Actually there's already one filed that you can vote on: Bug 26239 - Tray icon for controlling Zimbra Desktop service
What's happening in the background is all of your mail is continuing to sync that way when you open a web-browser again it's all ready to use.
Closing the prism browser or other web-client doesn't halt the service. This convo was just had over at: http://www.zimbra.com/forums/zimbra-...hone-home.html
For instance if you really want to halt it:
Windows: run > services.msc & stop the process.
Mac: launchctl stop com.zimbra.zdesktop & launchctl start com.zimbra.zdesktop
Linux: ./zdesktop stop & ./zdesktop start