Characteristics of 1D ordered arrays of optical centers in solid-state photonics

Coherent interaction of light with an ensemble of atoms is actively pursued for implementation of quantum optical platforms to generate, store and process quantum optical information. The strength of interaction between light and atoms is key to efficient and deterministic control of quantum information. As an example, distribution of quantum optical entanglement can be done more efficiently, in principle, when photonic qubits interact strongly with single atoms [1] compared to weak interaction with many atoms [2]. A conventional approach to increasing the light-atom interaction is to use low-loss and small-mode-volume optical resonators built around the atoms or optical centers [3].

In solids, an ensemble of atoms exhibits inhomogeneous broadening due to defects in the host material. High-bandwidth interaction has been achieved in small doped crystals with this property [4]. Using optical resonators to reach the strong-coupling regime, in this case, limits the interaction bandwidth due to the resonator linewidth. Therefore, exploring interaction mechanisms that enable strong light-atom coupling over a wide range of frequencies is desirable to advance quantum photonic technology and achieve broadband control of quantum optical information.

It has been proposed that an array of 2D or 3D identical atoms (or emitters) forming a periodic structure in space can coherently interact with light [5, 6] as an atomic mirror or they can superradiantly emit photons in certain directions. In 1D, laser-cooled atoms placed near a waveguide have also been shown to reach the strong coupling regime [7]. In these scenarios, atoms are considered to be identical (e.g. laser cooled atoms) and that imposes limitations on system's bandwidth (MHz) and its practical realization, which is due to the complex experimental setup required for laser trapping atoms. In this paper, we consider a mesoscopic 1D array of solid-state atoms coherently and collectively interacting with light over a wide range of frequencies. We discuss how such arrays can be implemented in solid-state photonics with large atom numbers and arbitrary geometries. Primarily, we present the theory to analyze the emission properties of such arrays and discuss its applications.

The rare-earth ions in bulk crystals have been used to demonstrate broadband quantum light storage and these systems have the potential to reach coherence time on the order of hours [8]. Rare earth ions randomly doped in a miniaturized optical resonator have also been used to show strong light-atom interactions over the resonator linewidth [9, 10].

Recently, our group has investigated interactions of photons with 1D arrays of rare-earth ions in solid-state crystals and photonic structures. In one experiment (see figure 1(a)), we used a randomly doped Er crystal to engineer an effective array inside the crystal [11]. We created the array by means of spatio-temporal hole burning to tailor an atomic profile with a periodic spatial distribution in one dimension. We showed that this effective array can behave like an atomic mirror coherently reflecting light ($E_\mathrm$) in the same direction as the input light ($E_\mathrm$).

Figure 1. (a) An standing-wave pump was used to create (via the holeburning process) an effective array in a randomly Er-doped Y2SiO5 (YSO) crystal [13]. The effective array could then create an atomic Bragg grating reflecting the probe light after the holeburning process was completed.(b) An array of Er ions was precisely implanted into a SiN microring resonator. Enhanced light collection (reduced loss) was observed at a wavelength commensurate with the lattice [14]. (c) An array of Tm ions implanted into a lithium niobate microring resonator can exhibit long-range coupling with superradiance signatures [13].

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In another experiment, we used precision ion implantation to deterministically create a periodic array of isotopically-pure Er ions inside a SiN microring resonator (see figure 1(b)). Microring resonators have been proposed as a scalable platform for light-atom interactions in solid-state systems [12]. We showed that in this platform, coherent interference between the scattered light inside the microring can be manifested as the reduced propagation loss at wavelengths commensurate with the array. Due to the amorphous structure of the SiN host, the inhomogeneous broadening and the decoherence rate of Er ions were large in this platform preventing us from observing of quantum interference or strong interactions.

To reach the quantum regime of light interaction with an ensemble of rare-earth ions, we also carried out experiments with an array of isotopically pure Tm3+ ions implanted inside a lithium niobate (LN) microresonator (see figure 1(c)). The crystalline structure of LN shows lower inhomogeneous broadening and decoherence for Tm3+ ions compared to Er3+ ions in SiN. Also, the large branching ratio of Tm3+ compared to Er3+ at the measured transition makes it easier to observe the collective decay effects. We were able to observe superradiance emission of photons from Tm3+ ions in LN resonators. The superradiance is the evidence of strong coupling of an array of solid-state optical centers with light. We note that the microresonator used in this study had large linewidth, and as the result a strong coupling could not be obtained by using the resonator alone. The geometry of ions enables interference between the light scattered by the ions, leading to directional and nonlinear emission. The emission between individual ions and its interference can be modeled as an effective coupling field between the ions that leads to long-range ion-ion coupling. The large inhomogeneous broadening of implanted ions and the relatively large cavity linewidth enables observation of collective light-atom coupling over a wide range of frequencies.

Typical microring resonators fabricated for photonic devices in LN have radii on the order of tens to hundreds of micrometers. These rings have a wide transmission window such that the intracavity fields can interface with Thulium or Erbium ions respectively. For our application, we fabricate rings with radius $R \approx 100\,\,\,\mu \text$ as shown in figure 2. The width of these waveguides is chosen to be sufficiently small to isolate the fundamental TE mode. The surface roughness of the ring dominates the scattering loss, modeled by a decay rate κ. Current state-of-the-art fabrication can produce rings with decay rates on the order of megahertz [15]. The ring is coupled to the rest of the optical elements via a bus waveguide, carrying driving fields E1 and E2.

Figure 2. Schematic representation of the physical system under study. Atoms or optical centers are confined in segments much smaller than the optical wavelength around a microring resonator. The segments contain atoms of different frequencies and form an array around the resonator.

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Into this microring, we have previously implanted Tm3+ ions in bands approximately 36 nanometers in width. The 3H6 to 3H4 transition of these ions corresponds to the desired 795 nanometer light, with a branching ratio close to unity for isotopically pure Thulium ions [16]. As a result, these ions can be considered as two-level systems coupled to our optical resonator modes. We then capture the decay to non-cavity modes by these ions as another decay rate Γ0.

When emitters such as rare-earth ions are incorporated into solid-state nanostructures, they experience an inhomogeneous broadening of the transition frequencies on the order of gigahertz [16]. Understanding the role of inhomogeneous broadening in cooperative coupling is of great practical importance and, to the best of our knowledge, it has not been studied in details. In this paper, we provide a theoretical model for studying light-atom interaction in solid-state media with inhomogeneous broadening. We discuss the influence of this broadening on collective behaviors in 1D microring structures as defined above. We show that the interplay between atomic and cavity broadening enables broadband cooperative effects.

Recently, there has been a great degree of interest in the effects of geometry on atomic ensembles coupled to a photonic medium. Atoms placed near a waveguide can induce an effective cavity mode, resulting in collective behaviors for atoms distributed in a lattice [17, 18]. Such systems have been used to engineer cooperatively enhanced atom-atom interactions [19] and coherent storage of photons in a lattice [20]. We consider an array of emitters inside a microring resonator coupled to counter-propagating modes. In the ring geometry considered, $\hat_1$ and $\hat_2$ denote the clockwise- and counterclockwise-propagating cavity modes, respectively. The atoms of uniform transition frequency ωa are distributed within the waveguide and confined in segments of width much less than the wavelength (λ), located at angles θn . The bare light-atom coupling strength is given by $g_0 = d\sqrt$ for dipole matrix element d, cross-section A, and driving field frequency ω. It is assumed to be identical for all atoms. In general, the cavity linewidth is assumed to be much greater than the linewidth of an atom. Thus, we model the intracavity field as a broadband field equation with mode frequency $\omega_0(k)$ in analogy to [21]. Initially, we will consider the case of a system of unbroadened atoms. The Hamiltonian of the system is given in terms of the Fourier components of the intracavity field as

Equation (1)

Here $\hat_^$ and $\hat_^$ are the operators for the atomic population and coherence of ith segment, respectively. $E^_j(k)$ denotes a classical driving field acting on the cavity mode k in direction j, and we assume that the intracavity field has a linear mode dispersion $\omega_0(k) = \pm v_p k$ for uniform phase velocity vp . We consider dissipation due to cavity decay at a rate κ and atomic decay to non-cavity modes at a rate Γ0,

Equation (2)

In its current form, equation (2) results in equations of motion of the atomic degrees of freedom containing field terms like

Equation (3)

The field-atom coupling in equation (3) contains the full expansion of the field terms. To eliminate these degrees of freedom, we define a field density operator $\hat^(\theta, t)$ as

Equation (4)

To determine the equations for the field mode operators, we integrate the equations of motion of each operator with respect to time. The clockwise and counterclockwise field mode operators $\hat_1(k, t)$ and $\hat_2(k, t)$ have their dynamics given by

Equation (5)Equation (6)

The solutions to equations (5) and (6) are then given by

Equation (7)Equation (8)

To simplify, we define $\hat^_1(\theta) \equiv \int_0^\infty e^ \hat_1(k, 0) \mathrmk$ and $\mathscr_1(\theta) = \frac} \int_0^\infty \frac(k) e^} - i v_p k} \mathrmk$. Performing the inverse Fourier transforms on equations (7) and (8) yields

Equation (9)

Resolving the time-integral component of equation (9), we find a pair of step-functions due to the cyclical nature of the ring resonator.

Equation (10)

The counter-clockwise component $\hat_2$ follows identically, with $\theta - \theta_m$ replaced by $\theta_m - \theta$ and $\hat_1^\left(\theta - \fract\right)$ replaced by $\hat_2^(\theta + \fract)$ in equation (10) due to the opposite direction of propagation. The term $A^_$ describes the delayed initial cavity fields at a position θ, and the term $\mathscr_(\theta)$ describes the time-independent effect of the classical driving field at a position θ. The field equation can then be viewed as two parts; an input term which describes the behavior of the input fields in the cavity, and a scattered field from each of the atoms. We also observe that the average time a photon spends in the cavity is given by $2\pi R q/v_p = 1/\kappa$. Applying the Markov approximation for the atomic degrees of freedom and adding the counterclockwise field component, we find

Equation (11)

Here, we have defined $\mathcal(\omega) \equiv \frac \sum_^ e^ + i\omega)\frac(2\pi p)}$, interpreted as the cavity effect for a mode of frequency ω. The $\mathscr(\omega)$ term contains both real and imaginary parts; this accounts for both the destructive and constructive interference from the photon traveling in the microring cavity, as well as the buildup of phase for photons off-resonance with the cavity. q represents the average number of revolutions a photon makes before decaying out of the cavity. Making a transformation to include $\mathcal(\omega)$ allows us to limit θ to the range $[0, 2\pi)$. The scattered field includes two terms as each atom scatters a field in both the clockwise and counterclockwise directions. To further simplify the equations, we will write the per-atom scattered field amplitude as

Equation (12)

This term encapsulates the effect of the phase between atoms at positions θn and θm on the coupling via a mode of frequency ωa . In the case where the resonator decay for a single revolution is negligible and the atomic frequency is sufficiently close to a cavity resonance, we can write

Equation (13)

When viewing the interatomic coupling, it is useful to express our coupling strength in terms of the cavity-enhanced cooperativity $\eta = \frac$. Grouping both the clockwise and counterclockwise initial fields into a single term $A^(\theta, t)$, we can write the equations of motion for the atomic population as

Equation (14)

In equation (14), the first line denotes the free-space evolution of the atomic population due to cavity-amplified decay to free space and repumping. The amplification term depends strongly on the atomic frequency; as the atoms are detuned away from the cavity resonance, the real part of $\mathcal(\omega)$ decays quickly in amplitude. The term in the second line is the result of initial population of the resonator at t = 0. Note that the terms $\hat^$ are explicitly time-dependent as the initial wave packet is not necessarily resonant with the cavity, and does not produce a standing wave. For a long-time steady state approximation, one can drop these terms as they decay exponentially. The final term describes an effective coupling between atoms mediated by the cavity field. The equations for the atomic coherence follow a similar form,

Equation (15)

The last term in equation (15) indicates an effect on the inter-atomic coherences that depends on the inversion of each atomic dipole. Note that this term vanishes for two atoms in the ground state. The second term functions like a source term for the inter-atomic coherences, and indicates that the long-time evolution of the system will experience a nonzero steady-state interatomic coherence as the steady-state population increases. Each atom in this ensemble experiences a cavity-induced enhancement to its emission rate given by $2\eta\Gamma_0\Re(\mathscr(\omega))$, as well as a frequency shift $\eta\Gamma_0\Im(\mathscr(\omega))$.

To better understand the role of the various terms in this equation, consider the case of all N atoms at the cavity-resonant frequency ω0. This corresponds to $k_0 R = \omega_0 \frac$ being an integer. Assuming that the atoms are placed such that each atom is located at an antinode of the field, then the phase component of the couplings are universally one. Assuming that the evolution time of the system is long relative to κ, then the constants simplify to

Equation (16)Equation (17)

In the resulting equations, we assume that the classical driving field is symmetric for all atoms; i.e. $\mathscr(\theta_i)$ is identical for all θi . Under these assumptions, all atoms behave identically regardless of position θi in the steady state. Thus, one has the symmetric expectation value relations $\langle \hat_^ \rangle = \langle \hat_^ \rangle$, $\langle \hat_^ \rangle = \langle \hat_^ \rangle$. Under these conditions and considering $\langle \hat_^ \rangle = -1$, the steady-state single-atom and atom-atom coherence can be written respectively as

Equation (18)

Note that the atomic decay rate, Γ0, is modified by a factor of $1+4N\eta$ due to the collective coupling enhancing the directional scattering into the cavity mode. The emission rate of the system then scales linearly with the number of participating atoms in a fully-cooperative array. This collectively enhanced emission is mediated by the intracavity field, and is maximized when the atoms are all located at antinodes. When this collective decay rate exceeds the original decay rate Γ0, the system becomes 'superradiant' [22]. Under these conditions, the field scattered by the surrounding ensemble of atoms induces directional scattering.

3.1. Correlation in emission from atomic arrays

To better understand the effect of the atomic geometry on the emissions from the ring resonator, we consider the correlations between field components. The second order-correlation function, $g^ (\tau)$ can be used to quantify the degree of directionality of emission from an array of emitters. We can write $g^ (\tau)$ for correlation between modes i and j observed at a position r0 in terms of the field operators as

Equation (19)

Let t denote the time at which the system reaches steady-state evolution. We consider exclusively the scattered field, ignoring the cavity driving field. Using the same time-evolution of the field modes as described above, we arrive at a spin-operator representation of equation (19),

Equation (20)

where $\alpha_$ denotes the coupling between an atom m and a field mode i. For the case of our ring resonator, this $\alpha_$ contains both the coupling strength g and a position-dependent effect $e^$. For τ = 0, in the case of complete atomic inversion these expectation values have well-known forms:

Equation (21)Equation (22)

For N unbroadened atoms at the resonant frequency of the cavity in a ring resonator with placements given by $\theta_1, \dots, \theta_N$, we find the following results for emissions into co- and counter-propagating modes the same frequency:

Equation (23)Equation (24)

For any N > 1, the upper and lower bounds for $g^(0)$ are given by

Equation (25)

The maximum value of equation (24) is achieved when the interatomic spacing is $\lambda/2$, as then $R(\theta_}} - \theta_m) = n \lambda/2 = n \pi/k_a$. In the special case of two atoms in a ring resonator, the $g^$ function can be simplified as $ g^_(0) = \cos^2(k_a R(\theta_m - \theta_}})) $. This case was recently studied experimentally for two defect centers in a SiC microdisk [6], and it was demonstrated that this system can allow for selection of only the copropagating modes by angular spacing of $\pi/2$. Note that the limits derived for the N-atom case implies that this effect is unique to the 2-atom regime. In the two-atom case, the atomic phase or position can be engineered such that perfect constructive or destructive interference between interactivity radiation fields can be observed. As the number of atoms increases, the atoms can no longer be positioned such that the interference between all atoms is completely destructive and one sees reduced emissions in the counterpropagating modes. The influence of geometry on the emergence of superradiant behavior has been previously evaluated in free space in [23, 24]. In the case of a resonator, the directionality of these bursts is replaced by a preference for a particular direction of propagation within the waveguide.

3.2. Effect of inhomogeneous broadening in atomic arrays

In the prior section, we developed a theory for handling atoms of any frequency coupled to a ring resonator. We now consider an ensemble of atoms with frequency distribution, e.g. inhomogeneous br

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