The multidimensional transporter device is a device from the mirror universe capable of transporting a person from one universe into another.
In 2371, Miles 'Smiley' O'Brien used the device to take Benjamin Sisko back to the mirror universe. (DS9: "Through the Looking Glass")
The mirror Jennifer Sisko visited the primary universe a year later using another device. She brought a device for Benjamin Sisko to join her and Jake Sisko in the mirror universe. It was programmed to only allow Sisko to use it. (DS9: "Shattered Mirror")
In 2374 the mirror Bareil Antos used the device to transport to the primary universe to steal an orb. (DS9: "Resurrection")
Miles O'Brien somehow obtained schematics for this device which he gave to Rom to study. In 2375, Rom left a PADD out containing the schematics; Grand Nagus Zek found the PADD and duplicated the interdimensional transporting process, entering the mirror universe in hopes of establishing commercial relations. (DS9: "The Emperor's New Cloak")