- Email: [email protected]
Momentum distribution of Bose-condensed atomic hydrogen
29 October 2001
Physics Letters A 289 (2001) 333–336 www.elsevier.com/locate/pla
Momentum distribution of Bose-condensed atomic hydrogen P. Vignolo ∗ , A. Minguzzi, M.P. Tosi INFM and Classe di Scienze, Scuola Normale Superiore, 56126 Pisa, Italy Received 18 July 2001; accepted 4 September 2001 Communicated by V.M. Agranovich
Abstract We evaluate the kinetic energy and the momentum distribution of a Bose-condensed cloud of hydrogen gas inside a Ioffe– Pritchard trap, using a semiclassical two-fluid approach. The relationship between temperature and mean kinetic energy of the gas depends sensitively on the shape of the confining potential. 2001 Published by Elsevier Science B.V. PACS: 03.75.Fi; 05.30.Jp Keywords: Bose–Einstein condensation; Semiclassical models
1. Introduction Bose–Einstein condensation (BEC) in a gas of spinpolarized atomic hydrogen was achieved [1] after a long quest [2]. At variance from other condensates of atomic vapours, hydrogen condensation could not be detected by the absorption imaging technique; rather, BEC was inferred through the measurement of the 1s–2s two-photon absorption spectrum. Both the Doppler-free and the Doppler-sensitive components of the spectrum were measured. The Doppler-free spectrum shows an asymmetric feature and a further peak which has been related to the presence of the condensate. Several explanations have been proposed for this spectrum [3,4], the interpretation being rooted to the fact that the condensate is experiencing strong fluctuations during the measurement process in the extremely elongated geometry of the confining Ioffe–Pritchard
* Corresponding author.
E-mail address: [email protected] (P. Vignolo).
magnetic trap [5]. These fluctuations affect much less the Doppler-sensitive portion of the spectrum, since it is already broadened in the momentum exchange process between photons and atoms. In this Letter we focus on the Doppler-sensitive portion of the spectrum, which is directly related to the momentum distribution of the cloud. The spectrum shows a large contribution from the thermal component and a small condensate peak on top of it. We do not treat here the condensate peak, since its width is dominated by the finite resolution of the measurement process [1]. We analyze instead the momentum distribution of the thermal component of the gas by a previously developed two-fluid model [6,7], fully taking into account the effect of the anharmonicity of the Ioffe–Pritchard trap [8]. We then extract the temperature of the gas from the width of the thermal cloud distribution, given the number of atoms in the cloud. The value that we obtain in this way for the temperature of the gas depends critically on the fraction of condensed atoms and differs significantly from an estimate based on the assumption of harmonic confinement.
0375-9601/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 6 2 5 - 9
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P. Vignolo et al. / Physics Letters A 289 (2001) 333–336
2. The model In the hydrogen experiment [1] a number of atoms of the order of N ∼ 1010 is trapped inside an axially symmetric Ioffe–Pritchard potential VIP (r) given by VIP (r) = (αr⊥ )2 + (βz2 + γ )2 − γ , (1) where r⊥ and z are cylindrical coordinates and α, β and γ are parameters which depend on the magnetic coil geometry and on the magnetic field intensity. For small displacements potential (1) reduces to harmonic oscillator confinement, 1 2 2 VIP (r) → Vho (r) = m ω⊥ r⊥ + ωz2 z2 , 2 r⊥ γ /α, z γ /2β, (2) √ where ω⊥ = √ α/ mγ is the radial oscillation frequency, ωz = 2β/m is the axial frequency and aho = (h¯ /mω)1/2 is the characteristic length of the system corresponding to the geometrical average of the ra2 ω )1/3 . The acdial and axial frequencies, ω = (ω⊥ z tual values of the parameters in the experiment are ω⊥ = 2π × 3.90 kHz and ωz = 2π × 10.2 Hz, and the number of atoms is such that the anharmonicity of the trap is explored in a significant manner. The hydrogen gas is cooled down to a temperature of the order of 0.8Tc , where Tc is the critical temperature for Bose–Einstein condensation. In these conditions both a thermal cloud and a condensate are present in the trap. In order to describe these two components we use a two-fluid approach, adopting the semiclassical Hartree–Fock scheme (HF) for the thermal cloud and the Thomas–Fermi approximation (TF) for the condensate. This mean-field approach is justified by the fact that in the hydrogen experiment the gas is in a very dilute regime: indeed, the dilution parameter is n0 a 3 10−9 , where n0 is the density at the center of the trap and a is the s-wave scattering length (a = 6.48 × 10−2 nm for hydrogen [9]). For an inhomogeneous gas the HF scheme yields results in agreement with the predictions of the full Bogoliubov theory [6]. More specifically, the semiclassical approximation is valid for a bosonic cloud when its temperature is appreciably larger than the quantum-level spacing. To estimate whether this condition is fulfilled in the hydrogen experiment we can use the quantumlevel spacing of the approximate harmonic oscillator
potential (Eq. (2)), and we obtain kB T 270h¯ ω⊥ 105hω ¯ z . The TF approximation for the condensate of a gas with repulsive interactions like hydrogen is valid in the strong coupling limit Nc a/aho 1, i.e., when the number Nc of atoms in the condensate is large. This approximation, which is therefore valid everywhere except in the critical region, corresponds to neglecting the kinetic energy term in the Gross–Pitaevskii equation. The use of the semiclassical two-fluid model to describe the equilibrium properties of the hydrogen gas in the experiment is therefore fully justified in the calculations that we present below. The momentum distributions of the thermal cloud fT (p) and of the condensate fc (p) are given by the Bose distribution in an effective potential and by the square modulus of the Fourier transform of the wavefunction of the condensate:
2 1 p exp + Veff (r) − µ fT (p) = kB T 2m −1 −1
d 3r
(3)
and
2 −ip·r 3 fc (p) = φc (r)e (4) d r . √ Here φc (r) = nc (r), the square root of the density profile of the condensate. In Eq. (3) the effective potential acting on the thermal cloud is given by Veff (r) = VIP (r) + 2gnc (r) + 2gnT (r),
(5)
nT (r) being the density profile of the thermal cloud. Finally, we have nc (r) =
and nT (r) =
1 µ − VIP (r) − 2gnT (r) g × θ µ − VIP (r) − 2gnT (r)
exp
1 kB T
−1 −1
p2 + Veff (r) − µ 2m
d 3p . (2π)3
(6)
(7)
The coupling constant g is determined by the s-wave scattering length according to g = 4π h¯ 2 a/m, and the
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chemical potential µ is fixed by the total number N of atoms through the relation
nc (r) + nT (r) d 3 r = N. (8) These equations complete the self-consistent closure of the model.
3. Results 3.1. Density of states The thermodynamic properties of a confined Bose– Einstein gas differ from those of a homogeneous gas and strongly depend on the shape of the confining potential [10]. The effect of the potential can most clearly be shown by calculating the single-particle density of states of the system, which in a semiclassical approximation reads [11] (2m)3/2 ρ(E) = (9) d 3 r E − V (r), 3 2 4π h¯ where V (r) is the confining potential. We have investigated how good is the harmonic approximation (2) to the Ioffe–Pritchard potential (1). As is shown in Fig. 1, the density of states of the Ioffe– Pritchard potential is substantially larger than that of the harmonic oscillator potential, implying a lower critical temperature for Bose–Einstein condensation.
Fig. 1. Density of states (in units of (hω ¯ ⊥ )−1 ) as function of energy , corresponding to a Ioffe–Pritchard (solid line) and a in units of hω ¯ ⊥ harmonic oscillator potential (dashed line). The parameters are those of the hydrogen experiment [1].
3.2. Thermal momentum distribution and kinetic energy In the experiments on atomic hydrogen [1] the momentum distribution has been deduced from the Doppler-sensitive spectrum as measured along the weakly confining axis of the trap. The temperature of the cloud can be inferred from the momentum distribution of the thermal component by taking its second moment, which gives the mean kinetic energy. We find that for a fixed number of atoms in the trap (which is known within large error-bars) the profile of the momentum distribution, the mean kinetic energy and the condensate fraction depend sensitively on the choice of the shape of the confinement. Therefore the extrapolation of the temperature of the gas from the
Fig. 2. Kinetic energy versus temperature for an interacting Bose gas in a Ioffe–Pritchard (full symbols) and a harmonic oscillator potential (open symbols) at different values of the total number of particles: N = 22 × 109 (triangles up), N = 18.3 × 109 (triangles down), N = 15 × 109 (squares) and N = 10.5 × 109 (circles). The horizontal line locates the kinetic energy reported from the hydrogen experiment [1].
experimental data will depend on the model chosen to describe the trap. Fig. 2 illustrates the role of both the total number of atoms and the trapping potential on the kinetic energy, which is shown as a function of temperature. As
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P. Vignolo et al. / Physics Letters A 289 (2001) 333–336
Fig. 3. Axial momentum distribution of the thermal component of a gas with N = 15 × 109 hydrogen atoms for different temperature T , kinetic energy Ekin = pz2 /2m/kB N h¯ ω⊥ and external potential Vext . The continuous line corresponds to T = 43 µK, Ekin = 40 µK and Vext = VIP (point 1 in Fig. 2), the dotted–dashed line to T = 53 µK, Ekin = 40 µK and Vext = Vho (point 2 in Fig. 2), and the dashed line to T = 43 µK, Ekin = 14 µK and Vext = Vho (point 3 in Fig. 2).
(h¯ mω⊥ )1/2 , with the values of m and ω⊥ from the experiment [1]. The continuous line in Fig. 3 corresponds to the case of the Ioffe–Pritchard potential and to the value of the kinetic energy measured in the hydrogen experiment (point 1 in Fig. 2). The dashed– dotted line and the dashed line give the momentum distributions of the gas in a harmonic potential having the same kinetic energy (point 2 in Fig. 2) or the same temperature (point 3 in Fig. 2) as the one in the Ioffe–Pritchard potential. In conclusion, we have shown how the choice of the model for the external confining potential influences the value of the temperature which is read from the curve of the momentum distribution. This is related to the dependence of the density of states on the external potential from which all the thermodynamic properties are derived. The estimate of the gas temperature and of the condensate fraction is important in understanding other properties of the gas, such as the collective excitation frequencies and damping rates. Acknowledgements
predicted from the study of the density of states, in the case of the Ioffe–Pritchard potential the critical temperature is lower than in that of a harmonic oscillator trap. Thus, for given kinetic energy and number of atoms, the temperature of the gas is lower in the Ioffe–Pritchard confinement. Using a total number of atoms equal to N = 15 × 109 and a mean kinetic energy corresponding to 40 µK from the experiment [1], in the case of the Ioffe–Pritchard potential we obtain Nc /N 7% and Nc = 1.1 × 109 , in good agreement with the experimental data Nc /N = 5+4 −2 %, Nc = (1.1 ± 0.6) × 109 . The value of the gas temperature extracted from the model is T = 43 µK for the Ioffe–Pritchard potential, while for the harmonic potential we would have T = 53 µK. The shape of the momentum distribution also depends on the model chosen to describe the confinement. In correspondence to the three points indicated by circles in Fig. 2, we report in Fig. 3 the axial momentum distribution of the thermal cloud, 2 d p⊥ fT (p), FT (pz ) = (10) (2π)2 at given total number of atoms. The units of length and momentum are aho ⊥ = (h¯ /mω⊥ )1/2 and pho ⊥ =
We acknowledge support from MIUR through PRIN2000 and from INFM through PRA2001.
References [1] D.G. Fried, T.C. Killian, L. Willmann, D. Landhuis, S.C. Moss, D. Kleppner, T.J. Greytak, Phys. Rev. Lett. 81 (1998) 3811. [2] A. Griffin, D.W. Snoke, S. Stringari (Eds.), Bose–Einstein Condensation, Cambridge University Press, Cambridge, 1995. [3] R. Coté, V. Kharchenko, Phys. Rev. Lett. 83 (1999) 2100. [4] C.J. Pethick, H.T.C. Stoof, Phys. Rev. A 64 (2001) 013618. [5] C.W. Gardiner, cond-mat/0009371. [6] A. Minguzzi, S. Conti, M.P. Tosi, J. Phys. Condens. Matter 9 (1997) L33. [7] A. Minguzzi, P. Vignolo, M.P. Tosi, Phys. Rev. A 62 (2000) 023604. [8] The effects of the anharmonicity of the external potential on the thermodynamic properties of the cloud were investigated by B. Pozzi, L. Salasnich, A. Parola, L. Reatto, Eur. Phys. J. D 11 (2000) 367. [9] M.J. Jamieson, A. Dalgarno, M. Kimura, Phys. Rev. A 51 (1995) 2626. [10] V.V. Goldman, I.F. Silvera, A.J. Leggett, Phys. Rev. B 24 (1981) 2870. [11] V. Bagnato, D.E. Pritchard, D. Kleppner, Phys. Rev. A 35 (1987) 4354.
Physics Letters A 289 (2001) 333–336 www.elsevier.com/locate/pla
Momentum distribution of Bose-condensed atomic hydrogen P. Vignolo ∗ , A. Minguzzi, M.P. Tosi INFM and Classe di Scienze, Scuola Normale Superiore, 56126 Pisa, Italy Received 18 July 2001; accepted 4 September 2001 Communicated by V.M. Agranovich
Abstract We evaluate the kinetic energy and the momentum distribution of a Bose-condensed cloud of hydrogen gas inside a Ioffe– Pritchard trap, using a semiclassical two-fluid approach. The relationship between temperature and mean kinetic energy of the gas depends sensitively on the shape of the confining potential. 2001 Published by Elsevier Science B.V. PACS: 03.75.Fi; 05.30.Jp Keywords: Bose–Einstein condensation; Semiclassical models
1. Introduction Bose–Einstein condensation (BEC) in a gas of spinpolarized atomic hydrogen was achieved [1] after a long quest [2]. At variance from other condensates of atomic vapours, hydrogen condensation could not be detected by the absorption imaging technique; rather, BEC was inferred through the measurement of the 1s–2s two-photon absorption spectrum. Both the Doppler-free and the Doppler-sensitive components of the spectrum were measured. The Doppler-free spectrum shows an asymmetric feature and a further peak which has been related to the presence of the condensate. Several explanations have been proposed for this spectrum [3,4], the interpretation being rooted to the fact that the condensate is experiencing strong fluctuations during the measurement process in the extremely elongated geometry of the confining Ioffe–Pritchard
* Corresponding author.
E-mail address: [email protected] (P. Vignolo).
magnetic trap [5]. These fluctuations affect much less the Doppler-sensitive portion of the spectrum, since it is already broadened in the momentum exchange process between photons and atoms. In this Letter we focus on the Doppler-sensitive portion of the spectrum, which is directly related to the momentum distribution of the cloud. The spectrum shows a large contribution from the thermal component and a small condensate peak on top of it. We do not treat here the condensate peak, since its width is dominated by the finite resolution of the measurement process [1]. We analyze instead the momentum distribution of the thermal component of the gas by a previously developed two-fluid model [6,7], fully taking into account the effect of the anharmonicity of the Ioffe–Pritchard trap [8]. We then extract the temperature of the gas from the width of the thermal cloud distribution, given the number of atoms in the cloud. The value that we obtain in this way for the temperature of the gas depends critically on the fraction of condensed atoms and differs significantly from an estimate based on the assumption of harmonic confinement.
0375-9601/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 6 2 5 - 9
334
P. Vignolo et al. / Physics Letters A 289 (2001) 333–336
2. The model In the hydrogen experiment [1] a number of atoms of the order of N ∼ 1010 is trapped inside an axially symmetric Ioffe–Pritchard potential VIP (r) given by VIP (r) = (αr⊥ )2 + (βz2 + γ )2 − γ , (1) where r⊥ and z are cylindrical coordinates and α, β and γ are parameters which depend on the magnetic coil geometry and on the magnetic field intensity. For small displacements potential (1) reduces to harmonic oscillator confinement, 1 2 2 VIP (r) → Vho (r) = m ω⊥ r⊥ + ωz2 z2 , 2 r⊥ γ /α, z γ /2β, (2) √ where ω⊥ = √ α/ mγ is the radial oscillation frequency, ωz = 2β/m is the axial frequency and aho = (h¯ /mω)1/2 is the characteristic length of the system corresponding to the geometrical average of the ra2 ω )1/3 . The acdial and axial frequencies, ω = (ω⊥ z tual values of the parameters in the experiment are ω⊥ = 2π × 3.90 kHz and ωz = 2π × 10.2 Hz, and the number of atoms is such that the anharmonicity of the trap is explored in a significant manner. The hydrogen gas is cooled down to a temperature of the order of 0.8Tc , where Tc is the critical temperature for Bose–Einstein condensation. In these conditions both a thermal cloud and a condensate are present in the trap. In order to describe these two components we use a two-fluid approach, adopting the semiclassical Hartree–Fock scheme (HF) for the thermal cloud and the Thomas–Fermi approximation (TF) for the condensate. This mean-field approach is justified by the fact that in the hydrogen experiment the gas is in a very dilute regime: indeed, the dilution parameter is n0 a 3 10−9 , where n0 is the density at the center of the trap and a is the s-wave scattering length (a = 6.48 × 10−2 nm for hydrogen [9]). For an inhomogeneous gas the HF scheme yields results in agreement with the predictions of the full Bogoliubov theory [6]. More specifically, the semiclassical approximation is valid for a bosonic cloud when its temperature is appreciably larger than the quantum-level spacing. To estimate whether this condition is fulfilled in the hydrogen experiment we can use the quantumlevel spacing of the approximate harmonic oscillator
potential (Eq. (2)), and we obtain kB T 270h¯ ω⊥ 105hω ¯ z . The TF approximation for the condensate of a gas with repulsive interactions like hydrogen is valid in the strong coupling limit Nc a/aho 1, i.e., when the number Nc of atoms in the condensate is large. This approximation, which is therefore valid everywhere except in the critical region, corresponds to neglecting the kinetic energy term in the Gross–Pitaevskii equation. The use of the semiclassical two-fluid model to describe the equilibrium properties of the hydrogen gas in the experiment is therefore fully justified in the calculations that we present below. The momentum distributions of the thermal cloud fT (p) and of the condensate fc (p) are given by the Bose distribution in an effective potential and by the square modulus of the Fourier transform of the wavefunction of the condensate:
2 1 p exp + Veff (r) − µ fT (p) = kB T 2m −1 −1
d 3r
(3)
and
2 −ip·r 3 fc (p) = φc (r)e (4) d r . √ Here φc (r) = nc (r), the square root of the density profile of the condensate. In Eq. (3) the effective potential acting on the thermal cloud is given by Veff (r) = VIP (r) + 2gnc (r) + 2gnT (r),
(5)
nT (r) being the density profile of the thermal cloud. Finally, we have nc (r) =
and nT (r) =
1 µ − VIP (r) − 2gnT (r) g × θ µ − VIP (r) − 2gnT (r)
exp
1 kB T
−1 −1
p2 + Veff (r) − µ 2m
d 3p . (2π)3
(6)
(7)
The coupling constant g is determined by the s-wave scattering length according to g = 4π h¯ 2 a/m, and the
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chemical potential µ is fixed by the total number N of atoms through the relation
nc (r) + nT (r) d 3 r = N. (8) These equations complete the self-consistent closure of the model.
3. Results 3.1. Density of states The thermodynamic properties of a confined Bose– Einstein gas differ from those of a homogeneous gas and strongly depend on the shape of the confining potential [10]. The effect of the potential can most clearly be shown by calculating the single-particle density of states of the system, which in a semiclassical approximation reads [11] (2m)3/2 ρ(E) = (9) d 3 r E − V (r), 3 2 4π h¯ where V (r) is the confining potential. We have investigated how good is the harmonic approximation (2) to the Ioffe–Pritchard potential (1). As is shown in Fig. 1, the density of states of the Ioffe– Pritchard potential is substantially larger than that of the harmonic oscillator potential, implying a lower critical temperature for Bose–Einstein condensation.
Fig. 1. Density of states (in units of (hω ¯ ⊥ )−1 ) as function of energy , corresponding to a Ioffe–Pritchard (solid line) and a in units of hω ¯ ⊥ harmonic oscillator potential (dashed line). The parameters are those of the hydrogen experiment [1].
3.2. Thermal momentum distribution and kinetic energy In the experiments on atomic hydrogen [1] the momentum distribution has been deduced from the Doppler-sensitive spectrum as measured along the weakly confining axis of the trap. The temperature of the cloud can be inferred from the momentum distribution of the thermal component by taking its second moment, which gives the mean kinetic energy. We find that for a fixed number of atoms in the trap (which is known within large error-bars) the profile of the momentum distribution, the mean kinetic energy and the condensate fraction depend sensitively on the choice of the shape of the confinement. Therefore the extrapolation of the temperature of the gas from the
Fig. 2. Kinetic energy versus temperature for an interacting Bose gas in a Ioffe–Pritchard (full symbols) and a harmonic oscillator potential (open symbols) at different values of the total number of particles: N = 22 × 109 (triangles up), N = 18.3 × 109 (triangles down), N = 15 × 109 (squares) and N = 10.5 × 109 (circles). The horizontal line locates the kinetic energy reported from the hydrogen experiment [1].
experimental data will depend on the model chosen to describe the trap. Fig. 2 illustrates the role of both the total number of atoms and the trapping potential on the kinetic energy, which is shown as a function of temperature. As
336
P. Vignolo et al. / Physics Letters A 289 (2001) 333–336
Fig. 3. Axial momentum distribution of the thermal component of a gas with N = 15 × 109 hydrogen atoms for different temperature T , kinetic energy Ekin = pz2 /2m/kB N h¯ ω⊥ and external potential Vext . The continuous line corresponds to T = 43 µK, Ekin = 40 µK and Vext = VIP (point 1 in Fig. 2), the dotted–dashed line to T = 53 µK, Ekin = 40 µK and Vext = Vho (point 2 in Fig. 2), and the dashed line to T = 43 µK, Ekin = 14 µK and Vext = Vho (point 3 in Fig. 2).
(h¯ mω⊥ )1/2 , with the values of m and ω⊥ from the experiment [1]. The continuous line in Fig. 3 corresponds to the case of the Ioffe–Pritchard potential and to the value of the kinetic energy measured in the hydrogen experiment (point 1 in Fig. 2). The dashed– dotted line and the dashed line give the momentum distributions of the gas in a harmonic potential having the same kinetic energy (point 2 in Fig. 2) or the same temperature (point 3 in Fig. 2) as the one in the Ioffe–Pritchard potential. In conclusion, we have shown how the choice of the model for the external confining potential influences the value of the temperature which is read from the curve of the momentum distribution. This is related to the dependence of the density of states on the external potential from which all the thermodynamic properties are derived. The estimate of the gas temperature and of the condensate fraction is important in understanding other properties of the gas, such as the collective excitation frequencies and damping rates. Acknowledgements
predicted from the study of the density of states, in the case of the Ioffe–Pritchard potential the critical temperature is lower than in that of a harmonic oscillator trap. Thus, for given kinetic energy and number of atoms, the temperature of the gas is lower in the Ioffe–Pritchard confinement. Using a total number of atoms equal to N = 15 × 109 and a mean kinetic energy corresponding to 40 µK from the experiment [1], in the case of the Ioffe–Pritchard potential we obtain Nc /N 7% and Nc = 1.1 × 109 , in good agreement with the experimental data Nc /N = 5+4 −2 %, Nc = (1.1 ± 0.6) × 109 . The value of the gas temperature extracted from the model is T = 43 µK for the Ioffe–Pritchard potential, while for the harmonic potential we would have T = 53 µK. The shape of the momentum distribution also depends on the model chosen to describe the confinement. In correspondence to the three points indicated by circles in Fig. 2, we report in Fig. 3 the axial momentum distribution of the thermal cloud, 2 d p⊥ fT (p), FT (pz ) = (10) (2π)2 at given total number of atoms. The units of length and momentum are aho ⊥ = (h¯ /mω⊥ )1/2 and pho ⊥ =
We acknowledge support from MIUR through PRIN2000 and from INFM through PRA2001.
References [1] D.G. Fried, T.C. Killian, L. Willmann, D. Landhuis, S.C. Moss, D. Kleppner, T.J. Greytak, Phys. Rev. Lett. 81 (1998) 3811. [2] A. Griffin, D.W. Snoke, S. Stringari (Eds.), Bose–Einstein Condensation, Cambridge University Press, Cambridge, 1995. [3] R. Coté, V. Kharchenko, Phys. Rev. Lett. 83 (1999) 2100. [4] C.J. Pethick, H.T.C. Stoof, Phys. Rev. A 64 (2001) 013618. [5] C.W. Gardiner, cond-mat/0009371. [6] A. Minguzzi, S. Conti, M.P. Tosi, J. Phys. Condens. Matter 9 (1997) L33. [7] A. Minguzzi, P. Vignolo, M.P. Tosi, Phys. Rev. A 62 (2000) 023604. [8] The effects of the anharmonicity of the external potential on the thermodynamic properties of the cloud were investigated by B. Pozzi, L. Salasnich, A. Parola, L. Reatto, Eur. Phys. J. D 11 (2000) 367. [9] M.J. Jamieson, A. Dalgarno, M. Kimura, Phys. Rev. A 51 (1995) 2626. [10] V.V. Goldman, I.F. Silvera, A.J. Leggett, Phys. Rev. B 24 (1981) 2870. [11] V. Bagnato, D.E. Pritchard, D. Kleppner, Phys. Rev. A 35 (1987) 4354.