Quantum Enhancement of the Index of Refraction in a Bose-Einstein Condensate

We study the index of refraction of an ultracold bosonic gas in the dilute regime. Using phase-contrast imaging with light detuned from resonance by several tens of linewidths, we image a single cloud of ultracold atoms for 100 consecutive shots, which enables the study of the scattering rate as a function of temperature and density using only a single cloud. We observe that the scattering rate is increased below the critical temperature for Bose-Einstein condensation by a factor of 3 compared to the single-atom scattering rate. We show that current atom-light interaction models to second order of the density show a similar increase, where the magnitude of the effect depends on the model that is used to calculate the pair-correlation function. This confirms that the effect of quantum statistics on the index of refraction is dominant in this regime.

Figure 1

A plot of the remaining fraction of particles on a logarithmic scale as a function of exposure time for a cloud above (red circles) and below (black crosses) the critical temperature $T_C$. The exposure time is the total time that the cloud is illuminated with light for PCI and the probe intensity is $67.7\mu \mathrm{W}/{\mathrm{cm}}^2$. The solid lines are linear fits assuming an exponential decay at $t=0$. The statistical error can be inferred from the spread in the data points. The dotted lines are independent calculations (as discussed in the text) of the loss rate for the condensed cloud based on the intensity of the probe light and the enhancement induced by the pair-correlation function for an ideal Bose gas (dotted green line) and Hartree-Fock model (dotted blue line).

Figure 2

The (a) real and (b) imaginary part of the index of refraction $n$ calculated as a function of temperature for a homogeneous density of $0.54\times {10}^{20}\mathrm{atoms}/{\mathrm{m}}^3$ and a detuning of $-350\mathrm{MHz}$. The results are for the Lorentz model (solid black line), the Lorentz model including the Lorentz-Lorenz correction (dotted black line), and Eq. (4), using the pair-correlation function of an ideal Bose gas (dotted green line) or the Hartree-Fock model (dotted blue line). The refractive index already starts to increase above $T_C$, acting as a precursor for Bose-Einstein condensation.

Figure 3

(a) The enhancement of the scattering rate as a function of $\mu$ and $T$ after averaging over the trap for the ideal, condensed Bose gas. The experimental conditions are shown by the black trace. The conditions change during the measurement due to the losses and heating induced by the probe light. (b) The enhancement as a function of the exposure time for our experimental conditions. The values follow the black trace in (a).