Exponential Improvement in Photon Storage Fidelities Using Subradiance and “Selective Radiance” in Atomic Arrays

A central goal within quantum optics is to realize efficient, controlled interactions between photons and atomic media. A fundamental limit in nearly all applications based on such systems arises from spontaneous emission, in which photons are absorbed by atoms and then rescattered into undesired channels. In typical theoretical treatments of atomic ensembles, it is assumed that this rescattering occurs independently, and at a rate given by a single isolated atom, which in turn gives rise to standard limits of fidelity in applications such as quantum memories for light or photonic quantum gates. However, this assumption can in fact be dramatically violated. In particular, it has long been known that spontaneous emission of a collective atomic excitation can be significantly suppressed through strong interference in emission between atoms. While this concept of “subradiance” is not new, thus far the techniques to exploit the effect have not been well understood. In this work, we provide a comprehensive treatment of this problem. First, we show that in ordered atomic arrays in free space, subradiant states acquire an elegant interpretation in terms of optical modes that are guided by the array, which only emit due to scattering from the ends of the finite system. We also go beyond the typically studied regime of a single atomic excitation and elucidate the properties of subradiant states in the many-excitation limit. Finally, we introduce the new concept of “selective radiance.” Whereas subradiant states experience a reduced coupling to all optical modes, selectively radiant states are tailored to simultaneously radiate efficiently into a desired channel while scattering into undesired channels is suppressed, thus enabling an enhanced atom-light interface. We show that these states naturally appear in chains of atoms coupled to nanophotonic structures, and we analyze the performance of photon storage exploiting such states. We find numerically that selectively radiant states allow for a photon storage error that scales exponentially better with the number of atoms than previously known bounds.

Popular Summary

Many quantum information-processing protocols can be realized by an ensemble of atoms interacting with light. The efficiency of this interaction is limited by the probability of the ensemble to emit a photon into an undesired direction rather than the mode of interest. The “standard” model is to assume that the emission into undesired channels occurs independently from surrounding atoms. When atoms are sufficiently close to each other, however, light emitted by different atoms can interfere and dramatically alter the undesired emission. We show that this interference can be exploited to a spectacular effect, yielding a platform for quantum optics far more powerful than the standard model predicts.

Using a mathematical description of an ordered one-dimensional array of atoms coupled to photons in the guided mode of a nanostructure, we identify the novel phenomenon of “selective radiance,” where certain collective atomic excitations emit strongly into the preferred guided modes, while their decay rate into free space is simultaneously suppressed because of an “impedance mismatch” with free-space electromagnetic radiation. This precisely fulfills the conditions needed to achieve an efficient atom-light interface. We show theoretically how these states can be utilized in a protocol for quantum memory for light, which, as the number of atoms increases, achieves an exponentially improved suppression of error over previously known bounds.

This result suggests that we should rethink the traditional paradigm used to describe atomic ensembles and more broadly explore the vast potential of collective atomic interference in quantum optics.

Figure 1

(a) Generic dispersion relation of frequency $\omega (k_z)$ versus Bloch wave vector $k_z$ for single-excitation modes of an infinite, one-dimensional chain. The Bloch vector $k_z$ is only uniquely defined within the first Brillouin zone $|k_z|\le \pi /d$. The dashed black line is the light line, and corresponds to the dispersion relation of light in vacuum propagating along the $\widehat{z}$ direction; i.e., $\omega =c|k_z|$. Atomic modes in the region enclosed within the light line (shaded) are generally unguided and radiate into free space. Outside the light line ($|k_z|>\omega /c$), the modes are guided and subradiant, as the electromagnetic field is evanescent in the directions transverse to the chain. The dispersion relation is generally expected to be rather flat and centered around the bare atomic resonance frequency ${\omega }_0$. (b) Collective frequency shifts and (c) decay rates for an atomic chain along $\widehat{z}$ with lattice constant $d/{\lambda }_0=0.2$, for parallel (blue) and transverse (red) atomic polarization. Circles correspond to the results for a finite system with $N=50\mathrm{atoms}$. The analytical expressions for the infinite chain are denoted by solid lines and approximate well the finite chain results, except for a small region close to the light line. In the infinite lattice case, the modes with $|k_z|>{\omega }_0/c$ are perfectly guided and the decay rate ${\mathrm{\Gamma }}_{k_z}$ is exactly zero. The light line (black dashed line) appears vertical over the very narrow frequency ranges plotted here.

Figure 2

(a) Illustration of a square lattice of atoms in the $\widehat{y}\text{−}\widehat{z}$ plane, with lattice constant $d$. (b) Corresponding reciprocal lattice in 2D, with lattice constant $2\pi /d$. The collective modes have well-defined quasimomentum $\mathbf{k}=(k_y,k_z)$ within the first Brillouin zone, which is indicated by the blue square. A circle of radius $|\mathbf{k}|=k_0$ defines the set of propagating electromagnetic modes in vacuum in the $\widehat{y}\text{−}\widehat{z}$ plane at the atomic frequency. Collective spin waves outside of this circle will be guided, with a decay rate ${\mathrm{\Gamma }}_{\mathbf{k}}=0$. For $k_0>k^{\max }=\sqrt{2}\pi /d$ (or equivalently, $d/{\lambda }_0>1/\sqrt{2}$) all collective eigenstates lie inside of the circle. (c),(d) Collective decay rates for the infinite square lattice for parallel and transverse atomic linear polarization, respectively. The lattice constant is set to $d/{\lambda }_0=0.2$. For transverse polarization, ${\mathrm{\Gamma }}_{\mathbf{k}}$ also vanishes for $\mathbf{k}\sim (0,0)$, provided that $d/{\lambda }_0<1$.

Figure 3

Single-excitation collective modes in a finite 1D chain of two-level atoms with polarization along the chain. (a) Decay rates of the $N$ modes at different lattice constants $d/{\lambda }_0$ for a finite chain of $N=50\mathrm{atoms}$. Subradiant modes arise only if $d/{\lambda }_0\le 1/2$. A vertical cut at the fixed value of $d/{\lambda }_0=0.2$ corresponds to the blue circles depicted in Fig. 1. (b) Field intensity (arb. units) in the $\widehat{y}\text{−}\widehat{z}$ plane ($x=5d)$ created by the most subradiant mode in a chain of $N=50$ atoms. The field is largely evanescent transverse to the bulk of the chain, while most of the energy is radiated out through scattering at the ends of the chain. White circles denote atomic positions. (c) Scaling with atom number of decay rates for the three most subradiant modes. A fit for large $N$ yields $\mathrm{\Gamma }\sim N^{-3}$. (d) Scaling with mode index $\xi$ of decay rates at fixed $N=50$. Here, $\xi$ is used to label the magnitude of the decay rates in increasing order ($\xi =1$ is the most subradiant state, while $\xi =N$ has the largest decay rate). A fit for small $\xi$ yields $\mathrm{\Gamma }\sim {\xi }^2$. Open and solid symbols denote the results obtained by exact diagonalization and from the ansatz of Eq. (11), respectively. The black dashed line corresponds to the eigenstate whose dominant wave vector $k$ crosses the light line ($k=k_0$). (b)–(d) are for $d/{\lambda }_0=0.3$.

Figure 4

Comparison between ansatz of Eq. (11) and exact single-excitation eigenstates in an atomic 1D chain. Here, we identify a selected number of modes based upon their dominant wave vector $k$, and compare the spatial wave-function coefficients $c^j$ with an ansatz built from the same wave vector: (a) $kd=\pi N/(N+1)$ (most subradiant state), (b) $k\sim k_0$ (close to light line), and (c) $kd=\pi /(N+1)$ (most radiant state), for parallel atomic polarization. Blue and red circles denote the coefficients of the exact state and ansatz, respectively, while the dashed red line indicates the function $\cos (k_{n}z)$ or $\sin (k_{n}z)$ associated with each mode. (d) Error in overlap $ϵ$ between exact state and ansatz as a function of $k$, for parallel (blue) and transverse (black) atomic polarization. The error decreases far from the light line (denoted by black dashed line). (e) Scaling of $ϵ$ with particle number $N$ for the most subradiant state. The lattice constant is set to $d/{\lambda }_0=0.3$ and [except in (e)] $N=20$.

Figure 5

Single-excitation collective modes in 2D square array of $50\times 50$ two-level atoms. Collective decay rates as a function of predominant wave vector ($k_y$, $k_z$) associated with the mode, for different values of $d/{\lambda }_0$: (a)–(c) are for parallel atomic polarization along $\widehat{y}$ axis; (d)–(f) are for transverse polarization. Subradiant and guided modes arise outside the circle defined by the light line ($|\mathbf{k}|=k_0$), if $d/{\lambda }_0<1/\sqrt{2}$. For $d/{\lambda }_0<1$ and transverse atomic polarization, a different class of subradiant states emerges at $\mathbf{k}=(0,0)$. (g) Scaling of decay rates with $N$ ($N^2$ being the number of atoms) for two particular modes $(k_y,k_z)=(\pi /d,0)$ (top) and $(k_y,k_z)=(\pi /d,\pi /d)$ (bottom). Different colors are for different values of $d/{\lambda }_0=0.25$, 0.3, 0.4, 0.5, 0.6, $1/\sqrt{2}$ (blue, green, red, cyan, purple, and brown, respectively). The dashed lines are guides to the eye for $N^{-3}$ and $N^{-6}$ scalings.

Figure 6

Decay rates of the $N^3$ modes at different lattice constants $d/{\lambda }_0$ for a cubic lattice of $10\times 10\times 10$ atoms with linear polarization along one of the lattice axes. The dashed red line corresponds to the particular value of $d/{\lambda }_0=\sqrt{3}/2$, where the light line touches the edge of the first Brillouin zone in 3D. In 3D, subradiant states can exist even beyond this value.

Figure 7

Decay rate of the most subradiant mode as a function of atom number $N$ in a circular ring of two-level atoms (black circles). Two consecutive atoms are separated a distance $d$ as shown in the sketch (here, $d/{\lambda }_0=0.3$), and the atomic polarization is transverse to the plane defined by the ring. For this geometry, there is an exponential suppression of the most subradiant decay rate with $N$. For comparison, the decay rate of the most subradiant mode of a linear chain with lattice constant $d$ is shown (blue circles), with only polynomial suppression with $N$.

Figure 8

Cavity in an atomic chain with slowly varying lattice constant. The chain along $\widehat{z}$ is divided into three regions: in the left and right regions, the lattice constant $d$ is uniform and equal to $d_{\max }$, while in the middle it changes slowly (from $d_{\max }$ to $d_{\min }=0.75d_{\max }$). (a) Dispersion relation for two infinite lattices with constant $d_{\max }$ (cyan) and $d_{\min }$ (dark blue) versus wave vector along the chain (in units of the corresponding lattice constant). The light line is indicated by the vertical dashed lines with the same color. A bandwidth of frequencies (shaded pink) emerges wherein propagating modes exist for lattice constant $d_{\min }$, but not for $d_{\max }$. This allows localized resonances to form within the nonuniform chain. An illustration of the chain is shown (top), where the red line represents how the lattice separation changes along the chain. (b) Excited-state population versus atom position corresponding to the fundamental mode in the cavity, illustrated for $N=90$ atoms. (c) Decay rate of the fundamental mode versus atom number, showing an exponential suppression. We choose $d_{\max }/{\lambda }_0=0.4$ and atomic polarization along the chain axis.

Figure 9

Two-excitation collective modes in a 1D chain of two-level atoms with polarization parallel to the chain. Probability $|c^{ij}{|}^2$ of atoms $i$ and $j$ to be excited (with $c^{ii}=0$) for (a) state resulting from occupying the most subradiant single-excitation mode twice, i.e., $(S_{\xi =1}^{†}{)}^2{|g⟩}^{\otimes N}$, (b) most subradiant mode $c_{\xi =1}^{ij}$ (obtained by exact diagonalization), (c) fermionic ansatz $c_{\mathrm{ans}}^{ij}$. The two axes denote the excited atoms position ($i$ and $j$). (d) Scaling of the collective decay ${\mathrm{\Gamma }}^{(2)}$ with atom number corresponding to the states (a) (orange circles), (b) (blue open circles), and (c) (blue solid circles). The lines are polynomial fits (to the last five points), close to $\sim 1/N$ (a) and $\sim 1/N^3$ (b),(c). (e) Decay rates as a function of the associated pair of quasimomentum values $(k_1,k_2)$ of the two excitations (by construction $k_1\ne k_2$). Subradiant states arise when both $k_1,k_2>k_0$, where $k_0$ corresponds to the light line (dotted lines). (f) Relative error $\delta \mathrm{\Gamma }/{\mathrm{\Gamma }}^{(2)}=|{\mathrm{\Gamma }}_{\mathrm{sum}}^{(2)}-{\mathrm{\Gamma }}^{(2)}|/{\mathrm{\Gamma }}^{(2)}$ between the numerically exact decay rate and the decay rate estimated from the sum of single-excitation decay rates ${\mathrm{\Gamma }}_{\mathrm{sum}}^{(2)}$. (g) Error in overlap between exact and antisymmetrized ansatz. In all plots $N$ is fixed to 50 atoms [except (d)] and $d/{\lambda }_0=0.3$.

Figure 10

Multiexcitation states in a 1D chain of two-level atoms with polarization parallel to the chain. (a) Scaling of the decay rate ${\mathrm{\Gamma }}_{\xi =1}^{(n)}/{\mathrm{\Gamma }}_0$ of the most subradiant $n$-excitation state with the atom number $N$ (blue open symbols). For comparison, the decay rates of the Fock state constructed with $n$ excitations in the most subradiant single-excitation mode ${\mathrm{\Gamma }}_b^{(n)}$ (orange solid symbols) and for the fermionic ansatz ${\mathrm{\Gamma }}_{\mathrm{ans}}^{(n)}$ (blue solid symbols) are shown ($n=2$, 3, 4 are denoted by circles, squares, and diamonds, respectively). The lines are polynomial fits close to ${\mathrm{\Gamma }}_{\xi =1}^{(n)}\sim N^{-3}$ and ${\mathrm{\Gamma }}_b^{(n)}\sim N^{-1}$, respectively. The sketches on top of the atomic chain represent the two-excitation density profile. (b) Scaling of the decay rate ${\mathrm{\Gamma }}_{\xi =1}^{(n)}/{\mathrm{\Gamma }}_0$ of the most subradiant state with the excitation density $n/N$. The dashed line corresponds to the predicted scaling $\sim (n/N{)}^3$ valid at low-excitation density ($n/N\ll 1$). Blue, red, and black are for $N=10$, $N=15$, and $N=20$, respectively. All plots are for $d/{\lambda }_0=0.3$ and atomic polarization parallel to the chain.

Figure 11

Schematic of the setup under consideration: $N$ two-level atoms are located in the vicinity of a dielectric nanofiber of dielectric constant $\epsilon$ and radius $r$, at a distance ${\rho }_a$ from the center of the fiber, and at a constant distance $d$ from each other. For the calculations in this paper, we take $k_{0}r=1.2$, ${\rho }_a=1.5r$, and $\epsilon =4$. The atoms interact with each other not only through the guided mode, but also through nonguided photons. The single-atom emission rates into the fiber and into free space are ${\mathrm{\Gamma }}_{1\mathrm{D}}$ and ${\mathrm{\Gamma }}^{\prime }$, respectively.

Figure 12

Linear optics for a chain of $N=20$ atoms coupled to a nanofiber, for $d={\lambda }_{1\mathrm{D}}/4$ ($k_{1\mathrm{D}}d=\pi /2$). (a) Transmittance, (b) reflectance, and (c) loss probability as a function of the atom-probe detuning. The blue curves are obtained by including the collective emission into free space [see Eq. (13)], and the red lines are produced within the “independent emission” model, where it is assumed that free-space emission is a single-atom effect [see Eq. (14)]. The parameters characterizing the nanofiber are given in Fig. 11.

Figure 13

Electromagnetically induced transparency scheme. The $|g⟩$ to $|e⟩$ transition is coupled to the guided mode, and the $|s⟩$ to $|e⟩$ transition is driven by an external, classical control field of Rabi frequency ${\mathrm{\Omega }}_c$. The distance between the atoms is a quarter of the guided-mode wavelength, $d={\lambda }_{1\mathrm{D}}/4$.

Figure 14

Electromagnetically induced transparency and photon storage efficiency, within the independent emission (red) and collective emission (blue) models. (a) Transmittance spectrum for a chain of $N=20$ atoms. The control field Rabi frequency is set to ${\mathrm{\Omega }}_c={\mathrm{\Gamma }}_0$, while other system parameters can be found in the main text. (b) Scaling of the bandwidth-delay product with the number of atoms $N$. The circles represent the numerical calculation. The red line shows the theoretically predicted answer within the independent emission model, $\mathcal{P}=\sqrt{D}\simeq 0.76\sqrt{N}$, and the blue curve represents the best fit of the numerical data to a linear scaling, $\mathcal{P}\simeq 0.30N$. The control field intensity is the same as in (a). (c) Infidelity in the retrieval of the spin excitation given by Eq. (26). The circles show the numerics. The red line represents the expected scaling derived in Ref. [38], within the independent emission model, $ϵ=5.8/D\simeq 10/N$, and the blue line is the best fit of the numerical results to a $\propto N^{-2}$ scaling, and shows $ϵ\simeq 26/N^2$ (the range for the fit is $N\in [30,200]$).

Figure 15

Selectively radiant states of the $s$ branch. (a) Guided [${\mathrm{\Gamma }}_{1\mathrm{D}}(k_z)$, light blue] and nonguided [${\mathrm{\Gamma }}^{\prime }(k_z)$, green] decay rates of the single-excitation eigenstates of ${\mathcal{H}}_{1\mathrm{D}}+{\mathcal{H}}^{\prime }+{\mathcal{H}}_c$ versus the dominant wave vector $k_z$ of each eigenstate, for a chain of $N=200$ atoms coupled to the fiber. The control field Rabi frequency is ${\mathrm{\Omega }}_c=4{\mathrm{\Gamma }}_0$, and the plot is restricted to eigenstates that consist mostly of population in the $s$ states (the $s$ branch). The gray shaded area represents the region within the light line, the dashed lines show the guided-mode wave vector $\pm k_{1\mathrm{D}}$, and the color lines are guides to the eye. (b) Scaling of the ratio ${\mathrm{\Gamma }}_{1\mathrm{D}}(k_z)/{\mathrm{\Gamma }}^{\prime }(k_z)$ with the number of atoms, at the wave vector $k_z$, where it is maximum. The dots are numerical results, and the curve represents the best quadratic fit, $\max \{{\mathrm{\Gamma }}_{1\mathrm{D}}(k_z)/{\mathrm{\Gamma }}^{\prime }(k_z)\}\simeq 0.0053N^2$.

Figure 16

Retrieval of a Gaussian spin wave given by Eq. (29), under a spatially uniform control field of Rabi frequency ${\mathrm{\Omega }}_c=0.1{\mathrm{\Gamma }}_0$, and within the collective emission model. (a) Evolution of the population in the $|s⟩$ states (upper plot) and $|e⟩$ states (lower plot) as a function of position and at several selected points in time $\tau$, for a chain of $N=200$ atoms (in arb. units). The time is normalized such that at $\tau =1$ all the spin population has completely decayed. (b) Instantaneous rate of photon scattering into free space ${\kappa }^{\prime }(\tau )$. The solid line is a numerical calculation based on Eq. (28), while the dashed line represents an estimate based on taking a Fourier decomposition of the spin amplitude $c_e(z_j,t)$ and weighting each component by a wave-vector-dependent decay rate. A large instantaneous scattering rate occurs when a large excited-state population is found at the end of the system [for example, at times $\tau =0.07$ and $\tau =0.9$ in (a)]. (c) Excited-state population $|c_e(k_z,\tau ){|}^2$ of the different wave vector components of the spin wave, for different evolution times [corresponding to the snapshots in (a)]. Only the region inside the light line is shown. (d) Scaling of the retrieval loss with the atom number $N$. The blue dots show the numerical calculation, whereas the blue line is the best fit to them and represents $ϵ=4.1/N$. The infidelities for the initial spin wave of Eq. (26) within the independent and collective emission models are shown by the dashed and dotted lines, respectively.

Figure 17

Schematic of the spatial profile of the control field ${\mathrm{\Omega }}_c(z_j)\sim \sqrt{N/(N+1-j)}$ (green curve, arb. units) for a chain of $N=200$ atoms. The initial spin wave, given by Eq. (29), is overlaid in blue.

Figure 18

Same as Fig. 16, but for a spatially varying control field of the form ${\mathrm{\Omega }}_c(z_j)=0.005\sqrt{N/(N+1-j)}{\mathrm{\Gamma }}_0$. In (a), the dotted lines show the analytical model. In (d), the blue line is a guide to the eye, and follows $ϵ=0.15e^{-N/23}$.

Figure 19

(a) The transfer matrix ${\mathcal{M}}_{\mathrm{sc}}$ relates the fields on one side and the other of the scatterer. The coefficients of ${\mathcal{M}}_{\mathrm{sc}}$ are determined by imposing continuity on the fields. (b) For a periodic array of scatterers the total transfer matrix is simply the product of matrices ${\mathcal{M}}_N={\mathcal{M}}^N$.

Figure 20

Integration contour for Eq. (c1) depicting the pole and branch cuts.

Figure 21

Linear optics for a chain of $N=20$ atoms coupled to a nanofiber, in the mirror configuration ($k_{1\mathrm{D}}d=\pi$). (a) Transmittance, (b) reflectance, and (c) loss probability as a function of the atom-probe detuning, within the collective (blue curves) and independent emission (red curves) models. (d) Spatial profile of the excited-state population (in arb. units) at two different detunings, as indicated by the arrows in (c). The green circles have been rescaled for the sake of clarity. The parameters characterizing the nanofiber are given in Fig. 11.