According to the 1870 census record, Bertha was born about 1861 in Prussia, and Anna was born about 1864 in Minnesota. Therefore, the immigration date is estimated between 1861 and 1864.
The 1870 census is the first record of the family in the US that I could find. I have not found any naturalization records that could be confirmed.
I did find that Sophia made a reappearance in the 1885 MN census. The family was listed as "Geart"...Fredericka (51), John (16), and Sophia (76).