Quantum dynamics of bosons in a two-ring ladder: Dynamical algebra, vortexlike excitations, and currents

We study the quantum dynamics of the Bose-Hubbard model on a ladder formed by two rings coupled by the tunneling effect. By implementing the Bogoliubov approximation scheme, we prove that, despite the presence of the inter-ring coupling term, the Hamiltonian decouples in many independent sub-Hamiltonians ${\widehat{H}}_k$ associated with momentum-mode pairs $\pm k$. Each sub-Hamiltonian ${\widehat{H}}_k$ is then shown to be part of a specific dynamical algebra. The properties of the latter allow us to perform the diagonalization process, to find the energy spectrum and the conserved quantities of the model, and to derive the time evolution of important physical observables. We then apply this solution scheme to the simplest possible closed ladder, the double trimer. After observing that the excitations of the system are weakly populated vortices, we explore the corresponding dynamics by varying the initial conditions and the model parameters. Finally, we show that the inter-ring tunneling determines a spectral collapse when approaching the border of the dynamical-stability region.

Figure 1

Schematic representation of the vortexlike weak excitations in our system. In each ring there can be both a clockwise current and an anticlockwise current. $J_{\perp }$ denotes the current of excited bosons between the two rings. The macroscopically occupied modes $n_0$ and $m_0$ (in gray), being semiclassic, can be considered a sort of reservoir.

Figure 2

Five different initial conditions. (a) No initial excitations, (b) a weak vortex in one ring, (c) a pair of counter-rotating weak vortices in the same ring, (d) two equal weak vortices in the two rings, and (e) a pair of counter-rotating weak vortices in the two rings.

Figure 3

Population and current dynamics for $T_{\parallel }=2,U=0.01,T=1,\hslash =1,N=1000,n_1\left(0\right)=10$, and $n_{-1}\left(0\right)=m_{\pm 1}\left(0\right)=0$. Top: $n_1\left(t\right)$ corresponds to the red dashed line, while $n_{-1}\left(t\right)$ corresponds to the blue solid line. $m_1\left(t\right)$ and $m_{-1}\left(t\right)$ feature the same behavior but are shifted by a semiperiod. Bottom: $J_{\mathrm{chir}}\left(t\right)$ is depicted by the black dashed line, and $J_{\perp }\left(t\right)$ is shown by the green solid line.

Figure 4

Population and current dynamics for $T_{\parallel }=2,U=0.01,T=1,\hslash =1,N=1000,n_1\left(0\right)=n_{-1}\left(0\right)=10$, and $m_{\pm 1}\left(0\right)=0$. Since the tunneling process always involves pairs of bosons, there cannot be momentum transfer between the two rings; hence, chiral current is identically zero. $m_1\left(t\right)$ and $m_{-1}\left(t\right)$ feature the same behavior of $n_{\pm 1}\left(t\right)$ but are shifted by a semiperiod.

Figure 5

Populations dynamics for $T_{\parallel }=2,U=0.01,T=1,\hslash =1,N=1000,n_1\left(0\right)=m_1\left(0\right)=10,n_{-1}\left(0\right)=m_{-1}\left(0\right)=0$, and ${\varphi }_1={\psi }_1=0$. $n_1\left(t\right)$ corresponds to the red dashed line, while $n_{-1}\left(t\right)$ corresponds to the blue solid line. As the tunneling is suppressed, $m_1\left(t\right)$ and $m_{-1}\left(t\right)$ feature the same behavior (fluctuations) as $n_1\left(t\right)$ and $n_{-1}\left(t\right)$, respectively. The chiral and the rung currents are identically zero.

Figure 6

Population and current dynamics for $T_{\parallel }=2,U=0.01,T=1,\hslash =1,N=1000,n_1\left(0\right)=m_1\left(0\right)=10,n_{-1}\left(0\right)=m_{-1}\left(0\right)=0$, and ${\psi }_1=0$ but ${\varphi }_1=\frac{\pi }{2}$. Top: $n_1\left(t\right)$ corresponds to the red dashed line, while $n_{-1}\left(t\right)$ is depicted by the blue solid line. $m_{-1}\left(t\right)$ features exactly the same behavior as $n_{-1}\left(t\right)$, while $m_1\left(t\right)$ is shifted by a semiperiod with respect to $n_1\left(t\right)$. Apart from quantum fluctuations, one can notice that the weakly populated vortex with $k=+1$ periodically completely transfers from one ring to the other and vice versa. Bottom: Chiral current (in black) and rung current (in green).

Figure 7

Population and current dynamics for $T_{\parallel }=2,U=0.01,T=1,\hslash =1,N=1000,n_1\left(0\right)=m_{-1}\left(0\right)=10,$ and $n_{-1}\left(0\right)=m_1\left(0\right)=0$. $n_1$ and $m_{-1}$ (dashed red line) have the same time evolution, and so do $m_1$ and $n_{-1}$ (blue solid line). The rung current is identically zero.

Figure 8

Energy levels as a function of the intertunneling parameter $T$. When $T=0$, the rings are decoupled; when $T\to \frac{3}{2}{T}_{\parallel }$, there is spectral collapse with respect to the first generalized harmonic oscillator.

Figure 9

Populations and currents dynamics for $T_{\parallel }=2,U=0.01,T=2.999,\hslash =1,N=1000,n_1\left(0\right)=10$, and $n_{-1}\left(0\right)=m_{\pm 1}\left(0\right)=0$. Top: $n_1\left(t\right)$ corresponds to the red dashed line, and $n_{-1}\left(t\right)$ is depicted by the blue solid line. $m_1\left(t\right)$ and $m_{-1}\left(t\right)$ essentially feature the same diverging behavior. The number of excited bosons rapidly increases and soon becomes nonphysical. Bottom: chiral (black dashed line) and rung (green solid line) currents exhibit a divergence too.