Optics Communications

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1 Optics Communications 283 (2) Contents lists available at ScienceDirect Optics Communications journal homepage: Monte Carlo method, classical fields and Bose statistics Emilia Witowsa a, Mariusz Gajda a,b, *, Kazimierz Rzą_zewsi b,c a Institute of Physics, Polish Academy of Sciences, Aleja Lotniów 32/46, Warsaw, Poland b Faculty of Mathematics and Sciences, Cardinal Stefan Wyszyńsi University, Warsaw, Poland c Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotniów 32/46, Warsaw, Poland article info abstract Article history: Received 6 July 29 Received in revised form 23 September 29 Accepted 9 October 29 In this paper we combine the classical fields method with the Monte Carlo approach for description of statistical properties of a Bose Einstein condensate. We show that the canonical ensemble of interacting classical fields can be generated using Metropolis algorithm. We obtain a probability distribution of the condensate occupation for the wealy interacting Bose gas at low temperatures and pay a particular attention to the two momenta of this distribution: mean and variance of the condensate occupation. Ó 29 Elsevier B.. All rights reserved.. Introduction * Corresponding author. Address: Institute of Physics, Polish Academy of Sciences, Aleja Lotniów 32/46, Warsaw, Poland. address: gajda@ifpan.edu.pl (M. Gajda). Statistics of a multiparticle quantum state is at the heart of quantum physics. Foc, coherent, thermal or squeezed states of photons are all studied both theoretically and experimentally. With the advent of experiments with quantum degenerate atomic the interest in the statistical properties of the Bose and Fermi gases has been revived. In this paper we concentrate our attention on the canonical ensemble of bosons confined by a trapping potential. It should be stressed from the very beginning that there are two major differences between photons and atoms. While photons are easily created and destroyed, the number of atoms is in the relevant ultra low temperatures strictly conserved. Besides, while photons are virtually noninteracting (except via host atoms, lie in nonlinear optical processes) atoms do interact and their gross statistical features are modified by interactions. Even the case of an ideal Bose gas was not fully understood when the atomic condensates were first created. A number of papers devoted to this problem demonstrated substantial differences between grand canonical (clearly not applicable to trapped atom experiments) canonical and microcanonical ensembles []. Exact results are available for the ideal Bose gas [2 4]. Understanding of equilibrium statistical properties of a wealy interacting Bose gas is still a challenge. Attention was mostly devoted to somewhat academic case of a finite number of atoms confined in a cubic volume with periodic boundary conditions for the multiparticle wave function. In D, and with the usual contact interaction, a lot is nown about the eigenvalues and the eigenvectors of the N-particle Hamiltonian [5]. In more realistic 3D we have to apply approximations. Still, in the box confinement with periodic boundary conditions we now at least the condensate wave function. It is simply a zero momentum state. We also now analytically the dispersion relation and the wave functions of quasiparticle excitations the Bogoliubov spectrum [6]. None of the two is analytically available for a realistic harmonic potential. In the box, for very low temperatures one can calculate analytically [7] the width of the probability distribution of the condensate occupation. The idea is rather simple: in the thermal state one assumes a free Bose statistics in every mode of the collective excitation. A generalization of the [7] approach, including the whole and not only the phonon branch of the spectrum, is presented in [8]. However, treating the quasiparticles in each mode as an ideal Bose gas one ignores their mutual interaction. At higher temperatures the Bogoliubov spectrum gets modified it goes to a form, nown as Bogoliubov Popov spectrum [9], which explicitly depends on the number of condensed atoms rather than their total number. One can still distribute the atoms among these quasiparticles according to the Bose statistics but the simplest approach [], in which the occupation of the condensate, entering the spectrum, is ept at its mean value runs into a contradiction: the object of the study the number of condensed atoms is a stochastic variable whose actual value depends on the number of excitations. In a recent paper [] this shortcoming has been overcome and the treatment presented accounts for the fluctuating number of the condensed atoms and thus fluctuating spectrum of quasiparticles in a self-consistent way. Moreover, a secular term has been extracted from that in the full hamiltonian, which describes the interaction between the quasiparticles. Still, the interaction between quasiparticles was treated in a rather crude approximation. In this paper we are going to present the first results for the fluctuating number of condensed atoms that fully account for the interaction between quasiparticles. Of course some other approximation will be made. 3-48/$ - see front matter Ó 29 Elsevier B.. All rights reserved. doi:.6/j.optcom.29..8

2 672 E. Witowsa et al. / Optics Communications 283 (2) The simplification in question is the classical field approximation. Several groups have extended the celebrated Bogoliubov approximation, in which the highly occupied condensate state is treated as a classical c-number function, to all highly occupied degrees of freedom [2]. This way one obtains a classical, nonlinear field theory with a finite number of modes. As such, in the thermal equilibrium, it exhibits the equipartition of energy. Thus it would suffer the ultraviolet catastrophe if not for the ultraviolet cut-off. There is a considerable debate over the best value of this cut-off. In our recent paper [3] we have shown that appropriate choice of the cut-off allows for matching the whole probability distribution obtained with the classical fields with the exact quantum solution for the ideal Bose gas. Equipped with this prescription of the cut-off we are going to present in this paper the first classical fields results for the distribution of the number of condensed atoms of wealy interacting Bose gas in a box with periodic boundary conditions. 2. The model In this paper we consider N bosons confined in a cubic box of length L with periodic boundary conditions and interacting via a zero range potential characterized by the s-wave scattering length a. Thus the full Hamiltonian in second quantization taes a form: Z¼L3 H ¼ drw y ðrþ p2 2m WðrÞþg 2 Z ¼L 3 drw y ðrþw y ðrþwðrþwðrþ; where the first term represents the inetic energy of the gas and the second is the interaction energy of the contact binary collisions. The atomic field operators obey the standard equal time Bose commutation relations with the only nonzero one: WðrÞ; W y ðr Þ ¼ dðr r Þ; ð2þ and the coupling constant expressed in terms of the scattering length a and mass m is: g ¼ 4ph2 a m : ð3þ It is convenient to expand the field operator in the plane wave basis: WðrÞ ¼ X a p ffiffiffi e ir ; ð4þ where a is the bosonic annihilation operator of particle with momentum h. The quantized plane waves are characterized by the wave vectors: ¼ 2p ð L n ; n 2 ; n 3 Þ; ð5þ where n ; n 2 ; n 3 are integers. The classical field approximation consists of: () replacement of the creation and annihilation operators of the plane waves by the complex amplitudes a, (2) restriction of the summation over modes to a finite number extended all the way to the momentum cut-off K max. This way the field operator WðrÞ is replaced by the classical field of well defined number of momenta modes: UðrÞ ¼ X p ffiffiffi e ir ; ð6þ jj6k max a and the energy given by: Z Z¼L3 E U ¼ dru ðrþ p2 2m UðrÞþg drjuðrþj 4 : 2 ¼L 3 ðþ ð7þ In our recent publication [3] we have shown that with a proper choice of the cut-off the statistical properties of the condensate calculated with the help of the classical fields is nearly identical with the exact quantum one for the ideal Bose gas. In thermodynamic limit the optimal cut-off satisfies: h 2 K 2 max 2m B T ¼ p fð3=2þ 2 ; ð8þ 4 where T is a temperature and B is the Boltzmann constant and fðxþ is the Riemann zeta function. For finite system the correction to the formula (8) depends wealy on the temperature [3]. Within this approximation, the classical probability distribution of finding the system in a given configuration of the mode amplitudes is: Pðfa gþ ¼ Z exp ½ E U= B TŠ: ð9þ Z is the canonical partition function and the restriction to the fixed number of N atoms is a constraint on the amplitudes: X ja j 2 ¼ N: ðþ jj6k max Our tas now is to generate numerically the probability distribution of the condensate population (zero momentum component occupation). To this end we shall use the powerful Monte Carlo algorithm. Similar method has been used to determine a shift of the critical temperature induced by the presence of interactions [4]. 3. The method The major problem in generating the canonical ensemble is to sample a phase space of all classical amplitudes fa g. The uniform sampling of this many dimensional phase space is evidently inefficient. The question of how to sample only the relevant phase space regions has been answered in the middle of the last century [5]. The main idea of the suggested Monte Carlo method is to generate a Marovian process of a random wal in the phase space. All states of the system visited during this wal are becoming the members of the statistical ensemble and are used in the ensemble averages. We adopt the Metropolis scheme [5] to the system of the classical fields as described above. In order to obtain a statistical average of any observable A A ¼ N X N s¼ hu s jaju s i: ðþ one should generate N copies of classical fields U s. The canonical average is obtained in the limit of N!provided that a number of members of the ensemble with energy E Us is proportional to the Boltzmann factor e E Us =BT. This goal can be achieved in a random wal where a single step of the Marov process is defined as follows:. A set of amplitudes fa ðsþ g determines a state selected to be a member of the canonical ensemble at the sth step of the random wal. The corresponding energy E Us of the classical field is calculated according to (7). As the initial condition (s ¼ ) any state satisfied the condition () may be chosen as a member of the ensemble. 2. A trial set of amplitudes f~a ðsþ g is generated by a random disturbance of fa ðsþ g!faðsþ þ dðsþ g followed by the normalization to account for the condition (). This way a trial classical field ~U s is obtained. The corresponding energy e E Us ~, the energy difference D s ¼ e E Us ~ E Us, as well as the Boltzmann factor p s ¼ e Ds= BT are then calculated.

3 E. Witowsa et al. / Optics Communications 283 (2) A new member of the Marov chain fa ðsþþ g is selected according to the following prescription: if D s < then the trial state becomes a new member of the ensemble, fa ðsþþ g¼f~a ðsþ g, if D s > then a random number < u < is generated. If u < p s then the trial state becomes a new member of the ensemble, fa ðsþþ g¼f~a ðsþ g. In the opposite case u > p s the initial state fa ðsþ g is once more included in the ensemble, fa ðsþþ g¼fa ðsþ g. In the limit of infinite Marov chain the selected states form the canonical ensemble. P(N ex ) K=4 K=5 K=8 A convergence of the procedure is the fastest if approximately every second trial state becomes a member of the ensemble. This factor depends on the assumed maximal value of the displacements d ðsþ which can be modified during the wal. Let us add that some number of the initial members of the ensemble should be ignored in order to avoid an influence of the arbitrarily selected initial state of the system. As a test of the numerical convergence of the algorithm we use the Bogoliubov method which is applicable at low temperatures, where excitations are of the form of independent quasiparticles of energy: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e ¼ ðe þ gnþ 2 ðgnþ 2 ; ð2þ where E ¼ h 2 2 =2m and n ¼ N= is a particle density. In the limit of macroscopic occupation of quasiparticle modes the thermal part of the classical fields, U T ðrþ ¼UðrÞ a, can be expressed as: U T ðrþ ¼ X U b p ffiffiffi e ir þ b p ffiffiffi e ir ; ð3þ where U 2 2 ¼ and =4 E U þ ¼ ; ð4þ E þ 2gn In Eq. (3) the summation is limited by the cut-off momentum because we restrict the field UðrÞ to the classical modes. b are amplitudes of the quasiparticles excitations which at a state of the thermal equilibrium are distributed according to the classical statistics [6]: P jb j 2 ¼ e p B T e e jb j 2 = B T : ð5þ N ex Fig.. Probability distribution of having N ex excited atoms for the ideal Bose gas in a D box potential with periodic boundary conditions for three different temperatures. Note that due to the constraint on the total number of atoms in the canonical ensemble the condensate occupation probability P ðn Þ is related to the probability distribution of occupation of excited modes P ðn Þ¼PðN N exþ. The dimensionless parameter b ¼ðh 2 =2mL 2 Þ= BT corresponds to the inverse temperature of the system. From left to right: b ¼ :36 (full circles), b ¼ :23 (full squares) b ¼ :9 (full triangles). (Dashed line) Exact quantum results. (Solid line) The analytic classical fields results, points the classical fields Monte Carlo simulations. The classical fields results (both the analytic and the Monte Carlo ones) were obtained for the optimal cut-off, K 2 max =ð2p=lþ ¼:58=b, i.e.: Kmax=ð2p=LÞ ¼4 for b ¼ :36, K max=ð2p=lþ ¼5 for b ¼ :23, and K max=ð2p=lþ ¼8 for b ¼ :9. Parameters of the simulations are: number of atoms N ¼ and number of Monte Carlo steps N ¼ 3 6. for various temperatures are shown in Fig.. The agreement between these three results is remarable. After gaining a confidence that the Monte Carlo algorithm for the classical fields was very efficient in reproducing statistical properties of the ideal Bose gas we may study a wealy interacting case. In Fig. 2 we present results for the 3D periodic box potential. We compare a probability of the thermal cloud occupation PðN ex Þ of the ideal Bose gas with the corresponding probabilities of the interacting system. Each curve corresponds to a different interaction strength parameter an =3 but the same temperature. The results for the ideal Bose gas are obtained in two ways: using the asymptotic quantum formula given in [] (full line) and according Our strategy is the following: (i) for a given temperature and the cut-off momentum we generate the amplitudes of the quasiparticles b according to (5), (ii) next we determine the plane wave amplitudes a ¼ U b þ b and a number of thermal particles N T ¼ P ja j 2, (iii) the occupation of the condensate is computed as N ¼ N N T and the data are collected to get the histogram corresponding to a condensate occupation probability distribution. P(N ex ).5. Q, IBG MC, IBG MC, an /3 =.33 MC, an /3 = The results.5 The probability distribution of the condensate occupation of the ideal Bose gas (IBG) trapped in a periodic box can be found analytically within the full quantum treatment in one spatial dimension only [3]. These results are the best benchmars for the classical fields Monte Carlo algorithm described in the previous section. In Fig. we compare results of the Monte Carlo calculations with the exact quantum formula, and with the analytic expression based on the classical fields method [3]. For the classical fields treatments i.e. the Monte Carlo and the analytic one, we chose the optimal cut-off of high momenta states [3]. The probability distributions of the occupation of excited modes (thermal cloud) N ex Fig. 2. Probability distribution of having N ex excited bosons in 3D box potential. (Solid line) Results of [] for the ideal Bose gas. Points correspond to the classical fields MC simulations: (full circles) the ideal gas, (full squares) an =3 ¼ :33, (full diamonds) an =3 ¼ :66. Parameters of the MC simulations are: number of bosons N ¼, number of MC steps N ¼ 4 7, temperature T=T c ¼ :59 where BT c ¼ 2ph 2 =mðn=l 3 fð3=2þþ 2=3 is the critical temperature of the ideal gas. The optimal cut-off is chosen according to h 2 K 2 max =2 m ¼ 2:7BT.

4 674 E. Witowsa et al. / Optics Communications 283 (2) N /N a n /3 d N /N a n /3 Fig. 3. A condensate fraction N =N (left) and condensate fluctuations dn =N(right) as a function of interaction strength an =3 for the 3D box potential. Points correspond to results of the Monte Carlo simulations while dashed lines are results obtained with the help of Bogoliubov method. Parameters of the simulations are: N ¼ 3 7 Monte Carlo steps and N B ¼ 3 6 realizations of the field in the Bogoliubov method. Number of bosons is N ¼ ; and the cut-off was chosen according to bh 2 K 2 max =2m ¼ 2:68. Two sets of data correspond to two different temperature: T=T c ¼ :57 (number of momenta modes M ¼ 4 3 ) full circles, and T=T c ¼ :29, (number of momenta modes M ¼ 6 3 ) full squares. The leading term of condensate fluctuation in the large N limit [] is dn =N :29 at T=T c ¼ :57 and dn =N :29 at T=T c ¼ :29 what is comparable with ours results N /N.6.4 d N /N T/T c T/T c Fig. 4. Condensate fraction (left) and condensate fluctuations (right) versus T=T c. Points are the results of the Monte Carlo simulations (with 3 7 steps) for the ideal gas (full circle) and for the interacting system an =3 ¼ :5 (crosses). Other parameters are: number of particles N ¼ ;, the cut-off bh 2 K 2 max =2m ¼ 2:68. The number of different momenta modes varies with the temperature from M ¼ 2 3 for the lowest one to M ¼ 2 3 for the highest temperature. to the classical fields Monte Carlo method studied here. Evidently, only the Monte Carlo method was used for the interacting systems. Maxima of probabilities of excited state occupations for the interacting systems are shifted towards smaller number of excited particles. Evidently interactions magnify the effect of Bose statistics. The mean condensate occupation hn i¼n P Nex N expðn ex Þ as well as the condensate fluctuations dn 2 ¼hðN hn iþ 2 i as functions of the interaction strength for two different temperatures are presented in Fig. 3. Both chosen temperatures are very low as compared with the critical temperature. This is the regime of applicability of the Bogoliubov method. Indeed, the results of simulations based on the Bogoliubov approach are in a perfect agreement with the MC simulations. The condensate fraction is a growing function of the interaction strength. In fact its value saturates for relatively small interaction parameter an =3. Simultaneously the condensate fluctuations are getting smaller with increasing interaction parameter and saturate at some value. In the thermodynamic limit the fluctuations of a condensate population for the interacting system should not depend on the interaction strength and according to [] their value should be equal to: dn =N :29 at T=T c ¼ :57, and dn =N :29 at T=T c ¼ :29. Our results for the finite system are a bit different, but yet close to the above predictions. The condensate population and its fluctuations in the whole region of relevant temperatures for a relatively small wealy interacting system of N ¼ bosons are presented in Fig. 4. Our results include both finite system and interaction induced corrections. The role of interactions is clearly visible the critical temperature is shifted towards higher temperatures and fluctuations of the condensate populations are decreased. 5. Conclusions In this paper we have presented a promising method of studying statistical properties of the wealy interacting Bose gas. It is based on the classical fields approximation. Combined with the Metropolis algorithm it offers an efficient scheme for generating numerically the canonical ensemble of the Bose system. A very similar method has already been used to calculate the shift of the critical temperature induced by interactions [4]. This notoriously difficult problem was studied by many authors and let them to vastly different predictions. The result of [4] is considered to be one of the most accurate. However, calculated there is the ratio of the shift to the critical temperature. This ratio is independent of the cut-off provided the cut-off is sufficiently large. Equipped with our earlier study of the optimal cut-off, we are now able to calculate the statistical properties of the interacting Bose gas as a function of temperature. Practically all results obtained to date for the statistics of the wealy interacting Bose gas (including the present paper) are obtained for the box confinement with the periodic boundary conditions. Our method may be extended to more realistic situations, such as the binding by the harmonic potential. The wor in this direction is in progress. Acnowledgements We would lie to than Paweł Ziń for his very helpful comments. The authors acnowledge financial support of the Polish Government Research Funds for 29 2.

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