Coherent phase slips in coupled matter-wave circuits

Quantum phase slips are a dual process of particle tunneling in coherent networks. Besides being of central interest to condensed matter physics, quantum phase slips are resources that are sought to be manipulated in quantum circuits. Here, we devise a specific matter-wave circuit enlightening quantum phase slips. Specifically, we investigate the quantum many-body dynamics of two side-by-side ring-shaped neutral bosonic systems coupled through a weak link. By imparting a suitable magnetic flux, persistent currents flow in each ring with given winding numbers. We demonstrate that coherent phase slips occur as winding number transfer among the two rings, with the populations in each ring remaining nearly constant. Such a phenomenon occurs as a result of a specific entanglement of circulating states, that, as such, cannot be captured by a mean-field treatment of the system. Our work can be relevant for the observation of quantum phase slips in cold-atom experiments and their manipulation in matter-wave circuits. To make contact with the field, we show that the phenomenon has clear signatures in the momentum distribution of the system providing the time-of-flight image of the condensate.

Figure 1

Schematic diagram of an optical lattice consisting of two-sided rings of five sites each. We represent the small link between rings, $t_l$, with a small interfering pattern connecting the rings. Both rings have an effective artificial gauge field denoted by ${\mathrm{\Omega }}_L$ and ${\mathrm{\Omega }}_R$.

Figure 2

Particle number's expectation value, standard deviation, and current oscillations after quenching a system of two rings of five sites, $N_s=5$, and $t_l=0.05$. The initial fluxes are ${\mathrm{\Omega }}_L=1, {\mathrm{\Omega }}_R=0$ and the final ones ${\mathrm{\Omega }}_L={\mathrm{\Omega }}_R=0.05$. (a) Noninteracting (single-particle) case, $U=0, N_p=1$. (b) Interactions set to $U=100$ and $N_p=2$ ($N_p>2$ is considered in Fig. 4). See Fig. 3 to compare the periods for different $t_l$.

Figure 3

Energy gap (such that periods $T=\frac{2\pi }{\mathrm{\Delta }E}$) as a function of the interring coupling $t_l$ and ${\mathrm{\Omega }}_L={\mathrm{\Omega }}_R=0.01,0.05,0.1$ for (a) one particle ($\mathrm{\Delta }E=\frac{2t_l}{N_s}$), (b) two particles and $U=100$ ($\mathrm{\Delta }E\propto t_l^2$). Dashed black lines in both plots mark the energy gaps for $t_l=0.05$, corresponding to $T=314$ (a) and $T=18793$ (b), the periods in Fig. 2.

Figure 4

Each plot compares QPSs where (excess) particles $N_e$ and holes $2N_s-N_e$ are exchanged in the strongly interacting regime, $U=1000$, and in rings of $N_s=3$ sites. Particle-hole symmetry is displayed for two different commensurate fillings, such that the total number of particles is $N_p=2N_s\times N_b+N_e$, with $N_b=0$ (left plots) and $N_b=1$ (right plots), and $N_e=\{2,4\}$ (top plots) and $N_e=\{1,5\}$ (bottom plots). We quench from ${\mathrm{\Omega }}_L=1, {\mathrm{\Omega }}_R=0$, to ${\mathrm{\Omega }}_L=0.01, {\mathrm{\Omega }}_R=0.01$, and from $t_l={10}^{-5}$ to $t_l=0.01$. Note that the period substantially changes for each commensurate filling $N_b=0,1$, despite having the same number of particles and holes. These changes appear because the energy levels shift proportionally to $N_b$.

Figure 5

Time-of-flight expansion of the left ring. Parameters are ${\mathrm{\Omega }}_L=1,{\mathrm{\Omega }}_R=0$ to ${\mathrm{\Omega }}_L=0.01,{\mathrm{\Omega }}_R=0.01, t_l=0$ to $t_l=0.05, N_p=2, N_s=5$ in each ring, $U=10$. Triangles indicate the times at which the TOF snapshots where taken.