User:Kccatfish93/sandbox/Rydber Atoms

Rydberg atoms have at least one electron in an excited state with a very high principle quantum number. Due to the high quantum numbers of the electrons, Rydberg atoms have some very remarkable properties. Perhaps the most important is the physical size of the resulting atom, due to the vast distance between the electron and atomic nucleus. This distance, which can be on the order of microns, creates a large dipole moment. . These large dipole moments increase the sensitivity of the atom to outside electric fields, making them ideal candidates for a multitude of problems in physics and quantum information science. As a downside, this increased sensitivity makes the atoms responsive to stray electromagnetic fields in the lab. It has been shown that Rydberg atoms have strong electrostatic interactions at longer ranges than are normally associated with neutral atoms, leading to advances in neutral atom quantum computing.

When Rydberg atoms lay spatially close to each other, the electric dipole-dipole interaction can shift the energy levels of nearby atoms The excitation of a single atom to a Rydberg state can “block” the excitation of large numbers of atoms surrounding it from absorbing radiation at the frequency of the incoming laser. This effect is known as the Rydberg Blockade.

Dipole Interaction
The dipole moment of a Rydberg atom can be written as
 * $$ P = e\mathbf{r} $$ ,

where $$e$$ is the elementary charge and r is its vector displacement with respect to the nucleus. Due to the very large distance between the valence electrons and the atomic nucleus, the dipole moment is an important characteristic of high quantum number Rydberg atoms. For the case of two close-lying, but non-overlapping atoms, the resulting dipole-dipole interaction that will shift the energy of the system by


 * $$ E_{DD}={{e^2\over{R^3}}(\mathbf{r^{(1)}}\cdot\mathbf{r^{(2)}}-3r1^{(1)}r1^{(2)})} $$

where R is the the interatomic distance, r1 is the component of r along the interatomic axis, and (1) and (2) indicate atom number 1 and 2. We can see that the function $$E_{DD}$$ is a function of the interatomic distance, so producing Rydberg atoms at the precise interatomic distance will be of utmost importance.

The total Hamiltonian of this two-atom system, omitting the fine and hyperfine structure, is

$$ H^{(Total)} = H^{(1)} + H^{(2)} + H^{(int)}$$ or, written in the quantum number basis,

$$ H = \sum_{n,l,m} E_{nlm}(|nlm\rangle \langle nlm|^{(1)} + |nlm\rangle  \langle nlm|^{(2)} + E_{DD}$$

Where $$ E_{nlm} $$is equal to $$ -1\over2(n-\delta_l)^2 $$ and $$ \delta_l

$$ is known as the "quantum defect." The quantum defect takes into account the fact that Rydberg electrons with small angular momenta have highly elliptical orbits that allow penetration into the core.

The Blockade


Lets look at a ground state $$ |g\rangle $$ and a Rydberg state $$ |r\rangle$$ of an atom with an energy separation $$ E$$ shown in Figure 1. When atoms 1 and 2 do not interact, we get two transitions at the frequency $$ E\over\hbar $$. In other words, the state $$ |gg\rangle$$ can go to $$ |rg\rangle$$  or $$ |gr\rangle$$ and eventually to $$ |rr\rangle$$. This allows excitation of both atoms to $$ |rr\rangle$$ simultaneously. However, if the two atoms interact, the energy level is shifted by an amount $$ E_{DD} $$ as described above, and the laser excitation can not bring the two atoms to state $$ |rr\rangle$$.

Let us now look at this interaction in more detail. Lets say that we have a neutral atom in the rotating frame coupled to a ryberg state, $$|r\rangle$$, by a laser with Rabi frequency $$\Omega$$ from two possible ground states $$|0\rangle $$ or $$|g\rangle$$ and detuning $$\Delta$$. After setting $$\hbar =1 $$, let the Hamiltonian for this atom be

$$H^{(1)} = -\Delta|r\rangle\langle r|^{(1)} + {\Omega\over 2}|g\rangle\langle r|^{(1)} - E_{HF}|0\rangle\langle 0|^{(1)} $$

where $$E_{HF} $$ is the hyperfine energy splitting. Now, we want to write a total Hamiltonian for this atom with the addition of another. We know that the total hamiltonian can be written as a sum of the bare atom hamiltonian plus the interaction term. So, the total hamiltonian for this state can be written as

$$H^{(Total)}=H^{(1)}+H^{(2)} + H^{int}  $$.

However, we know that the energy shift, $$E_{DD} $$, from the interaction between the two atoms affects the Rydberg state $$|rr \rangle$$, and thus can be written as

$$H^{(Total)}=H^{(1)}+H^{(2)} + E_{DD}|rr\rangle\langle rr|  $$

or in the expanded form$$H^{(Total)} = -\Delta|r\rangle\langle r|^{(2)} + {\Omega\over 2}|g\rangle\langle r|^{(2)} - E_{HF}|0\rangle\langle 0|^{(2)} -\Delta|r\rangle\langle r|^{(1)} + {\Omega\over 2}|g\rangle\langle r|^{(1)} - E_{HF}|0\rangle\langle 0|^{(1)}+E_{DD}|rr\rangle\langle rr| $$

However, we know that an atom in states $$|00\rangle, |0g\rangle  $$ and $$|g0 \rangle  $$doesn't feel the interaction term, and we will only need to deal with interactions involving

$$|gg\rangle, |rg\rangle, |gr\rangle $$, and $$|rr\rangle  $$. In order to rewrite the total hamiltonian, we can assume a subspace where the atoms are in one of these states. The total hamiltonian becomes

$$H^{(Total)} = -\Delta(|gr\rangle\langle gr|+|rg\rangle\langle rg| + 2|rr\rangle\langle rr|)+{\Omega\over 2}(|rg\rangle \langle gg| + |gr \rangle \langle gg| + hc) + E_{DD}|rr \rangle \langle rr| $$

where $$hc $$ is the hermetian conjugate. As is common when talking about Electromagnetically Induced Transparency (EIT), the total hamiltonian can be rewritten in a basis of Bright and Dark states. While in a Dark state, atoms do not interact with the field, and thus do not absorb radiation at a typically expected frequency. A Bright state is the orthogonal counterpart to the Dark state. Bright states, $$(|1 \rangle + |2 \rangle)\over \sqrt{2} $$, and Dark states, $$(|1 \rangle - |2 \rangle)\over \sqrt{2} $$ are states for a typical 3-level atom configuration. For our case, we can rewrite this in terms of our $$|rg \rangle$$ and $$|gr\rangle$$ basis states as $$B ={(|rg \rangle + |gr \rangle)\over \sqrt{2}} $$ and $$D ={(|rg \rangle - |gr \rangle)\over \sqrt{2}} $$ . So, now we can write our total hamiltonian as $$H^{(Total)} = -\Delta(|D\rangle\langle D|+|B\rangle\langle B| + 2|rr\rangle\langle rr|)+{\sqrt{2}\Omega\over 2}(|B\rangle \langle gg| + | rr\rangle \langle B| + hc) + E_{DD}|rr \rangle \langle rr| $$

As we can see from this hamiltonian, the $$|gg\rangle $$ state is coupled to the Bright state and the Dark states can be ignored. However, we need to have a dark state somewhere or we couldn't get the Blockade effect. Also, notice how the Rabi frequency is enhanced by a factor of $$\sqrt{2} $$. Since this is a simplified Hamiltonian, there are some additional terms that would need to be added to get an entirely accurate picture. The Dark states are actually found the errors of this Hamiltonian.

If we make the assumption that the interaction term, $$E_{DD}|rr \rangle \langle rr|  $$, is much greater than $$\Delta$$,  the transition from from the $$|gg\rangle $$ state to the $$|rr\rangle$$ state will be much more likely than the transition from the bright state,  $${(|rg \rangle + |gr \rangle)\over \sqrt{2}} $$, to $$|rr\rangle $$ . This process is conceptually shown in Figure 2.

Conceptually, when the first atom absorbs a photon, it suppresses  the second atom's ability to absorb light at the same frequency. This effect is known as the Rydberg Blockade.

If we want to look at interactions involving more than two atoms, our analysis can be readily extended to include atomic ensembles of N atoms. Instead of two bare atom hamiltonians, out total hamiltonian will now include N "copies" of it, as well as a correction to the Rabi frequency that scales as $$\sqrt{N}$$.

Applications
Multiple Demonstrations of Rydberg Blockades and their applications have materialized over the years.

Some of the earlier demonstrations were in clouds of atomic gasses. .

The Rydberg Blockade has been shown to have a significant upside in the field of quantum information processing. The dipole blockade was predicted by Lukin, et al. In the year 2001 to be a good candidate for QIP. In the year 2010, Isenhower et al. experimentally demonstrated that two individual neutral atoms could be used to create a Controlled-Not gate.

In 2009, a group at the University of Wisconsin Madison demonstrated experimentally that a single Rb atom excited to the $$ 79d_{5/2} $$  energy level resulted in a blockade on the second atom with a separation distance of $$ 10\mu m $$. The authors of the paper say their experiment is an important step in demonstrating "scalable neutral atom quantum logic devices".

In a 2017 paper published in PRL, a group demonstrated the entanglement of two different isotopes using a Rydberg Blockade. They succeeded in entangling  $$Rb^{87} $$and $$Rb^{85} $$ isotopes confined in separate single atom traps by $$3.8\mu m $$. They claim that their experiment "can be used for simulating any many--body system with multi-species interactions"

The effect has also been experimentally demonstrated in a BEC. A group at the Universität Stuttgart reported on a Blockade in $$Rb^{87} $$ BEC for multiple "excitation times" and "condensation temperatures."