Collapse of a Bose gas: Kinetic approach

Transcription

1 PRAMANA c Indian Academy of Sciences Vo. 79, No. 2 journa of August 2012 physics pp Coapse of a Bose gas: Kinetic approach SHYAMAL BISWAS Department of Physics, University of Cacutta, 92 A.P.C. Road, Kokata , India E-mai: sbiswas.phys.cu@gmai.com MS received 3 January 2012; revised 21 March 2012; accepted 28 March 2012 Abstract. We have anayticay expored the temperature dependence of critica number of partices for the coapse of a harmonicay trapped attractivey interacting Bose gas beow the condensation point by introducing a kinetic approach within the Hartree Fock approximation. The temperature dependence obtained by this easy approach is consistent with that obtained from the scaing theory. Keywords. Thermodynamica, statistica and static properties of condensates; Utracod and trapped gases; matter waves. PACS Nos Bc; d; Hh 1. Introduction For an utracod Bose gas, interpartice interaction is characterized by s-wave scattering ength (a s ) which can be tuned arbitrariy by the Feshbach resonance method [1]. For attractive (a s <0) interaction, a harmonicay trapped Bose gas tends to increase its density in the centra region of a trap. This tendency, of course, is opposed by the quantum and therma fuctuations. If the number of atoms is greater than the critica number (N c ), then the centra density increases strongy, and the zero-point and therma fuctuations are no onger abe to avoid the coapse of the gas. Consequenty, the gas becomes unstabe even for two-body interaction. The stabiity and coapse of the Bose Einstein condensates with negative scattering engths have aready been observed in the couds of utracod 7 Li [2] and 85 Rb [3] for temperatures (T ) cose to zero or we beow the condensation point (T c ). Soon after the observation, many theories for the coapse have been proposed for T 0 [4 10], as we as for T > 0 [11 14]. The remarkabe one among these (theories) was given by Baym and Pethick [4]. They proposed a scaing theory for T = 0, and we generaized their theory for 0 T T c within the Hartree Fock (H F) approximation [14]. In the generaized theory, different parts of the free energy (grand potentia) of our system were scaed by a parameter which reduces the ength scae of the system as a resut of attractive DOI: /s ; epubication: 20 Juy

2 Shyama Biswas interaction; and a critica number for the coapse was eventuay cacuated from a critica condition of existence of a metastabe minimum of the grand potentia [14]. In this brief report we sha cacuate the same, but in a kinetic approach which is supposed to be the easiest way. This time, for cacuating the critica number, we sha not start from the free energy, but sha adopt a mere kinetic theory ike approach based on the energy and pressure of the system. We sha start from the H F energy of the system, and sha pick up the kinetic energy and interaction energy parts of the H F energy. Whie the kinetic energy of the partices causes an outward pressure, the attractive interaction causes an inward pressure. For critica number of partices, magnitudes of the two pressures woud be the same. Beyond the critica number of partices, the inward pressure woud be arger than the outward one, and as a consequence, the whoe system woud coapse. Thus, we sha cacuate the critica number, and sha show its temperature dependence. Our present technique is easier than that of the aready existing theory [4,14] because outward and inward pressures appear as the first-order derivative of the two parts of energy with respect to the effective voume of the system, and the aready existing technique invoves a second-order derivative of the grand potentia with respect to the scaing parameter for obtaining the critica condition of existence of its metastabe minimum. 2. Quaitative resut Before going into the detais of the kinetic approach, et us estimate the critica number by quaitative manner. Our system consists of a arge number (N) of Bose partices, each of which is a three-dimensiona isotropic harmonic osciator with anguar frequency ω and mass m. The system, of course, is in thermodynamic equiibrium with its surroundings at temperature T.ForT 0, a the partices occupy the ground state, and the system can be we described by the ground state wave function 0 (r) = N/( 3 π 3/2 )e r 2 /2 2 in the position (r) space, where = /mω is the confining ength scae of the osciators [15]. Thus, for T = 0, the density of the condensed partices is given by [15] n 0 (r) = 0 (r) 2 = N 3 π 3/2 e r 2 / 2. (1) On the other hand, the number density of the excited partices (in the absence of interaction) is given by [14,16] n T (r) = 1 λ 3 T g 3/2 (e mω2 r 2 /2k B T ), (2) where λ T = 2π 2 /mk B T is the therma de Brogie waveength and g 3/2 (x) = x + x 2 /2 3/2 + x 3 /3 3/2 + is a Bose Einstein function of the rea variabe x. Let us consider the attractive interaction potentia as V int (r) = gδ 3 (r), where g = 4π 2 a/m is the couping constant and a = a s is the absoute vaue of the s-wave scattering ength [16 18]. Typica two-body interaction energy for N number of partices is N 2 g/2 3. Due to this, the gas tends to increase the density at the centra region of the trap. We beow T c (i.e. for T 0), this tendency is resisted by the zero-point motion of the atoms. At the critica situation, the energy (3N ω/2) for the zero-point 320 Pramana J. Phys., Vo. 79, No. 2, August 2012

3 Coapse of a Bose gas: Kinetic approach motion must be comparabe to the typica interaction energy. So, for T 0, we must have 3N c ω/2 Nc 2g/23 at the critica situation. From this reation we may have (/) 1. On the other hand, for 0 < T < T c, the typica energy of the system is 3 ω[kt/ ω] 4 ζ(4) N 4/3 ω [15]. So, for this range of temperatures, we must have Nc 4/3 ω Nc 2g/23 at the critica situation. From this reation, we may have (/) [/a] 1/2 > 1. However, near T c, the ength scae of the system is L Tc k B T c / ω N 1/6 [15] so that we can write N 4/3 c ω N 2 c g/2l3 T c at the critica situation, and consequenty, we may have (/) [/a] Quantitative resut from scaing theory From the above quaitative arguments we have got (/) 1 for T 0, (/) [/a] 1/2 for 0 < T < T c, and (/) [/a] 5 for T = T c. The proportionaity constants were determined by a scaing theory within H F approximation [14]. The scaing theory gives the proportionaity constants as for T = 0 [4,14], 1.210(T/T c ) 6 /(1 (T/T c ) 3 ) 3 for 0 < T < T c [14], and for T = T c [14]. In the foowing, we sha aso cacuate the same but in a different (kinetic) approach. This approach is supposed to be the easiest one, and, we want to know whether the easiest approach reproduces simiar resuts. 4. Kinetic approach Within the H F approximation we can express the energy functiona as [14,16] [ E = d 3 2 r 2m n 0 φ m n i φ i 2 i =0 + V (r)n 0 (r) + V (r)n T (r) + g 2 n2 0 (r) + 2gn 0 (r)n T (r) + gn 2 T (r) ], (3) where {φ i } represents the set of normaized H F wave functions and {n i } represent their occupations. To evauate the above energy functiona of eq. (3) we have to know n 0 (r) and n T (r) for the interacting case. But, for the simpicity of cacuation, we keep the forms of n 0 (r) and n T (r) as in eqs (1) and (2) [14,16]. Thus, we evauate the energy functiona in eq. (3) as [19] where E(t) =[(1 + 1)c 1 (t) c 2 (t)]n 2 m 2, (4) t = T/T c, c 1 (t) = 3 4 (1 t 3 ) N 1/3 ζ(4)t 4 [ζ(3)] 4/3, (5) Pramana J. Phys., Vo. 79, No. 2, August

4 and Shyama Biswas c 2 (t) = 1 Na 8ζ(3/2) [1 t 3 ] 2 N 1/2 a + t 3/2 [1 t 3 ] 2π π + S 2 N 1/2 a t 9/2, (6) π[ζ(3)] 3 S = i, j= (ij) 3/2 (i + j) 3/2 From eq. (4) we get the kinetic energy of the system as E k (t) = c 1 (t)n 2 m 2, (7) and the interaction energy of the system as where E int (t) = c 2 (t)n 2 m 2 = c 3(t)N 2 a 2 m 3, (8) c 3 (t) = c 2 (t) Na. (9) For 0 T < T c, the effective voume of the system is given by V = 4π 3 /3. The outward pressure is P out = E k / V, and the inward pressure is P in = E int / V.Forthe critica number of partices, P in woud be equa and opposite to P out. From this equaity reation, we can write the expression of the critica number as N c = 2c ( ) 1(t) 3V 1/3. (10) 3ac 3 (t) 4π 4.1 Resut for T 0 Equation (10) gives the expression of the critica number as = 2c 1(0) π 3c 3 (0) = = for T 0. (11) 2 Now we see that our new resut for T 0 is approximatey two times arger than the anaytic resut (0.671) obtained from the scaing theory [4,14]. The numerica simuation and experimenta resuts, for T 0, however, are [20] and [3] respectivey. 4.2 Resuts for 0 < T < T c To achieve the Bose Einstein condensation, the necessary condition is such that a/ 1 [16]. Thus, for 0 < T T c, we can write c 1 (t) 3 2 N 1/3 ζ(4)t 4 [ζ(3)] 4/3 322 Pramana J. Phys., Vo. 79, No. 2, August 2012

5 Coapse of a Bose gas: Kinetic approach T c Figure 1. Soid ine foows from eq. (12) and represents the pot for critica number of partices (in units of /a) with respect to temperature (in units of T c ). Here (/a) = [3]. Dotted ine represents the resuts of the scaing theory [14]. T from eq. (5). Simiary, for 0 < T T c, we can write c 3 (t) (1/ 2π)(1 t 3 ) 2 from eqs (9) and (6). Consequenty, for 0 < T < T c, we get the expression of N c from eq. (10) as N 1/3 c ζ(4)t 4 [ζ(3)] 4/ π [1 t 3 ]. 2 From this reation we can write [ ] 1/2 t 6 = a (1 t 3 ) ( ) a 1/4 3 1 t + O (12) 3 for 0 < T < T c up to a few eading orders in a/. We pot the right-hand side of eq. (12) with respect to T in figure 1 for a/ = [3]. Now, we see that the nature of the temperature dependence of the critica number is the same as that obtained from the scaing theory [14]. 4.3 Resut for T = T c For T = T c, the ength scae of the system is L Tc = N 1/6 [14,16], and the effective voume of the system is given by V = 4π 3 N 1/2 /3. For this reason, the critica condition woud be the same as that in eq. (10) except the factor V which is to be repaced by V/Nc 1/2. So, for T = T c, the critica number woud be obtained from N c = 2c ( ) 1(1) 3V 1/3 3ac 3 (1) 4π Nc 1/2. (13) Putting the vaues of c 1 (1) and c 3 (1) into eq. (13) we get [ ] 5 = for T = T c. (14) a Pramana J. Phys., Vo. 79, No. 2, August

6 Shyama Biswas Now we see that the critica number increases dramaticay as the condensation point is approached. Difference between our resut and the scaing resut [4,14] is appreciabe as because the scaing theory deas with a the terms of H F energy, and the scaed kinetic energy, athough is greater than the scaed potentia energy, yet comparabe in particuar for T 0. On the other hand, potentia energy of the harmonic trap has not been considered in the anaysis of our kinetic approach. 5. Concusion It is to be mentioned that we have considered the coapse of the condensate as we as the therma coud of the Bose gas ony for short-ranged attractive interaction. Coapse of condensates for short-ranged repusive and ong-ranged attractive interactions have aso been investigated experimentay [21] and theoreticay [22,23]. Initiay we expained the physics for the coapse of the attractive atomic Bose gas. Then we quaitativey estimated the critica number of partices for the coapse for 0 T T c. Finay, we cacuate the same by a kinetic approach within the Hartree Fock approximation. Our cacuations support the quaitative estimations and the quantitative resuts obtained from the previous scaing theory [14]. Athough the scaing theory was a more rigorous one, the nature of the temperature dependence of the critica number obtained by our kinetic theory (in figure 1) is simiar to that obtained from the scaing theory. These two theories are equa aprioryand sefconsistent with respect to the basic physics of the coapse. The scaing resut, for T 0, is coser to the experimenta data [3] which, however, is not avaiabe for the entire regime 0 < T T c. The power of our approach is its simpicity. Future experiment for 0 < T T c may justify our approach. Acknowedgements This work has been sponsored by the University Grants Commission (UGC) under DS Kothari Postdoctora Feowship Scheme [No. F.4-2/2006(BSR)/13-280/2008(BSR)]. Usefu discussions with J K Bhattacharjee of SNBNCBS are gratefuy acknowedged. Financia support and hospitaity of IACS are aso acknowedged for the initia part of this work. References [1] S Inouye, M R Andrews, J Stenger, H-J Miesner, D M S Kurn and W Kettere, Nature (1998) [2] C C Bradey, C A Sackett and R G Huet, Phys. Rev. Lett. 78, 985 (1997) [3] J L Roberts, N R Caussen, S L Cornish, E A Doney, E A Corne and C E Wieman, Phys. Rev. Lett. 86, 4211 (2001) [4] G Baym and C J Pethick, Phys. Rev. Lett. 76, 6 (1996) [5] L P Pitaevskii, Phys. Lett. A221, 14 (1996) [6] Y Kagan, E L Surkov and G V Shyapnikov, Phys. Rev. Lett. 54, R1753 (1996) [7] M Ueda and A J Leggett, Phys. Rev. Lett. 80, 1576 (1998) 324 Pramana J. Phys., Vo. 79, No. 2, August 2012

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