exited bosons, i.e. p 6= 0. X 1 N = n0 + p2 p6=0 exp β 2m − µ − 1 V Z 1 p2 = n + 4π dp 0 3 2 h 0 exp β p − µ − 1 2m (14) r p mk T 2 Z 1 x B p = n0 + V 2 dx −1 2π~ π 0 z exp(x) − 1 V = n0 + 3 g3=2(z) λth The V in the second term is a result of the spacial integration over the whole box and h3 simply a normalization factor.
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