Two-Point Phase Correlations of a One-Dimensional Bosonic Josephson Junction

We realize a one-dimensional Josephson junction using quantum degenerate Bose gases in a tunable double well potential on an atom chip. Matter wave interferometry gives direct access to the relative phase field, which reflects the interplay of thermally driven fluctuations and phase locking due to tunneling. The thermal equilibrium state is characterized by probing the full statistical distribution function of the two-point phase correlation. Comparison to a stochastic model allows us to measure the coupling strength and temperature and hence a full characterization of the system.

Figure 1

Two coupled Bose gases are released from a double well potential. The matter wave interference pattern emerging during expansion gives access to the relative phase $\phi (z)$ along the samples. Fluctuations in the absolute phase of the two Bose gases transform into visible density fluctuations [29, 30]. We characterize two-point phase correlations of the system by measuring the statistical properties of the difference of relative phases $\Delta \phi (\overline{z})=\phi (z)-\phi (z^{\prime })$.

Figure 2

Distributions of the difference of the relative phase $\Delta \phi (\overline{z})$ for decreasing tunnel coupling (from left to right, compare Table ). (a),(b) High coupling yields a narrow distribution and high coherence over the entire length of the system. (c)–(e) Decreasing coupling leads to an increasingly fast loss of spatial phase correlations and thus a randomization of relative phases. A slight bias of the distribution can be assigned to a small relative velocity ($20\mu \mathrm{m}/\mathrm{s}$) of the two samples during the expansion.

Figure 3

Real part of the phase correlation function $C(\overline{z})$ for different couplings and $T\simeq 155\mathrm{nK}$ (compare Table ). The symbols represent experimental values, derived from the data shown in Fig. 2. A stochastic OU model allows us to estimate the couplings $J_1$ to $J_5$ (compare Table ). The colored areas depict 2 standard deviations on the values of $C(\overline{z})$ obtained by repeating the analysis 30 times. A simulation of $C(\overline{z})$ for completely uncoupled Bose gases is displayed as a black dashed line and indicates the limits of our imaging resolution.

Figure 4

Real part of the phase correlation function $C(\overline{z})$ for fixed coupling $J_3$ and different temperatures $T_1$ to $T_3$. Comparison with the stochastic simulation allows thermometry of the 1D bosonic Josephson junction (compare Table ).

Figure 5

Distributions of $\Delta \phi (\overline{z})$ (a)–(c) and $\cos [\Delta \phi (\overline{z})]$ (d)–(f) for $\overline{z}=32\mu \mathrm{m}$ and three different couplings ($J_1$, $J_3$, $J_5$). Red histograms represent measured, blue lines simulated distributions for 500 realizations. (a)–(c) The mean value of $\Delta \phi (\overline{z})$ (dashed gray line) does not change while the phase spreads. (d)–(f) The mean value of $\cos \Delta \phi (\overline{z})$ (dashed gray line) decays from about 1 (strongly coupled) to 0 (uncoupled).