15_chan 71 Study of the Perturbation Science Diliman (January-June 2003) 15:1, 71-74 Study of the Perturbation to a Bose-Einstein Gas L. Chan National Institute of Physics College of Science, University of the Philippines Diliman 1101 Quezon City, Philippines E-mail: lchan@nip.upd.edu.ph ABSTRACT We developed a new approach to the perturbation theory for the effective Hamiltonian of condensate particles in Fock space. Using this new theory, we can easily analyze the effect of including a somewhat problematic term in the work of Ezawa et al. We thus showed that indeed, the inclusion of this term in the perturbation potential is justified. INTRODUCTION The phenomenon of the Bose-Einstein condensation, first observed (Anderson et al., 1995) for 57Rb at 170 K, followed by the cases (Davis et al., 1996; Bradley et al., 1997) of 23Na, 7Li, and 1H, has excited experimental and theoretical interests on different aspects of this quantum effect. In particular, Ezawa et al. (1998) studied the fluctuation of the condensate by modifying the Bogoliubov prescription (Bogoliubov, 1947) in replacing a 0 by 0N with (Ezawa & Luban, 1967; Ezawa, 1965) (1) where N 0 is the number of condensate particles and a 0 , the annihilation operator. This work was done with the modified Oppenheimer approach to perturbation theory in Fock space to obtain the effective Hamiltonian for the condensate, using as the strength parameter. It was then shown that the fluctuations are much less than N 0 , thus justifying Bogoliubov’s prescription. In this work, a term which is of zero-th order in was included in the perturbation, and the perturbation was carried out to second order in . This is unusual in perturbation work. We shall therefore consider the contributions of these terms to higher orders to seek justification for this work. THE SYSTEM HAMILTONIAN The Hamiltonian for a Bose-Einstein gas in a trap is (3) where v(x) is the trap potential and V(x) = V(-x), the interaction. The former, which varies much more slowly than the latter, is the chemical potential. In terms of the new field a 0 ', the field operator takes the form (4) where ( ) ( )n n n x a u xφ = ∑ . The operator a0 shall henceforth be taken to mean a 0 '. ' 0 0 0 .a N a→ + ( ) ( )(0) † † 0 0 BC no n n n n H J a a a a ≠ = + +∑ ( ) ( ) ( ) 2 † 3: 2 A AH x x x d x M φ ν µ φ ⎧ ⎫ = − ∆ + −⎨ ⎬ ⎩ ⎭ ∫ h ( ) ( ) ( ) ( ) ( )† † 3 31 ' ' ' ' 2 A A A Ax x V x x x x d xd xφ φ φ φ+ −∫ ( ) ( ) ( )0A x N u x xφ φ= + (2) ( ) 10Nλ − = λ λ 72 Chan Terms linear in (x) will arise in H and can be eliminated by adding and subtracting the Hartree potential (5) so that (6) The functions u n (x) are chosen to be real eigenfunctions of (7) In view of the short range of the interaction as compared to the wavelength of the atoms, people take the delta- function approximation and (8) where a is the scattering length of the atoms, so that Eq. (7) simplifies as (9) Although we have a set of non-linear eigenvalue equations, it is easy to reflect that we still have a complete orthonormal set of eigenfunctions. The Hamiltonian of the system can now be written as where and with J mn = L oomn and K lmn = L 0lmn . EZAWA’S PERTURBATION APPROACH Ezawa’s perturbation theory is formulated to solve for the effective Hamiltonian (12) in (13) where (14) with A n being an operator in the Hilbert space H B H C of the total system, and is an operator in H C . The perturbation problem is formulated by dividing H into three parts: H C , which acts only on the condensate; H B , which acts only on the out-ofcondensate particles; and H BC , which involves the interaction between condensate and out-of-condensate particles. The unperturbed Hamiltonian is then taken to be the terms down to the zero-th order term in H B and H C , given the names H B and H C . The rest of the terms are taken as the perturbation. We note that H BC contains a zero-order term in . In the lowest order, Ezawa et al. (1998) got and (15) where ( )†0 0 01 . 2 x a a= + Higher order terms are obtained after diagonalizing H B , a process which involves only an orthogonal transformation to handle mutual interaction between out-of-condensate particles. Using the perturbation formula , ( ) ( ) ( )2 30 0: ' 'H x N V x x u x d xν = −∫ ( ) ( ) ( ) ( ) 2 † 3 2 A H AH x x x x d x M φ ν ν µ φ ⎧ ⎫ = − + + −⎨ ⎬ ⎩ ⎭ ∫ h ( ) ( ) ( ) ( ) ( )† † 3 31 ' ' ' ' 2 A A A Ax x V x x x x d xd xφ φ φ φ+ −∫ ( ) ( ) ( )† 3A H Ax x x d xφ ν φ−∫ ( ) ( ) 2 . 2 Hh x x M ν ν= − ∆ + + h ( ) ( )' 'x x g x xν δ− = − 24 a g M π = h ( ) ( ) 2 2 0 0 2 h x gN u x M ν= − ∆ + + h ( ) ( ) ( )† †0 0n n n n n n n H E a a N a aε µ ε µ= + − + − +∑ ( ) ( )† † , 1 2 mn m m n n m n J a a a a+ + +∑ ( )† † † , , lmn l m n l m n l m n K a a a a a aλ+ +∑ ( ) ( ) ( ) ( ) 30klmn k l m nL gN u x u x u x u x d x= ∫ 1 ( 1) (0) (1) 2 ( 2 ) ...n n n n nλ λ λ − −Λ = Λ + Λ + Λ + Λ + ,n n nHψ ψ= Λ ( )(1) 2 ( 2)1 ...n n n nA n A A nψ λ λ= = + + + ( 1) 0 ,−Λ = ( )(0) † 20 00 0 0 0 00 01 , 2 W J a a W J xΛ = + + = + ( 2) (1) (1). . 0 0 , 0 .CH P VA H m A⎡ ⎤Λ = + ⎣ ⎦(11) (10) nΛ λ 2 † † , , , klmn k l m n k l m n L a a a aλ+ ∑ ( )2 4 30 0 0 0 0 1 1 2 2 nn n E N N g u x d x Jε= − − ∑∫ 73 Study of the Perturbation the result obtained was (16) where NEW PERTURBATION METHOD To investigate the higher order terms due to H BC (0), we shall assume that the perturbation consists of only this term V = H BC (0), so that (17) The perturbation approach is obtained by writing where (18) P is the projector to the condensate factor. We now define (19) The effective Hamiltonian can now be replaced by (20) which satisfies the equation (21) From this new eigenvalue equation, we break it up into two parts by projecting it with respect to P and giving and (23) This means that we can simplify the problem by finding a perturbation operator K = QKP satisfying corresponding operator equation (24) and (25) The P equation can be simplified into (26) which allows to be solved for once K is found. Furthermore, QKP can be left multiplied into this equation to give (27) from which the term containing can be eliminated with Eq. (25), yielding (28) The role of this equation is to determine K perturbatively, whether Eqs. (26) and (28) are the working equations of this perturbation approach. RESULTS OF PERTURBATION Using this new approach to perturbation, we get (29) the first order result of Ezawa et al., (30) ( ) ( )2 212 0 0 0 0 4 0 0 0 0 ,L x p p x L x p p x+ + + + ( )†0 0 01 . 2 p i a a= − − 2 4 ( 2) 3 40 0 3 0 4 0 2 42 2 p p K x K x M M Λ = + + + .B CH H H V= + + ( )1n i i n L n i Lψ ≠ = + + ∑ ( ) ( )1 1L P n K L n= + + + ( ) 1' 1 .i i i n i n K n i L i L L − ≠ ≠ = = +∑ ∑ ( ) 1' 1 'n n i i n L n i Lψ ψ − ≠ = + = + ∑ .P n K n= + ( ) ( ) 1' 1 1n nL L − Λ = + Λ + ' ' ' .n n nHψ ψ= Λ 1 ,Q P= − ( ) ( ) ( ) ' . B C nQ H H P K n QKP n QV P K n + + = Λ − + ( ) ( ) ( ) ( ) ' n C nP W H P K n P P K n PV P K n + + = + Λ − + (22) ( ) ( ) ( ) ( )'n C nP W H P K P P K PV P K+ + = + Λ − + ( ) ( )'B C nQ H H K QKP QV P K+ = Λ − + ( ) ( )'n C nP W H P P PV P K+ = Λ − + ( ) ( )'n C nQKP W H P QKP QKPV P K+ = Λ − + ( ) [ ] ( ) ( ),n CQ H W K Q H K P K Q V P K− + = − + ( ) ( )† †0 0 0 0 n n n n V J a a a a ≠ = + +∑ (1) 0 . B Q K VP H W = − − 'nΛ 'nΛ 74 Chan In fact we see that K will be a polynomial in so that the term Q[H C ,K]P vanishes to all orders, and the equation that determines K simplifies to (31) which is similar in form to the results of regular perturbation theory in operator form developed by Speisman (1957), and, therefore, we immediately get (32) and (33) Finally, the new Hamiltonian is given by substituting these expressions for K into Eq. (26). Explicitly, is proportional to x 0 2 and proportional to x 0 4. Since x 0 was estimated to be a small quantity in Ezawa’s work, we see that indeed, we have explicitly verified that the procedure to include H BC (0) in V is justified. ACKNOWLEDGMENT The author wishes to express his gratitude to Professor Hiroshi Ezawa for exposing him to this problem. REFERENCES Anderson, M.H., J.R. Einsher, M.R. Matthew, C.E. Wieman, & E.A. Cornell, 1995. Observation of Bose-Einstein condensation in a dilute atomic vapor. Science. 269: 198-201. Bradley, C.C., C.A. Sackett, & R.G. Hulet, 1997. Bose-Einstein condensation of lithium: Observation of limited condensate number. Phys. Rev. Lett. 78: 985-989. Bradley, C.C., C.A. Sackett, J.J. Tollet, & R.G. Hulet, 1995. Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions. Phys. Rev. Lett. 75: 1687-1690. Bogoliubov, N.N., 1947. On the theory of superfluidity. J. Phys. 11: 23. Davis, K.B., M.O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, & W. Ketterle, 1995. Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75: 3969-3973. Ezawa, H., 1965. Vestigial effects of singular potentials in diffusion theory and quantum mechanics. J. Math. Phys. 6: 380. Ezawa, H., K. Nakamura, K. Watanabe, & Y. Yamanaka, 1998. Fluctuation of the Bose-Einstein condensate in a trap. (unpublished). Ezawa, H. & M. Luban, 1967. Onset of ODLRO and the phase transition of an ideal Bose gas. J. Math. Phys. 8: 1285. Mewes, M.O., M.R. Andrews, N.J. van Druten, D.M. Kurn, D.S. Durfee, & W. Ketterle, 1996. Bose-Einstein condensation in a tightly confirming DC magnetic trap. Phys. Rev. Lett. 77: 416-419. Speisman, G., 1957. Convergent Schrödinger perturbation theory. Phys. Rev. 107: 1180-1192. ( )†0 0 02a a x+ = ( ) ( ) 0 C Q K Q K V P K W H = − + − ( 2) (1) (1) 0 B Q K VK K V W H ⎡ ⎤= −⎣ ⎦− ( 2) 0 0B B Q Q K V VP W H W H = − − (3) ( 2) ( 2) (1) (1) 0 B Q K VK K VP K VK W H ⎡ ⎤= − −⎣ ⎦− 0 0 0 0 0 B B B B B Q Q Q V V VP W H W H W H Q Q VPV VP W H W H ⎡ = ⎢− − −⎣ ⎤ − ⎥− − ⎦ ( )4Λ ( )2Λ