Will Bamboo integrate with Adobe to automate deployments?

You can have a look at this community article :- https://devpost.com/software/bamboo-aem-plugin#updates

//  Bamboo AEM plugin with short documentations as well.

    Download link:- https://code.tricode.nl/svbeek/bamboo-crx-plugin

Aslo check this post:-

Link:- http://help-forums.adobe.com/content/adobeforums/en/experience-manager-forum/adobe-experience-manager.topic.html/forum__knsj-i_am_tryingtowork.html

// Both Jenkins and Bamboo will get the job done. If cost is concern Jenkins is open source. However, I recommend Bamboo over Jenkins, if the team is using Jira and/or the rest of the Atlassian stack. I personally find a lot of value in Bamboo's ability to automatically generate a pipeline per branch.  At scale this permits a githubflow style of branch management. 

You should really consider the new bread build tools such as Wercker, Circle-CI and Travis which forgo the complex user interfaces in favor of a simple YML configuration file.

I'd also highly recommend Ansible for managing the provisioning and deployment of AEM infrastructure. Having your build server execute ansible playbooks really help makes things so much more manageable. Puppet, Chef are possible of course, the main point is to automate everything. Of course Ansible has Ansible Tower as a UI to help out.

More reference articles :- 

Link:- http://help-forums.adobe.com/content/adobeforums/en/experience-manager-forum/adobe-experience-manager.topic.html/forum__ig9w-hi_all_in_mypro.html

Link:- http://help-forums.adobe.com/content/adobeforums/en/experience-manager-forum/adobe-experience-manager.topic.html/forum__snsa-hi_am_planningt.html

~kautuk