Critical exponents for an impurity in a bosonic Josephson junction: Position measurement as a phase transition

We use fidelity susceptibility to calculate quantum critical scaling exponents for a system consisting of $N$ identical bosons interacting with a single impurity atom in a double-well potential (bosonic Josephson junction). Above a critical value of the boson-impurity interaction energy there is a spontaneous breaking of ${\mathbb{Z}}_2$ symmetry corresponding to a second-order quantum phase transition from a balanced to an imbalanced number of particles in either the left- or the right-hand well. We show that the exponents match those in the Lipkin-Meshkov-Glick and Dicke models, suggesting that the impurity model is in the same universality class. The phase transition can be interpreted as a measurement of the position of the impurity by the bosons.

Figure 1

Schematic of the proposed setup. A bosonic Josephson junction consists of $N$ identical bosons [represented by the small filled (blue) circles] which are able to tunnel between the two sides of a double-well potential. To this is added a single impurity atom [large filled (red) circle] which is also able to tunnel between the two wells.

Figure 2

Fidelity susceptibility as a function of $W$ for different system sizes: $N=200$ [dot-dashed (red) curve], $N=400$ [dotted (green) curve], $N=600$ [dashed orange) curve], and $N=800$ (solid black curve). Here $J^a=0.75J$, so $W_c=\sqrt{3}J$, which is shown by the vertical black line. It is clear that ${\chi }_F$ is not symmetric about the transition, and hence the need for two indices $\pm \alpha$ as indicated in Eq. (10).

Figure 3

A log-log plot of ${\chi }_{\mathrm{Fmax}}$ as a function of $N$ for different values of $J^a$: $0.75J$ [(red) squares], $1J$ [(blue) circles], and $1.25J$ (black triangles). Inset: Slopes of the log-log plot as a function of $1/N$ extrapolated in the $1/N\to 0$ limit. The range of system sizes is $1000\le N\le 3000$.

Figure 4

A plot of Eq. (15) for different system sizes. The parameters and values of $N$ are the same as those used in Fig. 2. A value of $\nu \simeq 3/2$ results in the optimal overlay of the curves. Inset: Magnification of the region around the origin.

Figure 5

Extrapolated values of $W_{\max }$ for different values of $J^a$: $0.75J$ [(red) squares], $1J$ [(blue) circles), and $1.25J$ (black triangles). Dashed lines show quadratic fits for $1/N\to 0$ and the system size range is $500\le N\le 2500$.