Collapse in ultracold Bose Josephson junctions

We investigate how ultracold atoms in double-well potentials can be used to study and put bounds on models describing wave-function collapse. We refer in particular to the continuous spontaneous localization (CSL) model, which is the most well studied among dynamical reduction models. It modifies the Schrödinger equation in order to include the collapse of the wave function in its dynamics. We consider Bose Josephson junctions, where ultracold bosons are trapped in a double-well potential, since they can be experimentally controlled with high accuracy and are suited and used to study macroscopic quantum phenomena on a scale of microns, with a number of particles typically ranging from $\sim {10}^2–{10}^3$ to $\sim {10}^5–{10}^6$. We study the CSL dynamics of three atomic states showing macroscopic quantum coherence: the atomic coherent state, the superposition of two atomic coherent states, and the NOON state. We show that for the last two states, the suppression of quantum coherence induced by the CSL model increases exponentially with the number of atoms. We observe that in the case of optically trapped atoms, the spontaneous photon emission of the atoms induces a dynamics similar to the CSL one, and we conclude that magnetically trapped atoms may be more convenient to experimentally test the CSL model. Finally, we discuss decoherence effects in order to provide reasonable estimates on the bounds that it is (or will be) possible to obtain for the parameters of the CSL model in such class of experiments. As an example, we show that a NOON state with $N\sim {10}^3$ with a coherence time of $\sim 1$ s can constrain the CSL parameters in a region where the other systems presently cannot.

Figure 1

Action of the collapse noise on an atom in a double-well potential (blue curve). The initial state (yellow curve) is delocalized over the two wells. The interaction of the atom with the collapse noise localizes the atomic state in one of the two wells (red curve).

Figure 2

Recent exclusion plot of the CSL model [64]. The upper colored regions represent the values of $\lambda$ and $r_C$ that have been excluded by experimental data. The lower gray region is a lower bound coming from the requirement that CSL must be effective in localizing macroscopic objects. The white region—that of interest—is still unexplored. The picture shows the bounds of a hypothetical experiment involving the macroscopically entangled states (11) or (13). Here, we made the hypothesis that their $N$-particle coherences are preserved for a time $t=1$ s, with three different number of particles: $N={10}^4$ (black dashed line), ${10}^6$ (blue dot-dashed line), and ${10}^8$ (red dotted line). The excluded regions are the upper part of the related lines.

Figure 3

Time evolution (25) of the normalized $N$-particle coherences ${\mathrm{\Gamma }}_t={\langle {\widehat{a}}_L^{†}{\widehat{a}}_R\rangle }_t^{\text{CSL}}/{\langle {\widehat{a}}_L^{†}{\widehat{a}}_R\rangle }_0^{\text{CSL}}$, for different number of atoms. Here we fix the CSL parameters $\lambda ={10}^{-11}{\text{s}}^{-1}$ and $r_C={10}^{-7}$ m, which are among the weakest parameters of the white region in Fig. 2.