Monte Carlo Simulation of Bose Einstein Condensation in Traps

Transcription

1 Monte Carlo Simulation of Bose Einstein Condensation in Traps J. L. DuBois, H. R. Glyde Department of Physics and Astronomy, University of Delaware Newark, Delaware 19716, USA 1. INTRODUCTION In this paper we explore the properties of N hard sphere bose particles confined in a harmonic potential at zero temperature. The aim is to determine the fraction of bosons condensed into one single particle natural orbital of the system (the condensate fraction, n 0 ) the shape of this condensate orbital in the trap, the shape of the total boson density in the trap including those bosons in states above the condensate and the total energy of the system. Specifically, each boson is represented by a hard sphere of diameter a. The density of bosons is n = N/V and V is an approximate volume of the trap. We take n = n(0), the calculated density at the centre of the trap. We evaluate the above properties over a wide range of the parameter, na 3 = Na 3 /V which is the ratio of the volume occupied by the bosons (Na 3 ) to the total volume V. We begin with the dilute limit na which is a typical density of 87 Rb atoms confined in a harmonic trap [1-4]. We consider intermediate density, na 3 10, the density of 85 Rb atoms in a harmonic magnetic trap in which a is increased taking advantage of Feshbach resonances [5]. We go to high density na which is actually somewhat higher than the density of liquid 4 He at Saturated Vapour Pressure (SVP), na 3 = 0.1. Our goal is to determine the condensate fraction n 0 and the condensate density over a wide density range, 10 6 na We see, for example, that in the dilute limit, the condensate is concentrated at the centre of the trap, as observed for 87 Rb atoms having a positive scattering length [4]. At intermediate na 3 10, the condensate density is almost uniformly spread (constant) throughout the trap and falls off rapidly at the edges. At high density (na 3 0.), there is little condensate density at the center of the trap. Rather, the condensate is at the edges of the trap where the total boson density is low as in liquid 4 He droplets [6]. The dramatic change in condensate distribution results from the interaction between the bosons both in and above the condensate.

2 Figure 1. Bose system densities showing typical densities for trapped 87 Rb, the highest density range reached experimentally for 85 Rb, and liquid Helium densities. In section we present the Hamiltonian and the trial variational wave functions. The results are given in section 3 followed by a conclusion.. TRAPPED BOSONS AND NATURAL ORBITALS The Hamiltonian for N bosons of mass m in a trapping potential V ext (r) and interacting via a two body potential V int ( r i r j ) is H = N ( h ) m i + V ext(r i ) + i N V int (r i, r j ). (1) i<j We consider spherically symmetric (S) and elliptical (E) harmonic trips V ext (r) = { 1 mω ho r (S), 1 mω ho (x + y + λ z ) (E) }. () Here ω ho defines the trap strength and λ = ω z/ω ho, the aspect ratio of the trap. The bosons interact via the potential V int (r) = { (r a), 0 (r > 0) }. (3) The 87 Rb 87 Rb interaction in the dilute limit can be well approximated [4] by a contact potential v(r) = 4π h a/m δ(r) = gδ(r). The scattering in this limit is purely s-wave with scattering length a = a Rb 5 Å. The scattering from a hard core potential is also pure s-wave with scattering length a. Thus the hard core and contact potential approximations are equivalent with a = a Rb so long as the S-wave approximation holds.

3 A characteristic trap length [4] is a ho = ( h/mω ho ) Å. Thus a Rb /a ho 1 and to be specific we take a definite value a/a ho = A typical atom density in a trap is n = N/V atoms/cm 3. This gives a typical interatom spacing l 1000 Å and na Since bose condensation takes place for nλ 3 T >.616, the thermal or de Broglie wavelength of atoms λ T 3000 Å is greater than the interatom space. Thus we have a dilute gas with a a ho, a l, a ho 10l and λ > T l so that quantum effects are extremely important. We use a trial wave function ψ (r 1,..., r N, α, β) = i g (r i, α, β) i<j f ( r i r j ) (4) where g (r, α, β) = exp [ α ( x + y + βz )] (5) is a Gaussian with variational parameters α and β and { f(r) = [1 a } r ] (r > a), 0 (r a) is a pair Jastrow wave function. This wave function is used to calculate the total energy and the one-body density matrix at T = 0 K. The condensate orbitals and condensate fraction are obtained by diagonalizing the OBDM as discussed by Löwdin [7] and Onsager and Penrose [8]. Further details can be found in Ref. [9]. (6) 3. RESULTS Fig. shows the shape of the condensate natural orbital φ 0 (r) for N = 18 HC bosons in a spherical trap as a function of the hard core diameter a. The hard core diameter is set at a = ma Rb for m = 1, 4,..., 64 where a Rb is the 87 Rb scattering length and a Rb /a ho = The dashed line in Fig. shows the non-interacting gas (m = 0) φ 0 (r) = π 3 4 exp[ r /] with r in units of a ho and φ 0 (r) normalized to unity. As the HC diameter a = ma ho increases, the condensate orbital spreads out to larger r in the trap and the condensate density at the centre of the trap (r = 0) decreases. At the largest a = 64a Rb shown, the condensate density is nearly uniform throughout the trap out to r 3a ho. At a = 64a Rb, the density at the center of the trap is n(0)a 3 = na This corresponds approximately to the densities achieved recently by Cornish et al. [5] using 85 Rb atoms and Feshbach resonances to increase the size of the scattering length. Thus for 85 Rb at na 3 0.0, we expect the condensate density to be quite flat rather than the parabolic shape predicted by the Thomas-Fermi approximation. Indeed at na 3 0.0, both the total density and the condensate density will be quite constant so that there is little to distinguish the total density distribution and the condensate distribution at T = 0 K. Fig. 3 shows the condensate fraction n 0 for the N = 18 HC bosons discussed above calculated using Monte Carlo (VMC) and the lowest order Bogoliubov approximation. For a uniform bose gas the Bogoliubov result [10] is n 0 = N 0 N = ( na 3 π ) 1. (7)

4 & & # & ' % # / &01# (*),+ '.- &0% &0 &0 " % "$#! Figure. Condensate natural orbital φ 0 (r) for N = 18 hard sphere (HS) bosons in a spherical, harmonic trap. The HS diameter a = ma Rb, a Rb /a ho = where a ho = h/mω ho is the trap length. Shown is φ 0 (r) for m = 0 (non-interacting gas), m = 1 ( 87 Rb atoms), and m = 4, 8, 16, 3, 64. The corresponding HS boson density na 3 at the center of the trap is: m = 1 ( ), m = 4 ( ), m = 16 (10 3 ), m = 3 ( ), m = 64 ( 10 ). At na 3 = 10 (m = 64), the condensate density is quite uniform in the trap out to r 3a ho. where n = N/V is the uniform density. For bosons in a spherical trap n 0 = N 0 N = 1 5π 3 ( nt F (0)a 3 π ) 1 (8) where n T F (0) is the density at the centre of the trap in the Thomas Fermi approximation (kinetic energy neglected). We see that at m = 16 (a/a ho and na ) the condensate fraction is n so that the depletion of the condensate arising from the interaction is approximately 5%. At this na 3 value we expect the Gross Pitaveskii (GP) equation [4,11,1] to be only approximately correct since the GP equation is an equation for the condensate only. At na , the Bogoliubov expression describes the depletion well. At m = 3 (n 0 = 0.90, na ) the Bogoliubov expression for the depletion is beginning to clearly overestimate the condensate fraction. In the uniform Bose gas [13], the Bogoliubov expression overestimates the condensate fraction by the same degree at na 3 10 ). The Bogoliubov result holds to somewhat higher na 3 values in the uniform case, probably because the density itself n = N/V cannot change in the uni-

5 VMC Bogoliubov trap Bogoliubov bulk n a/a ho 0.3 Figure 3. The condensate fraction n 0 in the condensate natural orbitals shown in Fig.. For HS diameter a = 4a Rb (na ) the depletion of the condensate is approximately 1%, n 0 = At a = 16a Rb (na ) depletion is significant (5%) but is well described by the Bogoliubov expression (Ref. 14). At a = 64a Rb (na 3 10 ), depletion is 5% and not well described by the Bogoliubov expression. form case but the density profile changes dramatically in a trap with na 3, as shown in Fig.. Fig. 4 shows the total density and condensate density of N = 18 atoms in a spherical trap for higher values of the HC diameter, a = ma Rb ; m = 18, 56 and 51. The condensate density is shown as N φ 0 (r) rather than N 0 φ 0 (r) so that it is more clearly visible. For m = 18 (n and na ), both the condensate density and the total density are quite constant for 0 r 3.5a ho and then both densities fall to zero rapidly by r 4a ho. At m = 56 (na and n 0 = 0.) the condensate density peaks at r 5a ho, i.e. at the edges of the total trap density. Thus given the geometric factors, the majority of the condensate is at the outer edges of the trap for na where n 0 = 0.. Essentially, the atoms above the condensate are playing a critical role in determining where the condensate is located the condensate is found at the edges of the trap where the total density is lowest and the condensate fraction n 0 (r) is highest. The condensate seeks the regions of lowest total density. At m = 51 (n 0 = 0.03 and na 3 = 0.6), the condensate is entirely at the edges of the trap. Also, we see correlations in the total density that arise from the inter-boson correlations induced by the hard core interactions. These kind of correlations are seen in the pair correlation function in uniform systems. They

6 - S 7 V U 8 -.0/ & ! "" # $ % &')( *$+, S WXS S WY7)S S W05TS )8 93: ; S WSS <>=@?BADCEA?GFIHKJ)LM3ONIPRQ 5TS Figure 4. Condensate density N φ 0 (r) (open circles) and the total boson density N( r ) (solid circles) for N = 18 bosons in a spherical trap for large HC diameter a = na Rb, m = 18, 56 and 51. For m = 18, (na and condensate fraction n 0 = 0.5), the condensate density is approximately uniform out to r/a ho 3.5 and then decreases rapidly. At m = 56, (na 3 = 0.19 and n 0 = 0.) the condensate density peaks at r/a ho 5, at the edges of trap density. At m = 56, the condensate density is predominantly at the edges of the trap. are seen in the total (one-body) density of a trap because the system is finite and has a high density at the centre and the correlations form around that high, central density. We now turn from the total and condensate densities to comparisons of the total and the condensate energies. The Gross-Pitaveskii (GP) equation for the condensate provides an energy for the condensate in a trap [4,15]. At small values of na 3 (e.g. na ) we expect this energy, E GP /N, to agree well with the total energy, E/N, since essentially all bosons are in the condensate. As na 3 increases and some bosons are excited to states above the condensate we expect the total energy E/N,

7 na n He N = 64 N = 18 N = a/a ho Figure 5. The condensate fraction n 0 for N = 64, 18 and 56 HS bosons in a spherical trap and large HS diameters a. At a = 56 a Rb (a Rb /a ho = ) and N = 18, n 0 0. and na where n = n(0) is the total density at the centre of the trap. Bulk liquid 4 He density is na 3 = 0.1 with a =.03 Å. which includes the bosons above the condensate, to lie above E GP /N. We compare E/N calculated using MC which includes all the bosons with E GP /N to explore how much the two differ as a function of na 3. Explicitly, we calculate δ (E/N) = (E MC E GP ) /N (9) for an elliptically trap with ω x = ω y = ω ho and ω z = ω ho 8 where again a = marb and a Rb /a ho = and m are integers. We use the GP energies calculated by Dalfovo and Stringari for this trap which are a function of the product Na/a ho only. We calculate the E/N using MC for N and a varying separately. For a homogeneous, dilute gas of bosons, the total energy is (in units of h /ma ), [ E N = 4πna ( na 3 π ) 1 ]. (10) where 4πna 3 is the Bogoliubov or GP energy of the condensate and δ(e/n) = E GP (18/15)(na 3 /π) 1 is the change in energy arising from exciting bosons out of

8 Figure 6. Variational Monte Carlo (MC) energy for N hard sphere bosons in an elliptical trap and the energy of the condensate in the same trap calculated using the Gross-Pitaevskii (GP) equation (Ref. 14). The HS diameter/scattering length is a = a Rb with a Rb /a ho = and a z = a / 4 8, a = a ho. the condensate to leading order in na 3. For bosons in a trap and calculated within the Thomas-Fermi approximation in which the kinetic energy is neglected, we have E G P N = 5 7 ( 15Na a ho ) [ π 8 ( nt F (0)a 3 π ) 1 ], (11) ( ) where E GP /N = Na 7 a ho is the GP energy of the condensate (in units of 1 hω ho) in the T F limit, n T F (0)a 3 = (15 5 /8π)N 5 (a/a ho ) 1 5 is the density at the center of the trap and δ(e/n) is proportional to N 3 5 (a/a ho ) 8 5. Our goal is to evaluate δ(e/n) by MC to see how well it is described by Eq. (5). We note that contributions to δ(e/n) in (3) could arise from E MC being too high since the VMC method gives an upper bound and from E GP being too low since E GP is a mean field theory as well as because of depletion. Fig. 6 shows the E MC and E GP in the dilute limit corresponding to 0 N 0, Rb atoms in an elliptical trap with a Rb /a ho = Clearly at small N, E MC and E GP agree well. There is some difference at large N with E MC N lying

9 δ E N % na Na/a ho Figure 7. Difference δ(e/n) = E MC /N E GP /N between MC and GP energies as a function of Na/a ho for four different values of a/a Rb = 1, 5, 10 and 0. The lines are fits of Eq. (1) to the date points. The values of na 3 corresponding to Na/ah = 86.6 for a/a ho = 1, 5.10 and 0 are shown on the RHS. approximately.5% above E GP at N = 0, 000 which corresponds to na and a depletion of the condensate of approximately 1%. In Fig. 7, we show the difference δ(e/n) = E MC /N E GP /N for larger values of na 3, for four values of the scattering lengths a = ma Rb, m = 1, 5, 10, 0. If the difference δ(e/n) arises from depletion of the condensate, then from (8), we expect δ(e/n) will scale with N and a as δ(e/n) N 3 5 (a/a ho ) 8 5. The lines in Fig. 7 are fits of δ(e/n) = η 1 N 3 5 (a/aho ) η, (1) where η 1 and η are fitting parameters. Clearly, the fits are good suggesting that the difference between E MC and E GP arises predominantly from depletion of the condensate and this energy difference is well described by the leading order correction in (5) up to na Thus the GP energy for the condensate plus the leading correction for depletion of the condensate appears to describe the total energy well up to the density range covered by 87 Rb. 4. CONCLUSIONS The condensate fraction, condensate density and total density of hard sphere bosons in a harmonic trap have been evaluated over a wide density range, 10 6 na using Monte Carlo methods. At high density, na 3 0.1, the condensate is

10 at the edges of the trap where the total density is low, as found in liquid 4 He droplets [6]. At intermediate density, na , as realized with 85 Rb atoms in a trap [5], the condensate density is quite uniformly distributed throughout the trap as is the total density. At low density, 10 6 na , the condensate is at the centre of the trap as predicted by the Gross Pitaevskii equation [4]. The Monte Carlo energy which includes the atoms in and above the condensate agrees (within 1%) with the GP energy for the condensate for densities up to na (n ). At values of na 3 up to na 3 10 the MC energy agrees with the GP energy for the condensate plus the leading mean field correction for excitation above the condensate. At higher na 3 value, the MC energy lies above the mean field value with corrections. ACKNOWLEDGMENTS It is a pleasure to acknowledge the support of the U.S. Army Research Office at Durham, North Carolina. REFERENCES [1] M. H. Anderson, J.R. Ensher, M.R. Matthews, C. E. Wieman, and E. A. Cornell, Science 69, 198 (1995). [] K. B. Davis, M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterie, Phys. Rev. Lett. 75, 3969 (1995). [3] C. C. Bradley, C. A. Sackett, J. J. Tolett, and R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995); C. C. Bradley, C. A. Sackett and R. G. Hulet, ibid. 78, 985 (1997). [4] F. Dalfovo, S. Giorgini, L. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999). [5] S. L. Cornish, N..R. Claussen, J..L. Roberts, E..A. Cornell and C. E. Wieman, Phys. Rev. Lett. 85, 1795 (000). [6] D. S. Lewart, V. R. Pandharipande, and S. C. Pieper, Phys. Rev. B 37, 4950 (1988). [7] P. O. Löwdin, Phys. Rev. 97, 1474 (1955). [8] L. Onsager and O. Penrose, Phys. Rev. 104, 576 (1956). [9] J. L. DuBois and H. R. Glyde, Phys. Rev. A, Dec. (000). [10] N. N. Bogoliubov, it J. Phys. (Moscow) 11, 3 (1947). [11] E. P. Gross, Nuovo Cimento 0, 454 (1961). [1] L. P. Pitaevskii, Ah. Eksp. Teor. Fiz. 40, 646 (1961) [Sov. Phys. JETP 13, 451 (1961)]. [13] S. Giorgini, J. Boronat and J. Casulleras, Phys. Rev. A 60, 519 (1999). [14] J. Javanainen, and S. M. Yoo, Phys. Rev. Lett. 76, 161 (1996). [15] F. Dalfovo and S. Stringari, Phys. Rev. A 53, 477 (1996).