Collective and single particle excitations of a trapped Bose gas

The density of states of a Bose-condensed gas confined in a harmonic trap is investigated. The predictions of Bogoliubov theory are compared with the ones of Hartree-Fock theory and of the hydrodynamic model. We show that the Hartree-Fock scheme provides an excellent description of the excitation spectrum in a wide range of energy, revealing a major role played by single-particle excitations in these confined systems. The crossover from the hydrodynamic regime, holding at low energies, to the independent particle regime is explicitly explored by studying the frequency of the surface mode as a function of their angular momentum. The applicability of the semiclassical approximation for the excited states is also discussed. We show that the semiclassical approach provides simple and accurate formulae for the density of states and the quantum depletion of the condensate.


I. INTRODUCTION
The collective modes of a Bose condensed gas confined by an external potential have been the object of extensive work in the last months. The successful agreement between experimental results [1,2] and theoretical predictions [3][4][5][6][7][8] for the collective frequencies at low temperature has stimulated an intensive research activity. Though only the modes with low multipolarity and frequency have been detected in experiments, the excitations at higher energy and angular momentum are also very important because they determine the statistical behavior of the system, including thermodynamics, transport phenomena and superfluid effects.
The excited states at high energy are expected to have single-particle nature. However, the transition from collective (phonon-like) to single-particle excitations in a inhomogeneous system can differ significantly from the case of a uniform Bose gas. In fact, the presence of a surface allows for the occurrence of single-particle states even at low energy (lower than the chemical potential). These states, of low energy but high multipolarity, are localized near the surface, where the condensate density becomes small. This behavior represents a peculiar and interesting feature of these confined systems; in a uniform Bose gas, in fact, only phonons are present at low energy. In a recent paper [9] we have already pointed out the effects of these single-particle states on the thermodynamic properties of the trapped gases.
In the present work we solve the equations for the excited states of a weakly interacting gas in a spherical trap at zero temperature within Bogoliubov theory. The main purpose is to investigate the collective (phonon-like) and single-particle character of the elementary excitations. This is accomplished by calculating key quantities, such as the density of states, the frequency of the surface modes and the quantum depletion of the condensate, and by comparing the predictions of Bogoliubov theory with the ones of different approximations, like Hartree-Fock theory and the hydrodynamic model. Finally we check the accuracy of the semiclassical approximation and we show that it provides simple and useful formulae for both the density of states and the quantum depletion of the condensate.

II. BOGOLIUBOV THEORY
The elementary excitations of a degenerate Bose gas are associated with the fluctuations of the condensate. At low temperature they are described by the time dependent Gross-Pitaevskii (GP) equation for the order parameter [10]: where dr|Ψ| 2 = N is the number of atoms in the condensate. At zero temperature N coincides with the total number of atoms, except for a very small difference δN ≪ N due to the quantum depletion of the condensate. The coupling constant g is proportional to the s-wave scattering length a through g = 4π 2 a/m. In the present work we will discuss the case of positive scattering length, a > 0, as in the experiments with Rubidium and Sodium, but the same formalism can be also applied to systems with negative scattering length. The trap is included through V ext , which is chosen here in the form of an isotropic harmonic potential: V ext (r) = (1/2)mω 2 HO r 2 . The harmonic trap provides also a typical length scale for the system, a HO = ( /mω HO ) 1/2 . Actually, the experimental traps have cylindrical symmetry, with different radial and axial frequencies, but the choice of a spherical trap, as we will discuss later, is not expected to affect the main conclusions of the present work, while reducing greatly the numerical effort.
The normal modes of the condensate can be found by linearizing the GP equations, i.e., looking for solutions of the form where µ is the chemical potential and functions u and v are the "particle" and "hole" components characterizing the Bogoliubov transformations. After inserting in (1) and retaining terms up to first order in u and v, one finds three equations. The first one is the nonlinear equation for the order parameter of the ground state [10], where H 0 = −( 2 /2m)∇ 2 + V ext (r); while u(r) and v(r) obey the following coupled equation: Numerical solutions of these equations have been recently found by different authors [4][5][6][7][8].
In the present work, we use them to calculate the density of states, the frequency of the surface modes, the quantum depletion of the condensate, in order to clarify the different role played by excitations having collective and single-particle character.
When the dimensionless parameter Na/a HO is large, the kinetic energy term in the ground state equation (3) becomes negligible with respect to the mean field term and one gets the Thomas-Fermi approximation: In the same limit the equations of motion (4)(5) coincide with the equations of the hydrodynamics (HD) of superfluids [3,11,12]. In the spherical case their eigenfrequencies take the analytic form [3] ω(n, ℓ) = ω HO (2n 2 + 2nℓ + 3n + ℓ) 1/2 , where ℓ and n are the angular momentum quantum number and the number of nodes in the radial solution, respectively. The deviations from the predictions of the noninteracting harmonic oscillator (HO) model, point out the effects of two-body interactions. These are particularly important for the so called "surface" modes (n = 0), where the HO prediction ω = ℓω H0 is significantly lowered to the hydrodynamic value ω = √ ℓ ω H0 . In general the HD prediction turns out to be very accurate for the low energy excitations of the system, while the ideal gas prediction is expected to be valid in the opposite case of high excitation energies. The exact solutions of the equations (4-5) provide the correct interpolation between the two limiting regimes.
A typical spectrum obtained from equations (4-5) is given in the upper part of Fig. 1 for a gas of N = 10000 atoms of Rubidium (scattering length a = 110a 0 , where a 0 is the Bohr radius). For the spherical trap we have chosen the frequency ω HO = 2πν HO = 2π187Hz, which is the average ω HO = (ω x ω y ω z ) 1/3 of the axial and radial frequencies of Ref. [1]. It corresponds to the oscillator length a HO = 0.791 × 10 −4 cm. Energy is given in units ω HO and the chemical potential is 8.41 in these units. The vertical bars have length (2ℓ + 1), so that the angular momentum of each state can be inferred from the figure. One clearly sees that, at energy much larger than the chemical potential, the excited states tend to be grouped into levels ω HO apart, as in the noninteracting HO model. Conversely, the energy of the lowest modes is close to the prediction of the HD equations [3]. For instance, the lowest ℓ = 2 and ℓ = 0 modes differ by less than 3% from the hydrodynamic values √ 2 and √ 5, respectively.

III. DENSITY OF STATES
Once the spectrum of excited states is calculated, one can count the number of states below a given energy ǫ: The density of states, g(ǫ) is simply the derivative dN(ǫ)/dǫ. Note that the quantity N(ǫ) is well defined even for a discretized spectrum, while the density of states implies averaging the number of states within small but finite energy intervals.
In Fig. 2 we show the quantity N(ǫ) obtained by counting the levels in the spectrum of However N(ǫ) remains well below the HD approximation, the latter being soon inadequate as ǫ increases. Indeed hydrodynamic theory accounts for collective phenomena and provides an excellent description of the low-lying elementary excitations of the system [3,8], but completely ignores single-particle effects. This is exactly the opposite of what Hartree-Fock (HF) theory does. For this reason it is interesting to compare the results of Bogoliubov theory with the predictions of HF theory in which one determines the eigenstates of the single-particle Hamiltonian [13,14] H The lowest eigenstates of the HF Hamiltonian are expected to be localized near the surface of the condensate. To understand better this point, let us take the large N limit. In this case one can use the Thomas-Fermi approximation (6) for the ground state density. The HF Hamiltonian then takes the simple form where R = [2µ T F /(mω 2 HO )] 1/2 is the classical radius of the condensate. The HF potential has a pronounced minimum at R. This potential well near the boundary persists in the HF Hamiltonian even for smaller values of N.
It is worth stressing that, in general, Hartree-Fock theory is expected to be correct for energies larger than the chemical potential. For these trapped bosons, however, it accounts also for the low energy excitations close to the boundary, where the density of the condensate is small. This can be seen in Fig. 1, where the Hartree-Fock and Bogoliubov spectra are compared. One notes that the two spectra are rather similar even below the chemical potential (µ = 8.41 in this case) except for the excitations having lowest angular momenta, i.e., shortest bars in the figure. Those collective excitations are instead correctly reproduced by the hydrodynamic model. The corresponding Hartree-Fock prediction for the quantity N(ǫ) is also given in Fig. 2 as a solid line. The agreement with the results of equations (4)(5) is remarkable in the whole range of energy. The figure shows the case of 10000 atoms but a similar agreement is found for all values of N relevant for the experiments (N ≃ 10 3 ÷ 10 7 ).
The above behavior represents a major difference with respect to the case of a homogeneous Bose gas where phonons have a crucial effect on the density of states.

IV. SURFACE EXCITATIONS
In order to better understand the transition from the collective to the single particle regime, we have explored in details the evolution of the excitation energy for the surface modes (n = 0) as a function of their angular momentum ℓ, as predicted by the solution of the equations (4-5). As already mentioned, the effects of the interaction are particularly important for such modes and are responsible for a significant lowering of their frequency.
In Fig. 3 we plot the quantity ω ℓ /ℓ, in units ω HO , for different numbers of atoms in the same trap. This ratio has an important physical meaning because, according to Landau's criterium for superfluidity, it provides the rotational frequency at which the ℓ-th surface excitation becomes unstable. The hydrodynamic prediction (8) is shown as a dashed line, the curve going asymptotically to zero for large ℓ. The figure shows that the Bogoliubov states first follow the HD curve, but, rather soon, they deviate from it, approaching asymptotically the non interacting value ω ℓ /ℓ = ω HO . The deviation from HD takes place at larger values of ℓ if N is increased, revealing that the HD approximation becomes applicable to a larger number of states in this limit.
A simple estimate of the typical value of ℓ at which the HD picture starts failing, one can take [15] ℓ c ≃ Rp c where R is the radius of the condensate, proportional to N 1/5 , and p c is of the order of the inverse of the surface thickness d = [a 4 HO /(2R)] 1/3 [16][17][18]. For larger values of ℓ, the wavelength of the excitations becomes shorter than d and one explores microscopic details of the boundary which can not be described by the Thomas-Fermi approximation (6) and by the HD equations. This yields l c ∝ (R/a HO ) 4/3 ∝ N 4/15 , corresponding to an excitation energy ω ∝ N 2/15 , smaller than the chemical potential, which instead behaves as N 2/5 . This explains why the crossover from the HD to the single particle regime takes place at energies smaller than µ.
For each value of N, the curves in Fig. 3 exhibit minima and one can define a critical frequency as Ω c = min(ω ℓ /ℓ). For rotational frequencies larger than Ω c the surface excitations become unstable. It is interesting to compare this value with the critical frequency needed to generate a vortex [19]. This is done in Fig. 4 where we compare the two critical frequencies as a function of N. We find that the lowest instability is always associated with the creation of a vortex. Note however that in order to generate a vortex one needs to transfer to the system a huge angular momentum (equal to N ), which is much higher than the value ℓ required to create a surface excitation.

V. SEMICLASSICAL APPROXIMATION AND SCALING BEHAVIOR
A good approximation for the density of states can be obtained by solving equations (4)(5) in the semiclassical approximation [9,20,21]. In this approximation, which is expected to hold for excitation energies much larger than the oscillator energy ω HO , the quantity N(ǫ) is a continuous function of ǫ defined by where corresponds to the semiclassical dispersion law. Here the quantity n 0 (r) = Ψ 2 0 (r) is the condensate density. In Fig. 5a  The use of the semiclassical approximation allows one to carry out the analysis of the density of states in a systematic way and to exploit the dependence on the relevant parameters of the system. In fact, when the number of atoms in the condensate is large enough to make the Thomas-Fermi approximation (6) accurate, the statistical properties of the system can be expressed in terms of a single scaling parameter η given by the ratio [22] between the chemical potential (7) In terms of the scaling parameter η and the dimensionless energyǫ = ǫ/(k b T c ), the number of states N(ǫ) predicted by the Bogoliubov semiclassical theory becomes This result has been obtained by using the Thomas-Fermi approximation (6) for the condensate density in (13)(14); this allows one to split the space integral into an inside region (first term in the square bracket) and an outside region (second term). It is worth stressing that equation (16), which are expected to hold in the scaling regime Na/a HO ≫ 1, provide a very good estimate of the semiclassical expression (13) even for relatively small N. For instance, the two predictions are indistinguishable in Fig 5a, being represented by the same solid line. In Fig. 5b we show the predictions for the density of states given by the semiclassical approximation (16) for three different values of η. The parameters of the recent experiments at Jila [1] and MIT [2], using very different traps, correspond to η ranging from 0.39 to 0.45.
Expression (16) can be also expanded at low energy, ǫ ≪ k B T c , still compatible with the assumption ǫ ≫ ω HO . One finds the law N(ǫ)/N ∝ ǫ 5/2 . This differs from the usual ǫ 3 law typical of the phonon regime, revealing the different behavior exhibited by these systems with respect to the homogeneous Bose gas.

VI. QUANTUM DEPLETION
In the last part of the paper we calculate the quantum depletion of the condensate which, according to Bogoliubov theory, is given by The "hole" components v j can be obtained by solving equations (4)(5). In semiclassical approximation [9] one replaces the sum over all the discrete states with the integral over p of the function v 2 (p, r) = 1 2ǫ(p, r) where ǫ(p, r) is the single particle energy (14). In a uniform gas this expression yields the most famous result δN/N = (8/3)(n 0 a 3 /π) 1/2 . In the trapped gas and in the limit Na/a HO ≫ 1, where TF approximation holds, the semiclassical approximation provides the simple analytic law [9] with η given in Eq. (15). Since the available experiment correspond to η ≃ 0.4, the quantum depletion turns out to be less than 1%, as already pointed out in Refs. [5,7]. In Fig. 6 we show the quantum depletion for 10000 and 50000 atoms of Rubidium obtained by summing over the Bogoliubov spectrum up to a given energy ǫ (solid lines). We compare it with the prediction of the semiclassical expression (18) (dashed lines), while the arrows indicate the asymptotic values (19), holding in the scaling limit. An important result emerging from the numerical calculation is the very slow convergency of the sum (17). This is not a surprise, since also in a homogeneous gas the convergency is slow due to the 1/p 4 tail in the momentum distribution at high momenta and one has to go up to ǫ = 100µ in order to saturate 90% of the sum (17).
The agreement between the quantum depletion obtained from the discretized sum (17) over the Bogoliubov states and from the semiclassical approximation (18) is satisfying and was not obvious a priori. Fig. 6 shows a discrepancy of the order of 5% between the two predictions for N = 10000, while for larger N the two curves tend to coincide. It is worth noticing that the two solid lines in Fig. 6 requires the summation of dr|v j (r)| 2 over up to 15000 different values of (n, ℓ) in the Bogoliubov spectrum; the calculation is then much heavier than the semiclassical one. The good accuracy of the semiclassical approach makes it useful in practical situations. This is especially true for the simple formula (19) which includes the case of anisotropic traps through the averaged frequency ω HO = (ω x ω y ω z ) 1/3 entering the Thomas-Fermi chemical potential µ T F and, hence, the scaling parameter η.
Finally, the large N semiclassical formula (19) shows the rather strong dependence of the depletion on the scattering length parameter a. If the magnetic tuning of the scattering length will become available, it will be possible in the future to increase significantly the value of η and consequently explore Bose gases where the quantum depletion is much larger.

VII. SUMMARY
We have investigated the elementary excitations of a dilute Bose gas in harmonic trap by solving the equations of Bogoliubov theory. Differently from the case of a uniform gas, where phonons dominate the system at energies of the order of, or lower than the chemical potential, the spectrum of the trapped gas shows an important particle-like behavior even at low energy. This fact has been here explored in detail. We have compared the results of Bo-goliubov theory for the density of states with the ones of Hartree-Fock theory finding a very good agreement on a wide range of energy. We have studied the behavior of surface modes, emphasizing the crossover from the low energy regime, well described by the hydrodynamic model, to the single-particle regime. This crossover provides also a critical frequency associated with a rotational instability and we have compared this frequency with the one needed to create a quantized vortex. Another important result emerging from our analysis is the high accuracy exhibited by the semiclassical approximation for the excited states.
Finally, we have calculated the quantum depletion of the condensate by summing the "hole" component dr|v(r)| 2 over all the states in the excitation spectrum of Bogoliubov theory.
The convergency of the sum turns out to be very slow, as expected by the analogy with the case of the uniform gas. Again we find an excellent agreement with the predictions of the semiclassical approximation. In the limit Na/a HO ≫ 1, the latter provides the simple and useful formula δN/N = 0.098 η 3 , in terms of the scaling parameter η = 1.57(N 1/6 a/a HO ) 2/5 .