User:Bsdvikas/sandbox

Micromasers are Microwave Cavities which can trap electromagnetic fields for long enough in side them to perform experiments by interacting them with atoms. The first work in this direction was initially proposed by Purcell in 1946 and was then developed theoretically. Experiments using this method was pioneered by S. Haroche in Paris, H. Walther in Garching in the last decades of 20th century and after. The Nobel prize in Physics in 2012 was awarded to Haroche for his work in Cavity QED along with DJ Wineland. = Theory = The Jaynes-Cummings model explains the theory behind the interaction between an atom with two energy levels, and a field of electromagnetic radiation. The atom is represented as a qubit consisting of an excited state $$|e\rangle\langle g| $$ and a ground state $$|g\rangle $$ seperated by an energy gap of $$ \Delta $$. The electromagnetic radiation is represented as the linear combination of Fock states or Photon number states $$(|n\rangle )$$ which follow the ladder algebra


 * $$ a^{\dagger}|n\rangle = \sqrt{n+1}|n+1\rangle \qquad  \qquad a|n\rangle = \sqrt{n}|n\rangle \qquad \qquad  a^{\dagger} a |n\rangle = n|n\rangle $$

The Hamiltonian of the full system has components corresponding to the atom, the field and the interaction between them.


 * $$\hat{H} = \hat{H}_{\text{field}} +\hat{H}_{\text{atom}} +\hat{H}_{\text{int}}$$

The atom and field pars of the Hamiltonian are obvious given the energy gap between photon states $$ \hbar \omega $$ and the energy gap of the atom $$ \hbar \Delta $$. The interaction Hamiltonian, on the other hand, has both counter rotating and co-rotating terms which simplifies after rotating wave approximation to two terms.



\begin{align} \hat{H}_\text{field} &= \hbar \omega \hat{a}^{\dagger}\hat{a}\\ \hat{H}_\text{atom} &= \hbar \Delta \frac{\hat{\sigma}_z}{2} = \frac{\hbar \omega_a}{2} (|e\rangle \langle e| - |g\rangle \langle g|) \\ \hat{H}_\text{Interaction} &= \hbar g (\hat{a} \hat{\sigma_+} + \hat{a}^\dagger \hat{\sigma_-}). \end{align}

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Where the coupling constant is given by $$ \hbar g = - \vec{d}^{eg}.\vec{u}_{\vec{k}\lambda}(\vec{r})\sqrt{2 \pi \hbar \omega} $$. Also notice that we have focused on a single photon-mode corresponding to $$ \hat{a}^\dagger and \hat{a} $$; to account for other modes, we would have to sum over individual $$ H_I $$s. Now if you consider the whole set of possible atom-field states it is easy to see that the interaction Hamiltonian can only connect $$|g, n+1\rangle $$ state with $$|e, n\rangle  $$


 * $$(\hat{a} \hat{\sigma_+} + \hat{a}^\dagger \hat{\sigma_-})|g, n+1\rangle = \sqrt{n+1}|e, n\rangle $$


 * $$(\hat{a} \hat{\sigma_+} + \hat{a}^\dagger \hat{\sigma_-})|e, n\rangle = \sqrt{n+1}|g, n+1\rangle $$

Hence we can define a new number operator $$ \hat{N} = \hat{a}^{\dagger}\hat{a} + |e\rangle \langle e| $$ which carries the same eigen-values for the two states $$|g, n+1\rangle $$and $$|e, n\rangle  $$ . Hence, we can re-write the Hamiltonian in terms of the Hamiltonians corresponding to constant $$ N $$ spaces


 * $$H^{(n)} = \hbar

\begin{pmatrix} n \omega +\frac{\Delta}{2} & g \sqrt{n+1} \\[8pt] g \sqrt{n+1} & (n+1)\omega -\frac{\Delta}{2} \end{pmatrix} $$ for n>1 and

H^{(0)} = 0 $$

and we can write the whole Hamiltonian as


 * $$H_I =

\begin{pmatrix} H^{(0)} & 0 & 0 &. \\ 0 & H^{(1)} & 0 &. \\ 0 & 0 & H^{(2)} &. \\ . & . & . & . & \\ \end{pmatrix} $$

In this picture, we can focus on a single constant $$ N $$ subspace to see the Rabi oscillations. Consider a system in the initial state $$ | e, n \rangle $$ which evolves under this hamiltonian for time $$ t $$. The state at time $$ t $$ is given by
 * $$ | \Psi (t) \rangle = e^{-\frac{i}{\hbar} \hat{H} t } | e, n \rangle = cos(\sqrt{n+1}gt)| e, n \rangle - i sin(\sqrt{n+1}gt)| g, n +1 \rangle $$

These Rabi oscillations are at the heart of all Cavity QED experiments in specific and all atom-photon interactions, like Ramsey interferometry, in general. We leverage this coherent rotation of atom-photon states to create quantum states, generate entanglement and so on.

Rydberg atoms
Rydberg atoms are atoms which are in an excited state with an elctron (or more) in a high principal quantum number (n) state. This means that the spatial distribution of the electron's positions is far away from the nucleus due to the orbital size scaling as $$ n^2 $$. These atoms, due to this weakly bound electron states, have a very large dipole moment compared to their ground state counterparts and hence are very responsive to Electric fields. This very large dipole moment means that the coupling constant $$ g $$ will be very large for Rydberg atoms, propelling us into a "strong coupling regime" where the large value of $$ g $$ helps us significantly affect the atom in very short times. Hence, we can now shoot these atoms through the electromagnetic cavities and the time of passage through the the cavity will be enough to to generate a few $$ \pi $$ rotation.

= Experimental setup = We saw that the microwave CQED experiments rely on the strong coupling between single two level atoms and a few microwave photons stored in a high Q superconducting cavity. Hence the basic experimental setup for these experiments consists of a beam of atoms (usually Rubidium) passing through a microwave Fabry-Perot-type cavity which will them be "measured" to see if it is in the excited or the ground state. We will the setup used by Serge Haroche and his group at Paris. While the finer details, like the species of the atom or the frequencies of lasers, might change, the overall protocol is the same in most of the Cavity QED experiments.

Atom qubit creation
A thermal beam of (in our case Rubidium) ions is produced from the oven (O in the picture) which come out with a distribution of velocities. These atoms then pass through a Circularization box (CB). Here these atoms are velocity selected using optical pumping and time of flight techniques which only let atoms with velocities in a certain small range to pass through. But the main task of the cicularization box is to turn these Rubidium atoms into Rydberg atoms. The valence electron in the Rubidium atom is pushed into the $$ n= 51 and 50 $$ states. It is a complicated process involving 52 photons where the valence electron is pushed into the $$ n= 52 $$ level which are then coupled with the  $$ n=50 $$ or  $$ n= 51 $$ selectively using a microwave pulse of either 49.647 GHz or 48.195 GHz respectively. The $$ n= 51 and 50 $$ states are chosen as our $$ | e \rangle $$ and $$ | g \rangle  $$ respectively. The transition frequency for this Atomic qubit is 51.099 GHz, which is what our Cavity will be tuned for. This atom has a diameter of about $$ 2500 au $$ and they can be created with a fidelity of more than 98%.

Cavity - The photon box
The atoms enter the Circularization box with their valence electron in $$ 5s_{1/2}, F=3 $$ state with an uncertainty in velocity of about 10 m/s and leave the CB as Rydberg atoms in the state of our choosing with a velocity width of 1.5 m/s. As these "mono-kinetic" atoms pass through the experiment, we can know their position within an uncertainty of $$ \pm 1mm $$. This is helpful in controlling the interaction of the atom with the cavity. \\ The cavity(C) is made of two niobium mirrors ground to the particular shape and smoothness. There is a small hole in the top mirror with a microwave source ($$ S_C $$) with which we can pump photons into the cavity. The electric field in the cavity, of the order of 1V/m, is switched on or off to generate a Stark shift in the atom which brings it in and out of resonance with the cavity. This, along with the knowledge of the electric field behaviour in the cavity, is used to control the amount of rotation accrued during the Rabi oscillations. These Cavities have very small leakage of photons, less than 1/ms which corresponds to a Q-factor of more than $$ 3x10^8 $$. This is a very important part of the experiment as any leakage of atoms during the experiment will simply cause it fail. Our modus operandi helps us here as the fast moving atoms get measured very fast, with a cavity crossing time of about $$ 20 \mu s $$. The cavity also has a circular aluminium covering around its waist which further increases the Q-factor. There are two holes on this Aluminium band to let the atoms through. We also have two beams of microwaves $$ S_R $$ before and after the cavity which let us apply rotations on the atoms before and after the interaction to perform various Quantum protocols.

Measurement
These atoms are then measured to see if they are in $$ | e \rangle $$ or $$ | g \rangle  $$ state. This is done by simple ionization experiments. The atoms pass in between two capacitor plates which have a small electric field between them. The first such capacitor is set so as to ionize only the atoms in $$ | e \rangle $$ state, which has a lower ionization energy. Then these atoms pass through another (not shown in the picture) such capacitor which now ionizes the atoms in $$ | g\rangle $$ states. The fields required for these measurement are very small, thanks to our Rydberg states. The electrons that get yanked out of the atoms hit one of the electrodes and generate a current pulse which is then measured = Applications = Microwave CQED was one of the first Quantum systems developed to perform Quantum experiments and were used to show many Quantum properties for the first time.

Rabi oscillations
As we saw with the Jaynes-Cummings model, if an atom initially $$ e $$ state passes through the cavity whose photon number state distribution goes as $$ P(n) $$, with an interaction time of $$ t$$, the final probability to measure $$g $$ state is given by


 * $$ \sum_n P(n)sin^2(g\sqrt{n+1}t) $$

That is, the various components of the photon number states drive the atom with different frequencies, which are proportional to square roots of integers. Hence if we plot this probability against the time of interaction $$ t $$, it's fourier transform should have components corresponding to square root of integer frequencies. And in deed that's what happens. The Fourier transform of the the signal for the case of an empty cavity will have but a single peak corresponding to $$ N=1 $$. But if we pump a few photons into the cavity and then take the Fourier transform of the signal, we will have peaks corresponding to multiple frequencies. If the amplitudes of these peaks were plotted, they also show a Poissonian distribution, with the mean being equal to the number of photons in the cavity. Even a nonzero photon number less than one gives visible peaks for higher frequencies and a great fit for the Poissonian curve. This is caused by "less than a photon", interacting with a system and a great Quantum signature.

Quantum memory
Now, by using $$ \pi $$ pulses during the interaction, we can reliably transmit a photon from atom to the cavity or vice versa. But if we send in an atom which not in either the excited or the ground state, but a coherent superposition, we generate entangled states. Consider a superposition state $$ c_g |g\rangle + c_e |e \rangle$$. Now if this were sent through an empty Cavity, with no photons, the ground state wouldn't interact with it, while the excited state would undergo a Rabi oscillation with the cavity. If we apply a $$ \pi $$ pulse, we rotate the state $$ |e,0\rangle -> |g,1\rangle $$ which means


 * $$c_g |g,0\rangle + c_e |e,0 \rangle ->c_g |g,0\rangle + c_e |g,1 \rangle = |g\rangle ( c_g |0\rangle + c_e |1 \rangle) $$

We have now transferred the atomic state into the Cavity, while preserving the coherence. This is basically storing the information from the atom into the cavity, turning the cavity into an information storage device. Now, if we send another atom in ground state through this cavity and apply a $$ \pi $$ pulse, we transfer this information into the new atom.
 * $$|g\rangle (c_g |0\rangle + c_e |1 \rangle) = c_g |g,0\rangle + c_e |g,1 \rangle -> |g,0\rangle + c_e |e,0 \rangle = ( c_g |g\rangle + c_e |e \rangle)|0\rangle $$

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Quantum Non-demolition measurement
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