Quantum and thermal fluctuations of trapped Bose-Einstein condensates

We quantize a semiclassical system defined by the Hamiltonian obtained from the asymptotic self-similar solution of the Gross-Pitaevskii equation for a trapped Bose-Einstein condensate with a linear gain term. On the basis of a Schrödinger equation derived in a space of ellipsoidal parameters, we analytically calculate the quantum mechanical and thermal variance in the ellipsoidal parameters for Bose-Einstein condensates in various shapes of trap. We show that, except for temperatures close to zero, dimensionless dispersions do not depend on the frequencies of the trap and they have the same dependence on dimensionless temperatures.

Figure 1

The dimensionless transverse dispersion ${(\Delta a_{\perp }∕l)}^2$ as a function of the dimensionless temperature $\Theta$ for different values of $\xi =0.1$, 1, and 10. A greater parameter $\xi$ gives the curve with a greater dispersion. There is very little dependence on $\xi$, with the upper (chain) line, the middle (broken) line, and the lower (full) line being for $\xi =10$, 1, and 0.1, respectively. The subfigure shows the same curves for $0⩽\Theta ⩽0.6$.

Figure 2

The dimensionless longitudinal dispersion ${(\Delta a_3∕l_0)}^2$ as a function of the dimensionless temperature ${\Theta }_0$. Here the upper (chain) line, the middle (broken) line, and the lower (full) line are for ${\xi }_0=10$, 1, and 0.1, respectively. The subfigure shows the same curves for $0⩽{\Theta }_0⩽0.6$.

Figure 3

The dimensionless transverse dispersion ${(\Delta a_{\perp }∕l)}^2$ as a function of the dimensionless parameter $\xi ={\omega }_0∕\omega$ for different values of dimensionless temperatures $\Theta =0$,0.1,…,0.6. A greater temperature gives a curve with a greater value of dispersion ${(\Delta a_{\perp }∕l)}^2$.

Figure 4

The dimensionless longitudinal dispersion ${(\Delta a_3∕l_0)}^2$ as a function of the dimensionless parameter ${\xi }_0=\omega ∕{\omega }_0$ for different values of dimensionless temperatures ${\Theta }_0=0$,0.1,…,0.6. A greater temperature gives a curve with a greater value of dispersion ${(\Delta a_3∕l_0)}^2$.