Experimental identification of interwell coherence and off-diagonal correlations in the superfluid-Mott-insulator transition of a one-dimensional optical lattice with double-well confinement

Jiang, Weng, and Xiang [Phys. Rev. B 76, 224515 (2007)] used density-matrix renormalization-group calculations to predict that, when a one-dimensional Bose-Hubbard optical lattice at commensurate filling is confined by a double-well trap envelope, the superfluid regions in the two wells remain coherently coupled through intervening Mott-insulator plateaus, and that this interwell off-diagonal long-range correlation has a direct momentum-space signature: satellite peaks in the momentum distribution $S(k)$ , together with a step-plateau structure of the interference visibility as $U/t$ is increased. They proposed these observables as an effective experimental probe of phase coherence between the two tunneling-coupled condensates. No experimental realization of this proposal has been reported. The problem is to design and perform experiments that unambiguously detect - or rule out - the predicted interwell coherence signatures in this double-well lattice geometry, and to establish the parameter window (temperature, trap smoothness, filling control) in which they survive.

Background

Ultracold bosonic atoms can be trapped in periodic light-shift potentials called optical lattices, whose lowest band realizes the Bose-Hubbard model: bosons hop between neighboring sites with amplitude $t$ and repel each other on site with strength $U$ , so that increasing $U/t$ drives a quantum phase transition between a superfluid (delocalized, phase-coherent) state and a Mott insulator (particles pinned one per site with suppressed number fluctuations) at commensurate filling (Greiner et al., Nature 415, 39 (2002)); in the Tonks-Girardeau limit $U\to\infty$ the bosons become impenetrable ('fermionized'). The momentum distribution $S(k)$ is measured by releasing the atoms and imaging the ballistically expanded cloud (time-of-flight imaging), and the contrast of the interference fringes defines the visibility $\nu=(S_{\max}-S_{\min})/(S_{\max}+S_{\min})$ , a standard measure of phase coherence. First-order coherence itself is carried by the off-diagonal correlation function $g(i,j)=\langle \hat a_i^\dagger \hat a_j\rangle$ , with $\hat a_i$ the annihilation operator on site $i$ ; its long-range part is the lattice analogue of off-diagonal long-range order.

Real experiments add a confining trap, making the system inhomogeneous: as $U/t$ grows, Mott-insulating plateaus at integer local density coexist with superfluid regions (a 'wedding-cake' density profile). For one-dimensional lattices, numerically exact methods - quantum Monte Carlo and the density-matrix renormalization group (DMRG), a controlled truncation method for ground states of long one-dimensional quantum lattices - predicted that this coexistence produces kinks in the visibility $\nu$ versus $U/t$ (Sengupta et al., Phys. Rev. Lett. 95, 220402 (2005)), and time-of-flight experiments on singly trapped one-dimensional lattices had already measured the loss of fringe visibility across the transition (Stöferle et al., Phys. Rev. Lett. 92, 130403 (2004)).

A separate experimental line realized a bosonic Josephson junction: two weakly linked condensates in a macroscopic double-well potential whose tunneling oscillations and relative phase are read out from interference fringes (Albiez et al., Phys. Rev. Lett. 95, 010402 (2005)). Jiang, Weng, and Xiang asked what happens when these ingredients are combined - a one-dimensional optical lattice whose overall confining potential is a single double well, so that two lattice segments separated by a central barrier share the atoms. Using DMRG on an 80-site Bose-Hubbard chain with open boundaries and the double-well envelope $V_T(i)=-0.024t(i-40.5)^2+2\times 10^{-5}t(i-40.5)^4$ at fillings $\rho=N/L=1/2$ and $\rho=1$ , they found the following. At $\rho=1/2$ , Mott plateaus form inside each well for $U/t\gtrsim 6.8$ , with visibility kinks near $U/t\approx 5.5$ and $6.4$ , and even in the Tonks-Girardeau limit a finite visibility $\nu\approx 0.44$ survives from the superfluid remnants. At $\rho=1$ , three superfluid regions separated by two Mott plateaus persist up to $U/t\approx 22$ ; the visibility and the integrated well population show a rich step-plateau structure, reflecting abrupt redistribution of bosons between local superfluid and Mott regions. Most importantly, whenever the separated superfluid regions retain strong interwell off-diagonal correlations $g(i,j)$ , the momentum distribution $S(k)$ develops satellite peaks in addition to its central peak, and these satellite peaks - like the underlying long-range correlations - vanish once the central Mott plateau disconnects the two wells. The authors concluded that measuring $S(k)$ is 'an effective probe for experiment to identify the off-diagonal long-range correlation and phase coherence due to the tunnelling between the two condensates'.

The source work is a zero-temperature calculation for one idealized geometry: it provides no experimental protocol, no finite-temperature analysis, and no study of how the predicted signatures depend on trap parameters or on dimensionality. Whether the interwell coherence it predicts can actually be identified in an experiment on this system is the open question.

Problem Statement

In the geometry studied by Jiang, Weng, and Xiang - a one-dimensional Bose-Hubbard optical lattice at overall commensurate filling ( $\rho=1$ ) confined by a double-well trap envelope, where superfluid regions in the two wells are separated by Mott-insulator plateaus - can the predicted interwell coherence be identified unambiguously in experiment? Concretely, can the off-diagonal long-range correlations between the spatially separated superfluid regions and their momentum-space signatures - satellite peaks in the time-of-flight momentum distribution $S(k)$ together with the step-plateau (kink) structure of the visibility $\nu$ as the ratio $U/t$ of on-site repulsion to hopping is swept - be identified unambiguously in experiment, with ordinary interference of two independent condensates excluded as a trivial explanation? Do these signatures survive realistic conditions - finite temperature, atom-number and trap-parameter fluctuations, imperfect adiabatic preparation - given that the source predictions are strictly zero-temperature results for an idealized trap, and over what window (temperature, trap smoothness, filling control) do they persist before being washed out?

References

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