arXiv:1208.5640 · Physical Review A

Breathing mode of two-dimensional atomic Fermi gases in harmonic traps

Chao Gao, Zhenhua Yu

Published Oct 9, 2012• 41 citations

Abstract

For two-dimensional (2D) atomic Fermi gases in harmonic traps, the SO(2,1) symmetry is broken by the interatomic interaction explicitly via the contact correlation operator. Consequently, the frequency of the breathing mode ${\ensuremath{\omega}}_{B}$ of the 2D Fermi gas can be different from $2{\ensuremath{\omega}}_{0}$, with ${\ensuremath{\omega}}_{0}$ the trapping frequency of harmonic potentials. At zero temperature, we use the sum rules of density correlation functions to yield upper bounds for ${\ensuremath{\omega}}_{B}$. We further calculate ${\ensuremath{\omega}}_{B}$ through the Euler equations in the hydrodynamic regime. The obtained value of ${\ensuremath{\omega}}_{B}$ satisfies the upper bounds and shows deviation from $2{\ensuremath{\omega}}_{0}$, which can be as large as about $8%$.

DOI: 10.1103/physreva.86.043609

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arXiv:1208.5640 · Physical Review A

Breathing mode of two-dimensional atomic Fermi gases in harmonic traps

Chao Gao, Zhenhua Yu

Published Oct 9, 2012• 41 citations

arXiv HTML PDF

Abstract

{\ensuremath{\omega}}_{B}

of the 2D Fermi gas can be different from

2{\ensuremath{\omega}}_{0}

, with

{\ensuremath{\omega}}_{0}

the trapping frequency of harmonic potentials. At zero temperature, we use the sum rules of density correlation functions to yield upper bounds for

{\ensuremath{\omega}}_{B}

. We further calculate

{\ensuremath{\omega}}_{B}

through the Euler equations in the hydrodynamic regime. The obtained value of

{\ensuremath{\omega}}_{B}

satisfies the upper bounds and shows deviation from

2{\ensuremath{\omega}}_{0}

, which can be as large as about

8%

DOI: 10.1103/physreva.86.043609

Related papers

Breathing-mode frequency of a strongly interacting Fermi gas across the two- to three-dimensional crossover

Umberto Toniolo, Brendan C. Mulkerin, Xia-Ji Liu +1 more

We address the interplay between dimension and quantum anomaly on the breathing mode frequency of a strongly interacting Fermi gas harmonically trapped at zero temperature. Using a beyond-mean-field Gaussian pair fluctuation theory, we employ periodic boundary conditions to simulate the dimensionality of the system and impose a local-density approximation, with two different schemes, to model different trapping potentials in the tightly confined axial direction. By using a sum-rule approach, we compute the breathing mode frequency associated with a small variation of the trapping frequency along the weakly confined transverse direction and describe its behavior as a function of the dimensionality, from two to three dimensions, and of the interaction strength. We compare our predictions with previous calculations on the two-dimensional breathing mode anomaly and discuss their possible observation in ultracold Fermi gases of $^{6}\mathrm{Li}$ and $^{40}\mathrm{K}$ atoms.We address the interplay between dimension and quantum anomaly on the breathing mode frequency of a strongly interacting Fermi gas harmonically trapped at zero temperature. Using a beyond-mean-field Gaussian pair fluctuation theory, we employ periodic boundary conditions to simulate the dimensionality of the system and impose a local-density approximation, with two different schemes, to model different trapping potentials in the tightly confined axial direction. By using a sum-rule approach, we compute the breathing mode frequency associated with a small variation of the trapping frequency along the weakly confined transverse direction and describe its behavior as a function of the dimensionality, from two to three dimensions, and of the interaction strength. We compare our predictions with previous calculations on the two-dimensional breathing mode anomaly and discuss their possible observation in ultracold Fermi gases of $^{6}\mathrm{Li}$ and $^{40}\mathrm{K}$ atoms.

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Hui Hu, Xia-Ji Liu

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Jeff Maki

An important tool for understanding the effects of interactions in harmonically trapped atomic gases is the examination of their collective modes. One such mode is the breathing or monopole mode, which is special as it is constrained to occur at twice the harmonic trapping frequency when the interactions are scale invariant. When the interactions are not scale invariant, the frequency of the breathing mode will deviate from twice the trap frequency. The deviation itself depends on the thermodynamic contacts, which describe how the energy changes with the interactions. In this work I examine how the thermodynamic contacts and the breathing-mode frequency of a spin-polarized one-dimensional (1D) $p$ -wave Fermi gas depend on the 1D scattering volume $\ensuremath{\ell}$ and the effective range $r$ in the high-temperature limit. Such dynamics can be studied in experiments and provide a tool for understanding how the dynamics depends on interactions with a finite effective range.An important tool for understanding the effects of interactions in harmonically trapped atomic gases is the examination of their collective modes. One such mode is the breathing or monopole mode, which is special as it is constrained to occur at twice the harmonic trapping frequency when the interactions are scale invariant. When the interactions are not scale invariant, the frequency of the breathing mode will deviate from twice the trap frequency. The deviation itself depends on the thermodynamic contacts, which describe how the energy changes with the interactions. In this work I examine how the thermodynamic contacts and the breathing-mode frequency of a spin-polarized one-dimensional (1D) $p$-wave Fermi gas depend on the 1D scattering volume $\ensuremath{\ell}$ and the effective range $r$ in the high-temperature limit. Such dynamics can be studied in experiments and provide a tool for understanding how the dynamics depends on interactions with a finite effective range.

arXiv:2110.05725Physical Review ADec 20, 20215 citations

{\ensuremath{\omega}}_{B}

of the 2D Fermi gas can be different from

2{\ensuremath{\omega}}_{0}

, with

{\ensuremath{\omega}}_{0}

the trapping frequency of harmonic potentials. At zero temperature, we use the sum rules of density correlation functions to yield upper bounds for

{\ensuremath{\omega}}_{B}

. We further calculate

{\ensuremath{\omega}}_{B}

through the Euler equations in the hydrodynamic regime. The obtained value of

{\ensuremath{\omega}}_{B}

satisfies the upper bounds and shows deviation from

2{\ensuremath{\omega}}_{0}

, which can be as large as about

8%

^{6}\mathrm{Li}

and

^{40}\mathrm{K}

atoms.

T

-matrix theories and the virial expansion, we obtain a hydrodynamic equation of the harmonically trapped Fermi gas (with trapping frequency ${\ensuremath{\omega}}_{0}$) through the local density approximation. By solving the hydrodynamic equations, we determine the breathing mode frequencies as a function of interaction strength and temperature. We find that the breathing mode anomaly depends sensitively on both interaction strength and temperature. In particular, in the strongly interacting regime, we predict a significant downshift of the breathing mode frequency, below the scale invariant value of $2{\ensuremath{\omega}}_{0}$, for temperatures of the order of the Fermi temperature.

Inspired by the renewed experimental activities on

p

-wave resonantly interacting atomic Fermi gases, we investigate theoretically some experimental observables of such systems at zero temperature in two dimensions, using both mean-field theory and Gaussian pair fluctuation theory. These observables include the two

p

-wave contact parameters and the breathing mode frequency, which can be readily measured in current cold-atom setups with

^{40}\mathrm{K}

and

^{6}\mathrm{Li}

atoms. We find that the many-body component of the two contact parameters exhibits a pronounced peak slightly above the resonance and consequently leads to a dip in the breathing mode frequency. In the resonance limit, we discuss the dependence of the equation of state and the breathing mode frequency on the dimensionless effective range of the interaction ${k}_{F}{R}_{p}\ensuremath{\ll}1$, where

{k}_{F}

is the Fermi wave vector and

{R}_{p}

is the effective range. The breathing mode frequency ${\ensuremath{\omega}}_{B}$ deviates from the scale-invariant prediction of ${\ensuremath{\omega}}_{c}=2{\ensuremath{\omega}}_{0}$, where ${\ensuremath{\omega}}_{0}$ is the trapping frequency of the harmonic potential. This frequency shift is caused by the necessary existence of the effective range. In the small range limit, we predict that the mode frequency deviation at the leading order is given by $\ensuremath{\delta}{\ensuremath{\omega}}_{B}\ensuremath{\simeq}\ensuremath{-}({\ensuremath{\omega}}_{0}/4){ln}^{\ensuremath{-}1}({k}_{F}{R}_{p})$.

p

-wave Fermi gas depend on the 1D scattering volume $\ensuremath{\ell}$ and the effective range

r

in the high-temperature limit. Such dynamics can be studied in experiments and provide a tool for understanding how the dynamics depends on interactions with a finite effective range.