Study of atomic geometry and its effect on photon generation and storage [Invited]

Keiichiro Furuya; Arindam Nandi; Arindam Nandi; Mahdi Hosseini; Mahdi Hosseini

doi:10.1364/OME.380715

1. Introduction

The study of collective excitations in atomic ensembles is an important research topic aiming to advance our understanding of ensemble dynamics in quantum systems. The framework for such study was initially formulated by Dicke [1]. It was shown that photons can mediate the interaction among atoms either confined in sub-wavelength regions in free space [2,3] or distributed inside optical resonators or waveguides [4,5]. The conditional emission of a single photon from an extended ensemble, $V>\lambda ^3$, can also give rise to cooperative emission (timed Dicke superradiance [6]). The spin of an ensemble of atoms, spontaneously emitting photons upon off-resonance excitation with a pump, can be projected into a single collective excitation when a single photon is detected in a pre-defined spatial mode. The collective excitation can give rise to the emission of a correlated photon through an inverse Raman process. Correltated bi-photons created in this way has been the center of research as a viable approach for entanglement distribution [7]. This approach has been used to entangle four quantum memories [8] and generate multi-dimensional correlated photons [9]. Recently, we have shown that by combining the storage process and projective measurement of single photons emitted from the ensemble one can probabilistically amplify coherent optical states [10]. Moreover, the effect of atomic distribution near waveguides [5,11] and cavities [12] has been investigated as a way to control photon emission and propagation through the waveguides. The system in which atoms are placed inside or nearby a ring cavity [13] is one of the popular ones to explore general dynamics in 1D and 2D quantum physics. Here, we consider atomic geometry to control the process of photon generation and storage in an ensemble of emitters or atoms. We show that by precision manipulation of position of atoms, forming a periodic or an aperiodic lattice, directional photon emission can be achieved to enhance correlation between the photons. Our approach can be used to describe and enhance photon generation processes from an ensemble of emitters [14], including neutral atoms, defect centers in solids, rare earth materials and quantum dots. In what follows, we first introduce the general model and describe light propagation through an atomic array in Sec. 2. We then study the effect of atomic geometry on light emission in Sec. 3. In Sec. 4 we explore the combined effect of propagation and emission by studying light storage and verify the influence of atomic geometry on the retrieval process.

2. Model: light propagation in a periodic lattice of atoms

We consider an array of atoms incorporated into an optical ring resonator as shown in Fig. 1(a). The atoms are distributed around the resonator with spacing between them chosen to study various collective behaviors. The positions of the atomic segments are denoted by $\theta _j$, as shown in Fig. 1(a). The distance, $\phi _j (j=1,2,\ldots, N),$ on the ring is the spacing between $j$th and $(j-1)$th atom. There should be a periodic condition on any distribution of atomic segments which is $\theta _N =2\pi$.

Fig. 1. (a) A schematic of an optical ring resonator hosting an array of atomic segments forming a lattice commensurate with the emission wavelength resonant with the cavity. (b) The input ($E_{in}$), transmitted ($E_f$) and reflected ($E_b$) light intensities are pulses propagating through the bus waveguide coupled to the resonator. An atomic lattice consists of 100 segments with spacing equal to the resonant wavelength, $\lambda$, is considered. (c) A pulse of light resonant with the resonator can be amplified or attenuated depending on the initial state of the atoms and the atomic spacing. The mean operator values $\sigma _{ee}$ and $\sigma _{gg}$ , respectively, represent the atomic population in the excited state $|e\rangle$ and ground state $|g\rangle$ , normalized to the total atom number. The simulation parameters used are : $g/\Gamma =0.5$, $\delta \omega =0$, $N=15$ considering a propagation time $c/r\phi _j\ll 2\pi /\Gamma$ where $c$ is the velocity of light in vacuum and $r$ is the radius of the ring resonator.

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The interaction Hamiltonian of the system is given in terms of atomic operators as

(1)$$H_{int} =\sum_j{\int{ \frac{\hbar}{2\pi} \{g(\theta) \frac{\hat{\sigma}_j}{ \sqrt{N}} (\hat{a}_c^{\dagger} e^{ikr\theta}+\hat{a}_{cc}^{\dagger} e^{-ikr\theta}) + h.c.\}d\theta}}$$

where $\hat {\sigma }_j=|g_j\rangle \langle e_j|$ is the atomic operator of $j-$th atom, $g(\theta )$ is the coupling strength and $\hat {a}_{c/cc}$ is the operator for the field propagating in the clockwise (c) or counter-clockwise (cc) direction, respectively. Assuming equal light-atom coupling strength, $\textbf {g}=g(\theta )$, at all discrete locations, we arrive at:

(2)$$H_{int} = \hbar \textbf{g} \big\{ \hat{S}(k) \hat{a}_{c}^{\dagger} + \hat{S}(-k) \hat{a}_{cc}^{\dagger} + h.c \big\} \\$$

where the collective atomic operator is defined as

(3)$$\hat{S}(k)= \frac{1}{\sqrt{N} }\sum_j{e^{-ikr\theta_j}\hat{\sigma}_j},$$

and $| g_1, g_2, \ldots, e_j, \cdots, g_N \rangle$ refers to the state with atom $j$ excited to the level $|e\rangle$ while other atoms remain in the ground state $|g\rangle$. Here $N$ is the total number of atoms or atomic segments in the ring resonator. Considering a single-pass propagation, the interaction dynamics can be obtained by numerically solving the semiclassical Maxwell-Bloch equations of motion. To this end, we write the time-dependent equations in Heisenberg picture and replace the operators with their mean values. We then numerically solve the following equations:

(4)$$\frac{d\hat{\mathcal{E}}_{c/cc}}{d\theta} = \pm i\mathcal{N}(\theta)\sum\hat{\sigma}_j$$

(5)$$\frac{d\hat{\sigma}_j}{dt} = -(\Gamma+i \delta\omega) \hat\sigma_j - i\textbf{g} (\hat{\mathcal{E}}_c^*+\hat{\mathcal{E}}_{cc}e^{2ikr\theta}) (\hat{\sigma}_{gg}-\hat{\sigma}_{ee})_j$$

where $\hat {\mathcal {E}}_{c/cc}(\theta ) = \hat {a}_{c/cc}e^{\mp ikr\theta }$ is the optical mode in the clockwise or counter-clockwise direction. Also, $\hat {\sigma }_{ee/gg}$ is the atomic population operator, and $\Gamma$ is the docoherence rate between two levels $|g\rangle$ and $|e\rangle$. The inhomogeneous broadening of the atomic transition is described by light-atom detuning $\delta \omega =\delta \omega (\theta _j)$ for individual atoms. The atomic density around the ring is described by a top-hat function as

(6)$$\mathcal{N}(\theta)= g r/4\pi c \sum_m{\{\tanh(\theta-m\phi_m+\delta\phi)- \tanh(\theta-m\phi_m-\delta\phi)\}}$$

with position uncertainty $\delta \phi$. By replacing operators with expected values of atomic, $\langle \hat {\sigma }\rangle =\sigma$, and optical, $\langle \hat {\mathcal {E}}\rangle =E$ fields, dynamics of the mean fields are achieved by integrating the equations over time and space. Figure 1(b) shows the result of numerical simulation of Eqs 4-5 where a single pulse enters the atomic array. Although the result is for single-pass propagation, the general dynamics holds for the ring resonator case as long as the cavity decay rate is much faster than light-atom coupling and the atomic decay rates. Later we will consider the cavity equations to solve for the intra-cavity field. For the case of large segment number, a significant portion of the incoming pulse is reflected due to the presence of an atomic Bragg resonance, which we will also discuss in Sec. 3.2. The atomic resonances of this kind has been predicted [15–17] and observed using cold atoms near waveguides and fibers [11]. The total amplification and attenuation of the pulse as a function of lattice constant or atom spacing are also shown in Fig. 1(c). The pulse can be amplified or attenuated depending on the initial atomic state and lattice spacing. The period of the oscillation in Fig. 1(c) is $\lambda /2N$ resulting from the spatial interference of the E-field amplitude.

3. Emission from the atomic array

We now use the model described above to study emission from the atomic ensemble geometrically arranged into a lattice. We shall see that in the limit of long coherence time, long-range cooperative emission is expected. When the coherence time is smaller than the cooperative decay time but still longer than the cavity lifetime, atomic Bragg resonances can be observed. In the single-photon limit, we see that atomic geometry can determine the directionality of the cooperative emission. We use these results to combine the propagation and emission dynamic of the array and study light storage in Sec. 4.

3.1 Long-range cooperative atomic resonance

The cooperative decay of an ensemble of emitters confined in a region much smaller than $\lambda ^3$ has been predicted [18,19] and experimentally observed in various platforms [2–5]. In a solid state environment, the presence of disorder such as inhomogeneous broadening is responsible for simultaneous appearance of sub and superradiant emission modes [20,21]. Moreover, it is known that atomic distribution affects the cooperative emission [21,22].

For observation of superradiance inside a cavity, the resonator decay rate, $\kappa$, must be large enough such that the intra-cavity photons leave the cavity before being reabsorbed by the atoms or before atoms decohere. Thus, the interactions need to be in the limit where $\kappa \gg \gamma +\Gamma$, where $\gamma$ is the excited state decay rate and $\Gamma$ is the decoherence rate. Also, the decoherence rate $\Gamma$ should be less than the cooperative decay rate, $N\gamma$, where $N$ is the atom number contributing to the emission. Note that we assume that there is a single atom per segment so that the atom number coincides with the number of atomic segments. The atomic lattice interacting via the intra-cavity light can exhibit mesoscopic cooperative interactions where cooperative coupling is extended to regions beyond the wavelength scale. Inside the resonator, the time evolution of the optical field, $\hat {a}_c$, in the clockwise direction is described by:

(7)$$\frac{d\hat{a}_c}{dt} = - \kappa \hat{a}_c + ig\tilde{\sigma} + \sqrt{\kappa/\tau} \mathcal{E}_{in}$$

where $\mathcal {E}_{in}$ represents the incoming field, $\tau$ is the round-trip time of photons inside the ring, and $\tilde {\sigma } =\int { \mathcal {N}(\theta )(\sum _j{ \hat {\sigma }_j} )d\theta }$. In writing the above equation, we ignore the effect of back scattering from the resonator surface coupling counter-propagating fields. Figure 2(a) shows the numerical solution to the pulsed emission from an ensemble of two-level atoms initially prepared in the excited state. In this regime, the pulse envelope has a width inversely proportional to the atom number, $N$, and its peak intensity scaling with $N^2$. The coherent emission is captured by the atomic coherence built-up ($\langle \tilde {\sigma }\rangle =\sigma$) shortly after atoms are initially prepared.

Fig. 2. (a)Intensity of the emission light from an ensemble of 60 and 100 atoms inside the resonator. Inset is a plot of atomic population and coherence as a function of time. (b) Maximum emission intensity as a function of effective light-atom coupling, $g^2/\kappa \gamma$ in the cavity for 100 atomic segments is plotted with different width of local atomic distribution, $\delta \phi$. Simulation parameters include: cavity decay rate, $\kappa =10^6\gamma$, atomic decoherence rate $\Gamma =0.01\kappa$, light-atom coupling strength, and $g/\Gamma =0.5$ (for (a)).

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The nonlinear scaling of peak intensity is evident in Fig. 2(b). As expected, the intensity increases quadratically with atom number beyond a threshold atom number (needed to overcome decoherence). We note that these results are also valid for the case where, instead of one atom per site, an ensemble of $N_0$ atoms are localized ($r\delta \phi \ll \lambda$) per segment. In such case, $g\to g\sqrt {N_0}$ and $r\delta \phi$ represents the atomic distribution per segment. As the atoms are delocalized with to the nodes of the intra-cavity standing wave, superradiance becomes less evident and a larger threshold is needed to overcome disorder. In presence of the inhomogeneous broadening, with width of $\Omega$, the requirement for the observation of cooperative decay is more stringent. The necessary condition for observation of superradiance then becomes $\Omega /N\ll \gamma$. With a resonator having a linewidth $\kappa <\Omega$, the effect of inhomogeneous broadening to cooperative emission can be evaluated by considering an effective atom number $N_{eff}=N\kappa /\Omega$ contributing to the superradiance emission. Overall, the broadening gives rise to sub and superradiance emission lines with non-exponential decay curves [3].

3.2 Emergence of Bragg atomic resonance and enhancement of excitation

When the cooperative emission conditions are not fully satisfied, atomic geometry can still be engineered to affect the light emission from the ensemble. Recently, we have performed an experiment studying Bragg resonances formed by a lattice of erbium ions in a silicon nitride ring resonator [12]. An array of atomic segments were implanted into the solid resonator and resonant excitation were used to probe the atomic resonance. Due to the relatively large inhomogeneous broadening and decoherence rate of the atoms in the systems, observation of Dicke superradiance was prohibited [20]. Nevertheless, the fast cavity decay rate, relative to the decoherence processes, enabled the investigation of other resonant effects. In an ordered lattice commensurate with the emission wavelength, a standing wave is formed by the emitted light from the discrete atomic segments. Because the lifetime of photons inside the resonator is longer than the decoherence time, interference between photons from spatially distributed atoms can occur within the time scale of the cavity lifetime. The maximum intra-cavity intensity (at the location of any segment) a short time after excitation scales with $\langle |\sum _j{E_{c/cc}(\theta _j)}|^2\rangle =\langle |NE_{max}|^2\rangle$, where $E_{max}$ is the maximum field amplitude from a single segment and $E_{c/cc}$ is the mean field propagating in either clockwise or counterclockwise directions. In a randomly distributed ensemble, on the other hand, the intensity at the segment $i$ is given by $\langle N|E_{max}|^2\rangle$. At the resonance condition in the lattice, the single-frequency intensity scales nonlinearly with the atom number. However, it does not reach the peak cooperative emission because the interaction time, $1/\kappa$, is smaller than the coherence time and light leaves the cavity before cooperative coherence is fully built up.

Nevertheless, a Bragg effect can be created by the 1D atomic array causing more efficient photoluminescence to occur. To see the resonant effect from the atoms, we consider single-pass propagation of light through the array. This is analogous to what described is Sec. 2. Here we show a different method to calculate the reflection from the array. The ratio between the spontaneous emission in direction of the array to that of the free-space, $C_j$, for each atom at position $j$, can be used to calculate the amount of light reflected or transmitted by the array. The normalized reflection through the ensemble can be found by multiplying the transfer matrix [23] of each atomic segment as, $P_{array}=|M(1,2)/M(2,2)|^2$, where $M_{array}=T_{a, N}.T_{f, N}.T_{a, N-1}.T_{f, N-1} \cdot T_{a,1}.T_{f,1}$ and

(8)$$\begin{aligned} T_{a, j}=\begin{pmatrix} 1-C_j/2 & -C_j/2 \\ C_j/2 & 1+C_j/2 \end{pmatrix}, ~ T_{f, j}=\begin{pmatrix} e^{ikr\phi_j} & 0 \\ 0 & e^{-ikr\phi_j} \end{pmatrix} \end{aligned}$$

are the transfer matrices of the atomic segment $j$ and free-space propagation from the $j-1$ to $j$ segment, respectively. Maximum reflection can be achieved when atom spacing is equal to multiple of $\lambda /2$ (see Fig. 3). Considering the ring geometry with counter-propagating fields, a resonant effect is expected enhancing the light-atom interactions. The enhancement in the interaction is not enough to change the rate of the spontaneous emission. However, the corresponding cavity effect (Bragg resonance) can increase the efficiency of the resonant photoluminescence. Taking advantage of the large inhomogeneous broadening in solids a continuous tuning of the atomic resonance can be achieved by varying the effective lattice spacing [12].

Fig. 3. Reflection from the atomic array as a function of lattice spacing in unit of the resonant wavelength for 100 and 10 atomic segments. Here $C_j=0.01$.

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3.3 Emission at the single-photon level

We now turn to single collective excitation where conditional measurement of a single photon emitted from the array can be used to generate a correlated photon propagating in the opposite direction to that of the original photon. In this low excitation limit, we are interested in the effect of atomic geometry on directionality of the second photon.

The collective emission from atomic ensembles has been the center of research for the generation of photon pairs from atomic gasses and solid state emitters [24–26]. As proposed by Duan-Lukin-Cirac-Zoller (DLCZ) [7] the memory-based bi-photon generation in this way can be used to implement entanglement distribution for long-distance secure communication [27]. The DLCZ protocol is one of the practical proposals, to date, for implementation of quantum repeaters [28]. Its proof of principle demonstration has been achieved using laser cooled atoms [8,29].

To study photon generation in the present case, we consider an array of three-level atoms hosted by a ring whose resonance wavelength matches the atomic transition, see Fig. 4(a). Figure 4(b) provides a 1-D representation of the system where the atomic segments form a commensurate or incommensurate lattice with light propagating along the lattice. We note that, instead of multiple atoms per segment, single atoms can be considered without change of the general discussion below.

Fig. 4. (a)The level diagram of a three-level atom considered with two pumps driving two $\Lambda$ transitions creating photons in the modes $a$ and $b$ in two directions. (b) 1-D representation of the light-atom interaction for two arrays of atoms one commensurate and one incommensurate with the intra-cavity light which is also resonant with the Raman transition of the atoms. (c) The normalized probability amplitude of emitting co-propagating bi-photons ($\Theta (k)$) in the same direction is plotted as a function of lattice constant for 6, 7 and 100 atoms interacting with the intra-cavity light.

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On the resonance condition, the atoms define fixed antinodes of a standing wave inside the resonator. They symmetrically emit light in both directions. Upon excitation with the pump light ($P_a$), a Stokes photon generated in the mode $a$ is indistinguishably emitted in the forward or backward (clockwise or counterclockwise) directions. In the single-photon regime, a collective excitation of atomic spins is formed as the footprint of the emitted photon. The collective excitation can then be retrieved using another pump ($P_b$) in a form of an anti-Stokes photon in mode $b$. In solids, bi-photon generation of this kind was achieved [26] by preparing atoms in an atomic frequency comb state.

A Hamiltonian similar to Eq. (2) can be written for the anti-Stokes photon in the mode $b$ with field operators denoted by $\hat {b}_c$ and $\hat {b}_{cc}$. Assuming a discrete distribution of atoms at locations $\theta _j$, we use second perturbation and trace out the atomic degrees of freedom to arrive at:

(9)$$\tilde{H}_{int} /\hbar^2\simeq J \big\{ \Theta(k) \hat{b}_{c}^{\dagger} \hat{a}_{c}^{\dagger} + \Theta(-k) \hat{b}_{cc}^{\dagger} \hat{a}_{cc}^{\dagger} + \hat{b}_{c}^{\dagger} \hat{a}_{cc}^{\dagger} + \hat{b}_{cc}^{\dagger} \hat{a}_{c}^{\dagger} + h.c \big\} \\$$

where $J$ is the effective coupling rate of each segment (atom), and the coefficient $\Theta (k)$, defined as

(10)$$\Theta(k) = \frac{1}{ N } \sum_{j=1}^{N}{e^{ 2 i kr\theta_j}} ,$$

is the collective phase factor controlled by the geometry of atoms. Applying this Hamiltonian to an initial vacuum state, returns the final (unnormalized) optical state given by

(11)$$| \Psi_f \rangle = \Theta(k) | 1_a \rangle_c | 1_b \rangle_c + \Theta(-k) | 1_a \rangle_{cc} | 1_b \rangle_{cc} + | 1_a \rangle_c | 1_b \rangle_{cc}+ | 1_a \rangle_{cc} | 1_b \rangle_c \\$$

where, for example, $|1_a\rangle _c$ represents one photon created in the mode $a$ propagating in the clockwise direction. The phase factor $\Theta (k)$ is related to the probability of generating two co-propagating photons and it is plotted in Fig. 4(c). When atomic spacing is a multiple of $\lambda /2$, this probability is maximum. The spacing can be chosen to minimize the probability of emitting co-propagating photons while enhancing the counter-propagating photon emission, even for a small number of atomic segments. When even number of segments are separated by $p\lambda /4$, where $p$ is an odd integer, the probability is always zero, resulting in directional emission of photons even in when low atomic segments considered. Such degree of control on directionality of the photon is important when correlated bi-photons are being generated from the atoms. Now lets consider an atomic lattice with spacing between the segments engineered to be $d_j=\lambda (1-A\zeta ^j)$, where $\lambda$ is the wavelength of the emitted photons, $A$ is a scaling constant and $\zeta$ is the incommensurability factor. As seen in Fig. 5(a) and (b), when the lattice becomes more and more incommensurate with the wavelength, the probability of generating co-propagating bi-photons decreases while a phase appears in the first term of Eq. (11), which is opposite to that of the second term. In the following section, we see how a proper choice of the $\Theta$ parameter may enable efficient storage in low atomic segment memories.

Fig. 5. The amplitude (a) and phase (b) of $\Theta (k)$ as a function of the incommensurability constant $\zeta$. The gray curve shows the amplitude and phase of $\Theta (k)$ for an atomic array off-resonant from the optical wavelength. Results plotted consider 100 segments with $A=0.2$. The gray curves are a good approximation for a randomly distributed ensemble.

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4. Storage in an ordered lattice

To see the effect of atomic geometry in protecting coherence in the system, in this section, we consider retrieval of a single-photon pulse after being stored inside the atoms as a collective excitation. We consider a lattice of two-level atoms initially prepared in the ground state, $|g\rangle$, arranged around the ring. A single photon pulse enters the cavity can be coherently absorbed by the atoms using, for example, a controlled reversible inhomogeneous broadening (CRIB) technique [30] and retrieved after time $\tau _s$. The Hamiltonian can be written as

(12)$$H_{int} / \hbar^2 \; \simeq \; J \big\{ \hat{S}\hat{S}^{\dagger} [\hat{a}_{c}^{\dagger}(t+\tau) + \hat{a}^{\dagger}_{cc}(t+\tau)][\hat{a}_{c}(t) + \hat{a}_{cc}(t)] +h.c \big\}.$$

For an input field (e.g. a photon wavepacket) initially propagating in the clockwise direction through an atomic lattice, the retrieved photon can be emitted in both directions introducing loss to the memory. We note that scattering into the free-space is determined by the spontaneous emission and is independent of the atomic geometry, in absence of cooperative effects. The free-space scattering introduces a constant loss to the process which we ignore in the following treatment. In absence of any phase noise and considering a random distribution of atoms in a large n ensemble, perfect retrieval can be achieved in the backward direction [31]. However, for a randomly distributed ensemble with a limited number of atomic segments, the directional retrieval is not guaranteed. Engineering the atomic position may enable to control the retrieval directionality even when a small number of atomic segments is used. In this case, considering the long-range excitation created among the atoms, a single photon efficiently stored inside the atoms results in non-local entanglement between atoms. Because atomic segments are separated by a distance on the order of the wavelength, they can be individually addressed and measured.

For an ordered lattice, the difference in transition probability between input and output photon modes is nonzero and its magnitude scales with $\Delta P= |J\int {e^{i\delta \theta (t)}dt}|^2- |J\int {\Theta (-k) e^{-i\delta \theta (t)}dt}|^2$, where $\delta \theta (t)$ is additional phase noise reducing the efficiency. To calculate this probability, we consider an input clockwise photon stored and then emitted into either a clockwise (second term) or a counter-clockwise (first term) photon. A zero transition probability, $\Delta P$, means equal possibility of emitting in both directions during the retrieval. As $\Delta P$ approaches one, the retrieved photon’s direction is well determined and opposite to that of the input photon. This probability is plotted in Fig. 6(a) for different atom number (or atomic segments) and spacing. It can be seen that the probability $\Delta P$ is always highest for an even number of segments separated by $\lambda /4$ while for a randomly distributed sample large number of segments are required to reach the same performance. By introducing phase fluctuation $\Delta P$ decreases (see inset of Fig. 6(a)). Away from the atomic resonance, the probability can increase thus recovering the maximum transition probability available. Moreover, when an incommensurate lattice is considered, the probability can increase near resonance. The result suggests the probability and thus the retrieval efficiency in a low atomic-segment memory can be increased by a careful choice of atomic distribution. To study light storage by a few atomic segments, we perform numerical simulation. We solve the following equations of motion for an input pulse, $\hat {\mathcal {E}}_{in}(t)$ being coherently absorbed by atoms inside the resonator. The absorption can be time reversed using an externally controlled field [32,33] or control of internal phase matching condition [34,35]. For the clockwise propagation we can write

(13)$$\frac{d\hat{\sigma}_c(\theta, t)}{dt} = (- \Gamma+ i\eta \theta + i \delta\theta(t))\sigma_c(\theta, t) +ig\hat{a}_c$$

(14)$$\begin{aligned}\frac{d\hat{a}_c(t)}{dt} = &- \frac{\kappa_c}{2} \hat{a}_c (t)+ i \int_{-\infty}^{t}{ \int_0^{2\pi}{\mathcal{N}(\theta) (\hat{\sigma}_c (\theta, t') + \hat{\sigma}_{cc}(\theta, t') e^{-2ikr\theta})d\theta}dt'} \\ &+\sqrt{\kappa_c/\tau} \hat{\mathcal{E}}_{in}(t) \end{aligned}$$

where $\eta$ is the spatial frequency gradient resulting in photon-echo retrieval [34], $\kappa _c$ is the cavity decay rate, $\tau$ is the photon lifetime inside the cavity and subscript $c$ and $cc$ refers to the clockwise and counterclockwise optical ($\hat {a}$) and atomic spin $\hat {\sigma }$ operators. We write similar equations for the counterclockwise fields and derive the output fields using the cavity input-output relationship. By numerically solving the above equations for mean-field operators , e.g. $\hat {\sigma }_c \to \langle \hat { \sigma }_c\rangle =\sigma _c$ and $\hat {a}_c \to \langle \hat {a}_c\rangle =a_c$, we calculate the light storage dynamics for different atomic geometries. Figure 6(b) shows the temporal profile of the input (black) and output (color) fields for both clockwise (solid color line) and counter-clockwise (dotted color line) fields. Multiple echoes are generated due to limited optical density of the medium. For atoms spaced by $\lambda /4$ the echo pulses alternate between the clockwise and counterclockwise directions, while for spacing of $\lambda /2$ the counter-propagating echoes are generated together. In the former scenario in the case of high optical density, most of the atomic excitation can be retrieved by the first echo. It can be seen that the ratio between the counter-propagating fields can be controlled using the atomic position even when low atomic-segment numbers considered. In this regime, a single photon stored among the atoms as a collective excitation will herald non-local entanglement between atoms featuring individually addressable sites.

Fig. 6. (a) The difference in transition probability $\Delta P$ of photon emission in opposite directions is plotted as a function of number of atomic segments for a randomly distributed ensemble (empty circles) and an ordered lattice with $\lambda /4$ spacing (filled circles). Inset shows the probability, $\Delta P$, as a function of spacing, for 6 segments spaced randomly, in a commensurate lattice, or an incommensurate lattice. (b) Result of semiclassical simulation of light storage in an array of six atomic segments separated by $\lambda /4$ or $\lambda /2$. The solid lines (dotted lines) represent the forward or clockwise (backward or counterclockwise) optical mode. The black pulse is the input and the color pulses are output pulses. Simulation parameters include: $\eta 2\pi /\Gamma =10^3$, $\Gamma =100\gamma$, and $N_0g^2/\kappa _c\gamma =10^4$.

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5. Conclusion

We investigated the role of atomic geometry on light emission and storage within a ring resonator. We showed that by designing atoms within a photonic waveguide or resonator, it is possible to boost the efficiency of the interactions. We showed that a lattice of atoms embedded in a ring resonator can be used to control the directionality of the retrieval process. Our results suggests the possibility of lattice engineering to study topological effects [36–39] where the clockwise and counterclockwise photons can be seen as pseudo-spin 1/2 particles. Moreover, light storage at a few number of atomic segments regime may result in creation of many-body entangled states [40] of distinguished modes using state of the art platforms [41]. Active elements including quantum dots [2,42,43], laser trapped atoms [5,44] and rare earth ions [12] integrated with photonic waveguides and resonators are suitable platforms for studying the above-mentioned cooperative effects for applications in classical and quantum photonics [45,46].

Acknowledgments

We would like to thank Qi Zhou for enlightening discussions.

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Study of atomic geometry and its effect on photon generation and storage [Invited]

Abstract

1. Introduction

2. Model: light propagation in a periodic lattice of atoms

3. Emission from the atomic array

3.1 Long-range cooperative atomic resonance

3.2 Emergence of Bragg atomic resonance and enhancement of excitation

3.3 Emission at the single-photon level

4. Storage in an ordered lattice

5. Conclusion

Acknowledgments

References

Cited By

Figures (6)

Equations (14)

Optical Materials Express