Equation of state of a laser-cooled gas

We experimentally determine the equation of state of a laser-cooled gas. By employing the Lane-Emden formalism, widely used in astrophysics, we derive the equilibrium atomic profiles in large magneto-optical traps where the thermodynamic effects are cast in a polytropic equation of state. The effects of multiple scattering of light are included, which results in a generalized Lane-Emden equation for the atomic profiles. A detailed experimental investigation reveals an excellent agreement with the model, with a twofold significance. On one hand, we can infer the details of the equation of state of the system, from an ideal gas to a correlated phase due to an effective electrical charge for the atoms, which is accurately described by a microscopical description of the effective electrostatic interaction. On the other hand, we are able map the effects of multiple scattering onto directly controllable experimental variables, which paves the way to subsequent experimental investigations of this collective interaction.

Figure 1

Integrated atomic density profiles (in false color code), and a one-dimensional horizontal cut passing through the center of the cloud (white points). The corresponding fitting curves to numerically generated solutions of the Lane-Emden equation in (5) are also displayed (black dashed line). The magnetic field gradient varies from top to bottom as $\mathbf{\nabla }B=7.5$, 10, and $12.5$ G/cm. From left to right, we vary the detuning as $\delta =-3.2, -2.8$, and $-2.4\mathrm{\Gamma }$. The error bars here correspond to standard deviations of the CCD pixel values, taken over 30 realizations.

Figure 2

In the left panel, we plot the normalized plasma frequency $\mathrm{\Omega }={\omega }_p^2/3{\omega }_0^2$ as a function of detuning $\delta$. The right panel depicts the polytropic exponent $\gamma$, also as a function of detuning $\delta$. The different curves correspond to $\mathbf{\nabla }B=7.5$ G/cm (red line), $\mathbf{\nabla }B=10$ G/cm (black line), and $\mathbf{\nabla }B=12.5$ G/cm (blue line).

Figure 3

Polytropic exponent, $\gamma$, as a function of the universal parameter $\xi$, for $\mathbf{\nabla }B=7.5\text{G}/\mathrm{cm}$ (red points), $\mathbf{\nabla }B=10$ G/cm (black points), and $\mathbf{\nabla }B=12.5$ G/cm (blue points). All points corresponding to different experimental conditions fall on the universal curve defined by Eq. (9). The inset corresponds to the same experimental points but binned in the $\xi$ parameter and equally spaced.