arXiv:cond-mat/9509166 · Physical Review Letters

Moment of Inertia and Superfluidity of a Trapped Bose Gas

S. Stringari

Published Feb 26, 1996• 63 citations

Abstract

The temperature dependence of the moment of inertia of a dilute Bose gas confined in a harmonic trap is determined. Deviations from the rigid value, due to the occurrence of Bose-Einstein condensation, reveal the superfluid behavior of the system. In the noninteracting gas these deviations become important at temperatures of the order of ${T}_{c}{N}^{\ensuremath{-}1/12}$. The role of interactions is also discussed.

DOI: 10.1103/physrevlett.76.1405

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arXiv:cond-mat/9509166 · Physical Review Letters

Moment of Inertia and Superfluidity of a Trapped Bose Gas

S. Stringari

Published Feb 26, 1996• 63 citations

arXiv HTML PDF

Abstract

{T}_{c}{N}^{\ensuremath{-}1/12}

. The role of interactions is also discussed.

DOI: 10.1103/physrevlett.76.1405

Related papers

Moment of inertia and quadrupole response function of a trapped superfluid

Francesca Zambelli, S. Stringari

arXiv:cond-mat/0004325Physical Review AFeb 5, 200149 citations

The distribution of the moment of inertia for harmonically trapped noninteracting Bosons at finite temperature: Large deviations

Manas Kulkarni, Satya N. Majumdar, Grégory Schehr

We compute the full probability distribution of the moment of inertia I \propto \sum_{i=1}^N \vec{r}_i^{\,2} I ∝ ∑ i = 1 N r → i 2 of a gas of N N noninteracting bosons trapped in a harmonic potential V(r) = (1/2)\, m\, \omega^2 r^2 V ( r ) = ( 1 / 2 ) m ω 2 r 2 , in all dimensions and at all temperature. The appropriate thermodynamic limit in a trapped Bose gas consists in taking the limit N\to ∞ N → ∞ and \omega\to 0 ω → 0 with their product \rho = N \omega^d ρ = N ω d fixed, where \rho ρ plays the role analogous to the density in a translationally invariant system. In this thermodynamic limit and in dimensions d;gt 1 d > 1 , the harmonically trapped Bose gas undergoes a Bose-Einstein condensation (BEC) transition as the density \rho ρ crosses a critical value \rho_c(\beta) ρ c ( β ) , where \beta β denotes the inverse temperature. We show that the nature of the condensation is different for 1<d≤ 2 1 < d ≤ 2 and d;gt 2We compute the full probability distribution of the moment of inertia I \propto \sum_{i=1}^N \vec{r}_i^{\,2} I ∝ ∑ i = 1 N r → i 2 of a gas of N N noninteracting bosons trapped in a harmonic potential V(r) = (1/2)\, m\, \omega^2 r^2 V ( r ) = ( 1 / 2 ) m ω 2 r 2 , in all dimensions and at all temperature. The appropriate thermodynamic limit in a trapped Bose gas consists in taking the limit N\to ∞ N → ∞ and \omega\to 0 ω → 0 with their product \rho = N \omega^d ρ = N ω d fixed, where \rho ρ plays the role analogous to the density in a translationally invariant system. In this thermodynamic limit and in dimensions d;gt 1 d > 1 , the harmonically trapped Bose gas undergoes a Bose-Einstein condensation (BEC) transition as the density \rho ρ crosses a critical value \rho_c(\beta) ρ c ( β ) , where \beta β denotes the inverse temperature. We show that the nature of the condensation is different for 1<d≤ 2 1 < d ≤ 2 and d;gt 2

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{T}_{c}{N}^{\ensuremath{-}1/12}

. The role of interactions is also discussed.

We investigate the thermodynamics of a Bose gas interacting with repulsive forces and confined in a harmonic trap. We show that the relevant parameters of the system (temperature, number

N

of atoms, harmonic oscillator length, deformation of the trap,

s

-wave scattering length) fix its thermodynamic behavior in the large

N

limit through two dimensionless scaling parameters. These are the reduced temperature

t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}T/{T}_{c}^{0}

and the ratio $\ensuremath{\eta}$ between the

T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0

value of the chemical potential, evaluated in the Thomas-Fermi limit, and the critical temperature

{T}_{c}^{0}

of the noninteracting model. The scaling functions relative to the condensate fraction, energy, chemical potential, and moment of inertia are calculated within the Popov approximation.

^{87}

Rb atoms as a function of the temperature. The frequency shifts that we measured show quenching of the moment of inertia of the Bose gas at temperatures below the transition temperature - the system has a lower moment of inertia that of a rigid body with the same mass distribution, because of superfluidity.

The moment of inertia \ensuremath{\Theta} of a trapped superfluid gas of atomic fermions

{(}^{6}\mathrm{Li})

is calculated as a function of the temperature. At zero temperature the moment of inertia takes on the irrotational-flow value. Only for T very close to

{T}_{c}

is rigid rotation attained. It is proposed that future measurements of the rotational energy will unambiguously reveal whether the system is in a superfluid state or not.