Integration of Jenkins and Atlassian Bamboo
Yes, you can trigger a build on jenkins via the remote access api of jenkins. Fairly straight forward. All you need is a tool to set off a http post e.g. curl. (see here https://wiki.jenkins-ci.org/display/JENKINS/Remote+access+API)
Then you only need your bamboo and jenkins instance to agree on a shared directory of where to put the artifacts of that jenkins build afterwards.