# Self-Organization in Cold Atoms Mediated by Diffractive Coupling

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## Abstract

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## 1. Introduction

## 2. Mechanism of Diffractive Coupling

#### 2.1. Single-Mirror Feedback Schemes and the Talbot Effect

#### 2.2. 2-Level Systems: Kerr and Saturable Nonlinearities

## 3. External Degrees of Freedom: Optomechanics

## 4. Internal Degrees of Freedom: Magnetic Ordering

#### 4.1. Optical Pumping Nonlinearity and Irreducible Tensor Components

#### 4.2. Dipolar Structures

#### 4.3. Hexagon Formation and Inversion Symmetry

#### 4.4. Quadrupole Structures

## 5. Light-Mediated Atomic Interaction

## 6. Self-Organization via Diffractive Coupling in Cavities

## 7. Conclusions and Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Scheme of feedback experiment: A coherent pump beam drives an atomic cloud of size L with a plane feedback mirror at a distance d. Pump photons are scattered by a spatial structure with period $\mathsf{\Lambda}$ in the cloud into sidebands at an angle $\theta $. Interference of these sidebands forms an optical lattice sustaining the spatial structure in the cloud. The sidebands can be visualized by looking at the far field of the transmitted light as obtained in the focal plane of a lens. The inset shows an example for an experimentally observed far field (FF) intensity distribution showing transverse Fourier space (displayed span −8.5 to 8.5 mrad around optical axis). The upper panel illustrate how the phase modulation arising after the cloud is diminishing during propagation whereas an amplitude modulation (lower panel) builds up by diffractive dephasing. At a distance $2d$ the amplitude modulation is fully developed and the phase is flat. Here, the transmitted light has the same spatial structure as the retro-reflected feedback beam re-entering the atomic cloud. This near field (NF) intensity distribution can be imaged onto a camera (example shown in lower right inset, displayed range 2 mm). Parameters for images: input intensity 129 mW/cm${}^{2}$, detuning $\Delta =7\mathsf{\Gamma}$, saturation parameter $s\approx 0.18$, dominantly optomechanical nonlinearity.

**Figure 2.**Illustration of feedback loop for single-mirror feedback. The x-axis shows transverse space in periods of the pattern. The y-axis denotes amplitude, but via the vertical displacements also propagation distance. The two light grey regions represent the atomic cloud as encountered by the forward beam (bottom) and the backwards beam reentering the cloud (top) after propagating to the mirror and back (distance $2d$, indicated by black arrow). The thick black line indicates a modulation of an atomic state variable, here for concreteness the population ${\rho}_{ee}$ of the excited state. The red line shows the resulting modulation of the refractive index. The green line indicates the phase of the transmitted beam just after traversing the cloud. The blue line indicates the amplitude modulation created in the backward beam by diffractive dephasing. (

**a**) Blue detuning: After a quarter of the Talbot distance intensity maxima in the backward beam are aligned with refractive index maxima (blue arrow) providing positive feedback for a self-focusing nonlinearity. (

**b**) Red detuning: After a three quarters of the Talbot distance intensity maxima in the backward beam are aligned with refractive index minima (red arrow) providing positive feedback for a self-defocusing nonlinearity. Hence, the period of the spatial modulation in the cloud experiencing maximum positive feedback for a given mirror distance d is reduced by a factor $\sqrt{3}$.

**Figure 3.**Examples of hexagonal patterns observed in a parameter range in which optomechanical and electronic nonlinearities are both present. The different structures appeared in different realizations of the experiment under nominally the same experimental conditions illustrating spontaneous symmetry breaking. (

**a**–

**c**) differ in the orientation of hexagonal axes. This difference is very small between (

**a**,

**b**) (the red line indicating an axis in (

**a**) is not aligned with a row of spots in (

**b**)), but very obvious between (

**a**)/(

**b**) and (

**c**). (

**d**) contains two different domains consisting of the hexagons in (

**a**,

**c**) with a defect line in between them. Parameters: ${I}_{\mathrm{in}}=250\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}$ mW/cm${}^{2}$, $\delta =+7\phantom{\rule{0.166667em}{0ex}}\mathsf{\Gamma}$, and $d=5$ mm.

**Figure 5.**Diffracted energy in sidebands vs. optical density in line centre for optomechanical (red circles), electronic 2-level (blue triangles) and optical pumping nonlinearities (black squares). Each subset of data is normalized to its maximal value. For the optomechanical and electronic case the raw data are from Figure S1 of [44] and pulse duration is used to distinguished between them (see Section 3): short pulses ($1\phantom{\rule{0.166667em}{0ex}}$ μs, blue triangles), long pulses ($200\phantom{\rule{0.166667em}{0ex}}$ μs, red circles). Parameters: $\Delta =+6\mathsf{\Gamma}$, $I=487\phantom{\rule{0.166667em}{0ex}}$ mW/cm${}^{2}$. Optical density is varied by varying duration of thermal expansion of the cloud by introducing a controlled delay between MOT turn-off and pump pulse. For the optical pumping nonlinearity, parameters are $\Delta =-12\phantom{\rule{0.166667em}{0ex}}\mathsf{\Gamma}$, $I=3.6$ mW/cm${}^{2}$, pump duration 0.5 ms. The optical density is varied by adjusting the repumper power before the MOT is turned off.

**Figure 6.**Illustration of feedback loop for single-mirror feedback for optomechanical interactions (see caption of Figure 2 for general explanation). Black line in atomic cloud denotes atomic density. (

**a**) Blue detuning ($\Delta >0$): Atoms are low field seekers and hence atomic and intensity structures are complementary (atomic troughs align with intensity peaks, situation of experiment described in [44]). (

**b**) Red detuning ($\Delta <0$): Atoms are high field seekers and hence atomic and intensity structures are aligned. Note that the nonlinearity is self-focusing also for red and blue detuning and the instability exists for both signs of the detuning (similar to [75] for the case of single-pass propagation).

**Figure 7.**(

**a**) Level structure of $F=2\to {F}^{\prime}=3$ hyperfine transition line of ${D}_{2}$-line of ${}^{87}$Rb. Red (blue) arrows indicate transitions via ${\sigma}_{+}$ (${\sigma}_{-}$) light with Rabi frequencies ${\mathsf{\Omega}}_{+}$ (${\mathsf{\Omega}}_{-}$) with strength relative to the strongest transitions between the stretched states. A longitudinal magnetic field with Larmor frequency ${\mathsf{\Omega}}_{z}$ splits the Zeeman sublevels. For simplicity, the $g-$factor for the ground state is assumed to apply also for the excited state. Transverse magnetic fields ${\mathsf{\Omega}}_{x}$, ${\mathsf{\Omega}}_{y}$ will induce coherences between Zeeman sublevels (for simplicity only one is shown). (

**b**) Reduced scheme of a $F=1\to {F}^{\prime}=2$ transition used in the theoretical description. It includes the possibility of orientation (irreducible tensor rank 1) and alignment (rank 2) states in the ground state. (

**c**) Further reduced scheme of a $F=1/2\to {F}^{\prime}=3/2$ transition keeping only the possibility of an orientation. In this situation, it is illustrated that an orientation with an increased population in the state with $m<0$ leads to a back-action on the light increasing absorption and phase shift for the ${\sigma}_{-}$-component. This orientation can be created by a dominance of ${\sigma}_{-}$- over ${\sigma}_{+}$-light, or for linearly polarized light by the incoherent Faraday effect.

**Figure 8.**Polarization states on Poincaré sphere (

**upper row**) and illustrations of magnetic states (

**lower row**) coupling to them. Red denotes south pole, blue north pole (for positive irreducible components). Leftmost column: A light beam with helicity, i.e., non-zero ${S}_{3}$ (out of the equatorial plane), will generate a magnetic dipole corresponding to an orientation w. The second left column illustrate a magnetic quadrupole with principle axis aligned to the z-axis corresponding to a longitudinal alignment X. It is driven by the total intensity ${S}_{0}$ independent of the polarization state. The remaining three columns illustrate quadrupoles in the x-y-plane driven by linearly polarized light in the equatorial plane of the Poincaré sphere. These correspond to the $\Delta m=2$-coherence $\mathsf{\Phi}$ in the atom representing a transverse alignment. Light polarized along the x-direction (centre column, pump polarization used in experiments) just drives the real part u of the coherence represented by a quadrupole with one (north) of the principal axes along the x-direction and the other along y. Light polarized at 45${}^{\circ}$ drives the imaginary part v with the north principal axis oriented at 45${}^{\circ}$. In general, light with a phase ${\varphi}_{L}$ between $\sigma $ components (second to right column) drives a quadrupole with north principle axis at the polarization angle ${\varphi}_{Q}={\varphi}_{p}={\varphi}_{L}/2$ (conventional optics notation, $({\varphi}_{L}-\pi )/2$ in convention used in our theoretical description).

**Figure 9.**Illustration of feedback loop for single-mirror feedback for magnetic dipole interactions (see caption of Figure 2 for general explanation). Black line: orientation w. For red detuning: red (magenta) line: refractive index for ${\sigma}_{-}$ (${\sigma}_{+}$) light, light (dark) green line: phase modulation of transmitted ${\sigma}_{-}$ (${\sigma}_{+}$) beam, dark (light) blue: intensity of ${\sigma}_{-}$ (${\sigma}_{+}$) beam after a quarter of the Talbot distance, orange: difference pump rate D.

**Figure 10.**Spontaneous antiferromagnetic ordering for ${B}_{x}={B}_{y}={B}_{z}=0$. (

**a**,

**b**): NF intensity pattern of the ${\sigma}^{+}$ and ${\sigma}^{-}$ components. (

**c**), NF intensity difference of the ${\sigma}^{+}$ and ${\sigma}^{-}$ components. (

**d**) Spatial structure of the orientation w obtained from numerical simulation. (

**e**) example for FF intensity distribution experimentally observed (not the same realization as in (

**a**–

**c**)). (

**f**) Numerically obtained Fourier spectrum of w. Parameter of experiment: ${b}_{0}=80$, $\delta =-8\mathsf{\Gamma}$, $I=10$ mW/cm${}^{2}$, $d=-20$ mm. (subpanels (

**a**–

**d**) adapted from [49]).

**Figure 11.**Illustration of feedback loop for single-mirror feedback for quadrupole interactions (see caption of Figure 2 for general explanation). Black line: modulation of imaginary part of coherences v. For red detuning: light (dark) green line: amplitude modulation of transmitted beam for ${\sigma}_{-}$ (${\sigma}_{+}$), light (dark) blue: phase of ${\sigma}_{-}$ (${\sigma}_{+}$) after a quarter of the Talbot distance, orange line: relative phase between circular polarization components ${\varphi}_{L}$ and modulated polarization direction.

**Figure 12.**Example for coherence-based structure detected in channel perpendicular to input polarization. (

**a**,

**b**): Near field intensity structures. (

**c**,

**d**): Far field intensity displaying Fourier spectrum (not obtained in same realizations as in (

**a**,

**b**)). Parameters: pump intensity 7 mW/cm${}^{2}$, detuning $\Delta =-12\mathsf{\Gamma}$, ${b}_{0}=110$. The field of view is 4.4 mm for the NF and 16 mrad for the FF images.

**Figure 13.**Examples for numerically obtained structures. (

**a**,

**c**): Near field intensity structures. (

**b**,

**d**): Corresponding far field intensity displaying Fourier spectra. Parameters ${B}_{x}=0.76$ G, ${B}_{y}={B}_{z}=0$, ${b}_{0}=130$, $\Delta =-12\mathsf{\Gamma}$, pump intensity 6.8 mW/cm${}^{2}$. The difference between the two realization is the size of the integration domain relative to the lattice period.

**Figure 14.**Upper row: Spatial distribution of atomic variables corresponding to the structures in Figure 13c: (

**a**) real part of coherence u, (

**b**) imaginary part of coherence v, (

**c**) orientation w, Lower row: Stokes parameters of backward beam, (

**d**) ${S}_{1}$, (

**e**) ${S}_{2}$, (

**f**) ${S}_{3}$.

**Figure 15.**(

**a**) 1D cut through 2D profiles of localized perturbation of atomic state (black, scaled to match the feedback pump rate in peak for presentation purposes) and resulting perturbation (i.e., without the homogeneous background) in the pump rate of backward beam (blue) after one round trip in feedback loop. The central lobe leads to the self-enhancement of the instability, the first side lobes to an enhancement at neighbouring lattice sites. Transverse space is measured in periods of the periodic lattice. The calculations are done in 2D and only profiles are presented. (

**b**) Feedback pump rate for different optical densities in line centre ${b}_{0}$ in logarithmic scale. (

**c**) Peak values of central (black squares) and first lopes (red circles) for different optical densities in line centre ${b}_{0}$ in logarithmic scale. For all panels the atomic perturbation has been small (0.1), homogeneous input pump rate $P=1$, detuning $7\mathsf{\Gamma}$, $R=1$. ${b}_{0}=1$ in (

**a**). (

**d**) 1D calculations of perturbation of backward pump rate being a factor of m narrower than in (

**a**). Results are then normalized by dividing by pump rate in centre $x=0$.

**Figure 16.**(

**a**) Scheme of a cavity of length L driven by a coherent input field and containing a cloud of cold atoms in the centre. A modulation of an atomic state variable of period $\mathsf{\Lambda}$ leads to scattering of the pump into sidebands at an angle $\theta $. The interference of these sidebands sustains the structure in the cloud. (

**b**) Light with a wavevector k larger than the resonant cavity wavevector ${k}_{c}$ can reestablish resonance by tilting to a suitable angle $\theta $. (

**c**) Plano-planar cavity with two afocal telescopes with diffractive properties equivalent to plano-planar cavity of reduced diffractive length.

**Figure 17.**Perturbation of atomic state variable (black line) proportional to refractive index and resulting perturbation (scaled to meet perturbation amplitude for purpose of visualization) in intra-cavity intensity after a time corresponding to five times the inverse of the cavity decay rate (see text and Figure 15 for details of the procedure). Parameters: pump intensity 1.05 above threshold; cavity detuning zero, $\Delta =-10\phantom{\rule{0.166667em}{0ex}}\mathsf{\Gamma}$, cooperativity parameter $C=0.0125$.

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Ackemann, T.; Labeyrie, G.; Baio, G.; Krešić, I.; Walker, J.G.M.; Costa Boquete, A.; Griffin, P.; Firth, W.J.; Kaiser, R.; Oppo, G.-L.;
et al. Self-Organization in Cold Atoms Mediated by Diffractive Coupling. *Atoms* **2021**, *9*, 35.
https://doi.org/10.3390/atoms9030035

**AMA Style**

Ackemann T, Labeyrie G, Baio G, Krešić I, Walker JGM, Costa Boquete A, Griffin P, Firth WJ, Kaiser R, Oppo G-L,
et al. Self-Organization in Cold Atoms Mediated by Diffractive Coupling. *Atoms*. 2021; 9(3):35.
https://doi.org/10.3390/atoms9030035

**Chicago/Turabian Style**

Ackemann, Thorsten, Guillaume Labeyrie, Giuseppe Baio, Ivor Krešić, Josh G. M. Walker, Adrian Costa Boquete, Paul Griffin, William J. Firth, Robin Kaiser, Gian-Luca Oppo,
and et al. 2021. "Self-Organization in Cold Atoms Mediated by Diffractive Coupling" *Atoms* 9, no. 3: 35.
https://doi.org/10.3390/atoms9030035