; hence, in general, for most of the systems it should be determined numerically. In this work, we employed Lumerical an FDTD-based commercial software to solve this equation numerically.
The Green's tensor is also an indication of the radiated power from the dipole in the system. Therefore, the larger the coupling to the modes the larger radiated power by the dipole. As mentioned in the previous section, for this nanobeam geometry, the cavity mode is mainly polarized along the y-direction, specifically in the XY plane of symmetry. This implies that the best coupling to various dipole orientations can be obtained for a y-directed dipole, the dipole oriented along the beam width. For the other two orientations, the radiated power is substantially lower, as the dipole and the cavity mode do not couple efficiently. This qualitative prediction can be clearly observed in the numerical calculations presented in Fig. 3a where the calculated Purcell factor for all of the x−, y -and z-oriented dipoles has been shown as a function of the dipole wavelength. As can be observed, the power emitted from a y-directed dipole substantially increases when the dipole energy approaches the cavity resonance wavelength. Moreover, the emitted power is noticeably higher compared to the power emitted from x, z-directed dipoles. Note that the radiated powers from the latter dipoles are not strictly zero since aside from the cavity mode, there are continuum of leaky modes that the dipoles could always couple to. The finite spikes in the radiation spectrum of the x−, z-oriented electric dipoles are due to some parasitic cavity effects in a finite-sized structure we considered in all of the simulations.

Casimir-Polder effect
When an emitter radiates in the vicinity of a large-index dielectric surface its radiation properties not only will be affected by the change of the local density of the optical states (LDOS) as discussed in the previous part, but also its electronic levels will be affected by a dipole-dipole interaction between the atom and its induced dipole image. This phenomenon known as Casimir-Polder potential is closely related to the modification of the emission properties of the dipole locally and is related to the Green's function via the following relation [41]: where sc (r, r; ) is the scattered part of the Green's tensor ( sc = − free space ) as defined in the previous section and n ( ) is the dynamical polarizability tensor of the emitter at frequency and in the �n⟩-state. Symbol P in front of the integral emphasizes that one has to calculate the Cauchy principal value of the integral. This is important as we need to calculate the Casimir-Polder potential of an excited atom in the vicinity of the structure, where P ( ) has a pole at D 2, corresponding to a real photon emission. Due to the large velocity of the atoms in a thermal gas the force exerted by U CP and its effect on altering the atomic trajectory can be ignored. Therefore, one only needs to include the effect of this potential on shifting the atomic resonance as ℏ CP = U CP. In general, the induced potential consists of resonant and non-resonant parts. While the former is related to all of the virtual transitions from the emitter the latter is related to the real transitions of an atom in the �n⟩-state. Unless the emitter is very close to the body, often the contribution from real photons exceeds the virtual photon parts [41]. In addition, as suggested by the results in Fig. 3a, the bodyassisted Green's function of this system is mainly peaked at the emitter resonance wavelength (i.e. res ), suggesting a further increase due to the LDOS. Therefore, for such a high-Q, single mode cavity we can limit ourselves to the resonant part and approximate the Casimir-Polder potential with the body-assisted Green's function in the vicinity of its resonance.
Employing this approximation and calculating the polarizability from the transition dipole moments and frequencies,  we have calculated the Casimir-Polder potential at several positions close to the device. In Fig. 3b, c, we show some of the calculated results of the frequency shift due to the Casimir-Polder effect ( CP ) away from the structure along the z-direction, and away from the cavity towards the hole edge along the y-direction, respectively. As can be seen, due to the induced dipole image, a Rb atom radiating at D 2 experiences about tens of MHz line shifts in the transition frequency. Also due to the attractive force between the dipole and its image the shift in the energy is always negative. Comparing these values with the transit broadening an atom experiences as it flies though the structure it can be concluded that the Casimir-Polder potential does not have a large effect. Indeed, the effect becomes more and more pronounced for slower atoms where the energy shift can lead to measurable detuning between the atom and cavity hence, diminishing the strong coupling between them.

Atom-light interactions and Monte-Carlo simulations
In the previous sections, we conducted some orders of magnitude estimation of the atom photon coupling as well as more quantitative investigations on the Casimir-Polder potential to explore the behavior of a moving atom in the vicinity of well-designed high-Q PCC with small mode volume. Those analyses suggest that the achievable coupling between a flying atom and a photon cavity is strong enough to expect quantum interference effects from this hybrid system even though moving atoms are typically subject to the large transit-time broadening. To fully evaluate the performance of the hybrid system, however, one needs to treat the atom-photon coupling more accurately. In this section, we employ a Monte Carlo scheme that combines the spatially varying photon field and the Casimir-Polder potential derived in the previous section with full quantum mechanical description of the atom-photon coupling to investigate the interaction of a moving atom with the nano-cavity. We investigate the motion of a thermal atom with a random velocity ⃗ v following a random trajectory ⃗ r(t) determined by its velocity. The evolution of the joint density matrix of the atom and photon at each time is determined via the following Liouville's equation: where (, Γ ) are the photon decay rate from the cavity and the atom lifetime in the excited state, respectively. a, a † are the annihilation and creation operators of the photon field, and −, + are the atomic lowering and raising operators, respectively. Atom-photon interaction can be properly studied with Fock states and we investigate the scenario where photons are injected with a weak, coherent excitation at the rate L. Therefore, the total Hamiltonian in the rotated frame of the laser is given via the following driven Jaynes-Cumming form as where c, a, L are the cavity, atom, and laser frequency, respectively, and g(t) is the vacuum Rabi coupling of the atom-cavity at atom position of r(t) and varies as the atom moves along the device. Also, due to the Casimir-Polder effect, the detuning between the atom and the laser is position and, therefore, time dependent as well. We would like to emphasize that using a complex Rabi coupling, as g(t), the effect of position dependence of the photonic mode phase has been inherently included. 1 Figure 4 shows the results of the Monte Carlo simulation for a moving atom in the vicinity of PCC for three different velocities and initial position. The joint density matrix has been evolved according to the above equation and the instantaneous expectation value of the intra-cavity photon numbers ⟨ a † a ⟩ has been calculated for different cavity-laser detuning values ( Δ cL ) while the atom moves in the vicinity of the cavity. The interaction time and the coupling strength between atom and cavity strongly depend on the atom velocity and its initial position; hence, the behavior drastically changes along different trajectories. The time-averaged photon number as a function of cavity-laser detuning has been shown in the second row for each case. The third row manifests the change of the intra-cavity photon number as a function of time when Δ cL =0. As can be observed the number of photons in the cavity drops when the coupling between the atom and the cavity photon is strong enough to induce a large detuning between the input laser frequency and the frequencies of the dressed states. In other words, the cavity becomes opaque to the input laser for a finite amount of time. The inverse scenario is observable as well. At large cavity-laser detuning, the presence of atom makes the cavity transparent for a while.
To produce a gated cloud of atoms one can employ the light-induced atomic-desorption (LIAD) process. In the 1 For a traveling wave with well-defined wave vector( ⃗ k ), hence a linear phase change in space, this is the well-known Doppler shift as However, due to the sub-wavelength features of the cavity field the phase is better to be included numerically and on the same ground as its amplitude.

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LIAD process, also known as the photo-electric effect for atoms, high-energy laser pulses hit the wall of an atomic vapor cell, which is coated with alkali atoms and desorb them from the surface [42][43][44][45]. When chosen properly atomic clouds as dense as 1000 atoms/ 3 can be generated [46]. The inset of Fig. 5 shows the typical trajectory of atoms released from the wall via LIAD process. Figure 5 shows the normalized intra-cavity photon number as a function of cavity-laser detuning after averaging over 100,000 atom trajectories. Due to the random paths, which give different atom-photon coupling, the dressed state splitting is trajectory dependent. However, the averaged vacuum Rabi coupling is still large enough to lead to the spitting of majority of fast moving atoms at Δ cL = 0 as