Introduction

A single atom coupled to an environment is usually subject to spontaneous emission and experiences a frequency shift referred to as the Lamb shift. In an aggregation of atoms coupled via the radiation field, collective effects can significantly alter the properties compared to a single emitter. For instance, this was realised by Dicke1,2, who showed that N identical atoms confined to a volume much smaller than a wavelength cubed collectively behave as one “super atom”. This leads to exaggerated properties such as an acceleration of spontaneous decay by a factor of χDicke = N (known as superradiance) or an enhanced frequency shift (sometimes also termed “collective Lamb shift”). Recently, also the correlated emission from extended ensembles of emitters has become the focus of experimental and theoretical3,4,5 investigations, where either the system size and/or the minimal interatomic distance a exceeds the scale of the characteristic wavelength λ0. The systems considered cover a wide range of possible realisations, including atoms near a nanofiber6, thin vapor layers7, cold atomic ensembles8,9,10,11,12, or thin-film cavities with embedded Mössbauer nuclei in the realm of x-ray quantum optics13,14,15,16,17,18,19,20.

The present work is motivated by the observation that in particular the latter experiments in the field of nuclear quantum optics exploited a deliberate control of superradiance properties, going beyond a mere characterisation. For instance, the observation of electromagnetically induced transparency at x-ray frequencies13 was enabled by the engineering of two distinct ensembles with different superradiance properties in a single sample. Another example is the implementation of spontaneously generated coherences14, which relied on the realization of a spatially anisotropic electromagnetic environment via superradiance. In both cases, superradiance was employed to design an artificial quantum system, which in turn enabled the observation of the desired effect.

This raises the question whether a systematic and constructive approach could be established to exploit superradiance for the design of artificial quantum systems. Such design capabilities could overcome the limited resources accessible in state-of-the-art experiments, and thereby enable more advanced level schemes required, e.g., for the exploration of non-linear and quantum effects at x-ray energies.

Here, we address this question by developing an analytical framework for superradiance in extended media encompassing different system dimensionalities, interatomic couplings, and environments. As our main result, we then derive expressions describing how collective decay rates and frequency shifts can be controlled in extended media, and show how they can be used for the design of an artificial optical transition.

We start with a single two-level atom (bare transition frequency ω0 = ck0, k0 = 2π/λ0, c is the speed of light) which is embedded in an electromagnetic environment (e. g., free space) and is characterised by its spontaneous decay rate Re(V0) ≡ γ0 (assuming Markovian reservoirs21). The coupling to the environment also results in a frequency shift Im(V0)/2 ≡ δω0 (single-atom Lamb shift). In the presence of an identical, second atom, photons can be exchanged between the two atoms. Due to irreversible loss to the reservoir, the inter-atomic coupling is complex21,22,23,24,25. Here, () represents the real-valued cross-damping (cross-coupling) term for two atoms located at positions ri and rj, respectively. Considering all pair-wise couplings in an ensemble of N ≫ 1 atoms, we find22,24

where E denotes the complex eigenenergy of the collective single-excitation atomic state ( and |0〉 signify the atomic raising operator for atom i and the vacuum state, respectively). Equation (1) is valid for all dimensions d of the atomic arrangement and for all (physically reasonable) couplings . Collective decay rates and frequency shifts are obtained via Γ ≡ −2Im(E) and Δ ≡ Re(E) − ω0, respectively22,24.

In Dicke’s small-volume limit, all atoms couple to each other with equal strength, leading to a collective decay rate Γ = Nγ0 = χDickeγ0 and a frequency shift Δ = χDickeδω0 with an enhancement factor χDicke (see methods). To describe an extended sample, we consider ordered atomic arrangements, and focus on chains (d = 1), square lattices (d = 2), and simple cubic lattices (d = 3), see Fig. 1. The smallest inter-atomic distance is given by the lattice constant a. Such ordered arrays are naturally provided by crystalline samples (e. g., solid state targets employed in x-ray quantum optics13,14,15,16,17,18, optical lattices of atoms26, or atom–cavity networks27). Furthermore, we consider a generic class of inter-atomic couplings

Figure 1: Design of an artificial optical transition through tailored superradiance.
figure 1

A d-dimensional lattice of atoms is embedded into an electromagnetic reservoir that mediates an inter-atomic coupling Vr ∝ 1/rα, where atoms are separated by a distance r and the coefficient α characterises the distance-dependence (see equation (2)). We show that the resulting collective eigenstates can be utilised for the implementation of an artificial transition with tunable decay rate and transition frequency.

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which depend on the distance r between atom pairs. Here, the coefficient α classifies the distance-dependence and Ad is a dimensionless coupling strength. We also assume the atomic dipole moments to be uniformly aligned along the x3 axis. This orientation dependence is taken into account by the angle θ (see methods for further details). Since multiple terms of type (2) can be accounted for by a linear combination, in particular also the three common implementations of three-dimensional free space22, atoms confined to two spatial dimensions28, or atoms coupled to a one-dimensional waveguide29 are covered. The coupling parameters for these three examples are specified in table 1. Note that in principle α can be artificially engineered and controlled as has recently been demonstrated at optical frequencies10.

Table 1 Dimension-dependent quantities.
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Results and Discussion

The solution of eigenproblem (1) (see methods) reveals that those eigenstates |k〉 whose wavevector’s magnitude matches the wavenumber set by the single atom transition, i. e., k = |k| = k0, exhibit the maximum possible decay rate Γmax = χmaxγ0 if the constraint

is fulfilled. This criterion is a necessary condition for the emergence of superradiance and represents bounds on the allowed power laws of the coupling terms (exponent α in equation (2)) as a function of the lattice dimension d. For the remainder, we assume that eq. (3) is satisfied. The enhancement factor χmax is (see table 1 for quantities bd and cd, and methods for the prefactor Ld(α))

In contrast to the maximum collective decay rate, we find that the collective frequency shift at k = k0 is always zero independent of the actual physical realisation. We thus conclude that the case of maximum superradiance is unsuitable for a control of both collective decay rates and frequency shifts.

To circumvent this problem, we also consider states with wavenumbers around k = k0. Indeed, for a large but finite system, also states with a wavenumber close to k0 can exhibit an enhanced decay rate. We illustrate this for the most relevant case α = (d − 1)/2 (which includes the three common implementations mentioned below equation (2)). For |k − k0|a ≪ 1, we find

where and sinc(ξ) ≡ sin(ξ)/ξ. Results are shown in Fig. 2, scaled in such a way that they encompass different dimensions and coupling types. As mentioned before, those states which are maximally superradiant at k = k0 do not exhibit a collective frequency shift. Rather, the frequency shift’s first two extrema around k0 occur at wavenumbers (where ). This finding represents a unique feature as it is independent of the actual realisation and provides a signature suitable for a direct experimental test.

Figure 2: Collective decay rates and frequency shifts.
figure 2

Decay rates (black dashed curve) and frequency shifts (blue solid curve) as function of the wavenumber for α = (d − 1)/2. The figure is valid independent of dimensionality and coupling type, due to the scaling of decay rate , shift and wavenumber . Note the offset h between the extrema of the frequency shift and the decay rate maximum.

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Equations (6) and (7) also offer means to design an artificial optical transition with desired decay rate and frequency shift. In fact, the enhancement factor χmax represents a characteristic scale for both decay rates and frequency shifts. As expected, we find that the particle number N and/or the sample volume can be used to control χmax. But additionally, eqs. (4) and (5) explain how the dimensionality d, the type of the inter-atomic coupling as described by α, as well as the coupling strength to the environment can be used to manipulate the enhancement factor. This is of particular relevance, since these parameters could also be tuned in situ10,30. However, as mentioned previously, these quantities are not sufficient to change the ratio between decay rate and frequency shift. This only becomes possible by also controlling the wave number k (see Fig. 2). Experimentally, the wavenumber could be adjusted via the excitation angle of the probing light field.

From a broader perspective, our results also enable us to understand how superradiant states from different realisations can be compared and categorised. This is important, e. g., if superradiant ensembles realised using different individual constituents are to be combined to an effective artificial quantum system. To this end, suppose that we can control the atom number and the volume such that and , respectively, where fN and are arbitrary positive real numbers. Under this transformation, the enhancement factor changes as

This behaviour allows us to classify superradiant states from systems with different dimensionality and types of coupling. For instance, we may say that two extended samples characterised by (d, α) and (d′, α′), respectively, are similar if they satisfy the same transformation rule (8) (leading to (α − 1/2)/d = (α′ − 1/2)/d′). As an example, if a one-dimensional system (d′ = 1) should “imitate” the superradiant state from three-dimensional free space (d = 3, α = 1), the electromagnetic environment would have to be “engineered”10 such that α′ = 2/3. Similarly, if an extended sample should realise small-volume superradiance, transformation (8) must reproduce the transformation of a Dicke system, which is simply with χDicke = N and . Hence, α = (1 − d)/2 ≥ 0, which reveals that only extended samples in one dimension (d = 1, otherwise we would have α < 0) can behave “Dicke-like”.

In conclusion, we have studied single-photon superradiance in extended media, and showed how superradiance can be engineered in such a way that an artificial optical transition with tunable decay rate and level shift is realised. This result provides the basic building block for a systematic approach towards engineering advanced artificial quantum systems via superradiance by design. A promising avenue for future studies is the extension of our work to coupled sub ensembles with the goal to design artificial multi-level atoms13.

Methods

For an extended lattice, the plane wave ansatz for eq. (1) yields the eigenstates’ decay rates Γk = −2Im(Ek) and frequency shifts Δk = Re(Ek) − ω0 as

Here, r = (x1, …, xd)T denotes a d-dimensional lattice vector with components xi = ani, i = 1, …, d, , and is even. Likewise, k = (k1, …, kd)T is the wavevector of the collective atomic excitation. The sum runs over all combinations of {ni} except n1 = … = nd = 0 and the couplings depend on the distance between atoms. We assume the atomic dipole moments to be uniformly aligned along the x3 axis (e. g., by applying a weak magnetic field). Thus, for d = 1, 2 the distance vector r (in the x1-x2 plane) is perpendicular to the dipole moments, and for d = 3 we have to take into account the polar angle θ = arccos(x3/r). Furthermore, we make use of the assumptions N ≫ 1 (many atoms) and k0a > 1 (extended sample). The decay rate eq. (9) can also be rewritten in terms of the enhancement factor

To arrive at the final expressions eqs (4, 5, 6, 7), we further manipulate eqs (9, 10, 11, 12) as follows. In this paper, we focus on the system’s eigenstates and—to keep the analysis general—do not consider geometric details or questions of how to excite and probe the system since such details vary from experiment to experiment. Around k ≡ |k| = k0, we can utilise a continuum formulation, rewrite the lattice sums in eq. (11) into an integral, and perform the angular integration for couplings of type (2) (see supplementary information for further technical details of the calculation), leading to

The dimension-dependent quantities Ad, bd, cd, and gd are listed in table 1 (for instance, Ad is real for d = 1, 2 and purely imaginary for d = 3). Note that the factor exp(±ik0r) from eq. (2) in the eigenproblem (1) can be understood as a radial translation in wavenumber space. In the shifted frame, a long-wavelength limit of the collective atomic excitation (which can be accounted for by a continuum description) corresponds to k → k0. This continuum formulation is applicable in the range . Further, in eq. (2), we have not included exponential damping of the form exp(−k0r/), where denotes a dimensionless absorption length that, for instance, empirically accounts for material imperfections. Such a damping factor in the integral in eq. (14) would lead to a broadening and modification of the k = k0-criterion for maximal superradiance, going beyond the scope of this paper. Details on the calculation of the integrals in eq. (14) can be found in the Supplementary Material.

The maximum enhancement factor (4) can be cast into the equivalent forms ( denotes the sample volume, is the number density, and since N ≫ 1)

Which formulation to choose from eqs. (16)–(18), depends on which quantities can be controlled in an experiment.

If for small volumes the length scale set by the inter-atomic distance a is effectively eliminated from the single-excitation eigenproblem (1) (possibly neglecting divergent contributions to the inter-atomic coupling2,31), all atoms couple to each other with equal strength V0. The resulting equation (which must hold for all ri) yields a maximal decay rate for a spatially constant wavefunction with equal relative phase between all atom pairs, representing the maximally symmetric Dicke state. For this state, Γ = −2Im(E) = Nγ0 and Δ = Re(E) − ω0 = Nδω0.

Additional Information

How to cite this article: Longo, P. et al. Tailoring superradiance to design artificial quantum systems. Sci. Rep. 6, 23628; doi: 10.1038/srep23628 (2016).