Single collective excitation of an atomic array trapped along a waveguide:
a study of cooperative emission for different atomic chain configurations

The quantization axis is taken along the axis of the waveguide, following the approach of Ref. [20], which is a practical configuration for a tripod scheme, and offers the benefits of a cylindrical symmetry. The atoms are all prepared in the $|F_{0},+1\rangle$ state (spin oriented along the waveguide).¹¹1This choice is different than in most experimental implementations, where the quantization axis is taken to be normal to the waveguide and typically in the plane of the atoms and waveguide. If instead of circular we used linear polarizations along either $x$ or $y$ axes, and superposed our atoms between $m=\pm 1$ states, we would obtain similar results to those discussed below. Subsequently, a weak and short coherent external light pulse excites the active atoms and creates a single collective excitation in a superradiant mode in the atomic array.

As a typical experimental example, in this study we consider the $|F_{0}=0\rangle\rightarrow|F=1\rangle$ transition of the ⁸⁷Rb D2 line, with a vacuum transition wavelength $\lambda_{0}~{}=~{}780$ nm, coupled to a silica nanofiber of radius $a\sim 200$ nm. The atoms are separated from the surface by half the fiber radius $a/2\sim 100$ nm. This configuration leads to a single-atom coupling ratio of $\beta=\gamma_{\textrm{1D}}/\gamma\sim 0.1$, where $\gamma_{\textrm{1D}}$ and $\gamma$ are the radiative decay rates into the guided mode and in free space (undisturbed), respectively [13, 21, 22].

where $|\psi(t)\rangle$ is a polariton-type entangled state that superposes a single atomic excitation and a single-photon occupation of the field modes. The decay of the excitation can be traced by the time-dependent probability $p=p(t)$ given by

The operator $\hat{P}$ defined as

projects the system density matrix $\hat{\rho}(t)$ onto the Hilbert subspace of the atomic excitation, where the photonic field is vacuum and the atomic state is any superposition of singly excited states with one specific $a$’th atom (with $a$ running from $1$ to $N$) occupying a Zeeman sublevel $|n\rangle$ of the excited state, and all $N-1$ other atoms occupying a Zeeman sublevel $|m_{j}\rangle$ ($j\neq a$) of the ground state. Note that the initial state is one of such states. The probability to obtain the system in its initial state can be found by projecting the final state $\hat{\rho}(t)$ onto its initial state with $\hat{P}\to\hat{P}_{0}=|\psi(0)\rangle\langle\psi(0)|$ in (2). In this case, we denote the observation probability as $p_{0}(t)$.

As $\hat{P}=\hat{P}^{2}$ and $\hat{P}|\psi(0)\rangle=|\psi(0)\rangle$, the excitation probability can be expressed in the following form

This basic equation describes the microscopic dynamics of the decay process in a multiatomic system, where the optical excitation can be virtually re-emitted many times. We considered here only the transformation of pure states, but the derivation can be straightforwardly generalized towards transformation of an arbitrary initial mixed state $\rho(0)$. The evolution operator $U(t,0)$ can be found in previous work [23, 24, 25, 22, 20] and is summarized in Appendix A.

Let us point out that the projection of the evolutionary operator on a single excitation subspace physically means that no doubly excited states play a role in the propagation dynamics through the waveguide. However including doubly excited states is necessary in our model to account for the near field virtual interaction giving rise to a static dipole-dipole coupling for any atom with its proximal neighbors.

From the point of view of scattering theory we deal here with an example of a process, where the wave-packet of an initial state of the compound system evolves into the final state of two uncoupled subsystems belonging to a particular scattering channel. In accordance with general principles of the scattering theory, the dynamics of such an incomplete scattering process can be described with the formalism of the Møller operators: the system’s states are transformed either from infinite past (operator $\hat{\Omega}^{(+)}$) or from infinite future (operator $\hat{\Omega}^{(-)}$) to a compound state considered at the conventional initial moment $t=0$ [26]. In the interaction representation, the corresponding asymptotic transformation from the initial state $|\psi(0)\rangle$ (excited atoms) to the outgoing state $|\psi\rangle_{\mathrm{out}}$ (emitted photon), is given by

where $H$ is the system Hamiltonian and $H_{0}$ is the channel Hamiltonian corresponding to the separated subsystems. In our case the output channel consists of the emitted photon and the de-excited atomic chain. Since the latter exists in its ground state, then in the rotating wave approximation (RWA), we can ignore in $H_{0}$ any coupling with the quantized field.

The Møller operators are nontrivial mathematical objects which map the states belonging to the continuous spectrum of the compound system onto the separated states for the particular scattering channel. Although the operators are constructed as a product of two unitary operators $U(\ldots)$ and $U_{0}(\ldots)$ describing the interacting and non-interacting dynamics respectively, if the compound system had a set of stable states belonging to its discrete spectrum, the product considered at infinite $\tau\to\infty$ generally would not tend to a unitary operator. In our system there are no stable and localized coupled excited states of the field and the atoms and the Møller operators are unitary. We detail the operator $\hat{\Omega}^{(-)\dagger}$, contributing to the transformation (5), in Appendix B.

The asymptotic state $|\psi\rangle_{\mathrm{out}}$ can be decomposed into the infinite set of the states $|f\rangle\equiv|G,s\rangle$. Here $|G\rangle=|m_{1},\ldots,m_{N}\rangle$ represents any of the collective atomic ground eigenstates, and the quantum number $s$ specifies the freely-propagating field modes modified by the presence of the waveguide. One particular mode is the guided mode $s=\sigma,k$ parameterized by an azimuthal number $\sigma=\pm 1$ and a longitudinal wavenumber $k$. Some details regarding the description of the field and of the atom-field interaction Hamiltonian near a nanostructure are discussed in Appendix C. The transition amplitude to a particular $f$-th final state is eventually given by

where $\omega_{s}$ is the frequency of the photon outgoing to the $s$-th mode. The mode, associated with the photon injected into the waveguide, contributes through the spatial profile of its displacement field $\mathbf{D}^{(s)}(\mathbf{r})$ subsequently taken at the position $\mathbf{r}=\mathbf{r}_{a}$ of each atom in the chain.

The spatial profile $\alpha=\alpha(z)$ for a pulse outgoing the system within any azimuthal guided mode, and when the atoms occupy any ground state, can be constructed as

where $f=f^{\prime}$ excepting the longitudinal wavenumbers $k$ and $k^{\prime}$. In accordance with the general concept of the scattering theory, the transformation (5) concerns only spatial profile of the asymptotic wave emerging the system. Physically we can associate it with free dynamics of the outgoing light wavepacket by substituting $z\to z-v_{g}\tau$, where $v_{g}$ is the group velocity.²²2Even with negligible dispersion the group velocity is an important parameter of the theory, see [22, 20], and typically it is about $v_{g}\sim 0.7\,c$ for a nanofiber.

For a separation between the atoms given by a multiple of a half-wavelength of the guided mode – the well-known atomic Bragg mirror case – the system is open from both sides and exhibits a cooperative exponential decay, with a rate given by $\Gamma_{C}\simeq N_{C}\,\gamma_{\textrm{1D}}/2$. The scaling rate $\gamma_{\textrm{1D}}$ coincides with the evaluation of the single particle self-energy part in [22]. Let us note that the $1/2$ factor results here from the polarization sensitivity of the cooperative emission on the $\sigma_{-}$ transition, see Fig. 1.³³3Here one should not confuse the $\sigma_{-}$ transition with the azimuthal mode $\sigma=-1$. In the emission process the atoms are coupled with both the modes $\sigma=\pm 1$ and each of the modes overlaps all three transitions $\sigma_{-},\pi,\sigma_{+}$. Nevertheless the emission of a single atom into the waveguide via the $\pi$-transition is typically much weaker than via the $\sigma_{\pm}$-transitions that justifies the $1/2$ factor, see [22] ( Eq. (3.5)) and related comments there.

We investigate how the emission by such a chain would change in different conditions. Specifically, this base case of a single excited chain will be compared with two other geometries, where we extend the atomic array by additional segments of atom arrays all in the ground state (see Fig. 1), to the left and/or to the right. In the following, we will refer to the additional segments as atomic mirrors, and to the single excitation collective state in the central segment as a collective emitter. In order to follow the decay process, we will provide three probabilities. The first one is $p(t)$, the excitation probability shared in the whole atomic array the sample as defined by (2) and (3). The second one is $p_{0}(t)$ that captures the decay of the original excited state, as defined in Section II.2 by swapping $\hat{P}$ for $\hat{P}_{0}$. Finally we will also compute the probability $p_{a}(t)$:

where $\hat{P}_{a}$ is defined as in (3), with the only exception that it projects only onto the central segment subspace. In the considered one-dimensional open system, we can expect the recurrent incoherent scattering to play only a negligible role. This translates to the balance $p_{0}(t)=p_{a}(t)$ being fulfilled, which we will verify for any atomic configuration, within our calculation precision.

We first note that, in this situation, the decay dynamics is insensitive to the actual position of the collective emitter within the full array. For a small enough number of atoms such that the Markovian approximation also applies for propagation along the full array from end to end, the retardation for the internal virtual processes is negligible. The initially prepared superradiant mode can thus be instantly re-expanded in the basis set of the collective eigenmodes of the complete resolvent operator and then obey the wavepacket dynamics driven by its complete self-energy part (effective Hamiltonian).

The beginning of the decay reveals a stage of fast cooperative emission into the waveguide, which arises from the projection of the initial state onto the superradiant mode of the complete atomic system. This projection has amplitude $\sqrt{N_{C}/N_{\mathrm{tot}}}$, hence the initial fast decay in Fig. 2 lasts only until $p(t)$ has decreased by $N_{C}/N_{\mathrm{tot}}$.

The asymptotes of $p_{0}(t)$ and $p(t)$ deviate from the natural spontaneous decay $\exp(-\gamma\,t)$. These later time dynamics result from the projection onto the $(N_{\mathrm{tot}}-1)$ other collective modes, which are subradiant into the waveguide and hence decay by spontaneous emission into the external modes with a decay rate $\gamma^{\mathrm{ext}}$. The presence of the waveguide restricts spontaneous emission to a smaller than $4\pi$ solid angle, resulting in $\gamma^{\mathrm{ext}}<\gamma$. ⁴⁴4If we combine the single-atom emission rate into the waveguide $\gamma_{\mathrm{1D}}$ with emission into the external modes $\gamma^{\mathrm{ext}}$ we get $\gamma^{\mathrm{ext}}+\gamma_{\mathrm{1D}}>\gamma$ (Purcell effect). Simultaneously we have $\gamma^{\mathrm{ext}}<\gamma$. This is a specific property of this one-dimensional and translationally invariant atomic configuration, arising from the ability to cooperatively emit the photon along the array direction via the Rayleigh channel with only a negligible probability to scatter via the Raman channel. We found that $p_{0}(t)=p_{a}(t)$ within our numerical precision, showing that there is no signature of recurrent scattering in the emission process and that the atomic excitation in the central segment only occupies the superradiant mode state (8).

The perfect ordering of atoms is a crucial requirement for the data presented in Fig. 2. Disorder in the left and right atomic mirrors, such as completely randomized positions, would break their ability for giving rise to coherent Bragg-type interference with the atoms in the central segment. In this case the decay dynamics is modified as shown in Fig. 3, where the atoms in the mirrors are disordered, following a uniform probability distribution along the length of the mirror with a density of $1$ atom per $\lambda^{\mathrm{wg}}/2$. The initially excited atoms remain fully ordered. Somewhat surprisingly, the beginning stage of the decay process in Fig. 3 appears similar to the case of the ordered system shown in Fig. 2. Even for the disordered configuration there is meaningful probability to project onto a cooperatively emitting collective state of the full atomic array involving all the atoms and with specific choice of the phase shifts $\vartheta_{a}$ in (8). That collective state exhibits fast superradiant emission from the full array into the waveguide observed at short times in Fig. 3.

The most striking difference with the ordered configuration concerns the population of the original state. In the tail of the process the dynamics changes and the disordered mirrors act as absorbers that incoherently re-scatter the photon into any mode. As a consequence we observe an exponential depopulation of the initial state nearly identical to the original undisturbed cooperative emission: the excitation leaves the collective emitter as if the atomic mirrors were absent. Just as above, no other collective state than the superradiant mode is populated within the active atoms, i.e. $p_{0}(t)=p_{a}(t)$.

The emission at long times, given by the asymptote of $p(t)$, is also different from the ordered case. In particular, a bump in the emission appears at long times in Fig. 3(b), that has its counterpart in $p(t)$ in Fig. 3(a), which originates mainly from the atomic mirrors as the excitation has depleted from the collective emitter ($p_{a}(t)$ is orders of magnitude smaller than $p(t)$ then). The long-term asymptote for $p(t)$ is an exponential decay $\sim\exp(-(\gamma^{\mathrm{ext}}+\gamma_{\mathrm{1D}})t)$, slightly faster than $\exp(-\gamma\,t)$, which originates from the Purcell enhancement of the single-atom emission by the presence of the waveguide. Interestingly, if the collective emitter is placed in the center of the full array rather than just on one end, the emission is delayed. Indeed if one end of the array is open for the emission from the initial state, the emitted light in that direction is directly lost and cannot be re-absorbed.

We presented our simulations for a particular random configuration of the atomic mirror, without ensemble averaging, so the emission of the sample indicates a slight signature of its configurational dependence. That is more visible for the approximately symmetric configuration showing slight beatings in its decay dynamics, see Fig. 3(c).

In Refs. [18, 19] this configuration has been called a quantum cavity, given that the central atoms sit at the anti-node of the photonic mode and therefore maximally couple to it. A mapping to the Jaynes-Cummings model was made [18], exemplified by the appearance of a vacuum Rabi splitting, which shows that the density of photonic states is indeed enhanced. However we emphasize the crucial difference with cavity-QED, already pointed out in [19], that in this process the left and right segments neither operate as classical light reflectors nor as a cavity trapping the light. This means that the photon lifetime in the system does not extend beyond the natural lifetime with the presence of the “atomic Bragg mirrors”.

In this geometry we eliminate the contribution of the global superradiant mode, which still exists but only has negligible overlap with the initially prepared state. But now each atomic segment, considered independently, possesses a superradiant mode (8) as one of the eigenstates of its own partial resolvent, and the complete resolvent operator can be decomposed into the basis of these states and reproduced by a separated matrix block in a part of the atomic Hilbert subspace. As a consequence the initial state, being a superposition of the eigenstates of this matrix block, would further evolve as a wave-packet, with periodic swapping of the excitation between the atomic segments. During this swapping process each segment has the ability for cooperative emission into the waveguide. The probability $p_{0}(t)$ should then reveal an oscillating behaviour, a signature of partial revival to the initial state. Physically it corresponds to a coupled dynamics between the atomic segments. During an excitation swapping cycle, a significant part of the optical excitation leaks into the waveguide. Since the emission into the waveguide has a superradiant nature from all segments, the total excitation probability $p(t)$ depletes faster than in the preceding examples, shown in Fig. 2 and Fig. 3.

Figures 4 and 5 present the result of our numerical simulations, supporting the above arguments for the same other calculation parameters as in Fig. 2. The strong difference with the results of Fig. 2 is evident, with the above mentioned oscillating behavior being observable in Fig. 4. These oscillations are the direct counterpart to the behavior observed in Refs. [18, 19] with a collective emitter instead of a single atom, here with full inclusion of dipole-dipole coupling and decay through free space. Noticeably the decay process is slowed down and its dynamics indicates the reviving oscillations for $p_{0}(t)=p_{a}(t)$ as well as certain deviations from exponential decay of $p(t)$.

When the atomic mirror is added to one side only, as shown in Fig. 5, the excitation exits the system faster than from the bare collective emitter. In the beginning stage, approximately one half of the excitation is emitted directly into the waveguide towards the empty direction. The other part of the excitation transfers to the atomic mirror and is then cooperatively emitted into the waveguide at a higher rate. In such a geometry the reviving oscillations of $p_{0}(t)=p_{a}(t)$ only appear in a long-term asymptote beyond the scale of the graph reproduced in the plot.

In some cases, like with the geometry of Fig. 5, the emission is found to decay faster than when the atomic mirrors are absent. This is a consequence of the collective effects from the full chain, which makes the cooperative emission into the waveguide generally stronger. The emission profiles thus further highlight the strong dependence of the system dynamics on the specifics of the atomic arrangement. Such directionality could for instance be useful for the implementation of efficient atom-based quantum optical protocols, as an alternative to intrinsic chiral systems [29, 30].