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Single component and binary mixtures of BECs in a double-well

Single component and binary mixtures of BECs in a double-well. Bruno Juli á -D í az Departament d’Estructura i Constituents de la Mat è ria Universitat de Barcelona (Spain). In collaboration with:

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Single component and binary mixtures of BECs in a double-well

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  1. Single component and binary mixtures of BECs in a double-well Bruno Juliá-Díaz Departament d’Estructura i Constituents de la Matèria Universitat de Barcelona (Spain) In collaboration with: D. Dagnino, M. Guilleumas, M. Lewenstein, J. Martorell, M. Melé-Messeguer, A. Polls

  2. Summary • Introduction • Single component case • Static properties of the two-site Bose-Hubbard hamiltonian • Mean-field vs exact dynamics • (brief) Binary mixtures of BECs

  3. Josephson (GP) • 1150 atoms 87Rb, trap conditions as in Albiez/Oberthaler • The initial phase is set to ZERO for all z • Definitions (standard): • Z(t) = (Nleft(t)-Nright(t))/Ntotal • Phase difference= =right-left

  4. Self trapping (GP) • 1150 atoms, trap conditions as before • If the initial imbalance is large enough, no Josephson oscillation occurs. Instead a self trapping regime appears Smerzi et al. (1997) • This is a genuine effect of having atom-atom contact interactions

  5. Experimentally… Albiez et al. (2005)

  6. A simple, but many-body H Lets consider the following two-site Bose-Hubbard model: J: hopping parameter >0 U: atom-atom interaction >0 (proportional to g)(attractive) Epsilon: Bias>0, promotes the left well The bias is here taken very small, Epsilon<<J It is customary to define, =NU/J Milburn et al (1997)

  7. One body density matrix The one body density matrix reads, Eigenvalues, n1+n2=1 If the system is fully condensed, then the eigenvalues are 1 and 0. The eigenvector corresponding to 1 is, Departure from 0,1 indicates the system is fragmented

  8. semiclassics The semiclassics is governed by the well known: z: population imbalance, (NL-NR)/N : phase difference, R-L 2J: Rabi time (the time it takes for the atoms to go from left to right and back in absence of atom- atom interactions) Smerzi et al. (1997)(Assuming a two mode ansatzfor the Gross Pitaevskii equation)

  9. Ground and highest excited state Black, ground state Red, highest excited |ck|2 With the usual base: |NL,NR>={|N-k,k>}= { |N,0>,|N-1,1>,…,|0,N>} The hamiltonian can be written as an N+1 square matrix (here 50+1) Any N particle vector can be written as, Cat-like state

  10. Properties of the whole spectrum In the plot, x-axis: k index y-axis: eigenvector index 1, ground, N+1 highest excited Color, proportional to |ck|2 =0 GS: binomial =4 GS: Cat-like =8 GS: Trapped =12 GS: Trapped N=50, bias=J/10^10

  11. Ground state: imbalance Population imbalance NU/J Blue dashed: Semiclassical prediction: sqrt(1-4/^2) Red solid: quantum result for the imbalance Band: dispersion of the imbalance N=50, bias=J/10^10 Julia-Diaz, Dagnino, Lewenstein, Martorell, Polls, PRA (2010)

  12. Variation with N The semiclassical behavior is the same in all cases (the bias is taken the same) The size of the highly disperse region decreases abruptly as N is increased For which value of  does the ‘quantum hop’ take place? It turns out to be an interesting interplay between N, U, J and the bias: Julia-Diaz, Martorell, Polls (2010)

  13. Dispersion of z versus N and 

  14. Occupations of the orbitals Blue dashed: Semiclassical prediction  1,0 Red solid: quantum result for the eigenvalues of the one body density matrix N=50, epsilon=J/10^10

  15. Experimentally observed… Beautiful experimental exploration ..but just of the mean field properties. Internal Josephson. (repulsive interactions) (N=500) T. Zibold et al. (Oberthaler’s group), arXiv 1008.3057

  16. G.S. evolution with Lambda Most occupied eigenstate of , normalized to its eigenvalue dotted: Semiclassical Red/Blue: quantum results N=50, epsilon=J/10^10 Less occupied eigenstate of , normalized to its eigenvalue

  17. Time evolution of |N,0 For fixed N and starting from a ‘mean-field’ like state: • The smaller the interaction, the better the mean-field describes the exact result. • Fragmentation builds up with time Population imbalance And orbital ocupations Blue solid: Semiclassical Black solid: quantum for the imbalance Red dashed: n1, black dotted, n2 N=50, epsilon=J/10^10

  18. Time evolution of |N,0 NU/J=1 NU/J=1.5 NU/J=5 Population imbalance And orbital ocupations Mele-Messeguer, M.Thesis. (2010)

  19. Time evolution of |N,0 When starting from a ‘mean-field state’: • For small N (here 10), clear deviations are quickly seen (less than a Rabi time here) between the mean-field/semiclassics results and the full quantum behavior • Correspondingly, the cloud is far from condensed as time evolves Population imbalance And orbitalocupations t/tRabi

  20. Time evolution of |N,0 When starting from a ‘mean-field state’: • For large N (here 1000), the mean field provides an excellent account of the full dynamics during long times (here almost two Rabi periods) • The cloud, thus, remains condensed for a while. Population imbalance And orbitalocupations t/tRabi

  21. Binary mixtures(just a taste)

  22. Binary Mixture 1150 atoms, trap conditions as the Heidelberg experiment. We consider now the other limit: (50%,0%,50%) configuration The initial phases are all ZERO Initial population imbalances are: z1 (0) = - z-1(0) Note there would be no josephson at all if both components were just one A longer oscillation is seen.

  23. Binary Mixture (II) 1150 atoms, We consider a (98%,0%,2%) configuration The initial phases are all ZERO Initial population imbalances are: z1 (0) = - z-1(0) The most populated component follows the usual Josephson oscillation The less populated one follows the most populated one (“anti-Josephson”).

  24. A novel way of extracting a0 and a2 We proposed an experimental way of accessing the spin independent and spin dependent scattering lenghts: a0 and a2 The key points are: • Consider a binary mixture made by populating the (F=1, m=1) and (F=1, m=-1) states of an F=1 spinor. In this way, gaa=gbb~gab • Perform two measurements: • highly polarized, Na>>Nb • Na~Nb and za(0)=-zb(0) • Extract from za(t) and zb(t) the scattering lengths. Julia-Diaz, Guilleumas, Lewenstein, Polls, Sanpera (PRA 2009)

  25. Beyond two-mode Mele-Messeguer, et al. arXiv: 1005.5272

  26. Summary Single component case Static properties of the Bose-Hubbard hamiltonian with small bias Existence of strongly correlated ‘cat-like’ ground states Cat-like states appear in the spectrum when the mean field predicts a bifurcation Can we see any ‘traces of the cats’? When does the mean-field description break down? J-D, Dagnino, Lewenstein, Martorell, Polls, PRA A 81, 023615 (2010) J-D, Martorell, Polls, PRA81, 063625 (2010) • Binary mixtures populating m=+/-1 of an F=1 spinor • Extraction of the scattering lengths • Complete analysis, beyond two-mode, … J-D, Guilleumas, Lewenstein, Polls, Sanpera, PRA 80, 023616 (2009) Mele-Messeguer, J-D, Guilleumas, Polls, Sanpera, arXiv: 1005.5272

  27. A measure of the “spread” Blue dashed: Semiclassical prediction: sqrt(1-4/^2) Red solid: quantum result for S= N=50, epsilon=J/10^10

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