Time-periodic quantum states of weakly interacting bosons in a harmonic trap

Abstract

We consider identical quantum bosons with weak contact interactions in a two-dimensional isotropic harmonic trap, and focus on states at the Lowest Landau Level (LLL). At linear order in the coupling parameter g, we exploit the rich algebraic structure of the problem to give an explicit construction of a large family of quantum states with energies of the form E0 + gE1/4 + O(g 2 ), where E0 and E1 are integers. As a result, any superposition of these states evolves periodically with a period of at most 8π/g until, at much longer time scales of order 1/g 2 , corrections to the energies of order g 2 become important and may upset this perfectly periodic behavior. We further construct coherent-like combinations of these states that naturally connect to classical dynamics in an appropriate regime, and explain how our findings relate to the known time-periodic features of the corresponding weakly nonlinear classical theory. We briefly comment on possible generalizations of our analysis to other numbers of spatial dimensions and other analogous physical systems.

Time-periodicities in holographic CFTs

“…In the next section, we will make the connection to the classical analysis more concrete and build coherent-like combinations of the ladder states that reproduce the time-periodic features of the invariant manifold of classical solutions described in section 2.4. The present analysis extends the techniques introduced in [41], applied to the simpler nonrelativistic version of the AdS system.…”

“…While the quantum resonant systems corresponding to our cases of interest cannot be fully solved analytically, our goal is to present their partial analytic solution: a subset of energy levels and their explicit wavefunctions. This solution builds on the previous work [40,41] for the simpler nonrelativistic analogs of the AdS systems. The explicit energy levels given by our solutions form simple ladders and provide clear quantum counterparts of the timeperiodic behaviors of the classical theory.…”

“…It is nonetheless worthwhile, given our initial motivation, to make the connection between the family of quantum 'ladder' states we find and the corresponding time-periodic classical solutions as explicit as possible. To this end (again, building on the analysis of [41] for the considerably simpler nonrelativistic case), we develop a construction of coherent-like states made entirely out of our ladder states (and not involving any other energy eigenstates of the Hamiltonian) that can approximate the time-periodic classical dynamics with arbitrary precision. This completes the circle and answers the initial question that had triggered this study, in addition to the explicit identification of the quantum level structure and its holographic CFT counterpart.…”

Growth of Sobolev norms for coupled Lowest Landau Level equations

“…Equation (1.1) is globally well-posed on E and we will see that it is also the case of equation (1.3) (see Theorem 1.1 below). We refer to [4,6,11] for more results on LLL and related equations.…”

Bounds on quantum evolution complexity via lattice cryptography

“…In contrast to the rich array of classical dynamical behaviors, the corresponding quantum theory is very economical in its structure and can be explored via an operation as simple as diagonalizing finite-sized numerical matrices [37]. (We mention in addition that (1.1) arises directly in the process of applying the standard Hamiltonian perturbation theory for the degenerate spectrum of quantum fields in strongly resonant domains at first order in the quartic interaction strength [69][70][71][72]. )…”