Thermodynamics of Bose-Einstein-condensed clouds using phase-contrast imaging

Phase contrast imaging is used to observe Bose-Einstein condensates at finite temperature in situ. The imaging technique is used to accurately derive the absolute phase shift of a probe laser beam due to both the condensate and the thermal cloud. The accuracy of the method is enhanced by using the periodicity of the intensity signal as a function of the accumulated phase. The measured density profiles can be described using a two-relevant-parameter fit, in which only the chemical potential and the temperature are to be determined. This allows us to directly compare the measured density profiles to different mean-field models in which the interaction between the condensed and the thermal atoms is taken into account to various degrees.

Figure 1

Schematic representation of the PCI setup. The atoms located at the center of the trap are imaged with lens $L_1$ (with focal distance $f_1$), which is placed at a distance $f_1$ from the center. A sharp image is created by placing lens $L_2$ (with focal length $f_2$) at a distance $d$ from $L_1$ and at a distance $f_2$ from the CCD camera. The phase spot (PS) is placed in the focal plain of the nondiffracted probe beam.

Figure 2

The phase-contrast signal $I/I_0$ as a function of the accumulated phase ${\varphi }_{\mathrm{atoms}}$ for different phases $\theta =\pi ,\pi /2,\pi /3,\pi /4$ of the phase spot.

Figure 3

The density distribution of a thermal cloud $n_{\mathrm{ex}}$ calculated using the noninteracting model (dashed line) and the semi-ideal model (solid line), for $T=0.9\mathrm{\hspace{0.5em}}\mu \mathrm{K}$, $N=1\times {10}^9\hspace{0.5em}\mathrm{atoms}$, for the typical experimental trap parameters. The inset shows the total density distribution $n_{\mathrm{ex}}+n_c$ for both models.

Figure 4

The combined density distribution of the thermal cloud and condensate $n_{\mathrm{ex}}+n_c$ calculated using the semi-ideal model (dashed line) and the Popov model (solid line) for $T=0.9\mathrm{\hspace{0.5em}}\mu \mathrm{K}$, $N=1\times {10}^9\mathrm{\hspace{0.5em}}\mathrm{atoms}$, and the typical experimental trap parameters. The left and right insets show an enlargement of the edges and the center of the condensate, respectively.

Figure 5

A set of four phase-contrast images during the final stage of the evaporation process: (a) $\mu /h=2.6\mathrm{\hspace{0.5em}}\mathrm{kHz}$, $T=1.01\mathrm{\hspace{0.5em}}\mu \mathrm{K}$, $N=7\times {10}^8\mathrm{\hspace{0.5em}}\mathrm{atoms}$, $N_c/N<1\mathrm{\%}$; (b) $\mu /h=4.4\mathrm{\hspace{0.5em}}\mathrm{kHz}$, $T=0.95\mathrm{\hspace{0.5em}}\mu \mathrm{K}$, $N=6\times {10}^8\mathrm{\hspace{0.5em}}\mathrm{atoms}$, $N_c/N\approx 15\mathrm{\%}$; (c) $\mu /h=5.2\mathrm{\hspace{0.5em}}\mathrm{kHz}$, $T=0.58\mathrm{\hspace{0.5em}}\mu \mathrm{K}$, $N=4\times {10}^8\mathrm{\hspace{0.5em}}\mathrm{atoms}$, $N_c/N\approx 50\mathrm{\%}$; (d) $\mu /h=4.8\mathrm{\hspace{0.5em}}\mathrm{kHz}$, $T=0.25\mathrm{\hspace{0.5em}}\mu \mathrm{K}$, $N=2\times {10}^8\mathrm{\hspace{0.5em}}\mathrm{atoms}$, $N_c/N>95\mathrm{\%}$.

Figure 6

The ratio ${\mu }_{\mathrm{ax}}/{\mu }_{\varphi }$ of the chemical potential based on the axial size $R_{\mathrm{ax}}$ and phase $\varphi$ as a function of the temperature. (a) $\eta =0.3$; (b) $\eta =0.25$. In the second series $\eta$ is reduced by reducing the number of atoms.

Figure 7

The number of thermal atoms $N_{\mathrm{ex}}$, number of condensed atoms $N_c$, and condensate fraction as a function of the temperature $T$. Each point is the result of a fit to the Popov model for a single measurement. The statistical uncertainty is smaller than the size of the symbol.

Figure 8

The chemical potential of a cold cloud derived from the fit to the Popov model as a function of temperature, both above and below $T_c$. The line is the result of a fit to Eq. (35) on the seven data points where $\mu /h<2.5\mathrm{\hspace{0.5em}}\mathrm{kHz}$. The data point enclosed in the box marks the observation of a small BEC. The data point marked by the cross is the point with the lowest temperature, for which no bimodal distribution is observed. The dashed line indicates $\mu =2U_0{n}_{\mathrm{ex}}$, the expected value at which condensation will set in, calculated using the average thermal density of the two accentuated data points.