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 revisits the well‑known delocalized (coherent) to self‑trapping transition of an interacting bosonic fluid confined in a double‑well potential, but it does so using fully quantum mechanical calculations rather than the usual mean‑field (Gross‑Pitaevski) approximation. The authors start from the two‑site Bose‑Hubbard Hamiltonian

( \hat H = -J(\hat a^\dagger_L\hat a_R + \hat a^\dagger_R\hat a_L) + \frac{U}{2}\sum_{i=L,R}\hat n_i(\hat n_i-1) ,)

where (J) is the tunnelling amplitude, (U) the on‑site interaction, and (N) the total particle number. They introduce the dimensionless interaction parameter (\Lambda = NU/2J), which controls the competition between kinetic delocalisation and interaction‑induced localisation.

In the mean‑field picture, for (\Lambda<\Lambda_c) the system exhibits Josephson‑type oscillations of the population imbalance, while for (\Lambda>\Lambda_c) the dynamics freezes into a self‑trapped state. The authors demonstrate that this picture is incomplete: when the full many‑body Hilbert space is diagonalised, the system evolves toward stationary states that are determined by two‑body correlations, not by a simple single‑particle order parameter.

The study proceeds by (i) exact diagonalisation of the Hamiltonian for moderate particle numbers, (ii) calculation of the one‑body reduced density matrix (\rho^{(1)}) and the two‑body correlation function

( g^{(2)}(i,j)=\frac{\langle \hat a_i^\dagger\hat a_j^\dagger\hat a_j\hat a_i\rangle}{\langle\hat n_i\rangle\langle\hat n_j\rangle},)

and (iii) time‑evolution of coherent initial states (|\theta,\phi\rangle = \sum_{n=0}^N C_n(\theta,\phi)|n,N-n\rangle).

The eigenstate analysis reveals a clear energy‑dependent bifurcation when (\Lambda) exceeds a critical value (\Lambda_c\approx 1). Low‑energy eigenstates have almost equal diagonal elements of (\rho^{(1)}) (delocalised) and (g^{(2)}(L,R)\approx 1), indicating strong inter‑well coherence. High‑energy eigenstates, by contrast, display a pronounced imbalance in (\rho^{(1)}) and a suppressed cross‑correlation (g^{(2)}(L,R)\ll 1), signalling that particles are predominantly localized in one well – the self‑trapped regime.

When coherent states are evolved in time, the mean‑field prediction of perpetual oscillations fails: the amplitude of the population imbalance decays, and the system settles into a stationary average value. This “self‑thermalisation” is driven by the two‑body interaction, which acts as an internal bath, effectively placing the system in a micro‑canonical ensemble of its own energy shell.

The order parameter for the transition is chosen as the normalised imbalance

( z = \frac{\langle \hat n_L-\hat n_R\rangle}{N}. )

As (\Lambda) is increased past (\Lambda_c), (z) grows continuously from zero to a finite value, indicating a continuous (second‑order) symmetry‑breaking phase transition. The symmetry broken is the exchange symmetry (L\leftrightarrow R); the two degenerate self‑trapped states are related by this operation. The authors also discuss the critical behaviour, noting that the transition is accompanied by a rapid increase in quantum entanglement (as measured by the von‑Neumann entropy of the reduced density matrix) and a marked change in the two‑body correlation landscape.

In the discussion, the authors argue that the transition should be viewed as a genuine quantum phase transition, not merely a dynamical crossover. They point out that the critical interaction strength and the energy dependence are experimentally accessible in ultracold‑atom setups using optical lattices or double‑well potentials, where one can tune (U) via Feshbach resonances and measure population imbalances with single‑site resolution.

Finally, the paper concludes that the delocalized‑to‑self‑trapped transition in a double‑well Bose fluid is a continuous symmetry‑breaking transition governed by many‑body correlations. The findings open the way to explore similar transitions in larger lattice geometries, in the presence of disorder, or under periodic driving, where the interplay of interaction, tunnelling, and external fields may generate richer phase diagrams.