There is no y dependence in your expression for the electric field, and hence there is no gradient force along the y-direction you are asking in this case. The expression of the electric field you gave is for a plane wave whose wave vector is in the xz plane. Any plane wave dose not change along the perpendicular direction of its propagation direction. This is kind of an ideal case because any beam of electric field has finite span in space, otherwise it will have infinite energy! If your electric field has finite span, for example a cylindrical beam with a Gaussian envelope, then your electric field has y dependence and you get a gradient force along y direction. I did not find the article you read but I guess they assumed finite span in space.

There is no $y$ dependence in your expression for the electric field, and hence there is no gradient force along the $y$-direction you are asking in this case. The expression of the electric field you gave is for a plane wave whose wave vector is in the xz plane. Any plane wave dose not change along the perpendicular direction of its propagation direction. This is kind of an ideal case because any beam of electric field has finite span in space, otherwise it will have infinite energy! If your electric field has finite span, for example a cylindrical beam with a Gaussian envelope, then your electric field has $y$ dependence and you get a gradient force along $y$ direction. I did not find the article you read but I guess they assumed finite span in space.