Generating mesoscopic entanglement of coherence and squeezed states in circuit QED system

Distributing entangled state using quantum repeater protocol: Trapped atomic ions in optomechanical cavities

“…Eight trapped ions labeled by (1, 2, • • • , 8) were considered, where the pairs (i, i + 1) with i = 1, 3, 5, 7 were initially prepared in Bell-like states. The entanglement between pairs (1,4) as well as (5,8) was produced by performing "interaction" between pairs (2,3) and (6,7) in two separate OMCs and then via operating proper measurements on trapped ions (2, 3) and (6, 7) (these measurements require appropriate projection operators). Finally, the entanglement was swapped between trapped ions (1,8) after performing the projective measurement with the appropriate Bell-like states on the states of trapped ions (1,4,5,8) in the last step of the introduced protocol.…”

“…Therefore, there exist two key points herein: initial entangled pairs and entanglement swapping process. At first, the entanglement can be generally generated by performing interaction [3][4][5][6][7][8][9] or using beam splitter [10,11]. Secondly, this generated entanglement can be swapped (to the separable pairs) using Bell state measurement (BSM) [12][13][14], beam splitter [15] or via performing interaction between independent particles (QED method) [16].…”

“…where the pairs (1, 2) and (3,4) are supposed to be prepared in the Bell-like states as below: Even though various entangled internal states and entangled states of vibrational modes typically nearly to our considered entangled state have been previously generated between two trapped atomic ions using different methods, we ourselves tried to produce the necessary entangled state in our protocol in Eq. (2) (see Appendix A).…”

“…[53], a scheme has been proposed for entanglement generation between two vibrational modes of two ions. Then, to swap the entanglement to trapped atomic ions (1,4), two separable trapped atomic ions (2,3) are sent to an OMC with one movable mirror (see figure 2(a)), while the initial states of optical and mechanical modes are supposed to be in their vacuum states. In this regard, the interaction between a two-level trapped atomic ion inside a single-mode optical cavity in the Lamb-Dick regime as well as the first vibrational sideband in the rotating wave approximation has been previously introduced [54,55].…”

“…In this Hamiltonian ω c (ω M ) is the frequency of optical (mechanical) mode with â, â † ( b, b † ) as the annihilation and creation operators associated with optical (vibrational) mode. Also, the third term in Ĥ0 corresponds to the trapped atomic ions (2,3), where ν is the frequency of the vibrational mode of ith ion with ĉi , ĉ † i as its annihilation, creation operators. σ(i) z is the ionic population inversion (the atomic levels of trapped ions have been shown in figure 2(b)), wherein the frequency of ionic transition has been indicated by ω 0 .…”