Quantum lattice solitons

Abstract

The number state method is used to study soliton bands for three anharmonic quantum lattices: i) The discrete nonlinear Schrödinger equation, ii) The Ablowitz-Ladik system, and iii) A fermionic polaron model. Each of these systems is assumed to have f -fold translational symmetry in one spatial dimension, where f is the number of freedoms (lattice points). At the second quantum level (n = 2) we calculate exact eigenfunctions and energies of pure quantum states, from which we determine binding energy (E b ), effective mass (m * ) and maximum group velocity (Vm) of the soliton bands as functions of the anharmonicity in the limit f → ∞. For arbitrary values of n we have asymptotic expressions for E b , m * , and Vm as functions of the anharmonicity in the limits of large and small anharmonicity. Using these expressions we discuss and describe wave packets of pure eigenstates that correspond to classical solitons.

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

“…This is a quantum version of the discrete nonlinear Schrödinger equation, which has been used to describe a great variety of systems [34]. The BH Hamiltonian is given by [35] …”

Quantum many-body dynamics of dark solitons in optical lattices

“…Below, we make connections with these works in our study. An early study of quantum lattice solitons is presented in [44]; a quantum theory of solitons in optical fibers was developed by Lai and Haus [45].…”

Boundary effects on quantum -breathers in a Bose–Hubbard chain

“…The eigenenergies are the sum of the two single-particle energies: When γ > 0, eigenvalues in the lower part of the spectrum are pushed down, and beyond γ ≈ 2 a band of f states splits off from the two-boson continuum. These are the two-boson bound states, with a high probability of finding the two bosons on the same lattice site, while the probability of them being separated by a distance r decreases exponentially with increasing r [16,15,14]. The critical value γ b = 2 for which the band of twoboson bound states splits off from the continuum may be explained as follows.…”

“…To describe quantum states, we use a number state basis |Φ n = |n 1 n 2 · · · n f [16], where n i = 0, 1, 2 represents the number of bosons at the i-th site of the lattice. |Φ n is an eigenstate of the number operatorN with eigenvalue n = f j=1 n j .…”

“…This is a quantum version of the discrete nonlinear Schrödinger equation, which has been used to describe a great variety of systems [15]. The BH Hamiltonian iŝ H =Ĥ 0 + γĤ 1 [16], witĥ…”