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arXiv:cond-mat/0004325 · Physical Review A

Moment of inertia and quadrupole response function of a trapped superfluid

Francesca Zambelli, S. Stringari

Published Feb 5, 2001• 49 citations
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Abstract

We derive an explicit relationship between the moment of inertia and the quadrupole response function of an interacting gas confined in a harmonic trap. The relationship holds for both Bose and Fermi systems and is well-suited to reveal the effects of irrotationality of the superfluid motion. Recent experimental results on the scissors mode are used to extract the value of the moment of inertia of a trapped Bose gas and to point out the deviations from the rigid value due to superfluidity.

DOI
10.1103/physreva.63.033602

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We derive an explicit relationship between the moment of inertia and the quadrupole response function of an interacting gas confined in a harmonic trap. The relationship holds for both Bose and Fermi systems and is well-suited to reveal the effects of irrotationality of the superfluid motion. Recent experimental results on the scissors mode are used to extract the value of the moment of inertia of a trapped Bose gas and to point out the deviations from the rigid value due to superfluidity.
We show that knowledge of the time-dependent response of a trapped gas, subject to a sudden rotation of a confining harmonic potential, allows for the determination of the moment of inertia of dipolar supersolid configurations. While in the presence of one-dimensional arrays of droplets the frequency of the resulting scissors oscillation provides accurate access to the value of the moment of inertia, two-dimensional-like configurations are characterized by low-frequency resonances in the rotating signal, reflecting the presence of significant rigid-body components in the rotational motion. Using the formalism of response-function theory and simulations based on the so-called extended time-dependent Gross-Pitaevskii equation, we point out the crucial role played by the low-frequency components in the determination of the moment of inertia and of its deviations from the irrotational value. We also propose a protocol based on the stationary rotation of the trap, followed by its sudden stop, which might provide a promising alternative to the experimental evaluation of the moment of inertia.
We use a generalized Gross-Pitaevskii equation for the condensate and a semiclassical kinetic equation for the noncondensate atoms to discuss the scissors mode in a trapped Bose-condensed gas at finite temperatures. Both equations include the effect of collisions between the condensate and noncondensate atoms. We solve the coupled moment equations describing oscillations of the quadrupole moments of the condensate and noncondensate components to find the collective-mode frequencies and collisional damping rates as a function of temperature. Our calculations extend those of Gu\'ery-Odelin and Stringari at T=0T=0T=0 and in the normal phase. They complement the numerical results of Jackson and Zaremba, although Landau damping is left out of our approach. Our results are also used to calculate the quadrupole response function, which is related to the moment of inertia. It is shown explicitly that the moment of inertia of a trapped Bose gas at finite temperatures involves a sum of an irrotational component from the condensate and a rotational component from the thermal-cloud atoms.
We relate the frequency of the scissors mode to the moment of inertia of a trapped Bose gas at finite temperature in a semi-classical approximation. We apply these theoretical results to the data obtained in our previous study of the properties of the scissors mode of a trapped Bose-Einstein condensate of 87^{87}87Rb 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 (6Li){(}^{6}\mathrm{Li})(6Li) 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 Tc{T}_{c}Tc​ 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.