Unveiling emergent internal time from entropy exchange in a cold-atom system

Abstract

We realize a cold-atom system to quantitatively test relational constructions of time. A well-isolated atomic Bose-Einstein condensate evolves in a conservative trap that is partitioned by a thin optical barrier into an observed and unobserved sector, with negligible dissipation on the experimental timescale. Motivated by relational-time approaches discussed in the Wheeler-DeWitt framework, we ask whether the dynamics of the observed sector can be ordered using only internal degrees of freedom. To this end, we construct an entropic time from an experimentally defined coarse-grained entropy, and demonstrate that it can robustly order the events in the observed sector across repeated cycles of expansion and recollapse. We finally derive an effective Schrödinger equation parameterized by this internal time and show that it is able to reproduce the measured evolution. These results establish a controlled experimental setting in which relational-time constructions can be quantitatively tested.

Figures (4)

• Figure 1: a) Experimental absorption images integrated along the $y$ axis showing the evolution of our system as a function of the external lab time. The colour scale indicates the integrated column density in arbitrary units. Our Bose-Einstein condensate evolves in a conservative harmonic trap that has a thin potential barrier at the bottom, as shown in panel b). The barrier separates the 'dark' (unobserved) from the 'bright' (observed) sector. In the experiment these correspond to the bottom (shaded) and top half of panel a). Depending on the height of the barrier, the condensate is able to cross over from one sector to the other. The moment the atoms start to populate the bright sector corresponds to the 'big bang' (blue stars), while the moment it comes back to the dark sector corresponds to the 'big crunch' (green stars).
• Figure 2: The solid symbols are the entropic internal time for the bright sector evaluated from experimental data using Eq. \ref{['eqtau']}, as a function of the external lab time. The experiment has been repeated for different values of the height of the potential barrier that separates the dark from the bright sector. The relative 1 $\sigma$ statistical uncertainty associated with each point (not shown) is approximately 5%.
• Figure 3: a) Experimentally measured values of the analogue massive scalar field $\phi$ of the bright sector as a function of the entropic time for different values of the height of the barrier potential. As described in the text, in the experiment $\phi$ corresponds to the $X$ component of the center of mass of the condensate. The horizontal line approximately corresponds to the edge of the potential barrier (centered at $\phi$=0). b) Experimentally measured value of the width of the condensate $\Sigma$ in the bright sector (corresponding to the size of the analogue universe) as a function of the entropic time, and for different values of $V$. The dotted curve is the result of the numerical simulations using Eq. (\ref{['SE']}) with $V\simeq0$. In both panels, the relative 1 $\sigma$ statistical uncertainty associated with each point (not shown) is approximately 5%.
• Figure 4: Density probability distribution of the bright sector $N(\tau)|\psi(\tau,a)|^2$ as a function of the entropic time $\tau$ and the scale parameter $a$, obtained by numerically solving Eq. \ref{['SE']} using the experimental parameters of the data set with $V\simeq$ 0 shown in Fig. \ref{['Fig2']} and \ref{['Fig3']}.