Macroscopic quantum self-trapping in bosonic Josephson junctions: an exact quantum treatment

Abstract

We investigate the fully quantum evolution of the population imbalance in a perfectly symmetric Bose-Josephson junction modeled by a two-mode Bose-Hubbard Hamiltonian, focusing on the validity of macroscopic quantum self-trapping beyond the mean-field theory. We show that for any finite number of particles the exact quantum dynamics leads to the breakdown of macroscopic quantum self-trapping after a finite time, regardless of the initial state. Using the symmetries of the Bose-Hubbard Hamiltonian, we provide a mathematical demonstration of this result and analyze the spectral properties governing the dynamics. We identify a branching behavior in the eigenvalues differences and a nontrivial structure of the population-imbalance amplitudes. These features allow us to distinguish two clearly different dynamical regimes and to elucidate the mechanism leading to the emergence of a quasi-MQST regime for large particle numbers. These findings bridge the gap between mean-field predictions and exact quantum dynamics and provide insight into the emergence of classical nonlinear behavior from finite quantum many-body systems.

Figures (8)

• Figure 1: Eigenvalues difference $E_{0,N}\equiv\lambda_{N/2}-\ell_{N/2}$ as a function of particle number $N$ for different values of the dimensionless interaction strength $\Lambda=UN/J\in[0,2]$ as indicated on the color bar, in semi-logarithmic scale, showing exponential decrease at large $\Lambda$. The inset shows the same data in a log-log scale to highlight the power-law decay at small $\Lambda$ values.
• Figure 2: Color map of the eigenvalue difference $E_{0,\sigma}$ defined in the text in units of the mean-field Josephson frequency $\Omega_J =J\sqrt{1+\Lambda}$, as a function of the interaction strength $\Lambda$ and the normalized sum index $\sigma/N$ for $N = 500$ particles. The solid gray curve corresponds to the branching transition line, and is given in Eq (\ref{['sigmacrit']}).
• Figure 3: Color map of the product of the coefficients $A_{0,\sigma}C_{0,\sigma}$ normalized by the initial population imbalance $z_0$, as a function of the dimensionless interaction strength $\Lambda$ and the normalized sum index $\sigma/N$ for a system with $N=200$ particles. The white dotted line indicates the value $\sigma^*/N$ corresponding to its maximum for each value of $\Lambda$. The lime dotted line shows the critical sum index \ref{['sigmacrit']}. The three solid lines correspond to the sum of $A_{0,\sigma}C_{0,\sigma}/z_0$ over all $\sigma$ (blue), over $\sigma > \sigma_c$ (red), and over $\sigma < \sigma_c$ (green). The vertical dashed lines indicate critical interaction strengths for MQST in mean field ($\Lambda_{c,\mathrm{MF}}$, black) and for quasi-MQST ($\bar{\Lambda}$, magenta; $\Lambda_1$, cyan; $\Lambda_2$ yellow). The initial state is prepared with $\Lambda_0=10$ and $\tilde{z}_0=0.6$.
• Figure 4: Fully quantum (left panel) and mean-field (right panel) time evolution of the time-averaged population imbalance $\langle z(t)\rangle_T / z_0$ (dimensionless) as a function of time, in unites of $J^{-1}$, for a system of $N=200$ particles and initial imbalance $z_0 = 0.57$, at various values of the interaction strength $\Lambda$ as indicated in the color-bar label. The running averaging time window is set to $JT = 25$. The initial state of the quantum problem is the ground state of the Hamiltonian with parameters $\Lambda_0 = 20$ and $\tilde{z}_0 = 0.6$, and for the mean-field we have taken $z(0)=z_0$ and $\phi(0)=0$.
• Figure S1: Eigenvalues as a function of the dimensionless interaction strength $\Lambda=UN/J$ for $N=200$ particles. The eigenvectors, ordered in ascending order, are presented by bold lines ($\lambda_i$) and dashed lines ($\ell_j)$.