The Quantum Mechanics Solver 20 potx

19 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. Consequently, the system possesses remarkable coherence properties. 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 We consider a particle of mass m placed in a harmonic potential with a fre- quency ω/2π. The Hamiltonian of the system is ˆ H = ˆ p 2 2m + 1 2 mω 2 ˆ r 2 , where ˆ r =(ˆx, ˆy, ˆz)and ˆ p =(ˆp x , ˆp y , ˆp z ) are respectively the position and momentum operators of the particle. We set a 0 =  ¯h/(mω). 19.1.1. Recall the energy levels of this system, and its ground state wave function φ 0 (r). 19.1.2. We wish to obtain an upper bound on this ground state energy by the variational method. We use a Gaussian trial wave function: ψ σ (r)= 1 (σ 2 π) 3/4 exp(−r 2 /(2σ 2 )) with σ>0 . (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. Compare the bound with the exact value, and comment on the result. 194 19 Properties of a Bose–Einstein Condensate Formulas:  |ψ σ (r)| 2 dx dy dz =1  |ψ σ (r)| 4 dx dy dz = 1 (2π) 3/2 1 σ 3  x 2 |ψ σ (r)| 2 dx dy dz = σ 2 2      ∂ψ σ (r) ∂x     2 dx dy dz = 1 2σ 2 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 . 19.2.1. In the absence of interactions between the particles, the Hamiltonian of the system is ˆ H = ˆ p 2 1 2m + ˆ p 2 2 2m + 1 2 mω 2 ˆ r 2 1 + 1 2 mω 2 ˆ r 2 2 . (a) What are the energy levels of this Hamiltonian? (b) What is the ground state wave function Φ 0 (r 1 , r 2 )? 19.2.2. We now suppose that the two particles interact via a potential v(r 1 − r 2 ). We assume that, on the scale of a 0 , this potential is of very short range and that it is peaked around the origin. Therefore, for two functions f(r)and g(r) which vary appreciably only over domains larger than a 0 , one has  f(r 1 ) g(r 2 ) v(r 1 − r 2 )d 3 r 1 d 3 r 2  4π¯h 2 a m  f(r) g(r)d 3 r. (19.2) The quantity a, which is called the scattering length, is a characteristic of the atomic species under consideration. It can be positive (repulsive interaction) or negative (attractive interaction). One can measure for instance that for sodium atoms (isotope 23 Na) a =3.4 nm, whereas a = −1.5 nm for lithium atoms (isotope 7 Li). (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. Comment on the sign of this energy shift. (b) Under what condition on a and a 0 is this perturbative approach expected to hold? 19.4 Condensates with Repulsive Interactions 195 19.3 Energy of a Bose–Einstein Condensate We now consider N particles confined in the same harmonic trap of angular frequency ω. The particles have pairwise interactions through the potential v(r) defined by (19.2). The Hamiltonian of the system is ˆ H = N  i=1  ˆ p 2 i 2m + 1 2 mω 2 ˆ r 2 i  + 1 2 N  i=1 N  j=1 j=i v( ˆ r i − ˆ r j ) . 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: Ψ σ (r 1 , r 2 , ,r N )=ψ σ (r 1 ) ψ σ (r 2 ) ψ σ (r N ) , where ψ σ (r) is defined in (19.1). 19.3.1. 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 |Ψ σ : E k (σ)=Ψ σ | N  i=1 ˆ p i 2 2m |Ψ σ  E p (σ)=Ψ σ | N  i=1 1 2 mω 2 ˆ r i 2 |Ψ σ  E int (σ)=Ψ σ | 1 2 N  i=1 N  j=1 j=i v( ˆ r i − ˆ r j )|Ψ σ  We set E(σ)=Ψ σ | ˆ H|Ψ σ . 19.3.2. We introduce the dimensionless quantities ˜ E(σ)=E(σ)/(N¯hω)and ˜σ = σ/a 0 . Give the expression of ˜ E in terms of ˜σ. Cast the result in the form ˜ E(σ)= 3 4  1 ˜σ 2 +˜σ 2  + η ˜σ 3 and express the quantity η as a function of N, a and a 0 . In all what follows, we shall assume that N  1. 19.3.3. For a = 0, recall the ground state energy of ˆ H. 19.4 Condensates with Repulsive Interactions In this part, we assume that the two-body interaction between the atoms is repulsive, i.e. a>0. 196 19 Properties of a Bose–Einstein Condensate 19.4.1. Draw qualitatively the value of ˜ E as a function of ˜σ. Discuss the variation with η of the position of its minimum ˜ E min . 19.4.2. We consider the case η  1. Show that the contribution of the kinetic energy to ˜ E is negligible. In that approximation, calculate an approximate value of ˜ E min . 19.4.3. 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. 19.1. 19.4.4. Figure 19.1 has been obtained with a sodium condensate (mass m = 3.8 ×10 −26 kg) in a harmonic trap of frequency ω/(2π) = 142 Hz. (a) Calculate a 0 and ¯hω for this potential. (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. 19.1. Compare the result with the value obtained in scattering experiments a =3.4nm.Isit possible a priori to improve the accuracy of the variational method? Fig. 19.1. 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 We now suppose that the scattering length a is negative. 19.5.1. Draw qualitatively ˜ E as a function of ˜σ. 19.5.2. Comment on the approximation (19.2) in the region σ → 0. 19.6 Solutions 197 19.5.3. 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. Calculate the corresponding size σ c as a function of a 0 . 19.5.4. In an experiment performed with lithium atoms (m =1.17 × 10 −26 kg), it has been noticed that the number of atoms in the condensate never exceeds 1200 for a trap of frequency ω/(2π) = 145 Hz. How can this result be explained? 19.6 Solutions Section 19.1: Particle in a Harmonic Trap 19.1.1. The Hamiltonian of a three-dimensional harmonic oscillator can be written ˆ H = ˆ H x + ˆ H y + ˆ H z , where ˆ H i represents a one dimensional harmonic oscillator of same frequency along the axis i = x, y, z. We therefore use a basis of eigenfunctions of ˆ H of the form φ(x, y, z)=χ n x (x) χ n y (y) χ n z (z), i.e. products of eigenfunctions of ˆ H x , ˆ H y , ˆ H z ,whereχ n (x)isthenth Hermite function. The eigenvalues of ˆ H can be written as E n =(n+3/2)¯hω,wheren = n x +n y +n z is a non-negative integer. The ground state wave function, of energy (3/2)¯hω, corresponds to n x = n y = n z = 0, i.e. φ 0 (r)= 1 (a 2 0 π) 3/4 exp[−r 2 /(2a 2 0 )] . 19.1.2. The trial wave functions ψ σ are normalized. In order to obtain an upper bound for the ground-state energy of ˆ H, we must calculate E(σ)= ψ σ | ˆ H|ψ σ  and minimize the result with respect to σ. Using the formulas given in the text, one obtains  ψ σ | ˆp 2 2m |ψ σ  =3 ¯h 2 2m 1 2σ 2  ψ σ | 1 2 mω 2 r 2 |ψ σ  =3 mω 2 2 σ 2 2 and E(σ)= 3 4 ¯hω  a 2 0 σ 2 + σ 2 a 2 0  . This quantity is minimum for σ = a 0 , and we find E min (σ)=(3/2) ¯hω.In this particular case, the upper bound coincides with the exact result. 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 19.2.1. (a) The Hamiltonian ˆ H can be written as ˆ H = ˆ H 1 + ˆ H 2 ,where ˆ H 1 and ˆ H 2 are respectively the Hamiltonians of particle 1 and particle 2. 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 ). The energy eigenvalues are E n =(n +3)¯hω,wheren is a non-negative integer. (b) The ground state of ˆ H is: Φ 0 (r 1 , r 2 )=φ 0 (r 1 ) φ 0 (r 2 )= 1 a 3 0 π 3/2 exp  −(r 2 1 + r 2 2 )/(2a 2 0 )  . 19.2.2. (a) Since the ground state of ˆ H is non-degenerate, its shift to first order in a can be written as ∆E = Φ 0 |˜v|Φ 0  =  |Φ 0 (r 1 , r 2 )| 2 v(r 1 − r 2 )d 3 r 1 d 3 r 2  4π¯h 2 a m  |φ 0 (r)| 4 d 3 r = 4π¯h 2 a m 1 (2π) 3/2 1 a 3 0 therefore ∆E ¯hω =  2 π a a 0 . For a repulsive interaction (a>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. (b) The perturbative approach yields a good approximation provided the energy shift ∆E is small compared to the level spacing ¯hω of ˆ H. Therefore, one must have |a|a 0 , i.e. 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: E k (σ)=N 3 4 ¯h 2 mσ 2 E p (σ)=N 3 4 mω 2 σ 2 E int (σ)= N(N − 1) 2  2 π ¯hω aa 2 0 σ 3 . Indeed, there are N kinetic energy and potential energy terms, and N (N −1)/2 pairs which contribute to the interaction energy. 19.6 Solutions 199 19.3.2. With the change of variables introduced in the text, one finds ˜ E(σ)= 3 4  1 ˜σ 2 +˜σ 2  + N − 1 √ 2π a a 0 1 ˜σ 3 so that η = N − 1 √ 2π a a 0 . 19.3.3. If the scattering length is zero, there is no interaction between the particles. 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 19.4.1. 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. For large ˜σ, the behavior of ˜ E does not depend on η. It is dominated by the potential energy term 3˜σ 2 /4. The minimum ˜ E min increases as η increases. This minimum corresponds the point where the potential energy term, which tends to favor small values of σ, matches the kinetic and interaction energy terms which, on the contrary, favor large sizes σ. 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. Fig. 19.2. Variation of ˜ E(˜σ)with˜σ for η =0, 10, 100, 1000 (from bottom to top) 19.4.2. Let us assume η is much larger than 1 and let us neglect a priori the kinetic energy term 1/˜σ 2 . The function (3/4)˜σ 2 + η/˜σ 3 is minimum for ˜σ min =(2η) 1/5 where its value is ˜ E min = 5 4 (2η) 2/5 . One can check a posteriori that it is legitimate to neglect the kinetic energy term 1/˜σ 2 . In fact it is always smaller than one of the two other contributions to ˜ E: 200 19 Properties of a Bose–Einstein Condensate • For ˜σ<˜σ min , one has 1/˜σ 2  η/˜σ 3 . • For ˜σ>˜σ min , one has 1/˜σ 2  ˜σ 2 . 19.4.3. For a number of atoms N  1, the energy of the system as calculated by the variational method is E N = 5 4 ¯hω   2 π N a a 0  2/5 . (19.3) This variation of E/N as N 2/5 is very well reproduced by the data. In Fig. 19.3 we have plotted a fit of the data with this law. One finds E/N  αN 2/5 with α =8.2 ×10 −33 Joule. Fig. 19.3. Fit of the experimental data with an N 2/5 law 19.4.4. (a) One finds a 0 =1.76 µmand¯hω =9.410 −32 Joule. (b) Consider the value a =3.4 nm given in the text. The approximation η  1 will hold if N  1300. This is clearly the case for the data of Fig. 19.1. (c) The coefficient α =8.2 ×10 −33 Joule found by fitting the data leads to a =2.8 nm. This value is somewhat lower than the expected value a =3.4nm. This is due to the fact that the result (19.3), E/(N¯hω)  1.142 (Na/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 (Na/a 0 ) 2/5 . The fit to the data is then in agreement with the experimental value of the scattering length. 19.6 Solutions 201 Section 19.5: Condensates with Attractive Interactions 19.5.1. The function ˜ E(˜σ) is represented in Fig. 19.4. We notice that it has a local minimum only for small enough values of η.Forη<0, there is always a minimum at 0, where the function tends to −∞. 0123 -10 -5 0 5 10 ( )E σ σ Fig. 19.4. Plot of ˜ E(˜σ)forη =0;η = −0.1; η = −0.27; η = −1(from top to bottom) 19.5.2. The absolute minimum at σ = 0 corresponds to a highly compressed atomic cloud. For such small sizes, approximation (19.2) for a “short range” potential loses its meaning. Physically, one must take into account the forma- tion of molecules and/or atomic aggregates which have not been considered here. 19.5.3. The local minimum at ˜σ = 0 disappears when ˜ E(˜σ) has an inflexion point where the derivative vanishes. This happens for a critical value of η determined by the two conditions: d ˜ E d˜σ =0 d 2 ˜ E d˜σ 2 =0. This leads to the system 0=− 1 ˜σ 4 +1− 2η ˜σ 5 0= 3 ˜σ 4 +1+ 8η ˜σ 5 from which we obtain |η c | = 2 5 5/4  0.27 ˜σ c = 1 5 1/4  0.67 202 19 Properties of a Bose–Einstein Condensate or σ c  0.67 a 0 . If the local minimum exists, i.e. for |η| < |η c |, 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. 19.5.4. For the given experimental data one finds a 0 =3.1 µm, and a critical number of atoms: N c = √ 2πη c a 0 |a| ∼ 1400 , in good agreement with experimental observations. 19.7 Comments The first Bose–Einstein condensate of a dilute atomic gas was observed in Boulder (USA) in 1995 (M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, Science 269, 198 (1995)) with rubidium atoms. The experimental data shown in this chapter for a sodium condensate come from the results published by M O. Mewes, M.R. Andrews, N.J. van Druten, D.M. Kurn, D.S. Durfee, and W. Ketterle, Phys. Rev. Lett. 77, 416 (1996). 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. The experimental results on lithium have been reported by C. Bradley, C.A. Sackett, and R.G. Hulet, Phys. Rev. Lett. 78, 985 (1997). The Nobel prize 2001 has been awarded to E. Cornell, W. Ketterle, and C. 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. . 1 √ 2π a a 0 . 19.3.3. If the scattering length is zero, there is no interaction between the particles. The ground state of the system is the product of the N functions φ 0 (r i ) and the ground state. (isotope 7 Li). (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. Comment on the sign of this energy shift. (b). in (19.1). 19.3.1. 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 |Ψ σ : E k (σ)=Ψ σ | N  i=1 ˆ p i 2 2m |Ψ σ 