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Fermionic Condensation in Ultracold Atoms, Nuclear Matter and Neutron Stars

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Fermionic Condensation in Ultracold Atoms, Nuclear Matter and Neutron Stars Journal of Physics: Conference Series OPEN ACCESS Related content - Condensate fraction of asymmetric three- Fermionic condensation in ultracold atoms, component Fermi gas Du Jia-Jia, Liang Jun-Jun and Liang Jiu- nuclear matter and neutron stars Qing - S-Wave Scattering Properties for Na—K Cold Collisions To cite this article: Luca Salasnich 2014 J. Phys.: Conf. Ser. 497 012026 Zhang Ji-Cai, Zhu Zun-Lue, Sun Jin-Feng et al. - Landau damping in a dipolar Bose–Fermi mixture in the Bose–Einstein condensation (BEC) limit View the article online for updates and enhancements. S M Moniri, H Yavari and E Darsheshdar This content was downloaded from IP address 170.106.40.139 on 26/09/2021 at 20:46 22nd International Laser Physics Worksop (LPHYS 13) IOP Publishing Journal of Physics: Conference Series 497 (2014) 012026 doi:10.1088/1742-6596/497/1/012026 Fermionic condensation in ultracold atoms, nuclear matter and neutron stars Luca Salasnich Dipartimento di Fisica e Astronomia “Galileo Galilei” and CNISM, Universit`a di Padova, Via Marzolo 8, 35131 Padova, Italy E-mail: [email protected] Abstract. We investigate the Bose-Einstein condensation of fermionic pairs in three different superfluid systems: ultracold and dilute atomic gases, bulk neutron matter, and neutron stars. In the case of dilute gases made of fermionic atoms the average distance between atoms is much larger than the effective radius of the inter-atomic potential. Here the condensation of fermionic pairs is analyzed as a function of the s-wave scattering length, which can be tuned in experiments by using the technique of Feshbach resonances from a small and negative value (corresponding to the Bardeen-Cooper-Schrieffer (BCS) regime of Cooper Fermi pairs) to a small and positive value (corresponding to the regime of the Bose-Einstein condensate (BEC) of molecular dimers), crossing the unitarity regime where the scattering length diverges. In the case of bulk neutron matter the s-wave scattering length of neutron-neutron potential is negative but fixed, and the condensate fraction of neutron-neutron pairs is studied as a function of the total neutron density. Our results clearly show a BCS-quasiunitary-BCS crossover by increasing the neutron density. Finally, in the case of neutron stars, where again the neutron-neutron scattering length is negative and fixed, we determine the condensate fraction as a function of the distance from the center of the neutron star, finding that the maximum condensate fraction appears in the crust of the neutron star. 1. Introduction In 1951 Penrose [1] and Onsager [2] introduced the idea of off-diagonal long-range order (ODLRO) of the one-body density matrix to determine the Bose-Einstein condensate fraction in a system of interacting bosons. In 1962 Yang [3] proved that for attractive fermions the Bose-Einstein condensation of fermionic pairs is instead related to the ODLRO of the two-body density matrix. It is now established that at zero temperature the Bose-Einstein condensate fraction of bosonic liquid 4He is below 10% [5], while for dilute and ultracold bosonic alkali-metal atoms it can reach 100% [4]. Some years ago, by using the ODLRO of the two-body density matrix, Salasnich, Manini and Parola [6] calculated the condensate fraction of fermionic pairs in the crossover from the Bardeen-Cooper-Schrieffer (BCS) state of Cooper Fermi pairs to the Bose-Einstein condensate (BEC) of molecular dimers at zero temperature (later in the same year there were other two papers [7, 8] on the same topic). In particular, it was found that the condensate fraction grows from zero to one in the BCS-BEC crossover [6, 7, 8]. These theoretical predictions are in quite good agreement with the data obtained in two experiments [9, 10] with Fermi vapors of 6Li atoms. Recently, the condensate fraction of bulk neutron matter has been calculated by Wlazlowski and Magierski [11, 12] and also by Salasnich [13]. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 22nd International Laser Physics Worksop (LPHYS 13) IOP Publishing Journal of Physics: Conference Series 497 (2014) 012026 doi:10.1088/1742-6596/497/1/012026 In this paper we review the zero-temperature mean-field theory we have used to extract the fermionic condensate fraction in ultracold atoms [6] and bulk neutron matter [13]. In addition we determine the condensate fraction of neutron-neutron pairs as a function of the distance from the center of a neutron star [14, 15]. In particular, we find the maximum condensate fraction at the distance r/R 0.96, where R is the star radius. 2. Bosonic and fermionic condensation A quantum system of interacting identical bosons can be described by the bosonic field operator φˆ(r), which satisfies the familiar commutation rules [16] φˆ(r), φˆ+(r) = δ(r − r) , φˆ(r), φˆ(r) = φˆ+(r), φˆ+(r) =0, (1) where A,ˆ Bˆ = AˆBˆ − BˆAˆ and δ(r) is the Dirac delta function. The bosonic one-body density matrix is given by n(r, r)=φˆ+(r) φˆ(r) , (2) where the average ···can be a thermal average or a zero-temperature average. Its diagonal part is the average local density of bosons, i.e. n(r)=n(r, r)=φˆ+(r) φˆ(r), while the average total number of bosons reads N = φˆ+(r) φˆ(r) d3r . (3) As previously discussed, Penrose and Onsager [1, 2] used the ODLRO of the one-body density matrix of a uniform bosonic system to determine the condensate number N0 of bosons. For a non- uniform bosonic system this condensate number N0 is nothing else than the largest eigenvalue of the one-body density matrix [16]. As shown by Yukalov [17], in the thermodynamic limit this is equivalent to the spontaneous symmetry breaking of U(1) gauge symmetry, which gives 2 3 N0 = |φˆ(r)| d r (4) as the average number of condensed bosons. A quantum system of interacting identical fermions with two spin components (σ =↑, ↓)can be described by the fermionic field operator ψˆσ(r), which satisfies the familiar anti-commutation rules [16] ˆ ˆ+ ˆ ˆ ˆ+ ˆ+ ψσ(r), ψσ (r ) = δ(r − r ) δσ,σ , ψσ(r), ψσ (r ) = ψσ (r), ψσ (r ) =0, (5) where A,ˆ Bˆ = AˆBˆ + BˆAˆ and δσ,σ is the Kronecher symbol. The fermionic one-body density matrix is given by + nσ,σ (r, r )=ψˆσ (r) ψˆσ (r ) . (6) Its diagonal part is the average local density of fermions with spin σ, i.e. nσ(r)=nσ,σ (r, r)= + ψˆσ (r) ψˆσ(r), while the average total number of fermions reads + 3 N = ψˆσ (r) ψˆσ(r) d r . (7) σ=↑,↓ Yang [3] suggested that for a uniform strongly-interacting fermionic system the number N0 of Bose-condensed fermions, that is twice the number of condensed fermionic pairs, is related by the ODLRO of the fermionic two-body density matrix, given by + + 1 2 ˆ 1 ˆ 2 ˆ ˆ nσ1,σ2,σ1,σ2 (r , r , r1, r2)= ψσ1 (r ) ψσ2 (r ) ψσ2 (r2) ψσ1 (r1) . (8) 2 22nd International Laser Physics Worksop (LPHYS 13) IOP Publishing Journal of Physics: Conference Series 497 (2014) 012026 doi:10.1088/1742-6596/497/1/012026 For a generic (non-uniform) strongly-interacting system of identical fermions this condensate number N0 is nothing else than twice the largest eigenvalue of the fermionic two-body density matrix [16]. In the thermodynamic limit this is equivalent [18] to the spontaneous symmetry breaking of SU(2) gauge symmetry, which gives 2 3 3 N0 =2 |ψˆσ(r) ψˆσ (r )| d r d r (9) σ,σ=↑,↓ as the average number of condensed fermions. It is important to stress that within the BCS theory of superconductors and fermionic superfluids the condensate number N0 fermions can be much smaller than the average total number of fermions N. Moreover, in a uniform system N0 is twice the average number of fermionic pairs in the (pseudo) Bose-Einstein condensate, i.e. the number of fermionic pairs which have their center of mass with zero linear momentum. 3. Ultracold and dilute atomic gases The shifted Hamiltonian of the uniform two-spin-component Fermi superfluid made of ultracold atoms is given by 2 ˆ 3 ˆ+ 2 ˆ ˆ+ ˆ+ ˆ ˆ H = d r ψσ (r) − ∇ − μ ψσ(r)+g ψ↑ (r) ψ↓ (r) ψ↓(r) ψ↑(r) , (10) σ=↑,↓ 2m where ψˆσ(r) is the field operator that annihilates a fermion of spin σ in the position r, while + ψˆσ (r) creates a fermion of spin σ in r.Hereg<0 is the strength of the attractive fermion- fermion interaction, which is approximated by a contract Fermi pseudo-potential [4] because for ultracold and dilute gases the average distance between atoms is much larger than the effective radius of the inter-atomic potential [4, 16]. The ground-state average of the number of fermions is given by Eq. (7). This total number N is fixed by the chemical potential μ which appears in Eq. (10). Within the Bogoliubov approach the mean-field Hamiltonian derived from Eq. (10) can be diagonalized by using the Bogoliubov-Valatin representation of the field operator ψˆσ(r)interms of the anti-commuting quasi-particle Bogoliubov operators ˆbkσ with amplitudes uk and vk and the quasi-particle energy Ek. In this way one finds familiar expressions for these quantities: 2 2 1/2 Ek = (k − μ) +Δ (11) and 2 uk =(1+(k − μ)/Ek) /2 (12) 2 vk =(1− (k − μ)/Ek) /2 , (13) 2 2 where k = k /(2m) is the single-particle energy.
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