Eur. Phys. J. D (2013) 67: 185 DOI: 10.1140/epjd/e2013-40243-x
THE EUROPEAN PHYSICAL JOURNAL D
Regular Article
Thermodynamic properties of a rotating Bose-Einstein condensation in a harmonic plus quartic trap Tharwat M. El-Sherbini1 , Doaa Hassan1,a , Abdelhamid A. Galal1,b , and Ahmed S. Hassan2 1 2
Department of Physics, Faculty of Science, Cairo University, Giza, Egypt Department of Physics, Faculty of Science, El Minia University, El Minia, Egypt Received 16 April 2013 / Received in final form 9 June 2013 c EDP Sciences, Societ` Published online 19 August 2013 – a Italiana di Fisica, Springer-Verlag 2013 Abstract. In this paper, the thermodynamic properties of a rotating Bose-Einstein condensate confined in a harmonic plus quartic potential are calculated using a modified semiclassical approximation. We determined the condensate fraction, critical temperature and heat capacity of the condensate using a method that takes into account deviations from ideal gas behavior due to the effect of finite size and the chemical potential when it is equal to the ground state energy (positive chemical potential). Predictions that can be directly compared with experiment have been found.
1 Introduction The investigation of fast rotating Bose-Einstein condensation (BEC) is one of the central topics in the study of many interesting problems, such as superfluid and quantum Hall physics [1–4]. In a harmonically trapped condensate that rotating at a frequency close to the trap frequency, the centerfugal force balances the trapping force, leads to a break of the confinement. To overcome this centerfugal barrier, Fetter [5–7] suggested adding a quartic term to increase the strength of the radial confinement. Such a trap was realized by the group of Dalibard in what has become known as the Paris experiment [8]. The theoretical investigation of this system has been considered the effective trapping potential [9–13]: Veff =
m 2 2 1 4 m 2 2 2 2 [ωz z + ω⊥ Ω r⊥ , r⊥ ] + κr⊥ − 2 4 2
(1)
2 where m is the mass of the atom, r⊥ = x2 + y 2 is the radial distance perpendicular to the rotation axis, ω⊥ {≡ ωx = ωy }, ωz are the effective trapping frequencies of the harmonic potential, and Ω is the rotation frequency. The parameter κ parametrizes the strength of the anharmonic quartic term, which plays an important role in exploring the region Ω ≥ ω⊥ . A positive value of the parameter κ ensures the confinement of the condensate at all rotation frequencies, as suggested by Fetter [5]. In a previous work, Kling and Pelster [14,15] used the semiclassical approximation to study the thermodynamic properties of a rotating condensate, using Paris’s parameters experiment. The main finding of their study are as a
e-mail:
[email protected] Present address: Physics Department, American University in Cairo, AUC Avenue, New Cairo, Egypt b
follows: condensation is possible even in the over-critical rotation regime, i.e. Ω ≥ ω⊥ . The condensate fraction varies as T ν , where T is the absolute temperature and ν ranged between 5/2 and 3. The heat capacity is discontinuous at the transition temperature. Moreover, it tends to zero in the low temperature limit, in agreement with the thermodynamics third law. While it approached the classical Dulong-Petit limit at high-temperature. Finally, the critical temperature was three times smaller than the temperature measured in the Paris experiment [8]. Stock et al. [10] showed that in the fast rotation regime Ω = ω⊥ the transition temperature T0 = 60 nK, while T0 = 120 nK for a non-rotating gas. However, the above studies did not consider the effects of the finite size and the inter-particle interactions [16–22]. In this respect, Gautam and Angom [23] found that the finite size effect is more pronounced for the quartic confining potential. A step further, we have shown that the effects of repulsive inter-particle interactions that are obtained in a mean field treatment can be realized in a single particle approach based on the density of states (DOS) by employing the positive chemical potential effect [24–27]. In the present work, we extend our modified semiclassical approximation to study the effects of finite size, trap anisotropy, and positive chemical potential on the thermodynamic parameters of a rotating gas in a harmonic plus quartic trap. In this approximation, we replace the sum over the discrete single-particle spectrum by an integral weighted by a density of state (DOS). The DOS was calculated using the high temperature expansion of the partition function [28,29]. Our numerical results are calculated by using the trap parameters of the Paris experiment. This paper is organized as follows: Section 2 outlines the physics of rotating trapped bosons in a combined
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quadratic plus quartic potential and the semiclassical approximation. The condensate fraction, critical temperature, heat capacity, and the Landau level approach for estimating the system temperature are given in ascending Sections from 3–6. In Section 7, discussion and conclusion are given.
2 The semiclassical approach 2.1 The single particle energy eigenvalue In the following, we follow Fetter [5] and Stock [30] to parametrize the energy eigenvalues through a perturbative 4 treatment of the quartic term, 14 κr⊥ . In the rotating frame for Ω ≤ ω⊥ the eigenvalues of the potential equation (1) up to the first order correction are given by [10]: E(nz , nm , mz ) = nz ωz + (2nm + |mz |)ω⊥ (1)
− mz Ω + E|mz | + E0
(2)
with E0 = 12 (2ω⊥ + ωz ), and (1)
E|mz | =
2 κ 2 2 (|mz | + 3|mz | + 2), 4m2 ω⊥
(3)
where nz = 0, 1. . . is the vibrational quantum number along z, nm = 0, 1. . . characterizes the excitation energy in the xy-plane for a state with a given mz , i.e. is the Landau level index, and mz = −nm , −nm + 1, −nm + 2, −nm + 3. . . is the angular momentum quantum number about the rotation axis z. The approximated result in m2 ω 3 equation (2) is valid for |mz | κ ⊥ , which is ∼500 for Paris experiment. The eigenvalues given in equations (2) and (3) can be written in equivalent form. Since mz = −nm , −nm + 1, −nm + 2. . ., consequently, −mz = |nm | for mz < 0. Following Cohen-Tannoudji et al. [31], we have to introduce a non-negative set of quantum numbers (pp. 727–741) for which equations (2) and (3) becomes, E(nz , nm , mz ) = nm (ω⊥ + Ω) −
2 κ 2 2 (nm + 3nm + 2) 4m2 ω⊥
+ mz (ω⊥ − Ω) + × (m2 z + 3mz + 2) + nz ωz + E0 ,
2 κ 2 4m2 ω⊥
(4)
with nz , nm , mz = 0, 1, 2. . . The same result can be obtained by using the classical analog for the radial part of the potential Fetter’s [5]. One of the methods which can be used to simplify considerably the eigenvalues is to restrict our calculation for first and second Landau levels, i.e. nm = 0, 1. Another unsatisfactory approximation which simplify the problem 2 is to neglect the term which contains m2 z and nm in equation (4). The possibility of using this approximation is
that: because the validity of the perturbation method requires that the magnitude of the correction term be small compared to the spacing between the unperturbed levels (in the present case ≤2ω⊥ ). Equation (4) shows that, the higher of the value of mz or nm the smaller of the parameter κ. Thus in order to keep the value of κ in its effective range in excited states we have to omit the term which 2 contains m2 z and nm . However, our primary interest is in the thermodynamic properties of the system which are insensitive to this approximation. Under the above mentioned approximation the energy eigenvalue of the single particle is given by: E(nz , nm , mz ) = nz ωz + mz (ω⊥ − Ω + δ) + nm (ω⊥ + Ω − δ) + E0 .
(5)
The parameter δ plays an important role in the thermodynamical properties behavior of this system. This parameter characterizes the relative strength of the quartic and the quadratic potential. For Paris experiment, it’s value is δ ≈ 0.71157, where κ = 2.6 × 10−11 J m−4 and ω⊥ = 2π × 75.5 Hz for this experiment set up.
2.2 The total number of particles In the grand-canonical ensemble, which we will use to calculate the thermodynamic quantities of interest for rotating BEC, the rotation-dependent chemical potential μ(Ω) is defined by the constraint [16–19,29], N=
∞
Nn (En ) = N0
n=0
+
∞ nz ,nm
ze−βE(nz ,mz ,nm ) , 1 − ze−βE(nz ,mz ,nm ) ,m =1
(6)
z
where Nn (En ) is the occupancy of a level with energy En , z and β = (1/KB T ). The quantity N0 = 1−z is the number of condensed particles in the single-particle ground state. The fugacity z is determined in terms of the rotationdependent chemical potential μ(Ω) and the ground state energy E0 , z = eβ(μ(Ω)−E0 ) . The sum in equation (6) cannot be performed directly and, provided the condition KB T >> (2Ω − δ) is satisfied. For a very large number of particles, it can be approximated by an integral over the single-particle energy , weighted by an appropriate smooth density of states (DOS) ρ( ). Hence, we write N = N0 +
∞ j=1
zj
∞
ρ( )e−jβ d .
(7)
0
Following the method outlined in our previous papers [25], an appropriate DOS for the energy spectrum (4) is
Eur. Phys. J. D (2013) 67: 185
given by:
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equation (11) becomes
μ(Ω) = ηkB T0 γ −11/15 ,
1 2 + 3 2 (ωg ) (ωg )2 ¯ r2 ) μ(Ω) ¯ 3 ω (9¯ ω2 − ω , × + 1 2 ωg 8ωg 2 ωs
ρ( ) = γ
where ωg T0 = kB (8)
with γ = (1 − (α − δ/ω⊥ )2 )−1 , α = Ω/ω⊥ 2 ω ¯ r = ω⊥ [(1/3)(2 + 2α2 + 2δ 2 /ω⊥ + (ωz /ω⊥ )2 )]1/2 2 ωg = [ω⊥ ωz ]1/3 , ω ¯ = (1/3)[2ω⊥ + ωz ] ωs = ω⊥ (1 + α − δ)(2ω⊥ + ωz + 4δ/3).
(13)
N ζ(3)
1/3
is the BEC transition temperature of the ideal Bose gas. In the thermodynamic limit the contribution from this scaling parameter is vanished [33]. In equation (13), μ(Ω) generalizes the well known Thomas-Fermi result holding for magnetically trapped condensate to include the effects of the rotation.
(9)
As illustrated in the subsequent section, this expression for the single-particle DOS implicitly includes inter-particle interaction effects through the dependence of μ(Ω) on the scattering length. 2.3 Chemical potential dependence of the interaction parameter Several authors have discussed the rotation dependence of the chemical potential μ(Ω) for finite number of particles N [5–7,14,15]. Another, simple method to calculate it, is to follow Campbell et al. [32]. In their recent work, they have identified a relevant interaction energy scale to explore the relationship between the non saturation of the ideal Bose gases and the interatomic interactions for the pure harmonically trapped gas. The identified energy scale is given by: 2/5 a ωg μ0 (ωg ) = 15N0 , (10) 2 ahar where a is the s-wave scattering length and ahar = mω g is the ground state spatial extension for the harmonic oscillator. It is clear that the energy in equation (10) is equivalent to the mean-field prediction for the nonrotating chemical potential μ0 (T = 0) of a harmonically trapped gas with N0 condensate atoms in the Thomas-Fermi limit. Since in our system the rotation leads to a shift in the harmonic oscillator frequency, then we have to generalize Hadzibabic result’s to calculate an accurate expression for μ(Ω). However, by using Fetter [5–7] results we have, i.e. 2/5 a ωg −2/5 μ(Ω) = μ(ωg )γ 15N0 = γ −2/5 (11) 2 ahar this results identify the relevant interaction energy scale for the rotating harmonically trapped gas. In terms of Dalfovo’s interaction scaling parameter η [21,22], 1/3
2/5 a 1 ζ(3) 15N0 , (12) η= 2 N ahar
3 Condensate fraction Now, it is straightforward to calculate the total number of particles from equations (7) and (8) 3 2 kB T kB T N = N0+γ g3 (z)+ g2 (z)R(Ω) , (14) ωg ωg
j k where gk (z) = ∞ j=1 (z /j ) is the polylgarithem function, and the parameter R(Ω) is given by:
¯ r2 ) μ(Ω) ¯ (9¯ ω2 − ω 3 ω . + R(Ω) = 2 ωg 4ωg ωs In terms of the reduced temperature t = T /T0 , the condensate fraction is given by: N0 = 1 − γt3 − R1 γt2 . N
(15)
The last term in equations (15) provides the correction to the rotating ideal gas result, which is given by: N 0 = 1 − γt3 . (16) N ideal The parameter R1 in equation (15) is given by 1 ω2 − ω ¯ r2 ) ¯ ζ(3) 3 ζ(2) 3 ω −11/15 (9¯ , (17) R1 = +ηγ ζ(3) 2 ωg N 4ωs where ζ is the Riemann zeta function. The parameter R1 provides simultaneously the main effects that can alter the ideal Bose gas in the same trap. In the bracket the first term gives the finite size effect while the second term accounts for the effect of the chemical potential when becomes equal to the energy of the lowest energy state. The latter is similar to the effect of repulsive interaction provided by the mean-field theory approach. In the thermodynamic limit, the parameter R1 → 0. The temperature dependence of the condensate fraction N0 /N following from equation (15) is shown in Figure 1. This figure shows that the condensate fraction depends crucially on the rotation frequency. It is clear that as
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Eur. Phys. J. D (2013) 67: 185
100 75 Tc
nK
50 25
1
0
0.75 0.25 0.5 0.5 0.25 0.75 1 0
Fig. 3. Critical temperature versus rotation rate α and anharmonic parameter δ. Fig. 1. Simultaneous finite size and the positive chemical potential effects in the condensed fraction as a function of the reduced temperature t. Results of Kling and Pelster of the ideal gas model are shown for comparison [14,15]. Solid, dash, dot, and dash dot lines are the results of Kling and Pelster. Dash dot dot, short dash, short dot, and short dash dot are the results of the present work. The interaction parameter for non-rotating gas is taken to be η = 0.49.
1 0.75
4 Critical temperature Let us now consider the effect of the finite size of the condensate and the positive chemical potential on the transition temperature of the rotating Bose gas trapped in a harmonic plus quartic potential. For both vanishing and non vanishing anharmonicity parameter δ, the critical temperature Tc (Ω) at which the condensation emerges can be obtain by setting N0 /N in equation (15) equal to zero [16–19,29]. Thus,
1 ideal Tc (Ω) = T0 R1 (18) (Ω) 1 − 3 where
N0 0.5 N 0.25
4
0
3 2
0.1 t
1 0.2
Fig. 2. The condensed fraction as a function of the reduced temperature t and the parameter δ for α = 1.0
the rotation frequency increases the characteristic shape of the condensate fraction dependence on temperature, N0 ∼ [1γt3 − R1 γt2 ], changes to N0 ∼ [1γt2 − R1 γt], (the shape of the system looks as a quasi two-dimensional system). This behavior is independent on the form of the DOS. In Figure 2, we represented the condensate fraction as a function of t and δ graphically for very fast rotation rate ω⊥ = Ω. This figure clarify the role of the quartic potential. For δ ∼ 0, the centrifugal force exactly cancels the confining effect of the harmonic potential. For δ = 0, the condensate fraction instead differ from zero for any value of t.
T0ideal (Ω) = T0 γ −1/3 .
(19)
In Figure 3, we use equation (18) to plot the critical temperature Tc as a function of the rotational frequency and anharmonic parameter. This figure shows that: in the absence of the quartic potential, i.e. δ = 0, and in the presence of very fast rotation (α = 1), the centrifugal force exactly cancels the confining effect of the harmonic potential and the condensate vanishes, Tc = 0. In the presence of the quartic potential, i.e. δ > 0, the confining potential is growing faster than r2 at large distance, in this case the critical temperature would instead differ from zero for any value of the rotating frequency. For small values of the rotation rate parameter α, Tc decreases almost linearly up to α = 0.7. In the fast rotation regime, Tc decreases rapidly with α. However, the quartic potential affects the critical temperature by only a few percent. In order to clarify the effects of finite size and the positive chemical potential, we plot in Figure 4 the transition temperature calculated from equation (19) for the ideal rotating boson gas (top graph) and from equation (18), which includes the effects of finite size and positive chemical potential (lower graph). The parameters for the experiment of Bretin et al. [8] are used. This figure shows that in a trapped rotating Bose gas, finite size and positive chemical potential have the effect of lowering the transition temperature. These effects are considerably for
Eur. Phys. J. D (2013) 67: 185
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60
Tc
nK
40 0.6
20 3 0.4 3.5 4
0.2
n 4.5
Cv 10 N kB 5
0.8 0.6
0 0
0.4 0.5
5
Fig. 4. The effects of finite size and positive chemical potential on the transition temperature of the rotating Bose gas. The calculations are made for α = .95. The graph on top shows the transition temperature for an infinite ideal gas. The lower graph shows the transition temperature for finite N = 10n and positive chemical potential, corresponding to non-zero η.
N < 104 and η = 0.3. Finally, the effects of finite size and the positive chemical potential on the condensation fraction is the same as the transition temperature.
5 Heat capacity We now turn to the condensate heat capacity CV (T ), which can be calculated by differentiating the total energy E with respect to the temperature T at fixed the particles number N and the volume V . ∞ ∞ ∂ ∂E j −jβ = E0 + CV (T ) = z ρ( )e d . ∂T ∂T 0 j=1 (20) At temperature T < T0 (condensed phase), the chemical potential remains equal the energy of the single particle ground state, the approximation z = 1 for the fugacity can be used. In this case, the heat capacity is given by: CV,T 1, the heat capacity drop suddenly to its asymptotic (T → ∞) value. The results previously obtained by Grossmann and Holthaus citegro for a gas with pure harmonic trapping are recovered for γ = 1. In Figure 5, we plot the heat capacity as a function of the reduced temperature and the rotation rate. The trap parameters of Paris experiment are used. This figure shows that the heat capacity is discontinuous at t = 1. According to Ehrenfests classification, the discontinuity at the transition temperature characterizes the phase transition as second order. In the thermodynamic limit N → ∞ (R1 = 0), the magnitude of the jump at t = 1 can be extracted from equations (21) and (22), ΔC (∞) ζ(3) |t=1 = 9γ ≈ 6.577γ. N kB ζ(2)
(23)
At very high temperature T → ∞, the heat capacity is given by: (∞)
lim
T →∞
while at T > T0 (gas phase), the situation is slightly more difficult. In this temperature regime, N0 = 0, and the fugacity z is a function of T , determined by the constraint (6). Following Grossmann and Holthaus [20], the heat capacity is given by, CV,T >T0 g4(z) 3 g3 (z) t +6 R1 t2 = γ 12 N kB ζ(3) ζ(3) 3g3 (z) 3 g2 (z) 2 t +2 R1 t − ζ(3) ζ(3) 3g3 (z) + 2R1 g2 (z)/t . (22) × g2 (z) + R1 g1 (z)/t
0.2
1
0
CV,T >T0 N kB
= 3γ.
(24)
Thus, the heat capacity deviates from its classical (high temperature) value ( CV = 3N kB , Dulong-Petit law) by a factor γ. While at very low temperature t → 0, the heat capacity tends to be zero, (∞)
lim
T →0
CV,T ω⊥ , the number of detectable vortices is reduced in spite of the gas still being in the condensed phase at ultra-cold temperature. The authors of this work pointed that the estimated system temperature T is of the order of the chemical potential extracted from the Thomas-Fermi distribution or the magnitude of the energy separation energy between Landau levels of the system. In this section, we calculate the estimated temperature T from the energy separation of the lowest Landau levels. When a rotating harmonically trapped BEC is strongly confined in the direction of the rotation axis z, the system essentially behaves as a quasi two-dimensional system in the (x, y)-plane. When the rotation frequency approaches the trapping frequencies in the x- and y-directions, the physics of the system can be described as a system of interacting bosons in the lowest Landau level. In the presence of the quartic confining term these levels are modified. In order to investigate this modification, it is convenient to introduce a set of quantum numbers, first introduced by Cohen-Tannoudji et al. [31]: j = mz + nm for the energy quantum number and k = mz − nm for the angular momentum quantum number. Furthermore, for simplicity, we consider a spherical harmonic oscillator, i.e. ω⊥ = ωz ≡ ω. Using the new set in equation (4) leads to: E(l, k) − E0 = lω − k(Ω − δ),
(26)
where, l = j + nz represents the energy quantum number. In the absence of rotation and quartic confinement, the harmonic oscillator spectrum is entirely discrete and the various degenerate energy levels are separated by one unit of ω according to the relation E(l) − E0 = lω, i.e. each
(27)
Thus, for each non-negative l there are 12 l2 + 23 l+1 semidegenerate angular momentum states corresponding to k values ranging from − 2l (l+3) to 2l (l+3) in steps of 2. States with k = 0 are un-shifted, states with positive k are shifted upward in energy by |k|δ, whereas those with negative k are shifted downward by the same amount. Thus, in this case the parameter δ plays an important role in controlling the magnitude of the splitting, and resultant lifting of the degeneracy, of angular momentum states. In this scenario, however, the condition for the validity of the perturbation method, δ 2ω must be borne in mind. 2. For 0 < Ω < ω⊥ , the excitation energy is given by equation (26). The energy quantum number l determines the large energy splitting and the angularmomentum quantum number k lifts the remaining degeneracy. In this case the Coriolis force eliminates the degeneracy of the non-rotating spectrum even in the absence of the quartic potential. States with k = 0 are still un-shifted, states with positive k are shifted downward in energy by |k|(Ω − δ), whereas those with negative k are shifted upward by the same amount. Thus, the levels are separated by (ω + Ω − δ), whereas the distance between two adjacent states in a given level is (ω − Ω + δ). 3. For Ω ∼ ω⊥ , a set of modified Landau levels are formed compared to the purely rotating harmonic case and the degeneracy is nearly restored. In this limit the quantum numbers mz and nm are most convenient to use. The excitation energy is given by: E(nz , mz , nm ) − E0 = nz ωz + mz δ + nm (2ω − δ). The integer nm becomes the Landau levels index, with Landau levels separated by (2ω − δ). The lowest Landau level has nm = 0 and k = mz . The first excited Landau level has nm = 1 and k = mz − 1, etc. The distance between two adjacent states in a given level is δ. 4. The system temperature at any rotation frequency can be estimate from the modified Landau levels, T =
(2ω⊥ − δ) · kB
(28)
In Figure 7, the above items are illustrated graphically.
7 Discussion and conclusion We have investigated the thermodynamical properties of BEC rotating in a quadratic plus quartic confining
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E - E0 l=3
Page 7 of 7
values. These two effects, as well as the anisotropy of the trap, should be taken into consideration for a proper estimation of the critical temperature and critical rotation frequency (the frequency which is required to achieve the vortex state). The behavior of the heat capacity show that the system has a second order phase transition. In the thermodynamical limit our results are in considerable agreement with the results of Kling and Pelster [14,15] and Stock et al. [10].
References E - E0
E - E0
(b)
(c)
Fig. 7. Excitation energy levels of Boson gases trapped in a rotating harmonic + quartic potential for different angular frequency Ω: (a) Ω = 0, (b) for moderate 0 < Ω < ω⊥ and (c) for Ω = ω⊥ .
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