User:PerezPhysics/sandbox/Quantum Memory with Vapor Atoms

Quantum Memory with Vapor Atoms
Quantum memory in vapor atoms systems is the process of storing and retrieving quantum information using the principles of quantum mechanics. Light itself carries information, and this information is often contained in its quantum state. When light interacts with matter, its quantum state can be effectively be transferred to the atom under appropriate conditions. During this moment, the information has effectively been “saved” in the atom, and at a later time, this quantum state (or quantum information) can be retrieved. This is the main concept behind quantum memory, also illustrated in the figure below.

This technique of storing and transferring quantum information from light to an atomic media has many application ranging from quantum information processing, quantum computing, and it is also used as the basis for quantum repeaters, which is a technology used for quantum information.

Quantum memory with vapor atoms is a unique process, in that it takes advantage of the properties of atoms that have three energy levels, also know as a lambda ($$ \Lambda $$) configuration, as well as using the principles of Electromagnetically Induced Transparency(EIT). These two main properties of light and matter, allow for the easy and efficient storage and retrieval of quantum information in the coupled system.

A few of the main challenges in the research area of quantum memory, is improving sensitivity, and the storage time. And, the storage time, is in practice limited by the coherence time.

Theory behind quantum memory with vapor atoms
Under ideal EIT conditions, assuming that the optical fields only interact with their corresponding transition paths, then the interaction Hamiltonian for a three level system is given as

$$\hat{H} =\begin{pmatrix} -\hbar \omega_{13} & 0 & -\mu_{13}E_1\\ 0 & -\hbar \omega_{23} & -\mu_{23}E_2\\ -\mu_{13}E_1 & -\mu_{23}E_2 & 0 \\ \end{pmatrix} $$

where $$ \omega_{13}$$ and $$ \omega_{23} $$ are the energy of the corresponding frequencies, and $$ E_{1,2} = \tilde{E}_{1,2}e^{-I\nu_{1,2} t} + c.c  $$  are the field interacting with the corresponding atomic transitions. $$\mu_{1,2} $$ and $$ \mu_{2,3} $$  are the corresponding dipole moment of the atomic transitions. Since we are usually interesting in effects that happen near resonance, then we must change to the rotating frame, and apply the rotating wave approximation(RWA), in this case, the interaction Hamiltonian becomes

$$ \begin{align} \hat{H}_R = \begin{pmatrix} -\hbar \Delta_1 & 0 & -\mu_{13} \tilde{E}_1\\ 0 & -\hbar \Delta_2 & -\mu_{23} \tilde{E}_2\\ -\mu_{13}\tilde{E}_1^* & - \mu_{23} \tilde{E}_2^* & 0 \end{pmatrix} \end{align} $$

where $$\Delta_i = \nu_i - \omega_{i3} $$, and the Rabi frequencies for each atomic transitions are defined as $$\Omega_p = \mu_{13}\tilde{E}_1/\hbar $$ and $$ \Omega_c = \mu_{23}\tilde{E}_2/\hbar $$, where the subscripts stand for probe and control field respectively.

At resonance, when $$\Delta_1 = \Delta_2 = 0 $$, the two branches $$ |1\rangle \to |3\rangle $$ and $$ |2 \rangle \to |3\rangle  $$ interfere destructively, and the light-atom coupled system is found to be in superposition of states $$|1\rangle $$ and $$|2\rangle$$, which is given by

$$ \begin{align} |D\rangle = \frac{\Omega_c |1\rangle - \Omega_p |2\rangle}{\sqrt{\Omega_c^2 + \Omega_p^2}} \end{align} $$

The state $$|D\rangle $$ is also known as a ``Dark state", because at any moment this cannot be observed as it has an eigenvalue of $$ 0$$, that is $$ \hat{H}_R |D\rangle = 0|D\rangle $$. We call this a ``Dark state" because during this moment the atom becomes invisible to the probe field, or in other words, in this moment the atom becomes transparent to the probe field. During this moment, any information about the probe and coupled fields, is contained in the amplitude of the Dark state, namely $$\Omega_c$$ and $$\Omega_p $$. If this state is to evolve in time due to a small detuning $$\delta $$, then we have

$$ \begin{align} |D(t)\rangle = \frac{\Omega_c |1\rangle - \Omega_p e^{i\delta t}|2 \rangle}{\sqrt{\Omega_c^2 + \Omega_p^2}} \end{align} $$

In this way we can see that the Dark state will eventually decay, and time that it will take is know as the coherence time $$\tau_{coh}$$. And for storing information purposes, the time limited to store information in the atom is directly related to the coherence time.

Experimental implementation: Rubidium vapor in high performance cavity
The figure below, represents a specific type of quantum memory experiment using Rubidium vapors, where the ground state is $$|g\rangle = |5S_{1/2}, F=1\rangle $$, the meta-stable state $$|m\rangle = |5S_{1/2}, F=2\rangle$$, and and excited state $$|e\rangle = |5P_{1/2}, F'=1\rangle $$, which are shown in the energy lambda configuration in figure 1a. The specific configuration of laser and mirror alignments was choosen to suppress the noise level. In figure 2a, we see the results of the photon flux from the input signal(or probe signal), red line indicates the signal before entering the vapor, and blue showing the signal exiting the vapor. In this case, it is shown that this has a storage time of $$100 $$ns.