Cirac - Zoller Scheme

Suppose we have cooled the ions situated along the z-axis so that their xy motion is minimized via
Laser cooling. However they are still free to move along the z-axis. Image the set of ions all moving
in unison so that the center of mass undergoes harmonic motion. In the quantum realm the excitations of this motion are called phonons. For vibrations with lowest energy is called the ground
state and no phonons are present, the next highest energy state has one phonon, and so on. We
can describe this by a ladder like stucture of energy levels

The ions are sufficently apart from each other so that they can be addressed by a laser individually.
The laser can excite, or de-excite the ion from its internal ground state |g⟩ to its excited internal
state |e⟩. A laser can also be used to add or emit a phonon. Then the internal ground state
of an ion can have several energy states 

|g ⟩|0⟩_ν     Ground internal state+no phonons    Energy = (Subscript[E, g])
|g ⟩|1⟩_ν     Ground internal state+1 phonon        Energy = (Subscript[E, g]) + hν
|g ⟩|2⟩_ν     Ground internal state+2 phonons       Energy = (Subscript[E, g]) + 2 hν
⋮
|e ⟩|0⟩_ν     Excited internal state+no phonons    Energy = (Subscript[E, g]) + ℏ ω
|e ⟩|1⟩_ν     Excited  internal state+1 phonon        Energy = (Subscript[E, g]) + ℏ ω+  hν
|e ⟩|2⟩_ν     Excited  internal state+2 phonons       Energy = (Subscript[E, g]) +  ℏ ω + 2 hν
⋮

Note we assume that the internal state energy difference ℏ ω >> hν

For a given ion m we let | g⟩ represent  the Qbit |0⟩  and |e ⟩  the Qbit |1⟩
so if we have a group of four ions and the first two are in the ground internal state
and the the last two are in an internal excited state they would represent the
computational basis state

|0011⟩

For a string of ions lets consider the m'th ion and label its internal states as
|g⟩_m  |e⟩_m  ( |0⟩_m  |1⟩_m).  Likewise for the n'th ion we use teh notation |g⟩_n  |e⟩_n  .

Imagine the system starting out as

|g⟩_m|g⟩_n |0⟩_ν     total energy  2 (Subscript[E, g])

Let' shine a laser on ion m with a frequency of  ω -hν.  This frequency is not sufficient
to excite the ion into its excited state |e⟩_m because you need at least an energy  ℏω to that
so the effect of this laser pulse, which we call shorthand (Subscript[U, m ])on this state is

(Subscript[U, m ])|g⟩_m|g⟩_n |0⟩_ν ⟶ |g⟩_m|g⟩_n |0⟩_ν

Likewise

(Subscript[U, m ])|g⟩_m|e⟩_n |0⟩_ν ⟶ |g⟩_m|e⟩_n |0⟩_ν

However

(Subscript[U, m ])|e⟩_m|g⟩_n |0⟩_ν ⟶ -i |g⟩_m|g⟩_n |1⟩_ν

That is the if (Subscript[U, m ]) acts on state |e⟩_m|g⟩_n |0⟩_ν it changes to a new state where the m'th Qubit changes
its value form 1  to 0 while a phonon is created. The energy of the initial configuration
|e⟩_m|g⟩_n |0⟩_ν   is  ℏω + 2(Subscript[E, g])  whereas the energy of
|g⟩_m|g⟩_n |1⟩_ν   is 2(Subscript[E, g]) + hν   and so the difference in energy is ℏω-hν  which is exactly the energy 
of the laser pulse photon applied on ion m. Thus by energetics alone we see why
(Subscript[U, m ])|e⟩_m|g⟩_n |0⟩_ν ⟶ -i |g⟩_m|g⟩_n |1⟩_ν is possible. But what about the -i factor, where does that come from ?  It turns out we can adjust the polarization properties of the laser in order the -i phase factor
appears (Deeper discussion of this requires a better understanding of quantum optics, which is beyond the scope of this course).
Finally  we also have

(Subscript[U, m ])|e⟩_m|e⟩_n |0⟩_ν ⟶ -i |g⟩_m|e⟩_n |1⟩_ν


In summary, after the first pulse

(Subscript[U, m ])|g⟩_m|g⟩_n |0⟩_ν ⟶ |g⟩_m|g⟩_n |0⟩_ν
(Subscript[U, m ])|g⟩_m|e⟩_n |0⟩_ν ⟶ |g⟩_m|e⟩_n |0⟩_ν
(Subscript[U, m ])|e⟩_m|g⟩_n |0⟩_ν ⟶ -i |g⟩_m|g⟩_n |1⟩_ν
(Subscript[U, m ])|e⟩_m|e⟩_n |0⟩_ν ⟶ -i |g⟩_m|e⟩_n |1⟩_ν


We now conisder a similar laser pulse, which we call (Subscript[V, n ]),  on ion n (this will have diffrenet polarization properties than the aforementioned pulse). We then have

(Subscript[V, n ])|g⟩_m|g⟩_n |0⟩_ν ⟶ |g⟩_m|g⟩_n |0⟩_ν
(Subscript[V, n ])|g⟩_m|e⟩_n |0⟩_ν ⟶ |g⟩_m|e⟩_n |0⟩_ν
(Subscript[V, n ])|g⟩_m|g⟩_n |1⟩_ν ⟶ - |g⟩_m|g⟩_n |1⟩_ν
(Subscript[V, n ])|g⟩_m|e⟩_n |1⟩_ν ⟶  |g⟩_m|e⟩_n |1⟩_ν

For each state in the first collumn the photon does not have enough energy to change the
state of Qbit n. 

Now we repeat pulse (Subscript[U, m]) once again so that 

(Subscript[U, m ]) |g⟩_m|g⟩_n |0⟩_ν  ⟶  |g⟩_m|g⟩_n |0⟩_ν
(Subscript[U, m ])|g⟩_m|e⟩_n |0⟩_ν  ⟶  |g⟩_m|e⟩_n |0⟩_ν
(Subscript[U, m ]) |g⟩_m|g⟩_n |1⟩_ν  ⟶  -i |e⟩_m|g⟩_n |0⟩_ν 
(Subscript[U, m ]) |g⟩_m|e⟩_n |1⟩_ν ⟶    i |e⟩_m|e⟩_n |0⟩_ν
 
 The three-pulses in this sequence  W = (Subscript[U, m ])(Subscript[V, n ]) (Subscript[U, m ]) then lead to the following
 transformations
 
 W |g⟩_m|g⟩_n |0⟩_ν ⟶ |g⟩_m|g⟩_n |0⟩_ν
 W |g⟩_m|e⟩_n |0⟩_ν ⟶ |g⟩_m|e⟩_n |0⟩_ν
 W |e⟩_m|g⟩_n |0⟩_ν ⟶ |e⟩_m|g⟩_n |0⟩_ν
 W |e⟩_m|g⟩_n |0⟩_ν ⟶ - |e⟩_m|e⟩_n |0⟩_ν
 
 Notice that all states return to their original configuartions, except the last state which
 picks up a negative sign.

Consider the  superposition 

(1/(Sqrt[2]))( |g⟩_n ±  |e⟩_n) 

According to the above table

W  |g⟩_m(1/(Sqrt[2]))( |g⟩_n ±  |e⟩_n) |  |0⟩_ν = |g⟩_m(1/(Sqrt[2]))( |g⟩_n ±  |e⟩_n) |  |0⟩_ν

W  |e⟩_m(1/(Sqrt[2]))( |g⟩_n ±  |e⟩_n) |  |0⟩_ν =  |e⟩_m(1/(Sqrt[2]))( |g⟩_n ∓  |e⟩_n) |  |0⟩_ν

So lets define the gate

(Subscript[H, n]) W (Subscript[H, n  ]) here (Subscript[H, n]) is the Hadamard gate acting on Qbit n.  Therefore

(Subscript[H, n]) W (Subscript[H, n  ])|g⟩_m |g⟩_n |0⟩_ν = (Subscript[H, n]) W |g⟩_m (1/(Sqrt[2]))( |g⟩_n +  |e⟩_n)  |0⟩_ν = 
(Subscript[H, n]) |g⟩_m (1/(Sqrt[2]))( |g⟩_n +  |e⟩_n)  |0⟩_ν = |g⟩_m|g⟩_n |0⟩_ν

(Subscript[H, n]) W (Subscript[H, n  ])|g⟩_m |e⟩_n |0⟩_ν = (Subscript[H, n]) W |g⟩_m (1/(Sqrt[2]))( |g⟩_n -  |e⟩_n)  |0⟩_ν = 
(Subscript[H, n]) |g⟩_m (1/(Sqrt[2]))( |g⟩_n -  |e⟩_n)  |0⟩_ν = |g⟩_m| e ⟩_n |0⟩_ν

(Subscript[H, n]) W (Subscript[H, n  ])|e⟩_m |g⟩_n |0⟩_ν = (Subscript[H, n]) W |e⟩_m (1/(Sqrt[2]))( |g⟩_n +  |e⟩_n)  |0⟩_ν = 
(Subscript[H, n]) |e⟩_m (1/(Sqrt[2]))( |g⟩_n -  |e⟩_n)  |0⟩_ν = |e⟩_m| e ⟩_n |0⟩_ν

(Subscript[H, n]) W (Subscript[H, n  ])|e⟩_m |e⟩_n |0⟩_ν = (Subscript[H, n]) W |e⟩_m (1/(Sqrt[2]))( |g⟩_n -  |e⟩_n)  |0⟩_ν = 
(Subscript[H, n]) |e⟩_m (1/(Sqrt[2]))( |g⟩_n +  |e⟩_n)  |0⟩_ν = |e⟩_m| g ⟩_n |0⟩_ν

In summary

(Subscript[H, n]) W (Subscript[H, n  ])|g⟩_m |g⟩_n |0⟩_ν = |g⟩_m|g⟩_n |0⟩_ν
(Subscript[H, n]) W (Subscript[H, n  ])|g⟩_m |e⟩_n |0⟩_ν = |g⟩_m| e ⟩_n |0⟩_ν
(Subscript[H, n]) W (Subscript[H, n  ])|e⟩_m |g⟩_n |0⟩_ν = |e⟩_m| e ⟩_n |0⟩_ν
(Subscript[H, n]) W (Subscript[H, n  ])|e⟩_m |e⟩_n |0⟩_ν = |e⟩_m| g ⟩_n |0⟩_ν

Thus we notice that the sequence of operations
(Subscript[H, n]) W (Subscript[H, n  ])is equivalent to a CNOT gate i.e. Qbit n is flipped if Qbit m is in state |e⟩
but is left alone if Qbit m is in state |g⟩.

Hyperlink["http://heart-c704.uibk.ac.at/publications/papers/qip04_schmidtkaler.pdf"]

(* Look at ~/week10/nature.pdf *)