In this note I will summarize the works by Polzik’s group. I will give the detials of the theoretical treatment in their works.
Atoms are good at storing and processing quantum information because of its long coherent time. Light is suitable for long-distance quantum state transform. Hence, the coupling between atomic ensemble and light is the key factor for realizing quantum network.
As the number of ensemble is huge, we can treat it continuously, just as we treat light. Because of the collective coupling, the quantum information processes, such as teleportation, can be archieved without cavity, which is greatly decreased the difficulty of the experiments.
1.1. continuous discription of light and matter
Let’s firstly discuss the light/matter interaction. We will use continuous discription of the light and matter. We quantized electromagnetic field in one dimension and only care about on polarization mode (x-polerization)
are continuous, have units of square root meters. The commutation is
The field Hamiltonian is
Now let’s transform the electric field into the spatial description.
where the commutation is . The dimension of and is . means the number of photons in the space between and .
1.2. Maxwell-Bloch equation
The operator ‘s motion follows Heisenberg equation. The Hamiltonian is . represents the Hamiltonian of pure radiation field. is the Hamiltonian of matter. is the interaction Hamiltonian between light and matter. Therefore we get
This is what to say Maxwell-Bloch equation, which describes the light field affected by atoms through the interaction .
In free space, we get time and space dependence . Let’s define , is the flux of photons at time . For the light in the vacuum state we get
1.3. Continuous Matter Operators
where is the dipole operator for the ‘th atom, is the location of the ‘th atom.
where . Together with the Heisenberg equation of motion for , we have the coupled quantum Maxwell-Bloch equations describing light/matter interaction.
We have following commutation
The dimension of is .
2. QND measurement induced entanglement
In this section we will discuss the scheme to realize QND measurement on matter and using the measurement to generate entanglement between atomic ensembles. The results we get in the last section will be widely used in this section.
The level struction of the atoms and schematic setup is shown in Lu-Ming Duan et al. PRL 85, 5643(2000).
The atomic ensemble is of a pencil shape with Fresnel number . The input laser pulse is linearly polarized with to orthogonal mode.
where is the frequency of the laser. is the photon number of th mode in the area . The dimension of is . Suppose that . The number of photons is .
For ouer coherent input, , . Because , we can treat classically. We define , . We get .
We use continuous atomic operators,
where . The Hamiltonian is
where . In the large detuing limit, we can adiabatically eliminated the upper state population from the equation of motion.
where is the number of atoms. Neglect the constant energy part, which only induces the global phase, The effective Hamiltonian is
The Heisenberg equation is
Here we neglect the loss terms, which is small.
Using input-output relation (20), we get
where . Is is easy to get the viarance of the operators
Here we use the fluctuation .
Suppose that , we get
After measurement on , the pile up noise in the variable is . The uncertainty relation must be hold on. Therefore, . We rotate the collective atomic spins, . Then we measure again. The QND measurement will be on and makes it squeezed, . We have prepared a two-mode squeezed state. The squeezing parameter is given by