Delocalization to self-trapping transition of a Bose fluid confined in a double well potential. An analysis via one- and two-body correlation properties

We revisit the coherent or delocalized to self-trapping transition in an interacting bosonic quantum fluid confined in a double well potential, in the context of full quantum calculations. We show that an $N$-particle Bose-Hubbard fluid reaches an stationary state through the two-body interactions. These stationary states are either delocalized or self-trapped in one of the wells, the former appearing as coherent oscillations in the mean-field approximation. By studying one- and two-body properties in the energy eigenstates and in a set of coherent states, we show that the delocalized to self-trapped transition occurs as a function of the energy of the fluid, provided the interparticle interaction is above a critical or threshold value. We argue that this is a type of symmetry-breaking continuous phase transition.

💡 Research Summary

The paper presents a comprehensive quantum‑mechanical study of a Bose‑Hubbard fluid confined in a symmetric double‑well potential, focusing on the transition from a delocalized (coherent‑oscillation) regime to a self‑trapped regime. Using the two‑mode Bose‑Hubbard Hamiltonian
( \hat H = -\Delta(\hat b_1^\dagger \hat b_2 + \hat b_2^\dagger \hat b_1) + U(\hat b_1^\dagger \hat b_1^\dagger \hat b_1 \hat b_1 + \hat b_2^\dagger \hat b_2^\dagger \hat b_2 \hat b_2) ),
the authors introduce the dimensionless interaction parameter (\Lambda = N U / \Delta) and the SU(2) spin operators (\hat J_x, \hat J_y, \hat J_z). The observables of interest are the particle number in well 1, (\hat N_1 = (\hat N + 2\hat J_z)/2), and the two‑body tunneling correlation (\hat J_x \hat J_x).

The study proceeds by exact diagonalisation and time‑evolution of the full N‑body wavefunction for particle numbers up to (N\sim10^4). Initial states are chosen from a family of coherent states (|\theta,\phi\rangle) that have well‑defined mean energy (\varepsilon) and small energy variance, allowing the authors to work effectively in a microcanonical ensemble (fixed N, fixed (\varepsilon)). From the full density matrix they construct the reduced one‑ and two‑body density matrices, from which (\langle \hat N_1(t)\rangle) and (\langle \hat J_x \hat J_x(t)\rangle) are obtained.

Key findings are:

1. Statistically stationary regime – For any (\Lambda) and any coherent initial state, the expectation values initially display the coherent Josephson oscillations predicted by mean‑field theory, but after a short transient they decay and settle into narrow distributions around time‑independent values. The width of these distributions shrinks with increasing N, indicating that the system reaches a “statistically stationary” state that depends only on N and the mean energy (\varepsilon).

2. Equivalence with energy eigenstates – The stationary averages coincide with the expectation values calculated in the exact energy eigenstates (|E_n\rangle). Thus, the long‑time behavior of the system can be described by a microcanonical ensemble, and the dynamics effectively “thermalises” with respect to one‑ and two‑body observables.

3. Interaction‑driven symmetry breaking – When (\Lambda) exceeds a critical value (\Lambda_c), the stationary values become asymmetric: (\langle \hat N_1\rangle) deviates from (N/2) and the system self‑traps in one well. Simultaneously, (\langle \hat J_x \hat J_x\rangle) remains large, signalling enhanced two‑body correlations. This marks a continuous quantum phase transition that breaks the underlying SU(2) symmetry.

4. Energy‑controlled transition – For a fixed (\Lambda > \Lambda_c), varying the mean energy (\varepsilon) (or equivalently the initial population imbalance) drives the system across the delocalized‑to‑self‑trapped boundary. Hence the transition can be viewed as occurring in the ((\Lambda,\varepsilon)) plane, rather than being solely a function of the interaction strength.

5. Revivals and decoherence – The authors observe that revivals of the initial coherent oscillations persist but become increasingly rare as N grows; the ratio of stationary‑time to revival‑time increases, reinforcing the notion of effective decoherence when only few‑body observables are measured.

The paper concludes that the delocalization‑to‑self‑trapping transition is a genuine continuous symmetry‑breaking quantum phase transition, characterized by a critical interaction strength and an energy‑dependent order parameter. It goes beyond mean‑field predictions by providing exact many‑body evidence, and it suggests that similar behavior should persist in asymmetric double wells, driven systems, and possibly in larger lattice configurations. Future directions include exploring finite‑temperature effects, coupling to external baths, and extensions to multi‑well lattices.