Bursts of polarised single photons from atom-cavity sources

We first present our proposal for polarisation controlled single photon production in an idealised three level system.

The atomic structure required for the production of polarised photons comprises two ground levels with total angular momenta F and F + 1, as well as one excited level with total angular momentum $F^{^{\prime}} = F$. Three of the individual states of the two ground levels are required for photon production, which we call $\{\left|g_2\right\rangle, \left|g_1^+\right\rangle, \left|g_1^-\right\rangle \}$. The latter two levels can be considered to be degenerate in the absence of a magnetic field. The excited state, $\left|x\right\rangle$, has dipole-allowed transitions to all three ground states (cf figure 1), yet the transition to $\left|g_1^0\right\rangle$ is dipole forbidden. A transition from $\left|x\right\rangle$ to either of the degenerate ground states $\left|g_1^{\pm}\right\rangle$ corresponds to the emission of photons of orthogonal circular polarisations $\sigma^{\mp}$ with respect to the chosen quantisation axis, while the transition $\left|g_2\right\rangle \to \left|x\right\rangle$ corresponds to the absorption of a photon of π-polarisation. This configuration arises naturally in alkali atoms and ions, both with and without hyperfine structure (for example $\mathrm{Rb}$, $\mathrm{Cs}$, $\mathrm{Ca}^{+}$, $\mathrm{Sr}^{+}$). Nevertheless, careful considerations with respect to the precise choice of atom must me made (cf section 3).

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The cornerstones of our scheme are a proper choice of quantisation axis and polarisation directions for the atomic driving, as well as a fast coherent re-preparation of the atom back to its original state. This enables the production of a burst of polarised single photons emitted with repetition rates on the MHz scale. Optical pumping can be used to initialise the atom at the outset and to re-initialise it after failed photon production. In what follows, we describe these cornerstones, as well as the implementation of the overall sequence.

An important aspect of our proposal is a choice of quantisation axis which is perpendicular to the cavity axis as shown in figure 2. In combination with the atomic selection rules, this uniquely defines the polarisation of the photon produced by the atom-cavity interaction.

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The cavity supports two degenerate polarisation modes, one is π-polarised and the other a linear superposition of $\sigma^{+}$ and $\sigma^-$. Without loss of generality, for a quantisation axis along the y axis, the polarisation modes supported by the cavity correspond to $\{\pi, H\}$ (in the lab frame, these correspond to polarisations parallel to the $\{x, y\}$ axes, cf figure 2). We aim to emit only into the linear polarisation modes supported by the cavity by enabling the atom to emit photons in a superposition state with respect to the quantisation axis, following ideas first introduced in [23] for H polarised photons from atom-photon entanglement.

A single atom is initially prepared in the state $\left|g_2\right\rangle$ and addressed by a laser with $\pi-$polarisation and total pulse duration T (polarised in the y direction in the lab frame, propagating along x cf figure 2), resonant with the transition from $\left|g_2\right\rangle$ to $\left|x\right\rangle$. The cavity is tuned to be resonant with the $\left|g_1^{\pm}\right\rangle \to \left|x\right\rangle$ transitions, supporting the dipole allowed $\sigma^{\mp}$ transitions while there is no dipole allowed π transition. The cavity therefore only supports the $H = (\sigma^{+} + \sigma^{-})/ \sqrt{2}$ polarisation mode. This paves the way for efficiently producing linearly polarised photons without a strong external magnetic field to address individual hyperfine states.

To produce photons, we consider a process known as V-STIRAP [24] whereupon the atomic population is adiabatically transferred from the initial state $\left|g_2\right\rangle$ to a superposition of states $\left|g_1^{\pm}\right\rangle$, leading to the following entangled atom-photon state, $\left|\Psi\right\rangle$:

$$\begin{align} \left|\Psi\right\rangle& = \frac{1}{\sqrt{2}}\left( \left|g_1^+, \sigma^-\right\rangle - \left|g_1^-, \sigma^+\right\rangle \right) \nonumber\\ & = \frac{1}{\sqrt{2}}\left(\left|\Phi_{\phi = \pi}\right\rangle\otimes\left|H\right\rangle-\mathrm i\left|\Phi_{\phi = 0}\right\rangle\otimes\left|V\right\rangle \right), \end{align} \tag{ 1 }$$

where $\left|\Phi_{\phi}\right\rangle = \frac{1}{\sqrt{2}}\left(\left|g_1^+\right\rangle+\mathrm e^{\mathrm i\phi}\left|g_1^-\right\rangle \right)$. ³ However, the cavity only supports one linear polarisation mode $H = (\sigma^{+} + \sigma^{-})/ \sqrt{2}$, where H is perpendicular to the cavity axis. V would correspond to an oscillation along the cavity axis and photons propagating perpendicular to the cavity axis. Therefore, the cavity acts like a polarisation filter, only supporting H-polarised light. As a consequence, the atom-cavity state collapses to the product state:

$$\begin{align} \left|\Psi_{H}\right\rangle = \left|\Phi_{\phi = \pi}\right\rangle\otimes\left|H\right\rangle, \end{align} \tag{ 2 }$$

which, upon photon leakage from the cavity, evolves to

$$\begin{align} \left|\Psi_{f}\right\rangle = \left|\Phi_{\phi = \pi}\right\rangle\otimes\left|0\right\rangle. \end{align} \tag{ 3 }$$

The efficiency of the V-STIRAP process is determined by estimating the cavity field decay rate with respect to the desired final state, given by an integral over the photon production time interval $[t_i, t_f]$:

$$\begin{align} \eta = 2\kappa \int_{t_{i}}^{t_f} \left\langle \Phi_{\pi}|\rho\left(t\right)\right\rangle \left\langle \rho\left(t\right)|\Phi_{\pi}\right\rangle \mathrm dt. \end{align} \tag{ 4 }$$

In the idealised state configuration, there is an equivalent analytic formulation which does not require an integral (cf the appendix for details).

We calculate various numerical photon production efficiencies in an idealised state configuration in figure 3, by assuming an atom is successfully initialised in $\left|g_2\right\rangle$ and addressed with a sinusoidal laser pulse of $\Omega_{\mathrm{VST}} = \Omega_{\mathrm{max}}\sin^2(\pi t / T )$, where the total pulse duration $T = 10 / \gamma$. The variation as a function of $\kappa/\gamma$, as well as $g/ \gamma$, where κ is the cavity field decay rate, g the atom cavity coupling strength ⁴ and γ the atomic depolarisation rate are shown in figure 3. Each data-point corresponds to a local optimisation problem for finding the optimal peak Rabi frequency $\Omega_{\mathrm{max}}$ which maximises the photon emission efficiency (cf the appendix for further details, where all the exact parameters are detailed).

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Physically, the constraints which limit the efficiency of photon production are two-fold: firstly, the need for adiabaticity in the population transfer process and second, the ability to extract the photons produced in the cavity. The strength of the atom cavity coupling g in relation to the atomic depolarisation rate γ, as well as the time-scale of the driving pulse $\Omega(t)$ with respect to the Rabi frequency Ω determine the extent to which the process can remain adiabatic. Moreover, the rate at which the photonic field decays, κ, is instrumental for extracting the photon efficiently from the cavity. As evident in figure 3 the efficiency initially increases in proportion to κ and approaches unity for higher g once $T \kappa \gt1$, because the extraction from the cavity no longer limits the efficiency of the photon production process. For shorter pulse durations, the process is also limited by a lack of adiabaticity and an increased likelihood of decoherence. We observe an efficiency trade-off exhibited by the photon production process: the desire for higher repetition rates competes with the efficiency, which will be discussed more extensively for a real atom in section 3.4.

A fundamental upper bound in the photon production efficiency obtained for V-STIRAP arises from the atom-cavity parameters. Controlling the time evolution of the Rabi frequency Ω enables the precise shaping of the time-dependent photon amplitude [14], as well as the efficiency with which it is produced. Even with lossless mirrors, the production of a temporally shaped photon is limited in efficiency by a theoretical maximum of $\eta = 2C/ (2C+1)$ [12, 14], where $C = g^2/(2\kappa \gamma)$ is the cooperativity of the cavity.

Coherent STIRAP re-preparation [25] enables fast and efficient re-preparation of the atom back into the desired initial state $\left|g_2\right\rangle$ with high re-preparation efficiencies. Unlike incoherent methods, which rely on eventual spontaneous emission into the desired state and are bounded by the atomic decay rate, coherent repumping over a time-scale T must satisfy the constraint of global adiabaticity [25], $\Omega_{\mathrm{rms}} T \gg 1$, where $\Omega_{\mathrm{rms}} = \sqrt{\hat{\Omega}_\mathrm S^2+\hat{\Omega}_\mathrm P^2}$, such that $\hat{\Omega}$ describes the time average amplitude over the pulse duration for the Stokes pulse $\Omega_\mathrm {S}$ and the Pump pulse $\Omega_\mathrm {P}$. The desire for short pulses with high repetition rates can be achieved with intense pulses with correspondingly high Rabi frequencies. Moreover, fully coherent atomic population transfer is reversible, essential for the implementation of quantum networks [26, 27] and for the preparation of highly entangled optical states with a single atomic memory [19].

After a photon has been successfully emitted, the atom, which is left in the state $\left|\Phi_{\pi}\right\rangle$, can be prepared back into the desired initial state $\left|g_2\right\rangle$ using a two pulse STIRAP [28] procedure which involves using two laser pulses (Stokes and Pump) to transfer the atomic population back to its initial state. Coherent methods like STIRAP are generally extremely good for achieving high population transfer efficiencies [29, 30] and are also fast compared to incoherent optical pumping.

To perform coherent repumping, the pump pulse must be resonant with the transitions from the groundstates $\left|g_{1}^{+}\right\rangle$ and $\left|g_{1}^{-}\right\rangle$, to $\left|x\right\rangle$ and a π polarised Stokes pulse must be resonant with respect to the $\left|g_2\right\rangle \rightarrow \left|x\right\rangle$ transition ⁵ . The pump pulse must be polarised in a linear combination of $\sigma^+$ and σ⁻ polarisation with respect to the quantisation axis, maintaining a particular phase relation between the circularly polarised components(cf equation (3)), such that it is linearly polarised perpendicularly both to the cavity and the quantisation axis ⁶ . As a result, the polarisation of the laser light and thus the phase between the two circularly polarised components of the light must match the phase of photon polarisation derived from equation (3). Both laser pulses must have equal effective Rabi frequencies ⁷ .

By carefully shaping the time dependent amplitudes of two pulses which address the transitions $\left|g_1^{\pm}\right\rangle \to \left|x\right\rangle$ and $\left|x\right\rangle \to \left|g_2\right\rangle$, it is possible to maximise re-preparation efficiency between the states $\left|g_1^{\pm}\right\rangle$ and $\left|g_2\right\rangle$. Various pulse-shaping techniques have been developed extensively, such as optimal control methods [30-32], reinforcement learning [33] or other numerical methods to minimise the non-adiabatic transitions [29]. It should be noted that for experimentally feasible pulse shapes, one requires $\Omega_\mathrm {S}(t = 0) = \Omega_\mathrm {P}(t = 0) = \Omega_\mathrm {S}(t = T) = \Omega_\mathrm {P}(t = T) = 0$, where T is the total pulse sequence duration ⁸ . With this in mind, the highest population transfer efficiency was attained by adopting a technique that involved employing two interleaved pulses of constant rms Rabi frequency $\Omega_{\mathrm{rms}}$, which were applied with a hyper-Gaussian mask [29]. The shape of the mask was numerically optimised for maximal population transfer efficiency (for more details cf the appendix). The simulated STIRAP re-preparation efficiency approach unity in the idealised system configuration ($\eta_{\mathrm{max}} \approx 0.997$ in figure 4), with losses parameterised by γ, as shown in figure 4.

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There is a trade-off however, between efficiency, repetition rate and driving laser intensity. The variation of the repumping efficiency with re-preparation time and peak Rabi frequencies is shown in figure 4. In general, faster re-preparation requires larger peak Rabi frequencies and in the idealised system configuration one can in principle arbitrarily increase the peak Rabi frequency to increase the efficiency, this is in stark contrast to many real atomic systems where these are limited.

The V-STIRAP process lies at the heart of the photon production scheme and its merits have been demonstrated for an idealised three level system in section 2.3. We now consider the full atomic structure of $^{87}\mathrm{Rb}$ and carefully determine the optimal parameters and efficiency trade-offs exemplified by this real atomic quantum system.

The relevant atomic states are shown in figure 5. After initialising the atom into the $\left|F = 2, m_F = 0\right\rangle$ ground state, a π polarised driving pulse, near resonant with the transition to $\left|F^{\prime} = 1, m_{F^{\prime}} = 0\right\rangle$ addresses the atom and the cavity is tuned to be resonant, such that one transfers the atom into the superposition state:

$$\begin{align} \left|\Phi_{\phi = \pi}\right\rangle = \frac{1}{\sqrt{2}}\left(\left|F = 1, m_F = 1\right\rangle-\left|F = 1, m_{F} = -1\right\rangle\right). \end{align} \tag{ 5 }$$

A linearly polarised photon is then emitted preferentially through one of the cavity mirrors.

It was shown in section 2.3 that in order to push the efficiencies near unity, one requires strong atom cavity coupling with respect to the atomic depolarisation rate and long photons or, alternatively, a fast photonic field decay. Some specific cavity regimes are explored in figure 6. We consider a fixed pulse of length 500 ns which takes the following form: $\Omega_{\mathrm{VST}} = \Omega_{\mathrm{max}}\sin^2(\pi t/T)$. Every particular cavity parameter configuration corresponds to a local optimisation problem with regards to the detuning Δ and peak Rabi frequency of the laser pulse $\Omega_{\mathrm{VST}}$. These are numerically optimised for maximising the photon production probability (cf equation (4)). In general, the optimal peak Rabi frequency increases with regards to an increase in atom-cavity coupling (cf the appendix for further details).

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Figure 6 shows that the D₁ transitions lead to higher efficiencies than the D₂ transitions. To compare the two possible transition lines more explicitly, we consider a realistic photonic field decay rate of $\kappa = 2 \times 2\pi$ MHz and vary only the atom-cavity coupling strength g as exemplified in figure 7. Importantly, unlike in an idealised three level system (cf figure 3), the efficiency does not asymptotically approach a theoretical maximum, as the atom cavity coupling strength (and cooperativity) further increase (cf figure 7). This can be attributed to the fact that, stronger coupling necessitates higher peak Rabi frequencies $\Omega_{\mathrm{VST}}$ which lead to undesirable off-resonant couplings to other excited levels. For the D₂ line, the excited levels $F^{\prime} = 1$ and $F^{\prime} = 2$ are split only by ${\approx}160$ MHz [34]. As the strength of the atom-cavity coupling increases beyond ${\approx}10 \times 2\pi$ MHz, we observe no further increase in V-STIRAP photon production efficiency on D₂ as shown in figure 7. The excited levels $F^{\prime} = 1,2$ for the D₁ transitions are significantly further detuned at ${\approx}816$ MHz. Thus, for the D₁ scheme, a decline in efficiency is not visible up to the couplings shown in figure 7.

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It shall also be stressed that the same physical cavity mirror parameters do not give rise to the same atom-cavity coupling strength for different atomic transitions, because the atom cavity coupling g is a function of the Clebsch–Gordan coefficients of the particular transitions between the states $\left|g_1^{\pm}\right\rangle, \left|x\right\rangle$ and $\left|g_2\right\rangle$. For the same physical cavity parameters, the coupling strengths scale as $g_{D_2}\approx g_{D_1} \times 2.3$. The trade-off between cavity mode volume and single photon production efficiency is also shown in figure 7 to compare existing cavity designs. There exists a cross-over point at a mode volume of around $2 \times 10^3\, \mathrm{nm}^3$ below which it becomes advantageous to use the D₁ line for photon production, but for most existing physical cavities [35], the D₂ line is preferential.

The coherent re-preparation scheme described in section 2.5 can be applied to the real atomic system of $^{87}\mathrm{Rb}$, which offers either the D₁ oder D₂ transitions for coherently re-preparing the atom back into the desired initial state with two counter-intuitively sequenced laser pulses of appropriate polarisations (cf the appendix for further details). In general, when simulating the full atomic structure we observe similar trade-offs noted in the previous section for polarised photon production with V-STIRAP.

We follow the same techniques introduced in section 2.5 and consider two interleaved pulses of constant rms Rabi frequency with a hyper-Gaussian mask [29] with numerically optimised shape and peak Rabi frequency (cf the appendix for further details). This leads to the highest population transfer efficiencies at the desired time-scales. As for the photon production process, off-resonant coupling to other excited levels is the main source of depolarisation and adversely affects the repumping efficiency, particularly for extremely fast repumping which requires high peak Rabi frequencies, as exemplified in figure 8 for both the D₁ and D₂ transitions. We observe that for re-preparation times of up to 150 ns the maximal population transfer efficiency for re-preparation on these lines is 0.95 and 0.83 respectively. Like in the V-STIRAP process, the relative proximity of the higher lying excited levels detrimentally affects the efficiency of the re-preparation with respect to the D₂ transitions. It should be noted, that the presence of the cavity may also adversely affect the re-preparation efficiency, as it produces photons on the same transitions used to coherently re-prepare the atom back into the initial state. However, significantly detuning the cavity from the resonantly driven transition to $F^{\prime} = 1, m_F = 0$ ensures that there is no such effect.

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Due to the reduced achievable efficiencies with the D₂ line (cf figure 8) and the adverse affect of a near-resonant cavity with respect to the re-preparation efficiency, we conclude that, given current state of the art cavity parameters [35], the optimal photon production scheme uses a combination of V-STIRAP with respect to the D₂ transitions for single photon production, as well as STIRAP re-preparation via the D₁ transitions. It exploits the optimal efficiency of the underlying processes and avoids the introduction of further losses in efficiency, as the cavity is now extremely far detuned from the transitions used for quickly re-preparing the atom back into the desired initial state.

An $^{87}\mathrm{Rb}$ atom loaded into an optical cavity is initially populated in a random ground state. Moreover, during the photon production cycle consisting of V-STIRAP and coherent re-preparation, the atom may spontaneously decay into any Zeeman state. For these reasons, it is necessary to use an optical pumping process to efficiently prepare the atom into the desired initial $\left|F = 2, m_F = 0\right\rangle$ state.

Optical pumping for the D₁ and D₂ transitions requires two π polarised lasers (with respect to the y axis, travelling along the x axis, as shown in figure 5) which are nearly resonant with the $F = 1 \rightarrow F^{\prime} = 2$ and $F = 2 \rightarrow F^{\prime} = 2$ transitions. Both lasers act simultaneously and through spontaneous emission all population eventually accumulates in the $F = 2, m_F = 0$ groundstate, because the transition from $F = 2, m_F = 0$ $\rightarrow$ $F^{\prime} = 2, m_F^{\prime} = 0$ is dipole forbidden. This is shown in figure 5.

Resonant pulses exhibit low population transfer efficiencies due to the creation of two-photon resonances which would repopulate other states of the F = 2 level, due to the formation of unwanted dark states formed by superpositions of individual sub states of F = 1 and F = 2. The detunings of both π polarised pulses from resonance, as well as their peak Rabi frequencies are numerically optimised for maximal population transfer to the desired state $\left|F = 2, m_F = 0\right\rangle$, confer the appendix for a detailed description of the parameters. The efficiency with which initialisation or re-preparation with optical pumping occurs over different time scales and various initial conditions is shown in figure 9 for the D₁ line. This is more efficient than using the D₂ transitions, which has additional excited atomic levels that adversely affect the population transfer efficiency. In principle, one can achieve arbitrarily high efficiencies with extremely long pulses. It should be stressed that the time scale at which optical pumping occurs is about an order of magnitude longer than that of coherent re-preparation because of the random walk of the population back into the desired initial state.

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Following the analysis of the previous two sections, we consider the combination of optical pumping, V-STIRAP, and STIRAP with respect to the D₂ and D₁ transitions, respectively, for polarised photon stream production.

We present simulated efficiencies for generating bursts of polarised photons with recurring cycles of D₂ V-STIRAP photon production and D₁ STIRAP re-preparation from a state of the art optical cavity [35] and $^{87}\mathrm{Rb}$ atoms—which for simplicity are assumed to be stationary ⁹ —in figure 10. It should be noted that the proposal outlined in section 2 and applied to $^{87}\mathrm{Rb}$ describes a perfectly degenerate state configuration, yet perfect degeneracy is nearly impossible to achieve due to the small B-field fluctuations. If these cause Larmor precessions on the time scales of the photons, indistinguishability is affected as the polarisation is no longer constant in time. One can mitigate this by imposing a small constant external field along the preferred quantisation axis (i.e. y) which induces a hyperfine state splitting smaller than the line-width of the transition or the cavity. In this case, the scheme is robust in the face of small external magnetic fields and a detailed technical discussion is provided in the appendix.

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The repetition rate of the V-STIRAP and STIRAP sequence can be altered by modifying the duration of either or both. In what follows we fix the coherent STIRAP re-preparation duration to 140 ns. Shorter re-preparation pulses lead to reduced efficiencies without significantly improving the overall photon production repetition rate and longer pulses only marginally increase efficiency. We do however consider two different V-STIRAP photon driving pulses of length 360 and 860 ns, which lead to overall repetition rates of 2 and 1 MHz respectively. Longer driving leads to higher single photon production efficiencies, as evident in figure 10, yet the expected number of single photon counts in a particular time interval is lower. However, when the photon burst length exceeds six or more consecutive photons, the expected count rate is larger for the photons which are driven more slowly at a repetition rate of 1 MHz.

Neither the V-STIRAP photon production, nor the STIRAP re-preparation are perfectly efficient. Thus, it is important to characterise the effects of decoherence on the produced photon stream. Atomic depolarisation during the V-STIRAP procedure may manifest itself, such that the atom is not prepared into a superposition of the states of the F = 1 level. This adversely affects the coherent STIRAP re-preparation efficiency which only correctly re-prepares the atom into the initial state if starting from the superposition state $\left|\Phi_{\pi}\right\rangle$. Moreoever, atomic depolarisation during STIRAP re-preparation can similarly lead to incoherent decay of the atom into the wrong Zeeman state. Simulations for a state of the art optical cavity [35] and optimal pulse parameters show that for 10 consecutive photon production attempts with STIRAP coherent re-pumping, a non-negligible accrual of population in other states of the F = 2 manifold is unavoidable. The population of undesired Zeeman states however leads to a small, efficiency gain with respect to the production of photon sequences with fixed linear polarisation. Even when the atomic population falls to the wrong Zeeman state after re-preparation, we can obtain a useful photon (of desired H polarisation). This is exemplified in table 1 (for further details see the appendix), as for instance when starting in the state $F = 2, m_F = -2$ the probability of emitting an H-polarised photon is 0.188.

Table 1. Probability of emitting a photon of particular polarisation, for a cavity with C ≈ 10 [35] when applying the optimised driving pulse when correctly initialised (i.e. $m_F = 0$) and when incorrectly initialised into the other sub-states. The purity, i.e. $p(H) / (p(H)+p(\pi))$ is also shown and decreases significantly for the $m_{F} = \pm 1$ sub-states.

$F = 2, m_F =$	−2	−1	0	1	2
p(H)	0.188	0.158	0.853	0.158	0.188
$p(\pi)$	0.049	0.710	0.009	0.710	0.049
Purity	0.8	0.182	0.99	0.182	0.8

Practically, undesired π polarised photons would be detected at the cavity output with a polarising beam splitter, indicating the failure of a particular coherent photon productions sequence. This could be used to immediately trigger incoherent re-preparation with optical pumping before initialising another coherent photon production sequence.

We also consider a possible sequence where the atom is re-prepared entirely incoherently with optical pumping lasting for 2.5 µs and with the V-STIRAP process lasting for 500 ns, yielding a single photon repetition rate of 0.33 MHz. This has the advantage of achieving repumping efficiencies close to unity for longer time scales, regardless of the initial atomic state after photon production. This of course incurs other costs; the repetition rate is significantly reduced and the atom does not maintain coherence throughout the process which precludes the performance of quantum information processing tasks with the atom as an entanglement mediator [19]. However, the likelihood to obtain counts for longer photon sequences (${\gt}7$) exceeds those achievable with coherent re-preparation.

B.1.1. V-STIRAP photon production.

In section 2 an idealised atomic emitter was considered with four configured states ${\left|x\right\rangle, \left|g_2\right\rangle, \left|g_1^-\right\rangle, \text{and} \left|g_1^+\right\rangle}$ and various cavity parameters for producing photons. The V-STIRAP driving pulse took the following form: $\Omega_{\mathrm{VST}}(t) = \Omega_{\mathrm{max}} \sin^2(\pi t / T )$, where $\Omega_{\mathrm{max}}$ is the peak Rabi frequency and $T = 10\gamma$ (different pulse lengths were omitted in favour of brevity and the trends for different pulse lengths were generally similar). The peak Rabi frequencies were numerically optimised for maximising photon production probability. Since the system is idealised, we normalised everything with respect to γ and then varied g and κ, the atom cavity coupling, as well as the photonic field decay rate κ. As evident in figure B.1, the optimal peak Rabi frequencies for these atom-cavity coupling strengths increase drastically as a function of g, especially for fast field decay. Changing the detuning from the state $\left|x\right\rangle$ has close to no effect on the efficiency calculation and hence all V-STIRAP processes considered here are resonant. It should also be noted that we only approach the theoretical upper limit in photon production efficiency (cf figure B.1) for higher cavity cooperativities in the regimes of high g and low κ or for very large κ, i.e. in the fast cavity regime.

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B.1.2. STIRAP re-preparation.

We follow a technique introduced in [29] and consider two pulses of constant rms Rabi frequency $\sqrt{(\Omega_\mathrm {S}^2+\Omega_\mathrm {P}^2)}$ with a hyper-Gaussian mask with numerically optimised shape. The analytic expressions for the pulse shapes of total duration T and peak Rabi frequency $\Omega_{\mathrm{max}}$ are as follows:

$$\begin{align} \Omega_{S}\left(t\right) & = \Omega_{\mathrm{max}}\cdot \mathrm e^{\left(-\left(\frac{t-T/2}{c}\right)^{2n}\right)}\sin\left(\frac{\pi}{2\cdot\left(1+\mathrm e^{-\left(\frac{a\left(t-T/2\right)}{T}\right)}\right)}\right)\nonumber\\ \Omega_{P}\left(t\right) & = \Omega_{\mathrm{max}}\cdot \mathrm e^{\left(-\left(\frac{t-T/2}{c}\right)^{2n}\right)}\cos\left(\frac{\pi}{2\cdot\left(1+\mathrm e^{-\left(\frac{a\left(t-T/2\right)}{T}\right)}\right)}\right) \end{align} \tag{ B.1 }$$

where (n, a) are control parameters and c = 0.05 is chosen such that both pulses exhibit (within the numerical limit of the solver) zero amplitude at t = 0 and t = T. This is a desiderata for making the simulation more realistic. The optimal values of (n, a) for the idealised system are $(6,14)$. The pulses exhibited shown in figure B.2 lead to re-preparation efficiencies which approach unity at large peak Rabi frequencies. However, this method does not take into account the entire structure of a real atom where further losses arise from off-resonant coupling to undesired levels.

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In section 3.1 we consider a real Rubidium atom with all the configured energy levels and their hyperfine states for both the D₁ and D₂ lines respectively. For D₁ this encompasses $F = (1,2)$ and $F^{\prime} = (1,2)$ levels, such that there are 8 ground and excited states respectively. For D₂ this requires the addition of $F^{\prime} = 0$ and $F^{\prime} = 3$ levels such that there are a total of 16 excited states. The couplings and state configuration used in all subsequent simulation, are for the perfectly degenerate case (excluding section B.2.1 where we explicitly vary the external magnetic field).

B.2.1. Quantisation axis & external magnetic field.

The proposed photon production scheme does not require a strong external magnetic field to significantly split the hyperfine states, such that they can be addressed individually. As shown in [22], in $^{87}\mathrm{Rb}$ this approach led to the apparition of non-linear Zeeman effects which strongly curtails achievable efficiencies. To ensure that the atomic quantisation does not vary erratically in time, we do however seek to impose a weak external magnetic field, otherwise background field fluctuations would adversely affect photon polarisation purity within a particular burst. The achievable simulated efficiencies in section 3 all assume perfectly degenerate ground-states, yet in reality we seek to impose a small external magnetic field, aligned with the quantisation axis which in turn imposes a small hyperfine state splitting. As demonstrated in figure B.3 this affects the phase φ of the atomic superposition state Φ:

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$$\begin{align} \left|\Phi_\phi\right\rangle = \frac{1}{\sqrt{2}}\left(\left|F = 1, m_F = 1\right\rangle+\mathrm e^{\mathrm i\phi}\left|F = 1, m_{F} = -1\right\rangle\right). \end{align} \tag{ B.2 }$$

Stronger hyperfine splitting eventually rotates the phase by an entire revolution and this does not only rotate the polarisation of the emitted photon, but also adversely affects the achievable photon production efficiencies as shown in figure B.3. A splitting of 1 MHz corresponds to a field of 1.43 G, therefore, in the mG regime, as demonstrated in [19], the phase is close to the desired phase of π and the efficiencies are very close to the limit achieved in a perfectly degenerate state configuration. Moreover, strong hyperfine splitting would require a modification of the polarisation of the STIRAP pump beam, such that its polarisation beam matches the polarisation of the emitted photon (cf equation (B.2)) where φ corresponds to the phase of the $\sigma^{\pm}$ beams.

B.2.2. V-STIRAP photon production.

As for the idealised system, there exists an optimisation problem at each particular cavity parameter configuration $(g, \kappa)$, to find the optimal peak Rabi frequency and detuning for maximising polarised photon production probability. The optimal peak Rabi frequencies at various atom cavity coupling strengths g and fixed $\kappa = 2 \times 2 \pi$ MHz are given in figure B.4. One generally observes an increase with g and Ω varies approximately linearly for low coupling strengths. Thereafter, Ω varies more erratically for higher coupling strengths as competing off-resonance effects adversely affect the photon production probability. Detuning the V-STIRAP process further from the atomic resonance had little effect on the efficiency, although detuning the cavity is important insofar as STIRAP re-preparation is performed on the same transitions used for photon production.

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B.2.3. STIRAP repumping.

The STIRAP re-preparation on either D₁ oder D₂ transitions comprises of a Stokes pulse which is π polarised and resonant with the F = 2 to $F^{\prime} = 1$ transition, as well as a pump pulse which comprises of two $\sigma_{\pm}$ polarised pulses which are resonant with the F = 1 to $F^{\prime} = 1$ transitions.

In a real atomic system, the peak Rabi frequency has to be treated as an additional control parameter due to the full atomic structure. We follow the same principles introduced in section B.1.2 and consider two pulses of constant rms Rabi frequency with a hyper-Gaussian mask with numerically optimised Gaussian shape and peak Rabi frequencies (cf equation (B.1)). Thus, $(n,a, \Omega_{\mathrm{max}})$ are free parameters and c = 0.05 is chosen such that both pulses exhibit (within the numerical limit of the solver) zero amplitude at t = 0 and t = T. We find the optimal parameters for maximal population transfer efficiency to be as follows: $(n,a) = (6,11)$ and $(\Omega^{D_1}_{\mathrm{max}}, \Omega^{D_2}_{\mathrm{max}}) \approx (41,49) \times 2\pi$. At a fixed repumping time of 150 ns, the maximal population transfer efficiencies with these pulse shapes are $\approx (0.95,0.83)$. The repumping pulse shapes for D₁ are depicted in figure B.5.

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It was found that for a time independent pulse detuning, resonant pulses gave the highest population transfer efficiency, but if one seeks to further improve the efficiency, a possible extension is to chirp the pulses by treating the pulse amplitude as a complex parameter.

B.2.4. Optical pumping.

Optical Pumping proceeds with two π polarised top-hat pulses which have different detunings from the $F = 1 \rightarrow F^{\prime} = 2$ and $F = 2 \rightarrow F^{\prime} = 2$ transitions, as well as different peak Rabi frequencies. Numerical optimisation with these control parameters was performed to determine the optimal population efficiency to the desired initial state $F = 2, m_F = 0$. The simulation results exhibited low pumping efficiencies for resonant pulses due to the creation of two-photon resonances which would repopulate other states of the F = 2 level, due to the formation of unwanted dark states formed by individual sub states of F = 1 and F = 2. Optimal detunings are determined to be $\Delta_1 = 4$ MHz and $\Delta_2 = -7.5$ MHz, for the F = 1 and F = 2 transitions respectively.

Both the D₁ and D₂ lines were considered for repumping, the latter is shown in figure B.6. The optimal peak Rabi frequencies for the D₁ transitions were $\Omega_1 / 2\pi = d_{(1, m_i) \rightarrow (2^{\prime}, m_i)}\times 34$ MHz which addresses F = 1 and $\Omega_2 / (2\pi) = d_{(2, m_i) \rightarrow (2^{\prime}, m_i)}\times 24$ MHz which addresses F = 2. The optimal peak Rabi frequencies for the D₂ transitions are $\Omega_1 / 2\pi = d_{(1, m_i) \rightarrow (2^{\prime}, m_i)}\times 57.5$ MHz which addresses F = 1 and $\Omega_2 /2\pi = d_{(2, m_i) \rightarrow (2^{\prime}, m_i)}\times 25.5$ MHz which addresses F = 2 and where $d_{i,j}$ describe the Clebsch–Gordan coefficients of the transition. The difference in the numerical factors can be attributed to the difference in Clebsch–Gordan coefficients $d_{i,j}$ of the two Fʹ levels.

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B.2.5. Photon production cycle.

It was important to analyse the distribution of of population after several V-STIRAP & STIRAP repumping cycles to consider the resulting photon polarisation when a V-STIRAP pulse addresses the atom, after it has been re-prepared to the wrong initial state. In table B.1 we show the distribution of atomic population in the F = 2 level after several V-STIRAP and STIRAP cycles. Significant accrual in undesired states adversely affects the efficiency for larger photon chains and since the π transitions are no longer dipole forbidden for the $m_F = \pm 1$ states, polarisation errors are introduced, as exemplified in figure B.7. This can however be experimentally mitigated by filtering conditional on photon polarisation at the cavity output. The optimal parameters for the simulations showcasing expected generation efficiencies for various photon chains, as well as the expected count rates in figure 10 for an existing cavity design [35], are as follows. $\Omega_{\mathrm{vst}}\approx105$ and $69\times2\pi$ for V-STIRAP pulses of duration $0.36\,\mu$s and $0.86\,\mu$s respectively, with no detuning with respect to the $\left|F = 2, m_F = 0\right\rangle \rightarrow \left|F^{\prime} = 1, m_{F^{{\prime}}} = 0\right\rangle$ transition and the cavity is resonant with the $\left|\Phi_{\pi}\right\rangle \rightarrow \left|F^{\prime} = 1, m_{F^{\prime}} = 0\right\rangle$. The optimal pulses are as shown in figure B.5 for re-preparation via the D₁ transition.

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Table B.1. Population after several repumping cycles.

$F = 2, m_F =$	−2	−1	0	1	2
n = 1	0	0.01	0.878	0.01	0
n = 10	0.121	0.078	0.44	0.078	0.121
n = 100	0.129	0.109	0.358	0.109	0.129