Manifestation of Classical Bifurcation in the Spectrum of the Integrable Quantum Dimer

Abstract

We analyze the classical and quantum properties of the integrable dimer problem. The classical version exhibits exactly one bifurcation in phase space, which gives birth to permutational symmetry broken trajectories and a separatrix. The quantum analysis yields all tunneling rates (splittings) in leading order of perturbation. In the semiclassical regime the eigenvalue spectrum obtained by numerically exact diagonalization allows one to conclude about the presence of a separatrix and a bifurcation in the corresponding classical model. PACS numbers: 05.45.+b, 03.20.+i, 03.65.Sq The problem of correspondence between classical and quantum-mechanical properties of nonlinear systems is currently an object of wide interest [1]. One interesting topic concerns Hamiltonian systems with a given symmetry (e.g., some permutational symmetry), where classical trajectories exist which are not invariant under the corresponding symmetry operation. This topic appears in analyzing selective bond excitation in chemistry and in the quantization of discrete breathers [2].We consider an integrable system with two degrees of freedom (TDF), whose classical version exhibits exactly one bifurcation (of periodic orbits) and separatrix manifold. This manifold cuts the phase space into three parts-one with invariant trajectories, and two with noninvariant trajectories, where the corresponding symmetry is the permutational one. By varying a single parameter it is possible to "switch" between these phase space parts by crossing the separatrix. It appears natural to expect in the quantum case a drastic change in the splittings of energy levels (which should be zero in the classical limit for the noninvariant phase space parts). However, the splittings are nonzero for any given value of the control parameter. The only way to avoid contradiction between the classical and quantum cases is to assume that the quantum level splittings tend to a steplike function (of, e.g., the level pair number) in the classical limit. The step should occur at the position of the classical separatrix. This problem can be coined also dynamical tunneling through a separatrix. There exist studies of the influence of classical chaos on dynamical tunneling [3]. This paper is an extension of previous studies on classical and quantum properties of the dimer system [4][5][6].We are able to trace the splittings of the level pairs using quantum perturbation methods. We consider the quasiclassical regime and show that the step indeed occurs. Therefore we are able to extract information about the classical separatrix and bifurcation. Further, we show that the quantum density of states (the second integral of motion is fixed) exhibits a sharp maximum at the separatrix energy. By calculating the corresponding classical quantity (with the help of Weyl's formula) we find that this singularity appears due to the integration over a part of the separatrix manifold which includes a hyperbolic isolated orbit.Let us consider the integrable dimer model with Hamiltonian [4]

Semiclassical quantization of anN-particle Bose-Hubbard model

“…To illustrate this issue, one can calculate the level density ̺(E) (normalized to unity) as a function of the energy [18]. Figure 3 shows a histogram of the level density for N = 1500 particles and different values of ε.…”

“…For a discussion of the relation between mean-field and Nparticle behavior see, e.g., [18,19] and references therein and [20] for its control by external driving fields. The self-trapping transition occurs at g = −v/N s and is connected to a bifurcation of the stationary states, the fixed points of the Hamiltonian (5), in the mean-field approximation.…”

Quantumq-breathers in a finite Bose-Hubbard chain: The case of two interacting bosons

“…In many cases quantum dynamics is important. Quantum breathers consist of superpositions of nearly degenerate many-quanta bound states, with very long times to tunnel from one lattice site to another [4,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. Remarkably quantum breathers, though being extended states in a translationally invariant system, are characterized by exponentially localized weight functions, in full analogy to their classical counterparts.…”

Dynamical Instability in a Trimeric Chain of Interacting Bose-Einstein Condensates

“…According to the standard mean-field treatment, which is fairly satisfactory when the average well populations are large [13], the dimer dynamics is integrable [14]. The latter is described by two macroscopic complex variables z i = |z i | exp(iϑ i ), accounting for the condensates' state (phase ϑ i and population |z i | 2 ), and exhibits two constants of motion, namely the total boson number and the energy.…”