Parity-Time (
The search for novel and powerful methods to control light using artificially engineered optical properties is a current driving force in photonics [1]. The possibility of exploiting gain and loss in photonic systems has led to interest in combined time and parity (
Metasurfaces [18] are thin nanostructured films of typically subwavelength-sized scatterers for the manipulation, detection, and control of light. Placing cold atoms in a two-dimensional (2D) planar array, e.g., by an optical lattice with unit occupancy has been proposed as a nanophotonic atom analogy to a metasurface [19], consisting of essentially point-like quantum scatterers. Transmission resonance narrowing due to giant Dicke subradiance below the fundamental quantum limit of a single-atom linewidth was recently experimentally observed [20] for a planar atom array, where all the atomic dipoles oscillated in phase, while incident fields can drive giant subradiance also in resonator metasurfaces [21]. These quantum-photonic [22] surfaces of atoms that have no dissipative losses due to absorption have several advantages over artificial resonator metasurfaces (even over low-dissipation all-dielectric ones [22]), as they could naturally operate in a single-photon limit [23], [24], [25], [26], [27], [28], [29], [30], [31], every atom has well-defined resonance parameters, and nonlinear response [32], [33] is easily achievable.
Here we show how the many-body dynamics of strongly coupled atoms in a planar array, with the dipole-dipole interactions mediated by resonant light, can be engineered to exhibit CPA, associated with an effective
Our model consists of a square array of cold atoms, in the yz plane, with one atom per lattice site, and lattice constant d. The dipole moment of atom j is
Here,
Inserting E_{ext} into Equation (1) leads to a set of coupled equations for the polarization amplitudes [37],
The dynamics of the atomic ensemble can be understood in terms of the collective excitation eigenmodes v_{n} of
The eigenvectors of the non-Hermitian
The eigenvectors obey
At such points, the right eigenvectors no longer form a basis, and
In the absence of Zeeman splitting, the atomic level structure is isotropic, and
We find in Section 2.3 that, while v_{j} are no longer eigenmodes of the full system when Zeeman splitting is turned on, it is still useful to describe the physics in this basis of eigenmodes of the unperturbed system.
There are two collective modes of the unperturbed system that can describe the optical response of the array [34]. They correspond to uniform excitations where the dipoles oscillate in phase and point either in the plane (in-plane mode
The collective in-plane excitation eigenmode has now been experimentally observed [20], with all atoms oscillating coherently in phase in an optical lattice of ^{87}Rb atoms with d = 532 nm and near unity filling fraction. According to Equation (10), this spacing with d = 0.68λ represents a subradiant collective state, resulting in a measured transmission linewidth ≈0.66γ, well below the single-atom linewidth. Experiments on illuminating atoms also in optical tweezer arrays are ongoing [46].
In the out-of-plane mode the dipoles radiate predominantly sideways, away from the dipole axis, and light is scattered many times before it can escape from the edges of the lattice. This leads to a dramatically narrowed linewidth with the mode becoming completely dark for large arrays,
The in-plane mode with a uniform phase profile can easily be excited by a wave incident in the x direction normal to the plane, as also shown experimentally [20]. Deeply subradiant modes, with
Similarly for the atomic lattice, the collective modes
The applicability of the two-mode model follows from the absence of coupling to all other collective modes due to phase mismatching. Figure 2 shows the accuracy of the effective two-mode model in a reflection lineshape of a Gaussian input beam from a 30 × 30 array, exhibiting a narrow Fano resonance and variation between complete transmission and total reflection [34]. The reflection amplitude is given by
As a first example of the physics of EPs, we consider such points away from
Figure 3(a) shows the decay of an initial state
The behavior as we transition through the EP is well described by the two-mode model (11) that yields
The solution of the full many-atom lattice dynamics similarly approaches the nonexponential solution at the EP, as higher-order terms in the exponential expansion are suppressed to longer and longer times (see Supplementary material for more details). Figure 3(b) shows the parameters extracted from fitting the full numeric simulation to the form predicted by Equation (13), with Ω → 0 and α diverging at the EP, as expected. The mean Petermann factor calculated from the two-mode model,
While nonexponential decay demonstrates a physical effect of EPs even in the absence of
Large arrays can respond collectively as a whole to incoming light normal to the lattice and the atomic excitations exhibit uniform phase profiles. The amplitudes of the incoming and outgoing (transmitted and reflected) fields, as well as each of these and the mode amplitudes are then related by linear transformations in the limit of low light intensity. This leads to quasi-1D physics, where transmission through even several stacked 2D arrays can be treated as a 1D process with each lattice responding as a single “superatom” with a collective resonance line shift and linewidth [45], [51]. Here we show how to prepare an array in a CPA phase that corresponds to the coherently scattered field perfectly canceling the external field, leading to no outgoing coherent light.
For zero Zeeman shifts, we can rewrite the equations of motion (11) in terms of the scalar amplitude of the y component of the uniform mode for a single planar array j
Here, we consider the response of an individual lattice plane to a general external field and drop the label j. We separate the fields according to the propagation direction
The second term denotes the scattered light
While an ideal lattice scatters light coherently, incoherent scattering can arise due to fluctuations in the atomic positions, quantum correlations, or defects in the lattice. In the following section, we consider a specific numerical example of incoherent light scattering due to position fluctuations. We now take Equation (17) to refer only the coherent contribution to light. Incoherently scattered light reduces the coherently scattered amplitude, the second term of Equation (17), and to incorporate this change, we replace η by
The difference between this and the evolution matrix H_{0} in Equation (11) is that the scattered light is expressed in terms of the mode amplitudes on the left-hand-side of Equation (21) (the
The scattering matrix S links the total outgoing fields to the incoming fields and can also be directly derived from Equation (17), or alternatively from Equation (21). Written in terms of the complex reflection amplitude r and transmission amplitude t, S is defined as
It follows that
As shown in the previous section,
We calculate directly the total coherent output intensity, averaged over 1500 stochastic realizations, for a 20 × 20 lattice under steady-state symmetric Gaussian beam illumination from both sides, with
Again this behavior is explained by the simple two-mode model. The eigenvalues λ_{±} of H_{T} are shown in Figure 4(b), for
The spatial variation of the total coherent outgoing field, for the same lattice parameters and illumination as Figure 4, is shown in Figure 5 for two cases, one where the lattice is strongly reflective, and one where CPA is achieved. In the second case, there is near total cancellation between the coherent scattering and the external field, resulting in a drastic reduction in intensity, with peak intensity around the edge of the lattice.
We propose two other realizations of
We now consider an array in the xy plane, where the quantization axis remains aligned with the z axis, such that the two in-plane uniform modes, with
Again, this
We have shown how an effective
Data used in this publication is available at https://doi.org/10.17635/lancaster/researchdata/408.
Funding source: Engineering and Physical Sciences Research Council
Award Identifier / Grant number: EP/P026133/1
Award Identifier / Grant number: EP/S002952/1
Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: We acknowledge financial support from the UK EPSRC (Grant Nos. EP/S002952/1, EP/P026133/1).
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
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The online version of this article offers supplementary material (https://doi.org/10.1515/nanoph-2020-0635).
© 2021 Kyle E. Ballantine and Janne Ruostekoski, published by De Gruyter, Berlin/Boston
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