Self-organization of atoms coupled to a chiral reservoir

Tightly confined modes of light, as in optical nanofibers or photonic crystal waveguides, can lead to large optical coupling in atomic systems, which mediates long-range interactions between atoms. These one-dimensional systems can naturally possess couplings that are asymmetric between modes propagating in different directions. Strong long-range interaction among atoms via these modes can drive them to a self-organized periodic distribution. In this paper, we examine the self-organizing behavior of atoms in one dimension coupled to a chiral reservoir. We determine the solution to the equations of motion in different parameter regimes, relative to both the detuning of the pump laser that initializes the atomic dipole-dipole interactions and the degree of reservoir chirality. In addition, we calculate possible experimental signatures such as reflectivity from self-organized atoms and motional sidebands.

Figure 1

(a) Atoms are free to move along the axis $z$ and are illuminated from the side by a pump field of real Rabi frequency $\mathrm{\Omega }$ and detuning $\delta$. They decay with different rates into left- and right-propagating guided modes (${\mathrm{\Gamma }}_L$ and ${\mathrm{\Gamma }}_R$) and into free space ($\gamma$). (b) The level diagram of the atom in question and the applied field. Although drawn here as positive (blue-detuned), $\delta$ may be of either sign.

Figure 2

Results of weak-scattering calculations, showing numerical results (points) for a range of chiralities, and analytic results (lines) for ${\chi }_r=0,1$. The $x$ axis enumerates the atoms while the $y$ axis shows their fractional positions as defined in the text.

Figure 3

Results of calculations near resonance at full chirality (${\chi }_r=1$), showing numerical results (points) and analytic results (lines). The weak-scattering case $\left(\delta \to \infty \right)$ has been included for comparison.

Figure 4

Time required to converge to equilibrium for each atom. This time is defined as the latest time at which the atomic momentum is more than 1% away from its equilibrium value. This data represents $10000$ separate instances of the weak-scattering limit with ${\chi }_r=1$ in random initial positions each time. For each trial, the time for the first atom to converge to its final position was calculated, and this was then subtracted from all data points to isolate the trends. Atoms further down the chain converge later in the process.

Figure 5

Evolution of the phase-slip configurations as ${\chi }_r$ increases from 0 to 0.27 at $\delta ={\mathrm{\Gamma }}_{\mathrm{tot}}$. These are produced by the numerical protocol outlined in Sec. 2b.

Figure 6

Effective potential resulting from integrating the force equation (11) in various equilibrium configurations, ranging from symmetric (${\chi }_r=0$) to just before collapse (${\chi }_r=0.25$) at detuning $\delta ={\mathrm{\Gamma }}_{\mathrm{tot}}$. This curve was generated by examining the force on the last atom before the phase slip (the rightmost atom in the left superatom), as illustrated in the inset. Note that a constant vertical offset on any of these lines has no physical significance.

Figure 7

Illustration of how a variable polarization of trapping light can be used to achieve the full range of chiralities. The central gray circle shows a cross section of the nanofiber. The elongated red-blue shape represents the trapping potential, which can be rotated by choosing a quasilinear polarization of the trapping light rotated by $\theta$ relative to the quantization axis. Since sweeping through this angle varies the coupling from one chirality to the next continuously, the value of ${\chi }_r$ will vary from $-1$ to 1. The positive $z$ axis points out of the page.

Figure 8

Illustration of how the transfer matrix can be used to calculate the fields on one side of an optical element given fields on the other side.

Figure 9

Reflection spectra for configurations near (${\chi }_r=0.15,0.21,0.24$) and just past $({\chi }_r=0.28)$ the superatom collapse ($\delta ={\mathrm{\Gamma }}_{\mathrm{tot}}$), calculated via transfer matrix technique. While these plots are quite similar for both directions, the quantity plotted here is $\left|R_L\right|$.

Figure 10

Reflection spectra for weak-scattering configurations, calculated via transfer matrix. Dashed lines represent $\left|R_L\right|$, the reflected amplitude of left-propagating light, while solid lines represent $\left|R_R\right|$.

Figure 11

Fourier transform (arbitrary units) of the motion of the leftmost five atoms in a fully chiral chain (${\chi }_r=1$) in the weak-scattering limit. Atoms were brought to equilibrium positions, damping was then turned off, and they were displaced slightly from equilibrium. Vertical dashed lines represent theoretical values where frequencies are expected, while numbers index both the frequencies and (by color) denote which line corresponds to which atom's motion. Motion of the leftmost atom ($\text{j}=0$) is not visible because it undergoes only uniform acceleration and is thus used as a reference point. The curves are shifted vertically relative to each other for ease of visualization.

Figure 12

Spatial configurations of self-organized solutions at several different strengths of ${\mathrm{\Gamma }}_{1\mathrm{D}}$. These calculations were made at $\delta =1{\mathrm{\Gamma }}_{\mathrm{tot}}$.