## Abstract

Confining light in open structures is a long-sought goal in nanophotonics and cavity quantum electrodynamics. Embedded eigenstates provide infinite lifetime despite the presence of available leakage channels, but in linear time-invariant systems they cannot be excited from the outside, due to reciprocity. Here, we investigate how atomic nonlinearities may support single-photon embedded eigenstates, which can be populated by a multi-photon excitation followed by internal relaxation. We calculate the system dynamics and show that photon trapping, as well as the reverse release process, can be achieved with arbitrarily high efficiencies. We also discuss the impact of loss, and a path toward the experimental verification of these concepts.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Trapping light for times much longer than dissipative and dephasing
timescales is essential to enhance light–matter interactions and to
create high-fidelity storage of quantum states. Light confinement is
conventionally achieved by suppressing unwanted radiation channels, e.g.,
with mirrors or photonic bandgap materials. However, radiation loss is
never totally suppressed in conventional open cavities. Moreover, due to
reciprocity, an idealized lossless cavity is necessarily decoupled from
the outside, thus making it impossible to inject energy or to detect its
internal state. Reciprocity, thus, prevents realizing linear
time-invariant systems that can both *efficiently collect*
and *perfectly confine* light.

Recently, there has been a significant interest in embedded eigenstates (EEs) [1], i.e., optical states in open resonators that are ideally confined despite the presence of available radiation channels. Several mechanisms to form EEs have been discussed [2–11], and their existence has been demonstrated in waveguides [4,5,7,9], photonic crystal slabs [6,12], and 3D nanostructures [8]. Despite their fundamentally interesting physics, EEs in linear systems are limited by reciprocity: ideal confinement implies that energy cannot be injected from the outside. A possible solution is to consider Kerr nonlinearities [10,13,14], such that the EE existence is dynamically triggered by the excitation, trapping part of the impinging radiation. However, the weakness of classical nonlinearities implies large optical powers and inherent inefficiencies, which hinders the application of these concepts for quantum information and low-power optoelectronics.

Here, we propose and investigate the possibility of creating and exciting EEs in coupled cavity–atom systems. Due to atomic nonlinearities, these EEs can host only a single photon. The system is therefore “transparent” upon single-photon excitation, but it can absorb multi-photon excitations, populating the EE. We show that upon two-photon excitation, the trapping process, whereby one photon is trapped in the EE and the other is re-emitted, occurs with arbitrarily high efficiencies upon suitable control of the pulse shape. Due to time invariance, the inverse process is also possible: a stored excitation can be released on-demand upon single-photon excitation. Even considering realistic losses, the proposed mechanism allows storing photons for times much longer than those obtainable with a single cavity, while preserving the excitation rate. Our protocol can be used to realize optical memories, and to achieve on-demand storing and release of single photons [15].

## 2. MODEL

#### A. Linear Resonators

A general condition to induce EEs in coupled systems has been introduced by Friedrich and Wintgen [1]. Following their scheme, we first consider a classical system [Fig. 1(a)] formed by two optical cavities resonating at frequencies ${\omega}_{1}$ and ${\omega}_{2}$, mutually coupled with rate $J$. The cavities interact with a single-mode waveguide with coupling strengths ${V}_{1}$ and ${V}_{2}$, respectively, such that ${\mathrm{\Gamma}}_{1,2}\equiv 2\pi {V}_{1,2}^{2}/{v}_{g}$ are the amplitude decay rates into the waveguide [16,17], and ${v}_{g}$ is the group velocity. We assume that within the spectral range of interest, the waveguide has a linear dispersion, and ${V}_{1/2}$ do not depend on frequency. The two-cavity system supports two coupled eigenmodes that, due to the presence of the waveguide, are lossy. However, when the destructive interference condition

is satisfied, one of the coupled modes becomes lossless [1], realizing an EE, while the amplitude decay rate of the other (bright) mode is $\mathrm{\Gamma}\equiv ({\mathrm{\Gamma}}_{1}+{\mathrm{\Gamma}}_{2})/2$ (see Supplement 1 for details). We note that light trapping based on destructive interference between cavities coupled to a waveguide was also investigated in Ref. [18]. Reciprocity dictates that this EE cannot be excited: when a quasi-monochromatic pulse [Fig. 1(b), blue shade], centered at the EE frequency, impinges on the system (set at the condition 1), it creates transient cavity fields (solid lines) that quickly vanish, showing no energy stored into the EE. Classical nonlinearities (such as the Kerr effect) inside the cavities may overcome this constraint [13,14], but at the price of high intensities and inefficiencies.#### B. Adding Atomic Nonlinearities

Rather than inserting a classical nonlinearity into the cavity, we
instead assume that one resonator is nonlinear at the few-photon
level, as in the case of natural or artificial atoms. We thus replace
cavity 1 with a two-level atom, and the system parameters are
re-labeled accordingly [Fig. 1(c)] (similar effects are obtained by replacing both cavities
with atoms; see Supplement
1). We assume that the cavity and atom
couple to each other at rate $J$, and they interact with the waveguide at
the same location $x=0$. A nonzero cavity–atom separation
would introduce an additional waveguide-mediated coupling; in this
case, an EE can be obtained when Eq. (1) holds and the
atom–cavity distance is properly chosen (see
Supplement
1 for details). The existence of these
bound states in pairs of atoms embedded in waveguides has been
recently discussed [19,20], but their trapping and release
dynamics were not investigated. We note that mixed
atomic–photonic bound states can also arise when a waveguide
with a bandgap is coupled to an atom whose emission frequency lies
within the bandgap [21]. These
bound states originate from the exponentially decaying field in the
waveguide, they can host multiple excitations, and they are not
*embedded* because their frequency lies outside of the
continuum of propagating mode. Their physics is therefore
fundamentally different from the EEs considered here in this work.

The Hamiltonian of the system in Fig. 1(c) reads ($\hslash =1$)

We verified the trapping behavior by applying input–output theory [23] to the Hamiltonian (2), and recasting the problem into a master equation for the density matrix of the atom–cavity system:

## 3. TRAPPING AND RELEASE OF SINGLE PHOTONS

We now verify that the EE excitation can occur for non-classical excitations, and at the single-photon level. In particular, we consider the trapping process where two photons impinge on the system and one gets trapped in the EE. We apply real-space formulation to the waveguide Hamiltonian (${v}_{g}=1$) [16,25]

Due to the system linearity for fixed excitation number, a trapped photon can be released by time reversal. Here, the initial state is $|\psi (t=0)\u27e9=\int \mathrm{d}xF(x){\widehat{c}}_{e}^{\u2020}(x)|{\mathrm{\Psi}}_{\mathrm{EE}}^{(1)}\u27e9$, i.e., the EE is excited and a single-photon wave packet, described by $F(x)[F(x>0)=0]$, impinges on the system. Figure 2(b) shows the system evolution when $F(x)$ is a Gaussian pulse [same parameters as in Fig. 2(a)]. After the single-photon pulse (not shown) reaches the system at $t=5/\mathrm{\Gamma}$, the energy stored in the EE is released almost completely, with a residual ${P}_{\mathrm{EE}}<0.05$.

Much higher efficiencies in the trapping process are obtained with
different two-photon pulse shapes. An optimal two-photon pulse can be
calculated by inspecting the (inverse) release process, again invoking
time reversal: if a single-photon pulse ${F}_{\mathrm{opt}}(x)$, impinging on an excited EE, produces a final
state with only two propagating photons [i.e., only
$\chi ({x}_{1},{x}_{2})\ne 0$ for $t\to +\infty $], this two-photon state ensures unitary
trapping efficiency when used as input. While this condition cannot be
satisfied *exactly* by any ${F}_{\mathrm{opt}}(x)$ (see Supplement
1 for further details), it can be
satisfied with an accuracy arbitrarily close to unity using, e.g.,
Gaussian pulses with increasing spatial widths. For example, when
${F}_{\mathrm{opt}}(x)\propto \mathrm{exp}[-{(x-{x}_{0})}^{2}/2{\sigma}^{2}+i{\omega}_{B}^{(1)}x]$ with $\sigma =5/\mathrm{\Gamma}$, after the release process, we obtain
${P}_{\mathrm{EE}}<0.03$. The corresponding final two-photon state
[Fig. 3(a), already mirrored
across the origin] is now used as input for the trapping process in
Fig. 3(b). The EE population
reaches ${P}_{\mathrm{EE}}(t\to \infty )\approx 0.98$, indicating a very high probability of
single-photon trapping. Differently from the two-photon Gaussian state,
the optimized ${\chi}_{\mathrm{IN}}({x}_{1},{x}_{2})$ [Fig. 3(a)] is squeezed along ${x}_{1}={x}_{2}$; spatial bunching of the two photons is
therefore beneficial to increase the EE excitation efficiency. A spatially
bunched two-photon input state can be realized via, e.g., spontaneous
parametric down-conversion [27] or
by modifying the photon statistics of a classical beam with atoms [28].

## 4. REALISTIC IMPLEMENTATIONS OF THE PROPOSED CONCEPT

Any realistic implementation of this concept will be affected by additional losses, which prevent infinite confinement. However, as long as the atom and cavity decay rates into the waveguide (${\mathrm{\Gamma}}_{A,C}\equiv 2\pi {V}_{A,C}^{2}/{v}_{g}$) are much larger than the additional decay rates ${\mathrm{\Gamma}}_{A,C}^{\prime}$, the EE can store an excitation for much longer than for isolated cavities or atoms. As an example, we compare [Fig. 4(a)] the two-photon excitation dynamics of the cavity–atom system (set at the EE condition) and a single cavity with same values of ${\mathrm{\Gamma}}_{C}$ and ${\mathrm{\Gamma}}_{C}^{\prime}$ (we neglect ${\mathrm{\Gamma}}_{A}^{\prime}$, since usually ${\mathrm{\Gamma}}_{A}^{\prime}\ll {\mathrm{\Gamma}}_{C}^{\prime}$). The excitation trapped in the EE (solid lines) has now a finite lifetime that decreases as ${\mathrm{\Gamma}}_{C}^{\prime}$ increases, but if ${\mathrm{\Gamma}}_{C}^{\prime}\ll {\mathrm{\Gamma}}_{C}$, the decay rate is much smaller than the decay rate of a single cavity (dashed lines), which is dominated by ${\mathrm{\Gamma}}_{C}$. The excitation rates are instead almost the same in the two cases.

So far, we have assumed that atom and cavity couple to each other and to the waveguide. This may be experimentally challenging, since, to have a relevant atom–cavity interaction, the atom is typically placed inside the cavity, which makes the atom–waveguide direct interaction weak. However, this mechanism can be extended to multi-atom or multi-cavity systems, which may be easier to implement. For example, by replacing also the second cavity by another (displaced or co-located) atom, a similar single-photon EE is obtained (see Supplement 1 for details). Another example is shown in Fig. 4(b): two interacting cavities couple to a waveguide [similar to Fig. 1(a)], and one cavity interacts with one atom at a rate $g$. With a similar analysis, we find that this system supports two single-photon EEs when the parameters satisfy

The experimental realization of this concept will require good control of the system parameters and small non-radiative decay rates. An ideal platform is constituted by superconducting circuits: recent experiments showed that superconducting qubits and radio-frequency cavities can couple to transmission lines with ${V}_{A/C}\approx (0.01/0.05)\sqrt{{v}_{g}{\omega}_{A/C}}$ (similar to those assumed here) and non-radiative decay rates ${\mathrm{\Gamma}}_{C}^{\prime}/{\mathrm{\Gamma}}_{C}\le 0.1$, and their frequency can be largely tuned by controlling the flux bias [30,31]. Moreover, their lumped nature allows to easily engineer the cavity–atom coupling and, importantly, to ensure that cavity and atom couple to the same point of the waveguide. Storage and release of single photons have been recently demonstrated with superconducting qubits [34]. There, however, the storage and release requires a dynamical control of the coupling elements with properly timed external pulses. The stored excitation decays in time due to the additional unwanted losses, similar to the case shown in Fig. 4(a). We note that, while a specific implementation will constrain the parameter ranges, Eq. (1) offers great flexibility: low values of $J$, e.g., can be counterbalanced by reducing ${\omega}_{A}-{\omega}_{C}$. The requirement ${\mathrm{\Gamma}}_{A/C}^{\prime}\ll {\mathrm{\Gamma}}_{A/C}$ has been demonstrated also in photonic crystal waveguides interfaced with cavities [29] and solid state [32] or natural [33] atoms. These platforms, however, offer less flexibility in terms of parameter tuning, and may be more suitable for the case of two distant resonators where $J=0$ (see details in Supplement 1). Moreover, we emphasize that, even when the parameters cannot be exactly matched to obtain a true EE [Eqs. (1) and (6)], one can still exploit this effect to create dark modes with very small decay rates, that can nonetheless be excited at a fast decay rate due to the atomic nonlinearity. The effect of parameter mismatch will therefore be similar to the presence of additional loss [Fig. 4(a)].

The proposed approach allows trapping single photons in an ideally lossless state, and releasing them on-demand. Due to the dependence of ${V}_{C}$ (${V}_{A}$) on the mutual polarization of the waveguide field and cavity field (atom dipole), the mechanism can be used to conditionally trap photons based on their polarization. Moreover, it can be used to realize an on-demand source of a single photon, with precisely controlled release time [15]. It offers several advantages compared to other storage protocols involving atom–cavity systems [34–36]: it does not require external control beams [34] or feedback mechanisms, and it uses a simple two-level system prepared in its ground state, rather than three- [35] or multi- [36] level systems prepared in particular superposition states.

## 5. CONCLUSION

To conclude, we proposed and theoretically investigated the excitation and release of single-photon EEs based on atomic nonlinearities. We demonstrated that the trapping of a single photon upon a two-photon excitation can occur with arbitrarily high efficiencies if the shape of the impinging pulse is suitably tailored. Even considering realistic losses [29–33], the proposed mechanism allows storing single photons for times much longer than the time needed to excite the EE, in contrast with single-cavity or single-atom configurations. The principle is extendable to experimentally feasible systems composed of, for example, two cavities with one of them containing the atom. These findings open exciting opportunities for quantum communications and computing, as well as for attojoule optoelectronic systems.

Note: after submission of this article, we became aware of a related scheme
by Calajó *et al.* [37], where the excitation of an embedded EE in a
pair of atoms or in a single atom in front of a mirror is
investigated.

## Funding

Air Force Office of Scientific Research (AFOSR) (MURI FA9550-17-1-0002); Simons Foundation.

## Acknowledgment

M.C. was partially supported by a Rubicon postdoctoral fellowship by The Netherlands Organization for Scientific Research (NWO).

See Supplement 1 for supporting content.

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