Phase diagram of Landau-Zener phenomena in coupled one-dimensional Bose quantum fluids

We study stationary and dynamical properties of the many-body Landau-Zener dynamics of a Bose quantum fluid confined in two coupled one-dimensional chains by using a many-body generalization recently reported Chen et al. [Nature Phys. 7, 61 (2011)], within the decoupling approximation and the one-level band scheme. The energy spectrum shows evidence of the structure of the avoided level crossings as a function of the onsite interparticle interaction strength. On the dynamical side, a phase diagram of the transfer efficiency across ground-state and inverse sweeps is presented. A totally different scenario with respect to the original single-particle Landau-Zener scheme is found for ground-state sweeps, in which a breakdown of the adiabatic region emerges as the sweep rate decreases. On the contrary, the transfer efficiency across inverse sweeps reveals consistent results with the single-particle Landau-Zener predictions. In the strong-coupling regime, we find that there is a critical value of the onsite interaction for which the transfer of particles starts to vanish independently of the sweep rate. Our results are in qualitative agreement with those of the experimental counterpart.

Figure 1

Schematic representation of the double-well trap chain potential. The coupling energies are intrachain $J^{\parallel }$ ($z$ direction) and interchain $J^{\perp }$ ($x$ direction). The time-dependent parameter $\Delta$ characterizes the tilt between the two wells of each site in the chain.

Figure 2

Schematic representation of the lattice sites in the chain of double wells.

Figure 3

Density plot of the optimal values of the order parameters ${\psi }_L$ and ${\psi }_R$ as functions of the dimensionless parameters $\tilde{U}$ and $\tilde{\Delta }$ for the ground state of Hamiltonian (3) for $N=6$.

Figure 4

Energy spectrum ${\tilde{E}}_{\tilde{U}}^n=E_n/\left(J\tilde{U}\right)$ of Hamiltonian (3), for $N=3$ and $\tilde{U}=1$ [(a) and (d)], 10 [(b) and (e)], 100 [(c) and (f)]. The values of $\tilde{\Delta }$ on the $x$ axis have been rescaled with respect to the dimensionless parameter $\tilde{U}$; that is, ${\tilde{\Delta }}_{\tilde{U}}=\tilde{\Delta }/\tilde{U}$. For $\tilde{U}=100$ the numbers on the right refer to the number of particles in the right well, while the circles indicate the tunneling resonances.

Figure 5

Instantaneous right evolution $n_R\left(t\right)$ from ground state for various rates $2\pi /\alpha =0.1$ (a), 1.0 (b), 2.0 (c), $\tilde{U}=0.75$, and $N=6$. The time is rescaled with respect to $T_{\text{max}}$, $\tilde{t}=t/T_{\text{max}}$, with ${\tilde{\Delta }}_0=20$. (c) Density plot of the normalized transfer efficiency $n_R$ at $t=T_{\text{max}}$. (d) Contour plots of the normalized transfer efficiency for several values of the the dimensionless parameter $\tilde{U}$ (see main text).

Figure 6

Density plot of the normalized transfer efficiency $n_R$ of ground-state sweeps as a function of dimensionless parameters $\tilde{U}$ and $\alpha$ at $t=T_{\text{max}}={\tilde{\Delta }}_0/\alpha$ for $N=2$ [(a) and (b)] and $N=8$ [(c) and (d)]. $-{\tilde{\Delta }}_0$ is the initial value of the tilt. Parameters are ${\tilde{\Delta }}_0=10$ [(a) and (c)] and ${\tilde{\Delta }}_0=100$ [(b) and (d)].

Figure 7

Density plot of normalized transfer efficiency $n_R$ of inverse sweeps as a function of dimensionless parameters $\tilde{U}$ and $\alpha$ for $2\pi /\alpha \in [0.1,10.0]$ and $\tilde{U}N/{\tilde{\Delta }}_0\in [0,3]$. Parameters are ${\tilde{\Delta }}_0=10$ (a), ${\tilde{\Delta }}_0=100$ (b), $N=2$.

Figure 8

Normalized transfer efficiency $n_R$ for ground-state (a) [inverse (b)] sweeps as a function of the final tilt ${\tilde{\Delta }}_f$ for $\tilde{U}=1.5$ and $2\pi /\alpha \in [0.1,3.0]$. Normalized transfer efficiency $n_R$ as a function of the final tilt ${\tilde{\Delta }}_f$, for $2\pi /\alpha =2.0$, [$2\pi /\alpha =1.0$] in the ground-state (c) [inverse (d)] case and $\tilde{U}\in [0,10]$. The number of particles is $N=2$. Dimensionless parameters are $n_R$, ${\tilde{\Delta }}_f$, $\tilde{U}$, and $\alpha$.

Figure 9

Energy spectrum of single particle in double-well potential. The asymmetry is determined by the value of the parameter $c$ in (A1). In these plots $c=0$, 0.04, and 0.08.