Composite traps with two-level atoms

[jamie.j1989]

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?

 

[eljose79]

I would like to commend you on your thorough and thoughtful questions. Let me address them one by one.

1) You are correct in your understanding that the initial state can only be written in terms of Fock states if the number of atoms in each trap is known beforehand. If you are unsure of the exact number of atoms in each trap, then the state would be described by a superposition of Fock states rather than a single Fock state. This means that you would need to make a measurement on the number of atoms in order to project the state into a single Fock state. However, this does not mean that you cannot write the initial state as a tensor product of Fock states. It simply means that the coefficients of the Fock states would be unknown until a measurement is made.

2) The way operators act on the state $|\Psi\rangle_2$ depends on the specific operator and the specific state. In general, operators act linearly on states, so they can be distributed over a tensor product. For example, let's consider the action of $\alpha_1^\dagger\alpha_1$ on the state $|N_1/2,b_1\rangle\otimes|N_2/2,b_2\rangle$. This can be written as

$$\begin{align*}\alpha_1^\dagger\alpha_1|N_1/2,b_1\rangle\otimes|N_2/2,b_2\rangle &= \alpha_1^\dagger\left(\frac{1}{2}\right)^{1/2}\left(|N_1,b_1\rangle+|N_1,a_1\rangle\right)\otimes|N_2/2,b_2\rangle \\ &= \frac{1}{2}\left(\alpha_1^\dagger|N_1,b_1\rangle+ \alpha_1^\dagger|N_1,a_1\rangle\right)\otimes|N_2/2,b_2\rangle \\ &= \frac{1}{2}\left(\sqrt{N_1+1}|N_1+1,b_1\rangle+ \sqrt{N_1}|N_1-1,a_1\rangle\right)\otimes|N_2/2,b_2\rangle \\ &= \frac