Optimal storage of a single photon by a single intra-cavity atom

The state of the system, composed of the cavity mode, the atom, and the modes of the transmission line, is described by the density operator $\hat{\rho }$. Its dynamics is governed by the master equation (ℏ = 1)

$$\begin{eqnarray}&&{\partial }_{t}\hat{\rho }=-{\rm{i}}[\hat{H}(t),\hat{\rho }]+{{ \mathcal L }}_{\mathrm{dis}}\hat{\rho },\end{eqnarray} \tag{ 1 }$$

where Hamiltonian $\hat{H}(t)$ describes the coherent dynamics of the modes of the electromagnetic field outside the resonator, of the single-mode cavity, of the atom's internal degrees of freedom, and of their mutual coupling. The incoherent dynamics, in turn, is given by superoperator ${{ \mathcal L }}_{\mathrm{dis}}$, and includes spontaneous decay of the atomic excited state, at rate γ, and cavity losses due to the finite transmittivity of the second cavity mirror as well as due to scattering and/or finite absorption of radiation at the mirror surfaces, at rate ${\kappa }_{\mathrm{loss}}$.

We first provide the details of the Hamiltonian. This is composed of two terms, $\hat{H}(t)={\hat{H}}_{\mathrm{fields}}+{\hat{H}}_{{\rm{I}}}(t)$. The first term, ${\hat{H}}_{\mathrm{fields}}$, describes the coherent dynamics of the fields in absence of the atom. It reads

$$\begin{eqnarray}&&{\hat{H}}_{\mathrm{fields}}=\displaystyle \sum _{k}({\omega }_{k}-{\omega }_{{\rm{c}}}){\hat{b}}_{k}^{\dagger }{\hat{b}}_{k}+\displaystyle \sum _{k}{\lambda }_{k}({\hat{a}}^{\dagger }{\hat{b}}_{k}+{\hat{b}}_{k}^{\dagger }\hat{a}),\end{eqnarray} \tag{ 2 }$$

and is reported in the reference frame of the cavity mode frequency ${\omega }_{{\rm{c}}}$. Here, operators ${\hat{b}}_{k}$ and ${\hat{b}}_{k}^{\dagger }$ annihilate and create, respectively, a photon at frequency ${\omega }_{k}$ in the transmission line, with $[{\hat{b}}_{k},{\hat{b}}_{k^{\prime} }^{\dagger }]={\delta }_{k,k^{\prime} }$. The modes ${\hat{b}}_{k}$ are formally obtained by quantizing the electromagnetic field in the transmission line and have the same polarization as the cavity mode. They couple with strength ${\lambda }_{k}$ to the cavity mode, which is described by a harmonic oscillator with annihilation and creation operators a and ${a}^{\dagger }$, where $[\hat{a},{\hat{a}}^{\dagger }]=1$ and $[\hat{a},{\hat{b}}_{k}]=[\hat{a},{\hat{b}}_{k}^{\dagger }]=0$. In the rotating-wave approximation the interaction is of beam-splitter type and conserves the total number of excitations. The coupling λ_k is related to the radiative damping rate of the cavity mode by the rate $\kappa =L| \lambda ({\omega }_{{\rm{c}}}){| }^{2}/c$, with $\lambda ({\omega }_{{\rm{c}}})$ the coupling strength at the cavity-mode resonance frequency [22] and L the length of the transmission line. Note that κ is the cavity decay rate because of transmission into the transmission line and is necessary for the storage, while ${\kappa }_{\mathrm{loss}}$ is the decay rate into other modes and is only detrimental.

The atom–photon interaction is treated in the dipole and rotating-wave approximation. The transition $| g\rangle \to | e\rangle$ couples with the cavity mode with strength (vacuum Rabi frequency) g. Transition $| r\rangle \to | e\rangle$ is driven by a classical laser with time-dependent Rabi frequency Ω(t), which is the function to be optimized in order to maximize the probability of transferring the excitation into state $| r\rangle$. The corresponding Hamiltonian reads

$$\begin{eqnarray}&&{\hat{H}}_{{\rm{I}}}=\delta | r\rangle \langle r| -{\rm{\Delta }}| e\rangle \langle e| +[g| e\rangle \langle g| \hat{a}+{\rm{\Omega }}(t)| e\rangle \langle r| +{\rm{h}}.{\rm{c}}.],\end{eqnarray} \tag{ 3 }$$

where ${\rm{\Delta }}={\omega }_{{\rm{c}}}-{\omega }_{e}$ is the detuning between the cavity frequency ${\omega }_{{\rm{c}}}$ and the frequency ω_e of the $| g\rangle -| e\rangle$ transition, while $\delta ={\omega }_{r}+{\omega }_{{\rm{L}}}-{\omega }_{{\rm{c}}}$ is the two-photon detuning which is evaluated using the central frequency ${\omega }_{{\rm{L}}}$ of the driving field Ω(t). Here, we denote by ${\omega }_{r}=({E}_{r}-{E}_{g})/{\hslash }$ the frequency difference (Bohr frequency) between the state $| r\rangle$ (of energy E_r) and the state $| g\rangle$ (of energy E_g). Unless otherwise stated, in the following we assume that the condition of two-photon resonance δ = 0 is fulfilled.

The irreversible processes that we consider in our theoretical description are (i) the radiative decay at rate γ from the excited state $| e\rangle$, where photons are emitted into free field modes other than the modes ${\hat{b}}_{k}$ introduced in equation (2), and (ii) the cavity losses at rate ${\kappa }_{\mathrm{loss}}$ due to absorption and scattering at the cavity mirrors and to the finite transmittivity of the second mirror. We model each of these phenomena by Born–Markov processes described by the superoperators ${{ \mathcal L }}_{\gamma }$ and ${{ \mathcal L }}_{{\kappa }_{\mathrm{loss}}}$, respectively, such that ${{ \mathcal L }}_{\mathrm{dis}}={{ \mathcal L }}_{\gamma }+{{ \mathcal L }}_{{\kappa }_{\mathrm{loss}}}$ and

$$\begin{eqnarray}&&{{ \mathcal L }}_{\gamma }\hat{\rho }=\gamma (2| {\xi }_{e}\rangle \langle e| \hat{\rho }| e\rangle \langle {\xi }_{e}| -| e\rangle \langle e| \hat{\rho }-\hat{\rho }| e\rangle \langle e| ),\end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray}&&{{ \mathcal L }}_{{\kappa }_{\mathrm{loss}}}\hat{\rho }={\kappa }_{\mathrm{loss}}(2\hat{a}\hat{\rho }{\hat{a}}^{\dagger }-{\hat{a}}^{\dagger }\hat{a}\hat{\rho }-\hat{\rho }{\hat{a}}^{\dagger }\hat{a}).\end{eqnarray} \tag{ 4b }$$

Here, $| {\xi }_{e}\rangle$ is an auxiliary atomic state where the losses of atomic population from the excited state $| e\rangle$ are collected.

The model is one-dimensional, the transmission line is at x < 0, and the cavity mirror is at position x = 0. The single incident photon is described by a superposition of single excitations of the modes of the external field [23]

$$\begin{eqnarray}&&| {\psi }_{\mathrm{sp}}\rangle =\displaystyle \sum _{k}{{ \mathcal E }}_{k}{\hat{b}}_{k}^{\dagger }| \mathrm{vac}\rangle ,\end{eqnarray} \tag{ 5 }$$

where $| \mathrm{vac}\rangle$ is the vacuum state and the amplitudes ${{ \mathcal E }}_{k}$ fulfill the normalization condition ${\sum }_{k}| {{ \mathcal E }}_{k}{| }^{2}=1$. For the studies performed in this work, we will consider the amplitudes

$$\begin{eqnarray}&&{{ \mathcal E }}_{k}=\sqrt{\displaystyle \frac{c}{2L}}{\int }_{-\infty }^{\infty }{\rm{d}}t{{\rm{e}}}^{{\rm{i}}({kc}-{\omega }_{{\rm{c}}})t}{{ \mathcal E }}_{\mathrm{in}}(t)\end{eqnarray} \tag{ 6 }$$

with c the speed of light, L the length of the transmission line, and

$$\begin{eqnarray}&&{{ \mathcal E }}_{\mathrm{in}}(t)=\displaystyle \frac{1}{\sqrt{T}}{\rm{{\rm{sech}} }}\left(\displaystyle \frac{2t}{T}\right)\end{eqnarray} \tag{ 7 }$$

the input amplitude at the position x = 0, with T the characteristic time determining the coherence time ${T}_{{\rm{c}}}$ of the photon, ${T}_{c}=\pi T/4\sqrt{3}$ (see definition in equation (10)). Our formalism applies to a generic input envelope, nevertheless the specific choice of equation (7) allows us to compare our results with previous studies, see [19–21]. The total state of the system at the initial time t = t₁ is given by the input photon in the transmission line, the empty resonator, and the atom in state $| g\rangle$. In particular, the dynamics is analyzed in the interval $t\in [{t}_{1},{t}_{2}]$, with ${t}_{1}\lt 0$, ${t}_{2}\gt 0$ and $| {t}_{1}| ,{t}_{2}\gg {T}_{{\rm{c}}}$, such that (i) at the initial time there is no spatial overlap between the single photon and the cavity mirror and (ii) assuming that the cavity mirror is perfectly reflecting, at $t={t}_{2}$ the photon has been reflected away from the mirror.

The initial state is described by the density operator $\hat{\rho }({t}_{0})=| {\psi }_{0} \rangle \langle {\psi }_{0}|$, where

$$\begin{eqnarray}&&| {\psi }_{0}\rangle =| g\rangle \otimes | 0{\rangle }_{c}\otimes | {\psi }_{\mathrm{sp}}\rangle ,\end{eqnarray} \tag{ 8 }$$

and $| 0{\rangle }_{c}$ is the Fock state of the resonator with zero photons.

Our target is to store the single photon into the atomic state $| r\rangle$ by shaping the laser field Ω(t). When comparing different storage approaches, it is essential to have a figure of merit characterizing the performance of the process. In accordance with [21] we define the efficiency η of the process as the ratio between the probability to find the excitation in the state $| {\psi }_{T}\rangle =| r\rangle \otimes | 0{\rangle }_{c}\otimes | \mathrm{vac}\rangle$ at time t and the number of impinging photons between t₁ and t, namely

$$\begin{eqnarray}&&\eta (t)=\displaystyle \frac{ \langle {\psi }_{T}| \hat{\rho }(t)| {\psi }_{T} \rangle }{{\int }_{{t}_{1}}^{t}| {{ \mathcal E }}_{\mathrm{in}}({t}^{{\rm{{\prime} }}}){| }^{2}{\rm{d}}{t}^{{\rm{{\prime} }}}},\end{eqnarray} \tag{ 9 }$$

where t > t₁ and the denominator is unity for t → t₂. We note that states $| {\psi }_{0}\rangle$ and $| {\psi }_{T}\rangle$ are connected by the coherent dynamics via the intermediate states $| e\rangle \otimes | 0{\rangle }_{c}\otimes | \mathrm{vac}\rangle$ and $| g\rangle \otimes | 1{\rangle }_{c}\otimes | \mathrm{vac}\rangle$. These states are unstable, since they can decay via spontaneous emission or via the parasitic cavity losses. Moreover, the incident photon can be reflected off the cavity. The latter is a unitary process, which results in a finite probability of finding a photon excitation in the transmission line after the photon has reached the mirror. The choice of Ω(t) shall maximize the transfer $| {\psi }_{0}\rangle \to | {\psi }_{T}\rangle$ by minimizing the losses as well as reflection at the cavity mirror.

The transmission line is here modeled by a cavity of length L, with a perfect mirror at x = −L. The second mirror at x = 0 coincides with the mirror of finite transmittivity, separating the transmission line from the optical cavity. The length L is chosen to be sufficiently large to simulate a continuum of modes for all practical purposes. This requires that the distance between neighboring frequencies is smaller than all characteristic frequencies of the problem. The smallest characteristic frequency is the bandwidth of the incident photon, which is the inverse of the photon duration in time. Since the initial state is assumed to be a single photon in a pure state, the latter coincides with the photon coherence time T_c [24] which is defined as

$$\begin{eqnarray}&&{T}_{{\rm{c}}}=\sqrt{\langle {t}^{2}\rangle -\langle t{\rangle }^{2}}\end{eqnarray} \tag{ 10 }$$

with $\langle {t}^{x}\rangle \equiv {\int }_{{t}_{1}}^{{t}_{2}}{t}^{x}| {{ \mathcal E }}_{\mathrm{in}}(t){| }^{2}{\rm{d}}t$, and

$$\begin{eqnarray}&&{\int }_{{t}_{1}}^{{t}_{2}}| {{ \mathcal E }}_{\mathrm{in}}(t){| }^{2}{\rm{d}}t=1-\varepsilon ,\end{eqnarray} \tag{ 11 }$$

where ε < 10⁻⁵ for the choice $| {t}_{1}| ={t}_{2}=6{T}_{{\rm{c}}}$ and $L=12{{cT}}_{{\rm{c}}}$. The modes of the transmission line are standing waves with wave vector along the x axis. For numerical purposes we take a finite number N of modes around the cavity wave number ${k}_{{\rm{c}}}=\tfrac{{\omega }_{{\rm{c}}}}{c}$. Their wave numbers are

$$\begin{eqnarray}&&{k}_{n}={k}_{{\rm{c}}}+\displaystyle \frac{n\pi }{L},\end{eqnarray} \tag{ 12 }$$

while $n=-(N-1)/2,\,\ldots ,\,(N-1)/2$, and the corresponding frequencies are ${\omega }_{n}={{ck}}_{n}$. We choose N and L so that our simulations are not significantly affected by the finite size of the transmission line and by the cutoff in the mode number N. We further choose N in order to appropriately describe spontaneous decay by the cavity mode. This is tested by initializing the system with no atom and one cavity photon and choosing the parameters so to reproduce the exponential damping of the cavity field.

Note that a single mode of the cavity is sufficient to describe the interaction with a single photon if the photon frequencies lie in a range which is smaller than the free spectral range of the cavity and is centered around the frequency of the cavity mode. In this work we choose the central frequency of the photon to coincide with the cavity mode frequency ${\omega }_{{\rm{p}}}={\omega }_{{\rm{c}}}$ and the spectrally broadest photon we consider (figures 5 and 6) spans about $16\times 2\pi \,\mathrm{MHz}$ around the cavity frequency ${\omega }_{{\rm{c}}}$. A cavity of $1\,\mathrm{cm}$ has a free spectral range of about 15 × 2π$\,{\text{}}\mathrm{GHz}$ which is three orders of magnitudes larger than the bandwidth of the photon. This justifies the approximation to a single mode cavity. The employed formalism can be applied to photons with other center frequencies as well, if the number of modes N is chosen sufficiently large and their center is appropriately shifted (see equation (12)).

Since the free field modes are included in the unitary evolution, it is possible to constantly monitor their state. The photon distribution in space at time t is given by

$$\begin{eqnarray}&&P(x,t)=\displaystyle \frac{2}{L}\displaystyle \sum _{n,m=1}^{N}{\rho }_{{nm}}(t)\sin \left(n\displaystyle \frac{\pi }{L}x\right)\sin \left(m\displaystyle \frac{\pi }{L}x\right),\end{eqnarray} \tag{ 13 }$$

where ${\rho }_{{nm}}(t)=\mathrm{Tr}\{\hat{\rho }(t)| {1}_{m}\rangle \langle {1}_{n}| \}$ and $| {1}_{n}\rangle ={b}_{{k}_{n}}^{\dagger }| \mathrm{vac}\rangle$.

A further important quantity characterizing the coupling between cavity mode and atom is the cooperativity C, which reads [21]

$$\begin{eqnarray}&&C=\displaystyle \frac{{g}^{2}}{\kappa \gamma }.\end{eqnarray} \tag{ 14 }$$

The cooperativity sets the maximum storage efficiency in the limit in which the cavity can be adiabatically eliminated from the dynamics of the system [21], which corresponds to assuming the condition

$$\begin{eqnarray}&&\gamma {{CT}}_{{\rm{c}}}\gg 1.\end{eqnarray} \tag{ 15 }$$

In this limit, in fact, the state $| g\rangle \otimes | 1{\rangle }_{{c}}\otimes | \mathrm{vac}\rangle$ can be eliminated from the dynamics. Then, the efficiency satisfies $\eta (t)\leqslant {\eta }_{\max }$ where the maximal efficiency η_max reads [21]

$$\begin{eqnarray}&&{\eta }_{\max }=\displaystyle \frac{C}{1+C}.\end{eqnarray} \tag{ 16 }$$

The maximal efficiency η_max is reached for any input photon envelope ${{ \mathcal E }}_{\mathrm{in}}(t)$ and detuning Δ, provided the adiabatic condition (15) is fulfilled.

In our study we also determine the probability that the photon is in the transmission line,

$$\begin{eqnarray}&&{P}_{r}(t)=\displaystyle \sum _{k}\mathrm{Tr}\{\hat{\rho }(t)| {1}_{k}\rangle \langle {1}_{k}| \},\end{eqnarray} \tag{ 17 }$$

the probability that spontaneous emission occurs,

$$\begin{eqnarray}&&{P}_{s}(t)=\mathrm{Tr}\{\hat{\rho }(t)| {\xi }_{e}\rangle \langle {\xi }_{e}| \},\end{eqnarray} \tag{ 18 }$$

and finally, the probability that cavity parasitic losses take place,

$$\begin{eqnarray}&&{P}_{\mathrm{loss}}(t)={\rm{Tr}}\{\hat{\rho }(t)| g,{0}_{c},\mathrm{vac}\rangle \langle g,{0}_{c},\mathrm{vac}| \}.\end{eqnarray} \tag{ 19 }$$

By means of these quantities we gain insight into the processes leading to optimal storage.

We first review the requirements and results of the individual protocols of [19–21] and investigate their efficiency for a single-atom quantum memory. The works of [19–21] determine the form of the optimal pulse Ω(t) for cavities with cooperativities C ≥ 1. The optimal pulse is found by imposing similar, but not equivalent requirements. In [19, 20] the authors determine Ω(t) by imposing impedance matching, namely, that there is no photon reflected back by the cavity mirror. In [21] the pulse Ω(t) warrants maximal storage, namely, maximal probability of transferring the photon into the atomic excitation $| r\rangle$. The latter requirement corresponds to maximizing the storage efficiency η defined in equation (9).

In detail, in [19] the authors determine the optimal pulse ${\rm{\Omega }}(t)$ that suppresses back-reflection from the cavity and warrants that the dynamics follows adiabatically the dark state of the system composed by cavity and atom. For this purpose the authors impose that the cavity field is resonant with the transition $| g\rangle \to | e\rangle$, namely Δ = 0. They further require that the coherence time ${T}_{{\rm{c}}}$ is larger than the cavity decay time, $\kappa {T}_{{\rm{c}}}\gtrsim 1$. Under these conditions the optimal pulse ${\rm{\Omega }}(t)={{\rm{\Omega }}}^{{\rm{F}}}(t)$ reads

$$\begin{eqnarray}&&{{\rm{\Omega }}}^{{\rm{F}}}(t)=\displaystyle \frac{g{{ \mathcal E }}_{\mathrm{in}}(t)}{\sqrt{{c}_{1}+2\kappa {\int }_{{t}_{1}}^{t}| {{ \mathcal E }}_{\mathrm{in}}(t^{\prime} ){| }^{2}{\rm{d}}t^{\prime} -| {{ \mathcal E }}_{\mathrm{in}}(t){| }^{2}}},\end{eqnarray} \tag{ 21 }$$

where c₁ regularize ${{\rm{\Omega }}}^{{\rm{F}}}(t)$ for $t\to {t}_{1}$. The work in [20] imposes the suppression of the back-reflected photon without any adiabatic approximation and finds the optimal pulse ${\rm{\Omega }}(t)={{\rm{\Omega }}}^{{\rm{D}}}(t)$, which takes the form

$$\begin{eqnarray}&&{{\rm{\Omega }}}^{{\rm{D}}}(t)=\displaystyle \frac{g{{ \mathcal E }}_{\mathrm{in}}(t)+(\dot{F}(t)+\gamma F(t))/g}{\sqrt{2\kappa {\rho }_{0}+2\kappa {\int }_{{t}_{1}}^{t}| {{ \mathcal E }}_{\mathrm{in}}({t}^{{\rm{{\prime} }}}){| }^{2}{\rm{d}}{t}^{{\rm{{\prime} }}}-| {{ \mathcal E }}_{\mathrm{in}}(t){| }^{2}-D(t)}},\end{eqnarray} \tag{ 22 }$$

with

$$\begin{eqnarray}&&{ \mathcal D }(t)=2\gamma {\int }_{{t}_{1}}^{t}| { \mathcal F }(t^{\prime} ){| }^{2}{\rm{d}}t^{\prime} +| { \mathcal F }(t){| }^{2}.\end{eqnarray} \tag{ 23 }$$

and ${ \mathcal F }(t)=\dot{{{ \mathcal E }}_{\mathrm{in}}}(t)-\kappa {{ \mathcal E }}_{\mathrm{in}}(t)$. Coefficient ${\rho }_{0}$ accounts for a small initial population in the target state $| r\rangle$ and it is relevant in order to avoid divergences in equation (22) for t → t₁, see [20] for an extensive discussion. The pulse ${{\rm{\Omega }}}^{{\rm{F}}}(t)$ of equation (21) can be recovered from equation (22) by imposing the conditions

$$\begin{eqnarray}&&\dot{{ \mathcal F }}(t)+\gamma { \mathcal F }(t)=0,\end{eqnarray} \tag{ 24a }$$

$$\begin{eqnarray}&&-| { \mathcal F }({t}_{1}){| }^{2}+2\kappa {\rho }_{0}={c}_{1}.\end{eqnarray} \tag{ 24b }$$

The control pulse ${{\rm{\Omega }}}^{{\rm{D}}}(t)$ can be considered as a generalization of ${{\rm{\Omega }}}^{{\rm{F}}}(t)$ since it is determined by solely imposing quantum impedance matching.

In [21] the authors determine the amplitude ${\rm{\Omega }}(t)$ that maximizes the efficiency η. This condition is not equivalent to imposing impedance matching. In fact, while in the case of impedance matching major losses through the excited state $| e\rangle$ are acceptable in order to minimize the probability of photon reflection, in the case of maximum transfer efficiency η those losses are detrimental and thus have to be minimized. The optimal pulse ${\rm{\Omega }}(t)={{\rm{\Omega }}}^{{\rm{G}}}(t)$ is determined for a generic detuning Δ by using an analytical model based on the adiabatic elimination of the excited state $| e\rangle$ of the atom and of the cavity field in the bad cavity limit $\kappa \gg g$. It reads

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Omega }}}^{{\rm{G}}}(t) & = & \displaystyle \frac{\gamma (1+C)+{\rm{i}}{\rm{\Delta }}}{\sqrt{2\gamma (1+C)}}\displaystyle \frac{{{ \mathcal E }}_{\mathrm{in}}(t)}{\sqrt{{\displaystyle \int }_{{t}_{1}}^{t}| {{ \mathcal E }}_{\mathrm{in}}(t^{\prime} ){| }^{2}{\rm{d}}t^{\prime} }}\times \exp \left(-{\rm{i}}\displaystyle \frac{{\rm{\Delta }}}{2\gamma (1+C)}\mathrm{ln}{\displaystyle \int }_{{t}_{1}}^{t}| {{ \mathcal E }}_{\mathrm{in}}(t^{\prime} ){| }^{2}{\rm{d}}t^{\prime} \right).\end{array}\end{eqnarray} \tag{ 25 }$$

In the limit in which the adiabatic conditions are fulfilled, this control pulse allows for storage with efficiency ${\eta }_{\max }$, equation (16). This efficiency approaches unity for cooperativities $C\gg 1$.

We start by integrating numerically the master equation for a single atom (1) after setting ${\kappa }_{\mathrm{loss}}=0$, namely, by neglecting parasitic losses. We determine the storage efficiency at the time t₂, which we identify by taking ${t}_{2}\gg {T}_{{\rm{c}}}$ for different choices of the control field ${\rm{\Omega }}={{\rm{\Omega }}}^{{\rm{G}}},{{\rm{\Omega }}}^{{\rm{F}}},{{\rm{\Omega }}}^{{\rm{D}}}$ in Hamiltonian (3). Numerically, t₂ corresponds to the time the photon would need to be reflected back into the initial position, assuming that the partially reflecting mirror is replaced by a perfect mirror. Our numerical simulations are performed for a single atom in a resonator in the good cavity limit.

Figure 2 display the efficiency and the losses as a function of κ, γ, and of the coherence time T_c of the photon (and thus of the adiabatic parameter $\gamma {T}_{{\rm{c}}}C$). Each curve corresponds to the different control pulses in the Hamiltonian (3) according to the three protocols. In subplot (a) we observe that the efficiency reached with the pulse ${{\rm{\Omega }}}^{{\rm{G}}}(t)$ corresponds to the maximum theoretical efficiency ${\eta }_{\max }$, while the efficiency with ${{\rm{\Omega }}}^{{\rm{D}}}$ is the smallest. In subplot (b) it is visible that the control pulse ${{\rm{\Omega }}}^{{\rm{G}}}(t)$ warrants the maximum efficiency even down to values of κ of the order of $\kappa \sim g/5$. Subplot (c) displays the efficiency as a function of the adiabatic parameter $\gamma {T}_{{\rm{c}}}C$: the protocol ${{\rm{\Omega }}}^{{\rm{G}}}(t)$ reaches the maximum theoretical efficiency ${\eta }_{\max }$ for $\gamma {T}_{{\rm{c}}}C\gtrsim 20$, while the other protocols have smaller efficiency for all values of ${T}_{{\rm{c}}}$. Figures 2(d)–(f) report the probability that the photon is reflected back into the transmission line, equation (17). It is evident that protocol ${{\rm{\Omega }}}^{{\rm{D}}}$ perfectly suppresses the back reflection probability in every regime here considered. However in the non-adiabatic regime (subplots (c), (f), (i), $\gamma {T}_{{\rm{c}}}C\lesssim 20$) the protocol ${{\rm{\Omega }}}^{{\rm{D}}}$, as well as the protocol ${{\rm{\Omega }}}^{{\rm{F}}}$, requires an increasing maximum Rabi frequency for decreasing ${T}_{{\rm{c}}}$. At the value of about $\gamma {T}_{{\rm{c}}}C\approx 3.74$ the Rabi frequency is so high that it is not anymore manageable by our numerical solver, for this reason the plots for the protocols ${{\rm{\Omega }}}^{{\rm{D}}}$ and ${{\rm{\Omega }}}^{{\rm{F}}}$ are reported for $\gamma {T}_{{\rm{c}}}C\gtrsim 3.74$. The same happens for small values of κ, subplots (b)(e)(h): in this case the plots for the protocols ${{\rm{\Omega }}}^{{\rm{D}}}$ and ${{\rm{\Omega }}}^{{\rm{F}}}$ are reported for $\kappa \gtrsim 0.3\times 2\pi \,\mathrm{MHz}$. The diverging Rabi frequency can be avoided by an appropriate choice of the parameters c₁ and ρ₀ in equations (21) and (22), respectively. Figures 2(g)–(i) report the losses via spontaneous emission of the atom, equation (18): while these losses are acceptable in order to minimize the back-reflected photon, they are detrimental for the intent of populating the target state $| r\rangle$. Protocol ${{\rm{\Omega }}}^{{\rm{D}}}$, which perfectly suppresses the back reflected photon, has the highest losses via spontaneous emission, which in the end leads to a lower efficiency η. Protocol ${{\rm{\Omega }}}^{{\rm{G}}}$ in turn, has the lowest radiative losses and it allows for the transfer with the maximal efficiency ${\eta }_{\max }$. Protocol ${{\rm{\Omega }}}^{{\rm{F}}}$ tries to minimize reflection of the photon at the cavity mirror. However, since ${{\rm{\Omega }}}^{{\rm{F}}}$ is derived with some approximations, it does not suppress completely the reflection and its final efficiency is between the ones of the other two protocols.

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An important general result of this study is that the bad cavity limit is not essential for reaching the maximal efficiency as long as the dynamics is adiabatic: the relevant parameter is in fact the cooperativity.

The protocols so far discussed assume an ideal optical resonator. In this section we analyze how their efficiency is modified by the presence of parasitic losses, here described by the superoperator ${{ \mathcal L }}_{{\kappa }_{\mathrm{loss}}}$ in equation (4b). In particular, we derive the maximal efficiency the protocols can reach as a function of ${\kappa }_{\mathrm{loss}}\gt 0$.

We first numerically determine the efficiency of the individual protocols as a function of ${\kappa }_{\mathrm{loss}}$ for ${T}_{{\rm{c}}}=0.5\,\mu {\rm{s}}$. Figure 3(a) displays η for ${\rm{\Omega }}={{\rm{\Omega }}}^{{\rm{G}}},{{\rm{\Omega }}}^{{\rm{D}}},{{\rm{\Omega }}}^{{\rm{F}}}$. It is evident that the effect of losses is detrimental, for instance it leads to a definite reduction of the maximal efficiency from η = 0.77 down to η = 0.68 for ${\kappa }_{\mathrm{loss}}\sim 0.1\kappa$. This result can be improved by identifying a control field ${\rm{\Omega }}={{\rm{\Omega }}}^{{\rm{X}}}$ which compensates, at least partially, the effects of these parasitic losses. The control field ${{\rm{\Omega }}}^{{\rm{X}}}(t)$ is derived in section 3.3 using the input–output formalism: it corresponds to performing the substitution $\kappa \to \kappa +{\kappa }_{\mathrm{loss}}$ in the functional form ${{\rm{\Omega }}}^{{\rm{G}}}(t)$ of equation (25). Specifically, it reads

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Omega }}}^{{\rm{X}}}(t) & = & \displaystyle \frac{\gamma (1+C^{\prime} )+{\rm{i}}{\rm{\Delta }}}{\sqrt{2\gamma (1+C^{\prime} )}}\displaystyle \frac{{{ \mathcal E }}_{\mathrm{in}}(t)}{\sqrt{{\displaystyle \int }_{{t}_{1}}^{t}| {{ \mathcal E }}_{\mathrm{in}}(t^{\prime} ){| }^{2}{\rm{d}}t^{\prime} }}\times \exp \left(-{\rm{i}}\displaystyle \frac{{\rm{\Delta }}}{2\gamma (1+C^{\prime} )}\mathrm{ln}{\displaystyle \int }_{{t}_{1}}^{t}| {{ \mathcal E }}_{\mathrm{in}}(t^{\prime} ){| }^{2}{\rm{d}}t^{\prime} \right),\end{array}\end{eqnarray} \tag{ 26 }$$

with the modified cooperativity

$$\begin{eqnarray}&&C^{\prime} =\displaystyle \frac{{g}^{2}}{\gamma (\kappa +{\kappa }_{\mathrm{loss}})}.\end{eqnarray} \tag{ 27 }$$

When the control pulse ${{\rm{\Omega }}}^{{\rm{X}}}(t)$ is used, the efficiency of the process corresponds to the maximal efficiency ${\eta }_{\max }^{{\prime} }$, which is now given by

$$\begin{eqnarray}&&{\eta }_{\max }^{{\prime} }=\displaystyle \frac{\kappa }{\kappa +{\kappa }_{\mathrm{loss}}}\displaystyle \frac{C^{\prime} }{1+C^{\prime} }.\end{eqnarray} \tag{ 28 }$$

Clearly, ${\eta }_{\max }^{{\prime} }\leqslant {\eta }_{\max }$, while the equality holds for ${\kappa }_{\mathrm{loss}}=0$.

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By inspecting the numerical results, we note that the efficiency obtained using ${{\rm{\Omega }}}^{{\rm{X}}}$ is always higher than the one reached by the other protocols. Even though for some values of ${\kappa }_{\mathrm{loss}}$ the efficiencies using different control fields may approach the one found with ${{\rm{\Omega }}}^{{\rm{X}}}$, yet the dynamics are substantially different. This is visible by inspecting the probability that the photon is reflected, the radiative losses, and the parasitic losses, as a function of ${\kappa }_{\mathrm{loss}}$ as shown in figures 3(b)–(d), respectively: each pulse distributes the losses in a different way, with ${{\rm{\Omega }}}^{{\rm{X}}}(t)$ interpolating among the different strategies in order to maximize the efficiency.

Figure 4 shows the evolution of the system for ${T}_{{\rm{c}}}=0.5\,\mu {\rm{s}}$. Figure 4(a) displays the envelope in time $| {{ \mathcal E }}_{\mathrm{in}}(t){| }^{2}$ for the photon given in equation (7), which is the one used also in this simulation. Figure 4(b) displays the control pulse shapes of the protocols ${{\rm{\Omega }}}^{{\rm{F}}}$, ${{\rm{\Omega }}}^{{\rm{G}}}$, ${{\rm{\Omega }}}^{{\rm{D}}}$, of [19–21] and ${{\rm{\Omega }}}^{{\rm{X}}}$ derived in this work (the pulse shapes are given analytically in equations (21), (22), (25), (26)). Figure 4(c) shows the population of the states and the losses during the evolution when the atom is driven by ${{\rm{\Omega }}}^{{\rm{X}}}(t)$. The efficiency of the transfer, equation (9), corresponds to the population of the state $| r\rangle$, ${\rho }_{{rr}}$. For the parameters of [17] the final efficiency is $\eta ({t}_{2})\approx {\eta }_{\max }^{{\prime} }\approx 0.653$.

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In the next subsection we report the derivation of ${{\rm{\Omega }}}^{{\rm{X}}}$ and ${\eta }_{\max }^{{\prime} }$ by means of the input–output formalism.

In this section we generalize the adiabatic protocol of [21] in order to identify the control field that maximizes the storage efficiency and to determine the maximum storage efficiency one can reach. The derivation presented in this section is based on the input–output formalism and it delivers equations (26) and (28).

We first justify the result for equation (28) using a time reversal argument applied in [21, 26]. Let us consider retrieval of the photon, assuming the atom is initially in state $| r\rangle$ and there is neither external nor cavity field. Then, in order to retrieve the photon, the control pulse ${\rm{\Omega }}(t)$ shall drive the transition $| r\rangle \to | e\rangle$ such that at the end of the process the state $| r\rangle$ is completely empty. The excited state $| e\rangle$ dissipates the excitation with probability $1/(1+C^{\prime} )$, while it can emit into the cavity mode with probability $C^{\prime} /(1+C^{\prime} )$. When the cavity mode is populated, a fraction ${\kappa }_{\mathrm{loss}}/(\kappa +{\kappa }_{\mathrm{loss}})$ is lost, while the fraction $\kappa /(\kappa +{\kappa }_{\mathrm{loss}})$ is emitted via the coupling mirror into the transmission line. From this argument one finds that the probability of retrieval is given by equation (28). Using the time reversal argument, this is also the efficiency of storage.

We now derive this result as well as ${{\rm{\Omega }}}^{{\rm{X}}}(t)$ starting from the retrieval process and then applying the time reversal argument. For this purpose, we restrict the dynamics to the Hilbert space ${ \mathcal H }$ composed by the states $\{| g,{1}_{c},\mathrm{vac}\rangle ,| e,{0}_{c},\mathrm{vac}\rangle ,| r,{0}_{c},\mathrm{vac}\rangle ,| g,{0}_{c},{1}_{k}\rangle :1\leqslant k\leqslant N\}$. In ${ \mathcal H }$ the probability is not conserved due to leakage via spontaneous decay and via parasitic cavity losses. Therefore, a generic state in ${ \mathcal H }$ takes the form $| \phi (t)\rangle =c(t)| g,{1}_{c},\mathrm{vac}\rangle +e(t)| e,{0}_{c},\mathrm{vac}\rangle +r(t)| r,{0}_{c},\mathrm{vac}\rangle +{\sum }_{k}{{ \mathcal E }}_{k}(t)| g,{0}_{c},{1}_{k}\rangle$, it evolves according to a non-Hermitian Hamiltonian and its norm decays exponentially with time [27]. We assume that at the initial time $t={t}_{1}$ the probability amplitude $r({t}_{1})$ equals 1, while all other probability amplitudes vanish. The equations of motion for the probability amplitudes read

$$\begin{eqnarray}&&\dot{c}(t)=-{\rm{i}}{ge}(t)-{\rm{i}}\sqrt{2\kappa }{{ \mathcal E }}_{\mathrm{in}}(t)-(\kappa +{\kappa }_{\mathrm{loss}})c(t),\end{eqnarray} \tag{ 29a }$$

$$\begin{eqnarray}&&\dot{e}(t)=({\rm{i}}{\rm{\Delta }}-\gamma )e(t)-{\rm{i}}{gc}(t)-{\rm{i}}{\rm{\Omega }}(t)r(t),\end{eqnarray} \tag{ 29b }$$

$$\begin{eqnarray}&&\dot{r}(t)=-{\rm{i}}{{\rm{\Omega }}}^{* }(t)e(t),\end{eqnarray} \tag{ 29c }$$

where we used the Markov approximation and the input–output formalism [25]. We now assume the bad-cavity limit $\kappa \gg g$ and adiabatically eliminate the cavity field from the equations of motion (which corresponds to assuming $\dot{c}(t)\approx 0$ over the typical time scales of the other variables). In this limit the input–output operator relation, ${\hat{{ \mathcal E }}}_{\mathrm{out}}(t)={\rm{i}}\sqrt{2\kappa }\hat{a}(t)-{\hat{{ \mathcal E }}}_{\mathrm{in}}(t)$, takes the form

$$\begin{eqnarray}&&{{ \mathcal E }}_{\mathrm{out}}(t)=G\sqrt{2\gamma C}e(t)+\displaystyle \frac{\kappa -{\kappa }_{\mathrm{loss}}}{\kappa +{\kappa }_{\mathrm{loss}}}{{ \mathcal E }}_{\mathrm{in}}(t),\end{eqnarray} \tag{ 30 }$$

where

$$\begin{eqnarray*}&&G=\kappa /(\kappa +{\kappa }_{\mathrm{loss}})\end{eqnarray*}$$

and C is given in equation (14). This equation has to be integrated together with the equations

$$\begin{eqnarray}&&\dot{e}(t)=[{\rm{i}}{\rm{\Delta }}-\gamma (1+{GC})]e(t)-{\rm{i}}{\rm{\Omega }}(t)r(t)-G\sqrt{2\gamma C}{{ \mathcal E }}_{\mathrm{in}}(t),\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&\dot{r}(t)=-{\rm{i}}{{\rm{\Omega }}}^{* }(t)e(t).\end{eqnarray} \tag{ 32 }$$

Our goal is to determine the retrieval efficiency assuming that at time t = 0 there is no input photonic excitation, thus ${{ \mathcal E }}_{\mathrm{in}}(t)=0$ at all times. Using these assumptions, the above equations can be cast into the form

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}(| e(t){| }^{2}+| r(t){| }^{2})=-2\gamma (1+C^{\prime} )| e(t){| }^{2}.\end{eqnarray} \tag{ 33 }$$

The probability that no excitations are left in the atom at time ${t}_{2}\gt 0$ (${t}_{2}\gg {T}_{{\rm{c}}}$) is the retrieval efficiency

$$\begin{eqnarray}\begin{array}{rcll}{\eta }_{\max }^{{\prime} } & = & {\displaystyle \int }_{{t}_{1}}^{{t}_{2}}| {{ \mathcal E }}_{\mathrm{out}}(t){| }^{2}{\rm{d}}t=2{G}^{2}\gamma C{\displaystyle \int }_{{t}_{1}}^{{t}_{2}}| e(t){| }^{2}{\rm{d}}t\,=\, & \displaystyle \frac{-{GC}^{\prime} }{1+C^{\prime} }{[| e(t){| }^{2}+| r(t){| }^{2}]}_{{t}_{1}}^{{t}_{2}}=\displaystyle \frac{{GC}^{\prime} }{1+C^{\prime} }.\end{array}\end{eqnarray} \tag{ 34 }$$

By means of the time reversal argument, this is also the storage efficiency.

The output field can be analytically determined by adiabatically eliminating the excited state from equations (30). This leads to the expression

$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal E }}_{\mathrm{out}}(t) & = & {\rm{i}}\sqrt{2\gamma {GC}^{\prime} }\displaystyle \frac{{\rm{\Omega }}(t)}{{\rm{i}}{\rm{\Delta }}-\gamma (1+C^{\prime} )}\times \exp \left({\displaystyle \int }_{{t}_{1}}^{t}\displaystyle \frac{| {\rm{\Omega }}(t^{\prime} ){| }^{2}}{{\rm{i}}{\rm{\Delta }}-\gamma (1+C^{\prime} )}{\rm{d}}t^{\prime} \right).\end{array}\end{eqnarray} \tag{ 35 }$$

Integrating the norm squared of equation (35) one obtains

$$\begin{eqnarray}&&\begin{array}{l}{\left(G\displaystyle \frac{C^{\prime} }{1+C^{\prime} }\right)}^{-1}{\displaystyle \int }_{{t}_{1}}^{t}| {{ \mathcal E }}_{\mathrm{out}}(t^{\prime} ){| }^{2}{\rm{d}}t^{\prime} =\,1-\exp \left[\displaystyle \frac{-2\gamma (1+C^{\prime} )}{{\gamma }^{2}{(1+C^{\prime} )}^{2}+{{\rm{\Delta }}}^{2}}{\displaystyle \int }_{{t}_{1}}^{t}| {\rm{\Omega }}(t^{\prime} ){| }^{2}{\rm{d}}t^{\prime} \right].\end{array}\end{eqnarray} \tag{ 36 }$$

We solve equation (36) to find $| {\rm{\Omega }}(t)|$, while the phase of ${\rm{\Omega }}(t)$ can be determined from equation (35). Finally, we obtain the control pulse ${{\rm{\Omega }}}^{{\rm{X}}}{}_{\mathrm{retr}}(t)$ which retrieves the photon with efficiency ${\eta }_{\max }^{{\prime} }$. It reads

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Omega }}}^{{\rm{X}}}{}_{\mathrm{retr}}(t) & = & \displaystyle \frac{\gamma (1+C^{\prime} )-{\rm{i}}{\rm{\Delta }}}{\sqrt{2\gamma (1+C^{\prime} )}}\displaystyle \frac{{{ \mathcal E }}_{\mathrm{out}}(t)}{\sqrt{{\displaystyle \int }_{t}^{{t}_{2}}| {{ \mathcal E }}_{\mathrm{out}}(t^{\prime} ){| }^{2}{\rm{d}}t^{\prime} }}\times \exp \left({\rm{i}}\displaystyle \frac{{\rm{\Delta }}}{2\gamma (1+C^{\prime} )}\mathrm{ln}{\displaystyle \int }_{t}^{{t}_{2}}(| {{ \mathcal E }}_{\mathrm{out}}(t^{\prime} ){| }^{2}{/\eta }_{\max }^{{\prime} }){\rm{d}}t^{\prime} \right).\end{array}\end{eqnarray} \tag{ 37 }$$

Using the time reversal argument, the control pulse ${{\rm{\Omega }}}^{{\rm{X}}}(t)={\rm{\Omega }}{{}^{{\rm{X}}}}_{\mathrm{retr}}^{* }(T-t)$ stores the time reversed input photon with ${{ \mathcal E }}_{\mathrm{in}}(t)={{ \mathcal E }}_{\mathrm{out}}^{* }(T-t)/\sqrt{{\eta }_{\max }^{{\prime} }}$ and $T={t}_{2}-{t}_{1}$, and it takes the form given in equation (26). This pulse has the same form as the pulse of equation (25), where now C has been replaced by $C^{\prime}$ (or equivalently $\kappa \to \kappa +{\kappa }_{\mathrm{loss}}$).

The generation of single photons with arbitrary shape of the wavepacket envelope in atom–cavity systems has been discussed theoretically in [21, 28] and demonstrated experimentally in [29, 30].

In [1, 26] it has been pointed out that photon storage and retrieval are connected by a time reversal transformation. This argument has profound implications. Consider for instance the pulse shape ${\rm{\Omega }}(t)$ which optimally stores an input photon with envelope ${{ \mathcal E }}_{\mathrm{in}}(t)$. This pulse shape is the time reversal of the pulse shape ${{\rm{\Omega }}}_{\mathrm{retr}}(t)={{\rm{\Omega }}}^{* }(T-t)$ which retrieves a photon with envelope ${{ \mathcal E }}_{\mathrm{out}}(t)={{ \mathcal E }}_{\mathrm{in}}^{* }(T-t)$ (here $T={t}_{2}-{t}_{1}$). In this case, the storage efficiency is equal to the efficiency of retrieval and is limited by the cooperativity through the relation in equation (28). We have numerically checked that this is fulfilled by considering adiabatic retrieval and storage of a single photon through 5 nodes, consisting of 5 identical cavity–atom systems. We applied ${{\rm{\Omega }}}_{\mathrm{retr}}(t)$ for the retrieval and the corresponding ${\rm{\Omega }}(t)$ for the storage. Within the numerical error, we verified that the storage efficiency of each retrieved photon remains constant and equal to the one of the first retrieved photon.

The protocol ${{\rm{\Omega }}}^{{\rm{G}}}(t)$ does not have any restriction on Δ: for every Δ there is a pulse ${{\rm{\Omega }}}^{{\rm{G}}}(t)$ that allows for storage with efficiency ${\eta }_{\max }$ (within the adiabatic regime), see equation (25). Figure A1 displays the storage efficiency and the losses for each protocol as a function of Δ, as expected the protocol ${{\rm{\Omega }}}^{{\rm{G}}}(t)$ performs in the same way for any values of Δ.

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A time-dependent phase $\chi (t)$ of the control pulse ${\rm{\Omega }}(t)=| {\rm{\Omega }}(t)| {{\rm{e}}}^{{\rm{i}}\chi (t)}$ can be implemented as a two-photon detuning

$$\begin{eqnarray}&&\delta =\dot{\chi }(t).\end{eqnarray} \tag{ A4 }$$

In fact, by applying the unitary transformation $\hat{U}(t)=\exp (-{\rm{i}}| r\rangle \langle r| \chi (t))$, the transformed Hamiltonian is $\hat{H}^{\prime} ={\hat{H}}_{{\rm{I}}}^{{\prime} }+{\hat{H}}_{\mathrm{fields}}$, where

$$\begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{{\rm{I}}}^{{\prime} } & = & \dot{\chi }(t)| r\rangle \langle r| -{\rm{\Delta }}| e\rangle \langle e| +(g| e\rangle \langle g| \hat{a}+| {\rm{\Omega }}(t)| | e\rangle \langle r| +{\rm{h}}.{\rm{c}}.).\end{array}\end{eqnarray} \tag{ A5 }$$

For ${{\rm{\Omega }}}^{{\rm{G}}}(t)$ we have

$$\begin{eqnarray}&&{\dot{\chi }}^{{\rm{G}}}(t)=\displaystyle \frac{-{\rm{\Delta }}}{2\gamma (1+C)}\cdot \displaystyle \frac{| {{ \mathcal E }}_{\mathrm{in}}(t){| }^{2}}{{\int }_{{t}_{1}}^{t}| {{ \mathcal E }}_{\mathrm{in}}(t^{\prime} ){| }^{2}\det ^{\prime} }\end{eqnarray} \tag{ A6a }$$

$$\begin{eqnarray}&&=\,\displaystyle \frac{-{\rm{\Delta }}| {{\rm{\Omega }}}^{{\rm{G}}}(t){| }^{2}}{{{\rm{\Delta }}}^{2}+{\gamma }^{2}{(1+C)}^{2}}.\end{eqnarray} \tag{ A6b }$$

Recall that also $| {{\rm{\Omega }}}^{{\rm{G}}}(t)|$ depends on Δ. This can be understood in terms of AC Stark shift: one-photon detuning ${\rm{\Delta }}\ne 0$ is a shift of the control laser out of resonance for the transition $| r\rangle -| e\rangle$ and thereby induces an AC Stark shift on the levels $| e\rangle$ and $| r\rangle$ of the atom; thus the condition of two-photon resonance does not hold anymore. In order to restore the latter, changes in frequency of the carrier and/or of the cavity and/or of the atomic levels are needed and they appear as a two-photon detuning in the Hamiltonian. This also explains why the reflected photon probability for the protocols ${{\rm{\Omega }}}^{{\rm{F}}}(t)$ and ${{\rm{\Omega }}}^{{\rm{D}}}(t)$ (see figure A1), which do not take into account the one-photon detuning, increases with increasing Δ: the input photon sees the system out of resonance and hence it is mostly reflected. Equation (A6b) gives the energy shift as a function of the Rabi frequency of the control pulse.