Quantum signatures of an oscillatory instability in the Bose-Hubbard trimer

We study the Bose-Hubbard model for three sites in a symmetric, triangular configuration and search for quantum signatures of the classical regime of oscillatory instabilities, known to appear through Hamiltonian Hopf bifurcations for the “single-depleted-well” family of stationary states in the discrete nonlinear Schrödinger equation. In the regimes of classical stability, single quantum eigenstates with properties analogous to those of the classical stationary states can be identified already for rather small particle numbers. On the other hand, in the instability regime the interaction with other eigenstates through avoided crossings leads to strong mixing, and no single eigenstate with classical-like properties can be seen. We compare the quantum dynamics resulting from initial conditions taken as perturbed quantum eigenstates and SU(3) coherent states, respectively, in a quantum-semiclassical transitional regime of 10–100 particles. While the perturbed quantum eigenstates do not show a classical-like behavior in the instability regime, a coherent state behaves analogously to a perturbed classical stationary state, and exhibits initially resonant oscillations with oscillation frequencies well described by classical internal-mode oscillations.

Figure 1

Energy spectrum of Eq. (1) vs $\alpha N$, for Bloch states with $k=0$ and $N=10,20,30,60$ particles, respectively. The state with the largest expansion coefficient for the basis function ${|N/2,N/2,0\rangle }_0$ (see text) is plotted with a bold (blue) line.

Figure 2

Left: Magnification of an avoided crossing corresponding to a strong resonance in Fig. 1 for $N=20$ particles. Right: comparison of the eigenstates on both sides of the crossing, at the points (i), (ii), (iii), and (iv) marked with arrows. In these figures, coefficients of the basis states (4) are arranged in the following order: ${|0,N,0\rangle }_0{,|1,N-1,0\rangle }_0{,|2,N-2,0\rangle }_0{,...,|N-1,1,0\rangle }_0{,|N-2,1,1\rangle }_0{,|N-3,2,1\rangle }_0{,\cdots ,|2,N-3,1\rangle }_0,{|N-4,2,2\rangle }_0,....$ (Note that, for instance, ${|1,N-2,1\rangle }_0$ did not follow ${|2,N-3,1\rangle }_0$ since ${|1,N-2,1\rangle }_0$ is a translation of ${|N-2,1,1\rangle }_0$ which occurred earlier in the series.) State number 11 is thus ${|10,10,0\rangle }_0$, and all states with numbers larger than 20 correspond to nonzero population on all three sites.

Figure 3

The maximum probability that, in any eigenstate, all particles are equally distributed between only two sites, plotted as a function of $\alpha N$ for $N$ between 10 and 80 particles. Blue (darker) and green (lighter) curves correspond to $k=0$ and $k=\pm 2\pi /3$, respectively. The shaded region marks the classical instability regime.

Figure 4

The quantity ${\rm Y}$ (5), measuring total overlap in eigenstates between symmetric compact SDW components and components which are not two-site localized, plotted as a function of $\alpha N$ for $N$ between 10 and 80 particles. The shaded region marks the classical instability regime.

Figure 5

Oscillation period $T_t=2\pi /\Delta E_k$ corrsponding to energy splitting between the eigenstate with maximum coefficient for the basis function ${|N/2,N/2,0\rangle }_0$ when $k=0$, and a corresponding eigenstate with $k=2\pi /3$ as described in the text vs (left) $\alpha N$ for fixed $N=60$ particles (the saturation around ${10}^{12}$ is due to using a limited numerical accuracy), and (right) particle number $N$ for fixed $\alpha N=10,11,12$ (from bottom to top). The shaded region in the left figure marks the classical instability regime.

Figure 6

${\langle n_i/N\rangle }_{\min }$ (left) and ${\langle n_i{n}_j/N^2\rangle }_{\max }$ (right) as a function of $\alpha N$, for linear combinations of eigenstates with $k=0,\pm 2\pi /3$ with maximum coefficient for the basis function ${|N/2,N/2,0\rangle }_0$, for $N=60$ (dotted blue), $N=80$ (light green), and $N=100$ (dark red), particles. The shaded region marks the classical instability regime.

Figure 7

Left: Difference $\Delta \langle n_3/N\rangle$ in occupation of site 3 (having initially smallest $\langle n_i\rangle$) between an exact and a perturbed tunneling pair of eigenstates, prepared as described in text. Right: Dynamics over the same time range of $\langle n_3/N\rangle$ for the unperturbed tunneling pair. $\alpha N=6$, $N=60$.

Figure 8

Projection of SU(3) coherent states onto tunneling pairs of eigenstates with maximal expansion coefficient for ${|N/2,N/2,0\rangle }_0$. Blue (dark) curves, eigenstates chosen as in Sec. 4a; green (light) curves, linear combinations of eigenstates with maximum coefficients for ${|N/2,N/2,0\rangle }_k$ for each $k$. [The blue (dark) curves coincide with the green (light) curves in regimes where only the latter can be seen.] The shaded region marks the classical instability regime.

Figure 9

Coefficients of dominating Fock basis states for SDW SU(3) coherent states (red crosses) and for the $k=0$ eigenstate with largest projection on the SU(3) coherent state (blue bars) when $N=60$ and (left) $\alpha N=1$; (middle) $\alpha N=12$; (right) $\alpha N=150$. The ordering of the basis states is the same as in Fig. 2, and thus state number 31 is ${|30,30,0\rangle }_0$, and all states with numbers higher than 60 correspond to nonzero population at all three sites.

Figure 10

Short-time dynamics with SDW SU(3) coherent state as initial condition, for $N=80$ particles and $\alpha N$ varying from 8 to 11.5 (i.e., crossing the classical upper stability transition). Green lines (lower at $t=0$) show relative population expectation values $\langle n_i/N\rangle$ for the initially empty site, and blue lines (upper at $t=0$) for one of the two initially excited sites.

Figure 11

Early dynamics showing the expectation value for the relative population of the initially unoccupied site, for a coherent-state initial condition with $N=80$ particles. Left: dynamics for $0<t<10$ and, from top to bottom, $\alpha N=11,11.5,12$; right: dynamics for $0<t<50$ and, from top to bottom, $\alpha N=11,12,13$.

Figure 12

Dynamics for the coherent-state initial condition for 20–90 particles, respectively, for $\alpha N=10$. The same quantities as in Fig. 10 are plotted.

Figure 13

Long-time dynamics for a system of 60 particles and $\alpha N=11$, for initial conditions given by (left) the SU(3) coherent state, and (right) a linear combination of exact eigenstates constituting a SDW tunneling pair as in Fig. 5, prepared to minimize $\langle n_3(t=0)\rangle$ (pseudoeigenstate).