Quantum dynamics in the bosonic Josephson junction Maya Chuchem $^{1}$, Katrina Smith-Mannschott2 , $^{3}$, Moritz Hiller$^{4}$, Tsampikos Kottos $^{2}$, Amichay Vardi5 , $^{6}$, and Doron Cohen1 $^{1}$Department of Physics, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 84105, Israel $^{2}$Department of Physics, Wesleyan University, Middletown, Connecticut 06459, USA $^{3}$MPI for Dynamics and Self-Organization, Bunsenstrasse10, D-37073G¨ ottingen, Germany $^{4}$Physics Institute, Albert-Ludwigs-Universit¨ at, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany $^{5}$Department of Chemistry, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 84105, Israel $^{6}$ITAMP, Harvard-Smithsonian CFA, 60 Garden St., Cambridge, Massachusetts 02138, USA We employ a semiclassical picture to study dynamics in a bosonic Josephson junction with vari- ous initial conditions. Phase-diffusion of coherent preparations in the Josephson regime is shown to depend on the initial relative phase between the two condensates. For initially incoherent conden- sates, we find a universal value for the buildup of coherence in the Josephson regime. In addition, we contrast two seemingly similar on-separatrix coherent preparations, finding striking differences in their convergence to classicality as the number of particles increases. Bose-Einstein condensates (BECs) of dilute, weakly- interacting gases offer a unique opportunity for exploring non-equilibrium many-body dynamics, far beyond small perturbations of the ground state. Highly excited states are naturally produced in BEC experiments and their dynamics can be traced with great precision and control. The most interesting possibilities lie in strong correlation effects, which imply a significant role of quantum fluctu- ations. between atoms lead to the collapse and revival of the rel- ative phase in a process known as phase diffusion [14–16]. (the appropriate term is in fact phase spreading ). Phase diffusion has been observed with astounding precision in an optical lattice in Refs.[17], in a double-BEC system in Refs.[18–20], and in a 1D spinor BEC in Ref. [21]. Typ- ically, the condensates are coherently prepared, held for a varying duration (‘hold time’) in which phase-diffusion takes place, and are then released and allowed to inter- fere, thus measuring the relative coherence through the many-realization fringe visibility. In order to establish this quantity, the experiment is repeated many times for each hold period. The importance of correlations and fluctuations may be enhanced by introducing an optical lattice, that can be controlled by tuning its depth. This tight confinement decreases the kinetic energy contribution with respect to the interactions between atoms. In the tight-binding limit, such systems are described by a Bose-Hubbard Hamiltonian (BHH), characterized by the hopping fre- quency K between adjacent lattice sites, the on-site in- teraction strength U , and the total atom number N . The strong correlation regime is attained when the character- istic coupling parameter u ≡ UN/K exceeds unity, as indicated by the quantum phase transition from a super- fluid to a Mott-insulator [2, 3]. Phase-diffusion experiments focus on the initial prepa- ration of a zero relative phase and its dispersion when no coupling is present between the condensates. However, in the presence of weak coupling during the hold time, the dynamics of phase diffusion is richer. It becomes sen- sitive to initial value of ϕ and the loss of coherence is most rapid for ϕ = π [22–24]. Here, we expand on a re- cent letter [24], showing that this quantum effect can be described to excellent accuracy by means of a semiclas- sical phase-space picture. Furthermore, exploiting the simplicity of the dimer phase space, we derive analytical expressions based on the classical phase-space propaga- tion [25]. The simplest BHH is obtained for two weakly coupled condensates (dimer). Its dynamics is readily mapped onto a SU(2) spin problem and is closely related to the physics of superconductor Josephson junctions [4, 5]. To the lowest order approximation, it may be described by a Gross-Pitaevskii mean-field theory, accurately account- ing for Josephson oscillations [6–8] and macroscopic self trapping [9], observed experimentally in Refs.[10, 11], as well as the equivalents of the ac and dc Josephson effect [12] observed in [13]. Phase-space methods [26] have been extensively ap- plied for the numerical simulation of quantum and ther- mal fluctuation effects in BECs [27–39]. Such meth- ods utilize the semiclassical propagation of phase-space distributions with quantum fluctuations emulated via stochastic noise terms, and using a cloud of initial condi- tions that reflects the uncertainty of the initial quantum wave-packet. One particular example is the truncated Wigner approximation [28, 31, 34, 36] where higher order derivatives in the equation of motion for the Wigner dis- tribution function are neglected, thus amounting to the propagation of an ensemble using the Gross-Pitaevskii equations. Due to the relative simplicity of the classical phase- space of the two-site BHH, it is possible to carry out its Both Josephson oscillations and macroscopic self trap- ping rely on coherent (Gaussian) preparations, with different initial population imbalance. The mean-field premise is that such states remain Gaussian throughout their evolution so that the relative phase ϕ between the two condensates remains defined. However, interactions arXiv:1001.2120v2 [quant-ph] 21 Oct 2010 I. INTRODUCTION