The behavior of collective oscillations of a trapped boson-fermion mixed condensate is studied in the sum rule approach. Mixing angle of bosonic and fermionic multipole operators is introduced so that the mixing characters of the low-lying collective modes are studied as functions of the bosonfermion interaction strength. For an attractive boson-fermion interaction, the low-lying monopole mode becomes a coherent oscillation of bosons and fermions and shows a rapid decrease in the excitation energy towards the instability point of the ground state. In contrast, the low-lying quadrupole mode keeps a bosonic character over a wide range of the interaction strengths. For the dipole mode the boson-fermion in-phase oscillation remains to be the eigenmode under the external oscillator potential. For weak repulsive values of the boson-fermion interaction strengths we found that an average energy of the out-of-phase dipole mode stays lower than the in-phase oscillation. Physical origin of the behavior of the multipole modes against boson-fermion interaction strength is discussed in some detail.Collective oscillation is one of the most prominent phenomena common to a variety of many-body systems. The realization of the Bose-Einstein condensates (BEC) for trapped Alkali atoms [1] offers a possibility to study such phenomena of quantum systems under ideal conditions. Up to now the experimental [2] as well as theoretical [3][4][5] studies of collective motions of BEC have been intensively performed. Quite recently the degenerate Fermi gas of trapped 40 K atoms has been realized [6], which motivates the study of collective motion also in Fermi gases [7]. These developments further encourage the study of possible boson-fermion mixed condensates of trapped atoms [8,9]. Now the static properties [10][11][12], stability conditions [13,14], and some dynamical properties [15,16] of trapped bosonfermion condensates have been investigated. In the present paper, we study the behavior of collective oscillations of a boson-fermion mixed condensate at T = 0 for both repulsive and attractive boson-fermion interactions. We adopt the sum rule approach [3,4] that has proved to be successful in the studies of collective excitations of Bose condensates. For the mixed condensate, we introduce a mixing angle of the boson/fermion excitation operators so as to allow the in-and out-of-phase oscillations of the bose and fermi condensates.In the sum rule approach we first calculate the energy weighted moments m p = j (E j − E 0 ) p | j|F |0 | 2 of the relevant multipole operator F , where |j 's represent the complete set of eigenstates of the Hamiltonian with energies E j , and |0 denotes the ground state. The excitation energy is expressed ashω = (m 3 /m 1 ) 1/2 which provides a useful expression for the average energy of the collective oscillation [3,7]. The moments are calculated from formulae m 1 =
The physics of ultracold atomic gases is making progress at a rapid pace, which has led to a realization of Bosonfermion mixtures of atomic gases ͓1-4͔. Boson-fermion ͑BF͒ mixtures may exhibit the richest variety of phenomena of all. They may show very different behavior from pure fermion or pure Bose gases ͓5,6͔. Especially interesting is a possible instability of the mixture when there is an attraction between bosons and fermions ͓5,7-9͔, as a recent experiment in fact suggests a collapse of the mixture ͓10͔.In the present work we propose and study quite a different scenario for an attractively interacting boson-fermion mixture. To simplify the problem in a first survey we shall consider the situation where there is no interaction between atoms of the same kind. As we will discuss at the end of the paper, this is not a severe approximation to cases where the interaction between like atoms is repulsive. More precisely we want to address the question of what happens to a mixture of free fermions and bosons when a ͑tunable͒ attraction is switched on between fermions and bosons. We imagine that correlated BF pairs will be created. These BF pairs are composite fermions and as such these BF pairs should form a Fermi gas of composites. Besides in ultracold atomic gases such a situation can exist in other branches of physics. For example, in nuclear systems ͑e.g., neutron stars͒ of highdensity K − mesons and nucleons may form a gas of ⌳'s and the ⌳'s may then form a Fermi gas of their own ͓11͔. Or in a quark-gluon plasma additional quarks may bind to preformed diquarks or color Cooper pairs ͑the "bosons"͒ ͓12͔ to form a gas of nucleons in the so-called hadronization transition. Further examples may be added to this list.For a numerical example, we take a mixture of 40 K ͑fer-mion͒ and 41 K ͑boson͒ atoms throughout the paper. They are known as candidates for a realization of this kind of quantum systems. While their scattering lengths are not well fixed at present, and different values have been reported experimentally ͓13͔, it is not crucial at the moment because our study will be mostly academic, elaborating on the basic phenomenon. Applications to realistic systems will be left for the future.Let us consider a single BF pair propagating in the background of a homogeneous gas of free one-component fermions and spinless bosons. We will formulate our approach for a situation at finite temperature T, though later on in our application we will concentrate on the T = 0 case. We have in mind an analogous study Cooper performed a long time ago ͓14͔ for the propagation of two fermions ͑spin up or down͒ in the background of a homogeneous gas of two-component free fermions. In other words we consider a situation where in the original Cooper problem one fermion type ͑let us say spin down͒ is replaced by spinless bosons. The BF propagator at finite temperature T and finite center-of-mass momentum P of the pair that is added to the system with momenta P /2+p ͑fermion͒ and P /2−p ͑boson͒ is
Spin asymmetry of the ground states is studied for the trapped spin-degenerate (two-component) gases of the fermionic atoms with the repulsive interaction between different components, and, for large particle number, the asymmetric (collective ferromagnetic) states are shown to be stable because it can be energetically favorable to increase the fermi energy of one component rather than the increase of the interaction energy between up-down components. We formulate the Thomas-Fermi equations and show the algebraic methods to solve them. From the Thomas-Fermi solutions, we find three kinds of ground states in finite system: 1) paramagnetic (spin-symmetric), 2) ferromagnetic (equilibrium) and 3) ferromagnetic (nonequilibrium) states. We show the density profiles and the critical atom numbers for these states obtained analytically, and, in ferromagnetic states, the spin-asymmetries are shown to occur in the central regions of the trapped gas, and grows up with increasing particle number. Based on the obtained results, we discuss the experimental conditions and current difficulties to realize the ferromagnetic states of the trapped atom gas, which should be overcome.
Instabilities for boson-fermion mixed condensates of trapped Alkali atoms due to the bosonfermion attractive interaction are studied using a variational method. Three regions are shown for their instabilities according to the boson-fermion interaction strength: stable, meta-stable and unstable ones. The stability condition is obtained analytically from the asymptotic expansion of the variational total energy. The life-time of metastable states is discussed for tunneling decay, and is estimated to be very long. It suggests that, except near the unstable border, meta-stable mixed condensate should be almost-stable against clusterizations. The critical border between meta-stable and unstable phases is calculated numerically and is shown to be consistent with the Mølmer scaling condition. [5][6][7], dynamical expansions after the removal of trapping potentials [8], and the instability changes of 7 Li-6 Li system with attractive boson-boson interaction [10]. Discoveries of theIn this paper, we study the instabilities and collapses of the polarized boson-fermion mixed condensates due to the attractive boson-fermion interaction. In BEC, such instabilities are observed in several systems: 1) the trapped meta-stable 7 Li BEC (observed experimentally [11]) equilibrated between the attractive boson-boson interaction and the kinetic pressure due to the finite confinement [12], 2) two-component uniform BEC (from bosons 1 and 2), whose stability condition is given by g 11 g 22 > g 2 12 (g ij : the coupling constant between bosons i and j). In the latter case, the stability condition depends only on the interaction strength, but the condition for the case 1) has also the particle number dependence [12].Let us consider the case of the trapped boson-fermion mixed condensate. In the present paper, we assume T = 0 and a spherical harmonic-oscillator for the trapping potential; extension to the deformed potential is straightforward. Using the Thomas-Fermi approximation for the fermion degree of freedom (appropriate for large N f condensate), the total energy of the system becomes a functional of the boson order-parameter Φ( r) and the fermion density distribution n f ( r): (1), the fermion-fermion interaction has been neglected; the elastic fermion-fermion s-wave scattering for a polarized gas is absent because of the Pauli blocking effects and the p-wave scattering is suppressed below 100 µK [14]. Evaluating the total energy with the variational method, we take the Gaussian ansatz for the boson order-parameter:where x = r/ξ is a radial distance scaled by the harmonic-oscillator length ξ = (h/mω) 1/2 , and N b = d 3 x|Φ(x; R)| 2 is total boson number. The variational parameter R in (2) corresponds to the root-mean-square (rms) radius of the boson density distribution. This kind of variational functions have also been used in the instability investigations of 1
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