Superfluidity in Bose-Hubbard circuits

A semiclassical theory is provided for the metastability regime diagram of atomtronic superfluid circuits. Such circuits typically exhibit high-dimensional chaos, and nonlinear resonances that couple the Bogoliubov excitations manifest. Contrary to the expectation, these resonances do not originate from the familiar Beliaev and Landau damping terms. Rather, they are described by a variant of the Cherry Hamiltonian of celestial mechanics. Consequently, we study the induced decay process and its dependence on the number of sites and condensed particles.

Figure 1

The superfluidity regime diagram. The panels are for rings with $M=4,5,10$ sites. Yellow and gray indicate the Landau-stability and linear-instability regions. Nonlinear resonances of the $V^A$ and $V^B$ types are indicted by red and gray lines, respectively. The explicit expression for the red resonance in (a) is Eq. (14). The additional thin red lines in(b) and (c) are fourth-order resonances. The green squares mark regions of interest that will be explored in Fig. 2. For the sake of generality we consider both positive and negative values of $u$.

Figure 2

The survival of a prepared coherent flow state. Quantum simulations are carried out in the region of interest, which has been indicted by a green square in Fig. 1. In each run all the particles are condensed initially in the same momentum orbital. The “survival” (see text) is plotted as a function of $\mathrm{\Phi }$ for different $u$ values. The constant $u$ slices are indicated in the insets by the color-coded horizontal lines. The systematic shift of the dips and their broadening reflect the $u$ dependence of the nonlinear resonances. (a) and (b) are for four-site and five-site rings, respectively.

Figure 3

The $N$ dependence of the decay process. Initially, all the particles are condensed in the same momentum orbital. (a) and (b) display the normalized occupation of the condensate as a function of time. The red lines are quantum simulations for $N=120$ particles, while the other lines (blue to gray) are based on semiclassical simulations for $N=120,500,1000,2000,4000$ particles. The interaction strength is $u=2.5$. (a) relates to $\mathrm{\Phi }\approx 0.27\pi$, for which the detuning is $\nu =0$, while (b) relates to $\mathrm{\Phi }=0.25\pi$. In the latter case we observe that the survival becomes $N$ dependent. This reflects the existence of a stability island, as illustrated in (c), which is a phase-space portrait of Eq. (15): the radial coordinate is $I\in [0,2.5]$, the polar angle is $\phi$, and the parameters are $\nu =2, \mu =1$, and $J=1$.

Figure 4

The escape dynamics for zero detuning. Using classical simulation, we plot the time dependence of the momentum-orbital occupation $n_k$ (red, orange, green, and yellow lines) as well as the quasiparticle occupations ${\tilde{n}}_q$ (blue, magenta, and cyan lines) for a single trajectory that starts at $n_0\sim 1$, while $n_k\ll 1$ for $k\ne 0$. Note that the initial semiclassical cloud is composed of many such starting points that occupy a Planck cell of radius $\sim 1/N$ in phase space. The dynamical scenario may consist of several steps, depending on the initial conditions. Typically, it will start with either the (a) exponential stage or (b) the parabolic stage. Then it will cross to the hyperbolic escape of Eq. (17) that terminates at $t_e$, possibly followed by (a) and (b) chaotic motion, or else (c) and (d) there might be either an intermediate or recurring reinjection scenario. The simulations are for an $M=4$ ring. In (a) and (b) the interaction is $u=3.84$, while in (c) and (d) it is $u=2.52,1.58$. The respective values $\mathrm{\Phi }=0.2\pi ,0.27\pi ,0.35\pi$ have been adjusted via the resonance condition (14).