Figure 1

Observation of thermal phase fluctuations. The experimental steps are depicted in (a). A Bose-Einstein condensate (solid line) is prepared in a double-well potential (dashed line) by adiabatically ramping up the barrier. The relative phase can be measured after a time-of-flight expansion by analyzing the resulting double-slit interference patterns (black line). (b) Polar plots of the relative phase obtained by repeating the experiment up to 60 times. The graphs show measurements for four different temperatures $T$ at constant tunneling coupling energy $E_j$, i.e., constant barrier height. The phase fluctuations increase with increasing temperature. (c) Polar plots of the relative phase for a constant temperature at four different tunneling coupling energies, i.e., different barrier heights. Here the fluctuations are reduced with increasing coherent tunneling coupling showing the stabilization.Reuse & Permissions

Figure 2

Loss of the coherence of the bosonic Josephson junction due to the coupling to a thermal environment. (a) The transition from coherent single realizations to incoherent ensemble averages. In the single realization a clear interference signal is observed, where the visibility is decreased due to the finite optical resolution of the imaging setup. After averaging over 10 realizations the visibility is reduced and after averaging over 50 realizations the coherence is lost ($\alpha =0.046$). In (b) the coherent evolution of the bosonic Josephson junction is depicted. The results shown are obtained by repeating the experiment at the same temperature as above but at a stronger tunneling coupling. Here the averaging leads to only a small degradation of the visibility ($\alpha =0.87$) showing how coherent coupling can counteract dephasing processes.Reuse & Permissions

Figure 3

Scaling behavior of the coherence factor of a bosonic Josephson junction. Each point is obtained by averaging over the cosine of the phases of at least 28 (in average 40) measurements at the same experimental conditions. The coherence factor $\alpha$ is plotted as a function of the scaling parameter $k_{B}T/E_j$, which is varied over 3 orders of magnitude ($49\mathrm{nK}<T<80\mathrm{nK}$, $0.6\mathrm{nK}<E_j/k_B<300\mathrm{nK}$). It shows good agreement with the theoretical prediction of the classical model Eq. (2) indicated by the solid line where also the uncertainty arising from the fitting error of the phase is taken into account. Typical error bars are shown which result from statistical errors and uncertainties of the experimental parameters (potential parameters, atom numbers, temperature).Reuse & Permissions

Figure 4

Heating up of a Bose gas. The filled circles correspond to measurements employing the phase fluctuation method and the open circles to the results obtained with the standard time-of-flight method. The gray shaded region shows the critical temperature expected from the experimental parameters and their uncertainties. The solid line is a fitting function assuming a power law for the heat capacity $C\propto (T/T_c{)}^d$ of the Bose gas below the critical temperature $T_c=59(4)\mathrm{nK}$, a constant heat capacity above the critical temperature, and a temperature independent transfer rate of energy. From this fit we deduce $d=2.7(6)$, which is consistent with the theoretical prediction of $d=3$ for an ideal Bose gas in a three-dimensional harmonic trap. The dashed line represents the expected behavior of an ideal classical gas for increasing temperature which makes the difference arising from quantum statistics evident.Reuse & Permissions