The Chachamim instituted bowing down in five places in the Amidah: in the beginning and end of Birkat Avot, in the beginning and end of Birkat Modim, and at the end of the Amidah, when one takes three steps backwards. They specifically chose those two berachot for they are the most important, and while reciting them one must try hard to concentrate properly (see Shulchan Aruch 101:1; Mishnah Berurah 3). If a person wants to bow down at the beginning or end of another berachah, he is taught not to, so as not to uproot the ruling of the Chachamim, and so that he will not look like an arrogant person who considers himself more righteous than others. However, in the middle of the berachot he is permitted to bow down (Shulchan Aruch 113:1; Mishnah Berurah 2).
One bows down when saying “Baruch Attah” and straightens himself when saying “Hashem.” At Modim, he bows down when he says “Modim Anachnu Lach” and straightens up when saying “Hashem” (Shulchan Aruch 113:7; Mishnah Berurah 12; for the laws on bowing down at the end of the Amidah, see further in this chapter, halachah 13).
The bow must be such that all the vertebrae in one’s spine move in place, one after the other, and the vertebrae protrude from his back. One bends his head and back until his face reaches the height between his heart and his waist, but he should not bend his head all the way until his belt, because that gives the appearance of arrogance (yohara). An elderly or ill person who has difficulty bending down lowers his head as much as he is able (Shulchan Aruch 113:5). One must bow down quickly to demonstrate his desire to bow before Hashem Blessed Be He, and when he straightens himself, he must do so slowly, as one who is interested in continuing to bow down before Him (Shulchan Aruch 113:6).
There are two minhagim regarding the manner of bowing. According to the minhag of the Ashkenazim, when one says “Baruch” he bends his knees and when he says “Attah” he bows until his vertebrae “click”. At Modim, in which the word “Baruch” is not recited, one bows immediately without initially bending his knees (Mishnah Berurah 113:12; and see Kitzur Shulchan Aruch 18:1). The Sephardim practice according to the Ari and bow down in two stages. First, a person bends his body (without bending his knees) and then his head. Similarly, when one straightens himself, first he straightens his body and afterwards his head (Kaf HaChaim 113:21).
Regarding bowing at Barchu, there are various customs. The crux of the issue lies in the reason chachamim taught not to bow at the aforementioned places. If the reason is because he is bowing at a place in which the Chachamim did not institute bowing, then it is forbidden to bow regularly at Barchu. That is what is written in Or L’Tzion, part 2, 5:13 and She’erit Yosef, part 2, p. 106, and that is the minhag of Sephardic rabbis; still, among the Sephardic communities, many are accustomed to bowing. However, if the reason not to bow is because by bowing at the expression of thanks (hoda’ah), he is demonstrating that he erred, (thinking that the word hoda’ah means prostration when it is really an expression of thanks), then if he is bowing in a different place in the prayer service in order to accept upon himself the yoke of Hashem’s kingdom, there is no flaw in that; therefore, he is permitted to bow regularly at Barchu. That is what is written in Shulchan Aruch HaRav 113:3, and that is the Ashkenazic custom. The Bei’ur Halachah brings support for this. In a place where all different Jewish ethnic groups pray together, it is proper that everyone bows slightly at Barchu so as not to magnify the division between the minhagim, for that is how most of Israel practices, thereby preserving the custom of prostration to a certain extent. Also, in that way, no additional bows are added to those instituted by the Chachamim, for this kind of bow is not a complete bow that requires the vertebrae in one’s spine to click.