Bose-Einstein condensation of ultracold atoms A contribution to the 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 , similar to superconductivity and superfluidity. Below the critical of Bose-Einstein condensation, the bosons of a occupy the energetic . 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 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 , ~ the , 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 |N; n0, n1,...i = p Q P φ0 ⊗ · · · ⊗ |φ0 i ⊗ φ1 ⊗ · · · ⊗ |φ1 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 |φii 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 ,... |N; n ,... = δ 0 δ 0 (4) 0 0 N ,N ni,ni i Q For each of the N! summations in |N; n0,...i there are exactly i ni! identical summations 0 0 in hN ; n0,...|, whereby the interchangeability of identical energy states was used. In this representation, the particle number operator of the i-th energy eigenvaluen ˆi is defined as follows: nˆi |N; n0, . . . , ni,... i = ni |N; n0, . . . , ni,... i (5) The particle number operator Nˆ and the Hamilton operator Hˆ can then be expressed by the i-th particle number operators. X Nˆ = nˆi (6) i X Hˆ = inˆi (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(Hˆ − µNˆ)), where µ is the 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(Hˆ − µNˆ))) ∞ X X D E = N; n0, n1,... | exp(−β(Hˆ − µNˆ))|N; n0, n1,... N=0 ni ∞ ∞ X X Y = ··· exp (−βni (i − µ)) (8) n0=0 n1=0 i ∞ ! 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 Hˆ =p ˆ2/2m. 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.

ϕ(x + L, y, z) = ϕ(x, y, z) ϕ(x, y + L, z) = ϕ(x, y, z) (12) ϕ(x, y, z + L) = ϕ(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 ∈ 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 ∞ p2 = n + 4π dp 0 3   2  h 0 exp β p − µ − 1 2m (14) r √ mk T 2 Z ∞ x B √ = 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. Further more we used the isotropy of the energy in 2 second term. For the third term the substitution p = 2mkBT x with its differential dp = 1 p 2 2mkBT/xdx is used. In the last term the fugacity z := exp (βµ) and the gp (z) are introduced. The polylogarithm is a generalization of the logarithm and is defined as follows: ∞ X zk 1 Z ∞ xp−1 g (z) := = dx , p ∈ , |z| < 1 (15) p kp Γ(p) z−1 exp(x) − 1 R>0 k=1 0 The polylogarithm is defined by the series, whereas the integral representation can be used for p and z in the given value ranges. The value ranges specified here are not the definition range of the polylogarithm, but those in which equality is valid with the integral representation. Since the chemical potential for the bosons is strictly less than zero, it follows that the fugacity is strictly less than one, as demanded for the polylogarithm. In addition we introduce the gamma function in (15): Z ∞ p−1 Γ(p) = dtt exp(−t) , p ∈ R>0 (16) 0 We note a relation between the Riemann zeta function and the polylogarithm, which is given by ζ(p) = gp(1). With the expression found here for the particle number, i.e. (14), the transition of the bose gase into the condensation phase can be investigated. For this purpose, the behaviour of the chemical potential with fixed particle count is examined at the thermodynamic limit N −→ ∞, V −→ ∞ and n := N/V = const. We expect that at very low temperatures some of the Bosons are in their ground state. In particular, we expect that at some point all particles are in the ground state, if there not enough thermal energy for the excitation of

4 the bosons. Therefore, the chemical potential must be zero, so that the number of particles in the ground state goes towards infinity in the thermodynamic limit. Conversely, for very high temperatures, all bosons are in excited states and thus the number of particles in the ground state is negligible. For this to happen, the chemical potential must be genuinely less than zero and must change according to the temperature changes. To understand how the transition between these two extreme situations occurs, consider our system at extremely low temperatures and warm it up slowly. At the beginning the chemical potential is therefore zero. When the temperature rises, the number of thermally exited bosons goes up, since 3/2 Nth, for µ = 0, is proportional to T . This is possible until all bosons are in thermally excited states. From there on the chemical potential must be genuinely less than zero so that the fugacity in the polylogarithm can compensate for the temperature dependence of the thermal de Broglie wavelength. This leads us directly to the general definition of the critical temperature Tcrit, as the temperature in which all bosons are thermally excited and the chemical potential is zero.

N = Nth (Tcrit, µ = 0) (17)

This makes it possible to determine the critical temperature for the previously considered case of the ideal bosegas in the box, which depends only on the mass of a single boson and the particle density: 2  2/3 2π~ n Tcrit = (18) mkB ζ(3/2) Below the critical temperature the number of ground state bosons are therefore given by: !  T 3/2 N0 = N 1 − , T < Tcrit (19) Tcrit

Figure 1 shows the relative number of bosons in the ground state against the scaled tem- 3/2 perature. The plot follows (19), i.e. N0/N ∝ (T/Tcrit) . However, this is only valid for the free ideal bose gas. In experiments one have to consider that the bose gas is not a perfect ideal bose gas, but rather a bose gas trapped in a potential. Figure 2 shows such experimental bose gas in a harmonic potential [7]. In this case the exponent in (19) is 3 different, i.e. N0/N ∝ (T/Tcrit) .

Figure 1: Theoretical ground state fraction N0/N against scaled temperature T/Tcrit.

5 Figure 2: Total number N (inset) and ground state fraction N0/N against scaled tem- perature T/Tcrit. The solid line shows the infinite N theory curve. The dashed line is a leastsquares fit to (19), but with 3 as the exponent.

To have a better idea why this phenomena is called condensation it is helpful to compute some of the thermodynamic state variables below and above the critical temperature, such as for pressure P = 2E/3V , internal energy U (see (11)) and heat capacity CV = ∂E/∂T .

3 V kBT CV 15 V 9 g3/2(z) U = kBT 3 g5/2(z) P = 3 g5/2(z) = 3 g5/2(z) − , T > Tcrit 2 λth λth kBN 4 Nλth 4 g1/2(z)

3 V kBT CV 15 V U = kBT 3 g5/2(1) P = 3 g5/2(1) = 3 g5/2(1) , T < Tcrit 2 λth λth kBN 4 Nλth (20) In all these quantities, one finds a kink or discontinuity at the critical temperature. This motivates the designation of this effect as condensation, since a typical characteristic of phase transitions is fulfilled by these discontinuities in derivatives of the state variables. However, this is only the case as long as the system is considered within the thermodynamic limit. With a finite size and particle number, smoothing effects occur in the state variables and the discontinuities disappear.

4 Gross-Pitaevskii equation

After the critical temperature at which Bose-Einstein condensation begins was discussed in the previous section, the focus is now on the distribution of the condensate in a external potential under the influence of weak interactions between the particles. First, we take a look at the assumptions that are made. As already mentioned, we consider only weak interactions, whose range r0 is much smaller than the mean free path. Because of this short range of the interaction, we will limit our considerations to two-particle interactions only. The bosons should already be in the condensate phase, i.e. T < Tcrit. From the scattering theory one knows that at low momentums the scattering amplitude is independent of energy and angle and depends only on the scattering length a. For the latter, we again assume that the mean free path is much bigger.

6 † To derive the Gross-Pitaevskii equation, the creation operatora ˆi of a particle in the i-th energy state and the annihilation operatora ˆj of the j-th energy state are defined. With these operators, single-particle operators such as the single-particle Hamiltonian operator and the external potential can be rewritten in the energy base.  2  ˆ X p † Hkin = i j aˆ aˆj 2m i i,j (21) ˆ X † Hext = hi |Vext| ji aˆi aˆj i,j Similarly, the two-particle operator of the interaction potential can be transformed into the energy base by four sums over two creation and two destruction operators each. 1 X Hˆ = hi, j |V | k, li aˆ†aˆ†aˆ aˆ (22) int 2 int i j k l i,j,k,l The next step is to describe these Hamiltonian operators by so-called field operators. We define the field operators with the base transformation from the energy base to the spacial base. X X Ψ(ˆ x) = hx|iiaˆi = φi(x)ˆai (23) i i ˆ † X † † Ψ (x) = φi (x)ˆai (24) i These operators create or annihilate a particle at position x and satisfy the same commu- tator relations as the creators and annihilators in the energy base. h i h i Ψ(ˆ x), Ψˆ x0 = 0 = Ψˆ †(x), Ψˆ † x0 (25) h i Ψ(ˆ x), Ψˆ † x0 = δ(3) x − x0 (26) One can now easily describe the previous Hamiltonian operators with field operators, using the commutator rules. X Z  2  Hˆ = d3xφ∗(x) − ~ ∆ φ (x)ˆa†aˆ kin i 2m j i j i,j (27) 2 Z = ~ d3(x)∇Ψˆ †(x)∇Ψ(x)ˆ 2m Z ˆ X 3 ∗ † Hext = d xφi (x)Vext(x)φj(x)ˆai aˆj i,j (28) Z 3 † = d xΨˆ (x)Vext(x)Ψ(ˆ x)

1 X ZZ Hˆ = d3xd3x0φ∗(x)ψ∗ x0 V x, x0 φ (x)φ x0 aˆ†aˆ†aˆ aˆ int 2 i j int k l i j k l i,j,k,l (29) 1 ZZ = d3xd3x0Ψˆ †(x)Ψˆ † x0 V x, x0 Ψ(ˆ x)Ψˆ x0 2 int We now calculate the time evolution of the field operator. For this we consider the time evolution of Ψ(x)ˆ which is given within the Heisenberg image as follows: ! ! iHtˆ −iHtˆ Ψ(ˆ x, t) = exp Ψ(ˆ x) exp (30) ~ ~

7 Since the field operator is considered in the Heisenberg picture, the time evolution is given by the following equation: ∂ i Ψ(ˆ x, t) = [Ψ(ˆ x, t), Hˆ ] (31) ~∂t One should keep in mind that the field operator without time evolution is not time- dependent and therefore the time derivative of the undeveloped operator is zero. To evaluate the equation, the commutator between the field operator and the Hamilton operator must be evaluated. To do this, each individual part of the Hamilton operator is evaluated separately. The commutator of the kinetic part of the Hamilton operator is used first: h i 2 Z   Ψ(ˆ x, t), Hˆ = ~ d3x0 Ψ(ˆ x, t)∇Ψˆ † x0, t ∇Ψˆ x0, t − ∇Ψˆ † x0, t ∇Ψˆ x0, t Ψ(ˆ x, t) kin 2m 2 Z   = − ~ d3x0 Ψ(ˆ x, t)Ψˆ † x0, t − Ψˆ † x0, t Ψ(ˆ x, t) ∆Ψˆ x0, t 2m 2 Z 2 = − ~ d3x0δ(3) x − x0 ∆Ψˆ x0, t = − ~ ∆Ψ(ˆ x, t) 2m 2m (32) By partially integrating the second part of the first line one can swap Ψ(ˆ x, t) and ∇Ψ(ˆ x0, t) and go back with a second partially integration. Therefore one can see that these two operators commutate within the integral. By using partially integration on the integral in the second line one gets the commutator from (26). For the commutator with the external part of the Hamilton operator, the commutator 31 can easily be found: Z h i 3 0 0  † 0  0  † 0  0   Ψ(ˆ x, t), Hˆext = d x Vext x Ψ(ˆ x, t)Ψˆ x , t Ψˆ x , t − Ψˆ x , t Ψˆ x , t Ψ(ˆ x, t) Z 3 0 0 h † 0 i 0  = d x Vext x Ψ(ˆ x, t), Ψˆ x , t Ψˆ x , t = Vext(x)Ψ(ˆ x, t) (33) The third commutator with the interaction part of the Hamilton operator can be calculated using the product rule for commutators: h i 1 ZZ   Ψ(ˆ x, t), Hˆ = d3x0d3x00 Ψ(ˆ x, t)Ψˆ † x0, t Ψˆ † x00, t − Ψˆ † x0, t Ψˆ † x00, t Ψ(ˆ x, t) int 2 00 0 0  00  Vint x , x Ψˆ x , t Ψˆ x , t 1 ZZ  h i h i  = d3x0d3x00 Ψˆ † x0, t Ψ(ˆ x, t), Ψˆ † x00, t + Ψ(ˆ x, t), Ψˆ † x0, t Ψˆ † x00, t 2 0 00 0  00  Vint x , x Ψˆ x , t Ψˆ x , t 1 Z = d3x0Ψˆ † x0, t V x0, x Ψˆ x0, t Ψ(ˆ x, t) 2 int 1 Z + d3x00Ψˆ † x00, t V x, x00 Ψ(ˆ x, t)Ψˆ x00, t 2 int Z 3 0 † 0  0  0  = d x Ψˆ x , t Vint x , x Ψˆ x , t Ψ(ˆ x, t) (34) With these three commutators the equation of motion can be written: ∂  2 Z  i Ψ(ˆ x, t) = − ~ ∆ + V (x) + d3x0Ψˆ † x0, t V x0, x Ψˆ x0, t Ψ(ˆ x, t) (35) ~∂t 2m ext int In order to finally arrive at the Gross-Pitaevskii equation, we now take our assumptions made above. Since we are at T < Tcrit we assume that all particles are in the ground state.

8 † Thus, the expected value haˆ0aˆ0i is equal to the total particle number N. This motivates us to replace the field operator as follows: √ √ ˆ aˆ0 N, Ψ Nφ0 =: Ψ0 (36) By doing this replacement the interaction potential must also be replaced by an effective 0 potential Veff . Furthermore, we use the short range of the interaction to replace Ψˆ 0 (x , t) with Ψˆ 0(x, t). After applying these substitutions, the following equation is obtained:

 2  ~ ∗ R 3 0 0 2m∆ + Vext(x) + Ψ0(x)Ψ0(x) d x Veff (x , x) Ψ0(x) (37)  2  ~ 2 ∂ = −2m∆ + Vext(x) + |Ψ0(x)| ga Ψ0(r) = i~∂tΨ0(x)

R 3 0 0 where ga := d x Veff (x , x) depends on the scattering length a and can be calculated within the frame of scattering theory, assuming a weak interaction. Equation (37) is the so-called Gross-Pitaevskii equation, also known as the non-linear Schr¨odingerequation. The latter term is explained if the limit g −→ 0 is considered and the Gross-Pitaevskii equation merges into the Schr¨odingerequation. The Gross-Pitaevskii equation is a non-linear differential equation which describes the distribution of the bosons in the condensate and their dynamics. Ψ0 can be regarded as the distribution function normalized to N or as the wave function of the condensate. Thus, the condensate behaves like a single-particle wave function that describes N particles.

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