# An Introduction to Rydberg atoms with ARC¶

Authors: Nikola Šibalić, Jonathan D. Pritchard, Charles S. Adams, Kevin J. Weatherill

Updated 08/06/2021

In this notebook:

For concise general overview of the current research directions in the Rydberg physics, and explanation of elementary underlying ideas, check new eBook below. It was made specifically for final year undergraduates and a new PhD reserchers and uses new electronic format to bring interactive figures for exploring different regimes and building intuition about processes (made with the help of ARC package):

The eBook above is interactive in web version and EPUB version if used with EPUB3 supporting device (e.g. iBooks on Mac/iPad, Gitden Reader in Android, or Document viewer Xreader on Linux). Under Windows we recommend using web version since the best currently available reader in Windows (Calibre) doesn't have full support for EPUB3.

### Configure Notebook¶

This notebook uses ARC package (Alkali Rydberg Calculator) that can be installed from Python Package Index (PyPI) with pip

pip install ARC-Alkali-Rydberg-Calculator .

For more details see GitHub. This installation can be done from notebook too, by prepending ! to the previous command.

# Rydberg Atom Energy Levels¶

Rydberg states are highly excited states of the outer valence electron where the properties can be scaled in terms of the principal quantum number, $n$. Originally observed in the spectral lines of hydrogen, the binding energy of the Rydberg series are given by

\begin{equation} E_{n\ell j} = - \frac{\mathrm{Ry}}{(n-\delta_{n\ell j})^2}, \end{equation}

where $n,\ell,j$ are the quantum numbers, $\mathrm{Ry}$ is the Rydberg constant and $\delta_{n\ell j}$ is the quantum defect. This defect describes the increase in binding energy for an alkali atom with respect to Hydrogen due to penetration and polarisation of the closed inner electron shells, which is most significant for the $\ell=0$ states with highly elliptical orbits, and can be neglected for states with $\ell>3$. The quantum defects are parameterised via

\begin{equation} \delta_{n\ell j} = \delta_0 + \frac{\delta_2}{(n-\delta_0)^2}+\frac{\delta_4}{(n-\delta_0)^4}+ \ldots, \end{equation}

with coefficients $\delta_{0,2,...}$ taken from measured energy levels.

The mass-corrected Rydberg constant is given by $\mathrm{Ry}=\mathrm{Ry}_\infty\times M/(M+m_e)$ where $m_e$ is the electron mass and $M$ is the atomic mass of the nucleus, with

\begin{equation} \mathrm{Ry}_\infty=\frac{e^4m_e}{16\pi^2\epsilon_0^2\hbar^2}. \end{equation}

To demonstrate the effect of quantum defects, the graph below shows the energy levels of Cesium highlighting the effect of the quantum defect between the different $\ell$ manifolds.

## Rydberg atom wavefunctions¶

Rydberg atom wavefunctions can be obtained from Schrödinger's equation,

\begin{equation} \left[-\frac{1}{2\mu}\left(\frac{\mathrm{d}^2}{\mathrm{d} r^2} +\frac{2}{r}\frac{\mathrm{d}}{\mathrm{d} r}\right) + \frac{\ell(\ell+1)}{2\mu r^2} + V(r)\right]R(r)= E_{n\ell j}R(r), \end{equation}

with reduced mass $\mu=Mm_e/(M+m_e)$ and $V(r)$ a model potential given by

\begin{equation} V(r) = -\frac{Z_{n\ell}(r)}{r}-\frac{\alpha_c}{2r^4}(1-e^{-(r/r_c)^6})+V_{\rm so}, \end{equation}

which at long range gives a $-1/r$ Colomb potential and at short range accounts for the finite size of the core, with radial charge given by

\begin{equation} Z_{n\ell}(r) = 1+(Z-1)e^{-a_1r}-r(a_3+a_4r)e^{-a_2r}. \end{equation}

Wavefunctions are calculated by numerical intergration of the model potential using parameters $a_{1..4},r_c$ and $\alpha_c$ taken from Marinescu et.al. Potential includes spin-orbit interaction $V_{\rm so}(r) = \alpha ~\mathbf{L}\cdot\mathbf{S}/(2r^3)$, where $\alpha$ is fine structure constant.

Rydberg atoms are large, with an average size $\langle r \rangle\propto n^2ea_0$, which leads to extremely large dipole matrix elements making them ideal for exploitng strong long range interactions. To illustrate this point, the probabiltiy distribution $\vert rR(r)\vert^2$ for $nS_{1/2}$ atomic wavefunctions are plotted below for increasing $n$ which highlights the dramatic scaling with atoms approaching $1~\mu$m in size for $n=100$ compared to $1~\unicode{x212B}$ for the groundstate!

## Matrix Elements¶

#### Dipole Matrix Elements¶

Relevant properties of the Rydberg states can be derived through evaluation of dipole matrix elements. Recalling the separability of the atomic wavefunctions into radial and spherical components $\psi(r,\theta,\phi)=Y_{\ell,m_\ell}(\theta,\phi)R_{n\ell}(r)$ in the uncoupled basis, the dipole matrix elements can be expressed as

\begin{equation} \langle n,\ell,m_\ell\vert r_q \vert n',\ell',m_\ell' \rangle = (-1)^{\ell-m_\ell}\begin{pmatrix}\ell&1&\ell'\\-m_\ell&q&m_\ell'\end{pmatrix}\langle\ell \vert\vert r\vert\vert\ell'\rangle, \end{equation}

where $q=-1,0,1$ corresponds to $\sigma^+,\pi$ and $\sigma^-$ transitions respectively with the reduced matrix element

\begin{equation} \langle\ell\vert\vert r\vert\vert\ell'\rangle = (-1)^\ell\sqrt{(2\ell+1)(2\ell'+1)}\begin{pmatrix}\ell&1&\ell'\\0&0&0\end{pmatrix}\mathcal{R}_{n\ell\rightarrow n'\ell'}, \end{equation}

where round braces denote Wigner-$3j$ symbols and the radial matrix element is evaluated from

\begin{equation} \mathcal{R}_{n\ell\rightarrow n'\ell'} = \int_{r_{\rm i}}^{r_{\rm o}} R_{n,\ell}(r)rR_{n',\ell'}(r)r^2\mathrm{d} r. \end{equation}

For highly excited states the hyperfine-structure splitting becomes negligible, however the fine-structure splitting $\hat{\mathcal{H}}=\beta \bf\hat{L}\cdot\hat{S}$ means the relevant basis is then fine structure basis ($j,m_j$) with matrix elements

\begin{equation} \langle n,\ell,j,m_j\vert r_q \vert n',\ell',j',m_j' \rangle = (-1)^{j-m_j}\begin{pmatrix}j&1&j'\\-m_j&q&m_j'\end{pmatrix}\langle j \vert\vert r\vert\vert j'\rangle, \end{equation}

and the reduced matrix element equal to

\begin{equation} \langle j \vert\vert r\vert\vert j'\rangle = (-1)^{\ell+s+j'+1}\delta_{s,s'}\sqrt{(2j+1)(2j'+1)} \begin{Bmatrix}j&1&j'\\\ell'&s&\ell\end{Bmatrix}\langle \ell\vert\vert r\vert\vert\ell'\rangle, \end{equation}

where the curly braces denote a Wigner-$6j$ symbol.

The cell below demonstrates how to extract the relevent matrix elements for different transitions and highlights the extremely large matrix elements of the Rydberg states for transitions to neighbouring states, which scale $\propto n^2$ due to the large electron radius shown above. Thus whilst the alkali $D$ line transitions have elements of order $~e a_0$, the Rydberg states can have $>1000~e a_0$ making them ideally suited for exploiting strong, long range interactions as will be explored below.

For accurate interaction potentials at short range, it is necessary to also consider quadrupole matrix elements for the Rydberg atoms. getQuadrupoleMatrixElement

\begin{equation} \mathcal{R}^Q_{n\ell\rightarrow n'\ell'} = \int_{r_{\rm i}}^{r_{\rm o}} R_{n',\ell}(r)r^2R_{n,\ell'}(r)r^2\mathrm{d} r. \end{equation}

For low-lying states, Numerov integration is usually not accurate enough in estimating dipole matrix element. Package also has a .csv table for each element, that can be manually expanded if needed, with various literature values for dipole matrix elements. These values are either obtained experimentally, or by more advanced theoretical calculations. If value is existing in the literature, package will by default use the literature value with the smallest error estimate.

Rydberg atoms have suprisingly long radiative lifetimes. For example lifetimes of $nS$ states are of the order of $10~\mu$s, for $n\sim30$. Lifetimes $\tau$ of $n$ states, with small orbital angular momenutum $\ell$ like $S$ or $P$ states, scales as $\tau\propto n_*^3$ where $n_*=n-\delta$ is the effective principal quantum number. Of all possible $\ell$ states, circular states $\ell=n-1$ have longest lifetimes, that scales as $\tau \propto n_*^5$.

The radiative lifetime is calculated from the Einstein-A coefficients as $1/\tau_0=\sum_{n'\ell'}A_{n\ell\rightarrow n'\ell'}$, with the summation over all dipole coupled states with energy $E_{n'\ell'}<E_{n\ell}$ and the A-coefficient calculated as

\begin{equation} A_{n\ell\rightarrow n'\ell'}=\frac{4\omega_{nn'}^3}{3c^3}\frac{\ell_\mathrm{max}}{2\ell+1}\mathcal{R}^2_{n\ell\rightarrow n'\ell'}. \end{equation}

Here the prefactor of $\omega_{nn'}^3$ means that although the dipole matrix element for transitions to nearby $n$ states is much larger than that for transitions to the ground state (e.g. Cs $\langle 60 S_{1/2}|er| 60 P_{3/2}\rangle = 4114 ~ea_0$ compared to $\langle 6 S_{1/2}|er| 60 P_{3/2}\rangle = 0.0038 ~ea_0$), the A-coefficients for these two transitions are $1.9\mathrm{s}^{-1}$ and $224\mathrm{s}^{-1}$ respectively, resulting in the lowest energy transitions being the dominant decay pathways for radiative decay from the Rydberg states.

At finite temperature, the black-body excitation spectrum causes a reduction in the lifetime due to driving transitions in the GHz and THz region to neighbouring Rydberg states both above and below the energy of the original Rydberg state. The black body loss is given by

\begin{equation} \frac{1}{\tau_\mathrm{BBR}}=\displaystyle\sum_{n'\ell'} \frac{A_{n\ell\rightarrow n'\ell'}}{\exp(\omega_{nn'}/k_\mathrm{B}T)-1}, \end{equation}

with the effective lifetime $1/\tau_\mathrm{eff}=1/\tau_0+1/\tau_\mathrm{BBR}$.

The figure below demonstrates the finite temperature effects by showing the strength of the diffent decay pathways for the $30S_{1/2}$ state in Rb at 300K showing the contribution from radiative decay (red columns) and black-body induced transitions (green). The effective lifetime is reduced by around a factor of 2 at room temperature compared to the natural lifetime.

This plot can be compared with Fig. 1 in Ref. I. I. Beterov et. al, PRA 79, 052504(2009). Similarly, we can reproduce black-body induced state depopulation rates (see Fig. 2 in the same reference):

Comparing lifetimes for different principal quantum numbers $n$ and orbital angular momentum states $l$, we see that lifetimes for $nP$ scale as $\propto n_*^3$, and lifetimes for circular states $n,l=n-1$ scale as $\propto n_*^5$

## Creating highly excited states¶

Rydberg atoms are highly excited atoms. If the excitation is done in single step necessary energy would typically correspond to UV radition. For example, in Caesium $6 S_{1/2} \rightarrow 60 P_{3/2}$ requred transition wavelength and frequencies are

#### Rabi Frequency¶

The transition strength is characterised by the Rabi frequency $\Omega=d\cdot\mathcal{E}/\hbar$, where $d$ is the dipole matrix element calcualte above and $\mathcal{E}$ is the electric field of the laser driving the excitation. Recalling that for a Gaussian beam with $1/e^2$ radius of $w_0$ and power $P$, the intensity is given by \begin{equation} I=\frac{2P}{\pi w_0^2}, \end{equation} which is related to the electric field by $I=(1/2)c\epsilon_0\vert \mathcal{E}\vert ^2$, resulting in the Rabi frequency of

\begin{equation} \Omega = \sqrt{\frac{4P}{\epsilon_0c\pi w_0^2}}\frac{\langle n,\ell,j,m_j\vert r_q \vert n',\ell',j',m_j' \rangle}{\hbar}. \end{equation}

Due to the weak overlap between the ground and highly excited Rydberg states (which scales as $\sim n^{*-3/2}$), the resulting matrix elements are relatively small meaning large intensities are required. As an example, consider a 1 W UV laser driving the transition above with a $1/e^2$ waist of 50 $\mu$m. The resulting Rabi frequency is only $\Omega/2\pi=10$ MHz!

An alternative way to reach these highly excited states is to use a mutli-step excitation scheme with a number of lasers of different wavelength. This enables use of transitions with favourable or convenient wavelengths (for example standard diode/fiber laser sources) with high power or strong transitions. The most commonly used is a two-step excitation via the D-line transition in the alkali's, with the added benefit of enabling electromagnetically induced transparency (EIT) to be engineered for mapping strong Rydberg interactions onto optical photons. In Rubidium, typically the $5 S_{1/2} \rightarrow 5 P_{3/2} \rightarrow 60 S_{1/2}$ transition is favoured, whose properties are

One can use even more involved schemes, for example in Ceasium three lasers can be used $6 S_{1/2} \rightarrow 6 P_{3/2} \rightarrow 7 S_{1/2} \rightarrow 60 P_{3/2}$. Note that now all the lasers are conveniantly in the near infrared (NIR) range!

Or, even four lasers can be achieved in order to couple to Rydberg states. For example [reference] $6 S_{1/2} \rightarrow 6P_{3/2} \rightarrow 7S_{1/2} \rightarrow 8 P_{1/2} \rightarrow 52 D_{3/2}$

Now even with moderate laser powers we can reach highly-excited states. For example for 10 mW laser focused down to 50 $\mu$m driving $8~P_{1/2}~m_j=1/2 \rightarrow 52~D_{3/2}~m_j=1/2$ we achieve

We can also calculate dipole matrix elements in hyperfine basis. Here we compare this element with the value from D. Steck, Cs D Line data

We can also obtain quadrupole matrix elements. For example, $5 S_{1/2}\rightarrow n D_{5/2}$ transitions, that have been also calculated in D. Tong et.al, PRA 79, 052509 (2009) (see Table I there), we can obtain by calling

## Rydberg Atom Stark Shifts¶

The large dipole moment of the Rydberg states makes them highly polarisable. For an atom in a d.c. electromagnetic field $\mathcal{E}$ directed along the $z$-axis, the interaction leads to a coupling given by

\begin{equation} \mathcal{H} = \mathcal{H}_0 + \mathcal{E}\hat{z}, \end{equation}

where $\mathcal{H}_0$ is the Hamiltonian for the unperturbed energy levels and $\hat{z}\equiv r_0$ in the spherical basis resulting in a selection rule $\Delta m_j=0$ and a shift dependent on $\vert m_j \vert$. For the low-$\ell$ states, the quantum defects cause the first-order Stark-shift to vanish, resulting in a second-order energy shift $\Delta E = -\frac{1}{2}\alpha_0\mathcal{E}^2$ where the polarizability for state $\vert n \ell j m_j\rangle$ is given by

\begin{equation} \alpha_0 = 2 e^2\displaystyle\sum_{n'\ell'j'\neq n\ell j} \frac{\vert \langle n,\ell,j,m_j\vert r_0 \vert n',\ell',j',m_j' \rangle \vert^2}{E_{n'\ell' j'}-E_{n\ell j}}. \end{equation}

Since the dipole matrix elements scale as $n^{2}$, and the energy difference scales as $n^{-3}$ the scalar polarizability $\alpha_0\propto n^7$. States with $\ell>3$ are degenerate, and therefore experience a linear Stark shift.

The evaluation of polarisability using second-order perturbation theory is limited, and only valid at weak fields. Instead, Stark maps accurate to all orders can be obtained by direct diagonalisatoin of the Hamiltonian $\mathcal{H}$, taking care to include a large enough set of basis states to achieve divergence. As well as revealing the exact energies of the Stark-coupled states, this permits extraction of the mixed electric field eigenstates which are relevant when considering optical excitation of a given Rydberg state which is now mixed into neighbouring energy levels.

The code below provides an example Stark map for the $35S_{1/2},\vert m_j\vert=1/2$ state in Cesium. After identifying the target state, a range of basis states are defined before running the diagonalisation for the range of E-fields specified, returning the desired Stark map. To show the effect of state mixing, the levels are coloured proportional to the fraction of the target state present in each eigenstate. This knowledge can also be used to determine the effective polarisability by extracting the energy level of the eigenstate with a contribution of at least 90% of the original target state.

For high $n$, the range of basis states may need to be increased due to the reduction in energy spacings but this can easily be checked for convergence by running twice with a larger basis set.

We can compare values for the polarizability with the measurements in the literature. For example, we can get theoretical predictions for results cited in Ref. M. S. O'Sullivan and B. P. Stoicheff, PRA 31, 2718 (1985) (see Table I there) by calling the following

This data can be compared also with analytical fit function that they recommend as a polarizability estimate.

We can also compare our calculations with recent calculations and measurements from J. Grimmel et.al, NJP 17, 053005 (2015) for Rubidium $35 S_{1/2}$ and $70 S_{1/2}$ states

# Rydberg atom interactions¶

As described above, the Rydberg states have extremely large dipole moments making them ideal candidates for controlled long-range interactions in quantum information processing and for studying effects such as resonant energy transfer or superradiance.

For two atoms separated by distance $\textbf{R}$ the dipole-dipole interaction is given by

\begin{equation} V(\textbf{R}) = \frac{\textbf{$\mu$}_1\cdot\textbf{$\mu$}_2}{R^3} -\frac{3(\textbf{$\mu$}_1\cdot\textbf{R})(\textbf{$\mu$}_2\cdot\textbf{R})}{R^5}, \end{equation}

where the dipole matrix elements $\mathbf{\mu}_{1,2}$ describe transitions from the initial Rydberg states $\vert r \rangle$ to other dipole-coupled states $\vert r'\rangle,\vert r''\rangle$ respectively. The dipole-coupled pair-states have an energy difference $\Delta$ give by

\begin{equation} \Delta = E_{r'}+E_{r''}-2E_{r}, \end{equation}

with the dominant contribution to the atom-atom interaction arising from the pair state of with smallest absolute defect $\vert \Delta \vert$. Reducing this to a simple two-pair problem, the Hamiltonian for the states $\vert r r\rangle,\vert r'r''\rangle$ is given by

\begin{equation} \mathcal{H} = \begin{pmatrix} 0 &V(R) \\ V(R)&\Delta \end{pmatrix}, \end{equation}

which has the eigenvalues

\begin{equation} \lambda_\pm = \frac{\Delta\pm\sqrt{\Delta^2+4V(R)^2}}{2}. \end{equation}

This results in two asympotitic limits for the atom-atom interactions:

i) Long-range ($V(R)\ll\Delta$)

In this limit, also known as the Van der Waals regime, pair-state energy is shifted by $-V(R)^2/\Delta\equiv - C_6/R^6$. The sign of the interaction (attractive/repulsive) is determined by $\Delta$, and the interaction scales as $C_6\propto n^{11}$.

ii) Short-range ($V(R)\gg\Delta$)

This is known as the resonant dipole-dipole regime with $\lambda=\pm C_3/R^3$ where $C_3\propto n^4$, resulting in significant state mixing.

The transition between these two regimes is known as the Van der Waals radius occuring at $V(R_\mathrm{vdW})=\Delta$, with $R_\mathrm{vdW}=\sqrt{C_6/\vert\Delta\vert}$.

Typically $\Delta\neq0$ for the Rydberg states, however for small $\Delta$ an external electric field can be used to Stark-shift the pair states into resonance known as a Förster resonance, resulting in resonant $C_3/R^3$ behaviour at all radii.