Resonant persistent currents for ultracold bosons on a lattice ring

We consider a one-dimensional bosonic gas on a ring lattice, in the presence of a localized barrier, and under the effect of an artificial gauge field. By means of exact diagonalization we study the persistent currents at varying interactions and barrier strength, for various values of lattice filling. While generically the persistent currents are strongly suppressed in the Mott insulator phase, they show a resonant behavior when the barrier strength becomes of the order of the interaction energy. We explain this phenomenon using an effective single-particle model. We show that this effect is robust at finite temperature, up the temperature scale where persistent currents vanish.

Figure 1

The persistent-current amplitude $\alpha$ as a function of $(w,u)$. Panels (a)–(d) are for rings with $(M,N)$ as follows: $(10,4)$; $(5,10)$; $(3,12)$; $(6,6)$. Panels (e) and (f) are zoomed versions of panels (b) and (c), respectively. The value plotted is calculated using Eq. (8) where the maximal current $I\left(\mathrm{\Phi }\right)$ is picked from all $\mathrm{\Phi }\in [0,\pi ]$ with a step size of $0.05\pi$. The dashed red lines are given by Eq. (14). The markers in panel (d) correspond to the spectra in Fig. 2.

Figure 2

Low-energy spectrum for an $M=6$ ring with $N=6$ particles and $\mathrm{\Phi }=0.9\pi$. Each point represents an eigenstate, positioned according to its energy $E$ (vertical axis) and its current $\mathcal{I}$ (horizontal axis). The current is in units of $2NJ/M$; note the different axis limits in panel (b). In all panels $u=3000$, while $w=300$, 500, and 700 from left to right. This corresponds to the large-$u$ regime, with $w$ values that are on the left, at the center, and to the right of the resonance. The $w$ values are marked in Fig. 1. Note that we only show here states with $E<200$ and not the entire spectrum.

Figure 3

The ground-state current (blue spiky lines), in units of $2NJ/M$, and the barrier occupation (red staircase lines), as a function of dimensionless barrier strength $w$ (upper panels) and dimensionless interaction strength $u$ (lower panels). The vertical red lines indicate the expected location of the resonances, using Eq. (14) (neglecting the $2J_q-\mathrm{\Delta }$ shift). In the rightmost panels we also plot the current using the effective single-particle Hamiltonian, Eq. (15) (green dashed), and the expected location of the resonances, including the $2J_q-\mathrm{\Delta }$ correction (vertical cyan lines). In all panels the flux is $\mathrm{\Phi }=0.9\pi$.

Figure 4

Finite-temperature persistent currents (in units of $2J)$ for a ring of $M=6$ and $N=6$ as a function of the dimensionless flux $\mathrm{\Phi }$. The dimensionless interaction and barrier strength are $u=3000$ and $w=495$. This corresponds to the center of the resonance that has been displayed in the upper right panel of Fig. 3. The black line is the ground-state current, while the colored lines correspond to different temperature values of $k_{B}T/2J=0.04$ (dashed blue line), $k_{B}T/2J=0.1$ (dotted red line), $k_{B}T/2J=0.2$ (dashed-dot yellow line), and $k_{B}T/2J=0.5$ (thin solid green line).

Figure 5

Thermal-averaged persistent-current amplitude as a function of $u$ and $k_{b}T$. Panels (a)–(d) are for rings with $(M,N)$ as follows: $(10,4)$; $(5,10)$; $(3,12)$; $(6,6)$. In panel (a) the barrier strength is $w=10$ and in panels (b)–(d) $w=100$. The value plotted is the maximal canonical average current, in units of $2NJ/M$, picked from all $\mathrm{\Phi }\in [0,\pi ]$ with a step size of $0.05\pi$. The red dotted lines are given by Eq. (14).

Figure 6

Weak-link setup for a ring of $M=6$ and $N=6$. Note that we treat the strong link regime $\left(J_w/J>1\right)$ on equal footing. (a) The persistent-current amplitude $\alpha$ as a function of $u$ and $J_w/J$ (note the reversed $x$ axis). The maximal current $I\left(\mathrm{\Phi }\right)$ is picked from all $\mathrm{\Phi }\in [0,\pi ]$ with a step size of $0.05\pi$. (b) The ground-state current (in units of $2J)$ and the combined occupation of the two strongly connected sites, as a function of $u$, with fixed $J_w/J=80$ and $\mathrm{\Phi }=0.9\pi$.