Bose-Einstein Condensates on slightly asymmetric double-well potentials Bruno Juliá-Díaz ICFO & Departament d Estructura i Constituents de la Matèria, Universitat de Barcelona (Spain) In collaboration with: D. Dagnino, M. Lewenstein, J. Martorell, A. Polls
Outline 1. BEC in a double-well (Simple system with already interesting correlated states 1. The two-site Bose-Hubbard Hamiltonian 2. Mean-field vs exact dynamics 3. Strongly correlated quantum states, role of the external small bias 2. Analytical insights 1. 1/N expansion 2. External bias vs tunneling strenght 3. Conclusions and outlook
A simple, but many-body H Lets consider the following two-site Bose-Hubbard model: J: Hopping parameter >0 U: Atom-atom interaction, proportional to the scattering length, U>0: attractive >0: Promotes the left well The bias is here taken very small, <<J It is customary to define, =NU/J Milburn et al (1997) The biasless case is mathematically similar to some versions of the Lipkin-Meshkov-Glick model (1960s)
Semiclassical results The semiclassics recovers the two-mode approximation to the mean field Gross Pitaevskii equations, The resulting semiclassical equations of motion are the well known z: Population imbalance, (N L -N R )/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 atomatom interactions) Smerzi et al. (1997) (Assuming a two mode ansatz for the Gross Pitaevskii equation)
Experimentally confirmed Albiez et al. (2005)
Experimentally observed Experimental exploration..but just of the semiclassical properties. Internal Josephson. (repulsive interactions) (N=500) Signatures of the strongly correlated state should be explored!! T. Zibold et al. (Oberthaler s group) PRL 2010
Back from the semiclassical The one body density matrix reads, GS 1 N aˆ aˆ L R aˆ aˆ L L aˆ aˆ L R aˆ aˆ R R GS Its two eigenvalues fulfill, n 1 +n 2 =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
Ground and highest excited state Black, ground state Red, highest excited c k 2 With the usual many-body base: N L,N R >= { N,0>, N-1,1>,, 0,N>} Any N particle vector can be written as, Cat-like state Cat-like structures are still present even in the presence of an external (small) bias in the system Julia-Diaz, Dagnino, Lewenstein, Martorell, Polls, PRA 81, 023615 (2010) Jaaskelainen and Meystre, PRA (2005, 2006)
Population imbalance Ground state: correlations Blue dashed: Semiclassical prediction: (1-4/ 2 ) 1/2 Red solid: quantum result for the imbalance Band: dispersion of the imbalance, N=50, =J/10 10 NU/J Julia-Diaz, Dagnino, Lewenstein, Martorell, Polls, PRA 81, 023615 (2010)
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, =J/10 10 Strongly correlated quantum states appear when the semiclassical equations have instabilities e.g. analogous to the nucleation of the first vortex in rotating clouds, e.g. Dagnino et al Nature Physics 2010
Variation with N The semiclassical behavior is the same in all cases (the bias is taken the same) The size of the strongly correlated 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.
Now some analytics The question we had in mind was, when does the system pop from the z=0 to the bifurcated branch in the quantum case?
Making k/n a continuous variable Slight change of notation to follow the work of Shchesnovich and Trippenbach (2008), L 1 R 2-2 Negative gamma Attraction /2 The time dependent Schrodinger equation and corresponding Hamiltonian read: Javanaien and Ivanov, PRA60, 2351 (1999) Jaaskelainen and Meystre, PRA71, 043603 (2005); PRA73, 013602 (2006) Shchesnovich and Trippenbach, PRA78, 023611 (2008) Critical =-1
Discrete equation for c k The many body state reads: With the coefficients obeying the following set of discrete equations:
Continuous limit Now lets expand in powers of h=1/n, with z=1-2x, assuming that c k is regular enough, i.e. c k+1 ~ c k And obtain a time-dependent-schrodinger-like equation with an effective z dependent mass Shchesnovich and Trippenbach, PRA78, 023611 (2008) Julia-Diaz, Martorell, Polls, PRA81, 063625 (2010)
Continuous limit (sample dynamics) The obtained equation is actually pretty accurate as compared to the original Bose-Hubbard Bose-Hubbard Schrodinger like equation Shchesnovich and Trippenbach, PRA78, 023611 (2008) Julia-Diaz, Martorell, Polls, PRA81, 063625 (2010)
Analytical model Interestingly enough we arrive to a pseudo-schroedinger equation, defined on a compact interval [-1,1] with a double-well potential! The non-linear interaction has been mapped into a barrier in the z-space The ground state energy can be estimated with good accuracy as the ground state of one of the wells Critical =-1
Two mode in Fock space We can now make a two-mode approximation to the dynamics in Fock space. 1. Simplify the effective mass: The two modes are labeled: z 2 2 2 1 z z 1 zm z 2. Expanding around the two minima of the bifurcation, z m, - z m The two mode Hamiltonian can then be written as: Diagonal terms are bias dependent How do we estimate the off diagonal t?
Estimating the tunneling 2t is the splitting between the ground (S) and first excited state (A) in the double-well in the biasless case (in fock space) Using a WKB approximation to compute the splitting and approximating the barrier by a parabola we get: Julia-Diaz, Martorell, Polls, PRA81, 063625 (2010)
Role of the external bias Bias dominated (almost condensed) Tunneling/Interaction dominated
Analytical picture Simple analytical formulas are obtained which provide a faithful description of the transition region. Eigenvalues of Transition point Julia-Diaz, Martorell, Polls, PRA81, 063625 (2010)
Summary and outlook BECs on double-wells Static properties of the Bose-Hubbard hamiltonian with small bias Existence of strongly correlated cat-like ground states even in the presence of an external bias Cat-like states appear in the spectrum when the semiclassics predicts a bifurcation Can we see any traces of the cats? J-D, Dagnino, Lewenstein, Martorell, Polls, PRA A 81, 023615 (2010) J-D, Martorell, Polls, PRA81, 063625 (2010) (current effort) Obtain simple analytic expressions using the simplified model for the dynamical generation of cat-like states Use the same model to predict the best squeezing conditions, both dynamical and static, for any initial coherent state