Creating heralded hyper-entangled photons using Rydberg atoms

Entangled photon pairs are a fundamental component for testing the foundations of quantum mechanics, and for modern quantum technologies such as teleportation and secured communication. Current state-of-the-art sources are based on nonlinear processes that are limited in their efficiency and wavelength tunability. This motivates the exploration of physical mechanisms for entangled photon generation, with a special interest in mechanisms that can be heralded, preferably at telecommunications wavelengths. Here we present a mechanism for the generation of heralded entangled photons from Rydberg atom cavity quantum electrodynamics (cavity QED). We propose a scheme to demonstrate the mechanism and quantify its expected performance. The heralding of the process enables non-destructive detection of the photon pairs. The entangled photons are produced by exciting a rubidium atom to a Rydberg state, from where the atom decays via two-photon emission (TPE). A Rydberg blockade helps to excite a single Rydberg excitation while the input light field is more efficiently collectively absorbed by all the atoms. The TPE rate is significantly enhanced by a designed photonic cavity, whose many resonances also translate into high-dimensional entanglement. The resulting high-dimensionally entangled photons are entangled in more than one degree of freedom: in all of their spectral components, in addition to the polarization—forming a hyper-entangled state, which is particularly interesting in high information capacity quantum communication. We characterize the photon comb states by analyzing the Hong-Ou-Mandel interference and propose proof-of-concept experiments.

In this section, we recall the QED formalism to calculate the two-photon spontaneous decay rate of an atom. We consider an atom initially in an excited state |i that decays to the final state |f .
The Hamiltonian representing the atom-light interaction is Here, we can neglect the A 2 term as its contribution for TPE rates is very small compared to a second order perturbation theory with the first term. The unitary operator for the atom-light interaction is given by: H int (t )dt + higher orders The first term in the above equation represents the zero order processes while the second term represents the first order processes which give the Fermi Golden rule for spontaneous emission rates. For the TPE rates, we consider the third term that represents the decay of atoms by two photons. The transition amplitude is written as: The state |m represents the contribution of all the energy levels.
In the interaction picture, the Hamiltonian is and ω 2 are the energies of the first and second photons and ω if is the energy difference between the transition level |i and |f . The two photon transition probability is given as: The TPE rate in terms of density of states ρ(ω) is given by:

II. TPE RATES OF AN ATOM IN FREE SPACE
In this section, we follow a known derivation for the TPE rates of an atom in free space. In Supplementary section V, we extend this derivation for an atom in a cavity. The density of vacuum modes (for each polarization) is given by: ρ(ω) = ω 2 dωdΩ (2πc) 3 /V , where dΩ is the solid angle covered by the emitted photons. From Eqn. S1, the TPE rate under the dipole approximation is given as: where k = ω/c and d f m is the dipole matrix for transition from state |m to |f . For simplicity, we The TPE rate is averaged over all the directions and polarizations.
Here, k 0 = 3Z 2 α 8a 0 is the energy difference of the hydrogen atom between states 1S and 2S, and a 0 is the Bohr radius. The coefficient of the TPE rate can be rewritten in terms of constants as: 4π 0 c = meca 0 is the fine-structure constant and Z is the atomic number. The total TPE rate is given by:

III. RUBIDIUM RYDBERG BOUND STATE WAVE FUNCTIONS
This section calculates the dipole matrix for different transitions by solving numerically the wave functions for the rubidium atom. The radial part of the wave function is calculated from the Schrodinger equation as: For the hydrogen atom, the V (r) represents the Coulomb potential (−1/r). For the rubidium atom, one has to add the effect due to the non-negligible penetration of the outer electron into the electronic cloud of the core. Therefore, V (r) is modeled as: 3 is the relativistic spin-orbit interaction. α c is the core polarizability, r c is a cutoff radius introduced to avoid divergence at short range and Z l (r) is the effective charge given by Z r (l) = 1 + (Z − 1)e −a 1 r − r(a 3 + a 4 r)e −a 2 r with l-dependent parameters a l which are obtained by fitting the model potential to measured energy levels for each element [1]. The modified energy levels are given by: where n * = n − δ nlj is the principle quantum number corrected by the quantum defect given by, Eqn. S3 is solved numerically by using a Numerov method. The integral in r starts from the outer radius at a distance r = 2n(n + 15)a 0 from the atomic core, where the wave function exponentially decays to zero. The integral is carried out inwards towards r = 0. An initial solution is assumed for the radial wave function which converges to the physical solution iteratively. Due to the nonzero quantum defect parameters δ, the wave function diverges at small r. Since the contribution at small r is negligible, the integral is stopped when it starts to diverge. In order to solve the wave function and to calculate the dipole matrix we have used various Python libraries developed by [1]. for momentum p = 0.001. As Z increases, the wave function oscillates fast, which effectively reduces its contribution for the TPE rates.

IV. THE EFFECT OF THE CONTINUUM STATES
The dipole matrix element contain the contributions of the bound states and continuum states. In case of the hydrogen atom, it has been shown that the contributions of continuum states are quite small but not negligible [2]. For calculating the continuum states we modelled the potential as V (r) = − Z r (this approximation is considered accurate for the continuum states; our model for the bound state does not rely on this approximation). The analytical expression for the continuum state for hydrogenic atom is [3]: R cont pl = C pl (2l + 1)! (2pr) l e −ipr F (iZ/p + l + 1, 2l + 2, 2ipr) where F represents a first order hypergeometric function and p = √ 2E represents the continuous momentum states in atomic units. For TPE rates, we calculate the overlap wave function of the initial and final bound states with the continuum states for the p-state l = 1. Through this calculation, we found that for rubidium atoms, the contributions of the continuum states are negligible. The reason is that the wave functions of the continuum states for rubidium oscillate much faster than for states with lower atomic number Z as shown in Fig. S1. Consequently, their contribution decreases with increasing Z as summarized in Table S1. We found that only Z = 1 contributes for TPE rates while the higher Z has no effect on TPE rates. For an ideal calculation, V (r) should be modified for the rubidium atom. But we have seen that for higher Z, the effect of the continuum states is negligible.
Thus, this should not affect the TPE rates.
where ω cn is the n th cavity mode frequency. The TPE rate is thus: The dipole moment matrix is in units of ea 0 . The coefficient of TPE can be rewritten in terms of constants as: The TPE rate can be finally written as: The photons emitted from the atom resonant with the cavity are emitted along the cavity axis. Also, the polarization of the emitted photons depends on the quantization axis which can be defined by applying an external magnetic field to the atoms. To add an additional degree of freedom to the entangled state, we propose to choose the quantization axis perpendicular to the cavity axis, so the cavity only supports σ + and σ − polarizations. In this section, we derive the analytical form for Hong-Ou Mandel interference with N-dimensional energy entangled states. The entanglement between different energy modes gives rise to interference which depends on the dimension of the entangled states. We present a concrete example for the required calculation with N = 4.
The total electric field at each of the input ports of the beam splitter BS are denoted by c and d. In each port the field is a superposition of all the energy modes with frequency ω The observation of quantum interference in two-photon coincidence measurements as a function of delay between the two photon paths confirms the entanglement between the energy modes. Thus, the time-resolved two-photon coincidence is detected when both detectors clicks; it is defined as: For N entangled pairs, the coincidence probability is generalized as For an atom inside a cavity, the frequency modes of the N-dimensional photon comb are separated by the cavity's free spectral range ν FSR . Therefore, the coincidence probability for N = 10 comb lines is P 2 (τ ) = 1 10 1 + 2 10 (9 cos(2πν FSR τ ) + 8 cos(4πν FSR τ ) + 7 cos(6πν FSR τ ) + 6 cos(8πν FSR τ ) + 5 cos(10πν FSR τ ) This equation is used in Fig. 4 to represent the coincidence probability for 10-dimensional entangled photons.

TRANSITIONS
The TPE processes can be mediated either by non-resonant processes where the atom decays through a higher energy level, or by resonant cascade processes where the atom decays through an energy state lower than the initial state. Usually, the non-resonant processes are very slow (in free space, without a cavity that enhances them) compared to the resonant ones. The resonant cascade processes involve transitions from real levels, which means that the photons are not emitted instantaneously. Instead, these processes generate energy-time correlations that depend on the lifetime of the intermediate states [4,5]. In our protocol, we suppress the contribution of the resonant cascade processes through an optical cavity where the cavity length is tuned (L = 98.417 µm) such that most of these resonant fast processes are suppressed. Table S2 summarizes the transitions from 60S 1/2 .
Thus nearly all TPE processes are due to the virtual transitions which lead to strongly entangled pairs. There are still some higher order cascade processes that are not suppressed, however their rates are lower than the TPE rates that produce our entangled photon pairs. Future designs can consider cavities with a broader spectral range that can be used to inhibit additional transitions. In Fig. S2, it