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- Questions about optical photon quantum computer

Hi. I'm learning the optical photon quantum computer from Nielsen's QCQI. Since I'm not familiar with quantum optics, I got some questions about it.

But, if I remember correctly, the mean energy should be ##\left < \alpha \right | H \left | \alpha \right > ##, the result is the same. Am I right?

However, by applying the definition of the coherent state, I got ##\left | \alpha = \sqrt {0.1} \right > =0.9512 \left | 0 \right > +0.3008 \left | 1 \right > +0.06726 \left | 2 \right > + \cdots##, which doesn't match the above expression. Where am I wrong?

Also, I have no idea how the two probabilities come out.

But I think, the time evolution should be ##P \left | 1 \right > =e^{-i \omega \Delta} \left | 1 \right > ##, so where did the ##\omega## go?

Then I tried to expand the expression of ##P## using the Hamiltonian: ##P=exp \left ( -i HL/c_0 \right )=exp \left ( -i \left ( n_0 -n \right ) Z L/c_0 \right )=e^{iZ \Delta}##. Then how can it be related to ##R_z \left ( \Delta \right )= e^{-iZ \Delta /2}##?

Still, I think the operation should be ##exp \left ( -i H t / \hbar \right )=exp \left [ \theta \left ( a^\dagger b - a b ^\dagger \right ) t/ \hbar \right ]##. Everything is fine, except where is the ##t / \hbar##? So in the beamsplitter, ##t / \hbar =1##?

Ok, I have post too many questions. Some of them may be silly. I'm looking forward to any helps. Thanks!

**Q1.**In page 288, the book reads: A laser outputs a state known as a coherent state ##\left | \alpha \right > = e^{- \left | \alpha \right | ^2 /2 } \sum_{n=0}^{\infty} \frac {\alpha ^ n} {\sqrt {n !}} \left | n \right >##, where ##\left | n \right >## is an ##n##-photon energy eigenstate. Then the mean energy is ##\left < \alpha \right | n \left | \alpha \right > = \left | \alpha \right | ^2##.But, if I remember correctly, the mean energy should be ##\left < \alpha \right | H \left | \alpha \right > ##, the result is the same. Am I right?

**Q2.**Still about the coherent state. The book reads: For example, for ##\alpha = \sqrt {0.1}##, we obtain the state ##\sqrt {0.90} \left | 0 \right > + \sqrt {0.09} \left | 1 \right > + \sqrt {0.002} \left | 2 \right > + \cdots##. Thus if light ever makes it through the attenuator, one know it is a single photon with probability better than 95%; the failure probability is thus 5%. After calculation, the state above is ##0.6154\left | 0 \right > +0.3 \left | 1 \right > +0.04472 \left | 2 \right > + \cdots##.However, by applying the definition of the coherent state, I got ##\left | \alpha = \sqrt {0.1} \right > =0.9512 \left | 0 \right > +0.3008 \left | 1 \right > +0.06726 \left | 2 \right > + \cdots##, which doesn't match the above expression. Where am I wrong?

Also, I have no idea how the two probabilities come out.

**Q3.**About the phase shifter, the book reads: A phase shifter ##P## acts just like normal time evolution, but at a different rate, and localized to only the modes going through it. ... . it takes ##\Delta = \left ( n-n_0 \right ) L / C_0## more time to propagate a distance ##L## in a medium with index of refraction ##n## than in vacuum. For example, the action of ##P## on the vacuum state is to do nothing: ##P \left | 0 \right > =\left | 0 \right >##, but on a single photon state, one obtains ##P \left | 1 \right > = e^{i \Delta} \left | 1 \right >##.But I think, the time evolution should be ##P \left | 1 \right > =e^{-i \omega \Delta} \left | 1 \right > ##, so where did the ##\omega## go?

**Q4.**Still about the phase shifter, the books reads: ##P## performs a transforms ... nothing more than a rotation ##R_z \left ( \Delta \right )= e^{-iZ \Delta /2}##. One can thus think of ##P## as resulting from time evolution under the Hamiltonian ##H=\left ( n_0 -n \right ) Z##, where ##P=exp \left ( -i HL/c_0 \right )##.Then I tried to expand the expression of ##P## using the Hamiltonian: ##P=exp \left ( -i HL/c_0 \right )=exp \left ( -i \left ( n_0 -n \right ) Z L/c_0 \right )=e^{iZ \Delta}##. Then how can it be related to ##R_z \left ( \Delta \right )= e^{-iZ \Delta /2}##?

**Q5.**About the Beamsplitter, the book reads: The Hamiltonian is ##H_{bs}=i \theta \left( a b^\dagger - a ^ \dagger b \right )##, and the beamsplitter performs the unitary operation ##B=exp \left [ \theta \left ( a^\dagger b - a b ^\dagger \right ) \right ]##.Still, I think the operation should be ##exp \left ( -i H t / \hbar \right )=exp \left [ \theta \left ( a^\dagger b - a b ^\dagger \right ) t/ \hbar \right ]##. Everything is fine, except where is the ##t / \hbar##? So in the beamsplitter, ##t / \hbar =1##?

Ok, I have post too many questions. Some of them may be silly. I'm looking forward to any helps. Thanks!