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arXiv:1106.0063 · Journal of Physics B Atomic Molecular and Optical Physics

Non-adiabatic transitions in a non-symmetric optical lattice

L. Morales-Molina, Sebastián A. Reyes

Published Sep 29, 2011• 9 citations
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Abstract

We study Landau-Zener interband transitions for a non-symmetric optical lattice in the presence of an external force. We show that gain and losses of the light beam, as well as the relative occupation probabilities of the bands involved in the transitions can be accurately managed upon tuning the amplitude of the non-Hermitian component of the lattice. Exact expressions for the transition and non-transition probabilities for a non-symmetric system obtained within a two-mode approximation are provided. These equations successfully account for the main features of the transitions in the optical lattice. We also interpret the non-conventional Bloch oscillations at criticality studied in Phys. Rev. Lett. 103, 123601 (2009) as a series of a Landau-Zener transitions.

DOI
10.1088/0953-4075/44/20/205403

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We study Landau-Zener interband transitions for a non-symmetric optical lattice in the presence of an external force. We show that gain and losses of the light beam, as well as the relative occupation probabilities of the bands involved in the transitions can be accurately managed upon tuning the amplitude of the non-Hermitian component of the lattice. Exact expressions for the transition and non-transition probabilities for a non-symmetric system obtained within a two-mode approximation are provided. These equations successfully account for the main features of the transitions in the optical lattice. We also interpret the non-conventional Bloch oscillations at criticality studied in Phys. Rev. Lett. 103, 123601 (2009) as a series of a Landau-Zener transitions.
Landau–Zener–Stückelberg interferometry in PT\mathcal{PT}PT-symmetric optical waveguides
We investigate theoretically the non-adiabatic transitions in a -symmetric lattice subject to a strong ac force. In an optical realization of the lattice, the curvature of the waveguides along the paraxial propagation direction plays the role of the external periodic force necessary to drive the system through an avoided level crossing. Analytical expressions for the transition probabilities after multiple passages are obtained within an effective two-mode approximation. We show that gain and losses of the light beam, as well as the relative occupation probabilities of the bands involved in the transitions, can be accurately managed upon tuning the parameters of the system and are particularly sensitive to the amplitude of the non-Hermitian component of the lattice. Numerical simulations for the complete system are found to agree very well with the approximate analytical results. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
We systematically investigate the non-Hermitian generalizations of the Landau-Zener (LZ) transition and the Landau-Zener-St\"uckelberg (LZS) interferometry. The LZ transition probabilities, or band populations, are calculated for a generic non-Hermitian model and their asymptotic behavior is analyzed. We then focus on non-Hermitian systems with a real adiabatic parameter and study the LZS interferometry formed out of two identical avoided level crossings. Four distinctive cases of interferometry are identified and the analytic formulas for the transition probabilities are calculated for each case. The differences and similarities between the non-Hermitian case and its Hermitian counterpart are emphasized. In particular, the geometrical phase originated from the sign change of the mass term at the two level crossings is still present in the non-Hermitian system, indicating its robustness against the non-Hermiticity. We further apply our non-Hermitian LZS theory to describing the Bloch oscillation in one-dimensional parity-time (PT)(\mathcal{PT})(PT) reversal symmetric non-Hermitian Su-Schrieffer-Heeger model and propose an experimental scheme to simulate such dynamics using photonic waveguide arrays. The Landau-Zener transition, as well as the LZS interferometry, can be visualized through the beam intensity profile and the transition probabilities measured by the center of mass of the profile.
We study the dynamics of ultracold atoms in an optical lattice under constant bias. After recapitulating the ideas underlying Bloch oscillations and Zener's formula for interband transitions, the Bloch-Zener scenario is tested by means of accurate numerical solutions to the time-dependent Schrödinger equation. It is shown how two shortcomings of the traditional Zener formula can be removed: the common weak-binding approximation can be circumvented by combining Kohn's insight into the structure of complex energy bands with the Dykhne-Davis-Pechukas description of transitions in terms of adiabatic excursions on analytically continued eigenvalue surfaces, and a usually neglected Stokes phenomenon comes into play when accounting for the finite width of the Brillouin zone. Treating Bose-Einstein condensates in optical lattices within the standard mean-field approximation at zero temperature, the ideal Bloch-Zener scenario turns out to be remarkably stable against the condensate's nonlinear self-interaction. Yet, under appropriate conditions a Bloch-oscillating Gross-Pitaevskii wavepacket reveals characteristic signatures of that nonlinearity, such as sudden phase jumps, slight shifts of the oscillation frequency or non-classical breathing modes. It is suggested that such experimentally detectable signatures will play an important role in future high-precision experiments aiming at the exploration of many-body dynamics in periodic potentials with condensates in optical lattices.
Bloch-Zener oscillations (BZO), i.e., the interplay between Bloch oscillations and Zener tunneling in two-band lattices under an external direct current (DC) force, are ubiquitous in different areas of wave physics, including photonics. While in Hermitian systems such oscillations are rather generally aperiodic and only accidentally periodic, in non-Hermitian (NH) lattices BZO can show a transition from aperiodic to periodic as a NH parameter in the system is varied. Remarkably, the phase transition can be either smooth or sharp, contrary to other types of NH phase transitions which are universally sharp. A discrete-time photonic quantum walk on a synthetic lattice is suggested for an experimental observation of smooth BZO phase transitions.
In the non-Hermitian Landau-Zener models, we investigate the dynamical transition both in parity-time symmetric and symmetry-broken regimes. Taking into account the complex nature of the energy of the non-Hermitian systems, the absolute value of the gap was used to determine the relaxation rate of the system. To show the dynamics of the phase transitions, the relative population is used to estimate the topological defect density in nonequilibrium phase transitions, rather than the excitations in the corresponding Hermitian systems. The result shows that the adiabatic-impulse approximation, which is fundamental to the Kibble-Zurek mechanism, may be adapted to the parity-time symmetric non-Hermitian Landau-Zener models to examine the dynamics near the critical point. The most basic non-Hermitian two-level models with an exact solution exhibiting the Kibble-Zurek mechanism are presented. It would be interesting to extend this scenario to quantum many-body models, such as the quantum phase transition in the Ising model.
The atomic current in the Fourier-synthesized optical lattices under a constant external force is investigated theoretically. Based on a two-band model, the atomic current is derived by solving the Boltzmann equations. We find that the stationary atomic current changes with the probability of Landau-Zener tunneling, depending on the adjustable energy structure of the optical lattices. In contrast to the classical results of an electron in superlattices given by the Esaki-Tsu equations, the relation between the stationary atomic current and the strength of the external force in optical lattices is modified significantly. Both these characteristics can be taken as an effective way to observe the Landau-Zener tunneling in the optical lattices.