- #1
jamie.j1989
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Let's say we have a system of spatially separated traps and two-level atoms where the levels are denoted as ##|a_i\rangle## and as ##|b_i\rangle##, and the subscript ##i## labels the trap number. The traps are initially loaded with ##N_i## atoms in the state ##|a\rangle##. If the two levels are internal states of an atom and all the atoms sit in the ground state of the traps I can write the state of the composite system as a tensor product of Fock states
$$|\Psi\rangle_1=|N_1,a_1\rangle\otimes|N_2,a_2\rangle.$$
Now we apply a beam splitter (pi/2 pulse) to this system and split the individual traps into 50/50 superpositions of the internal states.
$$|\Psi\rangle_2=\frac{1}{2}\left(|N_1/2,a_1\rangle+|N_1/2,b_1\rangle\right)\otimes\left(|N_2/2,a_2\rangle+|N_2/2,b_2\rangle\right)$$
Which can be expanded as
$$|\Psi\rangle_2=\frac{1}{2}\left(|\frac{N_1}{2},a_1\rangle\otimes|\frac{N_2}{2},a_2\rangle+|\frac{N_1}{2},a_1\rangle\otimes|\frac{N_2}{2},b_2\rangle+|\frac{N_1}{2},b_1\rangle\otimes|\frac{N_2}{2},a_2\rangle+|\frac{N_1}{2},b_1\rangle\otimes|\frac{N_2}{2},b_2\rangle\right)$$
I have two questions regarding this setup.
1) Am I only allowed to write the initial state ##|\Psi\rangle_1## in terms of Fock states if I know the exact number of atoms beforehand? For example, if I just have a procedure which loads these atoms into a trap, and I am not sure each time I load the traps precisely how many atoms are in each trap can I write this? Or can I only write it after I make some measurement on the number of atoms which would project what I think would be a coherent state into a Fock state?
2) How do operators act on the state ##|\Psi\rangle_2##, for example, if I have the particle annihilation and creation operators for each level and trap as, ##\alpha_i,\beta_i## and ##\alpha_i^\dagger,\beta_i^\dagger## repectively. How does ##\alpha_1^\dagger\alpha_1|N_1/2,b_1\rangle\otimes|N_2/2,b_2\rangle## behave? It seems like there are two options, being
$$\alpha_1^\dagger\alpha_1|N_1/2,b_1\rangle\otimes|N_2/2,b_2\rangle=0$$
or
$$\alpha_1^\dagger\alpha_1|N_1/2,b_1\rangle\otimes|N_2/2,b_2\rangle=|N_1/2,b_1\rangle\otimes|N_2/2,b_2\rangle$$
If the second is the case I don't know how to interpret a result such as the expectation of ##\alpha_1^\dagger\alpha_1##, which would be
$$_2\langle\alpha_1^\dagger\alpha_1\rangle_2=\frac{1}{4}\left(N_1+2\right)$$
Why would it be ##N_1/4+1/2##, this seems incorrect?
$$|\Psi\rangle_1=|N_1,a_1\rangle\otimes|N_2,a_2\rangle.$$
Now we apply a beam splitter (pi/2 pulse) to this system and split the individual traps into 50/50 superpositions of the internal states.
$$|\Psi\rangle_2=\frac{1}{2}\left(|N_1/2,a_1\rangle+|N_1/2,b_1\rangle\right)\otimes\left(|N_2/2,a_2\rangle+|N_2/2,b_2\rangle\right)$$
Which can be expanded as
$$|\Psi\rangle_2=\frac{1}{2}\left(|\frac{N_1}{2},a_1\rangle\otimes|\frac{N_2}{2},a_2\rangle+|\frac{N_1}{2},a_1\rangle\otimes|\frac{N_2}{2},b_2\rangle+|\frac{N_1}{2},b_1\rangle\otimes|\frac{N_2}{2},a_2\rangle+|\frac{N_1}{2},b_1\rangle\otimes|\frac{N_2}{2},b_2\rangle\right)$$
I have two questions regarding this setup.
1) Am I only allowed to write the initial state ##|\Psi\rangle_1## in terms of Fock states if I know the exact number of atoms beforehand? For example, if I just have a procedure which loads these atoms into a trap, and I am not sure each time I load the traps precisely how many atoms are in each trap can I write this? Or can I only write it after I make some measurement on the number of atoms which would project what I think would be a coherent state into a Fock state?
2) How do operators act on the state ##|\Psi\rangle_2##, for example, if I have the particle annihilation and creation operators for each level and trap as, ##\alpha_i,\beta_i## and ##\alpha_i^\dagger,\beta_i^\dagger## repectively. How does ##\alpha_1^\dagger\alpha_1|N_1/2,b_1\rangle\otimes|N_2/2,b_2\rangle## behave? It seems like there are two options, being
$$\alpha_1^\dagger\alpha_1|N_1/2,b_1\rangle\otimes|N_2/2,b_2\rangle=0$$
or
$$\alpha_1^\dagger\alpha_1|N_1/2,b_1\rangle\otimes|N_2/2,b_2\rangle=|N_1/2,b_1\rangle\otimes|N_2/2,b_2\rangle$$
If the second is the case I don't know how to interpret a result such as the expectation of ##\alpha_1^\dagger\alpha_1##, which would be
$$_2\langle\alpha_1^\dagger\alpha_1\rangle_2=\frac{1}{4}\left(N_1+2\right)$$
Why would it be ##N_1/4+1/2##, this seems incorrect?
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