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We consider a particle in a spatial symmetric/asymmetric potential driven by time periodic bichromatic AC fields of ratchet type. The associated time-dependent Schrödinger equation is conveniently tackled with the Floquet theory. We next proceed to investigate the ratchet effect induced by the driver, comparing the symmetric with the asymmetric cases. It turns out that the current in the asymmetric case is stronger than that of the symmetric one. Besides, we also investigate the case where the driver is a delta kicked acting on our spatial potential with more emphasis on its chaotic behaviour. Here we check that the current emerges as the phase space is mixed and that the system with asymmetric spatial potential becomes more chaotic than the symmetric one at low kicking strength.

The ratchet effect is the possibility of generating directed current from a fluctuating environment in the absence of external unbalanced forces or any perturbation of any sort [

This fluctuating environment is usually being induced by noisy dynamics governed by thermal fluctuations. This thermal fluctuation is usually brought about by nonlinear interaction between molecules. The Brownian ratchet has recently been discovered to be such a system that takes advantage of this noise as it converts random fluctuations into directed motion in the absence of external forces [

To observe the appearance of a DC current in a fluctuating environment of a dynamical system, the breaking of symmetries, that is, the spatial and temporal symmetries, is a necessary but not a sufficient condition. It has recently been shown that, in the area of atomic physics, directed transport has been achieved in driven ratchets with only the temporal symmetry being broken by dissipative processes [

Optical lattices are artificial crystals of light, consisting of hundreds and thousands of optical microtraps (potential wells) created by interfering optical laser beams. These optical lattices act as potential wells to trap cold quantum particles like Bosons and Fermions. Cold atoms in optical lattices provide an excellent experimental demonstration of the phenomenon of dynamical localization [

Let us start this section with a cloud of atoms equally populating the wells of double-well periodic potential exposed to periodically driven two-color lasers. The cloud of atoms is said to be at low density [

If the Floquet modes and quasienergies are known for a particular Hamiltonian

Now substituting for

This eigenvalue problem can be solved analytically or numerically.

The Floquet states

Now let us take a look at the time evolution operator for Schrödinger equation (

The Floquet modes at arbitrary time

By using the gauge transformation [

Integrating the above equation, we have that

A section of the Floquet mode spectrum of asymmetric potential with values of the quasienergies on the top right-hand side of the graph (see Figure

A section of the Floquet mode spectrum of asymmetric potential under periodic perturbation (see Figure

Solving the equation of motion for our Hamiltonian

This gives us

The current

For the purpose of simulation, we will restrict ourselves to the case where

In Figure

In Figure

In Figure

In Figure

The quantum Hamiltonian ratchet is based on the kicked rotor model. In this section, we are going to investigate a kicked rotor model for the Hamiltonian. By “kick” we mean a periodic impulse as forcing term [

The standard map is an area-preserving map of two generalised variables: position

Figure

In Figure

In Figure

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In Figure

In the asymmetric case, full chaos is achieved at

To conclude, we have studied the ratchet effect of cold atoms in periodically symmetric and asymmetric potentials driven by periodic bichromatic AC fields where it was found that the Floquet method has been preferred because of the periodicity of the time-dependent Schrödinger equation. In the classical limit, we show that the rectification effect of the ratchet current in asymmetric potential was more than that for a symmetric one. This result can be used to explain the concept of DC current enhancement due to incoherent electron tunnelling in an asymmetric potential, while, in the kicked ratchet, however, we found that the mix phase space of asymmetric potential becomes more chaotic at low kicking strength than of the symmetric case which equally explains why there is a great variation in their DC currents.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to extend their sincere thanks to AIMS-NEI network and AIMS-Cameroon in particular for their wonderful financial and material support and also wish to knowledge God for His strength and inspirations.