We consider a dilute homogeneous mixture of bosons and spin-polarized fermions at zero temperature. We first construct the formal scheme for carrying out systematic perturbation theory in terms of single particle Green's functions. We especially focus on the description of the boson-fermion interaction. To do so we need to introduce the renormalized boson-fermion $T$ matrix, which we determine to second order in the boson-fermion $s$ -wave scattering length. We also discuss how to incorporate the usual boson-boson $T$ matrix in mean field approximation to obtain the total ground-state properties of the system. The next-order term beyond mean field stems from the boson-fermion interaction and is proportional to ${a}_{\mathrm{BF}}{k}_{\mathrm{F}}.$ The total ground-state energy density to this order is the sum of the kinetic energy of the free fermions, the boson-boson mean-field interaction, the usual mean-field contribution to the boson-fermion interaction energy, and the first boson-fermion correction beyond mean field. We also compute the bosonic and the fermionic chemical potentials, the compressibilities, and the modification to the induced fermion-fermion interaction. We discuss the behavior of the total ground-state energy and the importance of the correction beyond mean field for various parameter regimes, in particular considering mixtures of ${}^{6}\mathrm{Li}$ and ${}^{7}\mathrm{Li}$ and of ${}^{3}\mathrm{He}$ and ${}^{4}\mathrm{He}.$ Moreover, we determine the modification of the induced fermion-fermion interaction due to the effects beyond mean field. We show that there is no effect on the depletion of the Bose condensate to first order in the boson-fermion scattering length ${a}_{\mathrm{BF}}.$ We consider a dilute homogeneous mixture of bosons and spin-polarized fermions at zero temperature. We first construct the formal scheme for carrying out systematic perturbation theory in terms of single particle Green's functions. We especially focus on the description of the boson-fermion interaction. To do so we need to introduce the renormalized boson-fermion $T$ matrix, which we determine to second order in the boson-fermion $s$-wave scattering length. We also discuss how to incorporate the usual boson-boson $T$ matrix in mean field approximation to obtain the total ground-state properties of the system. The next-order term beyond mean field stems from the boson-fermion interaction and is proportional to ${a}_{\mathrm{BF}}{k}_{\mathrm{F}}.$ The total ground-state energy density to this order is the sum of the kinetic energy of the free fermions, the boson-boson mean-field interaction, the usual mean-field contribution to the boson-fermion interaction energy, and the first boson-fermion correction beyond mean field. We also compute the bosonic and the fermionic chemical potentials, the compressibilities, and the modification to the induced fermion-fermion interaction. We discuss the behavior of the total ground-state energy and the importance of the correction beyond mean field for various parameter regimes, in particular considering mixtures of ${}^{6}\mathrm{Li}$ and ${}^{7}\mathrm{Li}$ and of ${}^{3}\mathrm{He}$ and ${}^{4}\mathrm{He}.$ Moreover, we determine the modification of the induced fermion-fermion interaction due to the effects beyond mean field. We show that there is no effect on the depletion of the Bose condensate to first order in the boson-fermion scattering length ${a}_{\mathrm{BF}}.$

arXiv:cond-mat/0201102Physical Review AApr 26, 200266 citations

Mean-Field Regime for Fermionic Systems

Niels Benedikter, Marcello Porta, Benjamin Schlein

SpringerBriefs in mathematical physicsOct 6, 20153 citations

Mean-field description of a dynamical collapse of a fermionic condensate in a trapped boson-fermion mixture

Sadhan K. Adhikari

We suggest a time-dependent dynamical mean-field-hydrodynamic model for the collapse of a trapped boson-fermion condensate and perform numerical simulation based on it to understand some aspects of the experiment by Modugno et al. [Science 297, 2240 (2002)] on the collapse of the fermionic condensate in the $^{40}\mathrm{K}$ - $^{87}\mathrm{R}\mathrm{b}$ mixture. We show that the mean-field model explains the formation of a stationary boson-fermion condensate at zero temperature with relative sizes compatible with experiment. This model is also found to yield a faithful representation of the collapse dynamics in qualitative agreement with experiment. In particular we consider the collapse of the fermionic condensate associated with (a) an increase of the number of bosonic atoms as in the experiment and (b) an increase of the attractive boson-fermion interaction using a Feshbach resonance. Suggestion for experiments of fermionic collapse using a Feshbach resonance is made.We suggest a time-dependent dynamical mean-field-hydrodynamic model for the collapse of a trapped boson-fermion condensate and perform numerical simulation based on it to understand some aspects of the experiment by Modugno et al. [Science 297, 2240 (2002)] on the collapse of the fermionic condensate in the $^{40}\mathrm{K}$- $^{87}\mathrm{R}\mathrm{b}$ mixture. We show that the mean-field model explains the formation of a stationary boson-fermion condensate at zero temperature with relative sizes compatible with experiment. This model is also found to yield a faithful representation of the collapse dynamics in qualitative agreement with experiment. In particular we consider the collapse of the fermionic condensate associated with (a) an increase of the number of bosonic atoms as in the experiment and (b) an increase of the attractive boson-fermion interaction using a Feshbach resonance. Suggestion for experiments of fermionic collapse using a Feshbach resonance is made.

arXiv:cond-mat/0407032Physical Review AOct 19, 200464 citations

Parametric control in coupled fermionic oscillators

Arnab Ghosh

A simple model of parametric coupling between two fermionic oscillators is considered. Statistical properties, in particular the mean and variance of quanta for a single mode, are described by means of a time-dependent reduced density operator for the system and the associated $P$ function. The density operator for fermionic fields as introduced by Cahill and Glauber [K. E. Cahill and R. J. Glauber, Phys. Rev. A 59, 1538 (1999)] thus can be shown to provide a quantum mechanical description of the fields closely resembling their bosonic counterpart. In doing so, special emphasis is given to population trapping, and quantum control over the states of the system.A simple model of parametric coupling between two fermionic oscillators is considered. Statistical properties, in particular the mean and variance of quanta for a single mode, are described by means of a time-dependent reduced density operator for the system and the associated $P$ function. The density operator for fermionic fields as introduced by Cahill and Glauber [K. E. Cahill and R. J. Glauber, Phys. Rev. A 59, 1538 (1999)] thus can be shown to provide a quantum mechanical description of the fields closely resembling their bosonic counterpart. In doing so, special emphasis is given to population trapping, and quantum control over the states of the system.

Physical Review AOct 14, 20145 citations

T

matrix, which we determine to second order in the boson-fermion

s

-wave scattering length. We also discuss how to incorporate the usual boson-boson

T

matrix in mean field approximation to obtain the total ground-state properties of the system. The next-order term beyond mean field stems from the boson-fermion interaction and is proportional to

{a}_{\mathrm{BF}}{k}_{\mathrm{F}}.

The total ground-state energy density to this order is the sum of the kinetic energy of the free fermions, the boson-boson mean-field interaction, the usual mean-field contribution to the boson-fermion interaction energy, and the first boson-fermion correction beyond mean field. We also compute the bosonic and the fermionic chemical potentials, the compressibilities, and the modification to the induced fermion-fermion interaction. We discuss the behavior of the total ground-state energy and the importance of the correction beyond mean field for various parameter regimes, in particular considering mixtures of

{}^{6}\mathrm{Li}

and

{}^{7}\mathrm{Li}

and of

{}^{3}\mathrm{He}

and

{}^{4}\mathrm{He}.

Moreover, we determine the modification of the induced fermion-fermion interaction due to the effects beyond mean field. We show that there is no effect on the depletion of the Bose condensate to first order in the boson-fermion scattering length

{a}_{\mathrm{BF}}.

^{40}\mathrm{K}

^{87}\mathrm{R}\mathrm{b}

mixture. We show that the mean-field model explains the formation of a stationary boson-fermion condensate at zero temperature with relative sizes compatible with experiment. This model is also found to yield a faithful representation of the collapse dynamics in qualitative agreement with experiment. In particular we consider the collapse of the fermionic condensate associated with (a) an increase of the number of bosonic atoms as in the experiment and (b) an increase of the attractive boson-fermion interaction using a Feshbach resonance. Suggestion for experiments of fermionic collapse using a Feshbach resonance is made.

P

function. The density operator for fermionic fields as introduced by Cahill and Glauber [K. E. Cahill and R. J. Glauber, Phys. Rev. A 59, 1538 (1999)] thus can be shown to provide a quantum mechanical description of the fields closely resembling their bosonic counterpart. In doing so, special emphasis is given to population trapping, and quantum control over the states of the system.