Quantum signatures of an oscillatory instability in the Bose-Hubbard trimer Magnus Johansson Department of Physics, Chemistry and Biology, Linköping University, Sweden Sevilla, July 12, 2012 Collaborators (PhD students): Peter Jason, Katarina Kirr Some ideas and inspiration: Chris Eilbeck (MOBIL, Sevilla, 2003!) P. Jason, M. Johansson, and K. Kirr, Phys. Rev. E (to appear 2012) Aim: Characterization of the classical transition to oscillatory instability (“Hamiltonian Hopf bifurcation”) in a fundamental quantum lattice model. “Simplest possible” model: “Single-depleted well” states in 3-site (triangular) DNLS ↔ quantum Bose-Hubbard trimer Numerical work:Exact diagonalization for moderate particle numbers (10-100) Analysis of eigenstates 'close' to classical states Comparison with SU(3) coherent states 1 Quantum dynamics close to classical instability transition
Oscillatory instabilities: Few words about classical DNLS trimer Stationary solutions: : φ = Particular family Termed “Single Depleted Well” (SDW) states by Franzosi/Penna, PRE 67 , 046227 (2003). Linear stability analysis: complex eigenvalues for intermediate nonlinearities. Carr/Eilbeck, Phys.Lett. A109 , 201 (1985) Oscillatory instabilities arise in Hamiltonian Hopf bifurcations through resonances between two internal oscillations whose contributions to the total energy are of opposite signs. (“Krein signatures”) (see Johansson, J. Phys. A 37 , 2201 (2004) and refs. therein; cf. also Goodman, J.Phys. A 44 , 425101 (2011)) 2
Illustrations of classical unstable dynamics (Johansson, J. Phys. A 37 , 2201 (2004)) Perturbed SDW state in instability interval 9.077... < γ N < 18: γ N = 10 γ N = 17 Trapping close to initial SDW “Intermittent population inversion” Phase-space pictures (Poincaré sections): 3 Strong chaos for 10.7.. < γ N < 18 SDW surrounded by phase-space dividing KAM tori for γ N < 10.7..
The Bose-Hubbard trimer + periodic boundary conditions Studied by MANY authors....: De Filippo et al, Nonlinearity 2 , 477 (1989); Cruzeiro et al, PRB 42 , 522 (1990); Wright et al, Physica D 69 , 18 (1993);...; Lee et al, PRL 97 , 180408 (2006); Buonsante et al, PRA 82 , 043615 (2010); PRA 84 , 061601(R) (2011); Viscondi/Furuya, J.Phys.A 44 , 175301 (2011);... The classical DNLS can be derived in the limit when the particle number goes to infinity for finite α N , in many different ways, e.g., ● Simply replace operators with c -numbers; ● Ansatz as tensor product of Glauber coherent states at each site; (e.g., Ellinas et al, Physica D 134 , 126 (1999); Amico/Penna, PRL 80 , 2189 (1998)) Conservation of total particle number only on average. ● SU(3) coherent state (“Hartree”) ansatz (e.g., Wright et al -93, Buonsante et al, PRA 72 , 043620 (2005); Viscondi/Furuya -11) Eigenstates of total number operator and thus conserves particle number exactly. 4
Energy spectrum For each particle number N , translational invariance splits spectrum into states with Bloch vector k = 0 and (the latter degenerate). → can use basis states (cf. Eilbeck, Scott,...) Spectrum for k = 0, increasing N : Thick blue line: state with largest coefficient for basis function | N /2, N /2, 0 > 0 . “Quantum SDW” states: eigenstates with exact zero population on one site at “anticontinuous” limit α N →∞. 5
Quantum SDW eigenstates Strong mixing with other eigenstates through avoided crossings in regimes of classical instability: N = 20 Basis states numbered so that 1-20 correspond to zero population on one site. Possible to identify “good” quantum SDW states as single eigenstates only away from strong resonances! 6
Some measures of the 'goodness' of SDW-like eigenstates Max probability that, in any eigenstate, all particles are equally distributed at only two sites ( k= 0, 2 π /3) Drops considerably in the classically unstable regime (grey) as N increases! Another, global measure: Measures total overlap in eigenstates between symmetric, compact SDW components and components which are not two-site localized. (If is small, an initially empty SDW-site can never grow large.) Develops a plateau of high values as 7 N increases!
Dynamics of 'tunneling pairs' (triplets) of eigenstates (cf. Bernstein et al, Nonlinearity 3, 293 (1990); Pinto/Flach, PRA 73 , 022717 (2006), and many others!) Since eigenstates are translationally invariant, a classical-like initial SDW state must contain components from all three subspaces Choose, e.g., for k = 0 the “quantum SDW” eigenstate with largest coefficient for the basis function | N /2, N /2, 0 > 0 , and corresponding 'suitable' states for k = (e.g., with max overlap). Energy splittings yields oscillation period for tunneling of SDW site ( if classical dynamics should be reproduced!) Tunneling times rapidly increase in classically stable regime, but remain of order 10-100 in unstable regime! 8 No single tunneling pair can capture the classical unstable dynamics.
SU(3) coherent states as initial conditions (cf. Nemoto et al., PRA 63 , 013604 (2000), Buonsante et al, PRA 72 , 043620 (2005), Lee et al., PRL 97, 180408 (2006), Trimborn et al., PRA 79 , 013608 (2009), Viscondi/Furuya, J.Phys.A 44, 175301 (2011) ,...) Explicit expansion in Fock states: (Viscondi/Furuya) gives 'classical-like' SDW-state with exact zero population at one site for each N Projection of SU(3) coherent states onto (two variants of) tunneling pairs of eigenstates: Pronounced dip develops in classically unstable regime as N increases! Particularly pronounced in the regime where the classically unstable solution is surrounded by strong chaos. 9
Few examples of dynamics with SU(3) coherent state initial conditions ● Crossing (upper) instability transition for fixed N = 80: Gradual transition from stable 'self-trapped' SDW to unstable population mixing. 10
● ”Quantum internal mode” oscillations in classically stable regime (fixed N = 80): The two fastest oscillations agree with classical internal modes, the longer modulation is pure quantum feature (increasing time-scales as N increases). 11
● Dynamics close to classical instability transition ( α N = 10) for increasing N: Dynamics approaches classically stable dynamics as N increases. 12
● Comparison of tunneling dynamics for initial SDW coherent-state and single “tunneling-pair” ( α N = 10, N = 60) ● Coherent state has initial SDW occupation exactly zero, but returns to considerably larger value after one oscillation ● Tunneling pair has small but nonzero initial SDW occupation, which it returns to almost perfectly after one period. 13
Conclusions : ● Classical oscillatory instability regime related to avoided crossings in quantum energy spectrum, with strong mixing between SDW-like eigenstate and other eigenstates. ● Several measures may be constructed giving clear signatures of classical instability regime already for 20-30 particles. ● Single “tunneling pairs” of quantum SDW eigenstates with a classical-like behaviour may be identified only in the classically stable regime. ● With SDW SU(3) coherent states as initial conditions, a gradual transition from stable oscillations to instability appears. Similar quantum broadenings of classical DNLS instability transitions earlier identified for: - Plane wave modulational instability (Altman, Polkovnikov et al, PRL 95 , 020402 (2005), PRA 71 , 063613 (2005)) - Self-trapping transition in the dimer (Wright et al -93, Bernstein -93,... very recently Buonsante et al, PRA 85 , 043625 (2012)) ● Open issue: is SDW scenario generic for other cases of oscillatory DNLS instabilities? (“Twisted” localized modes, dark solitons, standing waves, gap modes in diatomic chains, discrete vortices,...) ● Experimental observation for triple-well BEC configurations??? 14
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