We study many-body tunneling of a small Bose-Einstein condensate in a periodically modulated, tilted double-well potential. Periodic modulation of the trapping potential leads to an analog of photon-assisted tunneling, with distinct signatures of the interparticle interaction visible in the amount of particles transferred from one well to the other. In particular, under experimentally accessible conditions there exist well-developed half-integer Shapiro-like resonances.PACS numbers: 03.75. Lm, 74.50.+r In a recent experiment, it has become possible to monitor the Josephson-like oscillation of a sample of about 1000 Bose-Einstein-condensed atoms in an optical double-well potential, and to observe in situ both the evolution of the atomic densities, and of the relative phase between the condensates in both wells [1]. In the present Letter, we propose to extend such experiments by modulating the trapping potential periodically in time. As we will demonstrate, such modulated double-well condensates exhibit certain parallels to photon-assisted tunneling in microwave-driven superconducting Josephson junctions [2], although the "photons" now tend to fall into the lower Kilohertz regime.We consider a symmetric double-well trap filled with N Bose particles at temperature T = 0, as realized approximately in the Heidelberg experiment [1]. Denoting the tunneling splitting between the lowest pair of single-particle energy eigenstates by Ω, and adopting the common two-mode approximation [3,4,5], the system's Hamiltonian takes the idealized formwhere a ( †) i is the annihilation (creation) operator for a Boson in the i-th well, satisfying the commutation relation [a i , a † j ] = δ i,j (i, j = 1, 2). The on-site interaction energy of a single pair of Bosons occupying the same well is 2 κ, proportional to the s-wave scattering length of the particle species. This deceptively simple model (1), describing an unforced Bosonic Josephson junction, has recently been studied in considerable detail [6,7]. The key parameter governing its dynamics is the dimensionless ratio N κ/Ω. While the well-understood mean-field approximation [8, 9, 10] corresponds to the limit N → ∞ and κ → 0, taken such that the product N κ remains constant, present experiments start to explore the dynamics beyond the mean-field regime [1].We extend this system by introducing a twofold bias: We assume that the two wells are tilted such that their bottoms are misaligned in energy by an amount 2 µ 0 , which should be sufficiently large compared to the tunnel splitting Ω, so that the usual Josephson oscillations are strongly suppressed. In addition, we propose to modulate this tilted double well periodically in time, such that the individual wells are shifted sinusoidally up and down, in phase opposition to each other and with angular frequency ω, by an amount µ 1 . Such a modulation of an optical double well can be achieved by periodically shifting the focus of a blue-detuned laser, which creates the barrier between the two wells, with the help of a piezoactuated ...

For a weakly interacting Bose-Einstein condensate in a double well, an appropriate time-dependent modulation of the trapping potential counter-acts the "self-trapping" effects of the interactions, thereby allowing tunneling between the wells. It is demonstrated numerically that this modulation can be employed for transferring the condensate from one well to the other in a controlled way. Moreover it allows the production of mesoscopic entangled states on short time scales.

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